All-optical transistor based on Rydberg atom-assisted optomechanical system

Yi-Mou Liu; Xue-Dong Tian; Jing Wang; Chu-Hui Fan; Feng Gao; Qian-Qian Bao

doi:10.1364/OE.26.012330

1. Introduction

The invention of the electronic transistor has laid the cornerstone of the information age. With the development of quantum information technology, the research and fabrication of optical transistor have become an important branch. In general, the fabrication of optical transistors is based on optical non-reciprocity [1, 2]. Early, optical non-reciprocity has been achieved in optical waveguides [3–6] or optical nonlinear system [7–9] by breaking spatial inversion symmetry. Recently, several alternative schemes based on different principles have been proposed, such as photo-acoustic effects [10, 11], indirect inter-band photon transitions [12–14], space symmetric fracture structures [15–17], moving systems [18, 19] and parity time-symmetric structures [20–22]. In addition, for the potential applications in photonic quantum information processing, the abilities to be integrated on a chip, non-local control at long-range [23,24] and operated on a single-photon level [25,26] are desirable features for the realization of nonreciprocal all-optical devices in the future.

With the rapid development of new micro integrable devices, optomechanical systems have shown enormous potential for application in quantum information processing [27–29]. It has previously been shown that optomechanical systems can be used to induce nonreciprocal effects for light [30–32]. Multi-mode optomechanical systems have drawn much attention recently. Numbers of the novel and interesting phenomena are noted, such as high-fidelity quantum state transfer [33,34], enhanced quantum nonlinearities [35,36], phonon laser [37,38], coherent perfect absorption (CPA) [39,40] and so on. Especially, the CPA could be viewed as an inverse process of laser [41], which provides a new mechanism in optomechanical system for controlling optical nonreciprocity.

Currently, hybrid atom-optomechanics is a rapidly growing area of research in quantum optics [42–45]. While, most of the investigations on atom-assisted optomechanical cavities coupled with independent cold atoms are driven by quantum cavity modes and classical coherent control fields [46–50]. It is rare to report the study of multifunctional all-optical quantum devices using a hybrid system with long-range interaction between atoms. Recently, Rydberg atoms, coupled by dipole-dipole interactions (DDI), have been shown to be efficient nonlinear media in cavities in order to achieve additional control freedom of optomechanical interactions and applications [51–53]. An essential blockade effect based upon DDI prevents the excitation of more than one atom into a Rydberg state within a macroscopic volume of several micrometers in radius [54–57]. Meanwhile, based on dipole blockade, many promising proposals have been put forward for manipulating quantum states of atoms and photons [58–60], simulating many-body quantum systems [61,62], generating reliable single photons [63,64], and revealing some novel behaviors in EIT [65,66], etc.

In this paper, we study the optical response of a symmetric double-cavity optomechanical system assisted by Rydberg atoms with a movable mirror of perfect reflection and driven by two coupling fields and two probe fields. The transition from ground state to excited state of an atom is coupled by a cavity mode, and the transition from the excited state to Rydberg state is coupled with a classical control field. There are vdW interactions between the target atom in the cavity and the gate one outside.

We especially focus on the transmission and reflection properties of two side weak probe fields, by switching on/off the external control. In case (I_a), we find a controllable optical diode effect, that photons only pass through the system from one direction. In case (II_b) and (III_b), we achieve the function of rectifier, which can change the propagation direction of the photons, in two different ways. The first one (unidirectional coupling rectifier) is based on the coherent perfect transmission effect (CPT), and the other one (bidirectional coupling rectifier) is based on the coherent perfect synthesis effect (CPS). In case (IV_c), we obtain a controllable photon amplification, which is based on the four-wave mixing effect (FWM) of the atom-assisted optomechanical system. We expect that the all-optical controllable transistor (the optical correspondence of classical electrical transistor), which sets controllable photon diode, rectification, amplification and other functions as a whole, could be explored to build new tunable single-photon devices on quantum information networks

2. Theoretical model

We consider a hybrid optomechanical system with one movable mirror (membrane oscillator) of perfect reflection inserted between two fixed mirrors of partial transmission, which form two mechanical-coupled Fabry–Perot cavities [see Fig. 1]. We describe the two optical modes in the left or right cavity, respectively, by annihilation (creation) operators c_l ( $c_{l}^{†}$ ) or c_r ( $c_{r}^{†}$ ). And the only mechanical mode is described by b (b^†). These annihilation and creation operators Ô = {c_l, c_r, b} are bosons and satisfy the commutation relation $[{\hat{O}}_{i}, {\hat{O}}_{j}^{†}] = δ_{i, j}$ (i, j = c_l, c_r, b). Two probe (coupling) fields are used to drive the double-cavity system from either left or right fixed mirrors with their amplitudes denoted by $ε_{l} = \sqrt{2 κ ℘_{L} / (ℏ ω_{p})}$ and $ε_{r} = \sqrt{2 κ ℘_{R} / (ℏ ω_{p})}$ ( $ε_{cL} = \sqrt{2 κ ℘_{cL} / (ℏ ω_{c})}$ and $ε_{cR} = \sqrt{2 κ ℘_{cR} / (ℏ ω_{c})}$ ). Here κ is the decay rate of both cavity modes. ℘_L, ℘_R, ℘_cL and ℘_cR are the relevant field powers. ω_p (ω_c) is the probe (coupling) field frequency. The membrane oscillator has an eigenfrequency ω_m and a decay rate γ_m and thus exhibits a mechanical quality factor Q = γ_m/ω_m. Two identical optical cavities of lengths L and frequencies ω₀ are obtained when the membrane oscillator is at its equilibrium position in the absence of external excitation. And in the left Fabry–Perot cavity, cavity mode c_l also drives a cold target atom into three level ladder configuration together with a pump field with frequency ω_r. And there is another atom (gate atom) outside the system, with the van de Waals interaction (vdW) between them. Then the total Hamiltonian of our hybrid system in the rotating-wave frame can be written as

H = H_{0} + H_{I} .

