Optical response of highly excited particles in a strongly correlated system

We present a linear-response formalism for a system of correlated electrons out of equilibrium, as relevant for the probe optical absorption in pump-probe experiments. We consider the time dependent optical conductivity $\sigma (\omega ,t)$ and its nonequilibrium properties. As an application we numerically study a single highly excited charged particle in the spin background, as described within the two-dimensional $t$-$J$ model. Our results show that the optical sum rule approaches the equilibriumlike one very fast; however, the time evolution and the final asymptotic behavior of the absorption spectra in the finite systems considered still reveal dependence on the type of initial pump perturbation. This is observed in the evolution of its main features: the midinfrared peak and the Drude weight.

Figure 1

Sum rule $\langle \tau \rangle$ vs time $t$ for $J=0.4$ as obtained using the ED on $N=26$ sites as well as the EDLFS. Results are presented for initial states of (a) the localized hole and (b) the $\pi$-pulse excited hole. The g.s. $E_{\text{kin},x}^0$ value is also displayed.

Figure 2

Equilibrium kinetic energy $E_{\text{kin}}/2$ vs temperature $T$ for a single hole on a system with $N=26$ sites. Dots mark $E_{\text{kin}}(t\to \infty )/2$ for both types of excitations in $N=20,26$ systems and g.s.

Figure 3

Time dependent optical conductivity ${\tilde{\sigma }}^{\prime }(\omega ,t)$ (per hole) vs $\omega$ as calculated at different times $t=0,1.5,10$ for (a) the initially localized hole with the ED on $N=26$ sites, (b) the localized hole with the EDLFS, (c) the $\pi$-pulse excited hole with the ED, and (d) the $\pi$-pulse excited hole within EDLFS. For comparison the g.s. ${\tilde{\sigma }}_0\left(\omega \right)$ is shown. Broadening of spectra $\delta \omega =0.1$ is used.

Figure 4

Long-time response ${\tilde{\sigma }}^{\prime }(\omega ,\overline{t})$ vs $\omega$ as calculated with ED on $N=26$, obtained by average over responses in interval $t\in [20,60]\gg t_d$ for (a) the initially localized hole and (b) the $\pi$-pulse excited hole. Results are compared to the g.s. ${\tilde{\sigma }}_0\left(\omega \right)$ and the thermal ${\tilde{\sigma }}_{\text{th}}\left(\omega \right)$ at corresponding effective temperatures $T=0.35,0.5$. Spectra are broadened with $\delta \omega =0.1$.