Spinning microresonator-induced chiral optical transmission

Abstract

Chiral quantum optics is a new research area in light-matter interaction that depends on the direction of light propagation and offers a new path for the quantum regulation of light-matter interactions. In this paper, we study a spinning Kerr-type microresonator coupled with Λ-type atom ensembles, which are driven in opposite directions to generate asymmetric photon statistics. We find that a photon blockade can only be generated by driving the spinning resonator on right side without driving the spinning microresonator from the left side, resulting in chirality. The coupling strength between system modes can be precisely controlled by adjusting the detuning amount of the atomic pump field. Because of the splitting of the resonant frequency generated by the Fizeau drag, the destructive quantum interference generated in right side drive prevents the nonresonant transition path of state |1,0⟩ to state |2,0⟩. This direction-dependent chiral quantum optics is expected to be applied to chiral optical devices, single-photon sources and nonreciprocal quantum communications.

Graphical abstract

Keywords

chiral quantum optics / spinning microresonator / nonreciprocal / photon blockade

1 Introduction

Chirality is a special symmetry of existence in nature that describes the mirror-symmetry property of a system. It has received widespread attention and is involved in many physical phenomena, such as the quantum Hall effect [1], topological insulators [2, 3], and superconductors [4]. In advanced photonics nanostructured devices [5-11], strong light confinement can lock the local polarization of light in its propagation direction, which causes the emission, scattering, and absorption of photons by the quantum emitter to depend on the propagation direction. This research field, which depends on the propagation direction, that is, chirality combined with quantum optics, is called chiral quantum optics [12]. In chiral quantum optics, the result of light−matter interactions depends on the propagation of light; that is, forward and backward propagating photons and matter will have different interactions. Consequently, the light−matter interaction becomes nonreciprocal, and even photon blockades occur under extreme conditions.

Chiral quantum optics are widely used in photonic nanostructured devices, such as photonic waveguides that restrict the propagation direction [7], one-dimensional quantum emitters based on chiral coupling [13], and spin-momentum-locked light coupled to quantum emitters with polarization-dependent dipole transitions [14-17]. There are many methods used to achieve chiral optics, such as breaking the parity−time (PT) symmetry of the system to achieve nonreciprocality [18-20], chiral coupling of Rydberg atoms with microcavities [8, 13], using a nonlinear Kerr-type microcavity [21-23], optomagnonics [24-26], and using optomechanical devices [27-30]. Microcavities are an excellent way to achieve chiral quantum optics, as they greatly enhance light−matter interactions. In recent years, there have been significant advances in experimental microcavities with small mode volumes on the microscale and ultrahigh-quality factor Q (\sim {10}^8), which can greatly improve the performance of light−matter interactions [31-35]. They have been widely used in optomechanics [36-41], nonlinear optics [42-44], PT symmetric or antisymmetric optics [19, 45-47], cavity quantum electrodynamics (C-QED) [48], and ultrasensitive detection [49-53].

In particular, a spinning microresonator splits the resonant frequency of the countercirculating modes due to the Fizeau drag. In a recent experiment [54], based on the optical Sagnac effect, 99.6\mathrm{\%} optical isolation was achieved in a spinning resonator. There has been considerable research on microresonators because they can produce photon blockades (PBs) and thus play an important role in achieving nonreciprocal single photon sources, which are crucial elements for quantum information [55-58]. Moreover, by adjusting the amount of detuning and the rotation speed of the resonator, the chiral light−matter interaction has a high degree of controllability and good application prospects in quantum information processing. Meanwhile, symmetric mode-coupled nanoparticle-enhanced sensing by spinning microresonators has also been proposed [57]. However, this requirement that controlling coupling strength depending on nanoparticle location is difficult. Due to the characteristics of chiral quantum optics, we expect to obtain a chiral optical system with adjustable optical mode coupling strength in the resonator. Fortunately, we can adjust the atom-cavity mode coupling strength by adjusting the frequency and strength of the external control field [8].

