Here we first consider a spinning optical resonator evanescently coupled to a meandering optical waveguide at N coupling points, as shown in Figure 1. The resonator is rotated and the waveguide is stationary. The separation distance between different coupling points on the waveguide is denoted by L = x _m − x _n . We assume the coherence length of photons in the waveguide is larger than the smallest distance L _min, and therefore we can ignore the non-Markovian retarded effects [19, 52]. The nonspinning resonator, for example, a whispering-gallery-mode resonator with a resonant frequency ω _c , simultaneously supports both clockwise (CW) and counter-clockwise (CCW) travelling modes. The CW and CCW modes couple to each other through a scatterer or induced by surface roughness [53, 54], which results in an optical mode splitting. When the optical resonator rotates in one direction at an angular velocity Ω, the propagating effects of the CW and CCW modes are different, leading to an opposite Sagnac-Fizeau shift in resonant frequencies, i.e., ω _c → ω _c + Δ _F , with [55]. Δ F = ± n R Ω ω c c 1 − 1 n 2 − λ n d n d λ , (1) where n is the refractive index of the dielectric material, R is the radius of the optical resonator, and c (λ) is the velocity (wavelength) of light in vacuum. The dispersion term λdn/ndλ, denoting the relativistic origin of the Sagnac effect [51, 55], is very small in typical materials compared to the value of (1–1/n ²). In the following we assume the resonator rotates along the CCW direction, hence Δ _F > 0 (Δ _F < 0) represents the case of the driving field coming from the left-hand (right-hand) side. The resonant frequencies of the CW and CCW modes in this situation are ω _cw = ω _c + Δ _F and ω _ccw = ω _c − Δ _F , respectively.

In our consideration, the Hamiltonian of the spinning resonator can be written as (ℏ = 1) H c = ω c + Δ F c cw † c cw + ω c − Δ F c ccw † c ccw + J c cw † c ccw + c ccw † c cw . (2) Here c _cw and c _ccw ( c cw † and c ccw † ) are the annihilation (creation) operators of the CW and CCW modes, respectively. The coupling strength J denotes the interaction between these two modes induced by optical backscattering. The CW (CCW) mode can only be driven by an optical field coming from the left (right) side of the waveguide, own to the directionality of travelling wave modes in the resonator. The driving Hamiltonian is H d = i ∑ m = 1 N κ m , e c m , in c cw † − c cw + i ∑ m = 1 N κ m , e c m , in ′ c ccw † − c ccw , (3) where c _m,in and c m , in ′ are the input fields coming from the left and right sides at coupling point x _m , respectively. According to Fermi’s golden rule [56], κ m , e = 2 π g m 2 D ( ω ) describes the spontaneous emission of the resonator modes into the waveguide at coupling point x _m , with g _m being the resonator-waveguide coupling strength and D ( ω ) being the photon density of states in the waveguide. In the presence of decay channels, the effective non-Hermitian Hamiltonian of the whole system is given by H 1 = H c + H d − i Γ c c cw † c cw + c ccw † c ccw , (4) with Γ c = κ c 2 + ∑ m = 1 N κ m , e 2 , (5) where Γ _c is the total decay rate of the resonator mode, and κ _c is the intrinsic decay rate of the resonator.

According to the Heisenberg motional equations, the dynamic equations of the CW and CCW modes are yielded by d c cw d t = − i ω c + Δ F + Γ c c cw − i J c ccw + ∑ m = 1 N κ m , e c m , in , d c ccw d t = − i ω c − Δ F + Γ c c ccw − i J c cw + ∑ m = 1 N κ m , e c m , in ′ . (6) Note that k _cw = (ω _c + Δ _F )/c and k _ccw = (ω _c − Δ _F )/c are approximately regarded as the central mode vector of right-going and left-going photon in the waveguide emitted by the resonator [23], respectively. Different from the case without rotation, the accumulated phase shifts between neighbor coupling points for opposite propagation directions of the photons are distinct. As given in Refs. [17, 57–59], the local input-output relations for the CW and CCW modes at each coupling point x _m are written as c m , out = c m , in − κ m , e c cw , c m + 1 , in = c m , out e i k cw x m + 1 − x m , c m , out ′ = c m , in ′ − κ m , e c ccw , c m , in ′ = c m + 1 , out ′ e i k ccw x m + 1 − x m . (7) Substituting Eq. 7 into Eq. 6, we obtain the effective dynamic equations d c cw d t = − i ω c + Δ F + Γ c + ∑ m > n = 1 N κ m , e κ n , e e i k cw x m − x n c cw − i J c ccw + ∑ m = 1 N κ m , e e i k cw x m − x 1 c 1 , in , d c ccw d t = − i ω c − Δ F + Γ c + ∑ m > n = 1 N κ m , e κ n , e e i k ccw x m − x n c ccw − i J c cw + ∑ m = 1 N κ m , e e i k ccw x N − x m c N , in ′ . (8) The total input-output relations of this system take the form c N , out = c 1 , in e i k cw x N − x 1 − ∑ m = 1 N κ m , e e i k cw x N − x m c cw , c 1 , out ′ = c N , in ′ e i k ccw x N − x 1 − ∑ m = 1 N κ m , e e i k ccw x m − x 1 c ccw . (9)

