Photon blockade (PB) ^[1–5], 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 ^[6–8]. In experiments, PB has been demonstrated in cavity-QED or circuit-QED systems ^[4,5,9–12]. It has also been predicted in various nonlinear optical systems ^[13–15] and optomechanical (OM) devices ^[16–20]. 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 ^[21–24], have been proposed to achieve unconventional PB (UPB) even for arbitrarily weak nonlinearities ^[23–37]. 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 ^[41–43], Kerr resonators ^[44–46], thermo systems ^[47–49], devices with temporal modulation ^[50,51], and non-Hermitian systems ^[52–54]. 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 ^[58–60], unidirectional quantum amplifiers ^[61–65], 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 ^[37,70–72], providing a unique way to achieve not only UPB, but also phonon laser ^[72–76], 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.

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, K≈3κ (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., K≈0.04κ ^[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 (g≈0.63κ) in an auxiliary cavity that is well within the current experimental abilities ^[83–85]. In a spinning resonator, the refractive indices associated with the clockwise (+) and anticlockwise (−) optical modes are given as n±=n[1±nv(n−2−1)/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, ωR→ωR+ΔF, with 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.

In a rotating frame with respect to H0=ωd(aL†aL+aR†aR), the effective Hamiltonian of the system can be written as (see Appendix A for more details) 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=ωK−ωd, where K=L,R. The parameter J denotes the strength of the photon hopping interaction between the two cavity modes, and g=ωR/r[ℏ/(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/(ℏωd) 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 where p and q are dimensionless canonical position and momentum, with p=i(b†−b)/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] where 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] 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, δaL†δaL, the linearized equations of the fluctuation terms can be written as These equations can be solved in the frequency domain (see Appendix B). In particular, we find where and where we introduced the auxiliary functions

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)=⟨aL†2aL2⟩/⟨aL†aL⟩2 ^[29,89]. By taking the semi-classical approximation, i.e., aL=α+δaL, the correlation function gL(2)(0) can be given as ^[29] 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 where 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]: where L[o](ρ)=2oρo†−o†oρ−ρo†o are the Lindblad super-operators ^[89], for o=aL, aR, b, and b†, and n¯m=1/[exp(ℏωm/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=ΔR−δ=Δ and κL=κR=κ and use experimentally feasible parameters ^{[53,83,91–95]}, that is, λ=1550nm, QL=3×107, r=0.3mm, n=1.44, m=5×10−11kg, and Pin=2×10−17W. QL is typically 106–1012 ^[92,94,95], g is typically 103–106Hz ^[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)=6×10−3 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.

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/κ≈−0.29 and a peak at Δ/κ≈0.166, 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., 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.

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(b†−b)/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] and consider that Cmn≪Cm′n′≪C00 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): 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 T0≈4mK (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.

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,2−n (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,2−n 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.

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.