CCold Atoms in Driven OpticalLattices byMuhammad Ayub
SUBMITTED IN PARTIAL FULFILLMENT OF THEREQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYatDepartment of ElectronicsQuaid-i-Azam UniversityIslamabad, PakistanMarch 2012The work presented in this thesis is dedicatedto
My father, mother (late), wife and kids i Certificate
It is certified that the work contained in this thesis entitled “Cold Atoms inDriven Optical Lattices” is carried out by
Mr. Muhammad Ayub underthe supervision of
Dr. Farhan Saif .Research Supervisor: ————————————Dr. Farhan SaifPride of PerformanceAssociate Prof. Department of ElectronicsQuaid-i-Azam University, Islamabad.Chairman Department: ————————————Dr. Farhan SaifAssociate Prof. Department of ElectronicsQuaid-i-Azam University, Islamabad. March 2012ii
Abstract
The coherent control of matter waves in optical crystals is at the heartof many theoretical and experimental interests and has evolved as an ac-tive research area over the past two decades. Time-periodic modulationsin matter-wave optics provides an additional handle to control coherence ofmatter waves.In the thesis, the center of mass dynamics of cold atoms and the Bose-Einstein condensate in one dimensional optical lattice is considered both inthe absence and in the presence of external forcing. We discuss three sit-uations for matter waves: first, the cold atoms; second, sufficiently dilutecondensate or the condensate for which the inter-particle interaction can betuned to zero by exploiting Feshbach resonance, and condensate dynamicsare governed by the single particle wave packet dynamics; third, strong in-teraction regime, where, inter-atomic interaction can no longer be ignored.We show that wave packet evolution of the quantum particle probes para-metric regimes in the optical lattices which support classical period, quantummechanical revival and super revival phenomena. The analytical formalismdeveloped for the two regimes, namely, deep optical lattice and shallow opti-cal lattice. Parametric dependencies of energy spectrum and classical period,revival time and super revival are explained for the two regimes.The inherent non-linearity of the condensate due to inter-atomic interac-tions and Bragg scattering of the matter-wave by an optical lattice play theirrole in the condensate dynamics. The condensate shows anomalous disper-sion behavior at the edges of a Brillouin zone of the lattice and magnitudeof this dispersion can be controlled by tuning the lattice depth. Therefore,the condensate spreading can either be controlled by actively controlling thelattice parameters or by utilizing the interaction between atoms.The dynamics of condensate in driven optical lattice crystal are analyzedby studying dynamical stability of the condensate. The stability is deter-mined by the dispersion behavior of the condensate excited in driven opticallattice for a variable range of modulation and interaction parameters.vThe recurrence behavior of the condensate is analyzed as a function oftime close to the nonlinear resonances occurring in the classical counterpart.Our mathematical formalism for the recurrence time scales is presented as: delicate recurrences which take place for instance when lattice is weakly per-turbed; and robust recurrences which may manifest themself for sufficientlystrong external driving force. The analysis is not only valid for dilute con-densate but also applicable for strongly interacting homogeneous condensateprovided, the external modulation causes no significant change in densityprofile of the condensate. We explain parametric dependence of the dynam-ical recurrence times which can easily be realized in laboratory experiments.In addition, we find a good agreement between the obtained analytical resultsand numerical calculations.The stability of condensate is also explored in driven optical lattice numer-ically for rang of external parameters i.e. modulation strength, atom-atominteraction and lattice potential. ontents ii Appendices 131
Appendix A: Solution of an Arbitrary Potential . . . . . . . . . . . 131Appendix B: Anger function and distribution functions . . . . . . . 135
Bibliography 137List of Publications 162Acknowledgements 163 hapter 1Introduction
The discovery of light pressure has a long history dating back to Kepler ashe surmised that light exerts pressure on comets. Maxwell calculated ra-diation pressure in the framework of his own electromagnetic field theoryin 1873. First time, the light force on a thin metallic plate was experimen-tally measured by Lebedev [Lebedev 1901] followed by similar experiment byNicols [Nicols 1903]. Later, Lebedev [Lebedev 1910] tried to measure radia-tion pressure on gas molecules exerted by resonant field. Kapitza and Dirac[Kapitza and Dirac 1933] showed that a standing light field behave like adiffraction grating for electrons.The interaction of the external degree of freedom of an atom with electro-magnetic wave fields was observed first time in 1930, when Frisch measuredthe deflection of an atomic beam with resonant light from a sodium lamp[Frisch 1933]. The physical phenomenon responsible for mechanical action ofradiation is the momentum and energy exchange between field and particle[Einstein 1917]. The deflection is due to the recoil momentum, an atom gainswhile absorbing or emitting a single photon of light. When an atom absorbsa photon from a beam of light, it gains momentum in the direction of thefield. Since emission of photons is without preferred direction, the momen-tum acquired during the emission averages to zero over many cycles. Thisleads to a net force on the atom which is called the spontaneous force, orradiation pressure. The electric field induces a dipole moment in the atom. Ifthe laser is detuned to the red side of the atomic transition then the induced1 h 1. Introduction ψ ( x ) = | ψ ( x ) | exp [ iφ ( x )] h 1. Introduction | ψ ( x ) | gives the superfluid density, and φ ( x ),which is the phase of this wave function determines the superfluid velocity, ~ /M ∇ φ ( x ), here, M is mass of an atom and ~ is Planck’s constant. Thisis similar [Langer 1968] to the description of laser light by a coherent state[Glauber 1963] and is equally valid to the standard bosonic condensates andweakly bound fermion pairs, building blocks of the Bardeen-Cooper-Schrieffer(BCS) picture of super-fluidity in fermionic systems. The macroscopic wavefunction description in dilute gases is more simple than the conventionalsuper-fluids such as He, as the macroscopic wave function in later case pro-vides only a phenomenological description of the superfluid degrees of free-dom. In the presence of weak interactions, the BEC’s of dilute gases are es-sentially pure condensates sufficiently below the transition temperature. Themacroscopic description of wave function thus, has direct connection with themicroscopic degrees of freedom and provides a complete picture of static andtime dependent phenomena in nonlinear Schr¨odinger wave equation, knownas Gross-Pitaevskii equation [Gross 1961; Pitaevskii 1961]. Hence, the manybody effects of a BEC in dilute gases can effectively be described by single-particle wave packet and interactions can be understood as an additionalpotential which is proportional to the local particle density. An additionof small fluctuations around the zeroth order solutions develops a theoryof weakly interacting Bose gases known as Bogoliubov theory. Many bodyproblem then completely be solved as a set of non-interacting quasi-particleslike the closely related BCS-superfluid of weakly interacting fermions. Ul-tracold dilute gases serve as a reliable tool to understand the many bodyphysics and many characteristic effects have indeed been verified quantita-tively [Bloch et al 2008].In the last few years, two major experimental breakthrough were ob-served: first, the capability to control interaction strength in cold gasesthrough magnetic [Inouye et al 1998; Courteille et al 1998] or optical[Theis et al 1998] Feshbach resonances [Chin et al 2010] both for bosons orfermions; second, the ability to change the dimensionality with optical lat-tices, particularly to produce periodic potentials through the optical lattices. h 1. Introduction a s faces a strong decrease in the lifetime of the condensate caused by threebody losses and these losses are proportional to a s [Fedichev et al 1996;Petrov 2004]. The set back was encountered in strong correlated regimeby using a different technique [Greiner et al 2002a] to avoid the problem ofcondensate lifetime. The quantum phase transition from a superfluid to aMott-insulating phase was observed even in the regime where the averageinter-particle spacing is longer than the scattering length. On the otherhand, the strong confinement option with optical lattices opened the op-portunity to study the low dimensional systems with the possibility of theemergence of new phases. The first realization of a bosonic Luttinger liquid h 1. Introduction Cr, BEC’s withstrong dipolar interactions have been realized [Griesmaier et al 2005]. Inshort, combination of Feshbach resonances [Chin et al 2010] and dimension-ality control opens the ways to tune the nature and range of the interaction[Lahaye et al 2007], which enables to reach novel many body states accessibleeven in the fractional Quantum Hall effect.In Fermi gases, three body losses are strongly suppressed by Pauli ex-clusion principle. The rate of suppression decreases with increasing scat-tering length [Petrov et al 2004a] and Feshbach resonances enable to ex-plore the strong coupling regime in ultra-cold Fermi gases [OHara et al 2002;Bourdel et al]. Especially, there exist stable molecular states of weakly boundedfermion pairs in highly excited vibrational states [Strecker et al 2003][Cubizolles et al 2003]. The extra-ordinary stability of fermions near Fesh-bach resonances explore the transition phenomenon from a molecular BEC toa BCS-superfluid of weakly bound Cooper-pairs [Regal et al 2004; Zwierlein 2004].Particularly, the presence of pairing due to many body effects has beenprobed by spectroscopy of the gap [Chin et al 2004], or the closed chan-nel fraction [Partridge et al 2005] while super-fluidity has been verified bythe observation of quantized vortices [Zwierlein et al 2005]. These studieshave also been extended to Fermi gases for the spin down and spin up com-ponents [Zwierlein et al 2006; Partridge et al 2006], where, the difference inthe respective Fermi energies suppresses the pairing.The fermions with repulsive interaction and confined in an optical lat-tice, can be realized as an ideal and tunable version of the Hubbard model,a paradigm to understand strong correlation problems in condensed matterphysics. Experimentally, some of the fundamental properties of degeneratefermions in optical lattices like the Fermi surface existence and a band in-sulator with unit filling [K¨ohl et al 2005] have been observed. Since, it is h 1. Introduction h 1. Introduction s -wave scattering length can be tuned close h 1. Introduction super Bloch oscillations, withcondensates [Ivanov et al 2008; Gustavsson et al 2008; Alberti et al 2009][Halleret al 2010].In recent days, cold atoms in driven optical lattices, with an aim to co-herently control the matter waves, is emerging as a new area of researchboth in theoretical and experimental matter wave optics. We observe var-ious dynamical modes in a system modulated by time periodic forcing. Inthe corresponding classical systems, the stable nonlinear resonances are im-mersed in stochastic sea and the system may display global stochasticity be-yond a critical value of coupling or modulation strength [Lichtenberg 1992;Lichtenberg 2004; Ott 1993]. In the case of spatially periodic potentialsdriven by periodic force, the classical counterpart of the dynamical sys-tem displays dominant regular dynamics and dominant stochastic dynam-ics, one after the other, as a function of increasing modulation amplitude[Raizen 1999]. Due to spatial and temporal periodicity in the driven opticalcrystal, the quantum dynamics of a particle inside a nonlinear resonance iseffectively mapped on the Mathieu equation. For a general quantum sys-tem driven by a periodically time dependent external force, the Hamiltonianoperator is time periodic H ( t ) = H ( t + T ) where, T is period of external de-riving force. The discrete time translation t → t + T symmetry validate theFloquet formalism [Floquet 1883]. Floquet theory is an elegant formalismto study periodically driven system. In periodically driven systems, Floquetanalysis gives quasi-eigen states and quasi-energy eigen values.In such systems the role of the stationary states is taken over by theFloquet states [Zeldovich 1966; Ritus 1966]. If the operator H := H ( t ) − i ~ ∂ t , (1.1)acting on an extended Hilbert space of T -periodic functions [Sambe 1973],has T -periodic eigenfunctions, i.e., if the eigenvalue equation[ H ( t ) − i ~ ∂ t ] u α ( t ) = ε α u α ( t ) , (1.2) h 1. Introduction u α ( t ) = u α ( t + T ),then the Floquet states ψ α ( t ) := u α ( t ) exp( − iε α t/ ~ ) , (1.3)are solutions to the time-dependent Schr¨odinger equation. The eigenvalueEq. (1.2), now plays a role analogous to that of the stationary Schr¨odingerequation for time-independent systems, and the objective of a semiclassicaltheory is to approximate computation of the Floquet eigen-functions u α ( t )and the quasi-energies ε α , starting again from invariant objects in the clas-sical phase space.However, driven quantum systems pose some mathematical difficultiesthat are rarely met in the time-independent case. Consider a Hamiltonianoperator of the form H ( t ) = H + λH ( t ), where only H ( t ) = H ( t + T )carries the time dependence, and λ is a dimensionless coupling constant.Assume further that H possesses a discrete spectrum of eigen-values E n ( n = 1 , , . . . , ∞ ), with corresponding eigen-functions ϕ n . Then for λ = 0the Floquet states can be written as ψ ( n,m ) ( t ) = (cid:0) ϕ n e imω m t (cid:1) exp[ − i ( E n + m ~ ω m ) t/ ~ ] , (1.4)with ω m = 2 π/T . If m is an integer number, the Floquet function u ( n,m ) ( t ) = ϕ n e imω m t is T -periodic, as required and corresponding quasi-energies are ε ( n,m ) = E n + m ~ ω m . The index α in Eq. (1.2) thus becomes a doubleindex, α = ( n, m ), and the quasi energy spectrum is given by the energyeigenvalues modulo ~ ω m which can be mapped into first Brillouin zone ofwidth ~ ω m . For the Hermitian operator H ( x, t ), it is convenient to introducethe composite Hilbert space R ⊗ τ composed of the Hilbert space R of squareintegrable function on configuration space and space τ of functions periodicin time with period T = 2 π/ω m [Sambe 1973]. For the spatial part, the innerproduct is defined as < φ n | φ m > = Z dxφ ∗ n ( x ) φ m ( x ) = δ n,m . (1.5) h 1. Introduction < t | n > ≡ exp( imω m t ) , where, n = 0 , ± , ± , ± , ...... and innerproduct in τ is read as 1 T Z T dxφ ∗ n ( x ) φ m ( x ) = δ n,m . (1.6)Thus the eigen vectors of H obey the ortho-normality condition in the com-posite Hilbert space too.Floquet state formalism has been applied to a number of time-dependentproblems: from coherent states of driven Rydberg atoms [Vela-Arevalo 2005],chaotic quantum ratchets [Hur 2005], electron transmission in semiconduc-tor hetero-structures [Zhang 2006], selectively suppressing of tunneling inquantum-dot array [Villas-Bˆoas et al 2004] to frequency-comb laser fields[Son and Chu 2008].Time periodic modulation in matter wave optics have given birth bothto hybrid nano-opto-mechanical systems [Steinke et al 2011] and driven bil-liards [Leonel et al]. It was pointed out earlier that a metal-insulator tran-sition of ultra-cold atoms in quasi-periodic optical lattices is controllable byadjusting the amplitude of a sinusoidal external forcing [Eckardt 2005]. Theexperimental work in this direction gained the momentum recently, withthe novel observation of dynamical suppression of tunneling, and even re-versal of the sign of the tunneling matrix element induced by shaken op-tical lattices [Lignier et al 2007; Eckardt et al 2009]. An analog of photon-assisted tunneling with Bose-Einstein condensates in driven optical crystalshas been observed [Sias et al 2008]. Experimental demonstration of coher-ent control of superfluid to Mott-insulator transition of the matter wave inan optical lattice [Zenesini et al 2009], is also verified the theoretical pro-posal [Eckardt 2005]. The same principle reveals that this form of coherentcontrol has recently been explored successfully for frustrated magnetism indriven triangular optical lattices [Struck et al 2011]. Moreover, demonstra-tion of control over correlated tunneling is first time realized experimentallyin sinusoidally driven optical lattices [Chen et al 2011]. h 1. Introduction Outline:
In this thesis,first, we discuss center of mass dynamics of single particle wave packet inoptical crystal. This situation is attained in experiments with dilute conden-sates, or condensates for which the inter-particle s -wave scattering length h 1. Introduction chapter 2 , center of mass dynamics of single particle wave packet are dis-cussed in the absence of external modulation discussed. This chapter mostlyconsists of our published work in Ref: [Ayub et al 2009].In chapter 3 , for the sake of completeness, a tour d’horizon of the role ofinteraction in the dynamics of the condensate is given from the already pub-lished work. With numerical simulation, it is also shown that how we canexplain the nonlinear phenomena like solitons, truncated Bloch waves (self-trapped states) and solitonic trains by studying spatiotemporal behavior,position and momentum dispersion, and wave packet revivals of condensatein deep and shallow lattice in different nonlinear regimes.The main results of the thesis are discussed in chapter 4 , and chapter 5 .In chapter 4 , single particle wave packet dynamics is discussed in the pres-ence of external modulation. This chapter mostly consists of our publishedwork in Ref: [Ayub et al 2011]. In this chapter, we find analytical expres-sion for wave packet revivals in deep and shallow case. The spatio-temporaldynamics, position and momentum dispersion and auto-correlation confirmour analytical results numerically.In chapter 5 , we consider nonlinear dynamics of Bose-Einstein condensatein optical lattices in the presence of periodic external drive. Stability ofcondensate in driven optical lattice is studied versus nonlinear interactionand modulation numerically. Later, spatio-temporal behavior of the con-densate in optical lattice is studied. Chapter 5 consists of our published[Ayub and Saif 2012] and work in process of publication [Ayub and Saif 2012a].Results are discussed and concluded in chapter 6 . hapter 2Cold atoms in optical lattices The physical phenomenon responsible of mechanical action of radiation is themomentum and energy exchange between field and particle [Einstein 1917].The deflection is due to the recoil momentum, an atom gains when absorbingor emitting a single photon of light. When an atom absorbs a photon froma beam of light, it gains momentum in the direction of the field beam. Sinceemission of photons is without preferred direction, the momentum acquiredduring the emission averages to zero over many cycles. This leads to a netforce on the atom which is called the spontaneous force, or radiation pressure.The spontaneous force scales with the scattering rate and for large detuningfalls off quadratically with the detuning δ L , between the atomic transitionfrequency and the field frequency [Cohen-Tannoudji et al 1992b] F spont ∝ Iδ L , (2.1)here, I is laser intensity. There is another force named dipole force, whichis based on the coherent atom photon interaction. The electric field of lightinduces a dipole moment in the atom. If the induced dipole moment is inphase with the electric field, the interaction potential lowers in regions ofhigh field and the atoms experiences a force directing to those regions. Ifdipole moment is out of phase by π , a force pointing away from regions ofhigh field is experienced by the atom, consequently forced to the regions of13 h 2. Cold Atoms in Optical Lattices δ L , fromthe atomic resonance in the limit of large detuning, such that F dipole ∝ ∇ Iδ L . (2.2)The role of two forces is very important in manipulating and confiningneutral atoms. The spontaneous force F spont , used to cool atomic sample[J Holland et al 1999] while, dipole force is exploited to trap the atoms. Theexpressions of dipole and spontaneous forces show that with sufficient de-tuning the spontaneous force can be made negligibly very small while stillkeeping an appreciable dipole force. In this section, we derive the effective Hamiltonian for a two-level atomsinteracting with a far-detuned classical monochromatic standing wave field,ˆ E ( x, t ) . In the semi-classical derivation, we treat the electromagnetic fieldclassically, whereas, the atom is treated quantum mechanically with a groundstate | g i , and an excited state | e i , separated in energy by ~ ω o . The interactionof the atom with laser field ˆ E ( x, t ) is governed by the Hamiltonian whichconsists of three contributions, that is,ˆ H = ˆ H cm + ˆ H internal + ˆ H interaction , (2.3)where, ˆ H cm = ˆ p x M , (2.4)ˆ H internal = 12 ~ ω o σ z , (2.5)ˆ H interaction = − ˆ d · ˆ E ( x, t ) , = − ( h e | ˆ d. ˆ E | g i σ + + h g | ˆ d. ˆ E | e i σ − ) . (2.6)Here, σ ± are atomic raising and lowering operators, σ z is the Pauli spinmatrix and ˆ d is dipole moment. For linear polarization of electromagneticfield in y-direction, resonant Rabi frequency is defined asΩ = − h e | ˆ d. ˆ E | g i ~ = − h g | ˆ d. ˆ E | e i ~ = − h e | d. ˆ e y | g i ~ E. (2.7) h 2. Cold Atoms in Optical Lattices e y , is the unit vector along y-direction and we have applied dipoleapproximation which implies that amplitude, | ˆ E ( x, t ) | varies slowly on atomicsize scale. Counter propagating laser beams overlap spatially to create an optical lattice.We consider two linearly polarized counter propagating beams having samewave number k L and their polarization vectors are parallel. The electric filedfor the optical lattice isˆ E ( x, t ) = ˆ e y [ ε o cos( k L x ) e − iω L t + c.c. ] , (2.8)where, ω L , ε o are frequency and amplitude of laser field, respectively.The interaction Hamiltonian is modified as H interaction = ~ Ω ε o cos ( k L x ) e − iω L t σ + + H.c. (2.9)In last expression, we have used rotating wave approximation to eliminatethe counter rotating terms σ + e iω L t and σ − e − iω L t [Loudon 1983]. To separatethe centre of mass motion of atom and their internal states, we write theatomic state as | Ψ ( x, t ) i = Ψ g ( x, t ) | g i + Ψ e ( x, t ) e − iω L t | e i . (2.10)The Schr¨odinger equation for the physical system is − i ~ ∂∂t | Ψ ( x, t ) i = ˆ H | Ψ ( x, t ) i . (2.11)By substituting the Hamiltonian, ˆ H , in Eq. (2.3), and | Ψ i in above equationand later taking projection onto internal states | e i and | g i , we get two coupledequations, viz., i ~ ∂ Ψ g ∂t = − ~ M ∂ Ψ g ∂x − ~ Ω2 cos( k L x )Ψ e , and i ~ ∂ Ψ e ∂t = − ~ M ∂ Ψ e ∂x + ~ δ L Ψ e − ~ Ω2 cos( k L x )Ψ g . (2.12) h 2. Cold Atoms in Optical Lattices δ L = ω − ω L , spontaneous emis-sion can be neglected [Graham 1992]. Furthermore, the probability to findan atom in the excited state is negligible as a consequence of large detuning,the evolution properties of the atom in the field are completely determinedby the ground state amplitude which effectively describes the evolution ofthe atom in the electromagnetic field in its ground state, i ~ ∂ Ψ g ∂t = − ~ M ∂ Ψ g ∂x − ~ Ω eff ( k L x )Ψ g , (2.13)where, Ω eff = Ω /δ L is the effective Rabi frequency. The dynamics of anatom in its ground state, with energy shift ~ Ω eff , are governed by the effectiveHamiltonian, that is, H = p x M − V k L x ) , where, V = ~ Ω eff is an effective lattice potential amplitude. When the laserfield is red detuned to the atomic transition ( δ L > δ L < h 2. Cold Atoms in Optical Lattices I ,and the lattice depth, V , is expressed as [Dahan et al 1996; Morsch 2006] as V = ˜ ξ ~ II s Γ s δ L , (2.14)where, ˜ ξ is a correction related to level structure, I s is saturation intensityand Γ s is photon scattering rate [Grimm 2000]. The center of mass (CM) dynamics of an atomic wave packet in the presenceof a standing wave field for sufficiently large detuning is effectively controlledby the time independent Schr¨odinger wave equation, − ~ M ∂ ψ ( x ) ∂x − V k L x ) ψ ( x ) = E n ψ ( x ) , here, for simplicity, we have taken h x | Ψ g ( x, t ) i = e − iEt ψ ( x ). Substituting¯ x = k L x − π in above equation we find − ∂ ψ (¯ x ) ∂x + V E R cos(2¯ x ) ψ (¯ x ) = E n E R ψ (¯ x ) . where, E R = ~ k L M is recoil energy gained by the atom as it annihilates oremits a photon. The above equation is standard Mathieu equation, that is, ∂ ψ n ( x ) ∂x + [ a n − q cos(2 x )] ψ n ( x ) = 0 . (2.15)Here, and onward, for simplicity, the bar on variable x is dropped. We findMathieu characteristic parameters q = V E R , (2.16) Here, Mathieu characteristic parameter q and a n are scale by recoil energy and scalingis different than in [Ayub et al 2009]. We use symbol q for stationary optical lattice andsymbol q is reserved for driven optical lattice. h 2. Cold Atoms in Optical Lattices a n = E n E R . (2.17)Mathieu characteristic parameter q is scaled optical lattice potential and a n is scaled energy of n th lattice band both expressed in the units of recoilenergy.One of the most important characteristics of periodic function is emer-gence of band structure. The periodicity of potential dictates that eigenstates ψ n ( x ) can be chosen to have the form [Ashcroft and Mermin 1967] ψ n ( x ) = e ikx µ n ( x ) , (2.18)with periodicity condition, ψ n ( x ) = ψ n ( x + d ) , (2.19)where, k is quasi momentum, d =
