Schrieffer-Wolff transformations for experiments: Dynamically suppressing virtual doublon-hole excitations in a Fermi-Hubbard simulator

In strongly interacting systems with a separation of energy scales, low-energy effective Hamiltonians help provide insights into the relevant physics at low temperatures. The emergent interactions in the effective model are mediated by virtual excitations of high-energy states: For example, virtual doublon-hole excitations in the Fermi-Hubbard model mediate antiferromagnetic spin-exchange interactions in the derived effective model, known as the $t-J-3s$ model. Formally this procedure is described by performing a unitary Schrieffer-Wolff basis transformation. In the context of quantum simulation, it can be advantageous to consider the effective model to interpret experimental results. However, virtual excitations such as doublon-hole pairs can obfuscate the measurement of physical observables. Here we show that quantum simulators allow one to access the effective model even more directly by performing measurements in a rotated basis. We propose a protocol to perform a Schrieffer-Wolff transformation on Fermi-Hubbard low-energy eigenstates (or thermal states) to dynamically prepare approximate $t-J-3s$ model states using fermionic atoms in an optical lattice. Our protocol involves performing a linear ramp of the optical lattice depth, which is slow enough to eliminate the virtual doublon-hole fluctuations but fast enough to freeze out the dynamics in the effective model. We perform a numerical study using exact diagonalization and find an optimal ramp speed for which the state after the lattice ramp has maximal overlap with the $t-J-3s$ model state. We compare our numerics to experimental data from our Lithium-6 fermionic quantum gas microscope and show a proof-of-principle demonstration of this protocol. More generally, this protocol can be beneficial to studies of effective models by enabling the suppression of virtual excitations in a wide range of quantum simulation experiments.

Figure 1

Schematic of the protocol: (a) The protocol involves a linear ramp of the optical lattice depth $V\left(\tau \right)$ at a rate $\alpha \equiv dV/d\tau$, followed by an imaging sequence to measure correlators. Colors indicate different ramp speeds. (b) If the lattice ramp is very fast, i.e., diabatic, then the initial low-temperature Fermi-Hubbard state with doublon-hole virtual excitations and antiferromagnetic spin order is effectively frozen and remains the same after the ramp. If the lattice ramp is very slow, i.e., adiabatic, the quantum state flows towards vanishing virtual excitations but also possibly modified spin order. In the intermediate regime, for an optimal ramp speed virtual excitations are suppressed while also maintaining spin order, approximately mapping the initial state onto the effective model. See text for details.

Figure 2

(a) Doublon density $\langle {\widehat{\rho }}_D\rangle$ after the lattice ramp for the double-well case plotted as a function of ramp speed $\alpha$. Comparison of the analytical approximation of Eq. (9) (dashed lines) with numerical simulations (solid lines) for initial $U/t=8$ (purple, top) and $U/t=16$ (brown, bottom). Vertical dash-dotted lines indicate the critical ramp speed ${\alpha }^{*}$ from Eq. (10) below which the doublon density is strongly suppressed. The double-well Fermi-Hubbard Hamiltonian is schematically shown in the inset. (b) Doublon density $\langle {\rho }_D\rangle$ vs. normalized ramp speed $\alpha /{\alpha }^{*}$ for $U/t=8$. Solid lines are numerical results for a 12-site system at half-filling, 1D chain with periodic boundary conditions (orange dashed line) and 2D cluster, $4\times 3$ (green solid line). Experimental data points at half-filling shown in gray markers. The dashed line indicates the imaging fidelity limit to measuring doublon density in the experimental snapshots.

Figure 3

(a) Normalized spin correlation function ($z$ - component) for nearest neighbors ${\langle {\widehat{S}}_i^z{\widehat{S}}_j^z\rangle }_{\text{ramp}}/\left|{\langle {\widehat{S}}_i^z{\widehat{S}}_j^z\rangle }_{t-J}\right|$ plotted as a function of normalized ramp speed $\alpha /{\alpha }^{*}$ from numerical simulations at zero temperature. Colors (markers) indicate different doping levels. Dashed line indicates the normalized $t-J$ model correlator. Spin correlations are normalized by the magnitude of the $t-J-3s$ model correlators to plot different doping levels on the same plot. System size is 12 sites ($4\times 3$). (b) Spin correlation function for nearest neighbors ${\langle {\widehat{\mathit{S}}}_i·{\widehat{\mathit{S}}}_j\rangle }_C$ at finite temperature. Solid lines indicate numerical results for a nine-site ($3\times 3$) system at half-filling with temperatures ranging from $T/t=0$ at the bottom (purple) to $T/t=0.5$ at the top (red). $U_0/t_0=8$ for both plots.

Figure 4

(a) Fidelity of mapping to the $t-J-3s$ model ground state starting from the Fermi-Hubbard ground state, defined as $F=|〈{\mathrm{\Psi }}_0^{tJ}|{\widehat{U}}_{\text{ramp}}\left(\alpha \right){|{\mathrm{\Psi }}_0〉|}^2$. Solid lines show numerical results for a 12-site cluster ($4\times 3$) with colors (markers) indicating different doping levels. Dashed lines indicate the squared overlap of the initial state with the $t-J$ ground state, i.e., $|\langle {\mathrm{\Psi }}_0^{tJ}|{\mathrm{\Psi }}_0\rangle {|}^2$. (b) Comparison of the optimal lattice ramp unitary ${\widehat{U}}_{\text{ramp}}^{\text{opt}}$ and the Schrieffer-Wolff unitary transformation $e^{i\widehat{S}}$ for low-energy eigenstates of the Fermi-Hubbard model. Fidelity $F_1$ (blue circles) shows how close ${\widehat{U}}_{\text{ramp}}$ and $e^{i\widehat{S}}$ are when acting on Fermi-Hubbard eigenstates, $F_2$ (orange triangles) shows the fidelity of mapping to a $t-J-3s$ eigenstate when starting from the corresponding Fermi-Hubbard low-energy eigenstate, and $F_3$ (green crosses) shows the squared overlap of $t-J-3s$ eigenstates and corresponding Fermi-Hubbard eigenstates. $U_0/t_0=8$ for both plots.