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International Journal of Modern Physics B

Wave packet dynamics for Gross–Pitaevskii equation in one dimension: Dependence on initial conditions

Sukla Pal, Jayanta K. Bhattacharjee

Published Oct 22, 2015• 2 citations

Abstract

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.

DOI: 10.1142/s0217979215502161

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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.

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

g_c

beyond which there are no bound states scales as

D^{3/4}

(for large

D

). Studying the quantum evolution of wave packets, we observe that for

gg_c

, 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.

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).

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.

The dynamics of an initially excital Gaussian wave packet (harmonic oscillator coherent state, HOCS) in a general one-dimensional anharmonic system is studied analytically with the aid of a second quantization formalism for times shorter than a classical oscillation period. The initial state population probability PQ(t), its classical counterpart PC(t), and the overlap probability PQC(t) between a time evolved wave packet and a classically moving Gaussian with stationary shape are calculated in order to analyze the spreading of the wave packet. It was found that, in general, PQ(t) and PC(t) agree very well if the potential energy function has no strongly changing low order derivatives. This agreement is held even if the wave packet spreads drastically within the time scale considered as can be demonstrated by analyzing PQC(t). The analytical expressions for PQ(t), PC(t), and PQC(t) are applied to a dissociative third-order oscillator and a strongly anharmonic fourth-order oscillator. While for a harmonic system, the initial decay time of PQ(t) and PC(t) (which are identical in this case) are only dependent on the energy expectation value 〈H〉 with respect to the initial HOCS, differences up to an order of magnitude occur for HOCS with the same energy in the model systems.

The evolution of the center-of-mass wave function for a mesoscopic particle according to the Schr\"odinger-Newton equation can be approximated by a harmonic potential if the wave function is narrow compared to the size of the mesoscopic particle. It was noticed by Colin et al. [Phys. Rev. A 93, 062102 (2016).] that, in the regime where self-gravitational effects are weak, intermediate and wider wave functions may be approximated by a harmonic potential as well but with a width-dependent coupling, leading to a time evolution that is determined only by a differential equation for the width of a Gaussian wave function as a single parameter. Such an approximation results in considerably less computational effort in order to predict the self-gravitational effects on the wave-function dynamics. Here, we provide an alternative approach to this kind of approximation, including a rigorous derivation of the equations of motion for an initially Gaussian wave packet under the assumption that its shape is conserved. Our result deviates to some degree from the result by Colin et al. [Phys. Rev. A 93, 062102 (2016).], specifically in the limit of wide wave functions.

We present some physically interesting, in general non-stationary, one-dimensional solutions to the nonlinear phase modification of the Schrödinger equation proposed recently. The solutions include a coherent state, a phase-modified Gaussian wave packet in the potential of harmonic oscillator whose strength varies in time, a free Gaussian soliton, and a similar soliton in the potential of harmonic oscillator comoving with the soliton. The last of these solutions implies that there exist an energy level in the spectrum of harmonic oscillator which is not predicted by the linear theory. The free solitonic solution can be considered a model for a particle aspect of the wave-particle duality embodied in the quantum theory. The physical size of this particle is naturally rendered equal to its Compton wavelength in the subrelativistic framework in which the self-energy of the soliton is assumed to be equal to its rest-mass energy. The solitonic solutions exist only for the negative coupling constant for which the Gaussian wave packets must be larger than some critical finite size if their energy is to be bounded, i.e., they cannot be point-like objects.