Absolute Determination of the Single-Photon Optomechanical Coupling Rate via a Hopf Bifurcation

We establish a method for the determination of the single-photon optomechanical coupling rate, which characterizes the radiation pressure interaction in an optomechanical system. The estimation of the rate with which a mechanical oscillator, initially in a thermal state, undergoes a Hopf bifurcation, and reaches a limit cycle, allows us to determine the single-photon optomechanical coupling rate in a simple and consistent way. Most importantly, and in contrast to other methods, our method does not rely on knowledge of the system’s bath temperature and on a calibration of the signal. We provide the theoretical framework and experimentally validate this method, providing a procedure for the full characterization of an optomechanical system, which could be extended outside cavity optomechanics, whenever a resonator is driven into a limit cycle by the appropriate interaction with another degree of freedom.

Figure 1

(a) Determination of $g_0$ through the measurement of the stationary frequency noise fluctuations. Knowledge of the amplitude of the calibration tone, $\mathrm{\Delta }{V}_b$, becomes a ruler for the fluctuations of the cavity resonance frequency, $\mathrm{\Delta }{V}_m$. (b) Determination of $g_0$ through the measurement of the slope $\mathcal{S}$ of the temporal dynamics corresponding to reaching a limit cycle. The mechanical oscillator, which is excited by radiation pressure interaction with an optical cavity illuminated by a blue-detuned laser, undergoes a Hopf bifurcation, and reaches the limit cycle.

Figure 2

(a) Experimental setup. A probe beam, frequency modulated by an EOM, impinges on the optical cavity. The reflected beam is analyzed by homodyne detection in order to detect the mechanical motion. (b) VSN of the reflected field around the fundamental mode of the membrane. The orange area indicates the fundamental mode of the membrane under consideration; the blue area represents the calibration tone implemented by modulating the input field by means of EOM; the peak in the middle represents the fundamental mode of the second disregarded membrane.

Figure 3

(a) Experimental setup. The pump beam, blue detuned by $\mathrm{\Delta }=239.35$ kHz from the cavity resonance, is added to the experimental setup of Fig. 2 for engineering the optomechanical interaction. (b) Voltage noise $V_m$ acquired around 229.753 kHz in a bandwidth of 182 Hz, as a function of time. The time trace in the first 5 s corresponds to the thermal dynamics of the mechanical oscillator. After 5 s the pump beam is turned on, and the amplitude of the mechanical oscillator rises to a limit cycle. The orange and brown curves correspond to input powers $P_{\mathrm{pm}}=6.1(2)\mu \mathrm{W}$ and $P_{\mathrm{pm}}=21.0(5)\mu \mathrm{W}$, respectively.

Figure 4

(a) Dimensionless maximum slope, ${\mathcal{S}}^{\mathrm{mx}}=\mathcal{S}({\xi }^{\mathrm{mx}})$, as a function of the pump power $P_{\mathrm{pm}}$ and the single-photon optomechanical coupling rate $g_0$, evaluated for our experimental parameters. The blue line indicates the maximum slope for $g_0=2\pi \times 0.336$ Hz, and the blue dashed line the threshold pump power $P_{\mathrm{pm}}^{\mathrm{th}}\simeq 4.4\mu \mathrm{W}$, obtained for the experimental results presented in panel (b). (b) Maximum slope multiplied by a transduction constant coefficient $a(\mathrm{V}{\mathrm{s}}^{-1})$, ${\mathcal{S}}_V^{\mathrm{mx}}=a{\mathcal{S}}^{\mathrm{mx}}$, as a function of the pump power $P_{\mathrm{pm}}$. Filled circles represent experimental data; orange and brown circles correspond to the time traces reported in Fig. 3. The blue curve represents the best-fit result of Eq. (25) by using the Levenberg-Marquardt fitting method with fitting parameters $g_0$ and the transduction constant coefficient $a$; the best-fit values are given with $1-\sigma$ (approximately $68\mathrm{\%}$ confidence) uncertainty (light-blue area). The black dashed curve represents a linear fit for the determination of the threshold pump power.

Figure 5

Experimental setup for the characterization of the EOM. Two beams frequency detuned by $\mathrm{\Delta }$ by means of an AOM impinge on a cavity out of resonance. The function generator HP 8657B used for driving the AOM has a frequency resolution of 1 Hz and an accuracy of 2 ppm/year, and is hence negligible in our case. The beams both reflected, interfere on the polarizing beam splitter (PBS) before the cavity, and are finally collected on a PIN photodiode Thorlabs PDA10CF that has a flat bandwidth up to 150 MHz. The signal is analyzed with a lock-in amplifier. The powers of the probe and pump beams are $17$ and $350\mu \mathrm{W}$, respectively. The modulations ${\omega }_{\mathrm{PDH}}$ and ${\omega }_{\mathrm{cal}}$ are generated with two different function generators, a Siglent SDG 2122A and an Agilent 33220A, respectively. The two signals are then combined with a Minicircuit ZFSC-2-6+ and sent to the EOM with an approximate $6$ m cable. QWP is a quarter-wave plate.

Figure 6

(a) Ratio $J_1(\beta )/J_0(\beta )$ as a function of the frequency at which the EOM is driven. The RLC resonant frequency is around 5.015 MHz and the $Q$ factor is 6.4. The low-pass filter at low frequencies with cutoff frequency around 918.7 kHz is probably due to some parasitic components in the RLC circuit. The black dashed line represents the low-pass filter transfer function, the black dotted line is the RLC transfer function, and the black solid line is the sum of the dashed and dotted lines. (b) Ratio $J_1(\beta )/J_0(\beta )$ as a function of the voltage applied to the EOM at a frequency of 237 kHz, which is the frequency of the beating note to calibrate the optomechanical spectra. The transduction coefficient is 22.5(1) mrad/V. The black dotted line indicates our operating value, which is 19.5(1) rad. The black dashed line is a best fit and the blue shadow represents the error in the estimation of the modulation depth.

Figure 7

(a) VSN of the mechanical spectrum (light-yellow area) and cumulative function (brown line) for the mechanical mode. The light-brown line is a Lorentzian fit of the mechanical mode. The black dashed lines are linear fits of the bottom and top of the cumulative function. The dotted line represents the distance between the two linear fits and gives an area of $\mathrm{\Delta }{V}_m^2=2.8(4)\times {10}^{-10}{\mathrm{V}}^2$. (b) VSN of the calibration peak (light-blue area) and cumulative function (blue line) for the mechanical mode. The black dashed lines are linear fits of the bottom and top of the cumulative function. The dotted line represents the distance between the two linear fits and gives an area of $\mathrm{\Delta }{V}_b^2=1.669(1)\times {10}^{-8}{\mathrm{V}}^2$.

Figure 8

Maximum slope in $\mathrm{V}{\mathrm{s}}^{-1}$ as a function of the pump power $P_{\mathrm{pm}}$ reported in Fig. 4 of the main text. The right axis shows ${\mathcal{S}}_x$ measured in $\mathrm{pm}{\mathrm{s}}^{-1}$, and it is calibrated by matching the voltage signal $V_m$ (first 5 s in Fig. 3 of the main text) to the thermal displacement of the mechanical oscillator $q_T=\sqrt{2{\overline{n}}_m}{x}_{\mathrm{ZPF}}$.