## Abstract

We theoretically propose a scheme for realizing a quantum-limited directional amplifier in a triple-cavity optomechanical system, where one microwave cavity and two optical cavities are, respectively, coupled to a common mechanical resonator. Moreover, the two optical cavities are coupled directly to facilitate the directional amplification between microwave and optical photons. We find that directional amplification between the three cavity modes is achieved with two gain process and one conversion process, and the direction of amplification can be modulated by controlling the phase difference between the field-enhanced optomechanical coupling strengths. Furthermore, with increasing the optomechanical cooperativity, both gain and bandwidth of the directional amplifier can be enhanced, and the noise added to the amplifier can be suppressed to approach the standard quantum limit on the phase-preserving linear amplifier.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The electromagnetic wave transmission in media is generally invariant under exchange of source and detector [1], but it is one of the basic requirements to realize directional transmission in both classical and quantum information processing. Consequently, nonreciprocal devices, including isolators, circulators and directional amplifiers are of crucial importance, which can protect the signal source from spurious interferences. The conventional solution to achieve nonreciprocity is to bias the propagation channel by the external magnetic field, which requires the magneto-optic materials and therefore prevents the integration and miniaturization [2–4]. In the past decades, a variety of alternative strategies have been developed to break reciprocity without needing magneto-optical effects, such as refractive-index modulation [5,6], optical nonlinearity [7–9], angular momentum biasing in photonic or acoustic systems [10–12], and the quantum Hall effect [13].

Furthermore, cavity optomechanics studies the parametric coupling between an electromagnetic cavity and a mechanical resonator via radiation pressure [14–16]. Based on optomechanical couplings, significant achievements have been made in this field, including ground state cooling of the mechanical resonator [17,18], optomechanically induced transparency (OMIT) [19–24], microwave amplification [25–27], bidirectional state conversion between optical and microwave photons [28–31], phonon laser [32–35], and parity-time-symmetry-breaking chaos [36]. Recently, the radiation-pressure-induced optomechanical coupling has been exploited to break time-reversal symmetry and realize electromagnetic nonreciprocity at optical [37–48] and microwave frequencies [49–54]. Many of these works rely on controlling the relative phases of the field-enhanced optomechanical coupling strengths to achieve nonreciprocity. It is worth noting that directional amplifier can be realized with optomechanical systems by reservoir engineering [45], coherent mechanical drive [46], and blue-detuned optical pumping [47, 51]. Furthermore, directional amplifier has also been demonstrated in the superconducting microwave circuit with Josephson nonlinearity and parametric pumping [55–57].

Motivated by the above achievements, we theoretically investigate how to realize a quantum-limited directional amplifier based on a triple-cavity optomechanical system, where one microwave cavity and two directly coupled optical cavities are, respectively, coupled to a common mechanical resonator [49]. When the microwave cavity is pumped on its red sideband and the two optical cavities are pumped on their respective blue sidebands, directional amplification between microwave and optical photons can be achieved with two gain process and one conversion process. We show that the direction of the amplification depends on the relative phase between the field-enhanced optomechanical strengths, and the gain and bandwidth of the amplifier can be enhanced by increasing the optomechanical cooperativity. Moreover, the mechanical noise can be suppressed in the large cooperativity limit, and this phase-preserving amplifier can approach the quantum limit of a half added quanta.

Compared with the optical nonreciprocity in Ref. [37–48], our scheme can realize the directional amplification between optical and microwave photons without requiring the direct coupling between the optical and microwave cavities. Instead, another optical cavity is introduced to control the direction of amplification. In particular, the proposed directional amplifier is robust against mechanical noise and can reach the quantum-limited value of added noise. Note that quantum-limited directional amplifier for microwave signals has been proposed recently in an electromechanical setup comprising two microwave cavities and two mechanical resonators [51]. Compared with Ref. [51], our work can realize the directional amplification among the three cavities by combing two gain process and one conversion process, which has been demonstrated based on the Josephson Parametric Converter [56]. Furthermore, the phase difference at which isolation occurs depends on the detuning between the pump and cavity field in Ref. [51], but directional amplification can be achieved when the phase difference *ϕ* = ±*π*/2 in our work, which is easier to control.