Here H₀ describes the free Hamiltonian, including optical cavities H_c, movable mirror H_m, and atoms H_a, respectively. And H_I describes the interactions, including the driven and probe term H_cp, the coupling term between membrane and cavity mode H_mc, the coupling term between atoms and fields V_af, and the vdW potential V_vdW, with the following expressions:

\begin{array}{l} H_{c} & = ℏ Δ_{cm} [c_{l}^{†} c_{l} + c_{r}^{†} c_{r}], H_{m} = ℏ ω_{m} b^{†} b, \\ H_{a} & = ℏ [δ_{e g} σ_{e e} + δ_{r g} σ_{r r} + δ_{r^{'} g^{'}} σ_{r^{'} r^{'}}], \\ H_{c p} & = i ℏ ε_{cL} [c_{l}^{†} - c_{l}] + i ℏ [ε_{l} c_{l}^{†} e^{- i δ t} - ε_{l}^{*} c_{l} e^{i δ t}] \\ + i ℏ ε_{cR} [c_{r}^{†} - c_{r}] + i ℏ [ε_{cR} c_{r}^{†} e^{i θ} e^{- i δ t} - ε_{cR}^{*} c_{r} e^{i θ} e^{i δ t}], \\ H_{m c} & = ℏ g_{0} [c_{r}^{†} c_{r} - c_{l}^{†} c_{l}] [b^{†} + b], \\ V_{a f} & = - ℏ [Ω_{r} σ_{r e} + g_{ac} c_{l} σ_{e g} + Ω_{g} σ_{g^{'} r^{'}}] + H . C . \\ V_{v d W} & = ℏ \frac{C_{6}}{R_{tg}^{6}} σ_{r r} σ_{r^{'} r^{'}}, \end{array}

where we define Δ_cm = ω₀ − ω_m (δ = ω_p − ω_c, δ_eg = ω_eg − ω₀, δ_rg = ω_rg − ω₀ − ω_r, and δ_{r′ g′} = ω_{g′ r′} − ω_g) the detuning between cavity modes and phonon mode (cavity modes and coupling fields, atomic transition frequencies and coupling fields), θ the relative phase between left- and right-side probe fields,

g_{0} = \frac{ω_{0}}{L} \sqrt{ℏ / (2 m ω_{m})}

the hybrid coupling constant between mechanical and optical modes (m quality of the oscillator, V the volume of the cavity, and ε₀ the vacuum dielectric constant). The coupling constant (Rabi frequency) is defined as

g_{ac} = ℘_{ge} \sqrt{ω_{ge} / (2 ℏ V ε_{0})}

(

Ω_{r} = ℘_{er} \sqrt{ω_{er} / (2 ℏ V ε_{0})}

,

Ω_{g} = ℘_{g^{'} r^{'}} \sqrt{ω_{g^{'} r^{'}} / (2 ℏ V ε_{0})}

), and ℘_ge (℘_ge, ℘_{g′ r′}) is atomic transition dipole moment. In Eqs. (1), we focus on the Dipole-Dipole interaction as expressed by a vdW potential, where C₆ denotes the vdW coefficient, and R_tg is the interatomic distance between the target and the gate atom.

Fig. 1 Schematic diagram of the hybrid system double optomechanical cavity coupled with Rydberg atoms. (a₁) and (a₂) shows energy level structures of the target Rydberg atom coupled with the left cavity and the gate one which dominates the target one by vdW potential V_vdW between them. (b₁) and (b₂) show that the double-cavity decouple with atoms switching off the external control.

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Considering dissipation and quantum noise of the system, we can get the following Heisenberg-Langevin equations:

\begin{array}{l} \dot{b} & = - i ω_{m} b - i g_{0} [c_{l}^{†} c_{l} - c_{r}^{†} c_{r}] - \frac{γ_{m}}{2} b + \sqrt{γ_{m}} b^{i n}, \\ {\dot{c}}_{l} & = - [κ + i Δ_{c} - i g_{0} (b^{†} + b)] c_{l} + ε_{cL} + ε_{l} e^{i δ t} - i g_{ac} {\hat{σ}}_{e g} + \sqrt{2 κ} c_{l}^{in}, \\ {\dot{c}}_{r} & = - [κ + i Δ_{c} + i g_{0} (b^{†} + b)] c_{r} + ε_{cR} + ε_{r} e^{i θ} e^{- i δ t} + \sqrt{2 κ} c_{r}^{in}, \\ {\dot{\hat{σ}}}_{ge} & = - [i δ_{ge} + γ_{e}] {\hat{σ}}_{ge} + i Ω_{r}^{*} {\hat{σ}}_{gr} + i g_{ac} c_{l} + f_{1} (t), \\ {\dot{\hat{σ}}}_{gr} & = - [i δ_{gr} + i V_{t}^{s} + γ_{r}] {\hat{σ}}_{gr} + i Ω_{r} {\hat{σ}}_{ge} + f_{2} (t), \\ {\dot{\hat{σ}}}_{g^{'} r^{'}} & = - [i δ_{g^{'} r^{'}} + i V_{g}^{s} + γ_{r^{'}}] {\hat{σ}}_{g^{'} r^{'}} + i Ω_{g}^{*} + f_{3} (t) . \end{array}

