Measuring topological invariants in disordered discrete-time quantum walks

Barkhofen, Sonja; Nitsche, Thomas; Elster, Fabian; Lorz, Lennart; Gábris, Aurél; Jex, Igor; Silberhorn, Christine

doi:10.1103/physreva.96.033846

Cited by 94 publications

(92 citation statements)

References 46 publications

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“…It follows that dynamics of the system can be described as a discrete-time quantum walk which supports Floquet topological phases. Our experiment is in sharp contrast to previous studies of topological quantum walks in photonics [38][39][40][41][42][43][44][45][46], where dynamics are generated by Floquet operators rather than genuine Hamiltonians.Since the topological classification of our Floquet system is Z ⊕ Z, two distinct topological invariants exist [64]. We directly measure both topological invariants using time-averaged mean chiral displacement (A-MCD) by going to different time frames, and identity a topological phase transition as the driving parameters are tuned.…”

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confidence: 58%

Topological Quantum Walks in Momentum Space with a Bose-Einstein Condensate

et al. 2020

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We report the experimental implementation of discrete-time topological quantum walks of a Bose-Einstein condensate in momentum space. Introducing stroboscopic driving sequences to the generation of a momentum lattice, we show that the dynamics of atoms along the momentum lattice is dictated by a periodically driven Su-Schieffer-Heeger model, which is equivalent to a discretetime topological quantum walk. We directly measure the underlying topological invariants through time-averaged mean chiral displacements in different time frames, which are consistent with our experimental observation of topological phase transitions. The high tunability of the system further enables us to observe robust helical Floquet channels in the one-dimensional momentum lattice, which derive from the winding of Floquet quasienergy bands. Our experiment opens up the avenue of investigating discrete-time topological quantum walks using cold atoms, where the many-body environment and tunable interactions offer exciting new possibilities.Exploring topological phases is a main theme in modern physics. Characterized by topological invariants which reflect the global geometric properties of the system wave function, topological phases host a range of fascinating features, which are robust to local perturbations and are potentially useful for applications in quantum information and quantum computation [1,2]. Besides conventional topological materials in solid-state systems, topological phenomena also emerge away from equilibrium. For example, topological phases and emergent topological phenomena exist in non-Hermitian open systems [3][4][5][6][7][8][9][10][11][12][13][14], in periodically driven Floquet systems and quench processes [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33], which have stimulated intense interest recently due to the rapid progress in synthetic quantum-simulation platforms such as cold atoms [34][35][36][37], photonics [38][39][40][41][42][43][44][45][46][47][48][49][50][51][52], phononics [53], and superconducting qubits [54].A particularly interesting subject is topologies in periodically driven Floquet systems, which are shown to have a rich structure and host novel topological phases with no counterparts in static systems [20][21][22][23]. A paradigmatic example of topological Floquet dynamics is discrete-time quantum walks, which, besides potential applications in quantum information [55][56][57], have been widely used in photonics for the exploration of Floquet topological phases [38][39][40][41][42][43][44][45][46][47]. In cold atoms, whereas Floquet topological phases [34] and quantum walks [58] have been respectively implemented, quantum walks with topological properties are yet to be experimentally realized.Here we report the experimental implementation of discrete-time topological quantum walks in momentum space for a Bose-Einstein condensate (BEC). Combining the generation of momentum lattice [59-63] with stroboscopic driving sequences, dynamics of the conden-sate atoms is governed b...

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confidence: 58%

Topological Quantum Walks in Momentum Space with a Bose-Einstein Condensate

et al. 2020

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“…The scheme proposed above effectively realizes a discrete-time quantum walk with a four-dimensional (4D) coin, a generalization of the usual topological quantum walk with two-dimensional coin [21][22][23][24][25]27]. In order to completely characterize the topology of this driven model, we follow the method proposed in [27], and recently implemented in [24].…”

Section: Driven Ssh 4 Modelmentioning

confidence: 99%

“…The topological invariant of the bulk, the winding number  , allows one to predict the number of zero energy edge states. 1D chiral topological insulators have been realized in numerous platforms as ultracold atoms [6,11], photonic crystals [15], photonic quantum walks [21][22][23][24][25]. Let us notice that the 1D chiral Hamiltonian can be static or the effective Hamiltonian of a Floquet system.…”

