The Bogoliubov inequality and the nature of Bose-Einstein condensates for interacting atoms in spatial dimensions D≤2
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] S e p The Bogoliubov inequality and the nature of Bose-Einsteincondensates for interacting atoms in spatial dimensions D ≤ Moorad Alexanian ∗ Department of Physics and Physical OceanographyUniversity of North Carolina WilmingtonWilmington, NC 28403-5606 (Dated: March 12, 2018)
Abstract
We consider the restriction placed by the Bogoliubov inequality on the nature of the Bose-Einstein condensates (BECs) for interacting atoms in a spatial dimension D ≤ D ≤ PACS numbers: 05.30.Jp, 03.75.Lm, 03.75.Hh, 03.75.Nt ∗ [email protected] . INTRODUCTION Anderson disorder-induced localization (AL) describes the sudden transition of electronmobility from that of a conductor to that of an insulator owing to the disorder in the depthsof the potential wells in an otherwise periodic potential [1]. The noninteracting electronsare in single-particles states and it may be that the introduction of the interaction betweenelectrons may lead to delocalization. This has prompted the study of AL of ultracold atomsin 1 D disordered optical potentials. The ultracold atoms are considered to be noninteractingand in a BEC [2–5]. This bring to the fore the fundamental question of the existence orabsence of (gauge invariance) symmetry breaking terms in the Hamiltonian of interactingBose gases in spatial dimensionality D ≤ D ≤ p ,usually p = , is proven to be absent, or (2) a BEC in any single-particle state is shownto be absent. The former result has been established [13] completely rigorously; while inderiving the latter results [7–10], the Bogoliubov inequality is used without first proving itspecifically for the infinite-dimensional case of the Bose gas. II. INTERACTING BOSE GAS
Consider the Hamiltonian for an interacting Bose gasˆ H = Z d r ˆ ψ † ( r )( − ¯ h m ∇ ) ˆ ψ ( r )+ Z d r ˆ ψ † ( r ) V ext ( r ) ˆ ψ ( r )+ Z d r d r ′ ˆ ψ † ( r ) ˆ ψ † ( r ′ ) V ( r − r ′ ) ˆ ψ ( r ′ ) ˆ ψ ( r ) , (1)2here V ext ( r ) is the external potential, V ( r − r ′ ) is the two-particle, local interaction po-tential, and ˆ ψ ( r ) and ˆ ψ † ( r ) are bosonic field operators that destroy or create a particleat spatial position r . For the case of periodic potentials, we do not consider interactionsbetween the bosons and the crystal that result in the creation of phonons. Macroscopic oc-cupation in the single-particle state ψ ( r ) result in the non-vanishing [14] of the quasi-average ψ ( r ) = < ˆ ψ ( r ) > and so the boson field operatorˆ ψ ( r ) = ψ ( r ) + ˆ ϕ ( r ) , (2)with ψ ( r ) = s N V ( D ) X k ′ ξ k ′ e i k ′ · r ≡ s N V ( D ) f ( r ) , (3)and X k ′ | ξ k ′ | = 1 , (4)where N is the number of atoms in the condensate and V ( D ) is the D-dimensional ”volume”and < ˆ ϕ ( r ) > = 0. The operator ˆ ϕ ( r ) has no single-particle states that are in the condensateand so R d r ˆ ϕ † ( r ) ψ ( r ) = 0. The separation of ˆ ψ ( r ) into two parts gives rise to the following(gauge invariance) symmetry breaking term in the Hamiltonian (1)ˆ H symm = Z d r ˆ ϕ † ( r ) ψ ( r ) Z d r ′ [ V ( r − r ′ ) + V ( r ′ − r )] | ψ ( r ′ ) | + h.c. ≡ Z d r ˆ ϕ † ( r ) χ ( r ) + h.c. (5)The presence of this nonzero ˆ H symm in the Hamiltonian gives rise to further macroscopicoccupation in states other than the original state given by ψ ( r ) and so the condensate wave-function ψ ( r ) gets modified by augmenting the single-particles states where macroscopicoccupation occurs. In such a case, macroscopic occupation in the state b would give riseto macroscopic occupation in the states a , such that a = b , whenever the matrix element < ab | ˆ V | bb > of the potential ˆ V , which is the last term in Eq. (1), does not vanish. Fornoninteracting particles in an external potential, the Hamiltonian is diagonal in the repre-sentation of the eigenstates of the Hamiltonian and so macroscopic occupation in any giveneigenstate of the Hamiltonian does not give rise to further macroscopic occupation in anyother energy eigenstate. However, the effect of interparticle interactions can generate macro-scopic occupation in other energy eigenstates. For instance, for a harmonic trap, macroscopic3ccupation of the lowest energy state, the ground state, does not give rise to macroscopicoccupation of any of the higher energy eigenstates. However, interparticle interactions mayallow the generation of macroscopic occupation in other energy eigenstate thus modifyingthe original BEC. Note, however, that macroscopic occupation only in the single-particlestate with momentum p does not give rise to macroscopic occupation in any other momen-tum state since the matrix element in the momentum representation < qp | ˆ V | pp > vanishesby momentum conservation unless q = p .This consistency proviso requires that the correct condensate wavefunction ψ ( r ) corre-sponds to that which gives rise to no symmetry breaking term in the Hamiltonian. That isto say, ˆ H symm vanishes for the correct condensate wavefunction ψ ( r ). For instance, macro-scopic occupation in the single-particle states with momenta , ± k gives rise [15], with theaid of the symmetry breaking term ˆ H symm and owing to linear momentum conservation,to macroscopic occupation in the single-particle momenta states ± k ; therefore, ˆ ϕ † ( r ) isorthogonal to both ψ ( r ) and χ ( r ) and thus one has the possibility of macroscopic occupationin all the momentum states , ± k , ± k , ± k , · · · . In general, one can have also a finiteor an infinite sum [15] over the momentum variable k and so the momentum conservingtwo-particle interaction in the Hamiltonian (1) allows for condensates in the single-particlemomentum states k + P i m i k i , where m i = 0 , ± , ± , · · · [16]. The momenta { k i } representa set of vectors that are, in general, incommensurate thus giving rise to nonperiodic or ape-riodic condensates. However, if one or more of the momenta k i approaches zero, then thesequence of condensates over the momenta sequence { k ′ } with nonzero ξ k ′ has k ′ = 0 as alimit point (or accumulation point) as k i → /k –singularity [17] that isnecessary in all proofs of the absence of a BEC for D ≤
