Numerical Study of The Dynamical Casimir Effect and its Classical Analogue in a Double Cavity

Abstract

We study the time evolution of light fields inside a double cavity which is comprised of two perfect end mirrors and a parametrically driven, partially transmissive central mirror in both a classical and a quantum mechanical framework. It is common practise in the field of optomechanics to take a Hamiltonian approach \cite{aspelmeyer2014cavity} ignoring non-linear coupling terms between the light field and the moving mechanical element. By contrast, we start from the Maxwell wave equation which is second order in time and find that a first order in time Schr\"{o}dinger-type wave equation (equivalent to neglecting the non-linear coupling) is a valid approximation for low enough mirror reflectivity and speed and for large light frequencies. We also study adiabatic dynamics for the Maxwell wave equation and find it differs from the more familiar adiabaticity in the Schr\"{o}dinger equation.

Next, we numerically simulate the dynamical Casimir effect (DCE) in the double cavity with a sinusoidally driven central mirror following earlier numerical work on the perfect single cavity \cite{Ruser2006NumericalDCE,ruser2005vibrating,naylor2009dynamical}. Because our central mirror is partially transmissive it is physically more realistic and circumvents fundamental problems associated with having perfectly reflecting moving mirrors \cite{Moore1970DCESingleCavity,barton1993quantum}. The corresponding photon creation rates are drastically lower when compared to the perfectly reflective mirror case. Furthermore, if we make one of the cavities much longer than the other we can simulate the DCE for a single open cavity coupled to an environment without having to make the Markov approximation. The resultant asymmetric double cavity (ADC) model is valid for times short enough that only a negligible number of the photons that has leaked out of the open cavity has sloshed back in again. As for the symmetric case, one advantage of the ADC is that driven mirror is partially transmissive rather than perfectly reflecting.