Observation of dark edge states in parity-time-symmetric quantum dynamics

ABSTRACT Topological edge states arise in non-Hermitian parity-time (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\mathcal {PT}$\end{document})-symmetric systems, and manifest themselves as bright or dark edge states, depending on the imaginary components of their eigenenergies. As the spatial probabilities of dark edge states are suppressed during the non-unitary dynamics, it is a challenge to observe them experimentally. Here we report the experimental detection of dark edge states in photonic quantum walks with spontaneously broken \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\mathcal {PT}$\end{document} symmetry, thus providing a complete description of the topological phenomena therein. We experimentally confirm that the global Berry phase in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\mathcal {PT}$\end{document}-symmetric quantum-walk dynamics unambiguously defines topological invariants of the system in both the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\mathcal {PT}$\end{document}-symmetry-unbroken and -broken regimes. Our results establish a unified framework for characterizing topology in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\mathcal {PT}$\end{document}-symmetric quantum-walk dynamics, and provide a useful method to observe topological phenomena in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\mathcal {PT}$\end{document}-symmetric non-Hermitian systems in general.

With a single-photon source consisting of a β-barium-borate (BBO) nonlinear crystal pumped by a CW diode laser, we generate polarization-degenerate photon pairs at 801.6nm using a type-I spontaneous parametric down-conversion (SPDC) process. Upon detection of a trigger photon, the signal photon is heralded in the measurement setup. This trigger-signal photon pair is registered by a coincidence count at two APDs with a ∆t = 3ns time window. Total coincidence counts are about 10, 000 over a collection time of 2s.
The coin states |0 and |1 are respectively encoded in the horizontal |H and vertical |V polarizations of the heralded single photon, whose spatial modes represent the walker state. After passing through a polarizing beam splitter (PBS) followed by a HWP, the heralded single photon is projected into an arbitrary initial state and then proceeds through the quantum-walk interferometric network. We implement the coin operator R(θ) by HWPs with certain setting angles depending on the coin parameters (θ 1 , θ 2 ), and the shift operator S by a BD whose optical axis is cut so that the photons in |V are directly transmitted and those in |H undergo a lateral displacement into a neighboring spatial mode, respectively. The loss operator M is implemented by a sandwich-type HWP (at 22.5 • )-PPBS-HWP (at 22.5 • ) setup [4]. Here, the transmissivities of PPBS are (T H , T V ) = (1, 1 − p) for horizontally and vertically polarized photons, respectively.
We construct the raw probability distribution of the walker P R at time t by dividing the number of coincidence measurements at APDs using the total number of photon pairs, after correcting for the relative efficiencies of different APDs. The raw probability is then converted into the corrected probability P C (x, t) = γ 2t P R (x, t), which is obtained by multiplying the correction factor γ for the corresponding step t and represents the probability corresponding to parity-time (PT )-symmetric quantum walks (QWs) governed byŨ . Whereas, the normalized probability P N (x, t) is defined as P R (x, t)/ x P R (x, t).
The eigenvalues ofŨ are given by λ ± = d 0 ∓ i 1 − d 2 0 , where ± are band indices. Note that λ + λ − = 1, which is guaranteed by PT symmetry of the Floquet operatorŨ . As we define the effective Hamiltonian through U = exp(−iH eff ), the quasienergy spectrum of H eff is given by ± = i ln(λ ± ). Apparently, when d 2 0 < 1 for all k, the quasienergy spectrum is entirely real. In this case, the system is in the PT -symmetry-unbroken regime. In contrast, when d 2 0 > 1 is satisfied for a certain range of momenta k, the corresponding quasienergies in that range become complex. In this case, the system is in the PT -symmetry-broken regime. The transition between the above two scenarios, the so-called exceptional point, occurs when d 2 0 = 1 is satisfied at some discrete momenta while d 2 0 < 1 otherwise. At these momenta, the quasienergy band gap closes at = 0 (with + = − = 0) or = π (with We further divide the PT -symmetry-broken regime into the partially-broken and the completely-broken regimes, where complex quasienergies occur for part of or the whole first Brillioun zone, respectively. The important difference between the PT -symmetry-partially-broken and completely-broken cases is that the latter does not have quasienergy band gap closing, i.e., ( ± = 0, π) for any k in the symmetry completely-broken regime. In the left two columns of Fig. S1, we plot quasienergies ± and eigenvalues λ ± for the different scenarios above.