2. For a system confined to a box oflength L by an external potential, momenta is quantized and so k = πL n x ˆ x + πL n y ˆ y + πL n z ˆ z .Note, however, that recent papers still suppose that interacting Bose gases in D ≤ T > k = 0 a limit-point [17]. Similarly, limits on Bose-Einstein4ondensation in confined solid He, where large superfluid fractions have been reported, isbased [24] on supposing macroscopic occupation in only the single-particle momentum state k = 0, which is not the most general, possible BEC. In addition, it is supposed that for2 D , superfluidity is not a consequence of a BEC [21, 22] but is associated with the onsetof algebraically decaying off-diagonal long-range order (ODLRO) [25]. However, the exis-tence of ODLRO can be a consequence of macroscopic occupation in many single-particlemomenta states, which is equivalent to the existence of a BEC. In what follows, we con-sider macroscopic occupation for D ≤ k = 0 a point ofaccumulation. A. Example of BEC in D Consider the following one-dimensional example for a BEC with macroscopic occupationin the single-particle momentum states k = k n with n = 0 , ± , ± , · · · for a Bose system inthe presence of a nonperiodic potential, where k = 2 π/L . The BEC (3) becomes ψ ( x ) = s N π (cid:16) + ∞ X n = −∞ k ( k n + κ ) ν +1 (cid:17) − / ∞ X n = −∞ k cos( k nx )( k n + κ ) ν +1 / , (0 ≤ x ≤ π/k ) , (6)where ν > ψ (0) is finite. The BEC is normalized as follows R π/k d x | ψ ( x ) | = N .Expansion (6) represents a Fourier series and so ψ ( x + 2 π/k ) = ψ ( x ), which is a dynamicalconsequence since macroscopic occupation in the single-particle momentum states k = 0 , ± k implies macroscopic occupation in the states with momenta k = nk with n = 0 , ± , ± , · · · owing to the symmetry-breaking term (5). As a function of the complex variable k , theFourier series in (6) possesses simple or higher order poles at k = ± iκ/n , n = 1 , , , · · · ,for ν + 1 / k = ± iκ/n , n = 1 , , , · · · ,for ν + 1 / = positive integer. Therefore, k = 0 is an accumulation point of poles or branchpoint singularities depending on the value of ν .The sum (6) can be approximated with great accuracy by an integral when k is arbitrarilysmall. This would represent the passage of the Fourier series for the periodic function toa Riemann integral for a nonperiodic function. Accordingly, in the limit k → k = 0becomes a point of accumulation of condensates, the sum (6) may be converted into an5ntegral, and so ψ ( x ) = s N π (cid:16) Z + ∞−∞ d k k + κ ) ν +1 (cid:17) − / Z + ∞−∞ d k cos( kx )( k + κ ) ν +1 / = 12 ν − / π / vuut κ Γ(2 ν + 1) N Γ(2 ν + 1 / ( ν + 1 / κ ν | x | ν K ν ( κ | x | ) , ( −∞ < x < ∞ ) , (7)where Γ( z ) is the gamma function and K ν ( z ) is the modified Bessel function of the secondkind. The integral (7) converges for | x | > ν > − /
2; however, if the condensatewavefunction is required to be bounded at x = 0, then (7) converges for | x | ≥ ν > z ν K ν ( z )] / d z = − z ν K ν − ( z ), K ν ( z ) → ν − Γ( ν ) /z ν as z → ℜ ν >
0, and K − ν ( z ) = K ν ( z ). Therefore, z ν K ν ( z ) = 2 ν − Γ( ν ) − ν − Γ( ν − z + · · · as z → ν > z ν K ν ( z ) = 2 ν − Γ( ν ) − Γ(1 − ν ) z ν / (2 ν +1 ν ) + · · · as z → < ν < < ν < / x = 0 with the derivative tendingtoward ∞ or −∞ as one approaches the cusp. For ν = 1 /
2, the BEC (7) is continuousat x = 0 but the derivative is discontinuous there with slope of q π/ − q π/ x = 0. For 1 / < ν <
1, the BEC (7) attains its largest value of the centerof the localized BEC but it is not a local maximum. The extremal cases of ν = 0 , ν = 0, ψ ( x ) ∝ K ( κ | x | ) = − ln | x | + · · · thus there is a cusp singularitywith slope ∞ or −∞ as one traverses x = 0. Notice that BEC (7) is not bounded at thecenter of the localized condensate for ν = 0; nonetheless, the integral of the BEC density,which gives the total number of particles in the condensate, is finite. Finally, for ν = 1, zK ( z ) = 1 + ( z /
2) ln z + · · · as z → ψ ( x ) attains its largest value at z = 0 but itis not a local maximum.The standard deviations ∆ x and ∆ p follow directly from (7)∆ x = κ s ( ν + 1 / ν + 1 / ν + 1 , ∆ p = ¯ h κ s ν − / , (8)and so (∆ x )(∆ p ) = ¯ h vuut ( ν + 1 / ν + 1 / ν + 1)( ν − / ≥
12 ¯ h, (9)for ν > /
4. 6he sum in (6) can be carried out explicitly for ν = 1 / ψ ( x ) = A πκ s N π (cid:16) e πκ/k e − κx + e − πκ/k e κx e κπ/k − e − κπ/k (cid:17) , (0 ≤ x ≤ π/k ) , (10)where φ ( x + 2 π/k ) = φ ( x ) and the normalization constant A is A = (cid:16) π κ coth( π κk ) + π k κ csch ( π κk ) (cid:17) − / . (11)Note that for k →
0, one has that ψ ( x ) = q κN e − κ | x | , ( −∞ < x < ∞ ) , (12)which agrees with Eq. (7) since K / ( z ) = q π/ z e − z . Result (12) corresponds to a localizedBEC that decays exponentially and has a discontinuous derivative at the center.For ν > K ν ( z ) → ν − Γ( ν ) /z ν as z → ψ ( x )given by (7) is finite at x = 0. In addition, K ν ( z ) → q π/ z e − z as z → ∞ for ν > − / k →
