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Physical Review Letters

Influence of Dissipation on Quantum Coherence

Alan J. Bray, M A Moore

Published Nov 22, 1982• 406 citations
Publisher

Abstract

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 $T=0$ to spontaneous symmetry breaking.

DOI
10.1103/physrevlett.49.1545

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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 \ensuremathη\ensuremath{\eta}\ensuremathη, is considered. It is shown that, provided \ensuremathηq02\ensuremathℏ\frac{\ensuremath{\eta}{{q}_{0}}^{2}}{\ensuremath{\hbar}}\ensuremathℏ\ensuremathηq0​2​ exceeds a critical value of order unity (\ifmmode±\else\textpm\fiq0\ifmmode\pm\else\textpm\fi{}{q}_{0}\ifmmode±\else\textpm\fiq0​ 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.
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.
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.
Whether a quantum system with a double-well effective potential undergoes spontaneous symmetry breaking depends not only on the potential landscape but also on the kinetics and the coupling with additional degrees of freedom. Here we introduce a quasi-exactly solvable model to study this problem in the context of ferroelectrics, with results that apply to a broad class of quantum phase transitions. Exploiting the analytical solutions, we provide a strict definition of the quantum paraelectric regime and identify a distinct quantum ferroelectric regime in which symmetry breaking can be realized without tunneling features. We then show that explicit symmetry breaking cannot be inferred from the order-parameter Hamiltonian alone, but requires additional couplings. This leads us to identify a regime of \emph{quantum dissipative paraelectricity}, in which observable symmetry breaking is suppressed during the evolution toward the ground state, even when the double-well structure dominates over zero-point quantum fluctuations.
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 ${(\frac{T}{\ensuremath{\epsilon}})}^{2}$.