We experimentally confirm PT symmetry ofŨ by analyzing homogeneous QWs for both the PT -unbroken and broken states. We start with a homogeneous QW in PT -symmetry-unbroken regime, with the coin parameters (θ L,R 1 , θ L,R 2 ) = (−π/4, 3π/4 − 3ξ). We fix the parameter ξ = 0.1113 in our experiment. As illustrated in Fig. S1(a), all quasienergies are real and gaps are open for all momenta. Correspondingly, eigenvalues ofŨ all lie on a unit circle in the complex plane. The measured corrected probability distribution is ballistic, which agrees well with numerical simulations and is similar to that of a standard unitary QW.
We then change the coin parameters to (θ 1 , θ 2 ) = (−4π/9, 5π/9 + ξ). The resulting QW is at the exceptional point. As illustrated in Fig. S1(b), in this case all the quasienergies are still real, but the quasienergy gap closes at = 0. The measured corrected probability distribution is different from that of the standard unitary QW with a squeezed profile.

Global Berry phase of PT -symmetric QWs
We discuss the definition of topological invariants for PT -symmetric non-unitary QWsŨ . For the convenience of calculation, we apply a unitary transformation toŨ where V = e i π/2 2 σy , σ x,y,z are the Pauli matrices, 1 c is a two-by-two identity matrix, and the explicit expressions of d i (i = 0, 1, 2, 3) are given in the Supplemental Materials. Topological properties ofŨ is not changed under the unitary transformation. We will show that the winding number ν is defined through the global Berry phase as ν = ϕ B /2π. Here, ϕ B = ϕ Z+ + ϕ Z− , with the generalized Zak phases for the jth band (j = ±) Here, the integral is over the first Brillioun zone and χ j | and |ψ j are respectively the left and right eigenstates of W , defined through W † |χ j = λ * j |χ j and W |ψ j = λ j |ψ j , respectively. In the following, let us first evaluate the Berry connection which critically depends on whether d 2 0 is greater than 1 or not. As different momenta are decoupled, we will examine the Berry connection case by case.
Case I: the momentum region with d 2 0 < 1:-In the momentum regime with d 2 0 < 1, we have d 2 1 + d 2 2 + d 2 3 > 0, and the right and the left eigenvectors of W are with (θ1, θ2) = (−π/2 − 3ξ/8, −π/2 − 3ξ/8) in the completely broken PT -symmetric phase. The first column: the quasienergy as a function of quasimomentum where the solid (dashed) curves represent the real (imaginary) part of quasienergy. The second column: analytical results of the eigenvalues of the time-evolution operator in the complex plane. The third column: comparison between the measured (red bars) and the predicted (grey bars) probabilities after the seventh step with different coin parameters. The fourth column: the predicted probabilities after fifty steps with different coin parameters. Experimental errors are due to photon-counting statistics and represent the corresponding standard deviations.
Case II: the momentum region with d 2 0 > 1:-In the momentum regime with d 2 0 > 1, we have d 2 1 + d 2 2 + d 2 3 < 0, and the right and the left eigenvectors of W are where cosh 2Ξ = −id 1 /d, with Ξ ∈ (0, ∞). We then have where Ξ = dΞ/dk. The Berry connection is then The right and the left eigenvectors of W are As χ ± |ψ ± = χ ± |ψ ∓ = 0 and χ ± |χ ± = ψ ± |ψ ± = 1, the denominator in the Berry connection χ ± |ψ ± vanishes, giving rise to diverging A ± at these momenta. However, a closer examination reveals that the divergence in A + and A − cancels out in their summation A + + A − , giving rise to a well-defined "global Berry connection". For example, if the condition d 2 0 = 1 is approached in parameter space from the side with d 2 The imaginary parts of Berry connections A ± diverge, however, their summation is still ϑ and remains well-defined even at d 2 0 = 1. The situation is similar when the condition d 2 0 = 1 is approached in parameter space from the side with d 2 0 > 1, where A ± diverge but their sum is not. With the above analysis, we see that the global Berry phase is given by ϕ B = dϑ, regardless of whether the system is PT -symmetry unbroken or broken.
In contrast, at the topological phase boundary, the polar angle ϑ in the above expressions become ill-defined as d 2 = d 3 = 0. This occurs at k = 0, π for d 0 (k) = α, or at k = ±π/2 for d 0 (k) = −α. As such, the global Berry phase can no longer be defined on the topological phase boundary. This conclusion is the same as the Zak phase of a unitary QW.
