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 a1, a2, a3 are independently coupled to a common mechanical mode b via radiation pressure. Furthermore, cavities a2 and a3 are coupled directly via hopping interaction to facilitate the nonreciprocal transmission. Cavity a1 can have vastly different frequency from that of cavities a2 and a3, e.g., cavity a1 is a microwave cavity while cavities a2 and a3 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 a1 is a microwave cavity and a2 (a3) 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 a1 and two blue-detuned pump fields to drive the cavity modes a2 and a3, respectively. The Hamiltonian of this optomechanical system reads (ħ = 1)

H=k=13ωkakak+ωmbb+k=13gkakak(b+b)+J(a2a3+a3a2)+k=13εk(akeiωd,kt+akeiωd,kt),
where ak(ak) is the annihilation (creation) operator of the cavity mode ak (k = 1, 2, 3) with resonance frequency ωk, and b (b) is the annihilation (creation) operator of the mechanical mode b with resonance frequency ωm. The third term represents the intercation between the cavity mode ak and the mechanical mode b, where gk is the single-photon coupling strength between the cavity mode ak and the mechanical mode. The fourth term describes the interaction between the cavity mode a2 and a3 with the coupling strength J. The last term shows the interaction between the pump fields and the respective cavity modes, where εk and ωd,k are the amplitude and frequency of the pump field applied to the cavity mode ak. The driving terms in Eq. (1) can be made time independent by applying the unitary transformation U=exp(ik=13ωd,kakakt), and the new Hamiltonian Hrot = UHUiU∂U/∂t is then given by
Hrot=k=13Δkakak+ωmbb+k=13gkakak(b+b)+J(a2a3+a3a2)+k=13εk(ak+ak),
where we have set ωd,2 = ωd,3, and ∆k = ωkωd,k is the detuning between the cavity mode ak and the respective pump field.

 figure: Fig. 1

Fig. 1 Schematic diagram of the triple-cavity optomechanical system. One microwave cavity a1 and two optical cavities a2 and a3 are respectively coupled to the common mechanical resonator with the field-enhanced optomechanical coupling strength Gk(k = 1, 2, 3), where ϕ is the phase difference between Gk. Moreover, the two optical cavities are coupled directly via hopping interaction J to facilitate the nonreciprocal transmission. Cavity a1 is driven on its red sideband by a pump field at the frequency ωd,1, and cavities a2 and a3 are driven on their respective blue sidebands by the pump fields at the frequencies ωd,2 and ωd,3. The solid arrows represent the transmission elements between different modes.

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

a˙1=(κ1/2iΔ1)a1+ig1a1(b+b)+iε1+κ1a1,in,
a˙2=(κ2/2+iΔ2)a2ig2a2(b+b)iJa3iε2+κ2a2,in,
a˙3=(κ3/2+iΔ3)a3ig3a3(b+b)iJa2iε3+κ3a3,in,
b˙=(γm/2iωm)b+ikgkakak+γmbin.
where the decay and noise terms have been added phenomenologically. κ1, κ2 and κ3 are the decay rates of cavity mode ak (k = 1, 2, 3), and γm is the damping rate of the mechanical mode b. Furthermore, ak,in (k = 1, 2, 3) and bin are the noise operators for the cavity modes and mechanical mode with zero mean value and satisfy the following correlation functions
ak,in(t)ak,in(t)=δ(tt),ak,in(t)ak,in(t)=0,bin(t)bin(t)=(nm+1)δ(tt),bin(t)bin(t)=nmδ(tt),
where we have assumed that the thermal photon number equals to zero and the thermal phonon number nm of the mechanical resonator is given by nm=1/(eωm/kBTm1), with Tm being the environmental temperature and kB 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:

α1*=iε1κ1/2iΔ1,α2=iJα3+iε2κ2/2+iΔ2,α3=iJα2+iε3κ3/2+iΔ3,β*=ikgk|αk|2γm/2iωm,
where αk (k = 1, 2, 3) and β represent, respectively, the mean values of the cavity mode ak and mechanical mode b, and Δk=Δk+gk(β*+β) 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., ak = αk + δak and b = β + δb. The linearized QLEs are given by:
δa˙1=(κ1/2iΔ1)δa1+igα1*(δb+δb)+κ1a1,in,
δa˙2=(κ2/2+iΔ2)δa2ig2α2(δb+δb)iJδa3+κ2a2,in,
δa˙3=(κ3/2+iΔ3)δa3ig3α3(δb+δb)iJδa2+κ3a3,in,
δb˙=(γm/2iωm)δb+ikgk(αk*δak+αkδak)+γmbin.

To realize the directional amplification, we consider the case that Δ1=ωm and Δ2=Δ3=ωm. In the resolved sideband limit with ωmγm, κk (k = 1, 2, 3), the rotating wave approximation (RWA) can be applied. For simplicity, we can move into another interaction picture by introducing δakδakeΔkt, δbδbiωmt, ak,inak,ineΔkt, and binbineiωmt [62], and then Eqs. (9)(12) become

δa˙1=κ12δa1+iG1eiϕ1δb+κ1a1,in,
δa˙2=κ22δa2iG2eiϕ2δbiJδa3+κ2a2,in,
δa˙3=κ32δa3iG3eiϕ3δbiJδa2+κ3a3,in,
δb˙=γm2δb+i(G1eiϕ1δa1+G2eiϕ2δa2+G3eiϕ3δa3)+γmbin,
where we have set gkαk=gk|αk|eiϕk=Gkeiϕk with Gk being the field-enhanced coupling strength. The effective Hamiltonian associated with Eqs. (13)(16) is
Heff=G1δa1δbeiϕ1+G2δa2δbeiϕ2+G3δa3δbeiϕ3+Jδa2δa3+H.c..
The phases ϕk (k = 1, 2, 3) can be absorbed by redefining the operators δak and δb, and only the phase difference ϕ = ϕ3ϕ2 in the closed-loop interaction formed by G2, G3 and J has physical effects. For convenience, we can set ϕ1 = ϕ2 = 0 and ϕ3 = ϕ, and the symbol δ can be neglected, i.e., δakak and δbb. Subsequently, the QLEs (13)(16) can be written in the following matrix form:
μ˙=Mμ+Kμin,
where the vector μ=(a1,a2,a3,b)T, μin=(a1,in,a2,in,a3,in,bin)T, the diagonal matrix K = Diag[κ1, κ2, κ3, γm], and the coefficient matrix
M=(κ1/200iG10κ2/2iJiG20iJκ3/2iG3eiϕiG1iG2iG3eiϕγm/2).
The system is stable only if the real parts of all the eigenvalues of matrix 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

