Abstract

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.

© 2017 Optical Society of America

1. Introduction

With the rapid development of microfabrication technology, cavity optomechanical system [1–4] is becoming an appealing candidate to connect a broad spectrum of photonic, electronic, and atomic devices, besides being studied for fundamental questions of macroscopic systems in the quantum limit [5]. Recently, enormous progresses have been achieved that aim at the applications of optomechanical systems in ultra-high precision measurement [6–12], quantum information processing [13], quantum illumination [14] to optomechanically induced transparency [15–21], absorption [22,23], and amplification [24–28].

Among these applications, nonreciprocal transmission and amplification are of great interest in the study of the quantum analogue of photonic and electronic devices, such as diode, circulator, and transistor, which are crucial for scalable quantum information processing in integrated circuits [29]. In the past, nonreciprocal devices have been investigated broadly in optical systems [30–38]. In these devices, the occurrence of nonreciprocal light propagation is associated with the symmetry breaking induced by various mechanisms, such as magneto-optical Faraday effect [30], parametric modulation [31–34], optical nonlinearity [35, 36], and chiral light-matter interaction [37].

In recent years, it has been shown that the optomechanical system can be utilized to realize nonreciprocal effects for propagating light fields [39–43]. The nonreciprocal optical diodes are achieved in multi-mode optomechanical systems with effective breaking of time-reversal symmetry generated by on-demand gauge-invariant phases [44–48]. Nonreciprocal phenomena with directional amplification have been explored theoretically in general coupled-mode systems [49]. The phenomena of optical directional amplification have also been implemented experimentally very recently in multi-mode optomechanical systems [50–54].

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 [50–54] 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 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. 2, 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. 3. Conclusions are given in Sec. 4.

2. Model and transmission matrix

The optomechanical system under consideration consists of a mechanical oscillator with resonance frequency ωm and two optical cavities with resonance frequencies ω1 and ω2, 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

H=H0+HI+Hd.
The first term describes the free Hamiltonian of the cavity modes and the mechanical one with (ħ = 1)
H0=ω1a1a1+ω2a2a2+ωmbb,
where ai (ai) for i = 1, 2 and b (b) are the creation (annihilation) operators for the cavity modes and the mechanical one. The second term
HI=J(a1a2+a1a2)+igiaiai(b+b)
characterizes the linear coupling between the cavity modes with coupling strength J and the radiation-pressure force interaction between the cavities and the mechanical resonator with single-photon coupling strength gi. The third term Hd 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)
Hd=i(iεiaieiωdteiθi+h.c.)+(iεpa1eiωpt+iεbbeiωbt+h.c.),
where ωd is the frequency, εi is the amplitude, and θi is the phase of the two pump fields, ωp (ωb) is the frequency and εp (εb) 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 [55–58]. Here without loss of generality, we have assumed that J, g1,2, and ε1,2 are real numbers.

 figure: Fig. 1

Fig. 1 Schematic of a three-mode optomechanical system driven by two pump fields with the same frequency ωd. 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 ωb. The cavities and the mechanical resonator are coupled via radiation-pressure forces, and the cavities are directly coupled to each other.

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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 by

a˙1={γ1i[Δ1+g1(b+b)]}a1iJa2+ε1eiθ1+εpei(ωdωp)t+ξ1,
a˙2={γ2i[Δ2+g2(b+b)]}a2iJa1+ε2eiθ2+ξ2,
b˙=(γmiωm)bi(g1a1a1+g2a2a2)+εbeiωbt+ξm.
Here Δi = ωiωd (i = 1, 2) are the optical detunings of the cavities, γi (γm) are the decay rates of the two cavities (mechanical resonator), ξi (ξm) are the noise operators of the cavities (mechanical mode) with 〈ξi〉 = 〈ξm〉= 0.

We first derive the steady-state solution of the three-mode system under strong pump fields. Neglecting the effects of the optical probe field and mechanical drive, we can obtain the steady-state solution as

a1=(γ2+iΔ2)ε1eiθ1iJε2eiθ2(γ1+iΔ1)(γ2+iΔ2)+J2,
a2=(γ1+iΔ1)ε2eiθ2iJε1eiθ1(γ1+iΔ1)(γ2+iΔ2)+J2,
b=i(g1|a1|2+g2|a2|2)γm+iωm,
where 〈ai〉 (〈b〉) are the steady-state averages of the cavities (mechanical mode), and Δi=Δi+gi[b+b*] (i = 1, 2) are the cavity detunings shifted by the radiation-pressure 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 ai = 〈ai〉+ δai and b = 〈b〉 + δb, where δai are the fluctuations of the cavities and δb is that of the mechanical mode. Neglecting the nonlinear terms in the radiation-pressure interaction in Eqs. (8)(10), we obtain a set of linear QLEs for the fluctuation operators:

δa˙1=(γ1iΔ1)δa1iG1(δb+δb)iJδa2+εpei(ωdωp)t+ξ1,
δa˙2=(γ2iΔ2)δa2iG2(δb+δb)iJδa1+ξ2,
δb˙=(γmiωm)δbi(G1δa1+G1*δa1)i(G2δa2+G2*δa2)+εbeiωbt+ξm,
where Gi = giai〉 (i = 1, 2) represent the pump-enhanced linear optomechanical couplings.

