Optical unidirectional amplification in a three-mode optomechanical system

We study the directional amplification of an optical probe field in a three-mode optomechanical system, where the mechanical resonator interacts with two linearly-coupled optical cavities and the cavities are driven by strong optical pump fields. The optical probe field is injected into one of the cavity modes, and at the same time, the mechanical resonator is subject to a mechanical drive with the driving frequency equal to the frequency difference between the optical probe and pump fields. We show that the transmission of the probe field can be amplified in one direction and de-amplified in the opposite direction. This directional amplification or de-amplification results from the constructive or destruction interference between different transmission paths in this three-mode optomechanical system.

In this paper, we study a scheme to achieve directional amplification of an optical probe field in a three-mode optomechanical system, where a mechanical resonator is coupled to two optical modes that directly interact with each other. In this system, controllable phase difference between the linearized optomechanical couplings, which breaks the time-reversal symmetry of this three-mode system, is generated by the strong pump fields on the optical cavities. Meanwhile, the probe field is applied to one of the cavities and the mechanical resonator is subject to a mechanical drive with the driving frequency equal to the frequency difference between the optical probe and pump fields. The constructive (destructive) interference between the transmission paths for the optical probe field and its mechanical counterpart via the optomechanical interaction results in the amplification of the probe field [26]. Strong directional amplification of the optical field with high amplification ratio can be achieved in this system. In comparison with the previous works [50][51][52][53][54] in multi-mode optomechanical systems, where the directional amplification results from the blue-detuned pump fields, here we use the red-detuned pump fields as well as the additional mechanical drive to achieve the optical directional amplification in a three-mode optomechanical system. Since the blue-detuned (red-detuned) pump field will heat (cool) the motion of mechanical resonator in an optomechanical system, our scheme avoiding pumping with blue-detuned light can improve the stability of the amplification scheme in optomechanical systems. As a tradeoff, the additional mechanical drive with the driving frequency equal to the frequency difference between the optical probe and pump fields is required to achieve the directional amplification in our scheme. Our work Schematic of a three-mode optomechanical system driven by two pump fields with the same frequency ω d . A probe field with frequency ωp is applied to one of the two cavities, that is, incident in cavity 1 from the left side (the thin solid arrow) or incident in cavity 2 from the right side (the thin dashed arrow). The mechanical resonator is subject to a mechanical drive with the driving frequency ω b . The cavities and the mechanical resonator are coupled via radiationpressure forces, and the cavities are directly coupled to each other.
provides an alternative method to achieve the optical directional amplification in optomechanical systems, which could stimulate future studies of optomechanical interfaces in the implementation of nonreciprocal and nonlinear photonic devices. This paper is organized as follows. In Sec. II, we present the Hamiltonian of the three-mode optomechanical system for nonreciprocal amplification and our derivation of the transmission coefficients in this system. Details of the directional amplification and de-amplification of the optical probe field are studied in Sec. III. Conclusions are given in Sec. IV.

II. MODEL AND TRANSMISSION MATRIX
The optomechanical system under consideration consists of a mechanical oscillator with resonance frequency ω m and two optical cavities with resonance frequencies ω 1 and ω 2 , respectively, as illustrated in Fig. 1. We first focus on the case that the probe field is incident from the left side to the cavity 1. The total Hamiltonian of this system has the form (1) The first term describes the free Hamiltonian of the cavity modes and the mechanical one with ( = 1) where a † i (a i ) for i = 1, 2 and b † (b) are the creation (annihilation) operators for the cavity modes and the mechanical one. The second term characterizes the linear coupling between the cavity modes with coupling strength J and the radiationpressure force interaction between the cavities and the mechanical resonator with single-photon coupling strength g i . The third term H d describes the mechanical drive, the optical pump fields on the cavities, and the probe field (incident from the left side to cavity 1, see the thin solid arrow in Fig. 1) where ω d is the frequency, ε i is the amplitude, and θ i is the phase of the two pump fields, ω p (ω b ) is the frequency and ε p (ε b ) is the amplitude of the probe field on cavity 1 (the mechanical drive applied on the mechanical resonator). It is worth pointing out that the mechanical drive can be easily realized in experiments through an external electric drive [55][56][57][58]. Here without loss of generality, we have assumed that J, g 1,2 , and ε 1,2 are real numbers.
In the rotating frame with respect to the frequency of the pump fields, the quantum Langevin equations (QLEs) for the operators in the system are given bẏ Here ∆ i = ω i − ω d (i = 1, 2) are the optical detunings of the cavities, γ i (γ m ) are the decay rates of the two cavities (mechanical resonator), ξ i (ξ m ) are the noise operators of the cavities (mechanical mode) with ξ i = ξ m = 0. We first derive the steady-state solution of the threemode system under strong pump fields. Neglecting the effects of the optical probe field and mechanical drive, we can obtain the steady-state solution as where a i ( b ) are the steady-state averages of the cavi-ties (mechanical mode), and ∆ ′ i = ∆ i + g i [ b + b * ] (i = 1, 2) are the cavity detunings shifted by the radiationpressure force. These equations are coupled to each other and can be solved self-consistently.
Each operator of this system can be written as a sum of the steady-state solution and its fluctuation with a i = a i + δa i and b = b + δb, where δa i are the fluctuations of the cavities and δb is that of the mechanical mode. Neglecting the nonlinear terms in the radiationpressure interaction in Eqs. (8)-(10), we obtain a set of linear QLEs for the fluctuation operators: where G i = g i a i (i = 1, 2) represent the pumpenhanced linear optomechanical couplings.
In what follows, we fix ω b = ω p − ω d in our scheme, i.e., the frequency of the mechanical drive is always equal to the frequency difference between the optical probe and pump fields. To solve the above QLEs, we transform all the operators to another rotating frame with and ξ m → ξ m e −iω b t . In addition, we assume that the cavities are driven by the red-detuned pump fields and ∆ ′ i ∼ ω m . In this case, by using the rotating-wave approximation, one can neglect the fast-oscillating counterrotating terms and obtain the following linearized QLEs The cavity output fields δa out i (i = 1, 2) can be de-rived from the input-output theorem with where γ e i represents the cavity loss related to coupling between the cavity and the input (output) modes, and is part of the total cavity loss rate γ i with γ e i = η i γ i and η i ≤ 1. For simplicity of discussion, we focus on the case of over-coupled cavities with η i ≃ 1 and neglect cavity intrinsic dissipation [59][60][61]. With this assumption, δa in 1 = ε p / 2γ e 1 , δa in 2 = 0. The input field on the mechanical resonator can then be written in terms of the cavity input with δb in = γ e 1 /γ m (ye iϕ ) δa in 1 . The transmission coefficient that describes the dependence of the output field of cavity 2 on the input field δa in 1 can be defined as With Eqs. (18) and (20), we derive where we have defined the amplitude of the mechanical drive through ε b /ε p = ye iϕ (y > 0). Similarly, we can derive the transmission coefficient for a probe field applied to cavity 2 from the right side (see the thin dashed arrow in Fig. 1). In this case, we have δa in We derive that This equation shows that the propagation of the optical probe field in the three-mode optomechanical system depends strongly on the interference between various paths of the probe field via the optical cavity with amplitude ε p and the frequency-matched mechanical drive with amplitude ε b via the optomechanical interaction. And the transmission is not symmetric between cavities 1 and 2.

