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Physical Review Letters

Canonical formulation of quantum tunneling with dissipation

Kazuo Fujikawa, Satoshi Iso, Misao Sasaki, Hiroshi Suzuki

Published Feb 24, 1992• 44 citations
Publisher

Abstract

The 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. Several applications of this formulation for the Ohmic case are presented, and we show that the virtual mixing of excited states induced by dissipative interactions tends to enhance the tunneling rate. This enhancement effect becomes dominant for a super-Ohmic case.

DOI
10.1103/physrevlett.68.1093

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The 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. Several applications of this formulation for the Ohmic case are presented, and we show that the virtual mixing of excited states induced by dissipative interactions tends to enhance the tunneling rate. This enhancement effect becomes dominant for a super-Ohmic case.
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.
For the realistic theoretical descriptions of quantum devices, we have to take the unavoidable coupling to their surrounding environment into account. A lot of effort in theoretical physics has been made to describe decoherence effects and the corresponding quantum to classical transition in the past. In these studies, baths are coupled to certain observables making their behavior more classical. In the most general case, a quantum device is affected by environments coupled to different observables. In this thesis, we study the differences emerging from coupling the bath to different observables. In particular, we investigate the situation, where two baths are coupled simultaneously to two conjugate variables (e.g. xˆ and pˆ, denoted dissipative frustration). We study how the interplay of these two baths can lead to interesting effects. After an introduction of the topic, we discuss the results of this thesis in Part II and Part III. In Part II, we study the effect of quantum tunneling in presence of a dissipative interaction with two Ohmic environments to the position xˆ and/or to the momentum pˆ of a particle. In chapter 2, we consider a symmetric double well potential which is important in the context of coherent state preparation, as its ground state is a superposition of being in the left and in the right well. Using the semiclassical approximation to calculate instanton paths within the path integral method, we find that momentum dissipation enhances the coherent tunnel splitting. In presence of dissipative couplings to xˆ and pˆ, the momentum dissipation acts in favor of the delocalization of the particle shifting the critical coupling strength of the dissipative phase transition induced by the position dissipation. Chapter 3 contains a study of the escape rate out of a metastable well in presence of dissipation. Here we find that the incoherent tunneling rate is exponentially enhanced in presence of momentum dissipation. We further show that the influence of momentum dissipation depends on the slope of the part of barrier the particle escaping through. In presence of both dissipative couplings, we find a non-monotonic behavior of the escape rate as a function of the dissipative couplings strengths. In Part III, we show that it is possible to realize the situation of "dissipative frustration" in Josephson junction chains. In particular, we introduce charge dissipation as a physical realization of the afore introduced momentum coupling. After presenting the circuit elements needed for charge dissipation, we study a chain of Josephson junctions affected by Ohmic baths coupled to the charge and/or the phase. Using a self-consistent harmonic approximation, we determine the phase diagram at zero temperature which exhibits a quantum phase transition between an ordered phase, corresponding to the superconducting state, and a disordered phase, corresponding to the insulating state with localized superconducting charge. Interestingly, we find that the critical line separating the two phases has a non-monotonic behavior as a function of the dissipative coupling strength. Moreover, within the self-consistent harmonic approximation, we analyze the dissipation induced crossover between a first and second order phase transition, showing that dissipative frustration increases the range in which the phase transition is second order. The non-monotonic behavior is also reflected in the purity of the system that quantifies the degree of correlation between the system and the environment, and in the logarithmic negativity as an entanglement measure that encodes the internal quantum correlations in the chain.
Applying a technique developed recently [L. H. Yu and C. P. Sun, Phys. Rev. A 49, 592 (1994); L. H. Yu, Phys. Lett. 202, 167 (1995)] for a harmonic oscillator coupled to a bath of harmonic oscillators, we present an exact solution for the tunneling problem in an Ohmic dissipative system with inverted harmonic potential. The result shows that while the dissipation tends to suppress the tunneling, the Brownian motion tends to enhance the tunneling. Whether the tunneling rate increases or not would then depend on the initial conditions. We give a specific formula to calculate the tunneling probability determined by various parameters and the initial conditions. \textcopyright{} 1996 The American Physical Society.
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 ${(\frac{T}{\ensuremath{\epsilon}})}^{2}$.