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

Abstract

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...

Non-Hermitian Floquet Phases with Even-Integer Topological Invariants in a Periodically Quenched Two-Leg Ladder

“…The MCD was first introduced as a dynamical probe to the winding numbers of 1D topological insulators in the symmetry classes AIII and BDI [ 98 ], and later extended to Floquet systems [ 93 , 99 , 100 ], two-dimensional systems [ 101 ], many-body systems [ 102 ], systems in other symmetry classes [ 84 ], and recently also to non-Hermitian systems [ 50 , 51 , 53 ]. In the meantime, the MCD has also been measured experimentally in photonic [ 98 , 103 ] and cold atom [ 104 , 105 ] setups. In this section, we further generalize the MCD to non-Hermitian Floquet systems in the CII symmetry class, and employ it to dynamically characterize the topological phases found in the non-Hermitian PQTLL model.…”

“…Furthermore, a quantized jump of the MCD is observed every time when the system passes through a topological phase transition point. Experimentally, the MCDs have been measured in both the cold atom [ 104 , 105 ] and photonic systems [ 98 , 103 ], in which non-Hermiticity and driving fields can also be implemented [ 1 ]. Furthermore, the MCDs may also be detected directly in momentum space with the help of a recently proposed setup certaining the nitrogen-vacancy-center in diamond [ 36 ].…”

Periodic driving induced helical Floquet channels with ultracold atoms in momentum space

“…Now the periodically driven Hamiltonian for the system in Fig. 3(a) can be written as [60] H =v(t) where θ 2 = Ωτ 2 /2. Once we choose the duration τ 1 such that θ 1 = π/4, such a scheme forms a standard discrete-time quantum walk.…”

Bulk-edge correspondence in nonunitary Floquet systems with chiral symmetry

“…Since quantum walks are Floquet systems in which time evolves in a discrete manner, topological phases can be different from those which are described by time-independent Hamiltonians. Floquet topological phases of quantum walks have been intensively studied for the last decade [51][52][53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68][69], and topological edge states have been observed in experiments of both closed [70][71][72][73] and open [36,47,74,75] systems. Specifically, much attention has been paid to Floquet systems with chiral symmetry.…”