12 2 πk L = λ L is periodicity of optical lattice.Energy bands, and band gaps as a function of Mathieu characteristic pa-rameters q (scaled potential depth) are shown in Fig-2.2. Even Mathieusolutions, a m are blue curves and odd Mathieu solutions, b m are red curvesmostly overridden by blue curves. The dotted lines correspond to character-istic parameter a n = ± q i.e., E = ± V /E R . Note that for a given valueof q where q ≫ a n , the gap between the lowest energy state (lowest solidcurve) is roughly one-half of the spacing between solid and dashed curves,corresponding to the zero-point energy in the oscillator like limit.Eqs. (2.15-2.17) state that for a fixed value of q , there are countablyinfinite number of solutions, labeled by n . However, only for specific valuesof the parameter a n , the solutions will be periodic, with periods π or 2 π inthe variable x, which are denoted by a n , and b n , respectively for the even andodd solutions. Because of the intrinsic parity of the potential, the solutionscan be characterized as being even, ce n ( x, q ) or cosine-like for integral valuesof n , with n ≥ . Whereas, they are odd, se n ( x, q ) or sine-like for integralvalues of n , with n ≥
1. Limiting cases i.e., q = 0 , q . q ≫ h 2. Cold Atoms in Optical Lattices a m (for evensolutions, gray (on-line colour blue) curves) and b m (for odd solutions, dark(on-line colour red) curves, mostly overridden by blue curves) versus q for thequantum pendulum: The dotted lines correspond to characteristic parameter a n = ± q i.e., E = ± V /E R . Note that for a given value of q where q ≫ a n , the gap between the lowest energy state (lowest solid curve) is roughly one-half of the spacing between solid and dashed curves, corresponding to thezero-point energy in the oscillator like limit.Atom in an optical lattice may observe deep or shallow potential depthscorresponding to its energy. The atom observes a deep optical lattice poten-tial if effective lattice depth V is of the order of a few hundred single photonrecoil energies and temperature of the atom is around recoil temperature.In this case the dynamics of the quantum particle in the individual well isindependent and one obtains multiple realization of an-harmonic oscillators.On the other hand when the depth of the potential is just few recoil energiesand atom is at about recoil temperatures, it sees a shallow potential. In thiscase, the quantum mechanical effects caused by spatial periodicity of opticallattices such as formation of Bloch waves become important. Furthermore,the levels are broadened into bands due to resonant tunneling between adja- h 2. Cold Atoms in Optical Lattices V . A moderate values of V ,with an effective Plank’s constant of order unity, indicates the deep quan-tum regime. The semi-classical dynamics of the atom in the standing wavefield are observed however for large values of V and here, we find severaltightly bound energy bands. Quantum mechanical effects become importantonce the atomic de Broglie wavelength π ~ P significantly exceeds the latticeconstant d .Approximate expressions for the values of a n and b n both in q . q ≫ The regime in which the gap energy between lowest and first excited bandis larger than the width of the lowest band is usually referred as ”tight bindregime”. As linear Bloch waves are strongly localized in deep lattice potential(see Fig-2.4). In the tight binding limit
E << V , analytical results can easilybe derived. Following the Wannier states description, the time-dependentwave function for a state in the lowest energy band is written as ψ ( x, t ) = ∞ X j = −∞ φ j ( t ) ψ j ( x ) , (2.20)where, ψ j ( x ) = ψ j ( x − jd ) is the ground state of the j th lattice site and φ j ( x ) = q n j ( t ) e − iϕ j t , (2.21)is a complex function describing the amplitude, √ n j , and the phase, ϕ j , asso-ciated to the wave function in the j th site. It is safe to assume for deep latticepotential that the Wannier functions almost coincide with the ground state ofthe single potential well. The functions ψ j ( x ) are well localized at the latticesites, with a very small overlap among functions ψ j ( x ) and ψ j +1 ( x ) locatedat neighboring sites. With the assumption that next to nearest neighboring h 2. Cold Atoms in Optical Lattices φ j ( t ) iswritten as i ~ ddt φ j = − J ( φ j − + φ j +1 ) , (2.22)where, J = 8 √ √ π ( q o ) exp( − √ q o ) , (2.23)is hoping matrix element. We see that J is a function of the potential height,as the states ψ j ( x ) depend on the shape of the potential. Exact solutionsof Eq. (2.22) are the Bloch waves, in which the complex functions φ j ( t ) = e jkd − iEt/ ~ replaces the plane waves. In this case the phase difference betweenneighboring sites ∆ φ = φ j +1 − φ j = kd is constant across the entire latticeand dependent on the quasi-momentum k . Introducing this Bloch ansatzin Eq. (2.22), we find an analytical expression for the shape of the lowestenergy band [Kevrekidis 2009] E = ǫ − J cos ( kd ) , (2.24)where, ǫ is a constant energy. In Fig-2.3, lowest energy bands are shown for V = 2 E r (a) and V = 8 E r (b). Width of first excited band is indicated byblue arrows. Band gap between lowest and first excited band for V = 2 E r is shown in red arrows. The band width of lowest band, band gaps betweenlowest and first excited band and hopping matrix elements are shown inTable:2.1. From Fig-2.3 and Table-2.1, we see that as the lattice potential isincreased, band width of lower band and hopping matrix elements decreases,while, band gap between ground and first excited band increases. In this section, we discuss the energy spectrum and wave packet dynamicsof a single particle in deep optical lattice. An optical lattice is considereddeep when the hopping matrix element J , for next to nearest lattice sites isnegligible. In table-2.1 band width of lowest band, band gap between lowest h 2. Cold Atoms in Optical Lattices -1 0 148 -1 0 148121620 band gap E / E r band w i d t h V = r V = r k/k L Figure 2.3: Band characteristics of an optical lattice for V = 2 E r (a) and V = 8 E r (b).band and first excited band, nearest neighboring hopping matrix elements c and next to nearest neighboring hopping matrix elements, c , are shown fordifferent lattice depths. We note that band width is four times of hoppingmatrix elements for nearest neighbor lattice sites i.e., band width = 4 J . FromTable-2.1, we observe that for V = 6 E r and above, c is negligible. In limiting case, q ≫
1, the spectrum is given as a n ≈ b n +1 ≈ − q + 2 s √ q − s +12 − s +3 s √ q − .........., (2.25)where, s = 2 n + 1 . It has, thus, ( n + ) ~ ω h dependence in lower order,which resembles harmonic oscillator energy for ω h = 2 √ V . Here, in thedeep optical lattice limit, the band width is defined as [Abramowitz 1970] b n +1 − a n ≃ n +5 √ π q n exp( − √ q ) n ! , (2.26) h 2. Cold Atoms in Optical Lattices V ( E r ) band width ( E r ) band gap ( E r ) c c . . . − . . · − . . . − . . · − . . . − . . · − . . . − . . · − . . . − . . · − . . . − . . · − . . . − . . · − . . . − . . · − . . . − . . · − . . . − . . · − Table 2.1: Band width of lowest energy band, band gap between first excitedand lowest band, hopping matrix elements for nearest neighbor site c andnext to nearest neighbor site c .which shows that in the deep optical lattice ( q ≫ q near thebottom of the lattice, thin bands are seen, band width increases and bandgap decreases for excited bands with increasing band index. The hopingmatrix element, J, is related to band width in deep optical lattice case as band width = 4 J , which explains the tunneling between adjacent sites fordeep optical lattice. Eqs. (2.26) and (2.23) show that the width of the bandscorresponds to tunneling of the atom from one lattice site to the other. Inthe deep lattice potentials, this tunneling probability will be exponentiallysmall and band width therefore, reduces exponentially as a function of latticepotential. The time evolution of the particle, initially in state ψ ( x, , is obtained bytime evolution operator ˆ U , such that, ψ ( x, t ) = ˆ U ψ ( x,
0) = ∞ X n =0 c n φ n ( x ) exp( − i E n ~ t ) , (2.27) h 2. Cold Atoms in Optical Lattices E n , and φ n ( x ) , are energy eigen values and eigen states correspond-ing to quantum number, n . The probability amplitudes, c n , are definedas, h φ n ( x ) | ψ ( x, i . The quantum particle wave packet in optical potential,narrowly peaked around a mean quantum number ¯ n, displays quantum re-currences at different time scales, defined as T ( j ) = 2 π ( j ! ~ ) − E jn | n =¯ n , (2.28)where, E jn denotes the j th derivative of E n with respect to n . The time scale, T (1) , is termed as classical period as it provides a time at which the particlecompletes its evolution following the classical trajectory and reshapes itself.Whereas, at T (2) the particle reshapes itself as a consequence of quantuminterference in a nonlinear energy spectrum, which is purely a quantum phe-nomenon and thus named as quantum revival time. In the parametric regimeof a changing non-linearity with respect to quantum number, n , we find, thesuper revival time T (3) for the quantum particle [Leichtle et al 1996].Around the minima of lattice sites, harmonic evolution prevails and inthe presence of the higher order terms it gradually modified to the originalpotential. Microscopic investigation of the atom-optical system, using termby term contribution of cosine potential expansion reveals the dominant roleof the system’s particular parametric regime in the formation of eigen-statesand eigen energies. This leads to simplified analytical solutions around thepotential minima in the system, as discussed below. In the deep lattice limit, the potential near the minima can be approximatedas quadratic. Thus the particle placed near the minima of the lattice poten-tial, experiences a harmonic potential. The energy in this regime is obtainedas, E (0) n = (2 n + 1) ~ √ V − V , which can be identified in Eq. (2.25), by ig-noring square and higher powers in s. The eigen states of quadratic potentialare given as φ n ( x ) = q β n n ! √ π H n ( βx ) exp( − β x ) , where, H n ( βx ) are Hermitepolynomials and β = ( √ V ~ ) = √ q . h 2. Cold Atoms in Optical Lattices | A ( t ) | = ∞ X n =0 | c n | + 2 ∞ X n = m | c n | | c m | cos[( E n − E m ) t ~ ] . In the present parametric regime the square of the autocorrelation functionis written as | A ( t ) | = ∞ X n =0 | c n | + 2 ∞ X n = m | c n | | c m | cos(( n − m )2 p V t ) , (2.29)where, ∞ P n =0 | c n | is independent of time, and defines the interference free,averaged value of | A ( t ) | . The most dominant contribution to the | A ( t ) | comes from n − m = 1 in the second part at the right side of Eq. (2.29). Otherterms with, m − n > , have negligible role because their oscillation frequencyis an integral multiple of fundamental frequency 2 √ V , and are averaged outto zero. For the reason the square of the autocorrelation function in thepresent case oscillates following a cosine law with a frequency 2 √ V whichleads to a classical time period T (0) cl = π √ V as shown in Fig-2.4. Here, zeroin the superscript of T (0) cl defines the system’s classical period in the absenceof perturbation. At the integral multiples of classical period, | A ( t ) | is unity,whereas, at time, which is odd integral multiple of the half of the classicalperiod | A ( t ) | = ∞ X n =0 | c n | + 2 ∞ X n = m | c n | | c m | cos[( n − m ) π ] . (2.30)Here, cos[( n − m ) π ] has alternatively values +1 and − , when n − m becomeseven or odd, respectively. Thus, after the cancellation of positive terms withthe negative ones, second summation reduces to minimum value and | A ( t ) | attains its minima. We may write ψ ( x, t ) = exp[ − i (2 p V − V ~ ) t ] ∞ X n =0 c n φ n ( x ) exp( − i n p V t ) , (2.31)it is notable that the eigen states φ n ( x ) have parity ( − n . In case of evenparity, only the even terms c n are non vanishing and n -dependent exponent h 2. Cold Atoms in Optical Lattices π . In this case we see classical revivals of the initial atomic wave packet andquantum revivals take place at infinitely long time. Hence, we find revival ofthe atomic wave packet after each classical period only. Experimentally wemay realize the situation by placing very few recoil energy atoms deep in thecosine potential well. This reveals the information about the level spacingaround the bottom of the cosine potential. Interestingly we find an equalspacing between the energy levels from Fig-2.2, for large q and small valueof n . The spatio-temporal behavior of the wave packet in the quadraticpotential, as shown in Fig-2.4, confirms the above results.Figure 2.4: Time evolution of particle wave packet placed at the bottomof the cosine potential. The dimensions of the wave packet are ~ = 0 . , ∆ p = 0 . V = 10 . We show autocorrelation function vs time (right side)and spatio-temporal behavior of the material wave packet (left side). Thewave packet see equally spaced energy levels and rebuilds after every classicalperiod. Analytically calculated value of classical period and numerical resultsshow an excellent agreement.
Beyond harmonic oscillator limit, we find oscillator with non-linearity andenergy level spacing different from a constant value. The correction to theenergy of harmonic oscillator comes from the first order perturbation (see h 2. Cold Atoms in Optical Lattices E (4) n = − ~ (2 n + 2 n + 1) , whichagain can be identified in Eq. (2.25) by ignoring cubic and higher orderpowers in n . The atoms with little higher energy, which is equivalent toseveral recoil energies observe another time scale in which it reconstructsitself beyond classical period, i.e., quantum revival time. The behavior ofauto-correlation function for the wave packet exactly placed in this region,where only first order correction is sufficient, is shown in Fig-2.5. From thisfigure, we note that the wave packet displays revivals at quantum revivaltime. Thus, little above from the bottom of the of an optical lattice, thewave packet dynamics are modified due to nonlinear energy spacing. Theclassical time is modified as T (1) cl = α (1) T (0) cl , where α (1) = 1 + ¯ s √ q , andclassical periodicity for a particle in present situation is related to potentialheight and mean quantum number, here, ¯ s = 2¯ n + 1 . As ¯ n increases, classicalrevival time also increases, and the ratio, ¯ s √ q , is always less than unity inthe region where first correction is sufficient. The quantum revival time, T (1) rev = π ~ , is independent of mean quantum number ¯ n, whereas super revivaltime in this case is infinite. t |A(t)| Figure 2.5: Autocorrelation function for a particle undergoing quantum re-vival evolution in time. The parameters are same as in Fig-2.4. The wavepacket was placed close to the bottom of the potential well in the regimewhere first order correction is sufficient, it observes quantum revivals aftermany classical periods. h 2. Cold Atoms in Optical Lattices φ q n ( x ) = φ n ( x ) + φ (1 a ) n ( x ), where, φ (1 a ) n ( x ) is first order correction tothe harmonic oscillator wave function and is defined as φ (1 a ) n = D ( η φ n − + η φ n − − η φ n +2 − η φ n +4 ) . (2.32)Here, D , η , η , η and η are constants and defined in Appendix-A.In Fig-2.6, eigen states of quadratic, quartic, sixtic and octic oscillatorsare mapped on numerically obtained eigen states of cosine potential. Inthis figure for V = 10 and ~ = 1, first order correction to the eigen statesof unperturbed system matches with harmonic oscillator eigen states up to n = 3 , and mapping of quartic oscillator with exact solution is much improvedcompare to harmonic oscillator. Similarly mapping of sixtic oscillator isbetter than quartic oscillator and is quite good for octic oscillator for allbands with energy less than potential depth. In this case eight bands existin side the potential. From Fig-2.2, we note that as q increases, number ofbound bands also increases. Quantum Number
Harmonic Oscillator Quartic oscillator Sixtic OscillatorOctic Oscillator S Figure 2.6: comparison of cosine potential with the simplified potentials ismade by calculating the projection, S, of the eigen states of the cosine poten-tial on the eigen states of the simplified potentials . For first few quantumnumbers, the cosine potential very much resembles to the harmonic poten-tial, however, for higher quantum numbers the higher order corrections tothe harmonic oscillator are needed to make the resemblance. h 2. Cold Atoms in Optical Lattices | A ( t ) | = ∞ X n =0 | c n | + 2 ∞ X n = m | c n | | c m | cos[( n − m )2 p V t + ( m − n )( n + m + 1) ~ t ] . (2.33)Here, nonlinear dependence of energy eigen values on quantum number, n ,makes the argument of cosine function non-linear. The nonlinear term ( m − n )( n + m + 1) removes a degeneracy present for the harmonic case betweenthe cosine waves corresponding to nearest neighboring off diagonal termsand beyond. Hence, there overall evolution display a gradual dispersionleading to collapse which latter transforms in revival as the dispersion inwaves disappears. The values of | A ( t ) | in this regime at T rev = π ~ is simplifiedas | A ( t ) | = ∞ X n =0 | c n | + 2 ∞ X n = m | c n | | c m | cos[16 π √ q ( n − m )] . (2.34)From the Eq. (2.34), we infer that | A ( t ) | = 1 , when √ q is an integralmultiple of . In case √ q is not an integral multiple of , the wave packetrevival occurs little earlier than the revival time T rev = π ~ and also | A ( t ) | approaches to unity little earlier than T rev . Here, reconstruction of the wavepacket at T rev is out of phase by π, i.e., at that time all the waves are movingexactly in opposite directions compare to initial direction. But at T rev theyare all moving in the same direction and each wave is in phase not only withthere initial states but also with each other.Fig-2.7 shows spatio-temporal behavior of the wave packet in lattice po-tential. We see that wave packet spreads and oscillates in the lattice well,and after some time the original wave packet is divided into sub-wave-packetsFig-2.7-b. At a quantum revival time these sub-wave-packets constructivelyinterfere and the wave packet gains the original shape at the same initialposition Fig-2.7-c. At quantum revival time, the evolution is the same as atthe start of the time evolution shown in Fig-2.7-a. h 2. Cold Atoms in Optical Lattices T rev . Higher order non-linearity in the energy spectrum of the quantum pendulumshow up beyond quartic limit. In the presence of the second correction, theenergy is modified by the term E (6) n = − ~ √ V (2 n +3 n +3 n +1)32 . Second correctionto the energy modifies the time scales. The classical time period is now T (2) cl = α (2) T (0) cl , whereas the quantum revival time is modified as T (2) rev = | β (1) | T (1) rev . Here, α (2) = α (1) + s +1)2 q , β (1) = s q − T (2) spr = π √ V ~ , at whichreconstruction of original wave packet takes place. The other quantum revivaltimes occur at infinity.The energy eigen states in this regime are given as φ ( s ) n = φ n + φ (1 ,a ) n + φ (1 ,b ) n + φ (2 ,a ) n , where, φ (2 ,a ) n is perturbation in eigen states due to the H (6) termand φ (1 ,b ) n is second order perturbation caused by H (4) term [Doncheski 2003].The expressions φ (1 ,b ) n and φ (2 ,a ) n are given as φ (1 ,b ) n = D [ δ φ n − + δ φ n − + δ φ n − + δ φ n − + δ φ n +2 + δ φ n +4 + δ φ n +6 + δ φ n +8 ] , (2.35) h 2. Cold Atoms in Optical Lattices φ (2 ,a ) n = D [ χ φ n − + χ φ n − + χ φ n − + χ φ n +2 + χ φ n +4 + χ φ n +6 ] . The constants D , D , δ j ’ s and χ j ’ s are calculated in Appendix-A, where, j takes integral values. T r ev Mean Quantum Number
Figure 2.8: The quantum revival time vs mean quantum number is shownfor simplified potentials. In the presence of first order correction, i.e., forquartic potential, the quantum revival time is a constant. For higher ordercorrections it decreases with increasing mean quantum number.Again in this region classical time increases by increasing ¯ n and increasesfaster than as it was in quartic limit. However, in the present regime, thequantum revival time is not constant and decreases as ¯ n increases. The ¯ n dependence of quantum revival time is shown in Fig-2.8. The dependence ofrevival time on mean quantum number is associated with the non-linearityin the energy spectrum. For sufficiently deep lattice potential, near the lat-tice depth, non-linearity is almost zero and wave packet observes a linearspectrum which implies a quantum revival at infinity. However, as we moveaway from lowest energy band to the higher one, non-linearity emerges and h 2. Cold Atoms in Optical Lattices n increases. This behaviorof wave packet revival can be seen if we are away from the top of the latticepotential as well, where, continuum in energy is seen due to overlap of energybands and classical limit in reached at which recurrence behavior is expectedto vanish. The super revival time is independent of mean quantum number.It is directly proportional to square root of potential height and inverselyproportional to the square of scaled Plank’s constant. The temporal behav-ior of an atom in optical potential placed in this regime shows three timescales: classical periods enveloped in quantum revivals and quantum revivalsenveloped in super revivals as shown in Fig-2.9. At each super revival time,the atomic wave packet reconstructed.Similarly third correction in energy modifies the energy so that E (8) n = − ~ V (5 n +10 n +16 n +11 n +3)2 . The time scales in this case are T (3) cl = α (3) T (0) cl and T (3) rev = | β (2) | T (1) rev . Where, α (3) = α (2) + (5¯ s +17¯ s )2 q and β (2) = β (1) + s +172 q and super revival time is T (3) spr = | γ | T (2) spr where γ = s √ q − . Furthermore,the super quartic revival time, T , is independent of ¯ n . The higher ordercorrections in energy show that other time scales do exist in the system,but their times of recurrence are too large to consider them finite. In thepresence of third correction to energy, classical time increases as ¯ n increasesbut increases little faster than in the cases of quartic and sixtic corrections,whereas, the quantum revival time decreases faster as ¯ n increases comparedto the case of sixtic correction. The super revival time is not constant butdecreases as ¯ n increases.Energy corrections increase anharmonicity in the system. We discussedthat the first order correction to harmonic potential energy led to the quan-tum revivals, the second order energy correction led to super revivals andfourth correction led to quartic revival time. Comparison of revival times fordifferent energy corrections is shown in Fig-2.8. For first order energy cor-rection quantum revival time is constant, but for higher order corrections,revival time may decrease with increasing mean quantum number of the wavepacket. h 2. Cold Atoms in Optical Lattices q = 40 justifying q >> n satisfying the condition n << √ q andfor n >> √ q , Eq. (2.36), satisfy the numerically obtained energy spectrum.The intermediate range ( n ∼ √ q ), where, energy spectrum changes its char-acter from lower to high value is estimated as n c ≈ k p q k [Rey 2005],where, q y q denotes the closest integer to y . h 2. Cold Atoms in Optical Lattices In shallow optical lattice the condition q .
