2D Bose-Einstein Condensates Research Articles

Phase effects of a two-dimensional Bose–Einstein condensate

We investigate the phase effects of a periodically driven Bose–Einstein condensate (BEC) held in a spatially two-dimensional (2D) harmonic or inverted-harmonic potential. A formally exact solution of the time-dependent Gross–Pitaevskii equation is found, which depicts the shock wave with chaotic or periodic amplitude and phase. The atomic densities are illustrated numerically, and the circularly symmetric distributions in the separable phase and the axially symmetric BEC clusters in the inseparable phase are shown. It is demonstrated that the periodic driving may lead to chaos for both phases, which plays a role in avoiding the escape of the solution and restraining the BEC collapse and blast. The results suggest a method for controlling the directed transports of the 2D BEC.

Controlling the transverse instability of dark solitons and nucleation of vortices by a potential barrier

We study possibilities to suppress the transverse modulational instability (MI) of dark-soliton stripes in two-dimensional (2D) Bose-Einstein condensates (BECs) and self-defocusing bulk optical waveguides by means of quasi-1D structures. Adding an external repulsive barrier potential (which can be induced in BEC by a laser sheet, or by an embedded plate in optics), we demonstrate that it is possible to reduce the MI wavenumber band, and even render the dark-soliton stripe completely stable. Using this method, we demonstrate the control of the number of vortex pairs nucleated by each spatial period of the modulational perturbation. By means of the perturbation theory, we predict the number of the nucleated vortices per unit length. The analytical results are corroborated by the numerical computation of eigenmodes of small perturbations, as well as by direct simulations of the underlying Gross-Pitaevskii/nonlinear Schr\"{o}dinger equation.

Two-stage continuation algorithms for Bloch waves of Bose–Einstein condensates in optical lattices

Bloch waves of Bose–Einstein condensates (BEC) in optical lattices are extremum nonlinear eigenstates which satisfy the time-independent Gross–Pitaevskii equation (GPE). We describe an efficient Taylor predictor–Newton corrector continuation algorithm for tracing solution curves of parameter-dependent problems. Based on this algorithm, a novel two-stage continuation algorithm is developed for computing Bloch waves of 1D and 2D Bose–Einstein condensates (BEC) in optical lattices. We split the complex wave function into the sum of its real and imaginary parts. The original GPE becomes a couple of two nonlinear eigenvalue problems defined in the real domain with periodic boundary conditions. At the first stage we use the chemical potential μ as the continuation parameter. The Bloch wavenumber k ( k x , k y ) , and the coefficient of the cubic term are treated as the second and third continuation parameters, respectively. Then we compute the Bloch bands/surfaces for the 1D/2D problem with linear counterparts. At the second stage we use μ and k / k x or k y as the continuation parameters simultaneously with two constraint conditions. The states without linear counterparts in the GPE can be obtained via states with linear counterparts. Numerical results are reported for both 1D and 2D problems.

Bright and dark solitons in a quasi-1D Bose–Einstein condensates modelled by 1D Gross–Pitaevskii equation with time-dependent parameters

We investigate the exact bright and dark solitary wave solutions of an effective 1D Bose–Einstein condensate (BEC) by assuming that the interaction energy is much less than the kinetic energy in the transverse direction. In particular, following the earlier works in the literature Pérez-García et al. (2004) [50], Serkin et al. (2007) [51], Gurses (2007) [52] and Kundu (2009) [53], we point out that the effective 1D equation resulting from the Gross–Pitaevskii (GP) equation can be transformed into the standard soliton (bright/dark) possessing, completely integrable 1D nonlinear Schrödinger (NLS) equation by effecting a change of variables of the coordinates and the wave function. We consider both confining and expulsive harmonic trap potentials separately and treat the atomic scattering length, gain/loss term and trap frequency as the experimental control parameters by modulating them as a function of time. In the case when the trap frequency is kept constant, we show the existence of different kinds of soliton solutions, such as the periodic oscillating solitons, collapse and revival of condensate, snake-like solitons, stable solitons, soliton growth and decay and formation of two-soliton bound state, as the atomic scattering length and gain/loss term are varied. However, when the trap frequency is also modulated, we show the phenomena of collapse and revival of two-soliton like bound state formation of the condensate for double modulated periodic potential and bright and dark solitons for step-wise modulated potentials.