Eq. 8 exhibits a self-coupling in the CW or CCW mode, which arises from the self interference effects of reemitted photons between different connection points. Moreover, the Sagnac effect and the self-interference effects may significantly affect the optical properties of the system. We note that only when the resonator is nonspinning, the system is reciprocal. Based on these derivations, we will investigate the photon emission and transport properties in this system.

In the giant-atom waveguide-QED systems, the multiple coupling points result in a frequency-dependent decay rate and Lamb shift for a giant atom [23, 60]. Similarly, the interference effects induced by multiple coupling points in our system also give a modification of the frequency shift Δ _j and decay rate Γ _j for the CW and CCW mode. According to Eq. 8, we have Δ j = ∑ m > n = 1 N κ m , e κ n , e sin ϕ m n j , Γ j = Γ c + ∑ m > n = 1 N κ m , e κ n , e cos ϕ m n j . (10) where ϕ m n j = k j ( x m − x n ) with j = cw, ccw.

Here we consider the maximally symmetric case, in which decay rates of the resonator modes into the waveguide are the same at each coupling point with κ _m,e = κ _e and the distance between neighboring coupling points is identical with x _m+1 − x _m = d. Then we can set x _m − x _n = (m − n)d and θ _j = k _j d. Similar to the Lamb shift and decay rate in atomic physics, Eq. 10 becomes Δ j = κ e 2 N ⁡ sin θ j − sin N θ j 1 − cos θ j , Γ j = κ c 2 + κ e 2 1 − cos N θ j 1 − cos θ j . (11) We begin to discuss the effects of the rotation speed and number of coupling points on the emission properties under the condition of κ _c = 0. When the resonator is nonspinning with Ω = 0, the CW and CCW modes are degenerate with the Fizeau drag Δ _F = 0 and ω _ccw = ω _cw = ω _c . As increasing the rotation speed Ω, the Sagnac-Fizeau shift described by Eq. 1 linearly increases, as given in Supplementary Material. In our calculations, we choose the related parameters as follows: λ = 1,550 nm, R = 4.73 mm, and n = 1.4. For Ω = 0.97 GHz, we have Δ _F /ω _c = ±0.05 and (RΩ)/c ≈ 0.015. For the spinning resonator with a single coupling point (N = 1), Eq. 11 gives the results of Δ_cw = Δ_ccw = 0 and Γ_cw = Γ_ccw = (κ _c + κ _e )/2. When increasing the number of coupling points, the frequency shifts and decay rates for the CW and CCW modes have an opposite shift due to the rotation.

In Figures 2A,B, frequency shifts Δ _j and decay rates Γ _j are plotted as a function of the phase θ _c = ω _c d/c with N = 10 and Ω = 0.97 GHz. The frequency shifts Δ_cw and Δ_ccw take negative and positive values with the maximum at about 0.6Γ_max. Given that Δ_cw (Δ_ccw) is zero, the decay rate Γ_cw (Γ_ccw) reaches its highest magnitude at θ _c = 0.95 × 2π (θ _c = 1.05 × 2π). For θ _c = 0.95 × 2π, the accumulated phase of photons propagating along the CCW direction leads to Γ_ccw = 0, which arises from the destructive interference effects among the coupling points. In this case, the CCW mode of the resonator is decoupled from the waveguide. Moreover, there are a lot of additional lower and local maximum values in the decay rates. The phase θ _c of the local minima between these maxima scales with (1/N + Δ _F /ω _c ). Note that the rotation speed and number of coupling points make a big difference in the values of Γ_cw and Γ_ccw. Narrower resonances can be found in the decay rates when we consider more coupling points.

