Measuring High-Order Phonon Correlations in an Optomechanical Resonator

We use single photon detectors to probe the motional state of a superfluid ${}^4\mathrm{He}$ resonator of mass $\sim 1\mathrm{ng}$. The arrival times of Stokes and anti-Stokes photons (scattered by the resonator’s acoustic mode) are used to measure the resonator’s phonon coherences up to the fourth order. By postselecting on photon detection events, we also measure coherences in the resonator when $\le 3$ phonons have been added or subtracted. These measurements are found to be consistent with predictions that assume the acoustic mode to be in thermal equilibrium with a bath through a Markovian coupling.

Figure 1

(a) Device schematic: A fiber-based Fabry-Perot cavity is filled with superfluid ${}^4\mathrm{He}$. Blue shading denotes the instantaneous ${}^4\mathrm{He}$ density in an acoustic mode. Orange denotes the optical mode intensity. (b) Optical schematic showing the two drive lasers (red and blue paths), optomechanical cavity (OMC, black dashed box), acoustically scattered photons (green path), two signal filter cavities (green), and the two SPDs. The filter cavities (red and blue) before the OMC are used to suppress laser phase noise. (c) Optical spectrum showing the frequencies of the lasers, scattered photons, and filters, all with respect to the OMC’s optical resonance. (d) Photon count rate spectrum measured as a function of the drive laser detuning $\mathrm{\Delta }$, with $P_{\mathrm{in}}=400\mathrm{nW}$.

Figure 2

Phonon coherences: (a) The second-, (b) third-, and (c) fourth-order phonon coherences measured for $P_{\mathrm{in}}\approx 5\mu \mathrm{W}$, with photon arrival times binned in 2, 5, and $10\mu \mathrm{s}$ bins, respectively. In (a), the insets show the same data on a logarithmic scale. For the three-time dependent $g_{\mathrm{ac}}^{(4)}({\tau }_1,{\tau }_2,{\tau }_3)$ and $h_{\mathrm{ac}}^{(4)}({\tau }_1,{\tau }_2,{\tau }_3)$, we only show representative 2D slices of $g_{\mathrm{ac}}^{(4)}(0^{+},{\tau }_2,{\tau }_3)$ and $h_{\mathrm{ac}}^{(4)}(0^{+},{\tau }_2,{\tau }_3)$, where $\tau =0^{+}$ represents the bin with $5\mu \mathrm{s}<\tau <15\mu \mathrm{s}$. See Ref. [36] for other 2D slices. Solid lines or surfaces show the fits described in the text. Fits for (c) are to the entire 3D (i.e., ${\tau }_1$-, ${\tau }_2$-, ${\tau }_3$-dependent) dataset. Fit residuals are shown in black for (b) and (c).

Figure 3

The zero-delay second- and third-order coherences, and the coherence decay rates (${\overline{\gamma }}_{\mathrm{ac}}$), as a function of incident power $P_{\mathrm{in}}$. Data is extracted from fits to the second-order (circles) and third-order (squares) coherences. Solid lines show a fit to standard optomechanics theory.

Figure 4

(a) Dynamics of the mean phonon occupancy upon subtraction or addition of $k$ phonons at $\tau =0$. (b) Second order coherences of a 1-phonon subtracted (red) and added (blue) thermal state. Solid lines show the theoretical predictions, see Ref. [36]. Data shown for $P_{\mathrm{in}}\approx 5\mu \mathrm{W}$.