aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J a n Cold Bosons in Optical Lattices
V.I. Yukalov
Bogolubov Laboratory of Theoretical Physics,Joint Institute for Nuclear Research, Dubna 141980, Russia
Abstract
Basic properties of cold Bose atoms in optical lattices are reviewed. The mainprinciples of correct self-consistent description of arbitrary systems with Bose-Einsteincondensate are formulated. Theoretical methods for describing regular periodic latticesare presented. A special attention is paid to the discussion of Bose-atom properties inthe frame of the boson Hubbard model. Optical lattices with arbitrary strong disorder,induced by random potentials, are treated. Possible applications of cold atoms in opti-cal lattices are discussed, with an emphasis of their usefulness for quantum informationprocessing and quantum computing. An important feature of the present review ar-ticle, distinguishing it from other review works, is that theoretical fundamentals hereare not just mentioned in brief, but are thoroughly explained. This makes it easy forthe reader to follow the principal points without the immediate necessity of resortingto numerous publications in the field.
E-mail: [email protected] 1 ontents
1. Tools for Manipulating Atoms1.1 . Cold Atoms . Control Parameters . Atomic Fractions . System Classification . Cold Molecules
2. Systems with Bose-Einstein Condensate2.1 . Bose-Einstein Condensation . Penrose-Onsager Scheme . Order Indices . Representative Ensembles . Field Operators . Gauge Symmetry . Bogolubov Shift . Self-Consistent Approach . Condensate Existence . Superfluid Fraction . Equations of Motion . Uniform System . Anomalous Averages . Particle Fluctuations . Fragmented Condensates . Multicomponent Condensates . Model Condensates
3. Regular Optical Lattices3.1 . Optical Lattices . Periodic Structures . Condensate in Lattices . Operator of Momentum . Tight-Binding Approximation 2 .6 . Superfluidity in Lattices . Transverse Confinement . Bloch Spectrum . Spectrum Parameters . Elementary Excitations . Wave Stability . Moving Lattices . Soliton Formation . Transverse Resonances . Lagrange Variation
4. Boson Hubbard Model4.1 . Wannier Representation . Grand Hamiltonian . Bose-Condensed System . Thermodynamic Characteristics . Superfluid Fraction . Single-Site Approximation . Localized State
5. Phase States and Transitions5.1 . Existence of Pure Phases . Hard-Core Lattice Gas . Effective Interaction Parameter . Gutzwiller Single-Site Approximation . Dynamical Mean-Field Approximation . Small-System Numerical Diagonalization . Density-Matrix Renormalization Group . Strong-Coupling Perturbation Theory . Monte Carlo Simulations . Order of Phase Transition . Experiments on Superfluid-Insulator Transition . Layered Superfluid-Insulator Structure3 .13 . Models with Neighbor Interactions . Quasiperiodic Optical Lattices . Rotating Optical Lattices
6. Optical Lattices with Disorder6.1 . Random Potentials . Uniform Limit . Disordered Lattice . Disordered Superfluid . Phase Diagram
7. Nonstandard Lattice Models7.1 . Coexisting States . Vibrational Excitations . Phonon-Induced Interactions
8. Double-Well Optical Lattices8.1 . Effective Hamiltonians . Phase Transitions . Collective Excitations . Nonequilibrium States . Heterophase Lattices
9. Tools for Quantum Computing9.1 . Entanglement Production . Topological Modes . Coherent States . Coherent-Mode Register . Double-Well Register
10. Brief Concluding RemarksReferences Tools for Manipulating Atoms
Physics of cold trapped atoms has become nowadays a very fastly developing field of re-search, both theoretical and experimental. Magnetic, magneto-optical, and all optical trapsare employed for trapping atoms. Several atomic species have been cooled down to lowtemperatures, when their quantum degeneracy could be observed. The Bose-Einstein con-densation (BEC) of trapped atoms was experimentally realized [1–3]. At the present time,BEC has been achieved for 12 atomic species: H, He, Li, Na, K, K, Cr, Rb, Rb,
Cs,
Yb, and
Yb. Quantum degeneracy in trapped Fermi gases was achieved for Kand Li atoms [4–6]. Now there exist several books [7,8] and review articles treating theBose-Einstein condensation of Bose atoms [9–14] and the quantum properties of ultracoldFermi gases [15].An important development has been the realization of optical lattices, formed by inter-fering laser beams producing a standing wave. Cold atoms can be trapped for a long timein the minima of the created periodic potential [16]. There are several surveys consideringthe properties of cold atoms in optical lattices, e.g., [17–20].In the present review paper, the emphasis is made on the theory of cold Bose atoms inperiodic potentials. Such potentials are usually formed by optical lattices, though recentlymagnetic lattices have also been realized [21].There are two principal features making this review paper distinct from all other re-view articles. First, the basic theoretical points are thoroughly explained here, but not justmentioned in brief. This should allow the reader to better understand the theoretical fun-damentals and to easily follow the logic of the used mathematical methods. The materialof this paper can serve as a reference source for researchers in the field. Second, this papercovers the most recent theoretical results that have not yet been described in other reviewarticles.
Optical lattices with cold atoms provide an extraordinary possibility of creating systemswith a wide variety of properties, which can be manipulated in several ways. First of all, thelattice parameters themselves can be varied in a wide range. In experiments, optical latticescan be formed, having different spacing, depth, and filling factors. The latter can be eitherinteger or fractional and can vary between one and 10 atoms per lattice site [22,23]. Thenumber of lattice sites can also be different. One - two - and three - dimensional latticescan be formed. The lattices can be periodic and quasiperiodic. Different atomic species, ortheir mixtures, can be loaded in the lattice. The strength of interatomic interactions canbe regulated in a very wide range by employing the Feshbach resonance techniques [10,24].Varying temperature and/or lattice depth, it is possible to induce phase transitions betweenlocalized and delocalized states of atoms, as well as between the normal and superfluidphases.Lattice properties can also be regulated by imposing additional external potentials. Inparticular, random external fields can be used, producing disordered lattices. By means5f alternating external fields, one can manipulate the motion of atomic clouds. Employingspecial resonant alternating fields makes it possible to create an unusual state of matter, thenonground-state Bose-Einstein condensates.These rich potentialities of manipulating cold atoms in optical lattices make this objectof high importance for various applications. But the latter can become practicable onlybeing based on effective and correct theoretical investigations. The total number of atoms N , loaded into a lattice, can consist of several parts. An importantpart of a Bose system is that one forming Bose-Einstein condensate (BEC) of N atoms,which characterizes the coherent portion of atoms. As a rule the arising BEC leads to theappearance of superfluidity involving N sup atoms. There is no simple relation between N and N sup and even it is not compulsory that they be present simultaneously. The physicalorigins of N and N sup are different. The appearance of BEC manifests the existence ofcoherence in the system. Superfluidity demonstrates the presence of nontrivial response toa velocity boost, which is caused by strong atomic correlations.Atoms in a lattice can also be distinguished by the region of their motion. Atoms can belocalized in their lattice sites or can be delocalized and moving through the whole sample.The localized atoms are associated with the solid state of matter, possessing small compress-ibility and a gap in the single-particle spectrum. The number of atoms, forming a solid, willbe denoted by N sol . Delocalized atoms are typical of the liquid or gaseous state of matter,with gapless single-particle spectra.Since the total number of atoms N , as well as the atomic numbers N , N sup , and N sol ,can be very large, it is more appropriate to deal with the related atomic fractions, whichare:the condensate fraction n ≡ N N , superfluid fraction n sup ≡ N sup N , and the solid fraction n sol ≡ N sol N .
Similarly, the number of uncondensed atoms N defines the normal fraction n ≡ N /N .And one can define the fraction of atoms in fluid phase. But the fractions n , n sup , and n sol are the main for the classification of the major system features. The atomic fractions n , n sup , and n sol characterize the basic properties of systems formed byBose atoms in optical lattices. Thus, the existence of BEC means the presence of coherence,because of which such a system can be termed coherent, n > coherent ) . n = 0 ( incoherent ) . In the same way, the system is superfluid, when there is the superfluid fraction, n sup > superf luid ) . The absence of the superfluid fraction implies that the system is not superfluid, that is,normal, n sup = 0 ( normal ) . The presence of the solid fraction gives to the system rigidity typical of solids, n sol > solid ) . While, if there is no solid fraction, the system is either liquid or gaseous, generally speaking,fluid, n sol = 0 ( f luid ) . This terminology allows us to suggest the following classification of admissible systems,depending on the presence or absence of the related atomic fractions.(1)
Incoherent normal fluid : n = 0 , n sup = 0 , n sol = 0 . Ubiquitous examples are classical liquids and gases.(2)
Coherent normal fluid : n > , n sup = 0 , n sol = 0 . This case looks a bit exotic, though the situation, when there is BEC but there is no super-fluidity can be attributed to what one calls Bose glass, the state that may develop in thepresence of disorder.(3)
Incoherent superfluid : n = 0 , n sup > , n sol = 0 . The known examples are two-dimensional superfluid films without BEC.(4)
Coherent superfluid : n > , n sup > , n sol = 0 . This is superfluid He.(5)
Incoherent normal solid : n = 0 , n sup = 0 , n sol > . Coherent normal solid : n > , n sup = 0 , n sol > . This type of solids can also be attributed to the so-called Bose glass.(7)
Incoherent superfluid solid : n = 0 , n sup > , n sol > . The possibility of such solids is currently under discussion.(8)
Coherent superfluid solid : n > , n sup > , n sol > . This state looks admissible in optical lattices.Thus, there can exist 8 classes of systems, depending on the presence or absence of thefractions n , n sup , and n sol . These different states can be achieved by appropriately adjustingthe system parameters. Bose-Einstein condensate can, in principle, be created in different Bose systems. As ismentioned in subsection 1.1, at the present time, BEC has been achieved in 12 atomicspecies. The latest of them was
Yb [25]. In addition, there exist Bose molecules formedof either Bose or Fermi atoms [10,15,24,26,27]. In systems, composed of Bose molecules,BEC can also arise. Thus, BEC was produced in molecular systems, where the moleculeswere formed by Bose atoms ( Na , Rb , Rb , Cs ) as well as by Fermi atoms ( Li , K ). Among other systems, that could exhibit BEC, it is possible to mention boson quarkclusters and hadronic molecules [28,29]. Pion condensation in nuclear matter could be onemore example [30–34], though in this case the condensate itself possesses a periodic structure.The theory, presented in the following sections, is applicable to Bose systems of arbitrarynature, whether the constituents are atoms or molecules, or some kind of bosonic clusters.The sole thing is that these constituents are treated as Bose particles, characterized bytheir masses and interactions. Also, the main attention is paid to particles without internaldegrees of freedom. For instance, spins are assumed to be frozen, so that particles can betreated as spinless. The consideration of particles with spin degrees of freedom requires aseparate paper.Theoretical methods are general for describing any type of bosons, whether the latter areatoms or molecules. However, it is important to keep in mind that the possibility of creatingmolecules provides the way of enriching the system properties. Molecules can also be loadedin optical lattices [35–39]. 8 Systems with Bose-Einstein Condensate
Lattices can be periodic, quasiperiodic or even random, representing different external po-tentials making the system nonuniform. It is worth starting the consideration by formulatingthe general criteria characterizing the occurrence of BEC in nonuniform systems.Generally, BEC is the occupation of a single, or several, quantum states by a large numberof identical particles. For simplicity, we shall be talking about a single quantum state. Thegeneralization to several macroscopically occupied quantum states is straightforward andwill be done in Sec. 2.15.Historically, BEC was described by Einstein for ideal uniform Bose gas. The history andrelated historical references can be found in Ref. [40]. The quantum states of a uniform gasare characterized by the momentum k . Here and in what follows, we use the system of units,where the Planck constant and the Boltzmann constant are set to one, ~ ≡ k B ≡ k , whose concrete representation depends on the considered problem. There exists the stateoccupation number n k showing the number of particles in a quantum state k . Suppose thatamong all quantum states there occurs a single state k , for which the occupation number N ≡ n k (2.1)is large. Here ”large” means not merely that N is much larger than one, but that it iscomparable to the total number of particles N , such that N ∝ N . Then we can say thatthere occurs BEC into the state k .One says that the condensate state is macroscopically occupied. To make this phrasemathematically accurate, one resorts to the notion of the thermodynamic limit N → ∞ , V → ∞ , NV → const > , (2.2)where N is the total number of particles in the system of volume V . The state k is termed macroscopically occupied , when lim N →∞ N V > , (2.3)where the thermodynamic limit (2.2) is implied. Condition (2.3) is, actually, the Einsteincriterion of BEC.For trapped atoms, the system volume V may be not well defined. Then the thermo-dynamic limit can be specified in a different way [41]. If the system contains N trappedatoms, for which extensive observable quantities are defined, then the following limit can beconsidered. Let A N be an extensive observable quantity, then the effective thermodynamiclimit is N → ∞ , A N N → const . (2.4)For instance, taking for the observable quantity the internal energy E N of N particles, wehave [41] limit (2.4) as N → ∞ , E N N → const . (2.5)9n what follows, writing N → ∞ , we shall assume one of the forms of thermodynamic limit.Confined systems contain finite numbers of atoms N , though the latter is large. All finitesystems with N ≫ thermody-namic tests , not merely allowing for the simplification of calculations, but, which is the mostimportant, making it possible to check the correctness of theories. The Einstein criterion of BEC (2.3) is easily applicable to uniform ideal gases. But forinteracting systems, especially for nonuniform cases, to make criterion (2.3) useful requires,first, to specify how the quantum state occupation numbers n k could be found. Penrose andOnsager [42] suggested the following scheme.Assume that the single-particle density matrix ρ ( r , r ′ ) of the considered system is known.This matrix is a function of the real-space variables r and, generally, of time t . The latter doesnot enter ρ ( r , r ′ ) for equilibrium systems, but for nonequilibrium systems, ρ ( r , r ′ , t ) dependson time. In what follows, we shall omit, for the sake of brevity, the time dependence, wherethis is not important. However, we may keep in mind that the time variable can always beincluded, when the consideration concerns nonequilibrium cases.If the density matrix ρ ( r , r ′ ) is known, then one could solve the eigenvalue problem Z ρ ( r , r ′ ) ϕ k ( r ′ ) d r ′ = n k ϕ k ( r ) , (2.6)where the integration is over the whole volume specifying the system. The eigenfunctions ϕ k ( r ) are called the natural orbitals [43]. The family { ϕ k ( r ) } forms a complete orthonormalbasis, for which Z ϕ ∗ k ( r ) ϕ p ( r ) d r = δ kp . Since the single-particle density matrix is normalized to the total number of particles N = Z ρ ( r , r ) d r , the eigenvalues n k = Z ϕ ∗ k ( r ) ρ ( r , r ′ ) ϕ k ( r ′ ) d r d r ′ have the meaning of the occupation numbers of quantum states labelled by a multi-index k .In terms of the natural orbitals , the density matrix enjoys the diagonal expansion ρ ( r , r ′ ) = X k n k ϕ k ( r ) ϕ ∗ k ( r ′ ) . (2.7)Suppose that the maximal of the eigenvalues n k corresponds to a quantum state k , for whichwe may write N ≡ sup k n k = n k . (2.8)10eparating the state k from sum (2.7) gives ρ ( r , r ′ ) = N ϕ ( r ) ϕ ∗ ( r ′ ) + X k = k n k ϕ k ( r ) ϕ ∗ k ( r ′ ) , (2.9)where ϕ ( r ) ≡ ϕ k ( r ). Then the total number of particles can be written as the sum N = N + N , N ≡ X k = k n k (2.10)of the number of particles N in the state k and the number of all other particles N .One says that there happens BEC into the state k , if the latter is macroscopicallyoccupied, such that the largest eigenvalue (2.8) satisfies condition (2.3) in the sense of oneof the thermodynamic limits (2.2) or (2.4). Thus, we return to the Einstein criterion (2.3).The novelty in the Penrose-Onsager scheme is the interpretation of the number of condensedparticles (2.1) as the largest eigenvalue (2.8) of the single-particle density matrix.The Penrose-Onsager interpretation of the Einstein criterion for BEC is very general, be-ing applicable to arbitrary statistical systems, including confined systems of trapped atoms.This is contrary to the concept of the off-diagonal long-range order [44], introducing thenumber of condensed particles N through the limiting relation N V = lim | r − r ′ |→∞ ρ ( r , r ′ ) . This concept has a meaning solely for uniform infinite systems, while for trapped atoms itis not applicable [9], always strictly giving N = 0. The Penrose-Onsager scheme can be generalized by introducing the notion of order indices.The latter can be formulated for arbitrary operators [45]. Let ˆ A be an operator possessinga norm || ˆ A || and a trace Tr ˆ A . Then the operator order index of ˆ A is defined [45] as ω ( ˆ A ) ≡ log || ˆ A || log | Tr ˆ A | . Here the logarithm can be taken to any convenient base, for instance, it can be the naturallogarithm ln.The reduced density matrices can be treated as matrices with respect to their real-spacevariables [43]. Thus, the single-particle density matrix defines the first-order density matrixˆ ρ ≡ [ ρ ( r , r ′ )]. Then the order index of ˆ ρ is ω ( ˆ ρ ) = log || ˆ ρ || log Tr ˆ ρ . (2.11)Similarly, one can introduce the order indices of higher-order density matrices [46–49]. Forthe order index (2.11), since || ˆ ρ || = N , Tr ˆ ρ = N ,
11e obtain ω ( ˆ ρ ) = log N log N . (2.12)This index is not larger than one, ω ( ˆ ρ ) ≤
1, because N ≤ N .The order indices are convenient for classifying different types of order that can arise inthe system. For Bose systems, there can be three possibilities for the order indices (2.11) or(2.12). When ω ( ˆ ρ ) ≤ no order ) , (2.13)there is no ordering in the system, or at the most, a kind of short-range order may appear.In the interval 0 < ω ( ˆ ρ ) < mid − range ) , (2.14)the order index demonstrates the amount of mid-range order. There is no true BEC in thiscase, but there exists some ordering that can be associated with quasicondensate . The trueBEC corresponds to the index ω ( ˆ ρ ) = 1 ( long − range ) , (2.15)which happens in thermodynamic limit N → ∞ . In the finite uniform systems, as well as inlow-dimensional uniform systems, such as one- and two-dimensional systems, there can beno true BEC, but there can arise quasicondensate. In confined systems, BEC can happenin low-dimensional systems, depending on the type of the trapping potential [41]. The BECcriterion (2.3) is equivalent to condition (2.15) occurring in thermodynamic limit, ω ( ˆ ρ ) → N → ∞ ) . The notion of order indices is applicable to arbitrary statistical systems, whether finiteor infinite, uniform or nonuniform, equilibrium or nonequilibrium. The order indices retaintheir meaning, when the order parameters cannot be defined [45–49].
The BEC criteria of the previous sections signals the appearance of BEC. But these criteriaassume the knowledge of the density matrix supposed to be found beforehand. Such criteriado not prescribe the way of solving the problem.The very first step in considering any statistical system is the choice of a statisticalensemble to be used. The statistical ensemble is a triplet {F , ˆ ρ, ∂t } , in which F is the spaceof microstates, ˆ ρ = ˆ ρ (0) is the initial form of the statistical operator, and ∂t signifies theevolution law for the considered system. With the given F and ˆ ρ , one can find the statisticalaverage < ˆ A ( t ) > ≡ Tr F ˆ ρ ˆ A ( t ) (2.16)for an operator ˆ A ( t ). The prescribed evolution law makes it possible to define the temporalevolution of average (2.16), ∂∂t < ˆ A ( t ) > = Tr F ˆ ρ ∂ ˆ A ( t ) ∂t . (2.17)12he set of all operators ˆ A ( t ), corresponding to observable quantities, forms the algebra ofobservables O ≡ { ˆ A ( t ) } . The collection of the statistical averages (2.16) for all operatorsfrom the algebra of observables O is termed the statistical state < O > .When defining a statistical ensemble, it is necessary that it would correctly represent thestudied statistical system. This means that all conditions and constraints, uniquely definingthe system, must be taken into account when constructing the statistical operator ˆ ρ andformulating the evolution law. Such an ensemble is called representative.In the case of equilibrium systems, one usually tells that a statistical ensemble is definedby a Gibbs statistical operator, either canonical or grand canonical. One often calls thisthe ”Gibbs prescription”. In many situations, this is sufficient. However in general such apoint of view is a strong trivialization of the Gibbs ideas. Gibbs did write [50] that justprescribing a distribution, whether canonical or grand canonical, may be not sufficient, butthe description must be complimented by all those constraints and conditions that make thestatistical systems uniquely defined. Thus, the idea of representative statistical ensemblesis actually due to Gibbs [50]. The term ”representative ensembles” was employed by terHaar [51,52], who discussed the necessity of correctly representing statistical systems. Suchensembles, equipped with additional conditions, are also called conditional [53]. The generaltheory of equilibrium and quasiequilibrium representative ensembles was described in thereview article [54] and book [55]. Representative ensembles for Bose systems with brokengauge symmetry were covered in detail in Refs. [56,57].To specify the state of microstates, it is necessary to fix the system variables. Supposewe choose as the variables the field operators ψ ( r ) and ψ † ( r ), with the Bose commutationrelations (cid:2) ψ ( r ) , ψ † ( r ′ ) (cid:3) = δ ( r − r ′ ) , other relations being zero. The creation operator ψ † ( r ) generates the Fock space F ( ψ ),which is the space of microstates [55]. Then the statistical state is given by the averages(2.16), with the trace over F ( ψ ).In order to define the statistical operator ˆ ρ , we need to specify the conditions makingthe statistical ensemble representative. One evident condition is the normalization of thestatistical operator, < ˆ1 F > = 1 , (2.18)where ˆ1 F is the unity operator in F ( ψ ). The Hamiltonian energy operator ˆ H [ ψ ], which is afunctional of ψ and ψ † , defines the internal energy < ˆ H [ ψ ] > = E , (2.19)which is another statistical condition. The total number of particles N is given by theaverage < ˆ N [ ψ ] > = N , (2.20)of the number-of-particle operatorˆ N [ ψ ] ≡ Z ψ † ( r ) ψ ( r ) d r . (2.21)13imilarly, there can exist other condition operators ˆ C i [ ψ ], with i = 1 , , . . . , whose averagesdefine additional statistical conditions < ˆ C i [ ψ ] > = C i . (2.22)The statistical operator ˆ ρ of an equilibrium system is defined as the minimizer of theinformation functional [55] I [ ˆ ρ ] = Tr ˆ ρ ln ˆ ρ + λ (Tr ˆ ρ −
1) ++ β (cid:16) Tr ˆ ρ ˆ H [ ψ ] − E (cid:17) − βµ (cid:16) Tr ˆ ρ ˆ N [ ψ ] − N (cid:17) + β X i ν i (cid:16) Tr ˆ ρ ˆ C i [ ψ ] − C i (cid:17) , (2.23)in which λ , β , βµ , and βν i are the appropriate Lagrange multipliers. Minimizing (2.23)gives ˆ ρ = exp( − βH [ ψ ])Tr F ( ψ ) exp( − βH [ ψ ]) , (2.24)where the trace is over F ( ψ ) and H [ ψ ] ≡ ˆ H [ ψ ] − µ ˆ N [ ψ ] + X i ν i ˆ C i [ ψ ] (2.25)is the grand Hamiltonian . The Lagrange multiplier β = 1 /T is the inverse temperature and µ is called the chemical potential.After this, one can explicitly define what actually is the single-particle density matrix,which till now has appeared as an abstract notion. This density matrix is ρ ( r , r ′ ) ≡ < ψ † ( r ′ ) ψ ( r ) > . (2.26)The evolution equations for the field variables ψ ( r ) are obtained as follows [57]. Byintroducing the temporal energy operatorˆ E [ ψ ] ≡ Z ψ † ( r , t ) i ∂∂t ψ ( r , t ) d r , (2.27)we define the action functional Γ[ ψ ] ≡ Z (cid:16) ˆ E [ ψ ] − H [ ψ ] (cid:17) dt . (2.28)The evolution equations for ψ ( r , t ) and ψ † ( r , t ) are given by the extremization of the actionfunctional, δ Γ[ ψ ] δψ † ( r , t ) = 0 , (2.29)and by the Hermitian conjugation of the latter variational equation. In view of the actionfunctional (2.28), Eq. (2.29) yields i ∂∂t ψ ( r , t ) = δH [ ψ ] δψ † ( r , t ) . (2.30)14his equation is equivalent [55] to the Heisenberg equation i ∂∂t ψ ( r , t ) = [ ψ ( r , t ) , H [ ψ ]] . The initial condition for the evolution equation is ψ ( r ,
0) = ψ ( r ). The evolution is governedby the same grand Hamiltonian (2.25) as that characterizing the statistical operator (2.24).In the case of a nonequilibrium statistical system, additional conditions (2.22) shouldinclude the information on the initial values < ˆ A (0) > for the considered operators ˆ A ( t ).The procedure, described above, determines the standard way of characterizing a repre-sentative statistical ensemble. Here, it is the triplet of the Fock space of microstates F ( ψ ),the statistical operator (2.24), and of the evolution equations (2.29) or (2.30). For an equi-librium system, the grand Hamiltonian (2.25) may not need the last term with conditionaloperators. After a statistical ensemble has been constructed, we may pose the question whether BECoccurs in the system. Then we remember that BEC implies the macroscopic occupation ofa single quantum state. Quantum states, labelled by a multi-index k , are associated with anorthonormal basis { ϕ k ( r ) } . Expanding the field operator over this basis, we have ψ ( r ) = X k a k ϕ k ( r ) , (2.31)where the operators a k obey the commutation relations (cid:2) a k , a † p (cid:3) = δ kp , [ a k , a p ] = 0 . With expansion (2.31), the density matrix (2.26) takes the form ρ ( r , r ′ ) = X kp < a † k a p > ϕ p ( r ) ϕ ∗ k ( r ′ ) . (2.32)For a while, there is arbitrariness in choosing a basis in expansion (2.31). However, BECis a physical phenomenon and can occur not for an arbitrary chosen quantum state, but fora state naturally related to the considered physical system. This means that the expansionbasis { ϕ k ( r ) } is not arbitrary, but is to be formed by natural orbitals. In terms of the naturalorbitals, the density matrix (2.32) has to enjoy the diagonal expansion [43], which impliesthe quantum-number conservation condition < a † k a p > = δ kp n k , (2.33)where n k ≡ < a † k a k > (2.34)is the occupation number. 15f BEC is associated with a quantum state k , then the field operator (2.31) can beseparated into two parts, ψ ( r ) = ψ ( r ) + ψ ( r ) , (2.35)in which the first term is the operator of condensed particles, ψ ( r ) ≡ a ϕ ( r ) , (2.36)where a ≡ a k , and the second term is the operator of uncondensed particles, ψ ( r ) ≡ X k = k a k ϕ k ( r ) . (2.37)From the quantum-number conservation condition (2.33) it follows that < ψ † ( r ) ψ ( r ′ ) > = 0 , (2.38)since < a † a k > = 0 ( k = k ) . (2.39)And, because of the orthonormality of the basis { ϕ k ( r ) } , we have the orthogonality condition Z ψ † ( r ) ψ ( r ) d r = 0 . (2.40)The density matrix (2.26) takes the form ρ ( r , r ′ ) = < ψ † ( r ′ ) ψ ( r ) > + < ψ † ( r ′ ) ψ ( r ) > . (2.41)The number-of-particle operators are: for condensed particles,ˆ N [ ψ ] ≡ Z ψ † ( r ) ψ ( r ) d r = a † a , (2.42)and for uncondensed particlesˆ N [ ψ ] ≡ Z ψ † ( r ) ψ ( r ) d r = X k = k a † k a k . (2.43)In view of the orthogonality condition (2.40), the number-of-particle operator for the totalnumber of particles is ˆ N [ ψ ] = ˆ N [ ψ ] + ˆ N [ ψ ] . (2.44)The average number of particles in BEC is N = < ˆ N [ ψ ] > = < a † a > . (2.45)And the number of uncondensed particles is N = < ˆ N [ ψ ] > = X k = k n k . (2.46)16he above equations are valid for any system, whether uniform or not uniform, and for BECof arbitrary nature, related to a quantum state k .It is important to stress that operators (2.36) and (2.37) are not separate independentoperators, describing different particles, but ψ ( r ) and ψ ( r ) are simply two parts of oneBose field operator (2.35). This is evident from the commutation relations for ψ ( r ), h ψ ( r ) , ψ † ( r ′ ) i = ϕ ( r ) ϕ ∗ ( r ′ ) , (2.47)and for ψ ( r ), h ψ ( r ) , ψ † ( r ′ ) i = X k = k ϕ k ( r ) ϕ ∗ k ( r ′ ) , (2.48)which show that neither ψ ( r ) nor ψ ( r ) characterize Bose particles. There exists the solefield operator (2.35) enjoying the Bose commutation relations. This operator is defined onthe Fock space F ( ψ ) generated by ψ † .Using notation (2.42), the BEC criterion (2.3) can be written aslim N →∞ < ˆ N [ ψ ] >N > , (2.49)or, equivalently, as lim N →∞ < a † a >N > . (2.50)Calculating the averages, one employs the statistical ensemble with the grand Hamilto-nian (2.25) containing the chemical potential µ , which is the Lagrange multiplier guarantee-ing the normalization condition (2.20). There is here the sole normalization condition, sincethere exists only one field operator ψ ( r ) describing Bose particles. Phase transitions from a disordered phase to an ordered phase are usually accompanied bysome symmetry breaking [58]. BEC is associated with the global U (1) gauge symmetrybreaking. The fundamental question is whether the gauge symmetry breaking is necessaryand sufficient for the occurrence of BEC. In literature, one can meet controversial statements,some claiming that BEC does not require any symmetry breaking. This, however, is notcorrect. The gauge symmetry breaking is necessary and sufficient for the occurrence of BEC .The equivalence of BEC and gauge symmetry breaking has been discussed in recentpapers [59–62] and thoroughly explained in the review article [63]. Considering these phe-nomena, one should always keep in mind that, in finite systems, there are neither rigor-ously defined phase transitions nor symmetry breaking. Both of them can happen only inthermodynamic limit. So, the existence or absence of these phenomena acquires a correctmathematical meaning only under the thermodynamic limiting test, either in form (2.2) orin forms (2.4) and (2.5). However, one often talks about BEC or symmetry breaking evenin the case of a finite, but large, statistical system, with N ≫
1, keeping in mind that theproperties of the system are asymptotically close to those the system would possess in thethermodynamic limit. 17he U (1) gauge transformation can be represented as the transformation ψ ( r ) −→ ψ ( r ) e iα (2.51)for the field operator, with α being a real number. The Hamiltonian H [ ψ ] is assumed to beinvariant under the gauge transformation (2.51). The gauge symmetry of the system can bebroken by the Bogolubov method of infinitesimal sources [64,65], by defining H ε [ ψ ] ≡ H [ ϕ ] + ε √ ρ Z h ψ † ( r ) + ψ ( r ) i d r , (2.52)where ε is a real parameter and ρ is the mean particle density. The related statistical operatoris ˆ ρ ε ≡ exp( − βH ε [ ψ ])Tr exp( − βH ε [ ψ ]) , (2.53)with the trace over F ( ψ ). The operator averages are defined as < ˆ A > ε ≡ Tr ˆ ρ ε ˆ A . (2.54)According to the Bogolubov method of quasiaverages [64,65], one should, first, take thethermodynamic limit, after which the limit ε → ε a function ε N depending on N , such that it would appropriately tend to zerotogether with the thermodynamic limit [54,67,68]. But for the sake of clarity, we shall usehere the standard Bogolubov method [64,65] of quasiaverages.One can say that there happens the local spontaneous gauge-symmetry breaking , whenlim ε → lim N →∞ < ψ ( r ) > ε = 0 , (2.55)at least for some r . And the global spontaneous gauge-symmetry breaking implies thatlim ε → lim N →∞ N Z | < ψ ( r ) > ε | d r > . (2.56)Because of definition (2.36), one has Z | < ψ ( r ) > ε | d r = | < a > ε | . (2.57)Hence, inequality (2.56) of spontaneous gauge-symmetry breaking becomeslim ε → lim N →∞ | < a > ε | N > . (2.58)By the Cauchy-Schwarz inequality, | < a > ε | ≤ q < a † a > ε ε and N . Thereforelim ε → lim N →∞ | < a > ε | N ≤ lim ε → lim N →∞ < a † a > ε N . (2.59)This tells us, that the spontaneous gauge symmetry breaking (2.58) results in BEC, inagreement with condition (2.50).Moreover, inequality (2.59) can be made the equality. Recall that Hamiltonian (2.52) isa functional H ε [ ψ ] ≡ H ε [ ψ , ψ ]of ψ ( r ) and ψ ( r ). And let us introduce the statistical operatorˆ ρ ηε ≡ exp( − βH ε [ η, ψ ])Tr exp( − βH ε [ η, ψ ]) , (2.60)in which the operator ψ ( r ) is replaced by a function η ( r ). The related average of an operatorˆ A is < ˆ A > ηε ≡ Tr ˆ ρ ηε ˆ A . (2.61)Let us also define a class of correlation functions given by the form C ε ( ψ , ψ ) ≡ < . . . ψ † . . . ψ † . . . ψ . . . ψ > ε . (2.62)Replacing here all operators ψ † and ψ ( r ) by functions η ∗ ( r ) and η ( r ), we get the class ofcorrelation functions C ε ( η, ψ ) ≡ < . . . η ∗ . . . ψ † . . . η . . . ψ > ε . (2.63)It is assumed that the function η ( r ) is normalized to the same number of condensed particles,as ψ ( r ), such that Z < ψ † ( r ) ψ ( r ) > ε d r = Z | η ( r ) | d r = N . (2.64)Then the following statement holds [65]. Bogolubov theorem . In thermodynamic limit, the correlation functions (2.62) and(2.63), under normalization condition (2.64), coincide with each other,lim N →∞ C ε ( ψ , ψ ) = lim N →∞ C ε ( η, ψ ) , (2.65)for any real ε . In particular,lim ε → lim N →∞ C ε ( ψ , ψ ) = lim ε → lim N →∞ C ε ( η, ψ ) . (2.66)From the Bogolubov theorem it follows thatlim ε → lim N →∞ < ψ ( r ) > ε = η ( r ) . (2.67)19lso, we have lim ε → lim N →∞ < ψ † ( r ) ψ ( r ′ ) > ε = η ∗ ( r ) lim ε → lim N →∞ < ψ ( r ′ ) > ηε . (2.68)If η ( r ) is not identically zero, then the quantum-number conservation condition (2.38) ac-quires the form lim ε → lim N →∞ < ψ ( r ) > ηε = 0 . (2.69)Hence, for the field operator (2.35) one getslim ε → lim N →∞ < ψ ( r ) > ε = η ( r ) . (2.70)The condition (2.56) of global spontaneous gauge-symmetry breaking becomeslim N →∞ N Z | η ( r ) | d r > . (2.71)This, according to normalization (2.64), means the existence of BEC.To be more precise, we notice thatlim ε → lim N →∞ N Z | < ψ ( r ) > ε | d r == lim ε → lim N →∞ | < a > ε | N = lim N →∞ N Z | η ( r ) | d r . (2.72)At the same time, we findlim ε → lim N →∞ < ˆ N [ ψ ] > ε N = lim ε → lim N →∞ < a † a > ε N = lim N →∞ N Z | η ( r ) | d r . (2.73)Comparing the latter equations, we obtainlim ε → lim N →∞ N Z | < ψ ( r ) > ε | d r = lim ε → lim N →∞ < ˆ N [ ψ ] > ε N , (2.74)or in another form, lim ε → lim N →∞ | < a > ε | N = lim ε → lim N →∞ < a † a > ε N . (2.75)Equations (2.74) and (2.75) demonstrate that the spontaneous gauge symmetry breakingleads to the existence of BEC. This conclusion holds for any equilibrium system, whetheruniform or nonuniform. In the case of uniform systems, Eq. (2.75) was derived in Refs.[59–62].For uniform systems, there also exist the following theorem [60–62], first, proved byGinibre [69]. One considers the thermodynamic potentialsΩ ε ≡ − T ln Tr ( − βH ε [ ψ , ψ ]) (2.76)20nd Ω ηε ≡ − T ln Tr ( − βH ε [ η, ψ ]) , (2.77)where η is the minimizer of Eq. (2.77), such thatΩ ηε = inf x Ω xε . (2.78) Ginibre theorem . For the thermodynamic potentials (2.76) and (2.77), under condition(2.78), in the thermodynamic limit, one haslim N →∞ Ω ε N = lim N →∞ Ω ηε N (2.79)for any real ε , including ε → spontaneous gauge symmetry break-ing is a sufficient condition for the occurrence of BEC .The fact that the symmetry breaking is also a necessary condition for BEC was, first,proved by Roepstorff [70] and recently this proof was generalized by Lieb et al. [60,62]. Forthis purpose, one compares the average < a † a > for a uniform system without symmetrybreaking and the average < a > ε in the presence of the gauge symmetry breaking. Roepstorff theorem . In the thermodynamic limit,lim N →∞ < a † a >N ≤ lim ε → lim N →∞ | < a > ε | N . (2.80)This theorem shows that the occurrence of BEC necessarily leads to the gauge symmetrybreaking. More details on the relation between BEC and gauge symmetry breaking can befound in the review article [63].Thus, the conclusion is:
