Nonreciprocal unconventional photon blockade in a spinning optomechanical system

TL;DR

This paper demonstrates how a spinning optomechanical system can produce nonreciprocal unconventional photon blockade, enabling direction-dependent quantum light control with potential applications in quantum photonic devices.

Contribution

It introduces a novel nonreciprocal UPB mechanism in a spinning optomechanical system leveraging Fizeau drag effects, even with weak nonlinearity.

Findings

Nonreciprocal UPB achieved via Fizeau drag in spinning resonators.

Strong photon antibunching occurs only when driven from one side.

Potential applications in quantum diodes and chiral photonic devices.

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 an extremely 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 counter-circulating modes. Such nonreciprocal quantum UPB devices can be particularly useful in achieving e.g., few-photon diodes or circulators, and quantum chiral photonic engineering.

Equations

$${\Delta }_{\mathrm{F}}$$

$$\mathcal{H}=$$

$$+\hslash J(a_L^{†}{a}_R+a_R^{†}{a}_L)+\hslash g{a}_R^{†}{a}_R(b^{†}+b)$$

$$+i\hslash {ϵ}_d(a_L^{†}-a_L),$$

$$\frac{\mathrm{d}}{\mathrm{d}t}q=$$

$$\frac{\mathrm{d}}{\mathrm{d}t}p=$$

$$\frac{\mathrm{d}}{\mathrm{d}t}{a}_L=$$

$$\frac{\mathrm{d}}{\mathrm{d}t}{a}_R=$$

$$+\sqrt{{\kappa }_L}{a}_{R,\mathrm{in}},$$

$$⟨\xi (t)\xi (t^{\prime })⟩=\frac{1}{2\pi }\int \mathrm{d}\omega e^{-i\omega (t-t^{\prime })}{\Gamma }_m(\omega ),$$

$$\displaystyle\Gamma_{m}(\omega)=\frac{\omega\gamma_{m}}{2\omega_{m}}\left[1+\coth\Big{(}\frac{\hbar\omega}{2k_{B}T}\Big{)}\right],$$

$$⟨a_{K,\mathrm{in}}^{†}(t)a_{K,\mathrm{in}}(t^{\prime })⟩=$$

$$⟨a_{K,\mathrm{in}}(t)a_{K,\mathrm{in}}^{†}(t^{\prime })⟩=$$

$$\frac{\mathrm{d}}{\mathrm{d}t}\delta q=$$

$$\frac{\mathrm{d}}{\mathrm{d}t}\delta p=$$

$$\frac{\mathrm{d}}{\mathrm{d}t}\delta a_L=$$

$$\frac{\mathrm{d}}{\mathrm{d}t}\delta a_R=$$

$$-i{g}_b\beta \delta q+\sqrt{{\kappa }_R}{a}_{R,\mathrm{in}}.$$

$$\delta a_L(\omega )=$$

$$+H(\omega )a_{R,\mathrm{in}}^{†}(\omega )+Q(\omega )\xi (\omega ),$$

$$E(\omega )=$$

$$F(\omega )=$$

$$G(\omega )=$$

$$H(\omega )=$$

$$Q(\omega )=$$

$$A_1(\omega )=$$

$$-g_b^4|\beta {|}^4{\left(\frac{\chi (\omega )}{{\omega }_m}\right)}^2{V}_1^{-}(\omega )+J^2{V}_2^{+},$$

$$A_2(\omega )=$$

$$A_3(\omega )=$$

$$A_4(\omega )=$$

$$A_5(\omega )=$$

$${\Delta }_R^{\prime \prime }=$$

$$\chi (\omega )=$$

$$V_1^{\pm }(\omega )=$$

$$V_2^{\pm }(\omega )=$$

$$g_L^{(2)}(0)=$$

$$⟨\delta a_L^{\pm }(t)\delta a_L(t)⟩=$$

$$X_{a_L^{†}{a}_L}=$$

$$X_{a_L{a}_L}=$$

$$+G(\omega )H(-\omega ).$$

$$\dot{\rho }=$$

$$|1,0⟩\stackrel{\sqrt{2}{ϵ}_d}{⟶}|2,0⟩,$$

$$|1,0⟩\stackrel{J}{⟶}|0,1⟩\stackrel{{ϵ}_d}{⟶}|1,1⟩\stackrel{\sqrt{2}J}{⟶}|2,0⟩.$$

