Quantum Simulation Meets Nonequilibrium Dynamical Mean-Field Theory: Exploring the Periodically Driven, Strongly Correlated Fermi-Hubbard Model

We perform an ab initio comparison between nonequilibrium dynamical mean-field theory and optical lattice experiments by studying the time evolution of double occupations in the periodically driven Fermi-Hubbard model. For off-resonant driving, the range of validity of a description in terms of an effective static Hamiltonian is determined and its breakdown due to energy absorption close to resonance is demonstrated. For near-resonant driving, we investigate the response to a change in driving amplitude and discover an asymmetric excitation spectrum with respect to the detuning. In general, we find good agreement between experiment and theory, which cross validates the experimental and numerical approaches in a strongly correlated nonequilibrium system.

Figure 1

(a) Experiment: Three-dimensional brick wall structure in a trapping potential. The driving is applied in the $x$ direction. Theory: DMFT mapping of the lattice system to an effective impurity problem. It is characterized by the hybridization function $\mathrm{\Delta }(t,t^{\prime })$, which mimics the hopping of particles to neighboring sites in the lattice system. (b) Schematic illustration of the different effective Hamiltonians. In the off-resonant regime ($\hslash \mathrm{\Omega }\gg U$, $J$), the interaction $U$ is unaffected while the hopping parameter $J$ is renormalized. In the near-resonant regime ($\hslash \mathrm{\Omega }\approx U\gg J$), the interactions are reduced to $U-\hslash \mathrm{\Omega }$ and the hopping parameter depends on whether or not the tunneling process changes the number of doubly occupied sites. (c) DMFT simulations of the double occupation $\mathcal{D}$ in the off-resonant ($\mathrm{\Omega }/2\pi =3\mathrm{kHz}$, $U/h=750\mathrm{Hz}$, $J_x=200\mathrm{Hz}$, $J_y=J_z=40\mathrm{Hz}$) and near-resonant ($\mathrm{\Omega }/2\pi =U/h=3.5\mathrm{kHz}$, $J_x=200\mathrm{Hz}$, $J_y=J_z=100\mathrm{Hz}$) regimes. As in the experiment, the driving field $E(t)$ is ramped up linearly during a period $t_{\text{ramp}}$ and $\mathcal{D}$ is measured just after the ramp and averaged over one period of the excitation ($T=(2\pi /\mathrm{\Omega })$).

Figure 2

Off-resonant driving in the Fermi-Hubbard model at $U/h=0.75(3)$ and $W/h=0.71(7)\mathrm{kHz}$. Experimental (a) and DMFT (b) data display $\mathcal{D}$ measured after ramping up the drive on different timescales. The legend applies to both plots. In the off-resonant case (triangle left: $\mathrm{\Omega }/2\pi =5$, triangle right: $\mathrm{\Omega }/2\pi =3\mathrm{kHz}$) $\mathcal{D}$ is suppressed, whereas an increase is observed when $\mathrm{\Omega }/2\pi =1.5\mathrm{kHz}\simeq (U+W)/h$ (squares). Solid lines show data taken in an undriven system. The upper and lower lines are reference values for holding in the initial [$J_x/h=193(34)\mathrm{Hz}$] and final lattice [$J_x^{\mathrm{eff}}/h=81(13)\mathrm{Hz}$]. The red line displays data taken after ramping the lattice depth from $J_x$ to $J_x^{\mathrm{eff}}$ to mimic the driven data. The dashed red line indicates the saturation value reached at $t_{\text{ramp}}=5\mathrm{ms}$. Data in the shaded area of (a) are influenced by residual dynamics during detection and the finite bandwidth of the piezoelectric actuator. Error bars in (a) denote the standard error for 5 measurements and in (b) reflect the uncertainty of the entropy estimation in the experiment [39]. Panel (c) shows the local single-particle spectral function $A(\omega )$ (thin dashed) and its occupation $N(\omega )$ (solid) at $T/J_x=1.21$ in equilibrium at half-filling. The shaded area indicates the overlap between $N(\omega )$ and the hole occupation $\overline{N}(\omega -\mathrm{\Omega })$ shifted by the driving frequency $\mathrm{\Omega }/2\pi =1.5\mathrm{kHz}$ (dashed), corresponding to possible direct excitations. In (d) we plot the nonequilibrium spectra after ramping up the drive in 5 ms.

Figure 3

Resonant driving in the Fermi-Hubbard model for $J_{x,y,z}/h=[200(50),100(10),100(10)]\mathrm{Hz}$. Experimentally measured (a) and theoretically simulated (b) $\mathcal{D}$ for different ramp times and driving strengths at resonance [$\mathrm{\Omega }/(2\pi )=3.5\mathrm{kHz}$, $U/h=3.5(1)\mathrm{kHz}$]. Dynamics beyond $t_{\text{ramp}}=10\mathrm{ms}$ are influenced by trap effects and for $t_{\text{ramp}}=300\mathrm{ms}$ by heating [39] and is not considered in DMFT. Error bars are the same as in Fig. 2.

Figure 4

Near-resonant driving in the Fermi-Hubbard model for $J_{x,y,z}/h=[200(50),100(10),100(10)]\mathrm{Hz}$ at fixed strength $K_0=1.44(2)$ and frequency $\mathrm{\Omega }/(2\pi )=3.5\mathrm{kHz}$. Experimental (a) and numerical results (b) for $\mathcal{D}$ after different ramp times at interactions chosen symmetrically around the resonance. Dynamics beyond $t_{\text{ramp}}=10\mathrm{ms}$ are influenced by trap effects and for $t_{\text{ramp}}=300\mathrm{ms}$ by heating [39] and is not considered in DMFT. Error bars are the same as in Fig. 2. In (c) and (d), the occupations of the lower Hubbard band (solid lines) at $T/J_x=3.3$ are shown for symmetric detunings. The shaded area indicates the overlap with the hole occupation (dashed) shifted by the driving frequency. The data in (c) are for half filling and in (d) for lower filling.