Topological Floquet edge states in periodically curved waveguides
Bo Zhu, Honghua Zhong, Yongguan Ke, Xizhou Qin, Andrey A. Sukhorukov, Yuri S. Kivshar, Chaohong Lee
aa r X i v : . [ phy s i c s . op ti c s ] J un Topological Floquet edge states in periodically curved waveguides
Bo Zhu,
1, 2
Honghua Zhong,
1, 3
Yongguan Ke,
1, 2
Xizhou Qin, Andrey A. Sukhorukov, Yuri S. Kivshar, and Chaohong Lee
1, 2, 5, ∗ Laboratory of Quantum Engineering and Quantum Metrology, School of Physics and Astronomy,Sun Yat-Sen University (Zhuhai Campus), Zhuhai 519082, China State Key Laboratory of Optoelectronic Materials and Technologies,Sun Yat-Sen University (Guangzhou Campus), Guangzhou 510275, China Institute of Mathematics and Physics, Central South University of Forestry and Technology, Changsha 410004, China Nonlinear Physics Centre, Research School of Physics and Engineering,The Australian National University, Canberra ACT 2601, Australia Synergetic Innovation Center for Quantum Effects and Applications,Hunan Normal University, Changsha 410081, China (Dated: June 4, 2018)We study the Floquet edge states in arrays of periodically curved optical waveguides describedby the modulated Su-Schrieffer-Heeger model. Beyond the bulk-edge correspondence, our studyexplores the interplay between band topology and periodic modulations. By analysing the quasi-energy spectra and Zak phase, we reveal that, although topological and non-topological edge statescan exist for the same parameters, they can not appear in the same spectral gap . In the high-frequency limit, we find analytically all boundaries between the different phases and study thecoexistence of topological and non-topological edge states. In contrast to unmodulated systems, theedge states appear due to either band topology or modulation-induced defects. This means thatperiodic modulations may not only tune the parametric regions with nontrivial topology, but mayalso support novel edge states.
I. INTRODUCTION
Recently, topological photonics has emerged as a newapproach to manipulate properties of light under continu-ous deformations [1]. Electromagnetic topological stateshave been found in both microwave [2–4] and optical [5–7] regimes. Similar to topological insulators for electrons,photonic topological insulators have also been created [1–16]. Beyond conventional topological phenomena in lin-ear Hermitian systems, topological gap solitons have beenfound in nonlinear optical systems [17], and it was shownthat topological states can survive in non-Hermitian sys-tems [18]. Moreover, periodic modulations can bring sev-eral novel topological properties usually absent in theirnon-modulated analogues [9, 19–26].Bulk-edge correspondence [27, 28] is a well-establishedprinciple for two-dimensional (2D) topological systems.It establishes the exact correspondence between bulkstates subjected to periodic boundary conditions (PBCs)and edge states in the systems with open boundary con-ditions (OBCs). Up to now, topological edge states havebeen found in several 2D photonic systems [10, 13, 29, 30].However, for one-dimensional (1D) lattice models, edgestates have been shown to appear in periodically modu-lated but non-topological lattices [31, 32]. This suggeststhat edge states can be induced by either topology or pe-riodic modulations. Here, we wonder whether topologicaland non-topological edge states may coexist and, if they ∗ Corresponding author.Email: [email protected], [email protected]. may coexist, how to distinguish between topological andnon-topological edge states.In this work, we study the Floquet edge states (FESs)in arrays of periodically curved optical waveguides de-scribed by a periodically modulated Su-Schrieffer-Heeger(SSH) model [33]. We analyse, for the first time to ourknowledge, the interplay between band topology and pe-riodic modulations, and describe the coexistence of bothtopological and non-topological edge states supported bythe same parameters. Our results show that, for a specificgap, the Zak phase Z G m is either 0 or π , so that the topo-logical edge states appear only in the gap of Z G m = π .Through controlling both modulation frequency and am-plitude, we may drive the system from non-topologicalto topological regime, and vice versa. We demonstrateanalytically that periodic modulations induce a virtualdefect at the boundary , being the key mechanism for theformation of non-topological edge states.The paper is organized as follows. In Sec. II, we in-troduce our physical model and derive its coupled-modeequations. In Sec. III, we calculate the quasi-energy spec-tra under OBC. In Sec. IV, by employing the multi-scaleperturbation analysis, we give the effective coupled-modeequations and demonstrate the periodic modulations caninduce virtual defects at boundaries. The FESs includedefect-free surface states and Shockley-like surface states,which induced by virtual defects and the alternatingstrong and weak couplings between waveguides, respec-tively. In Sec. V, we analytically obtain the asymptoticphase boundary and numerically give the phase digramof appear FESs, respectively. We explore the topologi-cal nature of all FESs via calculate the bulk topologicalinvariant Zak phase. We find that Shockley-like surfacestates are topological FESs and defect-free surface statesare non-topological FESs. A brief summary is given inSec. VI. II. MODEL
