Dynamical preparation of the t-J model in a Fermi-Hubbard simulator via optical lattice ramps

The t-J model describes the low-energy properties of the doped Fermi-Hubbard model, relevant to the studies of high-Tc superconductivity. The "simpler" t-J model can be derived from the Fermi-Hubbard model by performing a Schrieffer-Wolff basis transformation and restricting the Hilbert space to exclude doubly occupied sites. We propose a protocol for cold atom experiments to dynamically prepare the t-J model ground (thermal) state in an optical lattice starting from the Fermi-Hubbard ground (thermal) state. Our simple protocol involves performing a slow linear ramp of the optical lattice depth just before fluorescence imaging, which acts as an approximate Schrieffer-Wolff transformation on low energy Fermi-Hubbard eigenstates and eliminates the doublon-hole fluctuations. This lattice ramp maps the Fermi-Hubbard eigenstates onto eigenstates of the t−J model which can then be imaged in the natural Fock basis of cold atom experiments. 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 model state and shows correlations approaching those for the t-J model. We also compare our numerics to existing experimental data from our Lithium-6 fermionic quantum gas microscope and show a proof-of-principle demonstration of this protocol. More generally, this protocol enables the study of low energy effective Hamiltonians derived via the Schrieffer-Wolff transformation frequently encountered in particle physics.