One-dimensional Bose-Einstein Condensate Research Articles

Effect of quasiparticle renormalization on the localization of the excitations of a one-dimensional Bose–Einstein condensate in a random potential

We investigate the effects of renormalization on the localization of the quasiparticle excitations of one-dimensional Bose–Einstein condensate in a random potential. Starting with a set of linearized equations of motion for the phases of superfluid grains coupled by Josephson interactions, we use mode-counting techniques to calculate the inverse localization length for large (10 8) arrays. Employing distributions for the interaction parameters that are the same as the initial (pre-renormalization) distributions used by Gurarie et al. (Phys. Rev. Lett. 101 (2008) 170407), we compare the initial-interaction results for the localization length with those obtained using renormalization group techniques.

Quantum pumping of interacting bosons

Quantum pumping in an interacting one-dimensional Bose-Einstein condensate is analyzed. It is shown that a spatially periodic potential, oscillating adiabatically in time with frequency ${\ensuremath{\omega}}_{0}$, acts as a quantum pump inducing an atom current from broken spatiotemporal symmetries of the driven potential. The current generated by the pump is strongly affected by the interactions. It has a power-law dependence on the frequency with the exponent determined by the interaction, while the coupling to the pump affects the amplitudes. It depends on the phase difference between two umklapp terms of the drive, providing evidence for the full quantum character of the boson transport. The results suggest the realization of a quantum pump with laser-cooled atoms.

Dark soliton in one-dimensional Bose—Einstein condensate under a periodic perturbation of trap

The perturbation of a confining trap leads to the collective oscillation of a Bose—Einstein condensate, thereby the propagation of a dark soliton in the condensate is affected. In this study, periodic perturbation is employed to match the soliton oscillation. We find that the soliton dynamics depends sensitively on the coupling between the moving direction of the trap and that of the soliton. The soliton energy/depth evolves periodically, and a relevant shift in the soliton trajectory occurs as compared with the unperturbed case. Overall, the soliton oscillation frequency changes little even if the perturbation amplitude and frequency vary.

Finite-size effects in Anderson localization of one-dimensional Bose-Einstein condensates

We investigate the disorder-induced localization transition in Bose-Einstein condensates for the Anderson and Aubry-Andr\'e models in the noninteracting limit using exact diagonalization. We show that, in addition to the standard superfluid fraction, other tools such as the entanglement and fidelity can provide clear signatures of the transition. Interestingly, the fidelity exhibits good sensitivity even for small lattices. Effects of the system size on these quantities are analyzed in detail, including the determination of a finite-size-scaling law for the critical disorder strength in the case of the Anderson model.

Chaos in a Bose—Einstein condensate

It is demonstrated that Smale-horseshoe chaos exists in the time evolution of the one-dimensional Bose—Einstein condensate driven by time-periodic harmonic or inverted-harmonic potential. A formally exact solution of the time-dependent Gross—Pitaevskii equation is constructed, which describes the matter shock waves with chaotic or periodic amplitudes and phases.

Nonautonomous matter-wave solitons near the Feshbach resonance

By means of analytical and numerical methods, we reveal the main features of nonautonomous matter-wave solitons near the Feshbach resonance in a one-dimensional Bose-Einstein condensate confined by a harmonic potential with a varying-in-time longitudinal trapping frequency. Based on the generalized nonautonomous Gross-Pitaevskii model, we show that solitons in nonautonomous physical systems exist only under certain conditions so that varying-in-time nonlinearity and confining harmonic potential cannot be chosen independently; they satisfy the exact integrability scenarios and complement each other. We focus on the most physically important examples where the applied magnetic field is either a linearly or a periodically varying-in-time function. In the case of periodically varying scattering length, variations of confining harmonic potential are found to be sign-reversible (periodic attractive and repulsive) to support the soliton-management regime. We investigate the losses of validity of one-dimensional (1D) approximation in the cases when, by the joint action of varying-in-time nonlinearity and confining potential, the atom cloud can be compressed from an initially elongated quasi-1D cigar-shaped geometry to a final ball-shaped three-dimensional geometry and the induced soliton collapse may occur.