We consider an array of coupled optical waveguides,where the waveguides are periodically curved along thelongitudinal propagation direction, see Fig. 1. The lightfield ψ ( x, y, z ) obeys the paraxial wave equation − i ∂ψ∂z = λ ′ πn ′ (cid:16) ∂ ∂x + ∂ ∂y (cid:17) ψ + 2 πλ ′ ν ( x, y, z ) ψ, (1)where λ ′ is the optical wavelength in vacuum, n ′ isthe medium refractive index, and ν ( x, y, z ) describesthe refractive index at ( x, y, z ). The waveguide centers x n ( z ) = x n ( z + T ) are periodically curved along thelongitudinal direction with the curving period T muchlarger than the inter-waveguide distance ∆ x . Here weset x n ( z ) = n ∆ x + A [cos( ωz ) −
1] with the modulationamplitude A and the modulation frequency ω . Figure 1. Schematic diagram of waveguide arrays curvedalong the propagation direction of light ( z -axis). The center-to-center spacing along the x-axis is fixed as ∆ x , and the onealong the y-axis is either 0 or ∆ y intermittently. The couplingstrength is either τ or τ intermittently. (a) τ /τ < τ = τ , and (b) τ /τ > τ = τ . By implementing the coordinate transformation: [ˆ z = z , ˆ y = y , ˆ x ( z ) = x − x ( z )], we have ∂ x = ∂ ˆ x , ∂ y = ∂ ˆ y and ∂ z = ∂ ˆ z − ˙ x ∂ ˆ x . Therefore the field ψ (ˆ x, ˆ y, ˆ z ) obeys − i ∂ψ∂ ˆ z = − i ˙ x ∂ψ∂ ˆ x + 2 πλ ′ νψ + λ ′ πn ′ (cid:16) ∂ ∂ ˆ x + ∂ ∂ ˆ y (cid:17) ψ. By applying the gauge transformation ψ = φ exp { i πn ′ λ ′ (2 ˙ x (ˆ z )ˆ x (ˆ z ) − Z ˆ z ˆ x ( ξ ) dξ ) } , the paraxial wave equation (1) can be written as − i ∂φ∂ ˆ z = λ ′ πn ′ (cid:16) ∂ ∂ ˆ x + ∂ ∂ ˆ y (cid:17) φ + 2 πλ ′ νφ − πn ′ λ ′ ¨ x ˆ xφ. Expanding the field into a superposition of the single-mode fields in individual waveguides φ (ˆ x, ˆ y, ˆ z ) = X n ϕ n (ˆ z ) a n (ˆ x, ˆ y ) , we obtain the coupled-mode equations − i dϕ n dz = τ n ϕ n +1 + τ n − ϕ n − + D n ϕ n − η ¨ x nϕ n , where η = 2 πn ′ /λ ′ as a normalized optical frequency, and τ n = 2 πλ ′ Z Z a ∗ n (ˆ x, ˆ y ) ν (ˆ x, ˆ y, ˆ z ) a n +1 (ˆ x, ˆ y ) d ˆ xd ˆ y,D n = 2 πλ ′ Z Z a ∗ n (ˆ x, ˆ y ) ν (ˆ x, ˆ y, ˆ z ) a n (ˆ x, ˆ y ) d ˆ xd ˆ y. By performing a transformation ϕ n = exp[ iηAω ˆ x n sin( ωz ) + iD n z ] u n , we derive the coupled-mode equations become as − i du n dz = τ n exp[ iηAω (ˆ x n +1 − ˆ x n ) sin( ωz )] u n +1 (2)+ τ n − exp[ − iηAω (ˆ x n − ˆ x n − ) sin( ωz )] u n − . Here η = 2 πn ′ /λ ′ , u n denotes the complex field ampli-tude for the n -th waveguide with n being the waveguideindex. As the center-to-center waveguide spacing alongthe x-axis is constant (i.e. ˆ x n +1 − ˆ x n = ˆ x n − ˆ x n − =∆ˆ x = 1) and one along the y-axis is either 0 or ∆ y intermittently, the hopping strengths can be written as τ n = { [1 − ( − n ] τ + [1 + ( − n ] τ } and the maximumhopping strength τ = max { τ , τ } is fixed. By adjustingthe distance ∆ˆ y , one may tune the values of τ n .Without loss of generality, we set η = 1 and τ = 1.Therefore the system can be described by the periodicallymodulated SSH-like Hamiltonian H ( z ) = N X n =1 ( τ n exp[ iAω sin( ωz )] u ∗ n u n +1 + h.c. ) , (3)with 2 N being the total number of optical waveguides.Chiral symmetry is represented by the sublattice oper-ator Γ = N P n u ∗ n − u n − − N P n u ∗ n u n , which is unitary,Hermitian and local. Obviously, Γ H Γ = − H , this meansthat this periodically modulated SSH-like Hamiltonianhas chiral symmetry [34]. On the other hand, the aboveHamiltonian also has time reversal symmetry, i.e. it isinvariant under the transformation [ z → − z, i → − i ]. III. FLOQUET ENERGY SPECTRUM
Since the system is invariant under z → z + T , accord-ing to the Floquet theorem [20], the steady states of thecoupled-mode equation (2) follow u n ( z ) = e − iEz + ∞ X χ = −∞ e − iχωz c n,χ , where c n,χ is the amplitude of the χ -th Floquet state.Substituting the above Floquet expansion into thecoupled-mode equations, one obtain quasi-energy equa-tion in the Floquet space Ec n,χ = + ∞ X χ ′ = −∞ τ n − e − iηAω sin( ωz ) e − i ( χ ′ − χ ) ωz c n − ,χ ′ + + ∞ X χ ′ = −∞ τ n e iηAω sin( ωz ) e − i ( χ ′ − χ ) ωz c n +1 ,χ ′ + + ∞ X χ ′ = −∞ χ ′ ωe − i ( χ ′ − χ ) ωz c n,χ ′ + X χ ′ = χ e − i ( χ ′ − χ ) ωz Ec n,χ ′ . We introduce the average over one modulation period forall z -dependent quantities and obtain the quasi-energyeigen mode equation Ec n,χ = + ∞ X χ ′ = −∞ τ n − J χ − χ ′ c n − ,χ ′ + + ∞ X χ ′ = −∞ τ n J χ ′ − χ c n +1 ,χ ′ + χωc n,χ , (4)where J χ ′ − χ is the Bessel function J χ ′ − χ ( Aω ). To obtainthe quasi-energy spectrum, one needs to truncate the Flo-quet space. In our calculation, we choose χ ′ , χ ∈ [ − X, X ]and Y = 2 X + 1 is the truncation number.Now we discuss the quasi-energy spectra under OBC.In Figs. 2(a) and 2(b), we show the scaled quasi-energy E/ω versus the scaled modulation amplitude
A/A . Inour calculation, A is given by the first zero-point of J ( A ω ), τ /τ = 1 .
