The study on manipulating the single-photon transport via the low-dimensional waveguides has motivated great interest due to the potential applications in quantum information processing [1][2] and quantum networks [3]. In the past decades, waveguide quantum electrodynamic (wQED) [4][5], which enables strong interactions between quantum emitters and light fields confined in one-dimensional waveguides, provides a powerful platform for realizing controllable single-photon scattering.

As an indispensable characteristic in the single-photon transmission, nonreciprocal photon transport, which allows the photon to propagate in one direction along a single waveguide but blocks light propagating in the reverse direction, plays an important role in preventing the information backflow in waveguide systems. Consequently, a wide variety of studies on nonreciprocal quantum devices with high contrast ratio have been implemented in various wQED systems, including atomic qubits [6][7], optical cavities [8][9], optomechanical or electromechanical systems [10]-[12], and chiral waveguide systems [13][14]. On the other hand, quantum routing, as another transport characteristic, can transfer quantum information from the input channel to different channels, and has also been demonstrated in numerous wQED systems [15]-[21], where multiple waveguides are utilized as quantum channels, and quantum emitters are used to distribute quantum information. However, in some proposed configurations, the low routing efficiency of no more than 0.5 for the single photons from the input waveguide channel to another may limit more potential applications, and thus it is of considerable interest to design a quantum router with high efficiency.

Optical resonators are useful components for wavelength switching, filtering, and routing applications. Due to their ultra-high quality factor, whispering gallery mode (WGM) optical microresonators have found widespread applications, ranging from sensitive sensing [22], optical switching and routing [23]-[25], coherent non-reciprocity transport [26][27], and so on. Usually, the microresonator is coupled to a straight waveguide at a single point. Interestingly, a recent report [28] shows that a spinning resonator can interact with a meandering waveguide at multiple coupling points, which results in the complete unidirectional transparency over the whole optical frequency range with no analogy in other quantum devices, due to the interference effects among different coupling points.

In this paper, we study the controllable single-photon transport in a meandering waveguide system coupled with a microresonator. In comparison with the rotary resonator scheme [28], our proposal is simple due to the static implementation without the resonator spinning. By using a real-space approach, the scattering amplitudes of single photons are derived analytically and the transport properties are discussed in detail. Numerical results show that the perfectly nonreciprocal transport and high-efficiency routing of single photons can be implemented via the joint effect of the local coupling phase and the accumulated phase of the photon propagating between two coupling points of the resonator, and the targeted routing efficiency with approximating 100% routing probability can be realized in two output ports owing to the phase-dependent interference effect. The proposed scattering system may be exploited potentially in designing high-efficiency quantum optical devices.

A schematic view of the composite system is displayed in Figure 1, where two one-dimensional waveguides is linked by a two-mode WGM resonator. The WGM resonator supports two degenerate propagating modes, described separately by the creation operator c 1 † of the clockwise mode and c 2 † of the counterclockwise mode, with the common frequency ω r and the intrinsic decay rate γ r . The resonator is coupled to the waveguide W a ( b ) through two connecting points located at x = 0 and x = d a ( b ) . When a single photon is injected into the waveguide, it will be scattered to four ports of the two waveguides by the resonator.

Figure 1. (Color online) Schematic diagram of single-photon routing between two waveguides. Two waveguides are simultaneously coupled to a two-mode WGM resonator via two separated coupling points located at x = 0 , 0 and x = d a , d b , respectively. When a single photon is incident from the waveguide, it will be routed to four ports by the resonator.