The spontaneous gauge-symmetry breaking is the necessary and sufficient condition forBose-Einstein condensation . Describing a system with BEC, one can follow the procedure of the previous sections, workingwith the field operator (2.35) defined on the Fock space F ( ψ ). This operator can formallybe partitioned into two terms. However, neither of these terms represents particles, sincethe commutation relations (2.47) and (2.48) are not of Bose type. Dealing with the fieldoperator (2.35), one should accomplish calculations in finite space, passing after this to thethermodynamic limit. Such an approach has three weak points.First, in practical calculations, it requires the use of perturbation theory with respectto atomic interactions, as has been done by Belyaev [71]. Hence, it is limited to weaklyinteracting Bose gases.Second, the mentioned perturbation theory is singular, being plagued by divergences. Sothat only the lowest orders of the perturbation theory are meaningful.Third, the commutation relations (2.47) and (2.48) are cumbersome and not convenientin calculations. 21ut we know from the theorems of the previous section that in the thermodynamic limitthe operator term ψ ( r ) reduces to a function η ( r ). Therefore, it is tempting to replace, fromthe very beginning, the operator ψ ( r ) in Eq. (2.35) by another operatorˆ ψ ( r ) ≡ η ( r ) + ψ ( r ) , (2.81)in which the operator ψ ( r ) has been replaced by a function η ( r ). The procedure of replacing ψ ( r ) by ˆ ψ ( r ) is called the Bogolubov shift [64,65,72,73].From the requirement that ˆ ψ ( r ) is a Bose operator, and because η ( r ) is a nonoperatorfunction, it follows that ψ ( r ) is a Bose field operator, with the standard Bose commutationrelations h ψ ( r ) , ψ † ( r ′ ) i = δ ( r − r ′ ) . (2.82)It is evident that to deal with these usual nice commutation relations is much simpler thanwith the awkward relations (2.47) and (2.48).In the shifted field operator (2.81), the term η ( r ) is named the condensate wave function ,and the term ψ ( r ) is the operator of uncondensed particles. To be correctly defined, theseterms are assumed to preserve the basic properties typical of ψ ( r ) and ψ ( r ). Thus, theorthogonality condition (2.40) now reads as Z η ∗ ( r ) ψ ( r ) d r = 0 . (2.83)And the quantum-number conservation condition (2.38), similarly to Eq. (2.69), now be-comes < ψ ( r ) > = 0 . (2.84)The number of condensed atoms (2.42) can be represented asˆ N = N ˆ1 F , (2.85)where, ˆ1 F is the unity operator in the appropriate Fock space and N , in agreement withnormalization (2.64), is N = Z | η ( r ) | d r . (2.86)By analogy with Eq. (2.45), we have N = < ˆ N > . (2.87)The number operator of noncondensed atoms (2.43) isˆ N = Z ψ † ( r ) ψ ( r ) d r . So that the number (2.46) of uncondensed atoms is N = < ˆ N > . (2.88)22ue to the orthogonality condition (2.83), the operator of the total number of particles isthe sum ˆ N ≡ Z ˆ ψ ( r ) ˆ ψ ( r ) d r = ˆ N + ˆ N . (2.89)Hence, the total number of particles is N = < ˆ N > = N + N . The convenience of the Bogolubov shift (2.81) is also in the fact that it explicitly breaksthe gauge symmetry since < ˆ ψ ( r ) > = η ( r ) , (2.90)which is necessary for correctly describing BEC. The latter equation makes it possible tocall the condensate wave function η ( r ) the system order parameter.The Bogolubov shift (2.81) is the basis for the majority of calculations for weakly non-ideal Bose gases at low temperatures, when almost all particles are in BEC [64,65,72,73].Perturbation theory for asymptotically weak interactions and low temperatures has beendeveloped for uniform [74,75] as well as for nonuniform [76,77] gases.However, as soon as one tries to describe not asymptotically weak interactions or highertemperatures, one encounters the Hohenberg-Martin dilemma [74]. Hohenberg and Mar-tin [74] showed that the theory, based on the standard grand canonical ensemble, wherethe gauge symmetry is broken by means of the Bogolubov shift, is internally inconsistent.Depending on the way of calculations, one gets either an unphysical gap in the spectrumof collective excitations, or local conservation laws, together with general thermodynamicrelations, become invalid. Recall that the excitation spectrum, according to the Hugenholtz-Pines theorem, must be gapless [65,78]. While conserving approximations [79] usually givea gap in the spectrum [80–82].The standard attempts to cure the problem are based on what Bogolubov [65] named”the mismatch of approximations”. One either arbitrarily adds some phenomenologicalterms or removes, without justification, other terms. The most popular trick is the omissionof anomalous averages, as first, was suggested by Shohno [83] and analysed by Reatto andStraley [84]. In recent years, the Shohno trick [83] is often ascribed to Popov, although, asis easy to infer from the Popov works [85–88], cited in this regard, he has never suggestedor used such an unjustified trick.Of course, all phenomenological attempts, involving the mismatch of approximations, ashas already been mentioned by Bogolubov [65], cannot cure the problem. Not self-consistentapproaches render the system unstable, spoil thermodynamic relations, and disrupt theBose-Einstein condensation phase transition from the second-order to the incorrect first-order transition [89–92]. A detailed analysis of this problem has been done in Ref. [57]. The origin of the Hohenberg-Martin dilemma is in the use of a nonrepresentative ensemblefor a Bose-condensed system with the gauge symmetry broken by the Bogolubov shift (2.81).Using a representative ensemble [54,55] cures the problem [56,57] and makes it possible todevelop a fully self-consistent theory, free of paradoxes [93–95]. This theory is conservingand gapless by construction, independently of the involved approximation [93–98].23o determine a representative ensemble, one should start with the specification of thespace of microstates, which depends on the choice of the accepted variables. When oneworks with the field operator ψ ( r ), as in Secs. 2.4. and 2.5, the corresponding space ofmicrostates is the Fock space F ( ψ ) generated by ψ † ( r ), as is explained in Ref. [55]. But,as soon as the Bogolubov shift (2.81) has been accomplished, the new field operator ˆ ψ ( r ) isdefined on another space, which is the Fock space F ( ψ ) generated by ψ † ( r ), with the spaces F ( ψ ) and F ( ψ ) being mutually orthogonal [99]. In the space F ( ψ ), there was the sole fieldvariable ψ ( r ), while in the space F ( ψ ) there are now two variables, the condensate wavefunction η ( r ) and the field operator of uncondensed particles ψ ( r ). Respectively, instead ofone normalization condition (2.20), there are two normalization conditions (2.87) and (2.88).The conservation-number condition (2.84) can be reduced to the standard form of thestatistical conditions (2.22) by defining the Hermitian operatorˆΛ ≡ Z h λ ( r ) ψ † ( r ) + λ ∗ ( r ) ψ ( r ) i d r , (2.91)which can be called the linear killer . This is because the Lagrange multiplier λ ( r ) has to bechosen so that < ˆΛ > = 0 , (2.92)which requires the absence of the terms linear in ψ ( r ) in the related grand Hamiltonian[57].Statistical averages of operators from the algebra of observables O ≡ { ˆ A ( t ) } are given as < ˆ A ( t ) > ≡ Tr F ( ψ ) ˆ ρ ˆ A ( t ) , (2.93)with a statistical operator ˆ ρ ≡ ˆ ρ (0). For instance, the energy Hamiltonianˆ H = ˆ H [ ˆ ψ ] ≡ ˆ H [ η, ψ ]defines the internal energy E = < ˆ H > . (2.94)In what follows, we shall omit the notation of spaces, over which the trace is taken, in orderto avoid cumbersome expressions.The statistical operator ˆ ρ is obtained from the minimization of the information functional I [ ˆ ρ ] under the statistical conditions (2.18), (2.94), (2.87), (2.88), and (2.92). The informationfunctional is I [ ˆ ρ ] = Tr ˆ ρ ln ˆ ρ + λ (Tr ˆ ρ −
1) ++ β (cid:16) Tr ˆ ρ ˆ H − E (cid:17) − βµ (cid:16) Tr ˆ ρ ˆ N − N (cid:17) − βµ (cid:16) Tr ˆ ρ ˆ N − N (cid:17) − β Tr ˆ ρ ˆΛ . (2.95)Its minimization yields the statistical operatorˆ ρ = exp( − βH )Tr exp( − βH ) , (2.96)with the grand Hamiltonian H ≡ ˆ H − µ ˆ N − µ ˆ N − ˆΛ . (2.97)24his Hamiltonian is, clearly, a functional H = H [ η, ψ ] of η ( r ) and ψ ( r ).The evolution laws are prescribed by extremizing the action functional, as is describedin Sec. 2.4. To this end, we define the temporal energy operatorˆ E ≡ Z ˆ ψ † ( r ) i ∂∂t ˆ ψ ( r ) d r . (2.98)With the Bogolubov shift (2.81), this takes the formˆ E = Z (cid:20) η ∗ ( r ) i ∂∂t η ( r ) + ψ † ( r ) i ∂∂t ψ ( r ) (cid:21) d r , (2.99)which shows that ˆ E = ˆ E [ η, ψ ] is a functional of η ( r ) and ψ ( r ). The effective action is alsoa functional of these variables, Γ[ η, ψ ] ≡ Z (cid:16) ˆ E − H (cid:17) dt . (2.100)The evolution laws are given by the extremization of the action functional (2.100) withrespect to the condensate wave function, δ Γ[ η, ψ ] δη ∗ ( r , t ) = 0 , (2.101)and with respect to the field operator of uncondensed particles, δ Γ[ η, ψ ] δψ † ( r , t ) = 0 . (2.102)These evolution equations, owing to the form of the action functional (2.100), are equivalentto the equation i ∂∂t η ( r , t ) = δH [ η, ψ ] δη ∗ ( r , t ) (2.103)for the condensate variable, and to the equation i ∂∂t ψ ( r , t ) = δH [ η, ψ ] δψ † ( r , t ) (2.104)for the field variable of uncondensed particles.Thus, the representative statistical ensemble for a Bose system with the gauge symmetrybreaking, induced by the Bogolubov shift (2.81), is the triplet {F ( ψ ) , ˆ ρ, ∂t } formed bythe Fock space F ( ψ ), generated by ψ † ( r ), the statistical operator (2.96), with the grandHamiltonian (2.97), and the evolution laws (2.103) and (2.104).In the case of an equilibrium system, we can introduce the grand thermodynamic potentialΩ = − T ln Tr e − βH , (2.105)defining all thermodynamics of the system. For example, the fraction of condensed atoms is n ≡ N N = − N ∂ Ω ∂µ , (2.106)25nd the fraction of uncondensed atoms is n ≡ N N = − N ∂ Ω ∂µ . (2.107)The equation for the condensate function is obtained from the statistical averaging of Eq.(2.103), under the condition that, for an equilibrium system, η ( r ) does not depend on time.Then we get the equation δ Ω δη ∗ ( r ) = < δH [ η, ψ ] δη ∗ ( r ) > = 0 , (2.108)which is equivalent to the Bogolubov minimization of the thermodynamic potential withrespect to the condensate variable [64,65,72,73].The free energy can be defined as F = Ω + µ N + µ N . (2.109)At the same time, keeping in mind the standard form of the free energy F = Ω + µN , (2.110)we find the expression for the system chemical potential µ = µ n + µ n . (2.111)The same form for the chemical potential (2.111) can be derived from the usual definition µ ≡ ∂F∂N . (2.112)The right-hand side here can be written as ∂F∂N = ∂F∂N ∂N ∂N + ∂F∂N ∂N ∂N , where ∂F∂N = µ , ∂F∂N = µ . Assuming that n and n are fixed in the thermodynamic limit N → ∞ , from the relations N = n N , N = n N , we have ∂N ∂N = n , ∂N ∂N = n . Combining these derivatives in definition (2.112), we get the same expression (2.111) for thechemical potential.It is possible to show [57] that the dispersion of the number-of-particle operator is givenby ∆ ( ˆ N ) = T ∂N∂µ , (2.113)26here the dispersion of a self-adjoint operator ˆ A is defined as∆ ( ˆ A ) ≡ < ˆ A > − < ˆ A > . At the end, the free energy can be represented as a function F = F ( T, V, N ) of temper-ature T , volume V , and the particle number N , with the differential dF = − S dT − P dV + µ dN , (2.114)in which S is entropy and P , pressure. And the grand potential (2.105) is a functionΩ = Ω( T, V, µ ), with the differential d Ω = − S dT − P dV − N dµ . (2.115)All thermodynamics follows from the above expressions. A more detailed discussion is givenin Ref. [57].
The existence of BEC, as such, requires the validity of an important necessary condition.Generally, the total number of atoms N is the sum P k n k of the occupation numbers forquantum states labelled by a multi-index k . The occurrence of BEC, by definition, meansthe microscopic occupation of a single quantum state k , when N ∝ N , in agreement withcondition (2.3). Only in such a case, it is meaningful to separate out of the sum P k n k asingle term, related to BEC, obtaining N = N + X k = k n k . (2.116)Mathematically, the possibility of that separation necessarily implies that, in thermodynamiclimit, the distribution n k over quantum states diverges when k → k . Hence, the necessarycondition for the BEC existence is lim N →∞ lim k → k n k = 0 , (2.117)where the thermodynamic limit is invoked in order to make the BEC rigorously defined.In the representative ensemble of the previous Sec. 2.8, the condensate wave functioncan be written as η ( r ) = p N ϕ ( r ) , (2.118)while the operator of uncondensed atoms can be expanded over the natural orbitals as ψ ( r ) = X k = k a k ϕ k ( r ) . (2.119)Hence, the occupation numbers are n k ≡ < a † k a k > , (2.120)27ith the statistical averaging defined in Eq. (2.93).If we turn to the terminology of Green functions, then the necessary condition (2.117) canbe connected with the properties of the poles of Green functions. Under the spontaneouslybroken gauge symmetry, the poles of the first-order and second-order Green functions coin-cide, that is, the single-particle spectrum coincides with the spectrum of collective excitations ε k [65,100]. For the latter, the necessary condition of condensate existence (2.117) translatesinto lim k → k ε k = 0 , (2.121)with the condition n k ≥ ε k ≥ , Im ε k ≤ . (2.122)Conditions (2.117) or (2.121) impose a constraint on the Lagrange multiplier µ , which hasto be such that to make the spectrum ε k gapless in the sense of the limit (2.121).To illustrate the properties (2.117) and (2.121), let us consider a uniform system, when k = 0. For a Bose-condensed uniform system, Bogolubov [65] rigorously proved the inequal-ities for the occupation numbers (2.120) in the case of nonzero temperature, n k ≥ mn T k − , (2.123)and at zero temperature, n k ≥ mn ε k k −
12 ( T = 0) , (2.124)where ε k is the real part of the spectrum of collective excitations. These inequalities can beslightly improved [101], resulting, for finite temperatures, in n k ≥ mn Tk −
12 (2.125)and for zero temperature, in n k ≥ mn ε k k −
12 ( T = 0) . (2.126)At zero temperature, one can use the Feynman relation [78,102,103] ε k = k mS ( k ) ( T = 0) , (2.127)in which S ( k ) is the structure factor. Then the Bogolubov inequality (2.126) takes the form n k ≥ n S ( k ) −
12 ( T = 0) . (2.128)At zero temperature, the structure factor possesses the long-wave limit [104] as S ( k ) ≃ k mc ( T = 0 , k → . (2.129)28ence, from the Feynman relation (2.127), one has ε k ≃ ck ( k → . (2.130)The same long-wave limit exists for the spectrum of collective excitations at finite temper-atures [65]. That is, limit (2.121) is valid for any T . From the above inequalities, it followsthat, for finite temperatures, n k ≥ mn Tk ( k → , and for zero temperature, n k ≥ mn c k ( T = 0 , k → . Therefore, in any case, the BEC existence condition (2.117) holds true.Condition (2.117) can also be generalized for nonequilibrium nonuniform systems. Butthe latter should, at least, be locally equilibrium in order that the meaning of thermodynamicphases be locally preserved. Then for ψ ( r , t ) one has the same expansion (2.119), but with a k ( t ) being a function of time. The occupation number (2.120) becomes a function of time, n k = n k ( t ). Then condition (2.117) defines the Lagrange multiplier µ as a function of time µ ( t ). If one employs the local-density approximation, then µ can also be a function of thespatial variable. Expressions, defining the superfluid fraction, can have different forms for uniform and nonuni-form systems. It is, therefore, important to recall the most general definition of the superfluidfraction, which could be applied to arbitrary systems, whether uniform or not. This generaldefinition is based on the calculation of the response to a velocity boost imposed on thesystem.Let ˆ H be the energy Hamiltonian of an immovable system, and let ˆ H v be the energyHamiltonian of the system moving, as a whole, with velocity v . The statistical operator,related to the moving system is denoted as ˆ ρ v . The corresponding statistical average of anoperator ˆ A is < ˆ A > v ≡ Tr ˆ ρ v ˆ A . (2.131)The return to the immovable system is realized through the limit < ˆ A > = lim v → < ˆ A > v . (2.132)The momentum operator of the total moving system can be represented asˆ P v ≡ ∂ ˆ H v ∂ v . (2.133)The momentum operator of the immovable system isˆ P = lim v → ˆ P v . (2.134)29he superfluid fraction can be defined as a fraction of particles nontrivially responding tothe velocity boost, n s ≡ mN lim v → ∂∂ v · < ˆ P v > v . (2.135)This is the most general definition, valid for arbitrary systems [10,57].For an equilibrium Bose system, with the grand Hamiltonian H v ≡ ˆ H v − µ ˆ N − µ ˆ N − ˆΛ , (2.136)which differs from Eq. (2.97) by the velocity boosted term ˆ H v , the statistical operator isˆ ρ v ≡ exp( − βH v )Tr exp( − βH v ) . (2.137)The differentiation in Eq. (2.135) is accomplished according to the rule of differentiationwith respect to parameters [105], which gives ∂∂ v · < ˆ P v > v = < ∂∂ v · ˆ P v > v − β cov (cid:18) ˆ P v , ∂H v ∂ v (cid:19) . Here the last term is the covariance defined ascov (cid:16) ˆ A, ˆ B (cid:17) ≡ < ˆ A ˆ B + ˆ B ˆ A > v − < ˆ A > v < ˆ B > v . Using definition (2.133), we obtain n s = 13 mN (cid:20) lim v → < ∂∂ v · ˆ P v > − β ∆ ( ˆ P ) (cid:21) , (2.138)where ∆ ( ˆ P ) ≡ < ˆ P > − < ˆ P > . The same form (2.138) can be derived from the definition [106,107] of the superfluidfraction n s ≡ mN lim v → ∂ Ω v ∂ v = 13 mN lim v → ∂ F v ∂ v , (2.139)applicable to equilibrium systems. Here the grand potential for the moving system isΩ v ≡ − T ln Tr exp( − βH v ) , (2.140)and the free energy for that system is F v ≡ − T ln Tr exp( − β ˆ H v ) . (2.141)The first derivative gives ∂F v ∂ v = ∂ Ω v ∂ v = < P v > v . (2.142)For an equilibrium system at rest, one has∆ ( ˆ P ) = < ˆ P > (cid:16) < ˆ P > = 0 (cid:17) . (2.143)30hen, one gets n s = 13 mN (cid:18) lim v → < ∂∂ v · ˆ P v > − β < ˆ P > (cid:19) . (2.144)Taking into account Eq. (2.143), we see that Eqs. (2.138) and (2.144) coincide.To specify the expression for the superfluid fraction, we can use the definition for theoperator of momentum ˆ P v ≡ Z ˆ ψ † v ( r , t )( − i ∇ ) ˆ ψ v ( r , t ) d r (2.145)of the moving system. The field operator of the moving system can be expressed throughthe field operator of the system at rest by means of the Galilean transformationˆ ψ v ( r , t ) = ˆ ψ ( r − v t, t ) exp (cid:26) i (cid:18) m v · r − mv t (cid:19)(cid:27) . (2.146)Consequently, ˆ P v = Z ˆ ψ † ( r ) ( − i ∇ + m v ) ˆ ψ ( r ) d r . (2.147)The energy Hamiltonian of the system at rest is ˆ H = ˆ H [ ˆ ψ ], while that of the moving systemis ˆ H v = ˆ H [ ˆ ψ v ]. Differentiating Eq. (2.147) gives ∂∂ v · ˆ P v = 3 m ˆ N .
We can define the dissipated heat of the considered quantum system as Q ≡ ∆ ( ˆ P )2 mN = < ˆ P > mN , (2.148)which is to be compared with the heat dissipated in a classical system, Q ≡ T . (2.149)Finally, the superfluid fraction (2.144) reduces to the form n s = 1 − QQ . (2.150)This formula is valid for arbitrary nonuniform equilibrium systems, including periodic latticepotentials.The superfluid fraction, as is known, is not directly related to the condensate fraction.A straightforward example is the liquid He, which at low temperature T → T c [116–118], when the superfluid and condensatefractions coincide [9]. Generally, superfluidity can exist without BEC, and vice versa.31 .11 Equations of Motion All observable quantities are functionals of the condensate function η ( r ) and the operatorof uncondensed particles ψ ( r ). These variables are defined by the evolution equations(2.103) and (2.104). To derive these equations explicitly, we need to specify the Hamiltonianˆ H = ˆ H [ η, ψ ].Let us take the energy Hamiltonian in the standard formˆ H = Z ˆ ψ ( r ) (cid:18) − ∇ m + U (cid:19) ˆ ψ ( r ) d r ++ 12 Z ˆ ψ † ( r ) ˆ ψ † ( r ′ )Φ( r − r ′ ) ˆ ψ ( r ′ ) ˆ ψ ( r ) d r d r ′ , (2.151)in which m is atomic mass, U = U ( r ) is an external potential, and Φ( r ) = Φ( − r ) is thebinary atomic interaction potential. After substituting here the Bogolubov shift (2.81),Hamiltonian (2.151) can be rewritten as the sum of five terms, depending on the number offactors of ψ . The same concerns the grand Hamiltonian (2.97), for which we obtain H = X n =0 H ( n ) . (2.152)Here the zero-order term is H (0) = Z η ∗ ( r ) (cid:18) − ∇ m + U − µ (cid:19) η ( r ) d r ++ 12 Z Φ( r − r ′ ) | η ( r ′ ) | | η ( r ) | d r d r ′ . (2.153)To satisfy the quantum-number conservation condition (2.84), the Hamiltonian should notcontain the terms linear in ψ [57]. For this purpose, the Lagrange multiplier λ ( r ) in operator(2.91) is to be taken such that to cancel all linear terms, resulting in H (1) = 0 . (2.154)For the second-order term, we have H (2) = Z ψ † ( r ) (cid:18) − ∇ m + U − µ (cid:19) ψ ( r ) d r ++ Z Φ( r − r ′ ) h | η ( r ) | ψ † ( r ′ ) ψ ( r ′ ) + η ∗ ( r ) η ( r ′ ) ψ † ( r ′ ) ψ ( r )++ 12 η ∗ ( r ) η ∗ ( r ′ ) ψ ( r ′ ) ψ ( r ) + 12 η ( r ) η ( r ′ ) ψ † ( r ′ ) ψ † ( r ) (cid:21) d r d r ′ . (2.155)The third-order term is H (3) = Z Φ( r − r ′ ) h η ∗ ( r ) ψ † ( r ′ ) ψ ( r ′ ) ψ ( r ) + ψ † ( r ) ψ † ( r ′ ) ψ ( r ′ ) η ( r ) i d r d r ′ . (2.156)32nd the fourth-order term is H (4) = 12 Z ψ † ( r ) ψ † ( r ′ )Φ( r − r ′ ) ψ ( r ′ ) ψ ( r ) d r d r ′ . (2.157)Inserting Hamiltonian (2.152) into the evolution equation (2.103) yields i ∂∂t η ( r , t ) = (cid:18) − ∇ m + U − µ (cid:19) η ( r , t ) ++ Z Φ( r − r ′ ) h | η ( r ′ ) | η ( r ) + ˆ X ( r , r ′ ) i d r ′ , (2.158)with the correlation operatorˆ X ( r , r ′ ) ≡ ψ † ( r ′ ) ψ ( r ′ ) η ( r ) + ψ † ( r ′ ) η ( r ′ ) ψ ( r ) + η ∗ ( r ′ ) ψ ( r ′ ) ψ ( r ) + ψ † ( r ′ ) ψ ( r ′ ) ψ ( r ) . And Eq. (2.104), with Hamiltonian (2.152), gives i ∂∂t ψ ( r , t ) = (cid:18) − ∇ m + U − µ (cid:19) ψ ( r , t ) ++ Z Φ( r − r ′ ) h | η ( r ′ ) | ψ ( r ) + η ∗ ( r ′ ) η ( r ) ψ ( r ′ ) + η ( r ′ ) η ( r ) ψ † ( r ′ ) + ˆ X ( r , r ′ ) i d r ′ . (2.159)The equation for the condensate wave function is obtained by averaging Eq. (2.158). Forthis purpose, let us define the normal density matrix ρ ( r , r ′ ) ≡ < ψ † ( r ′ ) ψ ( r ) > . (2.160)As soon as the gauge symmetry is broken, there arises the anomalous density matrix σ ( r , r ′ ) ≡ < ψ ( r ′ ) ψ ( r ) > . (2.161)The density of BEC is ρ ( r ) ≡ | η ( r ) | , (2.162)and the density of uncondensed atoms is ρ ( r ) ≡ ρ ( r , r ) = < ψ † ( r ) ψ ( r ) > . (2.163)The diagonal part of Eq. (2.161) is the anomalous average σ ( r ) ≡ σ ( r , r ) = < ψ ( r ) ψ ( r ) > . (2.164)The value | σ ( r ) | has the meaning of the density of pair-correlated particles [99]. The totaldensity of atoms ρ ( r ) = ρ ( r ) + ρ ( r ) (2.165)is the sum of densities (2.162) and (2.163). Also, we need the notation for the triple correlator ξ ( r , r ′ ) ≡ < ψ † ( r ′ ) ψ ( r ′ ) ψ ( r ) > . (2.166)33sing the above notation, we get < ˆ X ( r , r ′ ) > = ρ ( r ′ ) η ( r ) + ρ ( r , r ′ ) η ( r ′ ) + σ ( r , r ′ ) η ∗ ( r ′ ) + ξ ( r , r ′ ) . Finally, averaging Eq. (2.158), we obtain i ∂∂t η ( r , t ) = (cid:18) − ∇ m + U − µ (cid:19) η ( r , t ) ++ Z Φ( r − r ′ ) [ ρ ( r ′ ) η ( r ) + ρ ( r , r ′ ) η ( r ′ ) + σ ( r , r ′ ) η ∗ ( r ′ ) + ξ ( r , r ′ )] d r ′ . (2.167)This is a general equation for the condensate wave function in the case of an arbitrary Bosesystem. No approximation has been involved in deriving Eq. (2.167).For an equilibrium system, we have ∂∂t η ( r ) = 0 ( equilibrium ) . Then Eq. (2.167) becomes the eigenproblem (cid:20) − ∇ m + U ( r ) (cid:21) η ( r ) ++ Z Φ( r − r ′ ) [ ρ ( r ′ ) η ( r ) + ρ ( r , r ′ ) η ( r ′ ) + σ ( r , r ′ ) η ∗ ( r ′ ) + ξ ( r , r ′ )] d r ′ = µ η ( r ) (2.168)defining η ( r ) and µ .The above equations are valid for any interaction potential Φ( r ), with the sole restrictionthat it is integrable [55], so that (cid:12)(cid:12)(cid:12)(cid:12)Z Φ( r ) d r (cid:12)(cid:12)(cid:12)(cid:12) < ∞ . For dilute gases, when the interaction radius is much shorter than the mean interatomicdistance [7–14], one uses the local interaction potentialΦ( r ) = Φ δ ( r ) (cid:16) Φ ≡ π a s m (cid:17) , (2.169)in which a s is the scattering length. In that case, all equations simplify. For example, thecondensate-function equation (2.168) reads as (cid:20) − ∇ m + U ( r ) (cid:21) η ( r )++ Φ { [ ρ ( r ) + ρ ( r )] η ( r ) + σ ( r ) η ∗ ( r ) + ξ ( r , r ) } = µ η ( r ) . (2.170)This equation is valid for any nonuniform equilibrium Bose system, which can be treated asdilute.Note that a dilute gas can, at the same time, be strongly interacting. Really, the gas isdilute, when the interaction radius r is much shorter than the mean interatomic distance34 , that is, r ≪ a . Then the actual form of the interaction potential is not importantand this potential can be modelled by the local expression (2.169). The scattering lengthcharacterizes the interaction strength, sinceΦ = Z Φ( r ) d r = 4 π a s m . Nothing precludes the scattering length to be larger that the mean interatomic distance. If a s > a , then the average potential energy ρ Φ is larger than the effective kinetic energy, ρ Φ > ρ / m ( a s > a ) , which means that atoms strongly interact with each other. For a strongly interacting system,atomic correlations can be rather important [10,119–122]. But for dilute gases, the influenceof correlations can be taken into account through defining an effective scattering length a s . Though this review article is devoted to nonuniform systems, it is instructive to briefly touchthe uniform case. First, this will illustrate the self-consistency of the theory employing therepresentative statistical ensemble [56,57,93–99]. Second, the theory for uniform systems canbe used for generalizing the approach to nonuniform systems by means of the local-densityapproximation. Also, many formulas in the case of periodic potentials have the structurevery similar to that of expressions for the uniform system.In a uniform system, BEC occurs in the quantum state of zero momentum k = 0. Atomicdensities do not depend on the spatial variable. Thus, the condensate density ρ ( r ) = ρ ≡ N V , (2.171)the density of uncondensed atoms ρ ( r ) = ρ ≡ N V , (2.172)and the total density ρ ( r ) = ρ = ρ + ρ , (2.173)all are constants.For the interaction potential Φ( r ) the Fourier transformΦ k = Z Φ( r ) e − i k · r d r (2.174)is assumed to exist. Plane waves are the natural orbitals ϕ k ( r ) = e i k · r / √ V . Hence, it isconvenient to use everywhere the Fourier transforms.The problem can be explicitly solved by using the Hartree-Fock-Bogolubov (HFB) ap-proximation for Bose-condensed systems [94–98]. Then the thermodynamic potential (2.105)becomes Ω = E B + T V Z ln (cid:0) − e − βε k (cid:1) d k (2 π ) . (2.175)35ere the first term is the nonoperator expression E B = E HF B + 12 X k =0 ( ε k − ω k ) , (2.176)in which E HF B = H (0) − ρ Φ V − V X k =0 Φ k + p ( n k n p + σ k σ p ) ,H (0) = − N "
12 ( ρ + ρ )Φ + 1 V X p =0 ( n p + σ p )Φ p . (2.177)The Bogolubov spectrum is ε k = q ω k − ∆ k , (2.178)where ω k = k m + ρ Φ k + 1 V X p =0 ( n p Φ k + p − n p Φ p + σ p Φ p ) (2.179)and ∆ k = ρ Φ k + 1 V X p =0 σ p Φ k + p . (2.180)Equation (2.168) gives µ = ρ Φ + 1 V X p =0 ( n p + σ p )Φ p . (2.181)And the condition of the BEC existence (2.117) and (2.121) define µ = ρ Φ + 1 V X p =0 ( n p − σ p )Φ p . (2.182)The distributions n p and σ p are the Fourier transforms of the normal density matrix (2.160)and of the anomalous matrix (2.161), respectively. For the former, we have n k = ω k ε k coth (cid:16) ε k T (cid:17) − , (2.183)and for the latter, we find σ k = − ∆ k ε k coth (cid:16) ε k T (cid:17) . (2.184)The density of uncondensed atoms ρ = Z n k d k (2 π ) (2.185)defines the condensate density ρ = ρ − ρ . Consequently, the condensate fraction is n = 1 − ρ ρ . (2.186)36he dissipated heat (2.148) becomes Q = 1 ρ Z k m (cid:0) n k + n k − σ k (cid:1) d k (2 π ) , (2.187)which, according to Eq. (2.150), gives the superfluid fraction n s = 1 − Q T . (2.188)Again, these expressions are simplified in the case of dilute gases, with the local interac-tion potential (2.169). Then the Bogolubov spectrum (2.178) takes the standard form ε k = s ( ck ) + (cid:18) k m (cid:19) . (2.189)Notation (2.179) gives ω k = k m + mc , (2.190)while Eq. (2.180) reduces to ∆ k = ( ρ + σ )Φ . (2.191)The latter expression defines the sound velocity c through the relation∆ k ≡ ∆ ≡ mc . (2.192)And the anomalous average is σ ≡ Z σ k d k (2 π ) . (2.193)For the Lagrange multipliers (2.181) and (2.182) we obtain µ = ( ρ + ρ + σ )Φ (2.194)and, respectively, µ = ( ρ + ρ − σ )Φ . (2.195)Clearly, µ = µ .More details on the derivation of the above equations and on the investigation of theirproperties can be found in the original papers [57,94–98]. Anomalous averages of the type (2.161), (2.164), and (2.193) appear in all calculationsfor Bose systems with broken gauge symmetry. They always exist together with BEC,since both of them, the anomalous averages and the phenomenon of BEC, are caused bythe same reason, by the spontaneous breaking of symmetry [63]. And when the gaugesymmetry is restored, both n as well as σ become zero. Therefore, n and σ are eithersimultaneously nonzero, or simultaneously zero. It looks absolutely evident that setting37ne of them zero, while keeping another nonzero would be principally wrong. It is easyto check that the anomalous averages are often of the same order as the normal averages[123], hence omitting the latter, while keeping the former, is mathematically inappropriate.From these facts, it is clear that neglecting the anomalous averages (as one often does) isprincipally incorrect. It is also possible to check by direct calculations that the omission ofthe anomalous averages makes all calculations not self-consistent, dynamics not conserving,thermodynamics incorrect, disturbs the phase transition order, and moreover, renders thesystem unstable [57,123,124]. Therefore, it is absolutely compulsory to correctly keep accountof the anomalous averages.Dealing with the interaction potentials of finite interaction radii, one can use the resultsof the previous section, which requires to accomplish numerical calculations. The situationsimplifies for the local potential (2.169), when many calculations can be made analytically.The sole thing, however, which does not make the life easier, is that the anomalous average(2.193), for the local potential (2.169), becomes divergent. So, a regularization method isneeded.In the case of the local interaction potential (2.169), the anomalous average (2.193) readsas σ = − Z ∆2 ε k coth (cid:16) ε k T (cid:17) d k (2 π ) . (2.196)The integral diverges for all T < T c . The divergence is caused by the use of the local po-tential (2.169), resulting in ∆ k , in Eq. (2.192), containing no k -dependence. For interactionpotentials, whose Fourier transforms Φ k depend on k , one should use ∆ k from Eq. (2.180).Then, for Φ k diminishing at large k not slower thanΦ k ≤ constk α ( α > , k → ∞ ) , the integral in Eq. (2.196) converges. However, then, instead of simple Eqs. (2.189) to(2.195), one should return to much more complicated equations, based on expressions (2.178)to (2.184).Let us denote by σ ≡ lim T → σ (2.197)the zero temperature limit of the anomalous average. Equation (2.196) yields σ = − ∆ Z ε k d k (2 π ) . (2.198)Substituting here the Bogolubov spectrum (2.189), we meet the divergent integral Z ε k d k (2 π ) = m cπ Z ∞ x dx √ x . (2.199)The integral in Eq. (2.199) can be regularized by means of the dimensional regularization[11,125], which gives Z ∞ x dx √ x → − . →
0. Consequently, using this regularization presupposes that thevalue of c in Eq. (2.199) has also to be taken in the same weak-coupling limit. In this limit,Eqs. (2.191) and (2.192) give c ≃ c B √ n (Φ → , (2.200)where c B ≡ r ρ Φ m (2.201)is the Bogolubov expression for the sound velocity. With this condition Φ → Z ε k d k (2 π ) ≃ − m c B π √ n . (2.202)In this way, in the weak-coupling limit, for the zero-temperature form of the anomalousaverage (2.198), we obtain σ ≃ ∆ m c B π √ n (Φ → . (2.203)The standard prescription in using the dimensional regularization is to employ the latter inthe region of its applicability, after which to analytically continue the result to the wholeregion of parameters. Using Eq. (2.192), we finally obtain σ = ( mc ) π p mρ Φ . (2.204)One may notice that the procedure of the analytical continuation is not uniquely defined. For-tunately, its different variants do not differ much in the results, provided that the restoration-symmetry condition σ → n → n →
0, hence, simultaneously, it should be that σ → σ , we may rewrite Eq. (2.196) as σ = − Z ∆2 ε k d k (2 π ) − Z ∆2 ε k h coth (cid:16) ε k T (cid:17) − i d k (2 π ) . (2.205)The first term here, at low temperatures, can be replaced by form (2.204), which results in σ = ≃ σ − √ π ) ( mc ) Z ∞ ( √ x − / √ x (cid:20) coth (cid:18) mc T x (cid:19) − (cid:21) dx . (2.206)39t is worth stressing that Eqs. (2.205) and (2.206) are not identical. Equation (2.206) isvalid only for low temperatures, such that2 Tmc ≪ . (2.207)At these low temperatures, the main contribution to the integral in Eq. (2.206) comes fromthe region of small x . Then we can use the expansion √ √ x (cid:16) √ x − (cid:17) / ≃ x − x + 63128 x − x and the integral Z ∞ x n − [coth( px ) − dx = π n | B n | np n , where B n are the Bernoulli numbers. Let us introduce the notation α ≡ (cid:18) πT mc (cid:19) , (2.208)which is the squared ratio of the typical thermal energy πT to the characteristic kineticenergy k m = 2 mc ( k ≡ mc ) . From Eq. (2.206), we find the low-temperature expansion σ ≃ σ − ( mc ) π α (cid:18) − α + 6 α − α (cid:19) (2.209)for α →
0. In the lowest order in α , this gives σ ≃ ( mc ) π " c B c √ n − π (cid:18) Tmc (cid:19) ( T → . (2.210)Another asymptotic form of σ , which we can find, is its form at T → T c . The criticalpoint T c is the temperature, where n →
0, hence σ →
0. Respectively, from Eqs. (2.191)and (2.192), it follows that c →
0, as T → T c . Equation (2.196), for any T , can be identicallyrewritten as σ = − √ π ) ( mc ) Z ∞ (cid:0) √ x − (cid:1) / √ x coth (cid:18) mc T x (cid:19) dx . (2.211)When c →
0, we can use the asymptotic formcoth (cid:18) mc T x (cid:19) ≃ Tmc x ( c → , as a result of which, Eq. (2.211) gives σ ≃ − m cT π ( T → T c ) . (2.212)40hus, the correct anomalous average σ should interpolate between the low-temperaturebehavior (2.210) and the critical asymptotic form (2.212). In order to better illustrate theseasymptotic forms, it is convenient to introduce the dimensionless anomalous average σ ≡ σ ρ . (2.213)Also, let us define the dimensionless temperature t ≡ mTρ / (2.214)and the dimensionless sound velocity s ≡ mcρ / . (2.215)In this notation, the low-temperature expansion (2.209) becomes σ ≃ σ − s π α (cid:0) − α + 6 α (cid:1) , (2.216)when t →
0, with α = (cid:18) πt s (cid:19) . (2.217)In the case of the local potential (2.169), it is convenient to introduce the gas parameter γ ≡ ρ / a s . (2.218)Then the zero-temperature expression for the anomalous average (2.204) reads as σ = 2 s π √ πγn . (2.219)The critical limit (2.212), in dimensionless units takes the form σ ≃ − st π ( t → t c ) . (2.220)For the critical temperature, we obtain [57,97,98] t c = 3 . . (2.221)This coincides with the BEC temperature for the ideal Bose gas, as it should be in the caseof a mean-field picture [57].It is important to use the correct form for the anomalous average in order to get a self-consistent description of the system thermodynamics. At low temperatures, outside of thecritical region, expression (2.206) can be employed. But in the near vicinity of T c , the correctbehavior of the anomalous average is prescribed by Eq. (2.212). It is possible to check bydirect numerical calculations [98] that the asymptotic form (2.212) guarantees the secondorder of the BEC phase transition for any value of the gas parameter (2.218). While, if41ne takes another expression for the critical behavior of the anomalous average, one canget a first-order transition, which would be incorrect. For instance, omitting the anomalousaverage, as is done in the Shohno model [83], one always gets the wrong first order of theBEC transition. It is worth stressing that the BEC phase transition must be of second orderfor arbitrary interaction strength [57].To emphasize the second order of the BEC transition in the self-consistent theory de-scribed above, let us present some asymptotic expansions in powers of the relative temper-ature τ ≡ (cid:12)(cid:12)(cid:12)(cid:12) t − t c t c (cid:12)(cid:12)(cid:12)(cid:12) → . Using the asymptotic expression (2.220), we find the dimensionless sound velocity (2.215), s ≃ πt c τ + 9 π t c (cid:18) − πγt c (cid:19) τ , (2.222)the condensate fraction (2.186), n ≃ τ − τ , (2.223)the anomalous average (2.213), σ ≃ − τ + 38 (cid:18) πγt c (cid:19) τ , (2.224)and the superfluid fraction (2.188), n s ≃ τ − (cid:18) . t c (cid:19) τ = 32 τ − . τ . (2.225)These expansions explicitly demonstrate the second order of the BEC transition for arbitrarynonzero gas parameters γ > He is a strongly interacting system exhibiting superfluidphase transition of second order [108,126–130]. At low temperature, superfluid helium canbe characterized [130,131] by the gas parameter γ He ∼ = 0 .