0, which approximates the sum (6) with the integral (7), the condensatewavefunction is periodic, viz., ψ ( x + 2 π/k ) = ψ ( x ). However, in the limit k →
0, one hasan even, nonperiodic, localized condensate wavefunction.Actually, one can interchange the roles of x and k in Eq. (7) and so one has, instead,a condensate wavefunction ψ ( x ) ∝ ( x + a ) − ν − / with a corresponding momentum distri-bution ϕ ( k ) ∝ | k | ν K ν ( a | k | ), where a is a length scale. It is interesting that in experimentswith ultracold atoms (Ref. 28), BECs have been found that suggest a power-law decrease inthe wings of the atomic density with an exponent close to the value 2, viz., | ψ ( x ) | ∝ x − as | x | → ∞ . This would correspond to ν = 0 and so | ψ ( x ) | ∝ ( x + a ) − for ( −∞ < x < ∞ ).The momentum distribution associated with this BEC density is given by the modifiedBessel function of order zero ϕ ( k ) ∝ K ( a | k | ) for ( −∞ < k < ∞ ), where the singular point k = 0 is a logarithmic branch point since K ( z ) ∝ − ln z as z →
0. The standard deviation∆ p = ¯ h/ (2 a √
2) is finite even though ϕ ( k ) diverges logarithmically as k →
0; however, ∆ x is infinite even though | ψ ( x ) | is bounded everywhere. Note that the behavior of ψ ( x ) for x ≫ a is determined by the behavior of its Fourier transform ϕ ( k ) for ka ≪ χ ( x ) = s N π (cid:16) N/ X n = − N/ k ( k n + κ ) ν +1 (cid:17) − / N/ X n = − N/ k cos( k nx )( k n + κ ) ν +1 / , (0 ≤ x ≤ π/k ) , (13)7nd thus consider the combined limit k → N → ∞ in order to approximate the sumby an integral. If k N → ∞ as k → N → ∞ , then one obtains the previous resultgiven by Eq. (7). However, if k N/ → K < ∞ as k → N → ∞ , then χ ( x ) = s N π (cid:16) Z K − K d k k + κ ) ν +1 (cid:17) − / Z K − K d k cos( xk )( k + κ ) ν +1 / , ( −∞ < x < ∞ ) . (14)For one-dimensional crystals, K ≤ π/a , where a is the length of a unit cell. The condensatewavefunction χ ( x ) is normalized by R + ∞−∞ d x | χ ( x ) | = N . Note that for K | x | ≫
1, thesecond integral in (14) is dominated by the small values of k , viz., k ≪ K , and so therange of the integral in (14) can be extended to ±∞ when K | x | ≫ χ ( x ) ∝ | x | ν K ν ( κ | x | ) for K | x | ≫
1, which becomes χ ( x ) ∝ | x | ν − / e − κ | x | for κ | x | ≫ B. BEC in the harmonic trap
If the external potential is a one-dimensional harmonic oscillator, then the field operatorˆ ψ ( x ) is expanded in terms of the energy eigenstates ψ n ( x ) of the harmonic oscillator. If onehas macroscopic occupation in the ground state ψ ( x ), then ˆ ψ ( x ) = ψ ( x ) + ˆ ϕ ( x ), wherethe operator ˆ ϕ ( x ) is orthogonal to ψ ( x ) and so ˆ ϕ ( x ) has nonzero expansion coefficients foronly the creation operators of the higher energy harmonic oscillator eigenstates. It shouldbe noted that if one considers atom-atom interactions, then it may be that states other thanthe ground state may be also macroscopically occupied. The latter occurs if the matrixelement < l | ˆ V | > of the atom-atom interaction potential ˆ V does not vanish for l = 0 inwhich case there would be additional macroscopic occupations in the higher energy harmonicoscillator eigenstates ψ l ( x ).Consider the following illustration of a condensate in the ground state of a one-dimensional harmonic trap ψ ( x ) = s N π (cid:16) ∞ X n = −∞ k e − βk n (cid:17) − / ∞ X n = −∞ k e − βk n cos( k nx ) , − π/k ≤ x ≤ π/k . (15)The condensate (15) has macroscopic occupation in the single-particle momentum states k n , with n = 0 , ± , ± , · · · , ψ ( x ) is periodic ψ ( x + 2 π/k ) = ψ ( x ), and is normalizedas follows R π/k − π/k d x | ψ ( x ) | = N . Each Fourier coefficient of the series (15) possesses anessential singularity at k = ∞ and in the limit k →
0, may be approximated arbitrarily8ell by a Riemann integral and so ψ ( x ) = s N π (cid:16) Z ∞−∞ d k e − βk (cid:17) − / Z ∞−∞ d k e − βk cos( kx )= q N (cid:16) πβ (cid:17) / e − x / β , −∞ < x < ∞ . (16)Therefore, the condensate associated with the lowest energy state in the harmonic trapis represented by the macroscopic occupation of single-particle momentum states given by k = nk , where n = 0 , ± , ± , · · · , with k = 0 a point of accumulation as k → k , the BEC given by the series (15) con-verges for ℜ k = ( ℜ k ) − ( ℑ k ) > ℜ k ≤
0. The regions of convergenceand divergence resemble a Minkowski spacetime diagram. The series (15) converges in thespacelike regions and diverges in the timelike regions and the light cones. Along the realaxis, the series (15) converges except the singularity that it encounters at the origin k = 0. III. BOGOLIUBOV INEQUALITY
The absence or presence of a BEC in spatial dimensions D ≤ h{ ˆ A, ˆ A † }i ≥ k B T |h [ ˆ C, ˆ A ] i| / h [[ ˆ C, ˆ H ] , ˆ C † ] i , (17)where ˆ H is the Hamiltonian (1) of the system with arbitrary local interparticle and externalpotentials, the brackets denote thermal averages, and the operators ˆ A and ˆ C are arbitraryprovided all averages exist.Consider the following operators [17],ˆ C = Z d r e i k · r ˆ ψ † ( r ) ˆ ψ ( r ) (18)and ˆ A = Z d r Z d r ′ e − i k · r f ( r ) f ∗ ( r ′ ) ˆ ψ † ( r ) ˆ ψ ( r ′ ) , (19)where k is arbitrary. Now, h [ A † , A ] i = V ( D ) h [ ˆ C, ˆ A ] i = V ( D ) { N − N | A k | − X q h ˆ a † q ˆ a q i| ξ q + k | } , (20) h [[ ˆ C, ˆ H ] , ˆ C † ] i = ¯ h k m N, (21)9ith A k = X k ′ ξ k ′ ξ ∗ k ′ + k = 1 N Z d r | ψ ( r ) | e i k · r (22)with the aid of Eq. (3), where N is the total number of particles. Now, | A k | ≤ k = 0, that