In what follows, we fix ωb = ωpωd 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 δaiδaiei(ωpωd)t, ξiξiei(ωpωd)t, δbδbeiωbt, and ξmξmeiωbt. In addition, we assume that the cavities are driven by the red-detuned pump fields and Δi~ωm. In this case, by using the rotating-wave approximation, one can neglect the fast-oscillating counter-rotating terms and obtain the following linearized QLEs

δa˙1=Γ1δa1iG1δbiJδa2+εp+ξ1,
δa˙2=Γ2δa2iG2δbiJδa1+ξ2,
δb˙=ΓmδbiG1*δa1iG2*δa2+εb+ξm,
with Γi=γi+iΔi and Γm=γm+iΔm. Here Δi=Δi(ωpωd) and Δm = ωm − (ωpωd) are the detunings in the new rotating frame. The optical response of the cavities to the probe field can be obtained by solving the steady state of Eqs. (14)(16). By setting d〈…〉/dt = 0, we have
δa1=iG2εb(iJΓm+G1G2*)+(Γ2Γm+|G2|2)(εpΓmiG1εb)(Γ1Γm+|G1|2)(Γ2Γm+|G2|2)(iJΓm+G1G2*)(iJΓm+G1*G2),
δa2=(iJΓm+G1*G2)(εpΓmiG1εb)iG2εb(Γ1Γm+|G1|2)(Γ1Γm+|G1|2)(Γ2Γm+|G2|2)(iJΓm+G1G2*)(iJΓm+G1*G2),
δb=JΓm(εbJεpG2*)+Γ2Γm(εbΓ1iεpG1*)(Γ1Γm+|G1|2)(Γ2Γm+|G2|2)(iJΓm+G1G2*)(iJΓm+G1*G2).
The cavity output fields δaiout (i = 1, 2) can be derived from the input-output theorem with
δaiout+δaiin=2γieδai,
where γie represents the cavity loss related to coupling between the cavity and the input (output) modes, and is part of the total cavity loss rate γi with γie=ηiγi and ηi ≤ 1. For simplicity of discussion, we focus on the case of over-coupled cavities with ηi ≃ 1 and neglect cavity intrinsic dissipation [59–61]. With this assumption, δa1in=εp/2γ1e, δa2in=0. The input field on the mechanical resonator can then be written in terms of the cavity input with δbin=γ1e/γm(yeiφ)δa1in. The transmission coefficient that describes the dependence of the output field of cavity 2 on the input field δa1in can be defined as
t21δa2out/δa1in.
With Eqs. (18) and (20), we derive
t21=2γ1eγ2e[(iJΓm+G1*G2)(ΓmiG1yeiφ)+iG2yeiφ(Γ1Γm+|G1|2)(Γ1Γm+|G1|2)(Γ2Γm+|G2|2)(iJΓm+G1G2*)(iJΓm+G1*G2)],
where we have defined the amplitude of the mechanical drive through εbp = ye (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 δa1in=0, δa2in=εp/2γ2e, and still fix ωb = ωp − ωd. Here the transmission coefficient is defined as t12δa1out/δa2in. We derive that

t12=2γ1eγ2e[(iJΓm+G2*G1)(ΓmiG2yeiφ)+iG1yeiφ(Γ2Γm+|G2|2)(Γ2Γm+|G2|2)(Γ1Γm+|G1|2)(iJΓm+G2G1*)(iJΓm+G2*G1)].

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 εp and the frequency-matched mechanical drive with amplitude εb via the optomechanical interaction. And the transmission is not symmetric between cavities 1 and 2.

3. Directional amplification of optical probes

In this section, we will study the transmission of optical probe and the asymmetry in the transmission systematically. We will show that amplification of optical probe fields can be directional. Consider G1 = G > 0 and G2 = Geiθ for simplicity of discussion. The transmission coefficients can be rewritten as

t21=2γ1eγ2e[(iJΓm+G2eiθ)(ΓmiGyeiφ)+iG(Γ1Γm+G2)yei(θ+φ)(Γ1Γm+G2)(Γ2Γm+G2)(iJΓm+G2eiθ)(iJΓm+G2eiθ)],
and
t12=2γ1eγ2e[(iJΓm+G2eiθ)(ΓmiGyei(θ+φ))+iG(Γ2Γm+G2)yeiφ(Γ1Γm+G2)(Γ2Γm+G2)(iJΓm+G2eiθ)(iJΓm+G2eiθ)].

When εb = 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 G2 = 0), the system reduces to a standard two-mode optomechanical system. In this case, it was shown that the presence of the mechanical drive εb 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 εb 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 T21 ≡ |t21|2 and T12 ≡ |t12|2 as functions of Δm = ωm − (ωp ωd) at different values of the phases θ and φ. We observe that in general, the transmission of the probe field is asymmetric with T21T12, and T12 or T21 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, T21 → 0 and T12 ≫ 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].

 figure: Fig. 2

Fig. 2 The transmission probabilities T21 and T12 versus Δm = ωm (ωp − ωd) for different values of θ and φ: (a) θ = 0, φ = π/2; (b) θ = π/2, φ = 0; (c) θ−= π/2, φ = π/2. Other parameters are y = 20, η1,2 = 1, γ1 = 1.1γm, γ2 = 1.5γm, G = |G1,2| = J = γm, and Δ1,2=Δm.

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We plot the probability of the transmission T21 and T12 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 T12 is independent of φ, which can be given through Eq. (25).

 figure: Fig. 3

Fig. 3 Plot of the probability of transmission T21 and T12 as functions of θ and φ, respectively. (a) φ = π/2. (b) θ = π/2. Other parameters are y = 20, η1,2 = 1, G=|G1,2|=J=γm, Δm=Δ1,2=0, γ1 = 1.1γm, and γ2 = 1.5γm. One can see that at certain optimal values of θ and φ, e.g., θ = π/2, φ = π/2, T12 → 0 and T21 ≫ 1.