1, is satisfied and next to nearesthopping matrix elements also needed to take into account. In this section, wediscuss energy spectrum and wave packet dynamics of shallow optical lattice.
In shallow optical lattices energy spectrum is given as [Abramowitz 1970;McLachlan 1947] a ( q ) = − q q
128 + . . . ,a ( − q ) ≈ b ( q ) = 1 − q − q − q . . . ,a ( q ) = 4 + 5 q − q . . . ,b ( q ) = 4 − q
12 + 5 q . . . ,a ( − q ) ≈ b ( q ) = 9 + q − q
64 + . . . ,a ( q ) = 16 + q
30 + 433 q . . . ,b ( q ) = 16 + q − q . . . ,a n ≃ b n = n + q n −
1) + ... . for n > . (2.36)The above expression is not limited to integral value of n and is a very goodapproximation when n is of the form, m + . In case of integral value of n = m, the series holds only up to the terms not involving n − m in thedenominator. The difference between the characteristic values for even andodd solutions satisfy the relation a n − b n = O ( q n n n − ) as n → ∞ . h 2. Cold Atoms in Optical Lattices In shallow lattice potential limit, i.e., q .
1, neglecting the higher orderterms in q , the classical frequency and non-linearity are given by the ex-pressions ω = 2 n { − q n − } , ζ = 2 + q n +1( n − respectively. In the case ofshallow optical potential classical period is T ( cl )0 = { q n − } πn , (2.37)where, n is band index of lattice. The quantum revival time is T ( rev )0 = 2 π { − q n + 1)( n − } , (2.38)and super revival time is expressed as T ( spr )0 = π ( n − q n ( n + 1) , (2.39)Numerically dynamics are studied by evolving initially localized Gaussianwave packet in the lattice well. Fig-2.10, shows the square of auto-correlationfunction for Gaussian wave packet with ∆ x = ∆ p = 0 .
5, placed in an opticallattice with V = 2 E r . Wave packet revivals are clearly seen but revivalamplitude decrease with the passage of time due to tunneling to the adjacentlattice sites. While, this behavior of decreasing amplitude is not presentin the case of deep optical lattice as tunneling is suppressed due to strongconfinement. The spatio-temporal behavior of the same wave packet is shownin the Fig-2.11. Revival phenomenon of wave packet along with tunneling isalso seen. In this regime next to nearest tunneling is not negligible as givenin Table-2.1. Spatio-temporal behavior behavior explains the decrease in theamplitude with each revival in Fig-2.10. In this chapter, we have extended the understanding of eigen energy levelsand eigen states in optical lattices. Two asymptotic cases namely deep opti-cal lattice and shallow lattice are focused. We note that solutions obtained h 2. Cold Atoms in Optical Lattices | A ( t ) | t/T Figure 2.10: Autocorrelation function vs scaled time for wave packet initiallyplaced in shallow optical lattice near the bottom of potential well. V = 2 E r , ∆ p = 0 . T (1) rev = π ~ ) is independent of potential height, however has inverse propor-tionality with Plank’s constant, ~ . We show that for deep optical lattice,there is a region in which revivals are independent of lattice depth and thisregion expands with the increase in lattice depth, however beyond this re- h 2. Cold Atoms in Optical Lattices V = 2 E r . Other parameters are same as in Fig-2.10.Dark regions represent maximum probability.gion quantum revival time is no longer constant but decreases as ¯ n increases.The higher order time scale, super revival time T (2) spr = π √ V ~ exists in thisregion and is independent of ¯ n . Again this region expands as potential heightis increased but super revival time in this region is directly proportional tosquare root of potential height. Above this region other time scales also existwhere, super revival time is ¯ n dependent and decreases as ¯ n increases butthese time scales are too long to consider them finite.In shallow optical lattices, quantum tunneling plays dominant role andwe encounter wide energy bands. Two energy scales, band gap and bandwidth evolve in the system. If the band gap is larger than band width, tightbinding approximations can be applied in which next to nearest tunnelingcan safely be ignored. But in situations where, lattice is very shallow, nextto nearest tunneling can’t be ignored and wave packet disperses in positionspace with time causing in decrease in revival amplitude. hapter 3BEC in Optical Lattices Bose-Einstein condensation (BEC) is a pure quantum phenomenon consist-ing of the macroscopic occupation of a single-particle state by an ensemble ofidentical bosons in thermal equilibrium at finite temperature. In this chapterwe provide a review on the dynamics of BEC in optical lattices. In additionwe also explain our results related to BEC in shallow and deep optical lattices.Satyendra Nath Bose worked on the statistics of indistinguishable mass-lessparticles, that is, photons [Bose 1924], whereas, Albert Einstein extendedthe work to a gas of non-interacting mass particles and concluded that, be-low a critical temperature, a fraction of the total number of particles wouldoccupy the lowest energy single-particle state [Einstein 1924]. Such a systemundergoes a phase transition to form a BEC, which contains a macroscopicnumber of atoms occupying the lowest energy state. Bose-Einstein conden-sation in ultracold trapped atomic clouds for rubidium atoms was observedfirst time in 1995 [Anderson et al 1995]. The most fascinating applicationof these systems is the possibility to observe the quantum phenomena at amacroscopic scales.When an ideal quantum gas of particles with mass M , adopts Bose-Einstein statistics at temperature T , a phase transition occurs and the deBroglie wavelength, λ dB = p π ~ /M k B T , becomes comparable to the meaninter-particle separation (see Fig-3.1), r = n − / d , where, k B is the Boltzmann38 h 3. BEC in Optical Lattices n d is the atom number density. The particles density at thecentre of a condensed atomic sample is of the order of 10 -10 cm − , muchlower than the density of room-temperature molecules in air (10 cm − ),density of atoms in liquids or solids (10 cm − ) and density of nucleons inatomic nuclei (10 cm − ) [Pethick and Smith 2001]. Quantum phenomenaat such a low density corresponds to the temperature of the order of 10 − K or less.Rapid progress in laser cooling techniques [Cohen-Tannoudji 1992] pro-vide macroscopic population of atoms in the ground state at such a lowtemperature. A realistic Bose gas shows some level of inter particle interac-tion. Therefore, the system of an ideal Bose gas of non-interacting particlesis a fictitious system. However, this model provides the simplest example forthe realization of BEC. h 3. BEC in Optical Lattices In this section we discuss theoretical description of BEC in a periodic po-tential. Turning on an optical field introduces fragmentation of the wavefunction of the continuous BEC into local wave functions centred around thepotential minima. The fragmentation of BEC leads to a crystal like structurein which fragments mutually interact. Properly controlling the parameters ofthe lattice, macroscopic quantum interference phenomenon for vertical BECarray in gravitational field is observed [Anderson et al 1998].In quantum mechanics, time evolution of any finite wave packet in freespace shows dispersion, i.e., it expands with a velocity inversely proportionalto its initial spread. The presence of the periodic potential modifies the dis-persion. The dispersion relation (energy momentum relation E ( k )) can suf-ficiently be approximated with a parabola. In the linear case, E ( k ) relationis equivalent to the chemical potential ¯ µ (generally the chemical potentialincludes mean-field interaction energy). As the dispersion relation in freespace is quadratic in momentum, the parabola curvature can be associatedwith an effective mass description. This mass can be negative or positiveresulting in either anomalous or normal dispersion. In both cases the dis-persion leads to the spreading of the condensate, however, in the negativemass regions, the effective time arrow is reversed (due to the symmetry ofthe Schr¨odinger wave equation in free space: M < t > → M > t < h 3. BEC in Optical Lattices i ~ ∂∂t ψ ( r, t ) = (cid:20) − ~ M ▽ − V ( r ) (cid:21) ψ ( r, t ) + g D | ψ | ψ ( r, t ) . (3.1)Here, g D = 4 π ~ a s /M and a s is the s-wave scattering length which is inde-pendent of the details of two-body interaction potential and characterizes thecollisions. GPE is an appropriate description of condensate dynamics giventhat there is negligible depletion out of the condensed mode and n d | a s | ≪ n d is the density of condensate atom. The trapping potential is V ( r ) = V [sin ( k L x ) + sin ( k L y ) + sin ( k L z )] + M ω x x + ω y y + ω z z ) , (3.2)where, ω x , ω y and ω z are harmonic trapping frequencies. We focus our atten-tion on one dimensional (1-D) optical lattice potential. Eq. (3.1) can be ex-pressed in dimensionless parameters where, characteristic length, a L = d/π ,characteristic energy, E L = 2 E r and time with a scaling factor 2 ω r . With thisscaling, interaction term is g D = 4 π ( a s /a L ) and lattice potential is measuredin the unit of lattice recoil energy E r .For a condensate trapped in 1-D optical lattice along x-axis, with ω x and ω y , ω z ≡ ω ⊥ as the trapping frequencies, Gross-Pitaevskii Eq. (3.1) modelcan be reduced to a 1-D GPE with a periodic potential. The condition forthe 1-D optical lattice is given only when the linear density is smaller thanthe critical density n D < a s . For Rb atoms n D <
100 atoms µm − . Thetransverse wave function can be given by the ground state of a 2-D, radi-ally symmetric quantum version of harmonic oscillator. The normalizationcondition is given as, R ∞−∞ | Φ | dxdy = 1. The three dimensional wave func-tion then can be separated as ψ ( r, t ) = Φ( r ) ψ ( x, t ), to eliminate transversedimensions [P´erez-Gar´ca et al 1998], thus, yielding the 1-D GPE, i ∂∂t ψ ( z, t ) = (cid:20) − ∂ ∂z − V cos(2 z ) (cid:21) ψ ( z, t ) + g D | ψ | ψ ( z, t ) . (3.3)Here, g D = ( a s /d )( ω ⊥ /ω r ) and z = k L x . We have neglected the contributionof harmonic potential in the direction of the optical lattice and constantenergy shift in energy band. The stationary states of condensate in one h 3. BEC in Optical Lattices ψ ( z, t ) = φ ( z ) Exp [ − i ¯ µt ], here, ¯ µ is the chemical potential.The non-linear interaction term introduces an additional energy scale inthe system which enriches the linear stationary states. These states can becategorized as spatially extended non-linear Bloch waves, self-trapped states(truncated Bloch waves with an arbitrary localization length) or gap solitons(strongly localized matter-waves or non-spreading condensates). We have discussed the energy spectrum of single particle wave packet insection 2.6.1 for deep optical lattice and for shallow optical lattice in sec-tion 2.7.1. In this section, we analytically calculate the non-linear energyspectrum for well localized condensate in the presence of mean field inter-action term, which shows how energy spectrum is modified in the presenceof interaction term. Later, non-linear dynamics of the condensate initiallylocalized to many lattice sites are studied numerically. The Bloch spectrumfor condensate in optical lattice can be represented as E nk = Z L/ − L/ ϕ ∗ nk ( z ) H NLS [ ϕ nk ] ϕ nk ( z ) dz , (3.4)where, H NLS is Hamiltonian (3.3) can be written as H NLS [ ϕ ] = H L ( z ) + g D | ϕ | , (3.5)here, H L ( z ) ≡ − ∂ ∂z + V z ) , (3.6)is the linear lattice term (lattice Hamiltonian when interaction is absent).The chemical potential of condensate,¯ µ = lim k → E k , (3.7) h 3. BEC in Optical Lattices µ = ¯ µ . The particle group velocity, v nk , and the effective mass, M ∗ nk [Yukalov 2009], are respectively defined as v nk ≡ ∂E nk ∂k , (3.8)1 M ∗ nk ≡ ∂ E nk ∂k . (3.9)In tight-binding approximation, only the nearest-neighbor hoping matrixelements contribute to the tunneling. For the linear term Eq. (3.6), two typesof matrix elements over Wannier functions exist [Yukalov 2009], that is theon-site integral h ≡ Z L/ − L/ w ( z ) H L ( z ) w ( z ) dz, (3.10)and, the integral representing the nearest-neighbor overlap h ≡ Z L/ − L/ w ( z ) H L ( z ) w ( z − d ) dz . (3.11)The matrix element for non-linear term in Eq. (3.5) are generally 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 . (3.12)Considering tight-binding approximation, three kinds of integrals are encoun-tered. The on-site integral is I ≡ Z L/ − L/ w ( z ) dz , (3.13)the first-order overlap integral is I ≡ Z L/ − L/ w ( z ) w ( z − d ) dz , (3.14)and second-order overlap integral is given as I ≡ Z L/ − L/ w ( z ) w ( z − d ) dz . (3.15) h 3. BEC in Optical Lattices ε ≡ h + g D I , (3.16)tunneling parameter defined by first-order overlap integrals, J ≡ − h − I g D , (3.17)and the Bloch spectrum Eq. (3.4) defined as E k = ε − J cos( kd ) . (3.18)When the second-order overlap integrals Eq. (3.15) are considered, an addi-tional term with cos(2 kd ) appears.In this case, the chemical potential Eq. (3.7) is¯ µ ≡ lim k → E k = ε − J, (3.19)provided, only lowest band is considered. Substituting the Eqs. (3.16) and(3.17) in the above equation gives the expression of chemical potential as¯ µ = h + 2 h + g D ( I + 8 I ) . (3.20)The isothermal compressibility, κ T = d ¯ ν ∂ ¯ ν∂ ¯ µ = a ¯ ν ( ∂ ¯ µ∂ ¯ ν ) − , can be defined byconsidering only the dependence of the coupling parameter g D on the fillingfactor ¯ ν , and neglecting other dependencies on ¯ ν of the Wannier functions[Yukalov 2009]. Then κ T = M d l ⊥ a s ¯ ν ( I + 8 I ) = 1 ρ ( I + 8 I ) g D , (3.21)where, l ⊥ is transverse localization length. We note that, bosons in a lat-tice can be stable only in the presence of non-zero repulsive interactions.When atomic interactions are attractive (i.e., a s < g D → κ T would be infinite. In these two cases, the system would be unstable[Courteille et al 2001; Yukalov 2005a; Yukalov 2005b]. In the case of attrac-tive interactions, ( i.e., a s < h 3. BEC in Optical Lattices v k = 2 J d sin( kd ) , (3.22) M ∗ k = 12 J d cos( kd ) . (3.23)The group velocity and the effective mass in the limit k →
0, are simplifiedas M ∗ ≡ M ∗ = 12 J d ( k = 0) , (3.24) v k ≃ J d k (3.25)For tight-bind limit, the localized Wannier orbitals can be approximatedby a Gaussian function. However, the Gaussian ansatz neglects the smalloscillations characteristic in the tail of the Wannier function, the tunnelingmatrix element J is then underestimated by almost an order of magnitude.The corresponding Wannier functions for a one-dimensional lattice is w ( z − z ) = 1( √ π ∆ z ) / exp( − ( z − z ) z ) ) . (3.26)Using Eq. (3.26), the on-site integral given in Eq. (3.13) becomes I = 12∆ z √ π , (3.27)and the first order integral given in Eq. (3.14) becomes I = I ˜ γ , (3.28)where, ˜ γ = exp[ − d z ) ] . The second order overlap integral in Eq. (3.15) is I = I ˜ γ = I ˜ γ . (3.29)From above three expressions we note that I ≪ I ≪ I ( ∆ z d ≪ , (3.30) h 3. BEC in Optical Lattices h = − z ) + V − z ) ] , (3.31)and the overlap integral Eq. (3.11), we have h = ˜ γ (cid:20) − d − z ) z ) + V − z ) ] cos( d ) (cid:21) . (3.32)The local energy term (3.16) becomes ε = 12∆ z (cid:20) − z + V exp[ − z ) ] + g D √ π (cid:21) . (3.33)This leads to the nearest-neighbour hopping parameter in Eq. (3.17), as, J = ˜ γ " d z ) − − V − z ) ] cos( d ) − g D ˜ γ ∆ z √ π . (3.34)Hence the chemical potential defined in Eq. (3.20) reads as¯ µ = 2˜ γ (cid:20) − d − z ) z ) + V − z ) ] cos( d ) (cid:21) + g D z √ π h − ˜ γ i − z ) + V − z ) ] . (3.35)As a consequence the compressibility expression (3.21) gets the form κ T = 2∆ z √ πρg D [1 + 8˜ γ ] . (3.36)In the presence of non-linear atom-atom interaction, non-linear Blochwaves (spatially extended Bloch waves) exist with the periodicity of the lat-tice potential. For the case of weak non-linearity, the non-linear Bloch statesand linear ones are qualitatively similar. As the non-linearity (local conden-sate density) increases, the chemical potential corresponding to the Blochstate near the linear band edges shifts into the linear spectrum band gap[Ostrovskaya et al 2008]. The magnitude of this shift is directly proportionalto the non-linearity and its direction is determined by the scattering lengthsign as shown in Fig-3.2. h 3. BEC in Optical Lattices k = 0 (Grey) and k = 1 (red) for a repulsive [linesmarked with points (a) and (b)], g D = 0 . g D = − . V = 1 . k = 0) are stable. On the other hand,non-linear Bloch states representing the top of linear ground band ( k = 1)are dynamically unstable [Konotop and Salerno 2002]. This instability arisesdue to small fluctuations in condensate density and results in exponentialgrowth in Bogoliubov excitations. Experimentally dynamical instability wasdetected with BEC of large number of atoms, adiabatically driven to upperedge of the ground state, or non-adiabatically loading of BEC in k = 1 state[Fallani et al 2004; Sarlo et al 2005]. Dynamical instabilities lead to the lossof condensate atoms due to spatial fragmentation of the BEC density.We discuss roll of the non-linearity in the dynamics of condensate forthree different regimes namely: i) Non-linearity is the smallest energy scale,it is easy to realize experiments in this regime; ii) Non-linearity is larger than h 3. BEC in Optical Lattices V = 2 E r .In this case the width of lowest band is 0 . Er and band gap between firstexcited band and lowest band is 0 . E r . For shallow lattice case, dynamicsare explored for the cases i.e. in the absence of non-linearity ( g D = 0),for non-linearity g D = 0 . g D = 0 . g D = 1 . , (interaction energy is dominant). For deep lattice, an optical lattice withpotential V = 8 E r is considered. In this case the width of lowest band is0 . Er and band gap between first excited band and lowest band is 3 . E r .We study dynamics of condensate in deep lattice for non-linearity g D = 0 . g D = 1 . g D = 4 (non-linearity is dominant energy scale). In this regime, only small changes are expected in the single particle spectrumdue to the presence of non-linear term. Even though, stationary solutionshave no significant differences from the linear case, the dynamics in this caseis totally different. Formation of bright solitons when interaction is repulsive(when quasi-momentum is in the regime of negative mass) and dynamicalinstability are two important examples in this regard.
Shallow lattice potential
The condensate in the lowest band with a small momentum distributioncentred around a mean momentum k can be efficiently described by a slowly h 3. BEC in Optical Lattices A ( z, t ) , times Bloch statecorresponding to quasi-momentum k , ψ ( z, t ) = A ( z, t ) φ n =0 ,k ( z ) e − i ~ ¯ µ ( k ) t . (3.37)In the weakly interacting case, a differential equation for slowly varyingenvelope, A ( z, t ), can be derived which has the same form as the Gross-Pitaevskii equation with a modified interaction energy and dispersion, i ~ ( ∂∂t + v g ∂∂z ) A ( z, t ) = [ − ~ k L M eff ∂ ∂z + V ( z, t ) + g D α n | A | ] A ( z, t ) . (3.38)Here, α n = (1 /d ) R d/ − d/ dz | φ k | ∼ −
2, re-normalizes the interaction en-ergy as it increases with stronger localization in the lattice. The stationarysolutions of above Eq. (3.38) do not show significant difference from lin-ear case but dynamics shows some different characteristics i.e., formationof bright solitons (termed as gap-solitons by [Steel and Zhang 1998]) evenfor repulsive interaction with the condition that the mean quasi-momentumis in negative effective mass regime. Another very interesting phenomenonarising in the presence of non-linearity is dynamical instability, that is, ex-ponentially fast growth of small perturbation of condensate wave function[Konotop and Salerno 2002].A condensate with Gaussian profile is adiabatically loaded in the opticallattice by ramping up potential slowly. The adiabatic loading prepares thecondensate in positive mass regime (initially occupying the lower edge of theground band). Four parameters characterize the Gaussian condensate: (a)the width of the condensate, (b) the center-of-mass position, (c) the linearphase which describes the group velocity of the wave packet, and (d) thequadratic phase over the condensate which describes the linear evolutionof the condensate. The quadratic dispersion in momentum space directlytranslates into a quadratic phase in real space. Moreover, the non-linearenergy term also leads to a quadratic phase in first approximation since thedensity is quadratic near the Gaussian maximum.In Fig-3.3 and Fig-3.4, we show spatio-temporal dynamics of attractiveand repulsive condensate respectively, for non-linearity g D = 0, g D = 0 . h 3. BEC in Optical Lattices p = 0 . g D = 0, g D = − . g D = − . g D = − .
