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Physical Review A

Traversal time in macroscopic quantum tunneling

Paolo Moretti, Daniela Mugnai, Anedio Ranfagni, Maurizio Cetica

Published Dec 1, 1999• 8 citations
Publisher

Abstract

We present an interpretation of traversal time results in Josephson junctions, based on a stochastic model of tunneling processes which include dissipative effects. With the use of semiclassical analysis, the experiments performed up until now have supplied a time duration of the order of ${10}^{2}\mathrm{ps}$ (a remarkably long time, even considering the macroscopic nature of the systems under examination). According to our interpretation, this duration is only the imaginary part of a complex quantity, the real part of which (presumably the true physical duration) lies in the range of a few picoseconds.

DOI
10.1103/physreva.60.5087

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Macroscopic quantum tunneling with decoherence

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arXiv:quant-ph/0407185•Jul 23, 2004•0 citations
We present an interpretation of traversal time results in Josephson junctions, based on a stochastic model of tunneling processes which include dissipative effects. With the use of semiclassical analysis, the experiments performed up until now have supplied a time duration of the order of 102ps{10}^{2}\mathrm{ps}102ps (a remarkably long time, even considering the macroscopic nature of the systems under examination). According to our interpretation, this duration is only the imaginary part of a complex quantity, the real part of which (presumably the true physical duration) lies in the range of a few picoseconds.
The various ways of evaluating dissipative effects in macroscopic quantum tunneling are re-examined. The results obtained by using functional integration, while confirming those of previously given treatments, enable a comparison with available experimental results relative to Josephson junctions. A criterion based on the shortening of the semiclassical traversal time $\ensuremath{\tau}$ of the barrier with regard to dissipation can be established, according to which $\ensuremath{\Delta}\ensuremath{\tau}/\ensuremath{\tau}\ensuremath{\gtrsim}N/Q$, where QQQ is the quality factor of the junction and NNN is a numerical constant of order unity. The best agreement with the experiments is obtained for N=1.11N=1.11N=1.11, as it results from a semiempirical analysis based on an increase in the potential barrier caused by dissipative effects.
The problem of evaluating dissipative effects in macroscopic quantum tunneling is re-examined for the case of Josephson junctions, with the adoption of an alternative way with respect to several previously proposed and, in some cases, contradictory approaches. The system, which consists of a junction coupled to a transmission line, is analyzed both analytically and numerically. A test of the theoretical model, as compared to the experimental results available, is performed in accordance with a criterion based on a shortening of the traversal time.
Quantum tunneling is analyzed within the Everettian interpretation, where decoherence generates distinct reflected and transmitted branches. The tunneling probability is obtained from relative branch weights via the Born rule without invoking collapse. Tunneling time τTτ_T τT​ is reformulated as the branching duration τT∼NBτBτ_T \sim N_B τ_BτT​∼NB​τB​, where NB\ N_B  NB​ is the number of environmental degrees of freedom and τBτ_BτB​ is the decoherence time required to distinguish branches. Macroscopic tunneling in Josephson junctions is also analyzed.
The results relative to macroscopic quantum tunneling rate, out of the metastable state of Josephson junctions, are examined in view of determining the effect of dissipation. We adopt a simple criterion in accordance to which the effect of dissipation can be evaluated by analyzing the shortening of the semiclassical traversal time of the barrier. In almost all the considered cases, especially those with relatively large capacitance values, the relative time shortening turns out to be about 20% and with a corresponding quality factor Q ≃ 5.5. However, beyond the specific cases here considered, still in the regime of moderate dissipation, the method is applicable also to different situations with different values of the quality factor. The method allows, within the error limits, for a reliable determination of the load resistance RL, the less accessible quantity in the framework of the resistively and capacitively shunted junction model, provided that the characteristics of the junction (intrinsic capacitance, critical current, and the ratio of the bias current to the critical one) are known with sufficient accuracy.
The tunneling probability for a system modelling macroscopic quantum tunneling is computed. We consider an open quantum system with one degree of freedom consisting of a particle trapped in a cubic potential interacting with an environment characterized by a dissipative and a diffusion parameter. A representation based on the energy eigenfunctions of the isolated system, i. e. the system uncoupled to the environment, is used to write the dynamical master equation for the reduced Wigner function of the open quantum system. This equation becomes very simple in that representation. The use of the WKB approximation for the eigenfunctions suggests an approximation which allows an analytic computation of the tunneling rate, in this real time formalism, when the system is initially trapped in the false ground state. We find that the decoherence produced by the environment suppresses tunneling in agreement with results in other macroscopic quantum systems with different potentials. We confront our analytical predictions with an experiment where the escape rate from the zero voltage state was measured for a current-biased Josephson junction shunted with a resistor.