Quantifying and Controlling Prethermal Nonergodicity in Interacting Floquet Matter

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Quantifying and Controlling Prethermal Nonergodicity in Interacting Floquet Matter

K. Singh, C. J. Fujiwara, Z. A. Geiger, E. Q. Simmons, M. Lipatov, A. Cao, P. Dotti, S. V. Rajagopal, R. Senaratne, T. Shimasaki, M. Heyl, A. Eckardt, and D. M. Weld

Phys. Rev. X 9, 041021 – Published 29 October 2019

Abstract

The use of periodic driving for synthesizing many-body quantum states depends crucially on the existence of a prethermal regime, which exhibits drive-tunable properties while forestalling the effects of heating. This dependence motivates the search for direct experimental probes of the underlying localized nonergodic nature of the wave function in this metastable regime. We report experiments on a many-body Floquet system consisting of atoms in an optical lattice subjected to ultrastrong sign-changing amplitude modulation. Using a double-quench protocol, we measure an inverse participation ratio quantifying the degree of prethermal localization as a function of tunable drive parameters and interactions. We obtain a complete prethermal map of the drive-dependent properties of Floquet matter spanning four square decades of parameter space. Following the full time evolution, we observe sequential formation of two prethermal plateaux, interaction-driven ergodicity, and strongly frequency-dependent dynamics of long-time thermalization. The quantitative characterization of the prethermal Floquet matter realized in these experiments, along with the demonstration of control of its properties by variation of drive parameters and interactions, opens a new frontier for probing far-from-equilibrium quantum statistical mechanics and new possibilities for dynamical quantum engineering.

Received 19 June 2019
Revised 27 August 2019

DOI:https://doi.org/10.1103/PhysRevX.9.041021

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Cold gases in optical lattices Nonequilibrium statistical mechanics Quantum state engineering Quantum statistical mechanics

Ultracold gases

Optical lattices & traps

Atomic, Molecular & OpticalStatistical Physics & Thermodynamics

Authors & Affiliations

K. Singh ¹, C. J. Fujiwara¹, Z. A. Geiger¹, E. Q. Simmons¹, M. Lipatov¹, A. Cao ¹, P. Dotti¹, S. V. Rajagopal¹, R. Senaratne¹, T. Shimasaki ¹, M. Heyl², A. Eckardt ², and D. M. Weld^1,*

¹Department of Physics, University of California, Santa Barbara, California 93106, USA
²Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Strasse 38, 01187 Dresden, Germany

^*weld@ucsb.edu

Popular Summary

The ability to create and control quantum systems that are far out of equilibrium opens up intriguing possibilities for creating new states of matter whose properties are tuned via an external drive. However, this requires researchers to abandon the existing powerful tool set of equilibrium statistical mechanics, leaving them without effective theoretical and experimental techniques to characterize these quantum states. To overcome this challenge, we studied a “prethermal” ensemble of atoms, in which the initial state information is preserved against the effects of heating for macroscopic times. Our experiments allowed us to characterize some properties of the ensemble as well as to control other properties.

In our experiments, we work with a quantum gas of lithium atoms in an optical lattice (a trap built from counterpropagating laser beams). By modulating the amplitude of laser light in the lattice, we prepare a state of matter with properties that could be tuned by adjusting the drive amplitude and frequency. Images of the ensemble reveal a detailed map of state occupations, in good agreement with theoretical predictions. As the system evolves during the modulation, we observe the emergence and destruction of prethermal states, interaction-driven heating, and an unexpected dependence between the frequency of the drive and the dynamics of eventual thermalization.

The complete experimental control and quantitative characterization of prethermal-driven matter demonstrated here opens a new frontier for probing far-from-equilibrium quantum statistical mechanics and new possibilities for dynamical quantum engineering.

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Vol. 9, Iss. 4 — October - December 2019

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Figure 1
Mapping the prethermal state in the space of drive parameters. (a) Theoretical map of the projected static ground-band occupation fraction for a noninteracting periodic Gibbs ensemble, as a function of normalized drive frequency $Ω$ and normalized drive amplitude $α$ . Dotted line shows classical stability boundary of the equivalent driven pendulum. (b) Experimental measurement of normalized population in the center of the lowest band after a $500 − μ s$ hold of a noninteracting quantum gas in a modulated optical lattice, as a function of the same parameters. The $α > 1$ region to the right of the vertical dashed line is inaccessible without the sign-changing amplitude modulation introduced in this work. (c) Same measurement as (b) but in the presence of interatomic interactions ( $s$ -wave scattering length 30 nm). Color bar and axes are the same for all three panels.
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Figure 2
Characterizing the prethermal periodic Gibbs ensemble. (a) PGE prediction for occupation of excited even bands as a function of the drive parameters $α$ and $Ω$ . (b) Measured atom number fraction in the central 40% of the same bands. All axes and color bars are the same as for the theory panels. (c) Measured $f_{0}$ versus modulation time for the first $150 μ s$ of the drive for three different values of $(α, Ω)$ all for vanishing interactions. Dashed line shows PGE prediction; solid line shows TDSE prediction of the fluctuating evolution of a spatially homogeneous noninteracting system. Each point represents a single run; error bar shows representative estimated fractional error due to shot-to-shot number fluctuation. For these data, the optical lattice is immediately quenched back to the initial ten $E_{R}$ static lattices at the indicated modulation time and then band mapped.
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Figure 3
Entering the prethermal state. (a) Normalized ground-band occupation $f_{0}$ as a function of time during modulation with $α = 3$ , $Ω = 2.6$ , $a = 0$ . The system attains the prethermal value of $f_{0}$ on timescales comparable to a single drive cycle. These data are reproduced from the rightmost panel of Fig. 2 to facilitate direct comparison with Fig. 3. Inset shows a sample band-mapped image and its column integration for the indicated data point, with Brillouin zone boundaries and band indices labeled. (b) Population fraction in diffracted peaks after diabatic lattice snapoff, as a function of time during the same drive. $x$ axes are the same for the two plots. Gradual attainment of the prethermal steady state is apparent in the settling of the second-peak population.
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Figure 4
Effect of interaction on the long-time evolution of Floquet matter. (a) Short-time evolution of $f_{0}$ for a drive with $Ω = 2.6$ and $α = 3$ , at three different values of the $s$ -wave scattering length $a$ . Regardless of interaction strength, $f_{0}$ fluctuates near the PGE value (dashed line). Solid gray line shows the solution to the time-dependent Schrödinger equation for the noninteracting case. Top panel shows the driving waveform. (b) $f_{0}$ as a function of time for much longer times, for the same drive and interaction parameters. Data are boxcar averaged into $200 − μ s$ bins. The noninteracting and weakly interacting samples attain a second plateau (solid line) well below the PGE value (dashed line), while the strongest-interacting sample decays to a high-temperature state consistent with ergodicity. Shaded area indicates the estimated noise floor: At long times, we measure no significant ground-band occupation only for the strongest-interacting sample.
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Figure 5
Effect of drive frequency on the long-time evolution of Floquet matter. Panels show evolution of the participation ratio $1 / f_{0}$ for $a = 30 nm$ , $α = 3$ , and varying drive frequency $Ω$ as indicated in the inset. Dashed line indicates the PGE prediction for the constant- $f_{0}$ plateau which extends as the frequency increases. The time dependence of heating away from the prethermal plateau is observed to be strongly $Ω$ dependent: At low frequency, the participation ratio grows approximately as the square root of time (green solid line), while at the highest frequency, the heating is consistent with a $t^{1 / 4}$ dependence (red solid line).
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