o(ω)=+o(t)eiωtdt,
o(ω)=+o(t)eiωtdt,
the solution to the Eq. (18) in the frequency domain is
μ(ω)=(M+iωI)1Kμin(ω),
where I represents the unitary matrix. Upon substituting Eq. (22) into the standard input-output relation [65] μout(ω)=μin(ω)Kμ(ω), we can obtain
μout(ω)=T(ω)μin(ω),
where the output field vector µout(ω) is the Fourier transform of μout=(a1,out,a2,out,a3,out,bout)T, and the transmission matrix is given by
T(ω)=I+K(M+iωI)1K.
Here the matrix element Tij(ω) (i, j = 1, 2, 3, 4 correspond to a1, a2, a3, 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 a1 is driven on its red sideband while cavities a2 and a3 are driven on their respective blue sidebands. For convenience, we first discuss the condition of directional amplification between cavities a1 and a2 when an input field is resonant with the cavity frequency, i.e., ω = 0. According to Eqs. (19) and (24), the transmission matrix elements T21 and T12 can be given by

T12(ω)=κ1κ2A(ω)G1Γ3(G2+iJeiϕG3/Γ3),
T21(ω)=κ1κ2A(ω)G1Γ3(G2+iJeiϕG3/Γ3),
where A(ω)=Γ1(Γ2Γ3ΓmΓ2G32Γ3G22+ΓmJ22iJcosϕG2G3)+G12(Γ2Γ3+J2), Γ1 = −κ1/2 +, Γm = −γm/2 + , Γ2 = −κ2/2 +, and Γ3 = −κ3/2 +. Without loss of generality, we consider the case that the probe field incident on cavity a1 can be amplified when it is transmitted from cavity a2, but the probe field cannot be transmitted in the reverse direction, i.e., |T21|2 > 1 and |T12|2 = 0. According to Eqs. (25) and (26), it can be easily obtained that |T12|2 = 0 and |T21|2 ≠ 1 when ϕ = −π/2 and G3 = G2κ3/(2J), and we will show that |T21|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 |Ti1/T21| ≪ 1 (i = 3, 4), which can be realized by choosing J=κ2κ3/3. Under the above the conditions, the transmission amplitude T21 on resonance can be given by
T21(0)=8κ1κ2G1G24κ2G124κ1G22+κ1κ2γm=2C1C2C1C2+1,
with the optomechanical cooperativity Ck=4Gk2/(κkγm) for k = 1, 2. Furthermore, the full transmission matrix on resonance is
T(0)=(C1+C21C1C2+102iC1C2C1C2+12iC1C1C2+12C1C2C1C2+10iC1+C2+1C1C2+12iC2C1C2+10i002iC1C1C2+102C2C1C2+1C1C21C1C2+1),

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 κk =1–10 MHz and the field-enhanced coupling strength Gk can reach a few tens of MHz. We assume that κ1/2π = 2 MHz, κ2/2π = κ3/2π = 3 MHz, and the damping rate of the mechanical resonator γm/2π = 30 kHz. Because cavity modes a2 and a3 are driven on their blue sidebands, we first plot the stability diagram with respect to the cooperativities C1 and C2 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 C2 < C1 ≤ 60, and we can choose C2=C10.1C1 for a given C1 to ensure the system operates in the stable regime.

 figure: Fig. 2

Fig. 2 Stability diagram with respect to C1 and C2. Other parameters are κ1/2π = 2 MHz, κ2/2π = 3 MHz, κ3/2π = 3 MHz, γm = κ2/100, ω = 0, ϕ = −π/2, G3 = G2κ3/(2J), and Jκ2κ3/2.

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According to Eqs. (25) and (26), we can investigate the directional amplification between cavities a1 and a2. Figure 3 plots |T12|2 and |T21|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 |T21(0)|2 ≈ 23 dB while |T12(0)|2 = 0, which represents that the resonant probe field incident on cavity a1 can be greatly amplified when it is transmitted from cavity a2, but the transmission from cavity a2 to cavity a1 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 a1ba2 with a transmission amplitude proportional to G2, another is along a1ba3a2 with a transmission amplitude proportional to iJeG33. When ϕ = −π/2, ω = 0, and G3 = G2 κ3/(2J), constructive interference between the two paths results in the amplification from cavity a1 to cavity a2, but the transmission from a2 to a1 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 (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 |T21|2 is equal to |T12|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.

 figure: Fig. 3

Fig. 3 Transmission probabilities |T12|2 and |T21|2 as functions of the probe detuning ω for ϕ = −π/2, 0, π/2, and π, respectively. Other parameters are κ1/2π = 2 MHz, κ2/2π = 3 MHz, κ3/2π = 3 MHz, γm = κ2/100, C1 = 10, C2=C10.1C1, G3 = G2κ3/(2J), and Jκ2κ3/2.