is, A = 1 with theaid of Eq. (4). The vector q / ∈ { k ′ } , where { k ′ } is the set of condensate vectors for which ξ k ′ = 0. For a BEC at rest, | ξ k | = | ξ − k | . Note that ξ q + k = 0 for ( q + k ) ∈ { k ′ } and ξ q + k = 0 for q / ∈ { k ′ } and k ∈ { k ′ } .We sum the Bogoliubov inequality (17) over the single-particle momentum states in theset { k ′ } constituting the condensate, which includes an arbitrary neighborhood of the pointof accumulation of the condensate at k ′ = 0 that corresponds to a condensate at rest. Wewant to find an upper bound of the anticommutator h{ ˆ A, ˆ A † }i = 2 h ˆ A ˆ A † i + h [ ˆ A † , ˆ A ] i . Weextend the sum over the first term h ˆ A ˆ A † i over all values of k thus obtaining a larger upperbound M N N V ( D ) ≥ X k h ˆ A ˆ A † i , (23)where the condensate wavefunction ψ ( r ) is orthogonal to the operator ˆ ϕ † ( r ) and we assumethat the condensate density is bounded from above by | f ( r ) | ≤ M with 1 ≤ M < ∞ since R d r | f ( r ) | = V ( D ). In (23) use has been made of the completeness relation for themomentum eigenstates and a negative term resulting from a single commutation has beendropped. Note that we are considering a condensate where all the single-particle stateswith momentum k ′ are occupied macroscopically with k ′ = a point of accumulation. Inaddition, we are supposing that the number of particles in the “volume” V ( D ) is fixed, thatis, we are employing a canonical ensemble and so R d r ˆ ψ † ( r ) ˆ ψ ( r ) = P k ˆ a † k ˆ a k = ˆ N is actuallythe c-number N .Consider next the sum over k ′ of the commutator h [ ˆ A † , ˆ A ] i , X k ′ h [ ˆ A † , ˆ A ] i = N V ( D ) X k ′ [1 − | A k ′ | ] , (24)with the aid of (20) and where ξ q + k = 0 for q / ∈ { k ′ } and k ∈ { k ′ } . This sum over thecommutator is bounded from above provided the sum is restricted to values of k ′ that havea finite, upper bound. Now the right-hand side (RHS) of inequality (17) becomes k B T |h [ ˆ C, ˆ A ] i| / h [[ ˆ C, ˆ H ] , ˆ C † ] i = mk B T ¯ h V ( D ) N N X k ′ (cid:16) − | A k ′ | k ′ (cid:17) (25)10ith the aid of Eqs. (20) and (21). Note that the RHS is bounded in the upper limit of thesum; however, it is the lower limit as k ′ → D ≤ N = 0 for T >
0. Combining Eqs. (23)–(25), we have forthe Bogoliubov inequality,
M N + 12 X k ′ (1 − | A k ′ | ) ≥ mk B T ¯ h N N X k ′ (cid:16) − | A k ′ | k ′ (cid:17) . (26)Therefore, the existence of a BEC for T > k ′ of thecondensate. Note that the sums in (26) over the condensate momenta can be approximatedby integrals according to P k ′ → V ( D ) R d k ′ and so M NV ( D ) + 12 Z d k ′ (1 − | A k ′ | ) ≥ mk B T ¯ h N N Z d k ′ (cid:16) − | A k ′ | k ′ (cid:17) . (27)The integral on the RHS has no infrared divergence since by (22), (1 − | A k ′ | ) vanishesquadratically as k ′ → /k –singularity is removed thus allowing the existenceof a BEC for D ≤ A. Removal of /k –singularity in D To illustrate the removal of the 1 /k –singularity for the existence of a BEC for D ≤ D example given by Eq. (6) where ξ k n = B ( k n + κ ) ν +1 / ( n = 0 , ± , ± , · · · ) , (28)with the normalization constant B given by B = ( + ∞ X n = −∞ k n + κ ) ν +1 ) − / (29)with the aid of (4) and so by (22) A k l = A - k l = B ∞ X n = −∞ k n + κ ) ν +1 / k ( n − l ) + κ ) ν +1 / . (30)Now Γ( ν + 1 / k ( n − l ) + κ ] ν +1 / = Z ∞ d x x ν − / e − [ k ( n − l ) + κ ] x ∞ X j =0 ( − j Γ( j + ν + 1 / k j ( l − nl ) j j !( k n + κ ) j + ν +1 / (31)for ν > − /
2, which gives the following series expansion for (30) in the neighborhood of l = 0, A k l = 1 − B Γ( ν + 3 / ν + 1 / + ∞ X n = −∞ k l ( k n + κ ) ν +2 ++ 2 B Γ( ν + 5 / ν + 1 / + ∞ X n = −∞ k l n ( k n + κ ) ν +3 + O ( l ) . (32)The sum on the RHS of (26) is + ∞ X l = −∞ k l (1 − A k l ) (1 + A k l ) (33)and so the singularity at l = 0 in the summand is removed as indicated by the expansion(32) of A k l .Similarly for the BEC in the harmonic trap where (15) gives that A k l = (cid:16) ∞ X n = −∞ e − βk n (cid:17) − ∞ X n = −∞ e − βk n e − βk l cosh(2 nβlk ) , (34)which on expanding in powers of l about l = 0 removes the k l –singularity at k l = 0 inthe series on the RHS of the Bogoliubov inequality (26). IV. BECS IN PERIODIC AND DISORDERED POTENTIALS
The existence of superfluidity is usually associated with the existence of a BEC. Theexistence of superflow [26] in solid helium He has stimulated the search of a BEC in solidhelium thus establishing the existence of BECs in all three states of matter–gas, liquid, andsolid. It is to be noted that the proofs of the absence of a BEC in gases and liquids for D ≤ k , k ± q , then thesymmetry breaking term (5) gives rise to macroscopic occupation in the momenta states k ± n q with n = 0 , ± , ± , · · · . Accordingly, the condensate wavefunction is given by ψ k ( r ) = s N V ( D ) + ∞ X n = −∞ ξ k + n q e i ( k + n q ) · r ≡ e i k · r u k ( r ) , (35)which is of the Bloch form since u k ( r ) has the periodicity of the lattice, that is, u k ( r ) = u k ( r + t m ) since e i q · t m = 1. The primitive lattice translation vector t m = m a + m b + m c ,12here m i can take all integer values and a , b , and c are the edges of the unit cell, which formsa parallelepiped. Therefore, the possible values of the lattice condensate vector(s) q arein q -space or reciprocal space. Note that Bloch functions ψ k ( r ) and ψ k ′ ( r ) are orthogonalfor k ′ = k provided | k | < | q | / | k ′ | < | q | / R d r ψ ∗ k ′ ( r ) ψ k ( r ) = N δ k ′ , k . Theexpectation value of the momentum in the condensate follows from (35) and so Z d r ψ ∗ k ( r )( − i ¯ h ∇ ) ψ k ( r ) = N ¯ h ∞ X n = −∞ ( k + n q ) | ξ k + n q | = N ¯ h k (36)provided | ξ k + n q | = | ξ k − n q | . Therefore, k represents the energy dependent Bloch wavenumber or the quasi momentum per particle ¯ h k of the condensate.For the case of a 1 D crystal of length L = N a , where there are N primitive cells of length a one has that ψ k ( x ) = e ikx u k ( x ) , (37)with u k ( x + a ) = u k ( x ), where k = 2 πm/N a ( m = 0 , ± , ± , · · · ) owing to the boundarycondition ψ k ( x + N a ) = ψ k ( x ). The range for the momentum k is given by − π/a ≤ k ≤ π/a since if the Bloch condition holds for k it also holds for k ′ = k + 2 πm/a . One has from theSchr¨odinger equation that ψ − k ( x ) = ψ ∗ k ( x ) and so the negative values of k do not give riseto new solutions. In addition, the values of m are limited to m = 0 , , , · · · , N − m do not generate any new solutions and so we have only N solutionsper band. In fact, one needs an additional label, a band index, to be added to the Blochfunction to describe which band the function belongs. A. Atom-atom interactions and disordered potentials
The Bloch form for the condensate wavefunction ψ k ( r ) given by (35) is appropriate fornoninteracting particles in a perfect crystal for D = 3. However, the condensate (35) doesnot remove the 1 /k –singularity for finite q for spatial dimensions D ≤ T >
0, whichis required for the existence of a BEC for systems of noninteracting or interacting particleswhen embedded in periodic or in disordered potentials. It is interesting that the removalof the 1 /k –singularity leads to localization. Therefore, we suppose that the inclusion of aninterparticle potential, given by the third term in the RHS of (1), gives rise to a BEC provided q is arbitrarily small and so (35) ceases to be of the Bloch form. It is interesting that thisbehavior is equivalent to supposing a BEC that is a linear superposition of condensates with13iffering discrete, translational motion and so the resulting condensate wavefunction is givenby ψ ( r ) = X k ′ α k e i k · r u k ( r ) , (38)where the prime in the sum indicates that | k | < | q | /
2. The existence of a point of accumu-lation, or limit-point, at k = 0 is what is required for the existence of a BEC for systems withspatial dimensionality D ≤
2. Expression (38) represents a sort of wave packet constructednot by means of plane waves but instead by means of plane waves modulated by a Blochstate, which still leads to an expansion in terms of plane waves as the Fourier series given by(3). It is interesting that such types of states have been used in the study of the dynamicsof electron wave packets in crystals [27]. In addition, the wave packets so prepared remainin the same band at later times, in our case the lowest energy band, and are referred to asBloch-type states [27] since even though they are not Bloch states, viz. there is no singlemomentum ¯ h k such that ψ ( r + t m ) = e i k · t m ψ ( r ) even though u k ( r + t m ) = u k ( r ). Themomentum associated with the condensate (38) is Z d r ψ ∗ ( r )( − i ¯ h ∇ ) ψ ( r ) = N ¯ h X k ′ k | α k | = 0 , (39)where the average momentum of the condensate is zero, that is, we choose the system withrespect to which our condensate of N particles is at rest and so | α k | = | α − k | . Similarly,the kinetic energy T associated with the condensate (38) is T = Z d r ψ ∗ ( r )( − ¯ h ∇ µ ) ψ ( r ) = N X k ′ ¯ h k µ | α k | + N ¯ h k µ X k ′ | α k | ∞ X n = −∞ n | ξ k + n q | , (40)where µ is the particle mass and use has been made of the normalization conditions P k ′ | α k | = 1 and P ∞ n = −∞ | ξ k + n q | = 1.One would expect that the kinetic energy T of the condensate should be rather small,which suggest neglecting the second term on the RHS of (40). This corresponds to replacing u k ( r ) in (38) by a constant. Accordingly, ψ ( r ) ≈ X k ′ α k e i k · r . (41)Note that the BEC (41) is not a Bloch function since ψ ( r + t m ) = ψ ( r ). In addition, forcases where the sum over k is associated with a nonisolated singularity at k = 0, such a BECis equally applicable to Anderson localization of ultracold atoms in a disordered potential.14 . BECs in D crystals Consider the BEC in the lowest, or first Brillouin zone, energy band with macro-scopic occupation in the first, say, 2 M + 1, lowest energy states, viz. with k =0 , ± π/N a, ± π/N a, · · · , ± πM/N a with 0 < M < N/
2. Thus, the BEC (41) becomes ψ ( x ) = M X m = − M α m e πmxi/Na , (42)where α m = α − m with normalization condition N L = P Mm = − M | α m | . Therefore, in thedomain | ka | < πM/N , our condensate possesses macroscopic occupation of 2 M single-particle momentum states. Now ψ ( x ) in (42) satisfies the periodic boundary condition ψ ( x + L ) = ψ ( x ); however, ψ ( x ) does not satisfy the Bloch condition, viz., ψ ( x + la ) = ψ ( x )for 0 ≤ l < N . Notice that in the limit N → ∞ , M → ∞ such that M/N → K , one has, inany arbitrary neighborhood of the lowest energy state with k = 0, an unlimited number ofnonvanishing α m with k = 0 an nonisolated singularity. This behavior is what is requiredto remove the 1 /k –singularity that appears in the Bogoliubov inequality in order for thecrystalline system to possess a BEC in 1 D . C. BECs in disordered potentials
