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arXiv:2305.11802 · Physical review. B./Physical review. B

Engineering subharmonic responses beyond prethermalization via Floquet scar states

Ke Huang, Xiao Li

Published Feb 13, 2024• 3 citations
arXivHTMLPDF

Abstract

In this work we propose a scheme to engineer subharmonic responses via scar states in a generalized PXP model. We first show that the generalized PXP model also possesses a band of scar states like the pristine PXP model does. In addition, we reveal that a generalized forward-scattering approximation (FSA) still works for these scar states. We further argue that the FSA subspace exhibits an $SO(3)$ symmetry, which enables an $SO(3)\text{\ensuremath{-}}\mathrm{FSA}$ approach for the scar states. When such a model is placed under periodic driving, nontrivial Floquet scar states emerge in the quasienergy spectrum. One appealing feature of such Floquet scar states is that they can support subharmonic responses akin to the discrete time crystalline phase. In particular, such subharmonic responses can exist beyond the conventional prethermalization regime, where either large frequencies or large driving amplitudes are required.

DOI
10.1103/physrevb.109.064306

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In this work we propose a scheme to engineer subharmonic responses via scar states in a generalized PXP model. We first show that the generalized PXP model also possesses a band of scar states like the pristine PXP model does. In addition, we reveal that a generalized forward-scattering approximation (FSA) still works for these scar states. We further argue that the FSA subspace exhibits an SO(3)SO(3)SO(3) symmetry, which enables an SO(3)\ensuremath-FSASO(3)\text{\ensuremath{-}}\mathrm{FSA}SO(3)\ensuremath-FSA approach for the scar states. When such a model is placed under periodic driving, nontrivial Floquet scar states emerge in the quasienergy spectrum. One appealing feature of such Floquet scar states is that they can support subharmonic responses akin to the discrete time crystalline phase. In particular, such subharmonic responses can exist beyond the conventional prethermalization regime, where either large frequencies or large driving amplitudes are required.
We propose a scheme that generates period-doubled responses via periodically driving certain Hamiltonians hosting quantum many-body scars, akin to recent experimental observations in driven Rydberg atom arrays. Our construction takes advantage of an su(2) spectrum generating algebra associated with the static quantum-scarred Hamiltonian, which enacts a $\ensuremath{\pi}$ rotation within the scar subspace after one period of time evolution with appropriately chosen driving parameters. This yields period-doubled (subharmonic) responses in local observables for any choice of initial state residing in the scar subspace. The quasienergy spectrum features atypical $\ensuremath{\pi}$-paired eigenstates embedded in an otherwise fully thermal spectrum. The protocol requires neither a large driving frequency nor a large driving amplitude and is thus distinct from the prethermalization physics in previous investigations of the driven PXP model. We demonstrate our scheme using several spin-1/2 and spin-1 quantum scarred models possessing an exact su(2) spectrum generating algebra, as well as a symmetry-deformed PXP model, where the su(2) algebra is only approximate. Our results extend the class of models hosting quantum many-body scars that can be leveraged to yield time-crystalline behaviors under periodic driving.
The PXP model hosts a special set of nonergodic states, referred to as quantum many-body scars. One of the consequences of quantum scarring is the periodic revival of the wave function fidelity. It has been reported that quantum fidelity revival occurs in the PXP model for certain product states, and periodic driving of the chemical potential can enhance the magnitude of quantum revival and can even change the frequencies of revival showing the subharmonic response. Although the effect of the periodic driving in the PXP model has been studied in the limit of certain perturbative regimes, the general mechanism of such enhanced revival and frequency change has been barely studied. In this paper, we investigate how periodic driving in the PXP model can systematically control the fidelity revival. Particularly, focusing on a product state called the N\'eel state, we analyze the condition of driving to enhance the magnitude of revival or change the frequencies of revival. To clarify the reason for such control, we consider the similarities between the PXP model and the free spin-1/21/21/2 model in graph theoretical analysis and show that the quantum fidelity feature in the PXP model is well explained by the free spin-1/21/21/2 model. In addition, under a certain limit of the driving parameters, an analytic approach to explain the main features of the fidelity revival is also applied. Our results give insight into the scarring nature of the periodically driven PXP model and pave the way to understand the (sub)harmonic responses in these models and controls thereof.
