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arXiv:1711.06188 · Physical Review Letters

Localization Counteracts Decoherence in Noisy Floquet Topological Chains

Maria-Theresa Rieder, Lukas M. Sieberer, Mark H. Fischer, Ion Cosma Fulga

Published May 25, 2018• 24 citations
arXivHTMLPDF

Abstract

The topological phases of periodically driven, or Floquet systems, rely on a perfectly periodic modulation of system parameters in time. Even the smallest deviation from periodicity leads to decoherence, causing the boundary (end) states to leak into the system's bulk. Here, we show that in one dimension this decay of topologically protected end states depends fundamentally on the nature of the bulk states: a dispersive bulk results in an exponential decay, while a localized bulk slows the decay down to a diffusive process. The localization can be due to disorder, which remarkably counteracts decoherence even when it breaks the symmetry responsible for the topological protection. We derive this result analytically, using a novel, discrete-time Floquet-Lindblad formalism and confirm our findings with the help of numerical simulations. Our results are particularly relevant for experiments, where disorder can be tailored to protect Floquet topological phases from decoherence.

DOI
10.1103/physrevlett.120.216801

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The topological phases of periodically driven, or Floquet systems, rely on a perfectly periodic modulation of system parameters in time. Even the smallest deviation from periodicity leads to decoherence, causing the boundary (end) states to leak into the system's bulk. Here, we show that in one dimension this decay of topologically protected end states depends fundamentally on the nature of the bulk states: a dispersive bulk results in an exponential decay, while a localized bulk slows the decay down to a diffusive process. The localization can be due to disorder, which remarkably counteracts decoherence even when it breaks the symmetry responsible for the topological protection. We derive this result analytically, using a novel, discrete-time Floquet-Lindblad formalism and confirm our findings with the help of numerical simulations. Our results are particularly relevant for experiments, where disorder can be tailored to protect Floquet topological phases from decoherence.
We study the effect of noise on two-dimensional periodically driven topological phases, focusing on two examples: the anomalous Floquet-Anderson phase and the disordered Floquet-Chern phase. Both phases show an unexpected robustness against timing noise. The noise-induced decay of initially populated topological edge modes occurs in two stages: At short times, thermalization among edge modes leads to exponential decay. This is followed by slow algebraic decay ∼ n − 1 / 2 with the number of Floquet cycles n . The exponent of 1 / 2 is characteristic for one-dimensional diffusion, here occurring along the direction perpendicular to the edge. In contrast, localized modes in the bulk exhibit faster decay, ∼ n − 1 , corresponding to two-dimensional diffusion. We demonstrate these behaviors through full-scale numerical simulations and support our conclusions using analytical results based upon a phenomenological model. Our findings indicate that two-dimensional Floquet topological phases are ideal candidates for potential applications of Floquet topology, given the unavoidable presence of both quenched disorder and decoherence in experiments.
Periodically driven systems, also known as Floquet systems, can realize symmetry protected topological (SPT) phases that cannot be found in equilibrium. Here, we seek to understand the effects of strong disorder on such SPT phases, working with a one-dimensional Floquet system belonging to Altland-Zirnbauer class BDI. Using the transfer matrix method, we find that the disordered system hosts an array of trivial, topological, and Floquet topological phases. We then explore the phase diagram in the case of uniformly distributed bonds. Our analytic results are confirmed by a numeric computation of the bulk topological invariants. Although all states are generically localized, near phase transitions the localization length of topological edge states diverges. We derive the critical exponents governing these divergences, finding that they take on equilibrium values, despite the fact that we are considering phases unique to non-equilibrium. We also discuss how our work sets the stage for studying interacting Floquet SPT phases.
Much recent experimental effort has focused on the realization of exotic quantum states and dynamics predicted to occur in periodically driven systems. But how robust are the sought-after features, such as Floquet topological surface states, against unavoidable imperfections in the periodic driving? In this paper, we address this question in a broader context and study the dynamics of quantum systems subject to noise with periodically recurring statistics. We show that the stroboscopic time evolution of such systems is described by a noise-averaged Floquet superoperator. The eigenvectors and -values of this superoperator generalize the familiar concepts of Floquet states and quasienergies and allow us to describe decoherence due to noise efficiently. Applying the general formalism to the example of a noisy Floquet topological chain, we rederive and corroborate our recent findings on the noise-induced decay of topologically protected end states. These results follow directly from an expansion of the end state in eigenvectors of the Floquet superoperator.
We construct an example of a 1d quasiperiodically driven spin chain whose edge states can coherently store quantum information, protected by a combination of localization, dynamics, and topology. In a sharp departure from topological phases in static and periodically driven (Floquet) spin chains, this model does not rely upon microscopic symmetry protection: Instead, the edge states are protected purely by emergent dynamical symmetries. We explore the dynamical signatures of this emergent dynamical symmetry-protected topological (EDSPT) order through exact numerics, time evolving block decimation, and analytic high-frequency expansion, finding evidence that the EDSPT is a stable dynamical phase protected by bulk many-body localization up to (at least) stretched-exponentially long timescales, and possibly beyond. We argue that EDSPTs are special to the quasiperiodically driven setting, and cannot arise in Floquet systems. Moreover, we find evidence of a type of boundary critical with no known static or Floquet analogue, in which the edge spin dynamics transition from quasiperiodic to chaotic, leading to bulk thermalization.
Topologically protected edge states exactly at topological phase boundaries challenge the conventional belief that topological states must be associated with a bulk energy gap. Because periodically driven (Floquet) systems host unusually intricate topological phase boundaries, topological edge states can be prolific at such Floquet quantum criticality. Working on a class of chiral-symmetric, Floquet-driven Majorana fermion chains, we analytically and computationally show that the precise boundaries between different Floquet topological gapped phases can accommodate topological edge modes, including the so-called critical Majorana π modes. We also identify a general bulk-edge correspondence formula to predict and understand the emergence of topological edge modes at Floquet quantum criticality. Of direct interest to quantum simulation experiments, our work opens an avenue for exploring topological physics intrinsic to nonequilibrium phase boundaries. The authors develop a theoretical framework to characterize the topology and edge states at the nonequilibrium critical points of periodically driven (Floquet) quantum systems. The rule of bulk-edge correspondence applicable to both gapped and gapless Floquet topological phases in 1D chiral symmetric driven systems is proposed.
Discrete time crystals (DTCs) represent a nonequilibrium phase of matter characterized by spontaneousbreaking of discrete time-translational symmetry under periodic driving. While Floquet topological phases offer apromising route to stabilizing DTC behavior, their persistence in realistic open quantum systems remains an openproblem. In this work, we numerically investigate a periodically driven one-dimensional Ising chain locally coupledto a dissipative two-level system (TLS) defect, motivated by microscopic defects commonly observed in superconductingcircuit-QED devices. Using Lindblad master-equation simulations implemented in QuTiP, we show that weaklydissipative TLS defects can dynamically synchronize with the Floquet edge mode and inherit its subharmonic oscillations.As the defect coupling and dissipation strength increase, the system undergoes a crossover toward decoherence andFloquet thermalization, accompanied by the suppression of edge-localized period doubling. By combining quasienergyspectroscopy, boundary-condition analysis, rigidity diagnostics, and entropy evolution, we characterize the stabilitylimits of Floquet topological time-crystalline phases in realistic noisy quantum hardware.