Topological bound modes in anti-PT- symmetric optical waveguide arrays

We investigate the topological bound modes in a binary optical waveguide array with anti-parity-time (PT) symmetry. The anti-PT-symmetric arrays are realized by incorporating additional waveguides to the bare arrays, such that the effective coupling coefficients are imaginary. The systems experience two kinds of phase transition, including global topological order transition and quantum phase transition. As a result, the system supports two kinds of robust bound modes, which are protected by the global topological order and the quantum phase, respectively. The study provides a promising approach to realizing robust light transport by utilizing mediating components. © 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Studies of topological photonics have so far mostly dealt with Hermitian systems. Recently, a growing interest is paid to investigating topological properties of non-Hermitian systems with gain and loss, especially the Parity-Time (PT) -symmetric systems [18][19][20][21][22][23][24][25][26][27]. The PT symmetry is introduced to optics in 2008 for the first time [28], which requires the system Hamiltonian H obeys relation [PT, H] = 0. The system can possess real spectrum in the presence of gain and loss below PT-broken threshold [28][29][30][31]. The first experimental observation of PT symmetry is by utilizing two coupled semiconductor waveguides [29], where optical gain is provided though nonlinear two-wave mixing. Many fascinating phenomenon are reveled in PT-symmetric systems [31][32][33][34][35], such as double refraction [28], loss induced transparency [36], non-reciprocal propagation [29], and chiral mode switching [37][38][39]. Topological phase transition also emerges in PT-symmetric systems, which is closely related to the system degeneracy, known as exceptional point (EP) [19,21,23,40]. The topological bound modes with PT symmetry are firstly observed in coupled waveguide arrays [19], which are Hermitian-like and can be sustained with vanished gain and loss factor. Interestingly, PT-symmetric systems support unique robust bound mode with no Hermitian counterparts. topological or Anti-PT-s the anticomm PT symmetry later studies symmetric as circuits [49], state [45,46,4 For example demonstrated encircling an anti-PT symm symmetry [40 with C being non-Hermitian determining t coupled syste induced by c can be selecti utilized to enh efficiencies, a In this wo arrays with i incorporating coupling stren waveguides. W related to the the case in He can emerge b which the ima global topolog

Anti-PT s
We start by in coupling. The The bound m rder but with d ymmetric syst mutator, have al is in [44], whe demonstrate well [45,47] [58] and also in open ring resonator with asymmetric scattering [59]. However, the imaginary part of coupling coefficient is small compared with its real part. Complex coupling is also realized by dynamically modulating refractive index along wave propagation direction [60,61]. However, it increased difficulties for experimental implementation. We follow another approach by incorporating assistant waveguides into the bare array to realize imaginary coupling [45,62,63]. The static on-site complex potentials simplify its practical implementation and the coupling strength can be tuned by the amount of gain and loss. As shown in Fig. 1(a), the proposed weekly coupled array have four waveguides in each unit cell, which are labeled as A, B, C, and D with C and D being the assistant waveguides. The corresponding effective detuning of propagation constants is denoted as δ 1 , δ 2 , δ 3 , and δ 4 . Only nearest coupling is considered. The coupling is represented by J 1 and J 2 , which are assumed to be real and positive. The wave should evolve according to the coupled mode equation (CMT), which is given by (1) where a n , b n , c n , and d n denote the field amplitudes in respect waveguide in nth unit cell. If the absolute value of detuning |δ 3 | and |δ 4 | are much larger than the coupling J 1 and J 2 , the assistant waveguides C and D can be eliminated adiabatically [45], that is, (5) The equivalent system described by Eq. (5) is depicted in Fig. 1(b). Each unit cell contains two waveguides A and B with effective detuning of propagation constants being ± ∆/2, respectively. The intra-and inter-coupling coefficients are ic 1 and ic 2 , which are purely imaginary values. In order to realize the imaginary coupling, Eq. (4) indicates that the bare arrays should be with gain and the assistant waveguides are with loss. Moreover, the amount of loss significantly exceeds the amount of gain, that is, |δ 3 |, |δ 4 |  |δ 1 |, |δ 2 |. As the adiabatic elimination requires |δ 3 |, |δ 4 |  J 1 , J 2 , the bare coupling strength should also be much larger than effective imaginary coupling, that is, For periodic boundary, the Bloch theorem can be utilized and the system Hamiltonian is given by The time-reversal operation T transforms a z-independent operator to its complex conjugate, while the parity operator P exchanges locations of the modes [45]. Thus, the system Hamiltonian fulfills the relation (PT)H(PT) −1 = −H, which implies the system is anti-PTsymmetric. The anti-PT symmetry can be also defined by Pauli matrix σ i , which requires [23]. Chiral symmetry is also important in determining topological properties of the system, which is defined by σ z H(φ)σ z = −H(φ). The system possesses chiral symmetry only when the detuning is Δ = 0. This ensures the Berry phase of individual band is quantized. The eigenvalues of Eq. (6) are solved to be The corresponding right and left eigenvectors are figured out as where Δ k = arctan(−2iρ k /Δ) and ρ(φ) = c 1 + c 2 exp(−iφ) = ρ k exp(iφ k ). The system has three different phases according to anti-PT symmetry. When |c 1 − c 2 | > Δ/2, the system is in anti-PT-symmetric phase and eigenvalues are purely imaginary across the Brillion zone. When |c 1 − c 2 | < Δ/2, the system is in mixed anti-PT-symmetric and broken phases and eigenvalues become complex-valued at the edge of Brillion zone. When |c 1 + c 2 | < Δ/2, the anti-PT symmetry is fully broken and the eigenvalues develop into purely real values in the entire Brillion zone. We now investigate the topological invariant and show the topological transition arises in anti-PT-symmetric systems. In non-Hermitian systems, the Berry phase for individual band is defined by / , where ± labels the upper and lower bands and k represents Bloch momentum. The integration is taken over the 1D Brillouin zone. Substituting Eq. (8) into the expression for φ ± B, the Berry phase is derived as When the chiral symmetry holds, that is, Δ = 0, the second term of Eq. (9) is vanished. φ ± B is quantized to be an integer multiple of π, which is φ ± B = π for c 1 < c 2 and φ ± B = 0 for c 1 > c 2 . In this situation, the Berry phase of individual band can be still regarded as topological invariant and utilized to indicate the topological phase transition. However, as Δ ≠ 0, the second term is not vanished. The Berry phase of a given band is not quantized and becomes complex-valued. Interestingly, the global Berry phase φG Here we w that is, the mo the amplified input beam. T of Gaussian b energy of Gau is excited, the subject to mo and PT-symm . Here in anti the imaginary me. The system our study enri ucture is close ning at a band entire Hamilton n the role of Im tive Im(k z) , wou e can be extract part of band st −Ln(E out /E in )/ nd L presents t evolve analog function [68].
the mode profiles eld fidelity as a fun eguides C and D fo × 10 −3 μm −1 .