5. Lattice potential is V = 2 E r and condensate isadiabatically loaded in the lattice. In this figure dark colours regions showdensity maxima. h 3. BEC in Optical Lattices p = 0 . g D = 0, g D = 0 . g D = 0 . g D = 1 .
5. Lattice potential is V = 2 E r and condensate isadiabatically loaded in the lattice. In this figure dark colours regions showdensity maxima. h 3. BEC in Optical Lattices g D = 0 . g D = 1 . , (interaction energy is dom-inant) for shallow lattice. Here and in rest of the figures, dark shades repre-sent the maximum density for spatio-temporal structures. Spatial dispersionof the condensate both for attractive and repulsive case in shallow latticeis shown in Fig-3.5a and Fig-3.5b, respectively. Momentum dispersion forboth attractive and repulsive condensate is shown in Fig-3.6a and Fig-3.6b,respectively.From Fig-3.3 and Fig-3.4, we note that the dynamics for shallow lat-tice in the absence of non-linearity shows diffusion to the neighboring lat-tice sites. The spreading of condensate is due to Josephson like tunnelingfrom one lattice site to other. While, with the introduction of small non-linearity g D = − .
3, (attractive case), dispersion decreases and condensateis confined in few lattice sites only. On the other hand, for repulsive case( g D = 0 . n = 0 and k = 1 in this case), the effective mass ofmatter-wave becomes negative which effectively leads to the focusing of thematter-wave. A balance between the repulsive interaction and negative dis-persion near the top edge of the spectral band leads to the spatially localizednon-spreading wave packets with zero group velocity termed as matter-wavesolitons . The chemical potential corresponding to such localized waves liesin the gaps of the linear Bloch-wave spectrum and termed as gap solitons .In initially published work [Pelinovsky et al 2004], it is shown that gener-ally two type of solitons are bifurcated from each band edge. These arebright solitons centred on the minimum (on-site) of the lattice potential andmaximum (off-site) of the lattice potential. The gap solitons are well ap-proximated by a sech-like broad envelope A ( z ), with low-amplitude of thecorresponding Bloch wave, centered either off or on-site. The solitons arecomparatively weakly localized and have low density (number of particles).That is why, preparation of a very small condensate was required to observe h 3. BEC in Optical Lattices
53a one dimensional gap solitons. On the other hand low density also reducesthe chance of transverse excitations. In the dynamics, the distinction be-tween the on-site and off-site states does not play significant role because theoff-site states undergoes symmetry-breaking instability and transforms intoa stable on-site states [Eiermann et al 2004].The time evolution of spatial dispersion in Fig-3.5a and Fig-3.5b, showsthat dispersion also increases with time in the absence of non-linearity. How-ever, as the non-linearity ( g D ) is introduced, the spatial dispersion is mod-ified. Initially, it increases, later, in attractive case it decreases while, forrepulsive case, it keeps on increasing with time and it settles around a con-stant value. This suppression is due to formation of weakly localized solitons.These solitons are clearly seen in Fig-3.4 for g D = 0 .
3. Beyond t = 250, welllocalized soliton matter-waves appear as a balance between diffusion andfocusing (due to negative effective mass) is established.The momentum dispersion for the case of zero non-linearity ( g D = 0)initially, quickly increases and then fluctuates around a mean value showingcollapse and revivals in momentum dispersion. With the introduction of non-linearity, g D , the behavior of momentum dispersion is modified. Althoughthe ∆ p fluctuations around the same mean as in the case for zero non-linearitybut its collapse and revival behavior is changed for both attractive (Fig-3.6a)and repulsive (Fig-3.6b) condensate. Deep Optical Lattice
In the limit of deep periodic potential, condensate wave functions can bedescribed by Wannier wave functions localized at the potential minima. Thelocalization of condensate by deep optical lattice increases the on-site atom-atom interaction. The condensate wave function can be described with local-ized Wannier states associated with the lowest band [Chiofalo et al 2000]. Itis notable that the strong localization modifies the linear Wannier states dueto increase in atomic densities which enhance atom-atom interaction. Thedynamics of BEC is governed by discrete non-linear Schr¨odinger equation i ~ ddt ψ n = J ( ψ n − + ψ n +1 ) + g D | ψ n | ψ n + E n ψ n . (3.39) h 3. BEC in Optical Lattices =-1.5 0 10 20 g =-0.7 0 10 20 ∆ x g =-0.3 0 10 20 g = 0 (a) Spatial dispersion vs time with lattice potential V = 2 E r for g D = 0, g D = − . g D = − . g D = − .
5. Other parameters are ∆ z = 5 and condensate isadiabatically loaded in the lattice. =1.5 0 10 20 30 g =0.7 0 10 20 30 ∆ x g =0.3 0 10 20 30 g = 0 (b) Spatial dispersion vs time with lattice potential V = 2 E r for g D = 0, g D =0 . g D = 0 . g D = 1 .
5. Other parameters are ∆ z = 5 and condensate isadiabatically loaded in the lattice. Figure 3.5: Spatial dispersion vs time for shallow lattice. h 3. BEC in Optical Lattices =-1.5 0 1 g =-0.7 0 1 ∆ p g =-0.3 0 1 g = 0 (a) Momentum dispersion vs time with lattice potential V = 2 E r for g D = 0, g D = − . g D = − . g D = − .
5. Other parameters are , ∆ p = 0 . =1.5 0 1 g =0.7 0 1 ∆ p g =0.3 0 1 g = 0 (b) Momentum dispersion vs time with lattice potential V = 2 E r for g D = 0, g D =0 . g D = 0 . g D = 1 .
5. Other parameters are , ∆ p = 0 . Figure 3.6: Momentum dispersion vs time for shallow lattice. h 3. BEC in Optical Lattices k is propagated in a periodic potential dependson the quasi-momentum k and total atom number Λ ∝ N t . After sometime, the condensate expansion stops for large non-linearity and this behavioris independent of the initial quasi-momentum (self-trapping regime). Forsmall non-linearity (small atom number), the condensate expands indefinitely(diffusive regime). The solitonic propagation or breathers are only possiblefor quasi-momenta with negative mass regime. Since this excitation relies ona delicate balance between linear spreading and non-linearity, it only appearsfor very well-defined atom numbers.The dynamics is characterized by two basic parameters Λ and cos p ,where, Λ = g D / J and p = kd is quasi-momentum. In Fig-3.7, we show thepropagation characteristics which depend on these two parameters and theresulting evolution can be characterized by diffusion, self-trapping and soli-tonic propagation. The solitonic evolution is found by applying the conditionthat neither the quadratic phase nor the width is time dependent and Λ sol =2 √ π | cos p | exp {− z } / ∆ z . Here, ∆ z is initial width of Gaussian con-densate in units of the lattice constant. The number of atoms in a soliton isinversely proportional to the width of the soliton. These solitons very closely h 3. BEC in Optical Lattices g D = 0 .
1, whether, attractive or repulsive causes no signifi-cant difference. However, collapse and revival behavior of small fragments ofcondensate trapped on individual lattice sites is modified.The spatial dispersion does not show a significant difference with theintroduction of small non linearity g D = − .
1. The spatial dispersion inthis case is suppressed earlier compared to shallow lattice even for smallnon-linearity, for both attractive and repulsion condensates, as shown inFig-3.10b and Fig-3.10a. The instability arises due to non-linearity thatprompts non-linear Bloch states which decay into localized soliton trains[Ostrovskaya et al 2008].The momentum dispersion for the case of zero non-linearity ( g D = 0)quickly increases initially, and then fluctuates around a mean value showingcollapse and revivals in dispersion. With the introduction of non-linearity,the behavior of momentum dispersion is not modified significantly. Although∆ p fluctuates around the same mean as in the case of zero non-linearity butits collapse and revival behavior is slightly changed for both attractive (Fig-3.11a) and repulsive (Fig-3.11b) condensates, as shown for g D = 0 . The dynamics with intermediate non-linearity is an approximation which isgood as long as the non-linearity is such that 4
J < g D < E gap . h 3. BEC in Optical Lattices p = 0 .
1, non-linearity, g D = 0, g D = − . g D = − . g D = −
4. Lattice potential is V = 8 E r and conden-sate is adiabatically loaded in the lattice. The dark regions represent themaximum population.The self-trapping region shown in Fig-3.7 is explored when the width ofthe condensate remains finite for very very long time. The critical value of Λis Λ c = 2 √ π ∆ z | cos p | (1 − exp {− z } ) . When Λ < Λ c , on-site interactionenergy per particle is the smallest energy scale and the evolution is quali-tatively described with the assumption that the condensate having a meanquasi-momentum. In this limit the condensate is in the diffusive regime asshown in Fig-3.7. In the case, when Λ > Λ c , the band width is smaller thanthe on-site interaction energy and the description of a wave packet basedon a single central quasi-momentum is failed. The spreading of condensateis suppressed due to local dynamics at the band edges. Solitonic solutionsexist and show structures on the length scale of the periodicity, moreover, h 3. BEC in Optical Lattices p = 0 .
1, non-linearity, g D = 0, g D = 0 . g D =1 . g D = 4. Lattice potential is V = 8 E r and condensate is adiabaticallyloaded in the lattice. The darker regions represent the maximum energydensity.the solitonic solutions can be classified in terms of their symmetry with re-spect to the potential minima [Louis et al 2003]. Spatio-temporal dynamics(Fig-3.4) for repulsive condensate in shallow lattice shows that due to thestronger repulsion, cloud explodes to the band edge faster as compared tothe case where non-linearity is the smallest energy. Moreover, spatial disper-sion curve saturated earlier as compared to previous case. Fig-3.12a showsthe normalized population of the condensate at different times. From this fig-ure, we note that as the time passes steep edges appear in the density profileof the condensate. At t = 400 the steep edges are developed for the case ofrepulsive condensate with g D = 0 . h 3. BEC in Optical Lattices =-4 8 16 g =-1.5 8 16 ∆ x g =-0.1 8 16 g = 0 (a) Spatial dispersion vs time for lattice potential V = 8 E r , non-linearity, g D = 0, g D = − . g D = − . g D = −
4. Other parameters are ∆ p = 0 . =4 8 16 24 g =1.5 8 16 24 ∆ x g =0.1 8 16 24 g = 0 (b) Spatial dispersion vs time for lattice potential V = 8 E r , non-linearity, g D = 0, g D = 0 . g D = 1 . g D = 4. Other parameters are ∆ p = 0 . Figure 3.10: Spatial dispersion vs time for deep lattice. h 3. BEC in Optical Lattices =-4 0 1 2 g =-1.5 0 1 2 ∆ p g =-0.1 0 1 2 g = 0 (a) Momentum dispersion vs time for V = 8 E r , non-linearity, g D = 0, g D = − . g D = − . g D = −
4. Other parameters are ∆ p = 0 . =4 0 1 2 g =1.5 0 1 2 ∆ p g =0.1 0 1 2 g = 0 (b) Momentum dispersion vs time for lattice potential V = 8 E r , non-linearity, g D =0, g D = 0 . g D = 1 . g D = 4 and ∆ p = 0 . Figure 3.11: Momentum dispersion vs time for deep lattice. h 3. BEC in Optical Lattices N o r m a li z ed popu l a t i on Site Index t=0 t=100t=400 (a) Normalized population of repulsive condensed atoms at t = 0, t = 100 and t = 400.Other parameters are V = 2 E r and g D = 0 . N o r m a li z ed popu l a t i on Site Index t=0 t=100t=300 (b) Normalized population of repulsive condensed atoms at t = 0, t = 100 and t = 350.Other parameters are V = 2 E r and g D = 1 . Figure 3.12: Normalized population for repulsive condensate in optical lat-tice. h 3. BEC in Optical Lattices
In this regime, on-site interaction energy is largest energy scale, which impliesthat, it is larger than ground band width and band gap between lower andfirst excited band. Here, linear band concept is no longer applicable. Usingperturbation theory, an effective potential concept is introduced [Choi 1999]which simplifies the dynamics in this regime. Analytical solutions exist inthis regime [Bronski et al 2001] and energies obtained from these solutionsare function of quasi-momentum and reveal loop structures instead of flatbands in linear case.This approach explains the motion in an effective potential for each atom.The energy variations caused by non-linear term in the Gross-Pitaevskii equa-tion and the external periodic potential contribute to the effective potential.Due to periodic potential the atomic density is maximum at the poten-tial minima, the potential energy will be effectively increased (decreased)due to the attractive (repulsive) atom-atom interaction. The dynamics ofatoms can be described in an effective potential V eff without non-linear term[Choi 1999], so that, V eff = V g D cos(2 z ) . (3.40)This effective potential approximation remains quite valid provided con-densate density is nearly uniform which was experimentally confirmed forone dimensional potential [Morsch 2001]. This condition is fulfilled for weakexternal potential or strong atomic interactions. It was predicted that mo-tion of homogeneous condensate with large non-linearity is naively changedin the presence of periodic potential.Mostly, solutions in this regime can’t be derived analytically. However,in the homogeneous case, solutions are obtained analytically for the po-tential form V ( z ) = − V sn ( z, k e ) [Bronski et al 2001], where, the function h 3. BEC in Optical Lattices sn ( z, k e ) is Jacobian elliptic sine functions and k e is elliptic modulus, suchthat, 0 ≤ k e ≤ . When k e = 0, the potential describes an optical lattice. Thestationary solutions exist with and without a non-trivial phase. The stabilityof the solutions depends on the background density of atoms but an additionof a constant background of atoms leads to a homogeneous non-linear energyand stabilize the solutions. Loops in Band Structures
When the non-linear on-site interaction energy is dominant, the concept oflinear band structure is no longer applicable. However, the solutions, evenin the presence of non-linearity still show some resemblance to the linearband spectrum with new features such as loop formation as shown in Fig-3.13. Investigation of loop structures show that instabilities appear in theband structure for large non-linearity near the boundary of the first Brillouinzone [Wu et al 2003; Machholm et al 2003; Seaman 2004]. When interactionenergy per particle exceeds the lattice potential amplitude, the loop struc-ture appear in the lowest Bloch bands near the Brillouin zone boundary andnear the zone center in the higher bands. Fig-3.13 shows the loop structurein the energy bands. The swallow tail width increases with the interac-tion energy, which can extend into zone center. These loops seem as thecircumstance of the periodic potential, for the Brillouin zone center, theyare a general phenomenon which appear in the vanishing periodic potential[Machholm et al 2003]. In the vanishing potential limit, the loop formed be-tween the second and third bands are degenerated with a special excitedstate of a condensate, termed as dark soliton train.
Spatio-temporal dynamics in this regime for condensate in deep latticewith non-linearity g D = 4 shows that condensate is mostly trapped in fewlattice sites. A strong confining lattice potential suppresses Josephson tun-neling and during the time evolution, non-linear Bloch states decay into gapsoliton trains [Louis 2005]. Spatial dispersion in deep lattice case gains steadystate in much shorter time as compared to the shallow lattice due to shorterband width. An introduction of non-linearity quickly drives the cloud tothe band edge gap solitons appear in early evolution. For attractive case, h 3. BEC in Optical Lattices k [Machholm et al 2003]. The figure shows thatthe loop structure become less dominant as the lattice potential is increasedenriched collapse and revival behavior in each occupant site is seen. Spatialdispersion of condensate for this regime attains almost steady state earlierthan the other two regimes (non-linear energy is smallest and intermediateenergy). Momentum dispersion for attractive condensate in this regime re-veals collapse and revival in ∆ p with time and revival period in much shorterthan the other two regimes.For repulsive condensate with non-linearity g D = 4, matter waves re-main trapped in the potential wells where they were initially placed (Fig-3.9) dispersion in position space for this regime is smaller as compared tothe repulsive intermediate non-linearity regime. The dispersion in momen-tum shows fluctuations with very small amplitude and revival behavior of ∆ p is modified. Small fluctuations in ∆ p , and suppression in dispersion is dueto suppression of Josephson tunneling and formation of gap soliton trains[Louis 2005].For shallow lattice, spatio-temporal behavior for attractive condensatelattice in this regime for g D = − . t = 200, central lattice show-ing maximum density, neighboring lattice sites have negligible densities while,next to nearest neighboring lattice sites have non-negligible densities. After h 3. BEC in Optical Lattices t = 300 on both side of the condensate.These steep edges truncate the non-linear Bloch state [Alexander et al 2006].While, for intermediate non-linear regime it appeared at t = 400 and edgesare more steeper compared to the intermediate regime, indicating strongerself-trapping. Momentum dispersion (Fig-3.6b) shows that revival time doesnot change significantly but fluctuation of dispersion decreases in this case. The discussion on the dynamics of condensate in optical lattice will beincomplete without considering the instabilities that arise due to spatialconfinement of condensate in periodic potential and non-linearity. In thelattice potentials, two type of instabilities can exist: i) Landau instabil-ities, for which small perturbations lead to a lowering of the system en-ergy; ii) dynamical instabilities, due to exponential growth of perturbations[Wu 2000; Wu 2001; Wu et al 2003; Machholm et al 2003]. h 3. BEC in Optical Lattices v=0.01 / / q v=0.05 v=0.10 c = . / / o r q c = . / / o r q c = . k / / o r q k k c = . a.1 b.1 c.1a.2 b.2 c.2a.3 b.3 c.3a.4 b.4 c.4k d Q V o =0.01 V o =0.05 V o =0.1 U = . U = . U = . U = . g D / J g D / J g D / J g D / J Figure 3.14: Stability diagram obtained by [Wu et al 2003]. The parame-ters describing the physical situation are the potential modulation V , thenon-linearity g D for the homogeneous case, and the quasi-momentum, k ofthe homogeneous condensate. The parameter Q is the corresponding wavevector of the perturbation. It is important to note that k = 2 π/d impliesa modulation of the density with twice the period of the periodic potential.The regime in which the stationary states exhibit a Landau instability areindicated by the light shaded area. The results are symmetric in k and Q , soonly the parameter region, for 0 < k < / < Q < / k d . h 3. BEC in Optical Lattices Landau instability is generally discussed in the scenario of Bose liquids andtheir superfluidity, i.e., the Bose liquid can flow through tight spaces withoutfacing friction provided that its speed is below a critical value. Landau arguedthat a quantum current faces friction only when the creation of excitationson the liquid lowers the energy of the system. The same is true for a BEC’sconfined in an optical lattice. In order to find out the influence of smallperturbation on the energy of a given Bloch state e i k z φ k ( z ), the energy ofa slightly perturbed Bloch state, that is, ψ k ( z ) = e i k z (cid:8) φ k ( z ) + u k ( z, Q ) e iQ z + v ∗ k ( z, Q ) e iQ z (cid:9) , (3.41)gives the signatures of instability. The functions u k ( z, Q ) and v ∗ k ( z, Q ) havethe period of potential and Q ∈ [2 π/d, π/d ]. This perturbation can causeenergy deviation. A detailed mathematical method is given in references:[Wu et al 2003; Machholm et al 2003]. If the perturbation increases the en-ergy of the Bloch state, the condensate exhibits superflow as original Blochstate corresponds to a local energy minimum. If energy is negative, normalflow is expected. The numerical results are summarized in the Fig-3.14. Adetailed analysis giving the quasi-momentum, k d , for the existence of theboth type of instabilities as a function of non-linearity, g D , and potentialdepth, V , is given [Machholm et al 2003]. In the stability phase diagramsfor condensate, shown in the Fig-3.14, these results are symmetric in k and Q , so only the parameter region, for 0 < k < / < Q < / V ; ii) The non-linearity g D ; and iii) quasi-momentum k . Theenergy deviation is calculated as a function of the free parameter Q = π/d ,which describes a perturbation with the spatial period d . In Fig-3.14, thelight shaded area represents the Landau unstable region in which phononsemission can lower the systems energy. h 3. BEC in Optical Lattices Dynamical instability means that small deviations from the stationary solu-tion in the system exponentially grow in time. Dynamical instability occursin homogeneous condensates confined in an optical lattice only in the presenceof attractive interactions but can be induced by the presence of a periodicpotential even for repulsive interactions.To analyze dynamical stability of the condensate, same procedure isadopted as in the case of Landau instability but now the time-dependentGross-Pitaevskii equation is used. Taking only the linear term in the per-turbation, linear differential equation is obtained which describes the timeevolution of the small perturbation [Machholm et al 2003; Wu et al 2003].The Bloch states are stable if the corresponding eigenvalues are real. How-ever, complex eigenvalues are an indication that the perturbation may growexponentially. It is important to know that energetically unstable Blochstates are responsible of dynamical instability. The mode that is unstablefor the quasi-momentum of the condensate k = k d is specified by k = 2 π/d .This implies that the corresponding unstable (exponentially growing) modeshows a period doubling [Machholm et al 2004], since the functions v ( z ) and u ( z ) in Eq. (3.41) have the same period as the periodic potential.The Bloch states with k , lies outside the shaded area, represent superflowas they correspond to local energy minima. As g D is increased these super-flow regions expand and occupy the entire Brillouin zone for sufficiently large g D . Furthermore, the lattice potential V does not significantly influence thesuper-flow regions as it can be seen in each row. The phase boundaries for V << k = q Q + g D , for V = 0 which is shown as triangles symbols in the firstcolumn.Common feature of all the panels in Fig-3.14 is that there exists a criticalBloch wave number k d beyond which the Bloch states show dynamical insta-bilities. The onset of instability at k d always corresponds to Q = 1 /
2. Thisimplies that, if we drive the Bloch state from k = 0 to k = 1 /
2, at Q = ± / h 3. BEC in Optical Lattices k > k d , instabilities can occur. These unstable modesgrow, drives the system far away from the Bloch state and breaks the trans-lational symmetry of the system spontaneously. The dynamical instabilityhave also been investigated with the reference of an effective-mass approxi-mation [Konotop and Salerno 2002]. In chapter-2, we discussed the condensate revivals of cold atoms in opticallattices both for deep and shallow optical lattices. In this section we studythe role of non-linear interaction term on the time scales present in the systemfor deep and shallow lattices.
