Analytical exploration of the optomechanical attractor diagram and of limit cycles

TL;DR

This work advances the analytical understanding of optomechanical backaction by exactly summing the infinite Bessel-series that determine radiation-pressure forces, enabling tractable asymptotic analyses across amplitude and sideband regimes. By deriving closed forms for the time-averaged force, the drift and diffusion in a Fokker-Planck description, and the effective detunings in both static and dynamical settings, the authors illuminate the emergence and structure of attractor diagrams and optomechanical limit cycles beyond the resolved sideband approximation. The results reveal new, testable features such as resonance enhancements at integer detunings, suppression of oscillations at Δ_eff = n ω_m, and significant corrections to limit cycles when using full summations rather than truncated near-resonance approximations. The study also provides a versatile framework applicable to a broad class of Floquet problems, with potential experimental verification and extensions to multi-drive and Floquet-engineered systems.

Abstract

We analyse the interplay between mechanical and radiation pressure in an optomechanical cavity system. Our study is based on an analytical evaluation of the infinite Bessel summations involved, which previously had led to a numerical exploration of the so-called attractor diagram. The analytical expressions are then suitable for further asymptotic analysis in opposing regimes of the amplitude, which allows for a characterisation of the diagram in terms of elementary functions. Building on this framework, we investigate the emergence and properties of optomechanical limit cycles beyond the constraints of the resolved sideband approximation. By employing a Fokker-Planck formalism originally developed in the context of laser theory and then used in cavity optomechanics, we describe the quantum regime of these limit cycles, offering a more detailed and unified analytical perspective.

Figures (6)

• Figure 1: Generic optomechanical setting with detuning $\Delta = \omega_L - \omega_c$ and $G = \partial \omega_c / \partial x$ (frequency shift of the cavity resonance per unit mechanical displacement). The classical coupled equations of motion are given below in \ref{['alpha']} and \ref{['x']}.
• Figure 2: The maximum value of the ratio $f(r)\equiv \frac{\left\langle F\dot{x}\right\rangle}{P_{fric}}$ as a function of $c$ (in units of $2G|\alpha _{\mathrm{max}}|^{2}/\pi \Omega _{\mathrm{m}}$). The dashed lines correspond to the small $c$ and large $c$ behavior, governed by the formulas \ref{['ccsma']} and \ref{['hyt']}, respectively.
• Figure 3: Optical part of the Wigner diffusion from approximation \ref{['ccoo']} (blue) and for the exact formula \ref{['dtodo']} (red) for $\kappa=0.1\omega_m$. a) $\Delta_{\rm eff}=0.1\, \omega_m$. The approximation \ref{['ccoo']} reproduces with high accuracy the exact curve. b) $\Delta_{\rm eff}=0.6\, \omega_m$. The approximation \ref{['ccoo']} is no longer applicable as higher $n$ terms in the infinite sum \ref{['sumadd']} become important.
• Figure 4: Fractional differences as a function of $r$. For $\Delta_{\rm eff}=0.1\, \omega_m$ (brown curve) the difference reaches up to 20 per cent, whereas for $\Delta_{\rm eff}=0.6\, \omega_m$ (black curve) the difference is over 70 per cent for some values of $r$.
• Figure 5: The exact $\mu(r)$ for two different values of $\Delta$. Oscillations are suppressed by an extra factor $1/r$ when $\Delta=n \omega_m$. Here $\gamma=0$, $\kappa=\omega_m=1$. (a) In blue, $\Delta=0.8\omega_m$, in red $\Delta=\omega_m$. (b) Plot of $r\mu(r)$ for $\Delta=0.8 \omega_m$, exhibiting the $\mu\sim 1/r$ asymptotic behavior when $\Delta/\omega_m$ is not an integer. (c) Plot of $r^2\mu(r)$ exhibiting the $\mu\sim 1/r^2$ behavior for $\Delta = \omega_m$.