Mechanical cooling in the bistable regime of a dissipative optomechanical cavity with a Kerr medium

Ye Liu, Yang Liu, Chang-Sheng Hu, Yun-Kun Jiang, Huaizhi Wu, and Yong Li

Phys. Rev. A 108, 023503 – Published 4 August 2023

Abstract

In this paper, we study static bistability and mechanical cooling of a dissipative optomechanical cavity filled with a Kerr medium. The system exhibits optical bistability for a wide input-power range with the power threshold being greatly reduced, in contrast to the case of purely dissipative coupling. At the bistable regime, the membrane can be effectively cooled down to a few millikelvin from the room temperature under the unresolved sideband condition, where the effective mechanical temperature is a nonmonotonic function of intracavity intensity and reaches its minimum near the turning point of the upper stable branch. When the system is in the cryogenics environment, the effective mechanical temperature at the bistable regime shows a similar feature as in the room temperature case, but the optimal cooling appears at the monostable regime and approaches the mechanical ground state. Our results are of interest for further understanding bistable optomechanical systems, which have many applications in nonclassical state preparations and quantum information processing.

Received 28 May 2023
Accepted 21 July 2023

DOI:https://doi.org/10.1103/PhysRevA.108.023503

Physics Subject Headings (PhySH)

Optomechanics Squeezing of quantum noise

Atomic, Molecular & Optical

Authors & Affiliations

Ye Liu¹, Yang Liu ¹, Chang-Sheng Hu², Yun-Kun Jiang ¹, Huaizhi Wu ^1,*, and Yong Li ^3,†

¹Fujian Key Laboratory of Quantum Information and Quantum Optics and Department of Physics, Fuzhou University, Fuzhou 350116, China
²Department of Physics, Anhui Normal University, Wuhu 241000, China
³Center for Theoretical Physics, Hainan University, Haikou 570228, China

^*huaizhi.wu@fzu.edu.cn
^†yongli@hainanu.edu.cn

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Vol. 108, Iss. 2 — August 2023

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Figure 1
Schematics of the dissipative optomechanical setup. We consider a Michelson-Sagnac interferometer, which consists of three fixed perfect reflecting mirrors $M_{i}$ ( $i = 1, 2, 3$ ) and a fixed beam splitter (BS). A movable membrane and a Kerr medium are placed in the middle of $M_{2}$ and $M_{3}$ [54]. A strong classical driving field is input to the interferometer via the vertical ports of the BS. The part encircled by dashed line can be regarded as an effective end mirror $M_{dis}$ . The linewidth of the cavity depends on the membrane displacement, which leads to an effective dissipative coupling between the cavity mode and the mechanical motion.
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Figure 2
The normalized cavity intensity $U {\bar{n}}_{c} / κ$ (a) as a function of the dimensionless detuning $Δ / κ$ for driving power $P = 100$ mW and (b) as a function of the driving power $P$ for $Δ / κ = 3$ , with the nonlinearity strengths being $U = 50 μ Hz$ (orange), 100 $μ Hz$ (yellow), $150 μ Hz$ (purple), and $200 μ Hz$ (green) (from bottom to top). We consider the set of typical experimental parameters [21]: the wavelength of the input laser $λ = 2 π c / ω_{l} = 1064$ nm, the cavity decay rate $κ = 2 π \times 1.5 MHz$ , the dissipative optomechanical coupling rate $g = 2 π \times 0.1 Hz$ , the mechanical frequency $ω_{m} = 2 π \times 136 kHz$ , and the mechanical damping rate $γ_{m} = 2 π \times 0.23 Hz$ . The cavity intensity displays a characteristic $S$ -shaped curve, with the stable and unstable parts being indicated by the solid and dotted lines, respectively. The black triangles mark the parameter conditions plotted in Fig. 3.
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Figure 3
(a) The normalized effective resonance frequency $ω_{eff} (ω) / ω_{m}$ , (b) the normalized effective damping rate $γ_{eff} (ω) / γ_{m}$ , (c) the modulus of effective susceptibility $| χ_{eff} {(ω) |}^{2}$ , and (d) spectrum of the input vacuum noise as a function of the normalized frequency $ω / κ$ . The dotted line in (d) indicates the mechanical thermal noise, which for the environment of the room temperature $T = 293 K$ considered here is approximately frequency independent. The Kerr nonlinearity and the cavity detuning for the curves [from top to bottom in (a)] are $U = 50 μ Hz$ and $Δ = 1.43 κ$ (orange), $U = 100 μ Hz$ and $Δ = 2.51 κ$ (yellow), $U = 150 μ Hz$ and $Δ = 3.68 κ$ (purple), $U = 200 μ Hz$ and $Δ = 4.87 κ$ (green), respectively, corresponding to the parameters indicated by the black triangles in Fig. 2. Other parameters are the same as in Fig. 2.
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Figure 4
(a) Phase diagram of the stable regime (yellow) confirmed via the Routh-Hurwitz criterion, which requires $s_{1}$ , $s_{2}$ , and $s_{3}$ to be all positive. The unstable region in green corresponds to $s_{1} < 0$ (or $s_{2} < 0$ ), and the orange region corresponds to $s_{3} < 0$ . In the blue region, $s_{1}$ , $s_{2}$ , and $s_{3}$ are all negative. (b) Effective temperature of the membrane in the steady state versus $Δ / κ$ and $\tilde{U} / κ$ by cooling from the initial room temperature $T = 293 K$ . The black dash-dotted line indicates $λ (ω_{m}) = 0$ , which fits approximately with the boundary of the unstable region with $s_{1 (2)} < 0$ , corresponding to the green zone in (a). (c) Variance of the optical quadrature fluctuation $〈 δ x^{2} 〉$ and (d) the photon number fluctuation $δ n_{c}$ versus $Δ / κ$ and $\tilde{U} / κ$ . The inset shows $δ n_{c}$ versus $Δ / κ$ for $\tilde{U} / κ = 1$ and the photon number fluctuation diverges at the stability boundary with $s_{j} \to 0$ , $j = 1, 2, 3$ . In (b)–(d), the blank regions correspond to the classical unstable regime. The four curves [in (b,c)], and other parameters are the same as those in Fig. 2.
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Figure 5
Effective temperature of the membrane in the steady state versus $Δ / κ$ and $\tilde{U} / κ$ by cooling from the initial cryogenic temperature $T = 0.1 K$ . The blank regions correspond to the classical unstable regime confirmed via the Routh-Hurwitz criterion. The black dash-dotted line indicates $λ (ω_{m}) = 0$ . The inset shows the spectra of the input vacuum noise $S_{c} (ω)$ (dashed), the thermal noise $S_{Q}^{th} (ω)$ (dotted), and sum of them (solid). Other parameters are $(κ, g, ω_{m}, γ_{m}) / 2 π = (1.5 MHz$ , 0.35 Hz, 300 kHz, 0.1 Hz) and $P = 100$ mW.
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