Here κ, γ_m, γ_e and γ_r (γ_r′) correspond to the decay rate of cavities, oscillator, excited state |e〉 and that of Rydberg state |r〉 (|r′〉), respectively. $V_{t}^{s} = 〈 \frac{C_{6}}{R_{tg}^{6}} σ_{r^{'} r^{'}} 〉$ ( $V_{g}^{s} = 〈 \frac{C_{6}}{R_{tg}^{6}} σ_{r r} 〉$ ) is the mean value of vdW potential for the target (gate) atom. The bⁱⁿ is set to be the thermal noise on the movable, and $c_{l}^{in}$ ( $c_{r}^{in}$ ) is the input quantum vacuum noise from the left (right) cavity with zero mean value. Similarly, f_i (i = 1, 2, 3) is the quantum fluctuation of the operators above. Here, we are more interested in the mean optical response of the optomechanical system to probe field in the presence of both strong driving fields. In this regard, we can safely ignore the quantum fluctuations of all relevant operators and use the factorization assumption b〈c_i〉 = 〈b〉〈c_i〉 to generate the mean values in the steady state. In order to solve Eqs. (2), we write each operator as the sum of its mean value and its small fluctuation Ô₁ = 〈Ô₁〉 + δÔ₁, Ô₁ = {b, c_l, c_r, σ̂_ge, σ̂_gr, σ̂_{g′ r′}}, when both coupling fields are sufficiently strong. Then we get two series of equations about steady-state or fluctuation of operators, respectively,

\begin{array}{l} {\dot{b}}_{s} & = - i ω_{m} b_{s} - i g_{0} [c_{ls}^{*} c_{ls} - c_{rs}^{*} c_{rs}] - \frac{γ_{m}}{2} b_{s}, \\ {\dot{c}}_{ls} & = - [κ + i Δ_{c} - i g_{0} (b_{s}^{*} + b_{s})] c_{ls} + ε_{c_{L}} - i g_{ac} {\hat{σ}}_{ge}^{s}, \\ {\dot{c}}_{rs} & = - [κ + i Δ_{c} + i g_{0} (b_{s}^{*} + b_{s})] c_{rs} + ε_{c_{R}}, \\ {\dot{\hat{σ}}}_{ge}^{s} & = - [i δ_{ge} + γ_{e}] {\hat{σ}}_{ge}^{s} + i Ω_{r}^{*} {\hat{σ}}_{gr}^{s} + i g_{ac} c_{ls}, \\ {\dot{\hat{σ}}}_{gr}^{s} & = - [i δ_{gr} + i V_{t}^{s} + γ_{r}] {\hat{σ}}_{gr}^{s} + i Ω_{r} {\hat{σ}}_{ge}^{s}, \\ {\dot{\hat{σ}}}_{g^{'} r^{'}}^{s} & = - [i δ_{g^{'} r^{'}} + i V_{g}^{s} + γ_{r^{'}}] {\hat{σ}}_{g^{'} r^{'}}^{s} + i Ω_{g}^{*}, \end{array}

\begin{array}{l} δ \dot{b} & = - i ω_{m} δ b - i g_{0} [c_{ls}^{*} δ c_{l} + c_{ls} δ c_{l}^{†} - c_{rs}^{*} δ c_{r} - c_{rs} δ c_{r}^{†}] - \frac{γ_{m}}{2} δ b + \sqrt{γ_{m}} b^{in}, \\ δ {\dot{c}}_{l} & = - [κ + i {\tilde{Δ}}_{c}] δ c_{l} - i g_{0} c_{ls} (δ b^{†} + δ b) + ε_{l} e^{- i δ t} - i g_{ac} δ {\hat{σ}}_{ge} + \sqrt{2 κ} c_{l}^{in}, \\ δ {\dot{c}}_{r} & = - [κ + i {\tilde{Δ}}_{c}] δ c_{r} + i g_{0} c_{rs} (δ b^{†} + δ b) + ε_{R} e^{i θ} e^{- i δ t} \sqrt{2 κ} c_{r}^{in}, \\ δ {\dot{\hat{σ}}}_{ge} & = - [i δ_{ge} + γ_{e}] δ {\hat{σ}}_{ge} + i Ω_{r}^{*} δ {\hat{σ}}_{gr} + i g_{ac} δ c_{l} + f_{1} (t), \\ δ {\dot{\hat{σ}}}_{gr} & = - [i δ_{gr} + i V_{t}^{s} + γ_{r}] δ {\hat{σ}}_{gr} + i Ω_{r} δ {\hat{σ}}_{ge} + f_{2} (t), \\ δ {\dot{\hat{σ}}}_{g^{'} r^{'}} & = - [i δ_{g^{'} r^{'}} + i V_{g}^{s} + γ_{r^{'}}] δ {\hat{σ}}_{g^{'} r^{'}} + f_{3} (t), \end{array}

where the effective detuning is Δ̃_c = Δ_c − g_ob_s and c_is = 〈c_i〉 (i = l, r) is the steady state average of cavities.