In this paper, we demonstrate a precisely adjustable chiral optical system composed of a spinning Kerr-type resonator and \mathrm{\Lambda }-type atom ensembles. We can control the chirality of the system by changing the coupling strength between modes, the rotation speed of the resonator, and detuning amount of the light field. We find that destructive quantum interference between different excitation levels results in chirality even under weaker nonlinear conditions; that is, a photon blockade can only be produced by right side of the drive device and not left side. Our work can be used as a new method for realizing chiral optical devices, single-photon sources and nonreciprocal quantum communication.

2 The physical model and solution

The system considered here is a spinning resonator coupled with an add-drop-type taper fiber and a single three-level atom ensemble, as shown in Fig.1(a). The spinning resonator rotates at angular velocity \mathrm{\Omega } and supports both clockwise (CW) and counterclockwise (CCW) optical modes (with the same frequency {\omega }_c and dissipation rate \kappa). The light circulating in the resonator changes the Fizeau drag due to rotation, and the relationship between the Fizeau drag and the detuning amount is shown in Fig.1(b), i.e., {\omega }_0 \to {\omega }_0+{\mathrm{\Delta }}_F, with [59]

(1){\mathrm{\Delta }}_F =\pm {\displaystyle \frac{nr\mathrm{\Omega }{\omega }_0}{c}}\left(1-{\displaystyle \frac{1}{n^2}}-{\displaystyle \frac{\lambda }{n}}{\displaystyle \frac{\mathrm{d}n}{\mathrm{d}\lambda }}\right)=\pm \eta \mathrm{\Omega },

where {\omega }_0 is the resonance frequency of a nonspinning resonator, n is the refractive index, r is the resonator radius, c(\lambda ) is the speed (wavelength) of light in vacuum, and the dispersion term \mathrm{d}n/\mathrm{d}\lambda, characterizing the relativistic origin of the Sagnac effect, is relatively small. Therefore, in our model, where the resonator rotates counterclockwise, the CW mode exists at {\mathrm{\Delta }}_F> 0, and the CCW mode exists at . Fig.1(c) shows the \mathrm{\Lambda }-type energy levels of the atom, in which |e^k⟩\leftrightarrow |g^k⟩ represents the k-th atom transformation in the ensemble, and the transition is driven by an external control field with central frequency {\omega }_p. Because {\mathrm{\Delta }}_F brought by the Fizeau drag is much smaller than {\omega }_c, we assume that the |e^k⟩\leftrightarrow | f^k⟩ transition can simultaneously interact with the degenerate CW and CCW modes with the same coupling strength J, drive strength {\mathrm{\Omega }}_a and phase \beta. By switching off the beam and magnetic field of the magneto-optical trap (MOT) that traps ultracold atoms, we can make a few atoms fall from fixed positions to the microresonator and the coupling strength of which can be regarded as constant.

We consider the case where the resonator is driven on the right (or left) when the probe signal is input from port 1 (or port 2) or the transmission is detected from port 4 (or port 3). The Hamiltonian of the system driven by a weak laser on the left side can be described as \hat{H}={\hat{H}}_c+ {\hat{H}}_a +{\hat{H}}_I. {\hat{H}}_c denotes the Hamiltonian of the resonator, which is {\hat{H}}_c=({\omega }_c-{\mathrm{\Delta }}_F){\hat{a}}_{cw}^{†}{\hat{a}}_{cw}+({\omega }_c+ {\mathrm{\Delta }}_F){\hat{a}}_{ccw}^{†}{\hat{a}}_{ccw} by setting \hslash =1. The Hamiltonian of the \mathrm{\Lambda }-type energy level atom ensemble coupled to the resonator is {\hat{H}}_a= {\mathrm{\Sigma }}_k ({\omega }_e{\sigma }_{ee}^k+{\omega }_g{\sigma }_{gg}^k+{\omega }_f{\sigma }_{ff}^k), and {\sigma }_{ii}^k=|i^k⟩⟨i^k| stands for the corresponding transitions. The Hamiltonian {\hat{H}}_I of the interaction between the atom and the light field can be expressed as

