Entangling cavity optomechanical systems via a flying atom

Jun-Hao Liu; Yu-Bao Zhang; Ya-Fei Yu; Zhi-Ming Zhang

doi:10.1364/OE.25.007592

1. Introduction

Quantum entanglement [1] is an extraordinary phenomenon in quantum mechanics. It involves more than one system or more than one degree of freedom of a system. On one hand, quantum entanglement plays an important role in the study on the fundamental issues of quantum physics, for example, it can help us to reveal the boundary and the transition between classical word and quantum world [2–4]. On the other hand, quantum entanglement has a lot of potential applications in quantum science and technology, such as quantum computation [5, 6], quantum communication [7], quantum metrology [8], and so on [9,10]. Up to now, quantum entanglement has been realized with many physical objects, such as photons [11], atoms [12], ions [13], etc [14]. It has also been realized in many physical systems, including optical systems [15–17], cavity quantum electrodynamics (cavity QED) [18], circuit quantum electrodynamics (circuit QED) [19], ion traps [20,21] and so on [22].

Cavity optomechanics [23] is an active research field in recent years. It involves the radiation pressure force of light on objects. Cavity optomechanical systems (COMSs) provide solid platforms for exploring quantum effects in an extremely wide range from microscale to macroscale. Rich physical effects have been discovered in COMSs, for example, Kerr effect [24], bistability [25], dark mode [26], slow light [27], optomechanically induced transparency (OMIT) [28, 29], frequency conversion [30]. The COMSs can also be used to achieve the macroscopic superposition of quantum states. Bose et al. proposed a scheme to generate a large variety of nonclassical states of both the cavity field and the mechanical oscillator (the oscillating mirror) in COMS [31]. Marshall et al. performed a detailed study of the experimental requirements for the creation and observation of quantum superposition states of a mirror [32]. Meanwhile, many proposals have been brought forward to generate quantum entanglement in various COMSs [33–39]. The entanglement between mechanical modes and cavity modes [40, 41] have been proposed theoretically and demonstrated experimentally. Entanglement can also be generated between two mechanical oscillators by using a single-photon [42].

In this paper, we propose a scheme for generating entanglement among two COMSs. This paper is organized as follows. In Section II we introduce the theoretic model. In Section III we present the entanglement transfer and the entanglement swapping-like phenomenon between two COMSs in detail. In Section IV we discuss the measurement of the generated entangled states of the two COMSs, and analyze the decoherence of the generated entangled states of the two phonon modes. Finally, in Section V we discuss the experimental feasibility of our proposal and provide a brief summary.

2. Model and theory

Our proposed scheme is shown in Fig. 1, a two-level atom sequentially passes through and interacts with two standard COMSs. The atomic states are detected when the atom exits from the second COMS. In this way the entanglement of the COMSs can be generated, although there are no direct interaction between them. Now we analyze the procedures for entanglement generation in detail. The Hamiltonian for the interaction of the atom with the k-th COMS reads (ħ = 1)

H^{(k)} = \frac{ω_{a}}{2} σ_{z} + ω_{c} a_{k}^{†} a_{k} + ω_{m} b_{k}^{†} b_{k} - i G (a_{k} σ_{+} - a_{k}^{+} σ_{-}) - g a_{k}^{†} a_{k} (b_{k}^{†} + b_{k}),

where σ_z, σ₊, and σ₋ are the Pauli operators of the two-level atom with frequency ω_a, a_k (

a_{k}^{†}

) and b_k (

b_{k}^{†}

) are the annihilation (creation) operators of the cavity mode and the mechanical mode of the k-th COMS, respectively. For simplicity, we assume that the two COMSs are identical, i.e., ω_c (the cavity frequency), ω_m (the mechanical frequency), G (the Jaynes-Cummings coupling strength between the atom and the cavity field), and g (the single-photon optomechanical coupling strength between the cavity field and the mechanical mode) are the same for the two COMSs. We first make a transformation to the Hamiltonian with

U_{k} = exp [- i H_{0}^{(k)} t]

in which

H_{0}^{(k)} = \frac{ω_{a}}{2} σ_{z} + ω_{c} a_{k}^{†} a_{k}

, the Hamiltonian H^(k) can be transformed as

\begin{array}{l} H_{int}^{(k)} & = & i \frac{d U_{k}^{†}}{d t} U_{k} + U_{k}^{†} H^{(k)} U_{k} \\ = & ω_{m} b_{k}^{†} b_{k} - i G (a_{k} σ_{+} e^{i Δ t} - a_{k}^{†} σ_{-} e^{- i Δ t}) - g a_{k}^{†} a_{k} (b_{k}^{†} + b_{k}), \end{array}

where Δ = ω_a − ω_c is the frequency detuning between the atom and the cavity field. The first term of

H_{int}^{(k)}

describes the free Hamiltonian of the k-th mechanical mode. The second term describes the Jaynes-Cummings interaction between the atom with the k-th cavity field, and the third term describes the photon-phonon interaction of the k-th COMS. In this scheme, the atom interacts with each cavity field only for a period of time. Here we assume that the Jaynes-Cummings coupling strength is much larger than the optomechanical coupling strength, i.e., G ≫ g. Under this condition, we can ignore the photon-phonon interaction in (2) during the period when the atom passes through and interacts with the cavity. Then when the atom is in the k-th cavity, we can use the simplified Hamiltonian

H_{int}^{(k)} \approx H_{JC}^{(k)} = ω_{m} b_{k}^{†} b_{k} - i G (a_{k} σ_{+} - a_{k}^{†} σ_{-}),

and the corresponding time evolution operator

U_{JC}^{(k)} (t) = e^{- i H_{JC}^{(k)} t} = e^{- i ω_{m} b_{k}^{†} b_{k} t} e^{- G (a_{k} σ_{+} - a_{k}^{†} σ_{-}) t}

, in which we have assumed Δ = 0 for simplicity. When the atom leaves the cavity, there is no interaction between the atom and the COMSs, the Hamiltonian

H_{int}^{(k)}

(3) then becomes

H_{int}^{(k)} = H_{OM}^{(k)} = ω_{m} b_{k}^{†} b_{k} - g a_{k}^{†} a_{k} (b_{k}^{†} + b_{k}),

and the corresponding time evolution operator can be derived as [31]

U_{OM}^{(k)} (t) = e^{- i H_{OM}^{(k)} t} = e^{i Θ (t) {(a_{k}^{†} a_{k})}^{2}} D (δ) e^{- i ω_{m} t b_{k}^{†} b_{k}}

, where D(δ) = e^{δb^†−δ^*b} is the displacement operator in which

δ = a_{k}^{†} a_{k} β η (t)

, β = g/ω_m, η = 1 − e^−iω_mt, and Θ(t) = β²[ω_mt − sin(ω_mt)].

