To provide insight into the scattering of light by a molecule, we develop an analytical model describing the Raman process for a relatively simple scenario. We first consider that the electron–phonon coupling, described by the displacement value $d=\sqrt{S},$ is in the low range of possible values, as obtained for rigid molecules [56], [57] (*d*~10^{−1}). We further assume that the laser illumination is not too intense so that the linear response approximation is justified and that the Two-level system is not affected by pure dephasing processes.

The condition of weak electron–vibration coupling, considering that the time scale of the electron–vibration interaction is larger than that of the typically dominant relaxation processes, can be roughly estimated as *ω*_{m}d<*γa*/2, *g*. This assumption allows us to separate the full system according to the hierarchy of time scales: on the one hand, the fast-decaying and decohering reservoir part, which consists of the TLS strongly coupled with the plasmon, and on the other hand, the slowly varying part of the system represented by the vibrational mode [51]. This separation of time scales allows for using the Markov approximation and obtaining the effective vibrational dynamics under the influence of the reservoir. The derivation is discussed further in the Supplementary Information and yields the following set of equations:

$$\begin{array}{l}\frac{\text{d}}{\text{d}t}\u3008b\u3009=-\text{i}{\omega}_{\text{m}}\u3008b\u3009-\text{i}{\omega}_{\text{m}}d\u3008{\sigma}^{\u2020}\sigma {\u3009}_{\text{S}S}\\ \text{}-\mathrm{(}{\gamma}_{b}\mathrm{/}2\mathrm{+}{\Gamma}_{-}\mathrm{/}2\mathrm{-}{\Gamma}_{+}/2\mathrm{)}\u3008b\u3009\mathrm{,}\end{array}$$(9)

$$\begin{array}{c}\frac{\text{d}}{\text{d}t}\u3008{b}^{\u2020}b\u3009=-\text{i}{\omega}_{\text{m}}d\u3008{\sigma}^{\u2020}\sigma {\u3009}_{\text{S}S}\mathrm{(}\u3008{b}^{\u2020}\u3009-\u3008b\u3009\mathrm{)}\\ -\text{\hspace{0.17em}}\mathrm{(}{\gamma}_{b}+{\Gamma}_{-}-{\Gamma}_{+}\mathrm{)}\u3008{b}^{\u2020}b\u3009+{\Gamma}_{+}\mathrm{,}\end{array}$$(10)

where Γ_{+} and Γ_{−} are, respectively, the effective incoherent pumping and damping rates due to the reservoir. These rates are related to the steady-state reservoir spectral function $\mathcal{J}\mathrm{(}s\mathrm{)}=\mathrm{Re}\{{\displaystyle {\int}_{0}^{\infty}}\u3008\u3008{\sigma}^{\u2020}\sigma \mathrm{(}\tau \mathrm{)}{\sigma}^{\u2020}\sigma \mathrm{(}0\mathrm{)}\u3009\u3009{e}^{\text{i}s\tau}\text{d}\tau \}$ as follows:

$${\Gamma}_{-}=2{\omega}_{\text{m}}^{2}{d}^{2}\mathcal{J}\mathrm{(}{\omega}_{\text{m}}\mathrm{)}$$(11)

$${\Gamma}_{+}=2{\omega}_{\text{m}}^{2}{d}^{2}\mathcal{J}\mathrm{(}-{\omega}_{\text{m}}\mathrm{}\mathrm{)}\mathrm{,}$$(12)

with the following notation ⟨⟨*O*_{1}*O*_{2}⟩⟩=⟨*O*_{1}*O*_{2}⟩–⟨*O*_{1}⟩_{S}_{S}⟨*O*_{2}⟩_{S}_{S}. In the scheme considered above, the reservoir operators are obtained from the dynamics of the J-C system decoupled from the vibrations.

