Fast Feedback Control of Mechanical Motion Using Circuit Optomechanics

Cheng Wang, Louise Banniard, Laure Mercier de Lépinay, and Mika A. Sillanpää

Phys. Rev. Applied 19, 054091 – Published 30 May 2023

Abstract

Measurement-based control, using an active-feedback loop, is a standard tool in technology. Feedback control is also emerging as a useful and fundamental tool in quantum technology and in related fundamental studies, where it can be used to prepare and stabilize pure quantum states in various quantum systems. Feedback cooling of center-of-mass micromechanical oscillators, which typically exhibit high thermal noise far above the quantum regime, has been particularly actively studied and has recently been shown to allow ground-state cooling by means of optical measurements. Here we realize measurement-based feedback operations in an electromechanical system, cooling the mechanical thermal noise down to three quanta, limited by added amplifier noise. We also obtain significant cooling when the system is pumped at the blue optomechanical sideband, where the system is unstable without feedback.

Received 29 November 2022
Revised 17 February 2023
Accepted 28 April 2023

DOI:https://doi.org/10.1103/PhysRevApplied.19.054091

Physics Subject Headings (PhySH)

Light-matter interaction Optical forces Optomechanics Quantum control Quantum sensing

2-dimensional systems Micromechanical & nanomechanical oscillators

Microwave techniques

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & OpticalQuantum Information, Science & Technology

Authors & Affiliations

Cheng Wang, Louise Banniard, Laure Mercier de Lépinay, and Mika A. Sillanpää^*

QTF Centre of Excellence, Department of Applied Physics, Aalto University, FI-00076 Aalto, Finland

^*Mika.Sillanpaa@aalto.fi

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Issue

Vol. 19, Iss. 5 — May 2023

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Images

Figure 1
Feedback setup. (a) The basic frequency scheme of feedback cooling in cavity optomechanics, with the strong probe tone set at the cavity frequency ( $Δ = 0$ ). (b) The probe tone set alternatively to the blue mechanical sideband ( $Δ = ω_{m}$ ). (c) Optical micrograph of a similar circuit-electromechanical device. The aluminum-drumhead oscillator of diameter $13 μ m$ is connected to a meander inductor to form a cavity strongly coupled to a transmission line through a large external finger capacitor. An enlargement of the area of the drumhead is indicated by dashed red lines. (d) Simplified schematic of the microwave circuit around the electromechanical device; PM, phase modulator.
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Figure 2
Feedback-control measurements. We use a strong probe tone at the cavity center frequency ( $Δ = 0$ ), with effective coupling $G / 2 π ≃ 420 kHz$ . The data in each panel correspond to feedback gain $A_{0} / 2 π = 2.8 kHz$ (red), $A_{0} / 2 π = 6.7 kHz$ (blue), and $A_{0} / 2 π = 16.7 kHz$ (green). The solid lines are theoretical fits. (a) Mechanical frequency shift and (b) effective damping rate as functions of feedback-loop phase. (c) Area of the mechanical peak in the heterodyne output spectrum.
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Figure 3
Feedback cooling. The parameter values are $G / 2 π ≃ 427 kHz$ and $ϕ ≃ 143^{\circ}$ . (a),(b) In-loop heterodyne output spectra at the lower and upper sideband, respectively. The gain values are $A_{0} / 2 π ≃ 0$ , 10, 28, 76, 125, and $206 kHz$ from top to bottom. The solid lines are theoretical fits. (c) Damping rate extracted by our fitting Lorentzian curves to the spectra at lower gain values, together with a fit to Eq. (9b). The data are shown in the range where the peaks are roughly Lorentzian. (d) Mechanical occupation as a function of feedback gain. The arrow indicates the value $n_{m} ≃ 440$ at zero gain.
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Figure 4
Feedback stabilization and cooling of an intrinsically unstable system. The probe tone is set at the blue-sideband frequency ( $Δ = ω_{m}$ ), and the effective coupling is $G / 2 π ≃ 104 kHz$ . The gain values are $A_{0} / 2 π ≃ 189$ , 225, 267, and $378 kHz$ from top to bottom. The same quantities are presented as in the resonant-pumping case, Fig. 3. (a),(b) Lower-sideband and upper-sideband peaks around the probe tone, respectively. (c) Effective damping in the stable range. (d) Mechanical occupation as a function of feedback gain.
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Figure 5
Technical heating. Bath temperature of the mechanical oscillator as a function of feedback gain. Black circles represent resonant probing, and blue circles represent blue-sideband probing.
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