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Theoretical and Mathematical Physics

Friction in macroscopic tunneling

V. P. Krainov

Published Apr 1, 2026• 0 citations
Publisher

Abstract

Abstract Macroscopic tunneling of Cooper pairs of electrons in a system of two superconductors separated by an insulator was modeled by Caldeira and Leggett by quantum-mechanical motion of a particle through the potential barrier. Friction (conductivity of the Josephson contact in experiments) was modeled by a one-dimensional phonon thermostat that interacts with the particle. Then the tunneling exponent was expressed in terms of the friction coefficient determined by the thermostat parameters. Alternatively, we propose a direct introduction of the friction coefficient into a classical action in the tunneling exponent using the method of imaginary time. As a result, the numerical coefficient of the classical friction contribution to the tunneling exponent is obtained.

DOI
10.1134/s0040577926040082

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Abstract Macroscopic tunneling of Cooper pairs of electrons in a system of two superconductors separated by an insulator was modeled by Caldeira and Leggett by quantum-mechanical motion of a particle through the potential barrier. Friction (conductivity of the Josephson contact in experiments) was modeled by a one-dimensional phonon thermostat that interacts with the particle. Then the tunneling exponent was expressed in terms of the friction coefficient determined by the thermostat parameters. Alternatively, we propose a direct introduction of the friction coefficient into a classical action in the tunneling exponent using the method of imaginary time. As a result, the numerical coefficient of the classical friction contribution to the tunneling exponent is obtained.
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We present a comprehensive study of the effect of a transverse magnetic field on the macroscopic quantum tunneling of two identical coupled particles with dissipation. Two coupled particles of identical masses but opposite charges in the plane, in the presence of a constant transverse external magnetic field and an external potential interacting with a bath of harmonic oscillators are studied. In this case, the problem cannot be mapped to a one-dimensional problem, it strictly remains two-dimensional. We calculate the exact effective action both for the case of linear coupling to the bath and without a linear coupling to the bath using imaginary time path integral and Leggett's prescription at T=0. In the limit of zero magnetic field we recover a two dimensional version of the result found in \cite{em1} for the case of two identical particles. We find that in the limit of weak and strong dissipation, the effective action reduces to a two dimensional version of the Caldeira-Leggett form in terms of the reduced mass and magnetic field. The case of Ohmic dissipation with the motion of the two particles damped by the Ohmic frictional constant η\etaη is studied in detail.
Macroscopic quantum tunneling (MQT) for a single fluxon moving along a long Josephson junction is studied theoretically. To introduce a fluxon-pinning force, we consider inhomogeneities made by modifying thickness of an insulating layer locally. Two different situations are studied: one is the quantum tunneling from a metastable state caused by a single inhomogeneity, and the other is the quantum tunneling in a two-state system made by two inhomogeneities. In the quantum tunneling from a metastable state, the decay rate is estimated within the WKB approximation. Dissipation effects on a fluxon dynamics are taken into account by the Caldeira-Leggett theory. We propose a device to observe quantum tunneling of a fluxon experimentally. Required experimental resolutions to observe MQT of a fluxon seem attainable within the presently available micro-fabrication technique. For the two-state system, we study quantum resonance between two stable states, i.e., macroscopic quantum coherence (MQC). From the estimate for dissipation coefficients due to quasiparticle tunneling, the observation of MQC appears to be possible within the Caldeira-Leggett theory.
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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 $\ensuremath{\eta}$, 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.