III. DIRECTIONAL AMPLIFICATION OF OPTICAL PROBES
In this section, we will study the transmission of optical probe and the asymmetry in the transmission systemat-ically. We will show that amplification of optical probe fields can be directional. Consider G 1 = G > 0 and G 2 = Ge iθ for simplicity of discussion. The transmission coefficients can be rewritten as and When ε b = 0 (y = 0), the model reduces to that studied in [44], where the directional transmission of the probe field can be achieved under optimal parameters. In such a scheme, the introduction of the nontrivial phase θ breaks the time-reversal symmetry of this system and results in nonreciprocal propagation of the probe field. In contrast, in the presence of the frequency-matched mechanical drive and in the absence of the second cavity (J = 0 and G 2 = 0), the system reduces to a standard two-mode optomechanical system. In this case, it was shown that the presence of the mechanical drive ε b leads to the amplification of the output field [26]. The amplification and enhancement in energy arise from the phonon-photon parametric process in the presence of the frequency-matched mechanical drive.
Now we study the effect of the frequency-matched mechanical drive ε b on the propagation of the probe field in the three-mode optomechanical system in the general case of y = 0 and J = 0. In Fig. 2, we plot the probability of the transmission T 21 ≡ |t 21 | 2 and T 12 ≡ |t 12 | 2 as functions of ∆ m = ω m − (ω p − ω d ) at different values of the phases θ and ϕ. We observe that in general, the transmission of the probe field is asymmetric with T 21 = T 12 , and T 12 or T 21 can be much larger than 1. This result indicates nonreciprocity with amplification of the optical probe field. In particular, at certain optimal values of θ and ϕ, e.g., θ = π/2, ϕ = π/2, T 21 → 0 and T 12 ≫ 1, as shown in Figs. 2(c). The transmission from cavity 1 to cavity 2 is strongly amplified; whereas, the transmission on the opposite direction is suppressed. In this case, the amplification of the probe field results from phonon-photon parametric process due to the existence of the frequency-matched mechanical drive [26].
We plot the probability of the transmission T 21 and T 12 as functions of θ and ϕ in Fig. 3. It is also shown that the directional propagation can be achieved with θ = π/2 in Fig. 3(a) or ϕ = π/2 in Fig. 3(b). Note that, when θ = π/2 with other parameters given in the caption of Fig. 3(b), the probability of transmission T 12 is independent of ϕ, which can be given through Eq. (25).
probe field can be observed due to the presence of the mechanical drive frequency-matched to the probe field, and the direction of the amplification is opposite to that in the case of directional transmission in [44]. Strong amplification requires |y c | ≫ 1, i.e., the cavity damping rate γ 1 is approximately equal to the mechanical damping rate γ m .
To study the role of the mechanical drive, we plot T 21 and T 12 as functions of y in Fig. 4. This plot clearly demonstrates that the propagation of the optical field is strongly amplified with T 12 ∼ 600 when the mechanical drive becomes large (|y| ≫ 1). Meanwhile, when y ∼ y c = 20 under the parameters given in the caption of Fig. 4, the transmission in the opposite direction quickly drops with T 21 → 0.

IV. CONCLUSIONS
To conclude, we investigate the transmission of an optical probe field in a three-mode optomechanical system, where the mechanical resonator is subject to a mechanical drive with the driving frequency being equal to the frequency difference between the optical probe and pump fields. Under appropriate parameters, the directional amplification of the probe field resulting from the interference between different optical path and phonon-photon parametric process can be achieved. Amplification far exceeding unity can be achieved when the mechanical drive becomes strong. Such optomechanical setups could be used to switch and amplify weak probe signals in quantum networks.