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Physical Review Letters

Quantum tunneling, dissipation, and fluctuations

Robijn Bruinsma, Per Bak

Published Feb 3, 1986• 88 citations
Publisher

Abstract

A theory of quantum tunneling with dissipation is constructed. The frequency-dependent transmission coefficient is calculated at zero and finite temperatures. A fluctuation-dissipation theorem for tunneling particles is derived.
DOI
10.1103/physrevlett.56.420

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We suggest taking the fluctuation-dissipation theorem of Callen and Welton [Phys. Rev. 83, 34 (1951)] as a basis to study quantum dissipative phenomena (such as macroscopic quantum tunneling) in a manner analogous to the Nambu-Goldstone theorem for spontaneous symmetry breakdown [Phys. Rev. 122, 345 (1961); Nuovo Cimento 19, 154 (1961)]. It is shown that the essential physical contents of the Caldeira-Leggett model [Phys. Rev. Lett. 46, 211 (1981); Ann. Phys. (N.Y.) 149, 374 (1983)] such as the suppression of quantum coherence by Ohmic dissipation are derived from general principles only, namely, the fluctuation-dissipation theorem and unitarity and causality (i.e., dispersion relations), without referring to an explicit form of the Lagrangian. An interesting connection between quantum tunneling with Ohmic dissipation and Anderson's orthogonality theorem is also noted.
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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 $\mathrm{exp}[A(T)M{\ensuremath{\omega}}_{0}\frac{{q}_{0}^{2}}{\ensuremath{\hbar}}]$, where MMM is the mass of the system, ${\ensuremath{\omega}}_{0}$ 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, $A(T)\ensuremath{\propto}\mathrm{exp}(\frac{\ensuremath{-}\ensuremath{\hbar}{\ensuremath{\omega}}_{0}}{{k}_{B}T})$, whereas for a dissipative system A(T)A(T)A(T) grows algebraically with temperature.