Fig. 1 Schematic diagram of the system. A two-level atom sequentially passes through two cavity optomechanical systems (COMSs), and its states are detected when it exits from the second COMS. In this way the two COMSs can be entangled.

Download Full Size | PDF

The state of the whole system can be represented as |Ψ〉 = |A〉 |i〉_c₁ |j〉_m₁ |i〉_c₂ |j〉_m₂, where |A〉 ≡ |e〉 (|g〉) describes the excited (ground) state of the two-level atom, |i〉_c₁ (|i〉_c₂) and |j〉_m₁ (|j〉_m₂) represent the quantum state of the first (second) cavity field and the first (second) mechanical mode, respectively. Initially we prepare the system in the state

| Ψ_{0} 〉 = | e 〉 {| 0 〉}_{c_{1}} {| α 〉}_{m_{1}} {| 0 〉}_{c_{2}} {| α 〉}_{m_{2}},

i.e., the two cavity fields are both in the vacuum states {|0〉}, the mechanical modes are both in the coherent states {|α〉}, and the atom is in the exited state |e〉.

3. Entanglement between two cavity optomechanical systems

Now we discuss the entanglement of two COMSs in detail. As shown in Fig. 2, when the atom enters the first COMS, it will interact with the cavity field according to $U_{JC}^{(1)} (t)$ , meanwhile, the second COMS will evolve according to $U_{OM}^{(2)} (t)$ , therefore, the state of the system will evolve as follows

\begin{array}{l} | Ψ (t_{1}) 〉 & = & U_{JC}^{(1)} (τ_{1}) U_{OM}^{(2)} (τ_{1}) | Ψ_{0} 〉 \\ = & {sin (G τ_{1}) | g 〉 {| 1 〉}_{c_{1}} + cos (G τ_{1}) | e 〉 {| 0 〉}_{c_{1}}} {| α γ (τ_{1}) 〉}_{m_{1}} {| 0 〉}_{c_{2}} {| α γ (τ_{1}) 〉}_{m_{2}}, \end{array}

where τ₁ = t₁ − t₀, γ(τ) = e^−iω_mτ. It can be seen that the atom and cavity 1 evolve into an entangled state, the two mechanical modes evolve into the coherent state |αγ(τ₁)〉, while cavity 2 remains in the vacuum state. Obviously, the free evolution of the mechanical mode has periodicity and the period is T_m = 2π/ω_m. After the atom flies for an integer number of this period, the mirror will evolve to its original state |α〉. We assume that the atom leaves cavity 1 at time t₁, and τ₁ = t₁ − t₀ satisfies the conditions τ₁ = T_m and Gτ₁ = π/4, then the state of the system becomes

| Ψ (t_{1}) 〉 = \frac{1}{\sqrt{2}} (| g 〉 {| 1 〉}_{c_{1}} + | e 〉 {| 0 〉}_{c_{1}}) {| α 〉}_{m_{1}} {| 0 〉}_{c_{2}} {| α 〉}_{m_{2}} .

It can be seen that, at this moment, the atom and cavity 1 are in a maximum entangled state, while cavity 2 and the two mechanical modes return to their initial states. After the atom leaves the first COMS and before it enters into the second COMS, it does not interact with the two COMSs and flies freely, while the two COMSs evolve according to

U_{OM}^{(1)} (t)

and

U_{OM}^{(2)} (t)

, respectively. At time t₂ (at which the atom enters into COMS 2), the system will evolve into the state

\begin{array}{l} | Ψ (t_{2}) 〉 & = & U_{OM}^{(1)} (τ_{2}) U_{OM}^{(2)} (τ_{2}) | Ψ (t_{1}) 〉 \\ = & \frac{1}{\sqrt{2}} {e^{i θ_{+} (τ_{2})} | g 〉 {| 1 〉}_{c_{1}} {| ϕ_{+} (τ_{2}) 〉}_{m_{1}} + | e 〉 {| 0 〉}_{c_{1}} {| α γ (τ_{2}) 〉}_{m_{1}}} {| 0 〉}_{c_{2}} {| α γ (τ_{2}) 〉}_{m_{2}}, \end{array}

where the phase factor θ₊(τ) = Θ(τ) + Γ(τ), Γ(τ) = Im[βα^*(e^iω_mτ − 1)], ϕ₊(τ) = αγ(τ) + βη(τ), and τ₂ = t₂ − t₁ is the time during which the atom flies freely between the two COMSs.

Fig. 2 The temporal sequence of entangling two optomechanical systems. $U_{JC}^{(k)}$ and $U_{OM}^{(k)}$ are the time evolution operators defined in the text. The insets above time points represent the position of the atom.

