Derivation of the Gross-Pitaevskii dynamics through renormalized excitation number operators

CUDA programs for solving the time-dependent dipolar Gross–Pitaevskii equation in an anisotropic trap

OpenMP Fortran and C programs for solving the time-dependent Gross–Pitaevskii equation in an anisotropic trap

We study the time-evolution of initially trapped Bose-Einstein condensates in the Gross-Pitaevskii regime. Under a physically motivated assumption on the energy of the initial data, we show that condensation is preserved by the many-body evolution and that the dynamics of the condensate wave function can be described by the time-dependent Gross-Pitaevskii equation. With respect to previous works, we provide optimal bounds on the rate of condensation (i.e. on the number of excitations of the Bose-Einstein condensate). To reach this goal, we combine the method of \cite{LNS}, where fluctuations around the Hartree dynamics for $N$-particle initial data in the mean-field regime have been analyzed, with ideas from \cite{BDS}, where the evolution of Fock-space initial data in the Gross-Pitaevskii regime has been considered.

Hybrid OpenMP/MPI programs for solving the time-dependent Gross–Pitaevskii equation in a fully anisotropic trap

OpenMP Fortran programs for solving the time-dependent dipolar Gross-Pitaevskii equation

We study the dipole collective oscillations in the bose-fermi mixture using a dynamical time-dependent approach, which are formulated with the time-dependent Gross-Pitaevskii equation and the Vlasov equation. We find big difference in behaviors of fermion oscillation between the time-dependent approach and usual approaches such as the random-phase approximation and the sum-rule approach. While the bose gas oscillates monotonously, the fermion oscillation shows a beat and a damping. When the amplitude is not minimal, the dipole oscillation of the fermi gas cannot be described with a simple center-of-mass motion.

In this paper, we study the interaction-modulated tunneling dynamics of a Bose-Fermi superfluid mixture, where a Bose-Einstein condensate (BEC) with weak repulsive interaction is confined in a symmetric deep double-well potential and an equally populated two-component Fermi gas in a harmonic potential symmetrically is positioned in the center of the double-well potential. The tunneling between the two wells is modulated by fermions trapped in a harmonic potential. When the temperature is adequately low and the bosonic particle number is adequately large, we can employ the mean-field theory to describe the evolution of the BEC in the double-well potential through the time-dependent Gross-Pitaevskii equation. For the Fermi gas in the harmonic potential trap, we consider the case where the inter-fermion interaction is tuned on the deep Bose-Einstein condensate of the inter-fermion Feshbach resonance, where two fermions of spin-up and spin-down form a two-body bound state. Within the regime, the Fermi gas is well described by a condensate of these fermionic dimers, and hence can be simulated as well by a Gross-Pitaevskii equation of dimers. The inter-species interactions couple the dynamics of the two species, which results in interesting features in the tunneling oscillations. The dynamic equations of the BEC in the double-well potential is described by a two-mode approximation. Coupling it with time-dependent Gross-Pitaevskii equation of the harmonically potential trapped molecular BEC, we numerically investigate the dynamical evolution of the Boson-Fermi hybrid system under different initial conditions. It is found that the interaction among fermions in a harmonic potential leads to strong non-linearity in the oscillations of the bosons in the double-well potential and enriches the tunneling dynamics of the bosons. Especially, it strengthens macroscopic quantum self-trapping. And the macroscopic quantum self-trapping can be expressed in three forms: the phase tends to be negative and monotonically decreases with time, the phase evolves with time, and the phase tends to be positive and increases monotonically with time. This means that it is possible the tunneling dynamics of the BEC in double-well potential is adjustable. Our results can be verified experimentally in a Bose-Fermi superfluid mixture by varying different interaction parameters via Feshbach resonance and confinement-induced resonance.

A dilute gas of Bose-Einstein condensed atoms in a non-rotated and axially symmetric harmonic trap is modelled by the time dependent Gross-Pitaevskii equation. When the angular momentum carried by the condensate does not vanish, the minimum energy state describes vortices (or antivortices) that propagate around the trap center. The number of (anti)vortices increases with the angular momentum, and they repel each other to form Abrikosov lattices. Besides vortices and antivortices there are also stagnation points where the superflow vanishes; to our knowledge the stagnation points have not been analyzed previously, in the context of the Gross-Pitaevskii equation. The Poincaré index formula states that the difference in the number of vortices and stagnation points can never change. When the number of stagnation points is small, they tend to aggregate into degenerate propagating structures. But when the number becomes sufficiently large, the stagnation points tend to pair up with the vortex cores, to propagate around the trap center in regular lattice arrangements. There is an analogy with the geometry of the Kosterlitz-Thouless transition, with the angular momentum of the condensate as the external control parameter instead of the temperature.