2, 2 π/ω = 3, and the total latticenumber 2 N = 80. In the energy gap G , there ap-pear isolated zero-energy levels under some parametersranges. Because the quasi-energies have periodicity inFloquet space, so that similar isolated levels can also ap-pear in gaps G ± , ± ,... . In the energy gaps G − and G ,isolated nonzero-energy levels appear around A/A ∼ G ± , ± , ··· . Below, we concentrate our discussion on thequasi-energy ranges − / ≤ E/ω ≤ /
2. In particular,isolated zero- and nonzero-energy levels can coexist inthe same parametric region, see Fig. 2(b). The eigenstateprofiles, which localize at two edges, indicate that theseisolated levels are FESs [see Fig. 2(c)]. We know that the topological edge states in a static SSH model always ap-pear as zero-energy modes. However, in our modulatedsystem, there appear both zero- and nonzero-energy edgestates. Naturally, there arises an open question:
Are allFESs induced by topology? E / ω (a)0.96 1 1.04 1.08−0.0800.08 A/A E / ω (b) 01 (c)01 c n G G G −1 Figure 2. Quasi-energy spectra under open boundary condi-tion. (a) Scaled quasi-energy
E/ω vs. the scaled modulationamplitude
A/A . (b) Enlarged rectangular region of (a). (c)The Floquet edge states corresponding to the square, trian-gle, and diamond points in the three gaps ( G +1 , G , G − )at A/A = 0 .
98 marked in (b). The parameters are cho-sen as τ /τ = 1 .
2, 2 π/ω = 3, A ω ≃ .
405 [which gives J ( A ω ) = 0], the total lattice number 2 N = 80 and thetruncation number Y = 13. IV. MULTI-SCALE ANALYSIS
To understand how FESs appear in the high-frequencylimit, we employ the multi-scale perturbation analy-sis [31, 35]. We rewrite Eq.(2) as − i du n dz = X m W ( z ; n, m ) u m , (5)with W ( z ; n, m ) = 1 + ( − n δ n,m +1 τ e − iAω sin( ωz ) + δ n,m − τ e iAω sin( ωz ) ]+ 1 − ( − n δ n,m +1 τ e − iAω sin( ωz ) + δ n,m − τ e iAω sin( ωz ) ] . For the open boundary condition, we have u n< ≡ u n> N ≡
0, in which 2 N is the total lattice number.Therefore, W ( z ; n, m ) can be rewritten as W ( z ; n, m ) = 1 + ( − n δ n,m +1 τ e − iAω sin( ωz ) +(1 − δ n, N ) δ n,m − τ e iAω sin( ωz ) ]+ 1 − ( − n − δ n, ) δ n,m +1 τ e − iAω sin( ωz ) + δ n,m − τ e iAω sin( ωz ) ] . (6)Because the waveguide axes are periodically curvedalong the longitudinal propagation ( z -direction), we have W ( z ; n, m ) = W ( z + T ; n, m ), where T = 2 π/ω . In thehigh-frequency limit ( ω ≫ ε , which satisfy T = O ( ε ). Thus, the solutionof Eq.(5) can be given as the series expansion u n ( z ) = U n ( z , z , z , ... ) + εv n ( z − , z , z , z , ... )+ ε w n ( z − , z , z , z , ... )+ ε ζ n ( z − , z , z , z , ... ) + O ( ε ) , (7)where z l ′ = ε l ′ z . Then the differentiation is performedaccording to the usual convention: ddz = ε − ∂∂z − + ∂∂z + ε ∂∂z + ε ∂∂z + · · · . (8)In the series solution, the function U n describes the av-eraged behavior h u n i = U n ; h du n dz i = dU n dz , (9)in which the average notation h•i = εT − Z ε − ( z + T ) ε − z ( • )( z − ) dz − . It is worth to note that U n does not depend on the ‘fast’variable z − , this means that h U n i = U n ; h dU n dz i = dU n dz . (10)From Eqs.(9) and (10), we have h v n i = h w n i = h ζ n i ≡ h ∂v n ∂z l ′ i = h ∂w n ∂z l ′ i = h ∂ζ n ∂z l ′ i ≡ , (11)for l ′ = − , , , , · · · .Substituting Eq.(7) into Eq.(5) and collecting termswith different orders of ε , we obtain − i ∂U n ∂z = i ∂v n ∂z − + X m W ( z ; n, m ) U m , (12)for the order ε . Using the conditions Eq.(10) andEq.(11) and averaging Eq.(12), we have − i ∂U n ∂z = X m W ( n, m ) U m , (13) where W ( n, m ) = h W ( z ; n, m ) i . Then substitutingEq.(13) into Eq.(12), we can obtain the equation for v n − i ∂v n ∂z − = X m [ W ( z ; n, m ) − W ( n, m )] U m . (14)Thus through integrating the above equation, we derivean explicit expression for the function v n v n = iε − X m M ( z ; n, m ) U m , (15)with M ( z ; n, m ) = R [ W ( z ; n, m ) − W ( n, m )] dz . Here,the function M is periodic and has average zero value M ( z ; n, m ) ≡ M ( z + T ; n, m ); h M ( z ; n, m ) i = 0 . (16)For the order ε , we have − i ∂U n ∂z = i ∂v n ∂z + i ∂w n ∂z − + X m W ( z ; n, m ) v m . (17)Substituting Eqs.