Floquet dynamics in the quantum measurement of mechanical motion

The radiation-pressure interaction between one or more laser fields and a mechanical oscillator gives rise to a wide range of phenomena: From sideband cooling and backaction-evading measurements to ponderomotive and mechanical squeezing to entanglement and motional sideband asymmetry. In many protocols, such as dissipative mechanical squeezing, multiple lasers are utilized, giving rise to periodically driven optomechanical systems. Here we show that in this case Floquet dynamics can arise due to presence of Kerr-type nonlinearities, which are ubiquitous in optomechanical systems. Specifically, employing multiple probe tones, we perform sideband asymmetry measurements, a macroscopic quantum effect, on a silicon optomechanical crystal sideband cooled to 40% ground-state occupation. We show that the Floquet dynamics, resulting from the presence of multiple pump tones, gives rise to an artificially modified motional sideband asymmetry by redistributing thermal and quantum fluctuations among the initially independently scattered thermomechanical sidebands. For pump tones exhibiting large frequency separation, the dynamics is suppressed and accurate quantum noise thermometry demonstrated. We develop a theoretical model based on Floquet theory that accurately describes our observations. The resulting dynamics can be understood as resulting from a synthetic gauge field among the Fourier modes, which is created by the phase lag of the Kerr-type response. This phenomenon has wide-ranging implications for schemes utilizing several pumping tones, as commonly employed in backaction-evading measurements, dissipative optical squeezing, dissipative mechanical squeezing, and quantum noise thermometry. Our observation may equally well be used for optomechanical Floquet engineering, e.g., generation of topological phases of sound by periodic time modulation.

Figure 1

Optomechanical crystal and experimental setup. (a) False-color SEM image of the silicon optomechanical crystal cavity with a waveguide for laser input coupling. The path of the tapered fiber is indicated. The inset shows the simulated mechanical breathing mode and optical mode. (b) SEM image of the cavity. (c) Scheme for motional sideband asymmetry measurement using two probe tones and a cooling tone. (d) Experimental setup. PM, phase modulator; VOA, variable optical attenuator; AOM, acousto-optical modulator; BHD, balanced heterodyne detector; SA, spectrum analyzer; NA, network analyzer; PLL, phase-locked loop.

Figure 2

Artificial and quantum sideband asymmetry in optomechanical sideband cooling. In (a) and (b), the cooling tone is detuned $80\mathrm{MHz}$ from the red-detuned probe. (a) Inferred $\overline{n}$ ($\overline{n}+1$) from anti-Stokes (Stokes) mechanical sidebands, based on total thermomechanical noise. Overall cooling is observed, however a clear discrepancy exists since the cooling tone and the red-detuned probe should yield the same and be equal to the occupancy $\overline{n}$. Inset: Example of an actual observed noise spectrum, colored for different tones as in the main panel. Note that the choice of local oscillator frequency [Fig. 1] leads to the noise spectrum “folded over” and to the cooling tone sideband located close to the probe sidebands. The $\omega$ axis does not strictly correspond to Eq. (1) in this case. (b) The occupancy $\overline{n}$ obtained from motional sideband asymmetry, obtained both from the ratio of red- and blue-detuned probes and from the ratio of cooling tone and blue-detuned probe, which shows strong disagreement. Error bars represent $\pm 20\text{-MHz}$ tuning accuracy (see text for discussion). (c) and (d) show the data corresponding to (a) and (b), respectively, only with the cooling tone detuned $220\mathrm{MHz}$ from the red-detuned probe, where the effect of the thermal Kerr-type nonlinearity is strongly diminished. In (c), the lower curve is a fit according to the model of Appendix pp2 with the average asymmetry of the last two points, shown in (d), used for calibration; one quantum is added to result in the upper curve, coinciding with the Stokes sideband data. Inset (c) shows spectra of the last data point.

Figure 3

Observation of asymmetric noise spectra due to Kerr-type nonlinearity. (a) Pumping scheme with two equal red-detuned pumps. (b) The peak ratio of the observed spectra, relative to the ratio expected from bare optomechanical response, vs $\mathrm{\Omega }{}_{\mathrm{mod}}$ for constant power of $n{}_{\mathrm{c}}=640$ intracavity photons, showing decreased effect for higher modulation frequencies. The inset shows the spectra for data points of the same color in the main panel. (c) Increasing the pump power for a constant $\mathrm{\Omega }{}_{\mathrm{mod}}/2\pi =4\mathrm{MHz}$, showing higher-order Fourier modes. Also shown are fits to the analytic model of Sec. 3 (see text for details). (d) Calibration of sideband cooling using motional sideband asymmetry, with $\mathrm{\Omega }{}_{\mathrm{mod}}/2\pi =220\mathrm{MHz}$, free from Kerr-type artificial asymmetry. See text for more details. The oscillator is cooled down to $\overline{n}=1.5$ phonons.

