No friction, no tunnelling

M. Kraev

V. P. Krainov

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.

Masaaki Tokieda, K. Hagino

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.

Zhuangqi Cao, Cheng Yin

Peter Hänggi A, Waldemar Hontscha

The role of memory friction on dissipative quantum reaction rates is emphasized. The domain of low temperatures, bearing a variety of important tunnelling phenomena, is difficult to handle analytically. In order to fill this gap, we present new scaling relations for the bounce length τ B as a function of the dissipation renormalized barrier frequency μ. In addition, we propose a simple estimate for τ B , which in turn yields directly the thermal low temperature enhancement of the quantum decay rate for Ohmic friction, as well as for memory friction. Finally the range of validity of the theory is discussed and pictorally represented in a rate phase diagram.