(15) and (13) into Eq.(17), we obtain − i ∂U n ∂z = − iε − X m,j M ( z ; n, j ) W ( j, m ) U m + i ∂w n ∂z − + iε − X m,j W ( z ; n, j ) M ( z ; j, m ) U m . (18)Using the conditions (10), (11) and (16) and averagingEq.(18), we have − i ∂U n ∂z = iε − X m,j h W ( z ; n, j ) M ( z ; j, m ) i U m . (19)Substituting Eq.(19) into Eq.(18), we can obtain theequation for w n − i ∂w n ∂z − = − iε − X m,j M ( z ; n, j ) W ( j, m ) U m + iε − X m,j [ W ( z ; n, j ) M ( z ; j, m ) −h W ( z ; n, j ) M ( z ; j, m ) i ] U m . (20)Similarly, by performing integration, we can derive theexplicit expression for w n .For the order ε , we have − i ∂U n ∂z = i ∂v n ∂z + i ∂w n ∂z + i ∂ζ n ∂z − + X m W ( z ; n, m ) w m . (21)Using Eqs.(10) and (11) and averaging Eq.(21), we obtain − i ∂U n ∂z = X q h W ( z ; n, q ) w q i , (22)where the second term h W ( z ; n, q ) w q i = h [ W ( z ; n, q ) − W ( n, q )] w q i = − ε − h M ( z ; n, q ) ∂w q ∂ − i . (23)Then using Eqs.(20) and (11), we can rewrite Eq.(22) as − i ∂U n ∂z = ε − X q,m,j h M ( z ; n, q )[ W ( z ; q, j ) − W ( q, j )] M ( z : j, m ) i U m + ε − X q,m,j h M ( z ; n, q )[ W ( q, j ) M ( Z ; j, m ) − M ( z ; q, j ) W ( j, m )] i U m . (24)By combining Eqs.(13), (19) and (24) and usingEq. (8), we obtain a closed-form equation for U n − i dU n dz = X m W s ( n, m ) U m . (25)Here the effective coupling coefficients are given as W s ( n, m ) = W ( n, m ) + X j W ( n, j, m )+ X q,j W ( n, q, j, m ) , (26)with W ( n, m ) = h W ( z ; n, m ) i = 1 + ( − n δ n,m +1 τ +(1 − δ n, N ) δ n,m − τ ] J ( ηAω )+ 1 − ( − n − δ n, ) δ n,m +1 τ + δ n,m − τ ] J ( ηAω ) , X j W ( n, j, m ) = i X j h W ( z ; n, j ) M ( z ; j, m ) i = 0 , X q,j W ( n, q, j, m )= X q,j h M ( z ; n, q )[ W ( z ; q, j ) − W ( q, j )] M ( z ; j, m ) i + X q,j h M ( z ; n, q )[ W ( q, j ) M ( z ; j, m ) − M ( z ; q, j ) W ( j, m )] i = 1 + ( − n { δ n,m +1 [( τ /τ ) − τ /τ )+ δ n,m − [1 − ( τ /τ ) ] } ∆+ 1 − ( − n { δ n,m +1 [1 − ( τ /τ ) ]+ δ n,m − [( τ /τ ) − τ /τ ) } ∆+ τ τ ( δ n, δ m, + δ n, δ m, + δ n, N δ m, N − + δ n, N − δ m, N )∆ , with ∆ = − ω − τ X m =0 X j =0 , − m J j ( Aω ) J m ( Aω ) J j + m ( Aω ) j − m − . Finally, the effective equations for the slowly varyingfunctions U n ( z ) read as − i dU n − dz = τ a U n + τ b U n − + δ (2 n − , τ c U + δ (2 n − , N − τ c U N , − i dU n dz = τ b U n +1 + τ a U n − + δ (2 n, τ c U + δ (2 n, N ) τ c U N − . (27)with the Kronecker’s delta-function δ ( n,m ) . Here, the ef-fective couplings are given as τ a = τ J − ( τ /τ )Θ ,τ b = τ J + Θ ,τ c = τ ∆ / (2 τ ) . (28)with ∆ = − ω − τ P m =0 P j = { , − m } J j J m J j + m j − m − andΘ = [1 − ( τ /τ ) ]∆. The effective couplings τ c de-scribe the virtual defects at boundaries, as shown in theschematic diagram in Fig. 4.Based on the above discussions, the periodically mod-ulated system can be description by an effective staticSSH-like coupled-mode Eqs. (27). The major differencesis the existence of virtual defects at boundaries in theeffective model. Similar to a surface perturbation, thevirtual defects can form a defect-free surface states (orFESs) [31]. On the other hand, if τ c = 0, the static SSH-like coupled-mode equations reduce to conventional SSHmodel [33] and the defect-free surface states disappear.However, for the 1D conventional SSH model belongs tothe BDI symmetry class [36], which satisfy time rever-sal and chiral symmetry, can support an Z topologicalindex (the integer Z index can only take values 0 or 1)[37]. For | τ a | / | τ b | <
1, this system is topologically non-trivial and has one zero-energy mode localized at eachedge, the zero-energy edge mode also call Shockley-likesurface states [38]. For | τ a | / | τ b | >