(2){\hat{H}}_I ={\mathrm{\Sigma }}_k[J {\sigma }_{ef}^k({\hat{a}}_{cw}{\mathrm{e}}^{\mathrm{i}\beta }+ {\hat{a}}_{ccw}{\mathrm{e}}^{-\mathrm{i}\beta })+{\mathrm{\Omega }}_a {\sigma }_{eg}^k{\mathrm{e}}^{-\mathrm{i}{\omega }_{p}t}+ H.c.],

where {\sigma }_{ef}^k=|e^k⟩⟨f^k| corresponds to the k-th atom in ensemble transitions from |e^k⟩ to |f^k⟩, and the creation {\hat{a}}_{cw}^{†} ( {\hat{a}}_{ccw}^{†}) and the annihilation operator {\hat{a}}_{cw} ({\hat{a}}_{ccw}) are the bosonic operators of the cavity CW mode and CCW mode.

When the effective detunings are much larger than the corresponding coupling strength (\delta ,{\mathrm{\Delta }}_p\gg J,{\mathrm{\Omega }}_a) and the atom ensemble detuning {\mathrm{\Delta }}_a and the dissipation \gamma are larger than those of optical modes ({\mathrm{\Delta }}_a, \gamma \gg {\mathrm{\Delta }}_c,\kappa ), the excited states can be adiabatically eliminated. Considering the dissipation and the nonlinear term of this system, the effective Hamiltonian is non-Hermitian, which can be expressed as [8]

(3)\begin{array}{rl}{\hat{H}}_{eff}=& \mathrm{i}\chi G^2{\mathrm{e}}^{-\mathrm{i}2\beta }{\hat{a}}_{cw}^{†}{\hat{a}}_{ccw}+\mathrm{i}\chi G^2{\mathrm{e}}^{\mathrm{i}2\beta }{\hat{a}}_{cw}{\hat{a}}_{ccw}^{†}\\ & +({\mathrm{\Delta }}_{cw}-\mathrm{i}{\displaystyle \frac{{\kappa }^{\prime }}{2}}){\hat{a}}_{cw}^{†}{\hat{a}}_{cw}+\left({\mathrm{\Delta }}_{ccw}-\mathrm{i}{\displaystyle \frac{{\kappa }^{\prime }}{2}}\right){\hat{a}}_{ccw}^{†}{\hat{a}}_{ccw}.\end{array}

Here, G={\mathrm{\Omega }}_{a}J/{\mathrm{\Delta }}_p, \chi =4\mathrm{i}({\mathrm{\Delta }}_a-J^2/{\mathrm{\Delta }}_p)/{\gamma }^2-4/\gamma, and {\mathrm{\Delta }}_{cw(ccw)}={\mathrm{\Delta }}_c\mp {\mathrm{\Delta }}_F-G^2{\mathrm{\Delta }}_a /({\mathrm{\Delta }}_a^2+{\gamma }^2/4). The effective detuning of the atomic and cavity mode pump fields is {\mathrm{\Delta }}_a= {\omega }_p+ {\omega }_g- {\omega }_f+ {\mathrm{\Omega }}_a^2/{\mathrm{\Delta }}_p^2-{\omega }_c^{i}n, and the dissipation of optical modes is {\kappa }^{\prime }=\kappa + G^2\gamma /({\mathrm{\Delta }}_a^2+{\gamma }^2/4). Therefore, we can change the coupling strength between the optical mode {\hat{a}}_{cw} and {\hat{a}}_{ccw} by changing the Rabi frequency {\mathrm{\Omega }}_a and the detuning {\omega }_p of the atom ensemble, avoiding the difficulty of adjusting the nanoscale atom ensemble’s position.