The steady-state population and coherence follow directly from (9) and (10):

$${\u3008b\u3009}_{\text{S}S}=-\frac{\text{i}{\omega}_{\text{m}}d{\u3008{\sigma}^{\u2020}\sigma \u3009}_{\text{S}S}}{{\gamma}_{b}\mathrm{/}2\mathrm{+}{\Gamma}_{-}\mathrm{/}2\mathrm{-}{\Gamma}_{+}\mathrm{/}2\mathrm{+}\text{i}{\omega}_{\text{m}}}\mathrm{,}$$(13)

$${\u3008{b}^{\u2020}b\u3009}_{\text{S}S}=\mathrm{|}{\u3008b\u3009}_{\text{S}S}{\mathrm{|}}^{2}+\frac{{\Gamma}_{+}}{{\gamma}_{b}+{\Gamma}_{-}-{\Gamma}_{+}}\mathrm{,}$$(14)

As we have assumed that the pure dephasing of the TLS is negligible, we can write ⟨*σ*^{†}*σ*⟩_{S}_{S}≈|⟨*σ*⟩_{S}_{S}|^{2}. Next, we decompose the lowering operator of the TLS into its steady-state value ⟨*σ*⟩_{S}_{S} and the fluctuating part with zero mean *δσ*, as *σ*=⟨*σ*⟩_{S}_{S}+*δσ*, and thus we can approximate the population operator as *σ*^{†}*σ*≈|⟨*σ*⟩_{S}_{S}|^{2}+⟨*σ*⟩_{S}_{S}δσ^{†}*+*⟨*σ*^{†}⟩_{S}_{S}δσ.

We then transform the dependency of the spectral function J(*s*) from the correlation function of the full operator ⟨⟨*σ*^{†}*σ*(*τ*)*σ*^{†}*σ*(0)⟩⟩ to a simpler dependence with the correlation function ⟨⟨*δσ*(*τ*)*δσ*^{†}(0)⟩⟩=⟨⟨*σ*(*τ*)*σ*^{†}(0)⟩⟩:

$$\begin{array}{l}\mathcal{J}\mathrm{(}s\mathrm{)}=\mathrm{Re}\text{\hspace{0.17em}}\{{\displaystyle {\int}_{0}^{\infty}}\u3008\u3008{\sigma}^{\u2020}\sigma \mathrm{(}\tau \mathrm{)}{\sigma}^{\u2020}\sigma \mathrm{(}0\mathrm{)}\u3009\u3009{e}^{\text{i}s\tau}\text{d}\tau \}\\ \approx \text{\hspace{0.17em}}\mathrm{|}\u3008\sigma {\u3009}_{\text{S}S}{\mathrm{|}}^{2}\mathrm{Re}\text{\hspace{0.17em}}\{{\displaystyle {\int}_{0}^{\infty}}\u3008\u3008\sigma \mathrm{(}\tau \mathrm{)}{\sigma}^{\u2020}\mathrm{(}0\mathrm{)}\u3009\u3009{e}^{\text{i}\mathrm{(}s+{\omega}_{\text{L}}\mathrm{)}\tau}\text{d}\tau \}\\ =\text{\hspace{0.17em}}\mathrm{|}\u3008\sigma {\u3009}_{\text{S}S}{\mathrm{|}}^{2}\underset{\equiv {\mathcal{J}}_{0}\mathrm{(}s\mathrm{)}}{\underbrace{\frac{{g}^{2}\frac{{\gamma}_{a}}{2}+\frac{{\gamma}_{\sigma}}{2}\left[\frac{{\gamma}_{a}^{2}}{4}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\mathrm{(}\Delta \text{\hspace{0.17em}}-\text{\hspace{0.17em}}s\mathrm{)}}^{2}\right]}{{g}^{4}-2{g}^{2}\left[{\mathrm{(}\Delta -s\mathrm{)}}^{2}-\frac{{\gamma}_{\sigma}{\gamma}_{a}}{4}\right]+\left[\frac{{\gamma}_{\sigma}^{2}}{4}+{\mathrm{(}\Delta -s\mathrm{)}}^{2}\right]\left[\frac{{\gamma}_{a}^{2}}{4}+{\mathrm{(}\Delta -s\mathrm{)}}^{2}\right]}}}\mathrm{.}\end{array}$$(15)

