Statistics of work done in degenerate parametric amplification process

TL;DR

This paper analyzes the statistical properties of work done by classical pumps on a quantum optical oscillator, deriving work distribution functions and confirming fluctuation theorems, with insights into phase effects and symmetry breaking.

Contribution

It provides analytical and numerical methods to compute work statistics in a quantum oscillator driven by classical fields, revealing phase-dependent symmetry breaking effects.

Findings

Work distribution satisfies fluctuation theorems

Relative phase shift by π is required for symmetry

Analytical inversion of moment generating function when one pump is active

Abstract

We study statistics of work done by two classical electric field pumps (two-photon and one-photon resonant pumps) on a quantum optical oscillator. We compute moment generating function for the energy change of the oscillator, interpreted as work done by the classical drives on the quantum oscillator starting out in a thermalized Boltzmann state. The moment generating function is inverted, analytically when only one of the pumps is turned on and numerically when both the pumps are turned on, to get the probability function for the work. The resulting probability function for the work done by the classical drive is shown to satisfy transient detailed and integral work fluctuation theorems. Interestingly, we find that, in order for the work distribution function to satisfy the fluctuation theorem in presence of both the drivings, relative phase of drivings need to be shifted by $\pi$, this is related to the broken time reversal symmetry of the Hamiltonian.