Quantum tunnelling in a dissipative system

TY - JOUR

T1 - Quantum tunnelling in a dissipative system

AU - Caldeira, A. O.

AU - Leggett, A. J.

N1 - Funding Information: In this paper we have attempted to motivate, define, and discuss the question: What is the influence of dissipation on quantum tunnelling? Specifically, we have studied the quantum tunnelling behaviour of a system whose semiclassical dynamics is given by the dissipative Eq. (2.8), or a generalization of this. Our main conclusions are as follows: (1) The presence of dissipation always tends to suppress quantum tunnelling. (2) In the case of strictly linear dissipation (or more generally of a separable interaction), the suppression factor can be uniquely related to the phenomenological dissipation coefficient: in the general case a lower limit on this factor can be similarly related to it. (3) In the experimentally important case of strictly linear dissipation in a potential with cubic anharmonicity, the dominant part of the suppression factor can be written in the form exp - @(a) vqi/h, where q is the dissipation coefficient, q. the distance to be traversed under the barrier, and @(a) is a function of order 1 which we can estimate as explained in Section 5. To the extent that the behaviour of a given rfSQUID in the classical regime can be adequately described by the resistively shunted junction (RSJ) model, our results can be trivially transposed to it by the replacements q + @, V(q) + U(Q), M+ C, v -+ a, z R;‘. Our results can also be transposed if we assume that the correct description of the SQUID is some simple generalization of the RSJ model, e.g., such as to incorporate a nonlinear or frequency-dependent normal conductance a,. To go beyond this assumption would require an explicit discussion of the justification (for (a possibly generalised) Eq. (1.4), which in turn would lead us into detailed questions concerning the microscopic model of the junction, etc. These questions are unlikely, in our opinion, to affect the general nature of the results concerning the influence of dissipation on tunnelling, and are sufficiently technical that we have not attempted to discuss them here. Finally, it should of course be emphasized that all the calculations of this paper have been carried out within the conventional framework of quantum mechanics, that is, under the assumption that this framework can indeed be extrapolated to the macroscopic scale in the sense discussed in the Introduction. Should it eventually turn out that for a particular type of physical system quantum tunnelling is not observed under conditions where the theory predicts it should be, no doubt the most obvious inference would be that the calculations, or the model on which they are based, are wrong; however, an alternative inference, which it would be unwise to exclude totally a priori, would be that quantum mechanics cannot in fact be extrapolated in this way. In the course of this research, which has extended over the last four years, we have had many fruitful discussions with many colleagues both at the University of Sussex and elsewhere, some of whom are acknowledged in [28]. Over the last two years we have particularly benefited from discussions and/or correspondence about this problem with John Clarke, R. de Bruyn Ouboter, Roger Koch, Juhani Kurkijarvi, Gerd Schon, J. P. Sethna, Richard Voss, David Waxman, and Richard Webb. We are also grateful to John Bardeen for communicating to us his work on the related problem of tunnelling of charge density waves in quasi-one-dimensional solids, and Allen Goldman for suggestions concerning possible quantum tunnelling of the vortex-antivortex complex. We thank A. Widom and T. D. Clark for a preprint which induced us to elucidate the question of the “anomalous” cases mentioned in Appendix C. We are particularly grateful to Gabriel Barton for a critical reading of the first draft of the manuscript and helpful comments on it. One of us (AJL) gratefully acknowledges the hospitality of the Laboratory of Atomic and Solid State Physics at Cornell University during April 1980, when an important part of this work was done; the other (AOC) acknowledges financial support from CAPES (Coor-denecao de Aperfeicaomento de Pessoal de Nivel Superior) and from the Royal Society under the exchange agreement with CNPq (Conseilho National de Desen-volvimento Cientifico e Tecnologico).

PY - 1983/9

Y1 - 1983/9

UR - http://www.scopus.com/inward/record.url?scp=18344380226&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=18344380226&partnerID=8YFLogxK

U2 - 10.1016/0003-4916(83)90202-6

DO - 10.1016/0003-4916(83)90202-6

M3 - Article

AN - SCOPUS:18344380226

SN - 0003-4916

VL - 149

SP - 374

EP - 456

JO - Annals of Physics

JF - Annals of Physics

IS - 2

ER -