Abstract

We propose how to achieve quantum nonreciprocity via unconventional photon blockade (UPB) in a compound device consisting of an optical harmonic resonator and a spinning optomechanical resonator. We show that, even with very weak single-photon nonlinearity, nonreciprocal UPB can emerge in this system, i.e., strong photon antibunching can emerge only by driving the device from one side but not from the other side. This nonreciprocity results from the Fizeau drag, leading to different splitting of the resonance frequencies for the optical counter-circulating modes. Such quantum nonreciprocal devices can be particularly useful in achieving back-action-free quantum sensing or chiral photonic communications.

© 2019 Chinese Laser Press

1. INTRODUCTION

Photon blockade (PB) [15], i.e., the generation of the first photon in a nonlinear cavity, diminishes to almost zero the probability of generating another photon in the cavity; it plays a key role in single-photon control for quantum technology applications nowadays [68]. In experiments, PB has been demonstrated in cavity-QED or circuit-QED systems [4,5,912]. It has also been predicted in various nonlinear optical systems [1315] and optomechanical (OM) devices [1620]. Conventional PB occurs under the stringent condition of strong single-photon nonlinearities, which is highly challenging in practice.

To overcome this obstacle, coupled-resonator systems, with destructive interferences of different dissipative pathways [2124], have been proposed to achieve unconventional PB (UPB) even for arbitrarily weak nonlinearities [2337]. UPB provides a powerful tool to generate optimally sub-Poissonian light and also a way to reveal quantum correlations in weakly nonlinear devices [33,34]. Recently, UPB was demonstrated experimentally in coupled optical [36] or superconducting resonators [37].

It should be stressed that PB and UPB are very different phenomena, and, thus, their nonreciprocal generalizations are different as well. Indeed PB refers to a process where a single photon is blocking the entry (or generation) of more photons in a strongly nonlinear cavity. Thus, PB refers to state truncation, also referred to as nonlinear quantum scissors [38,39]. PB can be used as a source of single photons, since the PB light is sub-Poissonian (or photon antibunched) in second and higher orders, as characterized by the correlation functions g(n)(0)<1 for n=2,3,. By contrast to PB, UPB refers to the light that is optimally sub-Poissonian in the second order, g2(0)0, and is generated in a weakly nonlinear system allowing multi-path interference (e.g., two linearly coupled cavities, when one of them is also weakly coupled to a two-level atom). Thus, PB and UPB are induced by different effects: PB due to a large system nonlinearity and UPB via multipath interference even assuming extremely weak system nonlinearity. Note that light generated via UPB can exhibit higher-order super-Poissonian photon-number statistics, g(n)(0)>1 for some n>2. Thus, UPB is, in general, not a good source of single photons. This short comparison of PB and UPB indicates that the term UPB, as coined in Ref. [40] and now commonly accepted, is fundamentally different from PB, concerning their physical mechanisms and the properties of the light generated in them.

Here, we propose achieving and controlling nonreciprocal UPB with spinning devices. Nonreciprocal devices allow the flow of light from one side but block it from the other. Thus, such devices can be applied in noise-free quantum information signal processing and quantum communication for canceling interfering signals [41]. Nonreciprocal optical devices have been realized in OM devices [4143], Kerr resonators [4446], thermo systems [4749], devices with temporal modulation [50,51], and non-Hermitian systems [5254]. In a very recent experiment [55], 99.6% optical isolation was achieved in a spinning resonator based on the optical Sagnac effect. By using the spinning resonators, optomechanically induced transparency [56] and ultrasensitive nanoparticle sensing [57] have also been studied. However, these studies have mainly focused on the classical regimes, i.e., unidirectional control of transmission rates instead of quantum noises. We also note that in recent works, single-photon diodes [5860], unidirectional quantum amplifiers [6165], and one-way quantum routers [66] have been explored. In particular, nonreciprocal PB was predicted in a Kerr resonator [67] or a quadratic OM system [68], which, however, relies on the conventional condition of strong single-photon nonlinearity. These quantum nonreciprocal devices have potential applications for quantum control of light chiral and topological quantum technologies [69].

We also note that coupled-cavity systems have been studied extensively in experiments [37,7072], providing a unique way to achieve not only UPB, but also phonon laser [7276], slow light [77], and force sensing [70,71,78]. Here, we study nonreciprocal UPB in a coupled system with an optical harmonic cavity and a spinning OM resonator. We find that, by the spinning of an OM resonator, UPB can emerge nonreciprocally even with weak single-photon nonlinearity; that is, strongly antibunched photons can emerge only when the device is driven from one side but not the other side. Our work opens up a new route to engineer quantum chiral UPB devices, which can have practical applications in achieving, for example, photonic diodes or circulators, and nonreciprocal quantum communications at the few-photon level.

2. MODEL AND SOLUTIONS

We consider a compound system consisting of an optical harmonic resonator (with a resonance frequency ωL of the cavity field and a decay rate of κL) and a spinning anharmonic resonator (with a resonance frequency ωR of the cavity field and a decay rate of κR), as shown in Fig. 1. External light is coupled into and out of the resonator through a tapered fiber of frequency ωd and these two whispering-gallery-mode resonators are evanescently coupled to each other with a coupling strength of J [79]. Note that the required strong Kerr nonlinearity, K3κ (where κ is the cavity linewidth), in the previous proposal [67] is challenging for the current experiments. Here, we can use an experimentally feasible Kerr-nonlinear strength to realize nonreciprocal PB, i.e., K0.04κ [37], which is two orders of magnitude smaller than that in the former work [67]. Weak Kerr couplings can be achieved in cavity-atom systems [80], magnon devices [81], and OM systems [82], on which we focus here. We consider a weak OM coupling strength (g0.63κ) in an auxiliary cavity that is well within the current experimental abilities [8385]. In a spinning resonator, the refractive indices associated with the clockwise (+) and anticlockwise () optical modes are given as n±=n[1±nv(n21)/c], where v=rΩ is the tangential velocity with an angular velocity of Ω and radius r [55]. For light propagating in the spinning resonator, the optical mode experiences a Fizeau shift ΔF [86], that is, ωRωR+ΔF, with

ΔF=±nrΩωRc(11n2λndndλ)=±ηΩ,
where ωR=2πc/λ is the optical resonance frequency of the nonspinning OM resonator, c (λ) is the speed (wavelength) of light in vacuum, and n is the refractive index of the cavity. The dispersion term dn/dλ, characterizing the relativistic origin of the Sagnac effect, is relatively small in typical materials (1%) [55,86]. For convenience, we always assume counterclockwise rotation of the resonator. Hence, ±ΔF denote light propagating against (ΔF>0) and along (ΔF<0) the direction of the spinning OM resonator, respectively.

 

Fig. 1. Nonreciprocal UPB in a coupled-resonator system. Spinning the OM (Kerr-type) resonator results in different Fizeau drag ΔF for the counter-circulating whispering-gallery modes of the resonator. (a) By driving the system from the left-hand side, the direct excitation from state |1,0 to state |2,0 (red dotted arrow) will be forbidden by destructive quantum interference with the other paths drawn by green arrows, leading to photon antibunching. (b) Photon bunching occurs when the system is driven from the right side, due to lack of complete destructive quantum interference between the indicated levels (drawn by crossed green dotted arrows). Here, δ=g2/ωm is the energy shift induced by the OM nonlinearity.

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In a rotating frame with respect to H0=ωd(aLaL+aRaR), the effective Hamiltonian of the system can be written as (see Appendix A for more details)

H=ΔLaLaL+(ΔR+ΔF)aRaR+ωmbb+J(aLaR+aRaL)+gaRaR(b+b)+iϵd(aLaL),
where aL (aL) and aR (aR) are the photon annihilation (creation) operators for the cavity modes of the optical cavity (denoted with the subscript L) and the OM cavity (denoted with the subscript R), respectively. b (b) is the annihilation (creation) operator for the mechanical mode of the OM cavity. The frequency detuning between the cavity field in the left (right) cavity and the driving field is denoted as ΔK=ωKωd, where K=L,R. The parameter J denotes the strength of the photon hopping interaction between the two cavity modes, and g=ωR/r[/(2mωm)]1/2 describes the radiation-pressure coupling between the optical and vibrative modes in the OM resonator with frequency ωm and effective mass m. ϵd=κLPin/(ωd) denotes the driving strength that is coupled into the compound system through the optical fiber waveguide with a cavity loss rate of κL and driving power Pin.

The Heisenberg equations of motion of the system are then written as

ddtq=ωmp,ddtp=ωmqgbaRaRγm2p+ξ,ddtaL=(κL2+iΔL)aLiJaR+ϵd+κLaL,in,ddtaR=(κR2+iΔR)aRiJaLigbqaR+κLaR,in,
where p and q are dimensionless canonical position and momentum, with p=i(bb)/2 and q=(b+b)/2, respectively. ΔR=ΔR+ΔF and gb=2g, and κL=ωL/QL (κR=ωR/QR) is the dissipation rate and QL (QR) is the quality factor of the left (right) cavity. γm=ωm/QM is the damping rate with QM the quality factor of the mechanical mode. Moreover, ξ is the zero-mean Brownian stochastic operator, ξ(t)=0, resulting from the coupling of the mechanical resonator with the corresponding thermal environment and satisfying the correlation function [87]
ξ(t)ξ(t)=12πdωeiω(tt)Γm(ω),
where
Γm(ω)=ωγm2ωm[1+coth(ω2kBT)],
T is effective temperature of the environment of the mechanical resonator, and kB is the Boltzmann constant. The annihilation operators aL,in and aR,in are, respectively, the input vacuum noise operators of the optical cavity and the OM cavity with zero mean value, i.e., aL,in=aR,in=0, and they comply with the time-domain correlation functions [88,89]
aK,in(t)aK,in(t)=0,aK,in(t)aK,in(t)=δ(tt),
for K=L,R. Because the whole system interacts with a low-temperature environment (here we consider 0.1 mK), we neglect the mean thermal photon numbers at optical frequencies in the two cavities. In order to linearize the dynamics around the steady state of the system, we expend the operators as the sum of its steady-state mean values and a small fluctuation with zero mean value around it; that is, aL=α+δaL, aR=β+δaR, q=qs+δq, and p=ps+δp. By neglecting higher-order terms, δaLδaL, the linearized equations of the fluctuation terms can be written as
ddtδq=ωmδp,ddtδp=ωmδqgb(β*δaR+βδaR)γm2δp+ξ,ddtδaL=(κL2+iΔL)δaLiJδaR+κLaL,in,ddtδaR=(κR2+iΔR)δaRiJδaLigbqsδaRigbβδq+κRaR,in.
These equations can be solved in the frequency domain (see Appendix B). In particular, we find
δaL(ω)=E(ω)aL,in(ω)+F(ω)aL,in(ω)+G(ω)aR,in(ω)+H(ω)aR,in(ω)+Q(ω)ξ(ω),
where
E(ω)=κLA1(ω)A5(ω),F(ω)=κLA2(ω)A5(ω),G(ω)=κRA3(ω)A5(ω),H(ω)=κRA4(ω)A5(ω),Q(ω)=igbχ(ω)ωmA5(ω)[βA3(ω)+β*A4(ω)],
and
A1(ω)=[(κR2+iω)2+ΔR2]V1(ω)gb4|β|4[χ(ω)ωm]2V1(ω)+J2V2+,A2(ω)=iJ2gb2β2χ(ω)ωm,A3(ω)=iJV1(ω)V2iJ3,A4(ω)=Jgb2β2χ(ω)ωmV1(ω),A5(ω)=V1+A1(ω)+iJA3(ω),
where we introduced the auxiliary functions
ΔR=ΔR+gbqsgb2|β|2χ(ω),χ(ω)=ωm2/(ωm2ω2+iωγm2),V1±(ω)=κL2±i(ΔLω),V2±(ω)=κR2±i(ΔRω).

3. NONRECIPROCAL OPTICAL CORRELATIONS

Now, we focus on the statistical properties of photons in an optical cavity, which are described quantitatively via the normalized zero-time-delay second-order correlation function gL(2)(0)=aL2aL2/aLaL2 [29,89]. By taking the semi-classical approximation, i.e., aL=α+δaL, the correlation function gL(2)(0) can be given as [29]

gL(2)(0)=|α|4+4|α|2R1+2Re[α*2R2]+R3(|α|2+R1)2,
where R1=δaL(t)δaL(t), R2=[δaL(t)]2, and R3=δaL(t)δaL(t)δaL(t)δaL(t)=2R1+|R2|2.

From Eq. (8), the correlation between δaL(t) and δaL(t) can be calculated as

δaL±(t)δaL(t)=12π+XaL±aLdω,
where
δaL+(t)=δaL(t),δaL(t)=δaL(t),andXaLaL=|Q(ω)|2Γm(ω)+|F(ω)|2+|H(ω)|2,XaLaL=Q(ω)Q(ω)Γm(ω)+E(ω)F(ω)+G(ω)H(ω).
To obtain more accurate results, we introduce the density operator ρ(t) and numerically calculate the normalized zero-time-delay second-order correlation by the Lindblad master equation [90]:
ρ˙=1i[H,ρ]+κL2L[aL](ρ)+κR2L[aR](ρ)+γm2(n¯m+1)L[b](ρ)+γm2n¯mL[b](ρ),
where L[o](ρ)=2oρoooρρoo are the Lindblad super-operators [89], for o=aL, aR, b, and b, and n¯m=1/[exp(ωm/kBT)1] is the mean thermal phonon number of the mechanical mode at temperature T.

The second-order correlation function gL(2)(0) is shown in Fig. 2 as a function of optical detuning Δ/κ and angular velocity Ω. We assume ΔL=ΔRδ=Δ and κL=κR=κ and use experimentally feasible parameters [53,83,9195], that is, λ=1550nm, QL=3×107, r=0.3mm, n=1.44, m=5×1011kg, and Pin=2×1017W. QL is typically 1061012 [92,94,95], g is typically 103106Hz [83,91,92] in optical microresonators, and gL(2)(0)0.37 [36,37] was achieved experimentally. J can be adjusted by changing the distance of the double resonators [72]. In a recent experiment, autocorrelation measurements ranging from g(2)(0)=6×103 to 2 were achieved with an average fidelity of 0.998 in a photon-number-resolving detector [96]. Moreover, we set Ω=12kHz, which is experimentally feasible. The resonator with a radius of r=1.1mm can spin at an angular velocity of Ω=6.6kHz [55]. Using a levitated OM system [97,98], Ω can be increased even up to GHz values.

 

Fig. 2. Correlation function gL(2)(0) versus optical detuning Δ/κ (in units of cavity loss rate κL=κR=κ) with (a) Ω=0 and (b) Ω=12kHz, which is found numerically (solid curves) and analytically (dotted curve). The PB can be generated (red curves) or suppressed (blue curves) for different driving directions, which can be seen more clearly in panel (c). The other parameters are g/κ=0.63, ωm/κ=10 [91], J/κ=3, T=0.1mK (case 1), and g/κ=0.1 [28], ωm/κ=30 [92], J/κ=20, T=1mK (case 2).

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Our analytical results agree well with the numerical one. In the case of a nonspinning resonator, as shown in Fig. 2(a), gL(2)(0) is reciprocal regardless of the direction of the driving light, and always has a dip at Δ/κ0.29 and a peak at Δ/κ0.166, corresponding to strong photon antibunching and photon bunching, respectively [29]. The physical origin of the strong photon antibunching is the destructive interference between the direct and indirect paths of two-photon excitations, i.e.,

|1,02ϵd|2,0,|1,0J|0,1ϵd|1,12J|2,0.
In contrast, for a spinning device, gL(2)(0) exhibits giant nonreciprocity, which can be seen in Fig. 2(b). The PB can be generated, i.e., gL(2)(0)0.06, for ΔF<0, whereas it is significantly suppressed, i.e., gL(2)(0)4.72, for ΔF>0; this can be seen more clearly in Fig. 2(c). Nonreciprocal UPB induced by the Fizeau light-dragging effect, with difference in gL(2)(0) up to two orders of magnitude for opposite directions, can be achieved even with weak nonlinearity and, to our knowledge, has not been studied previously. Furthermore, in Fig. 2(b), we use two sets of parameters for solid (case 1) and dashed curves (case 2), respectively. It can be seen that nonreciprocity still exists in a parameter range closer to that in the experiment.

Since the anharmonicity of the system is very small, destructive quantum interference (rather than anharmonicity) is responsible for observing strong photon antibunching (referred to as UPB) and photon bunching (referred to as photon-induced tunneling) in the spinning devices, as shown in Fig. 1 and confirmed by our analytical calculations. Note that the role of complete (incomplete) destructive quantum interference is the same in both spinning and non-spinning UPB systems, and, thus, we refer to Ref. [24] where this interference-based mechanism was first explained in detail. Spinning the OM resonator results in different Fizeau drag ΔF for the counter-circulating whispering-gallery modes of the resonator. By driving the system from the left-hand side, direct excitation from state |1,0 to state |2,0 will be forbidden by destructive quantum interference with the indirect paths of two-photon excitations, leading to photon antibunching. In contrast, photon bunching occurs when the system is driven from the right side, due to lack of complete destructive quantum interference between the indicated levels [99]. Increasing the angular velocity results in an opposing frequency shift of ηΩ for light coming from opposite directions. gL(2)(0) also shifts linearly with Ω, but with different directions for ΔF<0 and ΔF>0; that is, we observe either a blue shift [see Fig. 3(a)] or a red shift [see Fig. 3(b)] with ΔF>0 or ΔF<0, respectively. A highly tunable nonreciprocal UPB device is thus achievable, by flexible tuning of Ω and Δ/κ. In addition, since gL(2)(0) is sensitive to Ω, this may also indicate a way for accurate measurements of velocity.

 

Fig. 3. Correlation function gL(2)(0) versus optical detuning Δ/κ (in units of cavity loss rate κL=κR=κ) at various angular velocities Ω upon driving the device from (a) the right-hand side or (b) the left-hand side. The dashed curves show our approximate analytical results, given in Eq. (12), whereas the solid curves are our numerical solutions. The other parameters are the same as those in Fig. 2 (case 1).

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4. OPTIMAL PARAMETERS FOR STRONG ANTIBUNCHING

As discussed above, UPB can be generated nonreciprocally. In this section, we analytically derive the optimal conditions for strong antibunching. Here we apply the method described in Ref. [24], which is based on the evolution of a complex non-Hermitian Hamiltonian, as given in Appendix C. Thus, our solution corresponds to only a semi-classical approximation of the solution of the quantum master equation, given in Eq. (15), where the terms corresponding to quantum jumps are ignored.

Since the phonon states can be decoupled from the photon states by using the unitary operator U=exp[g(bb)/ωm], the states of the system can be expressed as |ψ=|φ|ϕm, where |φ and |ϕm are the photon states and phonon states, respectively. Under the weak-driving condition, we make the ansatz [24]

|φ=C00|0,0+C10|1,0+C01|0,1+C20|2,0+C11|1,1+C02|0,2,
and consider that CmnCmnC00 for m+n=2, m+n=1, and the condition of C20=0; the optimal conditions are given by fixing J and κ (see Appendix C):
Δopta3+sgn(E)λ1λ24a4,gopt=ωm[Δopt(4Δopt2+5κ2)+ΔFλ3]2(2J2κ2)+2ΔFλ4,
the signal function sgn(E), a3=96ΔFκ, and λ1,2, which are defined in Appendix C, are related to the Fizeau drag ΔF. Physically, this means that the position of the minimum of gL(2)(0) is determined by the detuning between the two cavity fields. Thus, ΔF can lead to a shift of the minimum of gL(2)(0) to achieve nonreciprocity.

In order to visualize UPB more clearly, we show the contour plots of gL(2)(0) in logarithmic scale [i.e., log10gL(2)(0)] as a function of g/κ and Δ/κ in Fig. 4(a). By fixing Δ/κ=0.05, we obtain the function of gL(2)(0) in logarithmic scale versus the coupling strengths J/κ and g/κ of the resonators, as shown in Fig. 4(b). These plots show that strong photon antibunching occurs exactly at the values predicted by our analytical calculations in Eq. (17). Moreover, by computing gL(2)(0) as a function of Δ/κ and Ω with different mean thermal phonon numbers nth, as shown in Fig. 5, we confirm that rotation-induced nonreciprocity can still exist by considering thermal phonon noises. We note that thermal phonons greatly affect the correlation gL(2)(0) of photons and tend to destroy PB. Thus, to show this effect, in Fig. 6(a) we plot the correlation gL(2)(0) as a function of temperature T for various Fizeau shifts. We see that nonreciprocal UPB can be observed below the critical temperature T04mK (5 mK) for the spinning frequency of Ω=12kHz (Ω=50kHz) [see Fig. 6(b)]. By further increasing the optical dissipation of the OM cavity, as shown in Fig. 6(d), the critical temperature T0 can be made to reach a value of 10 mK.

 

Fig. 4. Correlation function gL(2)(0) in logarithmic scale [i.e., log10gL(2)(0)] versus (a) radiation-pressure coupling g/κ (in units of cavity loss rate κ=κL=κR) and optical detuning Δ/κ, and (b) coupling strength of the resonators J/κ and radiation-pressure coupling g/κ for optical detuning of Δ/κ=0.05. The angular velocity is Ω=12kHz and the white dashed curve corresponds to gL(2)(0)=1. The other parameters are the same as those in Fig. 3.

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Fig. 5. Correlation function gL(2)(0) versus optical detuning Δ/κ (in units of cavity loss rate κL=κR=κ) with varied mean thermal phonon numbers nth for various angular velocities Ω, and the resulting Fizeau shifts ΔF. The other parameters are the same as those in Fig. 4.

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Fig. 6. (a) Correlation function gL(2)(0) versus effective temperature T of the environment of the mechanical resonator for three values of Fizeau shift ΔF (ΔF>0, ΔF=0, and ΔF<0) for optimal values of Δopt and gopt. The other parameters are set the same as in case 2 in Fig. 2. Also shown is the correlation function gL(2)(0) versus T for various values of (b) spinning frequency, (c) mechanical decay, and (d) cavity decay, assuming the device is driven from the left-hand side and optical detuning is fixed at the optimal values.

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Finally, we note that a state (generated via UPB or another effect) with vanishing (or almost vanishing) second-order photon-number correlations, g(2)(0)0, is not necessarily a good single-photon source, i.e., the state might not be a (partially incoherent) superposition of only the vacuum and single-photon states. A good single-photon source is characterized not only by g(2)(0)0, but also by vanishing higher-order photon-number correlation functions, g(n)(0)0 for n>2. In UPB, g(n)(0) for n>2 can be greater than g(2)(0)0, or even greater than 1 [100]. Indeed, a standard analytical method for analyzing UPB, as proposed by Bamba et al. [24] and applied here, is based on expanding the wave function |φ of a two-resonator system in the power series |φ=Cn,m|n,m up to the terms Cn,2n (n=0,1,2) only, as given in Eq. (16). To obtain the optimal system parameters, which minimize g(2)(0) in UPB, this method requires setting C2,0=0 as set in Appendix C. Actually, the same expansion of |φ and the same ansatz are made in Ref. [24]. These assumptions imply that higher-order correlation functions g(n)(0) with n=3,4, vanish too. However, the truncation of the above expansion at the terms Cn,2n is often not justified for a system exhibiting UPB. Indeed, we find parameters for our system for which g(2)(0)0 and, simultaneously, g(3)(0)>1. We have confirmed this by precise numerical calculation of the steady states of our system based on the non-Hermitian Hamiltonian, given in Eq. (C1), in a Hilbert space larger than 4×4.