6. But at high temperatures, morerealistic potentials should be used. Such potentials contain, as a rule, hard cores, whichrequires to take into account short-range correlations [10,119–122]. The latter are oftendescribed in the frame of the Jastrow approximation [132–136].An important point is that the two-body scattering matrix [137] can be shown to bedirectly related to the total anomalous average (2.180). The latter, for a general nonuniformsystem, can be represented as∆ k = Z Φ( r − r ′ ) < ϕ ∗ k ( r ) ϕ − k ( r ′ ) ˆ ψ ( r ′ ) ˆ ψ ( r ) > d r d r ′ , (2.226)which describes the scattering of two particles. For a uniform system, the scattering processends with the plane waves ϕ k ( r ) = exp( i k · r ) / √ V . Then Eq. (2.226) becomes∆ k = Z Φ( r ) e i k · r < ˆ ψ ( r ) ˆ ψ (0) > d r . (2.227)42sing here the Bogolubov shift (2.81) gives exactly Eq. (2.180).On the other hand, the two-body scattering matrix can be defined as a solution of aLippman-Schwinger equation [138,139], which, in the limit of weak interactions, results inthe total anomalous average (2.180) evaluated in the same weak-coupling limit [140–142].For weak interactions, when ρ ≈ ρ , the anomalous average (2.180), in view of Eq.(2.184), can be rewritten as ∆ k ≃ ρ e Φ k ( ρ → ρ ) , (2.228)where the notation for an effective potential e Φ k = Φ k − V X p e Φ p Φ k + p ε p coth (cid:16) ε p T (cid:17) (2.229)is introduced. As is clear, Eq. (2.229) is nothing but a particular form of the Lippman-Schwinger equation. For a symmetric interaction potential, for which Φ − k = Φ k , the effectivepotential e Φ − k = e Φ k is also symmetric.Assuming that the potential Φ k fastly diminishes as k → ∞ , with the maximum of Φ k at the point k = 0, and keeping in mind weak interactions, we can invoke the followingapproximation: X p e Φ p Φ k + p ε p coth (cid:16) ε k T (cid:17) ∼ = e Φ k X k Φ k + p ε p h coth (cid:16) ε k T (cid:17) − i . (2.230)Then Eq. (2.229) is solved for the effective potential e Φ k = Φ k V P p Φ k + p ε p (cid:2) coth (cid:0) ε k T (cid:1) − (cid:3) . (2.231)For the local potential (2.169), the sum in the denominator of the above expression can berepresented as 1 V X p ε p h coth (cid:16) ε p T − (cid:17)i ≡ m cJ , (2.232)with the integral J = √ π ) Z ∞ ( √ x − / √ x (cid:20) coth (cid:18) mc T x (cid:19) − (cid:21) dx . (2.233)Thus, for the local potential (2.169) since Φ k = Φ , one gets e Φ = Φ m cJ Φ . (2.234)Defining an effective scattering length e a s through the notation e Φ ≡ π e a s m , (2.235)we have e a s = a s πa s mcJ . (2.236)43imilarly, one can introduce an effective gas parameter e γ ≡ ρ / e a s , (2.237)for which Eq. (2.236) gives e γ = γ πγsJ . (2.238)At low temperature, integral (2.233) yields J ≃ t s ( t → . And the effective scattering length (2.236) tends to the scattering length a s , e a s ≃ (cid:16) − πγ s t (cid:17) a s ( t → . (2.239)In the vicinity of the critical temperature, when t → t c , the sound velocity s tends tozero, s →
0, according to Eq. (2.222). Then integral (2.233) results in J ≃ t πs ( t → t c ) , hence Eq. (2.236) gives e a s ≃ s γt a s ( t → t c ) , (2.240)which tends to zero.The above analysis shows that the use of the two-body scattering matrix is equivalentto the HFB approximation in the weak-coupling limit. However, aiming at consideringstrong interactions, one is forced to return back to the anomalous average (2.196) expressedthrough a divergent integral. The latter can be regularized involving some kind of an analyticregularization, such as the dimensional regularization [11,125,143]. The latter gives the zero-temperature anomalous average (2.204). But near the critical temperature the anomalousaverage behaves as in Eq. (2.212). The correct overall behavior of the anomalous average(2.196) should interpolate between the noncritical form (2.210), valid outside of the criticalregion, and the critical asymptotic expression (2.212). The problem of particle fluctuations in Bose-condensed systems has attracted a great dealof attention provoking controversy in theoretical literature. Many tens of papers have beenpublished claiming the existence of thermodynamically anomalous particle fluctuations inBose-condensed systems everywhere below the critical temperature. A detailed accountof this trend, with many citations, can be found in the recent survey [144]. However, theoccurrence of such thermodynamically anomalous particle fluctuations, as has been explainedin Refs. [10,93,145,146], contradicts the rigorous theoretical relations as well as contravenesall known experiments. It is therefore important to pay some more attention to this problem.44irst of all, it is necessary to specify terminology. Observable quantities are representedby self-adjoint operators. Fluctuations of an observable quantity, associated with an operatorˆ A , are characterized by the operator dispersion∆ ( ˆ A ) ≡ < ˆ A > − < ˆ A > . (2.241)Generally, in statistical mechanics, one distinguishes intensive and extensive quantities [147].Fluctuations of intensive quantities are always finite, so that if ˆ A represents an intensivequantity, then its dispersion (2.241) is finite. Fluctuations of extensive quantities are de-scribed by dispersions (2.241) proportional to the system volume or to the total number ofparticles N . Fluctuations are termed thermodynamically normal , when0 ≤ ∆ ( ˆ A ) N < ∞ (2.242)for any N , including the thermodynamic limit N → ∞ . Condition (2.242) holds for any op-erators of observables, whether intensive or extensive, which guarantees the system stability[10,93,146].The number of particles in the system is represented by an operator ˆ N . So, particlefluctuations are characterized by the dispersion∆ ( ˆ N ) ≡ < ˆ N > − < ˆ N > . (2.243)Similarly to condition (2.242), particle fluctuations are thermodynamically normal, providedthat 0 ≤ ∆ ( ˆ N ) N < ∞ (2.244)for any N , including N → ∞ .The fact why conditions (2.242) or (2.244) are to be valid for any stable statistical sys-tem is that the reduced dispersions ∆ ( ˆ A ) /N describe the system susceptibilities, whichalso are observable quantities. More precisely, susceptibilities are intensive thermodynamiccharacteristics, hence, they have to be finite for any stable statistical system, except, maybe, the points of phase transitions, where the system is, actually, unstable. But the possibledivergence of susceptibilities at phase-transition points should not be confused with theirthermodynamic divergence. At a phase-transition point, a susceptibility could become di-vergent with respect to some thermodynamic parameter, such as temperature, pressure, etc.However it is never divergent with respect to the system volume or number of particles.Particle fluctuations are directly related to the isothermal compressibility κ T ≡ − V (cid:18) ∂P∂V (cid:19) − T = ∆ ( ˆ N ) ρT N , (2.245)where P is pressure, and to the hydrodynamic sound velocity s T , given by the equation s T ≡ m (cid:18) ∂P∂ρ (cid:19) T = 1 mρκ T = N Tm ∆ ( ˆ N ) . (2.246)45he structure factor S ( k ) = 1 + ρ Z [ g ( r ) − e − i k · r d r , (2.247)in which g ( r ) is a pair correlation function [148], is also expressed through the particledispersion (2.243), so that S (0) = ρT κ T = Tms T = ∆ ( ˆ N ) N . (2.248)As is evident, all these observable quantities, κ T , s T , and S ( k ), are finite then and only then,when the particle fluctuations are normal, satisfying condition (2.244).Fluctuations of an observable, represented by an operator ˆ A , are called thermodynamicallyanomalous , when condition (2.242) does not hold, as a result of which∆ ( ˆ A ) N → ∞ ( N → ∞ ) ( anomalous ) . Clearly, if particle fluctuations would be thermodynamically anomalous, then the isothermalcompressibility (2.245) would be infinite, sound velocity (2.246), zero, and the structurefactor (2.248) would also be infinite, all that manifesting the system instability [10,93,146].The thermodynamically normal properties of susceptibilities do not depend on the usedstatistical ensemble, provided that the representative ensembles are employed [54,56,57].The microcanonical ensemble can be considered as a projection of the canonical one, andthe canonical ensemble, as a projection of the grand canonical ensemble [149]. For anyrepresentative ensemble, susceptibilities should be finite almost everywhere, except the pointsof phase transitions. For example, in the grand canonical ensemble, the compressibility canbe found from the dispersion ∆ ( ˆ N ), as in Eq. (2.245). In the canonical ensemble, the totalnumber of particles is fixed. But this does not mean that the compressibility here becomeszero. One simply has to use another formula for calculating the compressibility, which inthe canonical ensemble can be calculated by means of the expression κ T = 1 V (cid:18) ∂ F∂V (cid:19) − T N , (2.249)where F is free energy. The compressibilities (2.245) and (2.249) have to be the same,defining the same observable quantities as the sound velocity (2.246) or the structure fac-tor (2.247). This concerns any statistical system, including Bose-condensed ones [150–152].In all experiments, whether with cold trapped atoms or with superfluid helium, all inten-sive quantities below T c are, of course, finite, including particle fluctuations measured as∆ ( ˆ N ) /N (see Ref. [153]). Divergencies can arise solely at the critical point itself [154].When one claims the occurrence of thermodynamically anomalous particle fluctuations inBose-condensed systems [144], one often tells that these anomalous fluctuations may happennot for the total number of particles but only separately for the number of condensed anduncondensed atoms. The operator of the total number of atoms is the sum ˆ N = ˆ N + ˆ N .One assumes that the fluctuations of ˆ N , given by the relative dispersion ∆ ( ˆ N ) /N could benormal, thus, not breaking the system stability, while the fluctuations of ˆ N and ˆ N couldbe anomalous. This assumption is, however, wrong [93,146].46et us consider two operators ˆ A and ˆ B , representing some observable quantities. Thedispersion of their sum ∆ ( ˆ A + ˆ B ) = ∆ ( ˆ A ) + ∆ ( ˆ B ) + 2cov (cid:16) ˆ A, ˆ B (cid:17) (2.250)is expressed through the particle dispersions ∆ ( ˆ A ) and ∆ ( ˆ B ) and the covariancecov( ˆ A + ˆ B ) ≡ < ˆ A ˆ B + ˆ B ˆ A > − < ˆ A >< ˆ B > .
The dispersions ∆ ( ˆ A ) and ∆ ( ˆ B ) are, by definition, positive or, at least, non-negative, whilethe covariance can be of any sign. However the covariance cannot compensate the partialdispersions, so that the total dispersion (2.250) is always governed by the largest partialdispersion. This rigorously follows from the theorem below. Theorem (Yukalov [93,146]) The dispersion of the sum of linearly independent self-adjoint operators (2.250) can be represented as∆ ( ˆ A + ˆ B ) = (cid:20)q ∆ ( ˆ A ) − q ∆ ( ˆ B ) (cid:21) + c AB q ∆ ( ˆ A )∆ ( ˆ B ) , (2.251)where 0 < c AB < . From here it follows that fluctuations of the sum of two operators ˆ A + ˆ B is thermodynam-ically anomalous then and only then, when at least one of the partial fluctuations of eitherˆ A or ˆ B is anomalous, the anomaly of the total dispersion ∆ ( ˆ A + ˆ B ) being governed by thelargest partial dispersion. Conversely, fluctuations of the sum ˆ A + ˆ B are thermodynamicallynormal if and only if all partial fluctuations are thermodynamically normal.Applying this theorem to the sum ˆ N = ˆ N + ˆ N , we see that, if the total dispersion ∆ ( ˆ N )is thermodynamically normal, which is compulsory for any stable system, then both partialdispersions, ∆ ( ˆ N ) as well as ∆ ( ˆ N ), must be normal. Thus, the normality of fluctuationsof the total number of particles ˆ N necessarily requires the normality of fluctuations of bothcondensed as well as uncondensed atoms.A very widespread misconception is that the condensate fluctuations in the grand canon-ical and canonical ensembles are different; in the grand canonical ensemble the fluctuationsare thermodynamically anomalous, such that ∆ ( ˆ N ) ∼ N , while in the canonical ensemblethey are normal. One even calls this ”the grand canonical catastrophe”. But there is no any”catastrophe” here. The seeming paradox comes about only because of the use of nonrepre-sentative ensembles [93]. The problem has been explained long time ago by ter Haar [155].The anomalous behavior ∆ ( ˆ N ) ∼ N appears in the grand canonical ensemble preservingthe gauge symmetry, while in the canonical ensemble, the gauge symmetry is effectively bro-ken. However, if the gauge symmetry is also broken in the grand canonical ensemble, thenno anomalous behavior of ∆ ( ˆ N ) arises, this dispersion being the same in both ensembles[93,155]. Recall that the gauge symmetry breaking is necessary and sufficient for describingstatistical systems with BEC [63]. 47reaking the gauge symmetry by means of the Bogolubov shift (2.81), we pass to theFock space F ( ψ ), where the number operator of condensed atoms is ˆ N = N ˆ1 F , as in Eq.(2.85). Then ∆ ( ˆ N ) becomes identically zero. If one prefers to work in the Fock space F ( ψ ), as is explained in Sec. 2.6, then the field operator ψ ( r ) becomes a function η ( r ) inthe thermodynamic limit. According to the Bogolubov theorem, it is easy to show [63] thatlim N →∞ ∆ ( ˆ N ) N ≡ F ( ψ ). The latter is orthogonal to the space F ( ψ ). The operator represen-tations on these spaces are unitary nonequivalent [99,156]. But in any case, the limit (2.252)holds true. Therefore, all particle fluctuations are, actually, caused by the uncondensedatoms, since ∆ ( ˆ N ) N ∼ = ∆ ( ˆ N ) N (2.253)for large N → ∞ .In the Bogolubov approximation, as well as in the HFB approximation, for a uniformsystem we have [10,145,146] ∆ ( ˆ N ) N = Tmc , (2.254)with the sound velocity c in the corresponding approximation. All these fluctuations, de-scribed by Eqs. (2.252), (2.253), and (2.254), are clearly thermodynamically normal.It is worth mentioning that particle fluctuations in uniform systems with BEC are normalfor interacting systems. For an ideal Bose gas, they are anomalous, which immediatelyfollows from Eq. (2.254), if one sets there c →
0, that is, reducing interactions to zero. Moreprecisely, for an ideal uniform Bose-condensed gas, it is easy to find [93] that∆ ( ˆ N ) N ∼ (cid:18) mTπ (cid:19) N / ρ / ( ideal gas ) , which means anomalous fluctuations. This implies that the ideal uniform gas with BECcannot exist, being an unstable object [10,93,145,146]. Fortunately, the purely ideal gascertainly does not exist, being just a cartoon of weakly interacting systems. Real atomsalways interact with each other, at least weakly. No matter how small the interaction, itdoes stabilize uniform Bose-condensed systems. External fields, forming trapping potentials,can also stabilize an ideal Bose gas, for instance, if the trapping is realized by harmonicpotentials [157] or power-law (though not all) potentials [41].Thus, any stable statistical system of interacting atoms, whether uniform or not, mustdisplay thermodynamically normal particle fluctuations. At the same time, as has beenemphasized at the beginning of the present section, many papers claim the occurrence ofanomalous particle fluctuations everywhere below T c , as is summarized in Ref. [144]. Theseanomalous fluctuations are claimed to be of the type ∆ ( ˆ N ) ∝ N / for interacting Bose-condensed systems of any nature, whether uniform or nonuniform trapped clouds. Moreover,since Bose-condensed systems are just one particular example of a very general class ofsystems with a broken continuous symmetry, the same type of fluctuations has to arise inall such systems [10,146]. This class of systems is really rather wide. In addition to cold48rapped atoms, it includes superfluid He, with broken gauge symmetry, isotropic magnets,with broken spin-rotational symmetry, and all solids, with broken transitional and rotationalsymmetries. If such systems would possess some divergent susceptibilities, this would meanthat they could not exist as a stable matter. There would be no superfluids, many magnets,and no solids. It is evident that such an exotic conclusion would be meaningless. - Weperfectly know that all that matter does exist and no one experiment has ever revealed anyanomalous susceptibility that would persistently be anomalous everywhere below the criticaltemperature [158,159]. The same concerns theoretical investigations for exactly solvablemodels [160] as well as correct calculations for other concrete models [58].It is instructive to consider the magnetic susceptibility of isotropic magnets. If thecompressibility of Bose systems would be really divergent as N / , when N → ∞ , then themagnetic susceptibility would also display the same divergence as N / .Let us turn, first, to the isotropic ferromagnet described by the Heisenberg model [161,162].The sample magnetization is given by the vector M = { M α } , with α = x, y, z , defined asthe statistical average M = < ˆ M > , ˆ M = µ S N ˆ S , in which µ S = g S µ B , g S is the gyromagnetic ratio for spin S , µ B is the Bohr magneton, and S ≡ N N X j =1 S j = { ˆ S α } is the reduced spin operator. The susceptibility tensor is defined as the response of themagnetization to the variation of an external field H = { h α } , so that χ αβ ≡ lim h → N ∂M α ∂h β = µ S T cov( ˆ S α , ˆ S β ) . (2.255)The diagonal magnetic susceptibility χ αα = µ S T ∆ ( ˆ S α ) (2.256)is expressed through the dispersion of the spin operator ˆ S α . Let M be directed along the z -axis, M = { M z , , } , with M z = N µ S Sσ , σ ≡ < S z >S . Then it is convenient to distinguish the longitudinal susceptibility χ || ≡ lim h → N ∂M z ∂h z = µ S T ∆ ( ˆ S z ) (2.257)and the transverse susceptibility χ ⊥ ≡ lim h → N ∂M z ∂h x = µ S T cov( ˆ S z , ˆ S x ) . (2.258)49n the mean-field approximation, the longitudinal susceptibility is χ || = Sµ S (1 − σ )2 T − J S (1 − σ ) , (2.259)where, for spin one-half, σ = tanh J Sσ + µ S h T ,J is an exchange integral (
J > h is an anisotropy field along the z -axis. Expression(2.259) is finite everywhere below T c , even when h →
0. This means, in view of Eq. (2.257),that spin fluctuations are normal.For the transverse susceptibility (2.258), one gets χ ⊥ = µ S Sσh . (2.260)In all real magnets, there always exists an anisotropy field, caused by one of many reasons,such as the natural magnetic lattice anisotropy, spin-orbital interactions, demagnetizingshape factors, and so on, including the Earth magnetic field. Therefore, in real life, thetransverse susceptibility (2.260) is finite, no matter how small the anisotropy field.Even if we consider an ideal situation with no anisotropy, when χ ⊥ diverges with re-spect to h →
0, this divergence does not make susceptibility (2.260) thermodynamicallyanomalous. The transverse susceptibility remains thermodynamically normal , since it doesnot diverge with respect to N → ∞ .Recall that the mathematically correct order of the limits, according to the Bogolubovmethod of quasi-averages [64,65], is in taking first the thermodynamic limit N → ∞ andonly after it, to consider the limit h →
0. Since susceptibility (2.260) does not diverge withrespect to N → ∞ , it is thermodynamically normal.Similarly, the divergence of the longitudinal susceptibility (2.257) with respect to T → T c ,when χ || ∝ (cid:12)(cid:12)(cid:12)(cid:12) − TT c (cid:12)(cid:12)(cid:12)(cid:12) − γ ( T → T c ) , with γ >
0, does not imply thermodynamically anomalous behavior. For any T = T c , thesusceptibility χ || is finite under the limit N → ∞ . So, the above expression χ || is alsothermodynamically normal. In addition, we should remember that, exactly at the criticaltemperature T c , the system is unstable.In the same way, one can check that the susceptibilities of isotropic antiferromagnets arealso thermodynamically normal [161,163].The appearance in some theoretical works of thermodynamically anomalous susceptibil-ities in magnets or thermodynamically anomalous particle fluctuations in Bose-condensedsystems is due solely to calculational mistakes. The standard such a mistake, as is explainedin Refs. [10,93,145,146], is as follows. One assumes the Bogolubov approximation [72,73],which is a second-order accuracy approximation with respect to the operators of uncondensedparticles, or one accepts a hydrodynamic approximation, which, actually, is mathematicallyequivalent to the Bogolubov approximation, also being a second-order approximation withrespect to some field operators. The higher-order terms, higher than two, are not well defined50n the second-order theory. Forgetting this, one calculates the fourth-order operator termsin the frame of a second-order theory. This inconsistency results in the arising anomalousexpressions, which are just calculational artifacts, and thus have no physical meaning. Thedetailed explanation can be found in Refs. [10,93,145,146].The fact that condition (2.242) has to be valid for any observable can be simply un-derstood in the following way. The average of an operator ˆ A , associated with an extensiveobservable quantity, is such that < ˆ A > ∝ N . Hence, condition (2.242) is equivalent to thecondition 0 ≤ ∆ ( ˆ A ) | < ˆ A > | < ∞ , meaning that the fluctuations of an observable cannot be infinitely larger than the observableitself.Thus, in any correct theory, all susceptibilities as well as fluctuations of observablesare always thermodynamically normal. Conditions (2.242), or (2.244), are necessary forthe stability of statistical systems. There are no experimental observations that woulddisplay thermodynamically anomalous susceptibilities in any system with broken continuoussymmetry, neither in trapped atoms, nor in liquid helium, nor in magnets, nor in solids. The consideration of the previous sections, treating a Bose system with a sole BEC, canbe generalized to the case, when several condensates arise in the system. There exist twoprincipally different situations for the appearance of multiple condensates, depending on howthe latter are distinguished. The distinction can be done according to two different kindsof the quantum numbers labelling quantum states. One type of the indices labels collectivequantum states. Such has been the index (multi-index) k labelling natural orbitals ϕ k ( r )in the previous sections. The indices of another type are those characterizing the internalstates of each particle, because of which such indices can be termed internal or individual.Examples of the individual indices are spin indices, hyperfine spin or isospin indices, andlike that. Briefly speaking, collective indices are associated with collective quantum statesof quasiparticles that are members of a statistical system, while individual indices describethe internal states of each separate particle. Respectively, there can be two types of multiplecondensates, depending on whether they are distinguished by collective or internal quantumnumbers.In a single-component statistical system, the arising BEC corresponds to the macroscopicoccupation of one of the quantum states characterized by natural orbitals, as has been de-scribed in Sections 2.1 and 2.2. One can assume that not one but several occupation numbers,related to natural orbitals, become macroscopic. Then the arising multiple condensates aredistinguished by the collective quantum index k , labelling the appropriate natural orbitals.This case of multiple condensates is what one calls the condensate fragmentation.The fragmented condensate is a multiple condensate consisting of several coexisting con-densates distinguished by the collective quantum index labelling the natural orbitals .Probably, the first example of the fragmented condensate was given by Pollock [164],who considered the coexistence of two condensates, one with the zero angular momentumand another with a nonzero angular momentum. The term ”fragmented condensates” was51oined by Nozi`eres and Saint James [165], who considered the coexisting condensates withzero and nonzero momenta. These condensates in equilibrium were shown [164,165] to beunstable. This conclusion looks rather clear, since the appearance of a condensate witheither a nonzero momentum or nonzero angular momentum rises the system free energy.In order to remain stable, the fragmented condensate should consist of degenerate parts,such that, though being described by different natural orbitals, they contribute to the systemthe same energy. An example is the fragmented quasicondensate whose parts possess thesame modulus of momentum but various arbitrary momentum directions [166–169]. Here, itis called the quasicondensate, since the order index ω ( ˆ ρ ), defined in Sec. 2.3., is ω ( ˆ ρ ) = 1 / ϕ k ( r ), associated withthe first-order density matrix, are labelled by a multi-index k . Let the total set { k } of allcollective indices k contain a subset { k ν } , such that { k ν } ⊂ { k } . The enumeration of themembers k ν of the subset { k ν } can be either continuous over a final interval 0 ≤ ν ≤ ν max or can be discrete, when ν = 0 , , , . . . .Expanding the field operator over natural orbitals, one has ψ ( r ) = X k a k ϕ k ( r ) = X ν ψ k ν ( r ) + ψ ( r ) . (2.261)Here the terms ψ k ν ( r ) ≡ a k ν ϕ k ν ( r ) (2.262)are assumed to be related to the expected fragmented condensates, while the term ψ ( r ) ≡ X k = k ν a k ϕ k ( r ) (2.263)corresponds to uncondensed particles. The condition k = k ν implies that k does not equalany k ν from the set { k ν } .For ϕ k ( r ) to represent natural orbitals, it is necessary and sufficient that the quantum-number conservation condition be valid, < a † k a p > = δ kp n k , n k ≡ < a † k a k > . (2.264)Then the density matrix ρ ( r , r ′ ) ≡ < ψ † ( r ′ ) ψ ( r ) > = X k n k ϕ k ( r ) ϕ ∗ k ( r ′ ) (2.265)52s diagonal in the expansion over natural orbitals.The terms (2.262) correspond to condensates whenlim N →∞ N k ν N > N k ν ≡ n k ν ) . (2.266)If so, the density matrix (2.265) can be separated into two parts, ρ ( r , r ′ ) = X ν N k ν ϕ k ν ( r ) ϕ ∗ k ν ( r ′ ) + X k = k ν n k ϕ k ( r ) ϕ ∗ k ( r ′ ) , (2.267)where the first sum represents the fragmented condensate.Realizing the Bogolubov shift, one has to replace the operator ψ ( r ) by the field operatorˆ ψ ( r ) = X ν η ν + ψ ( r ) . (2.268)The grand Hamiltonian, in the case of the fragmented condensate, is H = ˆ H [ ˆ ψ ] − X ν µ k ν ˆ N k ν − µ ˆ N − ˆΛ , (2.269)where ˆ H [ ˆ ψ ] is the Hamiltonian energy operator (2.151) and the linear killer ˆΛ = ˆΛ[ ψ ] isdefined as in Eq. (2.91). The system chemical potential generalizes form (2.111) to µ = X ν µ k ν n k ν + µ n , (2.270)where n k ν ≡ N k ν N , n ≡ N N .
To stress it again, the fragmented condensate, by definition [164,165], occurs, when sev-eral occupation numbers, associated with the natural orbitals , become macroscopic. Thisusually happens for nonequilibrium systems.
Contrary to fragmented condensates, which are rather rare and usually are not equilibrium,the multicomponent condensates are ubiquitous and may happen in any equilibrium systemconsisting of several kinds of Bose particles, distinguished by different internal numbers. Thesimplest case, studied long time ago [197–201], is the mixture of several components of Boseparticles with different masses and different interactions with each other. Another exampleis the mixture of atoms with different spins or hyperfine states [202]. And there are plenty ofother examples [203]. Among the systems formed by composite bosons, one can rememberthe systems of bipolarons composed of tightly bound electron pairs and of bound hole pairs[204] and also the mixtures of multiquark bosonic clusters [28,29].For a multicomponent system, one has several types of field operators ψ α ( r ), labelled byan individual index α , which can be either discrete or continuous. Respectively, there exist53everal orthonormal bases { ϕ αk ( r ) } , in which k = k ( α ) are the indices labelling collectivequantum states. Each component may possess BEC in a state labelled by k = k ( α ). Therelated field operator can be expanded over the given basis as ψ α ( r ) = X k a αk ϕ αk ( r ) = ψ α ( r ) + ψ α ( r ) , (2.271)where the first term ψ α ( r ) ≡ a α ϕ α ( r ) ≡ a αk ϕ αk ( r ) (2.272)is assumed to describe the BEC of the α -component, and the second term ψ α ( r ) ≡ X k = k a αk ϕ αk ( r ) (2.273)represents uncondensed particle of that component.If ϕ αk ( r ) are chosen as the natural orbitals, then the quantum-number conservationcondition must be valid, < a † αk a βp > = δ αβ δ kp n αk , n αk ≡ < a † αk a αk > , (2.274)similarly to condition (2.264). This guarantees that the density matrix ρ αβ ( r , r ′ ) ≡ < ψ † β ( r ′ ) ψ α ( r ) > (2.275)be diagonal, such that ρ αβ ( r , r ′ ) = δ αβ ρ α ( r , r ′ ) , (2.276)and that it would possess the diagonal expansion ρ α ( r , r ′ ) = X k n αk ϕ αk ( r ) ϕ ∗ αk ( r ′ ) . (2.277)The α -component enjoys condensation, whenlim N →∞ N α N > N α ≡ n αk ) . (2.278)Then matrix (2.277) can be separated into two parts as ρ α ( r , r ′ ) = N α ϕ α ( r ) ϕ ∗ α ( r ′ ) + X k = k n αk ϕ αk ( r ) ϕ ∗ αk ( r ′ ) . (2.279)The Bogolubov shift for a multicomponent system implies that each field operator ψ α ( r )is to be replaced by ˆ ψ α ( r ) = η α ( r ) + ψ α ( r ) . (2.280)The corresponding grand Hamiltonian becomes H = ˆ H hn ˆ ψ α oi − X α (cid:16) µ α ˆ N α + µ α ˆ N α + ˆΛ α (cid:17) , (2.281)54here ˆΛ α = ˆΛ α [ ψ α ].The total number of particles is N = X α N α , N α = N α + N α . (2.282)If there are no mutual transformations between different components, so that all N α arefixed, then the system has so many chemical potentials µ α = µ α n α + µ α n α ( N α = const ) (2.283)how many components it has. But if mutual transformations between components are al-lowed, so that only the total number of particles N is fixed, then all µ α = µ , and the systempossesses the sole chemical potential µ = µ α n α + µ α n α ( N = const ) . (2.284)Here, the particle concentrations are n α ≡ N α N , n α ≡ N α N . (2.285)Comparing Sections 2.15 and 2.16, we see that there are important differences betweenfragmented and multicomponent condensates. Condensate fragmentation occurs in a single-component system, whose several natural orbitals are macroscopically occupied. Multicom-ponent condensation happens in a multicomponent system, where several of the componentsacquire their own condensates. Of course, more complicated situations may occur, whenmulticomponent and fragmented condensates arise simultaneously.In the case of multicomponent systems, a special care has to be taken with respectto the stability of the considered multicomponent mixture. Depending on the kind of theinteractions in a multicomponent system, the latter can be unstable with reference to thecomponent stratification [198–201], when the components spatially separate from each other,rendering the system to a set of single-component parts.