Consider the Kerr nonlinear terms {\hat{a}}_{ccw}^{†}{\hat{a}}_{ccw}^{†}{\hat{a}}_{ccw}{\hat{a}}_{ccw} and {\hat{a}}_{cw}^{†}{\hat{a}}_{cw}^{†}{\hat{a}}_{cw}{\hat{a}}_{cw} and the cross Kerr nonlinear term {\hat{a}}_{cw}^{†}{\hat{a}}_{cw}{\hat{a}}_{ccw}^{†}{\hat{a}}_{ccw} that exist in the system. The cross Kerr nonlinear term is related to the quadratic term of J, after adiabatic elimination of atom. Under \gamma \gg {\mathrm{\Delta }}_a, the Taylor series expansion of \chi = \frac{4}{{\gamma }^2}[\mathrm{i}{\mathrm{\Delta }}_a- \mathrm{i}{J}^2/{\mathrm{\Delta }}_p-\gamma ] yields the nonlinear Kerr terms and the cross Kerr nonlinear term. At this moment, the effective Hamiltonian of the system can be written as \hat{H}={\hat{H}}_0 +{\hat{H}}_s+{\hat{H}}_d. Where {\hat{H}}_0= {\mathrm{\Delta }}_{cw}{\hat{a}}_{cw}^{†}{\hat{a}}_{cw} +{\mathrm{\Delta }}_{ccw}{\hat{a}}_{ccw}^{†}{\hat{a}}_{ccw}+ U({\hat{a}}_{cw}^{†} {\hat{a}}_{cw}^{†}{\hat{a}}_{cw}{\hat{a}}_{cw} + {\hat{a}}_{ccw}^{†}{\hat{a}}_{ccw}^{†}{\hat{a}}_{ccw}{\hat{a}}_{ccw}), {\hat{H}}_s= g_{12}{\hat{a}}_{cw}^{†}{\hat{a}}_{ccw} +g_{21}{\hat{a}}_{ccw}^{†}{\hat{a}}_{cw} + 2U{\hat{a}}_{cw}^{†}{\hat{a}}_{cw}{\hat{a}}_{ccw}^{†}{\hat{a}}_{ccw}, and {\hat{H}}_d=\epsilon ({\hat{a}}_{cw}^{†}+{\hat{a}}_{cw}). Here, the nonlinear term U is - 4G^2{J}^2/({\gamma }^2{\mathrm{\Delta }}_p), while the asymmetric coupling terms are g_{12}=4\mathrm{i}{G}^2(\mathrm{i}{\mathrm{\Delta }}_a-\gamma ){\mathrm{e}}^{-\mathrm{i}2\beta }/{\gamma }^2 and g_{21}=4\mathrm{i}{G}^2(\mathrm{i}{\mathrm{\Delta }}_a- \gamma ){\mathrm{e}}^{\mathrm{i}2\beta }/{\gamma }^2. And g_{12}= g_{21}=g when the phase of the atoms is the same \beta =0. When the excitation of the optical mode is weak enough, only lower excited states can be excited, and the average number of photons in the microcavity is relatively small. Assuming that the maximum number of photons is less than 2, the state of this system becomes

(4)\begin{array}{rl}|\phi ⟩=& C_{00}|00⟩+C_{10}|10⟩+C_{01}|01⟩\\ & +C_{20}|20⟩+C_{02}|02⟩+C_{11}|11⟩,\end{array}

where |n_{cw},n_{ccw}⟩ represents the Fock state in which this system has n_{cw} photons in the CW mode and n_{ccw} photons in the CCW mode. Due to the weak driving condition, we have C_{00}\gg C_{10},C_{01}\gg C_{11},C_{20},C_{02}. Wave function evolution satisfies the Schrödinger equation, and we obtain the dynamic equations for the coefficients C_{cw,ccw}.

First, we consider the one-photon excitation states. The evolution of these two states can be described as

(5)\begin{array}{r}\mathrm{i}{\displaystyle \frac{\mathrm{\partial }{C}_{10}}{\mathrm{\partial }t}}=({\mathrm{\Delta }}^{\prime }-{\mathrm{\Delta }}_F-\mathrm{i}\kappa /2)C_{10}+g{C}_{01}+\epsilon C_{00},\end{array}

(6)\begin{array}{r}\mathrm{i}{\displaystyle \frac{\mathrm{\partial }{C}_{01}}{\mathrm{\partial }t}}=({\mathrm{\Delta }}^{\prime }+{\mathrm{\Delta }}_F-\mathrm{i}\kappa /2)C_{01}+g{C}_{10}.\end{array}

