Quantum optomechanics with arbitrary mirror displacement: nonlinear Langevin equation with non-Markovian back-action noises

Abstract

Quantum optomechanics (QOM) explores the interaction between a quantum field, often confined in a cavity, and the quantum motion of mirrors, membranes or material media, usually assumed slow enough with negligible particle production, which is the central concern of dynamical Casimir effects, and a drive laser field for effective control. This rapidly developing field has a wide range of applications, from quantum sensing to the detection of gravitational waves. Because the opto-mechanical coupling is nonlinear, most theoretical investigations assume that the amplitude of mirror motion $x$ remains small. Recent years saw several papers treating $x^2, x^3$ orders in the mirror displacement. In our work we take a different route, deploying the functional perturbative method presented in [1] and enriched in [2], applicable for sufficiently weak opto-mechanical couplings. With this we can treat arbitrary displacement of the mirror, not just at a higher order in a power series expansion in $x$. We first derive the influence action of the quantum field and its non-Markovian noises back-reacting on the moving mirror with self-consistency. From it we derive a nonlinear Langevin equation driven by these back-action noises from the quantum field. We then show how results from our general modeling and treatment can be linked with the models and the Langevin equations presented in the literature, starting with the popular radiation pressure $Nx$ coupling, with $N$ the photon number, up to the recent $x^3$ order results. We hope this new approach and the results presented here can provide useful theoretical support for high precision QOM experimentation in the future.

The paper asks how to describe optomechanical back-action beyond a truncation in mirror displacement while retaining memory effects and quantum-field fluctuations. It considers one mechanical coordinate $Q$ coupled to multiple cavity modes through prescribed analytic functions $v_{nm}(Q)$ and $u_{nm}(Q)$ multiplying position–position and velocity–velocity products, including intermode terms. Using Schwinger–Keldysh and Feynman–Vernon methods, it expands the field influence action to second order in a weak interaction parameter $\lambda$, without expanding the coupling functions in $Q$. The resulting nonlinear Langevin equation contains potential renormalization, a history-dependent damping force, and Gaussian noise. Both damping and noise depend on $f_{nm}=v_{nm}-u_{nm}$; the vacuum kernels oscillate at mode-pair sum frequencies $\omega_n+\omega_m$. For $N$ cavity modes, the stochastic representation has $N(N+1)/2$ uncorrelated noise channels, one per unordered mode pair. Linear $f_{nm}(Q)$ gives additive noise and linear damping, whereas nonlinear functions produce multiplicative noise and nonlinear memory terms. When $v_{nm}=u_{nm}$, the derived noise and damping vanish, leaving potential corrections in the model considered. The framework is connected to photon-number radiation-pressure models and displacement expansions through cubic order. Its stated restriction is weak optomechanical coupling: arbitrary displacement dependence does not constitute a strong-coupling treatment. The calculations use vacuum field correlations; pumping and thermal reservoirs are discussed as additions rather than explicitly treated. Divergent local potential terms require renormalization, and higher-order, non-Gaussian back-action is left beyond the second-order calculation.