Observation and Interpretation of Motional Sideband Asymmetry in a Quantum Electromechanical Device

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Observation and Interpretation of Motional Sideband Asymmetry in a Quantum Electromechanical Device

A. J. Weinstein, C. U. Lei, E. E. Wollman, J. Suh, A. Metelmann, A. A. Clerk, and K. C. Schwab

Phys. Rev. X 4, 041003 – Published 7 October 2014

Abstract

Quantum electromechanical systems offer a unique opportunity to probe quantum noise properties in macroscopic devices, properties that ultimately stem from Heisenberg’s uncertainty relations. A simple example of this behavior is expected to occur in a microwave parametric transducer, where mechanical motion generates motional sidebands corresponding to the up-and-down frequency conversion of microwave photons. Because of quantum vacuum noise, the rates of these processes are expected to be unequal. We measure this fundamental imbalance in a microwave transducer coupled to a radio-frequency mechanical mode, cooled near the ground state of motion. We also discuss the subtle origin of this imbalance: depending on the measurement scheme, the imbalance is most naturally attributed to the quantum fluctuations of either the mechanical mode or of the electromagnetic field.

Received 15 May 2014

DOI:https://doi.org/10.1103/PhysRevX.4.041003

This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Authors & Affiliations

A. J. Weinstein^1,2, C. U. Lei^1,2, E. E. Wollman^1,2, J. Suh^1,2,†, A. Metelmann³, A. A. Clerk³, and K. C. Schwab^1,2,*

¹Applied Physics, Caltech, Pasadena, California 91125, USA
²Kavli Nanoscience Institute, Caltech, Pasadena, California 91125, USA
³Department of Physics, McGill University, Montreal, Quebec H3A 2T8, Canada

^*To whom all correspondence should be addressed. schwab@caltech.edu
^†Present address: Korea Research Institute of Standards and Science, Daejeon 305-340, Republic of Korea.

Popular Summary

Electromagnetic resonators have recently been used to prepare and detect macroscopic states of motion at levels approaching the limits set by quantum mechanics. However, few studies present clear and unambiguous evidence of quantum behavior of a macroscopic mechanical object. One distinctly quantum signature of a harmonic oscillator is an imbalance between the rates of energy absorption and emission from an external bath. This rate imbalance ultimately stems from Heisenberg uncertainty relations. We measure a rate imbalance between microwave signals generated from either up- or down-conversion of photons via scattering between mechanical and microwave resonators. Depending on the explicit measurement details, this imbalance stems from the quantum fluctuations of either the microwave or mechanical field.

We probe the vibrations of a radio-frequency mechanical membrane parametrically coupled to a superconducting microwave resonator with resonance frequency around 5 GHz. We mount the device in a dilution refrigerator to passively cool the microwave mode into its ground state and to suppress the thermal excitations of the mechanical mode. Pumping the system with detuned microwave drive fields, we monitor the microwave noise signals that are up- and down-converted from the drive fields via the parametric coupling between the modes. A careful comparison between the two signals reveals the rate imbalance due to both quantum and classical noise effects. For the linear detection scheme used in this work and recent related experiments, the imbalance is most naturally associated with the quantum fluctuations of the microwave field.

We expect that the sideband imbalance can serve as a calibrated thermometer for micron-scale mechanical systems.

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Vol. 4, Iss. 4 — October - December 2014

