Quantum enhanced feedback cooling of a mechanical oscillator
Laser cooling is a fundamental technique used in primary atomic frequency standards 1, quantum computers 2, quantum condensed matter physics 3 and tests of fundamental physics 4, among other areas. It has been known since the early 1990s that laser cooling can, in principle, be improved by using squeezed light as an electromagnetic reservoir 5; 6; while quantum feedback control using a squeezed light probe is also predicted to allow improved cooling. Here, we implement quantum feedback control of a micro-mechanical oscillator for the first time with a squeezed probe field. This allows quantum-enhanced feedback cooling with a measurement rate greater than it is possible with classical light, and a consequent reduction in the final oscillator temperature. Our results have significance for future applications in areas ranging from quantum information networks 7, to quantum-enhanced force and displacement measurements 8; 9 and fundamental tests of macroscopic quantum mechanics 10.
Real-time quantum feedback control 11 often involves time-continuous measurements of a quantum state followed by control of a physical system with the aim of steering that system into a desired state. Such control has been demonstrated in numerous experiments, including the preparation of microwave Fock states 12 and the generation of deterministic entanglement in quantum superconducting circuits 13. Quantum feedback control also allows the control of motional states via continuous measurements of a probe field followed by actuation back upon the motional degree of freedom. It has been used to steer single trapped atoms 14, levitated microspheres 15, micromechanical oscillators 16; 17 and suspended kilogram-scale mirrors 18 towards their motional ground states; and proposed as a means to prepare both mechanical Fock 19 and superposition states 20. However, to-date, all such experiments have used a classical probe field.
The use of a non-classical probe field for quantum control can both increase the effectiveness of steering to the desired quantum state and introduce new functionalities. For instance, it can be used to prepare a mechanical oscillator in a non-Gaussian quantum state even in the linear coupling regime 21; 22; 23, thereby circumventing the usual requirement to reach the single-photon strong coupling regime 24. Moreover, quantum feedback control using squeezed light can enable an efficient interface for quantum information networks 25, and allow the standard quantum limit of force measurements to be surpassed via squeezing-enhanced measurements 8. Here we demonstrate a first critical step towards such functionalities by realizing a quantum-enhanced cooling rate using a squeezed probe field and real-time feedback control of a mechanical mode.
The basic idea is to continuously monitor the position of the mechanical oscillator and subsequently use the measurement record to steer the oscillator towards the ground state with real-time feedback. The cooling rate is limited by the precision with which the position of the oscillator can be monitored, which in turn is often limited by the noise of the tracking laser beam 26. Therefore, by engineering the quantum state of the laser beam to reduce the measurement noise, an enhanced cooling rate can be expected. Using squeezed light with variance to continuously monitor the mechanical motion, a feedback loop with signal detection efficiency , an overall detection efficiency for the squeezed light, and feedback gain , can lower the temperature to
where is the detected variance , is the equilibrium temperature of the mechanical oscillator prior to feedback cooling, the corresponding mean phonon occupancy, and we have neglected radiation pressure induced heating due to the measurement process. Considering a cavity opto-mechanical system with cavity energy decay rate and mechanical dissipation rate , the linearised interaction is characterised by the cooperativity , where is the single-photon opto-mechanical coupling strength and the mean intra-cavity photon number. As the feedback gain is increased, the temperature drops, reaching a minimum for . At higher gain the oscillator begins to heat up due to the imparted measurement noise. In essence, the application of squeezed measurement noise acts to shift this onset of heating to higher gain values, thereby allowing cooling to temperatures below the shot noise imposed limit, as illustrated in Fig. 1d. Alternatively, the improved cooling efficiency can be appreciated as a consequence of an increased measurement rate , resulting from the squeezing-enhanced signal-to-noise ratio of the transduced mechanical motion 9. In comparison to a coherent state probe, the improvement is directly given by the degree of squeezing as . As the measurement rate is intimately related to the ability to achieve ground state cooling, requiring , squeezing-enhanced feedback cooling can greatly relax the structural requirements for the mechanical oscillator and thereby facilitate the preparation of macroscopic oscillators in their motional ground state. It should be noted that improved cooling can be also realized by increasing the power of the probe beam. However, it is challenging, in practice, to achieve cooling rates sufficient for ground state cooling due to localised bulk heating introduced by absorption of the probe field.