The general expression of BECs in a periodic potential is given by ψ k ( r ) = s N V ( D ) ∞ X n ,n , ··· = −∞ ξ k + n q + n q + ··· e i ( k + n q + n q + ··· ) · r ≡ e i k · r u k ( r ) , (43)with u k ( r ) = u k ( r + t m ) for any primitive lattice translation vector t m and where theset of vectors { q i } are in the reciprocal lattice space and so e q i · t m = 1 for i = 0 , , , · · · .The generation of macroscopic occupation in the momenta states in Eq. (43), which occurs,albeit, even in the presence of arbitrarily weak two-body interactions, is a direct consequenceof the symmetry breaking term (5) and the supposition that the single-particle momentastates , ± q , ± q , ± q , · · · are macroscopically occupied. Of course, if the interparticlepotentials are not negligible, then the BEC cannot be of the Bloch form (43). Therefore,in the presence of interparticle interactions, not all the vectors { q i } are in the reciprocallattice space, in which case the BEC can still be expressed in the form of Eq. (43) exceptthat now u k ( r ) = u k ( r + t m ) and so the BEC is not of the Bloch form. The latter is what15ne would expect for nonperiodic or disordered potentials for spatial dimensions D = 3. Itis clear that for D ≤
2, some of the vectors { q i } must vanish in a limiting process in orderto remove the 1 /k –singularity in the Bogoliubov inequality (26) required for the existenceof a BEC in spatial dimension D ≤ D ≤
2, which requires theremoval of the 1 /k –singularity by means of a point of accumulation at k = 0, is preciselythe same for both noninteracting atoms in a disordered potentials as well as for interactingatoms in the absence of an external potential. Therefore, the mathematical form of the BECin D ≤ T >
V. ANDERSON LOCALIZATION OF ULTRACOLD ATOMS
Anderson localization of matter waves has been observed with cold atoms from a nonin-teracting BEC in a one-dimensional disordered potential generated by a laser speckle pattern[28] and where the quasi-periodic lattice is the result of the addition of noncommensurateoptical periods [29]. The experiment consists in releasing a BEC in the 1 D disordered op-tical potential, where all the noninteracting atoms are originally in the same single-atomwavefunction, viz. the localized condensate. The atomic wavefunction initially expands andsubsequently stops expanding and the resulting wavepacket has wings that decay exponen-tially or as a power-law [28] and exponentially or Gaussian-like [29].The condensate of a nonideal Bose gas in 1 D and in the absence of an external potential isnot given by the macroscopic occupation of a single momentum state, as is the case in an idealBose gas in D = 3, owing to the 1 /k –singularity in the Bogoliubov inequality. Actually,for spatial dimensions D ≤
2, the condensate must to be nonuniform [17] and localized asshown above. Therefore, both noninteracting bosons in disordered potentials and interactingbosons in the absence of external potentials give rise to Anderson localization. It would beinteresting if one could experimentally discern the relative contribution to localization bythe disordered potential and by the interactions among atoms. In particular, if localizationwould persist in an ultracold, non-dilute atomic 1 D gas on expansion, where atom-atominteractions cannot be neglected, even in the absence of a disordered potential, viz., in theabsence of any external potential. The experimental proof that a theory, based on the16equirements imposed by the Bogoliubov inequality, allows for localization with repulsiveatom-atom interactions present would lend some support to the original work of Andersonof a sudden phase transition from conductor to insulator via the degree of disorder in thematerial [1]. It is interesting that the Mott metal-insulator transition leads to localizationwithout randomness owing to electron-electron interactions, which is somewhat similar tolocalization in an interacting Bose gas for D ≤ σ R and the high-momentumcutoff at the inverse healing length 1 /ξ in . For ξ in > σ R , the BEC wave function is expo-nentially localized, whereas for ξ in < σ R , the spatial decay is algebraic [30]. For the specklepotential considered, the Fourier transform of the correlation function vanishes for momenta k > σ − R resulting in a vanishing Lyapunov exponent for k > σ − R in the Born approxima-tion. In one-dimensional elastic scattering, the particle wave vector k in forward scatteringremains unchanged or changes sign in backscattering resulting in a momentum transfer of2 k . Therefore, the study of localization for k > σ − R requires going beyond the Born ap-proximation. The higher order corrections to the Born approximation give rise to ”effectivemobility edges” at k = pσ − R , where p is an integer that characterizes the successive correc-tions to the Born approximation [31, 32] . It is interesting that these higher-order terms ofthe Born series are necessary even for k < σ − R .The atom-atom interaction in the Hamiltonian (1) requires macroscopic occupation inmomenta k = n k , where n = 0 , ± , ± , · · · , owing to linear momentum conservation, ifthere is macroscopic occupation in the two momenta states k = 0 , k . Therefore, for 1 D ,forward scattering of momentum k requires backward scattering into the momentum state − k . Our example of a BEC in 1 D given by (6) reflects such requirements, which is a signaturethat one is dealing with a nonperturbative feature of the interparticle potential. Notice thatin obtaining the BEC density from the BEC wavefunction (6), one does not suppose thatthe phases for different momenta are uncorrelated since such supposition would give rise to aconstant BEC density, which would be equivalent to supposing macroscopic occupation in asingle momentum state and thus to a uniform BEC. The assumption that localized functionfor a given momentum k are uncorrelated is made in the case of random potentials [30].17he sum (3) representing the condensate wave function can be approximated arbitrarilywell by an integral in any spatial dimension. This is especially important for D ≤ k = 0 and so the BECdensity is spatially nonuniform. For instance, the example for 1 D given by the sum (6)with its corresponding approximate value given by integral (7). For the case ν = 1 /