Normal to
In the above s its topologica interface betw Now we study   On the oth the system is strength is a waveguides. T solely by gain previous stud (a n ', b n ') T = (a Both inter-an As Δ = 0, the zero, that is, energy of topo  Fig. 4(c), the shing amplitu waveguide B rotection on th a continuous d her hand, as th topological tr ctually contro Therefore, our n and loss con dy of SSH-like a n , ib n ) T Figure 5 p is simulated u 5(b), the incid confined at th different prop results shown is zero. The s and 5(d). The understood. W modes are ex modes, the bu the edge mod modes are mo

Abnorma
In spite of the symmetric ar refers to an a quantum phas  9) is van individual ban results can be symmetric. Th that is, c 1 − c 2 complex. In p and the system three phases transition den from one to unusual robus presents the pr using Rigorou dent fields are he ends of wav pagation distan n in Fig. 4(b) a situation is diff e wave diffuse When wave is l xcited simultan ulk modes dom des, one may p ostly confined a al topologica e general topol rays also expe abrupt change se transition em 6. The Berry phase phase. The red ive coupling stren (a) and 6(b) plo ults are numeric nished and the B nd is not quant e divided into th he eigenvalues 2 < Δ/2 < c 1 + phase III, that i m is in fully a have distinct notes the transit another. Ther st bound modes ropagation of t s coupled-wav the eigenmode eguide arrays d nce z is almost as the imaginar ferent when a s es into the str launched from neously. As so minated after e pump the two w at the terminati with increasing In the limit Δ = Δ is not vanis ing by increasi is, c 1 − c 2 < Δ/2 B ± are imagina complex and ues are real in Berry phase in hase is not co tem-associated three different transition inter The wave pro d [69]. In Fig.  4(d). The fields energy of two ed. This agrees ants of two edg As shown in F ion. The reaso ulk modes and amplified than tance. In order termination as has shown tha uantum phase t 43]. Here we e arrays as wel aginary part of the respectively. The g the detuning Δ = 0, the secon shed, the Berry ing the detunin 2, the system is ary as well. In the Berry pha the entire Bril this region is ontinuous. An d quantum phas t quantum pha rface.  The bound disorder. To v the structure. c 1 −δ c and c 2 8(b). The per geometric par bound modes waveguide, in Fig. 8(d)