Dark soliton in the Bose–Einstein condensates with nonlinearity and harmonic potential managements

The dynamics of dark solitons in one-dimensional Bose–Einstein condensates under the nonlinearity and harmonic potential managements is investigated. It is found that at the large particle limit the macroscopic wave function could evolve self-similarly, which provides a time-varying background for the propagation of dark solitons. The approximate dark soliton solution is derived and its center-of-mass motion is predicted analytically.

Bloch oscillations of Bose-Einstein condensates: Quantum counterpart of dynamical instability

We study the Bloch dynamics of a quasi one-dimensional Bose-Einstein condensate of cold atoms in a tilted optical lattice modeled by a Hamiltonian of Bose-Hubbard type: The corresponding mean-field system described by a discrete nonlinear Schr\"odinger equation can show a dynamical (or modulation) instability due to chaotic dynamics and equipartition over the quasimomentum modes. It is shown, that these phenomena are related to a depletion of the Floquet-Bogoliubov states and a decoherence of the condensate in the many-particle description. Three different types of dynamics are distinguished: (i) decaying oscillations in the region of dynamical instability, and (ii) persisting Bloch oscillations or (iii) periodic decay and revivals in the region of stability.

Dark soliton beats in the time-varying background of Bose-Einstein condensates

We investigate the dynamics of dark solitons in one-dimensional Bose-Einstein condensates. In the large particle limit, by introducing the lens-type transformation, we find that the macroscopic wave function evolves self-similarly when its initial profile strays from that of the equilibrium state, which provides a time-varying background for the propagation of dark solitons. The interaction of dark solitons with this kind of background is studied both analytically and numerically. We find that the center-of-mass motion of the dark soliton is deeply affected by the time-varying background, and the beating phenomena of dark soliton emerge when the intrinsic frequency of the dark soliton approaches that of the background. Lastly, we investigate the propagation of dark solitons in the freely expanding background.

Distillation of a one-dimensional Bose-Einstein condensate

We study the dynamics of a one-dimensional Bose-Einstein condensate located in a time-dependent double-well trapping potential. In particular we investigate the way the system discovers existence of a ground state created in the new deeper well. It was shown that the transfer of the system into the minimum of the potential is triggered by the appearance of the condensate in the new well. Only then the thermal cloud follows the condensed part. During the transfer the eigenvectors of the single-particle density matrix have components localized simultaneously in both wells, which indicates a partial coherence between the two parts of the system.

Soliton stability and collapse in the discrete nonpolynomial Schrödinger equation with dipole-dipole interactions

The stability and collapse of fundamental unstaggered bright solitons in the discrete Schrodinger equation with the nonpolynomial on-site nonlinearity, which models a nearly one-dimensional Bose-Einstein condensate trapped in a deep optical lattice, are studied in the presence of the long-range dipole-dipole (DD) interactions. The cases of both attractive and repulsive contact and DD interaction are considered. The results are summarized in the form of stability/collapse diagrams in the parametric space of the model, which demonstrate that the the attractive DD interactions stabilize the solitons and help to prevent the collapse. Mobility of the discrete solitons is briefly considered too.