Equation (15) thus can be factorised into two contributions: (i) the coherent population of the TLS, |⟨*σ*⟩_{S}_{S}|^{2}, which in the limit of low illumination intensity is given by the following expression:

$$\begin{array}{c}\mathrm{|}{\u3008\sigma \u3009}_{\text{S}S}{\mathrm{|}}^{2}\approx {\u3008{\sigma}^{\u2020}\sigma \u3009}_{\text{S}S}\\ \approx \frac{{g}^{2}\mathrm{|}\alpha {\mathrm{|}}^{2}\mathrm{(}{\Delta}^{2}+{\gamma}_{a}^{2}/4\mathrm{)}}{{\Delta}^{4}+{\mathrm{(}{g}^{2}+{\gamma}_{\sigma}{\gamma}_{a}/4\mathrm{)}}^{2}+{\Delta}^{2}\mathrm{(}{\gamma}_{\sigma}^{2}\mathrm{/}4\mathrm{+}{\gamma}_{a}^{2}\mathrm{/}4\mathrm{-}2{g}^{2}\mathrm{)}}\mathrm{,}\end{array}$$(16)

with $\alpha =\frac{-\mathcal{E}}{\Delta -\text{i}{\gamma}_{a}/2},$ and (ii) the absorption spectrum of the TLS (that contains the coupling of the TLS with the plasmon), J_{0}(*s*). An approximate version of the latter can be obtained analytically using the quantum regression theorem and considering the low pumping regime. The details of this derivation are given in the Supplementary Information, and the final result appears as the last expression in (15).

In the strong coupling regime, J_{0}(*s*) contains two peaks (i.e. maxima) approximately located at frequencies ${s}_{1}^{\mathcal{J}}\approx \Delta +\sqrt{{g}^{2}-\mathrm{(}{\gamma}_{\sigma}^{2}+{\gamma}_{a}^{2}\mathrm{)}/8}$ and ${s}_{2}^{\mathcal{J}}\approx \Delta -\sqrt{{g}^{2}-\mathrm{(}{\gamma}_{\sigma}^{2}+{\gamma}_{a}^{2}\mathrm{)}/8}$ (assuming a slowly varying numerator). We notice that the double-peaked structure can be resolved only if ${g}^{2}>\mathrm{(}{\gamma}_{\sigma}^{2}+{\gamma}_{a}^{2}\mathrm{)}/8.$

The expressions for the effective damping (11) and pumping (12), proportional to J(*s*), can then be interpreted as being the consequence of a two-step process. First, |⟨*σ*⟩_{S}_{S}|^{2} indicates that the pumping laser of frequency *ω*_{L} induces coherent population of the molecular excited state, which triggers the Raman process and thus increases Γ_{−} and Γ_{+}. In a second stage, the Stokes and anti-Stokes excitations appearing at frequencies *ω*_{L}±*ω*_{m} are absorbed and partially emitted to the far field via the resonance of the J-C system, given by J_{0}(*s*) [the emission at the Raman frequencies corresponds to J_{0}(±*ω*_{m})]. The enhancement of the Stokes scattering at frequency *ω*_{L}– *ω*_{m} leads to the enhancement of the vibrational pumping Γ_{+} (a vibrational quantum is created with each emitted Stokes photon), while the anti-Stokes emission at frequency *ω*_{L}+*ω*_{m} contributes to the vibrational damping Γ_{−}. A similar interpretation has also been shown to be valid for off-resonant SERS [17], [18], [58], [59].

Figure 3 demonstrates the validity of this description. We show in Figure 3A and B two-dimensional D colour-map plots of the incoherent part of the vibrational population, ⟨⟨*b*^{†}*b*⟩⟩_{S}_{S}=⟨*b*^{†}*b*⟩_{S}_{S}–|⟨*b*⟩_{S}_{S}|^{2}, as a function of the vibrational frequency *ω*_{m} and of the detuning Δ of the laser from the plasmon frequency. We display the incoherent part of the population as the coherent part |⟨*b*⟩_{S}_{S}|^{2} represents a constant static displacement of the vibration due to the coherent laser pumping.

