Addressing intramolecular vibrational redistribution in a single molecule through pump and probe surface-enhanced vibrational spectroscopy

TL;DR

The paper develops a quantum mechanical framework based on molecular optomechanics to model intramolecular vibrational redistribution (IVR) within surface-enhanced vibrational spectroscopy. It analyzes two pump–probe configurations: visible Stokes SERS pumping with anti-Stokes probing, and infrared-cavity pumping with anti-Stokes probing, using a Lindblad master equation to track vibrational populations $n_i$ and anti-Stokes signals $I_i^{aS}$. The main contributions include deriving a visible-cavity Hamiltonian, introducing a cubic anharmonic Fermi-resonance coupling $g_f$ between two vibrations, and predicting clear IVR signatures such as population transfer, Rabi-like oscillations, Fermi doublets, and infrared-driven incoherent contributions at single-molecule sensitivity with current plasmonic nanocavities. The results indicate that single-molecule IVR pathways can be resolved with existing SERS platforms, enabling targeted control of mode-specific chemistry.

Abstract

The development of accurate tools to characterize Intramolecular Vibrational Redistribution (IVR) is of major interest in chemistry. In this context, surface-enhanced vibrational spectroscopies stand up as well-established techniques to study molecular vibrational lines and populations with a sensitivity that can reach the singe-molecule level. However, to date, this possibility has not been fully developed to address IVR. Here, we establish a quantum mechanical framework based on molecular optomechanics that accounts for IVR, and adopt it to analyze strategies to optimize IVR characterization by vibrational spectroscopy. In particular, we model two different pump-and-probe configurations where the vibrational pumping is provided either by infrared laser illumination or by Stokes SERS. We show for the two pumping configurations the existence of clear signatures on the anti-Stokes SERS spectra of population transfer between coupled vibrational modes in a molecule. Our calculations adopt realistic molecular and SERS parameters, suggesting that these signatures of IVR are accessible at the single-molecule level with current experimental platforms.

Figures (5)