$$|\phi ⟩=$$

Full text

I Introduction

Photon blockade (PB) Tian92Quantum ; Leonski94Possibility ; Imamoglu97Strongly ; Birnbaum05Photon ; Muller15Coherent , i.e., the generation of the first photon in a nonlinear cavity diminishes almost to zero the probability of generating another photon in the cavity, plays a key role in single-photon control for quantum technology applications nowadays Gu17Microwave ; Scarani09The ; Buluta11Natural . In experiments, PB has been demonstrated in cavity-QED or circuit-QED systems Birnbaum05Photon ; Peyronel12Quantum ; Muller15Coherent ; Lang11Observation ; Hoffman11Dispersive ; Faraon08Coherent . PB has also been predicted in various nonlinear optical systems Ferretti10Photon ; Liao10Correlated ; Miranowicz13Two and optomechanical (OM) devices Rabl11Photon ; Nunnenkamp11Single ; Liao13Photon ; Xie16Photon ; Zhai18mechanical . Conventional PB happens 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 Leonski04Two ; Miranowicz06Kerr ; Liew10Single ; Bamba11Origin , have been proposed to achieve unconventional photon blockade (UPB) even for arbitrarily weak nonlinearities Liew10Single ; Bamba11Origin ; Majumdar12Loss ; Komar13Single ; Savona13Unconventional ; Xu13Antibunching ; Ferretti13Optimal ; Xu14Strong ; Zhang14Optimal ; Shen15Tunable ; Flayac17Unconventional ; Flayac17Nonclassical ; Zhou18Photon ; Snijders18Observation ; Vaneph18Observation . UPB provides a powerful tool to generate optimally sub-Poissonian light and also a way to reveal quantum correlations in weakly nonlinear devices Flayac17Unconventional ; Flayac17Nonclassical . Recently, UPB has been experimentally demonstrated in coupled optical Snijders18Observation or superconducting resonators Vaneph18Observation .

It should be stressed PB and UPB are very different phenomena, thus also their nonreciprocal generalizations are also different. Indeed PB refers to a process, when 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 Leonski01 . 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, which is optimally sub-Poissonian in second order, $g^{2}(0)\approx 0$, and is generated in a weakly-nonlinear system allowing for 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 multi-path interference assuming even an 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. Carusotto13 and now commonly accepted, is fundamentally different from PB, concerning their physical mechanisms and properties of their generated light.

Here, we propose to achieve and control nonreciprocal UPB with spinning devices. Nonreciprocal devices allow for 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 cancelling interfering signals Manipatruni09Optical . Nonreciprocal optical devices have been realized in OM devices Manipatruni09Optical ; Shen16Experimental ; Bernier17Nonreciprocal , Kerr resonators Cao17Experimental ; Bino18Microresonator ; Shi15Limitations , thermo systems Fan11An ; Zhang18Thermal ; Xia18Cavity , devices with temporal modulation Sounas17Non-reciprocal ; Caloz18Electromagnetic , and non-Hermitian systems Bender13Observation ; Peng14Parity ; Chang14Parity . In a very recent experiment Maayani18Flying , 99.6% optical isolation in a spinning resonator has been achieved based on the optical Sagnac effect. However, these studies have mainly focused on the classical regimes; that is, unidirectional control of transmission rates instead of quantum noises. We also note that in recent works, single-photon diodes Xia14Reversible ; Tang18An ; Scheucher16Quantum , unidirectional quantum amplifiers Abdo14Josephson ; Metelmann15Nonreciprocal ; Malz18Quantum ; Shen18Reconfigurable ; Song18Direction , and one-way quantum routers Barzanjeh18Manipulating have been explored. In particular, nonreciprocal PB was predicted in a Kerr resonator Huang18Nonreciprocal or a quadratic OM system Xu18arXiv , which, however, relies on the conventional condition of strong single-photon nonlinearity. These quantum nonreciprocal devices have potential applications for quantum control of light in chiral and topological quantum technologies Lodahl17Chiral .