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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 a1 can be directionally amplified when it is transmitted from cavity a2, and the signal field incident on cavity a2 can be transmitted to cavity a3 with unity gain, as shown in Fig. 4(b). Furthermore, Fig. 4(c) shows that the signal field incident on cavity a3 can be directionally amplified when it is transmitted from cavity a1, 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 a3 serves as the S port, cavity a1 is the I port, and a2 can be labeled as V port. The directional amplification is along the path a3Ga1Ga2Ca3, 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 a2Ga1Ga3Ca2. In this case, cavity a2 serves as the S port and cavity a3 is the V port. As discussed in Fig. 4, cavity a3 serves as the amplifier input port and cavity a1 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 a1, which is given by [53, 66]

S1,out(ω)=12dΩ2πa1,out(ω)a1,out(Ω)+a1,out(Ω)a1,out(ω)=12|T11(ω)|2+12|T12(ω)|2+12|T13(ω)|2+(nm+12)|T14(ω)|2,
where we have used the noise correlation function in the frequency domain and the relation o(ω) = [o(−ω)]. The noise added to the signal is defined as [51,66–68] N1(ω)=G1i3(ni+1/2)|T1i(ω)|2, where the power gain of the directional amplifier is given by
G=|T31(0)|2=4C1C2(C1C2+1)2.
On resonance, the noise added to the output of cavity a1 is given by
N1(0)=12(C1+C21)24C1C2+(nm+12)1C2,
where we have assumed that the thermal photon number n1 = n2 = n3 = 0 for the cavity modes and the thermal phonon number n4 = nm 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 C2. In addition, for large C1C2, we find at zero frequency N1(0)12, which is the quantum-limited value of the added noise for the phase-preserving linear amplifier [51,66–68].

 figure: Fig. 4

Fig. 4 Directional amplifier. (a)–(c) Transmission probabilities |Tij|2 (i, j = 1, 2, 3) as functions of the probe detuning ω/2π. (d) Graphical representation of the amplification process, where S, I, and V represent/the “Signal”, “Idler”, and “Vacuum” ports, respectively. G and C correspond to the gain and conversion process. Other parameters used are ϕ = −π/2, κ1/2π = 2 MHz, κ2/2π = 3 MHz, κ3/2π = 3 MHz, γm = κ2/100, C1 = 3, C2=C10.1C1, G3 = G2κ3/(2J), and Jκ2κ3/2.

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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 |T13|2 and (b) the added noise N1 with respect to the probe detuning ω for different values of cooperativity C1 with C2=C10.1C1. It can be seen from Fig. 5(a) that the peak value of |T13|2 and the bandwidth become larger when C1 increases from 1 to 50. Furthermore, Fig. 5(b) shows that the added number of noise quanta N1 decreases with increasing the cooperativity C1. In particular, when C1 and C2 are big enough, N1 on resonance can approach 1/2. Therefore, we can realize a quantum-limited directional amplifier based on this triple-cavity optomechanical system.

 figure: Fig. 5

Fig. 5 (a) Transmission probability |T13|2 and (b) added noise N1 of the cavity a1 as a function of the probe detuning ω with C1 = {1, 5, 10, 50} and C2=C10.1C1. Other parameters used are the same as those in Fig. 4 except nm = 50.

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Finally, Fig. 5 shows that the gain and the bandwidth of the directional amplifier are enhanced by increasing the cooperativity C1 with C2=C10.1C1. 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., κ1 = κ2 = κ3 = κ, then the denominator in Eq. (12) A(ω)18κ3γm(C1C2+1)14γmκ2(κ/γm+C12C2+2)iω, 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 (C1C2 + 1)/(κ/γm + C1 − 2C2 + 2)|. For κ/γm ≫ {1, C1, C2} the bandwidth Γ ≈ γm (C1C2 + 1), which is constrained by the damping rate of the mechanical resonator. Consequently, the gain-bandwidth product PΓG2γmC1C2. In addition, one should also consider the isolation bandwidth, where sufficient isolation is attained. Close to ω = 0, the reverse transmission amplitude T12(ω)iωG/κ. Thus, the isolation bandwidth is of order κ/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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17. J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature (London) 478, 89–92 (2011). [CrossRef]  

18. J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature (London) 475, 359–363 (2011). [CrossRef]  

19. G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803 (2010). [CrossRef]  

20. 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]  

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

22. X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nat. Phys. 9, 179–184 (2013). [CrossRef]  

23. H. Jing, Ş. K. Özdemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Sci. Rep. 5, 9663 (2015). [CrossRef]   [PubMed]  

24. Y. Jiao, H. Lü, J. Qian, Y. Li, and H. Jing, “Nonlinear optomechanics with gain and loss: amplifying higher-order sideband and group delay,” New J. Phys. 18, 083034 (2016). [CrossRef]  

25. F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. J. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature (London) 480, 351–354 (2011). [CrossRef]  

26. C. F. Ockeloen-Korppi, E. Damskägg, J.-M. Pirkkalainen, T. T. Heikkilä, F. Massel, and M. A. Sillanpää, “Low-noise amplification and frequency conversion with a multiport microwave optomechanical device,” Phys. Rev. X 6, 041024 (2016).

27. L. D. Tóth, N. R. Bernier, A. Nunnenkamp, A. K. Feofanov, and T. J. Kippenberg, “A dissipative quantum reservoir for microwave light using a mechanical oscillator,” Nat. Phys. 13, 787–793 (2017). [CrossRef]  

28. R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave and optical light,” Nat. Phys. 10, 321–326 (2014). [CrossRef]  

29. L. Tian, “Optoelectromechanical transducer: Reversible conversion between microwave and optical photons,” Ann. Phys. (Berlin) 527, 1 (2015). [CrossRef]  

30. C. Dong, V. Fiore, M. C. Kuzyk, L. Tian, and H. Wang, “Optical wavelength conversion via optomechanical coupling in a silica resonator,” Ann. Phys. (Berlin) 527, 100 (2015). [CrossRef]  

31. 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]  

32. I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two-level system,” Phys. Rev. Lett. 104, 083901 (2010). [CrossRef]   [PubMed]  

