Fig. 5
Pre-thermalization and transient Floquet steady states. a, b depict the time evolution of the period-averaged local double occupancy of the 12-site driven Hubbard model as a function of cycles under the pump, on and off the charge resonance with the upper Hubbard band, respectively. Dashed lines indicate onset of the pump plateau. c Extracted transient expectation values of the double occupancy at the pump plateau, as a function of pump strength A and detuning from the upper Hubbard band. Red arrows indicate values of t h used for a, b. Lines are guides to the eye. On resonance (gray region), the system heats rapidly, with \(\left\langle {{{\hat n}_ \uparrow }{{\hat n}_ \downarrow }} \right\rangle\) thermalizating independent of the pump strength A and approaching its infinite-temperature expectation value 1/4. Below resonance (white region), the system transiently realizes the TRS-breaking chiral quantum magnet, with a tunable Hubbard interaction U (as well as correspondingly tunable magnetic J, χ, see Fig. 4) as a function of pump strength. d Floquet fidelity \(F(T) = \left| {\left\langle {\Psi \left( {t + T} \right)\left| {\Psi (t)} \right.} \right\rangle } \right|\) on the pump plateau—below charge resonance, (1 − F) → 0, indicating the controlled preparation of a Floquet eigenstate. e Extracted stroboscopic heating rates per pump cycle and below resonance. Remarkably, heating is strongly suppressed close to the charge resonance, with the driven system requiring many thousands of pump cycles to finally absorb energy on the order of the Heisenberg exchange J