We also note that coupled-cavity systems have been extensively studied in experiments Vaneph18Observation ; Zhang18A ; Konotop16Nonlinear ; Ganainy18Non , providing a unique way to achieve not only UPB, but also phonon laser Grudinin10Phonon ; Jing14PT ; Zhang18A , slow light Zhang18Loss , and force sensing Liu16Metrology ; Konotop16Nonlinear ; Ganainy18Non . 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 in a nonreciprocal way even with a weak single-photon nonlinearity; that is, strongly antibunched photons can emerge only by driving the device 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.

II Model and Solutions

We consider a compound system consisting of an optical harmonic resonator (with the resonance frequency of the cavity field $\omega_{L}$ and the decay rate $\kappa_{L}$) and a spinning anharmonic resonator (with the resonance frequency of the cavity field $\omega_{R}$ and the decay rate $\kappa_{R}$), as shown in Fig. 1. An external light is coupled into and out of the resonator through a tapered fiber of frequency $\omega_{d}$ and these two whispering-gallery-mode resonators are evanescently coupled to each other with coupling strength $J$ Spillane03Ideality . Note that in the previous proposal Huang18Nonreciprocal , requiring the strong Kerr nonlinearity, $K\approx 3\kappa$ (where $\kappa$ is the cavity linewidth), is challenging for current experiments. Here we can use experimentally feasible Kerr-nonlinear strength to realize nonreciprocal PB; that is, $K\approx 0.04\kappa$ Vaneph18Observation , which is two orders of magnitude smaller than the former work Huang18Nonreciprocal . Weak Kerr couplings can be achieved in cavity-atom systems Schmidt96Giant , magnon devices Wang18Bistability , and OM systems Gong09Effective which we focus on here. We consider a weakly OM coupling strength ($g\approx 0.63\kappa$) in an auxiliary cavity which is well within current experimental abilities Ding11Wavelength ; Snijders16Purification ; Enzian19Observation . In a spinning resonator, the refractive indices associated with the clockwise ($\mathrm{+}$) and anticlockwise ($\mathrm{-}$) optical modes are given as $n_{\pm}=n\left[1\pm nv(n^{-2}-1)/c\right]$, where $v=r\Omega$ is the tangential velocity with the angular velocity $\Omega$ and radius $r$ Maayani18Flying . For light propagating in the spinning resonator, optical mode experiences a Fizeau shift $\Delta_{\mathrm{F}}$ Malykin00The ; that is, $\omega_{R}\rightarrow\omega_{R}+\Delta_{\mathrm{F}}$, with

[TABLE]

where $\omega_{R}=2\pi c/\lambda$ is the optical resonance frequency for the nonspinning OM resonator, $c$ ($\lambda$) is the speed (wavelength) of light in the vacuum, and $n$ is the refractive index of the cavity. The dispersion term ${\mathrm{d}n}/{\mathrm{d}\lambda}$, characterizing the relativistic origin of the Sagnac effect, is relatively small in typical materials ($\sim 1\%$) Malykin00The ; Maayani18Flying . For convenience, we always assume the counterclockwise rotation of the resonator. Hence the $\pm\Delta_{\mathrm{F}}$ denote the light propagating against ($\Delta_{\mathrm{F}}>0$) and along ($\Delta_{\mathrm{F}}<0$) the direction of the spinning OM resonator, respectively.

In a rotating frame with respect to $H_{0}=\omega_{d}(a_{L}^{\dagger}a_{L}+a_{R}^{\dagger}a_{R})$, the effective Hamiltonian of the system can be written as (see Appendix A for more details)

[TABLE]

where $a_{L}$ ($a_{L}^{\dagger}$) and $a_{R}$ ($a_{R}^{\dagger}$) are the photon annihilation (creation) operators for the cavity modes of the optical cavity (denoted by subscript $L$) and the OM cavity (denoted by subscript $R$), respectively; $b$ ($b^{\dagger}$) 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 by $\Delta_{K}=\omega_{K}-\omega_{d}$ where $K=L,R$; The parameter $J$ denotes the strength of the photon hopping interaction between the two cavity modes; $g=\omega_{R}/r\left[\hbar/(2m\omega_{m})\right]^{1/2}$ describes the radiation-pressure coupling between the optical and vibrative modes in the OM resonator with frequency $\omega_{m}$ and effective mass $m$; $\epsilon_{d}=\sqrt{\kappa_{L}P_{\mathrm{in}}/(\hbar\omega_{d})}$ denotes the driving strength that is coupled into the compound system through the optical fiber waveguide with cavity loss rate $\kappa_{L}$ and driving power $P_{\mathrm{in}}$.

