Enhancing Sideband Cooling by Feedback-Controlled Light

We realize a phase-sensitive closed-loop control scheme to engineer the fluctuations of the pump field which drives an optomechanical system and show that the corresponding cooling dynamics can be significantly improved. In particular, operating in the counterintuitive “antisquashing” regime of positive feedback and increased field fluctuations, sideband cooling of a nanomechanical membrane within an optical cavity can be improved by 7.5 dB with respect to the case without feedback. Close to the quantum regime of reduced thermal noise, such feedback-controlled light would allow going well below the quantum backaction cooling limit.

Figure 1

(a) A cavity is driven by a coherent field with amplitude quadrature ${\widehat{X}}_{\mathrm{in}}$ modified by an amplitude modulator fed with the output of the homodyne quadrature detection. The output field quadratures ${\widehat{X}}_{\mathrm{out}}$ and ${\widehat{Y}}_{\mathrm{out}}$ become correlated, depending on the homodyne phase $\varphi$ and on the nonresonant cavity driving at detuning $\mathrm{\Delta }$. (b) The radiation pressure interaction in an optomechanical cavity with decay rate $\kappa$ yields sidebands at mechanical frequencies $\pm {\omega }_m$, that correspond to processes which enhance ($A_{+}$, yellow) and reduce ($A_{-}$, blue) mechanical energy. (c)–(f) Theoretical results for the phonon number of the cooled resonator at a low temperature (when standard sideband cooling is limited by backaction noise), as a function of the feedback gain amplitude $g_{\mathrm{fb}}$ (see [25]) [panels (c) and (e)] and the homodyne phase $\varphi$ [panels (d) and (f)], for the parameters of the experiment of Ref. [24] [panels (c) and (d)] and of Ref. [19] [panels (e) and (f)]. The light-pink areas indicate results beyond the backaction limit. Solid blue curves represent results for perfect detection efficiency $\eta =1$, dashed blue curves for $\eta =0.42$ [27] in (c) and (d) and $\eta =0.36$ [28] in (e) and (f), and finally dotted blue curves for no feedback. Red lines are the best up-to-date results obtained for systems operating at the quantum backaction limit, in the optical [24] (with standard sideband cooling) and microwave [19] (where squeezing is employed to improve the system performance) regimes.

Figure 2

(a) Optomechanical cavity driven by the in-loop cavity mode (cooling beam). Dynamical backaction of the fundamental mechanical mode is provided by detuning the cooling beam by means of an AOM. Feedback is applied by AM the cooling laser with an electronically processed copy of the transmitted photocurrent. Inset: Image of the circular SiN membrane used, radius 0.615 mm and thickness 97 nm. The switches TG and FB allow the open- and closed-loop transfer functions to be measured [25]. A probe beam is used to monitor the cavity frequency fluctuations via balanced homodyne detection. (b) Current noise spectra $S_i$, normalized to the detection noise $S_n$, of the transmitted (bottom left) and reflected (bottom right) photocurrent measured placing the membrane in a position of zero optomechanical interaction $g_0=0$. Red and blue traces correspond to different signs of the amplifier output in the electronic filter. The vertical dashed gray line indicates the detuning $\mathrm{\Delta }=2\pi \times 330\mathrm{kHz}$. The light-blue (light-yellow) area represents the antisquashing (squashing) regime, where noise is amplified (reduced) below the detection noise. We refer to the gain for which the feedback based on transmission provides antisquashing around the cavity detuning (shaded area) as positive. The top part shows the corresponding measured amplitude (dashed lines) and phase (solid lines) of the complex open-loop feedback response function $\mathcal{T}(\omega )$ determined from the transmitted light detected with the switch FB open [25].

Figure 3

(a) Amplitude and (b) phase of the measured closed-loop transfer function for the transmitted cooling field detuned by $\mathrm{\Delta }=2\pi \times 330\mathrm{kHz}$ (dashed gray line) [25]. Black traces are acquired without the feedback loop. The dotted black line indicates the effective cavity frequency at instability. (c) Effective cavity decay rate ${\kappa }_{\mathrm{eff}}$, normalized to the out-of-loop decay rate $\kappa$, and (d) effective detuning shift $\delta {\mathrm{\Delta }}_{\mathrm{eff}}\equiv {\mathrm{\Delta }}_{\mathrm{eff}}-\mathrm{\Delta }$ as a function of the positive feedback gain. For positive feedback (from light blue to red) the effective cavity decay rate ${\kappa }_{\mathrm{eff}}$ decreases and the effective detuning ${\mathrm{\Delta }}_{\mathrm{eff}}$ increases, while for negative feedback (from light to dark blue) the situation is reversed. The gray area represents the instability region.

Figure 4

(a) Homodyne spectra of mechanical displacement noise $S_{xx}$. The blue trace represents the thermal fluctuations of the fundamental mechanical mode at 300 K; the gray trace is the detection noise. Dynamical backaction cools the mechanical motion down to 2 K, as shown by the red trace (cooling beam on at $33\mu \mathrm{W}$ and feedback off). From orange to light purple, the feedback is turned on and the gain increased. (b),(c) Effective energy reduction as a function of the gain (${\mathcal{G}}_{\mathrm{fb}}$) and normalized detuning ($\mathrm{\Delta }/{\omega }_m$) [25]. Dots are experimental data. Each dot in (b) corresponds to the spectrum of the same color in (a), the gray area represents the instability region, and the vertical gray line indicates the optimal gain value for cooling, used in plot (c). In (c), purple and red dots are results with and without feedback, respectively. In both panels, purple and red lines are theoretical results, computed using the measured parameters, with and without feedback, respectively.