Deep optical lattices
We consider a deep optical lattice with potential depth V = 8 E r , and aGaussian condensate comprising of lowest energy band with initial momen-tum spread ∆ p = 0 . E r around the mean p = 0.Autocorrelation function for attractive condensate is shown in Fig-3.15afor g D = 0, g D = − . g D = − . g D = −
4. and for repulsive inter-action in Fig-3.15b for the same non-linearity values. In the case of repulsiveinteraction the auto-correlation falls quickly as tunneling probability to thenearest neighbor and next to nearest neighbor is non-vanishing and later,the formation of solitonic trains as discussed in sec-3.3 suppresses tunnelingwave packet revivals. Stronger the repulsive interaction smaller the auto-correlation. For g D = 0 .
1, the revival structure doesn’t change significantlyexcept | A ( t ) | collapses to the lower value compared to the single particlewave packet dynamics ( g D = 0). When g D = 1 . g D = 4 (interaction energy is dominant), therevival structures changes significantly. For attractive interaction, a smallnon-linearity g D = 0 .
1, modifies the revival structures. For large attractiveinteraction ( g D = − . g D = −
4) revival time decreases. We study thecondensate distribution at collapse time and first revival time for g D = − . V = 8 E r in Fig-3.16. In this case collapse time t = 30 and quantum revival h 3. BEC in Optical Lattices =-4 0.5 1 g =-1.5 0.5 1 | A ( t ) | g =-0.1 0.5 1 g = 0 (a) Square of auto-correlation function of Gaussian condensate initially placed suchthat it spans lowest Bloch band of lattice with V = 8 E r , for g D = 0, g D = − . g D = − . g D = −
4. Other parameters are ∆ p = 0 . E r and mean initialmomentum p = 0 =4 0.5 1 g =1.5 0.5 1 | A ( t ) | g =0.1 0.5 1 g = 0 (b) Square of auto-correlation function of Gaussian condensate initially placed suchthat it spans lowest Bloch band of lattice with V = 8 E r , for g D = 0, g D = 0 . g D =0 . g D = 1 .
5. Other parameters are ∆ p = 0 . E r and mean initial momentum p = 0. Figure 3.15: Square of auto-correlation function vs time for Gaussian con-densate placed in deep lattice. h 3. BEC in Optical Lattices t = 90. At t = 30 condensate is localized in lattice wells and over-lap with initial condensate is minimum while at t = 90 overlap with initialcondensate is maximum.
45 50 550.000.150.300.45 P r obab ili t y Lattice Site index t=0 t=30 t=60
Figure 3.16: Distribution function for attractive BEC with g = − . t = 0, which defines initial distribution, t = 30, at which the condensatedisplays first collapse and t = 90, for first revival. The lattice potential is V = 8 E r . Shallow optical lattices
We consider a shallow optical lattice with potential depth V = 2 E r , andplot auto-correlation function for a condensate with momentum dispersion∆ p = 0 . E r .For strong homogeneous interaction (repulsive or attractive), an effectivepotential approximation given in Eq. (3.40) is valid in this regime and ex-pressions for classical periods, quantum revivals and super revivals given inSec-2.7 are valid. For attractive condensate with g D = 0 .
3, revival structurechanges and auto-correlation falls to a low value which is due to the initiallyJosephson tunneling of cloud and later suppression of tunneling due to for-mation of solitons near band edges. As interaction increases auto-correlationstays near the unity and for large non-linearity ( g D = − . g D = − . h 3. BEC in Optical Lattices =-1.5 0.5 1 g =-0.7 0.5 1 | A ( t ) | g =-0.3 0.5 1 g = 0 (a) Square of auto-correlation function of Gaussian condensate initially placed suchthat it spans lowest Bloch band of lattice with V = 2 E r , for g D = 0, g D = − . g D = − . g D = − .
5. Other parameters are ∆ p = 0 . E r and mean initialmomentum p = 0 =1.5 0.5 1 g =0.7 0.5 1 | A ( t ) | g =0.3 0.5 1 g = 0 (b) Square of auto-correlation function of Gaussian condensate initially placed suchthat it spans lowest Bloch band of lattice with V = 2 E r , for g D = 0, g D = 0 . g D =0 . g D = 1 .
5. Other parameters are ∆ p = 0 . E r and mean initial momentum p = 0 Figure 3.17: Square of auto-correlation function vs time for Gaussian con-densate placed in shallow lattice. h 3. BEC in Optical Lattices
74a decrease in revival time Fig-3.17a. For repulsive interaction, increase innon-linearity causes a decrease in auto-correlation due to the fact that ini-tially cloud has faster tunneling and latter due to steep edges tunnel is sup-pressed and auto-correlation falls with an increase in non-linearity due toself-trapping Fig-3.17b.
In this chapter, we summarize the dynamics of the condensate in optical lat-tice. It explains the influence of interaction term on the center of mass dy-namic of the condensate. The dynamics are investigated by studying spatio-temporal behavior, auto-correlation function, momentum and spatial disper-sion of the condensate. The inherent non-linearity of a condensate due toBragg scattering of a matter-wave from an optical lattice and repulsive atomicinteractions play its role in the dynamics. Under some circumstances, local-ization is possible as condensate shows anomalous dispersion at the edges of aBrillouin zone of the lattice and magnitude of this dispersion can be managedby tuning the lattice depth and interaction strength. Therefore, the conden-sate spreading either can be controlled by actively controlling the dispersionby varying the intensity of confining field or by utilizing the inter-atomic in-teraction. The later approach leads to non-linear localization of condensate.In shallow optical lattice, this non-linear localization is due to the formationof solitons or steep edges on both side of one-dimensional lattice which trun-cates Bloch states (self-trapping). While, for deep lattice case, localizationis due to the trains of localizes gap solitons which are formed due to decayof inhomogeneous unstable Bloch states. hapter 4Cold Atoms in Driven OpticalLattices
In this chapter, we explain the dynamics of cold atoms in driven optical lat-tice and focuses our attentions on classically integrable regions i.e., nonlinearresonances in phase space. In our analysis we employ Poincar´e surface ofsections, momentum dispersion and their parametric dependencies. Poincar´esurface of section reveals the stroboscopic map of regular and chaotic struc-tures of phase space and its parametric dependencies. Classical dynamics ofour system displays an intricate dominant regular and dominant stochasticdynamics, one after the other, as a function of increasing modulation ampli-tude in the limit where, lattice potential is small and modulation amplitudeis large enough [Moore 1994; Raizen 1999].For quantum wave packet dynamics, momentum dispersion, auto-correlationand spatio-temporal behavior are studied in the vicinity of nonlinear reso-nances. Due to spatial and temporal periodicity in driven optical lattices,the corresponding Schr¨odinger equation for driven lattice can be mappedon Mathieu equation which make the system a paradigm to understand theeffects of periodic modulation on wave packet evolution.Dynamical recurrences in quantum systems, which display chaotic dy-namics in their classical counterpart, are different due to their paramet-ric dependence on modulation effects and as they are limited to nonlinear75 h 4: Cold Atoms in Driven Optical Lattices
External drive in optical lattice introduces a phase modulation, hence, theelectric field in (2.8) is modified asˆ E ( x, t ) = ˆ e y [ ε o cos( k L x − ∆ Lk L sin ωt ) e − iω L t + c.c. ] , (4.1)where, ∆ L is amplitude and ω L is frequency of external modulation. Thedynamics of a cold atom of mass M in phase modulated optical lattice isgoverned by the Hamiltonian, H = p M + V k L { x − ∆ L sin( ω m t ) } ] , (4.2)where, k L is wave number and V define the potential depth of an opticallattice. Furthermore, ∆ L and ω m are amplitude and frequency. Introducingthe dimensionless parameters, we scale time by phase modulation frequencyand position x by natural length scale i.e., standing wave period, τ = ω m t,z = k L x, and λ = 2∆ Lk L . The momentum ˜ p can be related to the original momentum p as˜ p = k L M ω m p, with commutation relation[ z, ˜ p ] = k L M ω m i ~ = i ω r ω m ≡ ik − , (4.3) h 4: Cold Atoms in Driven Optical Lattices k − = 2 ω r ω m . The dimensionless Hamiltonian is written as H = p V { z − λ sin( τ ) } , (4.4)where, ˜ V = V k − / ~ ω m is effective potential depth. Here and in later discus-sion, for simplicity, the tilde sign on p have been ignored. In this section, we discuss the classical dynamics of particle in driven opti-cal lattice. The classical dynamics are governed by the following Hamiltonequations, ˙ z = ∂H∂p = p, and ˙ p = − ∂H∂z = − ˜ V { z − λ sin( τ ) } , or equivalently¨ z = − ˜ V { z − λ sin( τ ) } = − ˜ V ∞ X m = −∞ J m ( λ ) sin(2 z − mτ ) , (4.5)where, J m ( λ ) is m th order Bessel function. The expression at the right sidein last equation is written by invoking the Jocobi-Anger identity e iλ sin( ω m τ ) = ∞ X m = −∞ J m ( λ ) e imω m τ . The effect of potential on the atomic motion depends on the velocity ofthe atoms. The phase modulation velocity of lattice field is − λ cos( τ ). Themaximum velocity of potential is given by the phase modulation amplitude λ, which gives expectations that a particle can’t be accelerated to the velocitieslarger than λ . We confirm this argument by stationary phase analysis. When h 4: Cold Atoms in Driven Optical Lattices z ( τ ) − λ sin( τ ) be stationaryis ˙ φ ( τ ) = ˙ z τ − λ cos( τ ) = 0 . For a particle with momentum p , the above stationary phase condition isonly fulfilled if λ cos( τ ) = p or | p | < λ The classical Hamiltonian in Eq. (4.4), can be written as H ( m ) = p V J ( λ ) cos(2 z ) − ˜ V ∞ X m = −∞ J m ( λ )[cos(2 z − mτ )+( − m cos(2 z + mτ )] . (4.6)We can neglect the summation terms in Eq. (4.6) under the condition [Moore 1994],˜ V √ λ << , (4.7)and for small initial momentum, atoms follow the equation of motion,¨ z = − ˜ V J ( λ ) sin(2 z ) . (4.8)Which is equation of motion of an un-driven pendulum oscillating with fre-quency q ˜ V | J ( λ ) | and hence the expression for classical period is T cl = π q ˜ V | J ( λ ) | . Since the trapped atoms oscillate with large amplitude aroundlattice minima, the above harmonic approximation for classical period T cl , isnot quite valid. The energy conservation law gives, E = p − ˜ V J ( λ ) cos(2 z ) = − ˜ V J ( λ ) cos(2 z ) , (4.9)where, z represents the position of lattice well and it is considered thatmomentum changes very slowly, which yields p = 2 ˜ V J ( λ )[sin ( z ) − sin ( z )] . We find that dτ = dz q V J ( λ )[sin ( z ) − sin ( z )] , h 4: Cold Atoms in Driven Optical Lattices T = 2 q V | J ( λ ) | Z z dx p sin ( z ) − sin ( z ) ,T = 4 q V | J ( λ ) | F ( π , z ) , where, F ( π , z ) is complete elliptical integral of first kind. For z = 1, F ( π , .
5) = 1 . λ = 10 and ˜ V = 0 .
5, then T = 14 . . Numerically, the dynamics of classical particle are explored by studyingPoincar´e section (which is a stroboscopic map in zp − plane over one periodof external modulation) for fixed value of modulation amplitude. To plotPoincar´e surface section, 200 initial conditions randomly distributed in phasespace are taken and each initial point is evolved for the time, τ = 1000 π . Af-ter each period of phase modulation, the snap shot of position and momentumis taken. Taking advantage of periodicity of the potential V ( z ) = V ( z + π ),the position of the atom is also folded back to the interval [ − π/ , π/ λ , the hyperbolic fixed points survive, while there stable and unsta-ble manifolds don’t coincide. Rather, they intersect each other many timesgenerally in homoclinic points shaping as a homoclinic tangle and chaos isin a narrow chaotic layer around the separatrix which separate bounded andunbounded motion in the un-driven case. Moreover, all orbits with periods,a rational multiples of the period of external modulation are destroyed andreplaced by a chain of hyperbolic (unstable) and elliptic (stable) orbits. Theunstable and stable manifolds of the hyperbolic orbits intersect again andform a heteroclinic tangle in secondary separatrix layers around the ellipticislands i.e., in the domain of stability of the elliptic orbits.This structure is repeated again and again within each elliptic island onrapidly decreasing scale. As modulation strength increases, stochastic regionaround the separatrix increases on the expanse of regular region. While, large h 4: Cold Atoms in Driven Optical Lattices - - - -
1 0 1 -
1 0 1 zppp
Figure 4.1: Poincar´e surface of section for different modulation amplitudeswith ˜ V = 0 . p <
0) and right-running ( p >
0) solutions still existbut all oscillatory orbits, even those whose periods are the “most irrational”multiples of the driving period, are destroyed and the chaotic domain spreadgradually in the whole domain between the right-running ( p >
0) and left-running ( p <
0) regular solution. This is the domain of dominant chaos, witha threshold at some value λ = λ c . Beyond the λ c , almost all available phasespace in chaotic and only few islands of regular motion survive around theelliptic fixed point.However, in the case, where, the condition, ˜ V √ λ << , is satisfied, potential h 4: Cold Atoms in Driven Optical Lattices - - - -
1 0 1 -
1 0 1 zppp
Figure 4.2: Poincar´e surface of section for different modulation amplitudeswith ˜ V = 2.can be expressed by the approximation Eq. (4.7). The resonances reappearand their positions can be recognized by the sign of zero order Bessel function.In Fig-4.1 for modulation amplitudes λ = 7 and λ = 10, it is seen thatwhenever, the Bessel function, J ( λ ) is positive, the resonances are positionedat z = 0 and for negative J ( λ ), these elliptic fixed point are located at z = π .In the previous literature [Moore 1994; Raizen 1999], it is stated thatreappearance of classical resonances is only dependent on condition Eq. (4.7).While, extensive numerical study presented here, reveals that resonancesreappear only when ˜ V < h 4: Cold Atoms in Driven Optical Lattices - - - -
1 0 1 -
1 0 1 ppp z
Figure 4.3: Poincar´e surface of section for different modulation amplitudeswith ˜ V = 5 . λ constant and in-creasing ˜ V from a very small value. Increasing the effective potential ˜ V from zero, orbits with period rationally related to the period of driving forceare destroyed and replaced by resonances. The global chaos condition is nowattained at some critical value of ˜ V . h 4: Cold Atoms in Driven Optical Lattices We consider expansion Eq. (4.6), the terms in summation in this expansiondescribe resonance [Chirikov et al 1982] when p = ˙ z = m , (4.10)with width ∆ p m = 4 s ˜ V | J m ( λ ) | . (4.11)The resonance width exponentially decreases for | m | > λ , and approximatevalue for | J m ( λ ) | . √ πλ can be used when | m | < λ and λ >>
1. Thus amonginfinitely many resonances, only those have considerable size with m . λ .The resonance overlap criteria [Chirikov et al 1982] give the condition of on-set of chaos when ˜ V > √ πλ . (4.12)The effect of driving force on undriven resonance near ˙ z = p = 0 can alsobe understand by finding moving center of driven resonance as ˙ z = λ cos t .The resonance is now a moving region around p = 0 and oscillate withmodulation frequency and amplitude. The width of moving resonance remaintime independent [Chirikov et al 1982] i.e., ∆ p = p V . Thus every pointin the region | p | < λ is crossed by the resonance twice in each driving period.For the case when resonance crosses much faster than the system points, p changes by ∆ p ≃ − √ π ˜ V λ sin( z r ± π/ . (4.13)This gives the diffusion constant D ≃ < ∆ p >T / V λ , (4.14)and it is predicted that due to diffusive spreading of p over the domain | p | < λ , saturated momentum dispersion is < ∆ p > ≃ √ λ. (4.15) h 4: Cold Atoms in Driven Optical Lattices λ = 0 . , . ,
3. In theclassical case a Gaussian distribution of 10000 non-interacting particles isevolved. In both cases (classical and quantum mechanical) plots are averagedvalues computed for different initial positions such that z = − π , − π , − π ,0, π , π , π , initial momentum p = 0, initial momentum dispersion ∆ p = 0 . z = p . From Figs-4.6-4.9, it is seen thatclassical momentum dispersion increases quickly for small modulation andfluctuates around a mean value showing saturation behavior as for small λ ,the diffusion rate, D = ˜ V λ , is larger and velocity of atoms match with thevelocity of driving field and momentum transfer is more likely. As modulationis increased, the saturation time increases as diffusion decreases inverselywith modulation. While saturation value increases as classical resonanceboundary limits the momentum space ∆ p max ≈ λ √ . On the other hand, forfixed value of modulation, classical momentum dispersion, ∆ p cl , saturates athigher value for deep lattice potential. The fluctuation in the classical curvesis due to the presence of resonance (which can be seen in correspondingPoincar´e section) as it trap a part of the ensemble which oscillates inside theresonance with classical period.Classical evolution of non-interacting Gaussian distribution is shown inFig-4.4. In this figure, density plots, position and momentum distributionsof 10000 particles with initial momentum and position dispersion ∆ p = 0 . z = 1 respectively are shown for time τ = 0 (left column) and τ = 200for two different initial conditions: i) When initially distribution is placednear the center of resonance (middle column); ii) When it is located initiallyat stochastic region (right column). Classical dynamics show that when par-ticles are placed inside the resonance region they remain their and only thoseparticles diffuse which are part of distribution tail and initially find them-selves near the separatrix (Fig-4.4-b). Dispersion in position and momentumspace is almost negligible as seen from Figs-4.4-(e),(h). On the other hand,when particles are initially evolved in chaotic region, they quickly diffusesto all available phase space and only a small fraction of particles remains h 4: Cold Atoms in Driven Optical Lattices
2 0-2 -4 0 4 -20 0 20 (a) (e)(d) (c)(b) (f)(i)(h)(g) .08 .04 0 0.3 0 .006 .003 0 .04 .02 0 .02 .01 0 -4 0 4 -100 0 100 -500 0 500-4 0 4 -2 0 2 -4 0 4 z z zzzz ppp -6 0 6 p P ( z ) P ( p ) Figure 4.4: Density plots, position and momentum distributions of 10000non-interacting particles with initial momentum ∆ p = 0 . z = 1. Initial particle density (a), position distribution (d) andmomentum distribution (g) are shown in left column. For modulation λ = 3,when particle is initially placed near the center of resonance, particle density,position and momentum distribution are shown in (b), (e) and (h) respec-tively at τ = 200 in the middle column. While, (c), (f) and (i) show density,position and momentum distribution respectively, when particles are initiallyplaced in chaotic region at time, τ = 200. ˜ V = 2 in this case. h 4: Cold Atoms in Driven Optical Lattices In this section quantum dynamics of Gaussian wave packet are studied indriven optical lattices. Parametric dependencies of momentum dispersionand dependencies on initial excitation condition are studied.