Here {\mathrm{\Delta }}^{\prime } ={\mathrm{\Delta }}_c-G^2{\mathrm{\Delta }}_a /({\mathrm{\Delta }}_a^2+{\gamma }^2/4). Using steady-state conditions, we obtain C_{01}=- g{\mathrm{\Xi }}_2{C}_{10}, where {\mathrm{\Xi }}_2^{-1}= {\mathrm{\Delta }}^{\prime }+{\mathrm{\Delta }}_F- \mathrm{i}\kappa /2 and

(7)\begin{array}{r}{C}_{10}={\displaystyle \frac{\epsilon C_{00}}{{\mathrm{\Xi }}_0^{-1}-g^2{\mathrm{\Xi }}_2}},\end{array}

where {\mathrm{\Xi }}_0^{-1}={\mathrm{\Delta }}^{\prime }-{\mathrm{\Delta }}_F+ \mathrm{i}\kappa /2. Next, we consider the evolution of two-photon excitation states, and the evolution of these states can be given by the Schrödinger equation

(8)\begin{array}{rl}\mathrm{i}{\displaystyle \frac{\mathrm{\partial }{C}_{20}}{\mathrm{\partial }t}}=& (2{\mathrm{\Delta }}^{\prime }-{\mathrm{\Delta }}_F-\mathrm{i}\kappa +2U)C_{20}+\sqrt{2}\epsilon C_{10}\\ & +\sqrt{2}g{C}_{11},\end{array}

(9)\begin{array}{rl}\mathrm{i}{\displaystyle \frac{\mathrm{\partial }{C}_{11}}{\mathrm{\partial }t}}=& (2{\mathrm{\Delta }}^{\prime }-\mathrm{i}\kappa +2U)C_{11}+\sqrt{2}g{C}_{02}+\epsilon C_{01}\\ & +\sqrt{2}g{C}_{20},\end{array}

(10)\begin{array}{r}\mathrm{i}{\displaystyle \frac{\mathrm{\partial }{C}_{02}}{\mathrm{\partial }t}}=(2{\mathrm{\Delta }}^{\prime }+{\mathrm{\Delta }}_F-\mathrm{i}\kappa +2U)C_{02}+\sqrt{2}g{C}_{11}.\end{array}

Under adiabatic evolution, we have \frac{\mathrm{\partial }{C}_{02}}{\mathrm{\partial }t}=0 in the steady-state situation. Therefore, we obtain 2C_{02}= - \sqrt{2}g{\mathrm{\Xi }}_2^{\prime }{C}_{11}, {\mathrm{\Xi }}_2^{{}^{\prime }-1}={\mathrm{\Delta }}^{\prime }+{\mathrm{\Delta }}_F- \mathrm{i}\kappa /2 +U. Taking this formula into Eqs. (8) and (9) and using steady-state conditions, we have

(11)\begin{array}{r}0=2{\mathrm{\Xi }}_0^{{}^{\prime }-1}{C}_{20}+\sqrt{2}\epsilon C_{10}+\sqrt{2}g{C}_{11},\end{array}

(12)\begin{array}{r}0=(2{\mathrm{\Xi }}_1^{{}^{\prime }-1}-g^2{\mathrm{\Xi }}_2^{\prime })C_{11}+\sqrt{2}g{C}_{20}+\epsilon g{\mathrm{\Xi }}_2{C}_{10}.\end{array}

From Eqs. (11) and (12), we obtain

(13)\begin{array}{r}{C}_{11}={\displaystyle \frac{\epsilon g{\mathrm{\Xi }}_2+\epsilon g{\mathrm{\Xi }}_0}{2{\mathrm{\Xi }}_1^{{}^{\prime }-1}-g^2{\mathrm{\Xi }}_2^{\prime }-g^2{\mathrm{\Xi }}_0^{\prime }}}{C}_{10},\end{array}