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To further understand the effect of the frequency-matched mechanical drive on the transmission property of the probe field, we assume that the parameters G = |G1,2| = J = γm, Δm=Δ1,2=0, θ = π/2, and φ = π/2. Then the corresponding transmission coefficients T12 and T21 are simplified to be

T21=4γ1γ2(2γm(y+1)y(γ1+γm)(γ1+γm)(γ2+γm))2,
T12=4y2γ1γ2(γ1+γm)2.
In the absence of the mechanical drive (y → 0), the directional transmission of the probe field can occur with T12 → 0 and T21 > 0 (particularly, T21 = 1 at γ1,2 = γm) as shown in [44]. On the contrary, in the presence of frequency-matched mechanical drive with y = yc ≡ 2γm/(γ1γm) and |yc| ≫ 1, we have T21 → 0 and T12 ≫ 1. The directional amplification of the optical 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 |yc| ≫ 1, i.e., the cavity damping rate γ1 is approximately equal to the mechanical damping rate γm.

To study the role of the mechanical drive, we plot T21 and T12 as functions of y in Fig. 4. This plot clearly demonstrates that the propagation of the optical field is strongly amplified with T12 ~ 600 when the mechanical drive becomes large (|y| ≫ 1). Meanwhile, when y ~ yc = 20 under the parameters given in the caption of Fig. 4, the transmission in the opposite direction quickly drops with T21 → 0.

 figure: Fig. 4

Fig. 4 The transmission probabilities T21 and T12 versus y. Other parameters are θ = π/2, φ = π/2, G = |G1,2| = J = γm, Δm=Δ1,2=0, γ1 = 1.1γm, and γ2 = 1.5γm.

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4. 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.

Funding

National Natural Science Foundation of China (No. 11422437, No. 11505126, No. 11534002, No. 11421063, No. U1530401); Postdoctoral Science Foundation of China (Grant No. 2016M591055); PhD research startup foundation of Tianjin Normal University (Grant. No. 52XB1415); National Science Foundation (NSF) (No. DMR-0956064 and No. PHY-1720501); UC Multicampus-National Lab Collaborative Research and Training (Grant No. LFR-17-477237).

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40. M. Hafezi and P. Rabl, “Optomechanically induced non-reciprocity in microring resonators,” Opt. Express 20(7), 7672–7684 (2012). [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(10), 657–661 (2016). [CrossRef]  

42. J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Phys. 11(3), 275–280 (2015). [CrossRef]  

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

44. X. W. Xu and Y. Li, “Optical nonreciprocity and optomechanical circulator in three-mode optomechanical systems,” Phys. Rev. A 91(5), 053854 (2015). [CrossRef]  

45. 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(2), 023827 (2016). [CrossRef]  

46. L. Tian and Z. Li, “Nonreciprocal state conversion between microwave and optical Photons,” arXiv: 1610.09556 (2016).

47. A. Metelmann and A. A. Clerk, “Nonreciprocal photon transmission and amplification via reservoir engineering,” Phys. Rev. X 5(2), 021025 (2015).

48. Y. L. Zhang, C. H. Dong, C. L. Zou, X. B. Zou, Y. D. Wang, and G. C. Guo, “Optomechanical devices based on traveling-wave microresonators,” Phys. Rev. A 95(4), 043815 (2017). [CrossRef]  

49. L. Ranzani and J. Aumentado, “Graph-based analysis of nonreciprocity in coupled-mode systems,” New J. Physics , 17(2), 023024 (2015). [CrossRef]  

50. 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,” arXiv: 1703.05269.

51. D. Malz, L. D. Toth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” arXiv: 1705.00436.

52. 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]  

53. M. A. Miri, F. Ruesink, E. Verhagen, and A. Alù, “Fundamentals of optical non-reciprocity based on optomechanical coupling,” Phys. Rev. Applied 7(6), 064014 (2017). [CrossRef]  

54. N. R. Bernier, L. D. Tóth, A. Koottandavida, M. Ioannou, D. Malz, A. Nunnenkamp, A. K. Feofanov, and T. J. Kippenberg, “Nonreciprocal reconfigurable microwave optomechanical circuit,” arXiv: 1612.08223 (2016).

55. D. Rugar and P. Grütter, “Mechanical parametric amplification and thermomechanical noise squeezing,” Phys. Rev. Lett. 67(6), 699 (1991). [CrossRef]   [PubMed]  

56. H. Okamoto, A. Gourgout, C. Y. Chang, K. Onomitsu, I. Mahboob, E. Y. Chang, and H. Yamaguchi, “Coherent phonon manipulation in coupled mechanical resonators,” Nat. Phys. 9(8), 480–484 (2013). [CrossRef]  

57. T. Faust, J. Rieger, M. J. Seitner, J. P. Kotthaus, and E. M. Weig, “Coherent control of a classical nanomechanical two-level system,” Nat. Phys. 9(8), 485–488 (2013). [CrossRef]  

58. H. Fu, Z. Gong, T. Mao, C. Sun, S. Yi, Y. Li, and G. Cao, “Classical analog of Stuckelberg interferometry in a two-coupled-cantilever based optomechanical system,” Phys. Rev. A 94(4), 043855 (2016). [CrossRef]  

