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arXiv:2404.06536

Exact Floquet flat band and heating suppression via two-rate drive protocols

Banerjee, Tista, Sayan Choudhury, K. Sengupta

Published Apr 9, 2024• 0 citations
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Abstract

We demonstrate the existence of exact Floquet flat bands implying strong violation of the eigenstate thermalization hypothesis in a large class of closed quantum many-body systems in the presence of a two-rate drive characterized by frequencies $Ω_1$ and $Ω_2=νΩ_1$. We provide the exact analytic condition for this phenomenon to occur for a generic protocol; in particular, $ν=(2p+1)$, where $p$ is an integer, leads to such flat bands for both square-pulse and cosine drive protocols for arbitrary $Ω_1$. In the vicinity of these points, heating is suppressed up to very long timescales in such driven systems, leading to a prethermal regime; we demonstrate this by exact numerical studies of distribution and bandwidth of the Floquet eigenstates, spectral form factor, entanglement entropy, and correlation functions of an experimentally realizable finite driven Rydberg chain. The corresponding micromotion exhibits coherent reversal of excitations reminiscent of echoes. Our analysis constitutes a yet unexplored mechanism for heating suppression in driven closed quantum systems.

DOI
10.48550/arxiv.2404.06536

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We demonstrate the existence of exact Floquet flat bands implying strong violation of the eigenstate thermalization hypothesis in a large class of closed quantum many-body systems in the presence of a two-rate drive characterized by frequencies Ω1Ω_1Ω1​ and Ω2=νΩ1Ω_2=νΩ_1Ω2​=νΩ1​. We provide the exact analytic condition for this phenomenon to occur for a generic protocol; in particular, ν=(2p+1)ν=(2p+1)ν=(2p+1), where ppp is an integer, leads to such flat bands for both square-pulse and cosine drive protocols for arbitrary Ω1Ω_1Ω1​. In the vicinity of these points, heating is suppressed up to very long timescales in such driven systems, leading to a prethermal regime; we demonstrate this by exact numerical studies of distribution and bandwidth of the Floquet eigenstates, spectral form factor, entanglement entropy, and correlation functions of an experimentally realizable finite driven Rydberg chain. The corresponding micromotion exhibits coherent reversal of excitations reminiscent of echoes. Our analysis constitutes a yet unexplored mechanism for heating suppression in driven closed quantum systems.
Abstract We demonstrate the existence of exact Floquet flat bands in a class of non-integrable quantum many-body Hamiltonians leading to strong violation of the eigenstate thermalization hypothesis in the presence of a two-tone drive characterized by frequencies Ω1 and Ω 2 = ν Ω 1 , where ν is an integer. We provide the exact analytic condition for this phenomenon to occur for a generic protocol. A small departure from the flat band conditions leads to suppression of heating over a long prethermal timescale in an otherwise ergodic many-body system away from integrable or perturbative regimes; we elucidate the central role of the flat band behind such heating suppression. Our analysis indicates the existence of a perturbative Floquet Hamiltonian, away from high drive frequency/amplitude limit, which controls the dynamics of the driven system near such flat bands. We demonstrate this phenomenon by exact numerical studies of Floquet bandwidth, spectral form factor, entanglement and Shannon entropies, heat absorption, and correlation functions of a driven Rydberg chain. We also study the corresponding micromotion which exhibits coherent reversal of excitations reminiscent of echoes; in addition, it harbors an exact reflection symmetry about t = T 1 / 2 , where T 1 = 2 π / Ω 1 , which we elucidate. We discuss experiments which can test our theory.
This work presents a theoretical investigation of dynamic stabilization in interacting quantum many-body systems via high-frequency periodic driving. Using Floquet theory and the Magnus expansion, we derive the effective time-independent Hamiltonian governing the slow dynamics of a driven spin chain. We show that the external field induces a parametric renormalization of the transverse tunneling terms through a zeroth-order Bessel function dependence, leading to a critical drive amplitude at which transport is strongly suppressed. This suppression of the lowest-order hopping term gives rise to a long-lived prethermal regime in which many-body thermalization is significantly delayed. The resulting dynamics exhibit a sharp reduction in the short-to-intermediate time entanglement entropy growth rate. The framework is consistent with known results in Floquet engineering and coherent destruction of tunneling, and provides a unified interpretation of transport suppression and delayed scrambling in periodically driven systems. A falsifiable experimental protocol using ultracold atoms in optical lattices is proposed to validate the predicted threshold behavior and prethermal stabilization mechanism.