Download Full Size | PDF

3.1. Entanglement transfer

Here we assume that τ₂ = T_m/2. In this case, the state |Ψ(t₂)〉 in (8) becomes

\begin{array}{l} | Ψ_{T} (t_{2}) 〉 & = & U_{OM}^{(1)} (\frac{T_{m}}{2}) U_{OM}^{(2)} (\frac{T_{m}}{2}) | Ψ (t_{1}) 〉 \\ = & \frac{1}{\sqrt{2}} (e^{i Λ_{-}} | g 〉 {| 1 〉}_{c_{1}} {| 2 β - α 〉}_{m_{1}} + | e 〉 {| 0 〉}_{c_{1}} {| - α 〉}_{m_{1}}) {| 0 〉}_{c_{2}} {| - α 〉}_{m_{2}}, \end{array}

where Λ₋ = β²π − Im(2βα^*). It can be seen that, at this moment, the atom, cavity 1 and mechanical mode 1 are in a tripartite entangled state. At this time, the atom enters into the second COMS and interacts with cavity 2 according to

U_{JC}^{(2)} (t)

, meanwhile, the first COMS continue to evolve according to

U_{OM}^{(1)} (t)

. We assume that the atom leaves cavity 2 at time t₃, and the whole system will evolve to

\begin{array}{l} | Ψ_{T} (t_{3}) 〉 & = & U_{OM}^{(1)} (τ_{3}) U_{JC}^{(2)} (τ_{3}) Ψ_{T} (t_{2}) \\ = & \frac{1}{\sqrt{2}} {sin (G τ_{3}) | g 〉 {| 1 〉}_{c_{2}} + cos (G τ_{3}) | e 〉 {| 0 〉}_{c_{2}}} {| 0 〉}_{c_{1}} {| - α γ (τ_{3}) 〉}_{m_{1}} {| - α γ (τ_{3}) 〉}_{m_{2}} \\ + | g 〉 \frac{1}{\sqrt{2}} e^{i [Λ_{-} + θ^{'} (τ_{3})]} {| 1 〉}_{c_{1}} {| ξ (τ_{3}) 〉}_{m_{1}} {| 0 〉}_{c_{2}} {| - α γ (τ_{3}) 〉}_{m_{2}}, \end{array}

where τ₃ = t₃ − t₂, θ′(τ) = Θ(τ) + Γ′(τ), Γ′(τ) = Im[(2β² − βα^*)(e^iω_mτ − 1)], ξ(τ) = (2β − α)γ(τ) + βη(τ). As the two COMSs are identical, the time interval τ₃, during which the atom passes through the second COMS, should be the same as that of the atom passing through the first COMS, i.e., τ₃ = τ₁ = T_m, and Gτ₃ = Gτ₁ = π/4. Substituting these conditions into (10) we obtain

\begin{array}{l} | Ψ_{T} (t_{3}) 〉 & = & | g 〉 {\frac{1}{\sqrt{2}} e^{i [Λ_{-} + 2 π β^{2}]} {| 1 〉}_{c_{1}} {| 2 β - α 〉}_{m_{1}} {| 0 〉}_{c_{2}} {| - α 〉}_{m_{2}} \\ + \frac{1}{2} {| 1 〉}_{c_{1}} {| - α 〉}_{m_{2}} {| 0 〉}_{c_{2}} {| - α 〉}_{m_{1}}} + \frac{1}{2} | e 〉 {| 0 〉}_{c_{1}} {| - α 〉}_{m_{1}} {| 0 〉}_{c_{2}} {| - α 〉}_{m_{2}}, \end{array}

when the atom leaves cavity 2, the two COMSs will evolve according to

U_{OM}^{(1)} (t)

and

U_{OM}^{(2)} (t)

again, the time evolution of the whole system can be obtained as

\begin{array}{l} | Ψ_{T} (t_{d}) 〉 & = & U_{OM}^{(1)} (τ_{d}) U_{OM}^{(2)} (τ_{d}) | ψ_{T} (t_{3}) 〉 \\ = & | g 〉 {\frac{1}{\sqrt{2}} e^{i [Λ_{-} + 2 π β^{2} + θ^{'} (τ_{d})]} {| 1 〉}_{c_{1}} {| ξ (τ_{d}) 〉}_{m_{1}} {| 0 〉}_{c_{2}} {| - α γ (τ_{d}) 〉}_{m_{2}} \\ + \frac{1}{2} e^{i θ_{-} (τ_{d})} {| 0 〉}_{c_{2}} {| - α γ (τ_{d}) 〉}_{m_{1}} {| 1 〉}_{c_{1}} {| ϕ_{-} (τ_{d}) 〉}_{m_{2}}} \\ + \frac{1}{2} | e 〉 {| 0 〉}_{c_{1}} {| - α γ (τ_{d}) 〉}_{m_{1}} {| 0 〉}_{c_{2}} {| - α γ (τ_{d}) 〉}_{m_{2}}, \end{array}

where θ₋(τ) = Θ(τ) − Γ(τ), ϕ₋(τ) = −αγ(τ) + βη(τ), τ_d = t_d − t₃ is the time interval after the atom leaves COMS 2. Now if we detect the atomic state at τ_d = T_m/2, then depending on the measurement result, there are two possible states of the two COMSs. If the detecting result is |e〉, we will get a product state |0〉_c₁ |α〉_m₁ |0〉_c₂ |α〉_m₂, indicating that there is no entanglement in this system. However, if the detecting result is |g〉, then the two COMSs will collapse to the state

| ψ_{T} (\frac{T_{m}}{2} + t_{3}) 〉 = (\sqrt{\frac{2}{3}} e^{i 4 π β^{2}} {| 1 〉}_{c_{1}} {| 0 〉}_{c_{2}} {| α 〉}_{m_{2}} + \sqrt{\frac{2}{3}} e^{i Λ_{+}} {| 0 〉}_{c_{1}} {| 1 〉}_{c_{2}} {| α + 2 β 〉}_{m_{2}}) {| α 〉}_{m_{1}},

where Λ₊ = β²π + Im(2βα^*). It can be seen that cavity 1, cavity 2 and mechanical mode 2 are in a tripartite entangled state, while mechanical mode 1 does not entangle with them. Now the two COMSs will continue to evolve according to

U_{OM}^{(1)} (t)

and

U_{OM}^{(2)} (t)

