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Physical Review Letters

Influence of Dissipation on Quantum Tunneling in Macroscopic Systems

A. O. Caldeira, A. J. Leggett

Published Jan 26, 1981• 2,259 citations
Publisher

Abstract

A quantum system which can tunnel, at $T=0$, 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 $A$ is a numerical factor which is generally of order unity.

DOI
10.1103/physrevlett.46.211

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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 \ensuremathη\ensuremath{\eta}\ensuremathη, is considered. By only assuming that the environment response is linear, it is found that dissipation multiplies the tunneling probability by the factor exp[\ensuremath−A\ensuremathη(\ensuremathΔq)2\ensuremathℏ]\mathrm{exp}[\ensuremath{-}\frac{A\ensuremath{\eta}{(\ensuremath{\Delta}q)}^{2}}{\ensuremath{\hbar}}]exp[\ensuremath−\ensuremathℏA\ensuremathη(\ensuremathΔq)2​], where \ensuremathΔq\ensuremath{\Delta}q\ensuremathΔq is the "distance under the barrier" and AAA is a numerical factor which is generally of order unity.
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.
A quantum mechanical particle which moves in a symmetric double-well potential, and whose interaction with the environment is described in the classical regime by a phenomenological friction coefficient $\ensuremath{\eta}$, is considered. It is shown that, provided $\frac{\ensuremath{\eta}{{q}_{0}}^{2}}{\ensuremath{\hbar}}$ exceeds a critical value of order unity ($\ifmmode\pm\else\textpm\fi{}{q}_{0}$ are the locations of the potential minima), the mean rate of tunneling between the degenerate minima decreases with temperature, leading at T=0T=0T=0 to spontaneous symmetry breaking.
Quantum tunnelling induces transitions not only through potential barriers, but also through energy barriers of dynamical origin. The coupling to a noisy environment introduces dissipation which affects tunnelling. This effect is studied by an analytic non-perturbative solution of a problem involving tunnelling through a dynamical barrier in the presence of a noisy environment. The accessibility to the classically forbidden energy region above the dynamical barrier is reduced (enhanced) by the friction(diffusion) effect. The occupation of each level above the dynamical barrier is characterized by the succession in time of two different regimes. The transition between both regimes is ruled by an energy criterion which is discussed.
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.
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.