5. CONCLUSIONS

In summary, we studied nonreciprocal UPB in a system consisting of a purely optical resonator and a spinning OM resonator. Due to interference between two-photon excitation paths and the Sagnac effect, UPB can be generated nonreciprocally in our system; that is, UPB can occur when the system is driven from one direction but not from the other, even under weak OM interactions. The optimal conditions for one-way UPB were presented analytically. Moreover, we found that this quantum nonreciprocity can still exist by considering thermal phonon noises.

Concerning a possible experimental implementation of nonreciprocal UPB, it is worth noting that UPB for non-spinning devices has already been demonstrated experimentally in two recent works [36,37]. A number of experiments (including a very recent work [55]) have shown nonreciprocal quantum effects in spinning devices. So the main experimental task for achieving nonreciprocal UPB in a spinning device would be to combine the experimental setups of, e.g., Refs. [36,37,55] into a single spinning UPB setup.

Our proposal provides a feasible method to control the behavior of one-way photons, with potential applications in achieving, e.g., photonic diodes or circulators, quantum chiral communications, and nonreciprocal light engineering in the deep quantum regime.

APPENDIX A: DERIVATION OF EFFECTIVE HAMILTONIAN

The coupled system can be described by the Hamiltonian

H=H0+Hin+Hdr,H0=ωLaLaL+(ωR+ΔF)aRaR+ωmbb,Hin=J(aLaR+aRaL)+gaRaR(b+b),Hdr=iϵd(aLeiωdtaLeiωdt),
where aL (aL) and aR (aR) are the photon annihilation (creation) operators for the cavity modes of the optical cavity (denoted with the subscript L) and the OM cavity (denoted with the subscript R), respectively. b (b) is the annihilation (creation) operator for the mechanical mode of the OM cavity. The frequencies of the cavity fields are denoted with ωL and ωR. J is the coupling strength between the two resonators, and g=ωR/r[/(2mωm)]1/2 is the OM coupling strength between the optical mode and the mechanical mode in the OM cavity. ϵd=κLPin/(ωd) denotes the driving strength that is coupled into the compound system through the optical fiber waveguide.

Using the unitary operator U=exp[g(bb)/ωm] for the Hamiltonian (A1), we obtain a Kerr-type Hamiltonian [82]

Heff=UHU=ωLaLaL+(ωR+ΔF)aRaRδ(aRaR)2+J[aLaReδ(bb)+aLaReδ(bb)]+iϵd(aLeiωdtaLeiωdt),
where δ=g2/ωm. Under the conditions g/ωm1 and J<ωm/2, the Hamiltonian (A2) can be read as
Heff=ωLaLaL+(ωR+ΔF)aRaRδ(aRaR)2+J(aLaR+aLaR)+iϵd(aLeiωdtaLeiωdt).

APPENDIX B: FOURIER ANALYSIS OF FLUCTUATION TERMS

According to the Heisenberg equations of motion of Hamiltonian (2), and using the semi-classical approximation method, i.e., aL=α+δaL, aR=β+δaR, q=qs+δq, and p=ps+δp, the steady-state values of the system satisfy the equations

0=(κL2+iΔL)α+iJβϵd,0=[κR2+i(ΔR+gbqs)]βiJα,0=ωmqsgb|β|2.
Then we obtain
b3qs3+b2qs2+b1qs+b0=0,
where
b0=gbJ2ϵd2,b1=ωm(κLκR4+J2)2+ωm(κLΔR2+κRΔL2)2ωmΔLΔR(κLκR2+2J2ΔLΔR),b2=2ωmgb[κL2ΔR4+ΔL(ΔLΔRJ2)],b3=ωmgb2(κL24+ΔL2).
The fluctuation terms of the system can be written as
ddtδq=ωmδp,ddtδp=ωmδqgb(β*δaR+βδaR)γm2δp+ξ,ddtδaL=(κL2+iΔL)δaLiJδaR+κLaL,in,ddtδaR=(κR2+iΔR)δaRiJδaLigbqsδaRigbβδq+κRaR,in,
where we have neglected higher-order terms, δaLδaL. Here, the steady-state mean value qs is numerically solved from Eqs. (B2) and (B3).

By introducing the Fourier transform to the fluctuation equations, we find

iωδaL(ω)=(κL2+iΔL)δaL(ω)iJδaR(ω)+κLaL,in(ω),iωδaR(ω)=(κR2+iΔR)δaL(ω)iJδaR(ω)igbβδq(ω)+κRaR,in(ω),iωδq(ω)=ωmδp(ω),iωδp(ω)=ωmδq(ω)gb[β*δaR(ω)+βδaR(ω)]γm2δp(ω)+ξ(ω),
where ΔR=ΔR+gbqs; then we obtain
δq(ω)=gbβ*χ(ω)δaR(ω)gbβχ(ω)δaR(ω)+χ(ω)ξ(ω),
where
χ(ω)=ωmωm2ω2+iωγm/2.
Substituting Eq. (B6) into Eq. (B5), we have
M(ω)δaR(ω)=igb2β2χ(ω)δaR(ω)igbβχ(ω)ξ(ω)iJδL(ω)+κRaR,in(ω),
where
M(ω)=κR2+iω+iΔRi|β|2gb2χ(ω).
According to Eq. (B5), we obtain
iωδaL(ω)=(κL2iΔL)δaL(ω)+iJδaR(ω)+κLaL,in(ω),iωδaR(ω)=(κR2iΔR)δaR(ω)+iJδaR(ω)+igbβδq(ω)+κRaR,in(ω),iωδq(ω)=ωmδp(ω),iωδp(ω)=ωmδq(ω)gb[βδaR(ω)+β*δaR(ω)]γm2δp+ξ(ω),
then we have
N(ω)δaR(ω)=igb2β*2χ(ω)δaR(ω)+igbβ*χ(ω)ξ(ω)+iJδaL(ω)+κRaR,in(ω),
where
N(ω)=κR2+iωiΔR+i|β|2gb2χ(ω).
From Eq. (B10), we have
V(ω)δaL(ω)=iJδaR(ω)+κLaL,in(ω),
where V(ω)=κL/2+iωiΔL. Substituting Eq. (B13) into Eq. (B11), we find
T(ω)δaR(ω)=iχ(ω)gb2β*2V(ω)δaR(ω)+iχ(ω)gbβ*V(ω)ξ(ω)+iJκLaL,in(ω)+κRV(ω)aR,in(ω),
where T(ω)=N(ω)V(ω)+J2. Substituting Eq. (B14) into Eq. (B8), we obtain
FR(ω)δaR(ω)=χ2(ω)gb3β|β|2V(ω)ξ(ω)igbβχ(ω)T(ω)ξ(ω)Jgb2β2χ(ω)κLaL,in+igb2β2χ(ω)κRV(ω)aR,in(ω)iJT(ω)aL,inκRT(ω)aR,in,
where the auxiliary function is FR(ω)=M(ω)T(ω)χ2(ω)V(ω)gb4|β|4. Substituting Eq. (B15) into Eq. (B5), we have
FL(ω)δaL(ω)=iJχ2(ω)gb3β|β|2V(ω)ξ(ω)gbβχ(ω)JT(ω)ξ(ω)+iJ2gb2β2χ(ω)κLaL,in+Jgb2β2χ(ω)κRV(ω)aR,in(ω)iJκRT(ω)aR,inκL[M(ω)T(ω)U(ω)]aL,in,
where
FL(ω)=[M(ω)T(ω)U(ω)]V1(ω)+J2T(ω),U(ω)=χ2(ω)gb4|β|4(iω+κL2iΔL),V1(ω)=κL2+iω+iΔL.
Then we find
δaL(ω)=E(ω)aL,in(ω)+F(ω)aL,in(ω)+G(ω)aR,in(ω)+H(ω)aR,in(ω)+Q(ω)ξ(ω).
According to similar calculations, we find
δaL(ω)=E*(ω)aL,in(ω)+F*(ω)aL,in(ω)+G*(ω)aR,in(ω)+H*(ω)aR,in(ω)+Q*(ω)ξ(ω).
Using the Fourier transform, we obtain
aL,in(ω)aL,in(ω)=12πaL,in(t)eiωtdt×12πaL,in(t)eiωtdt=δ(ω+ω),
and
aR,in(ω)aR,in(ω)=δ(ω+ω).

APPENDIX C: DERIVATION OF OPTIMAL PARAMETERS

According to the quantum-trajectory method [101], the non-Hermitian Hamiltonian of the system containing the optical decay and mechanical damping terms is given by [101]

H=(ΔLiκL2)aLaL+(ΔRiκR2)aRaR+(ωmiγm2)bb+J(aLaR+aRaL)δ(aRaR)2+iϵd(aLaL),
where ΔR=ΔR+ΔF.

Under the weak-driving conditions, we can make the ansatz [24]

|φ=C00|0,0+C10|1,0+C01|0,1+C20|2,0+C11|1,1+C02|0,2.
Then we substitute the Hamiltonian [Eq. (C1)] and the general state [Eq. (C2)] into the Schrödinger equation
id|φdt=H|φ,
and then we have
HC00|0,0=iϵdC00|1,0,HC10|1,0=δLC10|1,0+JC10|0,1+iϵdC10(2|2,0|0,0),HC01|0,1=δRC01|0,1+JC01|1,0+iϵdC01|1,1,HC20|2,0=2δLC20|2,0+2JC20|1,1+iϵdC20(3|3,02|1,0),HC11|1,1=δLC11|1,1+δRC11|1,1+2JC11(|2,0+|0,2)+iϵdC11(2|2,1|0,1),HC02|0,2=2δRC02|0,22δC02|0,2+2JC02(|1,1+iϵdC02|1,2,
where the auxiliary functions are δL=ΔLiκL/2 and δR=ΔRiκR/2, and we have ignored the effects of the mechanical model because the phonon states are decoupled from the photon states [see Eq. (C1)]. By comparing the coefficients, we have
C00t=ϵdC10,iC10t=δLC10+JC012iϵdC20,iC01t=(δRδ)C01+JC10iϵdC11,iC11t=δLC11+(δRδ)C11+2J(C02+C20)+iϵdC01,iC02t=2(δRδ)C02+2JC112δC02,iC20t=2(δRδ)C20+2JC11+2iϵdC10.
Then the steady-state coefficients of the one- and two-particle states are given as
0=δLC10+JC01+iϵdC00,0=δRC01+JC10,
and
0=2δLC20+2JC11+i2ϵdC10,0=(δL+δR)C11+2JC20+2JC02+iϵdC01,0=2(δRδ)C02+2JC11,
where we have introduced the dissipative terms (proportional to κL and κR) and neglected the higher-order terms, as justified under the weak-driving conditions.

When we consider ΔL=ΔRδ=Δ, δ=g2/ωm, κL=κR=κ, and the condition of C20=0, we have

0=κ2(2δ6Δ5ΔF2)+4Δ2(2Δ2δ5δΔF2)+4ΔF(4ΔΔF3δΔδΔF+ΔF2)4J2δ,0=8δΔ12Δ2+κ2+ΔF(6δ20Δ8ΔF).
By eliminating δ, we obtain
a4Δ4+a3Δ3+a2Δ2+a1Δ+a0=0,
where
a0=κ(4J210ΔF2)(κ28ΔF2)2κ(κ444ΔF4),a1=8ΔF(6ΔF2κ+10J2κ+3),a2=8κ(2κ2+6J2+13ΔF2),a3=96ΔFκ,a4=32κ,
then we find the optimal conditions
Δopta3+sgn(E)λ1λ24a4,gopt=ωm[Δopt(4Δopt2+5κ2)+ΔFλ3]2(2J2κ2)+2ΔFλ4,
where
λ1=D+z13+z233,λ2=2Dz13z23+z333,λ3=20Δopt28ΔoptΔF4ΔF2+5κ2,λ4=10Δopt2+3Δopt+2ΔF,
and
sgn(E)={1(E>0),1(E<0),z1,2=AD+3B±B24AC2,z3=D2D(z13+z23)+(z13+z23)23A,A=D23F,B=DF9E2,C=F23DE2,D=3a328a4a2,E=a33+4a4a3a28a42a1,F=3a34+16a42a2216a4a32a2+16a42a3a164a43a0.

Funding

National Natural Science Foundation of China (NSFC) (11474087, 11774086).

REFERENCES

1. L. Tian and H. J. Carmichael, “Quantum trajectory simulations of two-state behavior in an optical cavity containing one atom,” Phys. Rev. A 46, R6801–R6804 (1992). [CrossRef]  

2. W. Leoński and R. Tanaś, “Possibility of producing the one-photon state in a kicked cavity with a nonlinear Kerr medium,” Phys. Rev. A 49, R20–R23 (1994). [CrossRef]  

3. A. Imamoḡlu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467–1470 (1997). [CrossRef]  

4. K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature (London) 436, 87–90 (2005). [CrossRef]  

5. K. Müller, A. Rundquist, K. A. Fischer, T. Sarmiento, K. G. Lagoudakis, Y. A. Kelaita, C. S. Muñoz, E. del Valle, F. P. Laussy, and J. Vučković, “Coherent generation of nonclassical light on chip via detuned photon blockade,” Phys. Rev. Lett. 114, 233601 (2015). [CrossRef]  

6. X. Gu, A. F. Kockum, A. Miranowicz, Y.-X. Liu, and F. Nori, “Microwave photonics with superconducting quantum circuits,” Phys. Rep. 718–719, 1–102 (2017). [CrossRef]  

7. V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009). [CrossRef]  

8. I. Buluta, S. Ashhab, and F. Nori, “Natural and artificial atoms for quantum computation,” Rep. Prog. Phys. 74, 104401 (2011). [CrossRef]  

9. T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletić, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature (London) 488, 57–60 (2012). [CrossRef]  

10. C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011). [CrossRef]  

11. A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Türeci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011). [CrossRef]  

12. A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. 4, 859–863 (2008). [CrossRef]  

13. S. Ferretti, L. C. Andreani, H. E. Türeci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841 (2010). [CrossRef]  

14. J.-Q. Liao and C. K. Law, “Correlated two-photon transport in a one-dimensional waveguide side-coupled to a nonlinear cavity,” Phys. Rev. A 82, 053836 (2010). [CrossRef]  

15. A. Miranowicz, M. Paprzycka, Y.-X. Liu, J. Bajer, and F. Nori, “Two-photon and three-photon blockades in driven nonlinear systems,” Phys. Rev. A 87, 023809 (2013). [CrossRef]  

16. P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107, 063601 (2011). [CrossRef]  

17. A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011). [CrossRef]  

18. J.-Q. Liao and F. Nori, “Photon blockade in quadratically coupled optomechanical systems,” Phys. Rev. A 88, 023853 (2013). [CrossRef]  

19. H. Xie, G.-W. Lin, X. Chen, Z.-H. Chen, and X.-M. Lin, “Single-photon nonlinearities in a strongly driven optomechanical system with quadratic coupling,” Phys. Rev. A 93, 063860 (2016). [CrossRef]  

20. C. Zhai, R. Huang, B. Li, H. Jing, and L.-M. Kuang, “Mechanical engineering of photon blockades in a cavity optomechanical system,” arXiv:1901.07654 (2019).

21. W. Leoński and A. Miranowicz, “Kerr nonlinear coupler and entanglement,” J. Opt. B 6, S37–S42 (2004). [CrossRef]  

22. A. Miranowicz and W. Leoński, “Two-mode optical state truncation and generation of maximally entangled states in pumped nonlinear couplers,” J. Phys. B 39, 1683–1700 (2006). [CrossRef]  

23. T. C. H. Liew and V. Savona, “Single photons from coupled quantum modes,” Phys. Rev. Lett. 104, 183601 (2010). [CrossRef]  

24. M. Bamba, A. Imamoğlu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802(R) (2011). [CrossRef]  

25. A. Majumdar, M. Bajcsy, A. Rundquist, and J. Vučković, “Loss-enabled sub-Poissonian light generation in a bimodal nanocavity,” Phys. Rev. Lett. 108, 163601 (2012). [CrossRef]  

26. S. Ferretti, V. Savona, and D. Gerace, “Optimal antibunching in passive photonic devices based on coupled nonlinear resonators,” New J. Phys. 15, 025012 (2013). [CrossRef]  

27. P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013). [CrossRef]  

28. V. Savona, “Unconventional photon blockade in coupled optomechanical systems,” arXiv:1302.5937 (2013).

29. X.-W. Xu and Y.-J. Li, “Antibunching photons in a cavity coupled to an optomechanical system,” J. Phys. B 46, 035502 (2013). [CrossRef]  

30. X.-W. Xu and Y. Li, “Strong photon antibunching of symmetric and antisymmetric modes in weakly nonlinear photonic molecules,” Phys. Rev. A 90, 033809 (2014). [CrossRef]  

31. W. Zhang, Z. Y. Yu, Y. M. Liu, and Y. W. Peng, “Optimal photon antibunching in a quantum-dot-bimodal-cavity system,” Phys. Rev. A 89, 043832 (2014). [CrossRef]  

32. H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Tunable photon blockade in coupled semiconductor cavities,” Phys. Rev. A 91, 063808 (2015). [CrossRef]  

33. H. Flayac and V. Savona, “Unconventional photon blockade,” Phys. Rev. A 96, 053810 (2017). [CrossRef]  

34. H. Flayac and V. Savona, “Nonclassical statistics from a polaritonic Josephson junction,” Phys. Rev. A 95, 043838 (2017). [CrossRef]  

35. F. Zhou, D.-G. Lai, and J.-Q. Liao, “Photon blockade effect in a coupled cavity system,” arXiv:1803.06642 (2018).

36. H. J. Snijders, J. A. Frey, J. Norman, H. Flayac, V. Savona, A. C. Gossard, J. E. Bowers, M. P. van Exter, D. Bouwmeester, and W. Löffler, “Observation of the unconventional photon blockade,” Phys. Rev. Lett. 121, 043601 (2018). [CrossRef]  

37. C. Vaneph, A. Morvan, G. Aiello, M. Féchant, M. Aprili, J. Gabelli, and J. Estève, “Observation of the unconventional photon blockade in the microwave domain,” Phys. Rev. Lett. 121, 043602 (2018). [CrossRef]  

38. A. Miranowicz, W. Leoński, and N. Imoto, “Quantum-optical states in finite-dimensional Hilbert space. I. General formalism,” in Modern Nonlinear Optics (Wiley, 2001), Vol. 119, pp. 195–213.

39. W. Leoński and A. Miranowicz, “Quantum-optical states in finite-dimensional Hilbert space. II. State generation,” Adv. Chem. Phys. 119, 155–193 (2003). [CrossRef]  

40. I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85, 299–366 (2013). [CrossRef]  

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

42. 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. Photonics 10, 657–661 (2016). [CrossRef]  

43. 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]  

44. Q.-T. Cao, H. Wang, C.-H. Dong, H. Jing, R.-S. Liu, X. Chen, L. Ge, Q. Gong, and Y.-F. Xiao, “Experimental demonstration of spontaneous chirality in a nonlinear microresonator,” Phys. Rev. Lett. 118, 033907 (2017). [CrossRef]  

45. L. D. Bino, J. M. Silver, M. T. M. Woodley, S. L. Stebbings, X. Zhao, and P. Del’Haye, “Microresonator isolators and circulators based on the intrinsic nonreciprocity of the Kerr effect,” Optica 5, 279–282 (2018). [CrossRef]  

46. Y. Shi, Z. Yu, and S. Fan, “Limitations of nonlinear optical isolators due to dynamic reciprocity,” Nat. Photonics 9, 388–392 (2015). [CrossRef]  

47. L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012). [CrossRef]  

48. S. Zhang, Y. Hu, G. Lin, Y. Niu, K. Xia, J. Gong, and S. Gong, “Thermal-motion-induced non-reciprocal quantum optical system,” Nat. Photonics 12, 744–748 (2018). [CrossRef]  

49. K. Y. Xia, F. Nori, and M. Xiao, “Cavity-free optical isolators and circulators using a chiral cross-Kerr nonlinearity,” Phys. Rev. Lett. 121, 203602 (2018). [CrossRef]  

50. D. L. Sounas and A. Alù, “Non-reciprocal photonics based on time modulation,” Nat. Photonics 11, 774–783 (2017). [CrossRef]  

51. C. Caloz, A. Alù, S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck-Léger, “Electromagnetic nonreciprocity,” Phys. Rev. Appl. 10, 047001 (2018). [CrossRef]  

52. N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013). [CrossRef]  

53. B. Peng, S. 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]  

54. 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. Photonics 8, 524–529 (2014). [CrossRef]  

55. S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature (London) 558, 569–572 (2018). [CrossRef]  

56. H. Lü, Y. Jiang, Y. Z. Wang, and H. Jing, “Optomechanically induced transparency in a spinning resonator,” Photon. Res. 5, 367–371 (2017). [CrossRef]  

57. H. Jing, H. Lü, S. K. Özdemir, T. Carmon, and F. Nori, “Nanoparticle sensing with a spinning resonator,” Optica 5, 1424–1430 (2018). [CrossRef]  

58. K. Y. Xia, G. W. Lu, G. W. Lin, Y. Q. Cheng, Y. P. Niu, S. Q. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90, 043802 (2014). [CrossRef]  

59. L. Tang, J. S. Tang, W. D. Zhang, G. W. Lu, Y. Zhang, K. Y. Xia, and M. Xiao, “An on-chip chiral single-photon interface: isolation and unidirectional emission,” arXiv:1811.02957 (2018).

60. M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354, 1577–1580 (2016). [CrossRef]  

61. B. Abdo, K. Sliwa, S. Shankar, M. Hatridge, L. Frunzio, R. Schoelkopf, and M. Devoret, “Josephson directional amplifier for quantum measurement of superconducting circuits,” Phys. Rev. Lett. 112, 167701 (2014). [CrossRef]  

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

63. 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]  

64. 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]  

65. A. Y. Song, Y. Shi, Q. Lin, and S. Fan, “Direction-dependent parity-time phase transition and non-reciprocal directional amplification with dynamic gain–loss modulation,” Phys. Rev. A 99, 013824 (2019). [CrossRef]  

66. S. Barzanjeh, M. Aquilina, and A. Xuereb, “Manipulating the flow of thermal noise in quantum devices,” Phys. Rev. Lett. 120, 060601 (2018). [CrossRef]  

67. R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121, 153601 (2018). [CrossRef]  

68. X.-W. Xu, Y.-J. Zhao, H. Wang, H. Jing, and A.-X. Chen, “Nonreciprocal photon blockade via quadratic optomechanical coupling,” arXiv:1809.07596 (2018).

