Coupling Enhancement and Symmetrization in Dissipative Optomechanical Systems

TL;DR

This work proposes a cross-Kerr–based strategy to enhance and symmetrize optomechanical coupling at the single-photon level using a dual-laser drive in a Fabry-Pérot cavity. By leveraging the cross-Kerr term $H_{CK}=\hbar\chi a^\dagger a b^\dagger b$ and a steady-state displacement $\beta_s$, the authors engineer an effective real coupling $g_s=\chi\beta_s$ and, under suitable conditions, realize symmetric photon–phonon fluctuations with an effective Hamiltonian $H_{eff}=\hbar\Delta''_c a^\dagger a + \hbar\Delta_m b^\dagger b - \hbar G_R (a^\dagger + a)(b^\dagger + b)$. The paper derives nonlinear and linearized quantum Langevin equations, identifies the real-coupling regime, and demonstrates strong-coupling access with surprisingly low driving powers in a circuit-QED setting. It then analyzes input–output scattering, showing that optimal reciprocal transmission occurs at $\omega=\Delta''_c=\Delta_m$, $\kappa_a=\gamma_b$, and $G_R=0.5\kappa_a$, with careful treatment of non-RWA contributions in dissipative regimes. Overall, the results establish a tunable route to single-photon optomechanical functionalities and pave the way for symmetric, ultrastrong-coupling devices in quantum technologies.

Abstract

Observing single-photon optomechanical effects remains a significant challenge in cavity optomechanics. To investigate intrinsic optomechanical nonlinear effects at the single-photon level, it is essential to strengthen the interaction between a single photon and a finite number of phonons. In this work, we enhance the optomechanical coupling by introducing a two-laser driving scheme together with a second-order nonlinear contribution to the cavity resonance frequency, which overtakes the original radiation-pressure nonlinearity as the dominant part in a prototypical Fabry-Pérot cavity. By properly tuning the two driving lasers and the cross-Kerr interaction so that the effective coupling becomes real, we theoretically establish a symmetric optomechanical model at the single-photon level where photon and phonon mode fluctuations exhibit identical dynamics. Within this framework, we analyze the optimal reciprocal transport of the input laser field. Through examination of the optimal signal transmission, we identify the critical boundary associated with the onset of the strong-coupling regime. Additionally, we compare the optical signal scattering behavior in both dissipative equilibrium and nonequilibrium symmetric optomechanical systems, with and without non-rotating-wave contributions. Our work achieves controllable enhancement of the optomechanical coupling and enables optimal signal transfer even in the ultrastrong-coupling regime, suggesting a promising route toward tunable single-photon optomechanical functionalities.

Figures (12)

• Figure 1: $\left( \rm{a} \right)$ A flowchart for obtaining an enhanced Hamiltonian ${H_{\rm{T}}}$. Initially, we consider a primitive (undriven) Fabry-Pérot cavity ${H_{\rm{FP}}}$. The first step is to introduce two laser beams ${H_{\rm{d}}}\left( t \right)$ to drive the initial system. Next, we extend the above closed system $H_{\rm{S}}^{\rm{P}}$ to an open quantum system $H_{\rm{T}}^{\rm{P}}$ by adding an environmental Hamiltonian $H_{\rm{B}}^{\rm{P}}$. Then, we perform a steady-state coherent displacement transformation on it. As the last step, we omit the CK interaction term to obtain an enhanced Hamiltonian ${H_{\rm{T}}}$ at the single-photon level. $\left( \rm{b} \right)$ Schematic illustration of a generalized optomechanical model $H_{\rm{T}}^{\rm{P}}$ [left side of $\left( \rm{a} \right)$], which is composed of a single-mode optical field $a$ (lower side) and a mechanical mode $b$ (upper side). The optical mode is coupled to the mechanical mode via both the RP interaction ${g_{\rm{0}}}$$\left( \leftrightarrow \right)$ and the CK interaction $\chi$$\left( \sim \right)$. The optical mode is driven by a low-power laser ${\Omega _a}$, while the mechanical mode is driven by a high-power laser ${\Omega _b}$. In addition, the near-resonant coupling between the system modes and their corresponding reservoir modes is also considered in this open quantum system.
• Figure 2: $\left( {\rm{a}} \right)$ Schematic diagram of a symmetric optomechanical system with dual laser drives. $\left( {\rm{b}} \right)$ The number of excited phonons in the system after the coherent displacement transformation at the steady state and the standard mean-field operation.
• Figure 3: $\left( \rm{a} \right)$ Average photon number ${\bar{N}_a}$ in the optical cavity varies with the laser power ${P_a} \in \left[ {0,0.5{\rm{fW}}} \right]$ and the phase ${\varphi _a} \in \left[ {0,0.5\pi } \right)$ of the optical driving laser field ${\Omega _a}$. $\left( \rm{b} \right)$ The average number of thermally excited phonons ${\bar{N}_b}$ varies with the laser power ${P_b} \in \left[ {0,5{\rm{fW}}} \right]$ and the phase ${\varphi _b} \in \left[ {\pi,1.5\pi } \right)$ of the mechanical driving laser field ${\Omega _b}$. Both $\left( \rm{a} \right)$ and $\left( \rm{b} \right)$ use the data of the circuit-QED in Table \ref{['tab-1']}: ${\Delta _{\rm{m}}} = {\omega _{{L_a}}} = {\omega _{{L_b}}} = 1.04\pi \times {10^{10}}{\rm{Hz}},{\kappa _a} = {\gamma _b} = 6.6\pi \times {10^7}{\rm{Hz}},\chi = 5\pi \times {10^6}{\rm{Hz}}$, and $\hbar = 1.05 \times {10^{ - 34}}J \cdot s$.
• Figure 4: Schematic diagram of a symmetric optomechanical system with dual laser drives at the single-photon level. The wavy line with $a \to b$ represents the transmission process of the photon signal from the optical cavity field to the mechanical oscillator, while the one with $b \to a$ describes the opposite process of the phonon signal.
• Figure 5: $\left( \rm{a} \right)$ Scattering probabilities $T_a^b$ after RWA (solid blue line) and ${\rm P}_a^b$ without RWA (orange dashed line) as functions of the effective optomechanical coupling ${G_R} \in \left[ {0,{\Delta _{\rm{m}}}} \right]$. $\left( \rm{b} \right)$ Output spectral contribution from the incoming optical vacuum field $\Theta _{{\rm{vac}}}^a$ as a function of the effective optomechanical coupling ${G_R} \in \left[ {0,{\Delta _{\rm{m}}}} \right]$. The other parameters for $\left( \rm{a} \right)$ and $\left( \rm{b} \right)$ are chosen as $\omega = {\Delta "_{\rm{c}}} = {\Delta _{\rm{m}}} = 158{\kappa _a} = 158{\gamma _b}$.