Simple models often are useful for quickly catching typical features of more complicatedrealistic systems. However, one should keep in mind that BEC might exist in a model,though could be absent in a realistic system that has been mimicked by the model. Or theproperties of BEC in a cartoon model could be essentially distorted, as compared to the realcase. As an example of such a situation, let us consider a very popular two-level model withBEC.Let us assume that N atoms can occupy only two energy levels, one with energy E andanother with energy E , such that E < E . The related field operators, a and a , satisfythe commutation relations h a , a † i = h a , a † i = 1 , [ a , a ] = [ a , a ] = h a , a † i = [ a , a ] = 0 . N atoms pertain to one of two levels is a † a + a † a = N . (2.286)Also suppose that the state, corresponding to energy E , is symmetric with respect to spa-tial inversion, while the state, associated with energy E , is antisymmetric. The standardsituation that is modelled by this picture is an ensemble of trapped atoms with discretespectrum. When temperature is low and atomic interactions are weak, one assumes that allatoms pile down to the lowest energy levels, say, to the lowest two levels. These assumptionsare typical when considering cold atoms in a double-well trap [203].The symmetric and antisymmetric states can be represented by linear combinations of”left” and ”right” field operators, c L and c R , respectively, so that a = 1 √ c L + c R ) , a = 1 √ c L − c R ) . (2.287)The operator a is symmetric with respect to the interchange of c L and c R , while the operator a is antisymmetric. The ”left” and ”right” field operators satisfy the same commutationrelations as the operators a and a , in particular, h c L , c † L i = h c R , c † R i = 1 , and other commutators being zero. Hence equations (2.287) and their converse, c L = 1 √ a + a ) , c R = 1 √ a − a ) , (2.288)represent canonical transformations. The new operators also obey the N -polarity condition c † L c L + c † R c R = N . (2.289)By assumption, the states, describing the two considered energy levels, correspond tonatural orbitals. Consequently, the quantum-number conservation condition is to be valid, < a † a > = 0 . (2.290)Then, from Eq. (2.288), we have < c † L c L > = < c † R c R > = N , (2.291)which means that the ”left” and ”right” sides are equally occupied. Also, we find < c † L c R > = 12 (cid:16) < a † a > − < a † a > (cid:17) . (2.292)In view of normalization (2.286), we get < c † L c R > = < c † R c L > = < a † a > − N . < a † a > depends on the strength of atomic interactions and temperature.These two factors deplete < a † a > from N , which can be represented as < a † a > = N (cid:18) − δ (cid:19) (0 ≤ δ ≤ , (2.293)where δ is a depletion factor. The occupation number of the second level becomes < a † a > = N δ . (2.294)And Eq. (2.292) yields < c † L c R > = N − δ ) . (2.295)In an ideal gas at zero temperature, there is no depletion, δ = 0, so that all atomscondense onto the lowest level, < a † a > = N , < a † a > = 0 ( δ = 0) . For high temperature or strong repulsive interactions, the depletion is maximal, δ = 1, whichgives < a † a > = < a † a > = N δ = 1) . The latter would mean that there appears the fragmented condensate. Such a conclusion,however, should not be treated seriously, if one remembers that the studied model hasmeaning only for very low temperatures and weak interactions. High temperatures andstrong interactions would destroy any condensate as such, so that the arising fragmentedcondensate at those conditions is nothing but an artifact of an oversimplified model.One could notice that the occupation numbers (2.293) and (2.294) are macroscopic forany finite depletion δ >
0. Thus one could hope that there exist such low temperaturesand weak interactions, when the depletion is already nonzero, however the two-level modelis still appropriate, hence, the fragmented condensate could occur. But this hope seemsto be invalid. The problem is that in any confining potential, including the double-wellpotential, the spectrum is countable, so that there are many other levels, except the twoconsidered. The stronger the interactions, the closer the double-well spectrum to that ofthe harmonic oscillator [205]. No matter how small the temperature and interactions, theywill spread atoms over higher levels, making all of them populated, so that the macroscopicpopulation could remain solely on the single lowest level. In any confining potential, thefragmented condensate could exist only during finite time, as a nonequilibrium substance,as it happens for coherent topological modes [174–196]. But an equilibrium fragmentedcondensate, corresponding to two or more levels, seems to be impossible in a confiningpotential.One sometimes calls the fragmented condensate the localized parts of the same condensatein a periodic potential, as in an optical lattice. This terminology, however, is not justified,since the lattice-site indices are not quantum numbers, in the same way as the ”left”nd”right” indices in Eq. (2.295) are not good quantum numbers. Till there exists any tunnelingbetween the lattice sites, there is no fragmentation, but there is the sole condensate with a57eriodic wave function. And if there is no tunneling, then again there is no fragmentation,but there can be merely several spatial separated condensates or no condensate at all, butan insulating Mott phase.Concluding, equilibrium fragmented condensates could arise in model considerations.However one should be cautious interpreting them as actually existing in real physical sys-tems. One should not forget the limitations of oversimplified models, sometimes yieldingartificial results having no counterparts in real physical systems.
Optical lattices are formed by standing waves created by laser beams. Optical potentials aredue to the interaction of the laser electric field with atomic transition dipoles correspondingto transitions between two internal atomic energy levels [16]. The laser frequency is takento be far detuned from the atomic resonance, which allows for the definition of an effectiveoptical potential. For large detuning, the excited level can be eliminated in the adiabaticapproximation [16–18,206].The optical potential , created by laser beams, in three dimensions, has the form V L ( r ) = X α =1 V α sin ( k α r α ) , (3.1)in which the wave vector k ≡ { k α } has the components k α = 2 πλ α = πa α (cid:18) a α = λ α (cid:19) , (3.2)related to the laser wavelength λ α . Potential (3.1) can be rewritten as V L ( r ) = 32 V − X α =1 V α cos (2 k α r α ) , (3.3)where V ≡ X α =1 V α . (3.4)The wave vector k defines the recoil energy E R ≡ k m ( k ≡ | k | ) . (3.5)The lattice spacing in the α -spatial direction is a α .It is possible to create optical lattices in one and two dimensions as well. For instance,the one-dimensional optical potential is V L ( z ) = V sin ( k z ) . (3.6)58hen the recoil energy (3.5) becomes E R = π ma (cid:16) k = πa (cid:17) , (3.7)where a is the lattice spacing.Keeping in mind, the local interaction potential (2.169), the energy Hamiltonian (2.151)is ˆ H = Z ˆ ψ † ( r ) (cid:18) − ∇ m + U + V L (cid:19) ˆ ψ ( r ) d r ++ 12 Φ Z ˆ ψ † ( r ) ˆ ψ † ( r ) ˆ ψ ( r ) ˆ ψ ( r ) d r , (3.8)in which U = U ( r ) is an external, say trapping, potential and V L = V L ( r ) is an opticalpotential. The field operator ˆ ψ ( r ) is the Bogolubov-shifted operator (2.81).For the system to be stable, the particle dispersion∆ ( ˆ N ) = T ∂N∂µ = N Tρ ∂ρ∂µ (3.9)must satisfy the stability condition (2.244), which guarantees that the isothermal compress-ibility κ T = ∆ ( ˆ N ) ρT N = 1 ρ ∂ρ∂µ (3.10)is positive and finite.In an optical lattice, the total number of particles N does not necessarily coincide withthe number of the lattice sites N L . Hence, the filling factor ν ≡ NN L (0 < ν < ∞ ) (3.11)can be any positive finite number. Defining the mean lattice spacing a ≡ (cid:18) VN L (cid:19) /d (3.12)for a d -dimensional lattice, the filling factor (3.11) can be represented as ρa d = ν . (3.13)Then, compressibility (3.10) takes the form κ T = a d ν ∂ν∂µ . (3.14)The general consideration of Bose systems with broken gauge symmetry, given in Sec.2, is applicable to systems with arbitrary external potentials. Therefore, similarly to Eq.(2.97), the grand Hamiltonian for Bose atoms in an optical lattice is H = ˆ H − µ ˆ N − µ ˆ N − ˆΛ , (3.15)59here ˆ H is the energy Hamiltonian (3.8) and ˆΛ is the linear killer (2.91).The lattice parameters can be varied in a rather wide range [16–18]. The typical exper-imental values for the laser wavelengths, used in creating optical lattices, are of the order λ ∼ − − − cm; then the mean lattice spacing is a a ∼ − − − cm and the recoilenergy is E R ∼ − Hz. The ratio of the potential depth V to the recoil energy is V /E R ∼ . − L ∼ − cm.Generally, it is possible to create not only periodic lattices, but also quasiperiodic lat-tices, in which at least one of the spatial directions is subject to the action of two or moreperiodic potentials with incommensurate periods. Such quasiperiodic lattices are similar toquasicrystals [207,208]. The periodic structure of a lattice is characterized by the lattice vector a = { a α } . The latticeis formed by the set { a i } of the vectors a i = { n i a α | n i = 0 , ± , ± , . . . } . (3.16)The optical potential is periodic with respect to vectors (3.16), V L ( r + a i ) = V L ( r ) . (3.17)For the description of periodic structures, one employs the Bloch functions ϕ nk ( r ) = e i k · r f nk ( r ) , (3.18)labelled by the band index n and quasimomentum k , with a periodic factor function f nk ( r + a i ) = f nk ( r ) . (3.19)The quasimomentum k pertains to the Brillouin zone B = (cid:26) k : − πa α ≤ k α ≤ πa α (cid:27) . (3.20)The number of k -points in the Brillouin zone (3.20) equals the number N L of the real-spacecells in the total lattice, X k ∈B N L . (3.21)A uniform system can be treated as a degenerate case of the periodic one, when the Blochfunction ϕ nk ( r ) reduces to the plane wave e i k · r / √ V and the periodic factor function f nk ( r ),to the constant 1 / √ V .Without the loss of generality, the factor function (3.19) can be chosen so that theproperty f ∗ nk ( r ) = f n, − k ( r ) (3.22)60e valid. As a result, the Bloch function (3.18) satisfies the equation ϕ ∗ nk ( r ) = ϕ n, − k ( r ) . (3.23)The real-space lattice { a i } can be related to the reciprocal lattice { g i } formed by thevectors g i defined by the condition g i · a i = g · a = 2 π . (3.24)The vectors of the reciprocal lattice and of quasimomentum pertain to different sets, since k ∈ B and g ∈ { g i } , because of which they, generally, do not coincide. Consequently, theproperty 1 V Z e i ( k − p + g ) · r d r = δ kp δ g (3.25)holds. Here and in what follows, the spatial integration is over the whole system volume V .The periodic function (3.19) can be expanded over the reciprocal lattice as f nk ( r ) = 1 √ V X g b gnk e i g · r , (3.26)where the summation is over all reciprocal vectors g ∈ { g i } . The coefficient in Eq. (3.26) is b gnk = 1 √ V Z f nk ( r ) e − i g · r d r . Using expansion (3.26), the Bloch function (3.18) can be written as ϕ nk ( r ) = 1 √ V X g b gnk e i ( g + k ) · r , (3.27)with b gnk = 1 √ V Z ϕ nk ( r ) e − i ( g + k ) · r d r . The Bloch functions are orthonormal, Z ϕ ∗ mk ( r ) ϕ np ( r ) d r = δ mn δ kp , (3.28)and generate a complete basis, for which X n X k ϕ nk ( r ) ϕ ∗ nk ( r ′ ) = δ ( r − r ′ ) . (3.29)Here and in what follows, summation over k implies the summation over the Brillouin zone(3.20). From Eq. (3.27), one has X g b ∗ gmk b gnp = δ mn δ kp . w nj ( r ) = 1 √ N L X k ϕ nk ( r ) e − i k · a j , (3.30)with the summation over k ∈ B , the inverse transform being ϕ nk ( r ) = 1 √ N L X j w nj ( r ) e i k · a j , (3.31)where the summation with respect to j is over the whole lattice { a j } , with j = 1 , , . . . , N L .The Wannier functions are defined up to a phase factor that can always be chosen suchthat to make the Wannier functions real and well localized [209,210]. The scalar productbetween the Bloch and Wannier functions Z ϕ ∗ mk ( r ) w nj ( r ) d r = δ mn √ N L e − i k · a j shows that these functions are asymptotically orthogonal for N L → ∞ ,lim N L →∞ Z ϕ ∗ mk ( r ) w nj ( r ) d r = 0 . (3.32)Using the equations1 N L X k e i k · ( a i − a j ) = δ ij , N L X j e i ( k − p ) · a j = δ kp , one can make it sure that Wannier functions are orthonormal, Z w ∗ mi ( r ) w nj ( r ) d r = δ mn δ ij , (3.33)and form a complete basis, since X nj w nj ( r ) w ∗ nj ( r ′ ) = δ ( r − r ′ ) . (3.34)As is mentioned above, Wannier functions can be made real, so that w ∗ nj ( r ) = w nj ( r ) , (3.35)which is connected with property (3.23) of Bloch functions.In view of Eqs. (3.30) and (3.27), one has w nj ( r ) = 1 √ V N L X g,k b gnk e i ( g + k ) · ( r − a j ) . (3.36)62his shows that the Wannier function can be represented as w nj ( r ) ≡ w n ( r − a j ) . (3.37)From here it follows that the coefficient b gnk = r N L V Z w n ( r ) e − i ( g + k ) · r d r enjoys the properties b gnk ≡ b n ( g + k ) , b ∗ n ( g + k ) = b n ( − g − k ) . The Bloch function (3.31) can be represented as ϕ nk ( r ) = 1 √ N L X j w n ( r − a j ) e i k · a j = 1 √ V X g b n ( g + k ) e i ( g + k ) · r . (3.38)From here, we see that ϕ n k + g ( r ) = ϕ nk ( r ) . (3.39)The bases of Bloch functions, { ϕ nk ( r ) } , and that of Wannier functions, { w n ( r − a j ) } ,seem to be equivalent for characterizing periodic structures. This, however, is not completelycorrect. As is discussed in Sec. 2.17, the indices of spatial localization are not good quantumnumbers. This concerns as well the lattice indices j = 1 , , . . . , N L . Because of the latter,Wannier functions cannot serve as natural orbitals. But Bloch functions can. Thus, theBloch function, corresponding to the lowest band n = 0 and to the zero quasimomentum k = 0, is the condensate natural orbital ϕ ( r ) ≡ lim k → ϕ k ( r ) . (3.40)According to relation (3.18), ϕ ( r ) = f ( r ) , hence, the condensate natural orbital is purely periodic. It is, by definition, normalized, Z | ϕ ( r ) | d r = 1 , (3.41)and, in compliance with Eq. (3.38), is of the form ϕ ( r ) = 1 √ V X g b ( g ) e i g · r . (3.42)Its relation to the Wannier functions, given by the equation ϕ ( r ) = 1 √ N L X j w ( r − a j ) , (3.43)63emonstrates that, if ϕ ( r ) is the condensate natural orbitals, then the associated Wannierfunction w ( r − a j ) = 1 √ N L X k ϕ k ( r ) e − i k · a j (3.44)is not a condensate natural orbital, since the latter involves the Bloch functions ϕ k ( r ) ofuncondensed atoms, with k = 0.In thermodynamic limit, when N → ∞ , V → ∞ , and N L → ∞ , the summation overquasimomenta is replaced by the integration according to the rule X k −→ V Z B d k (2 π ) d , (3.45)in which the integration is over the d -dimensional Brillouin zone B . Bloch functions can serve as natural orbitals. The lowest-band zero-quasimomentum Blochfunction corresponds to BEC. The condensate wave function, entering the Bogolubov shift(2.81), is η ( r ) = p N ϕ ( r ) , (3.46)with ϕ ( r ) being the condensate natural orbital (3.40). As usual, the condensate function(3.46) is normalized to the number of condensed particles N = Z | η ( r ) | d r . (3.47)In line with expansion (3.43), the condensate function (3.46) reads as η ( r ) = r N N L X j w ( r − a j ) . (3.48)This tells us again that w ( r − a j ) cannot be treated as the condensate wave function, butonly combination (3.48) forms the latter.The field operators of uncondensed atoms can be expanded over the Bloch functions, ψ ( r ) = X nk a nk ϕ nk ( r ) . (3.49)An expansion over Wannier functions is also admissible, ψ ( r ) = X nj c nj w n ( r − a j ) . (3.50)As is known from Chapter 2, the condensate function, by definition, is orthogonal to thefield operator of uncondensed atoms, Z η ∗ ( r ) ψ ( r ) d r = 0 . (3.51)64ubstituting here expansion (3.49) requires thatlim k → a k = 0 , (3.52)which means that the term with n = 0 and k = 0 is excluded from sum (3.49).From relations (3.30) and (3.31) between Bloch and Wannier functions, it follows thatlim k → a k = 1 √ N L X j c j . Therefore the property X j c j = 0 (3.53)must hold. Conditions (3.52) and (3.53) assure that the operator of the number of uncon-densed atoms ˆ N ≡ Z ψ † ( r ) ψ ( r ) d r = X nk a † nk a nk = X nj c † nj c nj enters additively into the operator ˆ N = N + ˆ N (3.54)of the total number of particles.Let us write explicitly the grand Hamiltonian (3.15), assuming that there are no externalpotentials disturbing the lattice, inserting there the Bogolubov shift (2.81), and involvingexpansion (3.49) over Bloch functions. This gives H = H (0) + H (2) + H (3) + H (4) , (3.55)where the terms linear in ψ are eliminated by the linear killer (2.91). The first term in sum(3.55) is H (0) = Z η ∗ ( r ) (cid:18) − ∇ m + V L − µ (cid:19) η ( r ) d r ++ 12 Φ Z | η ( r ) | d r . (3.56)In order to avoid too cumbersome notation, let us combine the two indices { n, k } into oneindex k . That is, the Bloch function ϕ nk ( r ) will be labelled simply as ϕ k ( r ), keeping in mindthat here k means n, k . Then the second term in sum (3.55) writes as H (2) = X kp (cid:20)Z ϕ ∗ k ( r ) (cid:18) − ∇ m + V L − µ + 2Φ | η | (cid:19) ϕ p ( r ) d r (cid:21) a † k a p ++ 12 X kp (cid:16) Φ kp a † k a † p + Φ ∗ kp a p a k (cid:17) , (3.57)where Φ kp ≡ Φ Z ϕ ∗ k ( r ) ϕ ∗ p ( r ) η ( r ) d r . H (3) = X kpq (cid:16) Φ kpq a † k a † p a q + Φ ∗ kpq a † q a p a k (cid:17) , (3.58)in which Φ kpq ≡ Φ Z ϕ ∗ k ( r ) ϕ ∗ p ( r ) ϕ q ( r ) η ( r ) d r . And the last term in Eq. (3.55) reads as H (4) = 12 X kpql Φ kpql a † k a † p a q a l , (3.59)with the notation Φ kpql ≡ Φ Z ϕ ∗ k ( r ) ϕ ∗ p ( r ) ϕ q ( r ) ϕ l ( r ) d r . Terms (3.58) and (3.59) can be simplified by invoking the Hartree-Fock-Bogolubov (HFB)approximation. The linear in a k terms, appearing in H (3) , are assumed to be cancelled bythe linear killer (2.91). The fourth term (3.59) becomes H (4) = 12 X kpq (cid:16) kqqp n q a † k a p + Φ kpqq σ q a † k a † p + Φ ∗ kpqq σ ∗ q a p a k (cid:17) −− X kp (2Φ kppk n k n p + Φ kkpp σ ∗ k σ p ) , (3.60)in which n k ≡ < a † k a k > , σ k ≡ < a k a − k > and the quantum-number conservation conditions, valid for natural orbitals, < a † k a p > = δ kp n k , < a k a p > = δ − kp σ k are taken into account, where − k means n , − k .By introducing the notation ω kp ≡ Z ϕ ∗ k ( r ) (cid:18) − ∇ m + V L + 2Φ | η | (cid:19) ϕ p ( r ) d r ++ 2 X q Φ kqqp n q − µ δ kp (3.61)and ∆ kp ≡ Φ kp + X q Φ kpqq σ q , (3.62)the grand Hamiltonian (3.55) acquires the form H = E HF B + X kp ω kp a † k a p +66 12 X kp (cid:16) ∆ kp a † k a † p + ∆ ∗ kp a p a k (cid:17) , (3.63)in which E HF B ≡ H (0) − X kp (2Φ kppk n k n p + Φ kkpp σ ∗ k σ p ) . (3.64)Hamiltonian (3.63) can be diagonalized by a canonical transformation. In order to sim-plify the consideration, we may assume that the diagonal elements in the summation over k and p give the main contribution in Eq. (3.63). This amounts to using the diagonalapproximation for ω kp and ∆ kp , so that ω kp = δ kp ω k , ∆ kp = δ − kp ∆ k , (3.65)where δ − kp implies δ mn δ − kp , since − k means n, − k . Then Eq. (3.61) reduces to ω k = Z ϕ ∗ k ( r ) (cid:18) − ∇ m + V L + 2Φ | η | (cid:19) ϕ k ( r ) d r ++ 2 X q Φ kqqk n q − µ (3.66)and Eq. (3.62) gives ∆ k = Φ − kk + X q Φ − kkqq σ q . (3.67)Hamiltonian (3.63) becomes H = E HF B + X k ω k a † k a k + 12 X k (cid:16) ∆ k a † k a †− k + ∆ ∗ k a − k a k (cid:17) . (3.68)This is in direct analogy with the grand Hamiltonian in the HFB approximation for uniformsystems [94–98], so that all calculations can be done in the same way. The difference fromthe uniform case is in different ω k and ∆ k defined in Eqs. (3.66) and (3.67) and in the factthat the quasimomentum pertains to the Brillouin zone.Following the same procedure as for the uniform system [94–98], and restoring the doubleindexation n, k for k , we obtain the Bogolubov-type spectrum ε nk = q ω nk − ∆ nk , (3.69)consisting of several branches labelled by the band index n = 0 , , , . . . .In agreement with the condensation condition (2.121), we require thatlim k → ε k = 0 , ε k ≥ . (3.70)This is equivalent to the condition lim k → ( ω k − ∆ k ) = 0 . µ = lim k → (cid:26)Z ϕ ∗ k ( r ) (cid:18) − ∇ m + V L + 2Φ | η | (cid:19) ϕ k ( r ) d r −− Φ − kk + X q (2Φ kqqk n q − Φ − kkqq σ q ) ) , (3.71)where again the short-hand notation is used, with k instead of n, k .The condensate-function equation is derived similarly to Eq. (2.170), which in the HFBapproximation for a lattice gives (cid:26) − ∇ m + V L ( r ) + Φ [ ρ ( r ) + 2 ρ ( r )] (cid:27) η ( r ) + Φ σ ( r ) η ∗ ( r ) = µ η ( r ) . (3.72)Here the densities of condensed and uncondensed atoms and the anomalous average are ρ ( r ) = | η ( r ) | , ρ ( r ) = X nk n nk | ϕ nk ( r ) | ,σ ( r ) = X nk σ nk ϕ nk ( r ) ϕ n, − k ( r ) . (3.73)The eigenproblem (3.72) defines the condensate function η ( r ) and the Lagrange multiplier µ that guarantees the validity of the normalization condition (3.47). Using the latter yields µ = 1 N Z η ∗ ( r ) (cid:26) − ∇ m + V L ( r ) + Φ [ ρ ( r ) + 2 ρ ( r )] (cid:27) η ( r ) d r ++ Φ N Z σ ( r ) ( η ∗ ( r )) d r . (3.74)This is to be compared with multiplier (3.71). Taking in the latter the limit k →
0, we havelim k → Φ − kk = Φ Z | η ( r ) | d r , lim k → Φ kppk = Φ N Z ρ ( r ) | ϕ p ( r ) | d r , lim k → Φ − kkpp = Φ N Z [ η ∗ ( r ) ϕ p ( r )] d r , since lim k → ϕ k ( r ) = η ( r ) √ N . Taking into consideration Eqs. (3.73), we find µ = 1 N Z η ∗ ( r ) (cid:26) − ∇ m + V L ( r ) + Φ [ ρ ( r ) + 2 ρ ( r ) ] (cid:27) η ( r ) d r − Φ N Z σ ( r ) ( η ∗ ( r )) d r . (3.75)As is seen, µ = µ . They coincide only in the limit of asymptotically weak interactions,when the Bogolubov approximation becomes applicable. In this approximation ρ ( r ) → , σ ( r ) → → , hence ρ ( r ) → ρ ( r ) ≡ ρ ( r ) + ρ ( r ) . As a result, µ ≃ µ ≃ N Z η ∗ ( r ) (cid:20) − ∇ m + V L ( r ) + Φ ρ ( r ) (cid:21) η ( r ) d r (Φ → . The Bogolubov approximation for weakly nonideal gas in tight-binding bands was consideredby Ramakumar and Das [211].
The operator of momentum defines the dissipated heat (2.148) and, respectively, the super-fluid fraction (2.150). According to Eqs. (2.133) and (2.134), it can be introduced throughthe relation ˆ P ≡ lim v → ∂ ˆ H v ∂ v , (3.76)in which ˆ H v is the energy Hamiltonianˆ H v = ˆ H + Z ˆ ψ ( r ) (cid:18) − i v · ∇ + mv (cid:19) ˆ ψ ( r ) d r , (3.77)obtained by substituting into ˆ H [ ˆ ψ ] the Galilean-transformed field operator (2.146). Thisgives the standard form ˆ P = Z ˆ ψ † ( r ) ( − i ∇ ) ˆ ψ ( r ) d r , (3.78)but with the Bogolubov-shifted field operator (2.81).We may notice that Z η ∗ ( r ) ( − i ∇ ) η ( r ) d r = N X g | b ( g ) | g = 0 , because of the property | b ( − g ) | = | b ( g ) | , derived in Sec. 3.2. Also, Z η ∗ ( r ) ( − i ∇ ) ψ ( r ) d r = 0 , P = Z ψ † ( r ) ( − i ∇ ) ψ ( r ) d r . (3.79)The field operator ψ ( r ) can be expanded either over Wannier functions or over Blochfunctions, as in Eqs. (3.49) and (3.50), with the relations c nj = 1 √ N L X k a nk e i k · a j , a nk = 1 √ N L X j c nj e − i k · a j . (3.80)Defining the matrix elements over Wannier functions, p mnij ≡ Z w m ( r − a i )( − i ∇ ) w n ( r − a j ) d r , (3.81)and over Bloch functions q mnkp ≡ Z ϕ ∗ mk ( r )( − i ∇ ) ϕ np ( r ) d r , (3.82)we get the representations of the momentum operator (3.79) in terms of the Wannier, c nj ,or Bloch, a nk , operators as ˆ P = X mn X ij p mnij c † mi c nj (3.83)and, respectively, ˆ P = X mn X kp q mnkp a † mk a np . (3.84)The matrix elements (3.81) and (3.82) are connected through the transformations p mnij = 1 N L X kp q mnkp e i k · a i − i p · a j , q mnkp = 1 N L X ij p mnij e − i k · a i + i p · a j . (3.85)In Eq. (3.81), we use the advantage of choosing Wannier functions as being real. Thisequation can also be written as p mnij = Z w m ( r − a ij )( − i ∇ ) w n ( r ) d r , (3.86)where a ij ≡ a i − a j , which allows for the use of the representation p mnij ≡ p mn ( a ij ) . (3.87)Matrix elements (3.81) have the property (cid:0) p mnij (cid:1) ∗ = p nmji = − p mnij , (3.88)70rom which it follows that p nnjj = p nn (0) = 0 . (3.89)Using Eq. (3.87), the second of the matrix elements (3.85) can be rewritten as q mnkp = 1 N L X ij p mn ( a ij ) e − i k · a ij − i ( k − p ) · a j . From here, we get q mnkp = δ kp q mnk , (3.90)where the diagonal element is q mnk = X j p mn ( a j ) e − i k · a j . (3.91)The latter enjoys the property ( q mnk ) ∗ = q nmk = − q mn − k . (3.92)Thence, momentum (3.84) takes the formˆ P = X mn X k q mnk a † mk a nk . (3.93)Using the quantum-number conservation condition < a † mk a np > = δ mn δ kp < a † nk a nk > , (3.94)we find that the total average momentum < ˆ P > = X nk q nmk < a † nk a nk > = 0 , (3.95)owing to property (3.92) and assuming that < a † nk a nk > is symmetric with respect to theinversion k to − k . This means that in the coordinate frame, coupled with the lattice, thetotal average momentum is zero, as it should be. If the lattice would be moving, then in thelaboratory frame the distribution < a † nk a nk > would not be symmetric with respect to theinversion of k . Since Wannier functions can be made well localized [209,210], one can assume that whenatoms are close to the lattice site a j , then they feel the potential V L ( r − a ) ≃ X α m ω α (cid:0) r α − a αj (cid:1) ( r ≈ a j ) , (3.96)which is an expansion of the lattice potential (3.1), so that ω α ≡ p E R V α . (3.97)71t is convenient to define the effective frequency ω ≡ d Y α =1 ω α ! /d , (3.98)in which d is space dimensionality, and the effective localization length l ≡ √ mω . (3.99)Good localization of atomic Wannier functions means that the localization length (3.99)is much smaller than the distance between the nearest neighbors a , which can be expressedin several equivalent inequalities, l a ≪ , k l ≪ , (3.100)where k is the modulus of the laser wave vector entering the recoil energy (3.5), which alsoallows us to write down the inequality E R ω ≪ . (3.101)For the harmonic potential (3.96), the lowest-band localized Wannier function can beapproximated by the Gaussian form w ( r − a j ) = (cid:16) mω π (cid:17) d/ Y α exp n − m ω α (cid:0) r α − a αj (cid:1) o . (3.102)For a cubic lattice, for which ω α = ω , this becomes w ( r − a j ) = 1( √ π l ) d/ exp (cid:26) − ( r − a j ) l (cid:27) . (3.103)The Wannier functions of higher bands could be approximated by the excited wave functionsof the harmonic oscillator. If one assumes that in the lattice there are no such strongexcitations that would transfer atoms to higher excited bands, then one can limit oneselfby considering only the lowest band characterized by the approximate Wannier functions(3.102) or (3.103).As a first example of using the tight-binding approximation, let us calculate the momen-tum operator (3.93), with the matrix element (3.91), in the single-band picture. Calculatingthe matrix element (3.81), we meet the integrals Z ∞ e − bx sinh x dx = r π b exp (cid:18) b (cid:19) Φ (cid:18) √ b (cid:19) , Z ∞ xe − bx cosh x dx = 12 b + 14 b r πb exp (cid:18) b (cid:19) Φ (cid:18) √ b (cid:19) , x ) ≡ √ π Z x e − t dt . The latter enjoys the property Φ( x ) ≃ x → ∞ ) . Therefore, for small b , we find Z ∞ e − bx ( x cosh x − sinh x ) dx ≃ b r πb exp (cid:18) b (cid:19) ( b ≪ . Then, for element (3.81), we obtain p ij ≡ p ( a ij ) = i a ij a ij exp (cid:18) − a ij l (cid:19) , (3.104)where a ij ≡ | a ij | and a ij ≡ a i − a j . The diagonal element p ii = 0, in agreement with property(3.89).As is seen, the value of element (3.104) exponentially decays for increasing a ij , whichmakes it possible to take into account only the nearest neighbors. Then, for Eq. (3.91), wehave q k = X
0. Therefore,according to relation (2.111), µ ≈ µ ( N ≈ N ) . Then, we may write κ T = aν ∂ν∂µ = aν (cid:18) ∂µ∂ν (cid:19) − . (3.135)It is worth noting the following important point. In this section, we have been consideringthe case of a practically completely condensed system, when N ≈ N . In the grand canonicalensemble with broken gauge symmetry [37,63] the condensate does not fluctuate, ∆ ( ˆ N ) = 0.Since N ≈ N and N is fixed, then ∆ ( ˆ N ) ≈
