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Journal of Physics B Atomic Molecular and Optical Physics

The cold-collision dynamics of two atomic wave packets

Yuri Alexandrovish Barbosa, Reginaldo de Jesus Napolitano

Published Jan 30, 2004• 0 citations
Publisher

Abstract

We present a thought experiment, based on recent experimental achievements, to manipulate the preparation of two Gaussian wave packets in collision at a specified relative speed. We establish the equations describing the collision dynamics in terms of centre-of-mass and relative coordinates. We show that, for cold collisions, there is always an effective classical time evolution giving the same asymptotic Wigner function as that obtained from the quantum scattered wave packet.

DOI
10.1088/0953-4075/37/4/006

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We present a thought experiment, based on recent experimental achievements, to manipulate the preparation of two Gaussian wave packets in collision at a specified relative speed. We establish the equations describing the collision dynamics in terms of centre-of-mass and relative coordinates. We show that, for cold collisions, there is always an effective classical time evolution giving the same asymptotic Wigner function as that obtained from the quantum scattered wave packet.
We consider multiple collisions of quantum wave packets in one dimension. The system under investigation consists of an impenetrable wall and two hard-core particles with very different masses. The lighter particle bounces between the heavier one and the wall. Both particles are initially represented by narrow Gaussian wave packets. A complete analytical solution of this problem is presented. The idea of the method used is to decompose the two-particle wave function into a continuous superposition of terms (channels), such that the multiple collisions within each channel do not lead to subsequent entanglement between the two particles. For each channel, the time evolution of the two-particle wave function is completely determined by the motion of the corresponding classical pointlike particles; therefore the whole quantum problem is reduced to a classical calculation. The calculation based on the above method reveals the following unexpected result: The entanglement between the two particles first increases with time due to the collisions, but then it begins to decrease, disappearing completely when the light particle becomes too slow to catch up with the heavy one.
We find a relationship between the dynamics of the Gaussian wave packet and the dynamics of the corresponding Gaussian Wigner function from the Hamiltonian/symplectic point of view. The main result states that the momentum map corresponding to the natural action of the symplectic group on the Siegel upper half space yields the covariance matrix of the corresponding Gaussian Wigner function. This fact, combined with Kostant’s coadjoint orbit covering theorem, establishes a symplectic/Poisson-geometric connection between the two dynamics. The Hamiltonian formulation naturally gives rise to corrections to the potential terms in the dynamics of both the wave packet and the Wigner function, thereby resulting in slightly different sets of equations from the conventional classical ones. We numerically investigate the effect of the correction term and demonstrate that it improves the accuracy of the dynamics as an approximation to the dynamics of expectation values of observables.
The scattering process of wave packets is often described recovering the simple monochromatic scheme supposing that the particle comes from −∞ with a sharp momentum distribution, thus allowing for approximations usually used in common calculations of physical quantities involved in the process. In this work, we study the scattering process of a Gaussian wave packet impinging on the step potential, finding out that its dynamics depends on the initial conditions of the incident particle, and in particular on the wave packet origin x0. We propose a semi-classical approximated model to describe the dynamics of the scattering wave packet, also defining a characteristic time interval tf as the time required for the formation of transmitted packet beyond the step. Through a comparison with the numerical solution of the SchrÖdinger equation, our model explains the tf as a function of the origin x0 and spread of the incident wave packet in coordinate space, giving rise to such dependence for a finite distance scattering processes.
Quantum computing provides a novel avenue towards simulating dynamical phenomena, and, in particular, scattering processes relevant for exploring the structure of matter. However, preparing and evolving particle wave packets on a quantum device is a nontrivial task. In this work, we propose a method to prepare Gaussian wave packets with momentum on top of the interacting ground state of a fermionic Hamiltonian. Using Givens rotation, we show how to efficiently obtain expectation values of observables throughout the evolution of the wave packets on digital quantum computers. We demonstrate our technique by applying it to the staggered lattice formulation of the Thirring model and studying the scattering of two wave packets. Monitoring the particle density and the entropy produced during the scattering process, we characterize the phenomenon and provide a first step towards studying more complicated collision processes on digital quantum computers. In addition, we perform a small-scale demonstration on IBM's quantum hardware, showing that our method is suitable for current and near-term quantum devices.
We develop methodology for performing time dependent quantum mechanical calculations by representing the wave function as a sum of Gaussian wave packets (GWP) each characterized by a set of parameters such as width, position, momentum, and phase. The problem of computing the time evolution of the wave function is thus reduced to that of finding the time evolution of the parameters in the Gaussians. This parameter motion is determined by minimizing the error made by replacing the exact wave function in the time dependent Schrödinger equation with its Gaussian representation approximant. This leads to first order differential equations for the time dependence of the parameters, and those describing the packet position and the momentum of each packet have some resemblance with the classical equations of motion. The paper studies numerically the strategy needed to achieve the best GWP representation of time dependent processes. The issues discussed are the representation of the initial wave function, the numerical stability and the solution of the differential equations giving the evolution of the parameters, and the analysis of the final wave function. Extensive comparisons are made with an approximate method which assumes that the Gaussians are independent and their width is smaller than the length scale over which the potential changes. This approximation greatly simplifies the calculations and has the advantage of a greater resemblance to classical mechanics, thus being more intuitive. We find, however, that its range of applications is limited to problems involving localized degrees of freedom that participate in the dynamic process for a very short time. Finally we give particular attention to the notion that the GWP representation of the wave function reduces the dynamics of one quantum degree of freedom to that of a set of pseudoparticles (each represented by one packet) moving according to a pseudoclassical (i.e., classical-like) mechanics whose ‘‘phase space’’ is described by a position and momentum as well as a complex phase and width.