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arXiv:2005.07715 · Physical review. B./Physical review. B

Dynamics of the vacuum state in a periodically driven Rydberg chain

Bhaskar Mukherjee, Arnab Sen, Diptiman Sen, K. Sengupta

Published Aug 17, 2020• 81 citations
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Abstract

We study the dynamics of a periodically driven Rydberg chain starting from the state with zero Rydberg excitations (vacuum state denoted by $|0\ensuremath{\rangle}$) using a square pulse protocol in the high drive amplitude limit. We show, using exact diagonalization for finite system sizes $(L\ensuremath{\le}26)$, that the Floquet Hamiltonian of the system, within a range of drive frequencies that we chart out, hosts a set of quantum scars that have large overlap with the $|0\ensuremath{\rangle}$ state. These scars are distinct from their counterparts having high overlap with the maximal Rydberg excitation state $(|{\mathbb{Z}}_{2}\ensuremath{\rangle})$; they coexist with the latter class of scars and lead to persistent coherent oscillations of the density-density correlator starting from the $|0\ensuremath{\rangle}$ state. We also identify special drive frequencies at which the system undergoes perfect dynamic freezing, and we provide an analytic explanation for this phenomenon. Finally, we demonstrate that for a wide range of drive frequencies, the system reaches a steady state with subthermal values of the density-density correlator. The presence of such subthermal steady states, which are absent for dynamics starting from the $|{\mathbb{Z}}_{2}\ensuremath{\rangle}$ state, imply a weak violation of the eigenstate thermalization hypothesis in finite-sized Rydberg chains distinct from that due to the scar-induced persistent oscillations reported earlier. We conjecture that in the thermodynamic limit, such states may exist as prethermal steady states that show anomalously slow relaxation. We supplement our numerical results by deriving an analytic expression for the Floquet Hamiltonian using a Floquet perturbation theory in the high amplitude limit, which provides an analytic, albeit qualitative, understanding of these phenomena at arbitrary drive frequencies. We discuss experiments that can test our theory.

DOI
10.1103/physrevb.102.075123

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We study the dynamics of a periodically driven Rydberg chain starting from the state with zero Rydberg excitations (vacuum state denoted by ∣0\ensuremath⟩|0\ensuremath{\rangle}∣0\ensuremath⟩) using a square pulse protocol in the high drive amplitude limit. We show, using exact diagonalization for finite system sizes (L\ensuremath≤26)(L\ensuremath{\le}26)(L\ensuremath≤26), that the Floquet Hamiltonian of the system, within a range of drive frequencies that we chart out, hosts a set of quantum scars that have large overlap with the ∣0\ensuremath⟩|0\ensuremath{\rangle}∣0\ensuremath⟩ state. These scars are distinct from their counterparts having high overlap with the maximal Rydberg excitation state (∣Z2\ensuremath⟩)(|{\mathbb{Z}}_{2}\ensuremath{\rangle})(∣Z2​\ensuremath⟩); they coexist with the latter class of scars and lead to persistent coherent oscillations of the density-density correlator starting from the ∣0\ensuremath⟩|0\ensuremath{\rangle}∣0\ensuremath⟩ state. We also identify special drive frequencies at which the system undergoes perfect dynamic freezing, and we provide an analytic explanation for this phenomenon. Finally, we demonstrate that for a wide range of drive frequencies, the system reaches a steady state with subthermal values of the density-density correlator. The presence of such subthermal steady states, which are absent for dynamics starting from the ∣Z2\ensuremath⟩|{\mathbb{Z}}_{2}\ensuremath{\rangle}∣Z2​\ensuremath⟩ state, imply a weak violation of the eigenstate thermalization hypothesis in finite-sized Rydberg chains distinct from that due to the scar-induced persistent oscillations reported earlier. We conjecture that in the thermodynamic limit, such states may exist as prethermal steady states that show anomalously slow relaxation. We supplement our numerical results by deriving an analytic expression for the Floquet Hamiltonian using a Floquet perturbation theory in the high amplitude limit, which provides an analytic, albeit qualitative, understanding of these phenomena at arbitrary drive frequencies. We discuss experiments that can test our theory.
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.
The presence of quantum scars, athermal eigenstates of a many-body Hamiltonian with finite-energy density, leads to an absence of ergodicity and long-time coherent dynamics starting from initial states that have a high overlap with scars as experimentally observed in a chain of ultracold Rydberg atoms. We show, via study of a periodically driven Rydberg chain, that the drive frequency acts as a tuning parameter for several reentrant transitions between ergodic and weak ergodicity breaking regimes. The former regime shows rapid thermalization of correlation functions and absence of scars in the Floquet spectrum of the system. The latter regime, in contrast, has scars; they lead to long-time coherent dynamics of correlation functions. We provide an analytical explanation of the existence of these regimes by going beyond a high-frequency Magnus expansion and using a novel perturbative approach valid at large drive amplitudes to derive a Floquet Hamiltonian which qualitatively explains the behavior of the driven system at arbitrary frequencies. We also discuss experiments involving finite Rydberg chains which can validate our theory. Our results demonstrate the possibility of drive-frequency-induced tuning between ergodic and weak ergodicity breaking dynamics in a disorder-free quantum many-body system.
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