In order to study the emission properties more clearly, for a special frequency we define the chirality parameter C as C = Γ cw − Γ ccw Γ cw + Γ ccw , (12) where C = 1 ( C = − 1 ) implies a truly unidirectional excitation of the right-going (left-going) photon, and C = 0 denotes the photon coupling into the waveguide without preference in both propagating directions. Figure 2C depicts the chiral factor C changing with number of coupling points N. When N = 1, the chiral factor is C = 0 . For θ _c = 0.95 × 2π, as increasing number of coupling points N, the chirality factor C first goes up and then oscillates slowly with a relative larger value around 1. Note that C = 1 is obtained for N = 10, corresponding to Γ_cw = 50κ _e and Γ_ccw = 0. The essence of the chirality is that accumulated phases for photons propagating in CW and CCW directions are different. By tuning the phase shift θ _c , for example, θ _c = 1.05 × 2π, the photon emission direction is totally switched. Figure 2D shows the chiral factor C as functions of number of coupling points N and rotation speed Ω for θ _c = 0.95 × 2π. By optimizing the rotation speed and number of coupling points, the chiral factor C can approach 1, and the chiral direction can be freely switched. Moreover, the directional emission will be realized in a large parameter regime.

Now we study how the rotation velocity and number of coupling points affect the optical response of the spinning resonator. We consider the resonator is excited by an external input signal in the CW direction with frequency ω _l and amplitude ɛ. In this case, the input signal from the left side is given by c 1 , in + ε e − i ω l t , with c _1,in being the vacuum input signal, while the input signal from the right side only contains the vacuum input field c N , in ′ . In the rotating frame at the driving frequency ω _l , the steady-state solutions of Eq. 8 can be written as c cw = i Δ c − Δ F + Δ ccw + Γ ccw ∑ m = 1 N κ m , e e i k cw x m − x 1 ε i Δ c − Δ F + Δ ccw + Γ ccw i Δ c + Δ F + Δ cw + Γ cw + J 2 . (13) Here Δ _c = ω _c − ω _l is the detuning between the resonator without rotation and the driving field. The transmission rate of the input signal is given by T L = c N , out ε 2 = 1 − i Δ c − Δ F + Δ ccw + Γ ccw ∑ m , n = 1 N κ m , e κ n , e e i k cw x m − x n i Δ c − Δ F + Δ ccw + Γ ccw i Δ c + Δ F + Δ cw + Γ cw + J 2 2 . (14)

Similarly, we also consider the case of an external input signal coming from the right side of the waveguide with ε ′ e − i ω l t . By solving the steady-state solutions of Eq. 8, we obtain c ccw = i Δ c + Δ F + Δ cw + Γ cw ∑ m = 1 N κ m , e e i k ccw x N − x m ε ′ i Δ c − Δ F + Δ ccw + Γ ccw i Δ c + Δ F + Δ cw + Γ cw + J 2 . (15) The transmission rate of the input signal is written as T R = c 1 , out ′ ε ′ 2 = 1 − i Δ c + Δ F + Δ cw + Γ cw ∑ m , n = 1 N κ m , e κ n , e e i k ccw x m − x n i Δ c − Δ F + Δ ccw + Γ ccw i Δ c + Δ F + Δ cw + Γ cw + J 2 2 . (16)

A nonreciprocal photon transmission with T _R ≠ T _L can be observed when the resonator is spinning. This fact is due to the different numerators in Eqs 13, 15. For the maximally symmetric case, we have Γ j ′ = ∑ m , n = 1 N κ m , e κ n , e e i k j x m − x n = κ e 1 − cos N θ j 1 − cos θ j . (17) For J = 0, the incident photon will be transmitted and absorbed with reflection being zero. In this scenario, the transmission curve T _L represents a Lorentzian line shape centered at Δ _c = −(Δ _F + Δ_cw) with a linewidth Γ_cw. For N = 1, we obtain Δ _c = −Δ _F and Γ_cw = (κ _c + κ _e )/2. The transmission dip is around 0. For multiple coupling points, as discussed above, Δ_cw and Γ_cw vary periodically with phase θ _c . The transmission rate T _L versus the detuning Δ _c and the phase θ _c are plotted in Figure 3A. It shows that θ _c will dramatically modify the transmission window. As we increase θ _c , the position of the transmission dip has a red-shift. When the phase θ _cw is 2π/N, the transmission dip disappears totally with T = 1, which means the resonator cannot be excited by the external field and corresponds to the optical dark state. This phenomenon arises from the destructive interferences in the multiple coupling points, which can be explained by Eq. 17. Moreover, the mode splitting is observed in some parameter range in Figure 3B when J = 5κ _e . The asymmetry of the two dips results from different decay rates and frequency shifts of these two modes.

