Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical review. B, Condensed matter

Dissipative quantum tunneling at finite temperatures

David Waxman, Anthony J. Leggett

Published Oct 1, 1985• 64 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.32.4450

Related papers

Quantum tunneling in the presence of an arbitrary linear dissipation mechanism

A. J. Leggett

This paper considers the tunneling out of a metastable state at $T=0$ of a system whose classical equation of motion is, in Fourier-transformed form, $K(\ensuremath{\omega})q(\ensuremath{\omega})=\ensuremath{-}[\frac{\ensuremath{\partial}V(q)}{\ensuremath{\partial}q}](\ensuremath{\omega})$ where $V(q)$ is a conservative potential and $K(\ensuremath{\omega})$ represents the effects of arbitrary linear dissipative and/or reactive elements. It is shown that, provided a few commonly satisfied conditions obtain, there is a simple prescription for writing down the imaginary-time effective action functional which determines the tunneling rate in the Wentzel-Kramers-Brillouin limit; namely, it contains the usual term in $V(q)$, plus a term of the form $(\frac{1}{2\ensuremath{\pi}}) \ensuremath{\int}{\ensuremath{-}\ensuremath{\infty}}^{\ensuremath{\infty}}\frac{1}{2} K(\ensuremath{-}i|\ensuremath{\omega}|){|\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{q}(\ensuremath{\omega})|}^{2}d\ensuremath{\omega}$, where $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{q}(\ensuremath{\omega})$ is the Fourier transform of the imaginary-time trajectory. Previously obtained results are special cases of this prescription. Applications are made to the case of "anomalous" dissipation (rate of dissipation proportional to the squared velocity of the momentum conjugate to the tunneling variable), to the "mixed" case (relaxation by collisions subject to a conservation law), and to more realistic models of a rf superconducting quantum-interference device.

Physical review. B, Condensed matter•Aug 1, 1984•300 citations

Quantum decay in a dissipative system

Lit-Deh Chang, Sudip Chakravarty

In view of recent interest in the problem of macroscopic quantum tunneling in systems involving the Josephson effect, we present an accurate numerical calculation of the tunneling rate of a system from a metastable well, at zero temperature, in the presence of dissipative coupling to the environment. Although we concentrate on a specific form of dissipation, as discussed by Caldeira and Leggett, we believe that such a numerical method can be extended to other forms of dissipation as well. Our method is based on the framework recently described by Caldeira and Leggett, and requires (a) a novel treatment of a nonlinear integro-differential equation and (b) an extension of the usual Fredholm scattering theory so as to be applicable to the present dissipative problem. We present explicit results for wide ranges of dissipation and estimate our error in the calculation of the exponent to be no larger than 0.1% and of the prefactor to be no larger than 2%.

Physical review. B, Condensed matter•Jan 1, 1984•85 citations

Dissipative quantum tunneling: A variational principle for phase shifts

Chang Han, Peter S. Riseborough

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.

Physical review. B, Condensed matter•Aug 1, 1989•1 citation

Heisenberg’s equations of motion for dissipative tunneling

Mohsen Razavy

The quantum-mechanical problem of tunneling of a particle coupled to a heat bath that consists of a large number of harmonic oscillators is studied in this paper. Here the equation of motion for the particle which is a nonlinear integro-differential equation is found by eliminating the degrees of freedom of the oscillators. The nonlinear operator equation may be solved by noting that the position operator at the time t can be expanded as a power series in terms of the initial position and momentum with time-dependent c-number coefficients. These coefficients also satisfy nonlinear integro-differential equations. For the problem of dissipative tunneling of a particle in a double-well potential, the first few terms of expansion have been calculated numerically. The expansion parameter in this formulation is a small dimensionless parameter which is inversely proportional to the product of the width and the square root of the height of the barrier.

Physical Review A•Jun 1, 1990•12 citations

Influence-functional theory of metastability in a dissipative quantum system

W. Zwerger

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 ${T}^{n+1}$ 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.