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

[TABLE]

where $p$ and $q$ are dimensionless canonical position and momentum with $p=i(b^{\dagger}-b)/\sqrt{2}$ and $q=(b+b^{\dagger})/\sqrt{2}$, respectively; $\Delta_{R}^{\prime}=\Delta_{R}+\Delta_{\mathrm{F}}$, and $g_{b}=\sqrt{2}g$; $\kappa_{L}=\omega_{L}/Q_{L}$ ($\kappa_{R}=\omega_{R}/Q_{R}$) is the dissipation rate and $Q_{L}$ ($Q_{R}$) is the quality factor of the left (right) cavity; $\gamma_{m}=\omega_{m}/Q_{M}$ is damping rate with the quality factor $Q_{M}$ of the mechanical mode. Moreover, $\xi$ is the zero-mean Brownian stochastic operator, $\langle\xi(t)\rangle=0$, resulting from the coupling of the mechanical resonator with corresponding thermal environment and satisfying the following correlation function Ford88Quantum :

[TABLE]

where

[TABLE]

and $T$ is effective temperature of the environment of the mechanical resonator and $k_{B}$ is the Boltzmann constant. The annihilation operators $a_{L,{\mathrm{in}}}$ and $a_{R,{\mathrm{in}}}$ are the input vacuum noise operators of the optical cavity and the OM cavity with zero mean value, respectively, i.e., $\langle a_{L,{\mathrm{in}}}\rangle=\langle a_{R,{\mathrm{in}}}\rangle=0$, and comply with time-domain correlation functions Gardiner00Quantum ; Walls94Quantum :

[TABLE]

for $K=L,R$. Because the whole system interacts with a low-temperature environment (here we consider 0.1 $\mathrm{mK}$), 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 fluctuations with zero mean value around it; that is, $a_{L}=\alpha+\delta a_{L}$, $a_{R}=\beta+\delta a_{R}$, $q=q_{s}+\delta q$, and $p=p_{s}+\delta p$. By neglecting higher-order terms, $\delta a_{L}^{\dagger}\delta a_{L}$, the linearized equations of the fluctuation terms can be written as:

[TABLE]

These equations can be solved in the frequency domain (see Appendix B). In particular, we find

[TABLE]

where

[TABLE]

and

[TABLE]

where we introduced the auxiliary functions:

[TABLE]

III Nonreciprocal Optical Correlations

Now, we focus on the statistical properties of photons in optical cavity, which are described quantitatively via normalized zero-time delay second-order correlation function $g_{L}^{(2)}(0)=\langle a_{L}^{\dagger 2}a_{L}^{2}\rangle/\langle a_{L}^{\dagger}a_{L}\rangle^{2}$ Walls94Quantum ; Xu13Antibunching . By taking the semiclassical approximation, i.e., $a_{L}=\alpha+\delta a_{L}$, the correlation function $g_{L}^{(2)}(0)$ can be given as Xu13Antibunching :

[TABLE]

where $\mathcal{R}_{1}=\langle\delta a_{L}^{\dagger}(t)\delta a_{L}(t)\rangle$, $\mathcal{R}_{2}=\langle[\delta a_{L}(t)]^{2}\rangle$, and $\mathcal{R}_{3}=\langle\delta a_{L}^{\dagger}(t)\delta a_{L}^{\dagger}(t)\delta a_{L}(t)\delta a_{L}(t)\rangle=2\mathcal{R}_{1}+|\mathcal{R}_{2}|^{2}$.