Figure 4

An illustration of the infinite array of coupled Fourier modes. The map to Fourier modes (10) introduces an infinite set of coupled Langevin equations, which has to be truncated at some order to obtain a solution. Due to the two pumps (cooling and red-detuned probe), thermomechanical noise incident on mode $\delta {\widehat{b}}^{\left(0\right)}$ is distributed onto $\delta {\widehat{a}}_r$ and $\delta {\widehat{a}}_c$, generating two sidebands. The cavity Fourier modes are coupled due to a cavity Kerr-type nonlinearity $[{\mathrm{\Delta }}_k$ (4)]. This effect modifies the sideband weight, which to lowest order (considering only the bold three-mode system) can be accounted for with effective optomechanical couplings (5), but in general leads to more sidebands [Fig. 3], modeled by including more Fourier modes in the description. Notably, due to the finite response time, the coupling between the Fourier modes is complex, with the phase ${\varphi }_{\text{th}}$ given in Eq. (9).

Figure 5

Sideband cooling of the nanomechanical oscillator with a single cooling tone. In this instance, with 4.5-K cryostat temperature and 160-mbar buffer gas pressure, the oscillator was cooled to occupancy $\overline{n}=1.9$, i.e., 34% ground-state occupation. (a) Mechanical occupation $\overline{n}$ derived from the area of the noise spectra vs intracavity photon number $n_c$. $\overline{n}$ is anchored to the cryostat temperature, $\overline{n}{}_{\mathrm{th}}=17$, at lowest cooling power. The solid line is a fit to the simple heating model (B2) with $\alpha$ and $\overline{n}{}_{\mathrm{th}}$ as free parameters, and the dashed line shows cooling in absence of absorption heating. The inset shows total mechanical damping $\mathrm{\Gamma }{}_{\mathrm{tot}}$ vs $n_c$, with a linear fit used to extract the bare mechanical damping rate ${\mathrm{\Gamma }}_m$ (intrinsic damping $\mathrm{\Gamma }{}_{\mathrm{int}}$ and the gas damping) in absence of dynamical backaction and the optomechanical coupling rate $g_0$. (b) Noise spectra ${\overline{S}}_I\left(\omega \right)$, normalized to the shot-noise level, at three different points highlighted with respective colors in the main panel. (c) Signal-to-noise ratio (SNR) vs the cooperativity $\mathcal{C}$ with a fit to the model (B3) with overall efficiency $\eta$ and heating $\alpha$ as free parameters. The highlighted points are the same as in (b).

Figure 6

Laser power induced heating of the mechanical mode. Mechanical mode temperature vs cryostat temperature at two different pumping powers which correspond to intracavity photon number $n_c=8.8$ and 35, where the sample pressure is kept at 25 mbar.

Figure 7

Effect of the buffer ${}^3\mathrm{He}$ gas on sideband cooling performance. (a) Mechanical damping showing increase with gas pressure. The linear field yields intrinsic damping $\mathrm{\Gamma }{}_{\mathrm{int}}/2\pi =84\mathrm{kHz}$. (b) Oscillator heating $\alpha$ showing decrease with gas pressure. Measurements were done at a cryostat temperature of $4.5\mathrm{K}$ except for low pressures, where insufficient thermalization leads to elevated bath temperatures, as indicated in (a). At a pressure of $\approx 0\mathrm{mbar}$, absorption heating leads to minimum $\overline{n}$ of 5.7, despite the low ${\mathrm{\Gamma }}_m$.

Figure 8

Design of the front input coupling mirror of the optomechanical crystal (OMC). (a) Simulated mechanical (top) and optical (bottom) field distribution of the OMC. (b) Simulated optical and mechanical external $Q$ vs number of holes of the front mirror, adjacent to the defect section of OMC. When the number of holes is increased, both optical (red dot) and mechanical (blue dot) external $Q$ increase. The dashed lines correspond to the measured intrinsic $Q$ of the optical (red) and mechanical (blue) mode.

Figure 9

Cavity frequency response measurement. (a) Experimental setup. AM, amplitude modulator. (b) Measurement scheme. (c) Response measurement at different pressures fitted with a three low-pass model.