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Journal of Physics B Atomic Molecular and Optical Physics

Interacting trajectory representation of quantum dynamics: influence of boundary conditions on the tunneling decay of resonant states

L. Cruz Rodríguez, Llinersy Uranga-Piña, A. Martínez-Mesa, C. Meier

Published Jan 5, 2023• 2 citations

Abstract

Abstract We perform quantum trajectory simulations of the decay dynamics of initially localized resonant states. Quantum dynamics is represented by a swarm of interacting trajectories which maps the originally quantum problem into the motion of an equivalent (higher-dimensional) classical system. We address two model problems, in which the decay of the initial resonance leads to either spatially confined or asymptotically free wave-packet dynamics, specifically on a double well potential and on a potential plain. The traditional choice of fixed boundary conditions in the interacting trajectory representation (ITR), set at infinity, is found to have a moderate influence on the accuracy of the ITR of quantum trajectory dynamics, for the motion on a double well potential, i.e. the results of the trajectory-based scheme are in good correspondence with those obtained via quantum wave-packet propagation up to several fundamental vibrational periods. On the other hand, standard boundary conditions have negligible effect on the interacting trajectory dynamics of a decaying shape resonance, whose predictions reproduce quantum mechanical results at long times.

DOI: 10.1088/1361-6455/acb0b0

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Abstract We perform quantum trajectory simulations of the decay dynamics of initially localized resonant states. Quantum dynamics is represented by a swarm of interacting trajectories which maps the originally quantum problem into the motion of an equivalent (higher-dimensional) classical system. We address two model problems, in which the decay of the initial resonance leads to either spatially confined or asymptotically free wave-packet dynamics, specifically on a double well potential and on a potential plain. The traditional choice of fixed boundary conditions in the interacting trajectory representation (ITR), set at infinity, is found to have a moderate influence on the accuracy of the ITR of quantum trajectory dynamics, for the motion on a double well potential, i.e. the results of the trajectory-based scheme are in good correspondence with those obtained via quantum wave-packet propagation up to several fundamental vibrational periods. On the other hand, standard boundary conditions have negligible effect on the interacting trajectory dynamics of a decaying shape resonance, whose predictions reproduce quantum mechanical results at long times.

A behavior of quantum states (superposition of two lowest eigenstates, Gaussian wave packet) in phase space is studied for one and two dimensional double well potential. Two dimensional potential is constructed from double well potential coupled linearly and quadratically to harmonic potential. Quantum trajectories are compared with classical ones. Preferable tunneling path in phase space is found. An influence of energy of initial Gaussian wave packet and trajectory initial condition on tunneling process is analyzed.. Coupling effect on trajectories and preferable tunneling path for two dimensional potential is studied.

A previous initial value coherent state representation is applied to Gaussian wave packet propagation on one-dimensional and two-dimensional double well potentials. The method uses a basis of trajectory guided frozen Gaussian coherent states, sampled from a Monte Carlo ensemble, for numerical solutions of the quantum time dependent integro-differential Schrödinger equation. Accurate tunneling splittings are obtained. Comparisons are also made with corresponding results obtained by the Herman–Kluk semiclassical initial value representation wave packet method.

Quantum tunneling effects of Gaussian wave packet in one- and two-dimensional double well potentials are investigated using quantum Liouville equation for time evolution of Wigner distribution in phase space. It is shown that a trajectory-based solution of this problem can be constructed by the entangled trajectory ensemble simulating the evolving quantum state. Quantum effects arise in this approach as a breakdown of the statistical independence of the trajectory ensemble. © 2014 Wiley Periodicals, Inc.

We demonstrate a different method of simulation of nonstationary quantum processes, considering the tunneling of two interacting identical particles, represented by wave packets. The used method of quantum molecular dynamics (WMD) is based on the Wigner representation of quantum mechanics. In the context of this method ensembles of classical trajectories are used to solve quantum Wigner-Liouville equation. These classical trajectories obey Hamiltonian-like equations, where the effective potential consists of the usual classical term and the quantum term, which depends on the Wigner function and its derivatives. The quantum term is calculated using local distribution of trajectories in phase space, therefore, classical trajectories are not independent, contrary to classical molecular dynamics. The developed WMD method takes into account the influence of exchange and interaction between particles. The role of direct and exchange interactions in tunneling is analyzed. The tunneling times for interacting particles are calculated.

In this paper, we describe a new method for simulating quantum processes using classical-like molecular dynamics. The approach is based on solving the quantum Liouville equation in the Wigner representation using ensembles of classical trajectories in phase space. The nonlocality of quantum mechanics is incorporated in the trajectory representation as nonclassical interactions between the members of the ensemble, leading to an entanglement of their evolution. The statistical independence of the individual trajectories making up an ensemble in the classical limit is lost when quantum effects are included, and the entire state of the system must be propagated as a unified whole. We develop the formalism and its numerical implementation, and illustrate its application on two model problems of quantum mechanical tunneling: escape from a metastable well and wave packet penetration of the Eckhart barrier.

We numerically investigate quantum dynamics in a one-dimensional double-well system emphasizing influence of a parametrically polychromatic perturbation on the dynamics. It is found that time dependence of transition probability for an initially localized wave packet between the wells shows two types of motion, coherent and incoherent motion, depending on the perturbation parameters. As the strength and/or the number of frequency components of the perturbation increase, coherent motion changes into incoherent one. The former is related to coherent tunneling of the wave packet due to coherence; the latter is related to a delocalized state caused by decoherence. In coherent motion, by virtue of coherence of the dynamics, the expectation value and the standard deviation of a dynamical variable such as the energy of the system show oscillatory time dependence around the initial values. On the contrary in incoherent motion, because of the decoherence, the time dependence fluctuates irregularly around a certain value after a rapid increase due to the resonance. We find that negativity of the Wigner function also show similar time dependence in each type of motion. We compare the classification of the quantum dynamics based on regularity of the time dependence with the one of corresponding classical dynamics based on the Lyapunov exponent. The classifications of the quantum and classical dynamics overlap well in the parameter space. Furthermore, we confirm decoherence of quantum dynamics in a kicked double-well system.