Coherent pumping of a Mott insulator: Fermi golden rule versus Rabi oscillations

Cold atoms provide a unique arena to study many-body systems far from equilibrium. Furthermore, phases in cold atom systems are conveniently investigated by dynamical probes pushing the system out of equilibrium. Here, we discuss the pumping of doubly occupied sites in a fermionic Mott insulator by a periodic modulation of the hopping amplitude. We show that deep in the insulating phase the many-body system can be mapped onto an effective two-level system which performs coherent Rabi oscillations due to the driving. Coupling of the two-level system to the remaining degrees of freedom renders the Rabi oscillations damped. We compare this scheme to an alternative description where the particles are incoherently pumped into a broad continuum.

Figure 1

(Color online) (a) Sketch of an experimental sequence for the generation of double occupancy by modulating the optical lattice during a time $\tau$. After an initial buildup of doubly occupied sites with a rate ${\Gamma }_{\text{up}}$, the fraction of particles residing in doubly occupied sites saturates at a value ${\mathcal{D}}_{\mathrm{sat}}$ after a time ${\tau }_{\mathrm{sat}}$. (b) Illustration of the effective Hilbert space ${\mathcal{H}}_{\mathrm{eff}}$ consisting of one bond embedded in an array of sites. The bath degrees of freedom consist of all doublon-hole configurations in which the pair resides on a different bond than where it was created.

Figure 2

Comparison of two possible descriptions: (a) A rate equation description where an incoherent pump rate $\left({\Gamma }_{\text{up}}^{\mathrm{FGR}}\right)$ into a broad doublon-hole continuum and subsequent decay $\left({\Gamma }_{\text{down}}^{\text{decay}}\right)$ or incoherent stimulated emission $\left({\Gamma }_{\text{up}}^{\mathrm{FGR}}\right)$ lead to a saturation. (b) Coherent Rabi oscillations (with frequency ${\Omega }_{\mathrm{R}}$) between the ground state and a single exited state where decay out of the effective Hilbert space is introduced via coupling to a bath $\left({\Gamma }_{\mathrm{in}\left(\mathrm{out}\right)}\right)$.

Figure 3

Fraction of particles residing in doubly occupied sites after modulation time $\tau$. After a buildup time corresponding to approximately one $\pi$ pulse $({\tau }_{\mathrm{R}}=\pi ∕{\Omega }_{\mathrm{R}}=\hslash \pi ∕2\delta t)$, the decay, with (weak) strength ${\Gamma }_{\mathrm{out}}=\delta t∕\hslash \approx 0.24t_0∕\hslash$ and ${\Gamma }_{\mathrm{in}}=0$, leads to a saturation at ${\mathcal{D}}_{\mathrm{sat}}=58\%$ corresponding to the fact that all “active” bonds, singlets, are excited in the system (solid line). A more realistic situation where the damping is increased by a factor of 2 renders the oscillations barely visible (dashed line). Taking into account a finite value of ${\Gamma }_{\mathrm{in}}$ leads to a decrease of the saturation value. Note that, irrespective of the saturation value, the time scale needed to pump the system into a doubly occupied state is given by the Rabi frequency.