Starting from coupled Gross-Pitaevskii equation (GPE) describing the dynamics of binary Bose Einstein Condensate (BEC), we have explored the dynamics of the width of initial wave-packets by following two different approaches. We have followed a recent coherent state based approach and have shown that in binary BEC the oscillation of the width of the wave packet can give rise to instability at a certain condition. Alternately we have proceeded by considering the trial ground state solutions for both the species and through the Ehrenfest theorem shown the existence of an instability which may lead to phase separation of the two species.
Starting from coupled Gross-Pitaevskii equation (GPE) describing the dynamics of binary Bose Einstein Condensate (BEC), we have explored the dynamics of the width of initial wave-packets by following two different approaches. We have followed a recent coherent state based approach and have shown that in binary BEC the oscillation of the width of the wave packet can give rise to instability at a certain condition. Alternately we have proceeded by considering the trial ground state solutions for both the species and through the Ehrenfest theorem shown the existence of an instability which may lead to phase separation of the two species.
We find a set of different orthonormalized states of a nonstationary harmonic oscillator and use them to expand the solution of the Gross-Pitaevskii equation with harmonic potential. The expansion series describes wave-packet trains of a Bose-Einstein condensate, which may be induced initially by the modulational instability. The center of any wave-packet train oscillates like a classical harmonic oscillator of frequency $\ensuremath{\omega}$. The width and height of the wave packet and the distance between two wave packets change simultaneously like an array of breathers with frequency $2\ensuremath{\omega}$. We demonstrate analytically and numerically that for a set of suitable parameters the wave-packet trains can be more exactly fitted to the matter-wave soliton trains observed by Strecker et al. and reported in Nature (London) 417, 150 (2002).
We consider systems governed by the Gross Pitaevskii equation (GPE) with the Morse potential V(x)=D(e−2ax−2e−ax) as the trapping potential. For positive values of the coupling constant g of the cubic term in GPE, we find that the critical value gc beyond which there are no bound states scales as D3/4 (for large D). Studying the quantum evolution of wave packets, we observe that for ggc, on the otherhand, all initial wave packets escape from the potential and the dynamics is like that of a quantum free particle. For g<0, we find that there can be initial conditions for which the escaping wave packet can propagate with very little change in width i,e., it remains almost shape invariant.
The propagation of an initially Gaussian wave packet of width [Formula: see text] in Gross–Pitaevskii equation is extensively studied both for attractive and repulsive interactions. It is predicted analytically and verified numerically that for a free particle with attractive interaction, the dynamics of the width is governed by an effective potential which is sensitive to initial conditions. If [Formula: see text] is equal to a corresponding critical width [Formula: see text], then the packet will propagate in time with very little change in shape. These are in essence like coherent states. Whereas, if [Formula: see text], depending on the nature of the effective potential for chosen [Formula: see text] and the interaction strength [Formula: see text], the width of the packet in course of time, either oscillates with bounded width or will spread like a free particle. For a simple harmonic oscillator (SHO) also, we find that for [Formula: see text] smaller than a critical value, there always exists a coupling strength for which the packet simply oscillates about the mean position without changing its shape, once again providing a resemblance to a coherent state. We also consider the Morse potential, which interpolates between the free particle and the oscillator. For large attractive interactions, the two limiting dynamics (free and simple harmonic) are indeed observed but in the intermediate form of the potential where the nonlinear terms dominate in the dynamics, an initial Gaussian wave packet does not retain its shape. For repulsive interaction, the Gaussian packet always changes shape no matter what the system parameters are.
In this paper, we mainly review recent results on mathematical theory andnumerical methods for Bose-Einstein condensation (BEC),based on the Gross-Pitaevskii equation (GPE). Starting from the simplest case with one-component BEC of the weakly interacting bosons, we study the reduction of GPE to lower dimensions, the ground states of BEC including the existence and uniqueness as well as nonexistence results, and the dynamics of GPE including dynamical laws, well-posedness of the Cauchy problem as well as the finite time blow-up. To compute the ground state, the gradient flow with discrete normalization (or imaginary time) method is reviewed and various full discretization methods are presented and compared. To simulate the dynamics, both finite difference methods and time splitting spectral methods are reviewed, and their error estimates are briefly outlined. When the GPE has symmetric properties, we show how to simplify the numerical methods. Then we compare two widely used scalings, i.e. physical scaling (commonly used) and semiclassical scaling, for BEC in strong repulsive interaction regime (Thomas-Fermi regime), and discuss semiclassical limits of the GPE. Extensions of these results for one-component BEC are thencarried out for rotating BEC by GPE with an angular momentum rotation, dipolar BEC by GPE with long range dipole-dipole interaction, and two-component BEC by coupled GPEs. Finally, as a perspective, we show briefly the mathematical models forspin-1 BEC, Bogoliubov excitation and BEC at finite temperature.
Quantum phenomena appear in a macroscopic scale in Bose-Einstein condensates. The Gross-Pitaevskii (GP) equation describes the dynamics of the weakly interacting Bose-Einstein condensates. The GP equation has a form of the Schroedinger equation with self-interaction. The coupled Gross-Pitaevskii equations are used to describe some mixtures of Bose-Einstein condensates. In this paper, we will show some numerical results of coupled Gross-Pitaevskii equations without self-interaction, which has a form of nonlinearly-coupled Schroedinger equations. We demonstrate that the macroscopic oscillation and the interference of two quantum wave packets decay in time owing to mutual interaction, which is analogous to the decoherence in quantum mechanics of many particles.
The achievement of Bose-Einstein condensation (BEC) in ultracold vapors of alkali atoms has given enormous impulse to the theoretical and experimental study of dilute atomic gases in condensed quantum states inside magnetic traps and optical lattices. This article offers a short survey on mathematical models and theories as well as numerical methods for BEC based on the mean field theory. We start with the Gross-Pitaevskii equation (GPE) in three dimensions (3D) for modeling one-component BEC of the weakly interacting bosons, scale it to obtain a three-parameter model and show how to reduce it to two dimensions (2D) and one dimension (1D) GPEs in certain limiting regimes. Mathematical theories and numerical methods for ground states and dynamics of BEC are provided. Extensions to GPE with an angular momentum rotation term for a rotating BEC, to GPE with long-range anisotropic dipole-dipole interaction for a dipolar BEC and to coupled GPEs for spin-orbit coupled BECs are discussed. Finally, some conclusions are drawn and future research perspectives are discussed.