Nonlinear quantum optomechanics in a Fano-mirror microcavity system

Abstract

We study a Fano-mirror optomechanical system in the quantum nonlinear regime. In this system, two strongly lossy optical modes hybridize through both coherent and dissipative couplings to form an effective optical mode with a drastically reduced linewidth. This linewidth reduction enables the system to access the single-photon strong-coupling and sideband-resolved regimes simultaneously. We formulate the system dynamics using an effective master-equation approach and benchmark it against quantum Langevin and dressed-state master-equation descriptions. With experimentally realistic parameters, we predict clear quantum signatures, including photon blockade and the generation of mechanical cat states. Our work establishes the Fano-mirror architecture as a promising platform for harnessing single-photon optomechanical nonlinearities for quantum state engineering under achievable experimental conditions.

Figures (7)

• Figure 1: (a) Schematic illustration of the Fano-mirror optomechanical system. (b) Diagram of the original Fano-mirror optomechanical setup, including the interactions among the two optical modes and the mechanical mode, see Sec. \ref{['SecModel']}. (c) Diagram of the system in the optical normal-mode basis, with all effective dissipation and interaction channels indicated, see Sec. \ref{['SecNormal']}.
• Figure 2: (a) Effective parameters of the optical normal modes (on a logarithmic scale), derived from the effective master equation (\ref{['optME']}), as functions of the dissipation ratio $\kappa_{d}/\kappa_{a}$. (b) Time evolution of the mean photon numbers in the optical normal modes $A_{\pm}$, under weak driving resonant with $A_{-}$, for $\kappa_{d}/\kappa_{a}=1.6\times10^{-3}$ and different values of the driving amplitude $\varepsilon_{p}$. (c) Eigenspectrum of the optical Liouvillian $\tilde{\mathcal{L}}_{\mathrm{opt}}$ for $\kappa_{d}/\kappa_{a}=1.6\times10^{-3}$. The left panel shows the full spectrum, while the right panel provides a zoom-in around the zero eigenvalue. The eigenvalues are denoted as $\lambda = \{\lambda_{1}, \lambda_{2}, \cdots\}$. (d) Steady-state mean phonon number $\langle b^{\dag}b \rangle_{ss}$ obtained from both the effective two-mode and the exact three-mode optomechanical models, as a function of $-g_{d,0}^{\omega}$ (normalized by the mechanical frequency), with $\varepsilon_{p}/\kappa_a=1\times10^{-8}$ and $\kappa_{d}/\kappa_{a}=1.6\times10^{-3}$. Other parameters: $\gamma_a/\kappa_a = 1\times10^{-4}$, $\omega_a/\kappa_a = 200$, $\omega_d/\kappa_a = 195$, $\Lambda/\kappa_a = 0.2$, $\Omega_{m}/\kappa_a=1.5\times10^{-6}$, $\gamma_{m}/\kappa_a=8\times10^{-14}$, $g_{a,0}^{\omega}/\Omega_{m}=0.25$, and $n_{\mathrm{th},b}=0$.
• Figure 3: Photon blockade in the dark normal mode. (a) Stead-state value $g^{(2)}(0)$ of the equal-time second-order correlation function versus the detuning $\Delta_{-}$ (between the $A_-$ mode and the driving field) for two values of $\kappa_{\mathrm{eff},-}$. For the parameters used here, $\kappa_{\mathrm{eff},-}/\Omega_m=0.0817$ and $11.51$ correspond to $\kappa_d/\kappa_a=1.6\times10^{-3}$ and $2.0\times10^{-3}$, respectively; see also Fig. \ref{['Eliminate']}. The red vertical line marks $\Delta_{-}=G_{-}^{2}/\Omega_{m}$, where $g^{(2)}(0)$ reaches its minimum. (b) $g^{(2)}(0)$ as a function of the normalized coupling strength $\eta=-G_{-}/\Omega_{m}$. (c) Time evolution of the equal-time second-order correlation function $g^{(2)}(t;0)$ for different mean thermal phonon numbers $n_{\mathrm{th},b}$. (d) $g^{(2)}(t;0)$ evaluated at $\Omega_{m}t=200$ as a function of $n_{\mathrm{th},b}$. The light blue areas in (a) and (c) represent the region where the photon antibunching effect arises. Parameters: $n_{\mathrm{th},b}=0$ in (a) and (b); $\eta=0.5$ in (a), (c), and (d); $\kappa_{d}/\kappa_{a}=1.6\times10^{-3}$ and $\Delta_{-}=G_{-}^{2}/\Omega_{m}$ in (b)--(d). Other parameters are $\gamma_a/\kappa_a = 1\times10^{-4}$, $\omega_a/\kappa_a = 200$, $\omega_d/\kappa_a = 195$, $\Lambda/\kappa_a = 0.2$, $\Omega_{m}/\kappa_a = 2\times10^{-6}$, $\gamma_{m}/\kappa_a = 8\times10^{-14}$, $\varepsilon_{p}/\kappa_a = 2\times10^{-9}$, and $g_{a,0}^{\omega}/\Omega_{m}=0.25$.
• Figure 4: Preparation processes of non-Gaussian mechanical states. (a)--(f) Snapshots of the Wigner function $W(x,p)$ of the mechanical mode at different time moments (normalized by $\tilde{\Gamma}=4|\tilde{G}_{-}|^{2}/\kappa_{\mathrm{eff},-}$), with [(a)--(c)] $\eta=0.15$ and [(d)--(f)] $\eta=0.35$. (g) and (h) Fidelity $\mathcal{F}$ and Wigner logarithmic negativity $\mathcal{W}$ of the prepared mechanical state with respect to the ideal even cat state [defined as $|\mathrm{cat},+\rangle$ in Eq. (\ref{['catstates']})], with (g) $\eta=0.15$ and (h) $\eta=0.35$. The inset in (g) compares the time evolution of the fidelity $\mathcal{F}$ with $n_{\mathrm{th},b}=0$ (blue solid) and $n_{\mathrm{th},b}=100$ (yellow dashed), with all other parameters identical to the main panel. Other parameters (unless otherwise specified) are $\gamma_a/\kappa_a = 1\times10^{-4}$, $\omega_a/\kappa_a = 200$, $\omega_d/\kappa_a = 195$, $\Lambda/\kappa_a = 0.2$, $\Omega_{m}/\kappa_a = 4\times10^{-6}$, $\gamma_{m}/\kappa_a = 8\times10^{-14}$, $g_{a,0}^{\omega}/\Omega_{m}=0.12$, $\alpha_{-}=\sqrt{0.1}$, $n_{\mathrm{th},b}=0$, and $\chi=1.54$.
• Figure 5: Comparison of the results with and without the dissipative optomechanical couplings. We assume $g_{a,0}^{\kappa}/\Omega_{m}=0.25$ and $g_{d,0}^{\kappa}/\Omega_{m}=0.92$ for dissipative optomechanical couplings. In panel (a), we compare the second-order correlation function $g^{(2)}(0)$ of the $A_{-}$ mode, with all other parameters identical to those used for the purple dot-solid line in Fig. \ref{['Blockade']}. In panel (b), we compare the fidelity $\mathcal{F}$ of the prepared mechanical state with respect to an ideal even cat state, with all other parameters identical to those used for Fig. \ref{['CatS']}. The two panels share the same legend.