• Figure 1: Sketch of the system under study. Top: Sketch of a single molecule inside a metallic nanoresonator. As a representative example of the system considered, a plasmonic nanocavity is made of an ultrathin gap (typically $\sim$ 1 nm) between a metallic tip and a planar metallic antenna. The plasmonic gap nanocavity sustains modes in the visible range that can enhance the incident electric field (green shadowing in the figure). The IR antenna sustains a mode in the infrared range corresponding to molecular vibrations ($\sim$ 3-50 µ m), which can enhance an IR incident field (dark red shadowing in the figure). Bottom inset: illustration of an Intramolecular vibrational redistribution pathway inside the molecule, where an excited high energy vibration (left) relaxes by transferring simultaneously its energy to two other vibrations to which it is coupled (right).
• Figure 2: Fermi resonance with $\omega_A=2\omega_B$. (a): Schematic representation of the energy levels of two vibrations A and B in which the fundamental of vibration A is resonantly coupled to the overtone of a vibration B by a Fermi resonance with coupling strength $g_{\textsc{f}}$ (gray arrows). The left part of the panel depicts the weak coupling point of view where modification of the energy levels by the coupling remains negligible ($2\sqrt{2}g_{\textsc{F}} \lesssim \gamma_A + \gamma_B$) while the right part depicts the strong coupling point of view where mode hybridization appears ($2\sqrt{2}g_{\textsc{F}} > \gamma_A + \gamma_B$). The blue arrows indicate decay channels, through non-radiative pathways at rates $\gamma_{\textsc{a}},\gamma_{\textsc{b}}$ or by anti-Stokes Raman scattering with optomechanical coupling strengths $g_{\textsc{vis,a}},g_{\textsc{vis,b}}$. (b): Sketch of anti-Stokes SERS spectrum with two vibrations strongly coupled by a Fermi resonance (case $2\sqrt{2}g_{\textsc{F}} > \gamma_A + \gamma_B$). The blue arrows indicate the possible Raman transitions. The equations on top indicate that the integrals of the anti-Stokes SERS intensity over the $\omega_B$ and $\omega_A$ peaks, respectively noted $I_{B}^{\text{aS}}$ and $I_{A}^{\text{aS}}$, are proportional to the product of the vibrational Raman coupling strengths square multiplied by vibrational populations.
• Figure 3: Signatures of IVR under cw Stokes SERS pumping and anti-Stokes SERS probing. (a): sketch of the configuration under study, which consists in a single molecule in a plasmonic gap nanocavity with a mode resonant at 633 nm (mode 1 in Section \ref{['sec:parameters']}), resonantly illuminated by continuous-wave laser light (wavelength 633 nm, green arrow). The red arrow represents SERS emission by Stokes scattering processes, that also induces a pumping of the molecular vibrations. The blue arrow stands for the measured anti-Stokes SERS signal. We do not consider here IR illumination nor the IR cavity mode. (b): Evolution of the population of state $|1,0\rangle$ (red continuous and dashed lines), state $|0,2\rangle$ (orange lines) and state $|0,1\rangle$ (blue lines) with the intensity of the pumping laser. The Fermi resonance coupling between states $|1,0\rangle$ and $|0,2\rangle$ is included ($g_F > 0$, solid lines) or neglected ($g_F =0$, dashed line). (c): anti-Stokes SERS spectra of two molecular vibrations coupled by the Fermi resonance (red lines) or uncoupled (black dashed lines), for two pumping intensities $I_{\textsc{vis}}= 7.1\times10^4$ µ W.µ m$^{-2}$ ($\Omega_{\textsc{vis}}=$ 300 meV) and $I_{\textsc{vis}}= 7.1\times10^6$ µ W.µ m$^{-2}$ ($\Omega_{\textsc{vis}}=$ 3000 meV). The dotted vertical gray lines indicate the fundamental frequency of vibrations A and B. The temperature of the molecule is set to $T=$ 100 K. The other parameters used in the simulations are introduced in Section \ref{['sec:parameters']}.
• Figure 4: Signatures of IVR under pulsed Stokes SERS pumping and pulsed anti-Stokes SERS probing. (a): Sketch of the configuration under study, which consists of a single molecule in a plasmonic gap nanocavity that supports two modes at visible wavelengths, one resonant at 633 nm (green shadowing on the figure, mode 1 in Section \ref{['sec:parameters']}) and another resonant at 785 nm (orange shadowing on the figure, mode 2 in Section \ref{['sec:parameters']}). The mode at 785 nm is resonantly illuminated with pulsed light (left orange arrow, pulse center at time $t=0$) that produces Stokes SERS light (red arrow) and vibrational pumping of the molecular vibrations. The cavity mode resonant at 633 nm is resonantly illuminated with pulsed light (right green arrow) arriving at a delayed time $t^{\text{probe}}$ that generates the measured