Quantum Dynamics of the Hubbard-Holstein Model in Equilibrium and Nonequilibrium: Application to Pump-Probe Phenomena

The spectral response and physical features of the 2D Hubbard-Holstein model are calculated both in equilibrium at zero and low chemical dopings, and after an ultrashort powerful light pulse, in undoped systems. At equilibrium and at strong charge-lattice couplings, the optical conductivity reveals a three-peak structure in agreement with experimental observations. After an ultrashort pulse and at nonzero electron-phonon interaction, phonon and spin subsystems oscillate with the phonon period $T_{\mathrm{ph}}\approx 80\mathrm{fs}$. The decay time of the phonon oscillations is about 150–200 fs, similar to the relaxation time of the charge system. We propose a criterion for observing these oscillations in high $T_c$ compounds: the time span of the pump light pulse ${\tau }_{\mathrm{pump}}$ has to be shorter than the phonon oscillation period $T_{\mathrm{ph}}$.

Figure 1

(a) Kinetic energy $K$ and (b) average number of phonons $N$ versus $\lambda$ at HF and in one-hole subspace.

Figure 2

OC at equilibrium (in units of ${\sigma }_0=e^2/\hslash a$) for different EPI values, at HF and in one-hole subspace.

Figure 3

Time evolution of (a) kinetic energy $K$, (b) and inset in (b) is the $U$ interaction contribution $E_U=⟨H_U⟩$, (c) and inset in (c) is the local spin-spin correlation, (d) average number of phonons $N$. Inset in (d) is EPI energy at $\lambda =0$ (dotted green line), $\lambda =0.1$ (solid black line), and $\lambda =0.5$ (dashed red line).

Figure 4

Nonequilibrium OC at three different times after the pump pulse at (a) $\lambda =0$, (b) $\lambda =0.1$, and (c) $\lambda =0.5$. Inset shows zoom of the bimagnon feature in the OC at HF and at equilibrium.