, and after T_m/2, the state of the system is

\begin{array}{l} | ψ_{T} (T_{m} + t_{3}) 〉 & = & U_{OM}^{(1)} (T_{M} / 2) U_{OM}^{(2)} (T_{m} / 2) | ψ_{T} (\frac{T_{m}}{2} + t_{3}) 〉 \\ = & (\sqrt{\frac{2}{3}} e^{i Λ_{-}} e^{i 4 π β^{2}} {| 1 〉}_{c_{1}} {| 2 β - α 〉}_{m_{1}} {| 0 〉}_{c_{2}} \\ + \sqrt{\frac{1}{3}} e^{i 2 π β^{2}} {| 0 〉}_{c_{1}} {| - α) 〉}_{m_{1}} | 1_{c_{2}} 〉) {| - α 〉}_{m_{2}}, \end{array}

now the cavity 1, cavity 2 and mechanical mode 1 are in a tripartite entangled state, while mechanical mode 2 does not entangle with them. We also calculate the situations of after T_m, 3T_m/2... and find that the mechanical mode 1 and 2 entangle with the two cavity modes alternatively.

3.2. Entanglement swapping-like process

Now we assume that τ₂ = T_m. In this case, the state |Ψ(t₂)〉 in (8) becomes

\begin{array}{l} | Ψ_{S} (t_{2}) 〉 & = & U_{OM}^{(1)} (T_{m}) U_{OM}^{(2)} (T_{m}) | Ψ (t_{1}) 〉 \\ = & \frac{1}{\sqrt{2}} (e^{i 2 π β^{2}} | g 〉 {| 1 〉}_{c_{1}} + | e 〉 {| 0 〉}_{c_{1}} {| α 〉}_{m_{1}}) {| 0 〉}_{c_{2}} {| α 〉}_{m_{2}}, \end{array}

similarly, during the time interval τ₃ = t₃ − t₂ the atom passes through COMS 2, the time evolution is

U_{OM}^{(1)} (t) U_{JC}^{(2)} (t)

, then the whole system will evolve to the state

\begin{array}{l} | Ψ_{S} (t_{3}) 〉 & = & U_{OM}^{(1)} (τ_{3}) U_{JC}^{(2)} (τ_{3}) | Ψ_{S} (t_{2}) 〉 \\ = & \frac{1}{\sqrt{2}} {cos (G τ_{3}) | e 〉 {| 0 〉}_{c_{2}} + sin (G τ_{3}) | g 〉 {| 1 〉}_{c_{2}}} {| α γ (τ_{3}) 〉}_{m_{2}} {| 0 〉}_{c_{1}} {| α γ (τ_{3}) 〉}_{m_{1}} \\ + \frac{1}{\sqrt{2}} e^{i [2 π β^{2} + θ_{+} (τ_{3})]} | g 〉 {| 1 〉}_{c_{1}} {| ϕ_{+} (τ_{3}) 〉}_{m_{1}} {| 0 〉}_{c_{2}} {| α γ (τ_{3}) 〉}_{m_{2}}, \end{array}

here we also have τ₃ = T_m and Gτ₃ = π/4, then (16) becomes

| Ψ_{S} (t_{3}) 〉 = {| g 〉 (\frac{1}{\sqrt{2}} e^{i 4 π β^{2}} {| 1 〉}_{c_{1}} {| 0 〉}_{c_{2}} + \frac{1}{2} {| 0 〉}_{c_{1}} {| 1 〉}_{c_{2}}) + \frac{1}{2} | e 〉 {| 0 〉}_{c_{1}} {| 0 〉}_{c_{2}}} {| α 〉}_{m_{1}} {| α 〉}_{m_{2}},

and after the atom leaves COMS 2, the two COMSs evolve according to

U_{OM}^{(1)} (t)

and

U_{OM}^{(2)} (t)

, respectively, the state of the whole system evolves as follows

\begin{array}{l} | Ψ_{S} (t_{d}) 〉 & = & U_{OM}^{(1)} (τ_{d}) U_{OM}^{(2)} (τ_{d}) | Ψ_{S} (t_{3}) 〉 \\ = & | g 〉 {\frac{1}{\sqrt{2}} e^{i [4 π β^{2} + θ_{+} (τ_{d})]} {| 1 〉}_{c_{1}} {| ϕ_{+} (τ_{d}) 〉}_{m_{1}} {| 0 〉}_{c_{2}} {| α γ (τ_{d}) 〉}_{m_{2}} \\ + \frac{1}{2} e^{i θ_{+} (τ_{d})} {| 0 〉}_{c_{1}} {| α γ (τ_{d}) 〉}_{m_{1}} {| 1 〉}_{c_{2}} {| ϕ_{+} (τ_{d}) 〉}_{m_{2}}} \\ + \frac{1}{2} | e 〉 {| 0 〉}_{c_{1}} {| α γ (τ_{d}) 〉}_{m_{1}} {| 0 〉}_{c_{2}} {| α γ (τ_{d}) 〉}_{m_{2}} . \end{array}

We detect the atomic state when τ_d = T_m. If the detecting result is |g〉, then the two COMSs will collapse to the state

| ψ_{S} (T_{m} / 2 + t_{3}) 〉 = (\sqrt{\frac{2}{3}} e^{i 6 π β^{2}} {| 1 〉}_{c_{1}} {| 0 〉}_{c_{2}} + \sqrt{\frac{1}{3}} e^{i 2 π β^{2}} {| 0 〉}_{c_{1}} {| 1 〉}_{c_{2}}) {| α 〉}_{m_{1}} {| α 〉}_{m_{2}} .