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Figure 1
Comparison between linear detection and photodetection. (a) Pump scheme. We consider a single microwave cavity (dotted line) pumped at $ω_{c} \pm (ω_{m} + δ)$ (green bars). The up-converted (red bar) and down-converted (blue bar) motional sidebands are placed tightly within the cavity linewidth. For figure clarity, the occupations of the microwave and mechanical modes are assumed to be zero. (b) Linear detection. The quantum contribution from the symmetrized motional noise ${\bar{S}}_{x x}$ is present in both sidebands. Microwave shot noise (brown band) and amplifier noise (beige band) combine to form the imprecision noise ${\bar{S}}_{I I}$ . This measurement is sensitive to noise correlation between the microwave and mechanical modes ( ${\bar{S}}_{I F}$ ), which results in asymmetric squashing (red region) and antisquashing (blue region) of the noise floor. (c) Photodetection. Normal-ordered detection is sensitive to the asymmetric motional noise spectrum $S_{x x}$ . The detector is not sensitive to microwave shot noise, and the noise floor ( $S_{I I}$ ) is from detector nonidealities (beige band), analogous to dark counts for a photodetector. Although the source is different, the sideband imbalance is identical in both photodetection and linear detection. For mathematical descriptions of $S_{I I}$ and $S_{I F}$ , refer to Appendix pp2.
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Figure 2
Device, calibration, and measurement scheme. (a) Electron micrograph of the measured device. A suspended aluminum (gray areas) membrane patterned on silicon (blue background) forms the electromechanical capacitor. It is connected to the surrounding spiral inductor to form a microwave resonator. Out of view, coupling capacitors on either side of the inductor couple the device to input and output coplanar waveguides. (b) Motional sideband calibration. The cryostat temperature is regulated while the mechanical mode is weakly probed with microwave tones set at $ω_{c} + ω_{m} + δ$ (blue) and at $ω_{c} - ω_{m} - δ$ (red), with detunings $δ = 2 π \times 500 Hz$ . The observed linear dependence provides the calibration between the normalized sideband power and the mechanical occupation factor. Insets: Up-converted motional sideband spectra collected at 20 mK (top) and 200 mK (bottom), with $Δ ω = ω - (ω_{c} - δ)$ . (c) Schematic of the microwave-measurement circuit.
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Figure 3
Sideband asymmetry. (a) Pump scheme. Three tones are placed about the microwave resonance. Two probe tones generate up-converted (red) and down-converted (blue) sidebands. An additional tone (green) cools the mechanical mode. (b) Sideband spectra. ${\bar{S}}_{I I, tot} [ω]$ measured at $n_{eff}^{th} = 0.60$ (blue) and 2.5 (orange) with ${\bar{n}}_{m} = 5.3$ and 7.1, respectively. (c) Sideband asymmetry. The ratio $n_{m}^{-} / n_{m}^{+}$ versus $n_{m}^{+}$ is plotted for increasing noise injection. (d) Sideband imbalance (blue) and sideband average (red) versus the measured noise increase $Δ η$ . Sideband imbalance $n_{m}^{-} - n_{m}^{+}$ and average $(n_{m}^{-} + n_{m}^{+}) / 2$ exhibit a linear trend with $Δ η$ . The imbalance at $Δ η = 0$ is the quantum imbalance due to the squashing of fluctuations of the microwave field.
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Figure 4
Effects of asymmetric parameters, the cooling tone, and the next-sideband contributions. The dotted blue (dashed red) lines correspond to the results for the (anti-)Stokes sideband at $ω - ω_{c} = + δ$ ( $- δ$ ). (a) Symmetrized output spectra as a function of frequency, for experimental parameters. The solid gray line (area) shows the non-RWA result, which exhibits two peaks at $ω = ω_{c} \pm δ$ . The dashed red and dotted blue lines correspond to two independent noise spectral densities, one valid for frequencies close to the anti-Stokes sideband [Eq. (a21a)] and the other one for frequencies close to the Stokes sideband [Eq. (a21b)], which each accurately describe the resonant peaks at $ω - ω_{c} = \mp δ$ . (b) Influence of the detuning parameter $δ$ on the output noise. For $δ ≫ γ_{tot} = γ_{M}$ , the results from the twin peak spectrum, i.e., from Eq. (a26), coincide with the single-peak approximations (dashed and dotted gray lines). (c) Effect of next-sideband contributions on the noise maximum. The gray lines include a 1% mismatch of the coupling strengths. (d) Symmetrized output spectra for zero-temperature baths with (solid light green line) and without (dashed black line) direct contributions from the cooling tone (experimental parameters). See the text for details. (e) Influence of the detuning $δ_{c}$ of the cooling beam on the sidebands. Plotted is $Δ n_{m}^{\pm} / n_{m}^{\pm} = (n_{m}^{\pm} - n_{m, 0}^{\pm}) / n_{m}^{\pm}$ , where $n_{m, 0}^{\pm}$ $(n_{m}^{\pm})$ corresponds to the sideband’s weight without (with) direct contributions from the cooling tone. Experimental parameters: $ω_{m} / (2 π) = 4 MHz$ , $γ_{M} / (2 π) = 360 Hz$ , $γ_{m} / (2 π) = 10 Hz$ , $δ / (2 π) = 5 kHz$ , $δ_{c} / (2 π) = 30 kHz$ , $κ / (2 π) = 870 kHz$ , $κ_{R} / (2 π) = 450 kHz$ , $κ_{L} / (2 π) = 155 kHz$ , and $γ_{opt} / (2 π) = 100 Hz$ . For the cavity baths, we assume $n_{R}^{th} = n_{L}^{th} = 0.3$ and $n_{c}^{th} = 0.24$ , and for the mechanical bath, we assume $n_{M}^{th} = 10$ .
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