Cooling of macroscopic mechanical resonators in hybrid atom-optomechanical systems

Xi Chen, Yong-Chun Liu, Pai Peng, Yanyan Zhi, and Yun-Feng Xiao

Phys. Rev. A 92, 033841 – Published 22 September 2015

Abstract

Cooling macroscopic objects is of importance for both fundamental and applied physics. Here we study the optomechanical cooling in a hybrid system which consists of a cloud of atoms coupled to a cavity optomechanical system. On one hand, the asymmetric Fano or electromagnetically induced transparency resonance is explored and the steady-state cooling limits of resonators with frequency $ω_{m}$ are analytically obtained, permitting ground-state cooling of massive low-frequency resonators beyond the resolved sideband limit. On the other hand, due to the excitation-saturation effect, the validity of cooling requires the number of atoms to be much larger than the number of steady-state excitations, which is proportional to $ω_{m}^{- 2}$ . Thus, this limitation plays a minor role in cooling higher-frequency resonators, but becomes important for macroscopic lower-frequency resonators. Under such limitation on the number of atoms, the optimal parameters are quantified. Our study can be a guideline for both theoretical and experimental study of cooling macroscopic objects in atom-optomechanical hybrid systems.

Received 25 July 2015

DOI:https://doi.org/10.1103/PhysRevA.92.033841

Authors & Affiliations

Xi Chen¹, Yong-Chun Liu¹, Pai Peng¹, Yanyan Zhi^1,2, and Yun-Feng Xiao^1,2,*

¹State Key Laboratory for Mesoscopic Physics and School of Physics, Peking University, Beijing 100871, People's Republic of China
²Collaborative Innovation Center of Quantum Matter, Beijing 100871, People's Republic of China

^*Corresponding author: yfxiao@pku.edu.cn; www.phy.pku.edu.cn/∼yfxiao/index.html

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Vol. 92, Iss. 3 — September 2015

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Images

Figure 1
(a) An optomechanical system with a cloud of ground-state atoms coupled to an optical cavity mode. The optical mode is coherently driven by an input laser. (b) The energy-level diagram of the system in the displaced frame. $| n_{a}, n_{c}, m 〉$ represents a state of the whole system with $n_{a}$ photons, $n_{c}$ atomic excitations, and $m$ phonons. The destructive quantum interference can take place between the excitation $| 1 〉 \to | 2 〉$ and $| 1 〉 \to | 2 〉 \to | 3 〉 \to | 2 〉$ to suppress heating, similar to the constructive interference between the two cooling excitation processes, $| 1 〉 \to | 2^{'} 〉$ and $| 1 〉 \to | 2^{'} 〉 \to | 3^{'} 〉 \to | 2^{'} 〉$ , to enhance cooling.
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Figure 2
EIT and Fano line shapes of the optical force spectrum $S_{FF} (ω)$ under $| Δ_{a} | ≪ κ$ and $| Δ_{a} | > κ$ , respectively. In [(a) or (b)], the blue dashed and red solid curves represent $S_{FF}^{EIT} (ω)$ [or $S_{FF}^{Fano} (ω)]$ and $S_{FF} (ω)$ . Insets: Sketches of the cavity profile modulated by the atoms. The black dashed curves denote the cavity line shape and the green solid curves represent the line shape of the atoms. The heights of red and blue peaks at $- ω_{m}$ and $ω_{m}$ correspond to the heating and cooling rates. The parameters are $κ = 10^{5} ω_{m}, γ_{c} = 0.1 ω_{m}, G_{0} = 100 ω_{m}, G = 100 ω_{m}, Δ_{a} = 5000 ω_{m}$ , and $Δ_{c} = ω_{m}$ in (a) and $κ = 10^{4} ω_{m}, G_{0} = 200 ω_{m}, G = 100 ω_{m}, Δ_{c} = 0.3 ω_{m}$ , and $Δ_{a} = J^{2} / (Δ_{c} + ω_{m})$ in (b).
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Figure 3
(a) Analytical and (b) numerical cooling limits $n_{f}$ for $γ_{c} = 10^{- 5} ω_{m} ≪ γ_{m} n_{th}$ . (c) Ground-state cooling boundaries of $κ_{eff}$ and $G_{eff}$ for different $γ_{m} n_{th}$ . The red solid, blue dashed, and orange dot-dashed curves, along with shaded regions, correspond to $n_{f} \leq 1$ for $γ_{m} n_{t h} / ω_{m} = 10^{- 4}, 10^{- 3}$ , and $5 \times 10^{- 3}$ . The scale of the $κ_{eff}$ axis is logarithmic. Other unspecified parameters are $κ = 10^{5} ω_{m}, γ_{m} = 10^{- 7} ω_{m}, n_{th} = 10^{4}, Δ_{c} = ω_{m}$ , and $Δ_{a} = 0$ .
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Figure 4
(a) Analytical and (b) numerical cooling limits $n_{f}$ for $γ_{c} = 0.1 ω_{m}$ , which satisfies $γ_{c} ≫ γ_{m} n_{th}$ and $γ_{c} ≪ ω_{m}$ . (c) Ground-state cooling boundaries of $κ_{eff}$ and $G_{eff}$ for different $γ_{m} n_{th}$ . The red solid, blue dashed, and orange dot-dashed curves, along with shaded regions, correspond to $n_{f} \leq 1$ for $γ_{m} n_{th} / ω_{m} = 10^{- 4}, 10^{- 3}$ , and $5 \times 10^{- 3}$ . Other unspecified parameters are the same as those in Fig. 3.
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Figure 5
Maximum mass of the membrane for ground-state cooling at $T = 1 mK$ . The red solid, blue dot-dashed, and orange dashed lines are the cases of $κ / g = 10^{4}, 10^{6}$ , and $10^{8}$ . In the light-blue shaded region, ground-state cooling of the membranes of $m > 1 mg$ can be achieved.
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Figure 6
Cooling limits for different detunings. The system parameters are $κ = 10^{5} ω_{m}, γ_{m} = 10^{- 7} ω_{m}, n_{th} = 10^{5}, G_{0} = 100 ω_{m}, G = 0.2 G_{0}$ , and $γ_{c} = 0.01 ω_{m}$ . Gray regions correspond to the final phonon number $n_{f} > 5$ , which is close to the unstable region. For this case, ground-state cooling can only be achieved in the region around $Δ_{c} = ω_{m}, Δ_{a} = 0.2 κ$ , which corresponds to the parameters for an EIT line shape.
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