Non-Markovian renormalization of optomechanical exceptional points

Abstract

We investigate how non-Markovian mechanical dissipation affects exceptional points in linearized optomechanical systems with red-sideband drive. For a chosen non-Ohmic mechanical bath, we derive analytical conditions for the memory-renormalized exceptional point by employing a pseudomode mapping, thereby demonstrating that structured environments displace the mode coalescence away from the Markovian prediction. Crucially, we reveal that failing to account for this memory-induced shift suppresses the divergent Petermann factor by orders of magnitude, showing that accurate bath modeling is essential for the successful operation of exceptional-point-based devices whenever reservoir-induced memory is non-negligible. We finally show that non-Markovianity modifies the cavity reflection spectrum, manifesting as a shallower optomechanically-induced-transparency dip, providing therefore an experimentally-accessible signature of structured mechanical environments.

Figures (5)

• Figure 1: Schematic of an optomechanical setup where a control laser enters the cavity from the left mirror and the intracavity mode $a$ couples with the micromechanical motion of the right mirror, represented by the oscillator-mode $b$. The mechanical eigenfrequency is $\omega_m$ and $G$ is the effective (linearized) optomechanical coupling. The optical decay rate is $\kappa$, while the bare mechanical decay rate is $\gamma$, associated with the structured (non-Ohmic) environment shaded in gray. We shall take $\omega_m \gg \kappa, G, \gamma$ in conformity with the resolved-sideband operation.
• Figure 2: (a) Real part ${\rm Re}[\lambda]/2\pi$ and (b) imaginary part ${\rm Im}[\lambda]/2\pi$ of the optomechanical eigenvalues in the presence of non-Markovian effects, demonstrating avoided crossing when evaluated at the Markovian resonance $\Delta^{(0)}_{\rm EP} = -\omega_m$. The vertical dotted line indicates the location of the conventional Markovian exceptional point at $G^{(0)}_{\rm EP} = (\kappa - \gamma)/4 \approx 48.75~{\rm kHz}$. Because the system is tuned to the bare mechanical frequency rather than the memory-renormalized detuning $\Delta_{{\rm EP}} \neq \Delta^{(0)}_{\rm EP}$, the modes fail to coalesce. The bare system parameters are $\omega_m/2\pi = 1~{\rm MHz}$, $\kappa/2\pi = 0.2~{\rm MHz}$, $\gamma/2\pi = 5~{\rm kHz}$, and $\Omega_c/2\pi = 1~{\rm MHz}$.
• Figure 3: (a) Real part ${\rm Re}[\lambda]/2\pi$ and (b) imaginary part ${\rm Im}[\lambda]/2\pi$ of the optomechanical eigenvalues as a function of the optomechanical coupling rate $G/2\pi$ for the bare system parameters $\omega_m/2\pi = 1~{\rm MHz}$, $\kappa/2\pi = 0.2~{\rm MHz}$, $\gamma/2\pi = 5~{\rm kHz}$, and $\Omega_c/2\pi = 1~{\rm MHz}$. The dashed gray curves represent the bare Markovian optomechanical modes, evaluated exactly at the red-sideband resonance $\Delta^{(0)}_{\rm EP} = -\omega_m$. The solid colored curves depict the optomechanical eigenvalues of the full non-Markovian system. To observe the true non-Markovian coalescence, these eigenvalues are evaluated at the numerically-determined shifted detuning $\Delta_{\rm EP}/2\pi \approx -998.69~{\rm kHz}$. The Markovian exceptional point is indicated by the vertical dotted gray line at $G = (\kappa - \gamma)/4$, and the memory-renormalized exceptional point is marked by the vertical dashed black line.
• Figure 4: Divergence of the Petermann factor $K_\pm$ demonstrating the extreme sensitivity of the optomechanical exceptional point to non-Markovian mechanical dissipation. The solid red curve represents the eigenmode nonorthogonality of the full $3 \times 3$ non-Markovian system evaluated at the numerical memory-renormalized resonance $\Delta_{\rm EP}/2\pi \approx -998.69~{\rm kHz}$. It exhibits a true mathematical singularity at the non-Markovian exceptional point $G_{\rm EP}/2\pi \approx 49.38~{\rm kHz}$ (vertical black dashed line), signaling perfect eigenvector coalescence. In stark contrast, the solid gray curve shows the same physical dynamics but evaluated at the standard Markovian red-sideband calibration $\Delta^{(0)}_{\rm EP} = -\omega_m$. Tuning to this bare resonance misses the singularity, suppressing the Petermann factor by several orders of magnitude and failing to achieve true mode coalescence. The vertical gray dashed line marks the location of the Markovian exceptional point $G^{(0)}_{\rm EP}/2\pi = 48.75~{\rm kHz}$.
• Figure 5: Cavity reflection spectrum $|r(\omega)|^2$ as a function of $\omega/2\pi$. The black dashed curve shows the prediction of the Markovian theory evaluated at the Markovian exceptional point $(\Delta^{(0)}_{\rm EP},G^{(0)}_{\rm EP})$, while the red solid curve shows the full non-Markovian response evaluated at the memory-renormalized exceptional point $(\Delta_{\rm EP},G_{\rm EP})$. The bare system parameters are $\omega_m/2\pi = 1~{\rm MHz}$, $\kappa/2\pi = 0.2~{\rm MHz}$, $\gamma/2\pi = 5~{\rm kHz}$, and $\Omega_c/2\pi = 1~{\rm MHz}$.