Two-dimensional quantum gas in a hybrid surface trap

We demonstrate a novel optical trapping scheme for ultracold atoms. Using a combination of evanescent wave, standing wave, and magnetic potentials we create a deeply 2D Bose-Einstein condensate (BEC) at a few microns from a glass surface. Using techniques such as broadband "white" light to create evanescent and standing waves, we realize a smooth potential with a trap frequency aspect ratio of 300:1:1 and long lifetimes. This makes the setup suitable for many-body quantum simulations and applications such as high precision measurements close to surfaces.

Superconvergence of high order FEMs for eigenvalue problems with periodic boundary conditions

We study Adini’s elements for nonlinear Schrödinger equations (NLS) defined in a square box with periodic boundary conditions. First we transform the time-dependent NLS to a time-independent stationary state equation, which is a nonlinear eigenvalue problem (NEP). A predictor–corrector continuation method is exploited to trace solution curves of the NEP. We are concerned with energy levels and superfluid densities of the NLS. We analyze superconvergence of the Adini elements for the linear Schrödinger equation defined in the unit square. The optimal convergence rate O ( h 6 ) is obtained for quasiuniform elements. For uniform rectangular elements, the superconvergence O ( h 6 + p ) is obtained for the minimal eigenvalue, where p = 1 or p = 2 . The theoretical analysis is confirmed by the numerical experiments. Other kinds of high order finite element methods (FEMs) and the superconvergence property are also investigated for the linear Schrödinger equation. Finally, the Adini elements-continuation method is exploited to compute energy levels and superfluid densities of a 2D Bose–Einstein condensates (BEC) in a periodic potential. Numerical results on the ground state as well as the first few excited-state solutions are reported.

An adaptive multigrid scheme for Bose–Einstein condensates in a periodic potential

We present a novel multigrid-continuation method for treating parameter-dependent problems. The proposed algorithm which can be flexibly implemented is a generalization of the two-grid discretization schemes [C.-S. Chien, B.-W. Jeng, A two-grid discretization scheme for semilinear elliptic eigenvalue problems, SIAM J. Sci. Comput. 27 (2006) 1287–1304]. That is, approximating points on a solution curve do not necessarily lie on the same fine grid. We apply the algorithm to compute energy levels and superfluid densities of Bose–Einstein condensates (BEC) in a periodic potential. Both positive and negative scattering lengths are considered in our numerical experiments. For positive scattering length, if the chemical potential is large enough, and the domain is properly chosen, the results show that the number of peaks of the first few energy states of the 2D BEC in a periodic potential depends on the wave number of the periodic potential. Moreover, for bright solitons the number of peaks of the ground state solutions is ( 1 d − 1 ) 2 and ( 1 d ) 2 , where the periodic potential is expressed in terms of the sine or the cosine functions, respectively. However, these formulae do not hold if the scattering length is negative. The numerical study is extended to the two-component, 1D and 2D BEC in a periodic potential.

A generalized-Laguerre–Hermite pseudospectral method for computing symmetric and central vortex states in Bose–Einstein condensates

A generalized-Laguerre–Hermite pseudospectral method is proposed for computing symmetric and central vortex states in Bose–Einstein condensates (BECs) in three dimensions with cylindrical symmetry. The new method is based on the properly scaled generalized-Laguerre–Hermite functions and a normalized gradient flow. It enjoys three important advantages: (i) it reduces a three-dimensional (3D) problem with cylindrical symmetry into an effective two-dimensional (2D) problem; (ii) it solves the problem in the whole space instead of in a truncated artificial computational domain and (iii) it is spectrally accurate. Extensive numerical results for computing symmetric and central vortex states in BECs are presented for one-dimensional (1D) BEC, 2D BEC with radial symmetry and 3D BEC with cylindrical symmetry.

Spontaneous symmetry breaking of gap solitons and phase transitions in double-well traps

We study stationary states of a two-dimensional (2D) Bose-Einstein condensate with both attractive and repulsive nonlinearities in a combination of a double-square-well potential in one direction and a perpendicular optical lattice. We look for dual-core solitons in this configuration, focusing on their symmetry-breaking bifurcations. For attractive interactions, without the lattice, a similar analysis was performed [M. Matuszewski et al., Phys. Rev. A 75, 063621 (2007)], where subcritical bifurcation transforming antisymmetric gap solitons into asymmetric ones was found. Here we focus on the effect of an optical lattice and the so created gap solitons. We discover that a phase transition occurs when the lattice depth increases and the additional dimension becomes strongly suppressed. The bifurcation type changes from subcritical (typical for a 2D system, with hysteresis) to supercritical (typical for a one-dimensional system). An additional advantage of the lattice is that gap solitons exist even for repulsive interactions. In this case we also discover bifurcation of a supercritical type. The analysis is based on a variational approximation, which is surprisingly well verified by numerical results.

