Floquet Dynamics in Driven Fermi-Hubbard Systems

Michael Messer, Kilian Sandholzer, Frederik Görg, Joaquín Minguzzi, Rémi Desbuquois, and Tilman Esslinger

Phys. Rev. Lett. 121, 233603 – Published 7 December 2018

Abstract

We study the dynamics and timescales of a periodically driven Fermi-Hubbard model in a three-dimensional hexagonal lattice. The evolution of the Floquet many-body state is analyzed by comparing it to an equivalent implementation in undriven systems. The dynamics of double occupancies for the near- and off-resonant driving regime indicate that the effective Hamiltonian picture is valid for several orders of magnitude in modulation time. Furthermore, we show that driving a hexagonal lattice compared to a simple cubic lattice allows us to modulate the system up to 1 s, corresponding to hundreds of tunneling times, with only minor atom loss. Here, driving at a frequency close to the interaction energy does not introduce resonant features to the atom loss.

Received 31 July 2018

DOI:https://doi.org/10.1103/PhysRevLett.121.233603

Physics Subject Headings (PhySH)

Cold gases in optical lattices Fermi gases

Floquet systems Strongly correlated systems

Extended Hubbard model

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Michael Messer, Kilian Sandholzer, Frederik Görg, Joaquín Minguzzi, Rémi Desbuquois, and Tilman Esslinger

Institute for Quantum Electronics, ETH Zurich, 8093 Zurich, Switzerland

Article Text (Subscription Required)

Click to Expand

Supplemental Material (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 121, Iss. 23 — 7 December 2018

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Periodically driven Fermi-Hubbard model. (a) Altering the effective Hamiltonian ( $H^{eff}$ ) affects the underlying many-body state and results in a change of the measured observable within different timescales of the system. A full time evolution of the observable is depicted in blue, while the time evolution under $H^{eff}$ is plotted in red. Deviations to the expected behavior in the effective Hamiltonian can arise for long modulation times when heating processes dominate. (b) Three-dimensional hexagonal structure to realize the driven Fermi-Hubbard model. (c) Schematics of the tight-binding model of the effective Hamiltonian. For off-resonant driving ( $ℏ ω ≫ U$ , $t$ ), the interactions $U$ are unaffected while the tunneling $t$ is renormalized. In contrast, for near-resonant driving ( $ℏ ω \approx U ≫ t$ ), the system exhibits a reduced effective interaction and a density assisted hopping process which is different from the single particle hopping.
Reuse & Permissions
Figure 2
Off-resonantly modulated Fermi-Hubbard model. (a) Starting from a [ $t_{x, y, z} / h = (200 (30), 40 (3), 40 (3)) Hz$ ] hexagonal lattice, we either increase the driving amplitude to $K_{0} = 1.69 (2)$ or perform a lattice ramp (undriven lattice) to imitate the same final $t_{x} / h = t_{x}^{eff} / h = 80 (10) Hz$ . The evolution of double occupancy fraction $D$ on a local timescale for the driven (red diamonds) and undriven lattices (blue squares) as a function of the ramp-up time $τ_{ramp}$ at $U / h = 500 (30) Hz$ . Evolution of $D$ when loading directly into the starting (final) lattice is shown in black (orange) circles. (b) For the global timescales, we start with a [ $t_{x, y, z} / h = (510 (90), 100 (6), 100 (10)) Hz$ ] lattice and ramp up the drive within 5 ms or directly decrease $t_{x}$ to $t_{x}^{eff} / h = 210 (40) Hz$ . Measured $D$ at $U / h = 700 (20) Hz$ as a function of the modulation time $τ_{hold}$ at constant $K_{0} = 1.69 (3)$ (red diamonds) and its counterpart in the undriven case (blue squares). Arrows indicating the reference values in the starting (final) lattice are shown in black (orange). The inset shows the corresponding number of atoms $N$ as a function of $τ_{hold}$ . Data points in a(b) are the mean and standard error of 5(10) individual measurements at different times within one driving period (see [42]).
Reuse & Permissions
Figure 3
Near-resonantly modulated Fermi-Hubbard model. (a) In the undriven lattice (blue squares) we change $U^{stat}$ in 2 ms using a Landau-Zener (LZ) transfer to a different spin state and, also, lower $t_{x}$ . For near-resonant driving, we either increase $K_{0}$ to 1.43(2) on a fixed timescale (2 ms) at $U_{load} / h = 4.63 (10) kHz$ and, then, tune the interactions in the driven system within a variable time $τ_{ramp}$ (green circles), or we, first, prepare the system at the final interaction $U_{final}$ and, then, vary $τ_{ramp}$ of the $K_{0}$ ramp (red diamonds). (b)–(d) Dynamics of $D$ for different $τ_{ramp}$ and three values of the final effective interaction $U_{final}^{eff} / h = [0.69 (8), - 0.02 (6), - 0.72 (5)] kHz$ or the corresponding static counterpart $U^{stat}$ . Arrows indicate the reference values when adiabatically loading the atoms in the starting lattice [ $t_{x, y, z} / h = (200 (30), 100 (10), 100 (10)) Hz$ ] (black) or the final lattice with reduced tunneling $t_{x}^{eff} / h = 110 (20) Hz$ and $U^{eff}$ (orange). Data points are the mean and standard error of five individual measurements at different times within one driving period [42].
Reuse & Permissions
Figure 4
Minimizing atom loss in driven 3D optical lattices. (a) Schematics of the first Brillouin zone (1st BZ) of the simple cubic (SC, blue) and hexagonal (HC, red) lattices and their real space geometry including an intermediate dimerized ( $D$ , green) lattice. All three lattice configurations have $t_{x} / h \approx 200 Hz$ but a different $t_{w} / h$ [SC: 200(10) Hz, $D$ : 36(2) Hz and HC: $< 1 Hz$ ]. (b) Calculated band structure ( $q_{z} = 0$ ) for the SC (HC) lattice plotted in blue-dashed (red). The HC lattice allows us to tune the band gap and dispersion of the higher band, while the bandwidth of the lowest band remains similar compared to the SC lattice (see inset). (c) Remaining fraction of atoms $N / N_{static}$ when driving for 1 s at fixed $K_{0} = 1.43 (2)$ for different frequencies $ω / (2 π)$ at $U / h = 710 (20) Hz$ , where $N_{static}$ is the remaining atom number after holding the system for 1 s in the undriven lattice. The two vertical dashed lines indicate the driving frequencies used in (d) and (e). (d) Normalized atom number $N / N_{static} (U = 0)$ in the HC lattice after a modulation of 1 s at $K_{0} = 1.43 (2)$ for two different frequencies $ω / (2 π) = 4.25 kHz$ (orange) and $ω / (2 π) = 2.5 kHz$ (cyan) and without drive (black) as a function of $U$ . (e) $D$ corresponding to the measurements in (d) but for a modulation time of 5 ms. Dashed vertical lines in (d) and (e) correspond to the resonance condition $U = \pm ℏ ω$ . Data points in c(d, e) are the mean and standard error of 3(5) individual measurements at different times within one driving period. Error bars in $U$ result from the uncertainty of the interaction calibration.
Reuse & Permissions

Physical Review Letters