Pulsed single-photon spectroscopy of an emitter with vibrational coupling

TL;DR

This work provides a complete analytic treatment of a single-photon pulse scattering from a two-level emitter coupled to a vibrational bath, including the full quadripartite system of emitter, vibrations, pulse, and environment. It derives the final quantum state of the scattered pulse, capturing vibrational correlations through the temporal density matrix and its frequency-domain spectral density matrix, and introduces a rigorous quantum Fisher information framework to bound and optimize spectroscopic precision. The study reveals Franck–Condon–induced suppression of linewidth-estimation precision and shows that frequency-resolved measurements can outperform time-resolved ones at strong vibrational coupling, a result that holds for both discrete and continuous vibrational spectra. Overall, the results establish fundamental limits and practical guidance for quantum-light-based spectroscopy of vibrationally coupled emitters, with implications for single-photon probes in molecular and solid-state systems.

Abstract

We analytically derive the quantum state of a single-photon pulse scattered from a single quantum two-level emitter interacting with a vibrational bath. This solution for the quadripartite system enables an information-theoretic characterization of vibrational effects in quantum light spectroscopy. We show that vibration-induced dephasing reduces the quantum Fisher information (QFI) for estimating the emitter's linewidth, largely reflecting the Franck-Condon suppression of light-matter coupling. Comparing time- and frequency-resolved photodetection, we find the latter to be more informative in estimating the emitter's linewidth for stronger vibrational coupling.

Figures (8)

• Figure 1: Single-photon scattering from a two-level emitter with vibrational coupling. A photon pulse $\ket{1_\xi}_\text{P}$ (a) excites the emitter from $\ket{g}_\text{T}$ to $\ket{e}_\text{T}$. The latter has a linewidth $\Gamma.$ Both levels feature vibrational manifolds (b). The square root of the Huang-Rhys factor $\sqrt{\lambda_0}\equiv\vert g_0\vert/\Omega_0$ sets the displacement between ground and excited vibrational manifolds. The emitter relaxes the vibrational reorganization energy $\lambda$ within the excited manifold before emitting a single photon. A fraction $\Gamma_\perp/(\Gamma_\perp + \Gamma)$ of this light is lost. The information in the detected fraction of the scattered pulse $\rho_\text{P}(t)$ is accessed via measurements (c).
• Figure 2: Fisher information (in the units of $\Gamma^2$) of estimating $\Gamma$ from the scattered pulse as a function of the Huang-Rhys factor $\lambda_0$. The curves display the QFI upper-bound $\mathcal{Q}_{\text{bound}}$ (black dashed), QFI of the scattered pulse $\mathcal{Q}(\rho_{\text{P}}(\infty))$ (blue solid) and CFI for time- (red small circles) and frequency-resolved (green small squares) photon counting. (c,d) plots the corresponding ratios of the CFI and QFI of the scattered pulse. The incident pulse-shape is a decaying exponential (in the time-domain) $\xi(t)= \exp(-t/(2T_\sigma))\Theta(t)/\sqrt{T_\sigma}$ with pulse-duration $T_\sigma=1/\Gamma$. We evaluate this state numerically via the convolution theorem supp. The system parameters are: $\Gamma=0.15\text{THz}$, $\Omega_0=1000\text{cm}^{-1}$.
• Figure 3: Fisher information (in the units of $\Gamma^2$) of estimating $\Gamma$ from the scattered pulse as a function of the Huang-Rhys factor $\lambda_0$. The curves display the QFI upper-bound $\mathcal{Q}_{\text{bound}}$ (black dashed), QFI of the scattered pulse $\mathcal{Q}(\rho_{\text{P}}(\infty))$ (blue solid) and CFI for time- (red small circles) and frequency-resolved (green small squares) photon counting. (c,d) plots the corresponding ratios of the CFI and QFI of the scattered pulse. The incident pulse-shape is a decaying exponential (in the time-domain) $\xi(t)= \exp(-t/(2T_\sigma))\Theta(t)/\sqrt{T_\sigma}$ with pulse-duration $T_\sigma=1/\Gamma$. We evaluate this state numerically via the convolution theorem supp. The system parameters are: $\Gamma=0.15\text{THz}$, $\Omega_0=100\text{cm}^{-1}$.
• Figure 4: (a) Fisher information (in units of $(\lambda_0)^2$) contained in the scattered pulse $\rho_\text{P}(\infty)$ for estimating the Huang–Rhys factor $\lambda_0$. The curves show the QFI (blue solid) and the CFI for time-resolved (red small circles) and frequency-resolved (green small squares) photon-counting measurements. (b) Ratio of CFI to QFI for these measurement schemes. The incident pulse is an exponentially decaying pulse with duration $T_\sigma=1/\Gamma$, $\Gamma=0.15$THz, $\Omega_0=100\text{cm}^{-1}$, temperature = $300$K.
• Figure 5: Fisher information (in units of $\Gamma^2$) contained in the scattered pulse $\rho_\text{P}(\infty)$ for estimating the linewidth $\Gamma$ with an imperfect detectors ($\Gamma_\perp>0$). The curves in the upper-panel (1a,2a,3a) show the QFI (blue solid) and the CFI for time-resolved (red small circles) and frequency-resolved (green small squares) photon-counting measurements. In the lower-panel (1b,2b,3b), the curves display the Ratio of CFI to QFI for these measurement schemes. The values for the detectors are chosen to be $\Gamma_\perp=0.0$ (1a,1b), $\Gamma_\perp=0.5\Gamma$ (2a,2b), and, $\Gamma_\perp=5\Gamma$ (3a,3b). The incident pulse is an exponentially decaying pulse with duration $T_\sigma=1/\Gamma$, $\Gamma=0.15$THz, $\Omega_0=100\text{cm}^{-1}$, temperature = $300$K.