Influence of Markovianity and self-consistency on time-resolved spectral functions of driven quantum systems

Abstract

We present a systematic comparison of the real-time Dyson expansion (RTDE) with established non-equilibrium Green's function approaches for simulating driven, interacting quantum systems. Focusing on density matrix dynamics, time-off-diagonal Green's functions, and time-resolved photoemission spectra, we benchmark RTDE against fully self-consistent Kadanoff-Baym equation (KBE) calculations, the generalized Kadanoff-Baym ansatz (GKBA), and exact diagonalization for small systems using second order many-body perturbation theory. Using a driven two-band Hubbard model, we show that mean-field single particle density matrix trajectories provide a reliable baseline for RTDE across a broad range of interaction strengths and excited-carrier populations. Further, RTDE accurately captures correlation effects in the Green's functions, including long-lived oscillations and revivals that are strongly suppressed by the overdamping inherent to self-consistent KBE schemes. As a consequence, RTDE resolves rich non-equilibrium spectral structure in time-resolved photoemission, such as interaction- and population-dependent quasiparticle splittings and bandgap renormalization, which are largely washed out in self-consistent approaches, yet are present in the exact solutions. Our results demonstrate that RTDE bridges the gap between mean-field propagation and full two-time KBE simulations, retaining favorable linear scaling while capturing essential dynamical correlations relevant for ultrafast spectroscopy.

Figures (8)

• Figure 1: Left: Two-legged Keldysh contour used to calculate non-equilibrium Green's functions. Right: Schematic of RTDE method, which first calculates the mean-field density matrix, which is subsequently used to calculate the off-diagonal Green's function in the region of the photoemission measurement, shown in gray.
• Figure 2: Dynamics of the momentum-resolved photodoped population $\rho_\mathbf{k}(t)$ at different $\mathbf{k}$-points after excitation by a pulse near resonance at $\mathbf{k}=\Gamma$. (a) Equilibrium band structure. The shaded area shows the bandgap, and the vertical lines show the above ($\omega_p=5.3$) and below ($\omega_p=4.8$) gap driving. On the right, the top (bottom) row corresponds to driving the system at $\omega_p=4.8$$(5.3)$, and the colors correspond to the $\mathbf{k}$-points in (a). All results are for intraband interaction $U=1$. The first column (b–c) uses interband interaction $V=0$, the second column (d–e) uses $V=0.1$, and the third column (f–g) uses $V=0.5$. (h) Percentage of electrons excited into the conduction band as the subgap pulse amplitude is varied. The different methods are shown by the full, dashed, and dotted lines (for TDHF, 2B-GKBA, and 2B-KBE).
• Figure 3: Heatmap plots of $\rho_\mathbf{k}^{HF}(t)$ (left), $\rho_\mathbf{k}^{2B\text{-}GKBA}(t)$ (middle), and $\rho_\mathbf{k}^{2B\text{-}KBE}(t)$ (right) for $V=0.5$, $U=1.0$, and $\omega_p=5.3$. The $E$ value of each plot corresponds to the one of the five lowest values that yield $N_c=5\%$ (defined by the intersection of the horizontal line with the curves in Fig. \ref{['fig:rho_trajectories']}h). Each descending plot corresponds to a higher value of $E$. Green dashed lines denote the reconstruction time window used in Sec. \ref{['sec:spect']} which has a probe width of 8$\sqrt{2}$
• Figure 4: The imaginary component of the lesser Green's function $\mathrm{Im}\left[G^<(T_M,t')\right]$ slice trajectory around the probing time and propagated from $t'=50$ to $t'=200$ using HF, 2B-RTDE, 2B-KBE, and ED with increasing values of $U=\{0.1$, $0.5$, $1.0\}$. Parameters used here are $V=0$, $\omega_p=5.3$, and the driving amplitude $E$ is chosen such that $N_c\approx5\%$.
• Figure 5: The imaginary component of the lesser Green's function $\mathrm{Im}\left[G^<(T_M,t')\right]$ slice trajectory around the probing time and propagated from $t'=50$ to $t'=200$ using HF, 2B-RTDE, 2B-KBE, and ED with increasing values of $N_c=\{2\%$, $5\%$, $10\%\}$ with fixed $U=0.5$. Parameters used here are $V=0$, $\omega_p=5.3$.