Statistics of work done in degenerate parametric amplification process

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.