The remainder of the paper is organized as follows. In Sec. 2, we introduce the theoretical model and derive the transmission matrix between the input and output operators. In Sec. 3, we investigate how to realize the quantum-limited directional amplifier based on this optomechanical system. The effects of phase difference and optomechanical cooperativity on the amplifier are discussed. We finally summarize our work in Sec. 4.

## 2. Model and theory

We consider the optomechanical system schematically shown in Fig. 1, where three cavity modes *a*_{1}, *a*_{2}, *a*_{3} are independently coupled to a common mechanical mode *b* via radiation pressure. Furthermore, cavities *a*_{2} and *a*_{3} are coupled directly via hopping interaction to facilitate the nonreciprocal transmission. Cavity *a*_{1} can have vastly different frequency from that of cavities *a*_{2} and *a*_{3}, e.g., cavity *a*_{1} is a microwave cavity while cavities *a*_{2} and *a*_{3} are two optical cavities, or vice versa. This model is still valid when all the three cavities are optical cavities, but here we focus on the directional amplification between optical and microwave photons based on this system, which can be realized with current experimental technology [28, 31, 58–60]. The system which consists of two cavity modes coupled to a common mechanical resonator has been realized [28,31]. When *a*_{1} is a microwave cavity and *a*_{2} (*a*_{3}) are optical cavities, time-dependent interaction between two optical cavities needs to be introduced, which can be generated by connecting the cavities to other cavity modes or waveguides [49, 60, 61]. In order to realize directional amplification, we apply a red-detuned pump field to drive the cavity mode *a*_{1} and two blue-detuned pump fields to drive the cavity modes *a*_{2} and *a*_{3}, respectively. The Hamiltonian of this optomechanical system reads (*ħ* = 1)

*a*(

_{k}*k*= 1, 2, 3) with resonance frequency

*ω*, and

_{k}*b*(

*b*

^{†}) is the annihilation (creation) operator of the mechanical mode

*b*with resonance frequency

*ω*. The third term represents the intercation between the cavity mode

_{m}*a*and the mechanical mode

_{k}*b*, where g

*is the single-photon coupling strength between the cavity mode*

_{k}*a*and the mechanical mode. The fourth term describes the interaction between the cavity mode

_{k}*a*

_{2}and

*a*

_{3}with the coupling strength

*J*. The last term shows the interaction between the pump fields and the respective cavity modes, where

*ε*and

_{k}*ω*,

_{d}*are the amplitude and frequency of the pump field applied to the cavity mode*

_{k}*a*. The driving terms in Eq. (1) can be made time independent by applying the unitary transformation $U=\mathrm{exp}\left(i{\displaystyle {\sum}_{k=1}^{3}{\omega}_{d,k}{a}_{k}^{\u2020}{a}_{k}t}\right)$, and the new Hamiltonian

_{k}*H*

_{rot}=

*UHU*

^{†}−

*iU∂U*

^{†}/

*∂t*is then given by

*ω*

_{d}_{,2}=

*ω*

_{d}_{,3}, and ∆

*=*

_{k}*ω*−

_{k}*ω*,

_{d}*is the detuning between the cavity mode*

_{k}*a*and the respective pump field.

_{k}According to Eq. (2), we can get the following quantum Langevin equations (QLEs):

*κ*

_{1},

*κ*

_{2}and

*κ*

_{3}are the decay rates of cavity mode

*a*(

_{k}*k*= 1, 2, 3), and

*γ*is the damping rate of the mechanical mode

_{m}*b*. Furthermore,

*a*

_{k}_{,in}(

*k*= 1, 2, 3) and

*b*

_{in}are the noise operators for the cavity modes and mechanical mode with zero mean value and satisfy the following correlation functions

*n*of the mechanical resonator is given by ${n}_{m}=1/({e}^{\hslash {\omega}_{m}/{k}_{\mathrm{B}}{T}_{m}}-1)$, with

_{m}*T*being the environmental temperature and

_{m}*k*

_{B}being the Boltzmann constant.