From Eq. (II), the correlation between $\delta a_{L}(t)$ and $\delta a_{L}^{\dagger}(t)$ can be calculated as

[TABLE]

where $\delta a_{L}^{+}(t)=\delta a_{L}^{\dagger}(t)$, $\delta a_{L}^{-}(t)=\delta a_{L}(t)$, and

[TABLE]

To obtain more accurate results, we introduce the density operator $\rho(t)$ and numerically calculate normalized zero-time delay second-order correlation by the Lindblad master equation Johansson13Qutip :

[TABLE]

where $\mathcal{L}[o](\rho)=2o\rho o^{\dagger}-o^{\dagger}o\rho-\rho o^{\dagger}o$ are the Lindblad superoperators Walls94Quantum , for $o=a_{L}$, $a_{R}$, $b$, $b^{\dagger}$, and $\bar{n}_{m}=1/\left[\exp(\hbar\omega_{m}/k_{B}T)-1\right]$ is the mean thermal phonon numbers of the mechanical mode at temperature $T$.

The second-order correlation function $g_{L}^{(2)}(0)$ is shown in Fig. 2 as function of the optical detuning $\Delta/\kappa$ and the angular velocity $\Omega$. We assume $\Delta_{L}=\Delta_{R}-\delta=\Delta$, $\kappa_{L}=\kappa_{R}=\kappa$ and use experimentally feasible parameters Vahala03Optical ; Peng14Parity ; Teufel11Sideband ; Ding11Wavelength ; Verhagen12Quantum ; Aspelmeyer14Cavity ; Huet16Millisecond ; that is, $\lambda=1550~{}\mathrm{nm}$, $Q_{L}=3\times 10^{7}$, $r=0.3~{}\mathrm{mm}$, $n=1.44$, $m=5\times 10^{-11}~{}\mathrm{kg}$, $P_{\mathrm{in}}=2\times 10^{-17}~{}\mathrm{W}$. $Q_{L}$ is typically $10^{6}-10^{12}$ Vahala03Optical ; Aspelmeyer14Cavity ; Huet16Millisecond , $g$ is typically $10^{3}-10^{6}~{}\mathrm{Hz}$ Ding11Wavelength ; Verhagen12Quantum ; Aspelmeyer14Cavity in optical microresonators, and $g_{L}^{(2)}(0)\sim 0.37$ Snijders18Observation ; Vaneph18Observation was experimentally achieved. $J$ can be adjusted by changing the distance of the double resonators Zhang18A . In a recent experiment, autocorrelation measurements range from $g^{(2)}(0)=6\times 10^{-3}$ to $2$ have been achieved with average fidelity $0.998$ in a photon-number-resolving detector Hlousek18Accurate . Moreover, we set $\Omega=12~{}\mathrm{kHz}$, which is experimentally feasible. The resonator with a radius of $r=1.1~{}\mathrm{mm}$ can spin at an angular velocity $\Omega=6.6~{}\mathrm{kHz}$ Maayani18Flying . By use of a levitated OM system Reimann18GHz ; Ahn18Optically , $\Omega$ can be increased even up to $\mathrm{GHz}$ values.

Our analytical results agree well with the numerical one. In the nonspinning-resonator case, as shown in Fig. 2(a), $g_{L}^{(2)}(0)$ is reciprocal regardless the direction of the driving light, and always has a dip at $\Delta/\kappa\approx-0.29$ and a peak at $\Delta/\kappa\approx 0.166$, corresponding to strong photon antibunching and photon bunching, respectively Xu13Antibunching . The physical origin of strong photon antibunching is the destructive interference between direct and indirect paths of two-photon excitations, i.e.,

[TABLE]

In contrast, for a spinning device, $g_{L}^{(2)}(0)$ exhibits giant nonreciprocity, which can be seen in Fig. 2(b). The PB can be generated, i.e., $g_{L}^{(2)}(0)\sim 0.06$, for $\Delta_{\mathrm{F}}<0$, while significantly suppressed, i.e., $g_{L}^{(2)}(0)\sim 4.72$, for $\Delta_{\mathrm{F}}>0$, which can be seen more clearly in Fig. 2(c). The nonreciprocal UPB induced by Fizeau light-dragging effect, with up to two orders of magnitude difference of $g_{L}^{(2)}(0)$ for opposite directions, can be achieved even with a weak nonlinearity and, to our knowledge, has not been studied. Furthermore, in Fig. 2(b), we use two sets of parameters for solid (case 1) and dashed curves (case 2), respectively. It is seen that nonreciprocity still exists in a parameter range closer to the experiment.

