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

Journals >Photonics Research >Volume 11 >Issue 9 >Page A26 > Article

Photonics Research
Vol. 11, Issue 9, A26 (2023)

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

Baiqiang Zhu^1、2, Keye Zhang^1、2、*, and Weiping Zhang^{2、3、4、5}

Author Affiliations

¹State Key Laboratory of Precision Spectroscopy, Department of Physics, School of Physics and Electronic Science, East China Normal University, Shanghai 200062, China

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

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

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

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

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DOI: 10.1364/PRJ.491788 Cite this Article Set citation alerts

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 Copy Citation Text

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Abstract

Photon number-squeezed states are of significant value in fundamental quantum research and have a wide range of applications in quantum metrology. Most of their preparation mechanisms require precise control of quantum dynamics and are less tolerant to dissipation. We propose a mechanism that is not subject to these restraints. In contrast to common approaches, we exploit the self-balancing between two types of dissipation induced by positive- and negative-temperature reservoirs to generate steady states with sub-Poissonian statistical distributions of photon numbers. We also show how to implement this mechanism with cavity optomechanical systems. The quality of the prepared photon number-squeezed state is estimated by our theoretical model combined with realistic parameters for various typical optomechanical systems.

\dot{ρ} = - \frac{i}{ℏ} [{\hat{H}}_{a}, ρ] + D [\hat{a} \sqrt{κ_{\hat{n}}^{+}}] ρ + D [\sqrt{κ_{\hat{n}}^{-}} {\hat{a}}^{†}] ρ,

(1)

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{\dot{P}}_{n} = κ_{n + 1}^{+} (n + 1) P_{n + 1} - κ_{n}^{+} n P_{n} + κ_{n}^{-} n P_{n - 1} - κ_{n + 1}^{-} (n + 1) P_{n},

(2)

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\frac{P_{n}}{P_{n - 1}} = \frac{κ_{n}^{-}}{κ_{n}^{+}}, n = 1, 2, \dots + \infty .

(3)

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Δ n^{2} \approx {[\frac{d}{d n} {(\frac{κ_{n}^{+}}{κ_{n}^{-}}) |}_{n = \bar{n}}]}^{- 1},

(4)

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κ_{\hat{n}}^{\pm} = \frac{κ_{0}}{1 + \exp [\pm k (n_{0} - \hat{n} + \frac{1}{2})]},

(5)

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P_{n} = e^{k \sum_{i = 1}^{n} (n_{0} - i + \frac{1}{2})} P_{0} = N e^{- \frac{k}{2} {(n - n_{0})}^{2}},

(6)

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\dot{ρ} = - \frac{i}{ℏ} [{\hat{H}}_{tot}, ρ] + D [\sqrt{κ_{\hat{x}}^{+}} \hat{a}] ρ + κ^{-} D [{\hat{a}}^{†}] ρ + L_{m} ρ,

(7)

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{\hat{H}}_{tot} = ℏ (ω_{a} - G \hat{x}) {\hat{a}}^{†} \hat{a} + \frac{{\hat{p}}^{2}}{2 m} + \frac{1}{2} m ω_{m}^{2} {\hat{x}}^{2} .

(8)

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κ_{\hat{x}}^{+} = κ_{v} + \frac{κ_{0}}{1 + \exp [4 (L - \hat{x}) / d]},

(9)

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L_{m} ρ = - \frac{i γ}{2 ℏ} [\hat{x}, {\hat{p}, ρ}] - \frac{γ}{2} (n_{th} + \frac{1}{2}) [\hat{x}, [\hat{x}, ρ]],

(10)

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κ_{n}^{+} = Tr [κ_{\hat{x}}^{+} ρ_{m} (n)] \approx κ_{v} + \frac{κ_{0}}{1 + \exp [4 (L - n x_{1}) / d^{'}]},

(11)

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\bar{n} \approx \frac{L}{x_{1}} - \frac{1}{2},

(12)

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Δ n \approx \sqrt{\frac{d^{'}}{4 x_{1}}} .

(13)

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κ_{n}^{+} \approx κ_{v} + \frac{κ_{0}}{1 + \exp [4 ξ (L - n x_{1}) / d^{'}]},

(14)

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Δ n \approx \sqrt{\frac{d^{'}}{4 ξ x_{1}}},

(15)

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ξ = 1 - \frac{1}{\cosh (\sqrt{\frac{Δ n^{2} γ}{\bar{n} κ^{-}}}) + \sqrt{\frac{γ}{\bar{n} κ^{-}}} \sinh (\sqrt{\frac{Δ n^{2} γ}{\bar{n} κ^{-}}})} .