0. However, as soon as N is fixed, theconsideration is reduced to the canonical ensemble. In the latter, the compressibility is notrelated to ∆ ( ˆ N ), as in Eq. (2.245), but has to be calculated differently. In the canonicalensemble, κ T can be expressed through the derivative of free energy, according to Eq. (2.249).Another ensemble, with a fixed number of particles N , is the Gibbs ensemble in which the roleof the thermodynamic potential is played by the chemical potential. Then the compressibilityis expressed through the derivative of the chemical potential, as in Eq. (3.155). It would notbe correct to say that, when the number of particles is fixed, so that ∆ ( ˆ N ) = 0, then thecompressibility would be zero because of relation (2.245). This relation has meaning only ifthe number of particles is not fixed. But if N is fixed, one has to invoke other definitions of κ T , such as Eqs. (2.249) or (3.135). The study of the Bloch spectrum is the standard problem of quantum solid-state physics[213,214]. The basic difficulty in the case of cold atoms in optical lattices is the existence ofinteractions between atoms, which makes the equations nonlinear. To illustrate the proper-ties of the Bloch spectrum, we shall analyze the quasi-one-dimensional optical lattice of theprevious Sec. 3.7.The Bloch spectrum, defined in Eq. (3.132), can be represented as E nk = Z L/ − L/ ϕ ∗ nk ( z ) H NLS [ ϕ nk ] ϕ nk ( z ) dz . (3.136)Its lowest value (3.134) gives the condensate chemical potential µ = lim k → min n E nk , (3.137)which, in the considered case of a fully condensed system, equals the system chemical po-tential, µ = µ . 78he Bloch spectrum is a single-particle spectrum, contrary to the Bogolubov spectrumof collective elementary excitations. For atoms in a lattice, these spectra are different [215].One defines the particle group velocity v nk ≡ ∂E nk ∂k (3.138)and the effective mass m ∗ nk , 1 m ∗ nk ≡ ∂ E nk ∂k . (3.139)The long-wave expression of the Bloch spectrum is E nk ≃ µ + v n k + k m ∗ n ( k → , (3.140)where v n ≡ lim k → v nk , m ∗ n ≡ lim k → m ∗ nk . Let us consider just one band. Using Eq. (3.136) and expanding the Bloch functionsover Wannier functions, as in Eq. (3.31), we have E k = 1 N L X ij Z L/ − L/ w i ( z ) H NLS [ ϕ k ] w j ( z ) e − ika ij dz , (3.141)where a ij ≡ a i − a j . For an ideal lattice, invoking representation (3.37), that is, w j ( z ) = w ( z − a j ), we get E k = X j Z L/ − L/ w ( z ) H NLS [ ϕ k ] w ( z − a j ) e ika j dz . (3.142)The nonlinear Schr¨odinger Hamiltonian (3.126) can be written as H NLS [ ϕ ] = H L ( z ) + N L Φ | ϕ | , (3.143)where the linear lattice term H L ( z ) ≡ − m ∂ ∂z + V L ( z ) (3.144)is separated out.Keeping in mind the tight-binding approximation of Sec. 3.5, we shall consider onlythe nearest-neighbor sites. Then, for the linear term (3.144), there are two types of matrixelements over Wannier functions, the single-site integral h ≡ Z L/ − L/ w ( z ) H L ( z ) w ( z ) dz (3.145)and the nearest-neighbor overlap integral h ≡ Z L/ − L/ w ( z ) H L ( z ) w ( z − a ) dz . (3.146)79he general form of the matrix element of the nonlinear term in Eq. (3.143) is propor-tional to the integral I j j j j ≡ Z L/ − L/ w j ( z ) w j ( z ) w j ( z ) w j ( z ) dz . (3.147)Treating again only the nearest neighbors, we have three kinds of integrals. The single-siteintegral is I ≡ Z L/ − L/ w ( z ) dz . (3.148)The first-order overlap integral is I ≡ Z L/ − L/ w ( z ) w ( z − a ) dz . (3.149)And also, we have the second-order overlap integral I ≡ Z L/ − L/ w ( z ) w ( z − a ) dz . (3.150)Taking only account of the nearest neighbors, Eq. (3.142) reads as E k = h + 2 h cos ka + Φ X j j j I j j j exp {− ik ( a j − a j − a j ) } . (3.151)The second-order overlap integral (3.150) is smaller than the zero-and first-order integrals(3.148) and (3.149). Therefore, retaining only the overlap integrals up to first order, one hasthe following terms. The single-site term is ε ≡ h + Φ I . (3.152)The first-order overlap integrals define the tunneling parameter J ≡ − h − I Φ . (3.153)So that the Bloch spectrum (3.151) becomes E k = ε − J cos( ka ) . (3.154)Taking account of the second-order overlap integral (3.150) would result in an additionalterm containing cos(2 ka ).The chemical potential (3.137) is µ ≡ lim k → E k = ε − J , (3.155)provided that the lowest band has been considered. This, with Eqs. (3.152) and (3.153),gives µ = h + 2 h + Φ ( I + 8 I ) . (3.156)80he isothermal compressibility, given in Eq. (3.135), can be approximately defined bytaking into account only the dependence of the coupling parameter (3.124) on the fillingfactor ν , but neglecting the possible dependence on ν of the Wannier functions. Then κ T = ma l ⊥ a s ν ( I + 8 I ) = 1 ρ ( I + 8 I )Φ . (3.157)We may notice that, similarly to the case of a uniform system, bosons in a lattice can bestable only in the presence of nonzero repulsive interactions. If atomic interactions wouldbe attractive, such that a s <
0, then compressibility (3.157) would be negative. And if theBose gas would be ideal, such that Φ →
0, then κ T would be infinite. In both these cases,the system would be unstable [93,145,146]. In the case of attractive atomic interactions,when a s <
0, the system stability can be restored for a finite number of atoms by imposinga trapping potential in all directions [9,174,216–218].With the Bloch spectrum (3.154), the group velocity (3.138) is v k = 2 J a sin( ka ) (3.158)and the effective mass, given by Eq. (3.139), reads as m ∗ k = 12 J a cos( ka ) . (3.159)In the limit of k →
0, the effective mass is m ∗ ≡ m ∗ = 12 J a ( k = 0) , (3.160)while the group velocity (3.158) behaves as v k ≃ J a k ( k → . To estimate the parameters entering the Bloch spectrum, let us consider the lowest band inthe tight-binding approximation. The corresponding Wannier function for a one-dimensionallattice is w ( z ) = 1( √ π l ) / exp (cid:18) − z l (cid:19) . (3.161)Using this, together with notation (3.99), we find the single-site integral (3.148), I = 1 √ π l = r mω π , (3.162)the first-order overlap integral (3.149), I = I exp (cid:18) − a l (cid:19) , (3.163)81nd the second-order overlap integral (3.150), I = I exp (cid:18) − a l (cid:19) = I exp (cid:18) − a l (cid:19) . (3.164)From these expressions, we see that I ≪ I ≪ I (cid:18) l a ≪ (cid:19) , (3.165)which justifies the neglect of the second-order overlap integral (3.164).Calculating Eqs. (3.145) and (3.146), we meet the integrals Z ∞ sin ( bx ) e − x dx = √ π (cid:16) − e − b (cid:17) , Z + ∞−∞ cos[ p ( x + a )] e − x dx = √ π cos ( pa ) exp (cid:18) − p (cid:19) . Then, for the single-site integral (3.145), we get h = ω V (cid:2) − exp (cid:0) − k l (cid:1)(cid:3) , (3.166)and for the overlap integral (3.146), we have h = − ω a l (cid:26) − l a − V l ω a (cid:2) (cid:0) − k l (cid:1)(cid:3)(cid:27) exp (cid:18) − a l (cid:19) . (3.167)These can be simplified remembering the conditions (3.100) of good localization, when l ≪ a and k l ≪
1. Then Eq. (3.166) reduces to h ∼ = ω V E R ω , (3.168)where E R ≡ k m = π ma . And Eq. (3.167) simplifies to h ∼ = − (cid:18) π ω E R − V (cid:19) exp (cid:18) − π ω E R (cid:19) . (3.169)The local energy term (3.152) becomes ε = ω V E R ω + Φ a r πω E R . (3.170)The effective frequency ω can be defined, in first approximation, as in Eq. (3.97), ω ≈ p V E R (cid:16) a s a ≪ (cid:17) , (3.171)82hich does not take into account atomic interactions. To find the dependence of ω on theinteraction of atoms, we may treat the effective frequency ω as a trial parameter defined inline with the optimized perturbation theory [219–221]. Accepting the optimization conditionas ∂ε ∂ω = 0 , (3.172)from Eq. (3.170), we get the equation ω (cid:18) a r πω E R (cid:19) = 4 V E R . (3.173)Invoking Eq. (3.171), we see that E R V ≈ (cid:18) E R ω (cid:19) ≪ , (3.174)according to condition (3.101). Using this in Eq. (3.173), we find ω ∼ = 2 p V E R " − Φ aE R (cid:18) π E R V (cid:19) / . (3.175)Taking account of atomic interactions diminishes the effective frequency ω .Recall that we consider weak interactions, since in the other case, the system could notbe almost completely condensed. Hence, to a good approximation, one can use the effectivefrequency estimated in Eq. (3.171). Then expression (3.168) becomes h ∼ = p V E R , (3.176)while Eq. (3.169) reduces to h ∼ = − (cid:18) π − (cid:19) V exp − π r V E R ! . (3.177)For the tunneling parameter (3.153), we obtain J ∼ = (cid:18) π − (cid:19) V exp − π r V E R ! −− √ π Φ a (cid:18) V E R (cid:19) / exp − π r V E R ! . (3.178)The chemical potential (3.156) reads as µ ∼ = p V E R − (cid:18) π − (cid:19) V exp − π r V E R ! +83 Φ √ π l " − π r V E R ! . (3.179)And compressibility (3.157) can be simplified to κ T ∼ = √ π l ρ Φ , (3.180)which means that the stability condition 0 < κ T < ∞ implies that interactions are repulsiveand finite, Φ > Elementary excitations characterize small deviations from the stationary solutions of Eq.(3.128) or Eq. (3.132). Suppose that ϕ ( z ), is an arbitrary stationary solution of Eq. (3.128),with an energy E . Small deviations from the stationary solution are described by the wavefunction ϕ ( z, t ) = (cid:2) ϕ ( z ) + u ( z ) e − iεt + v ∗ ( z ) e iεt (cid:3) e − i ( E − µ ) t . (3.181)Substituting this form into the nonlinear Schr¨odinger equation (3.125) gives the Bogolubovequations (cid:0) H NLS [ ϕ ] − E + N L Φ | ϕ ( z ) | − ε (cid:1) u ( z ) + N L Φ ϕ ( z ) v ( z ) = 0 , (cid:0) H NLS [ ϕ ] − E + N L Φ | ϕ ( z ) | + ε (cid:1) v ( z ) + N L Φ ( ϕ ∗ ( z )) u ( z ) = 0 . (3.182)As a stationary solution ϕ ( z ) with an energy E one can take any Bloch function ϕ nq withan energy E nq .For an equilibrium system, BEC corresponds to the lowest-energy Bloch function ϕ ( z )with the energy E = µ . Considering the elementary excitations above the condensaterequires to set as ϕ ( z ) in Eq. (3.182) the condensate Bloch function ϕ ( z ) = 1 √ N L X j w ( z − a j ) . (3.183)The Bogolubov functions u ( z ) and v ( z ) should be proportional to Bloch functions ϕ k ( z ) withnonzero k . Hence, we take u ( z ) ≡ u k ϕ k ( z ) , v ( z ) ≡ v k ϕ k ( z ) . (3.184)Let us introduce the notation ω k ≡ Z L/ − L/ ϕ ∗ k ( z ) H L ( z ) ϕ k ( z ) dz + 2∆ k − µ , (3.185)in which H L ( z ) is the linear lattice Hamiltonian (3.144) and∆ k ≡ Φ N L Z L/ − L/ | ϕ k ( z ) | ϕ ( z ) dz . (3.186)84he Bogolubov equations (3.182) reduce to( ω k − ε ) u k + ∆ k v k = 0 , ∆ k u k + ( ω k + ε ) v k = 0 , (3.187)which defines the Bogolubov spectrum of collective excitations ε k = q ω k − ∆ k . (3.188)Calculating quantities (3.185) and (3.186), we employ the tight-binding approximation.The Bloch functions ϕ k ( z ) are expanded over the Wannier functions w ( z − a j ), as in Eq.(3.31), and w ( z ) is taken in form (3.161). This yields for Eq. (3.185) ω k = ∆ + 4 J sin (cid:18) ka (cid:19) , (3.189)where ∆ ≡ ( I + 8 I )Φ , (3.190)while for Eq. (3.186), ∆ k = ∆ − I Φ sin (cid:18) ka (cid:19) . (3.191)Then the Bogolubov spectrum (3.188) is ε k = (cid:20) c a sin (cid:18) ka (cid:19) + 16 (cid:0) J − I Φ (cid:1) sin (cid:18) ka (cid:19)(cid:21) / , (3.192)where c is the sound velocity c ≡ p J + 2 I Φ ) a . (3.193)In the long-wave limit, when k →
0, one has ω k ≃ ∆ + J ( ka ) , ∆ k ≃ ∆ − I Φ ( ka ) . And Eq. (3.192) gives the gapless phonon spectrum ε k ≃ ck ( k → . (3.194)Comparing the Bloch spectrum (3.154) and the Bogolubov spectrum (3.192), we see thatthey are quite different [215,222,223]. To stress their difference, we may rewrite the Blochspectrum (3.154) in the form E k = µ + 4 J sin (cid:18) ka (cid:19) , (3.195)where µ is the chemical potential (3.155). In the long-wave limit, the Bloch spectrum(3.195) is E k ≃ µ + J ( ka ) ( k → , µ , contrary to the gapless phononspectrum (3.194).It is important to emphasize that for equilibrium bosons in lattices, in the presenceof BEC, when the gauge symmetry is broken, the single-particle spectrum of uncondensed atoms (3.69) coincides with the spectrum of elementary excitations (3.188), both of thembeing the Bogolubov spectra.Of the Bloch spectrum (3.154), or (3.195), solely one point, where n = 0 and k = 0, and E k = µ , corresponds to an equilibrium BEC. But the Bloch spectrum, in general, describes nonequilibrium condensates that are the analog of the coherent modes [174–176]. This is whythe Bloch spectrum does not need to coincide with the Bogolubov spectrum The condensate wave function ϕ ( z ) for an equilibrium system is assumed to correspond toa stable system. The related condition of thermodynamic stability is that compressibility(3.135), or (3.157), be positive and finite, 0 ≤ κ T < ∞ .Other Bloch functions ϕ nk ( z ), which are solutions to Eq. (3.132), as is stressed above,do not correspond to a thermodynamically equilibrium BEC. Though these functions ϕ nk ( z )are stationary solutions, but a statistical system with a condensate, characterized by such afunction is not in absolute equilibrium. It is, therefore, useful to study the stability of theBloch functions ϕ nk ( z ).There are, in general, several kinds of stability for solutions to differential equations [224–227]. The most often used is the notion of Lyapunov stability [228]. Let us recall this notionin general terms. Suppose we consider functions of the type ϕ ( x, t ), where x is a variable ofarbitrary nature, which can pertain to the continuous manifold R d , or to a discrete manifold,or to their combination, and where t ∈ [0 , ∞ ). Treating x as an enumeration index, one candefine the column function ϕ ( t ) ≡ [ ϕ ( x, t )]. Let ϕ ( t ) pertain to a Banach space (normed,complete space), where a norm || ϕ ( t ) || is defined. When ϕ ( t ) pertains to a Hilbert space, thenorm is naturally generated by the scalar product. Let us consider the evolution equation ∂ϕ ( t ) ∂t = F [ ϕ ] , (3.196)in which F [ ϕ ] is an operator functional in the same Banach space. It is assumed that, for agiven initial condition ϕ (0), the Cauchy problem (3.196) enjoys a unique solution.A solution ϕ ( t ) is Lyapunov stable , if for any other solution ϕ ( t ), such that || ϕ (0) − ϕ (0) || < δ , (3.197)with any δ >
0, there exists a positive number δ , for which || ϕ ( t ) − ϕ ( t ) || < δ ( t > . (3.198)The solution ϕ ( t ) is asymptotically stable , when there can be found such δ in Eq. (3.197)that lim t →∞ || ϕ ( t ) − ϕ ( t ) || = 0 . (3.199)86he solution ϕ ( t ) is exponentially stable , if there exists such δ in Eq. (3.197) thatlim t →∞ t ln || ϕ ( t ) − ϕ ( t ) || < . (3.200)In particular, ϕ ( t ) can be a stationary solution, for which F [ ϕ ] = 0. Exponential stability isa special case of asymptotic stability.Conversely, if, under condition (3.197): inequality (3.198) is not valid, the solution ϕ ( t )is Lyapunov unstable; when limit (3.199) does not follow for any δ , ϕ ( t ) is asymptoticallyunstable; and if the limit (3.200) becomes positive, then ϕ ( t ) is exponentially unstable.From these definitions, it is clear that the Lyapunov stability does not lead to the asymp-totic stability and, vice versa, the asymptotic stability does not imply the Lyapunov stability.Also, the Lyapunov instability does not forbid the asymptotic stability for some δ . And theasymptotic instability does not contradict to the Lyapunov stability.Lyapunov developed [228] two methods of controlling stability, the direct method andthe method of linearization. Lyapunov direct method is based on the existence of the Lyapunov functional, such that L [ ϕ ] ≥ ϕ ( t ) from the considered Banach space and which does not increase, ∂∂t L [ ϕ ] ≤ , (3.202)on the trajectories of Eq. (3.196). If such a Lyapunov functional exists, then the solution ϕ ( t ) is Lyapunov stable .The Lyapunov direct method is global, requiring the validity of condition (3.201) on thewhole Banach space. In many cases, one is interested not in the global stability, but in thelocal stability in the vicinity of a known solution ϕ ( t ). Then the direct Lyapunov method isreformulated as follows. Lyapunov local method assumes the existence of a functional L [ ϕ ], which does not increaseon the trajectories, as in Eq. (3.202), and which is minimal in the small vicinity of a givenfunction ϕ ( t ), that is, when for ϕ = ϕ + δϕ , one has δL [ ϕ ] = 0 , δ L [ ϕ ] > . (3.203)If such a functional exists, then the solution ϕ ( t ) is locally stable .In many cases, the role of the Lyapunov function is played by energy or by an effectiveenergy, if the energy is complimented by additional constraints. When the energy functional E [ ϕ ] is an integral of motion, then ∂E [ ϕ ] /∂t = 0, so that condition (3.202) holds. Hence,if E [ ϕ ] ≥
0, the motion is Lyapunov stable. This does not mean that the motion is locallystable in the vicinity of a given ϕ . To study the local stability near ϕ , one has to satisfyconditions (3.203) for the Lyapunov functional E [ ϕ ]. One says that a solution is energeticallystable , if ∂∂t E [ ϕ ] ≤ , δE [ ϕ ] = 0 , δ E [ ϕ ] > ϕ = ϕ + δϕ . That is, the energetic stability is just an example of the local stability , whenthe Lyapunov functional is represented by an energy functional.Above, we have considered the variants of the Lyapunov direct method for analysing thestability of solutions to the evolution equation (3.196). The second Lyapunov method isbased on the linearization of Eq. (3.196). Lyapunov linearization method requires the linearization of the evolution equation (3.196)with respect to small deviations from the given solution ϕ . Taking ϕ = ϕ + δϕ , one obtainslinear equations for δϕ , which are subject to the standard stability analysis [228,229]. Whenall Lyapunov exponents are negative, the solution ϕ is asymptotically stable . If at least oneof them is positive, then ϕ is asymptotically unstable. And when some of the Lyapunovexponents are zero, while others being negative, the considered solution is neutrally stable ,provided it remains finite for all t >
0. If the linearized equations show that the solution ϕ iseither asymptotically stable or neutrally stable, then such a solution is termed dynamicallystable .Suppose that one is interested in the stability of a stationary solution ϕ , for which ∂ϕ/∂t = 0. Considering small deviations from ϕ , when ϕ ( t ) = ϕ + δϕ ( t ), one can check helocal, or energetic stability by means of Eqs. (3.202) and (3.203), or (3.204). Alternatively,one can use the linearization method for the evolution equation (3.196). Then there existsthe following relation between different types of stability. Theorem . The local stability of a stationary solution yields its dynamic stability . Proof . Let ϕ be a stationary solution of Eq. (3.196), such that ∂ϕ/∂t = 0. Andlet this solution be locally stable, which implies that there exists a Lyapunov functional L [ ϕ ] satisfying conditions (3.202) and (3.203). Expanding the Lyapunov functional for theperturbed solution ϕ ( t ) = ϕ + δϕ ( t ), we have L [ ϕ ] = L [ ϕ ] + δL [ ϕ ] + δ L [ ϕ ] . From here, in view of Eqs. (3.202) and (3.203), it follows ∂∂t δ L [ ϕ ] = ∂∂t L [ ϕ ] ≤ . Taking || δϕ (0) || ≤ δ L [ ϕ (0)] , we find || δϕ ( t ) || ≤ δ L [ ϕ ( t )] ≤ δ L [ ϕ (0)] , which shows that the linear deviation δϕ ( t ) does not increase with time. Hence, ϕ is dy-namically stable.When the Lyapunov functional is the energy functional, the theorem tells us that theenergetic stability of a stationary solution yields its dynamic stability.The inverse, however, is not true. The dynamic stability does not necessarily yieldthe local, or energetic, stability. To illustrate this, let us consider a two-component field ϕ = { ϕ , ϕ } , where ϕ j = ϕ j ( x, t ), with the energy functional E [ ϕ ] = 12 Z (cid:0) ϕ | − | ϕ | − g | ϕ | | ϕ | (cid:1) dx . i ∂ϕ j ∂t = δE [ ϕ ] δϕ ∗ j , which gives i ∂ϕ ∂t = 12 ϕ (cid:0) − g | ϕ | (cid:1) , i ∂ϕ ∂t = − ϕ (cid:0) g | ϕ | (cid:1) . The stationary solutions of these equations are ϕ = ϕ = 0. The linearized equations give δϕ = c e − iωt , δϕ = c e iωt , where c j = c j ( x ) and ω = 1 /
2. Hence, the deviations δϕ j do not increase with time, whichmeans that the stationary solutions ϕ and ϕ are dynamically stable .For the energy functional, we have ∂∂t E [ ϕ ] = 0 , δE [ ϕ ] = 0 . However, the second variation δ E [ ϕ ] = 12 Z (cid:0) | c | − | c | (cid:1) dx is not positive defined, that is, the stationary solutions ϕ j = 0 do not provide a minimum of E [ ϕ ]. Hence these stationary solutions are energetically unstable .For an optical lattice, discussed in the previous sections, we may define the energy func-tional E [ ϕ ] ≡ N Z ϕ ∗ ( z )[ H L ( z ) − E ] ϕ ( z ) dz + N N L Φ Z | ϕ ( z ) | dz . (3.205)The stationarity condition δE [ ϕ ] δϕ ∗ ( z ) = 0 (3.206)results in the stationary nonlinear Schr¨odinger equation (3.128). The latter, due to its non-linearity, can possess different types of solutions, including Bloch waves, localized solitons,as well as density waves with a period differing from that of the optical potential [212].Limiting ourselves by the class of Bloch functions, we come to Eq. (3.132).The Bloch spectrum E nk , defined by Eq. (3.132), displays rather nontrivial behavior,caused by the nonlinearity of the eigenproblem. For sufficiently strong nonlinearity, thereappears the swallow-tail structure of E nk , when it is not uniquely defined as a function of k [230–234]. The swallow tails can appear at the edge of the lowest band, with n = 0 and k = π/a , and also in the middle of upper bands, with n ≥ k = 0. This happens whenthe interaction is sufficiently strong, so that aV Φ < . (3.207)89n the presence of nonlinearity, the optical lattice may provoke instability of Bloch waves ϕ nk ( z ) for some k . To find the region of stability, one considers small deviations in thevicinity of ϕ nn ( z ) by setting ϕ nk ( z, t ) = ϕ nk ( z ) + δϕ ( z, t ) , (3.208)where δϕ ( z, t ) = (cid:2) u nq ( z ) e i ( qz − εt ) + v ∗ nq ( z ) e − i ( qz − εt ) (cid:3) . (3.209)The energetic, or static, stability is defined by substituting Eq. (3.208) into the energyfunctional (3.205) and taking ε = 0 in Eq. (3.209). The dynamic stability is analyzed bylinearizing the evolution equation i ∂∂t ϕ nk ( z, t ) = ( H NLS [ ϕ nk ] − E nk ) ϕ nk ( z, t ) (3.210)with respect to the small deviation (3.209). The functions ϕ nk ( z ), u nq ( z ), and v nq ( z ) areBloch waves.Linearizing Eq. (3.210) gives for the energy ε the spectrum of collective excitations ε nkq around the Bloch spectrum E nk . Dynamic instability occurs when ε nkq becomes complex,since then there appears an exponentially increasing term in Eq. (3.209).From the general theory, expounded at the beginning of this section, it follows that theenergetic stability yields the dynamic stability. This means that, if the solution is dynam-ically unstable, it is also energetically unstable. However, the solution can be dynamicallystable, while being energetically unstable. All this is, of course, valid for optical lattices[230–234], as well as for vortex states [235].It is worth recalling that the Bloch functions ϕ nk , with n > k >
0, correspond toexcited nonequilibrium condensates. Therefore the stability, in any sense, of such nonequilib-rium condensates should be neither required nor expected. What is required is the stabilityof the equilibrium ground-state BEC, corresponding to the natural Bloch orbital ϕ , with n = 0 and k = 0. The stability of the latter is guaranteed by compressibility (3.135) be-ing positive and finite, which, in view of Eqs. (3.157) and (3.180), requires that atomicinteractions be repulsive and finite, that is, Φ > Different nonequilibrium states of BEC can be crated by moving the optical lattice. Toobtain an effective equation for BEC in a moving lattice, let us assume that the latter moveswith velocity v = v ( t ) along the z -axis. This means that, in the frame of the lattice, thecondensate moves with velocity − v ( t ). The wave function of a moving condensate ϕ v ( z, t )satisfies the nonlinear Schr¨odinger equation i ∂∂t ϕ v ( z, t ) = ( H NLS [ ϕ v ] − E ) ϕ v ( z, t ) (3.211)and is related to the wave function of an immovable condensate through the Galilean trans-formation ϕ v ( z, t ) = ϕ ( z + vt, t ) exp (cid:26) − i (cid:18) mvz + mv t (cid:19)(cid:27) . (3.212)90ubstituting function (3.212) into Eq. (3.211) and neglecting the term(ˆ p − mv ) ϕ ( z + vt, t ) ≈ (cid:18) ˆ p ≡ − i ∂∂z (cid:19) yields i ∂∂t ϕ ( z + vt, t ) = ( H NLS [ ϕ ] − E − m ˙ vz ) ϕ ( z + vt, t ) , (3.213)where ˙ v ≡ ∂v/∂t .There exist two different length scales in the system. One is the intersite distance a ,being the lattice period. The typical variation of Bloch functions is on the lattice scale a .And another scale is the effective size L of the studied atomic cloud, being much larger than a , aL ≪ . (3.214)The existence of such very different scales allows for the use of averaging techniques [236–239]and of the scale separation approach [240–243]. A similar procedure in electrodynamics iscalled the slowly varying amplitude approximation [244–249]. In line with such techniques,we may look for the solution of Eq. (3.213) in the form ϕ ( z + vt, t ) = A ( z, t ) ϕ q ( z ) , (3.215)where, for simplicity, we consider a single band and assume that the quasimomentum q = q ( t )is, generally speaking, a function of time. The factor A ( z, t ) in Eq. (3.215) is a slowly varyingamplitude, such that (cid:12)(cid:12)(cid:12)(cid:12) ∂A∂z (cid:12)(cid:12)(cid:12)(cid:12) ≪ (cid:12)(cid:12)(cid:12)(cid:12) ∂ϕ q ∂z (cid:12)(cid:12)(cid:12)(cid:12) . (3.216)While ϕ q ( z ) is a Bloch function given by the equation H L ( z ) ϕ q ( z ) = E q ϕ q ( z ) , (3.217)with the linear lattice Hamiltonian (3.144). According to condition (3.216), the function ϕ q ( z ) is fastly varying in space, as compared to the slow amplitude A ( z, t ). The latter isalso called the envelope. It is normalized as1 L Z L/ − L/ | A ( z, t ) | dz = 1 . (3.218)The Bloch function ϕ q ( z ) satisfies the usual normalization condition Z L/ − L/ | ϕ q ( z ) | dz = 1 . (3.219)An important point is the ansatz H L ( z ) A ( z, t ) ϕ q ( z ) = [ E q +ˆ p A ( z, t )] ϕ q ( z ) , (3.220)whose justification [250] is based on the averaging techniques.91ubstituting form (3.215) into Eq. (3.213) yields the equation i ∂A∂t = ( E q +ˆ p − E q ) A + α q | A | A + ( ˙ q − m ˙ v ) zA , (3.221)where A = A ( z, t ), the overdot means time derivative, and α q ≡ N L Φ Z L/ − L/ | ϕ q ( z ) | dz . (3.222)Deriving (3.221), we have also used the approximate equality ∂∂q ϕ q ( z ) ≈ izϕ q ( z )following from the fact that ϕ q ∝ e iqz .Expanding E q +ˆ p ≃ E q + v q ˆ p + 12 m ∗ q ˆ p (3.223)in powers of ˆ p = − i∂/∂z , with the group velocity v q and effective mass m ∗ q defined by therelations v q ≡ ∂E q ∂q m ∗ q ≡ ∂ E q ∂q , (3.224)we come to the evolution equation for the amplitude i (cid:18) ∂A∂t + v q ∂A∂z (cid:19) + 12 m ∗ q ∂ A∂z = α q | A | A + ( ˙ q − m ˙ v ) zA . (3.225) Suppose that, after moving the lattice and reaching a Bloch state with a quasimomentum q ,the motion has been stopped, so that ˙ q = ˙ v = 0. Then Eq. (3.225) reduces to i (cid:18) ∂A∂t + v q ∂A∂z (cid:19) + 12 m ∗ q ∂ A∂z = α q | A | A . (3.226)This is a nonlinear Schr¨odinger equation supporting soliton solutions [251]. Similar equationsare met in laser physics [252], in the theory of turbulent plasma [253,254], in the descriptionof magnetic matter [255], and in the theory of many other nonlinear materials [256,257]. ForBose condensates, Eq. (3.226) was derived by Lenz et al. [258].Equation (3.226) can be simplified by changing the variable to x ≡ z − v q tξ , (3.227)where ξ = 1 q | m ∗ q ε | (3.228)92s the healing length and ε is the soliton energy to be defined by the normalization condition(3.218). Let us introduce a function f ( x ) through the relation A ( z, t ) ≡ r εα q f ( x ) e − iεt . (3.229)The function f ( x ) can be chosen real, since Eq. (3.226) is invariant under the global gaugetransformation A → Ae iα . And let us define ζ ≡ sgn( m ∗ q ε ) . (3.230)Then Eq. (3.226) can be reduced to d fdx + ζ (cid:0) − f (cid:1) f = 0 . (3.231)Depending on the sign of ζ in Eq. (3.230), there are the following possibilities. Dark solitons correspond to ζ = 1 and the boundary conditionslim x →±∞ f ( x ) = ± . (3.232)The name comes from the fact that the density distribution | f ( x ) | has the lowest valueat x = 0. Dark solitons are called cavitons in the theory of plasma [253,254] and in laserphysics [252]. There can be two types of dark solitons. Normal dark soliton is formed by atoms with a positive effective mass, repulsive interac-tions, and with a positive soliton energy, m ∗ q > , α q > , ε > . (3.233) Dark gap soliton is characterized by a negative effective mass, attractive interactions,and a negative soliton energy, m ∗ q < , α q < , ε < . (3.234)It is worth stressing that the signs of the effective interaction α q and the soliton energy ε arechosen to be the same in order that the expression p ε/α q in Eq. (3.229) be real. This doesnot limit the generality, but simply takes into account that the amplitude (3.229) is definedup to a phase factor.The name of the gap soliton is due to the fact that, to achieve a negative effective mass,the atomic cloud has to be shifted to the edge of the Brillouin zone. This shift can be realizedby the appropriate motion of the lattice.The form of the dark soliton, being the solution of Eq. (3.231), with ζ = 1, under theboundary conditions (3.232), is f ( x ) = tanh x √ . (3.235)The normalization condition (3.218) for amplitude (3.229), with f ( x ) from Eq. (3.235), gives ε = α q = 12 m ∗ q ξ , (3.236)93he healing length (3.228) being ξ = 1 p m ∗ q α q . (3.237)Hence, the total dark envelope (3.229) becomes A ( z, t ) = tanh (cid:18) z − v q t √ ξ (cid:19) e − iεt , (3.238)with the healing length (3.237) and soliton energy (3.236). Bright solitons arise for ζ = −
1, under the boundary conditionslim x →±∞ f ( x ) = 0 . (3.239)Such solitons correspond to the maximum of the density distribution | f ( x ) | at x = 0. Theyare called as well the bell solitons. There are again two possibilities. Normal bright soliton is described by a positive effective mass, though attractive inter-actions and negative soliton energy, m ∗ q > , α q < , ε < . (3.240) Bright gap soliton possesses a negative effective mass, but repulsive interactions andpositive soliton energy, m ∗ q < , α q > , ε > . (3.241)Again, to make the effective mass negative, it is necessary to move the lattice so that theatomic cloud would acquire a quasimomentum at the edge of the first Brillouin zone.The bright soliton solution, resulting from Eq. (3.231), with ζ = −
1, under the boundaryconditions (3.239), is f ( x ) = √ x . (3.242)The normalization condition (3.218) yields the soliton energy ε = − m ∗ q α q L ) = − m ∗ q ξ , (3.243)with the healing length ξ = 2 | m ∗ q α q L | . (3.244)The total bright soliton envelope (3.229) takes the form A ( z, t ) = s L ξ sech (cid:18) z − v q tξ (cid:19) e − iεt , (3.245)with the healing length (3.244) and soliton energy (3.243). This form is analogous to Lang-mur solitons in plasma [253,254]. 94ne sometimes distinguishes solitons by their topological charge, defined aslim x →∞ [ f ( x ) − f ( − x )] . When the latter is nonzero, the solitons are termed topological. Thus, dark solitons aretopological. If the topological charge is zero, the solitons are called nontopological. Hence,bright solitons are nontopological.Dark solitons of BEC were generated for repulsive Rb atoms [259,260] and bright soli-tons in BEC were formed with attractive Li atoms [261,262]. The specific feature of gapsolitons is that they can be created only in the presence of a periodic lattice [263]. Gapsolitons were observed for Rb atoms [264].