We investigate the Gross-Pitaevskii equation for a Bose-Einstein condensate in a PT symmetric double-well potential by means of the time-dependent variational principle and numerically exact solutions. A one-dimensional and a fully three-dimensional setup are used. Stationary states are determined and the propagation of wave function is investigated using the time-dependent Gross-Pitaevskii equation. Due to the nonlinearity of the Gross-Pitaevskii equation the potential dependson the wave function and its solutions decide whether or not the Hamiltonian itself is PT symmetric. Stationary solutions with real energy eigenvalues fulfilling exact PT symmetry are found as well as PT broken eigenstates with complex energies. The latter describe decaying or growing probability amplitudes and are not true stationary solutions of the time-dependent Gross-Pitaevskii equation. However, they still provide qualitative information about the time evolution of the wave functions.

OpenMP GNU and Intel Fortran programs for solving the time-dependent Gross–Pitaevskii equation

A recent experiment [Deng et al., Nature 398, 218(1999)] demonstrated\nfour-wave mixing of matter wavepackets created from a Bose-Einstein condensate.\nThe experiment utilized light pulses to create two high-momentum wavepackets\nvia Bragg diffraction from a stationary Bose-Einstein condensate. The\nhigh-momentum components and the initial low momentum condensate interact to\nform a new momentum component due to the nonlinear self-interaction of the\nbosonic atoms. We develop a three-dimensional quantum mechanical description,\nbased on the slowly-varying-envelope approximation, for four-wave mixing in\nBose-Einstein condensates using the time-dependent Gross-Pitaevskii equation.\nWe apply this description to describe the experimental observations and to make\npredictions. We examine the role of phase-modulation, momentum and energy\nconservation (i.e., phase-matching), and particle number conservation in\nfour-wave mixing of matter waves, and develop simple models for understanding\nour numerical results.\n

We present a method for approximating the solution of the three-dimensional, time-dependent Gross-Pitaevskii equation (GPE) for Bose-Einstein-condensate systems where the confinement in one dimension is much tighter than in the other two. This method employs a hybrid Lagrangian variational technique whose trial wave function is the product of a completely unspecified function of the coordinates in the plane of weak confinement and a Gaussian in the strongly confined direction having a time-dependent width and quadratic phase. The hybrid Lagrangian variational method produces equations of motion that consist of (1) a two-dimensional (2D) effective GPE whose nonlinear coefficient contains the width of the Gaussian and (2) an equation of motion for the width that depends on the integral of the fourth power of the solution of the 2D effective GPE. We apply this method to the dynamics of Bose-Einstein condensates confined in ring-shaped potentials and compare the approximate solution to the numerical solution of the full 3D GPE.

C programs for solving the time-dependent Gross–Pitaevskii equation in a fully anisotropic trap

We applied the numerical combination of Runge-Kutta and Finite Difference (RKFD) scheme for a quantum reflection model of Bose-Einstein condensate (BEC) from a silicon surface. It is by the time-dependent Gross-Pitaevskii equation (GPE), a non-linear Schrödinger equation (NLSE) in the context of quantum mechanics. The role of cut-off potential δ and negative imaginary potential Vim is essential to estimating non-interacting BEC reflection models. Relying on these features, we performed a numerical simulation of the BEC quantum reflection model and calculated the effect of reflection probability R versus incident speed vx. The model is based on the three rapid potential variations: positive-step potential +Vstep, negative-step potential -Vstep, and Casimir-Polder potential VCP. As a result, the RKFP numerical scheme was successfully set up and applied to the quantum reflection model of BEC from the silicon surface. The numerical simulations results show that the reflection probability R decays exponentially to the incident speed vx.

We present a method to calculate the dynamics of very-low-temperature Bose-Einstein condensates in time-dependent traps. We consider a system with a well-defined number of particles, rather than a system in a coherent state with a well-defined phase. This preserves the $U(1)$ symmetry of the problem. We use a systematic asymptotic expansion in the square root of the fraction of noncondensed particles. In lowest order we recover the time-dependent Gross-Pitaevskii equation for the condensate wave function. The next order gives the linear dynamics of noncondensed particles. The higher order gives corrections to the time-dependent Gross-Pitaevskii equation including the effects of noncondensed particles on the condensate. We compare this method with the Bogoliubov--de Gennes approach.