WKB approach and quantum corrections to classical dynamics in the Josephson problem

TL;DR

This paper uses a many-body WKB method to calculate quantum corrections to the classical dynamics of the Josephson model, revealing new quantization conditions and quantum break times at critical energies.

Contribution

It introduces a WKB-based approach to derive quantum corrections and modified quantization conditions in the Josephson problem, especially at critical energies.

Findings

Derived a modified Bohr-Sommerfeld quantization condition.

Identified quantum break times with logarithmic scaling.

Extended WKB applicability to many-body quantum systems.

Abstract

We apply a many-body Wentzel-Kramers-Brillouin (WKB) approach to determine the leading quantum corrections to the semiclassical dynamics of the Josephson model, describing interacting bosons able to tunnel between two localized states. The semiclassical dynamics is known to divide between regular oscillations and self-trapped oscillations where the sign of the imbalance remains fixed. In both cases, the WKB wave functions are matched to Airy functions, yielding a modified Bohr-Sommerfeld quantization condition. At the critical energy dividing normal and self-trapped oscillations, the WKB wave functions should instead be matched to parabolic cylinder functions, leading to a quantization formula that is not just the Bohr-Sommerfeld formula, and recovering the known logarithmic quantum break times at this energy. This work thus provides another illustration of the usefulness of the WKB approach in certain many-body problems.