1, the system is topo-logically trivial with no edge modes. If change τ c = 0, thestatic SSH-like coupled-mode equations still satisfy timereversal and chiral symmetry, which illustrate that themulti-scale perturbation analysis do not change the sym-metry of the system. In similarly static system, the rela-tion between Shockley-like and Tamm-like surface stateshas been discussed [38–40]. Their results show that thetransitions between Shockley-like and Tamm-like surfacestates are observed by tuning the surface perturbation(embedded defects). In our system, without any embed-ded or nonlinearity-induced defects, the surface pertur-bation (virtual defects) is induced by periodical modu-lations. In the next section, we will give the parameterregions of FESs and explore their topological nature. V. NON-TOPOLOGICAL VS. TOPOLOGICALEDGE STATESA. Asymptotic phase boundary
To estimate the cutoff values (phase boundaries) forthe regions of FESs caused by virtual defects. We nowconsider stationary solutions in the form of U n ( z ) = U n (0) e iEz with E being the propagation constant. Sub-stituting it into Eq.(27), we obtain EU n − = τ a U n + τ b U n − +( δ n − , τ c U + δ n − , N − τ c U N ) EU n = τ b U n +1 + τ a U n − +( δ n, τ c U + δ n, N τ c U N − ) . (29)For an infinite lattice, we have EU n − = τ a U n + τ b U n − ,EU n = τ b U n +1 + τ a U n − . (30)The solution of Eqs.(30) can be given as the ansatz U n − = a Qe ikn + a P e − ikn ,U n = a P e ikn + a Qe − ikn , (31)where a and a are arbitrary nonzero constants. Sub-stituting Eqs.(31) into Eqs.(30), we obtain E (cid:20) PQ (cid:21) = (cid:20) τ a + τ b e ik τ a + τ b e − ik (cid:21) (cid:20) PQ (cid:21) , (32)then we can have PQ = Eτ a + τ b e − ik = τ a + τ b e ik E . (33)Therefore, the propagation constant is given as E = τ a + τ b + 2 τ a τ b cos( k ) , (34)for k ∈ [ − π, π ].For a finite but sufficiently large number of lattices(2 N = 80 in our calculation), consider the two edges, wehave EU = ( τ a + τ c ) U + τ b U ,EU = ( τ a + τ c ) U ,EU N = ( τ a + τ c ) U N − ,EU N − = ( τ a + τ c ) U N + τ b U N − . (35)Besides U and U N , the coupling equations is consistentwith the Eqs.(30). So that we should rewrite the ansatz,similarly the Eqs.(31), we have U n − = U ( n = 1) ,U n − = a Qe ikn + a P e − ikn (1 < n ≤ N ) ,U n = a P e ikn + a Qe − ikn (1 ≤ n < N ) ,U n = U N ( n = N ) . (36) First, we consider left boundary of lattices and we cangive a set of equation EU = ( τ a + τ c ) U + τ b U ,EU = ( τ a + τ c ) U ,EU N − = τ a U N − − + τ b U N − . (37)Combining Eqs.(36) and Eqs.(37), we have e − ik N − e ik N − =[ τ b PQ e − ik + ( τ a + τ c ) E − E ]( E PQ − τ b e ik − τ a e − ik )[ E PQ − τ b e ik − ( τ a + τ c ) E PQ ]( τ b PQ e − ik − E + τ a PQ e ik ) . (38)We set k = − i̺ have e − ik N − e ik N − = e − ̺ ( N − , where ̺ isreal number. If ̺ >
0, when N → ∞ have e − ̺ ( N − ≃ τ b PQ e − ̺ + ( τ a + τ c ) E − E ]( E PQ − τ b e ̺ − τ a e − ̺ ) ≃ . (39)If ̺ <
0, when N → ∞ have e − ̺ ( N − ≃ ∞ and equiva-lent to[ E PQ − τ b e ̺ − ( τ a + τ c ) E PQ ]( τ b PQ e − ̺ − E + τ a PQ e ̺ ) ≃ . (40)Combining Eq.(33) and Eq.(39), we have e ̺ = τ c ( τ c + 2 τ a ) τ a τ b = e ik = d. (41)Similarly, Combining Eq.(33) and Eq.(40), we have e − ̺ = τ a τ b τ c ( τ c + 2 τ a ) = e − ik = d − . (42)Thus in the vicinity of the self-collimation point[ J ( A ω ) = 0], as the couplings ( τ a , τ b ) are very weak,the edge states induced by the virtual defects with thequasi-energies E s is given as E s = τ a + τ b + τ a τ b [ e ik + e − ik ]= τ a + τ b + τ a τ b [ d + d − ] . (43)On the other hand, when we consider the right boundaryof lattices, we can also obtain the surface energy E s andwhich is agree with Eq.(43).Obviously, when E s > max( E ), FESs appear inthe energy gaps G − and G . Otherwise, when E s < min( E ), FESs appear in the gap G . Obviously,max( E ) and min( E ) are given by | cos( k ) | = 1. Fromcos( k ) = +1, one can obtain the cutoffs values A , cs /A ≃ − τ ˜ τ c ± F a τ τ . (44)From cos( k ) = −