33. J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Gröblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, “Phonon counting and intensity interferometry of a nanomechanical resonator,” Nature (London) 520, 522–525 (2015). [CrossRef]  

34. H. Jing, S. K. Özdemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “P T-symmetric phonon laser,” Phys. Rev. Lett. 113, 053604 (2014). [CrossRef]  

35. H. Lü, S. K. Özdemir, L.-M. Kuang, F. Nori, and H. Jing, “Exceptional points in random-defect phonon lasers,” Phys. Rev. Applied 8, 044020 (2017). [CrossRef]  

36. X.-Y. Lü, H. Jing, J.-Y. Ma, and Y. Wu, “P T-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015). [CrossRef]  

37. M. Hafezi and P. Rabl, “Optomechanically induced non-reciprocity in microring resonators,” Opt. Express 20, 7672–7684 (2012). [CrossRef]   [PubMed]  

38. A. Metelmann and A. A. Clerk, “Nonreciprocal photon transmission and amplification via reservoir engineering,” Phys. Rev. X 5, 021025 (2015).

39. J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Phys. 11, 275 (2015). [CrossRef]  

40. C. H. Dong, Z. Shen, C. L. Zou, Y. L. Zhang, W. Fu, and G. C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015). [CrossRef]   [PubMed]  

41. Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photon. 10, 657 (2016). [CrossRef]  

42. X.-W. Xu and Y. Li, “Optical nonreciprocity and optomechanical circulator in three-mode optomechanical systems,” Phys. Rev. A 91, 053854 (2015). [CrossRef]  

43. F. Ruesink, M.-A. Miri, A. Alù, and E. Verhagen, “Nonreciprocity and magnetic-free isolation based on optomechanical interactions,” Nat. Commun. 7, 13662 (2016). [CrossRef]   [PubMed]  

44. M.-A. Miri, F. Ruesink, E. Verhagen, and A. Alù, “Optical nonreciprocity based on optomechanical coupling,” Phys. Rev. Appl. 7, 064014 (2017). [CrossRef]  

45. K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, “Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13, 465–471 (2017). [CrossRef]  

46. Y. Li, Y. Y. Huang, X. Z. Zhang, and L. Tian, “Optical directional amplification in a three-mode optomechanical system,” Opt. Express 25, 18907–18916 (2017). [CrossRef]   [PubMed]  

47. Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9, 1797 (2018). [CrossRef]   [PubMed]  

48. D. B. Sohn, S. Kim, and Gaurav Bahl, “Time-reversal symmetry breaking with acoustic pumping of nanophotonic circuits,” Nat. Photon. 12, 91–97 (2018). [CrossRef]  

49. L. Tian and Z. Li, “Nonreciprocal quantum-state conversion between microwave and optical photons,” Phys. Rev. A 96, 013808 (2017). [CrossRef]  

50. X.-W. Xu, Y. Li, A.-X. Chen, and Y.-x. Liu, “Nonreciprocal conversion between microwave and optical photons in electro-optomechanical systems,” Phys. Rev. A 93, 023827 (2016). [CrossRef]  

51. D. Malz, L. D. Tóth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” Phys. Rev. Lett. 120, 023601 (2018). [CrossRef]   [PubMed]  

52. G. A. Peterson, F. Lecocq, K. Cicak, R. W. Simmonds, J. Aumentado, and J. D. Teufel, “Demonstration of efficient nonreciprocity in a microwave optomechanical circuit,” Phys. Rev. X 7, 031001 (2017).

53. N. R. Bernier, L. D. Tóth, A. Koottandavida, M. A. Ioannou, D. Malz, A. Nunnenkamp, A. K. Feofanov, and T. J. Kippenberg, “Nonreciprocal reconfigurable microwave optomechanical circuit,” Nat. Commun. 8, 604 (2017). [CrossRef]   [PubMed]  

54. S. Barzanjeh, M. Wulf, M. Peruzzo, M. Kalaee, P. B. Dieterle, O. Painter, and J. M. Fink, “Mechanical on-chip microwave circulator,” Nat. Commun. 8, 953 (2017). [CrossRef]   [PubMed]  

55. B. Abdo, K. Sliwa, L. Frunzio, and M. Devoret, “Directional amplification with a Josephson circuit,” Phys. Rev. X 3, 031001 (2013).

56. K. M. Sliwa, M. Hatridge, A. Narla, S. Shankar, L. Frunzio, R. J. Schoelkopf, and M. H. Devoret, “Reconfigurable Josephson circulator/directional amplifier,” Phys. Rev. X 5, 041020 (2015).

57. F. Lecocq, L. Ranzani, G. A. Peterson, K. Cicak, R. W. Simmonds, J. D. Teufel, and J. Aumentado, “Nonreciprocal microwave signal processing with a field-programmable Josephson amplifier,” Phys. Rev. Appl. 7, 024028 (2017). [CrossRef]  

58. J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, “Nanomechanical coupling between microwave and optical photons,” Nat. Phys. 9, 712–716 (2013). [CrossRef]  

59. 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]  

60. Y. Sato, Y. Tanaka, J. Upham, Y. Takahashi, T. Asano, and S. Noda, “Strong coupling between distant photonic nanocavities and its dynamic control,” Nat. Photon. 6, 56–61 (2012). [CrossRef]  

61. K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nat. Photon. 6, 782–787 (2012). [CrossRef]  

62. C. Genes, A. Mari, P. Tombesi, and D. Vitali, “Robust entanglement of a micromechanical resonator with output optical fields,” Phys. Rev. A 78, 032316 (2008). [CrossRef]  

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2018 (3)

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9, 1797 (2018).
[Crossref] [PubMed]

D. B. Sohn, S. Kim, and Gaurav Bahl, “Time-reversal symmetry breaking with acoustic pumping of nanophotonic circuits,” Nat. Photon. 12, 91–97 (2018).
[Crossref]

D. Malz, L. D. Tóth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” Phys. Rev. Lett. 120, 023601 (2018).
[Crossref] [PubMed]

2017 (11)

G. A. Peterson, F. Lecocq, K. Cicak, R. W. Simmonds, J. Aumentado, and J. D. Teufel, “Demonstration of efficient nonreciprocity in a microwave optomechanical circuit,” Phys. Rev. X 7, 031001 (2017).

