Time-resolved phase-space tomography of an optomechanical cavity

We experimentally study the phase-space distribution (PSD) of a mechanical resonator that is simultaneously coupled to two electromagnetic cavities. The first one, operating in the microwave band, is employed for inducing either cooling or self-excited oscillation (SEO), whereas the second one, operating in the optical band, is used for displacement detection. A tomography technique is employed for extracting the PSD from the signal reflected by the optical cavity. Measurements of PSD are performed in steady state near the threshold of SEO while sweeping the microwave cavity detuning. In addition, we monitor the time evolution of the transitions from an optomechanically cooled state to a state of self-excited oscillation. This transition is induced by abruptly switching the microwave driving frequency from the red-detuned region to the blue-detuned one. The experimental results are compared with theoretical predictions that are obtained by solving the Fokker-Planck equation. The feasibility of generating quantum superposition states in the system under study is briefly discussed.

Figure 1

Experimental setup. (a) Dual optomechanical cavity. The microwave cavity is a superconducting microstrip made of aluminum over a high-resistivity silicon wafer coated with a 100-nm-thick SiN layer. The optomechanical coupling is generated using a Nb-coated optical fiber that is positioned at a submicron distance from the trampoline. The optical setup (seen above the sample) allows using the optical cavity for displacement detection, whereas the microwave setup (seen below the sample) is employed for exciting the microwave cavity and measuring its response. For the measurements of PSD in steady state, a single microwave synthesizer is employed, whereas for the time-resolved measurements, two synthesizers (tuned to frequencies $f_{\mathrm{c}}$, and $f_{\mathrm{h}}$, respectively) are used together with a pulse generator in order to ensure a smooth switching from cooling to heating. (b) Off-reflected signals of both the optical and the microwave cavities, in the region where the system exhibits SEO in steady state. (c) The mechanical resonator at the end of the microstrip is shown in the electron micrograph. Several windows are opened in the Nb layer on the fiber tip using FIB, as can be seen in (d).

Figure 2

The PSD in steady state as a function of the normalized detuning $d$. (a) Measured PSD extracted using the technique of state tomography. (b) Calculated PSD obtained from the steady-state solution of Eq. (2). The lower parts of both panels with negative $A_r$ are mirror reflections of the upper parts with positive $A_r$. The following device parameters have been employed in the calculation: $G=0.013\text{Hz}$ (corresponding to frequency shift per displacement of $55\text{MHz}{\mu \text{m}}^{-1}$) and ${\gamma }_2{\delta }_{\mathrm{m}}^2{\gamma }_{\mathrm{m}}^{-1}=2.0\times {10}^{-4}$. The rate of nonlinear damping ${\gamma }_2$ is taken to be independent of the cavity driving parameters since the optomechanical contribution to ${\gamma }_2$ is found to be negligibly small.

Figure 3

(a1)–(d1) Measured and (a2)–(d2) calculated PSD in steady-state SEO. The normalized dwell time ${\gamma }_{\mathrm{m}}{t}_{\mathrm{d}}$ is (a) $0.12$, (b) $1.2$, (c) $7.5$, and (d) 12.

Figure 4

Time-resolved PSD as a function of the normalized elapsing time ${\gamma }_{\mathrm{m}}t$ since the switching from cooling to heating: (a) measured PSD and (b) calculated PSD, which is obtained by numerically integrating the Fokker-Planck equation (2). The lower parts of both panels with negative $A_r$ are mirror reflections of the upper parts with positive $A_r$.