# Self-consistent time-dependent harmonic approximation for the sine-Gordon model out of equilibrium

## TL;DR

This paper develops a self-consistent time-dependent harmonic approximation for the out-of-equilibrium sine-Gordon model, enabling analysis of quantum dynamics in tunnel-coupled Bose gases with comparisons to exact solutions.

## Contribution

It introduces a novel self-consistent harmonic approximation for the sine-Gordon model out of equilibrium, validated against exact solutions and applied to experimentally relevant systems.

## Key findings

- Derived the time evolution of phase fluctuations in Bose gases
- Validated the approximation against exact results at the free fermion point
- Provided insights into the dynamics of quantum sine-Gordon systems

## Abstract

We derive a self-consistent time-dependent harmonic approximation for the quantum sine-Gordon model out of equilibrium and apply the method to the dynamics of tunnel-coupled one-dimensional Bose gases. We determine the time evolution of experimentally relevant observables and in particular derive results for the probability distribution of subsystem phase fluctuations. We investigate the regime of validity of the approximation by applying it to the simpler case of a nonlinear harmonic oscillator, for which numerically exact results are available. We complement our self-consistent harmonic approximation by exact results at the free fermion point of the sine-Gordon model.