Closed-time-path approach to the optomechanical back-reaction problem

Abstract

We present a perturbative closed-time-path (in-in) formulation of an optomechanical system in which a quantum field interacts with a moving mirror via radiation pressure. We derive the effective action governing the dynamics of the moving mirror, incorporating the full back-reaction of the cavity field. These effects are encoded in fluctuation and dissipation kernels, that we show satisfy fluctuation-dissipation relations, and whose spectral structure reveals a direct connection with the underlying physical mechanism responsible for the back-reaction, that is particle creation by the dynamical Casimir effect. By deriving the semiclassical equations of motion for the moving mirror, and computing the energy radiated into the field within the in-out formalism of quantum field theory, we verify the energy balance between the mechanical energy dissipated by the optical back-reaction forces acting on the mirror and the energy carried by the particles created in the field.

Figures (3)

• Figure 1: Schematic representation of the optical cavity: the left mirror is fixed at the position $z=0$, while the right mirror, whose time-dependent position is $z=q(t)$, is free to move within a the potential $V(q)$. The mirrors interact with the scalar field $A(z,t)$ enclosed in the cavity.
• Figure 2: Illustrative representation of the closed-time trajectories that contribute to the influence functional for the moving mirror: The initial state of the field, described by the density operator $\rho_{\mathrm{A}}(Q_{ki},Q_{ki}';t_i)$, is evolved forward in time along the path $Q_{k+}(t)$, driven by the mirror motion $q_+(t)$, and backward in time along the path $Q_{k-}(t)$, driven by the motion $q_-(t)$. The forward and backward branches of the path integral meet in the same point $Q_{kf}$ at the final time $t_f$, thus closing the overall path. Integration over all possible final field amplitudes $Q_{kf}$ and against the variables $Q_{ki}$, and $Q_{ki}'$ sampling the initial state of the field, yields the influence functional, as define in Eq. \ref{["Fqq'"]} in the text.
• Figure 3: Diagrammatic representation of the decomposition of the fourth-order correlation function $D_k^{a_1 a_2 a_3 a_4}(t_1,t_2,t_3,t_4)$, in terms of connected correlators $G_k^{a_1\ldots a_n}(t_1,\ldots,t_n)$ with $n\leq 4$. The indices $a_i,a_j,a_m,a_n=\pm$ ($i,j,m,n=1,2,3,4$) specify whether the corresponding time arguments lie on the forward $(+)$ or backward $(-)$ branch of the closed-time-path contour. Since the connected generating functional $\mathcal{W}_{\mathrm{A}}$ for the free field in a Gaussian thermal state is quadratic in the sources $J_k$ [see Eq. \ref{['Sigma']}, in the text], all connected correlators with $n\neq 2$ vanish. Therefore, the fourth-order correlation function reduces to a sum of products of two-point closed-time-path propagators, namely the Feynman, Dyson, and Wightman functions.