The experimental setup for squeezed light enhanced feedback cooling is shown in Fig. 1a and a scanning electron micrograph of the opto-mechanical system – a micro-toroidal resonator – is displayed in Fig. 1b.
To demonstrate the principle of quantum-enhanced feedback cooling, we chose to work with the fundamental flexural mode of the micro-toroid. The mechanical resonance frequency of the mode was , well within the bandwidth of our feedback loop, and its effective mass was determined to be via finite element modelling. From the optically read out position power spectral density shown in Fig. 2c the damping rate was characterised to be . We note that, the predominantly vertical motion (see finite element model in Fig. 2c) of the fundamental flexural mode reduces the strength with which it couples to the optical field. While the coupling is sufficient for our proof-of-principle demonstration, stronger coupling – sufficient to enter the quantum-coherent coupling regime 27 – can be achieved using a radial breathing mode. This was precluded in our case, since the resonance frequency of this mode lays outside of our available control bandwidth.
In a first experiment, we implemented feedback cooling using coherent states of light with a power of . The results are illustrated in Fig. 2, where we cooled from a temperature of down to the limit set by the imprecision noise corresponding to the quantum noise of a coherent state. We carefully characterized the noise floor through attenuation measurements and balanced detection to verify that it is of true quantum origin, that is, pure vacuum noise. As the feedback gain is increased, the temperature of the mechanical mode is decreased to . Increasing the gain further leads to heating of the oscillator as the shot noise of the probing field dominates the control.
The demonstration of squeezed light enhanced cooling is presented in Fig. 2. Here, we used the same coherent excitation as for the coherent state cooling experiment, but the quantum fluctuations of the probe beam were now suppressed below vacuum noise. The actual noise suppression of the generated squeezed state was (see Supplementary information), but as a result of evanescent coupling and propagation losses in the tapered fibre, the measured noise reduction was limited to below the vacuum noise. Despite the losses a considerable increase in measurement rate of was achieved, and by applying the electronic feedback a clear suppression of the transduced thermally excited mechanical displacement fluctuations was observed, ultimately reaching the squeezed noise level well below shot noise. As a direct result of employing quantum-enhanced feedback cooling, the mechanical mode was cooled to an effective temperature of , more than lower than the minimum temperature achieved using coherent light.
A complementary characterisation of the demonstrated cooling scheme is provided by examining the time-domain phase space trajectory of the mechanical oscillator. Due to the large bandwidth of the homodyne detector compared to the mechanical dissipation rate, this could be monitored in real-time by simultaneous down-mixing of the homodyne photo-current with two in-quadrature signals and subsequent low-pass filtering 9. The recorded thermal trajectories are visualised in Fig. 2b. Both probing strategies result in a significant confinement of the oscillator’s random excursions in phase space when subject to feedback cooling. While the enhancement by using squeezed light is not directly obvious from the phase space trajectories (Fig. 2b) the effect is more pronounced by considering the marginal quadrature distributions (Fig. 2c), revealing a reduction in the variance of the cooled oscillator’s position for the squeezed light probe, which is in good agreement with the temperature reduction.
The resulting temperature estimates as function of inferred feedback gain are presented in Fig. 3a. A clear cooling improvement is observed for increasing gain until the thermal noise spectrum reaches the measurement imprecision noise level, after which the mechanical oscillator begins to heat up due to measurement noise being imprinted on its motion. As expected, the reduced imprecision noise of the squeezed probe shifts the onset of heating towards larger feedback gain values, allowing quantum-enabled cooling to temperatures below the limit set by shot noise.
The absolute cooling achieved in the present demonstration is only modest compared to state-of-the-art 17, the main limitations being the relatively poor opto-mechanical cooperativity provided by the flexural mode and the requirement for operating in the under-coupled regime to preserve squeezing.