We solve the Schr¨odinger equation using following ansatz for wave function ψ ( z, τ ) = exp[ ik − pz − ik − p τ ] φ ( z, τ ) . (4.16)with the assumption that amplitude φ ( z, τ ) is slowly varying. Using ouransatz we find ik − ∂ψ∂τ = [ p φ − ik − ∂φ∂τ ] exp[ ik − pz − ik − p τ ] , (4.17)and − k − ∂ ψ∂z = [ p φ − ik − p ∂φ∂z − k − ∂ φ∂z ] exp[ ik − pz − ik − p τ ] . (4.18)As φ ( z, τ ) is slowly varying amplitude, we can neglect its second order deriva-tive with respect to position space and get first order differential equation. ∂φ ( z, τ ) ∂τ + ∂φ ( z, τ ) ∂z = ik − ˜ V z − λ sin( τ )] φ ( z, τ ) , (4.19)The solution of this differential equation with initial condition φ ( z , t ) isgiven as φ ( z, τ ) = Z ∞ m = −∞ dz Q ( z, τ, z , τ ) φ ( z , τ ) , (4.20)where, Q ( z, τ, z , τ ) = δ [ z − ¯ z ( z , τ )] exp[ ik − ˜ V Z τ dτ cos[2¯ z ( τ ) − λ sin τ ]] . (4.21) h 4: Cold Atoms in Driven Optical Lattices z ( τ ) = z + p ( τ − τ ) is given by the solution of characteristic equation, d ¯ zdτ = p. (4.22)The exact solution of Eq. (4.19) is φ ( z, τ ) = exp[ ik − ˜ V Z τ dτ cos( z + p (¯ t − τ ) λ sin τ )] φ ( z − pτ, τ = 0) . (4.23)In last equation, we set τ = 0 and eliminated z using equations ¯ z ( τ ) = z + p ( τ − τ ) and z = z + p ( τ − τ ) . We consider a plane wave initially with momentum p and φ ( z, τ = 0) =(2 πk − ) − / . We find the solution ψ p ( z, τ ) = 1 √ πk − exp[ ik − ( p z − p τ + ˜ V Z τ dτ cos( z − p ( τ − ¯ t ) λ sin τ )] . (4.24)By taking Fourier transformation, wave function ψ p ( z, τ ) can be written inmomentum space ψ p ( p, τ ) (see Appendix-6). The momentum distribution | ψ p ( p, τ ) | for an integer number N of modulation periods can be written as P ( p = p + mk − , τ = 2 πN ) = J m ( π ˜ V k − sin( N πp )sin( πp ) J p ( − λ )) (4.25)where, J p is Anger function and momentum distribution is normalized as ∞ X m = −∞ P ( p + mk − ) = ∞ X m = −∞ J n = 1 (4.26)For the case, when p = 0, J ( λ ) = J ( − λ ) = J ( λ ) and J m (0) = δ n, . Inthis case momentum distribution modifies as P ( p = p + mk − , τ = 2 πN ) = J m ( ˜ V τ k − ) if λ = 0, J m ( ˜ V J ( λ ) τ k − ) if λ = 0 . Hence in the absence of modulation ( λ = 0), Bessel function envelope formomentum distribution is obtained. But in the presence of phase modula-tion, the interaction time is modified by the factor J ( λ ), which is depending h 4: Cold Atoms in Driven Optical Lattices p is an integer, J p ( − λ ) = J p ( − λ ) = ( − p J p ( λ ) where, momentum distribution is de-fined as P ( p = p + mk − , τ = 2 πN ) = δ m, if λ = 0, J m ( ˜ V J p ( λ ) τ k − ) if λ = 0 . Now we shall calculate the width of the momentum distribution ∆ p = ( ¯ p − ¯ p ) / ∆ p = ∞ X m = −∞ ( p + mk − ) P ( p + mk − ) − [ ∞ X m = −∞ ( p + mk − ) P ( p + mk − )] , = ∞ X m = −∞ ( p + m k − ) J m ( η ) − p [ ∞ X m = −∞ J m ( η )] , (4.27)here, we have used abbreviated Bessel argument η instead of the argumentin Eq. (4.25). As the summation rule gives ∞ X m = −∞ m J m ( η ) = η , (4.28)the expression for momentum dispersion is∆ p = ( p + k − η / − p . (4.29)Replacing the η by original expression, we get∆ p ( ˜ V , λ, τ = 2 N π ) = π ˜ V √ | sin( N πp )sin( πp ) J p ( − λ ) | . (4.30)The right hand side of the above expression is vanished for the zeros of J p ( − λ ). Actually the dynamics of an atom with momentum p are governedby a potential scaled by Anger function. For the zeros of Anger function,atoms undergo free evolution and ∆ p remains constant.In most experiments, the center of mass motion is approximated by Gaus-sian distribution. By considering Gaussian distribution as initial conditionwith ∆ z and ∆ p as initial dispersion in position and momentum respectively. h 4: Cold Atoms in Driven Optical Lattices
89A normalized minimum uncertainty wave packet initially located at position z with a mean momentum p in position space is given as ψ ( z,
0) = 1(2 π ∆ z ) exp[ − ( z − z ) z + ik − p z ] , (4.31)and in momentum representation ψ ( p,
0) = 1(2 π ∆ p ) exp[ − ( p − p ) p − ik − pz ] . (4.32)The wave function, evaluated by performing minor algebra is ψ ( z, τ ) = 1[2 π (∆ z ( τ )) ] / [ 2∆ z + ik − τ δz − ik − τ ] / exp[ ( z − z ) z − ik − ( z − z ) p + ik − p τ iτk − ∆ p ] × ∞ X m = −∞ b m ( τ ) e imz , (4.33)and position distribution P ( z, τ ) = 1 √ π ∆¯ z exp[ ( z − z − p τ ) z ) ] , (4.34)here, ∆¯ z = ∆ z [1 + ( ∆ p ∆ z ) ] / , (4.35)gives the dispersion in position space with time. The momentum distribution of a minimum uncertainty Gaussian wave packetwith initial width ∆ p = 0 .
5, mean momentum p = 0 is numerically explored.The wave packet is evolved for fixed time, τ = 200, and the results are shownin Fig-4.5 for lattice potentials ˜ V = 0 . , , . ,
8. In Figs-(4.5)-(4.9),results are averaged values computed for different initial positions such that x = − π , − π , − π , , π , π , π , such that wave packet is always adiabaticallyloaded in the lowest energy band of respective lattice. h 4: Cold Atoms in Driven Optical Lattices ∆ p λ V =0.36V = 2V = 5.7V = 8 ~ ~ ~~ Figure 4.5: Momentum distribution vs modulation of wave packet adiabat-ically loaded in the lattice with mean initial momentum p = 0. Otherparameters are ∆ p = 0 . k − = 1The momentum width for shallow optical lattice remain small and sat-urates earlier as compared to deep lattice. In deep lattice, stochastic re-gion grow quickly near the separatrix, resonance islands become smaller andsmaller and disappear. While, in the case of shallow lattice, chaotic regionsdevelop slowly around the separatrix, resonance islands reappear as seen inFig-4.6 for modulation amplitudes λ = 7 ,
10. For lattice potential wherequantum mechanical localization length π ˜ V λk − is smaller than classical reso-nance boundary (see Eq. (4.15)), diffusion in momentum space is limitedby quantum mechanical localization i.e., quantum mechanical suppression ofclassical diffusion. The ∆ p curve in Fig-4.5 for ˜ V = 0 .
36 saturates almostfor the value λ = 2 .
5, in the case of ˜ V = 2, it saturates at λ = 6 . h 4: Cold Atoms in Driven Optical Lattices λ =3 ∆ p cl ∆ p q ∆ p λ =1.5 ∆ p cl ∆ p q λ =0.5 ∆ p cl ∆ p q Figure 4.6: Classical and quantum momentum dispersion verses time for dif-ferent modulation amplitudes for shallow lattice ˜ V = 0 .
36. Other parametersare same as in Fig-4.5
The momentum dispersion is studied for fixed value of modulation ampli-tudes versus time both for shallow and deep lattice potentials. In Figs-4.6 to4.9, time evolution of momentum dispersion versus time is shown for effec-tive lattice potentials ˜ V = 0 .
36, 2, 5 .
7, 8, respectively for fixed modulationamplitudes λ = 0 . , . , p q is smallerthan classical momentum dispersion ∆ p cl as diffusion is suppressed due tolocalization effect of modulation [Bardroff 1995] and as modulation increases,saturation value of classical dispersion also increases. This classical effect canbe understand by studying the respective Poincar´e surface of section Fig-4.1,where, it is seen that global KAM-boundary surrounding stochastic sea and h 4: Cold Atoms in Driven Optical Lattices λ =3 ∆ p cl ∆ p q ∆ p λ =1.5 ∆ p cl ∆ p q λ =0.5 ∆ p cl ∆ p q Figure 4.7: Classical and quantum momentum dispersion verses time fordifferent modulation amplitudes for shallow lattice ˜ V = 2. Other parametersare same as in Fig-4.5regular regions is expanding with modulation and particles from the classicalensemble initially placed in stochastic region explore the the entire stochasticregion but unable to escape from KAM-surfaces. In Fig-4.7, as modulationamplitude increases saturation value of ∆ p q increases but at a value λ = 3,diffusion is suppressed earlier and at the lower value than λ = 1 .
5. It isanalytically shown that the quantum break time increases and localizationlength π ˜ V λk − decreases as λ increases. A similar behavior of classical and quan-tum dispersion in Fig-4.7 for λ = 1 . p q for fixed value ofmodulation also increases. When ˜ V < π ˜ V λk − and diffusion is limited by quantum h 4: Cold Atoms in Driven Optical Lattices λ =3 ∆ p cl ∆ p q ∆ p λ =1.5 ∆ p cl ∆ p q λ =0.5 ∆ p cl ∆ p q Figure 4.8: Classical and quantum momentum dispersion verses time fordifferent modulation amplitudes for the lattice ˜ V = 5 .
7. Other parametersare same as in Fig-4.5localization in this case and ∆ p q increases as λ increases; ii) The classicalresonance boundary is smaller than quantum localization length, dispersionin momentum space increases with modulation. Quantum localization effectdictates ∆ p to increase following localization length for small values of latticepotential but as soon as the classical boundary becomes smaller than quan-tum localization length, dispersion grows only due to resonance broadeningwith the growth in ˜ V . In this section, we extend the Floquet formalism discussed in chapter-1, tononlinear resonances and derive expressions for energy spectrum. In the pe-riodically driven potentials energy is no more a constant of motion. For thereason we solve the time dependent Schr¨odinger equation by using secular h 4: Cold Atoms in Driven Optical Lattices λ =3 ∆ p cl ∆ p q ∆ p λ =1.5 ∆ p cl ∆ p q λ =0.5 ∆ p cl ∆ p q Figure 4.9: Classical and quantum momentum dispersion verses time fordifferent modulation amplitudes for deep lattice ˜ V = 8. Other parametersare same as in Fig-4.5perturbation approximation as suggested by Max Born [Born 1960]. There-fore, the solution is obtained by averaging over rapidly changing variables.This leads us to find out a partial solution of the periodically driven systems.As a result, we find quasi energy eigen-states and the quasi eigen energies ofthe dynamical system for non-linear resonances.In order to study the quantum nonlinear resonances of the TDS, we con-sider the scaled Hamiltonian, H = H + λV ( x ) sin( τ ) . (4.36)Classically, this Hamiltonian is characterize by a single dimensionless param-eter λ . In quantum mechanical solutions, another scaled parameter, scalingPlank’s constant, ~ appears. The parameter λ controls the degree of non-integrability while, scaled Plank’s constant controls the scale at which itsquantum mechanical counterpart can resolve phase space structures. h 4: Cold Atoms in Driven Optical Lattices N th resonance can be written in the form [Saif 2001;Flatt´e 1996; Berman 1977], | ψ ( τ ) i = X n C n ( τ ) | n i exp {− i [ E ¯ n + ( n − ¯ n ) k − N ] τk − } , (4.37)here, E ¯ n is the mean energy, C n ( τ ) is time dependent probability amplitude,¯ n is mean quantum number, k − is scaled Plank’s constant and | n i are eigenstates of undriven system. On substituting (4.37) in the time dependentSchr¨odinger equation, we find that the probability amplitude C n ( τ ) , changeswith time following the equation, ik − ˙ C n ( τ ) = [ E n − E ¯ n − ( n − ¯ n ) k − N ] C n ( τ ) + λV i ( C n + N − C n − N ) . Where, V = V n − N ≈ V n + N are off-diagonal matrix ele-ments and are approximately constant near the potential minima for tight-binding approximations.We take the initial excitation such that it is narrowly peaked around themean value, ¯ n . For the reason, we take slow variations in the energy, E n ,around the ¯ n in a nonlinear resonance, and expand it up to second orderin Taylor expansion. Thus, the Schr¨odinger equation for the probabilityamplitudes, C n ( τ ), is ik − ˙ C n = k − ( n − ¯ n )( ω − N ) C n ( τ ) + 12 k − ( n − ¯ n ) ζ C n ( τ )+ λV i ( C n + N − C n − N ) . (4.38)In (4.38), fast oscillating terms are averaged out and only the resonant onesare kept. Parameters ω = ∂E n k − ∂n | n =¯ n , and ζ = ∂ E n k − ∂n | n =¯ n are the frequency andnon-linearity of the time independent system respectively.We introduce the Fourier representation for C n ( τ ) as, C n ( τ ) = 12 N π Z Nπ g ( θ, τ ) e − i ( n − ¯ n ) θ/N dθ, (4.39)which helps us to express (4.38) as the Schr¨odinger equation for g ( θ, τ ),such that ik − ˙ g ( θ, τ ) = H ( θ ) g ( θ, τ ). Here, the Hamiltonian H ( θ ) is givenas, H ( θ ) = − N k − ζ ∂ ∂θ − iN k − ( ω − N ) ∂∂θ − λV sin θ . In order to obtain thisequation, we consider the function g ( θ, τ ) as 2 N π periodic, in θ coordinate. h 4: Cold Atoms in Driven Optical Lattices g ( θ, τ ), as g ( θ, τ ) = ˜ g ( θ ) e − iετk − . Therefore, Schr¨odingerequation for g ( θ, t ) reduces to the standard Mathieu equation [Saif 2006],[ ∂ ∂θ + a − q cos 2 θ ]˜ g ( θ ) = 0 , (4.40)here, ˜ g = χ ( z ) exp ( − i N ω − z/N ζ k − ) and θ = 2 z + π/
2. The Mathieucharacteristic parameters [Abramowitz 1970; McLachlan 1947] are a = 8 N k − ζ [ ( N ω − N ζ + E µ,ν ] , (4.41)and q = 4 λVN k − ζ . (4.42)The π -periodic solutions of Eq. (4.40) correspond to even functions of theMathieu equation whose corresponding eigenvalues are real [Abramowitz 1970].These solutions are defined by Floquet states, i.e., ˜ g ( θ ) = e iµθ P µ ( θ ), where, P µ ( θ ) = P µ ( θ + π ), and µ is the characteristic exponent. In order to have π -periodic solutions in ˜ g ( θ ), we require µ to be defined as µ = µ ( j ) = 2 j/N ,where, j = 0 , , , ...., N − µ ( j ) can exist as a characteristic exponent of so-lution to the Mathieu equation for discrete ν (which takes integer values)only for certain value a ν ( µ ( j ) , q ), when q is fixed. Hence, with the help ofEq. (4.41), we obtain the values of unknown E . Therefore, we may expressthe quasi energy of the system as [Breuer and Holthaus 1991] E µ,ν = (cid:20) N k − ζ a ν ( µ ( j ) , q ) + k − ˜ αj (cid:21) mod k − ω, (4.43)where, the index ν takes the definition ν = n − ¯ n ) N , and ˜ α, defines the windingnumber. In this section we discuss two cases correspond to nonlinear resonances. Weconsider weakly coupled q < q ≫ h 4: Cold Atoms in Driven Optical Lattices q < ν [Saif 2001] and in q ≫ ν ,near the center of resonance matrix elements are constant. In the followingdiscussion, we analyze the wave packet dynamics in these regimes and showtheir parametric dependencies.The time scales, T ( j ) at which recurrences of an initially well-localizedwave packet occur depend on the quasi-energy of the respective system, Therecurrence times are obtained as, T ( j ) = 2 π/ Ω ( j ) , hence, the values of j as j = 1 , , ... , correspond to, respectively, classical, quantum, super, andhigher order revival times. With the help of Eq. (4.41) and Eq. (4.43), weobtain the frequencies Ω ( j ) asΩ (1) = 1 k − (cid:26) ∂ E µ,ν ∂µ + ∂ E µ,ν ∂ν (cid:27) , Ω (2) = 12! k − (cid:26) ∂ E µ,ν ∂µ + 2 ∂ E µ,ν ∂µ∂ν + ∂ E µ,ν ∂ν (cid:27) , (4.44)Ω (3) = 13! k − (cid:26) ∂ E µ,ν ∂µ + 3 ∂ E µ,ν ∂µ ∂ν + 3 ∂ E µ,ν ∂µ∂ν + ∂ E µ,ν ∂ν (cid:27) . After a few mathematical steps Eq. (4.44) reduce toΩ (1) = 1 k − (cid:26) ∂ E µ,ν ∂ν + αk − (cid:27) , Ω (2) = 12! k − (cid:26) ∂ E µ,ν ∂ν + αk − ∂ E µ,ν ∂ν (cid:27) , Ω (3) = 13! k − (cid:26) ∂ E µ,ν ∂ν + 3 αk − ∂ E µ,ν ∂ν (cid:27) . (4.45)which lead to classical period, T (1) = T ( cl ) λ , quantum revival time, T (2) = T ( rev ) λ , and super revival time, T (3) = T ( spr ) λ , when calculated at the meanvalues. Delicate Dynamical Recurrences
The condition, q <
1, may be satisfied in the presence of weak perturbationdue to external periodic force (and/or), for large nonlinearity and/or for largeeffective Plank’s constant [Saif 2001; Saif 2005a; Saif 2005; Iqbal 2006]. TheMathieu characteristic parameters, a ν and b ν are given [Abramowitz 1970],as h 4: Cold Atoms in Driven Optical Lattices a ν ≃ b ν = ν + q ν −
1) + ... . for ν > ν and are verygood approximations when ν is of the form, m + . In case of integral valueof ν , i.e., ν = m , the series holds only up to the terms not involving ν − m in the denominator.The energy spectrum for weakly modulated periodic potentials [Saif 2006]can be defined using Eq. (4.43) and Eq. (4.46). The relations obtained forclassical period, quantum revival time and super revival time, in the presenceof small perturbation for primary resonance N = 1, index j takes the value, j = 0, and time scale are simplified as T ( cl ) λ = (1 − M ( cl ) ) T ( cl )0 ∆ , , (4.47) T ( rev ) λ = (1 − M ( rev ) ) T ( rev )0 , , (4.48) T ( spr ) λ = πω λV ζ ∆ (1 − µ ) µ , (4.49)here, T ( cl )0 = πω is classical period and T ( rev )0 = 2 π/ ( k − ζ ) is quantum revivaltime in the absence of external modulation. Furthermore, ∆ = (1 − ω N ω ).The modification factors are given as M ( cl ) = −
12 ( λV ζ ∆ ω ) − µ ) , and M ( rev ) = 12 ( λV ζ ∆ ω ) µ (1 − µ ) , here, µ = k − ζ ∆2 ω is re-scaled non-linearity. Robust Dynamical Recurrences
The condition, q ≫
1, may be satisfied in the presence of large amplitude, λ ,of external modulation by periodic force in a dynamical system. In addition,we may get the regime by considering a system with very small linearity i.e., ζ ≈ k − be very small. Quasi h 4: Cold Atoms in Driven Optical Lattices a ν ≈ b ν +1 ≈ − q + 2 s √ q − s + 12 − s + 3 s √ q − ..., (4.50)where, s = 2 ν + 1 . The energy spectrum of nonlinear resonances for strongly modulated pe-riodic potentials can be defined using Eq. (4.43) and Eq. (4.50). Here, in thedeep potential limit, the band width is, b ν +1 − a ν ≃ ν +5 q π q ν + exp( − √ q ) ν ! . (4.51)Keeping lower order terms in s , in Eq. (4.50), we get quasi energy spectrumfor nonlinear resonances as ( ν + ) k − ω h , which resembles to the harmonicoscillator energy spectrum for ω h = 2 √ V .Similarly the relations for classical period, quantum revival time and su-per revival time for strongly driven case are obtained as, T ( cl ) λ = 4 πN k − ζ √ q n − v +18 √ q − v +1) +32 q o + 2 α , (4.52) T ( rev ) λ = 32 πN ζ n v +1)16 √ q + 8 αk − √ q − (2 v + 1) αk − o , (4.53) T ( spr ) λ = 32 πN ζ α [1 − αk − √ q { αk − (2 v + 1)2 } ] . (4.54)For primary resonance N = 1, index j takes the value, j = 0, and timescale are simplified as T ( cl ) λ = T ( cl )0 ∆8 µ (cid:20) − (4 + µ ) √ ζ µ √ q − (4 + µ ) ζ (8 µ ) q (cid:21) , (4.55) T ( rev ) λ = T ( rev )0 (cid:20) − µ + 4)16 µ √ q + 9(4 + µ ) (16 µ ) q (cid:21) , (4.56)and T ( spr ) λ = 2 π √ qk − ζ . (4.57) h 4: Cold Atoms in Driven Optical Lattices We consider ultracold atoms in standing wave field with phase modulationdue to acousto-optic modulator. The atoms in the presence of phase modu-lation experience an external force, thus exhibit dispersion both in classicaland quantum domain. In quantum dynamics atoms experience an additionalcontrol due to effective Plank’s constant. Here, we report in long time evo-lution, material wave packet display quantum recurrence phenomena.As time scales of driven optical lattice are expressed in terms of undrivensystem parameters, for the reason, we investigate, classical period, quan-tum revival time and super revival time scales of undriven optical lattice[Ayub et al 2009; Drese et al 1997; Dyrting 1993].In the case of undriven shallow optical potential classical period is T ( cl )0 = { q n − } π ¯ n , (4.58)where, q = V ω r is rescaled potential depth in units of recoil energy. Thequantum revival time is T ( rev )0 = 2 π { − q n + 1)(¯ n − } , (4.59)and super revival time is T ( spr )0 = π (¯ n − q ¯ n (¯ n + 1) , (4.60)In shallow lattice potential limit, i.e., q .
1, neglecting the higher orderterms in q the classical frequency, ω = 2¯ n { − q n − } and non-linearity, ζ = 2 + q n +1(¯ n − . On the other hand for deep optical lattice, the classicaltime period is T ( cl )0 = π √ q { s √ q + 3( s + 1)2 q } , (4.61)where, s = 2¯ n + 1. The quantum revival time T ( rev )0 = 4 π (1 − s q ) , (4.62) h 4: Cold Atoms in Driven Optical Lattices T ( spr )0 = 32 π √ q . (4.63)In deep optical potential limiting case, i.e., q >>
1, the classical frequencyis ω = 4( √ q − n +18 ) and non-linearity is ζ = | − − n +1)2 √ q | . Now for weakly driven shallow or deep lattice q .
1. Time scales for primaryresonance N = 1 are [Ayub et al 2011] T ( cl ) λ = T ( cl )0 (cid:20) q { l + β ) − } (cid:21) ∆ , (4.64) T ( rev ) λ = T ( rev )0 (cid:20) − q l + β ) + 1 { l + β ) − } (cid:21) , (4.65)and T ( spr ) λ = π { l + β ) − } ζ k − q ( l + β ) { l + β ) + 1 } . (4.66)Where, T ( cl )0 = πω ( l + β ) is the classical time period, T ( rev )0 = πk − ζ is quantumrevival time for unmodulated system and β = Nω − N k − ζ . In this case ω and ζ are frequency and non-linearity of respective undriven lattice. Here, timescales for weakly driven shallow lattice and weakly driven deep lattice looksimilar but there behavior is not similar as parameters ω and ζ are differentdue different energy spectrum corresponding to undriven lattices.Behavior of classical periods, quantum revivals and super revivals of mat-ter waves in modulated optical crystal in nonlinear resonances versus modu-lation is shown in Fig-4.10. In each plot of this figure, left vertical axis showsthe time scales when shallow optical lattice is modulated and right axis showsthe time scales when deep optical lattice is modulated. The upper row ofFig-4.10 represents the time scales for small q values i.e., delicate dynamicalrecurrences as a function of modulation λ , while, lower row represents thetime scales when q ≫
1, as a function of modulation λ , i.e., robust dynam-ical recurrences. Here, left column shows the results related to the classicalperiods. Quantum revival times are plotted in middle column, while, rightcolumn shows super revival times. h 4: Cold Atoms in Driven Optical Lattices T (spr) T (spr) T (rev) T (cl) T (cl) T (rev) Shallow LatticeDeep Lattice
Figure 4.10: Left panel: Classical time period versus λ for weak externalmodulation (a) and for strong external modulation (b). Middle panel: Quan-tum revival time versus λ for weak external modulation (c) and for strongexternal modulation (d). Right panel: Super revival time versus λ for weakmodulation (e) and for strong modulation (f). In this figure, for deep lattice V = ˜16 E r , for shallow lattice ˜ V = 2 E r and k − = 0 . h 4: Cold Atoms in Driven Optical Lattices λ = 0, (b) , .