59. M. Cai, O. J. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85(1), 74 (2000). [CrossRef]   [PubMed]  

60. S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91(4), 043902 (2003). [CrossRef]   [PubMed]  

61. L. Tian, “Optoelectromechanical transducer: Reversible conversion between microwave and optical photons,” Ann. Phys. (Berlin) 527(1–2), 1–14 (2015). [CrossRef]  

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    [Crossref]
  57. T. Faust, J. Rieger, M. J. Seitner, J. P. Kotthaus, and E. M. Weig, “Coherent control of a classical nanomechanical two-level system,” Nat. Phys. 9(8), 485–488 (2013).
    [Crossref]
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    [Crossref]
  59. M. Cai, O. J. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85(1), 74 (2000).
    [Crossref] [PubMed]
  60. S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91(4), 043902 (2003).
    [Crossref] [PubMed]
  61. L. Tian, “Optoelectromechanical transducer: Reversible conversion between microwave and optical photons,” Ann. Phys. (Berlin) 527(1–2), 1–14 (2015).
    [Crossref]

2017 (5)

L.-G. Si, H. Xiong, M. S. Zubairy, and Y. Wu, “Optomechanically induced opacity and amplification in a quadratically coupled optomechanical system,” Phys. Rev. A 95, 033803 (2017).
[Crossref]

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. Applied 7(2), 024028 (2017).
[Crossref]

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

Y. L. Zhang, C. H. Dong, C. L. Zou, X. B. Zou, Y. D. Wang, and G. C. Guo, “Optomechanical devices based on traveling-wave microresonators,” Phys. Rev. A 95(4), 043815 (2017).
[Crossref]

M. A. Miri, F. Ruesink, E. Verhagen, and A. Alù, “Fundamentals of optical non-reciprocity based on optomechanical coupling,” Phys. Rev. Applied 7(6), 064014 (2017).
[Crossref]

2016 (5)

H. Fu, Z. Gong, T. Mao, C. Sun, S. Yi, Y. Li, and G. Cao, “Classical analog of Stuckelberg interferometry in a two-coupled-cantilever based optomechanical system,” Phys. Rev. A 94(4), 043855 (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(2), 023827 (2016).
[Crossref]

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(10), 657–661 (2016).
[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]

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(12), 123902 (2016).
[Crossref] [PubMed]

2015 (9)

I. Söllner, S. Mahmoodian, S. L. Hansen, L. Midolo, A. Javadi, G. Kiršanskė, T. Pregnolato, H. El-Ella, E. H. Lee, J. D. Song, Søren Stobbe, and P. Lodahl, “Deterministic photon–emitter coupling in chiral photonic circuits,” Nat. Nanotechnol. 10(9), 775–778 (2015).
[Crossref] [PubMed]

W. Z. Jia, L. F. Wei, Y. Li, and Y. X. Liu, “Phase-dependent optical response properties in an optomechanical system by coherently driving the mechanical resonator,” Phys. Rev. A 91(4), 043843 (2015).
[Crossref]

X.-W. Xu and Y. Li, “Controllable optical output fields from an optomechanical system with mechanical driving,” Phys. Rev. A 92(2), 023855 (2015).
[Crossref]

Sh. Barzanjeh, S. Guha, C. Weedbrook, D. Vitali, J. H. Shapiro, and S. Pirandola, “Microwave quantum illumination,” Phys. Rev. Lett. 114(8), 080503 (2015).
[Crossref] [PubMed]

L. Tian, “Optoelectromechanical transducer: Reversible conversion between microwave and optical photons,” Ann. Phys. (Berlin) 527(1–2), 1–14 (2015).
[Crossref]

J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Phys. 11(3), 275–280 (2015).
[Crossref]

A. Metelmann and A. A. Clerk, “Nonreciprocal photon transmission and amplification via reservoir engineering,” Phys. Rev. X 5(2), 021025 (2015).

X. W. Xu and Y. Li, “Optical nonreciprocity and optomechanical circulator in three-mode optomechanical systems,” Phys. Rev. A 91(5), 053854 (2015).
[Crossref]

L. Ranzani and J. Aumentado, “Graph-based analysis of nonreciprocity in coupled-mode systems,” New J. Physics,  17(2), 023024 (2015).
[Crossref]

2014 (5)

A. Metelmann and A. A. Clerk, “Quantum-limited amplification via reservoir engineering,” Phys. Rev. Lett. 112(13), 133904 (2014).
[Crossref] [PubMed]

X. Xu and J. M. Taylor, “Squeezing in a coupled two-mode optomechanical system for force sensing below the standard quantum limit,” Phys. Rev. A 90(4), 043848 (2014).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391 (2014).
[Crossref]

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(12), 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(7), 524–529 (2014).
[Crossref]

2013 (9)

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(9), 093901 (2013).
[Crossref] [PubMed]

S. A. R. Horsley, J.-H. Wu, M. Artoni, and G. C. La Rocca, “Optical nonreciprocity of cold atom Bragg mirrors in motion,” Phys. Rev. Lett. 110(22), 223602 (2013).
[Crossref] [PubMed]

K. Qu and G. S. Agarwal, “Phonon-mediated electromagnetically induced absorption in hybrid optoelectromechanical systems,” Phys. Rev. A 87(3), 031802 (2013).
[Crossref]