2, theintegrand of the Fourier cosine transform in (7) possesses two simple poles at k = ± iκ and so for x > x < k = iκ ( k = − iκ ) that yields result(12). Therefore, the long-tail behavior of the BEC density is determined by the singularityin the complex k -space closest to the real axis. Note that for ν > − /
2, the nature ofthe singularities of the integrand in (7) is generally branch points or poles of higher ordersat k = ± iκ . Nonetheless, the asymptotic formula is still given by an exponential decay,which is a property of the Bessel function K ν ( z ) that tends exponentially to zero as z → ∞ through positive values. An exponentially decaying BEC density is a direct consequence ofFourier transform ϕ ( k ) of ψ ( x ) given by a meromorphic function of k , viz., ϕ ( k ) is analyticexcept at a set of isolated points, e.g., ratios of rational functions of k . If a pole of ϕ ( k )is off the imaginary axis, then the asymptotic exponential decay of ψ ( x ) is modulated bysinusoidal functions, which would represent a remnant periodicity in the system.In Section II.A, we mentioned examples of BEC in 1 D with algebraic localization, viz., ψ ( x ) ∝ x + a ) ν +1 / = 1(2 a ) ν √ π Γ( ν + 1 / Z ∞−∞ d k | k | ν K ν ( a | k | )cos( kx ) (44)for ν > − / a >
0, and −∞ < x < ∞ . The function z ν K ν ( z ) is an analytic function of z for ν = n + 1 /
2, ( n = 0 , , , · · · ), but the origin z = 0 is a branch point singularity for ν = n + 1 /
2, ( n = 0 , , , · · · ) [33]. However, the function that appears in the integrand in(44), viz., | z | ν K ν ( a | z | ), is not an analytic function of z since | z | is not an analytic function of z . In fact, K ν ( z ) is an analytic function of z throughout the z -plane cut along the negativereal axis since z = 0 is a branch point singularity. According to the Riemann–Lebesguelemma, the Fourier representation of the BEC ψ ( x ) goes to zero as x → ∞ and so thelarge distance behavior of the BEC is determined by the small k behavior of its Fouriertransform ϕ ( k ). Our example (44) of algebraic decay suggests that the behavior of theBEC ψ ( x ) ∝ / ( x + a ) ν +1 / is determined as x → ∞ by a branch point singularity at18 = 0. Note also that the value of the BEC ψ ( x ) for x ≈ ϕ ( k ) over a finite range of values of k near k = 0 for ν > z ν K ν ( z ) → ν − Γ( ν ) as z → ν >
0. However, for ν = 0, the range of values thatcontributes to the integral is very small and quite close to k = 0 owing to K ( z ) → − ln z as z →
0. For instance, the contribution to ψ (0) by the integral (44) for ν = 1 /
2, is 63%from the region 0 ≤ k < /a and 37% from the region k > /a . On the other hand, for ν = 0, the contribution to ψ (0) is 79% from 0 ≤ k < /a and 21% from k > /a . Note,however, that for the exponentially decaying BEC (7), ψ ( x ) ∝ ( κ | x | ) ν K ν ( κ | x | ) → ν − Γ( ν )as κ | x | → ν >
0, which can become arbitrarily large owing to the simple pole in Γ( ν )at ν = 0. However, for ν = 0, ψ ( x ) ∝ K ( κ | x | ) → − ln( κ | x | ) as κ | x | →
0. This logarithmicdivergence at x = 0 is a direct consequence of the large values of the momentum k , whichresults in the logarithmic divergence of the integral (7). This differs somewhat from thestudy [30] that suggests that, in general, it is the contribution of waves with very small k that is important for the accurate determination of ψ ( x ) in the center of the localized BEC.It may be, however, that the momentum k that appears in the Fourier integral (Eq. (6) inthe first of Ref. 30) may not be so directly connected with the variable k that appears inthe BEC density (Eq. (8) in the first of Ref. (30)) via the stationary, long-time momentumdistribution D ( k ) and the localized function φ k ( z ) of the plane-wave component e ikz .It is interesting that the behavior of the BEC density at large distances is determined bymomenta near the high-momentum cutoff k c owing to the localization of the independent k waves [30]. For ξ in > σ R , the Lyapunov exponent has a finite lower bound that leads to aBEC density that is exponentially localized. On the other hand, for ξ in < σ R , there is nosuch finite lower bound and so the localization is algebraic [30]. It is important to remarkthat the Bogoliubov inequality requires a limit point (or accumulation point) of plane wavemomenta at k = 0 for the existence of a BEC for D ≤ /k –singularity. VI. THE GROSS-PITAEVSKII EQUATION