It is difficult to solve Eqs. (3) and (4) directly. So we first solve the atomic part of the equations above, taking the part of the cavity system as variable parameter quantities [53]. The mathematical relationships between operators are calculated separately, aiming to simplify the calculation. In general, the optical response of the Rydberg atoms are affected by the vdW interactions. Note that $V_{t}^{s}$ tends to infinite for one Rydberg excitation 〈σ̂_rr〉 = 1 or vanishing for zero Rydberg excitation 〈σ̂_rr〉 = 0, if the distance R_tg is smaller than the blockade radius $R_{b} = \sqrt[6]{\frac{C_{6} (γ_{e} + i δ_{ge})}{Ω_{r}^{2}}}$ [66]. Therefore, the effective steady-state polarization of atomic system will be determined as

ϱ_{s} = 〈 ϱ_{2} 〉 〈 σ_{r^{'} r^{'}}^{s} 〉 + 〈 ϱ_{3 L} 〉 [1 - 〈 σ_{r^{'} r^{'}}^{s} 〉] .

Here 〈ϱ₂〉 is the steady-state mean value of polarization operator ${\hat{σ}}_{ge}^{s}$ in a two-level atomic system and 〈ϱ_3L〉 is that in a three-level Ladder type system,

\begin{array}{l} 〈 ϱ_{2} 〉 = \frac{i g_{ac} c_{ls}}{i δ_{ge} + γ_{e}}, \\ 〈 ϱ_{3 L} 〉 = \frac{i g_{ac} [i δ_{gr} + i V_{t}^{s} + γ_{r}] c_{ls}}{[i δ_{ge} + γ_{e}] [i δ_{gr} + i V_{t}^{s} + γ_{r}] + Ω_{r}^{*} Ω_{r}} . \end{array}

With

〈 σ_{r^{'} r^{'}}^{s} 〉 + 〈 σ_{g^{'} g^{'}}^{s} 〉 = 1

, we can also obtain that

〈 σ_{r^{'} r^{'}}^{s} 〉 ≃ \frac{γ_{r^{'}}^{2}}{Ω_{g}^{*} Ω_{g}} 〈 σ_{r^{'} g^{'}}^{s} σ_{g^{'} r^{'}}^{s} 〉 ≃ \frac{γ_{r^{'}}^{2}}{γ_{r^{'}}^{2} + δ_{g^{'} r^{'}}^{2}} .

So the steady-state solutions will be easily obtained,

\begin{array}{l} b_{s} = \frac{i g_{0} (c_{ls}^{*} c_{ls} - c_{rs}^{*} c_{rs})}{i Δ_{m} + \frac{γ_{m}}{2}}, \\ c_{ls} = \frac{ε_{cL} - i g_{ac} ϱ_{s}}{κ + i Δ_{c} - i g_{0} (b_{s}^{*} + b_{s})}, \\ c_{rs} = \frac{ε_{cR}}{κ + i Δ_{c} + i g_{0} (b_{s}^{*} + b_{s})} . \end{array}

Using the same method, from last third equations in Eqs. (4) we can get $δ ϱ_{s} = 〈 δ ϱ_{2} 〉〈 σ_{r^{'} r^{'}}^{s} 〉 + 〈 δ ϱ_{3 L} 〉 [1 - 〈 σ_{r^{'} r^{'}}^{s} 〉]$ , where

\begin{array}{l} 〈 δ ϱ_{2} 〉 & = \frac{i g_{ac} 〈 δ c_{l} 〉}{i δ_{ge} + γ_{e}}, \\ 〈 δ ϱ_{3 L} 〉 & = \frac{i g_{ac} [i δ_{gr} + i V_{t}^{s} + γ_{r}] 〈 δ c_{l} 〉}{[i δ_{ge} + γ_{e}] [i δ_{gr} + i V_{t}^{s} + γ_{r}] + Ω_{r}^{*} Ω_{r}} . \end{array}

Then keeping only the linear terms of fluctuation operators and moving into an interaction picture by introducing: δÔ → δÔe^−iΔ_it, δÔⁱⁿ → δÔⁱⁿe^−iΔ_it, Ô = {b, c_l, c_r}, $Δ_{i} = {δ_{ge}, δ_{gr} + V_{t}^{s}, {\tilde{Δ}}_{c}, ω_{m}}$ , we obtain the linearized quantum Langevin equations:

\begin{array}{l} δ \dot{b} = - i g_{0} (c_{ls}^{*} δ c_{l} - c_{rs}^{*} δ c_{r}) - \frac{γ_{m}}{2} δ b + \sqrt{γ_{m}} b^{in}, \\ δ {\dot{c}}_{l} = - κ δ c_{l} - i g_{0} c_{ls} δ b + ε_{L} e^{i x t} - i g_{ac} δ ϱ_{s} + \sqrt{2 κ} c_{l}^{in}, \\ δ {\dot{c}}_{r} = - κ δ c_{r} + i g_{0} c_{rs} δ b + ε_{R} e^{i θ} e^{- i x t} + \sqrt{2 κ} c_{r}^{in}, \end{array}