When creating gap solitons, it is necessary to shift an atomic cloud to the boundary ofthe Brillouin zone, where the effective mass becomes negative. But then the Bloch energyincreases, and it may happen that the transverse modes of atomic motion could be excited.In such a case, the quasi-one-dimensional picture for treating a single wave packet is notanymore appropriate, and one has to take into account transverse excitations [265]. Thiscan be done in the following way [250].When, despite of a strong transverse harmonic confinement, nevertheless, some transversemodes can be excited, then it is necessary, instead of Eq. (3.118), to consider the transversemotion described by the equation (cid:18) − ∇ ⊥ m + m ω ⊥ r ⊥ (cid:19) χ n ( r ⊥ ) = E ⊥ n χ n ( r ⊥ ) . (3.246)Under the harmonic transverse confinement, the eigenenergy spectrum of Eq. (3.246) is E ⊥ n = (2 n r + | m a | + 1) ω ⊥ , (3.247)where n r = 0 , , , . . . is a radial quantum number, m a = 0 , ± , ± , . . . is an azimuthalquantum number, and the multi-index n = { n r , m a } includes both of these numbers. Thelowest energy (3.247) is E ⊥ = ω ⊥ .When trapped atoms are in the lowest-energy transverse state, then their total spectrumis the sum of the lowest energy of transverse motion, ω ⊥ , and of the Bloch energy E q . For anexcited transverse state, with an energy E ⊥ n , the total spectrum is the sum E ⊥ n + E p . Although ω ⊥ < E ⊥ n , for n = 0, but if q ≈ π/a , there exists such a quasimomentum p ∈ [ − π/a, π/a ]that E q > E p . And the resonance condition ω ⊥ + E q = E ⊥ n + E p (3.248)can become valid. Then the transverse modes, with the energy E ⊥ n , become excited.In the presence of the excited transverse modes, the condensate wave function η ( r , t ),satisfying Eq. (3.115), has to be written as η ( r , t ) = X i χ n i ( r ⊥ ) B i ( z, t ) ϕ q i ( z ) exp (cid:8) − i (cid:0) E ⊥ n i − µ (cid:1) t (cid:9) , (3.249)95hich generalizes Eqs. (3.116), (3.127), and (3.215). The transverse wave functions χ n ( r ⊥ )are normalized to one, as in Eq. (3.117). The Bloch functions ϕ q ( z ) are normalized as inEq. (3.219). But the envelopes B i ( z, t ) satisfy the normalization conditions1 L Z L/ − L/ | B i ( z, t ) | dz = N i , (3.250)where N i = N i ( t ), generally, are functions of time. This normalization differs from that inEq. (3.218).Now, instead of one interaction parameter (3.222), there are several interaction parame-ters α ijkl ≡ N L Φ Z χ ∗ n i χ ∗ n j χ n k χ n l d r ⊥ Z ϕ ∗ q i ϕ ∗ q j ϕ q k ϕ q l dz . (3.251)Following the same way as in the previous sections, instead of Eq. (3.226), we obtain theset of equations i (cid:18) ∂B i ∂t + v i ∂B i ∂z (cid:19) + 12 m ∗ i ∂ B i ∂z = 1 N L X jkl α ijkl B ∗ j B k B l (3.252)for the mode envelopes B i ( z, t ), with i = 0 , ,
2. The mode envelope B ( z, t ) corresponds tothe central mode of a gap soliton, with q = π/a , while the envelopes B ( z, t ) and B ( z, t ),to the two side modes, for which the resonance condition (3.248) is valid. There are twotransverse modes, since the resonance condition (3.248) holds for two Bloch energies E q and E q , for which E q = E q (cid:16) q = πa − q , q = πa + q (cid:17) . (3.253)The form of parameters (3.251) shows that α ijkl = α jikl = α ijlk . (3.254)Also, for the quasimomenta q and q , related as in Eq. (3.253), one has ϕ ∗ q ( z ) = ϕ q ( z ) . (3.255)For the transverse wave functions, according to condition (3.248), one has χ n ( r ⊥ ) = χ n ( r ⊥ ) . Since the functions χ ( r ⊥ ) and χ ( r ⊥ ) possess different symmetries with respect to theinversion of r ⊥ , not all integrals in Eq. (3.251) are nonzero. These are α and α = α = α , α = α = α , as well as all those that are obtained from the above ones using symmetry (3.254) andproperty (3.255).An interesting solution of Eqs. (3.252) is represented by a triple solution [250], which isa triplet of solitons, one of which corresponds to q = π/a , that is, to a gap soliton, while two96thers are the side transverse modes. All three modes are bound with each other, so thatthey stay localized in space. Bound triplets of solitons are also called tritons [256].It is worth mentioning that quasi-one-dimensional gap solitons, as those that are consid-ered in Sec. 3.13, are usually unstable with respect to the formation of transverse modes[256]. Hence, gap solitons are, strictly speaking, quasisolitons, that is, the soliton-like so-lutions that in the long run are unstable, but can live sufficiently long to be observable.However the triple gap soliton can be stable [250]. Instead of solving the system of partial differential equations, it is possible to reduce theproblem to the solution of a set of ordinary differential equations by means of the Lagrangevariational method. To illustrate the latter, let us consider the system of three partialdifferential equations (3.252).Let us define the energy functional E [ B ] ≡ X j Z B ∗ j i ∂∂t B j dz , (3.256)the Hamiltonian functional H [ B ] ≡ X j Z B ∗ j (cid:18) v j ˆ p + ˆ p m (cid:19) B j dz + 12 L X ijkl α ijkl Z B ∗ i B ∗ j B k B l dz , (3.257)and the Lagrangian L [ B ] ≡ E [ B ] − H [ B ] . (3.258)Equations (3.252) follow from the Lagrange variational equations ddt δL [ B ] δ ˙ B j − δL [ B ] δB j . (3.259)Approximate solutions to Eqs. (3.252) can be constructed by invoking trial forms for thewave packets B i , for instance, as the Gaussian envelopes B j = C j ( √ π b j ) / exp (cid:26) − ( z − z j ) b j (cid:27) exp (cid:8) − i (cid:0) α j t − β j z − γ j z (cid:1)(cid:9) , (3.260)where all variables C j , b j , z j , α j , β j , and γ j are treated as functions of time. The evolutionequations for all these variables are obtained by applying the Lagrange equations to each ofthe variables C j , b j , z j , α j , β j , and γ j . Then, instead of three equations (3.252) in partialderivatives, one gets a set of 18 equations in ordinary derivatives [250]. The latter are mucheasier to solve numerically, as well as to analyze the stability of their solutions.97 Boson Hubbard Model
In Sec. 3.3 the representation of the grand Hamiltonian (3.15) is given by expanding the fieldoperators over Bloch functions. This results in the Bloch representation of the Hamiltonian(3.55) specified in Eq. (3.56) to (3.59). The field operators could also be expanded overthe basis of Wannier functions. Such a Wannier representation, leading to the Hubbardmodel [266], is widely employed for treating electrons in solid-state lattices and the relatedmetal-insulator phase transition [267–269]. In a particular case of half filling and neglectingdouble occupancies the Hubbard model can be reduced to the so-called t − J model [270].For a periodic Bose system with BEC, the field operator can be expanded over Wannierfunctions, ˆ ψ ( r ) = X nj ˆ c ij w n ( r − a j ) . (4.1)Keeping in mind the Bogolubov-shifted field operator (2.81), that isˆ ψ ( r ) = η ( r ) + ψ ( r ) , (4.2)we have the expansion η ( r ) = r N N L X j w ( r − a j ) (4.3)for the condensate wave function and the expansion ψ ( r ) = X nj c nj w n ( r − a j ) (4.4)for the operator of uncondensed atoms. This means thatˆ c nj = r N N L δ n + c nj . (4.5)Summing this over the lattice yields X nj ˆ c nj = p N N L + X nj c nj . (4.6)Remembering the orthogonality property (3.51) and Eq. (3.53), we have X j ˆ c j = p N N L , X j c j = 0 . (4.7)Substituting expansion (4.1) into the energy Hamiltonian (2.151), we meet the followingmatrix elements: the single-site term h mni ≡ Z w ∗ m ( r − a i ) H L ( r ) w n ( r − a i ) d r , (4.8)98he hopping, or tunneling, term J mnij ≡ − Z w ∗ m ( r − a i ) H L ( r ) w n ( r − a j ) d r , (4.9)where i = j , and the interaction term U n n n n j j j j ≡ Φ Z w ∗ n ( r − a j ) w ∗ n ( r − a j ) w n ( r − a j ) w n ( r − a j ) d r , (4.10)where the local interaction potential (2.169) is assumed and H L ( r ) ≡ − ∇ m + V L ( r )is the linear lattice Hamiltonian. Then Hamiltonian (2.151) becomesˆ H = − X i = j X mn J mnij ˆ c † mi ˆ c nj + X j X mn h mnj ˆ c † mj ˆ c nj ++ 12 X { j } X { n } U n n n n j j j j ˆ c † n j ˆ c † n j ˆ c n j ˆ c n j . (4.11)To simplify Eq. (4.11), one supposes that the main contribution here comes from thelowest band, so that the single-band approximation can be employed. In so doing, oneomits the band indices. Implying that Wannier functions are well localized, one retains inthe hopping term only the nearest neighbors, with the tunneling parameter J , and in theinteraction term, one keeps only the on-site interaction, with an interaction parameter U .Thus, one arrives at the Hubbard model ˆ H = − J X
12 (4.59)and the anomalous average σ k = − ∆ k ε k coth (cid:16) ε k T (cid:17) . (4.60)Transforming sums (4.43) to integrals according to Eq. (3.45), we find the fraction of un-condensed atoms n = 1 ρ Z B n k d k (2 π ) (4.61)and the anomalous average σ = 1 ρ Z B σ k d k (2 π ) , (4.62)where the integration is over the Brillouin zone.In the center of the Brillouin zone, functions (4.59) and (4.60) behave as n k ≃ T ∆( ck ) , σ k ≃ − T ∆( ck ) ( k → . At the boundary of the Brillouin zone, one has ω k ≃ ∆ + 2 z J , ε k ≃ p z J (∆ + z J ) (cid:18) k α → πa α (cid:19) . This shows that both n k and σ k are integrable, so that n as well as σ in Eqs. (4.61) and(4.62) are finite. That is, in the case of a lattice, there are no problems with a divergentanomalous average, as for a uniform system with the local interaction potential, which isdiscussed in Sec. 2.13. The lattice regularizes σ making it always finite.The condensate fraction n = 1 − n exists below the BEC phase transition T c . This isa second-order phase transition between the normal phase and Bose-condensed phase. At105 → T c , one has n → σ →
0, and ∆ →
0. Also, ε k → ω k , when T → T c . At the criticaltemperature, Eq. (4.55) becomes ω k = 4 J X α sin (cid:18) k α a (cid:19) ( T = T c ) . (4.63)The critical temperature T c , where n = 0 and n = 1, is given by the equation ρ = 12 Z B (cid:20) coth (cid:18) ω k T c (cid:19) − (cid:21) d k (2 π ) d , (4.64)in which ω k is defined in Eq. (4.63) and a d -dimensional lattice us considered.To estimate the critical temperature, let us keep in mind a cubic lattice, for which thefilling factor can be written as ν ≡ NN L = ρa d . (4.65)By introducing the dimensionless quasimomentum vector x ≡ (cid:26) x α = k α aπ : α = 1 , , . . . , d (cid:27) , Eq. (4.64) can be represented in the form2 d +1 ν = Z B ( coth " JT c d X α =1 sin (cid:16) π x α (cid:17) − ) d x , (4.66)in which the integration over each x α is between − B ≡ { x α ∈ [ − ,
1] : α = 1 , , . . . , d } . Noticing that the main contribution to integral (4.66) comes from the central region of theBrillouin zone, we can approximate this integral by considering the asymptotic behaviour ofthe integrand at small x α . This gives T c ∼ = 2 d π (cid:18)Z B d x P α x α (cid:19) − J ν . (4.67)In that approximation, BEC does not happen for one- and two-dimensional lattices, T c ≤ d ≤ . (4.68)For larger dimensionality d >
2, we can evaluate the integral in Eq. (4.67) by invoking theDebye-like approximation. To this end, the integral over the Brillouin zone is replaced bythe Debye sphere, whose radius is chosen so that to retain the normalization condition Z B d k (2 π ) d = N L V = ρν . (4.69)106he latter, in dimensionless units, reads as Z B d x = 2 d . (4.70)Then, the Debye approximation implies Z B d x = 2 π d/ Γ( d/ Z x D x d − dx = 2 d . (4.71)From here, the Debye radius is x D = 2 √ π (cid:20) d (cid:18) d (cid:19)(cid:21) /d . (4.72)In this approximation, Z B d x P α x α = 2 π d/ x d − D ( d − d/ . Equation (4.67) yields T c = 2 d π ( d − d/ π d/ x d − D J ν . (4.73)For a three-dimensional lattice, one has T c = 2 πx D J ν , x D = (cid:18) π (cid:19) / ( d = 3) . (4.74)This results in the BEC temperature T c ∼ = 5 J ν ( d = 3) . (4.75)A close estimate for the critical temperature of condensation in a lattice follows from theBogolubov approximation [275].Thus, in the HFB approximation, the BEC does not occur in one- and two-dimensionallattices. In a three-dimensional lattice, the BEC exists below T c given by Eq. (4.75). Thetransition temperature does not depend on the on-site interaction U . However, one shouldexpect that, for a sufficiently strong repulsion U , the system could go to an insulating state.This means that the HFB approximation for the boson Hubbard model is applicable only forthe Bose-condensed phase, but is not suitable for the insulating state. Other approximationswill be discussed in the following sections. All thermodynamic characteristics can be derived from the system grand potential. Forexample, when the system is in the Bose-condensed phase, for which the HFB approximationis applicable, the grand potential isΩ = E B + T V Z B ln (cid:0) − e − βε k (cid:1) d k (2 π ) , (4.76)107here ε k is the Bogolubov spectrum (4.52) and E B = H (0) − νN U (cid:0) n + σ (cid:1) + 12 X k ( ε k − ω k ) (4.77)follows from Eq. (4.51).The system free energy is F = Ω + µN , (4.78)with the chemical potential µ = µ n + µ n . (4.79)The latter, using the Lagrange multipliers (4.49) and (4.54), becomes µ = − z J + νU (1 + n + σ − n σ ) . (4.80)The internal energy is E ≡ < H B > + µN . Here H B is given by Eq. (4.50). Also, from Eqs.(4.25) and (4.49), we have H (0) = νn N U n − n + σ )] . (4.81)And Eq. (4.48) gives E HF B = νN U (cid:2) n − σ − n σ ) (cid:3) . (4.82)Then for the ground-state energy E ≡ E B + µN = E HF B + 12 X k ( ε k − ω k ) + µN , we find E N = − z J + ν U (cid:0) n − σ − n σ (cid:1) + 12 ρ Z B ( ε k − ω k ) d k (2 π ) . (4.83)Particle fluctuations are characterized by the dispersion ∆ ( ˆ N ) of the number-of-particleoperator, as defined in Eq. (2.243). For a Bose system, with the broken gauge symmetryby means of the Bogolubov shift, the condensate fluctuations are negligible [57,93], that is,∆ ( ˆ N ) →
0. Hence, all fluctuations are due to uncondensed particles,∆ ( ˆ N ) = ∆ ( ˆ N ) ˆ N = X k a † k a k ! . (4.84)Since the HFB approximation results in the Hamiltonian (4.47), quadratic with respect tothe filed operators a k , then calculating ∆ ( ˆ N ), one has to be in the frame of the quadraticapproximation, that is, neglecting the terms with n k and σ k . Similarly to the uniform case[10,93,145,146], for BEC in a lattice, we have∆ ( ˆ N ) = N [1 + 2 lim k → ( n k + σ k )] . (4.85)108rom Eqs. (4.59) and (4.60), we find n k ≃ T ∆ ε k + ∆12 T + T − ,σ k ≃ − T ∆ ε k − ∆12 T ( ε k → . (4.86)Therefore, lim k → ( n k + σ k ) = 12 (cid:18) T ∆ − (cid:19) . (4.87)Then Eq. (4.85) gives ∆ ( ˆ N ) = N T ∆ . (4.88)According to Eq. (4.46), one gets ∆ ( ˆ N ) = N TνU ( n + σ ) . (4.89)The isothermal compressibility reads as κ T = ∆ ( ˆ N ) ρT N = 1 ρνU ( n + σ ) . (4.90)Particle fluctuations are, of course, thermodynamically normal and the compressibilityis finite. The latter diverges only at the critical point T c , where n → σ →
0. Butbelow T c , the compressibility is finite everywhere for T < T c , provided that there is a finiteinteraction U . The superfluid fraction can be calculated by employing Eq. (3.106) which has been derivedin Sec. 2.10. Equation (3.106) is general and exact. For a lattice, the operator of momen-tum can be represented in forms (3.83) or (3.84). The latter reduces to Eq. (3.93). Intight-binding approximation, one can invoke Eq. (3.105). The dissipated heat in the HFBapproximation is given in Eq. (3.110).Another possibility is to rederive the superfluid fraction using explicitly the HubbardHamiltonian (4.12), in this derivation being based on the general definitions (2.135) or(2.139). For an equilibrium system, both these definitions yield Eq. (2.138). Generaliz-ing the latter for a d -dimensional system, we have n s = 1 mN d (cid:20) lim v → < ∂∂ v · ˆ P v > − β ∆ ( ˆ P ) (cid:21) . (4.91)In order to use Eq. (4.91), with the Hubbard Hamiltonian (4.12), it is necessary to findhow the Wannier field operators ˆ c j change under the velocity boost. Generally, if the system109s boosted with the velocity v , the field operator of the moving system can be expanded overthe Wannier functions as ˆ ψ v ( r , t ) = X j ˆ c j ( v , t ) w ( r − a j ) . (4.92)Being in the frame of the Hubbard model, the single-band case is considered here. Invertingexpansion (4.92) gives ˆ c j ( v , t ) = Z w ∗ ( r − a j ) ˆ ψ v ( r , t ) d r . (4.93)Substituting in Eq. (4.93) the Galilean transformation (2.146), with the expansionˆ ψ ( r − v t, t ) = X j ˆ c j (0 , t ) w ( r − v t − a j ) , (4.94)we get the relationˆ c i ( v , t ) = X j ˆ c j (0 , t ) exp (cid:18) − i mv t (cid:19) Z w ∗ ( r − a i ) w ( r − v t − a j ) e im v · r d r . (4.95)This is a general relation connecting the Wannier field operators ˆ c i ( v , t ) for a moving systemwith these operators ˆ c j (0 , t ) for an immovable lattice.Keeping in mind an equilibrium system, we can set time to zero, introducing the simplifiednotation ˆ c j ( v ) ≡ ˆ c j ( v , . (4.96)Diminishing the velocity to zero, we return to the old notation of the Wannier field operatorsof an immovable lattice, ˆ c j ≡ lim v → ˆ c j ( v ) = lim v → ˆ c j ( v , . (4.97)Then relation (4.95) becomesˆ c i ( v ) = X j ˆ c j Z w ∗ ( r − a i ) w ( r − a j ) e im v · r d r . (4.98)Since in the Hubbard model, one assumes well localized Wannier functions, one can use theapproximation Z w ∗ ( r − a j ) w ( r − a j ) e im v · r d r ∼ = δ ij e im v · a j . (4.99)Therefore, relation (4.98) simplifies toˆ c j ( v ) = ˆ c j e im v · a j . (4.100)The Hamiltonian (4.12) of the Hubbard model for an immovable system is the functionalˆ H = ˆ H [ˆ c j ]. For a moving lattice, the latter becomes ˆ H v = ˆ H [ˆ c j ( v )], which givesˆ H v = − J X
1. Thelatter, in turn, happens when atomic interactions are sufficiently weak. Strong interactions,as is known [94–98], deplete the condensed fraction, while enhance the superfluid fraction,resulting in the inequality n s ≫ n . In the presence of a lattice, strong interactions shouldlead to the destruction of both BEC and superfluidity and to the appearance of a localizedinsulating state. When atomic interactions are much larger than the tunneling rate, so that UJ ≫ , (4.141)than the intersite hopping of atoms is completely suppressed. This can be symbolized bythe localization condition c † i c j = δ ij c † i c i . (4.142)When all atoms are localized, global coherence cannot develop. There is no BEC and nogauge symmetry breaking, n = 0 , n = 1 . (4.143)The Hubbard Hamiltonian (4.12) reduces toˆ H = U X j c † j c † j c j c j . (4.144)Using the site-number operator, or filling operator ˆ n j ≡ c † j c j , (4.145)Hamiltonian (4.144) can be rewritten asˆ H = U X j ˆ n j (ˆ n j − . (4.146)Since there is just one kind of atoms, the sole chemical potential is sufficient. So, thegrand Hamiltonian is H ≡ ˆ H − µ ˆ N = X j H j , (4.147)where the site Hamiltonians are H j = U n j − (cid:18) U µ (cid:19) ˆ n j , (4.148)the number-of-particle operator being ˆ N = X j ˆ n j . (4.149)117or an ideal lattice, the average of the site-number operator (4.145) gives the filling factor ν = < ˆ n j > . (4.150)Explicitly, the latter reads as ν = Trˆ n j e − βH Tr e − βH . (4.151)This equation defines the chemical potential µ = µ ( ν, T ). Owing to the additive form of thegrand Hamiltonian (4.147), expression (4.151) becomes ν = Trˆ n j e − βH j Tr e − βH j . (4.152)The eigenproblem for the site-number operators (4.145),ˆ n j | n > = n | n > , (4.153)enjoys, as eigenvalues, the integers n = 0 , , , . . . , while | n > being the occupation-numberstate [55]. The eigenproblem for the site Hamiltonians (4.148), H j | n > = e n | n > , (4.154)gives the energy levels e n = U n − (cid:18) U µ (cid:19) n . (4.155)The levels are discrete. One can define the energy gap∆ e n ≡ e n +1 + e n − − e n , (4.156)which gives ∆ e n = U . (4.157)This is why one tells that the insulator energy spectrum possesses a gap.With the basis {| n > } of the occupation-number states, the filling factor (4.152) takesthe form ν = P ∞ n =0 ne − βe n P ∞ n =0 e − βe n . (4.158)For high temperature ( T ≫ U ), the sums in Eq. (4.158) can be replaced by integrals. Butfor low temperature ( T ≪ U ), the main contribution to the sums comes from the term withthe lowest energy e n .Let us consider low temperatures, such that TU ≪ . (4.159)The minimum of e n is defined by the conditions ∂e n ∂n = 0 , ∂ e n ∂n > . (4.160)118his yields the effective number n eff = 2 µ + U U ( U > . (4.161)For low temperatures, conditioned by inequality (4.159), the filling factor (4.158) is ν ≃ n eff = 2 µ + U U . (4.162)This defines the chemical potential µ ≃ (2 ν − U . (4.163)The grand thermodynamic potential isΩ ≡ − T ln ∞ X n =0 e − βe n ≃ e ν , (4.164)with e ν ≡ U ν − (cid:18) U µ (cid:19) ν . (4.165)Substituting here the chemical potential (4.163) yields e ν = − U ν . (4.166)The internal energy reads as E ≡ < H > + µN ≃ e ν N L + µN . (4.167)Hence, the ground-state energy per atom is EN = ( ν − U T = 0) . (4.168)The fluctuations of particles are characterized by the dispersion ∆ ( ˆ N ), as is described inSec. 2.14. With the number-of-particle operator (4.149), we have [93,94]∆ ( ˆ N ) = X j ∆ (ˆ n j ) + X i = j cov(ˆ n i , ˆ n j ) . For the considered localized state, < ˆ n i ˆ n j > = < ˆ n i >< ˆ n j > ( i = j ) , because of which cov(ˆ n i , ˆ n j ) = 0. The dispersion of the filling operator (4.145) is∆ (ˆ n j ) ≡ < ˆ n j > − < ˆ n j > = T ∂ν∂µ . (4.169)119his, according to Eq. (3.14), defines the compressibility κ T = a d ν ∂ν∂µ = a d ∆ (ˆ n j ) ν T . (4.170)At low temperatures, satisfying inequality (4.159), we have < ˆ n j > ≃ ν , < ˆ n j > ≃ ν . Therefore ∆ (ˆ n j ) ≃ κ T ≃
0. In that sense, one tells that the localized state isincompressible.
To locate the phase transition between the purely delocalized and localized states, one cancompare the corresponding thermodynamic potentials. For instance, one can consider thegrand potential of the Bose-condensed superfluid phase, Ω sup = Ω sup ( ν, J, U, T ), given, e.g.,by Eq. (4.76). From another side, one has the grand potential of the localized phase,Ω loc = Ω loc ( ν, U, T ), given, e.g., by Eq. (4.164). The phase boundary could be describedby the equality Ω sup = Ω loc . However, for finite temperature, the system cannot be com-pletely localized, but contains a portion of wandering atoms [283–285]. The same concernsthe noninteger filling factors ν , for which the system is not absolutely localized [286]. Com-plete localization can occur for integer filling factors ν = 1 , , . . . at zero temperature. Sucha completely localized state is called the Mott insulator [267,268]. The phase transitionbetween the superfluid state and Mott insulator, occurring for an integer filling at zero tem-perature, has been studied in various approximations. One usually starts with the HubbardHamiltonian ˆ H = − J X
2, hence, the potential energy per particle at one site is U/
4. Thetotal potential energy for the lattice is N L U/
4. So, the potential energy per particle is N L U/ N = U/ ν . The phase transition happens when the tunneling energy z J equals thepotential energy U/ ν . This defines the critical parameter (5.7) as u c = 4 ν . (5.8)Here it has been assumed that there are no external fields.121 .4 Gutzwiller Single-Site Approximation This approximation involves the use of variational wave functions of the Gutzwiller type,which are represented as products over the lattice sites [278–282,294]. At zero temperature,the critical parameter (5.8) is found to be u c = (cid:16) √ ν + √ ν (cid:17) . (5.9)For the unity filling ν = 1, this gives u c = 5 . ν = 1) . (5.10)The critical value (5.10) does not depend on the lattice dimensionality, which is usual formean-field-type approximations. Generally, the microscopic dynamics of the localization-delocalization transition is influenced by the space dimensionality [295]. Another type of a mean-field approximation was employed by Amico and Penna [296]. Theyfound that, at zero temperature, the superfluid-insulator transition is located at u c = 4 ν , (5.11)in exact agreement with the simple estimate (5.8). Again, the result does not depend onthe lattice dimensionality. One may notice that the Gutzwiller-approximation value (5.9)reduces to either Eq. (5.8) or Eq. (5.11) for large coordinate numbers ν . Direct numerical diagonalization can be done for small one-dimensional lattices, of about 10sites, at zero temperature [276]. This gives, for the unity filling factor, the critical value u c = 2 . ν = 1 , d = 1) . (5.12)For one-dimensional lattices, value (5.12) is lower than the values predicted by mean-filedestimates because of the stronger influence of fluctuations that are underestimated in mean-field approximations. This is another numerical method that can be applied to small one-dimensional lattices [297]giving u c = 1 . ν = 1 , d = 1) . (5.13)This is close to value (5.12) found by the numerical diagonalization.122 .8 Strong-Coupling Perturbation Theory Perturbation theory in powers of
J/U can be used, with the following summation of series,e.g., by means of Pad´e approximants [298-300]. One-, two-, and three-dimensional cubic(square) lattices have been considered. The results are u c = 1 . ν = 1 , d = 1 [299]) ,u c = 4 . ν = 1 , d = 2 [299]) ,u c = 4 . ν = 1 , d = 3 [298]) . (5.14)The value for the one-dimensional lattice is close to the numbers obtained by the small-system numerical diagonalization (5.12) and by the density-matrix renormalization group(5.13). The simulations are accomplished for a finite number of bosons, which can reach N ≃ atoms. One-, two-, and three-dimensional rectangular lattices have been investigated [301–305]. The most recent results are presented below: u c = 1 . ν = 1 , d = 1 [302]) ,u c = 4 . ν = 1 , d = 2 [305]) ,u c = 4 . ν = 1 , d = 3 [304]) . (5.15)All these quantities agree well with the strong-coupling perturbation theory. The finite-temperature phase diagram at filling factor ν = 1 is also found [304,305]. The criticaltemperature T c = T c ( U ) as a function of the on-site interaction displays a nonmonotonicbehavior, as has been suggested by Kleinert et al. [275]. When U increases from zero, T c ,first rises, reaches the maximum at around U/J ≈
5, and then diminishes to zero at thecritical value U c /J . The transition between the superfluid and Mott insulator phases, occurring at zero tem-perature and integer filling factors, is an example of the quantum phase transitions. Thisis a continuous transition, that is, a second-order phase transition. The role of an orderparameter is played by the condensate fraction n . Considering the latter as a function n = n ( u ) of the dimensionless parameter (5.7), one has the following behavior. In theabsence of interaction, n (0) = 1. Then, the condensate fraction diminishes with increasing u , and drops continuously to zero at the critical value u c . The continuous nature of thesuperfluid-Mott insulator quantum phase transition in any dimension follows from the factthat the d -dimensional Hubbard model with BEC pertains to the universality class of a d + 1-dimensional XY -model [306]. 123 .11 Experiments on Superfluid-Insulator Transition Several experiments observing the superfluid-insulator phase transition of cold bosons inoptical lattices have been accomplished. Actually, because of the finiteness of the lattices,it is not a sharp phase transition that has been observed, but a gradual crossover betweenthe superfluid and Mott insulator states, occurring around the critical values u c predictedby theory.The first experiment was by Greiner et al. [307] with Rb BEC at zero temperature. Athree-dimensional lattice was formed by three optical standing waves aligned orthogonal toeach other. The laser beams operated at a wavelength λ = 852 nm, forming the cubic three-dimensional lattice with the lattice spacing a = 4 . × − cm. The existence of coherencein the superfluid Bose-condensed phase and its absence in the insulating localized phase wasanalysed by studying the level of interference after suddenly turning off the trapping potentialand allowing atoms to expand freely. For the unity filling factor, the superfluid-insulatorcrossover was localized around u c ≈ Rb atoms. The optical lattice was formed by laser beams at a wavelength λ = 826nm, which translated into the lattice spacing a = 4 . × − cm. Bragg spectroscopy wasemployed for investigating the excitation spectrum. The superfluid insulator crossover, atthe unity filling factor, was observed close to u c ≈ .
8, though for one-dimensional latticesit should happen at u c ≈ .
8. This disagreement could be due to the finite size of the trap.A very important finding was that for strongly interacting Bose systems in optical latticesthe superfluid fraction could be significantly different from the coherent BEC fraction.Different types of lattices, one-, two-, and three-dimensional optical lattices were createdby K¨ohl et al. [309] for Rb atoms. The optical lattices were formed by retro-reflected laserbeams at a wavelength λ = 826 nm, which corresponded to the lattice spacing a = 4 . × − cm. The superfluid-insulator crossover was investigated by using Bragg spectroscopyfor studying the excitation spectra. The appearance of the discrete spectrum structure,associated with the Mott insulating phase, was observed between u = 4 and u = 8.A three-dimensional lattice, filled by sodium Na atoms, was formed by Xu et al. [310].A dye laser operated at λ = 594 . a = 2 . × − cm.The system properties were studied by the time-off-flight images. The filling factors couldbe varied between ν = 1 to ν = 5. The superfluid-insulator crossovers were observed aroundthe critical values u c given by the single-site approximation (5.9).In the experiment by Spielman et al. [311] the BEC of Rb atoms was loaded into atwo-dimensional optical lattice formed by laser light of λ = 820 nm. This gave the latticespacing a = 4 . × − cm. The system properties were analysed by studying the time-of-flight images. For the unity filling factor, the superfluid-insulator crossover was locatedaround the critical value u c ≈
4, which was in agreement with the Monte Carlo simulationsas could be seen from Eq. (5.15).
When, in addition to the optical lattice potential, there is an external trapping potential, thesystem becomes nonuniform, so that the lattice is no longer ideal. Then in some spatial parts124he conditions could be created for the occurrence of the insulating phase, while in otherspatial locations the superfluid phase would be preferable. This results in the formationof a shell structure, where the layers of Mott insulating phases alternate with the layers ofsuperfluid phases. This layered structure was studied by Monte Carlo methods [312], byemploying a pseudospin approximation [313], and was observed in experiment [314].
The Hubabrd model (5.1) contains only the on-site atomic interaction U . This is, of course,a simplification. When deriving the Hubabrd Hamiltonian in Sec. 4.1, we could see thatthe Wannier representation (4.11) of the general system Hamiltonian (2.151) includes inter-actions between all lattice sites. Strictly speaking, all these interactions are nonzero evenfor the local interaction potential, as follows from the matrix element (4.10). Consideringsolely the on-site interaction assumes that the interaction potential is of short-range typeand Wannier functions are well localized at their lattice sites. But when the lattice opticalpotential is shallow, the corresponding Wannier functions may be not so well localized. Or,if the pair atomic interaction potential is not short-range, then the interactions of atoms atneighboring sites can be sufficiently important and not negligible. For example, the dipolarquantum gas of Cr possesses long-range dipolar interactions [315,316]. Therefore, there arerealistic situations, when the different-site interactions could be important. This requires toconsider the extended Hubbard model ˆ H = − J X
0, butthere is no superfluidity, n s = 0.In order to demonstrate how the Hubbard model for a quasiperiodic lattice could beconstructed, let us consider the one-dimensional case with the bichromatic optical potential(5.17). The derivation of the Hubbard model can be done in the same way as in Sec. 4.1, ifone treats one of the sublattices as primary and accomplishes the expansions over Wannierfunctions associated with this primary sublattice. For instance, the first term in potential(5.17) can be treated as primary, hence, the primary sublattice having the period a , relatedto the laser wavevector k = π/a .The lattice local Hamiltonian can be separated into two parts, H L ( z ) = − ∇ m + V L ( z ) = H L ( z ) + ∆ H L ( z ) , (5.18)the first part including the primary potential, H L ( z ) ≡ − ∇ m + V sin ( k z ) , (5.19)and the addition being ∆ H L ( z ) ≡ V sin ( q z ) . (5.20)The primary term (5.19) is periodic over the sublattice { a j } , so that H L ( z + a j ) = H L ( z ) . The Hubbard parameters are defined in Eqs. (4.8), (4.9), and (4.10). As in Eq. (4.12), weshall consider the single-band Hubbard model.For the tunneling term, we have J ij = J ij + ∆ J ij , J ij ≡ − Z w ( z − a i ) H L ( z ) w ( z − a j ) dz , ∆ J ij ≡ − Z w ( z − a i )∆ H L ( z ) w ( z − a j ) dz . (5.21)Respectively, the single-site term h j becomes h j = h + ∆ h j , h ≡ Z w ( z ) H L ( z ) w ( z ) dz , h j ≡ Z w ( z − a j )∆ H L ( z ) w ( z − a j ) dz . (5.22)Using for Wannier functions the Gaussian approximation (3.161), we follow calculationssimilar to those in Sec. 3.9. Then we have the primary parameters J = (cid:18) π − (cid:19) V exp (cid:18) − a l (cid:19) , h = p V E R , (5.23)where J is J ij for nearest neighbors. The additional term for the tunneling parameter is∆ J ij = − V (cid:18) − a l (cid:19) (cid:8) − exp (cid:0) − q l (cid:1) cos[ q ( a i + a j )] (cid:9) , (5.24)and the addition to the single-site term is∆ h j = V (cid:2) − exp( − q l ) cos(2 q a j ) (cid:3) . (5.25)The latter equations can be simplified, when there is good localization, not only withrespect to the primary sublattice, such that k l ≪
1, but also with respect to the secondarysublattice, so that q l ≪ . (5.26)Then Eq. (5.24) becomes∆ J ij ∼ = − V exp (cid:18) − a l (cid:19) sin (cid:18) q a i + a j (cid:19) , (5.27)while Eq. (5.25) reduces to ∆ h j ∼ = V sin ( q a j ) . (5.28)The on-site interaction parameter U remains the same as in Eq. (4.14).Thus, the Hubbard parameters J and h acquire the site-dependent terms (5.27) and(5.28). Comparing these additional terms with the primary values J and h , we have (cid:12)(cid:12)(cid:12)(cid:12) ∆ J ij J (cid:12)(cid:12)(cid:12)(cid:12) ≤ . V V , ∆ h j h ≤ V V r V E R . (5.29)And the Hubbard Hamiltonian for the quasiperiodic bichromatic potential takes the formˆ H = − X
The presence of an externally incorporated disorder can essentially change the system proper-ties. And the possibility of varying the level of disorder presents a powerful tool for achievingdifferent states of matter. This problem has been considered in solid state physics for manyyears, with applications to conducting properties of materials [337–340], magnetic propertiesof spin glasses [341,342], to real-space glasses [343,344], amorphous alloys [345,346], and to128hysics of caking [347]. Disorder can also be introduced by the boundaries of finite systems[101,348] and by domain walls [349,350].Recently, several experiments [351–354] have studied Rb in random optical potentialscreated by optical speckles. It has been observed that the speckle randomness inducesdamping of collective excitations [351] and inhibition of expansion [352–354]. The effects ofa disordered optical potential on the transport and phase coherence of a BEC of Li atomshas also been studied, and inhibition of transport and damping of dipole excitations havebeen observed [355]. An ultracold bosonic gas of Rb in a 3-dimensional optical latticewas investigated, with disorder induced by a small contribution of fermionic K impurityatoms [356]. The random admixture was found to favor the localization of bosonic atoms.It was suggested [357] that a one-dimensional BEC in a weak random potential can exhibitAnderson localization [358].The Hamiltonian of atoms in an optical lattice potential V L ( r ), and also subject to theaction of an additional random potential ξ ( r ), has the formˆ H = Z ψ † ( r ) (cid:20) − ∇ m + V L ( r ) + ξ ( r ) (cid:21) ψ ( r ) d r + 12 Φ Z ψ † ( r ) ψ † ( r ) ψ ( r ) ψ ( r ) d r , (6.1)where Φ ≡ πa s /m . The random potential is real ξ ∗ ( r ) = ξ ( r ) . (6.2)Without the loss of generality, it can be taken as zero-centered, so that its stochastic aver-aging be ≪ ξ ( r ) ≫ = 0 . (6.3)The stochastic averaging of the pair product ≪ ξ ( r ) ξ ( r ′ ) ≫ = R ( r − r ′ ) (6.4)is characterized by a random-potential correlation function that is real and symmetric, suchthat R ∗ ( r ) = R ( r ) = R ( − r ) . (6.5)Details of defining the stochastic averaging can be found in the book [359].When using the Bloch representation for the field operator ψ ( r ) = X nk a nk ϕ nk ( r ) , it is convenient to introduce the matrix element of the random potential β mnkp ≡ Z ϕ ∗ mk ( r ) ξ ( r ) ϕ np ( r ) d r . (6.6)Then the term of Hamiltonian (6.1), containing the random potential, is Z ψ † ( r ) ξ ( r ) ψ ( r ) d r = X mn X kp β mnkp a † mk a np . ψ ( r ) = X nj c nj w n ( r − a j ) , then one meets the matrix element γ mnij ≡ Z w ∗ m ( r − a i ) ξ ( r ) w n ( r − a j ) d r . (6.7)And the random term in Eq. (6.1) becomes Z ψ † ( r ) ξ ( r ) ψ ( r ) d r = X mn X ij γ mnij c † mi c nj . The matrix elements (6.6) and (6.7) are connected through the Fourier transformations β mnkp = 1 N L X ij γ mnij e − i k · a i + i p · a j , γ mnij = 1 N L X kp β mnkp e i k · a i − i p · a j . (6.8)In the uniform limit, when the Bloch functions ϕ nk ( r ) tend to the plane waves e i k · r / √ V ,it is convenient to define the Fourier transform of the random potential ξ k ≡ √ V Z ξ ( r ) e − i k · r d r , ξ ( r ) ≡ √ V X k ξ k e i k · r . (6.9)Then one has β mnkp = 1 √ V ξ k − p . (6.10)In the presence of BEC, one has to break the global gauge symmetry [63], which can bedone by the Bogolubov shift of the field operator ψ ( r ) → ˆ ψ ( r ) ≡ η ( r ) + ψ ( r ) , (6.11)as is described in Sec. 2.7. This implies, in agreement with Sec. 3.3, the corresponding shiftof the Bloch field operator a nk → ˆ a nk ≡ δ n δ k p N + a nk (6.12)and of the Wannier field operator c nj → ˆ c nj ≡ δ n √ νn + c nj . (6.13)The Bogolubov shift realizes unitary nonequivalent operator representations [99,360–362].The grand Hamiltonian of a system with BEC is H = ˆ H − µ N − µ ˆ N . (6.14)130he type of disorder induced by an external random potential is termed the frozen disorderor quenched disorder. The corresponding grand thermodynamic potential is defined asΩ = − T ≪ ln Tr e − βH ≫ . (6.15)And the free energy is F = Ω + µN , µ = µ n + µ n . It is important to emphasize that there are two kinds of averaging for quantum randomsystems. One type of averaging is the stochastic averaging, denoted by the double anglebrackets ≪ . . . ≫ , characterizing the averaging over the distribution of random potentials.And there is the quantum statistical averaging, which for an operator ˆ A is defined as < ˆ A > H ≡ Tr ˆ ρ ˆ A , (6.16)with the statistical operator ˆ ρ = exp( − βH )Tr exp( − βH ) . (6.17)The total averaging < ˆ A > ≡ ≪
