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

Abstract

We establish a new 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, in a simple and consistent way, the single-photon optomechanical rate, without the need of knowing the bath temperature, and of calibrating the signal, compared to other methods. We provide the theoretical framework, and experimentally validate this method, providing a novel 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.

Two-Membrane Cavity Optomechanics: Linear and Non-Linear Dynamics

“…This analysis is validated considering the experimental time traces shown in Figure 7, where the displacement amplitudes of the two membranes q 1 and q 2 are determined by means of the calibration signal [34,56]. For t < 4 s (the pump beam is off), a value of the displacement amplitudes higher than the thermal ones is observed because of a slightly blue-detuned probe beam.…”

“…In the following we remove the index 1 when referring to the cavity pump field, that is α 1 → α, κ 1 → κ. We also ignore the Langevin equation of the weak probe beam, which is in resonance with the optomechanical cavity, and therefore realizes a perfectly non-invasive detection of the mechanical mode, without any backaction, as it occurs in a Michelson interferometer readout [34]. The cavity field and the mechanical mode are split into an average coherent amplitude and a fluctuating term α(t) = α 0 + δα(t) and β j (t) = β j,0 + δβ j (t).…”

“…In the first 10 s, we switched off the pump beam, and measured the thermal noise displacement of the fundamental modes of the two resonators. The magenta-dashed line highlights the external signal injected to estimate the single-photon optomechanical coupling g 1 2π × 0.43 Hz, and g 2 2π × 0.70 Hz, and to calibrate the VSN in displacement spectral noise (DSN) [34,56]. After 10 s, the blue detuned (∆ 1 /2π = 259.350 kHz) pump beam is switched on with a power of 4.25 µW, to study the non-linear dynamics of the optomechanical system.…”

“…We infer from the intersection point of Figure 6b, which corresponds to find the smallest root of γ e f f 1 (I st 1 ) = 0, a value ξ st 1 = 1.054, a steady-state displacement amplitude q st 1 = 2|A 1 |x zpf = 263.0 pm, and ∆ω e f f 1 = −2π • 0.04 Hz. Such behaviour has been investigated for the determination of g 0 [34]. For ξ 1, the sideband output field has a spectral amplitude linear with ξ, and it is possible to perform a direct measurement of the position coordinate q 1 ; for ξ ≥ 1 we should consider a correction factor because linearity is no more valid.…”

“…However, the radiation pressure interaction is proportional to the photon number and it may have non-linear effects on both the mechanical and optical degrees of freedom which become evident when the mechanical motion is excited [27] by means of laser driving on the blue sideband of the optical cavity. Optical backaction in this case counteracts the internal mechanical friction, and when the total effective damping becomes equal to zero, a Hopf bifurcation into a regime of self-induced mechanical oscillations takes place [23,[28][29][30][31][32][33][34]. A fixed amplitude limit cycle is established, with a free running oscillation phase, which may lock to external forces or to other optomechanical oscillators [35], leading to synchronization (see references [12,15,16,22,[36][37][38][39][40][41][42] for theoretical characterizations, and references [17][18][19][20][21]24,[43][44][45][46] for experimental demonstrations in optomechanical and electromechanical devices).…”