Scattering theory of frequency-entangled biphoton states facilitated by cavity polaritons

TL;DR

We develop a Green-function scattering framework for the interaction of frequency-entangled biphoton pairs with cavity polaritons described by the Tavis-Cummings model. The output joint spectral amplitude is expressed as a sum of coherent Rayleigh and incoherent redistribution contributions governed by polariton and bipolariton Green functions under steady-state conditions, enabling quantitative predictions of JSA modification across coupling regimes. The theory reveals how entanglement entropy of the scattered photons depends on input JSA, cavity population, and coupling strength, with distinctive spectral-filtering and bipolariton-correlation signatures that can be isolated by subtracting the coherent component. This approach provides a sensitive, low-photon-number spectroscopy method to probe polariton and bipolariton states in cavity quantum materials, spanning weak to ultrastrong coupling and potentially enabling insight into strongly correlated electronic and vibrational dynamics in nanostructures.

Abstract

The use of quantum light to probe exciton properties in semiconductor and molecular nanostructures typically occurs in the low-intensity regime. A substantial enhancement of exciton-photon coupling can be achieved with photonic cavities, where excitons hybridize with cavity modes to form polariton states. To provide a theoretical framework for interpreting experimental efforts in this direction, we develop a scattering theory describing the interaction of frequency-entangled photon pairs with cavity polariton and bipolariton states under various coupling regimes. Employing the Tavis-Cummings model in combination with our scattering approach, we present a quantitative analysis of how the interaction of the entangled photon pair with the polariton/bipolariton modifies its joint spectral amplitude (JSA). Specifically, we examine the effects of the cavity-mode steady-state population, exciton-cavity coupling strength, and different forms of the input photon JSA. Our results show that the entanglement entropy of the scattered photons is highly sensitive to the interplay between the input JSA and the spectral line shapes of the polariton resonances, emphasizing the cavity filtering effects. We suggest that biphoton scattering quantum light spectroscopy best serves as a sensitive probe of polariton and bipolariton states in the photon-vacuum cavity state.

Figures (7)

• Figure 1: Schematics of a photonic microcavity containing a semiconductor nanostructure, where the interaction between a cavity photon mode and nanostructured material excitons forms a hybrid light–matter quasiparticle — the cavity polariton. The polariton state is probed by scattering an input entangled photon pair, $|\Psi_\text{in}\rangle$, generated via spontaneous parametric down-conversion (SPDC), with the output state, $|\Psi_\text{out}\rangle$, detected after transmission through the TWINS interferometric setup.
• Figure 2: Feynman diagrams describing (a) single-photon scattering by the polariton mode and (b, c) biphoton scattering involving the steady-state propagation. The other diagrams describe the biphoton scattering events that involve (d) the bipolariton coherence and (e,f) the polariton-polariton excited state coherences. The total energy of the incoming and scattered photons during the scattering is conserved, as indicated in the blue caption below each diagram. The diagram correspondence rules are shown in (g) for single-polariton and the steady-state Green functions, (h) for the bipolariton and polariton-polariton coherence Green functions, and (k) for incoming and scattered photon lines with the vertex, respectively. The corresponding expression for each diagram (a)-(f) should by multiplied by the prefactor $2\pi\delta(\omega)$, where $\omega$ is substituted with the associated energy conservation expression and integrated over all frequency variables.
• Figure 3: (a) Calculated dispersion of the polariton energy $\hbar\omega_\pm$ (solid blue) and broadening $\hbar\gamma_\pm$ (dashed blue) for the cavity in vacuum state, $\bar{n}=0$ as a function of the normalized cooperative coupling parameter $\lambda\sqrt{N}/\omega_o$. The vertical dashed red line marks $\lambda\sqrt{N}/\omega_o = 0.0065$. The rest of the panels present calculation results for the empty cavity case, $\lambda\sqrt{N}=\bar{n}=0$: (b) $|G(\omega_s)|$, (c) $|G(\omega_s)G(\omega_i)|$, (d) $|G(\omega_s)\mathcal{G}(\omega_s + \omega_i)|$, (e) Absolute value of symmetrized input (SPDC) JSA, (f) Absolute value of symmetrized output JSA.
• Figure 4: Calculated (a) $|G(\omega_s)G(\omega_i)|$ and (b) absolute value of symmetrized output JSA, for the case of QEs in a strong coupling regime, $\lambda\sqrt{N}/\omega_o = 0.0065$, [designated by vertical red dashed line in Fig. \ref{['Fig:empty_cav']}(a)] with the cavity mode in vacuum state, $\bar{n}=0$. (c) Comparison of the entanglement entropy due to the symmetrized output JSA presented in panel (b) with the entanglement entropy of the symmetrized input JSA presented in Fig. \ref{['Fig:empty_cav']}(e) v.s. normalized cooperative coupling parameter $\lambda\sqrt{N}/\omega_o$.
• Figure 5: Calculation results for the case of cavity containing a single photon, $\bar{n}=1$: (a) Dispersion of the polariton energy $\hbar\omega_\pm$ (solid blue) and broadening $\hbar\gamma_\pm$ (dashed blue) as a function of the normalized cooperative coupling parameter $\lambda\sqrt{N}/\omega_o$. (b): Comparison of the entanglement entropy due to the symmetrized output JSA with the entanglement entropy of the symmetrized input JSA presented in Fig. \ref{['Fig:empty_cav']}(e).