In Figures 3C,D, we plot the transmission rates T _L and T _R when the incident photon coming from the left side and right sides versus the detuning Δ _c for θ = 0.95 × 2π. It shows that T _L can be larger or smaller than T _R for N = 5. In other words, the nonreciprocal transmission is clearly observed due to the rotation. The interference effects between coupling points enable the transmission dips asymmetric with different linewidths. For N = 10, the decay rate of the CCW mode is very small, which leads to the complete photon transmission with T _R = 1. Moreover, a sharp dip appears in the transmission spectra T _L for J = 5κ _e . Note that the phase θ _c can also be used to adjust the nonreciprocal transmission behavior.

The single-photon transport properties in a one-dimensional waveguide interacted with two giant atoms for three distinct topologies have been discussed in Ref. [61]. To study potential applications of the spinning resonator with multiple coupling points in large-scale quantum chiral networks, we now consider two separate spinning resonators evanescently coupled to a meandering waveguide at several different connection points. As shown in Figure 4, the optical resonator a (b) simultaneously supports both clockwise and counter-clockwise travelling optical modes. The creation operators of the CW and CCW modes are denoted by a cw † and a ccw † ( b cw † and b ccw † ), respectively. The optical resonator a (b), with stationary resonant frequency ω _a (ω _b ) and intrinsic decay rate κ _a (κ _b ), rotates along the CCW direction by an angular velocity Ω _a (Ω _b ). Owing to the rotation, the resonant frequencies of the CW and CCW modes in the resonator become ω _i,cw = ω _i + Δ _F,i and ω _i,ccw = ω _i − Δ _F,i with the subscript i = a, b, where Δ _F,i is given by Eq. 1. The resonator a (b) is coupled to the bent waveguide at connection points x 1 a and x 2 a ( x 1 b and x 2 b ). The phase factor ϕ _i is calculated as k ( x 1 i − x 2 i ) when an optical signal travelling between them, and the phase factor when photons travelling from resonator a to resonator b is ϕ L = k ( x 1 b − x 2 a ) . Here we note that there is no direct coupling between cavity a and cavity b due to the absence of the modal overlap.

The Hamiltonian of these two spinning resonators are given by H c ′ = ∑ j = cw , ccw ω a , j a j † a j + ω b , j b j † b j + J a a cw † a ccw + a ccw † a cw + J b b cw † b ccw + b ccw † b cw . (18) Here J _a (J _b ) is the coupling strength between the CW and CCW modes of the resonator a (b). The CCW (CW) modes in the resonators can only be driven by an optical field coming from the left (right) side of the waveguide. The amplitudes of the input fields at different coupling points are denoted by a _m,in, b _m,in, a m , in ′ , and b m , in ′ with m = 1, 2. The driving fields give the Hamiltonian H d ′ = i ∑ m = 1 2 κ m , e a a m , in a cw † − a cw + i ∑ m = 1 2 κ m , e a a m , in ′ a ccw † − a ccw + i ∑ m = 1 2 κ m , e b b m , in b cw † − b cw + i ∑ m = 1 2 κ m , e b b m , in ′ b ccw † − b ccw . (19) The non-Hermitian Hamiltonian of the whole system can be given by H 2 = H c ′ + H d ′ − i Γ a a cw † a cw + a ccw † a ccw − i Γ b b cw † b cw + b ccw † b ccw , (20) where Γ i = ( κ i + κ 1 , e i + κ 2 , e i ) / 2 and i = a, b. Note that κ _a (κ _b ) is the intrinsic optical loss of the resonator a (b), κ 1 , e i and κ 2 , e i are the waveguide-resonator coupling rates at coupling points x 1 i and x 2 i , respectively.

