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Physical Review A

Quantum tunneling in dissipative media: Intermediate-coupling-strength results

Peter S. Riseborough, Peter Hänggi A, E. Freidkin

Published Jul 1, 1985• 60 citations
Publisher

Abstract

We consider the motion of a quantum particle placed in a metastable minima of a potential well and examine the effect of interactions with the thermal reservoir at temperature T. We use the dilute, bounce approximation for the Green's function. We find a particular value of the dissipative coupling strength for which we can solve the Euler-Lagrange equations exactly, in the range of temperatures between zero and the crossover to the regime where thermal hopping dominates. We obtain analytic expressions for the tunneling rate in this temperature regime.

DOI
10.1103/physreva.32.489

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We consider the motion of a quantum particle placed in a metastable minima of a potential well and examine the effect of interactions with the thermal reservoir at temperature T. We use the dilute, bounce approximation for the Green's function. We find a particular value of the dissipative coupling strength for which we can solve the Euler-Lagrange equations exactly, in the range of temperatures between zero and the crossover to the regime where thermal hopping dominates. We obtain analytic expressions for the tunneling rate in this temperature regime.
The problem of metastability due to quantum and thermal fluctuations in a weakly damped system is treated within Feynman's influence-functional theory. It is shown that quite generally the nonequilibrium density matrix may be written as a single path integral. For a cubic metastable potential it has the form of a Feynman propagator with a complex Lagrangian containing second-order derivatives in time and terms noninvariant under time reversal. Dissipative tunneling and thermal activation emerge together as different solutions of the associated Lagrange equations. The low-temperature correction to the exponent of the tunneling rate behaves as Tn+1{T}^{n+1}Tn+1 for a spectrum of bath excitations vanishing as ${\ensuremath{\omega}}^{n}$. It is shown, moreover, how this result may be understood in terms of environment-assisted tunneling. Thus the low-temperature behavior of the tunneling rate provides a method to measure the dissipation mechanism in the quantum regime.
For a tunneling system we consider the coupling to the dissipative environment of the coordinate q, the momentum p and the potential function U(q). In the frame of the quantum theory of open systems we derive a quantum master equation, fundamental constraints and uncertainty relations. We obtain an analytical expression of the tunneling spectrum and show that at thermal equilibrium its shape entirely depends on the barrier characteristics while its amplitude is proportional to the strength of the dissipative coupling.
Abstract We explore dissipative quantum tunnelling, a phenomenon central to various physical and chemical processes, utilizing a model based on a double-well potential. This paper aims to bridge gaps in understanding the crossover from thermal activation to quantum tunnelling, a domain still shrouded in mystery despite extensive research. By numerically investigating a model derived from Caldeira–Leggett’s work on quantum Brownian motion, examining both Lindblad and stochastic Schrödinger dynamics, we offer new insights into the transition states in the crossover region. Contrary to a common belief that temperature strongly dampens all quantum effects, our findings reveal that under certain conditions temperature instead alters the nature of tunnelling from a deterministic and periodic process to a stochastic yet still very quantum phenomenon. This underscores the profound influence of quantum effects on transition rates and the critical role of temperature in modulating tunnelling behaviours. Additionally, we introduce a new model for quantum Brownian motion that takes Lindblad form and is formulated as a modification of the widely known model found in Breuer and Petruccione. In our approach, we remove the zero-temperature singularity resulting in a better description of low-temperature quantum Brownian motion near a potential minima. Despite these advancements, we recognize persistent challenges in accurately simulating the dynamics at extremely low temperatures for arbitrary potentials, particularly those that cannot be closely approximated locally by a quadratic function.
The tunneling process in dissipative quantum systems can be described in terms of a Feynman functional integral. The quasiclassical approximation entails the evaluation of solutions of the Euler-Lagrange equations in imaginary time and diagonalizing the small-amplitude excursions about the extremal trajectories. The latter process is equivalent to evaluating all the eigenvalues of integro-differential eigenvalue equations. We show that the exact eigenvalues are related to the phase shifts and derive a Hulth\'en-type of variational principle for the phase shifts appropriate for the above-described eigenvalue equation. We illustrate the method by applying it to the problem of the decay of a metastable state coupled to an Ohmic heat bath at T=0. We compare the accuracy with the Born approximation, rigorous results, and numerical evaluations.