In previous studies, both generalized Zak phases and generalized winding numbers have been proposed to serve as topological invariants for non-unitary FTPs. The generalized Zak phase, which is only valid in the PT -symmetryunbroken regime, can be written as Re (ϕ Z− ). As ϕ Z− = ϕ * Z+ in the PT -symmetry-unbroken regime, the generalized Zak phase defined in Refs. [1,2] is equivalent to the global Berry phase ϕ B in the PT -symmetry-unbroken regime.
On the other hand, according to Refs. [3,4], the generalized winding number for the Floquet operatorŨ is defined as where the unit vector n is a normalized projection of the vector d = (d 1 , d 2 , d 3 ) T in the y-z plane. As such, It is then straightforward to show that ϑ = (n × ∂n ∂k ) x , such that the generalized winding number is ν 1 = ϕ B /2π = ν . Topological invariants defined through the global Berry phase thus unify previous definitions in different contexts.
By fitting the Floquet operatorŨ in a different time framẽ we define another winding number ν through the global Berry phase ofŨ [1,5]. We then construct the topological numbers (ν 0 , ν π ) = ( ν −ν 2 , ν +ν 2 ). In the following section, we will confirm through numerical calculations that the topological numbers (ν 0 , ν π ) are directly related to the number of topological edge states at a given interface. Specifically, the number of edge states with quasienergy Re( ) = 0 [Re( ) = π] is equal to the difference of topological numbers ν 0 (ν π ) on either side of the boundary.

Topological number and topological edge states
In this section, we numerically confirm that localized topological edge states at a given boundary are dictated by the difference in topological numbers (ν 0 , ν π ) of the bulks on either side. More specifically, topological number ν 0 (ν π ) is associated with the number of topological edge states with Re( ) = 0 [Re( ) = π]. For convenience, we define ∆ν g = |ν L g − ν R g | (g = 0, π), where ν L g (ν R g ) is the topological number in the left (right) region. We numerically diagonalize Floquet operators of inhomogeneous QWs governed byŨ with N = 50. As illustrated in Fig. S2(a), when the difference in topological numbers is (∆ν 0 , ∆ν π ) = (2, 0), a pair of degenerate edge states, on odd and even sites respectively, exist at a given boundary (x = 0 or x = 50) with Re( ) = 0 [Re(λ) > 0]. We note that both regions belong to the PT -symmetry-unbroken regime. In this case, bright edge states with λ > 1 (red) appear near x = 0, while dark edge states with λ < 1 (black) appear near x = 50. In contrast, for bulk states with |λ| = 1, their spatial distributions are extended (green and orange).
In Fig. S2(c), we consider the case where both left and right regions are in the PT -symmetry-broken regime. The difference in topological numbers is (∆ν 0 , ∆ν π ) = (2, 0). In this case, while PT -symmetry-broken bulk states exist with |λ| = 1, localized edge states can still be identified as their eigenvalues λ deviate most from the unit circle. In terms of quasienergy, topological edge states in this case possess quasienergies with the largest imaginary parts. As shown in the central and right columns, localized edge states and extended bulk states are also differentiated by their distinct spatial probability distributions.
To summarize, we have numerically confirmed that the number of localized edge states is governed by ∆ν 0 and ∆ν π . We have further checked (not shown) that such a relation holds when both topological numbers are different. Such a bulk-boundary correspondence exists even when one or both of the bulks are in the PT -symmetry-broken regime. For all the cases shown in Fig. S2, bright (dark) edge states exist near x = 0 (x = 50). We note that their spatial location would be switched when we exchange the coin parameters of the left and right regions.

Edge-state wave functions
In this section, we solve for the wave function of topological edge states in PT -symmetry non-unitary QWs governed byŨ . We first consider the unitary QW with p = 0 (γ = 1), and derive the analytical solution of topological edge states near the boundary x = 0. We then demonstrate that topological edge states in the non-unitary case (γ = 1) have the same eigen wave functions as the unitary case. The difference lies in eigenvalues and hence the time evolution, where topological edge states in the non-unitary case acquire factors γ ±t , giving rise to the bright and dark edge states as discussed in the main text.