69. P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature 541, 473–480 (2017). [CrossRef]  

70. V. V. Konotop, J. K. Yang, and D. A. Zezyulin, “Nonlinear waves in PT-symmetric systems,” Rev. Mod. Phys. 88, 035002 (2016). [CrossRef]  

71. R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11–19 (2018). [CrossRef]  

72. J. Zhang, B. Peng, S. K. Özdemir, K. Pichler, D. O. Krimer, G. M. Zhao, F. Nori, Y.-X. Liu, S. Rotter, and L. Yang, “A phonon laser operating at an exceptional point,” Nat. Photonics 12, 479–484 (2018). [CrossRef]  

73. 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]  

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

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

76. Y. Jiang, S. Maayani, T. Carmon, F. Nori, and H. Jing, “Nonreciprocal phonon laser,” Phys. Rev. Appl. 10, 064037 (2018). [CrossRef]  

77. H. Zhang, F. Salf, Y. Jiao, and H. Jing, “Loss-induced transparency in optomechanics,” Opt. Express 26, 25199–25210 (2018). [CrossRef]  

78. Z.-P. Liu, J. Zhang, S. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-X. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016). [CrossRef]  

79. 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, 043902 (2003). [CrossRef]  

80. H. Schmidt and A. Imamoḡlu, “Giant Kerr nonlinearities obtained by electromagnetically induced transparency,” Opt. Lett. 21, 1936–1938 (1996). [CrossRef]  

81. Y.-P. Wang, G.-Q. Zhang, D. Zhang, T.-F. Li, C.-M. Hu, and J. Q. You, “Bistability of cavity magnon polaritons,” Phys. Rev. Lett. 120, 057202 (2018). [CrossRef]  

82. Z. R. Gong, H. Ian, Y.-X. Liu, C. P. Sun, and F. Nori, “Effective Hamiltonian approach to the Kerr nonlinearity in an optomechanical system,” Phys. Rev. A 80, 065801 (2009). [CrossRef]  

83. L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett. 98, 113108 (2011). [CrossRef]  

84. H. Snijders, J. A. Frey, J. Norman, M. P. Bakker, E. C. Langman, A. Gossard, J. E. Bowers, M. P. Van Exter, D. Bouwmeester, and W. Löffler, “Purification of a single-photon nonlinearity,” Nat. Commun. 7, 12578 (2016). [CrossRef]  

85. G. Enzian, M. Szczykulska, J. Silver, L. Del Bino, S. Zhang, I. A. Walmsley, P. Del’Haye, and M. R. Vanner, “Observation of Brillouin optomechanical strong coupling with an 11 GHz mechanical mode,” Optica 6, 7–14 (2019). [CrossRef]  

86. G. B. Malykin, “The Sagnac effect: correct and incorrect explanations,” Phys. Usp. 43, 1229–1252 (2000). [CrossRef]  

87. G. W. Ford, J. T. Lewis, and R. F. O’Connell, “Quantum Langevin equation,” Phys. Rev. A 37, 4419–4428 (1988). [CrossRef]  

88. C. W. Gardiner and P. Zoller, Quantum Noise (Springer, 2000).

89. D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 1994).

90. J. R. Johansson, P. D. Nation, and F. Nori, “Qutip 2: a Python framework for the dynamics of open quantum systems,” Comput. Phys. Commun. 184, 1234–1240 (2013). [CrossRef]  

91. E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature (London) 482, 63–67 (2012). [CrossRef]  

92. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014). [CrossRef]  

93. 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]  

94. K. J. Vahala, “Optical microcavities,” Nature (London) 424, 839–846 (2003). [CrossRef]  

95. V. Huet, A. Rasoloniaina, P. Guillemé, P. Rochard, P. Féron, M. Mortier, A. Levenson, K. Bencheikh, A. Yacomotti, and Y. Dumeige, “Millisecond photon lifetime in a slow-light microcavity,” Phys. Rev. Lett. 116, 133902 (2016). [CrossRef]  

96. J. Hloušek, M. Dudka, I. Straka, and M. Ježek, “Accurate detection of arbitrary photon statistics,” arXiv:1812.02262 (2018).

97. R. Reimann, M. Doderer, E. Hebestreit, R. Diehl, M. Frimmer, D. Windey, F. Tebbenjohanns, and L. Novotny, “GHz rotation of an optically trapped nanoparticle in vacuum,” Phys. Rev. Lett. 121, 033602 (2018). [CrossRef]  

98. J. Ahn, Z. Xu, J. Bang, Y.-H. Deng, T. M. Hoang, Q. Han, R.-M. Ma, and T. Li, “Optically levitated nanodumbbell torsion balance and GHz nanomechanical rotor,” Phys. Rev. Lett. 121, 033603 (2018). [CrossRef]  

99. F. Reiter, T. L. Nguyen, J. P. Home, and S. F. Yelin, “Cooperative breakdown of the oscillator blockade in the Dicke model,” arXiv:1807.06026 (2018).

100. M. Radulaski, K. A. Fischer, K. G. Lagoudakis, J. L. Zhang, and J. Vučković, “Photon blockade in two-emitter-cavity systems,” Phys. Rev. A 96, 011801(R) (2017). [CrossRef]  

101. M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. 70, 101–144 (1998). [CrossRef]  

References

  • View by:
  • |
  • |
  • |

  1. L. Tian and H. J. Carmichael, “Quantum trajectory simulations of two-state behavior in an optical cavity containing one atom,” Phys. Rev. A 46, R6801–R6804 (1992).
    [Crossref]
  2. W. Leoński and R. Tanaś, “Possibility of producing the one-photon state in a kicked cavity with a nonlinear Kerr medium,” Phys. Rev. A 49, R20–R23 (1994).
    [Crossref]
  3. A. Imamoḡlu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467–1470 (1997).
    [Crossref]
  4. K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature (London) 436, 87–90 (2005).
    [Crossref]
  5. K. Müller, A. Rundquist, K. A. Fischer, T. Sarmiento, K. G. Lagoudakis, Y. A. Kelaita, C. S. Muñoz, E. del Valle, F. P. Laussy, and J. Vučković, “Coherent generation of nonclassical light on chip via detuned photon blockade,” Phys. Rev. Lett. 114, 233601 (2015).
    [Crossref]
  6. X. Gu, A. F. Kockum, A. Miranowicz, Y.-X. Liu, and F. Nori, “Microwave photonics with superconducting quantum circuits,” Phys. Rep. 718–719, 1–102 (2017).
    [Crossref]
  7. V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
    [Crossref]
  8. I. Buluta, S. Ashhab, and F. Nori, “Natural and artificial atoms for quantum computation,” Rep. Prog. Phys. 74, 104401 (2011).
    [Crossref]
  9. T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletić, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature (London) 488, 57–60 (2012).
    [Crossref]
  10. C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
    [Crossref]
  11. A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Türeci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
    [Crossref]
  12. A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. 4, 859–863 (2008).
    [Crossref]
  13. S. Ferretti, L. C. Andreani, H. E. Türeci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841 (2010).
    [Crossref]
  14. J.-Q. Liao and C. K. Law, “Correlated two-photon transport in a one-dimensional waveguide side-coupled to a nonlinear cavity,” Phys. Rev. A 82, 053836 (2010).
    [Crossref]
  15. A. Miranowicz, M. Paprzycka, Y.-X. Liu, J. Bajer, and F. Nori, “Two-photon and three-photon blockades in driven nonlinear systems,” Phys. Rev. A 87, 023809 (2013).
    [Crossref]
  16. P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107, 063601 (2011).
    [Crossref]
  17. A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011).
    [Crossref]
  18. J.-Q. Liao and F. Nori, “Photon blockade in quadratically coupled optomechanical systems,” Phys. Rev. A 88, 023853 (2013).
    [Crossref]
  19. H. Xie, G.-W. Lin, X. Chen, Z.-H. Chen, and X.-M. Lin, “Single-photon nonlinearities in a strongly driven optomechanical system with quadratic coupling,” Phys. Rev. A 93, 063860 (2016).
    [Crossref]
  20. C. Zhai, R. Huang, B. Li, H. Jing, and L.-M. Kuang, “Mechanical engineering of photon blockades in a cavity optomechanical system,” arXiv:1901.07654 (2019).
  21. W. Leoński and A. Miranowicz, “Kerr nonlinear coupler and entanglement,” J. Opt. B 6, S37–S42 (2004).
    [Crossref]
  22. A. Miranowicz and W. Leoński, “Two-mode optical state truncation and generation of maximally entangled states in pumped nonlinear couplers,” J. Phys. B 39, 1683–1700 (2006).
    [Crossref]
  23. T. C. H. Liew and V. Savona, “Single photons from coupled quantum modes,” Phys. Rev. Lett. 104, 183601 (2010).
    [Crossref]
  24. M. Bamba, A. Imamoğlu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802(R) (2011).
    [Crossref]
  25. A. Majumdar, M. Bajcsy, A. Rundquist, and J. Vučković, “Loss-enabled sub-Poissonian light generation in a bimodal nanocavity,” Phys. Rev. Lett. 108, 163601 (2012).
    [Crossref]
  26. S. Ferretti, V. Savona, and D. Gerace, “Optimal antibunching in passive photonic devices based on coupled nonlinear resonators,” New J. Phys. 15, 025012 (2013).
    [Crossref]
  27. P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013).
    [Crossref]
  28. V. Savona, “Unconventional photon blockade in coupled optomechanical systems,” arXiv:1302.5937 (2013).
  29. X.-W. Xu and Y.-J. Li, “Antibunching photons in a cavity coupled to an optomechanical system,” J. Phys. B 46, 035502 (2013).
    [Crossref]
  30. X.-W. Xu and Y. Li, “Strong photon antibunching of symmetric and antisymmetric modes in weakly nonlinear photonic molecules,” Phys. Rev. A 90, 033809 (2014).
    [Crossref]
  31. W. Zhang, Z. Y. Yu, Y. M. Liu, and Y. W. Peng, “Optimal photon antibunching in a quantum-dot-bimodal-cavity system,” Phys. Rev. A 89, 043832 (2014).
    [Crossref]
  32. H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Tunable photon blockade in coupled semiconductor cavities,” Phys. Rev. A 91, 063808 (2015).
    [Crossref]
  33. H. Flayac and V. Savona, “Unconventional photon blockade,” Phys. Rev. A 96, 053810 (2017).
    [Crossref]
  34. H. Flayac and V. Savona, “Nonclassical statistics from a polaritonic Josephson junction,” Phys. Rev. A 95, 043838 (2017).
    [Crossref]
  35. F. Zhou, D.-G. Lai, and J.-Q. Liao, “Photon blockade effect in a coupled cavity system,” arXiv:1803.06642 (2018).
  36. H. J. Snijders, J. A. Frey, J. Norman, H. Flayac, V. Savona, A. C. Gossard, J. E. Bowers, M. P. van Exter, D. Bouwmeester, and W. Löffler, “Observation of the unconventional photon blockade,” Phys. Rev. Lett. 121, 043601 (2018).
    [Crossref]
  37. C. Vaneph, A. Morvan, G. Aiello, M. Féchant, M. Aprili, J. Gabelli, and J. Estève, “Observation of the unconventional photon blockade in the microwave domain,” Phys. Rev. Lett. 121, 043602 (2018).
    [Crossref]
  38. A. Miranowicz, W. Leoński, and N. Imoto, “Quantum-optical states in finite-dimensional Hilbert space. I. General formalism,” in Modern Nonlinear Optics (Wiley, 2001), Vol. 119, pp. 195–213.
  39. W. Leoński and A. Miranowicz, “Quantum-optical states in finite-dimensional Hilbert space. II. State generation,” Adv. Chem. Phys. 119, 155–193 (2003).
    [Crossref]
  40. I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85, 299–366 (2013).
    [Crossref]
  41. S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical nonreciprocity in optomechanical structures,” Phys. Rev. Lett. 102, 213903 (2009).
    [Crossref]
  42. 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. Photonics 10, 657–661 (2016).
    [Crossref]
  43. 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]
  44. Q.-T. Cao, H. Wang, C.-H. Dong, H. Jing, R.-S. Liu, X. Chen, L. Ge, Q. Gong, and Y.-F. Xiao, “Experimental demonstration of spontaneous chirality in a nonlinear microresonator,” Phys. Rev. Lett. 118, 033907 (2017).
    [Crossref]
  45. L. D. Bino, J. M. Silver, M. T. M. Woodley, S. L. Stebbings, X. Zhao, and P. Del’Haye, “Microresonator isolators and circulators based on the intrinsic nonreciprocity of the Kerr effect,” Optica 5, 279–282 (2018).
    [Crossref]
  46. Y. Shi, Z. Yu, and S. Fan, “Limitations of nonlinear optical isolators due to dynamic reciprocity,” Nat. Photonics 9, 388–392 (2015).
    [Crossref]
  47. L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
    [Crossref]
  48. S. Zhang, Y. Hu, G. Lin, Y. Niu, K. Xia, J. Gong, and S. Gong, “Thermal-motion-induced non-reciprocal quantum optical system,” Nat. Photonics 12, 744–748 (2018).
    [Crossref]
  49. K. Y. Xia, F. Nori, and M. Xiao, “Cavity-free optical isolators and circulators using a chiral cross-Kerr nonlinearity,” Phys. Rev. Lett. 121, 203602 (2018).
    [Crossref]
  50. D. L. Sounas and A. Alù, “Non-reciprocal photonics based on time modulation,” Nat. Photonics 11, 774–783 (2017).
    [Crossref]
  51. C. Caloz, A. Alù, S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck-Léger, “Electromagnetic nonreciprocity,” Phys. Rev. Appl. 10, 047001 (2018).
    [Crossref]
  52. N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
    [Crossref]
  53. B. Peng, S. 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]
  54. 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. Photonics 8, 524–529 (2014).
    [Crossref]
  55. S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature (London) 558, 569–572 (2018).
    [Crossref]
  56. H. Lü, Y. Jiang, Y. Z. Wang, and H. Jing, “Optomechanically induced transparency in a spinning resonator,” Photon. Res. 5, 367–371 (2017).
    [Crossref]
  57. H. Jing, H. Lü, S. K. Özdemir, T. Carmon, and F. Nori, “Nanoparticle sensing with a spinning resonator,” Optica 5, 1424–1430 (2018).
    [Crossref]
  58. K. Y. Xia, G. W. Lu, G. W. Lin, Y. Q. Cheng, Y. P. Niu, S. Q. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90, 043802 (2014).
    [Crossref]
  59. L. Tang, J. S. Tang, W. D. Zhang, G. W. Lu, Y. Zhang, K. Y. Xia, and M. Xiao, “An on-chip chiral single-photon interface: isolation and unidirectional emission,” arXiv:1811.02957 (2018).
  60. M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354, 1577–1580 (2016).
    [Crossref]
  61. B. Abdo, K. Sliwa, S. Shankar, M. Hatridge, L. Frunzio, R. Schoelkopf, and M. Devoret, “Josephson directional amplifier for quantum measurement of superconducting circuits,” Phys. Rev. Lett. 112, 167701 (2014).
    [Crossref]
  62. A. Metelmann and A. A. Clerk, “Nonreciprocal photon transmission and amplification via reservoir engineering,” Phys. Rev. X 5, 021025 (2015).
    [Crossref]
  63. 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]
  64. 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]
  65. A. Y. Song, Y. Shi, Q. Lin, and S. Fan, “Direction-dependent parity-time phase transition and non-reciprocal directional amplification with dynamic gain–loss modulation,” Phys. Rev. A 99, 013824 (2019).
    [Crossref]
  66. S. Barzanjeh, M. Aquilina, and A. Xuereb, “Manipulating the flow of thermal noise in quantum devices,” Phys. Rev. Lett. 120, 060601 (2018).
    [Crossref]
  67. R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121, 153601 (2018).
    [Crossref]
  68. X.-W. Xu, Y.-J. Zhao, H. Wang, H. Jing, and A.-X. Chen, “Nonreciprocal photon blockade via quadratic optomechanical coupling,” arXiv:1809.07596 (2018).
  69. P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature 541, 473–480 (2017).
    [Crossref]
  70. V. V. Konotop, J. K. Yang, and D. A. Zezyulin, “Nonlinear waves in PT-symmetric systems,” Rev. Mod. Phys. 88, 035002 (2016).
    [Crossref]
  71. R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11–19 (2018).
    [Crossref]
  72. J. Zhang, B. Peng, S. K. Özdemir, K. Pichler, D. O. Krimer, G. M. Zhao, F. Nori, Y.-X. Liu, S. Rotter, and L. Yang, “A phonon laser operating at an exceptional point,” Nat. Photonics 12, 479–484 (2018).
    [Crossref]
  73. 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]
  74. H. Jing, S. K. Özdemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113, 053604 (2014).
    [Crossref]
  75. H. Lü, S. K. Özdemir, L.-M. Kuang, F. Nori, and H. Jing, “Exceptional points in random-defect phonon lasers,” Phys. Rev. Appl. 8, 044020 (2017).
    [Crossref]
  76. Y. Jiang, S. Maayani, T. Carmon, F. Nori, and H. Jing, “Nonreciprocal phonon laser,” Phys. Rev. Appl. 10, 064037 (2018).
    [Crossref]
  77. H. Zhang, F. Salf, Y. Jiao, and H. Jing, “Loss-induced transparency in optomechanics,” Opt. Express 26, 25199–25210 (2018).
    [Crossref]
  78. Z.-P. Liu, J. Zhang, S. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-X. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
    [Crossref]
  79. 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, 043902 (2003).
    [Crossref]
  80. H. Schmidt and A. Imamoḡlu, “Giant Kerr nonlinearities obtained by electromagnetically induced transparency,” Opt. Lett. 21, 1936–1938 (1996).
    [Crossref]
  81. Y.-P. Wang, G.-Q. Zhang, D. Zhang, T.-F. Li, C.-M. Hu, and J. Q. You, “Bistability of cavity magnon polaritons,” Phys. Rev. Lett. 120, 057202 (2018).
    [Crossref]
  82. Z. R. Gong, H. Ian, Y.-X. Liu, C. P. Sun, and F. Nori, “Effective Hamiltonian approach to the Kerr nonlinearity in an optomechanical system,” Phys. Rev. A 80, 065801 (2009).
    [Crossref]
  83. L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett. 98, 113108 (2011).
    [Crossref]
  84. H. Snijders, J. A. Frey, J. Norman, M. P. Bakker, E. C. Langman, A. Gossard, J. E. Bowers, M. P. Van Exter, D. Bouwmeester, and W. Löffler, “Purification of a single-photon nonlinearity,” Nat. Commun. 7, 12578 (2016).
    [Crossref]
  85. G. Enzian, M. Szczykulska, J. Silver, L. Del Bino, S. Zhang, I. A. Walmsley, P. Del’Haye, and M. R. Vanner, “Observation of Brillouin optomechanical strong coupling with an 11  GHz mechanical mode,” Optica 6, 7–14 (2019).
    [Crossref]
  86. G. B. Malykin, “The Sagnac effect: correct and incorrect explanations,” Phys. Usp. 43, 1229–1252 (2000).
    [Crossref]
  87. G. W. Ford, J. T. Lewis, and R. F. O’Connell, “Quantum Langevin equation,” Phys. Rev. A 37, 4419–4428 (1988).
    [Crossref]
  88. C. W. Gardiner and P. Zoller, Quantum Noise (Springer, 2000).
  89. D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 1994).
  90. J. R. Johansson, P. D. Nation, and F. Nori, “Qutip 2: a Python framework for the dynamics of open quantum systems,” Comput. Phys. Commun. 184, 1234–1240 (2013).
    [Crossref]
  91. E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature (London) 482, 63–67 (2012).
    [Crossref]
  92. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
    [Crossref]
  93. 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]
  94. K. J. Vahala, “Optical microcavities,” Nature (London) 424, 839–846 (2003).
    [Crossref]
  95. V. Huet, A. Rasoloniaina, P. Guillemé, P. Rochard, P. Féron, M. Mortier, A. Levenson, K. Bencheikh, A. Yacomotti, and Y. Dumeige, “Millisecond photon lifetime in a slow-light microcavity,” Phys. Rev. Lett. 116, 133902 (2016).
    [Crossref]
  96. J. Hloušek, M. Dudka, I. Straka, and M. Ježek, “Accurate detection of arbitrary photon statistics,” arXiv:1812.02262 (2018).
  97. R. Reimann, M. Doderer, E. Hebestreit, R. Diehl, M. Frimmer, D. Windey, F. Tebbenjohanns, and L. Novotny, “GHz rotation of an optically trapped nanoparticle in vacuum,” Phys. Rev. Lett. 121, 033602 (2018).
    [Crossref]
  98. J. Ahn, Z. Xu, J. Bang, Y.-H. Deng, T. M. Hoang, Q. Han, R.-M. Ma, and T. Li, “Optically levitated nanodumbbell torsion balance and GHz nanomechanical rotor,” Phys. Rev. Lett. 121, 033603 (2018).
    [Crossref]
  99. F. Reiter, T. L. Nguyen, J. P. Home, and S. F. Yelin, “Cooperative breakdown of the oscillator blockade in the Dicke model,” arXiv:1807.06026 (2018).
  100. M. Radulaski, K. A. Fischer, K. G. Lagoudakis, J. L. Zhang, and J. Vučković, “Photon blockade in two-emitter-cavity systems,” Phys. Rev. A 96, 011801(R) (2017).
    [Crossref]
  101. M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. 70, 101–144 (1998).
    [Crossref]

2019 (2)

A. Y. Song, Y. Shi, Q. Lin, and S. Fan, “Direction-dependent parity-time phase transition and non-reciprocal directional amplification with dynamic gain–loss modulation,” Phys. Rev. A 99, 013824 (2019).
[Crossref]

G. Enzian, M. Szczykulska, J. Silver, L. Del Bino, S. Zhang, I. A. Walmsley, P. Del’Haye, and M. R. Vanner, “Observation of Brillouin optomechanical strong coupling with an 11  GHz mechanical mode,” Optica 6, 7–14 (2019).
[Crossref]

2018 (19)

Y.-P. Wang, G.-Q. Zhang, D. Zhang, T.-F. Li, C.-M. Hu, and J. Q. You, “Bistability of cavity magnon polaritons,” Phys. Rev. Lett. 120, 057202 (2018).
[Crossref]

R. Reimann, M. Doderer, E. Hebestreit, R. Diehl, M. Frimmer, D. Windey, F. Tebbenjohanns, and L. Novotny, “GHz rotation of an optically trapped nanoparticle in vacuum,” Phys. Rev. Lett. 121, 033602 (2018).
[Crossref]

J. Ahn, Z. Xu, J. Bang, Y.-H. Deng, T. M. Hoang, Q. Han, R.-M. Ma, and T. Li, “Optically levitated nanodumbbell torsion balance and GHz nanomechanical rotor,” Phys. Rev. Lett. 121, 033603 (2018).
[Crossref]

S. Barzanjeh, M. Aquilina, and A. Xuereb, “Manipulating the flow of thermal noise in quantum devices,” Phys. Rev. Lett. 120, 060601 (2018).
[Crossref]

R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121, 153601 (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]

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]

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11–19 (2018).
[Crossref]

J. Zhang, B. Peng, S. K. Özdemir, K. Pichler, D. O. Krimer, G. M. Zhao, F. Nori, Y.-X. Liu, S. Rotter, and L. Yang, “A phonon laser operating at an exceptional point,” Nat. Photonics 12, 479–484 (2018).
[Crossref]

Y. Jiang, S. Maayani, T. Carmon, F. Nori, and H. Jing, “Nonreciprocal phonon laser,” Phys. Rev. Appl. 10, 064037 (2018).
[Crossref]

H. Zhang, F. Salf, Y. Jiao, and H. Jing, “Loss-induced transparency in optomechanics,” Opt. Express 26, 25199–25210 (2018).
[Crossref]