Periodic driving has been established as a powerful technique for engineering novel phases of matter and intrinsically out-of-equilibrium phenomena such as time crystals. Recent paper by Bluvstein et al. [Science 371, 1355 (2021)] has demonstrated that periodic driving can also lead to a significant enhancement of quantum many-body scarring, whereby certain nonintegrable systems can display persistent quantum revivals from special initial states. Nevertheless, the mechanisms behind driving-induced scar enhancement remain poorly understood. Here we report a detailed study of the effect of periodic driving on the PXP model describing Rydberg atoms in the presence of a strong Rydberg blockade---the canonical static model of quantum many-body scarring. We show that periodic modulation of the chemical potential gives rise to a rich phase diagram, with at least two distinct types of scarring regimes that we distinguish by examining their Floquet spectra. We formulate a toy model, based on a sequence of square pulses, that accurately captures the details of the scarred dynamics and allows for analytical treatment in the large-amplitude and high-frequency driving regimes. Finally, we point out that driving with a spatially inhomogeneous chemical potential allows to stabilize quantum revivals from arbitrary initial states in the PXP model, via a mechanism similar to prethermalization.
The control of many-body quantum dynamics in complex systems is a key challenge in the quest to reliably produce and manipulate large-scale quantum entangled states. Recently, quench experiments in Rydberg atom arrays [Bluvstein et al. Science 371, 1355 (2021)SCIEAS0036-807510.1126/science.abg2530] demonstrated that coherent revivals associated with quantum many-body scars can be stabilized by periodic driving, generating stable subharmonic responses over a wide parameter regime. We analyze a simple, related model where these phenomena originate from spatiotemporal ordering in an effective Floquet unitary, corresponding to discrete time-crystalline behavior in a prethermal regime. Unlike conventional discrete time crystals, the subharmonic response exists only for Néel-like initial states, associated with quantum scars. We predict robustness to perturbations and identify emergent timescales that could be observed in future experiments. Our results suggest a route to controlling entanglement in interacting quantum systems by combining periodic driving with many-body scars.
We demonstrate that a continuous two frequency drive is a versatile and robust protocol to control the lifetime of quantum many body scars and to engineer non-equilibrium phases of driven quantum matter. By modulating the frequency ratio ccc (any rational number), we systematically explore prethermal features across a broad frequency range. For small integer values of ccc, we observe ergodicity breaking even at moderately low frequencies, signaling long-lived scarred dynamics. By continuously increasing ccc, one can generate non-monotonic transitions between ergodic and non-ergodic dynamics. These observations are consistent with the predictions of an effective Floquet Hamiltonian based approach. Furthermore, we exploit this tunability to engineer fractional subharmonic responses, highlighting the potential of two-frequency driving as a theoretical platform for controlling scars, prethermalization, and time crystal-like behavior.
Scar eigenstates in a many-body system refers to a small subset of non-thermal finite energy density eigenstates embedded into an otherwise thermal spectrum. This novel non-thermal behaviour has been seen in recent experiments simulating a one-dimensional PXP model with a kinetically-constrained local Hilbert space realised by a chain of Rydberg atoms. We probe these small sets of special eigenstates starting from particular initial states by computing the spread complexity associated to time evolution of the PXP hamiltonian. Since the scar subspace in this model is embedded only loosely, the scar states form a weakly broken representation of the Lie algebra. We demonstrate why a careful usage of the forward scattering approximation (FSA), instead of any other method, is required to extract the most appropriate set of Lanczos coefficients in this case as the consequence of this approximate symmetry. Only such a method leads to a well defined notion of a closed Krylov subspace and consequently, that of spread complexity. We show this using three separate initial states, namely|Z2⟩,|Z3⟩and the vacuum state, due to the disparate classes of scar states hosted by these sectors. We also discuss systematic methods of remedying the imperfections in the FSA setup stemming from these approximate symmetries.