Considering the linear dispersions of two waveguide modes and the rotating wave approximation, the real-space Hamiltonian [29] for the coupled system is given by ( ℏ = 1 hereafter)

where υ g is the group velocity of the moving photon, and C R s † ( x ) [ C L s † ( x ) ] stands for the creation operator of a right(left)-propagating photon in the waveguide W s ( s = a , b ) at the position x . η is the inter-mode backscattering strength between two degenerate modes. The coupling strengths are generally complex coefficients, which are denoted as V a j = V a e i ϕ j and V b j = V b e i φ j ( j = 1 , 2 ) , with the local coupling phases ϕ j and φ j . H.c. is the Hermitian conjugate. The interaction term in the Hamiltonian is described by direction-matching condition, due to the fact that the light fields of the two rotating modes of the resonator come from the left and right transmission of the incident waveguide modes, and then they are transferred in the opposite directions to the drop waveguide.

Assume that a photon is input from the left side of the waveguide with the energy E k = υ g k = ω . In the single-excitation subspace the scattering eigen-state of the Hamiltonian (1) are expressed as

where | ∅ 〉 represents the vacuum state with zero photon in the two waveguides and the resonator. ξ 1 and ξ 2 denote the excitation amplitude of two circular modes in the resonator. Φ R s ( x ) and Φ L s ( x ) describe the wave functions for the right and left guided-wave modes in two waveguides, and the corresponding expressions are

Here, θ ( x ) is the Heaviside step function, with θ ( 0 ) = 1 / 2 . t a ( b ) and r a ( b ) describe the transmission and reflection amplitudes of single photons in four ports of two waveguides, while t 1 ( 2 ) and r 1 ( 2 ) represent the transmission and reflection amplitudes between two coupling points in two waveguides.

Solving the eigen-equation H | ψ 〉 = E k | ψ 〉 , one can obtain the stationary-state expressions of the scattering coefficients as follows:

where Γ a = V a 2 / υ g and Γ b = V b 2 / υ g are the effective coupling strengths between the resonator and two waveguides. Q 2 = Δ ω + i γ r + i   Γ a [ 1 + e i ( α a + ϕ a ) ] + i   Γ b [ 1 + e i ( α b − ϕ b ) ] and Q 1 = Δ ω + i γ r + i   Γ a [ 1 + e i ( α a − ϕ a ) ] + i   Γ b [ 1 + e i ( α b + ϕ b ) ] . Δ ω = ω − ω r is the frequency detuning between the incident photon and the resonator. ϕ a = ϕ 2 − ϕ 1 and ϕ b = φ 2 − φ 1 are the local phase differences, and α a , b = k d a , b is the accumulated phase of the photon propagating between two coupling points, with d a , b being the effective propagation distance of two coupling locations due to the computation result of k ⋅ x [28].

In the double-waveguide system with four output ports, the transport characteristics of the incident photons are engineered by the scattering probabilities T a ( b ) = | t a ( b ) | 2 and R a ( b ) = | r a ( b ) | 2 . In this section, we investigate the nonreciprocal scattering properties of single photons in the single-waveguide structure, in which the second-point coupling Γ b = 0 is set. When a single photon is input from the right port of the bus waveguide, the propagation process is equivalent to the case that the forward photon coming from the left side of the waveguide. Therefore, applying the similar calculation methods, the backward-direction transmission amplitude t B of the incident photon from right to left can be obtained by replacing the opposite forward-direction transmission t F in Equation (4) with ϕ a → − ϕ a . The two transmission amplitudes are expressed as t F = t a and t B = 1 − i   Γ a Q 2 [ 1 + e i ( − ϕ a − α a ) ] [ 1 + e i ( α a + ϕ a ) ] / ( Q 1 Q 2 − η 2 ) , while two reflection amplitudes in the forward and backward directions are the same even exchanging the coupling phase ϕ a .

Clearly, for the zero dissipation with γ r = 0 , the single-photon transmission is reciprocal and has the same values of T B = | t B | 2 = T F = | t F | 2 . Thus, the optical nonreciprocity is caused by the difference between T B and T F , arising from the interference modification between backward propagation modes via the resonator dissipation, and the joint effects of two phases between two coupling points. To quantitatively describe the nonreciprocity in the system, the isolation contrast ratio is introduced as T I = T B − T F . It is easily observed that the system displays reciprocity with T B = T F when T I = 0 , while perfect nonreciprocal transmission is shown for T B = 0 , T F = 1 or T B = 1 , T F = 0 when T I = ± 1 .