L. D. Bino, J. M. Silver, M. T. M. Woodley, S. L. Stebbings, X. Zhao, and P. Del’Haye, “Microresonator isolators and circulators based on the intrinsic nonreciprocity of the Kerr effect,” Optica 5, 279–282 (2018).
[Crossref]

S. Zhang, Y. Hu, G. Lin, Y. Niu, K. Xia, J. Gong, and S. Gong, “Thermal-motion-induced non-reciprocal quantum optical system,” Nat. Photonics 12, 744–748 (2018).
[Crossref]

K. Y. Xia, F. Nori, and M. Xiao, “Cavity-free optical isolators and circulators using a chiral cross-Kerr nonlinearity,” Phys. Rev. Lett. 121, 203602 (2018).
[Crossref]

C. Caloz, A. Alù, S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck-Léger, “Electromagnetic nonreciprocity,” Phys. Rev. Appl. 10, 047001 (2018).
[Crossref]

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature (London) 558, 569–572 (2018).
[Crossref]

H. Jing, H. Lü, S. K. Özdemir, T. Carmon, and F. Nori, “Nanoparticle sensing with a spinning resonator,” Optica 5, 1424–1430 (2018).
[Crossref]

H. J. Snijders, J. A. Frey, J. Norman, H. Flayac, V. Savona, A. C. Gossard, J. E. Bowers, M. P. van Exter, D. Bouwmeester, and W. Löffler, “Observation of the unconventional photon blockade,” Phys. Rev. Lett. 121, 043601 (2018).
[Crossref]

C. Vaneph, A. Morvan, G. Aiello, M. Féchant, M. Aprili, J. Gabelli, and J. Estève, “Observation of the unconventional photon blockade in the microwave domain,” Phys. Rev. Lett. 121, 043602 (2018).
[Crossref]

2017 (10)

H. Flayac and V. Savona, “Unconventional photon blockade,” Phys. Rev. A 96, 053810 (2017).
[Crossref]

H. Flayac and V. Savona, “Nonclassical statistics from a polaritonic Josephson junction,” Phys. Rev. A 95, 043838 (2017).
[Crossref]

X. Gu, A. F. Kockum, A. Miranowicz, Y.-X. Liu, and F. Nori, “Microwave photonics with superconducting quantum circuits,” Phys. Rep. 718–719, 1–102 (2017).
[Crossref]

H. Lü, Y. Jiang, Y. Z. Wang, and H. Jing, “Optomechanically induced transparency in a spinning resonator,” Photon. Res. 5, 367–371 (2017).
[Crossref]

D. L. Sounas and A. Alù, “Non-reciprocal photonics based on time modulation,” Nat. Photonics 11, 774–783 (2017).
[Crossref]

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]

Q.-T. Cao, H. Wang, C.-H. Dong, H. Jing, R.-S. Liu, X. Chen, L. Ge, Q. Gong, and Y.-F. Xiao, “Experimental demonstration of spontaneous chirality in a nonlinear microresonator,” Phys. Rev. Lett. 118, 033907 (2017).
[Crossref]

P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature 541, 473–480 (2017).
[Crossref]

M. Radulaski, K. A. Fischer, K. G. Lagoudakis, J. L. Zhang, and J. Vučković, “Photon blockade in two-emitter-cavity systems,” Phys. Rev. A 96, 011801(R) (2017).
[Crossref]

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

2016 (7)

H. Snijders, J. A. Frey, J. Norman, M. P. Bakker, E. C. Langman, A. Gossard, J. E. Bowers, M. P. Van Exter, D. Bouwmeester, and W. Löffler, “Purification of a single-photon nonlinearity,” Nat. Commun. 7, 12578 (2016).
[Crossref]

V. Huet, A. Rasoloniaina, P. Guillemé, P. Rochard, P. Féron, M. Mortier, A. Levenson, K. Bencheikh, A. Yacomotti, and Y. Dumeige, “Millisecond photon lifetime in a slow-light microcavity,” Phys. Rev. Lett. 116, 133902 (2016).
[Crossref]

V. V. Konotop, J. K. Yang, and D. A. Zezyulin, “Nonlinear waves in PT-symmetric systems,” Rev. Mod. Phys. 88, 035002 (2016).
[Crossref]

M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354, 1577–1580 (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. Photonics 10, 657–661 (2016).
[Crossref]

Z.-P. Liu, J. Zhang, S. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-X. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
[Crossref]

H. Xie, G.-W. Lin, X. Chen, Z.-H. Chen, and X.-M. Lin, “Single-photon nonlinearities in a strongly driven optomechanical system with quadratic coupling,” Phys. Rev. A 93, 063860 (2016).
[Crossref]

2015 (4)

H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Tunable photon blockade in coupled semiconductor cavities,” Phys. Rev. A 91, 063808 (2015).
[Crossref]

K. Müller, A. Rundquist, K. A. Fischer, T. Sarmiento, K. G. Lagoudakis, Y. A. Kelaita, C. S. Muñoz, E. del Valle, F. P. Laussy, and J. Vučković, “Coherent generation of nonclassical light on chip via detuned photon blockade,” Phys. Rev. Lett. 114, 233601 (2015).
[Crossref]

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

Y. Shi, Z. Yu, and S. Fan, “Limitations of nonlinear optical isolators due to dynamic reciprocity,” Nat. Photonics 9, 388–392 (2015).
[Crossref]

2014 (8)

K. Y. Xia, G. W. Lu, G. W. Lin, Y. Q. Cheng, Y. P. Niu, S. Q. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90, 043802 (2014).
[Crossref]

B. Peng, S. 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]

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. Photonics 8, 524–529 (2014).
[Crossref]

B. Abdo, K. Sliwa, S. Shankar, M. Hatridge, L. Frunzio, R. Schoelkopf, and M. Devoret, “Josephson directional amplifier for quantum measurement of superconducting circuits,” Phys. Rev. Lett. 112, 167701 (2014).
[Crossref]

X.-W. Xu and Y. Li, “Strong photon antibunching of symmetric and antisymmetric modes in weakly nonlinear photonic molecules,” Phys. Rev. A 90, 033809 (2014).
[Crossref]

W. Zhang, Z. Y. Yu, Y. M. Liu, and Y. W. Peng, “Optimal photon antibunching in a quantum-dot-bimodal-cavity system,” Phys. Rev. A 89, 043832 (2014).
[Crossref]

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

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

2013 (8)

J. R. Johansson, P. D. Nation, and F. Nori, “Qutip 2: a Python framework for the dynamics of open quantum systems,” Comput. Phys. Commun. 184, 1234–1240 (2013).
[Crossref]

J.-Q. Liao and F. Nori, “Photon blockade in quadratically coupled optomechanical systems,” Phys. Rev. A 88, 023853 (2013).
[Crossref]

S. Ferretti, V. Savona, and D. Gerace, “Optimal antibunching in passive photonic devices based on coupled nonlinear resonators,” New J. Phys. 15, 025012 (2013).
[Crossref]

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013).
[Crossref]

X.-W. Xu and Y.-J. Li, “Antibunching photons in a cavity coupled to an optomechanical system,” J. Phys. B 46, 035502 (2013).
[Crossref]

A. Miranowicz, M. Paprzycka, Y.-X. Liu, J. Bajer, and F. Nori, “Two-photon and three-photon blockades in driven nonlinear systems,” Phys. Rev. A 87, 023809 (2013).
[Crossref]

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
[Crossref]

I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85, 299–366 (2013).
[Crossref]

2012 (4)

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[Crossref]

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletić, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature (London) 488, 57–60 (2012).
[Crossref]

A. Majumdar, M. Bajcsy, A. Rundquist, and J. Vučković, “Loss-enabled sub-Poissonian light generation in a bimodal nanocavity,” Phys. Rev. Lett. 108, 163601 (2012).
[Crossref]

E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature (London) 482, 63–67 (2012).
[Crossref]

2011 (8)

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]

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett. 98, 113108 (2011).
[Crossref]

M. Bamba, A. Imamoğlu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802(R) (2011).
[Crossref]

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref]

A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Türeci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
[Crossref]

P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107, 063601 (2011).
[Crossref]

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011).
[Crossref]

I. Buluta, S. Ashhab, and F. Nori, “Natural and artificial atoms for quantum computation,” Rep. Prog. Phys. 74, 104401 (2011).
[Crossref]

2010 (4)

S. Ferretti, L. C. Andreani, H. E. Türeci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841 (2010).
[Crossref]

J.-Q. Liao and C. K. Law, “Correlated two-photon transport in a one-dimensional waveguide side-coupled to a nonlinear cavity,” Phys. Rev. A 82, 053836 (2010).
[Crossref]

T. C. H. Liew and V. Savona, “Single photons from coupled quantum modes,” Phys. Rev. Lett. 104, 183601 (2010).
[Crossref]

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]

2009 (3)

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

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[Crossref]

Z. R. Gong, H. Ian, Y.-X. Liu, C. P. Sun, and F. Nori, “Effective Hamiltonian approach to the Kerr nonlinearity in an optomechanical system,” Phys. Rev. A 80, 065801 (2009).
[Crossref]

2008 (1)

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. 4, 859–863 (2008).
[Crossref]

2006 (1)

A. Miranowicz and W. Leoński, “Two-mode optical state truncation and generation of maximally entangled states in pumped nonlinear couplers,” J. Phys. B 39, 1683–1700 (2006).
[Crossref]

2005 (1)

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature (London) 436, 87–90 (2005).
[Crossref]

2004 (1)

W. Leoński and A. Miranowicz, “Kerr nonlinear coupler and entanglement,” J. Opt. B 6, S37–S42 (2004).
[Crossref]

2003 (3)

W. Leoński and A. Miranowicz, “Quantum-optical states in finite-dimensional Hilbert space. II. State generation,” Adv. Chem. Phys. 119, 155–193 (2003).
[Crossref]

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, 043902 (2003).
[Crossref]

K. J. Vahala, “Optical microcavities,” Nature (London) 424, 839–846 (2003).
[Crossref]

2000 (1)

G. B. Malykin, “The Sagnac effect: correct and incorrect explanations,” Phys. Usp. 43, 1229–1252 (2000).
[Crossref]

1998 (1)

M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. 70, 101–144 (1998).
[Crossref]

1997 (1)

A. Imamoḡlu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467–1470 (1997).
[Crossref]

1996 (1)

1994 (1)

W. Leoński and R. Tanaś, “Possibility of producing the one-photon state in a kicked cavity with a nonlinear Kerr medium,” Phys. Rev. A 49, R20–R23 (1994).
[Crossref]

1992 (1)

L. Tian and H. J. Carmichael, “Quantum trajectory simulations of two-state behavior in an optical cavity containing one atom,” Phys. Rev. A 46, R6801–R6804 (1992).
[Crossref]

1988 (1)

G. W. Ford, J. T. Lewis, and R. F. O’Connell, “Quantum Langevin equation,” Phys. Rev. A 37, 4419–4428 (1988).
[Crossref]

Abdo, B.

B. Abdo, K. Sliwa, S. Shankar, M. Hatridge, L. Frunzio, R. Schoelkopf, and M. Devoret, “Josephson directional amplifier for quantum measurement of superconducting circuits,” Phys. Rev. Lett. 112, 167701 (2014).
[Crossref]

Abdumalikov, A. A.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref]

Achouri, K.

C. Caloz, A. Alù, S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck-Léger, “Electromagnetic nonreciprocity,” Phys. Rev. Appl. 10, 047001 (2018).
[Crossref]

Ahn, J.

J. Ahn, Z. Xu, J. Bang, Y.-H. Deng, T. M. Hoang, Q. Han, R.-M. Ma, and T. Li, “Optically levitated nanodumbbell torsion balance and GHz nanomechanical rotor,” Phys. Rev. Lett. 121, 033603 (2018).
[Crossref]

Aiello, G.

C. Vaneph, A. Morvan, G. Aiello, M. Féchant, M. Aprili, J. Gabelli, and J. Estève, “Observation of the unconventional photon blockade in the microwave domain,” Phys. Rev. Lett. 121, 043602 (2018).
[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.

C. Caloz, A. Alù, S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck-Léger, “Electromagnetic nonreciprocity,” Phys. Rev. Appl. 10, 047001 (2018).
[Crossref]

D. L. Sounas and A. Alù, “Non-reciprocal photonics based on time modulation,” Nat. Photonics 11, 774–783 (2017).
[Crossref]

Andreani, L. C.

S. Ferretti, L. C. Andreani, H. E. Türeci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841 (2010).
[Crossref]

Aprili, M.

C. Vaneph, A. Morvan, G. Aiello, M. Féchant, M. Aprili, J. Gabelli, and J. Estève, “Observation of the unconventional photon blockade in the microwave domain,” Phys. Rev. Lett. 121, 043602 (2018).
[Crossref]

Aquilina, M.

S. Barzanjeh, M. Aquilina, and A. Xuereb, “Manipulating the flow of thermal noise in quantum devices,” Phys. Rev. Lett. 120, 060601 (2018).
[Crossref]

Ashhab, S.

I. Buluta, S. Ashhab, and F. Nori, “Natural and artificial atoms for quantum computation,” Rep. Prog. Phys. 74, 104401 (2011).
[Crossref]

Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

Aumentado, J.

A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Türeci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
[Crossref]

Bajcsy, M.

A. Majumdar, M. Bajcsy, A. Rundquist, and J. Vučković, “Loss-enabled sub-Poissonian light generation in a bimodal nanocavity,” Phys. Rev. Lett. 108, 163601 (2012).
[Crossref]

Bajer, J.

A. Miranowicz, M. Paprzycka, Y.-X. Liu, J. Bajer, and F. Nori, “Two-photon and three-photon blockades in driven nonlinear systems,” Phys. Rev. A 87, 023809 (2013).
[Crossref]

Baker, C.

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett. 98, 113108 (2011).
[Crossref]

Bakker, M. P.

H. Snijders, J. A. Frey, J. Norman, M. P. Bakker, E. C. Langman, A. Gossard, J. E. Bowers, M. P. Van Exter, D. Bouwmeester, and W. Löffler, “Purification of a single-photon nonlinearity,” Nat. Commun. 7, 12578 (2016).
[Crossref]

Bamba, M.

M. Bamba, A. Imamoğlu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802(R) (2011).
[Crossref]

Bang, J.

J. Ahn, Z. Xu, J. Bang, Y.-H. Deng, T. M. Hoang, Q. Han, R.-M. Ma, and T. Li, “Optically levitated nanodumbbell torsion balance and GHz nanomechanical rotor,” Phys. Rev. Lett. 121, 033603 (2018).
[Crossref]

Barzanjeh, S.

S. Barzanjeh, M. Aquilina, and A. Xuereb, “Manipulating the flow of thermal noise in quantum devices,” Phys. Rev. Lett. 120, 060601 (2018).
[Crossref]

Baur, M.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref]

Bechmann-Pasquinucci, H.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[Crossref]

Bencheikh, K.

V. Huet, A. Rasoloniaina, P. Guillemé, P. Rochard, P. Féron, M. Mortier, A. Levenson, K. Bencheikh, A. Yacomotti, and Y. Dumeige, “Millisecond photon lifetime in a slow-light microcavity,” Phys. Rev. Lett. 116, 133902 (2016).
[Crossref]

Bender, C. M.

B. Peng, S. 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]

Bender, N.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
[Crossref]

Bennett, S. D.

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013).
[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]

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]

Bino, L. D.

Birnbaum, K. M.

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature (London) 436, 87–90 (2005).
[Crossref]

Blais, A.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref]

Boca, A.

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature (London) 436, 87–90 (2005).
[Crossref]

Bodyfelt, J. D.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
[Crossref]

Boozer, A. D.

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature (London) 436, 87–90 (2005).
[Crossref]

Børkje, K.

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011).
[Crossref]

Bouwmeester, D.

H. J. Snijders, J. A. Frey, J. Norman, H. Flayac, V. Savona, A. C. Gossard, J. E. Bowers, M. P. van Exter, D. Bouwmeester, and W. Löffler, “Observation of the unconventional photon blockade,” Phys. Rev. Lett. 121, 043601 (2018).
[Crossref]

H. Snijders, J. A. Frey, J. Norman, M. P. Bakker, E. C. Langman, A. Gossard, J. E. Bowers, M. P. Van Exter, D. Bouwmeester, and W. Löffler, “Purification of a single-photon nonlinearity,” Nat. Commun. 7, 12578 (2016).
[Crossref]

Bowers, J. E.

H. J. Snijders, J. A. Frey, J. Norman, H. Flayac, V. Savona, A. C. Gossard, J. E. Bowers, M. P. van Exter, D. Bouwmeester, and W. Löffler, “Observation of the unconventional photon blockade,” Phys. Rev. Lett. 121, 043601 (2018).
[Crossref]

H. Snijders, J. A. Frey, J. Norman, M. P. Bakker, E. C. Langman, A. Gossard, J. E. Bowers, M. P. Van Exter, D. Bouwmeester, and W. Löffler, “Purification of a single-photon nonlinearity,” Nat. Commun. 7, 12578 (2016).
[Crossref]

Bozyigit, D.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref]

Buluta, I.

I. Buluta, S. Ashhab, and F. Nori, “Natural and artificial atoms for quantum computation,” Rep. Prog. Phys. 74, 104401 (2011).
[Crossref]

Caloz, C.

C. Caloz, A. Alù, S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck-Léger, “Electromagnetic nonreciprocity,” Phys. Rev. Appl. 10, 047001 (2018).
[Crossref]

Cao, Q.-T.

Q.-T. Cao, H. Wang, C.-H. Dong, H. Jing, R.-S. Liu, X. Chen, L. Ge, Q. Gong, and Y.-F. Xiao, “Experimental demonstration of spontaneous chirality in a nonlinear microresonator,” Phys. Rev. Lett. 118, 033907 (2017).
[Crossref]

Carmichael, H. J.

L. Tian and H. J. Carmichael, “Quantum trajectory simulations of two-state behavior in an optical cavity containing one atom,” Phys. Rev. A 46, R6801–R6804 (1992).
[Crossref]

Carmon, T.

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature (London) 558, 569–572 (2018).
[Crossref]

H. Jing, H. Lü, S. K. Özdemir, T. Carmon, and F. Nori, “Nanoparticle sensing with a spinning resonator,” Optica 5, 1424–1430 (2018).
[Crossref]

Y. Jiang, S. Maayani, T. Carmon, F. Nori, and H. Jing, “Nonreciprocal phonon laser,” Phys. Rev. Appl. 10, 064037 (2018).
[Crossref]

Carusotto, I.

I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85, 299–366 (2013).
[Crossref]

M. Bamba, A. Imamoğlu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802(R) (2011).
[Crossref]

Cerf, N. J.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[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. Photonics 8, 524–529 (2014).
[Crossref]

Chen, A.-X.

X.-W. Xu, Y.-J. Zhao, H. Wang, H. Jing, and A.-X. Chen, “Nonreciprocal photon blockade via quadratic optomechanical coupling,” arXiv:1809.07596 (2018).

Chen, X.

Q.-T. Cao, H. Wang, C.-H. Dong, H. Jing, R.-S. Liu, X. Chen, L. Ge, Q. Gong, and Y.-F. Xiao, “Experimental demonstration of spontaneous chirality in a nonlinear microresonator,” Phys. Rev. Lett. 118, 033907 (2017).
[Crossref]

H. Xie, G.-W. Lin, X. Chen, Z.-H. Chen, and X.-M. Lin, “Single-photon nonlinearities in a strongly driven optomechanical system with quadratic coupling,” Phys. Rev. A 93, 063860 (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]

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. Photonics 10, 657–661 (2016).
[Crossref]

Chen, Z.-H.

H. Xie, G.-W. Lin, X. Chen, Z.-H. Chen, and X.-M. Lin, “Single-photon nonlinearities in a strongly driven optomechanical system with quadratic coupling,” Phys. Rev. A 93, 063860 (2016).
[Crossref]

Cheng, Y. Q.

K. Y. Xia, G. W. Lu, G. W. Lin, Y. Q. Cheng, Y. P. Niu, S. Q. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90, 043802 (2014).
[Crossref]

Christodoulides, D. N.

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature (London) 558, 569–572 (2018).
[Crossref]

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11–19 (2018).
[Crossref]

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
[Crossref]

Cicak, K.

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]

Ciuti, C.

I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85, 299–366 (2013).
[Crossref]

M. Bamba, A. Imamoğlu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802(R) (2011).
[Crossref]

Clerk, A. A.

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

da Silva, M. P.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref]

Dahan, R.

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature (London) 558, 569–572 (2018).
[Crossref]

Deck-Léger, Z.-L.

C. Caloz, A. Alù, S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck-Léger, “Electromagnetic nonreciprocity,” Phys. Rev. Appl. 10, 047001 (2018).
[Crossref]

Del Bino, L.

del Valle, E.

K. Müller, A. Rundquist, K. A. Fischer, T. Sarmiento, K. G. Lagoudakis, Y. A. Kelaita, C. S. Muñoz, E. del Valle, F. P. Laussy, and J. Vučković, “Coherent generation of nonclassical light on chip via detuned photon blockade,” Phys. Rev. Lett. 114, 233601 (2015).
[Crossref]

Del’Haye, P.

Deléglise, S.

E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature (London) 482, 63–67 (2012).
[Crossref]

Deng, Y.-H.

J. Ahn, Z. Xu, J. Bang, Y.-H. Deng, T. M. Hoang, Q. Han, R.-M. Ma, and T. Li, “Optically levitated nanodumbbell torsion balance and GHz nanomechanical rotor,” Phys. Rev. Lett. 121, 033603 (2018).
[Crossref]

Deutsch, M.

A. Imamoḡlu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467–1470 (1997).
[Crossref]

Devoret, M.

B. Abdo, K. Sliwa, S. Shankar, M. Hatridge, L. Frunzio, R. Schoelkopf, and M. Devoret, “Josephson directional amplifier for quantum measurement of superconducting circuits,” Phys. Rev. Lett. 112, 167701 (2014).
[Crossref]

Diehl, R.

R. Reimann, M. Doderer, E. Hebestreit, R. Diehl, M. Frimmer, D. Windey, F. Tebbenjohanns, and L. Novotny, “GHz rotation of an optically trapped nanoparticle in vacuum,” Phys. Rev. Lett. 121, 033602 (2018).
[Crossref]

Ding, L.

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett. 98, 113108 (2011).
[Crossref]

Doderer, M.

R. Reimann, M. Doderer, E. Hebestreit, R. Diehl, M. Frimmer, D. Windey, F. Tebbenjohanns, and L. Novotny, “GHz rotation of an optically trapped nanoparticle in vacuum,” Phys. Rev. Lett. 121, 033602 (2018).
[Crossref]

Dong, C. H.

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]

Dong, C.-H.

Q.-T. Cao, H. Wang, C.-H. Dong, H. Jing, R.-S. Liu, X. Chen, L. Ge, Q. Gong, and Y.-F. Xiao, “Experimental demonstration of spontaneous chirality in a nonlinear microresonator,” Phys. Rev. Lett. 118, 033907 (2017).
[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. Photonics 10, 657–661 (2016).
[Crossref]

Donner, T.

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]

Ducci, S.

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett. 98, 113108 (2011).
[Crossref]

Dudka, M.

J. Hloušek, M. Dudka, I. Straka, and M. Ježek, “Accurate detection of arbitrary photon statistics,” arXiv:1812.02262 (2018).

Dumeige, Y.