The numerical results presented in Ref. (30) for the dynamic behavior of the BEC arebased on the time-dependent Gross-Pitaevskii equation (GPE) [34]. The GPE representsa mean-field description of the ground state and it is obtained by finding an extremum (a19inimum) of the energy as a functional of the BEC wave function [35]. The time-independentGPE for a conserved number of particles corresponding to the Hamiltonian (1) is − ¯ h m ∇ ψ ( r ) + V ext ( r ) ψ ( r ) + ψ ( r ) Z d r ′ [ V ( r − r ′ ) + V ( r ′ − r )] | ψ ( r ′ ) | − µψ ( r ) = 0 , (45)where µ is the chemical potential.It is important to remark that the dynamical symmetry-breaking term (5) that requiresmacroscopic occupation of many single-particle momentum states is determined solely bythe two-body interaction potential V ( r − r ′ ) and not at all by the external potential V ext ( r ).Accordingly, the consistency proviso that makes the symmetry-breaking term (5) vanish,viz., that the operator ˆ ϕ † ( r ) be orthogonal to both ψ ( r ) and χ ( r ), does not follow from theGPE (45). In fact, for the ground state, our consistency proviso, which requires the lasttwo terms in (45) be orthogonal to the operator ˆ ϕ † ( r ) , requires, therefore, that the sum − ¯ h m ∇ ψ ( r ) + V ext ( r ) ψ ( r ) in the GPE (45) be also orthogonal to ˆ ϕ † ( r ). The kinetic energyterm is certainly orthogonal to ˆ ϕ † ( r ) since the kinetic energy is diagonal in the single-particle momentum representation; however, the term V ext ( r ) ψ ( r ) in (45) is not diagonal inthe single-particle momentum representation and, therefore, the Fourier components V ext ( k )of V ext ( r ) must be in the set { k ′ } of condensate vectors, that is, k ∈ { k ′ } .For instance, for the BEC in the one-dimensional harmonic trap of Sec. II B, V ext ( x ) ψ ( x ) = s N L X q e iqx X k V ext ( k ) ξ q − k , (46)where L is the length of the one-dimensional “box.” Our consistency proviso requires that(46) be orthogonal to ˆ ϕ † ( r ); therefore, not only q ∈ { k ′ } but also k ∈ { k ′ } since otherwise ξ q − k would vanish since ξ q − k is nonzero only for ( q − p ) ∈ { k ′ } . Accordingly, the Fouriercoefficient V ext ( k ) of V ext ( x ) cannot have any nonzero Fourier components outside of thesingle-particle momenta that constitutes the condensate, viz., { k ′ } . The Fourier expansionof the harmonic trap is given by x = π k + 4 ∞ X n =1 ( − n cos( k nx ) k n − π/k ≤ x ≤ π/k (47)with nonzero coefficients for k = nk , n = 0 , ± , ± , · · · , which are the same single-particlemomenta with macroscopic occupation of the condensate wave function ψ ( x ) given by (15).It should be noted that we are considering the restrictions placed on BECs by the Bogoli-ubov inequality at finite temperatures. The determination of the condensate wave function20equires us to minimize the Helmholtz free energy with respect to the condensate wavefunction for fixed density and temperature. However, such thermodynamic potential is notavailable. Therefore, the study of BECs given by the GPE does not suffice since the GPEdescribes the ground state (zero temperature) of bosonic systems when all the particles arein the condensate. Nonetheless, we have shown that the symmetry breaking term (5), to-gether with the GPE, imposes conditions on the momenta of the Fourier coefficients of theexternal potential. VII. SUPERSOLID BEC
The analysis of the possible existence of BEC for D ≤ D = 3. Clearly, local interparticle potentialscannot give rise to a BEC of the Bloch form for D = 2 since the 1 /k –singularity in theBogoliubov inequality cannot be removed and still preserve the Bloch form for the BEC.The k behavior of the double commutator (21) follows from the kinetic energy term ofthe Hamiltonian since local interparticle potentials ˆ V do not contribute to the Bogoliubovcommutator, viz., h [[ ˆ C, ˆ V ] , ˆ C † ] i = 0. It is interesting that the latter is not the case fornonlocal potentials [36]. If, for instance, the two-particle potential is a sum of a localand a nonlocal potential, then the former potential does not contribute to the Bogoliubovcommutator while the latter does and if the decay of the nonlocal potential with distanceis sufficiently slow, then h [[ ˆ C, ˆ V ] , ˆ C † ] i ∝ k − ǫ with ǫ > k → D = 2 in the presence ofan infinitely long-range nonlocal potential between the condensate atoms.Recently, a Dicke quantum phase transition was realized in an open system formed bya BEC coupled to an optical cavity that gives rise to a self-organized supersolid phase[37]. It is interesting that the phase transition is driven by infinitely long-range interactionsbetween the condensed atoms. The analogy of that work to the Dicke model is based on theinteraction Hamiltonian that gives rise to a coupling of the pump and cavity fields to thezero-momentum states of the atoms to the symmetric superposition of atomic states thatcarry an additional unit of photon momentum. This is quite analogous to our dynamicallygenerated symmetry breaking term that allows condensation in atomic states that are integer21ultiples of a given condensate momentum. VIII. SUMMARY AND CONCLUSION
In random potentials, the underlying mechanism for AL is the suppression of particletransport due to destructive interference. The intriguing question is if such mechanism isundermined by the presence of interparticle interactions. For bosons, we have seen that thegeneration of periodic BECs is a direct consequence of the dynamical symmetry-breakingterm in the Hamiltonian that results from the macroscopic occupation of just two single-particle momentum states, viz., k = 0 , k . However, such types of BECs, albeit allowedfor D = 3, violate the Bogoliubov inequality for D ≤
2. Therefore, the presence of atom-atom interactions requires that k → k = 0 becomes an accumulation pointof condensates. Note that for D = 1, that suffices to remove the 1 /k –singularity in theBogoliubov inequality. However, for D = 2, the removal of the 1 /k –singularity requiresaugmenting the set { k } of condensed states so that the removal of the 1 /k –singularityoccurs for all approaches to the origin k = 0.Finally, the existence and nature of BECs for D ≤
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