1, one can obtain the cutoffs values A , cs /A ≃ − − τ ˜ τ c ± F b τ τ . (45)Here, A is the first root of the Bessel func-tion J ( Aω ) = 0, F a = p ( τ ˜ τ c ) + τ τ M + , F b = p ( τ ˜ τ c ) + τ τ M − , ˜ τ c = τ τ ˜∆, M ± = τ τ h − ( τ τ ) i ˜∆ nh − ( τ τ ) i ˜∆ ∓ τ c o ± ( ˜ τ c ) , and ˜∆ =∆ | A → A . These cutoff values define the boundaries be-tween the regions with and without FESs, see the dashedblue curves in Figs. 4(a) and (b), which also call defect-free surface states [31]. Since F b is a purely imaginarynumber for all 2 π/ω when τ /τ = 1 .
2, in Fig. 4(a), thereare no cutoff values A , cs /A . When 2 π/ω →
0, all cutoffvalues gradually converge into one point at
A/A = 1,and there are no FESs caused by the virtual defects.On the other hand, as the effective model Eq. (27) isan SSH-like model, the system changes from topologicalto non-topological when the effective coupling are tunedfrom | τ a | < | τ b | to | τ a | > | τ b | . The effective couplings( τ a , τ b ) depend on the original couplings ( τ , τ ) and thedriving parameters ( A, ω ). We show the effective cou-pling strengthes ( | τ a | , | τ b | ) versus the scaled modulationamplitude A/A for 2 π/ω = 2 and τ /τ = 1 .
2, see theinset in Fig. 4(a). There appear two intersection pointsat | τ a | = | τ b | when A/A increases. In the regions of | τ a | < | τ b | , topological FESs appear [the relevant cumula-tive phase being π ], which also call Shockley-like surfacestates [38]. The intersection points, where topologicalphase transition points occur, are given by A , ct /A ≃ ± τ /τ ) ˜∆ τ , (46)see the dashed blue curves 5 and 6 in inset of Fig. 4(b)Similarly, when 2 π/ω →
0, these two curves also gradu-ally converge into one point at
A/A = 1. Thus, in thelimit of 2 π/ω = 0, the effective couplings vanish when A/A = 1 and the modulation does not change the topo-logical feature when A/A is tuned through A/A = 1. B. Zak phase
To distinguish topological and non-topological FESs,we calculate the bulk topological invariant, the Zakphase [41]. Zak phase can be predict the existence (withthe relevant cumulative phase being π ) or absence (van-ishing cumulative phase) of topological FESs in specificgap.For a modulated SSH system of N cells (i.e. 2 N lat-tices) under PBC, by implementing a Fourier transform c n − ,χ = 1 √ N X k e ik (2 n − c ,k,χ ,c n,χ = 1 √ N X k e ik n c ,k,χ , (47) we obtain the quasi-energy spectra and the eigenstatesby diagonalizing the quasi-energy equation E l c ( l )1 ,k,χ c ( l )2 ,k,χ ! = X χ ′ b R ( k ) c ( l )1 ,k,χ ′ c ( l )2 ,k,χ ′ ! + χω ˆ I c ( l )1 ,k,χ c ( l )2 ,k,χ ! , with the 2 × I and the matrix b R ( k ) = (cid:18) P F ( k )˜ P F ( k ) 0 (cid:19) . Here, P F = τ J χ ′ − χ e ik + τ J χ − χ ′ e − ik , ˜ P F = τ J χ − χ ′ e − ik + τ J χ ′ − χ e ik , and k denotes the quasimo-mentum.To compute the Zak phase for the Floquet quasi-energyspectrum one needs to truncate the Floquet space. Thenumber of replicas needs to be chosen so that all relevanttransitions at the desired energy are kept. The Zak phase Z G m for a specific gap is given by summing up Z ( l ) for allbands below the gap, where Z ( l ) = i H k D c ( l ) k (cid:12)(cid:12)(cid:12) ∂ k (cid:12)(cid:12)(cid:12) c ( l ) k E dk with the eigenstates (cid:12)(cid:12)(cid:12) c ( l ) k E = P α,χ c ( l ) α,k,χ | α, k, χ i for the l -th band are superposition states of different Floquet-Bloch states | α, k, χ i . For a gap between the ( Y + m )-thand ( Y + m + 1)-th bands, its Zak phase Z G m is definedas Z G m = Y + m X l =1 Z ( l ) = Y + m X l =1 (cid:2) i I k D c ( l ) k (cid:12)(cid:12)(cid:12) ∂ k (cid:12)(cid:12)(cid:12) c ( l ) k E dk (cid:3) . (48)For example, the Zak phase Z G can be calculated bysumming up all Z ( l ) for the bands below the gap G , seein Fig. 3. −1 −0.5 0 0.5 1−1.5−1−0.500.511.5 k/( π /2) E / ω G G −1 χχχ =−1 χ =1 χ =0 G Z (Y+3) Z (Y+2) Z (Y+1) Z (Y) Z (Y−1) Z (Y−2) Z G =Z (Y+1) +Z (Y) +......+Z (1) Figure 3. Quasienergy spectrum in the quasi-momentumspace and the Zak phase for the gap G . Figure 4. Phase diagram of the Floquet edge states. Top: Schematic diagram for the effective model Eq. (27). (a,b) Phasediagrams for (a) τ /τ = 1 . τ /τ = 1 .
