Operational Criterion for Imaginary-Time Reconstruction in Time-Resolved Electronic Circular Dichroism

Abstract

Time-resolved electronic circular dichroism is commonly simulated by real-time propagation with a weak circular probe. Within linear response, such a probe measures the retarded mixed electric--magnetic susceptibility of the pump-prepared ensemble at delay $τ$, while finite probe envelopes enter through a known transfer function. Here we establish when this retarded kernel can be reconstructed from an imaginary-time correlator. We show that this replacement is justified only when the pump-prepared state is sufficiently stationary and approximately Kubo--Martin--Schwinger compliant with respect to an effective reference Hamiltonian. To assess this regime, we introduce a delay-resolved scalar mismatch between the greater and lesser mixed spectra and combine it with Kramers--Kronig and commutator sum-rule checks. Benchmarks on an exactly solvable driven model identify the delay windows in which Matsubara continuation is accurate and those in which a genuinely nonequilibrium treatment is required.

Figures (4)

• Figure 1: Cumulative integral $\frac{1}{\pi}\!\int_{0}^{\Omega}\!\Im\,\chi_{\mathrm{iso}}^{R}(\omega)\,d\omega$ versus $\Omega$. Ground truth (black dotted with markers), our Matsubara route (solid blue), and $\delta$-probe (orange dashed). Our curve tracks the truth, while the $\delta$-probe systematically undershoots, revealing finite-probe/window bias. This directly tests the commutator sum rule [Eq. \ref{['eq:sumrule']}].
• Figure 2: $\Delta(\text{Our Matsubara}) \equiv \Im\,(\chi_{\mathrm{iso}}^{R})_{\rm our}(\omega;\tau)-\Im\,(\chi_{\mathrm{iso}}^{R})_{\rm truth}(\omega;\tau)$ (solid blue) and $\Delta(\delta\text{-probe}) \equiv \Im\,(\chi_{\mathrm{iso}}^{R})_{\delta\text{-probe}}(\omega;\tau)-\Im\,(\chi_{\mathrm{iso}}^{R})_{\rm truth}(\omega;\tau)$ (orange dashed), with a zero baseline (black dotted). The $\delta$-probe shows large negative excursions at resonances, whereas our route fluctuates narrowly around zero.
• Figure 3: $\Im\chi_{\mathrm{iso}}^{R}(\omega)$ for $\omega\le 0.9~\text{eV}$: ground truth (black dotted), our Matsubara route (solid blue), and $\delta$-probe (orange dashed). Our route reproduces the $\omega\to0$ slope consistent with Kramers--Kronig [Eq. \ref{['eq:KK']}], while the $\delta$-probe drifts due to finite-window bias.
• Figure 4: $\Im\chi_{\mathrm{iso}}^{R}(\omega)$ from our Matsubara route (solid blue; shaded band = $\pm 2\sigma$), the $\delta$-probe reconstruction (orange dashed), and the ground truth (black dotted). The Matsubara curve tracks the reference across peaks and high-$\omega$ tails, while the $\delta$-probe shows peak broadening and systematic weight loss. On-plot metrics ($D_2$ and sum-rule residuals) quantify the improvement.