(14)\begin{array}{r}{C}_{20}=\left({\displaystyle \frac{1}{\sqrt{2}}}\right){\displaystyle \frac{2{\mathrm{\Xi }}_0^{\prime }{\mathrm{\Xi }}_1^{{}^{\prime }-1}+g^2({\mathrm{\Xi }}_2+{\mathrm{\Xi }}_2^{\prime }){\mathrm{\Xi }}_0^{\prime }}{2{\mathrm{\Xi }}_1^{{}^{\prime }-1}-g^2{\mathrm{\Xi }}_2^{\prime }-g^2{\mathrm{\Xi }}_0^{\prime }}}\epsilon C_{10},\end{array}

where {\mathrm{\Xi }}_1^{{}^{\prime }-1}={\mathrm{\Delta }}^{\prime }-\mathrm{i}\kappa /2+U and {\mathrm{\Xi }}_0^{{}^{\prime }-1}={\mathrm{\Delta }}^{\prime }-{\mathrm{\Delta }}_F-\mathrm{i}\kappa /2 +U.

In this paper, we mainly study the influence of the spin direction and velocity of the resonator on the statistical properties of the system photons. Therefore, we use the second-order correlation function g^{(2)} (0) to measure the statistical properties of photons. In the steady state, the second-order correlation function is expressed as

(15)\begin{array}{r}\begin{array}{rl}{g}_L^{(2)}(0)=& {\displaystyle \frac{⟨{\hat{a}}_{cw}^{†}{\hat{a}}_{cw}^{†}{\hat{a}}_{cw}{\hat{a}}_{cw}⟩}{⟨{\hat{a}}_{cw}^{†}{\hat{a}}_{cw}{⟩}^2}}.\end{array}\end{array}

Due to a weak pump, we obtain C_{00}\gg C_{01},C_{10}\sim \epsilon \gg C_{02},C_{20},C_{11}\sim {\epsilon }^2. We denote the probability of finding |n_{cw},n_{ccw}⟩ in the resonator by P_{cw,ccw}=|C_{cw,ccw}{|}^2. Then, we approximate g_L^{(2)}(0) \simeq \frac{2|C_{20}{|}^2}{|C_{10}{|}^4}.

3 Chiral light−matter interaction

In this section, we present an analysis of the various parameters of the second-order correlation function g_L^{(2)} (g_R^{(2)}). We assume a sufficiently weak drive field (\epsilon =0.01 \kappa) so that the number of photons in the microcavity is small enough for photon blockade. For convenience, we normalize all the parameters to the dissipation rate of the cavity modes \kappa.

We have realized a system of three-level atoms coupled with a microcavity with adjustable coupling strength between optical modes. Therefore, we studied the relationship between the coupling strength g_{12} and the second-order coherence functions g^{(2)}(0) when the effective detuning amount is 0 and \mathrm{\Omega } =5 kHz, as shown in Fig.2(a). We find that with increasing coupling strength, the second-order coherence function g^{(2)}(0) first decreases and then increases. Therefore, we can choose the appropriate coupling strength to achieve photon blockage. To understand the relationship between the photon statistical properties of the spinning resonator and the driving direction. We show the logarithmic plot (of base 10) of the equal-time second-order correlation functions g^{(2)}(0 ) as functions of the effective detuning {\mathrm{\Delta }}^{\prime }/\kappa and the rotation speed \mathrm{\Omega } in Fig.2. We show the equal-time second-order correlation functions g^{(2)} (0) plotted as functions of the detuning \mathrm{\Delta } /\kappa in Fig.2(b) for U=0.05\kappa, |g|=3\kappa, and resonator spinning speed \mathrm{\Omega } =5kHz. We find that g^{(2)} (0) driven by different sides are very different even under the same amount of detuning (i.e., g_R^{(2)}(0) \ll 1 at {\mathrm{\Delta }}^{\prime }/\kappa \simeq 0.085, while g_L^{(2)}(0) >1 at {\mathrm{\Delta }}^{\prime }/\kappa \simeq 0.085). These results indicate that the photons on the left-hand drive system ({\mathrm{\Delta }}^{\prime }/\kappa \simeq 0.085) exhibit the effect of a photon blockade (a photon enters the resonator and blocks subsequent photon entry). The system under the same conditions ({\mathrm{\Delta }}^{\prime }/\kappa \simeq 0.085) is driven on the right side, and photons exhibit photon bunching (a photon enters the resonator and induces subsequent photon entry). For a nonspinning resonator, g^{(2)}(0 ) always has a dip at {\mathrm{\Delta }}^{\prime } /\kappa \simeq 0.3 or peak at {\mathrm{\Delta }}^{\prime }/\kappa \simeq -0.19. We obtain results that the points for the strong antibunching numerically drift due to the Fizeau shift of the resonator spinning relative to the resonator without spinning, similar to the Doppler effect with Fig.2. Moreover, we also show the change in g^{(2)} (0) with rotation speed \mathrm{\Omega } in Fig.2(c). We find that as the rotation speed increases, the image of g^{(2)}(0 ) drifts toward the direction of increased detuning.

