Optomechanical preparation of photon number-squeezed states with a pair of thermal reservoirs of opposite temperatures

Baiqiang Zhu; Keye Zhang; Weiping Zhang

doi:doi:10.1364/PRJ.491788

Photonics Research, 2023, 11 (9): A26, Published Online: Aug. 28, 2023

Optomechanical preparation of photon number-squeezed states with a pair of thermal reservoirs of opposite temperatures

Baiqiang Zhu ^1,2Keye Zhang ^1,2,*Weiping Zhang ^2,3,4,5

Author Affiliations

¹ State Key Laboratory of Precision Spectroscopy, Department of Physics, School of Physics and Electronic Science, East China Normal Universityhttps://ror.org/02n96ep67, Shanghai 200062, China

² Shanghai Branch, Hefei National Laboratory, Shanghai 201315, China

³ School of Physics and Astronomy, and Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai 200240, China

⁴ Shanghai Research Center for Quantum Sciences, Shanghai 201315, China

⁵ Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China

Figures & Tables

Fig. 1. (a) Diagram of the population jump rates between neighboring Fock states. (b) Eigenvalues of the dissipation rate operators κn^± versus number n. (c) Number statistics distribution of the steady state. The probability Pn increases versus n in the region dominated by the negative-temperature dissipation and decays in the rest region dominated by the positive-temperature one, so a peak appears in the intermediate region.

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Fig. 2. (a) Cavity optomechanical scheme of feedback control. The optical cavity is coupled to the mechanical oscillator through dispersive and dissipative optomechanical interactions simultaneously. With the dispersive coupling, the oscillator undergoes a shift proportional to the radiation pressure force, i.e., to the photon number, and then changes the cavity dissipation rate κx^+ through the dissipative coupling. Except for the optomechanical dissipation, the cavity mode has a gain of rate κ− induced by the negative-temperature reservoir, and the oscillator is subjected to Brownian thermal noise. The high frequency of the optical mode makes our near-zero temperature assumption reasonable. (b) Dissipation control protocol. The positive-temperature dissipation rate κx^+ is smaller than the negative-temperature one in the region x<L but increases rapidly and overtakes it in the region x>L. The steep change occurs mainly in a region of width d.

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Fig. 3. Steady-state photon number statistics obtained by approximate solution [Eq. (13)] and numerical simulation of the master equation [Eq. (7)]. (a) Photon number fluctuation Δn versus dissipation ratio γ/κ−. The approximate solution is plotted in a red solid line, whereas, the numerical results are marked with a “+.” (b) Numerical results for the steady-state probability distribution of photon number for increasing γ/κ−. All results for g0=7.07×102ωm, (κ0,κ−,κv)=(10−1,10−2,10−3)ωm, and (d,L)=(14,7×104)xzpf.

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Table1. Experimental Parameters and Ideal Squeezing Degrees for Several Representative Optomechanical Systems

Setup	$m_{eff}$ (kg)	$ω_{m} / 2 π$ (Hz)	$g_{0} / 2 π$ (Hz)	$x_{1}$ (nm)	$d$ (nm)	$L$ (nm)	$Δ n$	$\bar{n}$	$Δ n^{2} / \bar{n}$ (dB)
Micromirror [79]	$1.1 \times 10^{- 10}$	$9.7 \times 10^{3}$	22	$1.27 \times 10^{- 8}$	${2.48}^{*}$	50 [80]	$7 \times 10^{3}$	$4 \times 10^{9}$	−19
SiN membrane [76]	$1 \times 10^{- 10}$	$1.03 \times 10^{5}$	0.57	$1 \times 10^{- 11}$	2.48	100 [81]	$2.5 \times 10^{5}$	$1 \times 10^{13}$	−22
Micro-disk [82]	$2 \times 10^{- 15}$	$2.5 \times 10^{7}$	26	$2.6 \times 10^{- 11}$	0.04	0.02 [83]	$1.9 \times 10^{4}$	$7.7 \times 10^{8}$	−3
Levitated particle [84]	$2.8 \times 10^{- 18}$	$3 \times 10^{5}$	3 [85,86]	$6.3 \times 10^{- 8}$	${0.3}^{*}$	30 [87,88]	$1.1 \times 10^{3}$	$4.8 \times 10^{8}$	−26
Photonic crystal [89]	$4 \times 10^{- 16}$	$4.9 \times 10^{6}$	$1.3 \times 10^{5}$	$3.5 \times 10^{- 6}$	10	100	$8.4 \times 10^{2}$	$2.9 \times 10^{7}$	−16
Cold atomic gases [90]	$2.4 \times 10^{- 22}$	$7 \times 10^{4}$	$3.5 \times 10^{6}$	70	$25^{*}$	2500	0.3	34.8	−26

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Baiqiang Zhu, Keye Zhang, Weiping Zhang. Optomechanical preparation of photon number-squeezed states with a pair of thermal reservoirs of opposite temperatures[J]. Photonics Research, 2023, 11(9): A26.

Optomechanical preparation of photon number-squeezed states with a pair of thermal reservoirs of opposite temperatures

Table1. Experimental Parameters and Ideal Squeezing Degrees for Several Representative Optomechanical Systems

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Optomechanical preparation of photon number-squeezed states with a pair of thermal reservoirs of opposite temperatures

Table1. Experimental Parameters and Ideal Squeezing Degrees for Several Representative Optomechanical Systems

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