In the homogeneous case, eigenstates ofŪ at a given momentum k can be written as In the inhomogeneous case with θ L 1,2 = θ R 1,2 , topological edge states with quasienergies E (0) = 0 or E (π) = π emerge near boundaries between different bulk topological phases. Wave functions of topological edge states near the boundary x = 0 can be constructed from Eq. (S22) by setting k = −iκ L and k = iκ R for the left and right regions, respectively. Note Re (κ L ) , Re (κ R ) > 0 so that probability distributions of the localized edge states vanish as |x| → ∞. We further notice that under the two-step QWŪ , wave functions on even sites and odd sites are decoupled.
The considerations above enable us to construct wave functions for topological edge states at the boundary near where |ψ o(e) (x) is the edge-state wave function on odd (even) sites, and r o(e) and t o(e) are the corresponding coefficients. We also have a ξ k b ξ k = − cos θ ξ 2 sin 2k i(cos θ ξ 2 sin θ ξ 1 cos 2k + cos θ ξ 1 sin θ ξ 2 ) (ξ = L, R), which, according to Eq. (S21), denotes coin states of edge-state wave functions with E (0) or E (π) . We will show in the following that all the coefficients above have analytical forms. From the dispersion relations, we first establish expressions for the spatial decay rates κ ξ (ξ = L, R) cosh 2κ ξ = cos E (0,π) + sin θ ξ 1 sin θ ξ the corresponding coin parameters θ L 1,2 (θ R 1,2 ), as well as the the quasienergy of the edge state E (0,π) . The edge-state spatial wave function is therefore typically asymmetric with respect to the boundary.
On the other hand,Ū has chiral symmetry with the symmetry operator Γ = x |x x| ⊗ σ x and ΓŪ Γ =Ū −1 . Edge states are therefore eigenstates of the chiral operator, such that Equivalently, the edge states are either in |+ or |− . Combining the expressions of a k and b k , we derive conditions for the coin states. Specifically, edge states with quasienergy E (0,π) are in |+ when And edge states are in |− when Finally, we show how to solve for the coefficients r o(e) and t o(e) by considering the Floquet operatorŪ = GF . As discussed in Ref. [5],Ū has chiral symmetry and support topological edge state at boundaries between regions with different topological numbers. Further, we notice thatŪ (θ ξ 1 , θ ξ 2 ) =Ū (θ ξ 2 , θ ξ 1 ), and that Eqs. (S25) and (S26) acquire different signs on exchanging θ ξ 1 and θ ξ 2 . We then establish that edge states underŪ andŪ have the same coin states at E (0) (λ = 1), and they have opposite coin states at E (π) (λ = −1).
Therefore, localized edge state ofŪ are also eigenstates ofŨ , with eigenvalues being ±γ or ±1/γ in the nonunitary case. The corresponding quasiernergies satisfy Re( ) = 0, π, which are exactly the conditions for topological edge states as required by pseudo-anti-unitarity ofŨ [6]. We therefore conclude that topological edge states underŨ have the same spatial and coin-state wave functions as those in unitary case. The difference lies in the quasienergies and hence the time evolution.
For edge states with eigenvalues ±γ, the corresponding quasienergies are i ln γ and π + i ln γ, respectively. Their probability distributions increase during the time evolution as γ 2t , and we identify them as the bright edge states. In contrast, for edge states with eigenvalues ±1/γ, the corresponding quasienergies are −i ln γ and π − i ln γ, respectively. Their probability distributions decrease during the time evolution as γ −2t , and we identify them as the dark edge states.
Due to PT symmetry ofŨ , eigenstates with the eigenvalues λ and λ −1 must appear in pairs. This implies that bright and dark edge states must also appear in pairs. From Eqs. (S25) and (S26), we see that at a given boundary, edges states associated with the same topological number (ν 0 or ν π ) have the same coin states and are of the same type (bright or dark). In fact, they only differ by the occupation of odd or even sites. Thus, edge states associated with the same topological number are two-fold degenerate. Further, bright edge states at a given boundary should change into dark ones, and vice versa, when we exchange the coin parameters on the two sides of the boundary. All these conclusions are consistent with numerical calculations, whereŨ is directly diagonalized.
Finally, we confirm the analytical edge-state wave functions derived above by comparing the normalized probability distributions from the analytical solution and from numerical simulations of QW dynamics governed byŨ . In Fig. S3, we see that as the time steps of the numerical simulation increase, the resulting normalized probability approaches that of the analytical solution. Apparently, it takes more time steps for the QW dynamics to converge to the edge-state distribution when at least one of the bulks is PT -symmetry broken.