Section: Introductionmentioning

confidence: 99%

“…The observation of edge states [11,21,30,31] and the measurement of the winding number through the mean chiral displacement [24] belong to the first category. The measurement of the winding number by interferometric architectures [6,25], by introducing losses [15,32,33], and by scattering measurements [22] belong to the second category. Here we generalize the notion of mean chiral displacement introduced in [24], and we present an intrinsic measurement of the topology for 1D chiral Hamiltonians with an arbitrary (even) number of sites per unit cell.…”

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confidence: 99%

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Topological characterization of chiral models through their long time dynamics

et al. 2018

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We study chiral models in one spatial dimension, both static and periodically driven. We demonstrate that their topological properties may be read out through the long time limit of a bulk observable, the mean chiral displacement. The derivation of this result is done in terms of spectral projectors, allowing for a detailed understanding of the physics. We show that the proposed detection converges rapidly and it can be implemented in a wide class of chiral systems. Furthermore, it can measure arbitrary winding numbers and topological boundaries, it applies to all non-interacting systems, independently of their quantum statistics, and it requires no additional elements, such as external fields, nor filled bands.Topological phases of matter constitute a new paradigm by escaping the standard Ginzburg-Landau theory of phase transitions. These exotic phases appear without any symmetry breaking and are not characterized by a local order parameter, but rather by a global topological order. In the last decade, topological insulators have attracted much interest [1]. These systems are insulators in their bulk but exhibit current carrying edge states protected by the topology. A classification of topological insulators in terms of their discrete symmetries and their spatial dimensionality has been obtained in the celebrated periodic table of topological insulators and superconductors [2]. The topological invariant characterizing these models can be derived from the bulk Hamiltonian and allows one to recover the so called bulk-edge correspondence, namely that the number of topologically protected edge states is proportional to the topological invariant. A famous example of this correspondence can be found in the Quantum-Hall effect where the quantization of the Hall conductance is rooted in the current-carrying protected edge states [3][4][5]. The ensemble of (natural and artificial) topological insulators is steadily growing, and these have been by now synthetically engineered in a multitude of physical systems such as atomic [6][7][8][9][10][11], superconducting [12], photonic [13][14][15][16][17] and acoustic platforms [18][19][20].This work focuses on one-dimensional (1D) topological insulators possessing chiral symmetry. As a consequence of the chiral symmetry, the different sites of the unit cell can always be regarded as part of two sublattices. The topological invariant of the bulk, the winding number  , allows one to predict the number of zero energy edge states. 1D chiral topological insulators have been realized in numerous platforms as ultracold atoms [6,11], photonic crystals [15], photonic quantum walks [21][22][23][24][25]. Let us notice that the 1D chiral Hamiltonian can be static or the effective Hamiltonian of a Floquet system. In the latter case, the topology can be richer than its static counterpart [26][27][28][29]. In both cases, two different approaches to characterize the topology of such systems have been proposed and implemented experimentally. The first one is to look at intrinsic prope...

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“…While these effects have been originally observed in semiconductor systems, experimental studies have been conducted on systems such as ultra cold atoms [4][5][6][7], photonic model systems [8][9][10][11][12], solid-state systems [13,14], superconducting circuits [15], mechanical oscillators [16] and microwave networks [17][18][19]. In photonic systems, topological phenomena can be accessed by implementing a split-step quantum walk on a 1D optical lattice [20][21][22].…”

Section: Introductionmentioning

confidence: 99%

Eigenvalue measurement of topologically protected edge states in split-step quantum walks

et al. 2019

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We study topological phenomena of quantum walks by implementing a novel protocol that extends the range of accessible properties to the eigenvalues of the walk operator. To this end, we experimentally realise for the first time a split-step quantum walk with decoupling, which allows for investigating the effect of a bulk-boundary while realising only a single bulk configuration. The experimental platform is implemented with the well-established time-multiplexing architecture based on fibre-loops and coherent input states. The symmetry protected edge states are approximated with high similarities and we read-out the phase relative to a reference for all modes. In this way we observe eigenvalues which are distinguished by the presence or absence of sign flips between steps. Furthermore, the results show that investigating a bulk-boundary with a single bulk is experimentally feasible when decoupling the walk beforehand.

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Measuring topological invariants in disordered discrete-time quantum walks

Cited by 94 publications

References 46 publications

Topological Quantum Walks in Momentum Space with a Bose-Einstein Condensate

Topological Quantum Walks in Momentum Space with a Bose-Einstein Condensate

Topological characterization of chiral models through their long time dynamics

Eigenvalue measurement of topologically protected edge states in split-step quantum walks

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