The steady-state solutions for the cavity and mechanical modes can be obtained by setting all the time derivatives in Eqs. (3)–(6) to be zero and neglecting all the noise terms, which obey the following algebraic equations:

*α*(

_{k}*k*= 1, 2, 3) and

*β*represent, respectively, the mean values of the cavity mode

*a*and mechanical mode

_{k}*b*, and ${\mathrm{\Delta}}_{k}^{\prime}={\mathrm{\Delta}}_{k}+{g}_{k}({\beta}^{*}+\beta )$ is the effective cavity detuning including radiation pressure effect. Subsequently, we can linearize the QLEs (3)–(6) by rewriting each Heisenberg operator as the sum of its steady-state mean value and a small fluctuation, i.e.,

*a*=

_{k}*α*+

_{k}*δa*and

_{k}*b*=

*β*+

*δb*. The linearized QLEs are given by:

To realize the directional amplification, we consider the case that ${\mathrm{\Delta}}_{1}^{\prime}={\omega}_{m}$ and ${\mathrm{\Delta}}_{2}^{\prime}={\mathrm{\Delta}}_{3}^{\prime}=-{\omega}_{m}$. In the resolved sideband limit with *ω _{m}* ≫

*γ*(

_{m}, κ_{k}*k*= 1, 2, 3), the rotating wave approximation (RWA) can be applied. For simplicity, we can move into another interaction picture by introducing $\delta {a}_{k}\to \delta {a}_{k}{e}^{-{\mathrm{\Delta}}_{k}^{\prime}t}$, $\delta b\to \delta {b}^{-i{\omega}_{m}t}$, ${a}_{k,\text{in}}\to {a}_{k,\text{in}}{e}^{-{\mathrm{\Delta}}_{k}^{\prime}t}$, and ${b}_{\text{in}}\to {b}_{\text{in}}{e}^{-i{\omega}_{m}t}$ [62], and then Eqs. (9)–(12) become

*G*being the field-enhanced coupling strength. The effective Hamiltonian associated with Eqs. (13)–(16) is

_{k}*ϕ*(

_{k}*k*= 1, 2, 3) can be absorbed by redefining the operators

*δa*and

_{k}*δb*, and only the phase difference

*ϕ*=

*ϕ*

_{3}−

*ϕ*

_{2}in the closed-loop interaction formed by

*G*

_{2},

*G*

_{3}and

*J*has physical effects. For convenience, we can set

*ϕ*

_{1}=

*ϕ*

_{2}= 0 and

*ϕ*

_{3}=

*ϕ*, and the symbol

*δ*can be neglected, i.e.,

*δa*→

_{k}*a*and

_{k}*δb*→

*b*. Subsequently, the QLEs (13)–(16) can be written in the following matrix form: where the vector $\mu ={\left({a}_{1}^{\u2020},{a}_{2},{a}_{3},{b}^{\u2020}\right)}^{\mathrm{T}}$, ${\mu}_{\text{in}}={\left({a}_{1,\text{in}}^{\u2020},{a}_{2,\text{in}},{a}_{3,\text{in}},{b}_{\text{in}}^{\u2020}\right)}^{\mathrm{T}}$, the diagonal matrix

*K*= Diag[

*κ*

_{1},

*κ*

_{2},

*κ*

_{3},

*γ*], and the coefficient matrix

_{m}*M*are negative, The stability condition can be derived analytically by the Routh-Hurwitz criterion [63, 64], whose general form is too cumbersome to give here. However, we will check the stability diagram numerically and choose the parameters that satisfy the stability condition.

By introducing the Fourier transform of the operators

*I*represents the unitary matrix. Upon substituting Eq. (22) into the standard input-output relation [65] ${\mu}_{\text{out}}(\omega )={\mu}_{\text{in}}(\omega )-\sqrt{K}\mu (\omega )$, we can obtain where the output field vector

*µ*

_{out}(

*ω*) is the Fourier transform of ${\mu}_{\text{out}}={\left({a}_{1,\text{out}}^{\u2020},{a}_{2,\text{out}},{a}_{3,\text{out}},{b}_{\text{out}}^{\u2020}\right)}^{\mathrm{T}}$, and the transmission matrix is given by Here the matrix element

*T*(

_{ij}*ω*) (

*i, j*= 1, 2, 3, 4 correspond to

*a*

_{1},

*a*

_{2},

*a*

_{3},

*b*, respectively) describes the transmission amplitude from mode

*j*to mode

*i*.

