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Physical Review Letters

Quantum Tunneling in Dissipative Systems at Finite Temperatures

Hermann Grabert, Ulrich Weiß, Peter Hänggi A

Published Jun 18, 1984• 231 citations
Publisher

Abstract

A quantum system which can tunnel out of a metastable state, and which interacts with an environment at temperature $T$, is considered. It is found that heat enhances the tunneling probability at $T=0$ by a factor $\mathrm{exp}[A(T)M{\ensuremath{\omega}}_{0}\frac{{q}_{0}^{2}}{\ensuremath{\hbar}}]$, where $M$ is the mass of the system, ${\ensuremath{\omega}}_{0}$ is the frequency of small oscillations about the metastable state, and ${q}_{0}$ is the tunneling distance. For an undamped system $A(T)$ is exponentially small, $A(T)\ensuremath{\propto}\mathrm{exp}(\frac{\ensuremath{-}\ensuremath{\hbar}{\ensuremath{\omega}}_{0}}{{k}_{B}T})$, whereas for a dissipative system $A(T)$ grows algebraically with temperature.

DOI
10.1103/physrevlett.52.2193

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A quantum system which can tunnel out of a metastable state, and which interacts with an environment at temperature TTT, is considered. It is found that heat enhances the tunneling probability at T=0T=0T=0 by a factor exp[A(T)M\ensuremathω0q02\ensuremathℏ]\mathrm{exp}[A(T)M{\ensuremath{\omega}}_{0}\frac{{q}_{0}^{2}}{\ensuremath{\hbar}}]exp[A(T)M\ensuremathω0​\ensuremathℏq02​​], where MMM is the mass of the system, \ensuremathω0{\ensuremath{\omega}}_{0}\ensuremathω0​ is the frequency of small oscillations about the metastable state, and q0{q}_{0}q0​ is the tunneling distance. For an undamped system A(T)A(T)A(T) is exponentially small, A(T)\ensuremath∝exp(\ensuremath−\ensuremathℏ\ensuremathω0kBT)A(T)\ensuremath{\propto}\mathrm{exp}(\frac{\ensuremath{-}\ensuremath{\hbar}{\ensuremath{\omega}}_{0}}{{k}_{B}T})A(T)\ensuremath∝exp(kB​T\ensuremath−\ensuremathℏ\ensuremathω0​​), whereas for a dissipative system A(T)A(T)A(T) grows algebraically with temperature.
A quantum system which can tunnel, at T=0T=0T=0, out of a metastable state and whose interaction with its environment is adequately described in the classically accessible region by a phenomenological friction coefficient $\ensuremath{\eta}$, is considered. By only assuming that the environment response is linear, it is found that dissipation multiplies the tunneling probability by the factor $\mathrm{exp}[\ensuremath{-}\frac{A\ensuremath{\eta}{(\ensuremath{\Delta}q)}^{2}}{\ensuremath{\hbar}}]$, where $\ensuremath{\Delta}q$ is the "distance under the barrier" and AAA is a numerical factor which is generally of order unity.
As a prelude to a discussion of tunneling in the presence of dissipation, we discuss the damped harmonic oscillator. We show that for a harmonic-oscillator potential, the Caldeira-Leggett Hamiltonian can be reduced to a sum of independent harmonic-oscillator Hamiltonians by a normal-mode transformation. We then discuss tunneling through a parabolic barrier and explain the apparently contradictory results in the literature. The decay of a metastable state is then discussed. We argue that much of the published work on tunneling in dissipative systems has assumed unphysical initial conditions. We calculate the tunneling out of a metastable state, on the assumption that the environment is initially at zero temperature, and find that quantum tunneling is enhanced by dissipation.
Finite temperatures are incorporated into the calculation of the tunneling rate in the presence of linear dissipation. A theory of finite-temperature tunneling is introduced which is based on the density matrix which provides an approximate description of the metastable state. Semiclassical functional integral methods are used to calculate the density matrix and the Wentzel-Kramers-Brillouin approximation is used to derive the tunnel current from the density matrix. Dissipation appropriate to superconducting quantum-interference devices is introduced into the calculation by the use of the existing model of a heat bath consisting of a prescribed distribution of harmonic oscillators linearly coupled to the tunneling variable. The finite-temperature tunnel escape rate obtained is of the form \ensuremath{\Gamma}=A exp(-S/\ensuremath{\Elzxh}). S is determined from the solution of a nonlinear integro-differential equation. An existing numerical technique was modified to achieve this. The quantity A is approximately evaluated.
We study quantum tunneling in a biased double-well potential in the presence of Ohmic dissipation with a friction coefficient $\ensuremath{\alpha}>1$. The tunneling rate out of the metastable well is calculated in real time as a function of the temperature TTT and bias energy $\ensuremath{\epsilon}$. The T=0T=0T=0 result agrees with a droplet-model calculation in imaginary time. At low temperatures we find an enhancement to the T=0T=0T=0 rate varying as ${(\frac{T}{\ensuremath{\epsilon}})}^{2}$.
The macroscopic quantum tunneling in a double-well potential with dissipation is formulated without referring to instantons. This formulation, in the domain of small dissipation, goes beyond the instanton approximation which corresponds to the two-state truncated system (or, the so-called spin-boson Hamiltonian). It is first confirmed that our formulation, when applied to the two-state truncated system, reproduces all the known results of the instanton approximation. It is then shown that the ground state is mixed with excited states by dissipative interactions and this effect tends to enhance the ground-state quantum tunneling. In the Ohmic dissipation case this enhancement factor, even for a shallow double-well potential, can be at most comparable to the suppression factor which has been known from the work by Caldeira and Leggett. In the super-Ohmic case, however, the enhancement factor generally exceeds the suppression factor and the two-state approximation fails. A salient feature in the choice of counter terms in macroscopic quantum tunneling, which is essential to the above conclusion, is emphasized.