V. Huet, A. Rasoloniaina, P. Guillemé, P. Rochard, P. Féron, M. Mortier, A. Levenson, K. Bencheikh, A. Yacomotti, and Y. Dumeige, “Millisecond photon lifetime in a slow-light microcavity,” Phys. Rev. Lett. 116, 133902 (2016).
[Crossref]

Dušek, M.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[Crossref]

Eichler, C.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref]

El-Ganainy, R.

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11–19 (2018).
[Crossref]

Ellis, F. M.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
[Crossref]

Englund, D.

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. 4, 859–863 (2008).
[Crossref]

Enzian, G.

Estève, J.

C. Vaneph, A. Morvan, G. Aiello, M. Féchant, M. Aprili, J. Gabelli, and J. Estève, “Observation of the unconventional photon blockade in the microwave domain,” Phys. Rev. Lett. 121, 043602 (2018).
[Crossref]

Factor, S.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
[Crossref]

Fan, L.

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[Crossref]

Fan, S.

A. Y. Song, Y. Shi, Q. Lin, and S. Fan, “Direction-dependent parity-time phase transition and non-reciprocal directional amplification with dynamic gain–loss modulation,” Phys. Rev. A 99, 013824 (2019).
[Crossref]

Y. Shi, Z. Yu, and S. Fan, “Limitations of nonlinear optical isolators due to dynamic reciprocity,” Nat. Photonics 9, 388–392 (2015).
[Crossref]

B. Peng, S. 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]

Faraon, A.

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. 4, 859–863 (2008).
[Crossref]

Favero, I.

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett. 98, 113108 (2011).
[Crossref]

Féchant, M.

C. Vaneph, A. Morvan, G. Aiello, M. Féchant, M. Aprili, J. Gabelli, and J. Estève, “Observation of the unconventional photon blockade in the microwave domain,” Phys. Rev. Lett. 121, 043602 (2018).
[Crossref]

Feofanov, A. K.

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]

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]

Féron, P.

V. Huet, A. Rasoloniaina, P. Guillemé, P. Rochard, P. Féron, M. Mortier, A. Levenson, K. Bencheikh, A. Yacomotti, and Y. Dumeige, “Millisecond photon lifetime in a slow-light microcavity,” Phys. Rev. Lett. 116, 133902 (2016).
[Crossref]

Ferretti, S.

S. Ferretti, V. Savona, and D. Gerace, “Optimal antibunching in passive photonic devices based on coupled nonlinear resonators,” New J. Phys. 15, 025012 (2013).
[Crossref]

S. Ferretti, L. C. Andreani, H. E. Türeci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841 (2010).
[Crossref]

Filipp, S.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref]

Fink, J. M.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref]

Firstenberg, O.

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletić, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature (London) 488, 57–60 (2012).
[Crossref]

Fischer, K. A.

M. Radulaski, K. A. Fischer, K. G. Lagoudakis, J. L. Zhang, and J. Vučković, “Photon blockade in two-emitter-cavity systems,” Phys. Rev. A 96, 011801(R) (2017).
[Crossref]

K. Müller, A. Rundquist, K. A. Fischer, T. Sarmiento, K. G. Lagoudakis, Y. A. Kelaita, C. S. Muñoz, E. del Valle, F. P. Laussy, and J. Vučković, “Coherent generation of nonclassical light on chip via detuned photon blockade,” Phys. Rev. Lett. 114, 233601 (2015).
[Crossref]

Flayac, H.

H. J. Snijders, J. A. Frey, J. Norman, H. Flayac, V. Savona, A. C. Gossard, J. E. Bowers, M. P. van Exter, D. Bouwmeester, and W. Löffler, “Observation of the unconventional photon blockade,” Phys. Rev. Lett. 121, 043601 (2018).
[Crossref]

H. Flayac and V. Savona, “Unconventional photon blockade,” Phys. Rev. A 96, 053810 (2017).
[Crossref]

H. Flayac and V. Savona, “Nonclassical statistics from a polaritonic Josephson junction,” Phys. Rev. A 95, 043838 (2017).
[Crossref]

Ford, G. W.

G. W. Ford, J. T. Lewis, and R. F. O’Connell, “Quantum Langevin equation,” Phys. Rev. A 37, 4419–4428 (1988).
[Crossref]

Frey, J. A.

H. J. Snijders, J. A. Frey, J. Norman, H. Flayac, V. Savona, A. C. Gossard, J. E. Bowers, M. P. van Exter, D. Bouwmeester, and W. Löffler, “Observation of the unconventional photon blockade,” Phys. Rev. Lett. 121, 043601 (2018).
[Crossref]

H. Snijders, J. A. Frey, J. Norman, M. P. Bakker, E. C. Langman, A. Gossard, J. E. Bowers, M. P. Van Exter, D. Bouwmeester, and W. Löffler, “Purification of a single-photon nonlinearity,” Nat. Commun. 7, 12578 (2016).
[Crossref]

Frimmer, M.

R. Reimann, M. Doderer, E. Hebestreit, R. Diehl, M. Frimmer, D. Windey, F. Tebbenjohanns, and L. Novotny, “GHz rotation of an optically trapped nanoparticle in vacuum,” Phys. Rev. Lett. 121, 033602 (2018).
[Crossref]

Frunzio, L.

B. Abdo, K. Sliwa, S. Shankar, M. Hatridge, L. Frunzio, R. Schoelkopf, and M. Devoret, “Josephson directional amplifier for quantum measurement of superconducting circuits,” Phys. Rev. Lett. 112, 167701 (2014).
[Crossref]

Fushman, I.

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. 4, 859–863 (2008).
[Crossref]

Gabelli, J.

C. Vaneph, A. Morvan, G. Aiello, M. Féchant, M. Aprili, J. Gabelli, and J. Estève, “Observation of the unconventional photon blockade in the microwave domain,” Phys. Rev. Lett. 121, 043602 (2018).
[Crossref]

Gardiner, C. W.

C. W. Gardiner and P. Zoller, Quantum Noise (Springer, 2000).

Ge, L.

Q.-T. Cao, H. Wang, C.-H. Dong, H. Jing, R.-S. Liu, X. Chen, L. Ge, Q. Gong, and Y.-F. Xiao, “Experimental demonstration of spontaneous chirality in a nonlinear microresonator,” Phys. Rev. Lett. 118, 033907 (2017).
[Crossref]

Gerace, D.

S. Ferretti, V. Savona, and D. Gerace, “Optimal antibunching in passive photonic devices based on coupled nonlinear resonators,” New J. Phys. 15, 025012 (2013).
[Crossref]

S. Ferretti, L. C. Andreani, H. E. Türeci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841 (2010).
[Crossref]

Gianfreda, M.

B. Peng, S. 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]

Girvin, S. M.

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011).
[Crossref]

Gong, J.

S. Zhang, Y. Hu, G. Lin, Y. Niu, K. Xia, J. Gong, and S. Gong, “Thermal-motion-induced non-reciprocal quantum optical system,” Nat. Photonics 12, 744–748 (2018).
[Crossref]

Gong, Q.

Q.-T. Cao, H. Wang, C.-H. Dong, H. Jing, R.-S. Liu, X. Chen, L. Ge, Q. Gong, and Y.-F. Xiao, “Experimental demonstration of spontaneous chirality in a nonlinear microresonator,” Phys. Rev. Lett. 118, 033907 (2017).
[Crossref]

Gong, S.

S. Zhang, Y. Hu, G. Lin, Y. Niu, K. Xia, J. Gong, and S. Gong, “Thermal-motion-induced non-reciprocal quantum optical system,” Nat. Photonics 12, 744–748 (2018).
[Crossref]

Gong, S. Q.

K. Y. Xia, G. W. Lu, G. W. Lin, Y. Q. Cheng, Y. P. Niu, S. Q. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90, 043802 (2014).
[Crossref]

Gong, Z. R.

Z. R. Gong, H. Ian, Y.-X. Liu, C. P. Sun, and F. Nori, “Effective Hamiltonian approach to the Kerr nonlinearity in an optomechanical system,” Phys. Rev. A 80, 065801 (2009).
[Crossref]

Gorshkov, A. V.

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletić, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature (London) 488, 57–60 (2012).
[Crossref]

Gossard, A.

H. Snijders, J. A. Frey, J. Norman, M. P. Bakker, E. C. Langman, A. Gossard, J. E. Bowers, M. P. Van Exter, D. Bouwmeester, and W. Löffler, “Purification of a single-photon nonlinearity,” Nat. Commun. 7, 12578 (2016).
[Crossref]

Gossard, A. C.

H. J. Snijders, J. A. Frey, J. Norman, H. Flayac, V. Savona, A. C. Gossard, J. E. Bowers, M. P. van Exter, D. Bouwmeester, and W. Löffler, “Observation of the unconventional photon blockade,” Phys. Rev. Lett. 121, 043601 (2018).
[Crossref]

Grudinin, I. S.

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]

Gu, X.

X. Gu, A. F. Kockum, A. Miranowicz, Y.-X. Liu, and F. Nori, “Microwave photonics with superconducting quantum circuits,” Phys. Rep. 718–719, 1–102 (2017).
[Crossref]

Guillemé, P.

V. Huet, A. Rasoloniaina, P. Guillemé, P. Rochard, P. Féron, M. Mortier, A. Levenson, K. Bencheikh, A. Yacomotti, and Y. Dumeige, “Millisecond photon lifetime in a slow-light microcavity,” Phys. Rev. Lett. 116, 133902 (2016).
[Crossref]

Guo, G. C.

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]

Guo, G.-C.

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. Photonics 10, 657–661 (2016).
[Crossref]

Habraken, S. J. M.

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013).
[Crossref]

Han, Q.

J. Ahn, Z. Xu, J. Bang, Y.-H. Deng, T. M. Hoang, Q. Han, R.-M. Ma, and T. Li, “Optically levitated nanodumbbell torsion balance and GHz nanomechanical rotor,” Phys. Rev. Lett. 121, 033603 (2018).
[Crossref]

Harlow, J. W.

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]

Hassan, A. U.

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature (London) 558, 569–572 (2018).
[Crossref]

Hatridge, M.

B. Abdo, K. Sliwa, S. Shankar, M. Hatridge, L. Frunzio, R. Schoelkopf, and M. Devoret, “Josephson directional amplifier for quantum measurement of superconducting circuits,” Phys. Rev. Lett. 112, 167701 (2014).
[Crossref]

Hebestreit, E.

R. Reimann, M. Doderer, E. Hebestreit, R. Diehl, M. Frimmer, D. Windey, F. Tebbenjohanns, and L. Novotny, “GHz rotation of an optically trapped nanoparticle in vacuum,” Phys. Rev. Lett. 121, 033602 (2018).
[Crossref]

Hilico, A.

M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354, 1577–1580 (2016).
[Crossref]

Hloušek, J.

J. Hloušek, M. Dudka, I. Straka, and M. Ježek, “Accurate detection of arbitrary photon statistics,” arXiv:1812.02262 (2018).

Hoang, T. M.

J. Ahn, Z. Xu, J. Bang, Y.-H. Deng, T. M. Hoang, Q. Han, R.-M. Ma, and T. Li, “Optically levitated nanodumbbell torsion balance and GHz nanomechanical rotor,” Phys. Rev. Lett. 121, 033603 (2018).
[Crossref]

Hofferberth, S.

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletić, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature (London) 488, 57–60 (2012).
[Crossref]

Hoffman, A. J.

A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Türeci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
[Crossref]

Home, J. P.

F. Reiter, T. L. Nguyen, J. P. Home, and S. F. Yelin, “Cooperative breakdown of the oscillator blockade in the Dicke model,” arXiv:1807.06026 (2018).

Houck, A. A.

A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Türeci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
[Crossref]

Hu, C.-M.

Y.-P. Wang, G.-Q. Zhang, D. Zhang, T.-F. Li, C.-M. Hu, and J. Q. You, “Bistability of cavity magnon polaritons,” Phys. Rev. Lett. 120, 057202 (2018).
[Crossref]

Hu, Y.

S. Zhang, Y. Hu, G. Lin, Y. Niu, K. Xia, J. Gong, and S. Gong, “Thermal-motion-induced non-reciprocal quantum optical system,” Nat. Photonics 12, 744–748 (2018).
[Crossref]

Hua, S.

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. Photonics 8, 524–529 (2014).
[Crossref]

Huang, R.

R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121, 153601 (2018).
[Crossref]

C. Zhai, R. Huang, B. Li, H. Jing, and L.-M. Kuang, “Mechanical engineering of photon blockades in a cavity optomechanical system,” arXiv:1901.07654 (2019).

Huet, V.

V. Huet, A. Rasoloniaina, P. Guillemé, P. Rochard, P. Féron, M. Mortier, A. Levenson, K. Bencheikh, A. Yacomotti, and Y. Dumeige, “Millisecond photon lifetime in a slow-light microcavity,” Phys. Rev. Lett. 116, 133902 (2016).
[Crossref]

Ian, H.

Z. R. Gong, H. Ian, Y.-X. Liu, C. P. Sun, and F. Nori, “Effective Hamiltonian approach to the Kerr nonlinearity in an optomechanical system,” Phys. Rev. A 80, 065801 (2009).
[Crossref]

Imamo?lu, A.

A. Imamoḡlu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467–1470 (1997).
[Crossref]

H. Schmidt and A. Imamoḡlu, “Giant Kerr nonlinearities obtained by electromagnetically induced transparency,” Opt. Lett. 21, 1936–1938 (1996).
[Crossref]

Imamoglu, A.

M. Bamba, A. Imamoğlu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802(R) (2011).
[Crossref]

Imoto, N.

A. Miranowicz, W. Leoński, and N. Imoto, “Quantum-optical states in finite-dimensional Hilbert space. I. General formalism,” in Modern Nonlinear Optics (Wiley, 2001), Vol. 119, pp. 195–213.

Ioannou, M. A.

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]

Ježek, M.

J. Hloušek, M. Dudka, I. Straka, and M. Ježek, “Accurate detection of arbitrary photon statistics,” arXiv:1812.02262 (2018).

Jiang, 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. Photonics 8, 524–529 (2014).
[Crossref]

Jiang, X.

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. Photonics 8, 524–529 (2014).
[Crossref]

Jiang, Y.

Y. Jiang, S. Maayani, T. Carmon, F. Nori, and H. Jing, “Nonreciprocal phonon laser,” Phys. Rev. Appl. 10, 064037 (2018).
[Crossref]

H. Lü, Y. Jiang, Y. Z. Wang, and H. Jing, “Optomechanically induced transparency in a spinning resonator,” Photon. Res. 5, 367–371 (2017).
[Crossref]

Jiao, Y.

Jing, H.

H. Zhang, F. Salf, Y. Jiao, and H. Jing, “Loss-induced transparency in optomechanics,” Opt. Express 26, 25199–25210 (2018).
[Crossref]

Y. Jiang, S. Maayani, T. Carmon, F. Nori, and H. Jing, “Nonreciprocal phonon laser,” Phys. Rev. Appl. 10, 064037 (2018).
[Crossref]

R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121, 153601 (2018).
[Crossref]

H. Jing, H. Lü, S. K. Özdemir, T. Carmon, and F. Nori, “Nanoparticle sensing with a spinning resonator,” Optica 5, 1424–1430 (2018).
[Crossref]

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature (London) 558, 569–572 (2018).
[Crossref]

H. Lü, Y. Jiang, Y. Z. Wang, and H. Jing, “Optomechanically induced transparency in a spinning resonator,” Photon. Res. 5, 367–371 (2017).
[Crossref]

Q.-T. Cao, H. Wang, C.-H. Dong, H. Jing, R.-S. Liu, X. Chen, L. Ge, Q. Gong, and Y.-F. Xiao, “Experimental demonstration of spontaneous chirality in a nonlinear microresonator,” Phys. Rev. Lett. 118, 033907 (2017).
[Crossref]

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

Z.-P. Liu, J. Zhang, S. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-X. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
[Crossref]

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

X.-W. Xu, Y.-J. Zhao, H. Wang, H. Jing, and A.-X. Chen, “Nonreciprocal photon blockade via quadratic optomechanical coupling,” arXiv:1809.07596 (2018).

C. Zhai, R. Huang, B. Li, H. Jing, and L.-M. Kuang, “Mechanical engineering of photon blockades in a cavity optomechanical system,” arXiv:1901.07654 (2019).

Johansson, J. R.

J. R. Johansson, P. D. Nation, and F. Nori, “Qutip 2: a Python framework for the dynamics of open quantum systems,” Comput. Phys. Commun. 184, 1234–1240 (2013).
[Crossref]

Kelaita, Y. A.

K. Müller, A. Rundquist, K. A. Fischer, T. Sarmiento, K. G. Lagoudakis, Y. A. Kelaita, C. S. Muñoz, E. del Valle, F. P. Laussy, and J. Vučković, “Coherent generation of nonclassical light on chip via detuned photon blockade,” Phys. Rev. Lett. 114, 233601 (2015).
[Crossref]

Khajavikhan, M.

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11–19 (2018).
[Crossref]

Kimble, H. J.

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature (London) 436, 87–90 (2005).
[Crossref]

Kippenberg, T. J.

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]

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]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature (London) 482, 63–67 (2012).
[Crossref]

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, 043902 (2003).
[Crossref]

Kligerman, Y.

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature (London) 558, 569–572 (2018).
[Crossref]

Knight, P. L.

M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. 70, 101–144 (1998).
[Crossref]

Kockum, A. F.

X. Gu, A. F. Kockum, A. Miranowicz, Y.-X. Liu, and F. Nori, “Microwave photonics with superconducting quantum circuits,” Phys. Rep. 718–719, 1–102 (2017).
[Crossref]

Kómár, P.

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013).
[Crossref]

Konotop, V. V.

V. V. Konotop, J. K. Yang, and D. A. Zezyulin, “Nonlinear waves in PT-symmetric systems,” Rev. Mod. Phys. 88, 035002 (2016).
[Crossref]

Koottandavida, A.

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]

Kottos, T.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
[Crossref]

Krimer, D. O.

J. Zhang, B. Peng, S. K. Özdemir, K. Pichler, D. O. Krimer, G. M. Zhao, F. Nori, Y.-X. Liu, S. Rotter, and L. Yang, “A phonon laser operating at an exceptional point,” Nat. Photonics 12, 479–484 (2018).
[Crossref]

Kuang, L.-M.

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

C. Zhai, R. Huang, B. Li, H. Jing, and L.-M. Kuang, “Mechanical engineering of photon blockades in a cavity optomechanical system,” arXiv:1901.07654 (2019).

Lagoudakis, K. G.

M. Radulaski, K. A. Fischer, K. G. Lagoudakis, J. L. Zhang, and J. Vučković, “Photon blockade in two-emitter-cavity systems,” Phys. Rev. A 96, 011801(R) (2017).
[Crossref]

K. Müller, A. Rundquist, K. A. Fischer, T. Sarmiento, K. G. Lagoudakis, Y. A. Kelaita, C. S. Muñoz, E. del Valle, F. P. Laussy, and J. Vučković, “Coherent generation of nonclassical light on chip via detuned photon blockade,” Phys. Rev. Lett. 114, 233601 (2015).
[Crossref]

Lai, D.-G.

F. Zhou, D.-G. Lai, and J.-Q. Liao, “Photon blockade effect in a coupled cavity system,” arXiv:1803.06642 (2018).

Lang, C.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref]

Langman, E. C.

H. Snijders, J. A. Frey, J. Norman, M. P. Bakker, E. C. Langman, A. Gossard, J. E. Bowers, M. P. Van Exter, D. Bouwmeester, and W. Löffler, “Purification of a single-photon nonlinearity,” Nat. Commun. 7, 12578 (2016).
[Crossref]

Laussy, F. P.

K. Müller, A. Rundquist, K. A. Fischer, T. Sarmiento, K. G. Lagoudakis, Y. A. Kelaita, C. S. Muñoz, E. del Valle, F. P. Laussy, and J. Vučković, “Coherent generation of nonclassical light on chip via detuned photon blockade,” Phys. Rev. Lett. 114, 233601 (2015).
[Crossref]

Law, C. K.

J.-Q. Liao and C. K. Law, “Correlated two-photon transport in a one-dimensional waveguide side-coupled to a nonlinear cavity,” Phys. Rev. A 82, 053836 (2010).
[Crossref]

Lee, H.

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]

Lehnert, K. W.

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]

Lei, F.

B. Peng, S. 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]

Lemaitre, A.

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett. 98, 113108 (2011).
[Crossref]

Leo, G.

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett. 98, 113108 (2011).
[Crossref]

Leonski, W.

A. Miranowicz and W. Leoński, “Two-mode optical state truncation and generation of maximally entangled states in pumped nonlinear couplers,” J. Phys. B 39, 1683–1700 (2006).
[Crossref]

W. Leoński and A. Miranowicz, “Kerr nonlinear coupler and entanglement,” J. Opt. B 6, S37–S42 (2004).
[Crossref]

W. Leoński and A. Miranowicz, “Quantum-optical states in finite-dimensional Hilbert space. II. State generation,” Adv. Chem. Phys. 119, 155–193 (2003).
[Crossref]

W. Leoński and R. Tanaś, “Possibility of producing the one-photon state in a kicked cavity with a nonlinear Kerr medium,” Phys. Rev. A 49, R20–R23 (1994).
[Crossref]

A. Miranowicz, W. Leoński, and N. Imoto, “Quantum-optical states in finite-dimensional Hilbert space. I. General formalism,” in Modern Nonlinear Optics (Wiley, 2001), Vol. 119, pp. 195–213.

Levenson, A.

V. Huet, A. Rasoloniaina, P. Guillemé, P. Rochard, P. Féron, M. Mortier, A. Levenson, K. Bencheikh, A. Yacomotti, and Y. Dumeige, “Millisecond photon lifetime in a slow-light microcavity,” Phys. Rev. Lett. 116, 133902 (2016).
[Crossref]

Lewis, J. T.

G. W. Ford, J. T. Lewis, and R. F. O’Connell, “Quantum Langevin equation,” Phys. Rev. A 37, 4419–4428 (1988).
[Crossref]

Li, B.

C. Zhai, R. Huang, B. Li, H. Jing, and L.-M. Kuang, “Mechanical engineering of photon blockades in a cavity optomechanical system,” arXiv:1901.07654 (2019).

Li, C.-W.

Z.-P. Liu, J. Zhang, S. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-X. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
[Crossref]

Li, D.

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]

Li, G.

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. Photonics 8, 524–529 (2014).
[Crossref]

Li, T.

J. Ahn, Z. Xu, J. Bang, Y.-H. Deng, T. M. Hoang, Q. Han, R.-M. Ma, and T. Li, “Optically levitated nanodumbbell torsion balance and GHz nanomechanical rotor,” Phys. Rev. Lett. 121, 033603 (2018).
[Crossref]

Li, T.-F.

Y.-P. Wang, G.-Q. Zhang, D. Zhang, T.-F. Li, C.-M. Hu, and J. Q. You, “Bistability of cavity magnon polaritons,” Phys. Rev. Lett. 120, 057202 (2018).
[Crossref]

Li, Y.

X.-W. Xu and Y. Li, “Strong photon antibunching of symmetric and antisymmetric modes in weakly nonlinear photonic molecules,” Phys. Rev. A 90, 033809 (2014).
[Crossref]

Li, Y.-J.