To show the influence of the input light direction on the statistical characteristics of the system photon, that is, whether the chiral effect is obvious, we introduce the chiral parameter

(16)\begin{array}{r}\mathrm{\Gamma }={\displaystyle \frac{g_L^{(2)}(0)-g_R^{(2)}(0)}{g_L^{(2)}(0)+g_R^{(2)}(0)}}.\end{array}

The parameter \mathrm{\Gamma } versus the spinning speed \mathrm{\Omega } and the effective detuning {\mathrm{\Delta }}^{\prime }/\kappa are shown in Fig.3(a). When g_L^{(2)}(0) (or g_R^{(2)}(0)) becomes zero, the chirality becomes 1 (or −1). In other words, when g_L^{(2)}(0) = 0 (or g_R^{(2)}(0) = 0), the system excited by the left side (right side) is in photon blockade, and we can think of it as a system with a strong chiral effect. Chirality generation is the cleft of the cavity mode by the Fizeau shift {\mathrm{\Delta }}_F generated by the spinning resonator. Photons are easily generated by excitation on the right side and suppressed by excitation on the left side in the blue highlighted area. A photon blockade can be simplified by changing the driving direction. We can obtain a chirality photon blockade by changing the spinning speed \mathrm{\Omega } of the spinning resonator and the effective detuning {\mathrm{\Delta }}^{\prime }/\kappa.

To obtain a system with the largest chiral value, the photon statistical property of driving on the right side is a photon blockade, and driving on the left side is a photon bunching. We give the relationship between the second-order correlation function of the left-hand drive and the right-hand drive at the same spinning speed of the resonator and the amount of detuning in Fig.3(b). We find that as the amount of detuning increases, the spinning speed \mathrm{\Omega } required to satisfy the optimal condition points of the systems driven on the right side and left side shifts in the opposite direction. In Fig.3(b), we find that a system with a strong chiral effect can be obtained at a specific detuning and spinning speed, and the chiral effect disappears when the spinning speed is very small. By adjusting the amount of detuning and the spinning speed of the resonator, we can obtain a system with a strong chiral effect.

4 Optimal parameters for photon blockade

In this section, we analytically drive the optimal conditions of a photo blockade. Previous studies have shown destructive quantum interference effects between different optical modes, so two coupled quantum modes with weak nonlinearity can be used as a single photon source [60, 61]. To date, more photon blockade properties in coupled-mode systems have been studied by theory and experimentation. Our system consists of optical modes with Kerr nonlinearity, which can also achieve unconventional photon blockades based on the results of previous studies. A chiral rotating microresonator can be used as a single photon source in the case of a photon blockade (PB) (i.e., g_L^2 (0)=0). From Eqs. (8)−(10) and setting C_{20} to approach zero, we obtain the optimal condition equation as

(17)\begin{array}{r}\left|\begin{array}{ccc}0& g_{12}& \epsilon \\ \sqrt{2}{\mathrm{\Xi }}_2^{{}^{\prime }-1}& g_{21}& 0\\ \sqrt{2}g& 2{\mathrm{\Xi }}_1^{{}^{\prime }-1}& -\epsilon g{\mathrm{\Xi }}_2\end{array}\right|=0,\end{array}