## 3. Quantum-limited directional amplifier

In this section, we discuss in detail how to realize the directional amplifier based on this triple-cavity optomechanical system. We have assumed that cavity *a*_{1} is driven on its red sideband while cavities *a*_{2} and *a*_{3} are driven on their respective blue sidebands. For convenience, we first discuss the condition of directional amplification between cavities *a*_{1} and *a*_{2} when an input field is resonant with the cavity frequency, i.e., *ω* = 0. According to Eqs. (19) and (24), the transmission matrix elements *T*_{21} and *T*_{12} can be given by

_{1}= −

*κ*

_{1}/2 +

*iω*, Γ

*= −*

_{m}*γ*/2 +

_{m}*iω*, Γ

_{2}= −

*κ*

_{2}/2 +

*iω*, and Γ

_{3}= −

*κ*

_{3}/2 +

*iω*. Without loss of generality, we consider the case that the probe field incident on cavity

*a*

_{1}can be amplified when it is transmitted from cavity

*a*

_{2}, but the probe field cannot be transmitted in the reverse direction, i.e., |

*T*

_{21}|

^{2}> 1 and |

*T*

_{12}|

^{2}= 0. According to Eqs. (25) and (26), it can be easily obtained that |

*T*

_{12}|

^{2}= 0 and |

*T*

_{21}|

^{2}≠ 1 when

*ϕ*= −

*π*/2 and

*G*

_{3}=

*G*

_{2}

*κ*

_{3}/(2

*J*), and we will show that |

*T*

_{21}|

^{2}can be much larger than 1 due to constructive interference in this hybrid system. Furthermore, to prevent loss of the input field to other modes, it requires that |

*T*

_{i}_{1}/

*T*

_{21}| ≪ 1 (

*i*= 3, 4), which can be realized by choosing $J=\sqrt{{\kappa}_{2}{\kappa}_{3}}/3$. Under the above the conditions, the transmission amplitude

*T*

_{21}on resonance can be given by

*k*= 1, 2. Furthermore, the full transmission matrix on resonance is

In the following, we demonstrate numerically the directional amplification between different cavity modes. The parameters are chosen from the recent experiments [17, 18,20]: the decay rates of both optical and microwave cavities can be *κ _{k}* =1–10 MHz and the field-enhanced coupling strength

*G*can reach a few tens of MHz. We assume that

_{k}*κ*

_{1}/2

*π*= 2 MHz,

*κ*

_{2}/2

*π*=

*κ*

_{3}/2

*π*= 3 MHz, and the damping rate of the mechanical resonator

*γ*/2

_{m}*π*= 30 kHz. Because cavity modes

*a*

_{2}and

*a*

_{3}are driven on their blue sidebands, we first plot the stability diagram with respect to the cooperativities

*C*

_{1}and

*C*

_{2}in Fig. 2, and we will choose the parameters in the stable regime to realize the directional amplifier. It can be seen that the system is stable when

*C*

_{2}<

*C*

_{1}≤ 60, and we can choose ${C}_{2}={C}_{1}-0.1\sqrt{{C}_{1}}$ for a given

*C*

_{1}to ensure the system operates in the stable regime.

According to Eqs. (25) and (26), we can investigate the directional amplification between cavities *a*_{1} and *a*_{2}. Figure 3 plots |*T*_{12}|^{2} and |*T*_{21}|^{2} as functions of the probe detuning *ω* for *ϕ* = −*π*/2, 0, *π/*2, and *π*, respectively. It can be seen from Fig. 3(a) that |*T*_{21}(0)|^{2} ≈ 23 dB while |*T*_{12}(0)|^{2} = 0, which represents that the resonant probe field incident on cavity *a*_{1} can be greatly amplified when it is transmitted from cavity *a*_{2}, but the transmission from cavity *a*_{2} to cavity *a*_{1} is totally forbidden. Therefore, directional amplification between microwave and optical photons is realized based on this optomechanical system. This phenomenon can be explained in terms of the interference between two possible paths, where one is along *a*_{1} → *b* → *a*_{2} with a transmission amplitude proportional to *G*_{2}, another is along *a*_{1} → *b* → *a*_{3} → *a*_{2} with a transmission amplitude proportional to *iJe ^{iϕ}G*