The effective dynamic evolution equations of the cavity modes can be written as d a cw d t = − i ω a + Δ F , a + Γ a + κ 1 , e a κ 2 , e a e i ϕ a , cw a cw − i J a a ccw − F cw b cw + κ 1 , e a e i ϕ a , cw + ϕ L , cw + ϕ b , cw + κ 2 , e a e i ϕ L , cw + ϕ b , cw b 2 , in , d a ccw d t = − i ω a − Δ F , a + Γ a + κ 1 , e a κ 2 , e a e i ϕ a , ccw a ccw − i J a a cw + κ 1 , e a + κ 2 , e a e i ϕ a , ccw a 1 , in ′ , d b cw d t = − i ω b + Δ F , b + Γ b + κ 1 , e b κ 2 , e b e i ϕ b , cw b cw − i J b b ccw + κ 1 , e b e i ϕ b , cw + κ 2 , e b b 2 , in , d b ccw d t = − i ω b − Δ F , b + Γ b + κ 1 , e b κ 2 , e b e i ϕ b , ccw b ccw − i J b b cw − F ccw a ccw + κ 1 , e b e i ϕ a , ccw + ϕ L , ccw + κ 2 , e b e i ϕ a , ccw + ϕ L , ccw + ϕ b , ccw a ′ 1 , in , (21) where F j = κ 1 , e a κ 1 , e b e i ϕ a , j + ϕ L , j + κ 1 , e a κ 2 , e b e i ϕ a , j + ϕ L , j + ϕ b , j + κ 2 , e a κ 1 , e b e i ϕ L , j + κ 2 , e a κ 2 , e b e i ϕ L , j + ϕ b , j . (22) Note that F _cw (F _ccw) denotes the effective unidirectional coupling strength between the CW (CCW) modes of these two resonators. The total input-output relations of this system take the form a 1 , out = b 2 , in e i ϕ a , cw + ϕ L , cw + ϕ b , cw − κ 1 , e a + κ 2 , e a e i ϕ a , cw a cw − κ 1 , e b e i ϕ a , cw + ϕ L , cw + κ 2 , e b e i ϕ a , cw + ϕ L , cw + ϕ b , cw b cw , b 2 , out ′ = a 1 , in ′ e i ϕ a , ccw + ϕ L , ccw + ϕ b , ccw − κ 2 , e b + κ 1 , e b e i ϕ b , ccw b ccw − κ 2 , e a e i ϕ L , ccw + ϕ b , ccw + κ 1 , e a e i ϕ a , ccw + ϕ L , ccw + ϕ b , ccw a ccw . (23) By using Eqs 21, 23, we can investigate the photon transport properties of this system in the steady state.

In the following, we consider the input signal only comes from one side of the waveguide. Supposed that an external input signal b _2,in is injected from the right side of the waveguide with ε e − i ω l t , where ɛ and ω _l are the amplitude and frequency of the driving field, respectively. In the rotating frame at the driving frequency ω _l , the steady-state solutions of the CW resonator modes in Eq. 21 are solved as 〈 a cw 〉 = U ccw V cw V ccw + J b 2 A cw − U ccw V ccw F cw B cw U cw U ccw + J a 2 V cw V ccw + J b 2 + J a J b F cw F ccw ε , 〈 b cw 〉 = V ccw U cw U ccw + J a 2 B cw + J a J b F ccw A cw U cw U ccw + J a 2 V cw V ccw + J b 2 + J a J b F cw F ccw ε , (24) where U cw = i Δ a + Δ F , a + Γ a + κ 1 , e a κ 2 , e a e i ϕ a , cw , U ccw = i Δ a − Δ F , a + Γ a + κ 1 , e a κ 2 , e a e i ϕ a , ccw , V cw = i Δ b + Δ F , b + Γ b + κ 1 , e b κ 2 , e b e i ϕ b , cw , V ccw = i Δ b − Δ F , b + Γ b + κ 1 , e b κ 2 , e b e i ϕ b , ccw , A cw = κ 1 , e a e i ϕ a , cw + ϕ L , cw + ϕ b , cw + κ 2 , e a e i ϕ L , cw + ϕ b , cw , A ccw = κ 1 , e a + κ 2 , e a e i ϕ a , ccw , B cw = κ 1 , e b e i ϕ b , cw + κ 2 , e b , B ccw = κ 1 , e b e i ϕ a , ccw + ϕ L , ccw + κ 2 , e b e i ϕ a , ccw + ϕ L , ccw + ϕ b , ccw . (25) Here, Δ _a = ω _a − ω _l (Δ _b = ω _b − ω _l ) is the detuning between the resonator a (b) without rotation and the driving field. According to Eq. 23, the transmission rate of the output port a _1,out for the input signal b _2,in can be defined as T R = a 1 , out / ε 2 .

