Thermalization of a pump-excited Mott insulator

We use nonequilibrium dynamical mean-field theory in combination with a recently implemented strong-coupling impurity solver to investigate the relaxation of a Mott insulator after a laser excitation with frequency comparable to the Hubbard gap. The time evolution of the double occupancy exhibits a crossover from a strongly damped transient at short times toward an exponential thermalization at long times. In the limit of strong interactions, the thermalization time is consistent with the exponentially small decay rate for artificially created doublons, which was measured in ultracold atomic gases. When the interaction is comparable to the bandwidth, however, the double occupancy thermalizes within a few times the inverse bandwidth along a rapid thermalization path in which the exponential tail is absent. Similar behavior can be observed in time-resolved photoemission spectroscopy. Our results show that a simple quasi-equilibrium description of the electronic state breaks down for pump-excited Mott insulators characterized by strong interactions.

Figure 1

Pump-excitation of a Mott insulator at $\beta =5$, $U=5$. (a) The electric field used for the data in this plot. The two frequencies $\Omega =5\pi /4\approx 3.927$ and $\Omega =5\pi /2\approx 7.854$ are comparable with the Hubbard gap and larger than the Hubbard gap, respectively, and the amplitudes $E_0=0.9679$ ($\Omega =5\pi /4$) and $E_0=3.6701$ ($\Omega =5\pi /2$) are chosen such that $T_{\text{eff}}=0.5$, i.e., the energy of the pump-excited system for $t>4$ is the same as in a system in thermal equilibrium at temperature $T=T_{\text{eff}}$. (b) Time-dependence of the total energy $\mathcal{E}$. (c) Time dependence of the double occupancy $d\left(t\right)$. The horizontal dotted line indicates the double occupancy $d\left(T_{\text{eff}}\right)$ in an equilibrium state at temperature $T=T_{\text{eff}}$.

Figure 2

Relaxation of the double occupancy $d\left(t\right)$ after an excitation with frequency $\Omega =U\pi /2$ ($\beta =5$). Panel (a): The difference $|d\left(t\right)-d\left(T_{\text{eff}}\right)|$, obtained from the OCA impurity solver. The pump-amplitudes $E_0$ are given by 3.6701 ($U=5$), 2.4397 ($U=3.5$), 2.1325 ($U=3$), 1.9123 ($U=2.5$), 1.8198 ($U=2.2$), 1.7761 ($U=2$), 1.6632 ($U=1.8$), 1.2765 ($U=1.5$), such that $T_{\text{eff}}=0.5$ in all cases. Lower panels: Parameters ${\tau }_{\text{th}}$ (b) and $d_0$ (c), obtained from fitting the curves in (a) for $t>12$ with Eq. (5) (square symbols). Stars and triangles show the result of the analogous investigation using NCA and the 3rd order hybridization expansion, respectively. The time-evolution in 3rd order was restricted to $t⩽15$. Black solid lines in (b) correspond to the exponential law[34] ${\tau }_{\text{th}}={\tau }_0\exp \left(\alpha U\log U\right)$, with $\alpha =1.01$ and ${\tau }_0=0.45\left(1.2\right)$ for OCA (NCA).

Figure 3

Time-resolved photoemission spectrum [Gaussian probe, $\delta =2$, $t_p=10$] of a pump-excited system ($U=5$, $\beta =5$), using the pump parameters of Fig. 1. The blue dotted line marked $T_{\text{eff}}=0.5$ is the photoemission spectrum of a system at temperature $T=0.5$, for the same Gaussian probe pulse as for the other two curves.

Figure 4

Time-resolved photoemission spectrum [Gaussian probe, $\delta =2$] of a pump-excited system ($U=2.5$, $\beta =5$), using the pump parameters of Fig. 2a ($\Omega =5\pi /4$, $E_0$=1.9123). Main plot: $I(\omega ,t_p)$ for various probe-times $t_p$, compared with the spectrum $I_{T_{\text{eff}}}\left(\omega \right)$ of the equilibrium state at $T_{\text{eff}}=0.5$. For better visibility of the upper Hubbard band, the data have been divided by a Fermi-function with temperature $T^{\prime }=0.55$. ($T^{\prime }$ is chosen by hand and is slightly larger than $T_{\text{eff}}$, in order to account for the broadening effect due to the finite pulse duration.) Inset: Relative difference $I(\omega ,t_p)$ to $I_{T_{\text{eff}}}\left(\omega \right)$ as a function of $t_p$, for three probe frequencies ${\omega }_{\left(1\right)}=2$, ${\omega }_{\left(2\right)}=0.5$, ${\omega }_{\left(3\right)}=-0.5$. Solid lines show exponential fits $I_{\Omega }\left(\omega \right)-I_{T_{\text{eff}}}\left(\omega \right)\propto \exp (-t/{\tau }_{\text{th}})$, with parameters ${\tau }_{\text{th}}$ = 4.70 [${\omega }_{\left(1\right)}$], 4.82 [${\omega }_{\left(2\right)}$], 4.77 [${\omega }_{\left(3\right)}$].