Considerable improvements in cooling performance can be expected by using an opto-mechanical system with enhanced cooperativity, and realising higher coupling and detection efficiencies for the squeezed mode. For instance, operating the system of Wilson et al. 17 in the under-coupled regime ( at a coupling efficiency of , where is the intra-cavity photon number of ) so that the injected squeezed light reflects from the cavity with high efficiency, with unity detection efficiency, and starting from , enables cooling to an occupation number phonons. Using a squeezed probe with an input squeezing of , the occupation number could be lowered to . We note that achieving close to unity detection efficiency is challenging for current quantum opto-mechanics experiments. This presents a serious constraint for quantum feedback control experiments. Indeed, with the experimentally realised efficiency reported by Wilson et al. of , even when achieving a cooperativity , the minimum mechanical occupancy that could be achieved with coherent light 9 is . Interestingly, while squeezed light is degraded by inefficiencies, it also provides a mechanism to mitigate them in feedback control experiments. If, amplitude-, rather than phase squeezing is used the contribution of vacuum noise entering the phase quadrature due to inefficiencies is suppressed relative to the amplified noise of the phase quadrature. This results in a higher effective efficiency for feedback control experiments of , where denotes the intra-cavity variance (see Supplementary information). We see that when , approaches unity. As an example, assuming an intra-cavity phase anti-squeezing variance above shot noise, the effective feedback cooling efficiency of Wilson et al. could be increased to , allowing, in principle, feedback cooling to an occupancy of .
We anticipate that the full benefit of squeezing-enhanced feedback cooling could be harnessed using recently developed tethered membrane mechanical oscillators 29 in conjunction with a high-quality optical cavity, or alternatively, by implementation in state-of-the-art opto-mechanical systems in the microwave regime 30.
Our demonstration of enhanced real-time quantum feedback exploiting squeezed state probing opens up a new avenue in the field of quantum opto-mechanics. Besides the demonstrated cooling effect, it lays the foundation for more advanced schemes including squeezing-enhanced quantum back-action evasion which in turn can be used to prepare mechanically squeezed states in the weak coupling regime and for generation of non-Gaussian mechanical quantum states. As an example, combining of squeezing with active feedback in a pulsed backaction evading scheme 23, it is possible to squeeze the mechanical oscillator by for an interaction strength of , while for a coherent state input, is required to squeeze the mechanics by the same amount. Moreover, supplementing the scheme with photon-subtraction, the mechanics can be driven into a quantum non-Gaussian state even for a modest interaction strength.
Bright squeezed light at for tracking the mechanical motion was provided by the nonlinear process of cavity-enhanced parametric down-conversion and injected into the toroidal optical cavity with decay rate via evanescent coupling using a tapered single-mode fibre. The total on-resonance transmission through the optical fibre and toroid was measured to be , mainly limited by scattering losses at the tapered region of the fibre. After interaction with the mechanical mode of the toroid, the phase quadrature of the probe beam was measured by a high efficiency homodyne detector (), and the output photocurrent was split to provide signals for both characterization of the mechanical motion and feedback. The feedback loop, consisting of bandpass filtering, delay, and amplification, was designed to exert a viscous damping force on a selected mechanical mode. The required velocity dependence of the applied force was controlled by a tunable signal delay. To actuate the mechanical motion of the toroid, an electric field was generated between a grounded aluminium plate supporting the toroid and a sharp electrode with a tip diameter of about 28. A bias voltage of was applied to the electrode to polarize the micro-toroid, in order to enhance the electrical transduction. Following the standard approach 31, the measured and calibrated Lorentzian spectra in Figs. 2a were fitted to a model including the noise squashing effect of in-loop measurements (see Supplementary information). For each measurement, the mechanical resonance frequency, feedback gain, and measurement noise level were used as free parameters. Substituting the fitted values into equation (1) the effective mode temperature was calculated.
The work was supported by the Lundbeck Foundation, the Danish Council for Independent Research (Sapere Aude 4184-00338B and 0602-01686B), the Australian Research Council Centre of Excellence CE110001013, and by the Air Force Office of Scientific Research and the Asian Office of Aerospace Research and Development. W.P.B. is supported by the Australian Research Council Future Fellowship FT140100650. Micro-toroid fabrication was undertaken within the Queensland Node of the Australian Nanofabrication Facility.
Iii Author contributions
The authors declare no competing financial interests. U.L.A. conceived the idea while C.S., H.K., T.G. and U.L.A. devised the experiment. C.S. performed the experiment with help from H.K., H.F., U.B.H., G.I.H., T.G. and J.B. C.S. and U.B.H. analysed the data with input from W.P.B. and U.L.A. C.S., U.B.H., T.G., W.P.B. and U.L.A. wrote the paper. C.S. and U.B.H. wrote the supplemental material with input from H.K. T.G., A.H. and U.L.A. supervised the work.
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