where x = δ − ω_m. And expectations of fluctuation operators satisfy the equations:

\begin{array}{l} 〈 δ \dot{b} 〉 = - i g_{0} (c_{ls}^{*} 〈 δ c_{l} 〉 - c_{rs}^{*} 〈 δ c_{r} 〉) - \frac{γ_{m}}{2} 〈 δ b 〉, \\ 〈 δ {\dot{c}}_{l} 〉 = - κ (δ c_{l}) - i g_{0} c_{ls} 〈 δ b 〉 + ε_{L} e^{- i x t} - i g_{ac} δ ϱ_{s}, \\ 〈 δ {\dot{c}}_{r} 〉 = - κ (δ c_{r}) + i g_{0} c_{rs} 〈 δ b 〉 + ε_{R} e^{i θ} e^{- i x t}, \end{array}

where the oscillating terms can be removed if steady-state solutions of Eqs. (8) are assumed to be form: 〈δs〉 = δs₊e^−ixt + δs₋e^ixt with s = b, c_l, c_r. Then it is easy to attain the following results,

\begin{array}{l} δ b_{+} = \frac{in G g_{ac}^{'} ε_{r} e^{i θ} - i G^{*} κ^{'} ε_{l}}{γ_{m}^{'} κ^{'} g_{ac}^{'} + G^{2} κ^{'} + G^{2} n^{2} g_{ac}^{'}}, \\ δ c_{l +} = \frac{G n ε_{r} e^{i θ t} + [G^{2} n^{2} + γ_{m}^{'} κ^{'}] ε_{l}}{γ_{m}^{'} κ^{'} g_{ac}^{'} + G^{2} κ^{'} + G^{2} n^{2} g_{ac}^{'}}, \\ δ c_{r +} = \frac{[γ_{m}^{'} g_{ac}^{'} + G^{2}] ε_{r} e^{i θ} + G^{2} n ε_{l}}{γ_{m}^{'} κ^{'} g_{ac}^{'} + G^{2} κ^{'} + G^{2} n^{2} g_{ac}^{'}}, \end{array}

where we set G = g₀c_ls as the effective optomechanical coupling rate and n² = |c_rs/c_ls|² as the photon number ratio of the two cavity modes, with κ′ = ix−κ,

γ_{m}^{'} = i x - \frac{γ_{m}}{2}

, g′_ac = ix−κ−ig_acϱ_s. In deriving Eqs. (10), we have also assumed that c_ls,rs is real-valued without loss of generality.

It is possible to determine the output fields ε_outl and ε_outr leaving from both cavities with the following input-output relation [40]

\begin{array}{r} ε_{outl} + ε_{l} e^{- i x t} = 2 κ 〈 δ c_{l} 〉, \\ ε_{outr} + ε_{r} e^{i θ} e^{- i x t} = 2 κ 〈 δ c r_{r} 〉, \end{array}

where the oscillating terms can be removed if we set ε_outl = ε_outl+e^−ixt + ε_outl−e^ixt and ε_outr = ε_outr+e^−ixt + ε_outr−e^ixt. Note that the output components ε_outl+ and ε_outr+ have the same Stokes frequency ω_p as the input probe fields ε_l and ε_r while the output components ε_outl− and ε_outr− are generated at the anti-Stokes frequency 2ω_c − ω_p in a nonlinear four-wave mixing process of optomechanical interaction. Then with Eqs. (11) we can obtain

\begin{array}{l} ε_{outl +} = 2 κ δ c_{l +} - ε_{l}, \\ ε_{outr +} = 2 κ δ c_{r +} - ε_{r} e^{i θ}, \end{array}

oscillating at the Stokes frequency of our special interest. So it is easy to find

\begin{array}{l} ε_{outl +} = \frac{2 κ n G^{2} ε_{r} e^{i θ} + [γ_{m}^{'} κ (2 κ - g_{ac}^{'}) + G^{2} (2 κ n^{2} - κ^{'} - n^{2} g_{ac}^{'})] ε_{l}}{γ_{m}^{'} κ g_{ac}^{'} + n^{2} g_{ac}^{'} G^{2} + κ^{'} G^{2}} \\ ε_{outr +} = \frac{2 κ n G^{2} ε_{l} - [(i x - 3 κ) (γ_{m}^{'} g_{ac}^{'} + G^{2}) + n^{2} g_{ac}^{'} G^{2}] ε_{r} e^{i θ}}{γ_{m}^{'} κ g_{ac}^{'} + n^{2} g_{ac}^{'} G^{2} + κ^{'} G^{2}} . \end{array}

3. Results and discussion

In this section, we numerically simulate the optical response of the hybrid system, which features controllable non-reciprocity and potential use of optical diode and transistor. The relevant parameters are set as γ_eg/2π=3.0MHz, Ω_r/2π=2.5MHz, Ω_g/2π=4.0MHz, δ_eg=100MHz and R_tg=10μm. And the important physical quantities that we study optical responses are the transmission coefficient. T_l = |ε_outr+/ε_l|² (T_r = |ε_outl+/ε_r|²) and reflection coefficient R_l = |ε_outl+/ε_l|² (R_r = |ε_outr+/ε_r|²), which are both the functions of the probe frequency x/κ. Then, we consider a specific situation that single-photon detuning is large δ_ge ≫ Ω_r ≃ Ω_g, but with two-photon resonance δ_gr = 0. With vdW potential $V_{t}^{s}$ , switching on the gate field Ω_g, the Rydberg blockade effect from gate atom ( $〈 {\hat{σ}}_{r^{'} r^{'}}^{s} 〉 = 1$ ) will lead to an effective detuning of target atom $δ_{eff} = [i δ_{gr} + i V_{t}^{s} + γ_{r}] (≫ Ω_{r}^{*} Ω_{r})$ to be infinite. And we can easily obtain effect atomic polarization ϱ_s ≃ 〈ϱ₂〉, which causes the decoupling between the target atom and the left cavity [see Eqs. (5)], with a large single photon detuning δ_ge.

