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Sideband cooling beyond the quantum backaction limit with squeezed light

Abstract

Quantum fluctuations of the electromagnetic vacuum produce measurable physical effects such as Casimir forces and the Lamb shift1. They also impose an observable limit—known as the quantum backaction limit—on the lowest temperatures that can be reached using conventional laser cooling techniques2,3. As laser cooling experiments continue to bring massive mechanical systems to unprecedentedly low temperatures4,5, this seemingly fundamental limit is increasingly important in the laboratory6. Fortunately, vacuum fluctuations are not immutable and can be ‘squeezed’, reducing amplitude fluctuations at the expense of phase fluctuations. Here we propose and experimentally demonstrate that squeezed light can be used to cool the motion of a macroscopic mechanical object below the quantum backaction limit. We first cool a microwave cavity optomechanical system using a coherent state of light to within 15 per cent of this limit. We then cool the system to more than two decibels below the quantum backaction limit using a squeezed microwave field generated by a Josephson parametric amplifier. From heterodyne spectroscopy of the mechanical sidebands, we measure a minimum thermal occupancy of 0.19 ± 0.01 phonons. With our technique, even low-frequency mechanical oscillators can in principle be cooled arbitrarily close to the motional ground state, enabling the exploration of quantum physics in larger, more massive systems.

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Figure 1: Experimental set-up.
Figure 2: Theoretical mechanical bath temperature in the presence of squeezed light.
Figure 3: Experimentally measured power spectral densities of the upper and lower mechanical sidebands.
Figure 4: Observed cooling enhancement with squeezed light.

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Acknowledgements

This work was financially supported by NIST and the DARPA QuASAR program. J.B.C. acknowledges the NIST National Research Council Postdoctoral Research Associateship Program for its financial support. Contribution of the US government, not subject to copyright.

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Authors

Contributions

J.B.C. and J.D.T. conceived and designed the experiment. F.L. fabricated the optomechanical circuit and contributed technical input for the experimental set-up. J.B.C. performed the measurements and data analysis, and developed the theory with assistance from J.D.T. All authors contributed to writing the manuscript.

Corresponding author

Correspondence to John D. Teufel.

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Competing interests

The authors declare no competing financial interests.

Additional information

Reviewer Information Nature thanks P. Tombesi and the other anonymous reviewer(s) for their contribution to the peer review of this work.

Extended data figures and tables

Extended Data Figure 1 Cryogenic measurement set-up.

LPF, low-pass filter; JPA, Josephson parametric amplifier; OM, optomechanical; HEMT, high-electron-mobility transistor amplifier; μ-wave, microwave. The components shaded in grey signify broadband microwave attenuators. The rectangular components shaded red indicate directional couplers.

Extended Data Figure 2 Calculated power spectral density of the ponderomotive squeezing and antisqueezing that would be detected by an ideal homodyne receiver.

The solid lines indicate pondermotive squeezing and the dashed lines antisqueezing. SNL denotes the shot-noise limit (vacuum fluctuations). The indicated mechanical resonance frequency Ω includes optical springing effects. The colour of each curve denotes the resultant power spectral density produced by either of two coherent-state pump detunings satisfying (Δ/Δ0) ± 1 = constant. Here, Δ0 denotes the traditional optimal drive detuning when sideband cooling with coherent states (equation (31)). All curves assume a mechanical quality factor Qm = 5 × 105, a constant (and strong) optical damping rate Γopt = Γ+ − Γ = 103Γ, a lossless optomechanical cavity and zero temperature. Additionally, a normalized cavity linewidth κ/Ω ≈ 2.85 has been assumed, which yields −10 dB of ponderomotive squeezing when Δ = Δ0 (see equation (30)).

Extended Data Figure 3 Calculated power spectral density (PSD) of the reflected drive field as would be detected by an ideal homodyne receiver.

The field quadrature being plotted is that which is maximally squeezed (ponderomotively). SNL denotes the shot-noise limit (vacuum fluctuations). All panels assume the same conditions as for Fig. 2, with the exception of a constant drive detuning (Δ = Δ0) and a constant drive cooperativity C = 4g2/(κΓ ) = 5,000. Each panel assumes a different level of injected squeezing, although the optimal squeezing phase θ = θ0 (equation (11)) is assumed throughout. The strength of the injected squeezing is parameterized by r, expressed in units of the critical squeezing parameter rc. The critical squeezing for this cavity is −5 dB.

Extended Data Figure 4 Modelling the impurity of the squeezing.

a, A pure squeezed state (with squeezing strength parameterized by rin) is subject to loss via a beamsplitter of transmittance ηin. Although the loss is to vacuum, the resulting squeezed state is impure and does not obey the minimum uncertainty relation: . b, Amplitude variance of a pure −10 dB squeezed state subject to loss as the squeezing phase θ is rotated (see inset). c, The Gaussian state in the right-hand panel of a can be equivalently represented with nl and r. The impurity is captured by the uncertainty product , which is maintained in the presence of an ideal entropy-preserving squeezing operation (parameterized by r). The dashed circles denote the standard vacuum fluctuations for reference. d, e, Computed values of r (d) and nl (e) given various values of input squeezing rin and ηin (see equations (39) and (40)).

Extended Data Figure 5 Numerically computed effective mechanical bath occupancy plotted against the normalized coupling rate g/κ for various pure squeezed states.

By numerically computing the full bath occupancy , we account for the role of ‘strong coupling’ physics discussed elsewhere. Various squeezing parameters r were considered (expressed in units of the critical squeezing parameter rc in equation (5)). All curves assume drive detuning Δ = Δ0 (equation (31)) and squeezing phase θ = θ0 (equation (11)). The optomechanical system is assumed to be deep in the resolved sideband cooling limit (κ/Ω = 0.1). The grey dashed line shows the strong coupling limit to the squeezing-enhanced cooling, which scales as (g/Ω)2.

Extended Data Figure 6 Comparison of the mechanical bath temperature with squeezed and unsqueenzed light in the presence of internal cavity loss.

The solid and dashed lines correspond to the mechanical bath temperature with squeezed (; equation (42)) and unsqueezed (; equation (2)) light, respectively. The bath occupancies are plotted as a function of the external coupling rate κext normalized by the internal loss rate of the cavity κ0. Each colour corresponds to a different value of κ0. All curves assume a drive detuning Δ0 (equation (31)) from cavity resonance and the corresponding optimal squeezing phase θ0 (equation (11)). and approach the same limit (equation (44)) as κext/κ0 → 0, indicating that internal cavity loss limits the coldest temperatures that can be reached when sideband cooling with any Gaussian state. For , diverges (equation (46)) whereas asymptotically approaches the ratio κ0/(4Ω) (equation (45)).

Extended Data Table 1 Summary of notation

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Clark, J., Lecocq, F., Simmonds, R. et al. Sideband cooling beyond the quantum backaction limit with squeezed light. Nature 541, 191–195 (2017). https://doi.org/10.1038/nature20604

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