X.-W. Xu and Y.-J. Li, “Antibunching photons in a cavity coupled to an optomechanical system,” J. Phys. B 46, 035502 (2013).
[Crossref]

Liang, Q.-Y.

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletić, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature (London) 488, 57–60 (2012).
[Crossref]

Liao, J.-Q.

R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121, 153601 (2018).
[Crossref]

J.-Q. Liao and F. Nori, “Photon blockade in quadratically coupled optomechanical systems,” Phys. Rev. A 88, 023853 (2013).
[Crossref]

J.-Q. Liao and C. K. Law, “Correlated two-photon transport in a one-dimensional waveguide side-coupled to a nonlinear cavity,” Phys. Rev. A 82, 053836 (2010).
[Crossref]

F. Zhou, D.-G. Lai, and J.-Q. Liao, “Photon blockade effect in a coupled cavity system,” arXiv:1803.06642 (2018).

Liew, T. C. H.

T. C. H. Liew and V. Savona, “Single photons from coupled quantum modes,” Phys. Rev. Lett. 104, 183601 (2010).
[Crossref]

Lin, G.

S. Zhang, Y. Hu, G. Lin, Y. Niu, K. Xia, J. Gong, and S. Gong, “Thermal-motion-induced non-reciprocal quantum optical system,” Nat. Photonics 12, 744–748 (2018).
[Crossref]

Lin, G. W.

K. Y. Xia, G. W. Lu, G. W. Lin, Y. Q. Cheng, Y. P. Niu, S. Q. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90, 043802 (2014).
[Crossref]

Lin, G.-W.

H. Xie, G.-W. Lin, X. Chen, Z.-H. Chen, and X.-M. Lin, “Single-photon nonlinearities in a strongly driven optomechanical system with quadratic coupling,” Phys. Rev. A 93, 063860 (2016).
[Crossref]

Lin, Q.

A. Y. Song, Y. Shi, Q. Lin, and S. Fan, “Direction-dependent parity-time phase transition and non-reciprocal directional amplification with dynamic gain–loss modulation,” Phys. Rev. A 99, 013824 (2019).
[Crossref]

Lin, X.-M.

H. Xie, G.-W. Lin, X. Chen, Z.-H. Chen, and X.-M. Lin, “Single-photon nonlinearities in a strongly driven optomechanical system with quadratic coupling,” Phys. Rev. A 93, 063860 (2016).
[Crossref]

Lipson, M.

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

Liu, R.-S.

Q.-T. Cao, H. Wang, C.-H. Dong, H. Jing, R.-S. Liu, X. Chen, L. Ge, Q. Gong, and Y.-F. Xiao, “Experimental demonstration of spontaneous chirality in a nonlinear microresonator,” Phys. Rev. Lett. 118, 033907 (2017).
[Crossref]

Liu, Y. M.

W. Zhang, Z. Y. Yu, Y. M. Liu, and Y. W. Peng, “Optimal photon antibunching in a quantum-dot-bimodal-cavity system,” Phys. Rev. A 89, 043832 (2014).
[Crossref]

Liu, Y.-X.

J. Zhang, B. Peng, S. K. Özdemir, K. Pichler, D. O. Krimer, G. M. Zhao, F. Nori, Y.-X. Liu, S. Rotter, and L. Yang, “A phonon laser operating at an exceptional point,” Nat. Photonics 12, 479–484 (2018).
[Crossref]

X. Gu, A. F. Kockum, A. Miranowicz, Y.-X. Liu, and F. Nori, “Microwave photonics with superconducting quantum circuits,” Phys. Rep. 718–719, 1–102 (2017).
[Crossref]

Z.-P. Liu, J. Zhang, S. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-X. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
[Crossref]

A. Miranowicz, M. Paprzycka, Y.-X. Liu, J. Bajer, and F. Nori, “Two-photon and three-photon blockades in driven nonlinear systems,” Phys. Rev. A 87, 023809 (2013).
[Crossref]

Z. R. Gong, H. Ian, Y.-X. Liu, C. P. Sun, and F. Nori, “Effective Hamiltonian approach to the Kerr nonlinearity in an optomechanical system,” Phys. Rev. A 80, 065801 (2009).
[Crossref]

Liu, Z.-P.

Z.-P. Liu, J. Zhang, S. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-X. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
[Crossref]

Lodahl, P.

P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature 541, 473–480 (2017).
[Crossref]

Löffler, W.

H. J. Snijders, J. A. Frey, J. Norman, H. Flayac, V. Savona, A. C. Gossard, J. E. Bowers, M. P. van Exter, D. Bouwmeester, and W. Löffler, “Observation of the unconventional photon blockade,” Phys. Rev. Lett. 121, 043601 (2018).
[Crossref]

H. Snijders, J. A. Frey, J. Norman, M. P. Bakker, E. C. Langman, A. Gossard, J. E. Bowers, M. P. Van Exter, D. Bouwmeester, and W. Löffler, “Purification of a single-photon nonlinearity,” Nat. Commun. 7, 12578 (2016).
[Crossref]

Long, G. L.

B. Peng, S. 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]

Lu, G. W.

K. Y. Xia, G. W. Lu, G. W. Lin, Y. Q. Cheng, Y. P. Niu, S. Q. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90, 043802 (2014).
[Crossref]

L. Tang, J. S. Tang, W. D. Zhang, G. W. Lu, Y. Zhang, K. Y. Xia, and M. Xiao, “An on-chip chiral single-photon interface: isolation and unidirectional emission,” arXiv:1811.02957 (2018).

Lü, H.

Lü, X.-Y.

Z.-P. Liu, J. Zhang, S. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-X. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
[Crossref]

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

Lukin, M. D.

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013).
[Crossref]

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletić, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature (London) 488, 57–60 (2012).
[Crossref]

Lütkenhaus, N.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[Crossref]

Ma, R.-M.

J. Ahn, Z. Xu, J. Bang, Y.-H. Deng, T. M. Hoang, Q. Han, R.-M. Ma, and T. Li, “Optically levitated nanodumbbell torsion balance and GHz nanomechanical rotor,” Phys. Rev. Lett. 121, 033603 (2018).
[Crossref]

Maayani, S.

Y. Jiang, S. Maayani, T. Carmon, F. Nori, and H. Jing, “Nonreciprocal phonon laser,” Phys. Rev. Appl. 10, 064037 (2018).
[Crossref]

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature (London) 558, 569–572 (2018).
[Crossref]

Mahmoodian, S.

P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature 541, 473–480 (2017).
[Crossref]

Majumdar, A.

A. Majumdar, M. Bajcsy, A. Rundquist, and J. Vučković, “Loss-enabled sub-Poissonian light generation in a bimodal nanocavity,” Phys. Rev. Lett. 108, 163601 (2012).
[Crossref]

Makris, K. G.

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11–19 (2018).
[Crossref]

Malykin, G. B.

G. B. Malykin, “The Sagnac effect: correct and incorrect explanations,” Phys. Usp. 43, 1229–1252 (2000).
[Crossref]

Malz, D.

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]

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]

Manipatruni, S.

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

Marquardt, F.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

Metelmann, A.

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

Milburn, G. J.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 1994).

Miller, R.

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature (London) 436, 87–90 (2005).
[Crossref]

Miranowicz, A.

R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121, 153601 (2018).
[Crossref]

X. Gu, A. F. Kockum, A. Miranowicz, Y.-X. Liu, and F. Nori, “Microwave photonics with superconducting quantum circuits,” Phys. Rep. 718–719, 1–102 (2017).
[Crossref]

A. Miranowicz, M. Paprzycka, Y.-X. Liu, J. Bajer, and F. Nori, “Two-photon and three-photon blockades in driven nonlinear systems,” Phys. Rev. A 87, 023809 (2013).
[Crossref]

A. Miranowicz and W. Leoński, “Two-mode optical state truncation and generation of maximally entangled states in pumped nonlinear couplers,” J. Phys. B 39, 1683–1700 (2006).
[Crossref]

W. Leoński and A. Miranowicz, “Kerr nonlinear coupler and entanglement,” J. Opt. B 6, S37–S42 (2004).
[Crossref]

W. Leoński and A. Miranowicz, “Quantum-optical states in finite-dimensional Hilbert space. II. State generation,” Adv. Chem. Phys. 119, 155–193 (2003).
[Crossref]

A. Miranowicz, W. Leoński, and N. Imoto, “Quantum-optical states in finite-dimensional Hilbert space. I. General formalism,” in Modern Nonlinear Optics (Wiley, 2001), Vol. 119, pp. 195–213.

Monifi, F.

B. Peng, S. 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]

Mortier, M.

V. Huet, A. Rasoloniaina, P. Guillemé, P. Rochard, P. Féron, M. Mortier, A. Levenson, K. Bencheikh, A. Yacomotti, and Y. Dumeige, “Millisecond photon lifetime in a slow-light microcavity,” Phys. Rev. Lett. 116, 133902 (2016).
[Crossref]

Morvan, A.

C. Vaneph, A. Morvan, G. Aiello, M. Féchant, M. Aprili, J. Gabelli, and J. Estève, “Observation of the unconventional photon blockade in the microwave domain,” Phys. Rev. Lett. 121, 043602 (2018).
[Crossref]

Moses, E.

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature (London) 558, 569–572 (2018).
[Crossref]

Müller, K.

K. Müller, A. Rundquist, K. A. Fischer, T. Sarmiento, K. G. Lagoudakis, Y. A. Kelaita, C. S. Muñoz, E. del Valle, F. P. Laussy, and J. Vučković, “Coherent generation of nonclassical light on chip via detuned photon blockade,” Phys. Rev. Lett. 114, 233601 (2015).
[Crossref]

Muñoz, C. S.

K. Müller, A. Rundquist, K. A. Fischer, T. Sarmiento, K. G. Lagoudakis, Y. A. Kelaita, C. S. Muñoz, E. del Valle, F. P. Laussy, and J. Vučković, “Coherent generation of nonclassical light on chip via detuned photon blockade,” Phys. Rev. Lett. 114, 233601 (2015).
[Crossref]

Musslimani, Z. H.

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11–19 (2018).
[Crossref]

Nation, P. D.

J. R. Johansson, P. D. Nation, and F. Nori, “Qutip 2: a Python framework for the dynamics of open quantum systems,” Comput. Phys. Commun. 184, 1234–1240 (2013).
[Crossref]

Nguyen, T. L.

F. Reiter, T. L. Nguyen, J. P. Home, and S. F. Yelin, “Cooperative breakdown of the oscillator blockade in the Dicke model,” arXiv:1807.06026 (2018).

Niu, B.

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[Crossref]

Niu, Y.

S. Zhang, Y. Hu, G. Lin, Y. Niu, K. Xia, J. Gong, and S. Gong, “Thermal-motion-induced non-reciprocal quantum optical system,” Nat. Photonics 12, 744–748 (2018).
[Crossref]

Niu, Y. P.

K. Y. Xia, G. W. Lu, G. W. Lin, Y. Q. Cheng, Y. P. Niu, S. Q. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90, 043802 (2014).
[Crossref]

Nori, F.

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature (London) 558, 569–572 (2018).
[Crossref]

H. Jing, H. Lü, S. K. Özdemir, T. Carmon, and F. Nori, “Nanoparticle sensing with a spinning resonator,” Optica 5, 1424–1430 (2018).
[Crossref]

K. Y. Xia, F. Nori, and M. Xiao, “Cavity-free optical isolators and circulators using a chiral cross-Kerr nonlinearity,” Phys. Rev. Lett. 121, 203602 (2018).
[Crossref]

J. Zhang, B. Peng, S. K. Özdemir, K. Pichler, D. O. Krimer, G. M. Zhao, F. Nori, Y.-X. Liu, S. Rotter, and L. Yang, “A phonon laser operating at an exceptional point,” Nat. Photonics 12, 479–484 (2018).
[Crossref]

R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121, 153601 (2018).
[Crossref]

Y. Jiang, S. Maayani, T. Carmon, F. Nori, and H. Jing, “Nonreciprocal phonon laser,” Phys. Rev. Appl. 10, 064037 (2018).
[Crossref]

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

X. Gu, A. F. Kockum, A. Miranowicz, Y.-X. Liu, and F. Nori, “Microwave photonics with superconducting quantum circuits,” Phys. Rep. 718–719, 1–102 (2017).
[Crossref]

Z.-P. Liu, J. Zhang, S. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-X. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
[Crossref]

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

B. Peng, S. 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. Miranowicz, M. Paprzycka, Y.-X. Liu, J. Bajer, and F. Nori, “Two-photon and three-photon blockades in driven nonlinear systems,” Phys. Rev. A 87, 023809 (2013).
[Crossref]

J.-Q. Liao and F. Nori, “Photon blockade in quadratically coupled optomechanical systems,” Phys. Rev. A 88, 023853 (2013).
[Crossref]

J. R. Johansson, P. D. Nation, and F. Nori, “Qutip 2: a Python framework for the dynamics of open quantum systems,” Comput. Phys. Commun. 184, 1234–1240 (2013).
[Crossref]

I. Buluta, S. Ashhab, and F. Nori, “Natural and artificial atoms for quantum computation,” Rep. Prog. Phys. 74, 104401 (2011).
[Crossref]

Z. R. Gong, H. Ian, Y.-X. Liu, C. P. Sun, and F. Nori, “Effective Hamiltonian approach to the Kerr nonlinearity in an optomechanical system,” Phys. Rev. A 80, 065801 (2009).
[Crossref]

Norman, J.

H. J. Snijders, J. A. Frey, J. Norman, H. Flayac, V. Savona, A. C. Gossard, J. E. Bowers, M. P. van Exter, D. Bouwmeester, and W. Löffler, “Observation of the unconventional photon blockade,” Phys. Rev. Lett. 121, 043601 (2018).
[Crossref]

H. Snijders, J. A. Frey, J. Norman, M. P. Bakker, E. C. Langman, A. Gossard, J. E. Bowers, M. P. Van Exter, D. Bouwmeester, and W. Löffler, “Purification of a single-photon nonlinearity,” Nat. Commun. 7, 12578 (2016).
[Crossref]

Northup, T. E.

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature (London) 436, 87–90 (2005).
[Crossref]

Novotny, L.

R. Reimann, M. Doderer, E. Hebestreit, R. Diehl, M. Frimmer, D. Windey, F. Tebbenjohanns, and L. Novotny, “GHz rotation of an optically trapped nanoparticle in vacuum,” Phys. Rev. Lett. 121, 033602 (2018).
[Crossref]

Nunnenkamp, A.

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]

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]

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011).
[Crossref]

O’Connell, R. F.

G. W. Ford, J. T. Lewis, and R. F. O’Connell, “Quantum Langevin equation,” Phys. Rev. A 37, 4419–4428 (1988).
[Crossref]

Özdemir, S. K.

J. Zhang, B. Peng, S. K. Özdemir, K. Pichler, D. O. Krimer, G. M. Zhao, F. Nori, Y.-X. Liu, S. Rotter, and L. Yang, “A phonon laser operating at an exceptional point,” Nat. Photonics 12, 479–484 (2018).
[Crossref]

H. Jing, H. Lü, S. K. Özdemir, T. Carmon, and F. Nori, “Nanoparticle sensing with a spinning resonator,” Optica 5, 1424–1430 (2018).
[Crossref]

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

Z.-P. Liu, J. Zhang, S. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-X. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
[Crossref]

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

B. Peng, S. 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]

Painter, O.

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]

Painter, O. J.

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, 043902 (2003).
[Crossref]

Paprzycka, M.

A. Miranowicz, M. Paprzycka, Y.-X. Liu, J. Bajer, and F. Nori, “Two-photon and three-photon blockades in driven nonlinear systems,” Phys. Rev. A 87, 023809 (2013).
[Crossref]

Peev, M.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[Crossref]

Peng, B.

J. Zhang, B. Peng, S. K. Özdemir, K. Pichler, D. O. Krimer, G. M. Zhao, F. Nori, Y.-X. Liu, S. Rotter, and L. Yang, “A phonon laser operating at an exceptional point,” Nat. Photonics 12, 479–484 (2018).
[Crossref]

Z.-P. Liu, J. Zhang, S. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-X. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
[Crossref]

B. Peng, S. 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]

Peng, Y. W.

W. Zhang, Z. Y. Yu, Y. M. Liu, and Y. W. Peng, “Optimal photon antibunching in a quantum-dot-bimodal-cavity system,” Phys. Rev. A 89, 043832 (2014).
[Crossref]

Petroff, P.

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. 4, 859–863 (2008).
[Crossref]

Peyronel, T.

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletić, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature (London) 488, 57–60 (2012).
[Crossref]

Pichler, H.

P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature 541, 473–480 (2017).
[Crossref]

Pichler, K.

J. Zhang, B. Peng, S. K. Özdemir, K. Pichler, D. O. Krimer, G. M. Zhao, F. Nori, Y.-X. Liu, S. Rotter, and L. Yang, “A phonon laser operating at an exceptional point,” Nat. Photonics 12, 479–484 (2018).
[Crossref]

Plenio, M. B.

M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. 70, 101–144 (1998).
[Crossref]

Pohl, T.

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletić, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature (London) 488, 57–60 (2012).
[Crossref]

Qi, M.

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[Crossref]

Rabl, P.

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013).
[Crossref]

P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107, 063601 (2011).
[Crossref]

Radulaski, M.

M. Radulaski, K. A. Fischer, K. G. Lagoudakis, J. L. Zhang, and J. Vučković, “Photon blockade in two-emitter-cavity systems,” Phys. Rev. A 96, 011801(R) (2017).
[Crossref]

Ramezani, H.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
[Crossref]

Rasoloniaina, A.

V. Huet, A. Rasoloniaina, P. Guillemé, P. Rochard, P. Féron, M. Mortier, A. Levenson, K. Bencheikh, A. Yacomotti, and Y. Dumeige, “Millisecond photon lifetime in a slow-light microcavity,” Phys. Rev. Lett. 116, 133902 (2016).
[Crossref]

Rauschenbeutel, A.

P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature 541, 473–480 (2017).
[Crossref]

M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354, 1577–1580 (2016).
[Crossref]

Reimann, R.

R. Reimann, M. Doderer, E. Hebestreit, R. Diehl, M. Frimmer, D. Windey, F. Tebbenjohanns, and L. Novotny, “GHz rotation of an optically trapped nanoparticle in vacuum,” Phys. Rev. Lett. 121, 033602 (2018).
[Crossref]

Reiter, F.

F. Reiter, T. L. Nguyen, J. P. Home, and S. F. Yelin, “Cooperative breakdown of the oscillator blockade in the Dicke model,” arXiv:1807.06026 (2018).

Robinson, J. T.

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

Rochard, P.

V. Huet, A. Rasoloniaina, P. Guillemé, P. Rochard, P. Féron, M. Mortier, A. Levenson, K. Bencheikh, A. Yacomotti, and Y. Dumeige, “Millisecond photon lifetime in a slow-light microcavity,” Phys. Rev. Lett. 116, 133902 (2016).
[Crossref]

Rotter, S.

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11–19 (2018).
[Crossref]

J. Zhang, B. Peng, S. K. Özdemir, K. Pichler, D. O. Krimer, G. M. Zhao, F. Nori, Y.-X. Liu, S. Rotter, and L. Yang, “A phonon laser operating at an exceptional point,” Nat. Photonics 12, 479–484 (2018).
[Crossref]

Rundquist, A.

K. Müller, A. Rundquist, K. A. Fischer, T. Sarmiento, K. G. Lagoudakis, Y. A. Kelaita, C. S. Muñoz, E. del Valle, F. P. Laussy, and J. Vučković, “Coherent generation of nonclassical light on chip via detuned photon blockade,” Phys. Rev. Lett. 114, 233601 (2015).
[Crossref]

A. Majumdar, M. Bajcsy, A. Rundquist, and J. Vučković, “Loss-enabled sub-Poissonian light generation in a bimodal nanocavity,” Phys. Rev. Lett. 108, 163601 (2012).
[Crossref]

Salf, F.

Sarmiento, T.

K. Müller, A. Rundquist, K. A. Fischer, T. Sarmiento, K. G. Lagoudakis, Y. A. Kelaita, C. S. Muñoz, E. del Valle, F. P. Laussy, and J. Vučković, “Coherent generation of nonclassical light on chip via detuned photon blockade,” Phys. Rev. Lett. 114, 233601 (2015).
[Crossref]

Savona, V.

H. J. Snijders, J. A. Frey, J. Norman, H. Flayac, V. Savona, A. C. Gossard, J. E. Bowers, M. P. van Exter, D. Bouwmeester, and W. Löffler, “Observation of the unconventional photon blockade,” Phys. Rev. Lett. 121, 043601 (2018).
[Crossref]

H. Flayac and V. Savona, “Nonclassical statistics from a polaritonic Josephson junction,” Phys. Rev. A 95, 043838 (2017).
[Crossref]

H. Flayac and V. Savona, “Unconventional photon blockade,” Phys. Rev. A 96, 053810 (2017).
[Crossref]

S. Ferretti, V. Savona, and D. Gerace, “Optimal antibunching in passive photonic devices based on coupled nonlinear resonators,” New J. Phys. 15, 025012 (2013).
[Crossref]

T. C. H. Liew and V. Savona, “Single photons from coupled quantum modes,” Phys. Rev. Lett. 104, 183601 (2010).
[Crossref]

V. Savona, “Unconventional photon blockade in coupled optomechanical systems,” arXiv:1302.5937 (2013).

Scarani, V.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[Crossref]

Scheucher, M.

M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354, 1577–1580 (2016).
[Crossref]

Schliesser, A.

E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature (London) 482, 63–67 (2012).
[Crossref]

Schmidt, H.

A. Imamoḡlu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467–1470 (1997).
[Crossref]

H. Schmidt and A. Imamoḡlu, “Giant Kerr nonlinearities obtained by electromagnetically induced transparency,” Opt. Lett. 21, 1936–1938 (1996).
[Crossref]

Schmidt, S.

A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Türeci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
[Crossref]

Schneeweiss, P.

P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature 541, 473–480 (2017).
[Crossref]

Schoelkopf, R.

B. Abdo, K. Sliwa, S. Shankar, M. Hatridge, L. Frunzio, R. Schoelkopf, and M. Devoret, “Josephson directional amplifier for quantum measurement of superconducting circuits,” Phys. Rev. Lett. 112, 167701 (2014).
[Crossref]

Senellart, P.

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett. 98, 113108 (2011).
[Crossref]

Shankar, S.

B. Abdo, K. Sliwa, S. Shankar, M. Hatridge, L. Frunzio, R. Schoelkopf, and M. Devoret, “Josephson directional amplifier for quantum measurement of superconducting circuits,” Phys. Rev. Lett. 112, 167701 (2014).
[Crossref]

Shen, H.

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[Crossref]

Shen, H. Z.

H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Tunable photon blockade in coupled semiconductor cavities,” Phys. Rev. A 91, 063808 (2015).
[Crossref]

Shen, Z.

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]

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. Photonics 10, 657–661 (2016).
[Crossref]

Shi, Y.

A. Y. Song, Y. Shi, Q. Lin, and S. Fan, “Direction-dependent parity-time phase transition and non-reciprocal directional amplification with dynamic gain–loss modulation,” Phys. Rev. A 99, 013824 (2019).
[Crossref]

Y. Shi, Z. Yu, and S. Fan, “Limitations of nonlinear optical isolators due to dynamic reciprocity,” Nat. Photonics 9, 388–392 (2015).
[Crossref]

Silver, J.