Now the two COMSs will continue to evolve according to

U_{OM}^{(1)} (t)

and

U_{OM}^{(2)} (t)

, and after T_m/2, the state of the system is

\begin{array}{l} | ψ_{S} (T_{m} + t_{3}) 〉 & = & U_{OM}^{(1)} (T_{m} / 2) U_{OM}^{(2)} (T_{m} / 2) | ψ_{S} (\frac{T_{m}}{2} + t_{3}) 〉 \\ = & \sqrt{\frac{2}{3}} e^{i (6 π β^{2} + Λ_{-})} {| 1 〉}_{c_{1}} {| 2 β - α 〉}_{m_{1}} {| 0 〉}_{c_{2}} {| - α 〉}_{m_{2}} \\ + \sqrt{\frac{1}{3}} e^{i (2 π β^{2} + Λ_{-})} {| 0 〉}_{c_{1}} {| - α 〉}_{m_{1}} {| 1 〉}_{c_{2}} {| 2 β - α 〉}_{m_{2}} . \end{array}

From (19) and (20), it can be seen that in our process we achieve the entanglement of the two mechanical mirrors after they respectively interact with the entangled cavity modes, even if they do not interact with each other. There is also no need for a measurement operation in our process. This is different from the entanglement swapping proposed by M. Żukowski et al. [43], so we name our process the entanglement swapping-like process.

It can be seen, from above discussions, that there are two important time-intervals in our scheme, one is that during which the atom passes through and interacts with a cavity, and the other is that during which the atom flies freely between two cavities. These two time-intervals can be controlled by adjusting the atomic velocity and/or the distance between two adjacent cavities.

4. Measurement and decoherence

We now discuss how to verify the generated entangled states of the two COMSs. For the system which contains two cavities, we can use flying atoms to realize the verification [42]. For example, for the entangled state (20), we assume that two measuring atoms, in their ground states |g〉₁ and |g〉₂, respectively fly through COMS 1 and COMS 2. After the measuring atoms leave the COMSs, the state of the system can be expressed as

\begin{array}{l} | ψ_{D} 〉 & = & U_{JC}^{(1)} (2 T_{m}) U_{JC}^{(2)} (2 T_{m}) {| g 〉}_{1} {| g 〉}_{2} | ψ_{S} (T_{m} + t_{3}) 〉 \\ = & (- \sqrt{\frac{2}{3}} e^{i (6 π β^{2} + Λ_{-})} {| e 〉}_{1} {| g 〉}_{2} {| 2 β - α 〉}_{m_{1}} {| - α 〉}_{m_{2}} \\ - \sqrt{\frac{1}{3}} e^{i (2 π β^{2} + Λ_{-})} | g_{1} 〉 {| e 〉}_{2} {| - α 〉}_{m_{1}} {| 2 β - α 〉}_{m_{2}}) {| 0 〉}_{c_{1}} {| 0 〉}_{c_{2}} . \end{array}

Since the photon has been absorbed by the atom, the two cavities will be in the vacuum states. Therefore, the two mechanical modes will experience free evolution in subsequent dynamics. We detect the states of the measuring atoms after they leave the COMSs. If one atom is found to be in the excited state, then according (21) and (20), the corresponding COMS will be in |1〉_c |2β − α〉_m, the other COMS will be in |0〉_c |−α〉_m. The probabilities of detecting the excited atom that passes through COMS 1 and COMS 2 are 2/3 and 1/3, respectively.

We can obtain the entangled states of the two mechanical mirrors by applying a π/2 Ramsey pulse to the measuring atoms before detecting the atomic states. The π/2 Ramsey pulse transforms the atomic states as $| g 〉 \to (| g 〉 + | e 〉) / \sqrt{2}$ , and $| e 〉 \to (- | g 〉 + | e 〉) / \sqrt{2}$ , then the state |ψ_D〉 becomes the following state (not normalized)

| ψ_{D}^{'} 〉 = | g_{1} 〉 | g_{2} 〉 | D_{+} 〉 + {| e 〉}_{1} | e_{2} 〉 | D_{-} 〉 + | g_{1} 〉 {| e 〉}_{2} | D_{+}^{'} 〉 + {| e 〉}_{1} {| g 〉}_{2} | D_{-}^{'} 〉

If the detected atomic states are |g〉₁ |g〉₂ or |e〉₁ |e〉₂, the two mechanical modes will collapse to the states

| D_{\pm} 〉 = \pm ℳ (\sqrt{\frac{1}{6}} e^{i (6 π β^{2} + Λ_{-})} {| 2 β - α 〉}_{m_{1}} {| - α 〉}_{m_{2}} + \sqrt{\frac{1}{12}} e^{i (2 π β^{2} + Λ_{-})} {| - α 〉}_{m_{1}} {| 2 β - α 〉}_{m_{2}}),

where the normalization constant

ℳ = {[1 / 4 + \frac{\sqrt{2}}{6} e^{- 4 β^{2}} cos (4 π β^{2})]}^{- 1 / 2}

, and the + and − signs correspond to the detected atomic states |g〉₁ |g〉₂ and |e〉₁ |e〉₂, respectively. If the detected atomic states are |g〉₁ |e〉₂ or |e〉₁ |g〉₂, the two mechanical modes will collapse to the states

| D_{\pm}^{'} 〉 = \pm ℳ^{'} (\sqrt{\frac{1}{6}} e^{i (6 π β^{2} + Λ_{-})} {| 2 β - α 〉}_{m_{1}} {| - α 〉}_{m_{2}} - \sqrt{\frac{1}{12}} e^{i (2 π β^{2} + Λ_{-})} {| - α 〉}_{m_{1}} {| 2 β - α 〉}_{m_{2}}),

where the normalization constant

ℳ^{'} = {[1 / 4 - \frac{\sqrt{2}}{6} e^{- 4 β^{2}} cos (4 π β^{2})]}^{- 1 / 2}

, and the + and − signs correspond to the detected atomic states |g〉₁ |e〉₂ and |e〉₁ |g〉₂, respectively.

All of the four states are the entangled coherent states of the two mechanical mirrors. The degree of entanglement can be measured with the concurrence [44]. We can obtain the concurrence of the four states as

C = \frac{\sqrt{2} 𝒩^{2}}{6} (1 - e^{- 4 β^{2}}),

where 𝒩 = ℳ for the states (23) and 𝒩 = ℳ′ for the states (24). Fig. 3 shows the variation of the concurrence C versus the parameter β. Since G ∼ ω_m, G ≫ g, β = g/ω_m, we have β ≪ 1. It can be seen that, in the considered region (0 to 0.1), the concurrence C increases with the increace of β, and the concurrence of the states |D′_±〉 is greater than the concurrence of the states |D_±〉 for a given β.

