Numerical study of metastability due to tunneling: The quantum string method

TL;DR

This paper extends the string method to quantum tunneling, enabling calculation of tunneling rates via minimal action paths, and applies it to Josephson junctions, showing good agreement with analytical results and experimental relevance.

Contribution

The paper introduces a quantum string method based on path-integral formalism to compute quantum tunneling rates, a novel approach for analyzing metastability.

Findings

Quantum string method accurately calculates tunneling rates.

Results agree with analytical predictions near critical bias.

Escape rates are measurable at bias currents below the critical value.

Abstract

We generalize the string method, originally designed for the study of thermally activated rare events, to the calculation of quantum tunneling rates. This generalization is based on the analogy between quantum mechanics and statistical mechanics in the path-integral formalism. The quantum string method first locates, in the space of imaginary-time trajectories, the minimal action path (MAP) between two minima of the imaginary-time action. From the MAP, the saddle-point (``bounce'') action associated with the exponential barrier penetration probability is obtained and the pre-exponential factor (the ratio of determinants) for the tunneling rate evaluated using stochastic simulation. The quantum string method is implemented to calculate the zero-temperature escape rates for the metastable zero-voltage states in the current-biased Josephson tunnel junction model. In the regime close to the critical bias current, direct comparison of the numerical and analytical results yields good agreement. Our calculations indicate that for the nanojunctions encountered in many experiments today, the (absolute) escape rates should be measurable at bias current much below the critical current.