N. R. Bernier, L. D. Tóth, A. Koottandavida, M. A. Ioannou, D. Malz, A. Nunnenkamp, A. K. Feofanov, and T. J. Kippenberg, “Nonreciprocal reconfigurable microwave optomechanical circuit,” Nat. Commun. 8, 604 (2017).
[Crossref] [PubMed]

S. Barzanjeh, M. Wulf, M. Peruzzo, M. Kalaee, P. B. Dieterle, O. Painter, and J. M. Fink, “Mechanical on-chip microwave circulator,” Nat. Commun. 8, 953 (2017).
[Crossref] [PubMed]

L. Tian and Z. Li, “Nonreciprocal quantum-state conversion between microwave and optical photons,” Phys. Rev. A 96, 013808 (2017).
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H. Lü, S. K. Özdemir, L.-M. Kuang, F. Nori, and H. Jing, “Exceptional points in random-defect phonon lasers,” Phys. Rev. Applied 8, 044020 (2017).
[Crossref]

M.-A. Miri, F. Ruesink, E. Verhagen, and A. Alù, “Optical nonreciprocity based on optomechanical coupling,” Phys. Rev. Appl. 7, 064014 (2017).
[Crossref]

K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, “Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13, 465–471 (2017).
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Y. Li, Y. Y. Huang, X. Z. Zhang, and L. Tian, “Optical directional amplification in a three-mode optomechanical system,” Opt. Express 25, 18907–18916 (2017).
[Crossref] [PubMed]

F. Lecocq, L. Ranzani, G. A. Peterson, K. Cicak, R. W. Simmonds, J. D. Teufel, and J. Aumentado, “Nonreciprocal microwave signal processing with a field-programmable Josephson amplifier,” Phys. Rev. Appl. 7, 024028 (2017).
[Crossref]

A. C. Mahoney, J. I. Colless, S. J. Pauka, J. M. Hornibrook, J. D. Watson, G. C. Gardner, M. J. Manfra, A. C. Doherty, and D. J. Reilly, “On-Chip microwave quantum Hall circulator,” Phys. Rev. X 7, 011007 (2017).

L. D. Tóth, N. R. Bernier, A. Nunnenkamp, A. K. Feofanov, and T. J. Kippenberg, “A dissipative quantum reservoir for microwave light using a mechanical oscillator,” Nat. Phys. 13, 787–793 (2017).
[Crossref]

2016 (6)

Y. Jiao, H. Lü, J. Qian, Y. Li, and H. Jing, “Nonlinear optomechanics with gain and loss: amplifying higher-order sideband and group delay,” New J. Phys. 18, 083034 (2016).
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C. F. Ockeloen-Korppi, E. Damskägg, J.-M. Pirkkalainen, T. T. Heikkilä, F. Massel, and M. A. Sillanpää, “Low-noise amplification and frequency conversion with a multiport microwave optomechanical device,” Phys. Rev. X 6, 041024 (2016).

X. Guo, C.-L. Zou, H. Jung, and H. X. Tang, “On-chip strong coupling and efficient frequency conversion between telecom and visible optical modes,” Phys. Rev. Lett. 117, 123902 (2016).
[Crossref] [PubMed]

F. Ruesink, M.-A. Miri, A. Alù, and E. Verhagen, “Nonreciprocity and magnetic-free isolation based on optomechanical interactions,” Nat. Commun. 7, 13662 (2016).
[Crossref] [PubMed]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photon. 10, 657 (2016).
[Crossref]

X.-W. Xu, Y. Li, A.-X. Chen, and Y.-x. Liu, “Nonreciprocal conversion between microwave and optical photons in electro-optomechanical systems,” Phys. Rev. A 93, 023827 (2016).
[Crossref]

2015 (11)

X.-W. Xu and Y. Li, “Optical nonreciprocity and optomechanical circulator in three-mode optomechanical systems,” Phys. Rev. A 91, 053854 (2015).
[Crossref]

X.-Y. Lü, H. Jing, J.-Y. Ma, and Y. Wu, “P T-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
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A. Metelmann and A. A. Clerk, “Nonreciprocal photon transmission and amplification via reservoir engineering,” Phys. Rev. X 5, 021025 (2015).

J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Phys. 11, 275 (2015).
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C. H. Dong, Z. Shen, C. L. Zou, Y. L. Zhang, W. Fu, and G. C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
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K. M. Sliwa, M. Hatridge, A. Narla, S. Shankar, L. Frunzio, R. J. Schoelkopf, and M. H. Devoret, “Reconfigurable Josephson circulator/directional amplifier,” Phys. Rev. X 5, 041020 (2015).

H. Xiong, L. G. Si, X. Y. Lv, X. X. Yang, and Y. Wu, “Review of cavity optomechanics in the weak-coupling regime: from linearization to intrinsic nonlinear interactions,” Sci. China Phys. Mech. Astron. 58, 1–13 (2015).
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H. Jing, Ş. K. Özdemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Sci. Rep. 5, 9663 (2015).
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J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Gröblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, “Phonon counting and intensity interferometry of a nanomechanical resonator,” Nature (London) 520, 522–525 (2015).
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L. Tian, “Optoelectromechanical transducer: Reversible conversion between microwave and optical photons,” Ann. Phys. (Berlin) 527, 1 (2015).
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C. Dong, V. Fiore, M. C. Kuzyk, L. Tian, and H. Wang, “Optical wavelength conversion via optomechanical coupling in a silica resonator,” Ann. Phys. (Berlin) 527, 100 (2015).
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2014 (8)