Fig. 3 The variation of the concurrence C versus the parameter β.

Download Full Size | PDF

In above discussion we do not take into account the dissipation and decoherence in the processes of the state-generation. Now we discuss the dissipation and decoherence after the states have been generated. For simplicity we only consider the dissipation and decoherence of the states (23) and (24) of the two mechanical mirrors. We assume that the two mechanical mirrors are coupled to their thermal bath at zero temperature, then the dissipation and decoherence can be described by the following master Eq. [45]

\frac{d ρ}{d t} = - i ω_{m} \sum_{k = 1}^{2} [b_{k}^{†} b_{k}, ρ] + \frac{γ_{m}}{2} \sum_{k = 1}^{2} (2 b_{k} ρ b_{k}^{†} - b_{k}^{†} b_{k} ρ - ρ b_{k}^{+} b_{k}),

where γ_m is the decay rate of the mechanical mirrors. We analyze the dissipation and decoherence process beginning from the state |D₊〉, i.e., we consider the initial state ρ(0) = |D₊〉〈D₊|. Then according to Eqs. (26) and (23), the density operator at time t will be (see Appendix)

\begin{array}{l} ρ (t) & = & ℳ^{2} {\frac{1}{6} {| ϑ^{'} (t) 〉}_{m_{1}} 〈 ϑ^{'} (t) | \otimes {| ϑ (t) 〉}_{m_{2}} 〈 ϑ (t) | \\ + \frac{1}{12} {| ϑ (t) 〉}_{m_{1}} 〈 ϑ (t) | \otimes {| ϑ^{'} (t) 〉}_{m_{2}} 〈 ϑ^{'} (t) | \\ + W (t) [\frac{\sqrt{2}}{12} e^{i 4 π β^{2}} {| ϑ^{'} (t) 〉}_{m_{1}} 〈 ϑ (t) | \otimes {| ϑ (t) 〉}_{m_{2}} 〈 ϑ^{'} (t) | \\ + \frac{\sqrt{2}}{12} e^{- i 4 π β^{2}} {| ϑ (t) 〉}_{m_{1}} 〈 ϑ^{'} (t) | \otimes {| ϑ^{'} (t) 〉}_{m_{2}} 〈 ϑ (t) |]}, \end{array}

where ϑ(t) = −αe^{−(iω_m+γ_m/2)t}, ϑ′(t) = (2β − α)e^{−(iω_m+γ_m/2)t}, W(t) = exp[−4β²(1 − e^−γ_mt)]. When t is small, W(t) ≈ exp(−4β²γ_mt), i.e., the decoherence rate is γ_decoh = 4β²γ_m, the energy decay rate is γ_decay = γ_m. Since β ≪ 1, therefore, γ_decoh < γ_decay, i.e., in this case, the decoherence rate is smaller than the energy decay rate.

5. Discussion and conclusion

Now we discuss the experimental feasibility of our scheme. We consider an ideal case for the entanglement generation process, in which the loss of the system is neglected. The relation between the atom-cavity coupling strength G and the cavity transit time τ₁ of the atom is Gτ₁ = π/4. We can neglect the loss during τ₁ when τ₁ ≪ 1/γ_c, 1/γ_a (γ_c and γ_a are the decay rates of the cavity and atom), i.e., G ≫ γ_c, γ_a, this is the atom-cavity strong coupling regime. The temporal separation of the cavities is τ₂ ∼ 2π/ω_m, we also need τ₂ ≪ 1/γ_c (in general, γ_m ≪ γ_c), i.e., ω_m ≫ γ_c, this is the resolved sideband condition, which has been achieved in some optomechanical systems [46, 47]. From τ₁ = π/4G and τ₁ = 2π/ω_m, we have G ∼ ω_m. Since we assume that G ≫ g, so we have ω_m ≫ g. There is no a requirement for the ratio of g to γ_c in our scheme.

In summary, we propose a scheme for generating the entangled states of cavity optomechanical systems (COMSs) via a flying two-level atom. We derive the analytical expressions for the generated entangled states. We find that by adjusting the distance between two COMSs, we have two ways to generate the the entangled states, one is the entanglement transfer between the two phonon-modes, and the other is the entanglement swapping-like phenomenon among the two photon-modes and two phonon-modes. Then we discuss the verification of the generated entangled states of the two COMSs, and we analyze the decoherence of the entangled states of the two mechanical mirrors. Our study paves the way to achieve entanglements involving macroscopic objects.

Appendix: Solving the master equation

The Eq. (26) can be rewritten as

\frac{d ρ}{d t} = - i ω_{m} \sum_{k = 1}^{2} [b_{k}^{†} b_{k}, ρ] + \sum_{k = 1}^{2} (J_{k} ρ + L_{k} ρ),

where the super-operators

J_{k} = γ_{m} b_{k} ρ b_{k}^{†}

,

L_{k} = - \frac{γ_{m}}{2} (b_{k}^{†} b_{k} ρ + ρ b_{k}^{†} b_{k})

. Equation (28) has the formal solution [48]

ρ (t) = exp [\sum_{k = 1}^{2} (J_{k} + L_{k} + F_{k}) t] ρ (0),

where

F_{k} ρ = - i ω_{m} [b_{k}^{+} b_{k}, ρ]

. The decoherence process can be divided into infinite time-segments, the quantum state of the system at moment t can be expressed as

ρ (t) = lim_{N \to \infty} \prod_{j = 1}^{N} {exp [\sum_{k = 1}^{2} (J_{k} + L_{k} + F_{k}) Δ t_{j}]} ρ (0) .

When N → ∞, Δt_j → 0, we have

ρ (t) = lim_{N \to \infty} \prod_{j = 1}^{N} {exp [\sum_{k = 1}^{2} (J_{k} + L_{k}) Δ t_{j}] exp [\sum_{k = 1}^{2} F_{k} Δ t_{j}]} ρ (0) .

