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arXiv:hep-ph/0008025

Quantum tunneling in thermal medium

Sh. Matsumoto, M. Yoshimura

Published Aug 3, 2000• 3 citations
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Abstract

Time evolution of tunneling phenomena in medium is studied using a standard model of environment interaction. A semiclassical formula valid at low, but finite temperatures is derived in the form of integral transform for the reduced Wigner function, and the tunneling probability in thermal medium is calculated for a general tunneling potential of one dimensional system. Effect of dissipation, its time evolution in particular, depends on the behavior of the potential far beyond the barrier.

DOI
10.48550/arxiv.hep-ph/0008025

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Time evolution of tunneling phenomena in medium is studied using a standard model of environment interaction. A semiclassical formula valid at low, but finite temperatures is derived in the form of integral transform for the reduced Wigner function, and the tunneling probability in thermal medium is calculated for a general tunneling potential of one dimensional system. Effect of dissipation, its time evolution in particular, depends on the behavior of the potential far beyond the barrier.
The time evolution of quantum tunneling is studied when the tunneling system is immersed in a thermal medium. We analyze in detail the behavior of the system after integrating out the environment. An exact result for the inverted harmonic oscillator of the tunneling potential is derived and the barrier penetration factor is explicitly worked out as a function of time. The quantum mechanical formula for the case without environment is modified both by the potential renormalization effect and by a dynamical factor which may differ appreciably from the one previously obtained in the time range of 1/(curvature at the top of potential barrier).
The time dependent density matrix of a system with potential barrier is studied using path integrals. The characterization of the initial state, which is assumed to be restricted to one side of the barrier, and the time evolution of the density matrix lead to a threefold path integral which is evaluated in the semiclassical limit. The semiclassical trajectories are found to move in the complex coordinate plane and barrier penetration only arises due to fluctuations. Both the form of the semiclassical paths and the relevant fluctuations change significantly as a function of temperature. The semiclassical analysis leads to a detailed picture of barrier penetration in the real time domain and the changeover from thermal activation to quantum tunneling. Deep tunneling is associated with quasizero modes in the fluctuation spectrum about the semiclassical orbits in the long time limit. The connection between this real time description of tunneling and the standard imaginary time instanton approach is established. Specific results are given for a double well potential and an Eckart barrier.
Macroscopic quantum tunnelling is described using the master equation for the reduced Wigner function of an open quantum system at zero temperature. Our model consists of a particle trapped in a cubic potential interacting with an environment characterized by dissipative and normal and anomalous diffusion coefficients. A representation based on the energy eigenfunctions of the isolated system, i.e. the system uncoupled to the environment, is used to write the reduced Wigner function, and the master equation becomes simpler in that representation. The energy eigenfunctions computed in a WKB approximation incorporate the tunnelling effect of the isolated system and the effect of the environment is described by an equation that is in many ways similar to a Fokker–Planck equation. Decoherence is easily identified from the master equation and we find that when the decoherence time is much shorter than the tunnelling time the master equation can be approximated by a Kramers-like equation describing thermal activation due to the zero point fluctuations of the quantum environment. The effect of anomalous diffusion can be dealt with perturbatively and its overall effect is to inhibit tunnelling.
The purpose of this paper is to study an effect of an environment on a quantum tunneling. Here, the system we consider is of a symmetric two-level type with “bare” tunneling frequency Δω0, and the correlation time of actions due to the environment, τc, is assumed to be much shorter than Δω0-1. In the investigations, the method of density matrix is used, and a coupling between the system and the environment is included up to lowest nontrivial order. After calculations, we reach the following conclusions: First, the possibility of the quantum tunneling is, in a high temperature range βℏΔω0 ≪1, washed out by the thermal actions from the environment. In the inverse case βℏΔω0 ≪1, where the quantum fluctuation of the environment plays a dominant role, the possibility of the quantum tunneling remains. Secondaly, in the case βℏΔω0 ≪1, the dissipation effect depresses the tunneling probability, while the potential barrier is effectively lowered by the actions due to the environment.
Using the phenomenological quantum friction models introduced by P. Caldirola [Nuovo Cimento 18, 393 (1941)] and E. Kanai [Prog. Theor. Phys. 3, 440 (1948)], M. D. Kostin [J. Chem. Phys. 57, 3589 (1972)], and K. Albrecht [Phys. Lett. B 56, 127 (1975)], we study quantum tunneling of a one-dimensional potential in the presence of energy dissipation. To this end, we calculate the tunneling probability using a time-dependent wave-packet method. The friction reduces the tunneling probability. We show that the three models provide similar penetrabilities to each other, among which the Caldirola-Kanai model requires the least numerical effort. We also discuss the effect of energy dissipation on quantum tunneling in terms of barrier distributions.