2. The red regions only support topological FESs, the yellow regions onlysupport non-topological FESs, and the mesh regions support both topological and non-topological FESs. The curves 1, 2, 3and 4 respectively correspond to the non-topological FESs cutoff values A cs /A , A cs /A , A cs /A and A cs /A . While the curves5 and 6 respectively correspond to the topological transition points A ct /A and A ct /A , where the inset in (b) is the enlargedregion nearby A/A ∼
0. The system changes from topological to non-topological when the effective couplings are tuned from | τ a | < | τ b | to | τ a | > | τ b | , see the inset in (a) for 2 π/ω = 2. C. Phase diagram
To verify the above analytical results, we numericallycalculate the quasi-energy spectra. From the quasi-energy spectra under OBC, we indeed find several FESsappear. We then calculate Zak phases of the correspond-ing bulk states under PBC, find that the Zak phase Z G m for a specific gap is either 0 or π and topological FESsonly appear in a gap of nonzero Z G m .In Fig. 4, we show the phase diagram of all possi-ble FESs in the parameter plane (2 π/ω, A/A ). Theappearance of topological FESs (red regions) and non-topological FESs (yellow regions) and their coexistence(mesh regions) sensitively depend on the coupling ratio τ /τ and the modulation parameters ( ω, A/A ). In theabsence of modulation, topological edge states appearonly if τ /τ <
1, otherwise no edge state appears. How-ever, by applying a proper modulation, topological FESsmay appear even if τ /τ > τ /τ <
1. In addition to the regions of topologicaland non-topological FESs, there exists the region of noedge states. When 2 π/ω →
0, topological FESs appearif τ /τ < J ( A ω ) = 0. Our numerical results clearly show all phase boundaries(the solid curves) gradually converge into one point at A/A = 1 when 2 π/ω →
0, which well agree with ouranalytical results (the dashed blue curves).
D. Non-coexistence of non-topological andtopological Floquet edge states in the same gap
Although non-topological and topological FESs can besupported by the same parameters, we find that they cannot appear in the same energy gap. In this section, weonly consider the quasi-energy ranges − / ≤ E/ω ≤ /
2, so that the topological FESs (Zak Phase Z G = π )only possible appear in gap G . We will prove that non-topological and topological FESs can not coexist in thegap G . For the whole Floquet spaces, due to the peri-odicity of quasi-energy. This prove indirect reflection thetopological FESs can not appear in the gap G ± , ± , ± ,... ,in addition the non-topological and topological FESs cannot coexist in the gap G , ± , ± ,... .If non-topological FESs appear in the gap G , the edgestate quasi-energy E s and the bulk-state quasi-energy E will satisfy the condition E s < min( E ). From Eq. (34),the condition E s < min( E ) reads E s < min( τ a + τ b − τ a τ b , τ a + τ b + 2 τ a τ b ) , (49)which requests the parameters obeying τ c ( τ c + 2 τ a ) < A/A ∼ τ c = τ τ ∆ < τ J ( Aω ) − τ τ (1 − ( τ τ ) )∆ + 12 ( τ τ )∆ > . (50)Below we separately discuss the two cases: ( I ) τ > τ > II ) τ > τ > Case-I : τ > τ >
0. Without loss of generality, onecan set τ = 1.As τ a < G requests τ a = τ J ( Aω ) − τ (1 − ( τ ) )∆ > ,τ b = J ( Aω ) + (1 − ( τ ) )∆ < , τ J ( Aω ) − τ (1 − ( τ ) )∆ + 12 τ ∆ > , (51)or τ a = τ J ( Aω ) − τ (1 − ( τ ) )∆ > ,τ b = J ( Aω ) + (1 − ( τ ) )∆ > , τ J ( Aω ) − τ (1 − ( τ ) )∆ + 12 τ ∆ > . (52)On the other hand, in the vicinity of A , we have J ( Aω ) < A → A +0 and J ( Aω ) > A → A − . Therefore, from the condition (51), one canobtain: (C1) (0 < τ < √ − ̥ ) ∩ (0 < τ < p / ̥ )when A → A − , and (C2) (0 < τ < p / ̥ )when A → A +0 . Here, the parameter ̥ is given as ̥ = ω J ( Aω )4 J ( Aω ) J ( Aω ) . However, under the condition (C2),one can find that E s <
0, this means that the condi-tion (C2) does not support non-topological FESs in thegap G . As we always have τ b < A → A +0 ,from the condition (52), we drive the condition (C3):( √ − ̥ < τ < p / ̥ ) when A → A − . Therefore,the appearance of non-topological FESs in the gap G always request A → A − (where τ a > | τ a | / | τ b | <
1, the topological FESsare zero-energy modes and always appear in the gap G .When A → A − (where τ a > | τ a | / | τ b | <
1, onecan obtain: (D1) (0 < τ < − √ ̥ ) for τ b < < τ < √ ̥ −
1) for τ b >
0. However, underthe conditions (D1) and (D2), one can find that E s < G . Case-II : τ > τ >