Now we consider the initial state ρ(0) = |D₊〉〈D₊|. For simplicity we take Δt_j = Δt = t/N, t_j = jΔt, and using the equation

exp [(J_{k} + L_{k}) Δ t] {| 2 β - α 〉}_{m_{k}} 〈 - α | = {(〈 - α | {| 2 β - α 〉}_{m_{k}})}^{Q} {| (2 β - α) B_{1} 〉}_{m_{k}} 〈 - α B_{1} |,

where Q = 1 − e^−γ_mΔt, B₁ = e^−γ_mΔt/2, we can obtain the state of the system at moment t₁

\begin{array}{l} ρ (t_{1}) & = & exp [\sum_{k = 1}^{2} (J_{k} + L_{k}) Δ t] exp [\sum_{k = 1}^{2} F_{k} Δ t] ρ (0) \\ = & ℳ^{2} {\frac{1}{6} {| η^{'} (Δ t) 〉}_{m_{1}} 〈 η^{'} (Δ t) | \otimes {| η (Δ t) 〉}_{m_{2}} 〈 η (Δ t) | \\ + \frac{1}{12} {| η (Δ t) 〉}_{m_{1}} 〈 η (Δ t) | \otimes {| η^{'} (Δ t) 〉}_{m_{2}} 〈 η^{'} (Δ t) | \\ + w_{1} [\frac{\sqrt{2}}{12} e^{i 4 π β^{2}} {| η^{'} (Δ t) 〉}_{m_{1}} 〈 η (Δ t) | \otimes {| η (Δ t) 〉}_{m_{2}} 〈 η^{'} (Δ t) | \\ + \frac{\sqrt{2}}{12} e^{- i 4 π β^{2}} {| η (Δ t) 〉}_{m_{1}} 〈 η^{'} (Δ t) | \otimes {| η^{'} (Δ t) 〉}_{m_{2}} 〈 η (Δ t) |]}, \end{array}

where η(Δt) = −B₁O₁α, η′(Δt) = B₁O₁(2β − α), O₁ = e^−iω_mΔt, and

w_{1} = {| 〈 η^{'} (Δ t) | | η (Δ t) 〉 |}^{2 Q} = exp [- 4 β^{2} (1 - e^{- γ_{m} Δ t})] .

We have

O_{2} = O_{3} = \dots = O_{N} = e^{i ω_{m} Δ t},

B_{2} = B_{3} = \dots = B_{N} = e^{- γ_{m} Δ t / 2},

and

\begin{array}{l} w_{2} & = & exp [- 4 β^{2} (1 - e^{- γ_{m} Δ t}) e^{- γ_{m} Δ t}], \\ w_{3} & = & exp [- 4 β^{2} (1 - e^{- γ_{m} Δ t}) e^{- 2 γ_{m} Δ t}], \\ \dots \\ w_{N} & = & exp [- 4 β^{2} (1 - e^{- γ_{m} Δ t}) e^{- (N - 1) γ_{m} Δ t}], \end{array}

then after the infinitesimal evolution for N times, the state of the system at moment t is

\begin{array}{l} ρ (t) & = & ℳ^{2} {\frac{1}{6} {| λ^{'} (t) 〉}_{m_{1}} 〈 λ^{'} (t) | \otimes {| λ (t) 〉}_{m_{2}} 〈 λ (t) | \\ + \frac{1}{12} {| λ (t) 〉}_{m_{1}} 〈 λ (t) | \otimes {| λ^{'} (t) 〉}_{m_{2}} 〈 λ^{'} (t) | \\ + w (t) [\frac{\sqrt{2}}{12} e^{i 4 π β^{2}} {| λ^{'} (t) 〉}_{m_{1}} 〈 λ (t) | \otimes {| λ (t) 〉}_{m_{2}} 〈 λ^{'} (t) | \\ + \frac{\sqrt{2}}{12} e^{- i 4 π β^{2}} {| λ (t) 〉}_{m_{1}} 〈 λ^{'} (t) | \otimes {| λ^{'} (t) 〉}_{m_{2}} 〈 λ (t) |]}, \end{array}

where λ(t) = −BOα, λ′(t) = BO(2β − α), and

O = O_{1} \times O_{2} \times \dots \times O_{N} = e^{i ω_{m} t},

B = B_{1} \times B_{2} \times \dots \times B_{N} = e^{- γ_{m} t / 2},

\begin{array}{l} w (t) & = & lim_{N \to \infty} w_{1} (t) \times w_{2} (t) \dots w_{N} (t) \\ = & lim_{N \to \infty} exp [- 4 β^{2} (1 - e^{- γ_{m} Δ t}) (1 + e^{- γ_{m} Δ t} + \dots + e^{- (N - 1) γ_{m} Δ t})] \\ = & exp [- 4 β^{2} (1 - e^{- γ_{m} t})] . \end{array}

Funding

National Natural Science Foundation of China (11574092, 61378012, 91121023, 60978009) and the National Basic Research Program of China (2013CB921804).