Density, phase and coherence properties of a low dimensional Bose-Einstein systems moving in a disordered potential

We present a detailed numerical study of the dynamics of a disordered one-dimensional Bose-Einstein condensates in position and momentum space. We particularly focus on the region where non-linearity and disorder simultaneously effect the time propagation of the condensate as well as the possible interference between various parts of the matter wave. We report oscillation between spatially extended and localized behavior for the propagating condensate which dies down with increasing non-linearity. We also report intriguing behavior of the phase fluctuation and the coherence properties of the matter wave. We also briefly compare these behavior with that of a two-dimensional condensate. We mention the relevance of our results to the related experiments on Anderson localization and indicate the possibility of future experiments

Chaotic shock waves of a Bose-Einstein condensate

It is demonstrated that the well-known Smale-horseshoe chaos exists in the time evolution of the one-dimensional Bose-Einstein condensate (BEC) driven by the time-periodic harmonic or inverted-harmonic potential. A formally exact solution of the time-dependent Gross-Pitaevskii equation is constructed, which describes the matter shock waves with chaotic or periodic amplitudes and phases. When the periodic driving is switched off and the number of condensed atoms is conserved, we obtained the exact stationary states and nonstationary states. The former contains the stable ``nonpropagated'' shock wave, and in the latter the shock wave alternately collapses and grows for the harmonic trapping or propagates with exponentially increased shock-front speed for the antitrapping. It is revealed that the existence of chaos plays a role for suppressing the blast of matter wave. The results suggest a method for preparing the exponentially accelerated BEC shock waves or the stable stationary states.

Dephasing in two decoupled one-dimensional Bose-Einstein condensates and the subexponential decay of the interwell coherence

We provide a simple physical picture of the loss of coherence between two coherently split one-dimensional Bose-Einstein condensates. The source of the dephasing is identified with nonlinear corrections to the elementary excitation energies in either of the two independent condensates. We retrieve the result by Burkov, Lukin and Demler [Phys. Rev. Lett. 98, 200404 (2007)] on the subexponential decay of the coherence ∝exp [-(t/t0)2/3] for the large time t, however, the scaling of t0 differs.

Quantum lattice Boltzmann simulation of expanding Bose-Einstein condensates in random potentials

The phenomenon of Anderson localization in expanding one-dimensional Bose-Einstein condensates is investigated by numerically solving the Gross-Pitaevskii equation with a random speckle potential. To this purpose, a quantum lattice Boltzmann (QLB) method is used, and compared with a standard Crank-Nicolson scheme. The QLB simulations show evidence of Anderson localization even for relatively low-energy condensates, with a healing length as large as one-tenth of the Thomas-Fermi length. Moreover, very long-time simulations, lasting up to 15 000 optical confinement periods, indicate that the Anderson localization degrades in time, although at a very slow pace. In particular, the inverse localization length is found to decay according to a t;{-1/3} law. This lends support to the idea that localized wave functions, although not strictly ground states, represent extremely long-lived metastable states of the expanding condensate.

Resonant energy transfer in Bose–Einstein condensates

We consider the dynamics of a dilute, magnetically-trapped one-dimensional Bose–Einstein condensate whose scattering length is periodically modulated with a frequency that linearly increases in time. We show that the response frequency of the condensate locks to its eigenfrequency for appropriate ranges of the parameters. The locking sets in at resonance, i.e., when the effective frequency of driving field is equal to the eigenfrequency, and is accompanied by a sudden increase of the oscillations amplitude due to resonant energy transfer. We show that the dynamics of the condensate is given, to leading order, by a driven harmonic oscillator on the time-dependent part of the width of the condensate. This equation captures accurately both the locking and the resonant energy transfer as it is evidenced by comparison with direct numerical simulations of original Gross–Pitaevskii equation.

Numerical study of one-dimensional and interacting Bose–Einstein condensates in a random potential

We present a detailed numerical study of the effect of a disordered potential on a confined one-dimensional Bose–Einstein condensate, in the framework of a mean-field description, using a highly efficient and fast converging numerical scheme. For repulsive interactions, we consider the Thomas–Fermi and Gaussian limits and for attractive interactions the behaviour of soliton solutions. We find that the average spatial extension of the stationary density profile decreases with an increasing disorder strength both for repulsive and attractive interactions among bosons. In the Thomas–Fermi limit, a strong localization of the bosons is obtained in momentum space around the state k = 0. The time-dependent density differs considerably in the cases we have considered. For attractive and disordered Bose–Einstein condensates, we show evidence of a bright soliton with an overall unchanged shape, but a disorder-dependent width. For weak disorder, the soliton is delocalized and for stronger disorder, it bounces back and forth between high potential barriers.