H. Jing, S. K. Özdemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “P T-symmetric phonon laser,” Phys. Rev. Lett. 113, 053604 (2014).
[Crossref]

R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave and optical light,” Nat. Phys. 10, 321–326 (2014).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391 (2014).
[Crossref]

R. Fleury, D. L. Sounas, C. F. Sieck, M. R. Haberman, and A. Alù, “Sound isolation and giant linear nonreciprocity in a compact acoustic circulator,” Science 343, 516–519 (2014).
[Crossref] [PubMed]

N. A. Estep, D. L. Sounas, J. Soric, and A. Alù, “Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops,” Nat. Phys. 10, 923–927 (2014).
[Crossref]

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524–529 (2014).
[Crossref]

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

A. Metelmann and A. A. Clerk, “Quantum-limited amplification via reservoir engineering,” Phys. Rev. Lett. 112, 133904 (2014).
[Crossref] [PubMed]

2013 (5)

J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, “Nanomechanical coupling between microwave and optical photons,” Nat. Phys. 9, 712–716 (2013).
[Crossref]

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]

B. Abdo, K. Sliwa, L. Frunzio, and M. Devoret, “Directional amplification with a Josephson circuit,” Phys. Rev. X 3, 031001 (2013).

D.-W. Wang, H.-T. Zhou, M.-J. Guo, J.-X. Zhang, J. Evers, and S.-Y. Zhu, “Optical diode made from a moving photonic crystal,” Phys. Rev. Lett. 110, 093901 (2013).
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X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nat. Phys. 9, 179–184 (2013).
[Crossref]

2012 (6)

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]

H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocity induced by interband photonic transition on a dilicon vhip,” Phys. Rev. Lett. 109, 033901 (2012).
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L. Deák and T. Fülöp, “Reciprocity in quantum, electromagnetic and other wave scattering,” Ann. Phys. 327, 1050–1077 (2012).
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M. Hafezi and P. Rabl, “Optomechanically induced non-reciprocity in microring resonators,” Opt. Express 20, 7672–7684 (2012).
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Y. Sato, Y. Tanaka, J. Upham, Y. Takahashi, T. Asano, and S. Noda, “Strong coupling between distant photonic nanocavities and its dynamic control,” Nat. Photon. 6, 56–61 (2012).
[Crossref]

K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nat. Photon. 6, 782–787 (2012).
[Crossref]

2011 (5)

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photon. 5, 758–762 (2011).
[Crossref]

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature (London) 478, 89–92 (2011).
[Crossref]

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature (London) 475, 359–363 (2011).
[Crossref]

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

F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. J. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature (London) 480, 351–354 (2011).
[Crossref]

2010 (4)

I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two-level system,” Phys. Rev. Lett. 104, 083901 (2010).
[Crossref] [PubMed]

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803 (2010).
[Crossref]

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]

A. A. Clerk, M. H. Devoret, S. M. Girvin, Florian Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155 (2010).
[Crossref]

2009 (2)

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photon. 3, 91–94 (2009).
[Crossref]

F. Marquardt and S. M. Girvin, “Optomechanics,” Physics 2, 40 (2009).
[Crossref]

2008 (1)

C. Genes, A. Mari, P. Tombesi, and D. Vitali, “Robust entanglement of a micromechanical resonator with output optical fields,” Phys. Rev. A 78, 032316 (2008).
[Crossref]

2004 (1)

R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717 (2004).
[Crossref]

1987 (1)

E. X. DeJesus and C. Kaufman, “Routh-Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations,” Phys. Rev. A 35, 5288 (1987).
[Crossref]

1985 (1)

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum system: quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761 (1985).
[Crossref]

1982 (1)

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26, 1817 (1982).
[Crossref]

1964 (1)

Abdo, B.

B. Abdo, K. Sliwa, L. Frunzio, and M. Devoret, “Directional amplification with a Josephson circuit,” Phys. Rev. X 3, 031001 (2013).

Agarwal, G. S.

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803 (2010).
[Crossref]

Alegre, T. P. M.

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature (London) 478, 89–92 (2011).
[Crossref]

Allman, M. S.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature (London) 475, 359–363 (2011).
[Crossref]

Alù, A.

M.-A. Miri, F. Ruesink, E. Verhagen, and A. Alù, “Optical nonreciprocity based on optomechanical coupling,” Phys. Rev. Appl. 7, 064014 (2017).
[Crossref]

F. Ruesink, M.-A. Miri, A. Alù, and E. Verhagen, “Nonreciprocity and magnetic-free isolation based on optomechanical interactions,” Nat. Commun. 7, 13662 (2016).
[Crossref] [PubMed]

R. Fleury, D. L. Sounas, C. F. Sieck, M. R. Haberman, and A. Alù, “Sound isolation and giant linear nonreciprocity in a compact acoustic circulator,” Science 343, 516–519 (2014).
[Crossref] [PubMed]

N. A. Estep, D. L. Sounas, J. Soric, and A. Alù, “Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops,” Nat. Phys. 10, 923–927 (2014).
[Crossref]

Andrews, R. W.

R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave and optical light,” Nat. Phys. 10, 321–326 (2014).
[Crossref]

Aplet, L. J.

Arcizet, O.

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]

Asano, T.

Y. Sato, Y. Tanaka, J. Upham, Y. Takahashi, T. Asano, and S. Noda, “Strong coupling between distant photonic nanocavities and its dynamic control,” Nat. Photon. 6, 56–61 (2012).
[Crossref]

Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391 (2014).
[Crossref]

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature (London) 478, 89–92 (2011).
[Crossref]

Aumentado, J.

G. A. Peterson, F. Lecocq, K. Cicak, R. W. Simmonds, J. Aumentado, and J. D. Teufel, “Demonstration of efficient nonreciprocity in a microwave optomechanical circuit,” Phys. Rev. X 7, 031001 (2017).