References and links

1. R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865–942 (2009). [CrossRef]

2. C. Monroe, D. M. Meekhof, B. E. King, and D. J. Wineland, “A Schrodinger cat superposition state of an atom,” Science 272, 1131 (1996). [CrossRef] [PubMed]

3. M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. Van der Zouw, and A. Zeilinger, “Wave–particle duality of C60 molecules,” Nature 401, 680–682 (1999). [CrossRef]

4. S. Gerlich, S. Eibenberger, M. Tomandl, S. Nimmrichter, K. Hornberger, P. J. Fagan, J. Tüen, M. Mayor, and M. Arndt, “Quantum interference of large organic molecules,” Nat. Commun. 2, 263 (2011). [CrossRef] [PubMed]

5. D. P. DiVincenzo, “Quantum computation,” Science 270, 255 (1995). [CrossRef]

6. A. Ekert and R. Jozsa, “Shor’s quantum algorithm for factorising numbers,” Rev. Mod. Phys. 68, 733–753 (1996). [CrossRef]

7. L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413–418 (2001). [CrossRef] [PubMed]

8. V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photon. 5, 222–229 (2011). [CrossRef]

9. A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67, 661 (1991). [CrossRef] [PubMed]

10. V. Scarani, S. Iblisdir, N. Gisin, and A. Acin, “Quantum cloning,” Rev. Mod. Phys. 77, 1225 (2005). [CrossRef]

11. P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337 (1995). [CrossRef] [PubMed]

12. E. Hagley, X. Maitre, G. Nogues, C. Wunderlich, M. Brune, J. M. Raimond, and S. Haroche, “Generation of Einstein-Podolsky-Rosen pairs of atoms,” Phys. Rev. Lett. 79, 1 (1997). [CrossRef]

13. Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, “Deterministic entanglement of two trapped ions,” Phys. Rev. Lett. 81, 3631 (1998). [CrossRef]

14. M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z. R. Wasilewski, and A. Forchel, “Coupling and entangling of quantum states in quantum dot molecules,” Science 291, 451–453 (2001). [CrossRef] [PubMed]

15. A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (2001). [CrossRef] [PubMed]

16. J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426, 264–267 (2003). [CrossRef]

17. Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Nonlocal entanglement concentration scheme for partially entangled multipartite systems with nonlinear optics,” Phys. Rev. A 77, 062325 (2008). [CrossRef]

18. S. Horoche and D. Kleppner, “Cavity quantum electrodynamics,” Phys. Today 42, 24 (1989). [CrossRef]

19. A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004). [CrossRef] [PubMed]

20. R. Blatt and D. Wineland, “Entangled states of trapped atomic ions,” Nature 453, 1008–1015 (2008). [CrossRef] [PubMed]

21. H. Häfner, W. Häsel, C. F. Roos, J. Benhelm, M. Chwalla, T. Köber, and O. Güne, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643–646 (2005). [CrossRef]

22. T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172–1176 (2008). [CrossRef] [PubMed]

23. P. Meystre, “A short walk through quantum optomechanics,” Ann. Phys. (Berlin) 525, 215 (2013). [CrossRef]

24. Z. R. Gong, H. Ian, Y. X. Liu, C. P. Sun, and F. Nori, “Effective Hamiltonian approach to the Kerr nonlinearity in an optomechanical system,” Phys. Rev. A 80, 065801 (2009). [CrossRef]

25. R. Ghobadi, A. R. Bahrampour, and C. Simon, “Quantum optomechanics in the bistable regime,” Phys. Rev. A 84, 033846 (2011). [CrossRef]

26. C. H. Dong, V. Fiore, M. C. Kuzyk, and H. L. Wang, “Optomechanical dark mode,” Science 338, 1609–1613 (2012). [CrossRef] [PubMed]

27. A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011). [CrossRef] [PubMed]

28. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010). [CrossRef] [PubMed]

29. M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013). [CrossRef]

30. J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3, 1196 (2012). [CrossRef] [PubMed]

31. S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A 56, 4175 (1997). [CrossRef]

32. W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. 91, 130401 (2010). [CrossRef]

33. S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002). [CrossRef] [PubMed]

34. F. Xue, Y. X. Liu, C. P. Sun, and F. Nori, “Two-mode squeezed states and entangled states of two mechanical resonators,” Phys. Rev. B 76, 064305 (2007). [CrossRef]

35. G. Vacanti, M. Paternostro, G. M. Palma, and V. Vedral, “Optomechanical to mechanical entanglement transformation,” New J. Phys. 10, 095014 (2008). [CrossRef]

36. S. Huang and G. S. Agarwal, “Entangling nanomechanical oscillators in a ring cavity by feeding squeezed light,” New J. Phys. 11, 103044 (2009). [CrossRef]

37. X. W. Xu, Y. J. Zhao, and Y. X. Liu, “Entangled-state engineering of vibrational modes in a multimembrane optomechanical system,” Phys. Rev. A 88, 022325 (2013). [CrossRef]

38. C. Joshi, J. Larson, M. Jonson, E. Andersson, and P. Öhberg, “Entanglement of distant optomechanical systems,” Phys. Rev. A 85, 033805 (2012). [CrossRef]

39. J. Zhang, K. Peng, and S. L. Braunstein, “Quantum-state transfer from light to macroscopic oscillators,” Phys. Rev. A 68, 013808 (2003). [CrossRef]

40. D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98, 030405 (2007). [CrossRef] [PubMed]

41. T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342, 710–713 (2013). [CrossRef] [PubMed]

42. J. Q. Liao, Q. Q. Wu, and F. Nori, “Entangling two macroscopic mechanical mirrors in a two-cavity optomechanical system,” Phys. Rev. A 89, 014302 (2014). [CrossRef]

43. M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, “Event-ready-detectors Bell experiment via entanglement swapping,” Phys. Rev. Lett. 71, 4287–4290 (1993). [CrossRef] [PubMed]

44. L. M. Kuang and Z. Lan, “Generation of atom-photon entangled states in atomic bose-einstein condensate via electromagnetically induced transparency,” Phys. Rev. A 68, 043606 (2004). [CrossRef]

45. C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge University Press, 2005), Chap. 8.

46. J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008). [CrossRef] [PubMed]

47. J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707–712 (2010). [CrossRef]

48. S. J. Phoenix, “Wave-packet evolution in the damped oscillator,” Phys. Rev. A 41, 5132 (1990). [CrossRef] [PubMed]

Entangling cavity optomechanical systems via a flying atom

Abstract

1. Introduction

2. Model and theory

3. Entanglement between two cavity optomechanical systems

3.1. Entanglement transfer

3.2. Entanglement swapping-like process

4. Measurement and decoherence

5. Discussion and conclusion

Appendix: Solving the master equation

Funding

References and links

Cited By

Figures (3)

Equations (41)

Optics Express