Reciprocal and unidirectional scattering of parity-time symmetric structures

Parity-time symmetry is of great interest. The reciprocal and unidirectional features are intriguing besides the symmetry phase transition. Recently, the reciprocal transmission, unidirectional reflectionless and invisibility are intensively studied. Here, we show the reciprocal reflection/transmission in -symmetric system is closely related to the type of symmetry, that is, the axial (reflection) symmetry leads to reciprocal reflection (transmission). The results are further elucidated by studying the scattering of rhombic ring form coupled resonators with enclosed synthetic magnetic flux. The nonreciprocal phase shift induced by the magnetic flux and gain/loss break the parity and time-reversal symmetry but keep the parity-time symmetry. The reciprocal reflection (transmission) and unidirectional transmission (reflection) are found in the axial (reflection) -symmetric ring centre. The explorations of symmetry and asymmetry from symmetry may shed light on novel one-way optical devices and application of -symmetric metamaterials.

Parity-time ( ) PT symmetric quantum system may possess entirely real spectrum although being non-Hermitian 1-13 . PT symmetric system is invariant under the combined PT operator in the presence of balanced gain and loss. In the past decade, PT -symmetric system has attracted tremendous interests as it possesses unintuitive but intriguing implications. Due to the similarity between the paraxial wave equation describing spatial light wave propagation and the temporal Schrödinger equation for quantum system, the complex refractive index distribution satisfying n * (x) = n(− x) mimics PT -symmetric potentials V * (x) = V(− x), PT -symmetric systems are proposed and realized in coupled optical waveguides through index guiding and a inclusion of balanced gain and loss regions [14][15][16][17] . A number of novel and non-trivial phenomena are found, such as power oscillation 17 , coherent perfect absorbers [18][19][20] , nonreciprocal light propagation 21 in coupled waveguides, and recently the PT -symmetric microcavity lasing [22][23][24] and gain induced large optical nonlinear [25][26][27][28][29] in coupled resonators.
The spectral singularity [30][31][32][33][34][35][36] and invisibility [37][38][39][40][41][42][43] in PT -symmetric system are hot topics, where reciprocal transmission and unidirectional reflectionless in PT -symmetric metamaterial are intriguing features for novel optical devises. These devices are useful for light transport, control and manipulation [44][45][46][47] . The symmetric scattering properties are usually attributed to certain internal symmetry of a scattering centre. For instance, the parity ( )  symmetry, or time-reversal  ( ) symmetry of a scattering centre leads to symmetric reflection and transmission 48 ( -symmetric system without unequal tunnelling amplitude is Hermitian, otherwise, only reciprocal reflection or transmission holds 49 ). Here, we report reciprocal reflection, similar as reciprocal transmission, are both related to the PT symmetry of a scattering centre: The axial (refection) PT symmetry, with respect to the input and output channels, induces reciprocal reflection (transmission). Recent efforts on photonic Aharonov-Bohm effect enable photons behaving like electrons in magnetic field. Effective magnetic field for photons can be introduced in coupled waveguides by bending the waveguides 50 , periodically modulating the refractive index 51 , and the photon-phonon interactions 52 ; or in coupled resonators by magneto-optical effect 53 , dynamic modulation 54 , and off-resonance coupling paths imbalance 55,56 . In this work, we focus on the PT -symmetric structure with balanced gain and loss threading by synthetic magnetic flux, where photons feel a nonreciprocal tunnelling phase between neighbour resonators. The nonreciprocal tunnellings and balanced gain and loss break the  and  symmetry but keep the PT symmetry of the scattering centre. The axial (reflection) PT symmetry will lead to reciprocal reflection (transmission) and unidirectional transmission (reflection). Our properties of a PT -symmetric structure are closely related to the classification of PT -symmetry. The parity operator  is the spatial reflection operator,  is the time-reversal operator. In Fig. 1a , the system is called reflection PT symmetric (Fig. 1b). The red plane indicates the up-to-down (left-to-right) spatial reflection correspondence of axial (reflection) PT symmetry.