_{3}/Γ

_{3}. When

*ϕ*= −

*π/*2,

*ω*= 0, and

*G*

_{3}=

*G*

_{2}

*κ*

_{3}/(2

*J*), constructive interference between the two paths results in the amplification from cavity

*a*

_{1}to cavity

*a*

_{2}, but the transmission from

*a*

_{2}to

*a*

_{1}is suppressed due to destructive interference. If the phase difference

*ϕ*is tuned to be

*π*/2, we can see from Fig. 3(c) that the direction of amplification is reversed compared with that in Fig. 3(a), which can be illustrated by changing the direction of all the arrows in Fig. 1. In both cases,

*φ*≠

*nπ*(

*n*is an integer), the time-reversal symmetry is broken and the nonreciprocal transmission appears in this optomechanical system. However, when

*φ*= ±

*π*, the transmission probability |

*T*

_{21}|

^{2}is equal to |

*T*

_{12}|

^{2}when the probe detuning

*ω*varies, as shown in Fig. 3(b) and 3(d). The transmission in different direction is reciprocal and there’s no directional amplification.

Furthermore, directional amplification between the three cavity modes is discussed in Fig. 4 with phase difference *ϕ* = −*π*/2. It can be seen from Fig. 4(a) that the signal field incident on cavity cavity *a*_{1} can be directionally amplified when it is transmitted from cavity *a*_{2}, and the signal field incident on cavity *a*_{2} can be transmitted to cavity *a*_{3} with unity gain, as shown in Fig. 4(b). Furthermore, Fig. 4(c) shows that the signal field incident on cavity *a*_{3} can be directionally amplified when it is transmitted from cavity *a*_{1}, but not vice versa. Therefore, directional amplification in this optomechanical system is realized by combining two gain (*G*) process and one conversion (*C*) process, which is similar to the directional amplifier based on the Josephson Parametric Converter (JPC) [56]. Here, the three cavities can be viewed as the three ports in the directional amplifier, which can be labeled as the Signal (S) input, Idler (I) input, and Vacuum (V) input, respectively. The signal field incident on S port (amplifier input) can be directionally amplified from I port (amplifier output), and the signal incident on V port is transmitted with unity gain back to the S port. When *ϕ* = −*π*/2, cavity *a*_{3} serves as the S port, cavity *a*_{1} is the I port, and *a*_{2} can be labeled as V port. The directional amplification is along the path ${a}_{3}\stackrel{G}{\to}{a}_{1}\stackrel{G}{\to}{a}_{2}\stackrel{C}{\to}{a}_{3}$, as shown in Fig. 4(d). In addition, if we tune the phase difference *ϕ* to be *π*/2, the direction of the amplification will be reversed, which is along ${a}_{2}\stackrel{G}{\to}{a}_{1}\stackrel{G}{\to}{a}_{3}\stackrel{C}{\to}{a}_{2}$. In this case, cavity *a*_{2} serves as the S port and cavity *a*_{3} is the V port. As discussed in Fig. 4, cavity *a*_{3} serves as the amplifier input port and cavity *a*_{1} can be viewed as the amplifier output port when the phase difference *ϕ* = −*π/*2. However, other modes such as the mechanical mode will inevitably add some noise to the directional amplifier. The added number of noise quanta of the amplifier can be obtained by calculating the output spectra of cavity *a*_{1}, which is given by [53, 66]

*o*

^{†}(

*ω*) = [

*o*(−

*ω*)]

^{†}. The noise added to the signal is defined as [51,66–68] ${\mathcal{N}}_{1}(\omega )={\mathcal{G}}^{-1}{\displaystyle {\sum}_{i\ne 3}({n}_{i}+1/2){\left|{T}_{1i}(\omega )\right|}^{2}}$, where the power gain of the directional amplifier is given by

*a*

_{1}is given by

*n*

_{1}=

*n*

_{2}=

*n*

_{3}= 0 for the cavity modes and the thermal phonon number

*n*

_{4}=

*n*for the mechanical mode. We can see from Eq. (31) that the thermal noise from the mechanical resonator can be suppressed by increasing the cooperativity

_{m}*C*

_{2}. In addition, for large

*C*

_{1}≥

*C*

_{2}, we find at zero frequency ${\mathcal{N}}_{1}(0)\to \frac{1}{2}$, which is the quantum-limited value of the added noise for the phase-preserving linear amplifier [51,66–68].