Tr ˆ ρ ˆ A ≫ (6.18)includes both, the quantum as well as stochastic averaging procedures.The condensate fraction n = 1 − n is expressed through the fraction of uncondencedatoms n ≡ N Z < ψ † ( r ) ψ ( r ) > d r = 1 N X nk < a † nk a nk > = 1 N X nj < c † nj c nj > . (6.19)The anomalous average is σ ≡ N Z < ψ ( r ) ψ ( r ) > d r = 1 N X nk < a nk a n, − k > = 1 N X nj < c nj c nj > . (6.20)For random systems, it is possible to define an additional order parameter, the glassyfraction n G ≡ N Z ≪ | < ψ ( r ) > H | ≫ d r , (6.21)which is analogous to the Edwards-Anderson order parameter for spin glasses [341,342].Though the total average < ψ ( r ) > = 0, the partial, quantum, average < ψ ( r ) > H is notnecessarily zero. Respectively, the quantum averages α nk ≡ < a nk > H , α nj ≡ < c nj > H (6.22)are not zero, contrary to the total averages < ψ ( r ) > = < a nk > = < c nj > = 0 . (6.23)From here it follows that ≪ α nk ≫ = ≪ α nj ≫ = 0 . (6.24)131he glassy fraction (6.21) can be rewritten as n G = 1 N X nk ≪ | α nk | ≫ = 1 N X nj ≪ | α nj | ≫ . (6.25)The presence of random fields induces in the system additional fluctuations, which istypical of random and chaotic systems [363]. Without disordering fields, the condensate isdepleted by atomic interactions and temperature. The inclusion of random fields depletesthe condensate even more and creates the glassy fraction (6.21). A uniform system is a limiting case of a very shallow lattice. Then, in the Bloch represen-tation, the Bloch functions are plane waves. Uniform Bose-condensed systems in randompotentials have been studied for asymptotically weak atomic interactions and asymptoticallyweak strength of disorder [364-366]. A theory for arbitrary strong random potentials andinteractions was developed in Refs. [367,368]. When a disordered potential is created insidea trap, then the BEC properties can be investigated by means of time-of -flight experiments[369].For a system, uniform on average, the condensate fraction is n = 1 ρ | η ( r ) | ( η ( r ) = √ ρ ) . (6.26)The fraction of uncondensed atoms (6.19) can be written as n = 1 N X k n k (cid:16) n k ≡ < a † k a k > (cid:17) . (6.27)The anomalous average (6.20) takes the form σ = 1 N X k σ k ( σ k ≡ < a k a − k > ) . (6.28)And the glassy fraction (6.21) becomes n G = 1 N X k ≪ | α k | ≫ , (6.29)where the notation α k ≡ < a k > H (6.30)is introduced.Accomplishing in Eq. (6.1) the Bogolubov shift (6.11) and passing to the Fourier-transformed field operators a k , for the grand Hamiltonian (6.14), we obtain H = X n =0 H ( n ) + H ext , (6.31)132n which the terms in the sum are H (0) = (cid:18) ρ Φ − µ (cid:19) N , H (1) = 0 ,H (2) = X k =0 (cid:18) k m + 2 ρ Φ − µ (cid:19) a † k a k + 12 X k =0 ρ Φ (cid:16) a † k a †− k + a − k a k (cid:17) ,H (3) = r ρ V X kp ( =0) Φ (cid:16) a † k a k + p a − p + a †− p a † k + p a k (cid:17) ,H (4) = 12 V X q X kp ( =0) Φ a † k a † p a k − q a p + q (6.32)and the last term, caused by the random potential, is H ext = ρ ξ √ V + √ ρ X k =0 (cid:16) a † k ξ k + ξ ∗ k a k (cid:17) + 1 √ V X kp ( =0) a † k a p ξ k − p . (6.33)The correlation function of a disordered potential, defined in Eq. (6.4), is assumed topossess a Fourier transform R k = Z R ( r ) e − i k · r d r , R ( r ) = 1 V X k R k e i k · r . (6.34)From properties (6.2) and (6.5), it follows that ξ ∗ k = ξ − k , R ∗ k = R − k = R k . (6.35)And the stochastic averaging (6.4) gives ≪ ξ ∗ k ξ p ≫ = δ kp R k . (6.36)In the case of white noise, one has R ( r ) = R δ ( r ) , R k = R . (6.37)Thence, Eq. (6.36) reduces to ≪ ξ ∗ k ξ p ≫ = δ kp R . (6.38)The Hamiltonian terms in Eq. (6.32) can be simplified by using the Hartree-Fock-Bogolubov approximation, as in Refs. [57,94–99]. But the random Hamiltonian term (6.33)should be treated with care. If one would use the standard mean-field decoupling for the lastterm in Eq. (6.33), one would kill in this term all quantum effects because of Eqs. (6.3) and(6.23). Therefore it is necessary to invoke a more delicate decoupling procedure. For thispurpose, it is convenient to employ the stochastic mean-field approximation suggested andused earlier for other physical systems [241,242,370–373]. This approximation in the presentcase yields a † k a p ξ k − p = (cid:16) a † k α p + α ∗ k a p − α ∗ k α p (cid:17) ξ k − p . (6.39)133hen Hamiltonian (6.31) can be diagonalized by the use of the nonuniform nonlinear canonical transformation a k = u k b k + v ∗− k b †− k + w k ϕ k , (6.40)which generalizes the standard uniform Bogolubov transformation. The diagonalizationimplies that the resulting Hamiltonian should be diagonal in terms of the operators b † k and b k , so that < b k > H = < b k b p > H = 0 . Hence, relation (6.40) gives < a k > H = w k ϕ k . (6.41)This diagonalization results in u k = ω k + ε k ε k , v k = ω k − ε k ε k , w k = − ω k + mc ,ω k = k m + mc , ω k = ε k + ( mc ) , (6.42)where ε k = s ( ck ) + (cid:18) k m (cid:19) (6.43)is the Bogolubov spectrum, with the sound velocity c defined by the equation mc = ( n + σ ) ρ Φ . (6.44)The random variable ϕ k in transformation (6.40) satisfies the Fredholm equation ϕ k = √ ρ ξ k − √ V X p ξ k − p ϕ p ω p + mc . (6.45)The nonuniform nonlinear transformation (6.40) reduces the grand Hamiltonian (6.31) tothe form H = E B + X k ε k b † k b k + ϕ p N , (6.46)with E B = 12 X k ( ε k − ω k ) − (cid:20) − n + 12 ( n + σ ) (cid:21) ρ Φ N . (6.47)The diagonal Hamiltonian (6.46) allows us to find the momentum distribution n k = ω k ε k coth (cid:16) ε k T (cid:17) −
12 + ≪ | α k | ≫ (6.48)and the anomalous average σ k = − mc ε k coth (cid:16) ε k T (cid:17) + ≪ | α k | ≫ . (6.49)134he random variable (6.30), in view of Eqs. (6.41) and (6.42), is α k = w k ϕ k = − ϕ k ω k + mc . (6.50)Therefore ≪ | α k | ≫ = ≪ | ϕ k | ≫ ( ω k + mc ) . (6.51)The fraction of uncondensed atoms (6.27) becomes a sum of two terms, n = n N + n G , (6.52)in which the first term n N = 12 ρ Z (cid:20) ω k ε k coth (cid:16) ε k T (cid:17) − (cid:21) d k (2 π ) (6.53)is the fraction of uncondensed atoms, due to finite temperature and interactions, while thesecond term n G = 1 ρ Z ≪ | ϕ k | ≫ ( ω k + mc ) d k (2 π ) (6.54)is the glassy fraction (6.29), caused by the random potential. The anomalous average (6.28)is also a sum of two terms, σ = σ N + n G , (6.55)with the first term σ N = − ρ Z mc ε k coth (cid:16) ε k T (cid:17) d k (2 π ) (6.56)and the second term (6.54).To find the superfluid fraction, we resort to the general definition of Sec. 2.10, n s = 1 − Q T (cid:18) Q ≡ < P > mN (cid:19) . (6.57)The dissipated heat consists of two parts, Q = Q N + Q G . (6.58)The first part Q N = 18 mρ Z k sinh ( ε k / T ) d k (2 π ) (6.59)is the heat dissipated by uncondensed atoms related to finite temperature and interactions.And the second part Q G = 12 mρ Z k ≪ | ϕ k | ≫ ε k ( ω k + mc ) coth (cid:16) ε k T (cid:17) d k (2 π ) (6.60)is the heat dissipated by the glassy fraction of atoms.135he analysis [367,368] of the derived equations shows that for weak interactions andsufficiently strong disorder, the superfluid fraction can become smaller than the condensatefraction, in agreement with Monte Carlo simulations [374]. But at relatively strong inter-actions, the superfluid fraction is larger than the condensate fraction for any strength ofdisorder. The condensate and superfluid fractions, and the glassy fraction always coexist,being together either nonzero or zero. In the presence of disorder, the condensate fractionbecomes a nonmonotonic function of the interaction strength, displaying an antidepletioneffect caused by the competition between the stabilizing role of the atomic interaction andthe destabilizing role of the disorder. An ideal Bose gas with BEC is stochastically unsta-ble, the BEC being destroyed by infinitesimally weak disorder. Finite atomic interactionsstabilize the system. When the strength of disorder is increased, reaching a critical value ζ c of the disorder parameter ζ ≡ m R πρ / = al L , in which l L ≡ π m R a ≡ ρ / , the condensate and superfluid fractions drop to zero by a first-order phase transition. Thecharacteristic length l L practically coincides with the Larkin length [375]. The critical valueof the disorder parameter ζ c , where the first-order phase transition occurs, depends on theinteraction strength and temperature. More details on the properties of the disorderedsuperfluid can be found in Refs. [367,368].After the strength of disorder reaches its critical value, characterized by the critical valueof the disorder parameter ζ c , the global coherence becomes destroyed and there is no globalBEC occupying the whole system. Also, there is no superfluidity. But for ζ > ζ c , thecondensate can decay into fragments each of the size of the Larkin length l L , so that thephase transition at ζ c could be interpreted as a spatial condensate fragmentation [376]. For ζ > ζ c , the local remnants of the condensate could remain, while superfluidity being absent.Such a state, according to the classification of Sec. 1.4, corresponds to the Bose glass. Thedestruction of superfluidity, with increasing disorder, through a first-order transition, wasalso found in Ref. [377]. Passing from the initial Hamiltonian (6.1) to the Wannier representation, we may, as usual,consider the single-band case. Then the matrix element (6.7) of the random potential ξ ( r )reads as γ ij = Z w ∗ ( r − a i ) ξ ( r ) w ( r − a j ) d r . (6.61)For the single-site effective potential, we have h j = h + γ j , (6.62)in which the constant term h ≡ Z w ∗ ( r ) H L ( r ) w ( r ) d r γ j ≡ γ jj = Z | w ( r − a j ) | ξ ( r ) d r . (6.63)This diagonal term is real because of Eq. (6.2), γ ∗ j = γ j . (6.64)From Hamiltonian (6.1), the Hubbard Hamiltonian followsˆ H = − X
25) systems [390], strong-coupling expansions[298], density matrix renormalization group [391], mean-field single-site approximation [392–394], and Monte Carlo simulations [395–404].The general physical properties of disordered boson lattices have been described by Fisheret al. [306]. Depending on the system parameters, several states can be realized. There existsthe usual superfluid coherent phase, where n s > n >
0. This phase possesses a finitecompressibility and has no gap in the spectrum of collective excitations. The Mott insulatingphase can occur at zero temperature and integer filling. This phase has zero compressibilityand a gap for particle-hole excitations. The superfluid-Mott insulator phase transition is asecond-order transition being in the universality class of the ( d + 1)-dimensional XY model.When an additional trapping potential is imposed on the lattice, the phase transition pointcould be slightly shifted [405].Except these two phases that also exist in regular optical lattices, for disordered lattices,there can arise a novel phase called the Bose glass [306]. This third phase, due to thepresence of disorder, is insulating because of the localization effects of the randomness, it ischaracterized by a finite compressibility and no gap in the spectrum. The Bose glass exhibitsno superfluidity ( n s = 0), but possesses local remnants of BEC ( n > − D < γ j < D ,so that the related stochastic distribution is p ( γ j ) = D , | γ j | < D , | γ j | > D . The most often discussed phase diagram concerns the case of zero temperature and integerfilling, when the pure Mott insulating phase can exist. The disorder is introduced through auniformly distributed on-site potential, as is explained above. It is generally accepted thatwhen increasing disorder, the Mott insulating phase transforms, through a first-order phasetransition, into the Bose glass phase. While increasing the tunneling parameter yields thesuperfluid phase with BEC. Under sufficiently strong disorder, the Bose glass transforms intosuperfluid through a second-order transition, provided the uniformly distributed disorder isbounded by a finite D .There exist, however, a controversy with regard to the transformation of the Mott insu-lator into superfluid. Fisher et al. [306] argued that, in the presence of any finite disorder,the transition from the Mott insulator to superfluid occurs only through the intermedi-ate Bose-glass phase. This picture has been supported by several numerical calculations144298,391,399,401]. The corresponding quantitative phase diagram is shown in Fig. 1. Pre-cise numerical values depend on the lattice dimensionality and can be found in the citedreferences. J/U
D/U
Coherent superfluid Bose glassn > 0 n s = 0Mott insulatorn = 0 n s = 0n > 0 n s > 0 Figure 1: Possible qualitative phase portrait for a disordered lattice with an integer fillingfactor at zero temperature.Other researches claim that there are two different regimes in a disordered boson Hubbardmodel. For weak disorder, the Mott insulating phase is sustained up to the direct transitioninto a superfluid. Strong disorder changes the nature of the transition to that of the Boseglass to superfluid transition. Thus, contrary to the above cited works, it is stated that atweak disorder the direct Mott insulator to superfluid phase transition does occur, without anintervening Bose glass phase. This picture is also based on several numerical investigations[389,393,396,398,400,402,403]. The related qualitative phase diagram is presented in Fig. 2. J/U D/U
Coherent superfluidn > 0 n s > 0Mott insulatorn = 0 n s = 0 Bose glassn > 0 n s = 0 Figure 2: Possible qualitative phase diagram for a disordered lattice with an integer fillingfactor at zero temperature. 145
Nonstandard Lattice Models
For noncommenssurate filling, a mixture of localized and delocalized atoms exists in thelattice [407] at all temperatures, including zero. Such a mixture of coexisting localized anddelocalized states occurs at finite temperatures for any filling, including integer. Finally,there are indications [408] that, even at zero temperature and integer filling, close to theboundary between the superfluid and Mott insulating phases, there can arise an itinerant-localized dual structure, where the localized and itinerant states coexist.The coexistence of two different states could be described phenomenologically, involvingthe Ginzburg-Landau functional [409] generalized to the mixture of several states [54]. But,of course, it is more important to have a microscopic model characterizing such a coexistenceof delocalized and localized atoms.Let us assume that a superfluid state coexists with a portion of localized atoms. Then, inaddition to the Bogolubov shifted field operator η ( r ) + ψ ( r ), there exists the field operator ψ ( r ), so that η ( r ) is the condensate wave function, ψ ( r ) is the field operator of uncondenseddelocalized atoms, and ψ ( r ) is the field operator of uncondensed localized atoms. Each ofthese operators can be expanded over Wannier functions, η ( r ) = √ νn X j w ( r − a j ) , ψ ( r ) = X j c j w ( r − a j ) ,ψ ( r ) = X j b j w ( r − a j ) . (7.1)The field operators of delocalized and localized atoms commute with each other,[ ψ ( r ) , ψ ( r ′ )] = [ ψ ( r ) , ψ † ( r ′ )] = 0 , [ c i , b j ] = [ c i , b † j ] = 0 . (7.2)The Bose commutation relations are valid for each type of the field operators.Thus, the starting point is the assumption of the existence of two types of atoms differingfrom each other by their localization property. The delocalized atoms are characterized by anoperator c j , while the localized atoms, by b j . This is somewhat close to the consideration ofa two-component system [288]. Another analogy is the picture of a crystal where a portion ofatoms are localized, while a part of atoms can jump between lattice sites [220,410]. Probably,the most direct interpretation of the existence of two types of atoms is the consideration ofthe multiband Hamiltonian (4.11), in which two bands are taken into account. One bandis the conducting band, whose atoms are itinerant, being characterized by the field operatorˆ c j . Another band is that of bound states, with localized atoms described by an operator b j .The localized atoms of bound states are considered as normal, such that < b j > = 0 . (7.3)This requires [57] that the related Hamiltonian be invariant under the global gauge trans-formation b j → b j e iα , with a real α . So, the two-band Hubabrd Hamiltonian can be writtendown asˆ H = − X
1. So that the intersite inter-actions are negligible as compared to the on-site interactions. These short-range interactionscould be comparable only for very shallow lattices, for which, however, the Hubbard modelas such would be not a good approximation to reality. But for long-range interactions, theextended Hubbard model is well justified.As soon as there are atomic interactions between different sites, there appear collectivevibrational excitations, that is, phonons. Their introduction into the extended Hubbard152odel can be done similarly to the quantization of collective coordinates for extended quan-tum systems [415,416].In the Hubbard model (5.16), there is the single-site term containing the quantity h j ,defined in Eq. (4.8), which for a single-band model becomes h j ≡ Z w ∗ ( r − a j ) (cid:20) − ∇ m + V L ( r ) (cid:21) w ( r − a j ) d r . (7.46)Let us introduce the notation p j ≡ Z w ∗ ( r − a j )( −∇ ) w ( r − a j ) d r . (7.47)Then Eq. (7.46) writes as h j = p j m + Z w ∗ ( r ) V L ( r ) w ( r ) d r . (7.48)The second term in Eq. (7.48) is a constant and can be omitted. The extended Hubbardmodel takes the formˆ H = − J X
12 ˆ u αi ˆ u βi −
12 ˆ u αj ˆ u βj (cid:19) , (7.55)where U ij ≡ U ( a ij ) ( a ij ≡ a i − a j ) , αij ≡ ∂U ij ∂a αi , Φ αβij ≡ ∂ U ij ∂a αi ∂a βj . (7.56)Atomic interactions, as usual, are symmetric with respect to spatial inversion, U ( a ji ) = U ( a ij ) , U ji = U ij . (7.57)From here, it follows thatΦ αij = − Φ αji , Φ αβij = Φ βαij = Φ βαji = Φ αβji . (7.58)For an ideal lattice, the sum X j ( = i ) U ij = X j ( = i ) U ( a ij )does not depend on a i . Therefore X j ( = i ) Φ αij = ∂∂a αi X j ( = i ) U ij = 0 , X j ( = i ) Φ αβij = − ∂ ∂a αi ∂a βi X j ( = i ) U ij = 0 , (7.59)where the property ∂U ( a ij ) ∂a αi = − ∂U ( a ij ) ∂a αj is employed. Consequently, Eq. (7.55) satisfies the relation X i = j U (ˆ r ij ) = X i = j U ij + X i = j X αβ Φ αβij ˆ u αi ˆ u βj . (7.60)Also, because of condition (7.54), one has < ˆ U ij > = U ij + X αβ Φ αβij (cid:16) < ˆ u αi ˆ u βj > − < ˆ u αj ˆ u βj > (cid:17) . (7.61)To simplify the problem, let us resort to the mean-field decoupling < c † i c i c † j c j > = < c † i c i >< c † j c j > ( i = j ) (7.62)for the atomic operators. And let us decouple the atomic and phonon variables as follows:ˆ p j c † j c j = ˆ p j < c † j c j > + < ˆ p j > c † j c j − < ˆ p j >< c † j c j > , ˆ U ij c † i c † j c j c i = ˆ U ij < c † i c † j c j c i > + < ˆ U ij > c † i c † j c j c i − < ˆ U ij >< c † i c † j c j c i > . (7.63)Also, let us recall the notation for the filling factor ν ≡ NN L = 1 N L X j < c † j c j > = < c † j c j > , (7.64)154here again an ideal lattice is assumed.Accomplishing quantization (7.50) in Hamiltonian (7.49) and decoupling the atomic andphonon variables according to Eq. (7.63), we come to the Hamiltonianˆ H = E sh + ˆ H at + ˆ H ph (7.65)with the separated variables. Here the first term shifts Hamiltonian (7.65) by the nonoperatorquantity E sh = − ν X j < ˆ p j > m − ν X i = j < ˆ U ij > . (7.66)The second term in Eq. (7.65) is an effective atomic Hamiltonianˆ H at = − J X
0. We may note thatΦ αβij ≡ ∂ U ( a ij ) ∂a αi ∂a βj = − ∂ U ( a ij ) ∂a αi ∂a βi . Therefore ω ks ≥
0, tending to zero in the limit k →
0. For a d -dimensional lattice, thepolarization index s = 1 , , . . . , d . Hence Eq. (7.73) defines d phonon branches.For Hamiltonian (7.71), we have < b † ks b ks > = h exp (cid:16) ω ks T (cid:17) − i − . (7.74)This yields < ˆ p j > = m N X ks ω ks coth (cid:16) ω ks T (cid:17) . Then the mean phonon kinetic energy per particle is K ≡ < p j > m = 14 N X ks ω ks coth (cid:16) ω ks T (cid:17) . (7.75)For the correlation function of atomic displacements, we have < ˆ u αi ˆ u βj > = δ ij ν N X ks e αks e βks mω ks coth (cid:16) ω ks T (cid:17) . (7.76)156he average total phonon energy is < ˆ H ph > = 2 N K = 12 X ks ω ks coth (cid:16) ω ks T (cid:17) . (7.77)Denoting the interaction average (7.61) as˜ U ij ≡ < ˆ U ij > = U ij + ∆ U ij , (7.78)we have ∆ U ij = − ν N X ks X αβ Φ αβij e αks e βks mω ks coth (cid:16) ω ks T (cid:17) . (7.79)The latter quantity shows how atomic interactions change in the presence of phonon excita-tions.The energy shift (7.66), in view of properties (7.59), becomes E sh = − (cid:16) K + ν (cid:17) N , (7.80)where the notation Φ ≡ N L X i = j U ij is employed. The atomic Hamiltonian (7.67), with Eqs. (7.75), (7.78), and (7.79), takes the formˆ H at = − J X
0, ifthe space dimensionality is d ≤
2. This means that, for these low dimensions, atoms cannot159e localized in a lattice. Their mean-square deviations (7.97) become infinite. And thephonon-induced interaction (7.98) is also infinite. It is important to stress that interaction(7.98) becomes infinite for any finite D ij . And the dynamical matrix D ij is always finite forany nonvanishing atomic interactions. As is discussed in Sec. 7.2, the matrix D ij is finite,though may be small, even for the local interactions, proportional to δ ( r ). Thus, for suchlocal interactions and well localized atoms, for which l ≪ a , one has U ( a ij ) ≈ U exp (cid:18) − a ij d l (cid:19) , from where D ij ≈ a d l U ( a ij ) . Therefore the localized states of the low-dimensional lattices, with d ≤
2, at finite tempera-ture
T >
0, are unstable with respect to vibrational excitations. That is, in such lattices, apurely insulating stable phase cannot exist.At zero temperature, the localized state in a one-dimensional lattice also cannot exist,since integral (7.97) diverges for d = 1 even when T = 0. Thence, the Mott insulatingphase cannot be a stable phase in a one-dimensional lattice. However, it can exist in quasi-one-dimensional lattices, which, actually, are just three-dimensional lattices elongated inone direction and tightly confined in two other directions. What one realizes in experi-ments are always quasi-low-dimensional lattices, but never purely one-dimensional or purelytwo-dimensional ones. So, what is measured in experiments with low-dimensional latticesdoes not need to exactly coincide with numerical calculations accomplished for purely low-dimensional lattices.For d -dimensional lattices at zero temperature, the Debye approximation gives Z B ω k d k (2 π ) d = ρd ( d − νT D , where T D is the Debye temperature (7.95). Then Eq. (7.97) yields r = νd d − mT D ( T = 0) . (7.99)Hence the phonon-induced interaction (7.98) is∆ U ij = νd d − mT D D ij ( T = 0) . (7.100)These formulas again confirm that the localized states in one-dimensional lattices cannotexist. But the localized states in two-dimensional lattices can occur at zero temperature.Three-dimensional lattices with localized atoms are also stable at zero temperature.At finite temperatures, such that T ≫ T D , the Debye approximation gives Z B ω k coth (cid:16) ω k T (cid:17) d k (2 π ) d ≃ ρT d ( d − νT D . r ≃ νT d ( d − mT D . (7.101)Then the phonon-induced interaction (7.98) is∆ U ij ≃ νT d ( d − mT D D ij . (7.102)From Eqs. (7.101) and (7.102), we again see that the two-dimensional lattices with localizedatoms are unstable at nonzero temperature. But such localized states can arise for three-dimensional lattices.To evaluate how strong the phonon-induced interaction ∆ U ij is, being compared to thebare interaction U ij , let us consider a cubic lattice with nearest-neighbor interactions, when D ij = D = ∂ U ( a ) ∂a . (7.103)Suppose that atoms interact through dipole forces, for which U ( a ) ∼ a − . As a result, D = 12 a U ( a ) . (7.104)Then the phonon sound velocity (7.90) is c = r m ν U ( a ) . (7.105)At zero temperature, the phonon-induced interaction (7.100) writes as∆ U ij = d ( d − k D a r νm U ( a ) . (7.106)Using the relation E R = k m = π ma , we find ∆ U ij U ij = √ d π ( d − k D a s E R νU ij , (7.107)where k D a = √ π (cid:20) d (cid:18) d (cid:19)(cid:21) /d . For two-and three-dimensional lattices we have k D a = √ π = 3 .
545 ( d = 2) ,k D a = 3 .
898 ( d = 3) . U ij U ij = 0 . s E R νU ij ( T = 0; d = 2 , . (7.108)At finite temperatures, such that T ≫ T D , the phonon-induced interaction is given byEq. (7.102), from where ∆ U ij U ij ≃ d ( d − k D a ) (cid:18) TνU ij (cid:19) . (7.109)For a three-dimensional lattice, this gives∆ U ij U ij ≃ . TνU ij ( T > T D , d = 3) . (7.110)Equations (7.108) and (7.110) show that phonons can substantially renormalize atomic in-teractions.The extended Hubbard model (5.16) has been studied theoretically for one-dimensional[317–322] and two-dimensional [417] lattices. However, it is necessary to be cautious in-terpreting the results of numerical calculations. As follows from the above analysis, thelocalized states in one-dimensional lattices are unstable with respect to phonon excitationsat any temperature. The Mott insulating phase, strictly speaking, cannot be realized insuch lattices even at zero temperature. In two- and three-dimensional lattices, the Mottinsulating phase can happen at zero temperature. But, investigating the phase diagram, oneshould take into account the phonon-induced renormalization of atomic interactions.The boundary between the insulating and Bose-condensed phases can be defined bystudying the behavior of the condensate fraction [418]. One should keep in mind that thisboundary can be shifted because of the influence of the phonon-induced interactions.For optical lattices with disorder, the phase diagram essentially depends on the presenceof the order-parameter fluctuations [419]. The existence of the vibrational atomic fluctua-tions can also strongly influence the phase portrait of disordered optical lattices. For thelatter, the phonon excitations can occur to be even more dramatic than for ideal lattices. Recently, a double-well optical lattice was realized experimentally [420], being a lattice eachsite of which is represented by a double-well potential. Dynamics of cold atoms in a separatedouble-well has been considered in several publications [421–423]. But to study the propertiesof the whole double-well lattice, it is necessary to have an appropriate lattice Hamiltonian.We should start with the general Hamiltonian in the Wannier representation, given byEq. (4.11). Contrary to the case of the standard Hubbard model (4.12), for the double-well lattice, it is impossible to resort to the single-band approximation. This is because thetunneling of atoms between the wells of a double well results in the splitting of the ground-state level onto two energy levels that can be very close to each other [205]. Without taking162ccount of this splitting there would be no atomic tunneling between the wells. Since, inaddition, atoms interact with each other, this tunneling is essentially nonlinear [424,425].Thus, for the double-well lattices, we have to retain not less than two lowest energy levels,that is, we have to deal with at least a two-band case. It is important to stress that in theexpansion of the field operator ψ ( r ) = X nj c nj w n ( r − a j ) (8.1)the index n enumerates the quantum bands, that is, the quantum energy levels, but not”left” or ”right” positions. The latter, as is explained in Sec. 2.17, are not good quantumnumbers. The necessity of taking into account several energy levels is typical of atoms incomplex multi-well configurations [426–428] as well as can occur for some metastable systems[429].Introducing the notation E mnij ≡ Z w ∗ m ( r − a i ) H L ( r ) w n ( r − a j ) d r , (8.2)in which H L ( r ) ≡ − ∇ m + V L ( r ) , Hamiltonian (4.11) can be rewritten asˆ H = X ij X mn E mnij c † mi c nj + 12 X { j } X { n } U n n n n j j j j c † n j c † n j c n j c n j . (8.3)The indices m and n here have to take at least two values, m, n = 1 ,
2. When there is BECin the lattice, so that the gauge symmetry becomes broken, the field operators c nj should bereplaced by the Bogolubov-shifted operators (4.5).In order to find the relation between the matrix element (8.2) and the Bloch energy E nk ,which is the eigenvalue of the equation H L ( r ) ϕ nk ( r ) = E nk ϕ nk ( r ) , we can employ the expansion of the Bloch functions ϕ nk ( r ) over the Wannier functions. Thenthe above eigenproblem transforms into the equation H L ( r ) w n ( r − a i ) = 1 N L X jk E nk e − i k · a ij w n ( r − a j ) . This shows that the Wannier functions, strictly speaking, are not the eigenfunctions of thelattice Hamiltonian H L ( r ). Using this equation, for the matrix element (8.2), we find E mnij = δ mn E nij , where E nij = 1 N L X k E nk e i k · a ij . E nij = δ ij E n + (1 − δ ij ) J nij , (8.4)in which E n = Z w ∗ n ( r − a j ) H L ( r ) w n ( r − a j ) d r = 1 N L X k E nk ,J nij = Z w ∗ n ( r − a i ) H L ( r ) w n ( r − a j ) d r = 1 N L X k E nk e i k · a ij , ( i = j ) . There is a temptation to reduce the number of these parameters by assuming that the abovequantities do not depend on the band indices. This, however, is not a good idea. Thetunneling between the wells of a single double-well depends on the nonzero value of thedifference E jj − E jj . But this difference would be zero, if the values E mnij would not dependon the band indices. In oder not to kill the tunneling between the wells of a double well,one has to retain the dependence on the band indices. The consideration of model (8.3),with many independent parameters, is rather complicated and can be done by setting someof these parameters to zero [430].There is, however, a case, when Hamiltonian (8.3) can be essentially simplified. This iswhen the filling factor is strictly fixed to one, so that the double occupancy of a lattice siteis prohibited, which is manifested by the unipolarity conditions X n c † nj c nj = 1 , c nj c nj = 0 . (8.5)Let us also assume that the lattice is in the insulating state, such that the atomic hoppingbetween different lattice sites is negligible, (cid:12)(cid:12)(cid:12)(cid:12) J ij E n (cid:12)(cid:12)(cid:12)(cid:12) ≪ i = j ) . (8.6)Under this condition, Wannier functions become approximate eigenfunctions of H L ( r ), inthe sense that H L ( r ) w n ( r − a j ) ≃ E n w n ( r − a j ) . Under conditions (8.5) and (8.6), Hamiltonian (8.3) reduces to the formˆ H = X nj E n c † nj c nj + 12 X i = j X mnm ′ n ′ V mnm ′ n ′ ij c † mi c † nj c m ′ j c n ′ i , (8.7)in which V mnm ′ n ′ ij ≡ V mnm ′ n ′ ijji + V mnn ′ m ′ ijij . Retaining only the two lowest bands implies that n = 1 ,
2. Then the unipolarity conditions(8.5) are c † j c j + c † j c j = 1 , c j c j = c j c j = 0 . For this two-band case, it is possible to resort to the pseudospin representation, similarto that used for some ferroelectrics [431,432]. The pseudospin operators are defined as S xj = 12 (cid:16) c † j c j − c † j c j (cid:17) , S yj = i (cid:16) c † j c j − c † j c j (cid:17) , zj = 12 (cid:16) c † j c j + c † j c j (cid:17) . (8.8)This gives c † j c j = 12 + S xj , c † j c j = 12 − S xj ,c † j c j = S zj − iS yj , c † j c j = S zj + iS yj . (8.9)To clarify the physical meaning of the pseudospin operators (8.8), one can introduce the left, c jL , and the right, c jR , operators by the relations c j = 1 √ c jL + c jR ) , c j = 1 √ c jL − c jR ) ,c jL = 1 √ c j + c j ) , c jR = 1 √ c j − c j ) . (8.10)Then operators (8.8) become S xj = 12 (cid:16) c † jL c jR + c † jR c jL (cid:17) , S yj = − i (cid:16) c † jL c jR − c † jR c jL (cid:17) ,S zj = 12 (cid:16) c † jL c jL − c † jR c jR (cid:17) . (8.11)These equations demonstrate that S xj describes the tunneling intensity between the left andright wells of a double-well potential centered at the j -site; S yj characterizes the Josephsoncurrent between the left and right wells; and S zj is the displacement operator for the imbalancebetween the wells.The ground-state wave function and that of the first excited state in a double-well possessdifferent symmetry properties and differing topology [205]. For instance, the ground-statewave function w ( r ) is symmetric with respect to spatial inversion, while the excited-statewave function w ( r ) is antisymmetric, w ( − r ) = w ( r ) , w ( − r ) = − w ( r ) . (8.12)Due to the symmetry properties (8.12), some of the matrix elements V mnm ′ n ′ ij can becomezero. Concretely, these are the matrix elements that are nondiagonal with respect to theband indices.Let us consider the integralΦ ij ≡ Z Φ( r − r ′ ) w ∗ ( r − a i ) | w ( r ′ − a j ) | w ( r − a i ) d r d r ′ , in which Φ( − r ) = Φ( r ) is any symmetric pair-interaction potential and i = j . Shifting here r by a i and r ′ by a j gives an equivalent formΦ ij ≡ Z Φ( r − r ′ + a ij ) w ∗ ( r ) | w ( r ′ ) | w ( r ) d r d r ′ . r and r ′ , using the symmetry of the interactionpotential, and invoking the symmetry properties (8.12), we haveΦ ij = − Z Φ( r − r ′ − a ij ) w ∗ ( r ) | w ( r ′ ) | w ( r ) d r d r ′ . From the last two equations, keeping in mind that the Wannier functions form an orthonor-mal basis and can be made well localized, we findΦ ij ∼ = − Φ ij ∼ = 0 . In the same way, it is easy to show that other nondiagonal matrix elements of the interactionpotential are practically zero. As a result, we obtain V ij = V ij = V ij = V ij = V ij = V ij = V ij = V ij = 0 . (8.13)Choosing real Wannier functions yields V ij = V ij = V ij = V ij , V ij = V ij . (8.14)These matrix elements are nonzero.Let us introduce the notation E ≡
12 ( E + E ) (8.15)and the following combinations of the interaction matrix elements: A ij ≡ (cid:0) V ij + V ij + 2 V ij (cid:1) , B ij ≡ (cid:0) V ij + V ij − V ij (cid:1) ,C ij ≡ (cid:0) V ij − V ij (cid:1) , I ij ≡ − V ij . (8.16)Also, let us define the quantity Ω ≡ E − E + X j ( = i ) C ij , (8.17)playing the role of a tunneling parameter characterizing the tunneling between the wells ofa double-well potential.As a result, Hamiltonian (8.7) reduces to the pseudospin formˆ H = E N + 12 X i = j A ij − Ω X j S xj + X i = j B ij S xi S xj − X i = j I ij S zi S zj . (8.18)By their definitions, the diagonal matrix elements V ij , V ij , and V ij , by their absolutevalues, can be close to each other, but much larger than the exchange matrix element I ij , sothat | V ij | ≪ | V ij | . (8.19)166hen, from Eqs. (8.16), it follows that | C ij | ≪ | A ij | , | I ij | ≪ | A ij | . (8.20)However the term, containing A ij , is not an operator, hence can be omitted from the Hamil-tonian, as well as the term E N . The remaining terms, with B ij , I ij , and Ω, can be ofthe same order. By varying the shape of a double well, it is possible to make the energydifference E − E quite large or exponentially small [205]. Thence the tunneling parameterΩ in Eq. (8.17) can be varied in a wide range. That is, in general, the term with B ij cannotbe omitted. It can be neglected only when the tunneling parameter Ω is sufficiently large,such that it is much larger than B ij . Note that the tunneling between different lattice sitescan be modulated by shaking the lattice [433]. In a similar way, one could also modulatethe effective tunneling between the adjacent wells of a double-well potential. To study what kind of phase transitions occurs for Hamiltonian (8.18), let us resort to themean-field approximation S αi S βj = < S αi > S βj + S αi < S βj > − < S αi >< S βj > ( i = j ) . (8.21)And let us introduce the notation A ≡ N L X i = j A ij , B ≡ N L X i = j B ij , I ≡ N L X i = j I ij . (8.22)Under the unipolarity conditions (8.5), the filling factor is strictly one, and N L = N . Also,we define the effective tunneling Ω ≡ Ω − B < S xj > . (8.23)Then Hamiltonian (8.18) acquires the formˆ H = H non − Ω X j S xj − I X j < S zi > S zj , (8.24)in which the first term is the nonoperator quantity H non = N E + N (cid:0) A − B < S xi > +2 I < S zi > (cid:1) . (8.25)By introducing an effective ”magnetic” field H eff ≡ { H αeff } ≡ { Ω , , I < S zj > } , (8.26)Hamiltonian (8.24) can be rewritten asˆ H = H non − X j H eff · S j . (8.27)167he corresponding free energy is F = H non − N T ln (cid:18) H eff T (cid:19) , (8.28)where H eff ≡ | H eff | , which gives H eff = q Ω + 4 I < S zj > . (8.29)The average values < S αj > can be found from the equation < S αj > = − N ∂F∂H αeff . This yields the equations for the x -component (tunneling intensity) < S xj > = Ω2 H eff tanh (cid:18) H eff T (cid:19) , (8.30)the y -component (Josephson current) < S yj > = 0 , (8.31)and the z -component (well imbalance) < S zj > = < S zj > IH eff tanh (cid:18) H eff T (cid:19) . (8.32)Let us define the averages x ≡ < S xj > , y ≡ < S yj > , z ≡ < S zj > . (8.33)It is convenient to introduce the dimensionless quantities ω ≡ Ω I + B , b ≡ BI + B . (8.34)Using these, we have Ω I + B = ω − bx , II + B = 1 − b . (8.35)Also, let us define the dimensionless quantity h ≡ H eff I + B , (8.36)which is h = p ( ω − bx ) + (1 − b ) z . (8.37)The nonoperator part (8.25) of Hamiltonian (8.24) reads as H non N = E + A I + B (cid:2) (1 − b ) z − bx (cid:3) . (8.38)168mploying the dimensionless quantities, defined above, the temperature T will be measuredin units of I + B .In the dimensionless notation, the averages (8.30), (8.31), and (8.32) yield the tunnelingintensity x = ω − bxh tanh (cid:18) h T (cid:19) , (8.39)the Josephson current y = 0 , (8.40)and the well imbalance z = z − bh tanh (cid:18) h T (cid:19) . (8.41)These variables satisfy the condition x + y + z = tanh (cid:18) h T (cid:19) . (8.42)Equations (8.39) and (8.41) are invariant under the replacement x → − x , ω → − ω , z → − z . Therefore, without the loss of generality, we can consider only the case, when x ≥ ω ≥ z ≥
0. The inequality ω ≥ E < E .Equation (8.41) shows that there can be two types of solutions, when z = 0 and when z = 0. The well-imbalance z plays the role of an order parameter. If z = 0, this meansthat atoms are mainly shifted to one of the double wells. While if z = 0, then atoms onthe average equally populate both wells. The thermodynamic phase with z = 0 is calledordered, while that with z = 0 is termed disordered.For the ordered phase, when z = 0, Eq. (8.41) gives1 − bh tanh (cid:18) h T (cid:19) = 1 , (8.43)which defines z . Using Eq. (8.43) in Eq. (8.39) yields x = ω − bx − b , from where it follows x = ω . (8.44)Since, by definition, the variable x is positive and less than one, we see that the orderedphase can exist if 0 ≤ ω ≤ ≤ x ≤ . (8.45)Equation (8.43), with h = (1 − b ) √ ω + z , (8.46)169efines z > T < T c . The critical temperature is T c = (1 − b ) ω ω , (8.47)where artanh ω = 12 ln 1 + ω − ω . When ω →
0, then T c ≃ − b ω ≪ , which in dimensional units becomes T c ≃ I/
2. And if ω →
1, then T c →
0. Thus, theordered phase, with z >
0, can exist when both
T < T c and ω ≤ z = 0 . (8.48)Equation (8.37) gives h = ω − bx . (8.49)Then Eq. (8.39) yields x = tanh (cid:18) ω − bx T (cid:19) . (8.50)For x to be non-negative, it should be that ω ≥ bx ( x ≥ . (8.51)The disordered phase arises, when the ordered phase cannot exist, that is, when either T > T c or ω > T = 0,the ordered phase has x = ω , z = √ − ω ( ω < . (8.52)While the disordered phase is described by x = 1 , z = 0 , ( ω > . (8.53)At the value ω = 1, the quantum phase transition occurs.The reduced internal energy E ≡ N < ˆ H > is E = H non N − I + B (cid:2) ( ω − bx ) x + (1 − b ) z (cid:3) . (8.54)In view of Eq. (8.38), this gives E = E + A I + B (cid:2) bx − ω − (1 − b ) z (cid:3) . (8.55)170he energy of the ordered phase, E = E + A − I + B (cid:0) − b + ω (cid:1) ( ω < E = E + A I + B b − ω ) ( ω > . These energies coincide at the critical value ω = 1. The dynamics of the pseudospin operators follows from the Heisenberg equations i dS αj dt = h S αj , ˆ H i , with the commutation relations[ S xi , S yj ] = iδ ij S zj , [ S yi , S zj ] = iδ ij S xj , [ S zi , S xj ] = iδ ij S yj . Using the pseudospin Hamiltonian (8.18) results in the equations of motion dS xi dt = 2 S yi X j ( = i ) I ij S zj , dS yi dt = Ω S zi − S xi X j ( = i ) I ij S zj − S zi X j ( = i ) B ij S xj ,dS zi dt = − Ω S yi + 2 S yi X j ( = i ) B ij S xj . (8.56)Collective excitations in the random-phase approximation can be found by representingthe pseudospin operators as S αj = < S αj > + δS αj (8.57)and considering δS αj as a small deviation from an equilibrium average. Equation (8.57) is tobe substituted into Eqs. (8.56), which are linearized with respect to δS αj . In the zero order,one gets dxdt = 0 , dydt = (Ω − Ix ) z , dzdt = 0 . (8.58)For the ordered phase, when z = 0, one has x = ω ; and for the disordered phase, z = 0.Therefore the second equation in Eqs. (8.58) becomes dy/dt = 0. So, all averages, x, y , and z , do not depend on time, as it should be for equilibrium quantities. In the first order, onehas ddt δS xi = IzδS yi ,ddt δS yi = − IzδS xi − z X j ( = i ) B ij δS xj + Ω δS zi − x X j ( = i ) I ij δS zj , dt δS zi = − Ω δS yi . (8.59)One defines the Fourier transforms δS αj = 1 N L X k σ αk e i ( k · a j − εt ) , σ αk = X j δS αj e − i ( k · a j − εt ) . (8.60)Similarly, Fourier transforms are introduced for the interaction functions, B ij = 1 N L X k B k e i k · a ij , B k = X i B ij e − i k · a ij ,I ij = 1 N L X k I k e i k · a ij , I k = X i I ij e − i k · a ij . (8.61)Then Eqs. (8.59) yield iεσ xk + zIσ yk = 0 , ( I + B k ) zσ xk − iεσ yk + ( xI k − Ω) σ zk = 0 , Ω σ yk − iεσ zk = 0 . (8.62)The condition for the existence of nontrivial solutions to Eqs. (8.62) gives the equation ε (cid:2) ε − (Ω − xI k ) − I ( I + B k ) z (cid:3) = 0 , (8.63)in which Ω = Ω − Bx .