0. Without loss of generality, onecan set τ = 1.As τ a < G requests τ a >
0. On the other hand, in the vicinity of A , wehave J ( Aω ) < A → A +0 and J ( Aω ) > A → A − . When τ > τ >
0, we always have τ a < A → A +0 , so that the appearance of non-topological FESsin the gap G always requests A → A − . Moreover, when A → A − , we always have τ b = τ J ( Aω )+(1 − ( τ ) )∆ >
0. Thus the appearance of non-topological FESs in thegap G requests τ a = J ( Aω ) − τ (1 − ( 1 τ ) )∆ > ,τ b = τ J ( Aω ) + (1 − ( 1 τ ) )∆ > , J ( Aω ) − τ (1 − ( 1 τ ) )∆ + 12 ( 1 τ )∆ > . (53)The above condition (53) requests ( √ √ − ̥ < τ < E s < G when τ > τ > −1 0 1−0.0500.05 E / ω (a) −1 0 1−0.0500.05 E / ω (b)−1 0 100.1 log ( τ / τ ) | τ a | , | τ b | (c) −1 0 101 Z G / π log ( τ / τ )(d) G G −1 G G G −1 G | τ b | e f g| τ a | Figure 5. Scaled quasi-energy
E/ω vs. coupling ratio τ /τ .(a) Band-gap structure of the effective model (27). (b) Band-gap structure of the original model (2). (c) Effective couplingstrengths ( | τ a | , | τ b | ) vs. the coupling ratio τ /τ . (d) The Zakphases for the gap G , in which the black and dashed bluelines correspond to the effective and original models, respec-tively. The parameters are chosen as A/A = 0 .
98, 2 π/ω = 3, A ω ≃ . N = 80 and the truncationnumber Y = 13. In order to explore how the ratio τ /τ affects theFESs, we show how the scaled quasi-energy spectrumdepends on τ /τ . The quasi-energy spectra and Zakphases show that, even when the modulation frequencyis not very high, the effective model may well explainthe behaviors in the original system. The deviation be-tween the effective and original models decreases withthe modulation frequency and gradually vanishes in thehigh-frequency limit. In Fig. 5, we show the quasi-energy spectra, the effective couplings and Zak phases for0 A/A = 0 .
98 and 2 π/ω = 3. Although the quasi-energieshave small differences, the band-gap structures are al-most the same, in which both zero and nonzero FESsmay appear in different gaps, see Figs. 5(a) and 5(b).From the effective model, topological FESs are alwayszero-energy modes and only appear in the gap G when | τ a | / | τ b | <
1, see Figs. 5(a) and 5(c). In addition to thetopological FESs, due to the modulation-induced virtualdefects, there also exist non-topological FESs in differentgaps. Moreover, the band-gap structures show that topo-logical and non-topological FESs can not appear in thesame gap, which confirms our previous analytical analy-sis. From the Zak phases, the effective and original mod-els show similar topological phase transitions, but thetransition points show small deviations dependent uponthe modulation frequency, see Fig. 5(d).
VI. CONCLUSION
In summary, we have studied the Floquet edge statesin arrays of curved optical waveguides described by theperiodically modulated SSH model. According to theFloquet theorem, we give the quasi-energy spectra underOBC and find several FESs. To understand how FESsappear, we employ the multi-scale perturbation analysisand find the periodic modulations can induce virtual de-fects at boundaries. Similar to a surface perturbation,the virtual defects can form a FESs (defect-free surfacestates) [31]. On the other hand, by changing the ratioof | τ a | / | τ b | , one can also obtain a FESs (Shockley-like surface states).In order to explore the topological nature of all FESs,we have calculated the quasi-energy spectra and theZak phases. Our results indicate that Shockley-like sur-face state is a topological FES and defect-free surfacestate is a non-topological FES. The topological and non-topological FESs may be supported by the same param-eters, but they always appear in different energy gaps.Without any embedded or nonlinearity-induced defects,these edge states originate from the interplay betweenthe bulk band topology and periodic modulations. Wehave derived analytically the boundaries between differ-ent topological phases, and have verified these resultsnumerically. We believe our work provides new perspec-tives for topological photonics govern by periodic modu-lations, and it can be employed for a control of topolog-ical phase transitions. Although our analysis has beenperformed for arrays of periodically curved optical wave-guides, it can be applicable to other lattice systems suchas ultracold atoms in optical lattices [42, 43], photoniccrystals [18], and discrete quantum walks [44, 45]. ACKNOWLEDGMENTS
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