Schrieffer-Wolff Transformations for Experiments: Dynamically Suppressing Virtual Doublon-Hole Excitations in a Fermi-Hubbard Simulator

TL;DR

This paper introduces a protocol using a lattice ramp in quantum simulators to suppress virtual doublon-hole excitations in the Fermi-Hubbard model, enabling more direct access to effective low-energy models like the $t-J-3s$ model.

Contribution

It proposes a practical method to perform a Schrieffer-Wolff transformation dynamically in experiments, improving the study of effective models in quantum simulation.

Findings

Optimal ramp speed maximizes overlap with the effective model state

Numerical simulations validate the protocol's effectiveness

Experimental demonstration confirms feasibility

Abstract

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.