P. Meystre, “A short walk through quantum optomechanics,” Ann. Phys. (Berlin) 525(3), 215–233 (2013).
[Crossref]

A. Arvanitaki and A. A. Geraci, “Detecting high-frequency gravitational waves with optically levitated sensors,” Phys. Rev. Lett. 110(7), 071105 (2013).
[Crossref] [PubMed]

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88(1), 013804 (2013).
[Crossref]

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(3), 179–184 (2013).
[Crossref]

H. Okamoto, A. Gourgout, C. Y. Chang, K. Onomitsu, I. Mahboob, E. Y. Chang, and H. Yamaguchi, “Coherent phonon manipulation in coupled mechanical resonators,” Nat. Phys. 9(8), 480–484 (2013).
[Crossref]

T. Faust, J. Rieger, M. J. Seitner, J. P. Kotthaus, and E. M. Weig, “Coherent control of a classical nanomechanical two-level system,” Nat. Phys. 9(8), 485–488 (2013).
[Crossref]

2012 (5)

M. Hafezi and P. Rabl, “Optomechanically induced non-reciprocity in microring resonators,” Opt. Express 20(7), 7672–7684 (2012).
[Crossref] [PubMed]

A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and O. Painter, “A high-resolution microchip optomechanical accelerometer,” Nat. Photon. 6(11), 768–772 (2012).
[Crossref]

M. Aspelmeyer, P. Meystre, and K. C. Schwab, “Quantum optomechanics,” Phys. Today 65(7), 29–35 (2012).
[Crossref]

S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swaim, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. Rubinsztein-Dunlop, “Cavity optomechanical magnetometer,” Phys. Rev. Lett. 108(12), 120801 (2012).
[Crossref] [PubMed]

F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-electromechanical system,” New J. Phys. 14(12), 123037 (2012).
[Crossref]

2011 (5)

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(7377), 351–354 (2011).
[Crossref]

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(12), 758–762 (2011).
[Crossref]

D. E. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter, “Slowing and stopping light using an optomechanical crystal array,” New J. Phys. 13(2), 023003 (2011).
[Crossref]

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature (London),  471(7337), 204–208 (2011).
[Crossref]

A. H. Safavi-Naeini, T. P. M. 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(7341), 69–73 (2011).
[Crossref]

2010 (2)

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010).
[Crossref]

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref] [PubMed]

2009 (3)

J. D. Teufel, T. Donner, M. A. Castellanos-Beltran, J. W. Harlow, and K. W. Lehnert, “Nanomechanical motion measured with an imprecision below that at the standard quantum limit,” Nat. Nanotechnol. 4(12), 820–823 (2009).
[Crossref] [PubMed]

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photon. 3(2), 91–94 (2009).
[Crossref]

S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical nonreciprocity in optomechanical structures,” Phys. Rev. Lett. 102(21), 213903 (2009).
[Crossref] [PubMed]

2008 (2)

C. A. Regal, J. D. Teufel, and K. W. Lehnert, “Measuring nanomechanical motion with a microwave cavity interferometer,” Nat. Phys. 4(7), 555–560 (2008).
[Crossref]

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321(5893), 1172–1176 (2008).
[Crossref] [PubMed]

2007 (1)

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref] [PubMed]

2004 (1)

D. Rugar, R. Budakian, H. J. Mamin, and B. W. Chui, “Single spin detection by magnetic resonance force microscopy,” Nature (London) 430(6997), 329–332 (2004).
[Crossref]

2003 (2)

S. Mancini, D. Vitali, and P. Tombesi, “Scheme for teleportation of quantum states onto a mechanical resonator,” Phys. Rev. Lett. 90(13), 137901 (2003).
[Crossref] [PubMed]

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91(4), 043902 (2003).
[Crossref] [PubMed]

2000 (1)

M. Cai, O. J. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85(1), 74 (2000).
[Crossref] [PubMed]

1991 (1)

D. Rugar and P. Grütter, “Mechanical parametric amplification and thermomechanical noise squeezing,” Phys. Rev. Lett. 67(6), 699 (1991).
[Crossref] [PubMed]

Agarwal, G. S.

K. Qu and G. S. Agarwal, “Phonon-mediated electromagnetically induced absorption in hybrid optoelectromechanical systems,” Phys. Rev. A 87(3), 031802 (2013).
[Crossref]

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010).
[Crossref]

Alegre, T. P. M.

A. H. Safavi-Naeini, T. P. M. 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(7341), 69–73 (2011).
[Crossref]

Allman, M. S.

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature (London),  471(7337), 204–208 (2011).
[Crossref]

Alù, A.

M. A. Miri, F. Ruesink, E. Verhagen, and A. Alù, “Fundamentals of optical non-reciprocity based on optomechanical coupling,” Phys. Rev. Applied 7(6), 064014 (2017).
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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]

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(12), 923–927 (2014).
[Crossref]

Arcizet, O.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref] [PubMed]

Artoni, M.

S. A. R. Horsley, J.-H. Wu, M. Artoni, and G. C. La Rocca, “Optical nonreciprocity of cold atom Bragg mirrors in motion,” Phys. Rev. Lett. 110(22), 223602 (2013).
[Crossref] [PubMed]

Arvanitaki, A.

A. Arvanitaki and A. A. Geraci, “Detecting high-frequency gravitational waves with optically levitated sensors,” Phys. Rev. Lett. 110(7), 071105 (2013).
[Crossref] [PubMed]

Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391 (2014).
[Crossref]

M. Aspelmeyer, P. Meystre, and K. C. Schwab, “Quantum optomechanics,” Phys. Today 65(7), 29–35 (2012).
[Crossref]

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref] [PubMed]

Aumentado, J.