Figure 3: Resonant surface-enhanced Raman scattering for weak illumination (*ħ*ℰ=1 meV), no pure dephasing, and weak coupling between the electronic and vibrational levels of the molecule, *d*=0.1.

(A, B) Incoherent population of the vibrational mode under laser excitation as a function of the laser detuning Δ and vibrational frequency *ω*_{m}, calculated (A) from the analytical model and (B) from the full numerical calculation. The lines mark the conditions when the frequency of one of the bare J-C absorption peaks coincides with the frequency of the first-order Stokes line (red line) or the incident laser (green line). (C) Numerically calculated surface-enhanced resonant Raman scattering spectra *s*_{E}(*ω*, Δ) at emission frequency *ω* as a function of the plasmon detuning Δ, for a vibrational mode energy *ħω*_{m}=50 meV. The black dashed lines mark positions of the first- and second-order Stokes lines. The white dashed lines mark the condition for which either the frequency of the first-order Stokes line or the frequency of the incident laser matches with one of the Jaynes–Cummings peaks. (D) Maximum intensity of the Stokes Raman line (at *ω*=*ω*_{L}–*ω*_{m}) as a function of the detuning of the incident laser from the plasmon Δ, calculated with the analytical model [(17) (red line)] and with the full numerical model (black crosses). The other parameters used for the calculations in (A–D) are *ħγ*_{b}=2 meV, *ħγ*_{σ}=2×10^{−5} eV, *ħγ*_{a}=150 meV, *ħg*=100 meV, ħ*γ*_{ϕ}=0 eV, Δ=*δ*.

The approximate (Figure 3A) and exact (Figure 3B) results are very similar, both exhibiting a clear population maximum for a detuning *ħ*Δ≈–100 meV and a frequency of the vibrational mode *ħω*_{m}≈180 meV. This maximum can be understood as a consequence of the two-step process described above. Pumping, |⟨*σ*⟩_{S}_{S}|^{2}, is optimised when the incident laser excites the absorption peaks near the resonant condition *ω*_{L}≈*ω*_{pl}±*g* (*ω*_{L}≈*ω*_{pl}±*g* marked by the vertical green line). On the other hand, the Raman–Stokes process, leading to vibrational pumping, is maximised when the Stokes–Raman line coincides with one of the J-C absorption peaks (red lines, marking the condition *ω*_{L}–*ω*_{m}≈*ω*_{pl}–*g*). Figure 3A and B show that the strongest incoherent vibrational population is indeed achieved when both conditions are fulfilled (region near the intersection of the green and the red line). We remark that for the regime under consideration, *ħ*ℰ=1 meV and temperature *T*≈0 K, the enhancement of the anti-Stokes lines does not significantly influence the incoherent vibrational population.

Although the incoherent vibrational populations provide important information about the system dynamics, the quantity typically accessible in experiments is the inelastic Raman emission spectrum [given by (5)]. Using the separation of the system’s dynamics into the slow vibration and the fast J-C reservoir, we can isolate the Raman contribution to the scattering spectrum from the fluorescence background (as discussed in Supplementary Information). By doing this, the resulting expressions for the Stokes (*s*_{e,}_{S}) and anti-Stokes (*s*_{e,}_{aS}) emission lines are as follows:

$$\begin{array}{l}{s}_{\text{e}\mathrm{,}S}\mathrm{(}\omega \mathrm{)}={\omega}_{\text{m}}^{2}{d}^{2}\mathrm{|}\u3008\sigma {\u3009}_{\text{S}S}{\mathrm{|}}^{2}\mathrm{|}\mathcal{A}\mathrm{(}-{\omega}_{\text{m}}\mathrm{)}{|}^{2}\\ \text{}\times \text{\hspace{0.17em}}\text{\hspace{0.17em}}2\mathrm{Re}\{{\displaystyle {\int}_{0}^{\infty}}\u3008\u3008b\mathrm{(}0\mathrm{)}{b}^{\u2020}\mathrm{(}\tau \mathrm{)}\u3009\u3009{e}^{\text{i}\mathrm{(}\omega -{\omega}_{\text{L}}\mathrm{)}\tau}\text{d}\tau \}\mathrm{,}\end{array}$$(17)

