Wave-Function Tomography of Topological Dimer Chains with Long-Range Couplings

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Wave-Function Tomography of Topological Dimer Chains with Long-Range Couplings

F. Pellerin, R. Houvenaghel, W. A. Coish, I. Carusotto, and P. St-Jean

Phys. Rev. Lett. 132, 183802 – Published 1 May 2024

See synopsis: A Photonic Emulator of Topological Matter

Abstract

The ability to tailor with a high accuracy the intersite connectivity in a lattice is a crucial tool for realizing novel topological phases of matter. Here, we report the experimental realization of photonic dimer chains with long-range hopping terms of arbitrary strength and phase, providing a rich generalization of the Su-Schrieffer-Heeger model which, in its conventional form, is limited to nearest-neighbor couplings only. Our experiment is based on a synthetic dimension scheme involving the frequency modes of an optical fiber loop platform. This setup provides direct access to both the band dispersion and the geometry of the Bloch wave functions throughout the entire Brillouin zone allowing us to extract the winding number for any possible configuration. Finally, we highlight a topological phase transition solely driven by a time-reversal-breaking synthetic gauge field associated with the phase of the long-range hopping, providing a route for engineering topological bands in photonic lattices belonging to the AIII symmetry class.

Received 5 July 2023
Accepted 12 March 2024

DOI:https://doi.org/10.1103/PhysRevLett.132.183802

Physics Subject Headings (PhySH)

Photonics

Atomic, Molecular & Optical

synopsis

A Photonic Emulator of Topological Matter

Published 1 May 2024

A method for freely adjusting the parameters of a loop of optical fiber enables the exploration of exotic topological phases of matter.

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Authors & Affiliations

F. Pellerin¹, R. Houvenaghel ², W. A. Coish ³, I. Carusotto⁴, and P. St-Jean ^1,5

¹Département de Physique, Université de Montréal, C.P. 6128, Succursale Centre-Ville, Montréal, Québec H3C 3J7, Canada
²Département de Physique, Ecole Normale Supérieure de Lyon, 46 allée d’Italie, F69007 Lyon, France
³Department of Physics, McGill University, 3600 rue University, Montreal, Québec H3A 2T8, Canada
⁴Pitaevskii BEC Center, INO-CNR and Dipartimento di Fisica, Università di Trento, via Sommarive 14, I-38123 Trento, Italy
⁵Institut Courtois, Université de Montréal, Montréal, Quebec H2V 0B3, Canada

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Issue

Vol. 132, Iss. 18 — 3 May 2024

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Images

Figure 1
(a) Schematic representation of a topological dimer chain. The unit cell is defined by the dotted rectangle and the arrows describe the forward- ( $a$ , $b$ , $c$ , $d$ ) and backward- $(a^{*}, b^{*}, c^{*}, d^{*})$ propagating hopping amplitudes. (b)–(c) Topological phase diagrams presenting the winding number as a function of the different couplings for the SSH model (b) and the dimer chain with long-range couplings (c) in a case where $a = b^{*}$ and the ratios $c / a$ and $d^{*} / a$ are real. (d) Experimental setup: clockwise (blue) and counterclockwise (red) modes of an optical fiber loop are coupled with a $75 ∶ 25$ fiber coupler. The system is probed with an optical fiber acting as a transmission line weakly coupled to the cavity with a $99 ∶ 1$ FC. On top, circulators are used to independently modulate CW and CCW modes with a pair of electro-optical phase modulators. (e) The energy spectrum of the cavity has a FSR of $Ω$ and consists of symmetric and antisymmetric superpositions of CW and CCW modes ( $| m, \pm ⟩$ ), separated by a frequency splitting $δ$ .
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Figure 2
(a)–(b) Band structure measurements for the trivial (a) and topological phases (b) obtained by probing the transmitted intensity of the cavity as a function of time for different values of the laser detuning. Effective hopping terms (in units of V) are provided above each panel. (c)-(d) Trajectory of $g (k)$ in the complex plane for $k$ moving across the Brillouin zone in the trivial and topological cases exhibiting a winding number $W = 0$ and 1, respectively. (e) Slow temporal modulation of the amplitude of the transmitted intensity peaks associated with the lower band in the trivial $W = 0$ (top) and topological $W = 1$ (bottom) cases. Vertical dashed lines indicate the periodicity of this modulation.
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Figure 3
Measurements (top row) and numerical simulations (bottom row) of the band structure (left column) and of the trajectory of $g (k)$ (central column) for dimer chains with long-range hopping amplitudes. Each panel shows results for each possible value of the winding number: $W = - 1$ (a), $W = 0$ (b), $W = + 1$ (c), and $W = + 2$ (d). On the right side of each panel, we reproduce the phase diagram of Fig. 1, with a star indicating in which region of the phase diagram each measurement is realized, and provide the effective hopping amplitudes, normalized by $η$ (in units of V).
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Figure 4
(a) Schematic representation of a topological dimer chain with $a = b^{*}$ , $c = 0$ , and complex-valued $d^{*} / a$ . The two sublattices are identified with $+$ and $-$ as in Fig. 1. (b) Schematic representation of the trajectory in the topological phase diagram followed by the Hamiltonian as the phase of the 3rd-nearest-neighbor hopping amplitudes ( $θ$ ) is scanned. (c)–(e) Measurements of the band structure (top left) and $g (k)$ trajectory (top right) as the phase of the long-range hopping is tuned from $θ = 0$ (c), to $θ = π / 2$ (d), and $θ = π$ (e). The bottom row presents the corresponding theoretical calculation of the band structure (left) and $g (k)$ (right). For each panel, the effective hopping terms normalized by $η$ are provided (in units of V).
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