Since the anharmonicity of the system is very small, destructive quantum interference (rather then the anharmonicity) is responsible for observing strong photon antibunching (referred to as UPB) and photon bunching (as referred to photon-induced tunnelling) 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, thus we refer to Ref. Bamba11Origin , where this interference-based mechanism was first explained in detail. Spinning the OM resonator results in different Fizeau drag $\Delta_{\mathrm{F}}$ for the counter-circulating whispering-gallery modes of the resonator. By driving the system from the left-hand side, the direct excitation from state $|1,0\rangle$ to state $|2,0\rangle$ will be forbidden by the destructive quantum interference with the indirect paths of two-photon excitations, which leads to photon antibunching. In contrast, photon bunching occurs by driving the system from the right side, due to the lack of the complete destructive quantum interference between the indicated levels Reiter18Cooperative . Increasing the angular velocity results in an opposing frequency shift of $\eta\Omega$ for light coming from opposite directions. $g_{L}^{(2)}(0)$ also experiences linearly shifts with $\Omega$, but with different directions for $\Delta_{\mathrm{F}}<0$ or $\Delta_{\mathrm{F}}>0$; that is, we observe either a blueshift [see Fig. 3(a)] or a redshift [see Fig. 3(b)] with $\Delta_{\mathrm{F}}>0$ or $\Delta_{\mathrm{F}}<0$, respectively. A highly-tunable nonreciprocal UPB device is thus achievable, by flexible tuning of $\Omega$ and $\Delta/\kappa$. In addition, since $g_{L}^{(2)}(0)$ is sensitive to $\Omega$, this may also indicate a way for accurate measurements of velocity.

IV Optimal Parameters for Strong Antibunching

As discussed above, UPB can be generated nonreciprocally. In this section, we analytically derive the optimal conditions of strong antibunching. We apply here the method described in Ref. Bamba11Origin , which is based on the evolution of a complex non-Hermitian Hamiltonian, as given in Appendix C. Thus, our solution corresponds to only a semiclassical 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 $U=\exp\left[-g(b^{\dagger}-b)/\omega_{m}\right]$, the states of the system can be expressed as $|\psi\rangle=|\varphi\rangle|\phi_{m}\rangle$, where $|\varphi\rangle$ and $|\phi_{\mathrm{m}}\rangle$ are the photon states and the phonon states, respectively. Under the weak-driving condition, we make the ansatz Bamba11Origin

[TABLE]

and consider that $C_{mn}\ll C_{m^{\prime}n^{\prime}}\ll C_{00}$ for $m+n=2$, $m^{\prime}+n^{\prime}=1$, and the condition of $C_{20}=0$, the optimal conditions are given by fixing $J$ and $\kappa$ (see Appendix C)

[TABLE]

where $\mathrm{sgn}(E)$ is the signal function, $a_{3}=-96\Delta_{\mathrm{F}}\kappa$, and $\lambda_{1,2}$, which are defined in Appendix C, are related to the Fizeau drag $\Delta_{\mathrm{F}}$. Physically, this means that the position of the minimum of $g^{(2)}_{L}(0)$ is determined by the detuning between the two cavity fields. Thus, $\Delta_{\mathrm{F}}$ can lead to a shift of the minimum of $g^{(2)}_{L}(0)$ to achieve nonreciprocity.