We first examine the influence of the phase modulation on the nonreciprocal single-photon transport properties. The transmission T B , T F and the contrast ratio T I versus the detuning Δ ω and the phases are plotted in Figure 2. When ϕ a = π / 2 and α a = π / 4 in Figure 2(a), one can find that T B and T F display different transmission behaviors with two separate line shapes. T F represents a Lorentzian line shape and has larger transmission probability than T B , while the complete transmission blockading with T B = 0 occurs at the resonance point Δ ω = 0 , and the peak of T I is less than 1 in this situation. Especially, for ϕ a = α a = π / 2 in Figure 2(b), the backward-direction moving photons maintain complete transmission over the entire range of the frequency detuning, as displayed by the straight red dashed lines with T B = 1 . This can be observed from the product term of the numerator of [ 1 + e i ( α a − ϕ a ) ] = 0 in the front formula of T B . It means that the left-going photons propagate directly through the waveguide without absorption, since the resonator is decoupled from the waveguide, and the

Figure 2. (Color online) Transmission T F (solid blue curves), T B (dashed red curves) and the contrast ratio T I (dash-dotted green curves) versus the detuning Δ ω for different phases in (a, b). T I as a function of the phase ϕ a or α a in (c) and (d), respectively. (a) ϕ a = π / 4 and α a = π / 2 , (b) ϕ a = π / 2 and α a = π / 2 , (c) α a = π / 2 , and (d) ϕ a = π / 2 . Here, γ r = 2 and η = 1 are set. All parameters except the phases are in units of Γ a .

resonator-waveguide coupling strength becomes zero via the phase modulation. Accordingly, the right-moving photons can obtain the zero transmission of T F = 0 , whereas the perfect nonreciprocal transport for the contrast ratio T I with the maximum value of 1 emerge at Δ ω = 0 , arising from the constructive and destructive interferences from two phases. The contrast ratio T I as a function of the local phase and the accumulated phase are plotted separately in Figure 2(c) and Figure 2(d), and the two phase modulations are similar when fixing one of the two phase parameters. Note that T I = 1 ( T I = − 1 ) emerges corresponding to ϕ a = π / 2 ( 3 π / 2 ) or α a = π / 2 ( 3 π / 2 ) , it implies that the symmetrical processes in two directions can be effectively regulated by the phase modulation.

In the numerical calculations, all these parameters except the phases are in units of the coupling Γ a for simplicity. Here, γ r , η , and Γ a are of the same order of magnitude. These parameters are adopted to be close to the experimental results in a coupled waveguide-resonator system [30], where they are around the same order of magnitude, such as Γ a / 2 π = 6.9 MHz, η / 2 π = 5.5 MHz, and γ r / 2 π = 2.5 MHz. Although ω r = 5.94 GHz is large and not on the same order of magnitude of MHz, we focus on the detuning Δ ω = ω − ω r between the incident photon frequency and the resonator’s intrinsic frequency.

To get a deeper insight into the control of the nonreciprocal behaviors via the phases, Figure 3 displays these transmission varying with the phase parameters. By comparing Figure 3(a) and Figure 3(b), one can find from the inverse peak and dip locations that the two transmission probabilities T B and T F have opposite responses for the local phase difference ϕ a . For example, for the fixed phase of α a = π / 2 , T F = 1 and T B = 0 for ϕ a = 3 π / 2 , while T F = 0 and T B = 1 for ϕ a = π / 2 . Figure 3(c) represents the contrast ratio of the transmission probabilities for the left-going and right-going photons. As expected, two optimal nonreciprocal transmission windows are presented in two similar bright red or blue regions. The maximum T I = ± 1 is obtained by adjusting ϕ a to be π / 2 and 3 π / 2 , respectively. Symmetrically, similar conclusions can be drawn for the accumulated phase α a . Figure 3(d) shows that four symmetric dark-red elliptical region with high-value ratio of T I appears in the ( α , ϕ ) plane, and two peaks and two dips in the centers of the four areas indicate that perfect single-photon nonreciprocal transmission of T I = ± 1 can be achieved only when the phases meet the condition of α a = ( π / 2 , 3 π / 2 ) and ϕ a = ( π / 2 , 3 π / 2 ) .