Silver, J. M.

Simmonds, R. W.

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]

Sirois, A. J.

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]

Sliwa, K.

B. Abdo, K. Sliwa, S. Shankar, M. Hatridge, L. Frunzio, R. Schoelkopf, and M. Devoret, “Josephson directional amplifier for quantum measurement of superconducting circuits,” Phys. Rev. Lett. 112, 167701 (2014).
[Crossref]

Snijders, H.

H. Snijders, J. A. Frey, J. Norman, M. P. Bakker, E. C. Langman, A. Gossard, J. E. Bowers, M. P. Van Exter, D. Bouwmeester, and W. Löffler, “Purification of a single-photon nonlinearity,” Nat. Commun. 7, 12578 (2016).
[Crossref]

Snijders, H. J.

H. J. Snijders, J. A. Frey, J. Norman, H. Flayac, V. Savona, A. C. Gossard, J. E. Bowers, M. P. van Exter, D. Bouwmeester, and W. Löffler, “Observation of the unconventional photon blockade,” Phys. Rev. Lett. 121, 043601 (2018).
[Crossref]

Song, A. Y.

A. Y. Song, Y. Shi, Q. Lin, and S. Fan, “Direction-dependent parity-time phase transition and non-reciprocal directional amplification with dynamic gain–loss modulation,” Phys. Rev. A 99, 013824 (2019).
[Crossref]

Sounas, D.

C. Caloz, A. Alù, S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck-Léger, “Electromagnetic nonreciprocity,” Phys. Rev. Appl. 10, 047001 (2018).
[Crossref]

Sounas, D. L.

D. L. Sounas and A. Alù, “Non-reciprocal photonics based on time modulation,” Nat. Photonics 11, 774–783 (2017).
[Crossref]

Spietz, L.

A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Türeci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
[Crossref]

Spillane, S. M.

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, 043902 (2003).
[Crossref]

Srinivasan, S. J.

A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Türeci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
[Crossref]

Stannigel, K.

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013).
[Crossref]

Stebbings, S. L.

Steffen, L.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref]

Stobbe, S.

P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature 541, 473–480 (2017).
[Crossref]

Stoltz, N.

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. 4, 859–863 (2008).
[Crossref]

Straka, I.

J. Hloušek, M. Dudka, I. Straka, and M. Ježek, “Accurate detection of arbitrary photon statistics,” arXiv:1812.02262 (2018).

Sun, C. P.

Z. R. Gong, H. Ian, Y.-X. Liu, C. P. Sun, and F. Nori, “Effective Hamiltonian approach to the Kerr nonlinearity in an optomechanical system,” Phys. Rev. A 80, 065801 (2009).
[Crossref]

Sun, F.-W.

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]

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. Photonics 10, 657–661 (2016).
[Crossref]

Szczykulska, M.

Tanas, R.

W. Leoński and R. Tanaś, “Possibility of producing the one-photon state in a kicked cavity with a nonlinear Kerr medium,” Phys. Rev. A 49, R20–R23 (1994).
[Crossref]

Tang, J. S.

L. Tang, J. S. Tang, W. D. Zhang, G. W. Lu, Y. Zhang, K. Y. Xia, and M. Xiao, “An on-chip chiral single-photon interface: isolation and unidirectional emission,” arXiv:1811.02957 (2018).

Tang, L.

L. Tang, J. S. Tang, W. D. Zhang, G. W. Lu, Y. Zhang, K. Y. Xia, and M. Xiao, “An on-chip chiral single-photon interface: isolation and unidirectional emission,” arXiv:1811.02957 (2018).

Tebbenjohanns, F.

R. Reimann, M. Doderer, E. Hebestreit, R. Diehl, M. Frimmer, D. Windey, F. Tebbenjohanns, and L. Novotny, “GHz rotation of an optically trapped nanoparticle in vacuum,” Phys. Rev. Lett. 121, 033602 (2018).
[Crossref]

Teufel, J. D.

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]

Tian, L.

L. Tian and H. J. Carmichael, “Quantum trajectory simulations of two-state behavior in an optical cavity containing one atom,” Phys. Rev. A 46, R6801–R6804 (1992).
[Crossref]

Tóth, L. D.

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]

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]

Tretyakov, S.

C. Caloz, A. Alù, S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck-Léger, “Electromagnetic nonreciprocity,” Phys. Rev. Appl. 10, 047001 (2018).
[Crossref]

Türeci, H. E.

A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Türeci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
[Crossref]

S. Ferretti, L. C. Andreani, H. E. Türeci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841 (2010).
[Crossref]

Twamley, J.

K. Y. Xia, G. W. Lu, G. W. Lin, Y. Q. Cheng, Y. P. Niu, S. Q. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90, 043802 (2014).
[Crossref]

Vahala, K. J.

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]

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, 043902 (2003).
[Crossref]

K. J. Vahala, “Optical microcavities,” Nature (London) 424, 839–846 (2003).
[Crossref]

van Exter, M. P.

H. J. Snijders, J. A. Frey, J. Norman, H. Flayac, V. Savona, A. C. Gossard, J. E. Bowers, M. P. van Exter, D. Bouwmeester, and W. Löffler, “Observation of the unconventional photon blockade,” Phys. Rev. Lett. 121, 043601 (2018).
[Crossref]

H. Snijders, J. A. Frey, J. Norman, M. P. Bakker, E. C. Langman, A. Gossard, J. E. Bowers, M. P. Van Exter, D. Bouwmeester, and W. Löffler, “Purification of a single-photon nonlinearity,” Nat. Commun. 7, 12578 (2016).
[Crossref]

Vaneph, C.

C. Vaneph, A. Morvan, G. Aiello, M. Féchant, M. Aprili, J. Gabelli, and J. Estève, “Observation of the unconventional photon blockade in the microwave domain,” Phys. Rev. Lett. 121, 043602 (2018).
[Crossref]

Vanner, M. R.

Varghese, L. T.

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[Crossref]

Verhagen, E.

E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature (London) 482, 63–67 (2012).
[Crossref]

Volz, J.

P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature 541, 473–480 (2017).
[Crossref]

M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354, 1577–1580 (2016).
[Crossref]

Vuckovic, J.

M. Radulaski, K. A. Fischer, K. G. Lagoudakis, J. L. Zhang, and J. Vučković, “Photon blockade in two-emitter-cavity systems,” Phys. Rev. A 96, 011801(R) (2017).
[Crossref]

K. Müller, A. Rundquist, K. A. Fischer, T. Sarmiento, K. G. Lagoudakis, Y. A. Kelaita, C. S. Muñoz, E. del Valle, F. P. Laussy, and J. Vučković, “Coherent generation of nonclassical light on chip via detuned photon blockade,” Phys. Rev. Lett. 114, 233601 (2015).
[Crossref]

A. Majumdar, M. Bajcsy, A. Rundquist, and J. Vučković, “Loss-enabled sub-Poissonian light generation in a bimodal nanocavity,” Phys. Rev. Lett. 108, 163601 (2012).
[Crossref]

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. 4, 859–863 (2008).
[Crossref]

Vuletic, V.

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletić, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature (London) 488, 57–60 (2012).
[Crossref]

Wallraff, A.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref]

Walls, D. F.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 1994).

Walmsley, I. A.

Wang, G.

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. Photonics 8, 524–529 (2014).
[Crossref]

Wang, H.

Q.-T. Cao, H. Wang, C.-H. Dong, H. Jing, R.-S. Liu, X. Chen, L. Ge, Q. Gong, and Y.-F. Xiao, “Experimental demonstration of spontaneous chirality in a nonlinear microresonator,” Phys. Rev. Lett. 118, 033907 (2017).
[Crossref]

X.-W. Xu, Y.-J. Zhao, H. Wang, H. Jing, and A.-X. Chen, “Nonreciprocal photon blockade via quadratic optomechanical coupling,” arXiv:1809.07596 (2018).

Wang, J.

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[Crossref]

Wang, Y. Z.

Wang, Y.-P.

Y.-P. Wang, G.-Q. Zhang, D. Zhang, T.-F. Li, C.-M. Hu, and J. Q. You, “Bistability of cavity magnon polaritons,” Phys. Rev. Lett. 120, 057202 (2018).
[Crossref]

Weiner, A. M.

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[Crossref]

Weis, S.

E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature (London) 482, 63–67 (2012).
[Crossref]

Wen, J.

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. Photonics 8, 524–529 (2014).
[Crossref]

Whittaker, J. D.

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]

Will, E.

M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354, 1577–1580 (2016).
[Crossref]

Windey, D.

R. Reimann, M. Doderer, E. Hebestreit, R. Diehl, M. Frimmer, D. Windey, F. Tebbenjohanns, and L. Novotny, “GHz rotation of an optically trapped nanoparticle in vacuum,” Phys. Rev. Lett. 121, 033602 (2018).
[Crossref]

Woodley, M. T. M.

Woods, G.

A. Imamoḡlu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467–1470 (1997).
[Crossref]

Xia, K.

S. Zhang, Y. Hu, G. Lin, Y. Niu, K. Xia, J. Gong, and S. Gong, “Thermal-motion-induced non-reciprocal quantum optical system,” Nat. Photonics 12, 744–748 (2018).
[Crossref]

Xia, K. Y.

K. Y. Xia, F. Nori, and M. Xiao, “Cavity-free optical isolators and circulators using a chiral cross-Kerr nonlinearity,” Phys. Rev. Lett. 121, 203602 (2018).
[Crossref]

K. Y. Xia, G. W. Lu, G. W. Lin, Y. Q. Cheng, Y. P. Niu, S. Q. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90, 043802 (2014).
[Crossref]

L. Tang, J. S. Tang, W. D. Zhang, G. W. Lu, Y. Zhang, K. Y. Xia, and M. Xiao, “An on-chip chiral single-photon interface: isolation and unidirectional emission,” arXiv:1811.02957 (2018).

Xiao, M.

K. Y. Xia, F. Nori, and M. Xiao, “Cavity-free optical isolators and circulators using a chiral cross-Kerr nonlinearity,” Phys. Rev. Lett. 121, 203602 (2018).
[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. Photonics 8, 524–529 (2014).
[Crossref]

L. Tang, J. S. Tang, W. D. Zhang, G. W. Lu, Y. Zhang, K. Y. Xia, and M. Xiao, “An on-chip chiral single-photon interface: isolation and unidirectional emission,” arXiv:1811.02957 (2018).

Xiao, Y.-F.

Q.-T. Cao, H. Wang, C.-H. Dong, H. Jing, R.-S. Liu, X. Chen, L. Ge, Q. Gong, and Y.-F. Xiao, “Experimental demonstration of spontaneous chirality in a nonlinear microresonator,” Phys. Rev. Lett. 118, 033907 (2017).
[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. Photonics 10, 657–661 (2016).
[Crossref]

Xie, H.

H. Xie, G.-W. Lin, X. Chen, Z.-H. Chen, and X.-M. Lin, “Single-photon nonlinearities in a strongly driven optomechanical system with quadratic coupling,” Phys. Rev. A 93, 063860 (2016).
[Crossref]

Xu, X.-W.

X.-W. Xu and Y. Li, “Strong photon antibunching of symmetric and antisymmetric modes in weakly nonlinear photonic molecules,” Phys. Rev. A 90, 033809 (2014).
[Crossref]

X.-W. Xu and Y.-J. Li, “Antibunching photons in a cavity coupled to an optomechanical system,” J. Phys. B 46, 035502 (2013).
[Crossref]

X.-W. Xu, Y.-J. Zhao, H. Wang, H. Jing, and A.-X. Chen, “Nonreciprocal photon blockade via quadratic optomechanical coupling,” arXiv:1809.07596 (2018).

Xu, Z.

J. Ahn, Z. Xu, J. Bang, Y.-H. Deng, T. M. Hoang, Q. Han, R.-M. Ma, and T. Li, “Optically levitated nanodumbbell torsion balance and GHz nanomechanical rotor,” Phys. Rev. Lett. 121, 033603 (2018).
[Crossref]

Xuan, Y.

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[Crossref]

Xuereb, A.

S. Barzanjeh, M. Aquilina, and A. Xuereb, “Manipulating the flow of thermal noise in quantum devices,” Phys. Rev. Lett. 120, 060601 (2018).
[Crossref]

Yacomotti, A.

V. Huet, A. Rasoloniaina, P. Guillemé, P. Rochard, P. Féron, M. Mortier, A. Levenson, K. Bencheikh, A. Yacomotti, and Y. Dumeige, “Millisecond photon lifetime in a slow-light microcavity,” Phys. Rev. Lett. 116, 133902 (2016).
[Crossref]

Yang, C.

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. Photonics 8, 524–529 (2014).
[Crossref]

Yang, J. K.

V. V. Konotop, J. K. Yang, and D. A. Zezyulin, “Nonlinear waves in PT-symmetric systems,” Rev. Mod. Phys. 88, 035002 (2016).
[Crossref]

Yang, L.

J. Zhang, B. Peng, S. K. Özdemir, K. Pichler, D. O. Krimer, G. M. Zhao, F. Nori, Y.-X. Liu, S. Rotter, and L. Yang, “A phonon laser operating at an exceptional point,” Nat. Photonics 12, 479–484 (2018).
[Crossref]

Z.-P. Liu, J. Zhang, S. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-X. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
[Crossref]

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

B. Peng, S. 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]

Yelin, S. F.

F. Reiter, T. L. Nguyen, J. P. Home, and S. F. Yelin, “Cooperative breakdown of the oscillator blockade in the Dicke model,” arXiv:1807.06026 (2018).

Yi, X. X.

H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Tunable photon blockade in coupled semiconductor cavities,” Phys. Rev. A 91, 063808 (2015).
[Crossref]

You, J. Q.

Y.-P. Wang, G.-Q. Zhang, D. Zhang, T.-F. Li, C.-M. Hu, and J. Q. You, “Bistability of cavity magnon polaritons,” Phys. Rev. Lett. 120, 057202 (2018).
[Crossref]

Yu, Z.

Y. Shi, Z. Yu, and S. Fan, “Limitations of nonlinear optical isolators due to dynamic reciprocity,” Nat. Photonics 9, 388–392 (2015).
[Crossref]

Yu, Z. Y.

W. Zhang, Z. Y. Yu, Y. M. Liu, and Y. W. Peng, “Optimal photon antibunching in a quantum-dot-bimodal-cavity system,” Phys. Rev. A 89, 043832 (2014).
[Crossref]

Zezyulin, D. A.

V. V. Konotop, J. K. Yang, and D. A. Zezyulin, “Nonlinear waves in PT-symmetric systems,” Rev. Mod. Phys. 88, 035002 (2016).
[Crossref]

Zhai, C.

C. Zhai, R. Huang, B. Li, H. Jing, and L.-M. Kuang, “Mechanical engineering of photon blockades in a cavity optomechanical system,” arXiv:1901.07654 (2019).

Zhang, D.

Y.-P. Wang, G.-Q. Zhang, D. Zhang, T.-F. Li, C.-M. Hu, and J. Q. You, “Bistability of cavity magnon polaritons,” Phys. Rev. Lett. 120, 057202 (2018).
[Crossref]

Zhang, G.-Q.

Y.-P. Wang, G.-Q. Zhang, D. Zhang, T.-F. Li, C.-M. Hu, and J. Q. You, “Bistability of cavity magnon polaritons,” Phys. Rev. Lett. 120, 057202 (2018).
[Crossref]

Zhang, H.

Zhang, J.

J. Zhang, B. Peng, S. K. Özdemir, K. Pichler, D. O. Krimer, G. M. Zhao, F. Nori, Y.-X. Liu, S. Rotter, and L. Yang, “A phonon laser operating at an exceptional point,” Nat. Photonics 12, 479–484 (2018).
[Crossref]

Z.-P. Liu, J. Zhang, S. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-X. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
[Crossref]

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

Zhang, J. L.

M. Radulaski, K. A. Fischer, K. G. Lagoudakis, J. L. Zhang, and J. Vučković, “Photon blockade in two-emitter-cavity systems,” Phys. Rev. A 96, 011801(R) (2017).
[Crossref]

Zhang, S.

G. Enzian, M. Szczykulska, J. Silver, L. Del Bino, S. Zhang, I. A. Walmsley, P. Del’Haye, and M. R. Vanner, “Observation of Brillouin optomechanical strong coupling with an 11  GHz mechanical mode,” Optica 6, 7–14 (2019).
[Crossref]

S. Zhang, Y. Hu, G. Lin, Y. Niu, K. Xia, J. Gong, and S. Gong, “Thermal-motion-induced non-reciprocal quantum optical system,” Nat. Photonics 12, 744–748 (2018).
[Crossref]

Zhang, W.

W. Zhang, Z. Y. Yu, Y. M. Liu, and Y. W. Peng, “Optimal photon antibunching in a quantum-dot-bimodal-cavity system,” Phys. Rev. A 89, 043832 (2014).
[Crossref]

Zhang, W. D.

L. Tang, J. S. Tang, W. D. Zhang, G. W. Lu, Y. Zhang, K. Y. Xia, and M. Xiao, “An on-chip chiral single-photon interface: isolation and unidirectional emission,” arXiv:1811.02957 (2018).

Zhang, Y.

L. Tang, J. S. Tang, W. D. Zhang, G. W. Lu, Y. Zhang, K. Y. Xia, and M. Xiao, “An on-chip chiral single-photon interface: isolation and unidirectional emission,” arXiv:1811.02957 (2018).

Zhang, Y.-L.

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]

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. Photonics 10, 657–661 (2016).
[Crossref]

Zhao, G. M.

J. Zhang, B. Peng, S. K. Özdemir, K. Pichler, D. O. Krimer, G. M. Zhao, F. Nori, Y.-X. Liu, S. Rotter, and L. Yang, “A phonon laser operating at an exceptional point,” Nat. Photonics 12, 479–484 (2018).
[Crossref]

Zhao, X.

Zhao, Y.-J.

X.-W. Xu, Y.-J. Zhao, H. Wang, H. Jing, and A.-X. Chen, “Nonreciprocal photon blockade via quadratic optomechanical coupling,” arXiv:1809.07596 (2018).

Zhou, F.

F. Zhou, D.-G. Lai, and J.-Q. Liao, “Photon blockade effect in a coupled cavity system,” arXiv:1803.06642 (2018).

Zhou, Y. H.

H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Tunable photon blockade in coupled semiconductor cavities,” Phys. Rev. A 91, 063808 (2015).
[Crossref]

Zoller, P.

P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature 541, 473–480 (2017).
[Crossref]

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013).
[Crossref]

C. W. Gardiner and P. Zoller, Quantum Noise (Springer, 2000).

Zou, C.-L.

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]

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. Photonics 10, 657–661 (2016).
[Crossref]

Zou, X. B.

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]

Zou, X.-B.

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. Photonics 10, 657–661 (2016).
[Crossref]

Adv. Chem. Phys. (1)

W. Leoński and A. Miranowicz, “Quantum-optical states in finite-dimensional Hilbert space. II. State generation,” Adv. Chem. Phys. 119, 155–193 (2003).
[Crossref]

Appl. Phys. Lett. (1)

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett. 98, 113108 (2011).
[Crossref]

Comput. Phys. Commun. (1)

J. R. Johansson, P. D. Nation, and F. Nori, “Qutip 2: a Python framework for the dynamics of open quantum systems,” Comput. Phys. Commun. 184, 1234–1240 (2013).
[Crossref]

J. Opt. B (1)

W. Leoński and A. Miranowicz, “Kerr nonlinear coupler and entanglement,” J. Opt. B 6, S37–S42 (2004).
[Crossref]

J. Phys. B (2)

A. Miranowicz and W. Leoński, “Two-mode optical state truncation and generation of maximally entangled states in pumped nonlinear couplers,” J. Phys. B 39, 1683–1700 (2006).
[Crossref]

X.-W. Xu and Y.-J. Li, “Antibunching photons in a cavity coupled to an optomechanical system,” J. Phys. B 46, 035502 (2013).
[Crossref]

Nat. Commun. (3)

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]

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]

H. Snijders, J. A. Frey, J. Norman, M. P. Bakker, E. C. Langman, A. Gossard, J. E. Bowers, M. P. Van Exter, D. Bouwmeester, and W. Löffler, “Purification of a single-photon nonlinearity,” Nat. Commun. 7, 12578 (2016).
[Crossref]

Nat. Photonics (6)

J. Zhang, B. Peng, S. K. Özdemir, K. Pichler, D. O. Krimer, G. M. Zhao, F. Nori, Y.-X. Liu, S. Rotter, and L. Yang, “A phonon laser operating at an exceptional point,” Nat. Photonics 12, 479–484 (2018).
[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. Photonics 10, 657–661 (2016).
[Crossref]

Y. Shi, Z. Yu, and S. Fan, “Limitations of nonlinear optical isolators due to dynamic reciprocity,” Nat. Photonics 9, 388–392 (2015).
[Crossref]

S. Zhang, Y. Hu, G. Lin, Y. Niu, K. Xia, J. Gong, and S. Gong, “Thermal-motion-induced non-reciprocal quantum optical system,” Nat. Photonics 12, 744–748 (2018).
[Crossref]

D. L. Sounas and A. Alù, “Non-reciprocal photonics based on time modulation,” Nat. Photonics 11, 774–783 (2017).
[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. Photonics 8, 524–529 (2014).
[Crossref]

Nat. Phys. (3)

B. Peng, S. 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]

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11–19 (2018).
[Crossref]

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. 4, 859–863 (2008).
[Crossref]

Nature (1)

P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature 541, 473–480 (2017).
[Crossref]

Nature (London) (6)

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature (London) 558, 569–572 (2018).
[Crossref]

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature (London) 436, 87–90 (2005).
[Crossref]

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletić, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature (London) 488, 57–60 (2012).
[Crossref]

E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature (London) 482, 63–67 (2012).
[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]

K. J. Vahala, “Optical microcavities,” Nature (London) 424, 839–846 (2003).
[Crossref]

New J. Phys. (1)

S. Ferretti, V. Savona, and D. Gerace, “Optimal antibunching in passive photonic devices based on coupled nonlinear resonators,” New J. Phys. 15, 025012 (2013).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Optica (3)

Photon. Res. (1)

Phys. Rep. (1)

X. Gu, A. F. Kockum, A. Miranowicz, Y.-X. Liu, and F. Nori, “Microwave photonics with superconducting quantum circuits,” Phys. Rep. 718–719, 1–102 (2017).
[Crossref]

Phys. Rev. A (19)

L. Tian and H. J. Carmichael, “Quantum trajectory simulations of two-state behavior in an optical cavity containing one atom,” Phys. Rev. A 46, R6801–R6804 (1992).
[Crossref]

W. Leoński and R. Tanaś, “Possibility of producing the one-photon state in a kicked cavity with a nonlinear Kerr medium,” Phys. Rev. A 49, R20–R23 (1994).
[Crossref]

S. Ferretti, L. C. Andreani, H. E. Türeci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841 (2010).
[Crossref]

J.-Q. Liao and C. K. Law, “Correlated two-photon transport in a one-dimensional waveguide side-coupled to a nonlinear cavity,” Phys. Rev. A 82, 053836 (2010).
[Crossref]