Physical Review A•Mar 1, 1985•28 citations

Decay of a metastable state: A variational approach

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

We evaluate the exponential suppression of the quantum tunneling rate due to the coupling with a thermal reservoir. The exponential dependence is calculated by a variational method, in which an approximate saddle point of the effective action is found. The effect of finite temperatures is accounted for. We calculate the exponential part of the rate for several potentials, some of which have not been considered previously. The method gives excellent agreement with numerically calculated values, where they exist. We also consider the effect of Ohmic damping mechanisms of Drude form (memory damping). This latter case may be of importance for experiments on rf superconducting quantum interference devices and Josephson junctions where frequency-dependent damping mechanisms have been observed. The trends exhibited by the variational approximation can be understood qualitatively on the basis of physical arguments.

Physical review. B, Condensed matter•Aug 1, 1986•16 citations
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.
This paper considers the tunneling out of a metastable state at T=0T=0T=0 of a system whose classical equation of motion is, in Fourier-transformed form, $K(\ensuremath{\omega})q(\ensuremath{\omega})=\ensuremath{-}[\frac{\ensuremath{\partial}V(q)}{\ensuremath{\partial}q}](\ensuremath{\omega})$ where V(q)V(q)V(q) is a conservative potential and $K(\ensuremath{\omega})$ represents the effects of arbitrary linear dissipative and/or reactive elements. It is shown that, provided a few commonly satisfied conditions obtain, there is a simple prescription for writing down the imaginary-time effective action functional which determines the tunneling rate in the Wentzel-Kramers-Brillouin limit; namely, it contains the usual term in V(q)V(q)V(q), plus a term of the form $(\frac{1}{2\ensuremath{\pi}}) \ensuremath{\int}{\ensuremath{-}\ensuremath{\infty}}^{\ensuremath{\infty}}\frac{1}{2} K(\ensuremath{-}i|\ensuremath{\omega}|){|\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{q}(\ensuremath{\omega})|}^{2}d\ensuremath{\omega}$, where $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{q}(\ensuremath{\omega})$ is the Fourier transform of the imaginary-time trajectory. Previously obtained results are special cases of this prescription. Applications are made to the case of "anomalous" dissipation (rate of dissipation proportional to the squared velocity of the momentum conjugate to the tunneling variable), to the "mixed" case (relaxation by collisions subject to a conservation law), and to more realistic models of a rf superconducting quantum-interference device.
In view of recent interest in the problem of macroscopic quantum tunneling in systems involving the Josephson effect, we present an accurate numerical calculation of the tunneling rate of a system from a metastable well, at zero temperature, in the presence of dissipative coupling to the environment. Although we concentrate on a specific form of dissipation, as discussed by Caldeira and Leggett, we believe that such a numerical method can be extended to other forms of dissipation as well. Our method is based on the framework recently described by Caldeira and Leggett, and requires (a) a novel treatment of a nonlinear integro-differential equation and (b) an extension of the usual Fredholm scattering theory so as to be applicable to the present dissipative problem. We present explicit results for wide ranges of dissipation and estimate our error in the calculation of the exponent to be no larger than 0.1% and of the prefactor to be no larger than 2%.
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.
The quantum-mechanical problem of tunneling of a particle coupled to a heat bath that consists of a large number of harmonic oscillators is studied in this paper. Here the equation of motion for the particle which is a nonlinear integro-differential equation is found by eliminating the degrees of freedom of the oscillators. The nonlinear operator equation may be solved by noting that the position operator at the time t can be expanded as a power series in terms of the initial position and momentum with time-dependent c-number coefficients. These coefficients also satisfy nonlinear integro-differential equations. For the problem of dissipative tunneling of a particle in a double-well potential, the first few terms of expansion have been calculated numerically. The expansion parameter in this formulation is a small dimensionless parameter which is inversely proportional to the product of the width and the square root of the height of the barrier.
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.
We evaluate the exponential suppression of the quantum tunneling rate due to the coupling with a thermal reservoir. The exponential dependence is calculated by a variational method, in which an approximate saddle point of the effective action is found. The effect of finite temperatures is accounted for. We calculate the exponential part of the rate for several potentials, some of which have not been considered previously. The method gives excellent agreement with numerically calculated values, where they exist. We also consider the effect of Ohmic damping mechanisms of Drude form (memory damping). This latter case may be of importance for experiments on rf superconducting quantum interference devices and Josephson junctions where frequency-dependent damping mechanisms have been observed. The trends exhibited by the variational approximation can be understood qualitatively on the basis of physical arguments.