In the following discussion, we only consider the two extreme situations of the system: ℐ. Optical reciprocity and ℐℐ. Optical non-reciprocity. Situation ℐ corresponds to decoupling between the target atom and left cavity [see Fig. 1(a) and Fig. 2(c₂)]. Under this condition, we only give reflection of cavities for simplicity (T_l and T_r are also easily obtained),

\begin{array}{l} R_{l} = {| \frac{2 κ n^{2} G^{2} e^{i θ} + [γ_{m}^{'} κ κ^{'} + 2 κ G^{2} - (n^{2} + 1) κ^{'} G^{2}] ε_{l} / ε_{r}}{γ_{m}^{'} κ κ^{'} + (n^{2} + 1) κ^{'} G^{2}} |}^{2}, \\ R_{r} = {| \frac{2 κ n^{2} G^{2} + [γ_{m}^{'} κ κ^{'} + 2 κ G^{2} - (n^{2} + 1) κ^{'} G^{2}] ε_{r} e^{i θ} / ε_{l}}{γ_{m}^{'} κ κ^{'} + (n^{2} + 1) κ^{'} G^{2}} |}^{2}, \end{array}

with g′_ac ≃ κ′. The system satisfies the space reversal symmetry, because of R_l = R_r with θ = nπ [See Fig. 2(c₁)].

Fig. 2 Energy level diagram (a) and external control timing flowchart (b). Standardization energy of the output probe field (c) R_r = |ε_outR+/ε_R|² (reflection coefficient of left cavity, blue) and R_l = |ε_outL+/ε_L|² (reflection coefficient of right cavity, red circle) as the function of x/κ, (c₁) switching off the external control (coupled with atoms) and (c₂) switching on that (decoupled with atoms), with G = κ, n = 1 and θ = π.

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On the contrary, in Situation ℐℐ [See Fig. 2(b) Stage I/ III and Fig. 2(c₁)], the coupling between the target atom and left cavity breaks the space reversal symmetry of the system. Based on the two cases above, three functions of an optical transistor can be achieved in the hybrid system under different conditions. Before discussing the achievement of these functions, we first focus on external control form for switching between Situation ℐ and Situation ℐℐ.

External control

Here, we define external control as a manipulation, switching on which can block the coupling between the target atom and cavity system. We can achieve this goal in two ways here. Firstly, we consider the control switching on/off the pumping field Ω_r [see Fig 2(b) Stage IV]. We set δ_ge ≫ g_ac with κ ≈ δ_gr ≈ 0, which is typical Rydberg-EIT resonance with large single photon detuning condition. Under this condition, Ω_r ≠ 0 is the necessary condition for effective coupling between cavity system and Rydberg atomic ensembles, because large single photon detuning δ_ge will result in decoupling with Ω_r = 0. It is a typical and efficient coherent control, which is easy in the experiments. However, it is obviously not a nonlocal or single photon level quantum control.

Secondly, we consider the control scheme that double-cavity optomechanical system with the long-range Rydberg vdW blockade effect between two Rydberg atoms [see Fig. 1(b) and Fig 2(b) Stage II]. We set |r〉 ≡ 84s and |r′〉 ≡ 83s with $C_{6}^{d} (r, r^{'}) / 2 π ≃ 9.7 \times 10^{12} s^{- 1} μ m$ and $C_{6}^{e} (r, r^{'}) / 2 π ≃ 7.5 \times 10^{12} s^{- 1} μ m$ , so relationship between blockade radius and the atom distance is R_b>12.9μm>R_tg. Similar to self-interaction, vdW interactions owing to Rydberg excitation of |r′〉 can lead to a large frequency shift of |r〉. Under the case, we can get $i δ_{gr} + i V_{t}^{s} + γ_{r} \to \infty$ and 〈ϱ_3L〉 ≈ 〈ϱ₂〉 by switching on the gate pulse to pump the Rydberg state |r′〉. With a large single photon detuning δ_ge ≫ g_ac, the target atom is also decoupled with the system. In other words, it is possible to manipulate the optical response of this hybrid system, by control the Rydberg excitation of gate atom σ_r′r′, which is a long range nonlocal optical manipulation. Moreover, the manipulation can reach single-photon level: The wavelength of gate pulse is at near ultraviolet band, and it is replaced by two-step excitation with 780nm and 480nm lasers, generally.

Then, we will discuss the functions with our hybrid system below.