Figure 3. (Color online) Transmission T F , T B and the contrast ratio T I as a function of ϕ a and Δ ω in (a), (b) and (c), respectively. T I as a function of ϕ a and α a in (d). α a = π / 2 in (a, b, c). Other parameters are same as that in Figure 2.

Note that the resonator dissipation is also a crucial parameter to function the system nonreciprocity. Further, the influence of the dissipation γ r and the inter-mode backscattering strength η on the nonreciprocal transmission T I is also researched. When α a = ϕ a = π / 2 is fixed, these parameters to obtain the maximum value of T I = 1 are governed by the equation of γ r 2 − 2 γ r + η 2 − Δ ω 2 = 0 or Δ ω ( γ r − 1 ) = 0 , which is reduced by T F = 0 . As shown in Figure 4(a), at the resonance point Δ ω = 0 , a dark-red narrow arc-shaped region emerges in the ( η , γ r ) plane, which means that the maximum T I can be achieved via multiple parameter values, since there is a series of solutions related to dissipation and backscattering strength for the result of γ r = 1 ± 1 − η 2 . However, for the nonzero detuning, γ r = 1 is determined, and thus η can get a single definite solution η = 1 + Δ ω 2 , which is related to the detuning. As an example, Figure 4(b) displays a maximum peak with ( η , γ r ) = ( 3.16 , 1 ) located at the center of the dull-red circular region, when Δ ω = 3 is set. Therefore, over a wide frequency range, perfect nonreciprocity can be regulated by tuning the dissipation and the inter-mode backscattering strength.

Figure 4. (Color online) T I as a function of η and γ r for different detunings: (a) Δ ω = 0 and (b) Δ ω = 3 . Other parameters are same as that in Figure 2(b).

As a comparison with the photon transport in the single-waveguide case, quantum routing of single photons in the coupled double-waveguide waveguide is also examined. When a photon enters into the bus waveguide, it will be scattered by the resonator, and the photon probability flow will be distributed at four ports. Note that the total sum of the photon flow will remain unchanged (see the green straight line), when the dissipations are not taken into account, since the conservation relation T m + R m + T n + R n = 1 holds on. Note that the sum of the photon flow will remain unchanged in the absence of the resonator dissipation, because the conservation relation T a + R a + T b + R b = 1 holds, as shown by the green straight lines in Figure 5(a) and Figure 5(b). For simplicity, γ r = 0 and Γ b = Γ a are assumed in the numerical calculations. Then, we consider how these parameters affect the routing features transferring between two different waveguide channels, and thus we focus on the routing efficiency from the bus waveguide to another. For comparison, we first consider the single-point coupling in the coupled system for the straight waveguides, corresponding to V a 2 = V b 2 = 0 . In this case, the routing amplitudes in Equation (4) can be simplified as

Figure 5. (Color online) Four-port routing S ( T a for dash dotted blue curves, R a for dotted red curves, T b for solid cyan curves, and R b for dashed pink curves) versus the frequency detuning Δ ω . (a) the single-point coupling case independent of the phases, (b) ϕ a = ϕ b = π / 4 , (c) ϕ a = ϕ b = π / 2 , and (d) ϕ a = π / 2 , ϕ b = 3 π / 2 . T b = R a in (a), R b = R a = 0 in (c), and T b = R a = 0 in (d). These curves corresponding to equal transmission overlap completely in the scattering spectra. The green straight line denotes the total sum of the photon flow. T b as a function of α a and ϕ a in (e), and R b as a function of ϕ a and ϕ b in (f). ϕ a = ϕ b and α a = α b are set in (e), while only α a = α b is taken in (f). These parameters are Γ b = 1 , γ r = 0 , η = 2 , and α a = α b = π / 2 .