We study the stroboscopic dynamics of a periodically driven finite Rydberg chain with staggered ($\mathrm{\ensuremath{\Delta}}$) and time-dependent uniform $[\ensuremath{\lambda}(t)]$ detuning terms using exact diagonalization. We show that at intermediate drive frequencies (${\ensuremath{\omega}}_{D}$), the presence of a finite $\mathrm{\ensuremath{\Delta}}$ results in violation of the eigenstate thermalization hypothesis (ETH) via clustering of Floquet eigenstates. Such clustering is lost at special commensurate drive frequencies for which $\ensuremath{\hbar}{\ensuremath{\omega}}_{d}=n\mathrm{\ensuremath{\Delta}}$ ($n\ensuremath{\in}Z$) leading to restoration of ergodicity. The violation of ETH in these driven finite-sized chains is also evident from the dynamical freezing displayed by the density-density correlation between Rydberg excitations at even sites of the chain for specific ${\ensuremath{\omega}}_{D}$. Such a correlator exhibits stable oscillations with perfect revivals when driven close to the freezing frequencies for initial all spin-down ($|0\ensuremath{\rangle}$) or Neel ($|{\mathbb{Z}}_{2}\ensuremath{\rangle}$, with up spins on even sites) states. In contrast, for the $|{\overline{\mathbb{Z}}}_{2}\ensuremath{\rangle}$ (time-reversed partner of $|{\mathbb{Z}}_{2}\ensuremath{\rangle}$) initial state, we find complete absence of such oscillations leading to freezing for a range of ${\ensuremath{\omega}}_{D}$; this range increases with $\mathrm{\ensuremath{\Delta}}$. We also study the properties of quantum many-body scars in the Floquet spectrum of the model as a function of $\mathrm{\ensuremath{\Delta}}$ and show the existence of mid-spectrum scars at large $\mathrm{\ensuremath{\Delta}}$ which do not have overlap with either $|0\ensuremath{\rangle}$ or $|{\mathbb{Z}}_{2}\ensuremath{\rangle}$ states. We supplement our numerical results with those from an analytic Floquet Hamiltonian computed using Floquet perturbation theory which allows us to provide qualitative analytical explanations of the above-mentioned numerical results.
We study a Rydberg chain subjected to a two-tone periodic drive of the detuning Δ(t)Δ(t)Δ(t) and Rabi frequency Ω(t)Ω(t)Ω(t) at frequencies ω1ω_1ω1​ and ω2=νω1ω_2=νω_1ω2​=νω1​ respectively with ν∈Zν\in \mathbf{Z}ν∈Z. In the strong Rydberg blockade limit, such a system reduces to the PXP model and exhibits exact Floquet flat band. Here we determine, using a time-dependent Schrieffer-Wolff transformation, the domain of validity of such PXP dynamics once the blockade-violating processes are taken into account. We study the melting of such flat bands in the presence of strong, but finite, blockade strength. Using Floquet perturbation theory (FPT), we identify two mechanisms for such melting; the first occurs due to the presence of many-body resonances involving resonant production of nearest-neighbor Rydberg excitation pairs while the second involves virtual processes that access intermediate states with such pairs. Our analytical FPT results match with those obtained using exact diagonalization on a finite chain. We study the dynamics for ∣vac⟩|{\rm vac}\rangle∣vac⟩ (Fock state with no Rydberg excitation), ∣Z2⟩|{\mathbb {Z}}_2\rangle∣Z2​⟩, and ∣Z3⟩|{\mathbb {Z}}_3\rangle∣Z3​⟩ (Fock states with Rydberg excitations on alternate and every third site respectively) initial states. We find that ∣vac⟩|{\rm vac}\rangle∣vac⟩ shows exceptional stability against resonant pair formation while ∣Z3⟩|{\mathbb {Z}}_3\rangle∣Z3​⟩ exhibits flat band melting via virtual processes away from the direct resonances. We find oscillatory dynamics with large time period for the ∣Z2⟩|{\mathbb {Z}}_2\rangle∣Z2​⟩ initial state and the presence of Floquet scars that have large overlap with ∣Z3⟩|{\mathbb {Z}}_3\rangle∣Z3​⟩ state; these phenomena occur due to processes that are absent in the PXP limit. We propose measurement of nearest neighbor pair correlation functions using Rydberg atom quantum simulator platforms as an accessible experimental probe for these effects.
Periodic driving has emerged as a powerful tool in the quest to engineer new and exotic quantum phases. While driven many-body systems are generically expected to absorb energy indefinitely and reach an infinite-temperature state, the rate of heating can be exponentially suppressed when the drive frequency is large compared to the local energy scales of the system -- leading to long-lived 'prethermal' regimes. In this work, we experimentally study a bosonic cloud of ultracold atoms in a driven optical lattice and identify such a prethermal regime in the Bose-Hubbard model. By measuring the energy absorption of the cloud as the driving frequency is increased, we observe an exponential-in-frequency reduction of the heating rate persisting over more than 2 orders of magnitude. The tunability of the lattice potentials allows us to explore one- and two-dimensional systems in a range of different interacting regimes. Alongside the exponential decrease, the dependence of the heating rate on the frequency displays features characteristic of the phase diagram of the Bose-Hubbard model, whose understanding is additionally supported by numerical simulations in one dimension. Our results show experimental evidence of the phenomenon of Floquet prethermalization, and provide insight into the characterization of heating for driven bosonic systems.
We study a chain of periodically driven Rydberg atoms and identify a class of drive protocols for which the system exhibits emergent prethermal Bethe integrability at special drive frequencies. We provide a perturbative analytic expression of its Floquet Hamiltonian in the large drive amplitude regime. We demonstrate integrability of the leading term of this Floquet Hamiltonian at special drive frequencies, which we identify, by mapping it to the Hamiltonian of the paradigmatic spin-1/21/21/2 XXZ{\rm XXZ}XXZ chain. We support our analytical results by exact diagonalization studies on finite chains. Our numerical results on level statistics, half-chain entanglement entropy, and longitudinal magnetization of the driven chain brings out its emergent integrable nature at the special drive frequencies which persists up to a large prethermal timescale.