5, (c) 1 . V = 2 E r ;Lower row: (d) λ = 0, (e) 0 .
5, (f) 1 . V = 16 E r .itative and quantitative behavior of revival time is almost similar. Classicalperiod and quantum revival time for delicate dynamical recurrences showgood numerical and analytical resemblance for the system with our previouswork [Saif 2005a; Iqbal 2006; Saif 2006].To have an idea about the classical dynamics of the system, we plot thePoincar´e surface of section for shallow optical lattice ( ˜ V = 2 E r ) with modula-tion strengths λ = 0, 0 .
5, 1 and for deep lattice ( ˜ V = 16 E r ) with modulationstrengths λ = 0, 0 .
5, 1 . λ >
0. This resonance emerges when thetime period of external force matches with period of unperturbed system.One effect of an external modulation is the development of stochastic regionnear the separatrix. As modulation is increased, while the frequency is fixed,the size of stochastic region increases at the cost of regular region.In order to observe the numerical dynamics of a quantum particle insidea resonance, we evolve a well localized Gaussian wave packet in the driven h 4: Cold Atoms in Driven Optical Lattices N = 1. This can be realized in experimental setup of Mark Raizen at Austin, Texas. The authors [Moore 1994] worked withsodium atoms to observe the quantum mechanical suppression of classicaldiffusive motion and employed the (3 S , F = 2) → (3 P , F = 3) transition at589 nm , with ω / π = 5 . × Hz.
The detuning was δ L / π = 5 . × Hz.
The recoil frequency of sodium atoms was ω r / π = 25 kHz for selected laserfrequency and modulation frequency was chosen ω m / π = 1 . M Hz , whereas,the other parameters were k − = 0 .
038 and q = 55 (or ˜ V = 0 . S → P transition in cesium atoms setting λ L =852 nm and ω / π = 3 . × Hz . The detuning was δ L / π = 3 × Hz, with q up to 1 .
5. The recoil frequency was ω r / π = 2 . kHz , so that adriving frequency ω m / π = 10 Hz , three orders of magnitude lower thanin the former experiment, gives k − ≈
4, taking the dynamics to the deepquantum regime. We numerically investigate the validity of our results withknown theoretical and experimental [Moore 1994] results with dimensionlessrescaled Plank’s constant k − = 0 .
16, ˜ V = 0 . , which is a case of strongmodulation to a deep optical lattice of potential depth V = 28 . E r . Thewave packet is initially well localized in such a way that localization lengthis less than or order of lattice spacing.Fig-4.12 shows spatio-temporal evolution of an initially well localizedwave packet in a lattice potential well. Fig-4.12-(a) is spatio-temporal dy-namics of atomic wave packet for λ = 0 .
05, while Fig-4.12-(b) presents thecase, for external modulation λ = 0 .
1. Spatiotemporal evolution of wavepacket in optical lattice shows that wave packet diffuses to the neighboringlattice sites by tunneling and splits into small wavelets. Later, these waveletsconstructively interfere and wave packet revival takes place. It is clear thatas the modulation increases, revival time changes as shown in Fig-4.10 whichis due to a change in interference pattern and confirms analytical results asdiscussed above. h 4: Cold Atoms in Driven Optical Lattices λ = 0 . λ = 0 . V = 2 ∆ p = 0 . k − = 0 . . Dark regions show represent the maximum probability to find the particle.
On the other hand, for strongly driven optical lattice q ≫
1. Time scalesfor the atomic wave packet for primary resonance with N = 1 are given as[Ayub et al 2011] T ( cl ) λ = 2 πk − ζ {√ q − l + β ) + 18 } , (4.67) T ( rev ) λ = 8 πk − ζ (cid:20) − { l + β ) + 1 } √ q (cid:21) , (4.68)and T ( spr ) λ = 32 π √ qk − ζ . (4.69)In case of strongly driven lattice, when external modulation frequency isclose to the harmonic frequency, matrix elements, V can be approximated by h 4: Cold Atoms in Driven Optical Lattices q can be approximatedas q ≈ √ n +1 λq k − ζ [Drese et al 1997] . Under this approximation time scales are T ( cl ) λ = 16 πq / √ ζ n + 1) √ λ − { l + β ) + 1 } q k − √ ζ , (4.70) T ( rev ) λ = 8 πk − ζ " − { l + β ) + 1 } q k − √ ζ n + 1) √ λ , and T ( spr ) λ = 64 π (¯ n + 1) √ λk − ζ q . (4.71)When lattice is strongly modulated by an external periodic force, the clas-sical period decreases as modulation increases. Classical period for stronglydriven optical lattice is given by Eq. (4.67). The behavior of classical periodfor strongly driven lattice versus modulation is qualitatively of the same or-der for both strongly driven shallow lattice and strongly driven deep latticeas shown in Fig-4.10(b). The behavior of classical period in strongly drivenlattice case is understandable as strong modulation influence more energybands of undriven lattice to follow the external frequency and near the cen-ter of nonlinear resonance the energy spectrum is almost linear, as can beinferred from Eq. (4.43) and Eq. (4.50) with assumptions q ≫ l issmall, i.e., wave packet is placed near the center of resonance.Quantum revival time in nonlinear resonances versus modulation is shownin middle column of Fig-4.10. For robust dynamical recurrences, the behav-ior of quantum revival time is given by Eq. (4.68). The qualitative behaviorof revival time for strongly driven shallow lattice is different from that ofstrongly driven deep lattice Fig-4.10(d), as in the later case, change in re-vival time is almost one order of magnitude larger than the former case, forequal changes in modulation. Here, revival time differently depends on λ for shallow and deep lattice in the case of strong modulation. The energyspectrum of deep and shallow optical lattices are different i.e. frequenciesassociated with the lattice spectrum are different for the two cases. For deeplattice case energy spectrum (band structure) near the bottom is almost h 4: Cold Atoms in Driven Optical Lattices λ = 3 . Other parameters are ˜ V = 0 .
36 ∆ p = 0 . k − = 0 .
16. Here, white regions show the maximum probability to find the particle andregions with dark color has higher probability than the lighter colors.equally spaced and spacing decreases as band index increases (shown in Fig-2.2). In addition, band gaps are wider near the bottom of the lattice potentialcompared to shallow lattice and β = Nω − N k − ζ is positive near the center ofprimary resonance ( N = 1). Here, ω is transition frequency between thetwo lowest energy levels. The contribution from second term in Eq. (4.68)decreases as λ or q increases and revival time increases. This can also beexplained as follow. When an external modulation of some fixed frequency ω m is applied and modulation is strong enough then lattice frequencies whichare slightly different from modulation frequency, follow modulation frequencyand spectrum near the nonlinear resonances is linear as more and more bandsfollow the external modulation and energy spectrum near the resonance cen-ter have minimum non-linearity. Hence, stronger the modulation, smaller the h 4: Cold Atoms in Driven Optical Lattices β for N = 1 is negative and sign before sec-ond term in Eq. (4.68)) becomes positive. Now as λ or q is increased, secondterm decreases and hence revival time decreases in this case.Fig-4.13 shows spatio-temporal evolution of an initially well localizedwave packet in a lattice potential well. Fig-4.13-(a) is for the spatio-temporaldynamics of atomic wave packet in the absence of periodic modulation, whileFig-4.13-(b) presents the case when external modulation, λ = 3. Spatio-temporal dynamics of atomic wave packet shows that revival time changesas given analytical expression (4.68) plotted in Fig-4.10.The super revival time behavior versus modulation, shown in right col-umn of Fig-4.10. For robust dynamical recurrences Eq. (4.69) gives the timescale for super revivals. The super revival for robust dynamical recurrencesincreases with modulation as shown in the Fig-4.10(f). Here, qualitative be-havior of super revival time is same for strongly driven shallow lattice andstrongly driven deep lattice but quantitatively super revival time increasesalmost two times faster.Square of auto-correlation function ( | A ( t ) | ) for the minimum uncer-tainty wave packet is plotted as a function of time in Fig-4.14 and Fig-4.15for λ = 1 . λ = 0 . q = 27 . p = ∆ z = 0 . k − = 0 . V = 16 E r . In these figures upper inset isan enlarged view which displays classical periods, while lower inset of thefigures show quantum revivals. Lower panel shows the existence of superrevivals. The classical period, quantum revival time and super revival timeare indicated by arrows and showing there characteristics. Numerical resultsare in good agreement with analytical expressions. Figures-4.12 and 4.13show spatiotemporal dynamics of Gaussian wave packet inside a resonancefor V = 16 E r and V = 2 E r respectively. For other parameters, see figurecaption. These numerical results are in agreement with analytical results. h 4: Cold Atoms in Driven Optical Lattices λ = 0 . k − = 0 .
5, ˜ V = 16 E r , ∆ z = ∆ p = 0 . In this chapter, classical and quantum dynamics are studied. Classical dy-namics are explored by studying Poincar´e sections, classical momentum dis-persion and time evolution of Gaussian distribution. It is noted that for˜ V <
1, the periodically driven lattice have classical dynamics, neither reg-ular nor completely chaotic but displays an intricate dominant regular dy-namics and dominant stochastic dynamics one after the other as a functionof increasing modulation amplitude. While, for effective lattice potential,˜ V ≥
1, the intricate dominant regular and dominant stochastic dynamics h 4: Cold Atoms in Driven Optical Lattices λ = 1 . k − = 0 .
5, ˜ V = 16 E r , ∆ z = ∆ p = 0 . p <
1) and right running( p >
1) regular solutions, increases with modulation. These regular solutionsbounding stochastic region have an upper bound on momentum dispersion.Quantum dynamics are explored by studying quantum momentum dis-persion, temporal and spatio-temporal behavior of wave packet in inside non-linear resonances that exist in the classical counterpart. Dynamics of wavepacket inside nonlinear resonances show dynamical recurrences behavior. Asa function of time the recurrence behavior of wave packet is analyzed close to h 4: Cold Atoms in Driven Optical Lattices q ≫ hapter 5Condensate in Driven OpticalLattices In this chapter, we study the dynamics of condensate in driven optical latticesby analyzing dispersive behavior in position and momentum space. The dis-persion behavior gives the parametric limits where, the condensate is stablein driven lattices. These stability limits are verified by studying the spatio-temporal dynamics of the condensate in driven lattice numerically. Latter,revival times are studies for different interaction regimes.The dynamics of a condensate in driven one dimensional optical lattice,strongly confined by radial trap is governed by the Hamiltonian [Staliunas et al 2006;Eckardt 2010], H = p M + V o k L { x − ∆ L sin( ω m t ) } ] + g D | ˜ ψ | , where, k L is wave number, V o defines the potential depth of an optical lattice,∆ L and ω m are amplitude and frequency of external derive, whereas, M isthe mass of an atom. Furthermore, g D = ~ k L ω ⊥ a s defines the effective twobody interaction coefficient, ω ⊥ is radial trap frequency and a s is inter-atomics-wave scattering length.The unitary transformation (a special case of Kramers-Hennebergertransformation) ˜ ψ = ψ ( x, t )( x, t ) exp[ i ~ { ω m M ∆ L cos( ω m t ) x + β ( t ) } ] , where, The unitary transformation is time periodic and preserves the quasi-energy spectrum. h 5: Condensate in Driven Optical Lattices β ( t ) = ω m ∆ L M [ sin(2 ω m t )2 ω m + t ] , to a frame co-moving with the lattice, modifiesthe Hamiltonian as H = p M + V o k L x − F x sin ω m t + g D | ψ | , (5.1)where, F = M ∆ Lω m is amplitude of inertial force emerging in the oscillat-ing frame. To examine the dynamics of condensate in driven optical latticesnumerically, Hamiltonian (5.1) is expressed in dimensionless quantities. Wescale the quantities so that z = k L x, τ = ω m t , ψ = ψ √ n , where, n is aver-age density of a condensate. Multiplying the Schr¨odinger wave equation by ω r ~ ω m , where, ω r = ~ k L M is single photon recoil frequency, we get dimensionlessHamiltonian, viz.,˜ H = − k − ∂ ∂z + ˜ V o z + λz sin τ + G | ψ | . (5.2)Here, G = g D n k − ~ ω m , and λ = F d k − ~ ω m = k L ∆ L, is scaled modulation amplitude.Also ˜ V o = V o k − ~ ω m , and τ is scaled time in the units of modulation frequency, ω m .In the tight binding regime, wave function ˜ ψ can be be expressed inWannier states. Furthermore, as condensate is loaded in lowest vibrationalstate of optical lattice, ˜ ψ can be expanded in terms of lower band Wannierfunctions, ˜ ψ = X j c j ( z, τ ) w o ( z − jd ) , (5.3)where, w o ( z − jd ) is first band Wannier function centred at jth lattice siteand c j is complex amplitude of mini condensate associate with jth latticesite.Substituting Eq. (5.3) in Eq. (5.2), we get discrete GPE i ∂c j ∂τ = − J ( c j +1 + c j − ) + λ sin τ j c j + G | c j | , (5.4)with k − = 1, J = − R dzw ∗ o ( z ) H o w o ( z − d ) dz, (5.5) H o = − ∂ ∂z + ˜ V o sin(2 z ) . (5.6) h 5: Condensate in Driven Optical Lattices J is tunneling or the hoping matrix element between nearest neigh-boring lattice sites. In Eq. (5.4) the overall energy shift ε o is given by ε o = R dz w ∗ o ( z ) H o w o ( z ) dz, (5.7)which is set to zero.In the case of dilute condensate, i.e., G = 0, Eq. (5.4) describe the dy-namics of single-particle whose analytical solutions can explicitly be derivedusing the following gauge transformation [Kolovsky et al], c j ( τ ) → exp [ − i j ( λ sin( τ ))] ˜ c j ( τ ) . (5.8)Then the Eq. (5.4) becomes i ∂ ˜ c j ( τ ) ∂t = − J ( e − iD ( τ ) ˜ c j +1 ( τ ) + e iD ( τ ) ˜ c j − ( τ )) , (5.9)with, D ( τ ) = λ sin( τ ). Sine this gauge transformation retrieves back thetranslation symmetry in the above equation, we can consider the spatiallyperiodic boundary conditions, i.e., ˜ c j ( τ ) = ˜ c L + j ( τ ). Now the correspondingsemi-classical Hamiltonian is H ( τ ) = − J X j (cid:0) e − iD ( τ ) ˜ c ∗ j ˜ c j +1 ( t ) + e iD ( τ ) ˜ c ∗ j +1 ˜ c j ( τ ) (cid:1) . (5.10)We use the Bloch-wave representation to obtain the solutions for ˜ c j ( τ ),˜ c j ( τ ) = L − / X k e ikj b k , (5.11)where, the quasi-momentum is defined as k = 2 πn/L and L represents latticelength or number of lattice sites in discrete version of GPE such that − π ≤ k < π , n = 0 , ± , . . . , ± ( L − / L is odd, and n = 0 , ± , . . . , ± L/ L is even. Now equations showing the time evolution of b k take thesimple form i ∂b k ∂τ = − J cos [ k − λ sin( τ )] b k , (5.12)with solutions expressed as b k ( τ ) = b k (0) exp { i J [cos( k ) S ( τ ) − sin( k ) C ( τ )] } . (5.13) h 5: Condensate in Driven Optical Lattices b k (0) is the integral constant, and S ( τ ) and C ( τ ) are defined as follow S ( τ ) = + ∞ X m = −∞ sin( mτ ) m J m ( λ ) , (5.14) C ( τ ) = + ∞ X m = −∞ cos( mτ ) − m J m ( λ ) , (5.15)where, J m is the m th order Bessel function of the first kind.The the inverse Fourier transformation can also give analytical solutions c j for G = 0. These solution can be the starting points to derive the trialsolutions for G = 0. Our concern is about b k (0) = δ k,p case, which gives thefinal expressions of Eq. (5.4) as c j ( τ ) = exp { i j [ p − λ sin( τ )] }× exp { i J [cos( p ) S ( τ ) − sin( p ) C ( τ )] − igτ } . (5.16)These solutions are applicable for L → ∞ in which the boundary conditionsmake no difference. The stability of BEC in driven optical lattice is studied taking into accountthe mean field dynamics of condensate by [Creffield 2009] for large modu-lation frequency with the condition that ~ ω m is less than the energy gapbetween two lowest bands. For the undriven case with repulsive interaction,the dynamical instability does not occur [Wu et al 2003]. As λ →
0, the crit-ical value of interaction G cr at which instability occur approaches to infinity.With the increase of modulation strength λ , the critical value for interaction, G cr , abruptly drops and touches a wide local minimum value. Then it at-tains a maximum value at first root ( λ = 2 . J , as in tight binding regime tunneling of condensate is suppressed andstability is regained. Beyond the first root of zeroth-order Bessel function,the condensate becomes unstable for any positive value of interaction. Nearthe second root of Bessel function another maximum of G cr value is attained h 5: Condensate in Driven Optical Lattices (a) ˜ V = 0 .
36 (b) ˜ V = 2 Figure 5.1: Momentum and position dispersion of condensate vs modulation.The dark blue region corresponds to minimum dispersion and thus maximumdensity, whereas, bright red colour displays maximum dispersion leading tominimum stability. A systematic colour drops from dark red to light redindicates the variation of dispersion from minimum to maximum value inmomentum space (upper row) and position space (lower row). The dispersionis measured at τ = 100 π (after 50 periods of modulation) for k − = 1, ∆ p = 0 . J ( λ ) and dotted line showsfunction given in Eq. (5.17). h 5: Condensate in Driven Optical Lattices J eff = J J ( λ ). When λ is larger than 2.4048, then J eff isnegative and J eff G cr becomes negative and dynamical instability can occur.The physical significance of this change in sign is experimentally realized[Lignier et al 2007], as it caused the momentum distribution function to bediscretely shifted by π . Hence, the tunneling not only renormalized in am-plitude, but also acquires a phase-factor of e iπ [Creffield and Sols 2008]. Asimilar feature was explored in the analysis of a condensate in acceleratedlattice [Zheng et al 2004a].In the next section, we explore dispersion of condensate in driven lat-tice as a function of modulation and interaction strength for a fixed timeboth in position and momentum space. Suppression of dispersion in posi-tion and momentum displays the localization and hence the stability of thecondensate. the momentum and position dispersion of repulsive condensate in drivenoptical lattice as a function of interaction and modulation amplitude. Weprovide density plots to explain dispersion as function of modulation andinteraction. In these density plots, the dark blue regions corresponds tominimum dispersion and thus maximum density, whereas, bright red colourdisplays maximum dispersion leading to minimum stability. A systematiccolour drops from dark red to light red indicates the variation of dispersionfrom minimum to maximum value in momentum space (upper row) andposition space (lower row). In Fig-5.1, we plot momentum and positiondispersion for effective potential ˜ V = 0 .
36 and ˜ V = 2. In Fig-5.1a, weconsider ˜ V = 0 .
36 and in Fig-5.1b, we take ˜ V o = 2 . In this figure the darkline is a plot of zeroth-order Bessel function while dotted line represents the h 5: Condensate in Driven Optical Lattices f ( λ ), which is sum of first three order Bessel functions of first kindand defined as f ( λ ) = J ( λ ) + J ( λ ) + J ( λ ) + J ( λ ) . (5.17)For small value of ˜ V , where, the condition, ˜ V √ λ , is satisfied, dispersion isminimum at the zeroes of zeroth-order Bessel. For the values of modulationand effective potential where condition ˜ V √ λ is not satisfied, zeros of the func-tion expressed by the Eq. (5.17) define dispersion minima. Near the zeros ofthe Bessel function, momentum and position dispersion is very small whichshows that instabilities in this parameter regime are suppressed due to dy-namical localization (DL) [Creffield 2009]. In DL and coherent destructionof tunneling (CDT), the initial localized quantum state never diffuses undera periodic external field. In this aspect, these phenomena are similar. How-ever, the dissimilarities are: i) CDT is derived approximately in an extremecase of a small value of the transfer matrix element while DL is an exactresult obtained in an infinite driven system and is valid irrespective of themagnitudes of the transfer matrix element. On the other hand, in DL theinitial distribution oscillates around the initial value whereas, in the CDT,the initial distribution is frozen [Kayanuma and Saito 2008].Now as the interaction is increased, the dispersion stays constant forsome critical value of G cr beyond which dispersion increases with interac-tion. Beside stable regions, there exists regions which display maximum dis-persion. These are unstable regions for condensate. Our findings are in goodagreement with earlier obtained results, for example in Ref: [Creffield 2009],where, it is shown that for small external modulation, G cr falls quickly tozero while in our case dispersion remain small for small modulation strengthand small interaction G cr . Moreover, the position dispersion has large val-ues for large G and dispersion minima disappear for small critical value ofinteraction G cr .In Fig-5.4, momentum and position dispersion versus modulation andinteraction is shown for ˜ V = 5 . h 5: Condensate in Driven Optical Lattices (a) ˜ V = 0 .
36 (b) ˜ V = 2 Figure 5.2: Momentum and position dispersion of condensate vs modulationfor ω m = 2. The dispersion is measured at τ = 100 π (after 50 periods ofmodulation) for k − = 1, ∆ p = 0 .
1. In this graph, solid line shows zero orderBessel function of first kind J ( λ ) and dotted line shows behavior of firstthree terms in function given by Eq. (5.17). h 5: Condensate in Driven Optical Lattices V , is scaled by modulationfrequency, ω m , and for large modulation frequency, ω m , effective potential issmall. If we consider the large modulation frequency, near the roots of Besselfunction the value of critical interaction G cr , increases and windows of mini-mum dispersion is widen up as shown in Fig-5.2a and Fig-5.2b provided that ~ ω m is not larger than the energy gap between the lowest energy bands. InFig-5.2a, the number of dispersion minima both in momentum and positionare large compared to Fig-5.1a. It is noted that these minima exit at the ze-ros of J ( λ ) + J ( λ ) + J ( λ ). With increase in modulation frequency ω m , therole of higher order terms are decreased and for sufficient large modulation,potential minima exist at the zeros of first order Bessel function as dicussedin [Creffield 2009].To understand the dispersion behavior fully, we excite minimum un-certainty condensate for different initial conditions i.e., the points in Fig-5.1a, where, the dispersion is minimum and a point where, dispersion ismaximum. Fig-5.3 shows time evolution of momentum and position dis-persion for modulation and interaction, ( λ, G ) = (7 . , , (7 . , , (8 . , λ, G ) = (8 . , , (8 . , . λ, G ) = (7 . , λ, G ) = (7 . , λ, G ) = (8 . ,
0) or ( λ, G ) = (8 . , .