Equation (8.63) defines three branches for the spectrum of collective excitations. One branchis trivial, ε ( k ) = 0 . (8.64)And two other branches are given by the equation ε , ( k ) = Ω(Ω − xI k ) + I ( I + B k ) z . (8.65)The branches of spectrum (8.65) describe the pseudospin oscillations. These branches possessgaps for both ordered as well as disordered phases, and, in the long-wave limit, they vary as k . Nonequilibrium states in lattices appear, when atoms are subject to temporal external fields.This is possible to accomplish by varying the shape of the lattice, for instance, by changingthe configuration of the double-well potential in each lattice site. Atomic interactions canbe modulated by means of the Feshbach resonance techniques. A nonequilibrium situationarises in the process of loading atoms into the lattice.The nonequilibrium behavior of atoms in a double well lattice can be characterized byconsidering the temporal evolution of the average quantities (8.33). The evolution equationsfor these quantities are to be obtained from averaging the operator equations (8.56). When172ccomplishing such an averaging, it is customary to invoke the mean-field approximation(8.21). This standard way has a principal defect of not taking into account atomic collisionsresulting in the appearance of damping. Not taking account of the latter can lead to incorrectdynamics and wrong physical conclusions.The existence of atomic collisions yielding the arising attenuation, can be included in thedynamics by employing the local-field approximation. This approximation is based on thefact that a kind of local equilibrium exists even in strongly nonequilibrium systems [434-436]. Then one can consider atomic collisions as occurring in an effective local field of otherparticles [437]. The resulting attenuation effects are included into the evolution equationsthrough the damping parameters, characterizing the longitudinal, γ , and transverse, γ ,relaxation. The values of these parameters can be calculated in the same way as is done formagnetic systems [438].The local fields for variables (8.33) are defined through expressions (8.39), (8.40), and(8.41) as x t ≡ ω − bxh tanh (cid:18) h T (cid:19) , y t = 0 , z t ≡ − bh z tanh (cid:18) h T (cid:19) , (8.66)where h is given by Eq. (8.37). Averaging Eqs. (8.56) in the local-field approximationresults in the evolution equations for the tunneling intensity dxdt = (1 − b ) yz − γ ( x − x t ) , (8.67)Josephson current dydt = ( ω − x ) z − γ ( y − y t ) , (8.68)and the well imbalance dzdt = ( bx − ω ) y − γ ( z − z t ) , (8.69)with the local fields (8.66). These equations describe the time evolution of x = x ( t ), y = y ( t ),and z = z ( t ) under the given initial conditions x = x (0) , y = y (0) , z = z (0) . (8.70)Here the dimensionless parameters (8.34) are employed and time is measured in units of1 / ( I + B ).The evolution equations (8.67), (8.68), and (8.69), depending on the parameters b and ω ,can show two types of behavior, attenuating to two different fixed points, one correspondingto the ordered stationary solution and another, to the disordered stationary solution. It may happen that a double-well lattice is neither completely ordered nor completely dis-ordered, but consists of the ordered regions intermixed with disordered parts. The spatialdistribution of these differently ordered regions is random, as well as their sizes and shapes.Such an object composed of a random mixture of different phases is called the heterophase ystem . In condensed matter, there are plenty of examples of such systems, as has beenreviewed in the articles [54,66]. A schematic picture of a heterophase double-well latticeis shown in Fig. 3, where the ordered regions are marked by arrows, while the disorderedregions are left empty. Because of their random distribution in space and because they oftenarise randomly, that is, in a fluctuational way, in time, the heterophase regions inside astatistical system are called heterophase fluctuations. Such fluctuations can be provoked bythe environmental randomness [439], even when the latter corresponds to a very weak noise.But they can also be due to intrinsic causes, such as local fluctuations of entropy or tem-perature [440], fluctuations of density or composition fluctuations [441], and other internalperturbations generated by the system itself [442]. The nuclei of one phase inside anotherare also termed droplets or clusters [443]. The system with heterophase fluctuations can bemore stable than a pure-phase sample. What kind of internal structure is more profitablefor a system is chosen by the system itself, which self-organizes for reaching an optimal state[444,445]. A thorough description of possible origins of heterophase fluctuations is givenin the review articles [54,66]. A general microscopic theory of statistical systems with het-erophase fluctuations has been developed [67,446–453] and reviewed in Refs. [54,66]. Below,this theory is applied to describing the heterophase double-well lattices. =) =)=) =) =) =) =)=) =) =) =) =) =)=) =) =) =) =) =) =) =)=) =) =) =) =)=) =) =) =) =)=) =) =) =) =)=) =) =) =) =) =)=) =) =) =) =) =) =)=) =) =) =) =) =) =)=) =) =) =) =)=) =) =) =)=) =) =) =) =) =) =)=) =) =) =)=) =) =)=) =) =) =) =) =) =) =) =)=) =) =) =) =) =)=) =) =) =) =)=) =) =)=) =) =) =)=) =)
Figure 3: Schematic picture of a heterophase double-well lattice. Arrows show the regionsof the ordered phase. Empty space corresponds to the regions of the disordered phaseEach thermodynamic phase is characterized by its typical states forming a Hilbert space.The coexistence of several phases corresponds to the direct sum of the Hilbert spaces, relatedto particular phases. The Fock space over the direct sum of Hilbert spaces is identified withthe tensor product of Fock spaces over each of the Hilbert space [454].Suppose that the considered system is a mixture of several thermodynamic phases enu-merated by the index f = 1 , , . . . Each phase is characterized by a Fock space F f of micro-scopic states typical of the given phase [54]. The total Fock space of the whole system is thefiber space ˜ F = O f F f . (8.71)After averaging over heterophase configurations [54], the system is described by a renormal-174zed Hamiltonian ˜ H = M f ˆ H f , (8.72)which is a direct sum of the partial Hamiltonians associated with the related phases. In adouble-well lattice, there can be two different thermodynamic phases, the ordered phase andthe disordered one, as is described in Sec. 8.2. Therefore the fiber space is˜ F = F O F , (8.73)while the renormalized Hamiltonian is˜ H = ˆ H M ˆ H . (8.74)The ordered and disordered phases are characterized by different values of the averages ofthe imbalance operator S zj . The average of an operator ˆ A , related to an f -phase, is definedas < ˆ A > f ≡ Tr F f ˆ ρ f ˆ A f , (8.75)where the statistical operator is ˆ ρ f = exp( − β ˆ H f )Tr F f exp( − β ˆ H f ) (8.76)and ˆ A f is a representation of the operator ˆ A on the Fock space F f . Let us ascribe the index f = 1 to the ordered phase, while the index f = 2, to the disordered phase. Then thesephases are defined as those for which < S zj > = 0 , < S zj > ≡ . (8.77)As a result of averaging over heterophase configurations, the renormalzied Hamiltonian(8.74) depends on the geometric probabilities of the phases, w f , which satisfy the conditions w + w = 1 , ≤ w f ≤ . (8.78)Following the general procedure [54], for the considered case of the double-well lattice, wehaveˆ H f = w f E N + w f X i = j A ij − w f Ω X j S xj + w f X i = j B ij S xi S xj − w f X i = j I ij S zi S zj . (8.79)Resorting again to the mean-field approximation (8.21) and introducing the notationΩ f ≡ Ω − w f B < S xj > f (8.80)yields ˆ H f = H nonf − w f Ω f X j S xj − w f I X j < S zi > f S zj , (8.81)175ith the first, nonoperator, term being H nonf = w f E N + w f (cid:0) A − B < S xi > f + 2 I < S zi > f (cid:1) N . (8.82)By defining the effective field H efff ≡ { w f Ω f , , w f I < S zj > f } reduces Hamiltonian (8.81) to the formˆ H f = H nonf − X j H efff · S j . (8.83)The free energy of the whole system is F = F + F , (8.84)with F f = H nonf − N T ln H efff T ! , (8.85)where H efff = w f q Ω f + 4 w f I < S zj > f . (8.86)For the averages of the pseudospin operators, we get < S xj > f = w f Ω f H efff tanh H efff T ! ,< S yj > f = 0 , < S zj > f = w f < S zj > f IH efff tanh H efff T ! . (8.87)It is convenient to employ the reduced variables x f ≡ < S xj > f , y f ≡ < S yj > f , z f ≡ < S zj > f (8.88)and use the dimensionless quantities (8.34). We define the effective tunneling frequency ω f ≡ ω − bw f x f (8.89)and introduce the notation h f ≡ w f q ω f + (1 − b ) w f z f . (8.90)Then averages (8.87) transform into x f = w f ω f h f tanh (cid:18) h f T (cid:19) , y f = 0 , z f = w f z f − bh f tanh (cid:18) h f T (cid:19) , (8.91)176here the temperature T is measured in units of I + B .Condition (8.77), distinguishing the ordered and disordered phases, becomes z = 0 , z ≡ . (8.92)For the ordered phase, z is defined by the equation w − bh tanh (cid:18) h T (cid:19) = 1 , (8.93)while x , by the equation x = ω (1 − b ) w . The latter, in view of Eq. (8.89), gives x = ωw . (8.94)From here, it follows that the ordered component can exist if0 ≤ ω ≤ w (0 ≤ x ≤ . (8.95)Expressions (8.89) and (8.90) reduce to ω = (1 − b ) ω , h = (1 − b ) w q ω + w z . (8.96)When z tends to zero, this can happen at the temperature T c = w (1 − b ) ω ω/w ) , (8.97)where w = w ( T c ).For the disordered component, for which z = 0, we have ω = ω − bw x , h = w ω . (8.98)And x is defined by the equation x = tanh (cid:16) w ω T (cid:17) . (8.99)The proportions of the phases are prescribed by the system stability. The equationsfor the phase probabilities w f can be found by minimizing the free energy (8.84) under thenormalization condition (8.78). For that purpose, we define w ≡ w , w ≡ − w . (8.100)Then, the free energy F = F ( w ) is minimized with respect to w . From the equation ∂F ( w ) ∂w = 0 , (8.101)177e find w = 2 u + ω x − ω x u − (1 − b ) z ,w = 2 u − ω x + ω x − (1 − b ) z u − (1 − b ) z , (8.102)where the notation u ≡ AI + B (8.103)is used.Let us analyze the obtained equations for the case of zero temperature. Then we have x = ωw , x = 1 ,z = s − ω w , z = 0 ( T = 0) . (8.104)Also, ω = (1 − b ) ω , ω = ω − bx , h = (1 − b ) w , h = w ( ω − bw ) . (8.105)Probabilities (8.102) reduce to w = 2 u + b − ω u + 2 b − , w = 2 u + b + ω − u + 2 b − . (8.106)By definition, 0 ≤ w f ≤
1. This imposes the constraints under which the heterophasemixture can exist, 1 − b − u < ω < u + b . (8.107)In particular, one can notice that w = w = 12 , z = 0 (cid:18) ω = 12 (cid:19) . (8.108)To check whether the energy of the heterophase mixture is lower than that of a purephase, let us consider the internal energy E mix ≡ N < ˜ H > = E + E , (8.109)in which E f ≡ N < ˆ H f > f . (8.110)For the Hamiltonian (8.81), we find E f = H nonf N − I + B w f (cid:2) ω f x f + (1 − b ) w f z f (cid:3) , H nonf N = w f E + I + B w f (cid:2) u − bx f + (1 − b ) z f (cid:3) . Combining the latter two expressions, we get E f = w f E − I + B ωw f x f + I + B w f (cid:2) u + bx f − (1 − b ) z f (cid:3) . (8.111)For the ordered and disordered components, Eq. (8.111) gives, respectively E = w E + I + B (cid:2) w (2 u + b − − ω (cid:3) ,E = w E + I + B (cid:2) w (2 u + b ) − ωw (cid:3) . (8.112)The total sum (8.109) becomes E mix = E + I + B (cid:2) u + b − ω − ω − w (2 u + b − ω ) + w (4 u + 2 b − (cid:3) , (8.113)where notation (8.100) is employed.Minimizing Eq. (8.113) with respect to w implies that ∂E mix ∂w = 0 , ∂ E mix ∂w > . (8.114)From Eq. (8.113), we have ∂E mix ∂w = I + B w (4 u + 2 b − − u − b + ω ] , ∂ E mix ∂w = I + B u + 2 b − . The first of Eqs. (8.114) yields the expressions for the phase probabilities (8.106). And thesecond condition in Eqs. (8.114) requires that4 u + 2 b − > . (8.115)If one compares the energy (8.113) of the heterophase mixture with the energy of the pureordered phase E ord ≡ E mix ( w = 1 , w = 0) , (8.116)then one gets the difference E mix − E ord = − (2 u + b + ω − u + 2 b −
1) ( I + B ) , (8.117)which shows that the energy of the mixture is lower than that of the pure ordered phaseunder the same condition (8.115).The difference of the ordered-phase energy (8.116) with the disordered-phase energy E dis ≡ E mix ( w = 0 , w = 1) (8.118)179eads as E ord − E dis = − ( ω − I + B ) . (8.119)Hence, E ord ≤ E dis for all ω . Therefore the energy of the heterophase mixture (8.113)satisfies the inequality E mix < E ord ≤ E dis , (8.120)provided that conditions (8.107) and (8.115) are valid. That is, under these conditions, themixed state is more profitable than the pure phases. Cold atoms in optical lattices are considered as a very promising tool for realizing quantuminformation processing and quantum computation [17,455]. General problems of quantumcomputation and information are described in the books [456,457] and reviews [458,459].Here we concentrate our attention on the possibility of employing cold atoms in opticallattices as a tool for this purpose.Probably, the main advantage of quantum devices for information processing and com-putation is the feasibility of creating entanglement. This purely quantum property, whichis absent in classical devices, should make quantum processors much more powerful andminiature.The notion of entanglement has two sides. The entanglement of a quantum state describesthe structure of this state. However, quantum states, as such, are not measurable quantities,so that their entanglement could be used only indirectly. Also, there is no uniquely definedmeasure of entanglement for quantum states, especially when the latter are mixed [456–459].The other notion is the entanglement production , which shows how much entanglementis generated by a quantum operation. There exists a general measure of entanglementproduction, valid for arbitrary systems [460,461]. This measure of entanglement productionis introduced below, with keeping in mind its application to lattices.Let us have a lattice whose lattice sites are enumerated with the index j = 1 , , . . . , N L .For the purpose of information processing, it is necessary to have a deep lattice potential,so that atoms be well localized in the lattice sites. The appearance of BEC diminishes thelevel of entanglement [461]. Therefore the insulating state is preferable. Finite temperaturereduces the feasibility of manipulating atoms. Hence the system is to be deeply cooled down,so that atoms be almost at zero temperature.Suppose that atoms in a j -site can have different quantum numbers labelled by the index n j , such that these states | n j > form a basis {| n j > } . The closed linear envelope over thisbasis is the Hilbert space H j ≡ L{| n j > } . (9.1)Denoting a given set { n j } of the indices n j by n , one can define the states | n > ≡ O j | n j > ( n ≡ { n j } ) . (9.2)180sing states (9.2) as a basis {| n j > } makes it possible to construct the closed linear envelopeover this basis, which yields the Hilbert space H ≡ L{ | n > } = O j H j . (9.3)The states of space (9.3) can be represented as ϕ = X n c n | n > ( ϕ ∈ H ) . (9.4)Generally, these states do not have the form of a tensor product ⊗ j ϕ j , with ϕ j ∈ H j . Letus separate out of the space H the disentangled set D ≡ { f = O j ϕ j | ϕ j ∈ H j } , (9.5)whose members have the form of the tensor products. Then the compliment H \ D is theset of entangled states.For any quantum operation, represented by an operator ˆ A on H , we can introduce thenorm over D , || ˆ A || D ≡ sup f,f ′ | ( f, ˆ Af ′ ) | , (9.6)where f ∈ D , f ′ ∈ D , || f || = || f ′ || = 1 , with the norm || f || ≡ √ f, f generated by the scalar product.It is worth noting that the restricted norm (9.6) over the set D can also be defined asa norm over a weighted Hilbert space [54,66]. With this aim in view, we can introduce theweighted Hilbert space H D as a projected space, in which the scalar product is defined as( f, f ′ ) H D ≡ ( P f ϕ, P f ′ ϕ ′ ) H , where P f is a projector, such that P f ϕ ≡ f ∈ D . The norm of ˆ A over the weighted Hilbert space H D is defined as || ˆ A || H D ≡ || P D ˆ AP D || H , where P D = { P f | P f ϕ = f ∈ D} . By this definition || ˆ A || D = || ˆ A || H D . Let us introduce the compliment space
H \ H j = O i ( = j ) H i (9.7)181nd define the partially traced operator ˆ A j ≡ Tr H\H j ˆ A . (9.8)Then we construct the factor operator ˆ A ⊗ ≡ Tr H ˆ A Tr D N j ˆ A j O j ˆ A j , (9.9)for which Tr D ˆ A ⊗ = Tr H ˆ A .
The measure of entanglement generated by the operator ˆ A is defined [460,461] as ε ( ˆ A ) ≡ log || ˆ A || D || ˆ A ⊗ || D , (9.10)where log is to the base two. This measure can be employed for any operator possessing atrace. For example, one can consider how much entanglement is produced by a Hamiltonianin a finite Hilbert space. More often, one is interested in the level of entanglement producedby a statistical operator. To realize any scheme of information processing it is necessary to possess objects that couldbe transferred into different quantum states. In the case of cold atoms, these could be internalatomic states [17]. Another possibility is to generate topological coherent modes in BEC,as suggested in Ref. [174]. Various properties of these modes, representing nonground-stateBose condensates, have been studied in several papers [174–196,462-466]. The generationof such modes can be accomplished in two ways. One method is the modulation of thetrapping potential with the frequency in resonance with the transition frequency betweentwo coherent modes [174–176]. The other way, as has been mentioned in Refs. [189,190,192]and analysed in Ref. [464], is by the resonant modulation of the atomic scattering length.Both these techniques are illustrated below.Let us consider a deep lattice, in each site of which there are many Bose atoms. Opticallattices with large filling factors, reaching ν ∼ , are readily available in experiment [22,23].All atoms inside a lattice site can be made well localized, with the intersite tunneling almostcompletely suppressed. Temperature can be kept low, so that practically all atoms pilingdown to BEC.Since lattice sites are very deep, we can start the consideration from a single site, repre-senting a kind of a trap. At low temperature and weak interactions, the system inside thetrap is described by the condensate wave function satisfying the Gross-Pitaevskii equation i ∂∂t η ( r , t ) = (cid:20) − ∇ m + U ( r , t ) − µ + Φ ( t ) | η ( r , t ) | (cid:21) η ( r , t ) . (9.11)The condensate wave function is normalized to the number of atoms inside the trap, that is,to the filling factor Z | η ( r , t ) | d r = ν . (9.12)182he external potential U ( r , t ) = U ( r ) + V ( r , t ) (9.13)consists of a trapping potential U ( r ), characterizing the optical potential at the consideredlattice site, and of an additional modulating potential V ( r , t ). The interaction strength canalso be made time-dependent by means of the Feshbach resonance techniques.For convenience, one can use the relation η ( r , t ) ≡ √ ν ϕ ( r , t ) , (9.14)defining the function ϕ ( r , t ) normalized to one, Z | ϕ ( r , t ) | d r = 1 . (9.15)Then Eq. (9.11) reads as i ∂∂t ϕ ( r , t ) = (cid:20) − ∇ m + U ( r , t ) − µ + ν Φ ( t ) | ϕ ( r , t ) | (cid:21) ϕ ( r , t ) . (9.16)For a stationary case, when there is no modulating field, V ( r , t ) = 0, and atomic inter-actions are constant, Φ ( t ) = Φ , Eq. (9.16) becomes i ∂∂t ϕ n ( r , t ) = (cid:20) − ∇ m + U ( r ) − µ + ν Φ | ϕ n ( r , t ) | (cid:21) ϕ n ( r , t ) , (9.17)where a multi-index n enumerates quantum states. In stationary solutions, the spatial andtemporal variables can be separated as follows: ϕ n ( r , t ) = ϕ n ( r ) e − iω n t . (9.18)As a result, Eq. (9.17) reduces to the eigenproblem (cid:20) − ∇ m + U ( r ) + ν Φ | ϕ n ( r ) | (cid:21) ϕ n ( r ) = E n ϕ n ( r ) , (9.19)in which the eigenvalues E n ≡ ω n + µ (9.20)have the property min n E n = µ , min n ω n = 0 . The eigenfunctions ϕ n ( r ) of Eq. (9.19) are the topological coherent modes [174–176]. Equally,the condensate functions η n ( r ) = √ ν ϕ n ( r ) (9.21)can also be called the topological coherent modes. The condensate function (9.21), corre-sponding to the lowest energy E n = µ , characterizes the standard BEC. The higher modesof η n ( r ) describe the nonground-state condensates [174–176]. The functions ϕ n ( r ) and η n ( r )differ solely by their normalizations Z | ϕ n ( r ) | d r = 1 , Z | η n ( r ) | d r = ν . V ( r , t ) = V ( r ) cos ωt + V ( r ) sin ωt . (9.22)Similarly, the interaction strength can be made time-dependent asΦ ( t ) = Φ + ε cos ωt + ε sin ωt . (9.23)It is of principal importance to choose the frequency of the alternating potentials to betuned to a resonance with a transition frequency related to the energy levels we wish toconnect. Let us consider two energy levels, E and E , with the transition frequency being ω ≡ E − E . (9.24)Then the resonance condition is (cid:12)(cid:12)(cid:12)(cid:12) ∆ ωω (cid:12)(cid:12)(cid:12)(cid:12) ≪ ω ≡ ω − ω ) . (9.25)For instance, E can correspond to the lowest energy level, equal to µ .We can look for the solution to the temporal Eq. (9.16) in the form of the expansionover the coherent modes, ϕ ( r , t ) = X n c n ( t ) ϕ n ( r , t ) . (9.26)The coefficient functions can be treated as slow in time, compared to the exponential oscil-lations in Eq. (9.18), such that 1 ω n (cid:12)(cid:12)(cid:12)(cid:12) dc n dt (cid:12)(cid:12)(cid:12)(cid:12) ≪ . (9.27)The latter condition requires that atomic interactions and the pumping alternating fieldswould not be too strong, which is easily realized in experiment [174–176,182]. The normal-ization condition X n | c n ( t ) | = 1 (9.28)is assumed.When there are two time scales, one can resort to the averaging techniques [236,237]and to the scale separation approach [240–242]. To this end, we substitute expansion (9.26)into Eq. (9.16), multiply the latter by ϕ ∗ n ( r , t ), integrate over r , and accomplish the timeaveraging according to the rule { f ( t ) } t ≡ lim τ →∞ τ Z τ f ( t ) dt , (9.29)where the slow variables are kept as quasi-integrals of motion [240–242]. For example,averaging (9.29) gives (cid:8) e i ( ω m − ω n ) t (cid:9) t = δ mn , { ϕ ∗ m ( r , t ) ϕ n ( r , t ) } t = δ mn | ϕ n ( r ) | . Therefore, the functions ϕ n ( r , t ) are orthogonal on average, though the functions ϕ n ( r ) canbe not orthogonal. Also, we have (cid:8) e i ( ω m + ω n − ω k − ω p ) t (cid:9) t = δ mk δ np + δ mp δ nk − δ mk δ np δ mn . Let us introduce the notation for the matrix elements of the interaction α mn ≡ ν Φ Z | ϕ m ( r ) | (cid:2) | ϕ n ( r ) | − | ϕ m ( r ) | (cid:3) d r , (9.30)of the pumping potential β mn ≡ Z ϕ ∗ m ( r )[ V ( r ) − iV ( r )] ϕ n ( r ) d r , (9.31)and of the interaction modulation γ n ≡ ν ( ε − iε ) Z ϕ ∗ ( r ) | ϕ n ( r ) | ϕ ( r ) d r . (9.32)Then Eq. (9.16) yields i dc n dt = X m ( = n ) α nm | c m | c n ++ 12 δ n e i ∆ ωt X m ( =2) γ m | c m | c + γ | c | c + β c + 12 δ n e − i ∆ ωt γ ∗ c ∗ c ++ 12 δ n e − i ∆ ωt X m ( =1) γ ∗ m | c m | c + γ ∗ | c | c + β ∗ c + 12 δ n e i ∆ ωt γ c ∗ c . (9.33)It is not difficult to notice [182] that, if at the initial time t = 0, c n (0) = 0 for n = 1 ,
2, then c n ( t ) = 0 for all t ≥ n = 1 ,
2. Hence Eq. (9.33) can be separated into two equations i dc dt = α | c | c + 12 e i ∆ ωt (cid:0) γ | c | c + γ | c | c + β c (cid:1) + 12 e − i ∆ ωt γ ∗ c ∗ c ,i dc dt = α | c | c + 12 e − i ∆ ωt (cid:0) γ ∗ | c | c + γ ∗ | c | c + β ∗ c (cid:1) + 12 e i ∆ ωt γ c ∗ c . (9.34)The solutions to these equations define the temporal behavior of the fractional mode popu-lations p n ( t ) ≡ | c n ( t ) | . (9.35)The properties of Eqs. (9.34) have been studied in detail for the case of the modegeneration by means of the trapping-potential modulation, when β = 0 while γ n = 0, inRefs. [174–176,179,180,182,184–186,192–196]. The generalization for the case of the multiplemode generation has been given [189,190]. It has also been shown that the generation of thenonground-state condensate is achievable at nonzero temperature [193,465]. The creation ofthe topological coherent modes by the modulation of the interaction strength, when β = 0but γ n = 0, is considered in Ref. [464]. 185 .3 Coherent States Now let us turn to the situation, when there is a lattice with N L sites. In each site a deepwell is formed by an optical potential. The number of atoms in a j -lattice site is ν j ≫ ν j does not depend on the site index. But, in general,the lattice can be nonideal. Then ν j can be different for different sites.By employing the resonant generation, described in the previous section, one can excitein the j -site the topological coherent modes labelled by a multi-index n j . Suppose that η n j are the coherent fields associated with the j -site and normalized to the correspondingoccupation number Z | η n j ( r ) | d r = ν j . (9.36)Similarly to Eq. (9.21), we can also define the functions ϕ n j ( r ) normalized to one, such that η n j ( r ) = √ ν j ϕ n j ( r ) . (9.37)Being the solutions to the nonlinear Schr¨odinger equation of type (9.19), the coherent fields η n i ( r ) and η n j ( r ), with i = j , are not, generally, orthogonal, that is, the scalar product Z η ∗ n i ( r ) η n j ( r ) d r ≡ ν ij (9.38)is not necessarily zero for i = j . The diagonal quantity ν jj = ν j (9.39)is the occupation number of the j -site.In the Fock space, the coherent state, associated with the n j -mode, is given by the column | n j > = " exp( − ν j / √ n ! n Y k =0 η n j ( r k ) , (9.40)where n = 0 , , , . . . . Expression (9.40) is the short-hand notation for the column state ofthe type | m > = e − ν/ η m ( r ) √ η m ( r ) η m ( r ) ··· √ n ! η m ( r ) η m ( r ) . . . η m ( r n ) ··· . (9.41)The coherent states (9.40) are not necessarily orthogonal to each other, so that the scalarproduct < n i | n j > = exp (cid:18) − ν i + ν j ν ij (cid:19) (9.42)186s not, in general, zero for i = j . But the coherent states (9.40) are normalized to one, since < n j | n j > = 1 . (9.43)However, as follows from Eqs. (9.38) and (9.42), the coherent states are asymptoticallyorthogonal [467,468] in the sense that < n i | n j > ≃ δ ij ( ν i + ν j ≫ . (9.44)They also are asymptotically complete in the weak sense, X n j | n j >< n j | ≃ ν i + ν j ≫ . (9.45)Therefore the states | n > , defined in Eq. (9.2), form the asymptotically orthogonal andcomplete basis {| n > } in the Hilbert space (9.3). The topological coherent modes can be used as a tool for a quantum register of informationprocessing. These modes possess a rich variety of interesting properties [174–196,462–466].The most important, for the purpose of quantum information processing, is the feasibil-ity of producing entangled states [192,467–469]. Entanglement production with topologicalcoherent modes and its temporal evolution can be regulated by external fields [465,467–469].The statistical state of a lattice with coherent modes generated in its lattice sites, ischaracterized by the statistical operator ˆ ρ which can be expanded over the basis {| n > } ofthe coherent states, ˆ ρ = X n p n | n >< n | . (9.46)The normalization condition Tr H ˆ ρ = X n p n = 1is assumed. Following the procedure of Sec. 9.1, we construct the factor operatorˆ ρ ⊗ ≡ O j ˆ ρ j , ˆ ρ j ≡ Tr H\H j ˆ ρ . (9.47)The normalization conditions are valid:Tr H j ˆ ρ j = 1 Tr H ˆ ρ ⊗ = Y j Tr H j ˆ ρ j = 1 , where ˆ ρ j = X n p n | n j >< n j | . Using the measure of entanglement production (9.10), we define the level of entanglementproduced by the statistical operator (9.46), ε ( ˆ ρ ) = log || ˆ ρ || D || ˆ ρ ⊗ || D . (9.48)187ere || ˆ ρ || D = sup n p n , || ˆ ρ j || H j = sup n j X n ( = n j ) p n , || ˆ ρ ⊗ || D = Y j || ˆ ρ j || H j . As a result, Eq. (9.48) yields ε ( ˆ ρ ) = log sup n p n Q j sup n j P n ( = n j ) p n . (9.49)Entanglement in the lattice is generated whensup n p n = Y j sup n j X n ( = n j ) p n , that is, when the lattice sites are somehow correlated. There are several sources of theircorrelation. First, this is the common history of the condensate preparation. Second, thelattice sites are never completely independent, but there always exists at least a weak tun-neling. Third, atoms from different sites do interact, even though this interaction can berather weak. Finally, the modulating resonant fields, producing the coherent modes, canbe common for all sites of the lattice. The maximal correlation between the modes fromdifferent sites happens when all sites are identical and modulated synchronously, so that p n = p n Y j δ nn j . Then the statistical operator (9.46) isˆ ρ = X n p n | nn . . . n >< nn . . . n | . And we have sup n p n = p n , X n ( = n j ) p n = p n δ nn j , sup n j p n δ nn j = p n . The entanglement-production measure (9.49) reduces to ε ( ˆ ρ ) = − ( N L −
1) log sup n p n . (9.50)The quantity p n = p n ( t ) is the same as in Eq. (9.35), hence, is defined by the evolutionequations (9.34). If the number of sites N L is large, measure (9.50) can be made very large.Since the value p n ( t ) can be regulated, the evolution of measure (9.50) can also be regulated[465,467,469], thus, allowing for the realization of the coherent-mode lattice register.The specific features of the coherent-mode register are:(i) The working objects, multimode condensates, are mesoscopic. Entanglement is ac-complished for these mesoscopic objects, but not for separate particles.(ii) A very strong level of entanglement can be produced, when ε ( ˆ ρ ) ∼ N L ≫ N L lattice sites, with M modes ineach, the computation dimension is M N L . Thus, for two modes ( M = 2) in a lattice of N L = 100 sites, the computation dimension is 10 .(iv) The properties of the lattice, the strength of atomic interactions, and the resonantmodulating fields can be varied in a very wide range, thus, making the mode register highlycontrollable.(v) It is feasible to organize parallel computation by producing operations in differentparts of the lattice.(vi) Erasing memory is a simple process that can be done by appropriately varying themodulating fields.(vii) The decoherence time is sufficiently long. Estimates [182,465] give it of the order of10 −
100 seconds.
Double-well lattices are considered as a very promising tool for quantum information pro-cessing and quantum computing. Recently, such double-well lattices have been realized ex-perimentally in two-dimensional [420] and three-dimensional [470–474] configurations. Thelattices were loaded by Rb atoms. The total number of lattice sites was around 3 × .The filling factor could be varied between one and about 200 atoms. The properties of thedouble wells, such as the barrier height, the distance between the wells, and the relativeenergy offset, could be dynamically controlled. The atoms could be transferred between theleft and right wells in a controllable way.The possibility of dynamically varying the properties of the double-well lattices allowsfor the regulation of their states and dynamics [475]. This controllable regulation is ofhigh importance for realizing quantum information processing and quantum computing withdouble-well lattices. For the latter purpose, the lattices with the filling factor one seem to bethe most appropriate. The properties of such double-well lattices are described in Chapter8. Note that the double-well potentials can be made asymmetric, which provides additionalpossibilities for regulating the system properties [476].Quantum information protocols hold the promise of technological applications unattain-able by purely classical means. In order to realize both, the storage of quantum informationand the faithful long-distance communication, combined systems of atoms interacting withphotons seem to be good candidates [477]. It would be interesting to consider the interactionof coherent electromagnetic fields with atoms located at the cites of a double-well lattice.
10 Brief Concluding Remarks
The material, covered in the present review article, is so extensive that it would take too muchspace for a more or less detailed concluding discussion. And listing in short the consideredtopics would duplicate the Contents. Therefore, instead of having a concluding summary,the reader is advised to survey again the Contents.189t the present time, optical lattices is a fastly developing field of research. There per-manently appear new interesting results. For instance, density modulations in an elongatedBEC with a disorder potential were observed [478]. The direct observation of the Andersonlocalization [358] of boson matter waves in a one-dimensional non-interacting BEC with dis-order was announced [479,480]. The Anderson localization is a phenomenon typical of theideal gases, while rather weak interactions destroy this effect [481].The phenomenon of the Anderson localization occurs in real space. There exists ananalogous effect, called the dynamical localization [482], happening in momentum space.Such a dynamical localization can be realized by means of the quasiperiodic kicked-rotatormodel [483].Despite the variety of novel experimental observations, the basic theoretical points re-main the same. In this review, the emphasis was exactly on the main theoretical ideasand methods. Therefore the material of this review should remain useful in future for anyresearcher in the field of optical lattices.Many techniques, related to periodic potentials, like those treated in the review, areactually common for Bose as well as for Fermi systems. Although the physics of the latter,in many respects, is different. The most detailed description of the state of the art of ultracoldFermi gases has recently been given by Ketterle and Zwierlein [484] (see also [15,485]).In conclusion, it is worth mentioning that many properties of trapped atoms are similarto those of particles in quantum dots, finite nuclei, and clusters. The discussion of the lattersystems can be found in the review articles [486–490].
Acknowledgements
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