Similarly, when an external input signal is injected from the left side of the waveguide with ε ′ e − i ω l t , the steady-state solutions of the CCW resonator modes in Eq. 21 are also solved as 〈 a ccw 〉 = U cw V cw V ccw + J b 2 A ccw + J a J b F cw B ccw U cw U ccw + J a 2 V cw V ccw + J b 2 + J a J b F cw F ccw ε ′ , 〈 b ccw 〉 = V cw U cw U ccw + J a 2 B ccw − U cw V cw F ccw A ccw U cw U ccw + J a 2 V cw V ccw + J b 2 + J a J b F cw F ccw ε ′ , (26) Once again, the transmission rate of the ouput port b 2 , out ′ is given by T L = | ⟨ b 2 , out ′ ⟩ / ε ′ | 2 .

In the following, we choose the related parameters as follows: ω _a = ω _b = ω _c , Ω _a = Ω _b = 0.97 GHz, κ m , e a = κ m , e b = κ , κ _a = κ _b = 0.5κ, κ = 5 × 10⁻³ ω _c and ϕ _L,cw = π. Thus, Δ _a = Δ _b = Δ _c and Δ _F,a = Δ _F,b . We first consider the CW and CCW modes decoupling, i.e., J _a = J _b = 0. In Figures 5A,B, we plot the transmission rates T _R and T _L versus the detuning Δ _c /κ and the phase ϕ _b,cw/π for ϕ _a,cw = π. According to Eqs 23, 24, the transmission rate T _R represents a Lorentzian line shape centered at Δ _c = −Δ_F − κ sin (ϕ _b,cw) with a linewidth Γ _b + κ cos (ϕ _b,cw). However, the behavior of transmission rate T _L is different. A mode splitting may appear around Δ _c = Δ _F , which implies indirect coherent coupling between the CCW modes of these two resonators is achieved. The reason behind this phenomenon is that the phase ϕ _a,ccw is not equal to π own to the rotation. Moreover, the phase ϕ _b,cw can significantly change the transmission windows with a period 2π. To give more details, in Figures 5C,D we plot the profiles of T _R and T _L changing with Δ _c /κ for ϕ _b,cw = π and ϕ _b,cw = 1.5π. By contrast, one finds that for ϕ _b,cw = π, the CW modes decouple to the waveguide corresponding to an optical dark state with T _R = 1, while the CCW modes are excited with a transmission dip in T _L . For ϕ _b,cw = 1.5π, strong coupling with a double-dip-type curve in T _L can be realized. The photon nonreciprocal transmission behavior is observed due to the Sagnac effects and the interference effects among multiple coupling points. Note that for ϕ _b,cw = π, similar results are obtained by tuning the phase ϕ _a,cw.

In Figures 6A,B, we plot the transmission rates T _R and T _L versus the detuning Δ _c /κ for different J _b . For J _b = 10κ, the transmission spectra display an asymmetric four-dips structure. When decreasing J _b , the transmission dips can be suppressed. Moreover, T _L is always larger (smaller) than T _R in the region of Δ _c < 0 (Δ _c > 0). In order to describe the nonreciprocity clearly, we define the isolation ratio as I dB = − 10 × log 10 T L T R . (27) In Figure 7, the isolation ratio I changing with the detuning Δ _c /κ and the coupling strength J _b is plotted. It shows that for J _b = 0 the ratio achieves I ≈ 10 dB ( I ≈ − 5 dB ) when fixing Δ _c = 11κ (Δ _c = −11.5κ). As we increase J _b , a larger mode splitting for Δ _c > 0 is observed. For J _b = 10κ, the ratio reaches I ≈ 17 dB when Δ _c is set as 15κ. In this case, the photons coming from the left side are blocked, which implies a directional photon transfer between different coupling points. Therefore, the nonreciprocal transmission behavior is also controlled by adjusting the coupling strengths between the CW and CCW modes and the detuning Δ _c .