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. Applied 7(2), 024028 (2017).
[Crossref]

L. Ranzani and J. Aumentado, “Graph-based analysis of nonreciprocity in coupled-mode systems,” New J. Physics,  17(2), 023024 (2015).
[Crossref]

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,” arXiv: 1703.05269.

Bahl, G.

J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Phys. 11(3), 275–280 (2015).
[Crossref]

Barzanjeh, Sh.

Sh. Barzanjeh, S. Guha, C. Weedbrook, D. Vitali, J. H. Shapiro, and S. Pirandola, “Microwave quantum illumination,” Phys. Rev. Lett. 114(8), 080503 (2015).
[Crossref] [PubMed]

Bawaj, M.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88(1), 013804 (2013).
[Crossref]

Bernier, N. R.

D. Malz, L. D. Toth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” arXiv: 1705.00436.

N. R. Bernier, L. D. Tóth, A. Koottandavida, M. Ioannou, D. Malz, A. Nunnenkamp, A. K. Feofanov, and T. J. Kippenberg, “Nonreciprocal reconfigurable microwave optomechanical circuit,” arXiv: 1612.08223 (2016).

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(12), 758–762 (2011).
[Crossref]

Biancofiore, C.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88(1), 013804 (2013).
[Crossref]

Blasius, T. D.

A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and O. Painter, “A high-resolution microchip optomechanical accelerometer,” Nat. Photon. 6(11), 768–772 (2012).
[Crossref]

Böhm, H. R.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref] [PubMed]

Bowen, W. P.

S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swaim, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. Rubinsztein-Dunlop, “Cavity optomechanical magnetometer,” Phys. Rev. Lett. 108(12), 120801 (2012).
[Crossref] [PubMed]

Budakian, R.

D. Rugar, R. Budakian, H. J. Mamin, and B. W. Chui, “Single spin detection by magnetic resonance force microscopy,” Nature (London) 430(6997), 329–332 (2004).
[Crossref]

Cai, M.

M. Cai, O. J. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85(1), 74 (2000).
[Crossref] [PubMed]

Cao, G.

H. Fu, Z. Gong, T. Mao, C. Sun, S. Yi, Y. Li, and G. Cao, “Classical analog of Stuckelberg interferometry in a two-coupled-cantilever based optomechanical system,” Phys. Rev. A 94(4), 043855 (2016).
[Crossref]

Castellanos-Beltran, M. A.

J. D. Teufel, T. Donner, M. A. Castellanos-Beltran, J. W. Harlow, and K. W. Lehnert, “Nanomechanical motion measured with an imprecision below that at the standard quantum limit,” Nat. Nanotechnol. 4(12), 820–823 (2009).
[Crossref] [PubMed]

Chan, J.

A. H. Safavi-Naeini, T. P. M. 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(7341), 69–73 (2011).
[Crossref]

Chang, C. Y.

H. Okamoto, A. Gourgout, C. Y. Chang, K. Onomitsu, I. Mahboob, E. Y. Chang, and H. Yamaguchi, “Coherent phonon manipulation in coupled mechanical resonators,” Nat. Phys. 9(8), 480–484 (2013).
[Crossref]

Chang, D. E.

A. H. Safavi-Naeini, T. P. M. 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(7341), 69–73 (2011).
[Crossref]

D. E. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter, “Slowing and stopping light using an optomechanical crystal array,” New J. Phys. 13(2), 023003 (2011).
[Crossref]

Chang, E. Y.

H. Okamoto, A. Gourgout, C. Y. Chang, K. Onomitsu, I. Mahboob, E. Y. Chang, and H. Yamaguchi, “Coherent phonon manipulation in coupled mechanical resonators,” Nat. Phys. 9(8), 480–484 (2013).
[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(7), 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(2), 023827 (2016).
[Crossref]

Chen, Y.

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(10), 657–661 (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(7377), 351–354 (2011).
[Crossref]

Chui, B. W.

D. Rugar, R. Budakian, H. J. Mamin, and B. W. Chui, “Single spin detection by magnetic resonance force microscopy,” Nature (London) 430(6997), 329–332 (2004).
[Crossref]

Cicak, K.

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. Applied 7(2), 024028 (2017).
[Crossref]

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature (London),  471(7337), 204–208 (2011).
[Crossref]

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,” arXiv: 1703.05269.

Clerk, A. A.

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

A. Metelmann and A. A. Clerk, “Nonreciprocal photon transmission and amplification via reservoir engineering,” Phys. Rev. X 5(2), 021025 (2015).

A. Metelmann and A. A. Clerk, “Quantum-limited amplification via reservoir engineering,” Phys. Rev. Lett. 112(13), 133904 (2014).
[Crossref] [PubMed]

Deléglise, S.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref] [PubMed]

Dionne, G. F.

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(12), 758–762 (2011).
[Crossref]

Dong, C. H.

Y. L. Zhang, C. H. Dong, C. L. Zou, X. B. Zou, Y. D. Wang, and G. C. Guo, “Optomechanical devices based on traveling-wave microresonators,” Phys. Rev. A 95(4), 043815 (2017).
[Crossref]

Dong, C.-H.

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(10), 657–661 (2016).
[Crossref]

Donner, T.

