  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.
