Observation of a Topological Edge State Stabilized by Dissipation

Helene Wetter, Michael Fleischhauer, Stefan Linden, and Julian Schmitt

Phys. Rev. Lett. 131, 083801 – Published 24 August 2023

Abstract

Robust states emerging at the boundary of a system constitute a hallmark for topological band structures. Other than in closed systems, topologically protected states can occur even in systems with a trivial band structure, if exposed to suitably modulated losses. Here, we study the dissipation-induced emergence of a topological band structure in a non-Hermitian one-dimensional lattice system, realized by arrays of plasmonic waveguides with tailored loss. We obtain direct evidence for a topological edge state that resides in the center of the band gap. By tuning dissipation and hopping, the formation and breakdown of an interface state between topologically distinct regions is demonstrated.

Received 15 March 2023
Accepted 20 July 2023

DOI:https://doi.org/10.1103/PhysRevLett.131.083801

Physics Subject Headings (PhySH)

Open quantum systems Topological effects in photonic systems Topological phase transition

Nonequilibrium systems Topological materials Waveguide arrays

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Helene Wetter ¹, Michael Fleischhauer ², Stefan Linden ¹, and Julian Schmitt ^3,*

¹Physikalisches Institut, Universität Bonn, Nussallee 12, 53115 Bonn, Germany
²Department of Physics and Research Center OPTIMAS, RPTU Kaiserslautern-Landau, 67663 Kaiserslautern, Germany
³Institut für Angewandte Physik, Universität Bonn, Wegelerstrasse 8, 53115 Bonn, Germany

^*Corresponding author: schmitt@iap.uni-bonn.de

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Vol. 131, Iss. 8 — 25 August 2023

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Images

Figure 1
Experimental scheme. (a) Lattice system with nontrivial topology induced solely by dissipation (top) and experimental realization with DLSPPWs spaced by distance $d$ , where losses are induced by chromium stripes of width $w$ (bottom). (b) Complex-valued energy spectrum for 40 lattice sites with $g_{1} = | g_{2} |$ . For $g_{2} < 0$ , the system is topologically trivial and the probability density $| Ψ |^{2}$ is concentrated in the first two lattice sites (top left). In the topologically nontrivial regime ( $g_{2} > 0$ ), $| Ψ |^{2}$ is exponentially localized at the edges (top right) and associated with midgap states at zero energy (red lines). (c) Left: Waveguide sample with chromium stripes (dark gray) arranged to realize trivial (topological) domains in the left (right) sample half. The chromium-free region on top is used to excite the waveguides by grating coupling of laser light. Right: Measured dissipation in a single waveguide $Im β$ versus $w$ and hopping $J$ between two waveguides versus $d$ , along with fits (lines).
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Figure 2
SPP evolution for metallike (left, I), dissipative topologically trivial (middle, II), and dissipation-induced topological domains (right, III). (a) Real-space intensity distribution for injection at the edge (top) and at the first site of a unit cell in the bulk (bottom), for a sample of 48 waveguides spaced by $d = 1.4 μ m$ , corresponding to $J = 0.045 (3) μ m^{- 1}$ . Solid and white dashed lines indicate sample boundaries and unit cells, and black dashed lines give the center of mass. For the used $w = 0.7 μ m$ in (II) and (III), $Im β = 0.1 μ m^{- 1}$ , and $g_{1} = | g_{2} | = 1.1$ . (b) Simulated intensity evolution in the arrays from (a) with $J = 0.045 μ m^{- 1}$ and $Im β = 0.1 μ m^{- 1}$ for the lossy [red in (a)] and $Im β = 0.01 μ m^{- 1}$ for the low-loss [blue in (a)] waveguides, in good agreement with the experimental results. (c) Momentum-resolved energy spectra with cos-shaped band for bulk excitation, band gap in the trivial, and flat band in the topological domains, respectively, for edge excitation [35], along with theory prediction (top). Dashed lines indicate the energy of the topological zero mode (right), which lies between the bulk bands of case (II) (middle).
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Figure 3
Dissipation-induced emergence of interface state between distinct topological domains. (a) Interface between phase-(II) and phase-(III) domains realized by different loss patterns, and real-space intensity distribution $I (x, z)$ upon excitation of the interface waveguide (red) for increasing losses $Im β = {0, 0.06, 0.09, 0.1} μ m^{- 1}$ from larger chromium widths $w$ . Losses correspond to $g_{1} = | g_{2} | = {0, 0.7, 1, 1.1}$ and, as before, $J = 0.045 (3) μ m^{- 1}$ . (b) Line profiles of the intensity averaged around $z = 50 μ m$ . (c) Decay length $ℓ$ of intensity in interface waveguide for cases in (a); for fitted line profiles $I (x = 0, z)$ , see Ref. [35]. Despite increased chromium absorption, an enhanced propagation distance is observed, providing evidence for a topologically robust state. Shaded areas show simulations for the interface (red) and bulk (gray), where the enhanced $ℓ$ indicates the interface robustness beyond $g_{2} \approx 0.7$ . (d) Advantage of topological interface over isolated waveguide, seen in reduction of calculated $Im E$ when induced losses and hopping are of the same order $g_{2} \sim 1$ . For small $g_{2}$ (shaded) the edge state delocalizes, for large $g_{2}$ Zeno-like decoupling dominates.
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Figure 4
Breaking the topological protection at an exceptional point. (a) SPP evolution for increasing hopping $J$ , realized by reduced waveguide spacings $d = {1.8, 1.6, 1.4, 1.2, 1.0} μ m$ at $w = 0.7 μ m$ ( $Im β = 0.1 μ m^{- 1}$ ). For excitation at the trivial edge (A), the population oscillates faster as $J$ increases. Upon injecting light into the (II)-(III) interface (B), a quasistationary edge state is observed, which at $J \approx 0.09 μ m^{- 1}$ becomes delocalized in the bulk, in agreement with simulations (right). (b) Oscillation frequency $k_{z}$ at excited waveguide fitted with $I (z) = a_{1} \cos (k_{z} z + ϕ) e^{- z / ℓ} + a_{0}$ . Solid line and bars give numerical results, and error bars denote uncertainties of fit parameter. (c) Left: Calculated probability density at boundaries, along with bulk modes (gray). A delocalization of the interface mode (red) is visible for too large $J$ , indicating the loss of topological character. Right: Complex eigenenergies of the two topological (colored) and 38 bulk (gray) states, as $J$ is varied. For $J \approx 0.095 μ m^{- 1}$ , the eigenvalues coalesce at an exceptional point (yellow).
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