5) as these points lie in minimum momen-tum dispersion regions. The dispersion in position increases slowly with timefor ( λ, G ) = (8 . , .
5) as this point doesn’t lie in the minimum dispersion re-gion. The small wiggling behavior seen in momentum dispersion is due tocollapse and revival of mini-condensates trapped in each effective potentialminima due to tunneling suppression. This suppression in tunneling is dueto coherent destruction of tunneling around the zeros of Bessel function. h 5: Condensate in Driven Optical Lattices ∆ x τλ = 7.7, G=0 λ = 7.7, G=2 λ = 8.4, G=0 λ = 8.4, G=0.5 λ = 8.4, G=2 0.5 1 1.5 ∆ p Figure 5.3: Momentum and position dispersion of condensate vs timefor modulation and interaction, ( λ, G ) = (7 . , , (7 . , , (8 . ,
2) where,dispersion is maximum in position and momentum space. Whereas, for( λ, G ) = (8 . , , (8 . , .
5) dispersion is negligible in momentum space whileit is slower in position space compare to the previous case as shown in Fig-5.1.Other parameters are same as in Fig-5.1a. h 5: Condensate in Driven Optical Lattices τ = 100 π (after 50 periods of modulation) for˜ V = 5 .
7. Other parameters are same as in Fig-5.1
In this section, spatio-temporal dynamics of condensate are studied by ini-tializing a Gaussian condensate in driven optical lattice. Fig-5.5 shows thespatio-temporal dynamics of condensate initially excited in localized win-dows for (
G, λ ) = (0 , . , ( G, λ ) = (2 , . G, λ ) = (0 , , ( G, λ ) = (2 , h 5: Condensate in Driven Optical Lattices z is plotted at time, τ = 0 , ,
100 for the two contrasting regions with λ = 1 . λ = 9. When condensate is excited with parameters belong to the regionwhere relevant density plot show minimum dispersion for λ = 1 .
5, no signif-icant change in spatio-temporal profile and probability is noted with time.On the other hand, when condensate is excited with parameters belong tothe region where relevant density plot show maximum dispersion for λ = 9, itshows dispersive behavior both in spatio-temporal profile and density plots. In this section, we study dynamical recurrences of a Bose-Einstein condensatein optical crystal subject to periodic external driving force. The recurrencebehavior of the condensate is analyzed as a function of time close to nonlinearresonances occurring in the classical counterpart. Our mathematical formal-ism for the recurrence time scales is presented as: delicate recurrences whichtake place for instance when lattice is weakly perturbed; and, robust recur-rences which may manifest themself for sufficiently strong external drivingforce. The analysis is not only valid for dilute condensate but also applica-ble for strongly interacting homogeneous condensate provided, the externalmodulation causes no significant change in density profile of the condensate.We explain parametric dependence of the dynamical recurrence times whichcan easily be realized in laboratory experiments. In addition, we find a goodagreement between the obtained analytical results and numerical calcula-tions.For the dynamics of weakly interacting BECs in optical lattice the in-teraction term can safely be neglected [Eckardt 2010], which is a situation h 5: Condensate in Driven Optical Lattices (a) Spatio-temporal dynamics of condensate for λ = 1 .
5, interaction G = 0 , λ = 9,interaction G = 0 ,
2. Other parameters are same as in Fig-5.4(b) Wave function snapshot for different times t = 0 , ,
100 for the two cases: leftside λ = 1 . λ = 9. Figure 5.5: Spatio-temporal dynamics of condensate (upper row) and stro-boscopic plots of probability at different times (lower row). h 5: Condensate in Driven Optical Lattices ˜ V o cos 2 z + G | ψ | seenby each atom may as well be written [Choi 1999] as, V eff = ´ V z ) + const, where, ´ V = ˜ V o G .The analytical result, calculated using perturbation theory [Choi 1999], isvalid as long as the condensate density is nearly uniform, i.e., ´ V << V o or a strong atomic interaction G . The condition is experimentally confirmed for one dimensional potential[Morsch 2001]. t (b)(a)(c) (ms) Figure 5.6: Spatio-temporal behavior of atomic condensate for q = 0( λ = 0),´ V = 2 (a) q = 0 . β , ´ V = 2(b) and q = 0 . β , ´ V = 0 . p = 0 . β = k − ζ V and k − = 1 . Dark regions in this figure show higherdensity.Fig-5.6 shows spatio-temporal evolution of an initially well localized con-densate in a crystal potential well. Fig-5.6a is spatio-temporal dynamics in h 5: Condensate in Driven Optical Lattices q = 0 . β and different value of q o . Spatio-temporal evolutionof the condensate in optical crystal shows that condensate diffuses to theneighboring lattice sites by tunneling and splits into smaller wavelets. Later,these wavelets constructively interfere and condensate revival takes place.Recurrence time calculated numerically is the same as obtained from ana-lytical results. Keeping modulation constant as q = 0 . β but for different´ V , which may be a consequence of varied atom-atom interaction, recurrencetime is modified. We note that in the absence of interaction term, G , revivaltime changes with ´ V and interference pattern is similar. But with the intro-duction of interaction term not only revival time is modified due to changein ´ V , interference pattern is also modified as seen in Fig-5.6c.Fig. 5.7 shows spatio-temporal evolution of an initially well localizedwave packet in a lattice potential well inside a resonance for ´ V = 16. Fig-5.7a is for the spatio-temporal dynamics of atomic condensate in the absenceof periodic modulation, Fig-5.7b presents the case for external modulation, q = 3 β , while, Fig-5.7c shows the behavior of condensate for q = 9 β . In theFig-5.7c only classical periods are seen as in deep effective potential higherorder time scales are very large. The quantum revival times in Fig-5.7 seennumerically are the same as obtained analytically in Eq. (4.68).The suggested theoretical results may be realized in experimental setup of recently performed experiments at Pisa [Lignier et al 2007], where,dynamical control of matter wave tunneling is studied in strongly shakenoptical crystals. A BECs of about 5 × Rb atoms was evolved in adipole trap which was realized using two intersecting Gaussian laser beamsat 1030 nm wavelength and a power of around 1 W per beam focused to waistsof 50 µ m. After obtaining pure condensate, trap beams were readjusted toobtain elongated condensates with the trap frequencies (80 Hz in radial and ≈
20 Hz in the longitudinal direction). Along the axis of one of the dipole trapbeams a one-dimensional optical crystal potential was introduced and thepower of the lattice beams ramped up in 50 ms in order to avoid excitations ofthe BEC to the non-condensed atoms. The optical crystals used was createdusing two counter-propagating Gaussian laser beams ( λ L = 852 nm) with h 5: Condensate in Driven Optical Lattices q = 3 β (b)and when modulation q = 9 β (c). Other parameters are ´ V = 0 .
36, ∆ p = 0 . k − = 0 .
16. Here, white regions show the maximum density and regionswith dark color has higher density than the lighter colors.120 µ m waist and a resulting optical crystal spacing d L = λ L / . µ m.The depth V o of the resulting periodic potential is measured in units of E r = ~ π / (2 M d L ). In laboratory, accessible quantity like scaled optical latticedepth ´ V ranges from 1 to 20.For optical crystal with potential depth ´ V = 2, and ´ V = 16, the meanseparation of the two lowest bands is ≈ . kHz and 20 . kHz , respectively.For the driving frequency, ω m / π , ranging from 3 kHz − kHz , the rescaledPlank’s constant k − ranges from 0 .
668 to 2 . hapter 6Discussion In this thesis, we considered the center of mass dynamics of ultra-cold atomsand Bose-Einstein condensate in one dimensional optical lattice, both in theabsence and presence of external modulation.In chapter-2, we discuss the center of mass dynamics of the single particlewave packet in the absence of external modulation. We provide analyticalstudy in two regimes, that is deep optical lattice and shallow lattice poten-tial. In this chapter, we first time discuss the wave packet behavior and theirparametric dependence both analytically and numerically. In deep latticecase, the energy bands collapse to a single level as tunneling is suppressednear the lattice potential minima. In this regime, we observe collapse andrevival behavior in wave packet dynamics. As for sufficiently deep lattice, en-ergy spectrum is linear near the bottom of lattice potential, a single particlewave packet revives after each classical period. Beyond linear regime, en-ergy dependence is quadratic and wave packet dynamics in this region showscomplete quantum revivals enveloping many classical periods, however, be-yond this region quantum revival time is no longer constant but decreasesas ¯ n increases, where, ¯ n , is mean band index. This dependence of revivaltime on mean quantum number can be associated with the non-linearity inthe energy spectrum. For sufficiently deep lattice potential, near the lat-tice depth, non-linearity is almost zero as we move up from lowest energyband to the higher one, non-linearity emerges and dictates the revival time.Since, revival time is inversely proportional to non-linearity, the quantum128 h 6: Discussion n increases. This behavior of wave packet revivalcan only be seen in sufficiently deep lattice if we are away from the top of thelattice potential where, continuum in energy is seen and semi-classical limitin reached at which according to Correspondence Principle quantum revivaltime excepted to diverge. The higher order time scale exist in the region ofchanging non-linearity. super revival time in this region can be seen. Againthis region expands as potential height is increased but super revival time inthis region is directly proportional to square root of potential height. Abovethis region other time scales also exist where, super revival time decreases as¯ n increases but these time scales are too long to consider them finite.In shallow optical lattices, quantum tunneling plays its role and we en-counter with wide energy bands instead of energy levels. Two energy scales,band gap and band width play active role in the dynamics of the system. Ifthe band gap is larger than band width, tight binding approximations canbe applied. But in situations where, lattice is very shallow, next to nearesttunneling can’t be ignored and tight binding is no longer valid. The wavepacket in this regime shows dispersive behavior in position space with timecausing a decrease in revival amplitude.In chapter-3, we discuss center of mass dynamics of the condensate inan optical lattice for the sake of completeness of the discussion. It is firsttime shown that now nonlinear phenomena like solitonic behavior and self-trapping of condensate can be explained by studying momentum and spatialdispersion, spatio-temporal behavior and wave packet revivals of the conden-sate. The inherent non-linearity of a condensate due to interatomic inter-actions and Bragg scattering of a matter-wave by an optical lattice play itsrole in the dynamics. Localization is possible as condensate shows anomalousdispersion at the edges of a Brillouin zone of the lattice and magnitude of thisdispersion can be managed. Therefore, the condensate spreading either canbe controlled by actively controlling the lattice parameters or by utilizing theatom-atom interaction. The second approach leads to non-linearly localizedstates.For periodically driven optical lattice, classical and quantum dynamicsare explored. Classical dynamics are explained by studying Poincar´e surface h 6: Discussion V , and square root ofmodulation amplitude, λ is much smaller than unity and can effectively beexplained by zeroth-order Bessel function of first kind.Quantum dynamics of a wave packet inside non-linear resonances showsdynamical recurrences. As a function of time the recurrence behavior ofwave packet is analyzed close to nonlinear resonances. Our mathematicalframework based on Floquet theory is developed in two classes: delicaterecurrences which take place when lattice is weakly perturbed; and robustrecurrences which manifest for sufficiently strong external driving force. An-alytical expression for the classical period, revival time and super revivaltime are developed. Classical periods of a wave packet initially localizednear the center of resonance increases with modulation for delicate dynam-ics. While, quantum revivals and super revivals decreases with modulation.Similarly, robust dynamics condition is satisfied when shallow or deep po-tential is strongly modulated. Classical periods in robust case decreases andsuper revival times increase with modulation. Here, quantum revival timefor the case when deep lattice is strongly modulated increases with modu-lation, while, it decreases when shallow lattice is strongly modulated. Thedifference in the behavior is due to the contrast in energy spectrum of un-driven lattices. When modulation is increased in deep lattice case more andmore energy levels are influenced by external modulation non-linearity inthe energy spectrum near the center of resonance decreases and revival timeincreases. Parametric dependence of analytical results are confirmed by ex-act numerical solutions both for delicate and robust dynamical recurrences.Temporal and spatio-temporal dynamics shows that the non-linearity in theenergy spectrum of the undriven system, and the initial conditions on the h 6: Discussion G cr . Now as the interaction is increased, the dispersion stays constantfor some critical value of G cr beyond which dispersion increases with interac-tion. This critical value of interaction increases as the modulation frequencyincreases. Beside stable regions, there exists regions which display maximumdispersion. These are the regions where condensate is dynamically unstable.Our analysis for classical period, quantum revival time and super revivaltime represented for single particle wave packet is also valid for sufficientlydilute BECs or the condensate for which the inter-particle interaction canbe tuned to zero by exploiting Feshbach resonance. In addition, we suggestthat the analysis is valid for strongly interacting homogeneous condensatesas well where the nonlinear term can be replaced by an effective potentialprovided the external modulation causes slight changes in the density profileof the condensate. ppendices Appendix A: Solution of an Arbitrary Poten-tial
An arbitrary potential U ( r ) , around its minima can be solved by taking itsTaylor’s expansion [Liboff 2002], that is U ( r ) = U ( r m ) + G (1) ( r − r m ) + G (2) ( r − r m ) + G (3) ( r − r m ) + .., (6.1)where, G ( j ) = ( j !) − ∂ j U ( r = r m ) /∂r j , and j is an integer. The value of G ( j ) for odd j is zero as potential is cos( x ) and it is calculated at the potentialminima r = r m . Thus in the presence of weak non-linearity the term G (6) < 1) ( n − 2) ( n − 3) ( n − 4) ( n − 5) ( n − 6) ( n − ,δ = (6 n − p n ( n − 1) ( n − 2) ( n − 3) ( n − 4) ( n − ,δ = (cid:0) n − n + 7 (cid:1) p n ( n − 1) ( n − 2) ( n − ,δ = (56 n − n + 214 n − p n ( n − ,δ = (56 n + 396 n + 838 n + 645)8 p ( n + 1) ( n + 2) ,δ = (31 n + 197 n + 258)16 p ( n + 1) ( n + 2) ( n + 3) ( n + 4) ,δ = (11 n +27)24 p ( n + 1) ( n + 2) ( n + 3) ( n + 4) ( n + 5) ( n + 6) , and δ = √ ( n +1)( n +2)( n +3)( n +4)( n +5)( n +6)( n +7)( n +8)32 . Hence, following the same procedure, the first order correction due to H (6) term changes the Hamiltonian of the system as H = H + H (4) + H (6) . (6.6)Hence the corrected eigen function in presence of correction due to H (4) and H (6) terms appear as φ ( s ) n = φ n + φ (1 ,a ) n + φ (1 ,b ) n + φ (2 ,a ) n , where, φ (2 ,a ) n = D [ χ φ n − + χ φ n − + χ φ n − + χ φ n +2 + χ φ n +4 + χ φ n +6 . Here, D = G (6) ( √ q ) ~ ω h ,χ = 6 p n ( n − n − n − n − n − ,χ = 34 (2 n − p n ( n − n − n − ,χ = 152 ( n − n + 1) p n ( n − ,χ = 152 ( n + 3 n + 3) p ( n + 1)( n + 2) ,χ = 34 (2 n + 5) p ( n + 1) ( n + 2) ( n + 3) ( n + 4) ,χ = 2 p ( n + 1) ( n + 2) ( n + 3) ( n + 4) ( n + 5) ( n + 6) . ppendix A H (4) using first order and second order perturbation theoryrespectively. The result is as under: E (4) n = ( α n + α n + α ) ~ ω h , where, α = 3 D a , α = 6 D a , α = 6 D a , C b = ( G (3) ) ( √ q ) ~ ω h and D a = G (4) ( √ q ) . At next order, the first order perturbation of H (6) and secondorder perturbation of H (4) contribute. The result can be written as: E (6) n = ( β n + β n + β n + β ) ~ ω h , where, β = 3 I a − J b , β = 8 I a − J b , β = 6 I a − J b , β = 4 I a − J b , and I a = 5 G (6) ( √ q ) ~ ω h , J b = 2( G (4) ) ( √ q ) ( ~ ω h ) . At the next higher order, we need to evaluate three contributions; H (8) infirst order, H (6) and H (4) in second order and H (4) in third order. The energyexpression is E (8) n = ( γ n + γ n + γ n + γ n + γ ) ~ ω h , where, γ = 3 X − Y − Z, γ = 8 X − Y − Z, γ = 10 X − Y − Z,γ = 4 X − Y − Z, γ = 2 X − Y − Z, and X = 35 G (8) ( √ q ) ~ ω h , Y =30 G (6) G (4) ( √ q ) ( ~ ω h ) , Z = 48( G (4) ) ( √ q ) ( ~ ω h ) . Now the energy of the system is E n = E (0) n + E (4) n + E (6) n + E (8) n , or E n = ( κ n + κ n + κ n + κ n + κ ) ~ ω h + U ( r m ) , (6.7)where, κ = α + β + γ + , κ = α + β + γ + 1 , κ = α + β + γ ,κ = β + γ , and κ = γ . ppendix-B Appendix B: Anger function and distributionfunctions In this appendix, we solve the equation, ψ p ( z, t ) = 1 √ πk − exp[ ik − ( p z − p t ] exp[ i ˜ V k − ϕ ( z, t )] , (6.8)where, phase ϕ ( z, t ) is defined as ϕ ( z, t ) = Z t dτ cos( z − p ( t − τ ) λ sin τ ) . (6.9)We solve the above equation to find momentum distribution by consideringinteraction time as a integer multiple of 2 π . We decompose the integral into integrals of interval 2 π and find ϕ ( z, t = 2 N π ) = Re [ N − X ν =0 e − ip πN Z π ( ν +1)2 πν dτ exp[ i ( p ( τ ) − λ sin τ ] e iz , (6.10)which is modified when we replace τ ′ = τ − πνϕ ( z, t = 2 N π ) = Re [ N − X ν =0 e ip πν e − ip πN Z π dτ ′ exp[ i ( p ( τ ′ ) − λ sin τ ′ ] e iz . (6.11)As N − X ν =0 e ip πν = e ip πN − e ip πν − e ip π ( N − sin( N πp )sin( N πp ) , (6.12) ϕ ( z, t = 2 N π ) = sin( N πp )sin( πp ) Re [ Z π dτ ′ exp[ i { p ( τ ′ − π ) − λ sin τ ′ } ] e i ( z − p πN ) ] . (6.13)Substituting τ = τ ′ − π, in above equation yields Anger function , J p ( ξ ) = 12 π Z π − π dτ exp[ i { p τ − ξ sin τ } = 1 π Z π dτ cos[ p τ − ξ sin τ ] , (6.14)and hence the phase ϕ ( z, t = 2 N π ) = sin( N πp )sin( πp ) J p ( − λ ) cos( z − p πN ) . (6.15) ppendix-B t =2 N π . We place Eq. (6.14) in equation (6.8) and exploit the Jocobi-Anger identity (4.3), we find the wave function ψ p ( z, t = 2 N π ) = 1 √ πk − ∞ X m = −∞ B m exp[ ik − ( p + mk − ) z ] exp[ − ik − p N π ] , (6.16)where, b − m = ( ie − ip Nπ ) n J m ( π ˜ V k − sin( N πp )sin( πp ) J p ( − λ )) . (6.17)The Fourier transformation ψ p ( p, t ) = 1 √ πk − Z ∞∞ dz exp[ ik − z ] ψ p ( z, t ) , (6.18)of wave function in Eq. 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Ketterle, Vortices and superfluidity in a stronglyinteracting Fermi gas, Nature , 1047 (2005). ibliography Fermionic superfluidity with imbalanced spin populations, Science , 492 (2006). ist of Publications Muhammad Ayub , Khalid Naseer, Manzoor Ali and Farhan Saif, Atom optics quantum pendulum, J. Rus. Las. Res. , 205 (2009).2. Muhammad Ayub , Khalid Naseer and Farhan Saif, Dynamical recurrences based on Floquet theory of nonlinear resonances, Euro Phys. J. D , 491 (2011).3. Muhammad Ayub and Farhan Saif, Delicate and robust dynamical recurrences of matter waves in driven opticalcrystals, Phys. Rev. A , 023634, (2012).4. Muhammad Ayub and Farhan Saif, Spatio-temporal dynamics of condensate in driven optical lattices, to be submitted.5. Muhammad Ayub and Farhan Saif, Quantum chaos with cold atoms in driven optical lattices, submitted.6. Khalid Naseer, Muhammad Ayub and Farhan Saif, Atomic bullets: Coherent, non-dispersive, and accelerated matter waves, submitted.7. Muhammad Ayub , Kashif Ammar and Farhan Saif, Dynamic localization of Bose-Einstein condensate in optomechanics, submitted. 166 cknowledgements First and for most, I am thankful to my thesis supervisor Dr. Farhan Saif onmany accounts, including his continual encouragement and persistent guid-ance into this fascinating and encapsulating field of atom optics. I am alsogreatly indebted to Prof. Dr. Azhar Abbas Rizvi and Dr. Qaiser AbbasNaqvi, Chairman Department of Electronics, for their inspirational teach-ings, kind and cooperative attitude.Fruitful discussions with the group fellows, namely, Dr. Rameez, Dr.Shahid, Khalid, Inam, Tasawar, Manzoor, Javed, Jamil, Adnan, Umar, Kashifand Asjad have helped a lot to enrich my conceptual horizons and I submitmy sincere gratitude to all of them. I would also like to acknowledge otherPh.D fellows namely, Dr. Ghaffar, Dr. Shakeel, Mustansar, Dr. Ehsan andArshad for general discussions on research areas other than quantum optics.I feel strongly indebted to my parents, my family for their love, bestwishes, encouragement and unconditional support without which I wouldhave been unable to complete anything worth while.I am grateful to Higher Education Commission, Government of Pakistanas well for the financial support through Indigious Scholarship Scheme.