Solitons in two-dimensional Bose-Einstein condensates

The excitations of a two-dimensional 2D Bose-Einstein condensate in the presence of a soliton are studied by solving the Kadomtsev-Petviashvili equation which is valid when the velocity of the soliton approaches the speed of sound. The excitation spectrum is found to contain states which are localized near the soliton and have a dispersion law similar to the one of the stable branch of transverse oscillations of a 1D gray soliton in a 2D condensate. By using the stabilization method we show that these localized excitations behave as resonant states coupled to the continuum of free excitations of the condensate.

A finite difference continuation method for computing energy levels of Bose–Einstein condensates

We study a finite difference continuation (FDC) method for computing energy levels and wave functions of Bose–Einstein condensates (BEC), which is governed by the Gross–Pitaevskii equation (GPE). We choose the chemical potential λ as the continuation parameter so that the proposed algorithm can compute all energy levels of the discrete GPE. The GPE is discretized using the second-order finite difference method (FDM), which is treated as a special case of finite element methods (FEM) using the piecewise bilinear and linear interpolatory functions. Thus the mathematical theory of FEM for elliptic eigenvalue problems (EEP) also holds for the Schrödinger eigenvalue problem (SEP) associated with the GPE. This guarantees the existence of discrete numerical solutions for the ground-state as well as excited-states of the SEP in the variational form. We also study superconvergence of FDM for solution derivatives of parameter-dependent problems (PDP). It is proved that the superconvergence O ( h t ) in the discrete H 1 norm is achieved, where t = 2 and t = 1.5 for rectangular and polygonal domains, respectively, and h is the maximal boundary length of difference grids. Moreover, the FDC algorithm can be implemented very efficiently using a simplified two-grid scheme for computing energy levels of the BEC. Numerical results are reported for the ground-state of two-coupled NLS defined in a large square domain, and in particular, for the second-excited state solutions of the 2D BEC in a periodic potential.

Ground-State Properties and Collective Excitations in a 2D Bose-Einstein Condensate with Gravity-Like Interatomic Attraction

We study the ground-state properties of a Bose-Einstein condensate (BEC) with the short-range repulsion and gravitylike 1/r interatomic attraction in two-dimensions (2D). Using the variational approach, we obtain the ground-state energy and show that the condensate is stable for all interaction strenghts in 2D. We also determine the collective excitations at zero temperature using the time-dependent variational method. We analyze the properties of the Thomas-Fermi-gravity (TF-G) and gravity (G) regimes.

Vortices in rotating Bose–Einstein condensates confined in homogeneous traps

We investigate analytically the thermodynamical stability of vortices in the ground state of rotating 2D Bose–Einstein condensates confined in asymptotically homogeneous trapping potentials in the Thomas–Fermi regime. Our starting point is the Gross–Pitaevskii energy functional in the rotating frame. By estimating lower and upper bounds for this energy, we show that the leading order in energy and density can be described by the corresponding Thomas–Fermi quantities and we derive the next order contributions due to vortices. As an application, we consider a general potential of the form V(x,y)=(x2+λ2y2)s/2 with slope s∈[2,∞) and anisotropy λ∈(0,1] which includes the harmonic (s=2) and ‘flat’ (s→∞) traps, respectively. For this potential, we derive the critical angular velocities for the existence of vortices and show that all vortices are single-quantized. Moreover, we derive relations which determine the distribution of the vortices in the condensate i.e. the vortex pattern.

Dissipative dynamics of matter-wave solitons in a nonlinear optical lattice

Dynamics and stability of solitons in two-dimensional (2D) Bose-Einstein condensates (BEC), with one-dimensional (1D) conservative plus dissipative nonlinear optical lattices, are investigated. In the case of focusing media (with attractive atomic systems), the collapse of the wave packet is arrested by the dissipative periodic nonlinearity. The adiabatic variation of the background scattering length leads to metastable matter-wave solitons. When the atom feeding mechanism is used, a dissipative soliton can exist in focusing 2D media with 1D periodic nonlinearity. In the defocusing media (repulsive BEC case) with harmonic trap in one direction and nonlinear optical lattice in the other direction, the stable soliton can exist. Variational approach simulations are confirmed by full numerical results for the 2D Gross-Pitaevskii equation.