Explicit solutions to an effective Gross–Pitaevskii equation: One-dimensional Bose–Einstein condensate in specific traps

An effective Gross–Pitaevskii equation, which describes the dynamics of quasi-one-dimensional Bose–Einstein condensates in specific potential traps, is considered, and new families of exact solutions are reported, which include periodic and solitary waves. The solutions are applied to the description of BEC patterns trapped in optical-lattice potentials.

Ground state of the time-independent Gross–Pitaevskii equation

We present a suite of programs to determine the ground state of the time-independent Gross–Pitaevskii equation, used in the simulation of Bose–Einstein condensates. The calculation is based on the Optimal Damping Algorithm, ensuring a fast convergence to the true ground state. Versions are given for the one-, two-, and three-dimensional equation, using either a spectral method, well suited for harmonic trapping potentials, or a spatial grid. Program summary Program title: GPODA Catalogue identifier: ADZN_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADZN_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 5339 No. of bytes in distributed program, including test data, etc.: 19 426 Distribution format: tar.gz Programming language: Fortran 90 Computer: ANY (Compilers under which the program has been tested: Absoft Pro Fortran, The Portland Group Fortran 90/95 compiler, Intel Fortran Compiler) RAM: From <1 MB in 1D to ∼ 10 2 MB for a large 3D grid Classification: 2.7, 4.9 External routines: LAPACK, BLAS, DFFTPACK Nature of problem: The order parameter (or wave function) of a Bose–Einstein condensate (BEC) is obtained, in a mean field approximation, by the Gross–Pitaevskii equation (GPE) [F. Dalfovo, S. Giorgini, L.P. Pitaevskii, S. Stringari, Rev. Mod. Phys. 71 (1999) 463]. The GPE is a nonlinear Schrödinger-like equation, including here a confining potential. The stationary state of a BEC is obtained by finding the ground state of the time-independent GPE, i.e., the order parameter that minimizes the energy. In addition to the standard three-dimensional GPE, tight traps can lead to effective two- or even one-dimensional BECs, so the 2D and 1D GPEs are also considered. Solution method: The ground state of the time-independent of the GPE is calculated using the Optimal Damping Algorithm [E. Cancès, C. Le Bris, Int. J. Quantum Chem. 79 (2000) 82]. Two sets of programs are given, using either a spectral representation of the order parameter [C.M. Dion, E. Cancès, Phys. Rev. E 67 (2003) 046706], suitable for a (quasi) harmonic trapping potential, or by discretizing the order parameter on a spatial grid. Running time: From seconds in 1D to a few hours for large 3D grids

Modulated amplitude waves in collisionally inhomogeneous Bose–Einstein condensates

We investigate the dynamics of an effectively one-dimensional Bose–Einstein condensate (BEC) with scattering length a subjected to a spatially periodic modulation, a = a ( x ) = a ( x + L ) . This “collisionally inhomogeneous” BEC is described by a Gross–Pitaevskii (GP) equation whose nonlinearity coefficient is a periodic function of x . We transform this equation into a GP equation with a constant coefficient and an additional effective potential and study a class of extended wave solutions of the transformed equation. For weak underlying inhomogeneity, the effective potential takes a form resembling a superlattice, and the amplitude dynamics of the solutions of the constant-coefficient GP equation obey a nonlinear generalization of the Ince equation. In the small-amplitude limit, we use averaging to construct analytical solutions for modulated amplitude waves (MAWs), whose stability we subsequently examine using both numerical simulations of the original GP equation and fixed-point computations with the MAWs as numerically exact solutions. We show that “on-site” solutions, whose maxima correspond to maxima of a ( x ) , are more robust and likely to be observed than their “off-site” counterparts.