or

(18)\begin{array}{rl}& 2U^2({\mathrm{\Delta }}^{\prime }+{\mathrm{\Delta }}_F)+U(g_{12}{g}_{21}-{\kappa }^{\prime 2}+4{\mathrm{\Delta }}^{\prime 2}+6{\mathrm{\Delta }}^{\prime }{\mathrm{\Delta }}_F+2{\mathrm{\Delta }}_F^2)\\ & +2{\mathrm{\Delta }}^{\prime }({\mathrm{\Delta }}^{\prime }+{\mathrm{\Delta }}_F{)}^2-\left({\displaystyle \frac{3{\mathrm{\Delta }}^{\prime }}{2}}+{\mathrm{\Delta }}_F\right){\kappa }^{\prime 2}-\mathrm{i}U{\kappa }^{\prime }(4{\mathrm{\Delta }}^{\prime }+3{\mathrm{\Delta }}_F)\\ & -\mathrm{i}{U}^2{\kappa }^{\prime }-\mathrm{i}{\kappa }^{\prime }(3{\mathrm{\Delta }}^{\prime }+{\mathrm{\Delta }}_F)({\mathrm{\Delta }}^{\prime }+{\mathrm{\Delta }}_F)+\mathrm{i}{\displaystyle \frac{{\kappa }^{\prime 3}}{4}}=0.\end{array}

As the Kerr nonlinearity U is weak enough to ignore the second-order term, the amount of detuning should be satisfied under the optimal solution,

(19)\begin{array}{r}{\mathrm{\Delta }}_{opt}\simeq -{\displaystyle \frac{2}{3}}{\mathrm{\Delta }}_F\pm {\displaystyle \frac{{\kappa }^{\prime }}{2\sqrt{3}}},\end{array}

(20)\begin{array}{r}{U}_{opt}\simeq \pm {\displaystyle \frac{2}{\sqrt{3}}}{\displaystyle \frac{{\kappa }^{\prime 3}}{3g^2-2{\kappa }^{\prime 2}}}.\end{array}

The change in g_L^{(2)}(0) with the coupling strength |g_{12}| (or |g_{21}|) and the Kerr nonlinearity U is shown in Fig.4. It can be observed that as g increases, the optimal U_{opt} required for the minimum of g_{cw}^{(2)}(0) decreases simultaneously. In other words, we can increase the coupling strength g in the experiment to reduce the required nonlinear Kerr term U. As noted above, an increase in g enhances quantum interference between different states and thus enhances the photon blockade effect.

We show the energy-level diagram and the transition paths in Fig.5. We can get the eigenenergy of the supermodel {\omega }_{\pm }={\omega }_R\pm \sqrt{g^2+{\mathrm{\Delta }}_F} from Eq. (4). When U=2\sqrt{g^2+{\mathrm{\Delta }}_F}, the energy of the left-driven system just satisfies the single-photon excitation, and the detuning of 2U suppresses the further transition of the photon, resulting in photon blockade. Meanwhile, the left-driven system just satisfies the two-photon resonance, resulting in photon-induced tunneling (PIT). In the quantum transition pathway, the optimal nonlinear interaction strength is inversely proportional to the coupling strength when \mathrm{\Delta }= {\mathrm{\Delta }}_{opt}, and the increase in coupling strength results in the suppression of the nonresonant transition from state |1,0⟩ to state |2,0⟩. The right-hand drive system produces a destructive quantum interference path, and the system produces photon blockade, as shown in Fig.5(b). Therefore, the system has excellent chiral quantum optical properties due to photon blockade by right-side driving and photon-induced tunneling by the left-side driving.

5 Conclusions

In summary, we studied a precisely controlled spinning microresonator driven by a weak field. By adjusting the coupling strength between modes and the detuning amount of the light field, asymmetric photon statistics are generated even under the weaker nonlinear Kerr term. Due to chiral light-matter interactions, the destructive quantum interference generated in the right side drive prevents the nonresonant transition path of state |1,0⟩ to state |2,0⟩, so photon blockade can only be generated by driving the spinning resonator on right side but not from the left side that drives the spinning microresonator to generate. Moreover, we can achieve a controllable photon blockade by adjusting the coupling strength between modes and the rotational speed in the chiral interaction. Our work can be applied to chiral optics, single photon sources and nonreciprocal quantum communication.

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