J. D. Teufel, T. Donner, M. A. Castellanos-Beltran, J. W. Harlow, and K. W. Lehnert, “Nanomechanical motion measured with an imprecision below that at the standard quantum limit,” Nat. Nanotechnol. 4(12), 820–823 (2009).
[Crossref] [PubMed]

Eichenfield, M.

A. H. Safavi-Naeini, T. P. M. 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(7341), 69–73 (2011).
[Crossref]

El-Ella, H.

I. Söllner, S. Mahmoodian, S. L. Hansen, L. Midolo, A. Javadi, G. Kiršanskė, T. Pregnolato, H. El-Ella, E. H. Lee, J. D. Song, Søren Stobbe, and P. Lodahl, “Deterministic photon–emitter coupling in chiral photonic circuits,” Nat. Nanotechnol. 10(9), 775–778 (2015).
[Crossref] [PubMed]

Estep, N. A.

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(12), 923–927 (2014).
[Crossref]

Evers, J.

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(9), 093901 (2013).
[Crossref] [PubMed]

Fan, S.

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photon. 3(2), 91–94 (2009).
[Crossref]

Fang, K.

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

Faust, T.

T. Faust, J. Rieger, M. J. Seitner, J. P. Kotthaus, and E. M. Weig, “Coherent control of a classical nanomechanical two-level system,” Nat. Phys. 9(8), 485–488 (2013).
[Crossref]

Feofanov, A. K.

N. R. Bernier, L. D. Tóth, A. Koottandavida, M. Ioannou, D. Malz, A. Nunnenkamp, A. K. Feofanov, and T. J. Kippenberg, “Nonreciprocal reconfigurable microwave optomechanical circuit,” arXiv: 1612.08223 (2016).

D. Malz, L. D. Toth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” arXiv: 1705.00436.

Ferreira, A.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref] [PubMed]

Forstner, S.

S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swaim, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. Rubinsztein-Dunlop, “Cavity optomechanical magnetometer,” Phys. Rev. Lett. 108(12), 120801 (2012).
[Crossref] [PubMed]

Fu, H.

H. Fu, Z. Gong, T. Mao, C. Sun, S. Yi, Y. Li, and G. Cao, “Classical analog of Stuckelberg interferometry in a two-coupled-cantilever based optomechanical system,” Phys. Rev. A 94(4), 043855 (2016).
[Crossref]

Galassi, M.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88(1), 013804 (2013).
[Crossref]

Gavartin, E.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref] [PubMed]

Geraci, A. A.

A. Arvanitaki and A. A. Geraci, “Detecting high-frequency gravitational waves with optically levitated sensors,” Phys. Rev. Lett. 110(7), 071105 (2013).
[Crossref] [PubMed]

Gigan, S.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref] [PubMed]

Giuseppe, G. Di

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88(1), 013804 (2013).
[Crossref]

Gong, Z.

H. Fu, Z. Gong, T. Mao, C. Sun, S. Yi, Y. Li, and G. Cao, “Classical analog of Stuckelberg interferometry in a two-coupled-cantilever based optomechanical system,” Phys. Rev. A 94(4), 043855 (2016).
[Crossref]

Gourgout, A.

H. Okamoto, A. Gourgout, C. Y. Chang, K. Onomitsu, I. Mahboob, E. Y. Chang, and H. Yamaguchi, “Coherent phonon manipulation in coupled mechanical resonators,” Nat. Phys. 9(8), 480–484 (2013).
[Crossref]

Gross, R.

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Figures (4)

Fig. 1
Fig. 1 Schematic of a three-mode optomechanical system driven by two pump fields with the same frequency ωd. 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 ωb. The cavities and the mechanical resonator are coupled via radiation-pressure forces, and the cavities are directly coupled to each other.
Fig. 2
Fig. 2 The transmission probabilities T21 and T12 versus Δm = ωm (ωp − ωd) for different values of θ and φ: (a) θ = 0, φ = π/2; (b) θ = π/2, φ = 0; (c) θ−= π/2, φ = π/2. Other parameters are y = 20, η1,2 = 1, γ1 = 1.1γm, γ2 = 1.5γm, G = |G1,2| = J = γm, and Δ 1 , 2 = Δ m.
Fig. 3
Fig. 3 Plot of the probability of transmission T21 and T12 as functions of θ and φ, respectively. (a) φ = π/2. (b) θ = π/2. Other parameters are y = 20, η1,2 = 1, G = | G 1 , 2 | = J = γ m, Δ m = Δ 1 , 2 = 0, γ1 = 1.1γm, and γ2 = 1.5γm. One can see that at certain optimal values of θ and φ, e.g., θ = π/2, φ = π/2, T12 → 0 and T21 ≫ 1.
Fig. 4
Fig. 4 The transmission probabilities T21 and T12 versus y. Other parameters are θ = π/2, φ = π/2, G = |G1,2| = J = γm, Δ m = Δ 1 , 2 = 0, γ1 = 1.1γm, and γ2 = 1.5γm.

Equations (27)

Equations on this page are rendered with MathJax. Learn more.