In order to address the reciprocal and unidirectional scattering behavior. We study the reflection and transmission of a scattering centre for the left side and right side inputs, respectively. The Hamiltonian of the scattering system is H = H L + H c + H R with H L (H R ) being the Hamiltonian of the left (right) lead. We denote the two scattering states as ψ k L and ψ k R for the input with wave vector k. The forward going and backward going waves are in form of ± e ikj . Combining with the reflection and transmission coefficient, we assume the scattering state wave function on the leads (not at the spectral singularities, see Methods) of left side input as,   In other words, the axial PT symmetry leads to the reciprocal reflection. Notice that we have reciprocal reflec- at any transmission zero t L,R = 0, where one-way pass through is possible. Furthermore,considering the waves with vectors k and − k, the reflection and transmission coefficients further Reciprocal transmission under reflection PT symmetry. As shown in Fig From equations (6,7), we notice the transmission probabilities for the left side and right side inputs are the same, i.e., This indicates the reflection PT symmetry leads to the reciprocal transmission, as observed in Bragg gratings and other PT -symmetric structures [45][46][47] . Notice that we have reciprocal transmission at any reflection zero r L,R = 0, where unidirectional reflectionless is possible [37][38][39][40][41][42][43] . Furthermore, considering the waves with vectors k and − k, the reflection and transmission coefficients further We show that in the present of PT symmetry, the reciprocal reflection or transmission in a scattering centre is protected when the axial or reflection PT symmetry holds even though the  and  symmetry are absent. Moreover, PT symmetry structure may exhibit unidirectional scattering behavior.
PT -symmetric rhombic ring structures. We use a rhombic ring structure (Fig. 1c,d) to elucidate the results. The scattering centre encloses with an effective magnetic flux Φ, photons moving along the rhombic ring structure in clockwise (counterclockwise) direction will acquire an additional direction-dependent phase factor

thus photons tunnelling is nonreciprocal except when
. This is an effective photon Aharonov-Bohm effect creating by synthetic magnetic field [50][51][52][53][54][55][56] . The phase factor ± Φ e i is an analytical function of Φ with period of 2π, it is sufficient to understand the influence of magnetic flux on the scattering by studying Φ in the region π , ) [0 2 . To realize a synthetic magnetic field, two ring resonators are coupled through an auxiliary off-resonant ring resonator. The auxiliary resonator introduces optical paths imbalance when coupling to two primary resonators, the auxiliary resonator can be effectively reduced and create a coupling phase factor between two primary resonators. The coupled resonators under synthesized magnetic field is described by a magnetic tight-binding Hamiltonian 55,56 , where φ = Φ /4 is a nonreciprocal phase shift induced by the magnetic flux in the tunnelling constant. In Fig. 1c, the Hamiltonian of the scattering centre is , where γ is the gain/loss rate. The balanced gain and loss are the origin of the non-Hermiticity realized in the optical systems [14][15][16][17][23][24][25][26][27] . The configuration is axial PT -symmetric with the parity operator acting on the rhombic ring sites defined as Fig. 1d, the Hamiltonian of the scattering centre is , and the configuration is reflection PT -symmetric. In the system, the magnetic flux is inverted meanwhile the gain and loss are switched under the  or  operation. However, the system is invariant under the combined PT operator, i.e., the presence of non-trivial magnetic flux as well as balanced gain and loss both break the  and  symmetry but keep the PT symmetry of the system.
The scattering centre is actually a two-arm Aharonov-Bohm interferometer. Light wave propagates through two pathways (A and B) between the connection sites − 1, 1 and interfere with each other. The interference generates the output which varies as the enclosed magnetic flux. The effective magnetic field is gauge invariant and the magnetic flux acts globally, thus the reflection and transmission are not affected by the nonreciprocal phase Scientific RepoRts | 6:20976 | DOI: 10.1038/srep20976 distribution in the tunnellings for fixed magnetic flux. In the following, we discuss the reflection and transmission of the PT -symmetric rhombic ring structures in details.