Stability and Excitations of Solitons in 2D Bose-Einstein Condensates

The small oscillations of solitons in 2D Bose-Einstein condensates are investigated by solving the Kadomtsev-Petviashvili equation which is valid when the velocity of the soliton approaches the speed of sound. We show that the soliton is stable and that the lowest excited states obey the same dispersion law as the one of the stable branch of excitations of a 1D gray soliton in a 2D condensate. The role of these states in thermodynamics is discussed.

Anderson Localization of Expanding Bose-Einstein Condensates in Random Potentials

We show that the expansion of an initially confined interacting 1D Bose-Einstein condensate can exhibit Anderson localization in a weak random potential with correlation length sigma(R). For speckle potentials the Fourier transform of the correlation function vanishes for momenta k>2/sigma(R) so that the Lyapunov exponent vanishes in the Born approximation for k>1/sigma(R). Then, for the initial healing length of the condensate xi(in)>sigma(R) the localization is exponential, and for xi(in)<sigma(R) it changes to algebraic.

Collective excitations of a 1D Bose–Einstein condensate in an anharmonic trap

Abstract We investigate the collective excitations of a one-dimension Bose–Einstein condensate trapped in an anharmonic potential by solving the time-dependent Tonks–Girardeau equation. The governing equations of motions for the low-energy excitations are obtained using variational approaches. The motion of a 1D BEC in a harmonic trap is just like the motion of one particle in a harmonic trap. And quartic distortion of the potential causes the blue-shift and red-shift on the excitation frequency while cube distortion only causes the red-shift.

Soliton trains and vortex streets as a form of Cerenkov radiation in trapped Bose–Einstein condensates

We numerically study the nucleation of gray solitons and vortex–antivortex pairs created by a moving impurity in, respectively, 1D and 2D Bose–Einstein condensates (BECs) confined by a parabolic potential. The simulations emulate the motion of a localized laser-beam spot through the trapped condensate. Our results for the 1D case indicate that, due to the inhomogeneity of the BEC density, the critical speed for nucleation, as a function of the condensate density displays two distinct dependences. In particular, the square root of the critical density for nucleation as a function of speed displays two different linear regimes corresponding to small and large velocities. Effectively, the emission of gray solitons and vortex–antivortex pairs occurs for any velocity of the impurity, as any given velocity will be supercritical in a region with a sufficiently small density. At longer times, the first nucleation is followed by generation of an array of solitons in 1D (“soliton train”) or vortex pairs in 2D (“vortex street”) by the moving object.

Waves in Nonlinear Lattices: Ultrashort Optical Pulses and Bose-Einstein Condensates

The nonlinear Schrödinger equation i (partial differential)(z)A(z,x,t)+(inverted Delta)(2)(x,t)A+[1+m(kappax)]|A|2A=0 models the propagation of ultrashort laser pulses in a planar waveguide for which the Kerr nonlinearity varies along the transverse coordinate x, and also the evolution of 2D Bose-Einstein condensates in which the scattering length varies in one dimension. Stability of bound states depends on the value of kappa=beamwidth/lattice period. Wide (kappa>>1) and kappa=O(1) bound states centered at a maximum of m(x) are unstable, as they violate the slope condition. Bound states centered at a minimum of m(x) violate the spectral condition, resulting in a drift instability. Thus, a nonlinear lattice can only stabilize narrow bound states centered at a maximum of m(x). Even in that case, the stability region is so small that these bound states are "mathematically stable" but "physically unstable."

Propagation of a Dark Soliton in a Disordered Bose-Einstein Condensate

We consider the propagation of a dark soliton in a quasi-1D Bose-Einstein condensate in presence of a random potential. This configuration involves nonlinear effects and disorder, and we argue that, contrarily to the study of stationary transmission coefficients through a nonlinear disordered slab, it is a well-defined problem. It is found that a dark soliton decays algebraically, over a characteristic length which is independent of its initial velocity, and much larger than both the healing length and the 1D scattering length of the system. We also determine the characteristic decay time.