$$\begin{array}{l}{s}_{\text{e}\mathrm{,}aS}\mathrm{(}\omega \mathrm{)}={\omega}_{\text{m}}^{2}{d}^{2}\mathrm{|}\u3008\sigma {\u3009}_{\text{S}S}{\mathrm{|}}^{2}\mathrm{|}\mathcal{A}\mathrm{(}{\omega}_{\text{m}}\mathrm{)}{|}^{2}\\ \text{}\times \text{\hspace{0.17em}}\text{\hspace{0.17em}}2\mathrm{Re}\{{\displaystyle {\int}_{0}^{\infty}}\u3008\u3008{b}^{\u2020}\mathrm{(}0\mathrm{)}b\mathrm{(}\tau \mathrm{)}\u3009\u3009{e}^{\text{i}\mathrm{(}\omega -{\omega}_{\text{L}}\mathrm{)}\tau}\text{d}\tau \}\mathrm{,}\end{array}$$(18)

where |⟨*σ*⟩_{S}_{S}|^{2} is given by (16), and |A|^{2} is the reservoir spectral function that describes the enhancement of the emission of the Raman photons into the far field due to the presence of the J-C system (we assume that the emission from the molecule via the plasmon is much stronger than its direct far-field emission):

$$\begin{array}{l}\mathrm{|}\mathcal{A}\mathrm{(}s\mathrm{)}{|}^{2}\\ =\frac{{g}^{2}}{{g}^{4}-2{g}^{2}\mathrm{[}\mathrm{(}\Delta -s{\mathrm{)}}^{2}-{\gamma}_{\sigma}{\gamma}_{a}/4]+\mathrm{[}{\gamma}_{\sigma}^{2}\mathrm{/}4\mathrm{+}{\mathrm{(}\Delta -s\mathrm{)}}^{2}][{\gamma}_{a}^{2}\mathrm{/}4\mathrm{+}{\mathrm{(}\Delta -s\mathrm{)}}^{2}\mathrm{]}}\mathrm{.}\end{array}$$(19)

This function is double peaked, and for the conditions considered in this article, the peak positions, ${s}_{1}^{\mathcal{A}}=\Delta +\sqrt{{g}^{2}-\mathrm{(}{\gamma}_{\sigma}^{2}+{\gamma}_{a}^{2}\mathrm{)}/8}$ and ${s}_{2}^{\mathcal{A}}=\Delta -\sqrt{{g}^{2}-\mathrm{(}{\gamma}_{\sigma}^{2}+{\gamma}_{a}^{2}\mathrm{)}/8},$ are similar as ${s}_{1}^{\mathcal{J}}$ and ${s}_{2}^{\mathcal{J}}$ for the spectral function J_{0}(*s*).

Last, the integrals of the vibrational correlation functions in (17) and (18) represent the emission line shape that can be obtained from the equations describing the dynamics of the vibration coupled with the reservoir [17] (9, 10). Applying the quantum regression theorem to these equations, we obtain

$$\begin{array}{l}\mathrm{Re}\{{\displaystyle {\int}_{0}^{\infty}}\u3008\u3008b\mathrm{(}0\mathrm{)}{b}^{\u2020}\mathrm{(}t\mathrm{)}\u3009\u3009{e}^{\text{i}\mathrm{(}\omega -{\omega}_{\text{L}}\mathrm{)}t}\}\\ \text{}=\frac{{\gamma}_{\text{vdp}}/2}{{\mathrm{(}\omega -{\omega}_{\text{L}}+{\omega}_{\text{m}}\mathrm{)}}^{2}+{\gamma}_{\text{vdp}}^{2}/4}\mathrm{(}1+\u3008\u3008{b}^{\u2020}b\u3009{\u3009}_{\text{S}S}\mathrm{)}\mathrm{,}\end{array}$$(20)