The reflection and transmission coefficients for the axial PT -symmetric rhombic ring structure (Fig. 1c)  The reflection and transmission probabilities are functions of the magnetic flux Φ , gain/loss rate γ, and wave vector k. They satisfy Fig. 2b,c), and Fig. 2e,f). In the absence of non-trivial magnetic flux Φ , or gain/loss γ, the system is -symmetric (reflection--symmetric for Φ absence, i.e., left to right by mirror imaging; inversion--symmetric for γ absence, i.e., left to right by 180° rotation), the reflection and transmission are both reciprocal. The non-trivial magnetic flux Φ together with balanced gain and loss γ break the  symmetry. The symmetric transmission in the PT -symmetric system at π ≠ / k 2 is broken, i.e., the transmission is unidirectional at π ≠ / k 2. Moreover, the axial PT symmetry protects the symmetric reflection, therefore, the reflection is reciprocal but the transmission is unidirectional. The white curves in Fig. 2 show the reflection/transmission zeros. At γ = ± (Φ/ ) ≠ k 2 cos cot 2 0, we have t L = 0 or t R = 0 with total reflection = , r 1 L R 2 . This indicates that we only have a non-zero transmission for the right side or left side input, thus the axial PT -symmetric rhombic ring structure allows one-way pass through.
Photons circle in the scattering centre either in a clockwise direction or in a counterclockwise direction, we schematically illustrate the two pathways in Supplementary Fig. 1. The phase difference between two pathways affects the interference in the scattering centre, thus the transmission varies as the effective magnetic flux induced phase difference. The phase difference between clockwise direction and counterclockwise direction of transmission pathways is Φ for the left side input (Supplementary Fig. 1c) or − Φ for the right side input ( Supplementary  Fig. 1d). The transmission pathways are not equivalent in the presence of gain/loss, the interference of phase difference being Φ is different from the interference of phase difference being − Φ . Therefore, the unidirectional ). In Fig. 3, we plot the reflection and transmission probabilities for an axial PT -symmetric rhombic ring structure (Fig. 1c) at several set parameters. Figure 3a is for a system with balanced gain and loss in the absence of magnetic flux, i.e., γ = 1/2, Φ = 0. The gain/loss, being non-Hermitian, plays the role of on-site potentials and is the origin of unidirectional behavior. However, the presence of balanced gain and loss alone does not ensure unidirectional scattering. We notice the reflection and transmission in Fig. 3a are both reciprocal. The scattering is unitary even though the system is non-Hermitian (the balanced gain and loss of this rhombic ring structure γ ( − ) † †

i A A B B can be reduced to an anti-Hermitian interaction
, and the non-Hermiticity of the scattering centre only arises from the anti-Hermitian interaction between ′, ′ A B , which is proved to have unitary scattering 57 ). By introducing magnetic flux to the system, the  and  symmetry is destroyed but the PT symmetry holds. The interference between light waves from the loss arm and the gain arm generates unidirectional transmission for non-trivial magnetic flux. Figure 3b is for a system in the presence of non-trivial magnetic flux, i.e., γ = 1/2, Φ = π/2. The unidirectional transmission zero happens at π = (± / ) k arccos 1 4 , i.e., at k ≈ 0.420π, , which indicates a one-way pass through. Figure 3c is for a Hermitian scattering centre in the presence of non-trivial magnetic flux, i.e., γ = 0, Φ = π/2, we have Hermitian scattering without unidirectional behavior.