In order to visualize UPB more clearly, we show the contour plots of $g_{L}^{(2)}(0)$ in logarithmic scale [i.e., $\log_{10}g_{L}^{(2)}(0)$] as function of $g/\kappa$ and $\Delta/\kappa$ in Fig. 4(a). By fixing $\Delta/\kappa=-0.05$, we obtain the function of $g_{L}^{(2)}(0)$ in logarithmic scale versus the coupling strength of the resonators $J/\kappa$ and $g/\kappa$, as shown in Fig. 4(b). These plots show that strong photon antibunching occurs exactly at the values predicted from our analytical calculations in Eq. (IV). Moreover, by computing $g_{L}^{(2)}(0)$ as the function of $\Delta/\kappa$ and $\Omega$ with different mean thermal phonon numbers $\mathrm{n_{\mathrm{th}}}$, 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 $g_{L}^{(2)}(0)$ of photons and tend to destroy photon blockade. Thus, to show this effect, in Fig. 6(a), we plot the correlation $g_{L}^{(2)}(0)$ as a function of temperature $T$ for various Fizeau shifts. We see that nonreciprocal UPB can be observed below the critical temperature $T_{0}\approx 4~{}\mathrm{mK}$ ($5~{}\mathrm{mK}$) for the spinning frequency $\Omega=12~{}\mathrm{kHz}$ ($\Omega=50~{}\mathrm{kHz}$) [see Fig. 6(b)]. By further increasing the optical dissipation of the optomechanical cavity, as shown in Fig. 6(d), the critical temperature $T_{0}$ can reach the value of $10~{}\mathrm{mK}$.

Finally, we note that a state (generated via UPB or another effect) with vanishing (or almost vanishing) second-order photon-number correlations, $g^{(2)}(0)\approx 0$, 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 $g^{(2)}(0)\approx 0$, but also by vanishing higher-order photon-number correlation functions, $g^{(n)}(0)\approx 0$ for $n>2$. In UPB, $g^{(n)}(0)$ for $n>2$ can be greater than $g^{(2)}(0)\approx 0$, or even greater than 1 Radulaski17 . Indeed a standard analytical method for analyzing UPB, as proposed by Bamba et al. Bamba11Origin and applied here, is based on expanding the wave function $|\varphi\rangle$ of a two-resonator system in power series $|\varphi\rangle=\sum C_{n,m}|n,m\rangle$ up to the terms $C{n,2-n}$ ($n=0,1,2$) only, as given in Eq. (IV). To obtain the optimal system parameters, which minimize $g^{(2)}(0)$ in UPB, this method requires to set $C_{2,0}=0$ as set in Appendix C. Actually, the same expansion of $|\varphi\rangle$ and same ansatz are made in Ref. Bamba11Origin . 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 $C_{n,2-n}$ is often not justified for a system exhibiting UPB. Indeed, we find parameters for our system, for which $g^{(2)}(0)\approx 0$ and simultaneously $g^{(3)}(0)>1$. We have confirmed this by a precise numerical calculation of the steady states of our system based on the non-Hermitian Hamiltonian, given in Eq. (C), in the Hilbert space larger than $4\times 4$.

V Conclusions

In summary, we studied nonreciprocal UPB in a system consisting of a purely optical resonator and a spinning OM resonator. Due to the interference between two-photon excitations 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 the weak OM interactions. The optimal conditions for one-way UPB were given analytically. Moreover, we found this quantum nonreciprocity can still exist by considering thermal phonon noises.

Concerning a possible experimental implementation of nonreciprocal UP, it is worth noting that UPB for non-spinning devices has already been demonstrated experimentally in two recent works Snijders18Observation ; Vaneph18Observation . A number of experiments (including the very recent work Maayani18Flying ) have shown non-reciprocal quantum effects in spinning devices. So the main experimental task for achieving non-reciprocal UPB in a spinning device would be to combine experimental setups of, e.g., Refs. Snijders18Observation ; Vaneph18Observation ; Maayani18Flying into a single spinning UPB setup.

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

Appendix A Derivation of Effective Hamiltonian

The coupled system can be described by the following Hamiltonian

[TABLE]

where $a_{L}$ ($a_{L}^{\dagger}$) and $a_{R}$ ($a_{R}^{\dagger}$) are the photon annihilation (creation) operators for the cavity modes of the optical cavity (denoted by subscript $L$) and the OM cavity (denoted by subscript $R$), respectively; $b$ ($b^{\dagger}$) is the annihilation (creation) operator for the mechanical mode of the OM cavity. The frequencies of the cavity fields are denoted by $\omega_{L}$ and $\omega_{R}$. $J$ is the coupling strength between the two resonators, $g=\omega_{R}/r\left[\hbar/(2m\omega_{m})\right]^{1/2}$ is the OM coupling strength between the optical mode and the mechanical mode in the OM cavity, $\epsilon_{d}=\sqrt{\kappa_{L}P_{\mathrm{in}}/(\hbar\omega_{d})}$ denotes the driving strength which is coupled into the compound system through the optical fiber waveguide.

