Mode Competition and Anomalous Cooling in a Multimode Phonon Laser

We study mode competition in a multimode “phonon laser” comprised of an optical cavity employing a highly reflective membrane as the output coupler. Mechanical gain is provided by the intracavity radiation pressure, to which many mechanical modes are coupled. We calculate the gain and find that strong oscillation in one mode suppresses the gain in other modes. For sufficiently strong oscillation, the gain of the other modes actually switches sign and becomes damping, a process we call “anomalous cooling.” We demonstrate that mode competition leads to single-mode operation and find excellent agreement with our theory, including anomalous cooling.

Figure 1

A Fabry-Perot cavity employing a highly reflective subwavelength grating in a silicon nitride membrane, operated in vacuum, forms the optomechanical system. A “probe” laser is locked to a cavity resonance, while a “pump” laser is blue detuned relative to the adjacent longitudinal cavity resonance and provides mechanical gain. A third laser is used for a Michelson interferometer to sense membrane displacements. PID is the proportional-integral-derivative controller, HPF is high-pass filter, and LPF is low-pass filter.

Figure 2

Gain saturation vs oscillation amplitude, calculated for detuning $\delta \omega =0.67\gamma$. Red curve: single-mode normalized antidamping ${\mathrm{\Gamma }}_{21}^{\text{RP}}({\chi }_{21})/{\mathrm{\Gamma }}_{21}^{\text{RP}}(0)$ of mode (2,1) vs dimensionless amplitude ${\chi }_{21}$. Blue curve: two-mode normalized (anti)damping of mode (1,2) when it is oscillating weakly ${\mathrm{\Gamma }}_{12}^{\text{RP}}(0,{\chi }_{21})/{\mathrm{\Gamma }}_{12}^{\text{RP}}(0,0)$ for arbitrary ${\chi }_{21}$. The gain of the (1,2) mode switches sign for large ${\chi }_{21}$.

Figure 3

(a),(b) Time dependence of mode competition between (1,2) and (2,1) modes, where the outcome is unpredictable. Blue curves: (1,2) mode. Red curves: (2,1) mode. Solid lines: experiment; the pump is turned on at $t\approx 50\mathrm{s}$ and turned off at $t\approx 175\mathrm{s}$. Dashed lines: simulations; the pump on and off times are slightly different for clarity. (c) Set of simulations for slightly different initial conditions; no thermal excitation. (d) Measured set of 13 trajectories such as those shown in (a) and (b).

Figure 4

Effective temperature of the (1,2) mode while the (2,1) mode oscillates, illustrating anomalous cooling for large pump powers. Black dashed line and circles: temperature inferred without pump $T_0=278(21)\mathrm{K}$. Blue circles: effective temperature with pump. Solid red line: analytic prediction $T_{\text{eff}}=293\mathrm{K}\times {\mathrm{\Gamma }}_{12}^{\text{intr}}/({\mathrm{\Gamma }}_{12}^{\text{intr}}+{\mathrm{\Gamma }}_{12}^{\text{RP}})$. Inset: histograms of the amplitude of the (1,2) mode. Black curve: no pump. Green curve: weak pump $P_{\mathrm{\text{in}}}=1.3P_{2,1}^{\text{thresh}}$. Violet curve: strong pump $P_{\mathrm{\text{in}}}=6.4P_{2,1}^{\text{thresh}}$.