In the rhombic ring structure under axial PT symmetry (Fig. 1c), the reflection and transmission coefficients r L , r R , t L , t R diverge at the spectral singularities 30 59 . Now, we turn to discuss the rhombic ring structure under reflection PT symmetry. The configuration is shown in Fig. 1d. Supplementary Fig. 2 schematically illustrates the pathways of photons. The connection sites are linked by two same pathways. In the presence of magnetic flux Φ , photons travelling from left lead to right lead in clockwise direction and counterclockwise direction acquire additional phases + Φ /2 and − Φ /2 in the two pathways ( Supplementary Fig. 2c), respectively. The situation is unchanged for photons travelling inversely from right lead to left lead ( Supplementary Fig. 2d). Equivalently, the upper and lower pathways are undistinguishable. Therefore, only relative phase difference Φ matters (affecting the transmission coefficient) and the transmission is directionless. The reflection and transmission coefficients are calculated as (see Methods) The reflection and transmission coefficients are functions of the magnetic flux Φ , gain/loss rate γ, and wave vector k. They satisfy Figure 4 implies a reciprocal transmission (Fig. 4a,d) and unidirectional reflection (Fig. 4b,c,e,f). In this configuration, the Figure 3. Symmetric reflection under axial PT symmetry. (a) γ = 1/2, Φ = 0, (b) γ = 1/2, Φ = π/2, (c) γ = 0, Φ = π/2. scattering with both reflection and transmission being reciprocal happens in the absence of gain and loss (γ = 0), that is when the system is -symmetric. In the presence of gain and loss (γ ≠ 0), the reflection probability is unidirectional, but the reflection PT symmetry protects the reciprocal transmission. Due to the presence of gain and loss, the probability of the total reflection and transmission after scattering is not unitary, being balanced gain and loss rate dependent. The white curves in Fig. 4 show the reflection and transmission zeros. At k = π/2 and Φ = 0, , the system exhibits unidirectional reflectionless with reciprocal transmission.
In Fig. 5, we plot the reflection and transmission probabilities for a reflection PT -symmetric rhombic ring structure (Fig. 1d) at several set parameters. Figure 5a,b are for balanced gain and loss rate γ = 1/2 with two different magnetic flux Φ = 0 and π/2, respectively. The reciprocal transmission and unidirectional reflection are clearly seen. In Fig. 5a, the unidirectional reflectionless happens at k ≈ 0.27π, 0.73π, . In Fig. 5b, the unidirectional reflectionless happens at k ≈ 0.072π, 0.928π, ; or at k ≈ 0.310π, 0.690π, = . r 0 634 In the absence of gain and loss γ = 0, the reflection and axial PT -symmetric rhombic ring configurations reduce to an identical system. In Fig. 5c, we plots the reflection and transmission probabilities of a scattering centre in the absence of both gain and loss and magnetic flux, i.e., γ = 0, Φ = 0, we observe Hermitian scattering behavior of reciprocal reflection and transmission similar as γ = 0, Φ = π/2 shown in Fig. 3c. Notice that no spectral singularity emerges in the scattering of reflection PT -symmetric rhombic ring system. The system with Φ = 0 leads to input with wave vector k = π/2 both sides invisible that r L = r R = 0, t L = t R = 1 (black crosses in Fig. 4a-c); The system with Φ ≠ 0 leads to input with vector k = π/2 both sides opaque that r L = r R = 1, t L = t R = 0. For the input with wave vector k = π/2, the scattering behavior is very sensitive to the magnetic flux.

Conclusion
We investigate the reciprocal and unidirectional scattering of PT -symmetric structures. We show an insightful understanding of the symmetric scattering behavior, that is associated with the type of PT symmetry, defined as the PT symmetry of the connection sites on the PT -symmetric structures. We find that the axial (reflection) PT symmetry leads to reciprocal reflection (transmission). The transmission (reflection) is unidirectional affected by the magnetic flux and gain/loss, this is because the magnetic flux induced nonreciprocal phase and the gain/loss break the  or  symmetry of the scattering centre. The results are further elucidated using a PT -symmetric rhombic ring structure with enclosed effective magnetic flux describing by tight-binding model. The physical realization of such scattering centre is possible in optical systems such as coupled waveguides array and coupled resonators. Notice that our conclusions are also applicable to the system with nonreciprocal tunnelling being unequal tunnelling amplitude 60 . We believe our findings may shed light on coherent light transport and would be useful for applications of quantum devices with inherent symmetry, in particular, for novel unidirectional optical device designs that not limited to optical diodes using synthetic PT -symmetric metamaterial.

Methods
Schrödinger equations. The input and output leads are described by two semi-infinite tight-binding chain.