3.1. (I_a) controllable photon diode

The first function achieved with the system is controlled optical diode, which allows photons pass through only in one direction. It is coherent perfect absorption that is the basis for implementing of this function above. This CPA effect can be seen as a reverse process of laser. In other words, both probe fields experience a complete absorption without yielding any energy output from the two cavity mirrors when the membrane oscillator has a very large decay rate, is caused by quantum interference. And then, the parameters should be set as γ = 2κ, θ = nπ and ε_l = ε_r ≠ 0. From Eqs. (12), we can find that CPA will occur at $x_{\pm} = \pm \sqrt{(n^{2} + 1) G^{2} - κ^{2}}$ [39,40].

Figures 3 displays the transmission T_l and T_r of the hybrid system vs. detuning κ/x and G, with Situation ℐ and Situation ℐℐ. It is obvious to find that CPA effect also appears and the transmissions of the two sides are absolutely the same T_l = T_r [see Fig. 3 (a₁) and Fig. 3 (b₁)], switching off the external control. The physics is that the target atom is decoupled with the left cavity, because of Rydberg blockade effect from gate atom, which makes the system in Situation ℐ. Photons can flow symmetrically through the hybrid system in two directions. And then, switching on the external control, which activates atom-cavity coupling, will break the optical symmetry (Situation ℐℐ T_l ≠ T_r). In particular, T_l ≃ 1 [Fig. 2 (a₂)] in the range −4 < x/κ < 4, g′_ac >> κ′ owing to enough large g′_ac >> κ and |κ| ≃ |3κ − ix| << g′_ac,

T_{l} \approx {| \frac{(3 κ - i x) (γ_{m}^{'} g_{ac}^{'} + G^{2}) + n^{2} g_{ac}^{'} G^{2}]}{γ_{m}^{'} κ g_{ac}^{'} + n^{2} g_{ac}^{'} G^{2} + κ^{'} G^{2}} |}^{2} \approx 1 .

However, T_r also can be equal to zero [Fig. 3 (b₂)]. Furthermore, T_l = 1 and T_r = 0 means it is possible to find the range to achieve the photon diode framework which is a controllable unidirectional type optical device [see Fig. 3 (c₁) and Fig. 3 (c₂)].

Fig. 3 Transmission coefficient of left side T_l (in a₁/b₁) and right side T_r (in a₂/b₂) of the hybrid system vs. detuning x/κ and G, without (with) external control, when γ = 2κ and θ = nπ with n = 1. c₁ and c₂ are the functional sketch maps.

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3.2. (II_b) and (III_b) controllable photon rectifier

Secondly, aiming to achieve optical rectification, we will discuss two different cases, (II_b) unidirectional rectification and (III_b) bidirectional rectification which is based on coherent perfect transmission effect (CPT) and coherent perfect synthesis effect (CPS).

(II_b) unidirectional coupling rectifier

We set γ_m = 0.1κ, θ = nπ, ε_l ≠ 0 and ε_r = 0 (without right side input field). From the reference [39], coherent perfect transmission effect (CPT) is the basis for implementing this function. In CPT, one probe field travels through the optomechanical system without suffering any energy loss in the absence of the other probe field if the membrane oscillator has a very small decay rate.

Under this situation, if there is no coupling between atoms and cavity, we can find R_l = 0 [Fig. 4 (a₁)] and T_l = 1 [Fig. 4 (b₁)] near the range −0.5 < x < 0.5 with G > κ. Photons flow through the hybrid system perfectly from the left side without reflection [Fig. 4 (c₁)]. This perfect transmission is due to quantum coherence of the double optomechanical system, which can be controlled by phase θ. However, the optical properties of the system have changed, once there are effective coupling between atoms and cavity system. We find

\begin{array}{l} T_{l} \approx {| \frac{2 κ n G^{2}}{γ_{m}^{'} κ g_{ac}^{'} + n^{2} g_{ac}^{'} G^{2} + κ^{'} G^{2}} |}^{2} \approx 0, \\ R_{l} \approx {| \frac{[(i x - 3 κ) (γ_{m}^{'} g_{ac}^{'} + G^{2}) + n^{2} g_{ac}^{'} G^{2}]}{γ_{m}^{'} κ g_{ac}^{'} + n^{2} g_{ac}^{'} G^{2} + κ^{'} G^{2}} |}^{2} \approx 1, \end{array}

[Fig. 4 (a₂) and Fig. 4 (b₂)]. Photons are reflected back from the hybrid system around the same range with 2κ << g′_ac. It will be a framework for photon rectifier, if the manipulation has been achieved to control the direction of photon flow effectively.

Fig. 4 Reflection coefficient of left side R_l (in a₁(a₂)) and Transmission coefficient of right side T_r (in b₁(b₂)) of the hybrid system vs. detuning x/κ and G, without (with) external control, when γ = 2κ, γ_m → 0 and θ = nπ with ε_l ≠ 0, ε_r = 0. c₁ and c₂ are the functional sketch maps only with left side input probe field ε_L.

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(III_b) bidirectional coupling rectifier

In addition to the photon rectification, optical transistors should also have the function of an amplifier. We will implement this amplification function in two different ways. Firstly, we set γ_m = 0.1κ, G = 0.2κ and θ = (2n + 1)π/2. In Fig. 5 (a₁) and Fig. 5 (b₁), we find T_l = 2 and R_l = 0, when the target atom decouples from the left cavity. The specific performance is that photons output only from one side, entering from both sides. And the number of the output photons is twice as much as the input photon number, which is coherent perfect synthesis studied by Xiao-Bo Yan et al. [39]. In CPS, one probe field is totally transmitted while the other one is totally reflected to generate a perfect synthesis at one cavity mirror when the membrane oscillator has a very small decay rate. Obviously, γ_m < G < κ << g′_ac and ε_Re^iθ = −iε_r result T_l = Tr ≅ R_l = R_r = 1 [Fig. 5 (a₂)], which will be back to the symmetric structure [Fig. 5 (b₂)], if we switch on the external control to break the symmetry of the system.

