Optomechanical Backaction in the Bistable Regime

Abstract

With a variety of realisations, optomechanics utilizes its light matter interaction to test fundamental physics. By coupling the phonons of a mechanical resonator to the photons in a high quality cavity, control of increasingly macroscopic objects has become feasible. In such systems, state manipulation of the mechanical mode is achieved by driving the cavity. To be able to achieve high drive powers the system is typically designed such that it remains in a linear response regime when driven. A nonlinear response and especially bistability in a driven cavity is often considered detrimentally to cooling and state preparation in optomechanical systems and is avoided in experiments. Here we show, that with an intrinsic nonlinear cavity backaction cooling of a mechanical resonator is feasible operating deeply within the nonlinear regime of the cavity. With our theory taking the nonlinearity into account, precise predictions on backaction cooling can be achieved even with a cavity beyond the bifurcation point, where the cavity photon number spectrum starts to deviate from a typical Lorentzian shape.

Figures (10)

• Figure 1: a) Photon number spectrum $g_0^2 S_\mathrm{nn}[\omega]$ for cavities with $\mathcal{K}_0/2\pi = 0 \,\mathrm{kHz}$, $\mathcal{K}_1/2\pi = 8\,\mathrm{kHz}$ and $\mathcal{K}_2/2\pi = 16\,\mathrm{kHz}$. Other parameters for the plots are linewidth $\kappa/2\pi = 3\,\mathrm{MHz}$, mechanical frequency $\omega_\mathrm{m}/2\pi = 300\,\mathrm{kHz}$ and a coupling strength of $g_0/2\pi = 1.7 \,\mathrm{kHz}$. The drive is detuned by $\Delta/\omega_m = - 11$ from the bare frequency of the cavity $\omega_c$ and set to a constant power $n_{in}/n_{bi}=1.5$ with respect to $\mathcal{K}_2$. Gray dashed lines indicate the (anti-)/Stokes scattering processes. The photon induced frequency shift due to $\mathcal{K}$ is evident and becomes more pronounced with increasing $\mathcal{K}$. For $\mathcal{K}_2$, $S_\mathrm{nn}[\omega]$ shows two stable solutions indicating the bifurcation due to the splitting of the photon number $n_c$ shown in the inset. Same color scheme as in b). b) Imbalance of the Stoke and anti-Stokes rates for different detunings $\Delta/\omega_m$. The larger imbalance shows the enhanced cooling capability for a nonlinear cavity in the mono and bistable regime compared to a linear system. For $\mathcal{K}_2$ both solutions feature a discontinuity close to switching to the other branch.
• Figure 2: Setup and characterisation. a) Schematic circuit diagram of the setup with junction inductance $L_J$, cavity inductance $L$ and $C$, $C_c$ and $C_g$ as the self, coupling and ground capacitance, respectively. b) Picture of the sample. Silicon substrate in gold, U-shaped cavity in silver, cantilever in gray and magnet false colored in purple (a,b taken from zoepflKerrEnhances2023). c) Cavity response for different input powers. For intermediate powers the effects of the nonlinearity become visible and for highest power the cavity shows the splitting into low (red) and high (green) photon number branch. For this power the arrows indicate the direction of the frequency sweep.
• Figure 3: a) Relaxation of the frequency shift $\delta\mspace{-4mu}f$ after driving the cavity at different powers, for the cavity being initialised in the high photon number branch. Solid lines represent an exponential fit to extract the timescale $\overline{\tau} = (0.96\pm 0.02) \,\mathrm{s}$. Inset shows a zoom in on the data plotted on a log-scale. b) Frequency shift $\delta\mspace{-4mu}f$ against probe-cavity detuning $\Delta$ for a drive strength $n_{in}/n_{bi} = 3.0$. The differen branches are probed by tuning the probe towards the cavity from low/high frequencies to scan the low (red) and high (green) branch.
• Figure 4: Cooling traces at $g_0/2\pi = 99\,\mathrm{Hz}$. a) Phonon number $\langle n_m \rangle$, b) linewidth $\Gamma_m/2\pi$ and c) frequency shift $\delta \omega_m /2\pi$ for intermediate powers $n_{in}/n_{bi}=0.5$. Gray lines display a steady state at an effective bath temperature $T_\mathrm{eff} = 267\,\mathrm{mK}$. Phonon number $\langle n_m \rangle$, linewidth $\Gamma_m/2\pi$ and frequency shift $\delta \omega_m /2\pi$ for higher powers of $n_{in}/n_{bi}=1.9$ in d-f) and $n_{in}/n_{bi}=3.0$ in g-i), respectively. Data and theoretical extrapolation from $n_{in}/n_{bi}=0.5$ in red for the low and green for the high photon number branch. Filled circles correspond to data taken with the pulse tube of the cryostat turned off to further reduce vibrations.
• Figure S. 1: Schematic drawing of the setup. The dashed circle indicates the positions of the eddy current damping tube not shown in Fig. \ref{['fig:susp_fridge']}a). The spring is connected to the setup via a tower structure inside a brass tube attached to the base plate of the cryostat. For the damping we then attach several NdFeB magnets (grey blocks in the zoom in).