Numerical method to compute optical conductivity based on pump-probe simulations

A numerical method to calculate optical conductivity based on a pump-probe setup is presented. Its validity and limits are tested and demonstrated via concrete numerical simulations on the half-filled one-dimensional extended Hubbard model both in and out of equilibrium. By employing either a steplike or a Gaussian-like probing vector potential, it is found that in nonequilibrium, the method in the narrow-probe-pulse limit can be identified with variant types of linear-response theory, which, in equilibrium, produce identical results. The observation reveals the underlying probe-pulse dependence of the optical conductivity calculations in nonequilibrium, which may have applications in the theoretical analysis of ultrafast spectroscopy measurements.

Figure 1

A schematic diagram of the current-current correlations considered in NLR and VNLR, respectively.

Figure 2

Schematic illustrations of the vector potentials and the corresponding electric fields for the Gaussian pulse [(a) and (b)] and the steplike pulse [(c) and (d)] with various widths.

Figure 3

The calculated $\text{Re}\sigma \left(\omega \right)$ in equilibrium (zero temperature) for the Hamiltonian (1) with different $t_{d,\text{probe}}$. Parameters: $L=10, U=10$, and $V=4.5$. (a) By PP-Gaussian with ${\omega }_{\text{probe}}=10$ and $A_{0,\text{probe}}=1.0\times {10}^{-6}$. (b) By PP-ramp with $A_{0,\text{step}}=1.0\times {10}^{-4}$. Note that the difference is negligible between the results of $t_{\text{d,probe}}=0.2$ (red curves) and those of $t_{\text{d,probe}}=0.02$ (blue curves).

Figure 4

$\text{Re}\sigma \left(\omega \right)$ in equilibrium (zero temperature) for the Hamiltonian (1) obtained by the KF, NLR, and PP methods in (a) the open boundary condition (open BC) and (b) the periodic BC. Parameters: $L=10, U=10$, and $V=4.5$.

Figure 5

The time-dependent optical conductivity $\text{Re}\sigma (\omega ,t)$ at $t=10$ for the Hamiltonian (1) obtained using different methods in the periodic BC, where $t$ is the time delay between the probe and pump (quench) time. In (a) and (d), a pump pulse is applied with parameters $A_{0,\text{pump}}=0.2, {\omega }_{\text{pump}}=6.29$ (resonant frequency), and $t_{d,\text{pump}}=0.5$. In (b) and (e), a $\pi$-quench is applied. In (c) and (f), a $U$-quench with the $U$ value changed from 10 to 4 is applied. The blue solid lines represent the results of the VNLR in the left column [(a), (b), and (c)] and NLR in the right column [(d), (e), and (f)], respectively, while the red dashed lines represent the results of the PP-Gaussian and PP-step in the two columns. The insets in the left column display the details in the vicinity of $\omega =0$ obtained by NKF, VNLR, and PP-Gaussian. Model parameters: $L=10, U=10$, and $V=3$.

Figure 6

The time-dependent optical conductivity $\text{Re}\sigma (\omega ,t)$ for the Hamiltonian (1) obtained by NLR and VNRL in a $U$-quench case, where $U$ is switched from 0 to 2 at $t=0$. The probing time is set at $t=0$ in (a) and $t=5$ in (b), respectively. Model parameters: $L=10$ and $V=0$.