(16)

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\frac{d ⟨ \hat{n} ⟩}{d t} = κ_{\bar{n}}^{-} (⟨ \hat{n} ⟩ + 1) - κ_{\bar{n}}^{+} ⟨ \hat{n} ⟩,

(A1)

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\frac{d ⟨ {\hat{n}}^{2} ⟩}{d t} = κ_{\bar{n}}^{-} (⟨ \hat{n} ⟩ - 2 ⟨ {\hat{n}}^{2} ⟩) + κ_{\bar{n}}^{+} (2 ⟨ {\hat{n}}^{2} ⟩ + 3 ⟨ \hat{n} ⟩ + 1),

(A2)

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{\dot{P}}_{n} = ⟨ n | \dot{ρ} | n ⟩ = ⟨ n | (- \frac{i}{ℏ} [{\hat{H}}_{a}, ρ] + D [\hat{a} \sqrt{κ_{\hat{n}}^{+}}] ρ + D [\sqrt{κ_{\hat{n}}^{-}} {\hat{a}}^{†}] ρ) | n ⟩,

(B1)

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⟨ n | \hat{a} \sqrt{κ_{\hat{n}}^{+}} ρ \sqrt{κ_{\hat{n}}^{+}} {\hat{a}}^{†} | n ⟩ = κ_{n + 1}^{+} (n + 1) ⟨ n + 1 | ρ | n + 1 ⟩,

(B2)

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⟨ n | \sqrt{κ_{\hat{n}}^{+}} {\hat{a}}^{†} \hat{a} \sqrt{κ_{\hat{n}}^{+}} ρ | n ⟩ = κ_{n}^{+} n ⟨ n | ρ | n ⟩,

(B3)

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⟨ n | \sqrt{κ_{\hat{n}}^{-}} {\hat{a}}^{†} ρ \hat{a} \sqrt{κ_{\hat{n}}^{-}} | n ⟩ = κ_{n}^{-} n ⟨ n - 1 | ρ | n - 1 ⟩,

(B4)

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⟨ n | \hat{a} κ_{\hat{n}}^{-} {\hat{a}}^{†} ρ | n ⟩ = κ_{n + 1}^{-} (n + 1) ⟨ n | ρ | n ⟩ .

(B5)

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var (x) = var (θ) = - {[\partial_{θ}^{2} \ln L {(θ | x_{i}) |}_{θ = x_{0} - x_{i}}]}^{- 1} = - {[\partial_{θ}^{2} \ln P {(x_{i} + θ) |}_{θ = x_{0} - x_{i}}]}^{- 1},

(C1)

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ρ_{s} = \sum_{n} P_{n} | n ⟩ ⟨ n | \otimes ρ_{m} (n),

(D1)

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{\dot{P}}_{n} = (n + 1) P_{n + 1} κ_{n + 1}^{+} - n P_{n} κ_{n}^{+} + n P_{n - 1} κ^{-} - (n + 1) P_{n} κ^{-},

(D2)

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{\dot{ρ}}_{m} (n) = - \frac{i}{ℏ} [{\hat{H}}_{n}, ρ_{m} (n)] + L_{m} ρ_{m} (n) + \frac{P_{n + 1}}{P_{n}} (n + 1) κ_{n + 1}^{+} [ρ_{m} (n + 1) - ρ_{m} (n)] + \frac{P_{n - 1}}{P_{n}} n κ^{-} [ρ_{m} (n - 1) - ρ_{m} (n)],

(D3)

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κ_{n}^{+} = Tr [κ_{\hat{x}}^{+} ρ_{m} (n)] \approx \int {κ_{v} + \frac{κ_{0}}{1 + \exp [4 (L - x) / d]}} p_{n} (x) d x,

(D4)

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p_{n} (x) = ⟨ x | ρ_{m} (n) | x ⟩ \approx \frac{1}{\sqrt{2 π} Δ x} \exp [- \frac{{(x - x_{n})}^{2}}{2 Δ x^{2}}],

(D5)

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κ_{n}^{+} \approx κ_{v} + \frac{κ_{0}}{1 + \exp [4 (L - n x_{1}) / d^{'}]} .

(D6)

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{\dot{x}}_{n} = Tr [\hat{x} {\dot{ρ}}_{m} (n)] = - γ (x_{n} - n x_{1}) + \frac{P_{n + 1}}{P_{n}} (n + 1) κ_{n + 1}^{+} (x_{n + 1} - x_{n}) + \frac{P_{n - 1}}{P_{n}} n κ^{-} (x_{n - 1} - x_{n}),

(D7)

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ξ = 1 - \frac{1}{\cosh (\sqrt{\frac{Δ n^{2} γ}{\bar{n} κ^{-}}}) + \sqrt{\frac{γ}{\bar{n} κ^{-}}} \sinh (\sqrt{\frac{Δ n^{2} γ}{\bar{n} κ^{-}}})} .

(D8)

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κ_{n}^{+} \approx κ_{v} + \frac{κ_{0}}{1 + \exp [4 ξ (L - n x_{1}) / d^{'}]},

(D9)

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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

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