Fig. 5 Reflection (Transmission) coefficient of left side R_l (T_r) of the hybrid system vs. detuning x/κ, without (with) external control (in a₁, a₂ (b₁, b₂)), when γ = 2κ, γ_m ≃ G → 0 and θ = (2n + 1)π/2 with n = 1. a₂ (b₂) is the functional sketch maps with both sides input probe field ε_L, ε_Re^iθ, corresponding to a₁ (b₁).

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3.3. (IV_c) controllable photon amplification

Then, the case of photon amplifier occurs when we set parameters as Case (I_a), but with different G. Switching on the control, if we increase G the effective optomechanical coupling rate of the left cavity at the CPA point of the right cavity, with $x_{\pm} \approx \pm \sqrt{(n^{2} + 1) G^{2} - κ^{2}} \approx n G$ and κ′ ≈ κ − inG, we will get the transmission rate

T_{l} \approx \frac{{(3 κ g_{ac}^{'} - G)}^{2} + g_{ac}^{2'} {(1 + G)}^{2}}{{(κ g_{ac}^{'} - G)}^{2} + g_{ac}^{2'}} > 1,

with n = 1 and G ≫ κ ≈ γ_m. We can increase G to enhance the magnification of the amplifier. The four-wave mixing process (FWM) involved in atomic transition leads to the gain of output the left cavity output [Fig. 6 (b₁) and Fig. 6 (b₂)] (ω_cl − ω_cr = 2ω_m). We know the atom transition frequency is near the one of the left cavity mode (ω_cl ≈ ω_ge), so this frequency conversion process can be represented as 2ω_cl − ω_ge − ω_cr = 2ω_m. Generally, the decay rate of a Rydberg state is large, so the frequency conversion process above is almost irreversible, which leads the photon number in the right cavity and ε_r+ increasing. However, the magnitude of this type gain is limited, which depends on the number of atom (only one) and the coupling rate g₀. In other words, a large G value is in need. But the larger average photon number in the left cavity c_ls will lead strong nonlinearity and instability of the system. And failure for linearization and instability of system will be also caused.

Fig. 6 Standardization energy of the output probe field T_l = |ε_outL+/ε_R|² (transmission coefficient of left cavity, blue) and T_l = |ε_outR+/ε_L|² (transmission coefficient of right cavity, red circle) as the function of x/κ, (a₁, a₂) switching off the external control (coupled with atoms) and (b₁, b₂) switching on that (decoupled with atoms), with G = κ, 2κ, n = 1 and θ = nπ

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4. Conclusions

In summary, we have studied the optical response of the Rydberg-atom-assisted double-cavity optomechanical system. With external control, we can switch this hybrid system between Situation ℐ (the target atom decoupling from the cavity system) and situation ℐℐ (the target atom coupling to the left cavity). And whether the optical reciprocity of the original symmetric system will be destroyed is controllable. We consider four special cases depending on the choice of actual parameters: (I_a) with θ = nπ and ε_l = ε_r ≠ 0; (II_b) with θ = nπ, γ_m → 0, ε_l ≠ 0 and ε_r = 0; (III_b) with θ = (2n + 1)π/2, γ_m ≃ G → 0 and ε_l = ε_r ≠ 0; (IV_c) θ = nπ, ε_l = ε_r ≠ 0 and G > κ.

In case (I_a), our numerical calculations show that the target atom coupled with single cavity breaks the optical reciprocity of the symmetric system. Photons can only flow through the system in one direction, which is the optical diode effect, when we switch on the external control. Then, we propose two types of optical rectifier schemes with our hybrid system. In case (II_b), when we turn off the input field of the right side, the system has become a controllable photon unidirectional coupling rectifier, which controls propagation behavior of the photons. In other words, photons that should have transmitted have been reflected back absolutely. And the other type of the rectifier, in case (III_b), is achieved basing on the synthesis effect (CPS), that is one side output with input fields from both sides (bidirectional coupling rectifier). In the case (IV_c), we obtained an amplifier based on FWM effect with atom coupled.

Though recent study on optical transistors has achieved great progress, most of them have only achieved optical control. There is no optical correspondence of classical electric transistors which set controller, rectification, amplification or other functions as a whole. Owing to the blockade effect between Rydberg atoms, it is great promising to make this type of control to the single photon level. We hope our work can provide a new way of thinking and a substantial role in promoting the study of the optical transistor and other all-optical devices.

Funding

National Natural Science Foundation of China (NSFC) (11704063); China Postdoctoral Science Foundation (2016M601362); Fundamental Research Funds for Central Universities (2412017QD004); Subject Construction Project of School of Physics NENU (111715014).

Acknowledgments

We thank Prof. Jin-Hui Wu, Prof. Xiao-Bo Yan, Ir. You-Jun Liu, and Ir. Ping Qi for fruitful discussions.

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All-optical transistor based on Rydberg atom-assisted optomechanical system

Abstract

1. Introduction

2. Theoretical model