anti-Stokes SERS signal (blue arrow). We do not consider here IR illumination nor the IR cavity mode. (b): Time dynamics of the integrated anti-Stokes SERS signals from molecular vibrations A and B, $I^{\text{aS,int}}_{A}(t)$ (red line) and $I^{\text{aS,int}}_{B}(t)$ (brown line), respectively. Both curves are normalized to the maximum of $I^{\text{aS,int}}_{A}$ for $g_{\textsc{f}}=0$. (c): Dynamics of the population of the states $|1,0\rangle$ (red line), $|0,2\rangle$ (orange line) and $|0,1\rangle$ (blue line) without probe pulse ($\Omega_{\textsc{vis}}^{\text{probe,max}}=$ 0) and without Fermi resonance coupling ($g_{\textsc{f}}=0$). The vertical black dotted line indicates the time $\sim$ 240 fs at which all the states approximately reach their maximum population. The diagonal black dash-dotted line is a guide to the eyes showing an exponential relaxation of vibrational mode A with rate $\gamma_A$. The gray shaded area indicates the time interval during which the pump pulse intensity is greater than half its maximum value (FWHM). (d): Same as in (c) but when the Fermi resonance coupling $g_{\textsc{f}}>0$ is included. Parameters: in all panels the Gaussian pump pulse duration has a full-width at half maximum (of intensity) $\Delta\tau^{\text{pump}}=$ 500 fs and a peak intensity $I_{\textsc{vis}}^{\text{pump,max}}=$ 5.7$\times 10^{6}$µ W.µ m$^{-2}$ (i.e. $\Omega_{\textsc{vis}}^{\text{pump,max}}=$ 3000 meV). The Gaussian probe pulse is turned on in (b) with duration $\Delta \tau^{\text{probe}}=$ 500 fs and peak intensity $I_{\textsc{vis}}^{\text{probe,max}}=$ 1.3$\times 10^{5}$ µ W.µ m$^{-2}$ (i.e. $\Omega_{\textsc{vis}}^{\text{probe,max}}=$ 400 meV), and is turned off ($I_{\textsc{vis}}^{\text{probe,max}}=0$) in (c),(d). The temperature of the molecule is set to $T=$ 100 K. The other parameters used in the simulations are introduced in Section \ref{['sec:parameters']}.
• Figure 5: Signatures of IVR under cw IR pumping and cw anti-Stokes SERS probing. (a): sketch of the configuration under study, which consists of a single molecule in a plasmonic gap nanocavity that supports a mode resonant at 633 nm (green shadowing on the figure, mode 1 in Section \ref{['sec:parameters']}). The bottom mirror of the nanocavity is a flat metallic disk that supports a plasmonic resonance in the infrared range (mode at 9.1 µ m, that is $\hbar \omega=$ 136 meV, IR mode in Section \ref{['sec:parameters']}) responsible for an enhancement of the IR electric field at the position of the molecule (red shadowing on the figure). The IR mode is resonantly illuminated with continuous-wave IR light (red arrow) that resonantly pumps a single molecular vibration of frequency $\hbar \omega_A=$ 136 meV. The visible cavity mode is resonantly illuminated by continuous-wave visible light (green arrow) that produces a measured anti-Stokes SERS signal revealing vibrational populations (blue arrow). (b): evolution of the populations of states $|1,0\rangle$ (red line), $|0,2\rangle$ (orange line) and $|0,1\rangle$ (blue lines) with the intensity of the pumping IR laser. The red dots show the quantity $|\langle 0,0|\hat{\rho}| 1,0 \rangle|^2$ that is the coherent population of state $|1,0\rangle$ (here it is almost equal to the coherent population $|\langle \hat{b}_A \rangle|^2$ of vibration A). The orange dots show the quantity $2|\langle 0,0|\hat{\rho}| 0,2 \rangle|^2$ that is the coherent population of state $|0,2\rangle$. (c): anti-Stokes SERS spectra of the molecule for several cw IR pumping intensities $I_{\textsc{ir}}=$ 0 µ W.µ m$^{-2}$ (blue line), $I_{\textsc{ir}}=$ 10 µ W.µ m$^{-2}$ ($\Omega_{\textsc{ir}}=$ 4.7 meV, green line)) and $I_{\textsc{ir}}=$ 1000 µ W.µ m$^{-2}$ ($\Omega_{\textsc{ir}}=$ 46.8 meV, red line). The dotted vertical gray lines are guides to the eye and indicate the frequencies of vibration A and B. In panels (b) and (c), the Fermi resonance coupling is $g_{\textsc{f}}=$ 1.0 meV and the Raman probe intensity is $I_{\textsc{vis}}=$ 100 µ W.µ m$^{-2}$ ($\Omega_{\textsc{vis}}=$ 11.3 meV). The temperature of the molecule is set to $T=$ 150 K. The other parameters used in the simulations are introduced in Section \ref{['sec:parameters']}. In panel (c), the coherent contribution to the spectrum (nearly vertical red lines at 136 meV) is plotted using a Lorentzian lineshape with very small width (0.003 meV) and with an amplitude defined such that the spectral integral of the Lorentzian is equal to $(\omega_{\textsc{vis}} + \omega_A)^4 \Gamma^{-}_A |\langle 0,0|\hat{\rho}| 1,0 \rangle|^2$ (see Eq. (\ref{['eq:anti_stoke_SERS_semiclassical']})). The value 0.003 meV has been chosen to mimic the linewidth of the excitation laser.