$$\begin{array}{l}\mathrm{Re}\{{\displaystyle {\int}_{0}^{\infty}}\u3008\u3008{b}^{\u2020}\mathrm{(}0\mathrm{)}b\mathrm{(}t\mathrm{)}\u3009\u3009{e}^{\text{i}\mathrm{(}\omega -{\omega}_{\text{L}}\mathrm{)}t}\}\\ \text{}=\frac{{\gamma}_{\text{vdp}}/2}{{\mathrm{(}\omega -{\omega}_{\text{L}}-{\omega}_{\text{m}}\mathrm{)}}^{2}+{\gamma}_{\text{vdp}}^{2}/4}\u3008\u3008{b}^{\u2020}b\u3009{\u3009}_{\text{S}S}\mathrm{,}\end{array}$$(21)

with *γ*_{vdp}=*γ*_{b}+Γ_{−}–Γ_{+}. The emitted signal (17 and 18) is thus proportional to the efficiency of the excitation of the system, ∝|⟨*σ*⟩_{S}_{S}|^{2}, and to the emission enhancement due to the coupled TLS plasmon, |A(*s*)|^{2}. Furthermore, the anti-Stokes and Stokes lines are proportional to the incoherent population of the vibrations ⟨⟨*b*^{†}*b*⟩⟩_{S}_{S} and to 1+⟨⟨*b*^{†}*b*⟩⟩_{S}_{S} [60], respectively, where ⟨⟨*b*^{†}*b*⟩⟩_{S}_{S} itself is enhanced by the effect of the reservoir on the excitation and emission process (11–15). The line width of the Raman peaks is finally given by the interplay between the damping rates, *γ*_{b} together with Γ_{−}, both broadening the peak, and the pumping rate Γ_{+} that narrows the peak. These properties again mirror the behaviour that has been discussed for off-resonant Raman [16], [17], [18], except for the existence of a more complex reservoir.

Figure 3C and D illustrate the behaviour of the resulting Raman emission for a vibrational mode of energy *ħω*_{m}=50 meV and $d=\sqrt{S}=\mathrm{0.1.}$ We first show in Figure 3C the emission spectra obtained from the numerical solution of the full system [Hamiltonian in (8) and corresponding loss terms], as a function of the detuning Δ of the laser from the plasmon frequency. The inelastic emission can be split into two components, the emission resulting from the resonance fluorescence of the J-C system and the Raman emission yielding the vibrational lines. The former is present in the emission spectra of Figure 3C as a broad background symmetrical around the zero detuning *ħ*Δ=0 meV, in agreement with previous studies [28], [48]. The strong Raman–Stokes line is also clearly distinguished on top of the background at *ω*_{L}–*ω*_{m}, accompanied by a second-order Stokes transition line (not described by the analytical model that accounts only for the first-order transition) at *ω*_{L}–2*ω*_{m} and a very weak anti-Stokes line at *ω*_{L}+*ω*_{m}.

To better analyse the Δ dependence of the Raman signal and further test the validity of our model, we focus in Figure 3D on the maximum of the Stokes line, which we plot as a function of the laser detuning Δ as calculated from the full numerical model (black dots) and as given by the analytical expression in (17) (red line). The agreement between the two is excellent. We observe a double-peaked structure that is, however, not symmetric with respect to Δ=0, but with the central minimum blue-detuned by 25 meV. This blue detuning can be understood as the result of having to optimise the product of the enhancement at the excitation *ω*_{L} and emission *ω*_{L}–*ω*_{m} frequencies, as given by |⟨*σ*⟩_{S}_{S}|^{2} and |A(*s*)|^{2}, respectively (assuming that the incoherent vibrational population remains small ⟨⟨*b*^{†}*b*⟩⟩_{S}_{S}≪1). The individual contributions of |⟨*σ*⟩_{S}_{S}|^{2} and |A(*s*)|^{2} are shown in Figure S1 of the Supplementary Information.

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