Hybrid Einstein-Podolsky-Rosen steering in an atom-optomechanical system

Huatang Tan and Lihui Sun

Phys. Rev. A 92, 063812 – Published 8 December 2015

Abstract

Einstein-Podolsky-Rosen steering manifests a type of quantum correlations intermediate between entanglement and Bell nonlocality. In this paper we propose a scheme for realizing hybrid atom-mechanical quantum steering in the steady-state regime. In our scheme, an optomechanical two-mode cavity is coupled to a distant ensemble of double- $λ$ atoms in a cascade way, which induces the dissipative interaction between the mechanics and the internal atomic states. We show that strong cavity dissipation can lead to two-way atom-mechanical steering and the optimal steering exhibited in an approximate two-mode squeezed-vacuum atom-mechanical state. Moreover, we find that for the present cascade coupling scheme, the one-way steering from the mechanical oscillator to the atoms is achievable under certain conditions, while the reverse one-way steering is impossible to obtain. The reason for achieving the asymmetric steering is analyzed and the effect of mechanical thermal noise on the steering is also studied.

Received 12 June 2015
Revised 24 September 2015

DOI:https://doi.org/10.1103/PhysRevA.92.063812

Authors & Affiliations

Huatang Tan^1,* and Lihui Sun^2,3,†

¹Department of Physics, Huazhong Normal University, Wuhan 430079, China
²Center for Controlled Quantum Systems and Department of Physics and Engineering Physics, Stevens Institute of Technology, Hoboken, New Jersey 07030, USA
³Institute of Quantum Optics and Information Photonics, School of Physics and Optoelectronic Engineering, Yangtze University, Jingzhou 434023, China

^*tht@mail.ccnu.edu.cn
^†lhsun@yangtzeu.edu.cn

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Vol. 92, Iss. 6 — December 2015

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Images

Figure 1
(a) Laser-driven optomechanical two-mode cavity that is unidirectionally coupled to an ensemble of atoms in free space. The mechanical oscillator at frequency $ω_{m}$ mediates two cavity fields of $ω_{1}$ and $ω_{2}$ that are driven respectively by a red-detuned laser of frequency $ω_{l 1}$ and a blue-detuned laser at frequency $ω_{l 2}$ . The sidebands and carriers in the cavity outputs are separated and subjected to polarization modulation. (b) Carriers and sidebands are used to couple the corresponding atomic transitions. Here we assume that the atomic frequency $Ω_{a}$ is equal to the mechanical frequency $Ω_{m}$ , which can be realized by adjusting the strength of the magnetic field.
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Figure 2
Dependence of the long-time atom-mechanical entanglement $E_{N}$ , steering $S_{m | a}$ and $S_{a | m}$ , and purity $P$ of the atom-mechanical states on the cavity dissipation rate $κ$ for different coupling $g_{2}$ . The other parameters are $g_{1} = 1.2 g_{2}, γ_{1} = g_{2}^{2} / κ$ , and $γ_{2} = g_{1}^{2} / κ$ . In (a)–(c) $γ_{m} = 0$ and in (d)–(f) $γ_{m} = 2 \times 10^{- 6} Ω_{m}$ and $n_{th} = 20$ , which corresponds to the temperature $T ≃ 0.01 K$ for a mechanical oscillator with frequency $Ω_{m} / 2 π = 10 MHz$ and quality factor $Q = 5 \times 10^{5}$ .
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Figure 3
Dependence of the difference $n_{m a} = 〈 {\hat{b}}_{m}^{†} {\hat{b}}_{m} 〉 - 〈 {\hat{b}}_{a}^{†} {\hat{b}}_{a} 〉$ in the long-time regime on the cavity dissipation rate $κ$ . The other parameters are the same as in Fig. 2 for $γ_{m} = 2 \times 10^{- 6} Ω_{m}$ and $n_{th} = 20$ .
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Figure 4
Time evolution of the steering and entanglement for $g_{2} = 0.02 Ω_{m}, κ = 0.2 Ω_{m}, g_{1} = 1.2 g_{2}, γ_{1} = g_{2}^{2} / κ, γ_{2} = g_{1}^{2} / κ, γ_{m} = 2 \times 10^{- 6} Ω_{m}$ , and $n_{th} = 20$ . The thick and thin curves are plotted for initial ground states, apart from the initial mechanical thermal state with thermal phonon number $n_{t h 0} = 10$ for the thin curves.
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Figure 5
Dependence of the entanglement $E_{N}$ and steering $S_{m | a}$ and $S_{a | m}$ in the long-time regime on (a) the cavity dissipation rate $κ$ and (b) and (c) the coupling $γ_{2}$ . The parameters in (a) are the same as those in Fig. 2 for $g_{2} = 0.2 Ω_{m}$ . In (b) and (c) the other parameters are $κ = 5 g_{2}, g_{1} = 1.2 g_{2}, γ_{1} = 0.2 g_{2}, γ_{m} = 2 \times 10^{- 6} Ω_{m}$ , and $n_{th} = 20$ . The insets plot the corresponding difference $n_{m a} = 〈 {\hat{b}}_{m}^{†} {\hat{b}}_{m} 〉 - 〈 {\hat{b}}_{a}^{†} {\hat{b}}_{a} 〉$ .
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Figure 6
Dependence of the steering $S_{m | a}$ and $S_{a | m}$ in the long-time region on the environment temperature $T$ for a mechanical oscillator with frequency $Ω_{m} / 2 π = 10$ MHz and quality factor $Q_{m} = ω_{m} / γ_{m} = 5 \times 10^{5}$ . The other parameters are (a) $κ = 0.25 Ω_{m}, g_{2} = 0.02 Ω_{m}, g_{1} = 1.2 g_{2}, γ_{1} = g_{2}^{2} / κ$ , and $γ_{2} = g_{1}^{2} / κ$ and (b) $κ = Ω_{m}, g_{2} = 0.2 Ω_{m}, g_{1} = 1.2 g_{2}, γ_{1} = 0.04 Ω_{m}$ , and $γ_{2} = 0.12 Ω_{m}$ .
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