Quantum Signature of a Squeezed Mechanical Oscillator

A. Chowdhury, P. Vezio, M. Bonaldi, A. Borrielli, F. Marino, B. Morana, G. A. Prodi, P. M. Sarro, E. Serra, and F. Marin

Phys. Rev. Lett. 124, 023601 – Published 13 January 2020

Abstract

Recent optomechanical experiments have observed nonclassical properties in macroscopic mechanical oscillators. A key indicator of such properties is the asymmetry in the strength of the motional sidebands produced in the probe electromagnetic field, which is originated by the noncommutativity between the oscillator ladder operators. Here we extend the analysis to a squeezed state of an oscillator embedded in an optical cavity, produced by the parametric effect originated by a suitable combination of optical fields. The motional sidebands assume a peculiar shape, related to the modified system dynamics, with asymmetric features revealing and quantifying the quantum component of the squeezed oscillator motion.

Received 17 July 2019

DOI:https://doi.org/10.1103/PhysRevLett.124.023601

Physics Subject Headings (PhySH)

Optomechanics Quantum optics Squeezing of quantum noise

Atomic, Molecular & Optical

Authors & Affiliations

A. Chowdhury¹, P. Vezio², M. Bonaldi ^3,4, A. Borrielli ^3,4, F. Marino^5,1, B. Morana^3,6, G. A. Prodi ^4,7, P. M. Sarro⁶, E. Serra ^4,6, and F. Marin ^1,2,5,8,*

¹CNR-INO, L.go Enrico Fermi 6, I-50125 Firenze, Italy
²European Laboratory for Non-Linear Spectroscopy (LENS), Via Carrara 1, I-50019 Sesto Fiorentino (FI), Italy
³Institute of Materials for Electronics and Magnetism, Nanoscience-Trento-FBK Division, 38123 Povo, Trento, Italy
⁴Istituto Nazionale di Fisica Nucleare (INFN), Trento Institute for Fundamental Physics and Application, I-38123 Povo, Trento, Italy
⁵INFN, Sezione di Firenze, Via Sansone 1, I-50019 Sesto Fiorentino (FI), Italy
⁶Dept. of Microelectronics and Computer Engineering /ECTM/DIMES, Delft University of Technology, Feldmanweg 17, 2628 CT Delft, Netherlands
⁷Dipartimento di Fisica, Università di Trento, I-38123 Povo, Trento, Italy
⁸Dipartimento di Fisica e Astronomia, Università di Firenze, Via Sansone 1, I-50019 Sesto Fiorentino (FI), Italy

^*marin@fi.infn.it

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Issue

Vol. 124, Iss. 2 — 17 January 2020

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Images

Figure 1
Sketch of the experimental setup (see text) and conceptual scheme of the field frequencies. The LO is placed on the blue side of the probe ( $ω_{p}$ ) and detuned by $Δ_{LO} ≪ Ω_{m}$ , therefore the Stokes lines are on the red side of the LO, while the anti-Stokes lines are on the blue side. In the heterodyne spectra, they are located, respectively, at $Ω_{m} + Δ_{LO}$ (Stokes) and $Ω_{m} - Δ_{LO}$ (anti-Stokes).
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Figure 2
(a),(c) Heterodyne spectra (without demodulation) around the (0,2) membrane mode at $Ω_{m} / 2 π ≃ 530 kHz$ , (a) without parametric drive and (c) with parametric drive. In (a) the spectrum is fitted by Lorentzian curves (solid line). In (c) the fitting function (dark green line) is the superposition of a broad and a narrow Lorentzian shape, whose contributions are shown with blue and red lines. (b),(d) Spectra of the fluctuations in two quadratures, obtained by phase-sensitive demodulation of the heterodyne signal at $Ω_{par} / 2$ , (b) without parametric drive and (d) with parametric drive. In (b) the two spectra (dark and light green symbols) are not distinguishable, and one single Lorentzian fit is shown (solid line). In (d) the two spectra (orange and blue symbols) are fitted with different Lorentzian curves (red and blue solid lines). The full vertical scales correspond, respectively, to $2 {Hz}^{2} / Hz$ (a),(b), $4 {Hz}^{2} / Hz$ (c), $8 {Hz}^{2} / Hz$ (d).
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Figure 3
Green symbols: sideband asymmetry $R$ with no parametric drive (i.e., with detuned modulation tone), for increasing power in the modulation tone. Sideband ratios $R_{+}$ (blue circles) and $R_{-}$ (red circles) with coherent parametric drive. The values of $s$ in the abscissa are extracted from the fitted widths $Γ_{+} = Γ_{eff} (1 + s)$ and $Γ_{-} = Γ_{eff} (1 - s)$ . Dashed lines show the corresponding theoretical behavior, with shadowed areas given by the uncertainty in the system parameters (in particular, 5% in the cavity width and 0.5 K in the temperature). Error bars reflect the standard deviations in 5 consecutive independent measurements, each one lasting 100 s. The total pump power is kept constant during all the measurements.
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Figure 4
Variance in the $X$ (orange squares) and $Y$ (cyan squares) quadratures, normalized to $σ_{0}^{2}$ , as a function of the ratio between modulation and cooling tones, for constant total pump power. Dashed lines show the theoretical behavior. Magenta and blue circles are the correspondent expected values, calculated, respectively, as $1 / (1 - s)$ and $1 / (1 + s)$ , where $s$ is extracted from the width of the broad and narrow Lorentzian contributions in the heterodyne spectra. Error bars reflect the standard deviations in 5 consecutive independent measurements, each one lasting 100 s.
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