To see more clearly the effect of the cooperativity on the gain of the amplifier and the added noise, in Fig. 5, we plot (a) the transmission probability |*T*_{13}|^{2} and (b) the added noise ${\mathcal{N}}_{1}$ with respect to the probe detuning *ω* for different values of cooperativity *C*_{1} with ${C}_{2}={C}_{1}-0.1\sqrt{{C}_{1}}$. It can be seen from Fig. 5(a) that the peak value of |*T*_{13}|^{2} and the bandwidth become larger when *C*_{1} increases from 1 to 50. Furthermore, Fig. 5(b) shows that the added number of noise quanta ${\mathcal{N}}_{1}$ decreases with increasing the cooperativity *C*_{1}. In particular, when *C*_{1} and *C*_{2} are big enough, ${\mathcal{N}}_{1}$ on resonance can approach 1/2. Therefore, we can realize a quantum-limited directional amplifier based on this triple-cavity optomechanical system.

Finally, Fig. 5 shows that the gain and the bandwidth of the directional amplifier are enhanced by increasing the cooperativity *C*_{1} with ${C}_{2}={C}_{1}-0.1\sqrt{{C}_{1}}$. The expressions of bandwidth and gain-bandwidth product can be approximated obtained as follows. If we assume the decay rates of the three cavities are the same, i.e., *κ*_{1} = *κ*_{2} = *κ*_{3} = *κ*, then the denominator in Eq. (12) $A(\omega )\approx \frac{1}{8}{\kappa}^{3}{\gamma}_{m}({C}_{1}-{C}_{2}+1)-\frac{1}{4}{\gamma}_{m}{\kappa}^{2}(\kappa /{\gamma}_{m}+{C}_{1}-2{C}_{2}+2)i\omega $, where only the terms to the first order of *ω* is kept [51]. The bandwidth Γ is approximated by the smallest |*ω*| at which 2|*A*(0)|^{2} = |*A*(*ω*)|^{2} and we obtain the bandwidth Γ = |*κ* (*C*_{1} − *C*_{2} + 1)/(*κ*/*γ _{m}* +

*C*

_{1}− 2

*C*

_{2}+ 2)|. For

*κ*/

*γ*≫ {1,

_{m}*C*

_{1},

*C*

_{2}} the bandwidth Γ ≈

*γ*(

_{m}*C*

_{1}−

*C*

_{2}+ 1), which is constrained by the damping rate of the mechanical resonator. Consequently, the gain-bandwidth product $P\equiv \mathrm{\Gamma}\sqrt{\mathcal{G}}\approx 2{\gamma}_{m}\sqrt{{C}_{1}{C}_{2}}$. In addition, one should also consider the isolation bandwidth, where sufficient isolation is attained. Close to

*ω*= 0, the reverse transmission amplitude ${T}_{12}(\omega )\approx -i\omega \sqrt{\mathcal{G}}/\kappa $. Thus, the isolation bandwidth is of order $\kappa /\sqrt{\mathcal{G}}$, which can be broadened by increasing the cavity decay rate

*κ*. In our scheme, the isolation bandwidth is larger than the gain bandwidth, and therefore the signal field can be directionally amplified within the gain bandwidth.

## 4. Conclusion

To conclude, we have presented a quantum-limited directional amplifier based on a triple-cavity optomechanical system, where one microwave cavity is pumped on its red sideband and two coupled optical cavities are pumped on their respective blue sidebands. Quantum interference between two possible paths in this optomechanical systems results in the directional amplification between microwave and optical photons, where the phase difference between the optomechanical coupling strengths plays a vital role. Furthermore, we have shown that the gain and bandwidth of the amplifier can be improved by increasing the optomechanical cooperativity, and the noise added to this amplifier can approach the quantum limit of half a quanta in the large cooperativity.

## Funding

Natural Science Foundation of China (No. 11304110, No. 11604115); Postdoctoral Science Foundation of China (Grant No. 2017M620593); Qing Lan Project of Universities in Jiangsu Province.

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