Optically tunable nonlinear mechanical damping in an optomechanical resonator

TL;DR

This work reveals optically tunable nonlinear mechanical damping in a cavity optomechanical resonator by exploiting delayed backaction in a partly resolved sideband regime. A memory-kernel formalism is developed to capture history-dependent nonlinearities, and explicit expressions for first-, second-, and third-order optomechanical couplings yield detuning-controlled Duffing and nonlinear-damping terms. Experimentally, a microbottle resonator demonstrates self-nonlinear damping with strength $oldsymbol{eta}_{ ext{opt}}$, tunable by laser detuning, and cross-nonlinear damping between two mechanical modes via a common optical mode. The results establish optomechanical platforms as versatile tools for engineering nonlinear dissipation and nonequilibrium dynamics, with potential implications for dissipative phase phenomena and optomechanical Floquet engineering.

Abstract

We theoretically propose and experimentally demonstrate optically tunable nonlinear mechanical damping in a cavity optomechanical system utilizing a partly resolved sideband regime. Optomechanical coupling provides a delayed nonlinear backaction to the mechanical modes, resulting in nonlinear mechanical damping. This optically induced nonlinear damping is observed in the frequency and time domains, and we show using both theory and experiment that it can be tuned via laser detuning. We also observe optically mediated cross-nonlinear damping between two mechanical modes: the amplitude of one mode modulates the damping of the other. The presented results show a fully tunable scheme of nonlinear mechanical damping that will be applicable to various non-trivial systems, governed by nonlinear, nonequilibrium, and non-Hermitian phenomena.

Figures (7)

• Figure 1: (a) Concept of nonlinear mechanical damping induced by delayed backaction in cavity optomechanics, where cavity decay rate $\kappa_{\mathrm{cav}}$ becomes comparable to mechanical frequency $\Omega_{\mathrm{m}}$. (b) Theoretical detuning dependence of the mechanical nonlinearities near cavity resonance. The orange curve shows nonlinear damping strength $\beta_{\mathrm{opt}}(\Delta)\Omega_{\mathrm{m}}$ and the purple curve represents Duffing nonlinearity $\alpha_{\mathrm{opt}}(\Delta)$. The light-blue dashed line indicates the optical cavity transmittance. (c) Theoretically calculated maximum strength of the nonlinearities optimized over detuning $\Delta$ as a function of $\kappa_{\mathrm{cav}}/\Omega_\mathrm{m}$. The orange curve represents maximum nonlinear damping strength $\max_{\Delta \in \mathbb{R}}\abs{\beta_{\mathrm{opt}}(\Delta)}\Omega_{\mathrm{m}}$ and the purple curve shows maximum Duffing nonlinearity $\max_{\Delta \in \mathbb{R}}\abs{\alpha_{\mathrm{opt}}(\Delta)}$. Schematic illustrations highlight the distinct optomechanical backaction mechanisms in different parameter regimes. All calculations use parameters corresponding to the employed experimental system (a microbottle optomechanical resonator, as discussed later in Fig. \ref{['fig:2']}) assuming an input optical power of 50 mW.
• Figure 2: (a) Schematic of microbottle optomechanical resonator and its optical micrograph, in which the bottle region is highlighted in light yellow. Two representative radial breathing modes are illustrated, anticipating the multimode discussion in Fig. \ref{['fig:4']}. (b) Optical transmission spectrum (at $\lambda = 1.55~\mathrm{\mu m}$) with a Lorentzian fit yielding $\kappa_{\mathrm{cav}}/2\pi = 194$ MHz ($Q_{\mathrm{cav}} = 1.0 \times 10^{6}$). (c) Mechanical resonance spectrum of a representative radial breathing mode (RBM) under weak excitation with a Lorentzian fit yielding $\Gamma_{\mathrm{m}}/2\pi = 0.83$ kHz at $\Omega_{\mathrm{m}}/2\pi = 48.476$ MHz. (d) Mechanical resonance spectra at different excitation strengths parameterized by the voltage applied to the EOM, $V_{\mathrm{EOM}}$. Inset: Corresponding normalized spectra at 10.0 V and 2.0 V. (e) Ring-down traces for different drive strengths. Black dashed curves show nonlinear-damping fits while blue dashed curves show linear-damping fits. (f) Damping rate versus oscillation amplitude, exhibiting quadratic scaling consistent with cubic nonlinear damping. Fit yields the linear damping rate, $\Gamma_{\mathrm{0}}/2\pi=0.89\ \mathrm{kHz}$, and $\beta_{\mathrm{opt}}/2\pi=6.2\times10^{23}\ \mathrm{Hz}/\mathrm{m}^2$. The absolute amplitude is calibrated from the thermal motion of the RBM using effective mass $m_{\mathrm{eff}} \simeq 7\times 10^{-9}$ kg (See Supplemental Material for details).
• Figure 3: Mechanical damping change as a function of laser detuning. The plots show the experimentally measured damping change extracted from the linewidth difference between strong and weak excitation ($V_{\mathrm{EOM}} = 9.0$ and $3.0$ V). The orange curve represents the calculated damping increase using the theoretically estimated nonlinear damping coefficient [orange curve in Fig. \ref{['fig:1']}(b)]. The calculation is calibrated using a detuning-dependent radiation-pressure drive expressed through oscillation amplitude $X(\Delta)$, which scales with the intracavity photon number as $X(\Delta)\propto n_{\mathrm{cav}}(\Delta)$. The light-blue dashed line represents the optical cavity resonance as shown in Fig. \ref{['fig:1']}(b).
• Figure 4: (a) Concept of cross-nonlinear damping in a multimode optomechanical system, where a single optical mode ($\mathrm{O}$) is coupled to two mechanical modes: a probe mode ($\mathrm{M}_{\mathrm{p}}$) and a control mode ($\mathrm{M}_{\mathrm{c}}$). The probe and control modes were resonantly driven at frequencies $f_{\mathrm{pro}} = 48.0$ MHz and $f_{\mathrm{con}} = 48.5$ MHz, respectively. (b) Ring-down measurements of the probe mode with the control mode strongly driven (ON, red) or undriven (OFF, blue) showing an approximate 0.6 kHz increase in damping. The data were obtained with drive voltages $V_{\mathrm{pro}} = 6.0$ V and $V_{\mathrm{con}} = 10.0$ V, where $V_{\mathrm{pro}}$ and $V_{\mathrm{con}}$ denote the drive voltages for the probe and control modes, respectively. (c) Increase in probe-mode damping rate $\Gamma_{\mathrm{pro}}/2\pi$ as a function of drive voltages reveals both self-nonlinear damping (dependence on $V_{\mathrm{pro}}$) and cross-nonlinear damping (dependence on $V_{\mathrm{con}}$). The damping rates are extracted from linear fits to the ring-down traces.
• Figure S1: Overview of experimental configuration. PC: Polarization controller, PD: Photodetector, SA: Spectrum analyzer.