A. Miranowicz, M. Paprzycka, Y.-X. Liu, J. Bajer, and F. Nori, “Two-photon and three-photon blockades in driven nonlinear systems,” Phys. Rev. A 87, 023809 (2013).
[Crossref]

J.-Q. Liao and F. Nori, “Photon blockade in quadratically coupled optomechanical systems,” Phys. Rev. A 88, 023853 (2013).
[Crossref]

H. Xie, G.-W. Lin, X. Chen, Z.-H. Chen, and X.-M. Lin, “Single-photon nonlinearities in a strongly driven optomechanical system with quadratic coupling,” Phys. Rev. A 93, 063860 (2016).
[Crossref]

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013).
[Crossref]

M. Bamba, A. Imamoğlu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802(R) (2011).
[Crossref]

X.-W. Xu and Y. Li, “Strong photon antibunching of symmetric and antisymmetric modes in weakly nonlinear photonic molecules,” Phys. Rev. A 90, 033809 (2014).
[Crossref]

W. Zhang, Z. Y. Yu, Y. M. Liu, and Y. W. Peng, “Optimal photon antibunching in a quantum-dot-bimodal-cavity system,” Phys. Rev. A 89, 043832 (2014).
[Crossref]

H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Tunable photon blockade in coupled semiconductor cavities,” Phys. Rev. A 91, 063808 (2015).
[Crossref]

H. Flayac and V. Savona, “Unconventional photon blockade,” Phys. Rev. A 96, 053810 (2017).
[Crossref]

H. Flayac and V. Savona, “Nonclassical statistics from a polaritonic Josephson junction,” Phys. Rev. A 95, 043838 (2017).
[Crossref]

K. Y. Xia, G. W. Lu, G. W. Lin, Y. Q. Cheng, Y. P. Niu, S. Q. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90, 043802 (2014).
[Crossref]

A. Y. Song, Y. Shi, Q. Lin, and S. Fan, “Direction-dependent parity-time phase transition and non-reciprocal directional amplification with dynamic gain–loss modulation,” Phys. Rev. A 99, 013824 (2019).
[Crossref]

G. W. Ford, J. T. Lewis, and R. F. O’Connell, “Quantum Langevin equation,” Phys. Rev. A 37, 4419–4428 (1988).
[Crossref]

M. Radulaski, K. A. Fischer, K. G. Lagoudakis, J. L. Zhang, and J. Vučković, “Photon blockade in two-emitter-cavity systems,” Phys. Rev. A 96, 011801(R) (2017).
[Crossref]

Z. R. Gong, H. Ian, Y.-X. Liu, C. P. Sun, and F. Nori, “Effective Hamiltonian approach to the Kerr nonlinearity in an optomechanical system,” Phys. Rev. A 80, 065801 (2009).
[Crossref]

Phys. Rev. Appl. (3)

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

Y. Jiang, S. Maayani, T. Carmon, F. Nori, and H. Jing, “Nonreciprocal phonon laser,” Phys. Rev. Appl. 10, 064037 (2018).
[Crossref]

C. Caloz, A. Alù, S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck-Léger, “Electromagnetic nonreciprocity,” Phys. Rev. Appl. 10, 047001 (2018).
[Crossref]

Phys. Rev. Lett. (26)

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
[Crossref]

K. Y. Xia, F. Nori, and M. Xiao, “Cavity-free optical isolators and circulators using a chiral cross-Kerr nonlinearity,” Phys. Rev. Lett. 121, 203602 (2018).
[Crossref]

B. Abdo, K. Sliwa, S. Shankar, M. Hatridge, L. Frunzio, R. Schoelkopf, and M. Devoret, “Josephson directional amplifier for quantum measurement of superconducting circuits,” Phys. Rev. Lett. 112, 167701 (2014).
[Crossref]

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]

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

Y.-P. Wang, G.-Q. Zhang, D. Zhang, T.-F. Li, C.-M. Hu, and J. Q. You, “Bistability of cavity magnon polaritons,” Phys. Rev. Lett. 120, 057202 (2018).
[Crossref]

Z.-P. Liu, J. Zhang, S. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-X. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
[Crossref]

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, 043902 (2003).
[Crossref]

S. Barzanjeh, M. Aquilina, and A. Xuereb, “Manipulating the flow of thermal noise in quantum devices,” Phys. Rev. Lett. 120, 060601 (2018).
[Crossref]

R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121, 153601 (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]

A. Majumdar, M. Bajcsy, A. Rundquist, and J. Vučković, “Loss-enabled sub-Poissonian light generation in a bimodal nanocavity,” Phys. Rev. Lett. 108, 163601 (2012).
[Crossref]

Q.-T. Cao, H. Wang, C.-H. Dong, H. Jing, R.-S. Liu, X. Chen, L. Ge, Q. Gong, and Y.-F. Xiao, “Experimental demonstration of spontaneous chirality in a nonlinear microresonator,” Phys. Rev. Lett. 118, 033907 (2017).
[Crossref]

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

H. J. Snijders, J. A. Frey, J. Norman, H. Flayac, V. Savona, A. C. Gossard, J. E. Bowers, M. P. van Exter, D. Bouwmeester, and W. Löffler, “Observation of the unconventional photon blockade,” Phys. Rev. Lett. 121, 043601 (2018).
[Crossref]

C. Vaneph, A. Morvan, G. Aiello, M. Féchant, M. Aprili, J. Gabelli, and J. Estève, “Observation of the unconventional photon blockade in the microwave domain,” Phys. Rev. Lett. 121, 043602 (2018).
[Crossref]

T. C. H. Liew and V. Savona, “Single photons from coupled quantum modes,” Phys. Rev. Lett. 104, 183601 (2010).
[Crossref]

P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107, 063601 (2011).
[Crossref]

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011).
[Crossref]

A. Imamoḡlu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467–1470 (1997).
[Crossref]

K. Müller, A. Rundquist, K. A. Fischer, T. Sarmiento, K. G. Lagoudakis, Y. A. Kelaita, C. S. Muñoz, E. del Valle, F. P. Laussy, and J. Vučković, “Coherent generation of nonclassical light on chip via detuned photon blockade,” Phys. Rev. Lett. 114, 233601 (2015).
[Crossref]

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref]

A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Türeci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
[Crossref]

R. Reimann, M. Doderer, E. Hebestreit, R. Diehl, M. Frimmer, D. Windey, F. Tebbenjohanns, and L. Novotny, “GHz rotation of an optically trapped nanoparticle in vacuum,” Phys. Rev. Lett. 121, 033602 (2018).
[Crossref]

J. Ahn, Z. Xu, J. Bang, Y.-H. Deng, T. M. Hoang, Q. Han, R.-M. Ma, and T. Li, “Optically levitated nanodumbbell torsion balance and GHz nanomechanical rotor,” Phys. Rev. Lett. 121, 033603 (2018).
[Crossref]

V. Huet, A. Rasoloniaina, P. Guillemé, P. Rochard, P. Féron, M. Mortier, A. Levenson, K. Bencheikh, A. Yacomotti, and Y. Dumeige, “Millisecond photon lifetime in a slow-light microcavity,” Phys. Rev. Lett. 116, 133902 (2016).
[Crossref]

Phys. Rev. X (1)

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

Phys. Usp. (1)

G. B. Malykin, “The Sagnac effect: correct and incorrect explanations,” Phys. Usp. 43, 1229–1252 (2000).
[Crossref]

Rep. Prog. Phys. (1)

I. Buluta, S. Ashhab, and F. Nori, “Natural and artificial atoms for quantum computation,” Rep. Prog. Phys. 74, 104401 (2011).
[Crossref]

Rev. Mod. Phys. (5)

I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85, 299–366 (2013).
[Crossref]

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[Crossref]

V. V. Konotop, J. K. Yang, and D. A. Zezyulin, “Nonlinear waves in PT-symmetric systems,” Rev. Mod. Phys. 88, 035002 (2016).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. 70, 101–144 (1998).
[Crossref]

Science (2)

M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354, 1577–1580 (2016).
[Crossref]

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[Crossref]

Other (10)

L. Tang, J. S. Tang, W. D. Zhang, G. W. Lu, Y. Zhang, K. Y. Xia, and M. Xiao, “An on-chip chiral single-photon interface: isolation and unidirectional emission,” arXiv:1811.02957 (2018).

X.-W. Xu, Y.-J. Zhao, H. Wang, H. Jing, and A.-X. Chen, “Nonreciprocal photon blockade via quadratic optomechanical coupling,” arXiv:1809.07596 (2018).

C. Zhai, R. Huang, B. Li, H. Jing, and L.-M. Kuang, “Mechanical engineering of photon blockades in a cavity optomechanical system,” arXiv:1901.07654 (2019).

A. Miranowicz, W. Leoński, and N. Imoto, “Quantum-optical states in finite-dimensional Hilbert space. I. General formalism,” in Modern Nonlinear Optics (Wiley, 2001), Vol. 119, pp. 195–213.

V. Savona, “Unconventional photon blockade in coupled optomechanical systems,” arXiv:1302.5937 (2013).

F. Zhou, D.-G. Lai, and J.-Q. Liao, “Photon blockade effect in a coupled cavity system,” arXiv:1803.06642 (2018).

F. Reiter, T. L. Nguyen, J. P. Home, and S. F. Yelin, “Cooperative breakdown of the oscillator blockade in the Dicke model,” arXiv:1807.06026 (2018).

J. Hloušek, M. Dudka, I. Straka, and M. Ježek, “Accurate detection of arbitrary photon statistics,” arXiv:1812.02262 (2018).

C. W. Gardiner and P. Zoller, Quantum Noise (Springer, 2000).

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 1994).

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Figures (6)

Fig. 1.
Fig. 1. Nonreciprocal UPB in a coupled-resonator system. Spinning the OM (Kerr-type) resonator results in different Fizeau drag ΔF for the counter-circulating whispering-gallery modes of the resonator. (a) By driving the system from the left-hand side, the direct excitation from state |1,0 to state |2,0 (red dotted arrow) will be forbidden by destructive quantum interference with the other paths drawn by green arrows, leading to photon antibunching. (b) Photon bunching occurs when the system is driven from the right side, due to lack of complete destructive quantum interference between the indicated levels (drawn by crossed green dotted arrows). Here, δ=g2/ωm is the energy shift induced by the OM nonlinearity.
Fig. 2.
Fig. 2. Correlation function gL(2)(0) versus optical detuning Δ/κ (in units of cavity loss rate κL=κR=κ) with (a) Ω=0 and (b) Ω=12kHz, which is found numerically (solid curves) and analytically (dotted curve). The PB can be generated (red curves) or suppressed (blue curves) for different driving directions, which can be seen more clearly in panel (c). The other parameters are g/κ=0.63, ωm/κ=10 [91], J/κ=3, T=0.1mK (case 1), and g/κ=0.1 [28], ωm/κ=30 [92], J/κ=20, T=1mK (case 2).
Fig. 3.
Fig. 3. Correlation function gL(2)(0) versus optical detuning Δ/κ (in units of cavity loss rate κL=κR=κ) at various angular velocities Ω upon driving the device from (a) the right-hand side or (b) the left-hand side. The dashed curves show our approximate analytical results, given in Eq. (12), whereas the solid curves are our numerical solutions. The other parameters are the same as those in Fig. 2 (case 1).
Fig. 4.
Fig. 4. Correlation function gL(2)(0) in logarithmic scale [i.e., log10gL(2)(0)] versus (a) radiation-pressure coupling g/κ (in units of cavity loss rate κ=κL=κR) and optical detuning Δ/κ, and (b) coupling strength of the resonators J/κ and radiation-pressure coupling g/κ for optical detuning of Δ/κ=0.05. The angular velocity is Ω=12kHz and the white dashed curve corresponds to gL(2)(0)=1. The other parameters are the same as those in Fig. 3.
Fig. 5.
Fig. 5. Correlation function gL(2)(0) versus optical detuning Δ/κ (in units of cavity loss rate κL=κR=κ) with varied mean thermal phonon numbers nth for various angular velocities Ω, and the resulting Fizeau shifts ΔF. The other parameters are the same as those in Fig. 4.
Fig. 6.
Fig. 6. (a) Correlation function gL(2)(0) versus effective temperature T of the environment of the mechanical resonator for three values of Fizeau shift ΔF (ΔF>0, ΔF=0, and ΔF<0) for optimal values of Δopt and gopt. The other parameters are set the same as in case 2 in Fig. 2. Also shown is the correlation function gL(2)(0) versus T for various values of (b) spinning frequency, (c) mechanical decay, and (d) cavity decay, assuming the device is driven from the left-hand side and optical detuning is fixed at the optimal values.

Equations (55)

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ΔF=±nrΩωRc(11n2λndndλ)=±ηΩ,
H=ΔLaLaL+(ΔR+ΔF)aRaR+ωmbb+J(aLaR+aRaL)+gaRaR(b+b)+iϵd(aLaL),
ddtq=ωmp,ddtp=ωmqgbaRaRγm2p+ξ,ddtaL=(κL2+iΔL)aLiJaR+ϵd+κLaL,in,ddtaR=(κR2+iΔR)aRiJaLigbqaR+κLaR,in,
ξ(t)ξ(t)=12πdωeiω(tt)Γm(ω),
Γm(ω)=ωγm2ωm[1+coth(ω2kBT)],
aK,in(t)aK,in(t)=0,aK,in(t)aK,in(t)=δ(tt),
ddtδq=ωmδp,ddtδp=ωmδqgb(β*δaR+βδaR)γm2δp+ξ,ddtδaL=(κL2+iΔL)δaLiJδaR+κLaL,in,ddtδaR=(κR2+iΔR)δaRiJδaLigbqsδaRigbβδq+κRaR,in.
δaL(ω)=E(ω)aL,in(ω)+F(ω)aL,in(ω)+G(ω)aR,in(ω)+H(ω)aR,in(ω)+Q(ω)ξ(ω),
E(ω)=κLA1(ω)A5(ω),F(ω)=κLA2(ω)A5(ω),G(ω)=κRA3(ω)A5(ω),H(ω)=κRA4(ω)A5(ω),Q(ω)=igbχ(ω)ωmA5(ω)[βA3(ω)+β*A4(ω)],
A1(ω)=[(κR2+iω)2+ΔR2]V1(ω)gb4|β|4[χ(ω)ωm]2V1(ω)+J2V2+,A2(ω)=iJ2gb2β2χ(ω)ωm,A3(ω)=iJV1(ω)V2iJ3,A4(ω)=Jgb2β2χ(ω)ωmV1(ω),A5(ω)=V1+A1(ω)+iJA3(ω),
ΔR=ΔR+gbqsgb2|β|2χ(ω),χ(ω)=ωm2/(ωm2ω2+iωγm2),V1±(ω)=κL2±i(ΔLω),V2±(ω)=κR2±i(ΔRω).
gL(2)(0)=|α|4+4|α|2R1+2Re[α*2R2]+R3(|α|2+R1)2,
δaL±(t)δaL(t)=12π+XaL±aLdω,
δaL+(t)=δaL(t),δaL(t)=δaL(t),andXaLaL=|Q(ω)|2Γm(ω)+|F(ω)|2+|H(ω)|2,XaLaL=Q(ω)Q(ω)Γm(ω)+E(ω)F(ω)+G(ω)H(ω).
ρ˙=1i[H,ρ]+κL2L[aL](ρ)+κR2L[aR](ρ)+γm2(n¯m+1)L[b](ρ)+γm2n¯mL[b](ρ),
|1,02ϵd|2,0,|1,0J|0,1ϵd|1,12J|2,0.
|φ=C00|0,0+C10|1,0+C01|0,1+C20|2,0+C11|1,1+C02|0,2,
Δopta3+sgn(E)λ1λ24a4,gopt=ωm[Δopt(4Δopt2+5κ2)+ΔFλ3]2(2J2κ2)+2ΔFλ4,
H=H0+Hin+Hdr,H0=ωLaLaL+(ωR+ΔF)aRaR+ωmbb,Hin=J(aLaR+aRaL)+gaRaR(b+b),Hdr=iϵd(aLeiωdtaLeiωdt),
Heff=UHU=ωLaLaL+(ωR+ΔF)aRaRδ(aRaR)2+J[aLaReδ(bb)+aLaReδ(bb)]+iϵd(aLeiωdtaLeiωdt),
Heff=ωLaLaL+(ωR+ΔF)aRaRδ(aRaR)2+J(aLaR+aLaR)+iϵd(aLeiωdtaLeiωdt).
0=(κL2+iΔL)α+iJβϵd,0=[κR2+i(ΔR+gbqs)]βiJα,0=ωmqsgb|β|2.
b3qs3+b2qs2+b1qs+b0=0,
b0=gbJ2ϵd2,b1=ωm(κLκR4+J2)2+ωm(κLΔR2+κRΔL2)2ωmΔLΔR(κLκR2+2J2ΔLΔR),b2=2ωmgb[κL2ΔR4+ΔL(ΔLΔRJ2)],b3=ωmgb2(κL24+ΔL2).
ddtδq=ωmδp,ddtδp=ωmδqgb(β*δaR+βδaR)γm2δp+ξ,ddtδaL=(κL2+iΔL)δaLiJδaR+κLaL,in,ddtδaR=(κR2+iΔR)δaRiJδaLigbqsδaRigbβδq+κRaR,in,
iωδaL(ω)=(κL2+iΔL)δaL(ω)iJδaR(ω)+κLaL,in(ω),iωδaR(ω)=(κR2+iΔR)δaL(ω)iJδaR(ω)igbβδq(ω)+κRaR,in(ω),iωδq(ω)=ωmδp(ω),iωδp(ω)=ωmδq(ω)gb[β*δaR(ω)+βδaR(ω)]γm2δp(ω)+ξ(ω),
δq(ω)=gbβ*χ(ω)δaR(ω)gbβχ(ω)δaR(ω)+χ(ω)ξ(ω),
χ(ω)=ωmωm2ω2+iωγm/2.
M(ω)δaR(ω)=igb2β2χ(ω)δaR(ω)igbβχ(ω)ξ(ω)iJδL(ω)+κRaR,in(ω),
M(ω)=κR2+iω+iΔRi|β|2gb2χ(ω).
iωδaL(ω)=(κL2iΔL)δaL(ω)+iJδaR(ω)+κLaL,in(ω),iωδaR(ω)=(κR2iΔR)δaR(ω)+iJδaR(ω)+igbβδq(ω)+κRaR,in(ω),iωδq(ω)=ωmδp(ω),iωδp(ω)=ωmδq(ω)gb[βδaR(ω)+β*δaR(ω)]γm2δp+ξ(ω),
N(ω)δaR(ω)=igb2β*2χ(ω)δaR(ω)+igbβ*χ(ω)ξ(ω)+iJδaL(ω)+κRaR,in(ω),
N(ω)=κR2+iωiΔR+i|β|2gb2χ(ω).
V(ω)δaL(ω)=iJδaR(ω)+κLaL,in(ω),
T(ω)δaR(ω)=iχ(ω)gb2β*2V(ω)δaR(ω)+iχ(ω)gbβ*V(ω)ξ(ω)+iJκLaL,in(ω)+κRV(ω)aR,in(ω),
FR(ω)δaR(ω)=χ2(ω)gb3β|β|2V(ω)ξ(ω)igbβχ(ω)T(ω)ξ(ω)Jgb2β2χ(ω)κLaL,in+igb2β2χ(ω)κRV(ω)aR,in(ω)iJT(ω)aL,inκRT(ω)aR,in,
FL(ω)δaL(ω)=iJχ2(ω)gb3β|β|2V(ω)ξ(ω)gbβχ(ω)JT(ω)ξ(ω)+iJ2gb2β2χ(ω)κLaL,in+Jgb2β2χ(ω)κRV(ω)aR,in(ω)iJκRT(ω)aR,inκL[M(ω)T(ω)U(ω)]aL,in,
FL(ω)=[M(ω)T(ω)U(ω)]V1(ω)+J2T(ω),U(ω)=χ2(ω)gb4|β|4(iω+κL2iΔL),V1(ω)=κL2+iω+iΔL.
δaL(ω)=E(ω)aL,in(ω)+F(ω)aL,in(ω)+G(ω)aR,in(ω)+H(ω)aR,in(ω)+Q(ω)ξ(ω).
δaL(ω)=E*(ω)aL,in(ω)+F*(ω)aL,in(ω)+G*(ω)aR,in(ω)+H*(ω)aR,in(ω)+Q*(ω)ξ(ω).
aL,in(ω)aL,in(ω)=12πaL,in(t)eiωtdt×12πaL,in(t)eiωtdt=δ(ω+ω),
aR,in(ω)aR,in(ω)=δ(ω+ω).
H=(ΔLiκL2)aLaL+(ΔRiκR2)aRaR+(ωmiγm2)bb+J(aLaR+aRaL)δ(aRaR)2+iϵd(aLaL),
|φ=C00|0,0+C10|1,0+C01|0,1+C20|2,0+C11|1,1+C02|0,2.
id|φdt=H|φ,
HC00|0,0=iϵdC00|1,0,HC10|1,0=δLC10|1,0+JC10|0,1+iϵdC10(2|2,0|0,0),HC01|0,1=δRC01|0,1+JC01|1,0+iϵdC01|1,1,HC20|2,0=2δLC20|2,0+2JC20|1,1+iϵdC20(3|3,02|1,0),HC11|1,1=δLC11|1,1+δRC11|1,1+2JC11(|2,0+|0,2)+iϵdC11(2|2,1|0,1),HC02|0,2=2δRC02|0,22δC02|0,2+2JC02(|1,1+iϵdC02|1,2,
C00t=ϵdC10,iC10t=δLC10+JC012iϵdC20,iC01t=(δRδ)C01+JC10iϵdC11,iC11t=δLC11+(δRδ)C11+2J(C02+C20)+iϵdC01,iC02t=2(δRδ)C02+2JC112δC02,iC20t=2(δRδ)C20+2JC11+2iϵdC10.
0=δLC10+JC01+iϵdC00,0=δRC01+JC10,
0=2δLC20+2JC11+i2ϵdC10,0=(δL+δR)C11+2JC20+2JC02+iϵdC01,0=2(δRδ)C02+2JC11,
0=κ2(2δ6Δ5ΔF2)+4Δ2(2Δ2δ5δΔF2)+4ΔF(4ΔΔF3δΔδΔF+ΔF2)4J2δ,0=8δΔ12Δ2+κ2+ΔF(6δ20Δ8ΔF).
a4Δ4+a3Δ3+a2Δ2+a1Δ+a0=0,
a0=κ(4J210ΔF2)(κ28ΔF2)2κ(κ444ΔF4),a1=8ΔF(6ΔF2κ+10J2κ+3),a2=8κ(2κ2+6J2+13ΔF2),a3=96ΔFκ,a4=32κ,
Δopta3+sgn(E)λ1λ24a4,gopt=ωm[Δopt(4Δopt2+5κ2)+ΔFλ3]2(2J2κ2)+2ΔFλ4,
λ1=D+z13+z233,λ2=2Dz13z23+z333,λ3=20Δopt28ΔoptΔF4ΔF2+5κ2,λ4=10Δopt2+3Δopt+2ΔF,
sgn(E)={1(E>0),1(E<0),z1,2=AD+3B±B24AC2,z3=D2D(z13+z23)+(z13+z23)23A,A=D23F,B=DF9E2,C=F23DE2,D=3a328a4a2,E=a33+4a4a3a28a42a1,F=3a34+16a42a2216a4a32a2+16a42a3a164a43a0.

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