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Physical Review Research

Experimental classification of quenched quantum walks by dynamical Chern number

Xiao‐Ye Xu, Qinqin Wang, Si-Jing Tao, Wei-Wei Pan, Zhe Chen, Munsif Jan, Yongtao Zhan, Kai Sun, Jin‐Shi Xu, Yong‐Jian Han, Chuan‐Feng Li, Guang‐Can Guo

Published Oct 22, 2019• 12 citations
Publisher

Abstract

The authors experimentally elucidate the complete classification of quenched quantum walks. This paper extends the use of the dynamical Chern number, originally introduced in quenches of static systems, to periodically driven systems. These results show that the class of the quenches is related closely to the relevant quasi-equilibrium topological invariants.

DOI
10.1103/physrevresearch.1.033039

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The authors experimentally elucidate the complete classification of quenched quantum walks. This paper extends the use of the dynamical Chern number, originally introduced in quenches of static systems, to periodically driven systems. These results show that the class of the quenches is related closely to the relevant quasi-equilibrium topological invariants.
In this review, we give a brief review on the recent progress in the theoretical research of quench dynamics in topological band systems. Beginning with two band models, we introduce conception of dynamical Chern number and give the connection between the dynamical Chern number and topological invariant in the corresponding equilibrium systems. Then by studying the 1 + 1 dimensional parent Hamiltonian, we show the complete dynamical classification of Altland-Zirnbauer classes, and show the crossing of entanglement spectrum as a feature of dynamical bulk edge correspondence. Furthermore, we consider the impact of the disorder and band dispersion. At last, we show the experimental simulation of dynamical Chern number by a superconducting qubit system.
Integer-valued topological indices, characterizing nonlocal properties of quantum states of matter, are known to directly predict robust physical properties of equilibrium systems. The Chern number, e.g., determines the quantized Hall conductivity of an insulator. Using non-interacting fermionic atoms in a periodically driven optical lattice, here we demonstrate experimentally that the Chern number determines also the far-from-equilibrium dynamics of a quantum system. Extending a respective proposal to Floquet systems, we measure the linking number that characterizes the trajectories of momentum-space vortices emerging after a strong quench. We observe that it directly corresponds to the ground-state Chern number. This one-to-one relation between a dynamical and a static topological index allows us to experimentally map out the phase diagram of our system. Furthermore, we measure the instantaneous Chern number and show that it remains zero under the unitary dynamics.
We study the effect of strong disorder on topology and entanglement in quench dynamics. Although disorder-induced topological phases have been well studied in equilibrium, the disorder-induced topology in quench dynamics has not been explored. In this paper, we predict a disorder-induced topology of postquench states characterized by the quantized dynamical Chern number and the crossings in the entanglement spectrum in (1+1)(1+1)(1+1) dimensions. The dynamical Chern number undergoes transitions from zero to unity and back to zero when increasing the disorder strength. The boundaries between different dynamical Chern numbers are determined by delocalized critical points in the postquench Hamiltonian with the strong disorder. An experimental realization in quantum walks is discussed.
Quantum walks (QWs) provide a powerful tool as a quantum simulator to study topological phases. However, some interesting topological phenomena in the two dimensional (2D) QW that do not exist in the one dimensional case, e.g., the edge-state-enhanced transport, have not been demonstrated experimentally. Here we have observed 2D topological bound states with vanishing Chern numbers and confirmed the robustness of these bound states with respect to perturbations and disorder, which go beyond what has been known in static systems and are unique to periodically driven systems. Our studies open up an avenue to explore topological properties in multidimensional QWs.
We study the dynamical quantum phase transition of a critical quantum quench, in which the pre-quenched Hamiltonian, or the post quenched Hamiltonian, or both of them are set to be the critical points of equilibrium quantum phase transitions. We find a half-quantized or unquantized dynamical topological order parameter and dynamical Chern number; these results and also the existence of dynamical quantum phase transition are all closely related to the singularity of the Bogoliubov angle at the gap-closing momentum. The effects of the singularity may also be canceled out if both the prequenched and postquenched Hamiltonians are critical; then the dynamical topological order parameter and dynamical Chern number restore to integer ones. Our findings show that the widely accepted definitions of the dynamical topological order parameter and dynamical Chern number are problematic for the critical quenches in the perspective of topology, which call for new definitions of them.
Recent experiments began to explore the topological properties of quench dynamics, i.e. the time evolution following a sudden change in the Hamiltonian, via tomography of quantum gases in optical lattices. In contrast to the well established theory for static band insulators or periodically driven systems, at present it is not clear whether, and how, topological invariants can be defined for a general quench of band insulators. Previous work solved a special case of this problem beautifully using Hopf mapping of two-band Hamiltonians in two dimensions. But it only works for topologically trivial initial state and is hard to generalize to multiband systems or other dimensions. Here we introduce the concept of loop unitary constructed from the unitary time-evolution operator, and show its homotopy invariant fully characterizes the dynamical topology. For two-band systems in two dimensions, we prove that the invariant is precisely equal to the change in the Chern number across the quench regardless of the initial state. We further show that the nontrivial dynamical topology manifests as hedgehog defects in the loop unitary, and also as winding and linking of its eigenvectors along a curve where dynamical quantum phase transition occurs. This opens up a systematic route to classify and characterize quantum quench dynamics.