F. Lecocq, L. Ranzani, G. A. Peterson, K. Cicak, R. W. Simmonds, J. D. Teufel, and J. Aumentado, “Nonreciprocal microwave signal processing with a field-programmable Josephson amplifier,” Phys. Rev. Appl. 7, 024028 (2017).
[Crossref]

Awschalom, D. D.

J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, “Nanomechanical coupling between microwave and optical photons,” Nat. Phys. 9, 712–716 (2013).
[Crossref]

Bahl, G.

J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Phys. 11, 275 (2015).
[Crossref]

Bahl, Gaurav

D. B. Sohn, S. Kim, and Gaurav Bahl, “Time-reversal symmetry breaking with acoustic pumping of nanophotonic circuits,” Nat. Photon. 12, 91–97 (2018).
[Crossref]

Barzanjeh, S.

S. Barzanjeh, M. Wulf, M. Peruzzo, M. Kalaee, P. B. Dieterle, O. Painter, and J. M. Fink, “Mechanical on-chip microwave circulator,” Nat. Commun. 8, 953 (2017).
[Crossref] [PubMed]

Bender, C. M.

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Bernier, N. R.

D. Malz, L. D. Tóth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” Phys. Rev. Lett. 120, 023601 (2018).
[Crossref] [PubMed]

N. R. Bernier, L. D. Tóth, A. Koottandavida, M. A. Ioannou, D. Malz, A. Nunnenkamp, A. K. Feofanov, and T. J. Kippenberg, “Nonreciprocal reconfigurable microwave optomechanical circuit,” Nat. Commun. 8, 604 (2017).
[Crossref] [PubMed]

L. D. Tóth, N. R. Bernier, A. Nunnenkamp, A. K. Feofanov, and T. J. Kippenberg, “A dissipative quantum reservoir for microwave light using a mechanical oscillator,” Nat. Phys. 13, 787–793 (2017).
[Crossref]

Bi, L.

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photon. 5, 758–762 (2011).
[Crossref]

Bochmann, J.

J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, “Nanomechanical coupling between microwave and optical photons,” Nat. Phys. 9, 712–716 (2013).
[Crossref]

Carson, J. W.

Caves, C. M.

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26, 1817 (1982).
[Crossref]

Chan, J.

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]

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature (London) 478, 89–92 (2011).
[Crossref]

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

Chang, D. E.

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

Chang, L.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524–529 (2014).
[Crossref]

Chen, A.-X.

X.-W. Xu, Y. Li, A.-X. Chen, and Y.-x. Liu, “Nonreciprocal conversion between microwave and optical photons in electro-optomechanical systems,” Phys. Rev. A 93, 023827 (2016).
[Crossref]

Chen, Y.

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9, 1797 (2018).
[Crossref] [PubMed]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photon. 10, 657 (2016).
[Crossref]

Cho, S. U.

F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. J. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature (London) 480, 351–354 (2011).
[Crossref]

Cicak, K.

G. A. Peterson, F. Lecocq, K. Cicak, R. W. Simmonds, J. Aumentado, and J. D. Teufel, “Demonstration of efficient nonreciprocity in a microwave optomechanical circuit,” Phys. Rev. X 7, 031001 (2017).

F. Lecocq, L. Ranzani, G. A. Peterson, K. Cicak, R. W. Simmonds, J. D. Teufel, and J. Aumentado, “Nonreciprocal microwave signal processing with a field-programmable Josephson amplifier,” Phys. Rev. Appl. 7, 024028 (2017).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 Schematic diagram of the triple-cavity optomechanical system. One microwave cavity a1 and two optical cavities a2 and a3 are respectively coupled to the common mechanical resonator with the field-enhanced optomechanical coupling strength Gk(k = 1, 2, 3), where ϕ is the phase difference between Gk. Moreover, the two optical cavities are coupled directly via hopping interaction J to facilitate the nonreciprocal transmission. Cavity a1 is driven on its red sideband by a pump field at the frequency ωd,1, and cavities a2 and a3 are driven on their respective blue sidebands by the pump fields at the frequencies ωd,2 and ωd,3. The solid arrows represent the transmission elements between different modes.
Fig. 2
Fig. 2 Stability diagram with respect to C1 and C2. Other parameters are κ1/2π = 2 MHz, κ2/2π = 3 MHz, κ3/2π = 3 MHz, γm = κ2/100, ω = 0, ϕ = −π/2, G3 = G2κ3/(2J), and J κ 2 κ 3 / 2.
Fig. 3
Fig. 3 Transmission probabilities |T12|2 and |T21|2 as functions of the probe detuning ω for ϕ = −π/2, 0, π/2, and π, respectively. Other parameters are κ1/2π = 2 MHz, κ2/2π = 3 MHz, κ3/2π = 3 MHz, γm = κ2/100, C1 = 10, C 2 = C 1 0.1 C 1, G3 = G2κ3/(2J), and J κ 2 κ 3 / 2.
Fig. 4
Fig. 4 Directional amplifier. (a)–(c) Transmission probabilities |Tij|2 (i, j = 1, 2, 3) as functions of the probe detuning ω/2π. (d) Graphical representation of the amplification process, where S, I, and V represent/the “Signal”, “Idler”, and “Vacuum” ports, respectively. G and C correspond to the gain and conversion process. Other parameters used are ϕ = −π/2, κ1/2π = 2 MHz, κ2/2π = 3 MHz, κ3/2π = 3 MHz, γm = κ2/100, C1 = 3, C 2 = C 1 0.1 C 1, G3 = G2κ3/(2J), and J κ 2 κ 3 / 2.
Fig. 5
Fig. 5 (a) Transmission probability |T13|2 and (b) added noise N 1 of the cavity a1 as a function of the probe detuning ω with C1 = {1, 5, 10, 50} and C 2 = C 1 0.1 C 1. Other parameters used are the same as those in Fig. 4 except nm = 50.