with Q = Δ ω + i γ r + i Γ a / 2 + i Γ b / 2 . It is evident that these scattering probabilities are strongly independent of the phases. Due to the direction-matching effect, the scattering of incident light through the symmetrical coupling channels of Γ b = Γ a into two left and right ports in the two waveguides leads to the same probability, i.e., T b = R a . The routing spectra of these two ports overlap as shown in Figure 5(a), and they have the equal maximum peak values of 0.25. Correspondingly, another port routing R b scattered by the resonator is bigger than T b , but the routing efficiency is still low with less than 0.5.

While for the two-point coupling case of the resonator and the waveguides, routing efficiency is improved by the phase modulation. Figure 5(b) shows that the routing efficiency of T b increases compared to the single-point coupling case when ϕ a = ϕ b = π / 4 and α a = α b = π / 2 . Specially, when ϕ a = ϕ b = π / 2 and γ r = 0 in Figure 5(c), the routing efficiency of T b can reach the maximum value of unity at the resonance point Δ ω = 0 via the asymmetrical waveguide-resonator coupling, meanwhile the two left routing ports are decoupled from waveguides with the zero couplings, which means the transfer to the two ports are blocked, and R b = R a = 0 for all frequencies. Therefore, the directional routing to the right port of the waveguide W b can be implemented by modulating the phases, under the phase-dependent interference effects. Similarly, the directional routing to the left port of the waveguide W b can be also performed by modulating the phase as ϕ a = π / 2 and ϕ b = 3 π / 2 . In the case T b = R a = 0 for all frequencies, and two peaks with 100% routing probability emerge in Figure 5(d). These results clearly show that the proposed routing scheme can transfer photons from the input port to another arbitrarily selected port in the drop waveguide with a perfect routing through the appropriate phase modulation. Moreover, Figure 5(e) and Figure 5(f) display the dependence of the routing efficiency of T b on the phases, respectively. It is observed that for α a = α b = π / 2 , the maximum value of targeted single-photon router with 100% routing probability T b emerges at ϕ a = ϕ b = π / 2 or ϕ a = ϕ b = 3 π / 2 , while the maximum value of R n appears only when the phases meet the condition of ϕ a = π / 2 and ϕ b = 3 π / 2 when α a = α b = π / 2 is fixed.

It is worth noting that inter-mode backscattering strength η also plays a significant role in achieving directional routing. As seen from Figure 6(a), for η = 1 , two peaks for R b with the maximum routing probability of unity are presented, while a single peak for T b with the peak value less than unity appears at the resonance point. Increasing η from 1 to 3, two peaks for T b emerge, which indicates that the the maximum routing probability of T b is sensitive to η . Figure 6(b) also shows the continuous variation trend of two routing maximum in a wide frequency range for the varying η . It shows that the maximum routing probability R b m = 1 independent of η arises, while there is the maximum routing probability T b m = 1 when η exceeds 2. For comparison, Figure 6(c) and Figure 6(d) also display separately the dependence of the routing probability of T b and R b on η . It can be found that two similar V-shaped regions denoting high routing efficiency appear in two planes, but both have significantly different base positions of 0 and 2. Through the analysis of Equation (4), the maximum transfer rate T b = 1 can be realized when satisfying the condition Δ ω = ± η 2 − 4 . Mathematically η ≥ 2 is required to get the two real number solutions.

