Pump-intensity-scaling of Two-Photon-Absorption and Photon Statistics of Entangled-Photon Fields

TL;DR

This work develops a non-perturbative framework for describing entangled-photon fields generated by PDC under strong pump illumination, deriving linear integro-differential equations for field correlations within a Wigner-representation formalism to handle finite pump bandwidth. By decomposing the two-photon absorption signal into inter-mode and intra-mode contributions and linking these to Glauber g^{(2)} statistics, the study shows that the inter-mode (entanglement-bearing) component can scale linearly with pump intensity over a broad range, while intra-mode contributions increasingly dominate at higher intensities; increasing pump bandwidth further extends the linear regime. A practical experimental scheme is proposed to filter out intra-mode contributions by using two identical PDC crystals to generate uncorrelated pulses, allowing isolation of entanglement effects at higher intensities. The results provide a pathway to leverage entangled light in nonlinear spectroscopy by controlling pump bandwidth and employing filtering to reveal quantum correlations in TPA signals, with implications for high-resolution spectroscopy and quantum-light–matter interactions.

Abstract

We use a non-perturbative theoretical approach to the parametric down-conversion (PDC) process, which generates entangled-photon field for an arbitrarily strong pump-pulse. This approach can be used to evaluate multi-point field correlation functions to compute nonlinear spectroscopic signals induced by a strong pump. The entangled-photon statistics is studied using Glauber's $g^{(2)}$ function, which helps understand the significance of the photon entanglement-time and the pump-pulse intensity on spectroscopic signals. Under the non-perturbative treatment of the entangled field, the two-photon absorption (TPA) signal shows linear to strongly non-linear growth with the pump intensity, rather than linear to quadratic scaling reported previously. An increase in the range of pump intensity for the linear scaling is observed as the pump band-width is increased. We propose an experimental scheme that can select contributions to the TPA signal that arise solely from interactions with the entangled photons, and filter out unentangled photon contributions, which are dominant at higher pump intensities, paving a way to explore the entanglement effects at higher intensities.

Figures (7)

• Figure 1: Diagrams contributing to the first term (top panel), the second term (middle panel), and the third term (bottom panel) in Eq. (\ref{['eq-1a']}). Diagrams for the last term in Eq. (\ref{['eq-1a']}) are obtained by interchanging the left and right interactions at $\tau_i, i=1,2,3,4$, in the bottom diagram. Interactions at times $t$ and $\tau$ represented by the horizontal arrows correspond to the observed fluorescent mode. Red (blue) curves with two arrowheads denote ordinary (anomalous), $D^{-+} (D^{--}, D^{++})$ entangled-field propagators (see discussion below Eq. (\ref{['eq-1b']})). Arrowheads pointing (out) into the vertical black lines represent field $(E^\dag) E$.
• Figure 2: Energy level scheme used in our simulations with energies $e_1$=1.9 fs$^{-1}$, $e_2$=2 fs$^{-1}$, $e_3$=2.1 fs$^{-1}$, and $e_4$=4 fs$^{-1}$.
• Figure 3: Time-dependence of the inter-mode ($g^{(2)}_1$, solid) and intra-mode ($g^{(2)}_0-1$, dashed) parts of $g^{(2)}(\tau)$ for different entanglement times, as indicated. Upper(lower) panel: for high (low) pump intensity, $I_p$=3.35$\times$10$^{16}$ (3.35$\times$10$^{14}$) W/cm$^2$. Other parameters are: $\sigma_p$=0.3 fs$^{-1}$, $\omega_p$=4 fs$^{-1}$, $\omega_s$=2 fs$^{-1}$, and PDC crystal length $l=20$$\mu$m. The inset compares the $g^{(2)}(\tau)$ function calculated numerically (solid) with the semi-analytical (dashed) solution.
• Figure 4: The time-dependence of the inter-mode ($g^{(2)}_1$, solid) and the intra-mode ($g^{(2)}_0$, dashed) contributions for different pulse bandwidths $\sigma_p$. The upper (lower) panel shows the inter-mode and intra-mode contributions for a strong (weak) pump with the field strength of $I_p$=3.35$\times$10$^{16}$ (3.35$\times$10$^{14}$) W/cm$^2$ for the entanglement time $\Delta T=8$fs. The inset compares the total $g^{(2)}$ calculated numerically (solid) with semi-analytical results (dashed) for $I_p$=3.35$\times$10$^{14}$ W/cm$^{2}$.
• Figure 5: Normalized TPA signal from the model molecule (Eq. \ref{['eq-4']}). The dashed-red (grey) curve is the inter-mode contribution for $I_p$=3.73$\times$10$^{13}$ (9.33$\times$10$^{12}$) W/cm$^{2}$. Here $E_f=4 fs^{-1}, E_{e1}=1.9 fs^{-1},E_{e_2}= 2.0 fs^{-1}, E_{e_3}=2.1 fs^{-1}$, $\sigma_p=0.1 fs^{-1}$, $\eta$=0.015, $T_e$=13.33 fs.