H = H 0 + H I + H d .
H 0 = ω 1 a 1 a 1 + ω 2 a 2 a 2 + ω m b b ,
H I = J ( a 1 a 2 + a 1 a 2 ) + i g i a i a i ( b + b )
H d = i ( i ε i a i e i ω d t e i θ i + h . c . ) + ( i ε p a 1 e i ω p t + i ε b b e i ω b t + h . c . ) ,
a ˙ 1 = { γ 1 i [ Δ 1 + g 1 ( b + b ) ] } a 1 i J a 2 + ε 1 e i θ 1 + ε p e i ( ω d ω p ) t + ξ 1 ,
a ˙ 2 = { γ 2 i [ Δ 2 + g 2 ( b + b ) ] } a 2 i J a 1 + ε 2 e i θ 2 + ξ 2 ,
b ˙ = ( γ m i ω m ) b i ( g 1 a 1 a 1 + g 2 a 2 a 2 ) + ε b e i ω b t + ξ m .
a 1 = ( γ 2 + i Δ 2 ) ε 1 e i θ 1 i J ε 2 e i θ 2 ( γ 1 + i Δ 1 ) ( γ 2 + i Δ 2 ) + J 2 ,
a 2 = ( γ 1 + i Δ 1 ) ε 2 e i θ 2 i J ε 1 e i θ 1 ( γ 1 + i Δ 1 ) ( γ 2 + i Δ 2 ) + J 2 ,
b = i ( g 1 | a 1 | 2 + g 2 | a 2 | 2 ) γ m + i ω m ,
δ a ˙ 1 = ( γ 1 i Δ 1 ) δ a 1 i G 1 ( δ b + δ b ) i J δ a 2 + ε p e i ( ω d ω p ) t + ξ 1 ,
δ a ˙ 2 = ( γ 2 i Δ 2 ) δ a 2 i G 2 ( δ b + δ b ) i J δ a 1 + ξ 2 ,
δ b ˙ = ( γ m i ω m ) δ b i ( G 1 δ a 1 + G 1 * δ a 1 ) i ( G 2 δ a 2 + G 2 * δ a 2 ) + ε b e i ω b t + ξ m ,
δ a ˙ 1 = Γ 1 δ a 1 i G 1 δ b i J δ a 2 + ε p + ξ 1 ,
δ a ˙ 2 = Γ 2 δ a 2 i G 2 δ b i J δ a 1 + ξ 2 ,
δ b ˙ = Γ m δ b i G 1 * δ a 1 i G 2 * δ a 2 + ε b + ξ m ,
δ a 1 = i G 2 ε b ( i J Γ m + G 1 G 2 * ) + ( Γ 2 Γ m + | G 2 | 2 ) ( ε p Γ m i G 1 ε b ) ( Γ 1 Γ m + | G 1 | 2 ) ( Γ 2 Γ m + | G 2 | 2 ) ( i J Γ m + G 1 G 2 * ) ( i J Γ m + G 1 * G 2 ) ,
δ a 2 = ( i J Γ m + G 1 * G 2 ) ( ε p Γ m i G 1 ε b ) i G 2 ε b ( Γ 1 Γ m + | G 1 | 2 ) ( Γ 1 Γ m + | G 1 | 2 ) ( Γ 2 Γ m + | G 2 | 2 ) ( i J Γ m + G 1 G 2 * ) ( i J Γ m + G 1 * G 2 ) ,
δ b = J Γ m ( ε b J ε p G 2 * ) + Γ 2 Γ m ( ε b Γ 1 i ε p G 1 * ) ( Γ 1 Γ m + | G 1 | 2 ) ( Γ 2 Γ m + | G 2 | 2 ) ( i J Γ m + G 1 G 2 * ) ( i J Γ m + G 1 * G 2 ) .
δ a i o u t + δ a i i n = 2 γ i e δ a i ,
t 21 δ a 2 o u t / δ a 1 i n .
t 21 = 2 γ 1 e γ 2 e [ ( i J Γ m + G 1 * G 2 ) ( Γ m i G 1 y e i φ ) + i G 2 y e i φ ( Γ 1 Γ m + | G 1 | 2 ) ( Γ 1 Γ m + | G 1 | 2 ) ( Γ 2 Γ m + | G 2 | 2 ) ( i J Γ m + G 1 G 2 * ) ( i J Γ m + G 1 * G 2 ) ] ,
t 12 = 2 γ 1 e γ 2 e [ ( i J Γ m + G 2 * G 1 ) ( Γ m i G 2 y e i φ ) + i G 1 y e i φ ( Γ 2 Γ m + | G 2 | 2 ) ( Γ 2 Γ m + | G 2 | 2 ) ( Γ 1 Γ m + | G 1 | 2 ) ( i J Γ m + G 2 G 1 * ) ( i J Γ m + G 2 * G 1 ) ] .
t 21 = 2 γ 1 e γ 2 e [ ( i J Γ m + G 2 e i θ ) ( Γ m i G y e i φ ) + i G ( Γ 1 Γ m + G 2 ) y e i ( θ + φ ) ( Γ 1 Γ m + G 2 ) ( Γ 2 Γ m + G 2 ) ( i J Γ m + G 2 e i θ ) ( i J Γ m + G 2 e i θ ) ] ,
t 12 = 2 γ 1 e γ 2 e [ ( i J Γ m + G 2 e i θ ) ( Γ m i G y e i ( θ + φ ) ) + i G ( Γ 2 Γ m + G 2 ) y e i φ ( Γ 1 Γ m + G 2 ) ( Γ 2 Γ m + G 2 ) ( i J Γ m + G 2 e i θ ) ( i J Γ m + G 2 e i θ ) ] .
T 21 = 4 γ 1 γ 2 ( 2 γ m ( y + 1 ) y ( γ 1 + γ m ) ( γ 1 + γ m ) ( γ 2 + γ m ) ) 2 ,
T 12 = 4 y 2 γ 1 γ 2 ( γ 1 + γ m ) 2 .

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