Figure 6. (Color online) (a) Routig efficiency T b (thin curves) and R b (thick curves) as a function of the detuning Δ ω for different backscattering strengths: η = 1 (solid lines) and η = 3 (dashed lines). (b) The maximum routing peak T b m (solid blue line) and R b m (dashed red curve) in the frequency range versus η . (c) T b and (d) R b as a function of η and Δ ω . Γ b = 1 , γ r = 0 , and α a = α b = ϕ a = π / 2 . ϕ b = π / 2 for T b and ϕ b = 3 π / 2 for R b .

Physically, the scenario results from the mode direction-matching transmission. The routing photon into the left port of waveguide W b is redirected by the counter-clockwise mode for the directly incident photon, and the routing efficiency can reach 1 at some frequencies for these straight-through photons. However, the routing photon into the right port of waveguide W b originates from the photon transmitted along the clockwise mode, which is coupled through backscattering. When the backscattering strength is weak, not all photons can be transferred to the right port, and thus the routing efficiency can not reach 1. When increasing the backscattering strength, more photons enter this channel, thereby reaching the maximum efficiency value of 1. Experimentally, the resonator-waveguide coupling strength presented by the phases can be tuned by the their separation and mode polarization direction. The backscattering strength is adjustable via a single subwavelength scatterer, which is attached on the resonator surface and results in a modification of the mode density by varying the Rayleigh scattering rate [31].

The effects of some other parameters on the quantum routing are remarked in Figure 7. When the different couplings are taken into accounted, it can be found in Figure 7(a) that the maximum routing efficiency emerges for the equal coupling Γ b / Γ a = 1 . Figure 7(b) further shows that the maximum value in the wide frequency scope occurs at the equal coupling, since the complete photon transfer between two waveguides appears only over the identical coupling channels. When the inevitable dissipation is considered in Figure 7(c), the routing peaks decrease

Figure 7. (Color online) T b (thin curves) and R b (thick curves) as a function of the detuning Δ ω for different parameters. (a) Γ b = 0.5 (solid lines), Γ b = 1 (dashed lines), and Γ b = 1.5 (dash dotted lines). (c) γ r = 0 (solid lines), γ r = 0.2 (dashed lines), and γ r = 1 (dash dotted lines). The maximum routing peak T b m (solid blue line) and R b m (dashed red curve) in the frequency range versus Γ b in (b). Other parameters are the same as that in Figure 5(c).

with the increase of dissipation due to the photon leakage of the resonator. However, high routing efficiency is available for the general loss, such as T b = 0.83 at γ = 0.2 .

In summary, we have examined the tunable single-photon nonreciprocal transmission and directional routing in a microresonator-waveguide structure with multiple coupling points. Applying a full quantum approach related to the real-space Hamiltonian, the routing probabilities of single photons from the incident waveguide to another are obtained exactly. Our results show that the perfectly nonreciprocal transmission of single photons can be manipulated by the combinational effect of the local coupling phase and the accumulated phase between two coupling points of the resonator and the bent waveguides. Similarly, the high-efficiency targeted quantum routing in two ports of the drop waveguide can be performed by varying the related phases of the resonator due to the phase-dependent interference effect. The optimal parameter sets of the perfectly nonreciprocal transmission and the directional routing with the unity probability in the ideal case are presented as ( α a = α b = ϕ a , ϕ b , η , γ ) = ( π / 2 , π / 2 , 2 , 2 ) and ( α a = α b = ϕ a , ϕ b , η , γ ) = [ π / 2 , π / 2 ( 3 π / 2 ) , 2 , 0 ] , respectively. Of course, the bending loss or manufacturing tolerance in the coupling distance and the approximate linear dispersion are ignored for simplicity, which indeed reduces faintly the scattering performance of the system. These results are expected to provide potential applications in quantum photonic circuits, integrated optical chips and quantum information processing.

This work was supported by Key Laboratory of Low Dimensional Quantum Materials and Sensor Devices of Jiangxi Education Institutes (No. GanJiaoKeZi-20241301), and Jiangxi Provincial Natural Science Foundation (Grant No. 20212BAB201014).