Spontaneous Raman Scattering under Vibrational Strong Coupling: The Critical Role of Polariton Spatial Mode Coherence

Abstract

Resonant coupling of a vibration to a cavity mode has been reported to dramatically modify spontaneous Raman scattering, but subsequent studies have produced conflicting results. In this Letter, we develop a microscopic quantum framework that captures the spatial structure of polaritonic modes. In a homogeneously filled cavity, spatial overlap between polaritons and cavity resonances enforces selection rules that suppress the initially reported polaritonic Raman peaks, consistent with most experiments. In contrast, for a quasi-two-dimensional (2d) molecular layer, these rules are lifted, yielding Raman peaks at the polariton energies. Our work clarifies that the Raman response under vibrational strong coupling is determined by cavity-vibration spatial mode overlap and offers a framework for Raman studies of strongly coupled quasi-2d systems.

Figures (3)

• Figure 1: $N$ molecules are confined in a Fabry-Perot cavity of thickness $L$. (a) The cavity field is quantized along $\hat{\mathbf{z}}$, yielding discrete mode indices $n$. Raman scattering is driven by a laser of frequency $\omega_\mathrm{L}$ and detected at frequency $\omega_\mathrm{S}$ and angle $\theta_\mathrm{S}$, corresponding to polariton modes with indices $n_\mathrm{L}$ and $n_\mathrm{S}$. Selection rules suppress the resonant polariton Raman peaks with $n = 1$. (b) Each molecule $j$ has electronic and vibrational degrees of freedom: a ground-state vibrational mode $\ket{v}_j$ of frequency $\omega_0$ strongly coupled to the cavity, and excited-state vibrational modes $\ket{w}_j$ of frequencies $\omega_w$ weakly coupled to the cavity.
• Figure 2: Absence of resonant polaritonic peaks in the Raman spectra for a Fabry-Perot cavity filled with molecules. Panels (a) and (c) display the polaritonic dispersion relations for two coupling strengths $\nu$, with color indicating their vibrational weight $( x^{\pm}_{q,n})^2-( z^{\pm}_{q,n})^2$ (see text). At fixed scattered angle $\theta_\mathrm{S}$, each dashed line corresponds to a Raman spectra in panels (b) and (d). The intersections between the dashed lines and the polaritonic branches are marked by points (red points for resonant $n=1$ polaritons, black for the "dark" polaritons $n>1$), which denote the predicted Raman peaks solving the energy conservation condition of Eq. \ref{['eq:raman_rate']}. These peaks are indicated by vertical dashed lines, with colors matching those of the points, in panels (b) and (d). Each peak is broadened by a Lorentzian of width $0.008\,\omega_0$ for clarity. The insets display enlarged regions of the dispersion. Parameters: $\omega_\mathrm{L} = 10.9\, \omega_0$, $L = \lambda_0/2$, with $\lambda_0 = 2\pi c / \omega_0$ and analogous definitions for $\lambda_\mathrm{L}$ and $\lambda_\mathrm{S}$. The highest-energy cavity mode considered in the calculations is $n=100$.
• Figure 3: Raman spectra for a Fabry-Perot cavity containing a single molecular layer. A schematic of this configuration is shown in the inset of panel (a). (a) Polaritonic dispersion, with colored dashed lines indicating the Raman spectra shown in panel (b) for corresponding scattered angles $\theta_\mathrm{S}$. Intersections between dashed lines and polaritonic branches in (a) denote the predicted Raman shifts, determined by the energy conservation condition in Eq. \ref{['eq:raman_rate_slice']}, and are marked as vertical dashed lines of matching color in panel (b). Here, we take a coupling strength of $\nu = 0.066\, \omega_0$ and fix the layer height at $h = 0.48\, L$. Other parameters are identical to Fig. \ref{['fig:cavity_bulk']}. The highest-energy cavity mode considered in the calculations is $n=5$.