Equations (31)

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H = k = 1 3 ω k a k a k + ω m b b + k = 1 3 g k a k a k ( b + b ) + J ( a 2 a 3 + a 3 a 2 ) + k = 1 3 ε k ( a k e i ω d , k t + a k e i ω d , k t ) ,
H rot = k = 1 3 Δ k a k a k + ω m b b + k = 1 3 g k a k a k ( b + b ) + J ( a 2 a 3 + a 3 a 2 ) + k = 1 3 ε k ( a k + a k ) ,
a ˙ 1 = ( κ 1 / 2 i Δ 1 ) a 1 + i g 1 a 1 ( b + b ) + i ε 1 + κ 1 a 1 , in ,
a ˙ 2 = ( κ 2 / 2 + i Δ 2 ) a 2 i g 2 a 2 ( b + b ) i J a 3 i ε 2 + κ 2 a 2 , in ,
a ˙ 3 = ( κ 3 / 2 + i Δ 3 ) a 3 i g 3 a 3 ( b + b ) i J a 2 i ε 3 + κ 3 a 3 , in ,
b ˙ = ( γ m / 2 i ω m ) b + i k g k a k a k + γ m b in .
a k , in ( t ) a k , in ( t ) = δ ( t t ) , a k , in ( t ) a k , in ( t ) = 0 , b in ( t ) b in ( t ) = ( n m + 1 ) δ ( t t ) , b in ( t ) b in ( t ) = n m δ ( t t ) ,
α 1 * = i ε 1 κ 1 / 2 i Δ 1 , α 2 = i J α 3 + i ε 2 κ 2 / 2 + i Δ 2 , α 3 = i J α 2 + i ε 3 κ 3 / 2 + i Δ 3 , β * = i k g k | α k | 2 γ m / 2 i ω m ,
δ a ˙ 1 = ( κ 1 / 2 i Δ 1 ) δ a 1 + i g α 1 * ( δ b + δ b ) + κ 1 a 1 , in ,
δ a ˙ 2 = ( κ 2 / 2 + i Δ 2 ) δ a 2 i g 2 α 2 ( δ b + δ b ) i J δ a 3 + κ 2 a 2 , in ,
δ a ˙ 3 = ( κ 3 / 2 + i Δ 3 ) δ a 3 i g 3 α 3 ( δ b + δ b ) i J δ a 2 + κ 3 a 3 , in ,
δ b ˙ = ( γ m / 2 i ω m ) δ b + i k g k ( α k * δ a k + α k δ a k ) + γ m b in .
δ a ˙ 1 = κ 1 2 δ a 1 + i G 1 e i ϕ 1 δ b + κ 1 a 1 , in ,
δ a ˙ 2 = κ 2 2 δ a 2 i G 2 e i ϕ 2 δ b i J δ a 3 + κ 2 a 2 , in ,
δ a ˙ 3 = κ 3 2 δ a 3 i G 3 e i ϕ 3 δ b i J δ a 2 + κ 3 a 3 , in ,
δ b ˙ = γ m 2 δ b + i ( G 1 e i ϕ 1 δ a 1 + G 2 e i ϕ 2 δ a 2 + G 3 e i ϕ 3 δ a 3 ) + γ m b i n ,
H eff = G 1 δ a 1 δ b e i ϕ 1 + G 2 δ a 2 δ b e i ϕ 2 + G 3 δ a 3 δ b e i ϕ 3 + J δ a 2 δ a 3 + H . c ..
μ ˙ = M μ + K μ i n ,
M = ( κ 1 / 2 0 0 i G 1 0 κ 2 / 2 i J i G 2 0 i J κ 3 / 2 i G 3 e i ϕ i G 1 i G 2 i G 3 e i ϕ γ m / 2 ) .
o ( ω ) = + o ( t ) e i ω t d t ,
o ( ω ) = + o ( t ) e i ω t d t ,
μ ( ω ) = ( M + i ω I ) 1 K μ in ( ω ) ,
μ out ( ω ) = T ( ω ) μ in ( ω ) ,
T ( ω ) = I + K ( M + i ω I ) 1 K .
T 12 ( ω ) = κ 1 κ 2 A ( ω ) G 1 Γ 3 ( G 2 + i J e i ϕ G 3 / Γ 3 ) ,
T 21 ( ω ) = κ 1 κ 2 A ( ω ) G 1 Γ 3 ( G 2 + i J e i ϕ G 3 / Γ 3 ) ,
T 21 ( 0 ) = 8 κ 1 κ 2 G 1 G 2 4 κ 2 G 1 2 4 κ 1 G 2 2 + κ 1 κ 2 γ m = 2 C 1 C 2 C 1 C 2 + 1 ,
T ( 0 ) = ( C 1 + C 2 1 C 1 C 2 + 1 0 2 i C 1 C 2 C 1 C 2 + 1 2 i C 1 C 1 C 2 + 1 2 C 1 C 2 C 1 C 2 + 1 0 i C 1 + C 2 + 1 C 1 C 2 + 1 2 i C 2 C 1 C 2 + 1 0 i 0 0 2 i C 1 C 1 C 2 + 1 0 2 C 2 C 1 C 2 + 1 C 1 C 2 1 C 1 C 2 + 1 ) ,
S 1 , out ( ω ) = 1 2 d Ω 2 π a 1 , out ( ω ) a 1 , out ( Ω ) + a 1 , out ( Ω ) a 1 , out ( ω ) = 1 2 | T 11 ( ω ) | 2 + 1 2 | T 12 ( ω ) | 2 + 1 2 | T 13 ( ω ) | 2 + ( n m + 1 2 ) | T 14 ( ω ) | 2 ,
G = | T 31 ( 0 ) | 2 = 4 C 1 C 2 ( C 1 C 2 + 1 ) 2 .
N 1 ( 0 ) = 1 2 ( C 1 + C 2 1 ) 2 4 C 1 C 2 + ( n m + 1 2 ) 1 C 2 ,

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