Using the unitary operator $U=\exp\left[-g(b^{\dagger}-b)/\omega_{m}\right]$ to Hamiltonian (A), we obtain a Kerr-type one Gong09Effective

[TABLE]

where $\delta=g^{2}/\omega_{m}$. Under the conditions, $g/\omega_{m}\ll 1$ and $J<\omega_{m}/2$, the Hamiltonian (A) can be read as

[TABLE]

Appendix B The Fourier Analysis of Fluctuation Terms

According to the Heisenberg equations of motion of Hamiltonian (II), and using semiclassical approximation method, i.e., $a_{L}=\alpha+\delta a_{L}$, $a_{R}=\beta+\delta a_{R}$, $q=q_{s}+\delta q$, and $p=p_{s}+\delta p$, the steady-state values of the system satisfy the following equations:

[TABLE]

Then we obtain

[TABLE]

where

[TABLE]

The fluctuation terms of the system can be written as:

[TABLE]

where we have neglected higher-order terms, $\delta a_{L}^{\dagger}\delta a_{L}$. Here, the steady-state mean value $q_{s}$ is numerically solved from Eqs. (22) and (B).

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

[TABLE]

where $\Delta_{R}^{\prime\prime\prime}=\Delta_{R}^{\prime}+g_{b}q_{s}$, then we obtain

[TABLE]

where

[TABLE]

Substituting Eq. (B) into Eq. (B), we have

[TABLE]

where

[TABLE]

According to Eq. (B), we obtain

[TABLE]

then we have

[TABLE]

where

[TABLE]

From Eq. (B), we have

[TABLE]

where $V(\omega)=\kappa_{L}/2+i\omega-i\Delta_{L}$. Substituting Eq. (33) into Eq. (B), we find

[TABLE]

where $T(\omega)=N(\omega)V(\omega)+J^{2}$. Substituting Eq. (B) into Eq. (B), we obtain

[TABLE]

where the auxiliary function are $F_{R}(\omega)=M(\omega)T(\omega)-\chi^{2}(\omega)V(\omega)g_{b}^{4}|\beta|^{4}$. Substituting Eq. (B) into Eq. (B), we have

[TABLE]

where

[TABLE]

Then we find

[TABLE]

According to similar calculations, we find

[TABLE]

Using the Fourier transform, we obtain

[TABLE]

and

[TABLE]

Appendix C Derivation of Optimal Parameters

According to the quantum-trajectory method Plenio98The , the non-Hermitian Hamiltonian of the system containing the optical decay and mechanical damping terms given by Plenio98The

[TABLE]

where $\Delta_{R}^{\prime}=\Delta_{R}+\Delta_{\mathrm{F}}$.

Under the weak-driving conditions, we can make the ansatz Bamba11Origin :

[TABLE]

Then we substitute Hamiltonian (C) and the general state (C) into the Schrödinger equation

[TABLE]

then we have:

[TABLE]

where the auxiliary functions are $\delta_{L}=\Delta_{L}-i\kappa_{L}/2$ and $\delta_{R}=\Delta_{R}^{\prime}-i\kappa_{R}/2$, and we have ignored the effects of the mechanical model, because the phonon states are decoupled from the photon states [see Eq. C]. By comparing the coefficients, we have

[TABLE]

Then the steady-state coefficients of the one- and two-particle states are given as

[TABLE]

and

[TABLE]

where we have introduced the dissipative terms (proportional to $\kappa_{L}$ and $\kappa_{R}$) and neglected the higher-order terms, as justified under the weak-driving conditions.

When we consider $\Delta_{L}=\Delta_{R}-\delta=\Delta$, $\delta=g^{2}/\omega_{m}$, $\kappa_{L}=\kappa_{R}=\kappa$, and the condition of $C_{20}=0$, wehave

[TABLE]

By eliminating $\delta$, we obtain

[TABLE]

where

[TABLE]

then we find the optimal conditions

[TABLE]

where

[TABLE]

and

[TABLE]

Funding. NSF of China under Grants No. 11474087 and No. 11774086, and the HuNU Program for Talented Youth.