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arXiv:2401.16789

Analysis of Time-Evolution of Gaussian Wavepackets in Non-Hermitian Systems

Amartya Bose

Published Jan 30, 2024• 0 citations
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Abstract

Simulation and analysis of multidimensional dynamics of a quantum non-Hmeritian system is a challenging problem. Gaussian wavepacket dynamics has proven to be an intuitive semiclassical approach to approximately solving the dynamics of quantum systems. A Gaussian wavepacket approach is proposed for a continuous space extension to the Hatano-Nelson model that enables transparent analysis of the dynamics in terms of complex classical trajectories. We demonstrate certain cases where the configuration space trajectory can be made fully real by transforming the initial conditions to account for the non-Hermiticity appropriately through the momentum coordinates. However, in general the complex phase space is unavoidable. For the cases where the trajectory is real, the effective force can be decomposed into that due to the potential energy surface and that due to the imaginary vector potential. The impact of the vector potential on the trajectory of the wavepacket is directly proportional to both the strength of the vector potential and the width of the wavepacket.

DOI
10.48550/arxiv.2401.16789

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Simulation and analysis of multidimensional dynamics of a quantum non-Hmeritian system is a challenging problem. Gaussian wavepacket dynamics has proven to be an intuitive semiclassical approach to approximately solving the dynamics of quantum systems. A Gaussian wavepacket approach is proposed for a continuous space extension to the Hatano-Nelson model that enables transparent analysis of the dynamics in terms of complex classical trajectories. We demonstrate certain cases where the configuration space trajectory can be made fully real by transforming the initial conditions to account for the non-Hermiticity appropriately through the momentum coordinates. However, in general the complex phase space is unavoidable. For the cases where the trajectory is real, the effective force can be decomposed into that due to the potential energy surface and that due to the imaginary vector potential. The impact of the vector potential on the trajectory of the wavepacket is directly proportional to both the strength of the vector potential and the width of the wavepacket.
The semiclassical Herman–Kluk initial value approximation of the quantum-mechanical propagator, which is heavily used in dynamical studies of atomic and molecular systems, takes into account nonlinearity of the underlying classical dynamics by using multiple classical initial value solutions. In applications to the propagation of Gaussian wavepackets non-Gaussian distortions in the course of time and therefore a realistic description of the quantum dynamics are possible. This is in contrast to the single-trajectory approximations investigated in a recent paper by Baranger et al , one of which was erroneously termed Herman–Kluk approximation.
We examine an extension to the theory of Gaussian wavepacket dynamics in a one-dimensional potential by means of a sequence of time-dependent displacement and squeezing transformations. Exact expressions for the quantum dynamics are found, and relationships are explored between the squeezed system, Gaussian wavepacket dynamics, the time-dependent harmonic oscillator, and wavepacket dynamics in a Gauss-Hermite basis. Expressions are given for the matrix elements of the potential in some simple cases. Several examples are given, including the propagation of a non-Gaussian initial state in a Morse potential.
Many features of Bloch oscillations in one-dimensional quantum lattices with a static force can be described by quasiclassical considerations for example by means of the acceleration theorem, at least for Hermitian systems. Here the quasiclassical approach is extended to non-Hermitian lattices, which are of increasing interest. The analysis is based on a generalised non-Hermitian phase space dynamics developed recently. Applications to a single-band tight-binding system demonstrate that many features of the quantum dynamics can be understood from this classical description qualitatively and even quantitatively. Two non-Hermitian and PT -symmetric examples are studied, a Hatano–Nelson lattice with real coupling constants and a system with purely imaginary couplings, both for initially localised states in space or in momentum. It is shown that the time-evolution of the norm of the wave packet and the expectation values of position and momentum can be described in a classical picture.
We consider a semiclassical approximation, first derived by Heller and coworkers, for the time evolution of an originally Gaussian wave packet in terms of complex trajectories. We also derive additional approximations replacing the complex trajectories by real ones. These yield three different semiclassical formulae involving different real trajectories. One of these formulae is Heller's thawed Gaussian approximation. The other approximations are non-Gaussian and may involve several trajectories determined by mixed initial–final conditions. These different formulae are tested for the cases of scattering by a hard wall, scattering by an attractive Gaussian potential and bound motion in a quartic oscillator. The formula with complex trajectories gives good results in all cases. The non-Gaussian approximations with real trajectories work well in some cases, whereas the thawed Gaussian works only in very simple situations.
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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.