Reservoir Dynamics in Vibrational Polaritons Revealed by Detuning-Dependent Ultrafast Spectroscopy

Abstract

There is an emerging consensus that transient spectral features of cavity vibrational polaritons are dominated by molecular population dynamics of the optically dark reservoir rather than by excited-state transitions within the polariton manifold. In this work, we use ultrafast mid-infrared pump–probe spectroscopy to study vibrational polaritons of ammonium thiocyanate (NH₄SCN) in dimethylformamide (DMF), with Rabi splitting values of 38 cm^–1 to 95 cm^–1 determined by concentration. The observed polariton line widths show clear robustness to molecular inhomogeneity. By applying second-order kinetic modeling, we extract the dynamics of reservoir states in a zero-detuned cavity from transient polaritonic responses. We find that the reservoir lifetimes are similar to those of the uncoupled molecules. In contrast, under detuned conditions the reservoir relaxes faster regardless of the sign of detuning, suggesting that the cavity modes may open additional relaxation channels. These findings show how cavity properties such as the sample concentration, inhomogeneity, and detunings shape transient polariton spectra.

More than a decade after the first report of a chemical reaction modified by vibrational strong coupling (VSC), the underlying mechanisms driving the phenomenon are still poorly understood. VSC occurs when an infrared-active molecular vibration interacts with the optical mode of a cavity (Figure a). Strong coupling is defined as the regime where energy exchange between the vibration and cavity photon exceeds their individual dissipation rates. − Strong coupling produces two hybrid light-matter normal modesmolecular vibrational polaritons (MVPs)that appear as new peaks in the transmission spectrum. In addition to these two bright modes called upper and lower polaritons (UP and LP), there exists a reservoir of “dark” states (DS) that vastly outnumber the polariton states and lack the appropriate symmetry to contribute to the light-matter hybridization (Figure b). Molecular inhomogeneity in disordered systems such as liquids can also localize excitations, creating a population of effectively uncoupled molecules distinct from the collective dark states of the idealized Tavis-Cummings model. Despite the dominance of reservoir states, numerous reports have shown altered chemical reactivity under strong coupling. Many experiments show slowed ,− or accelerated − reaction rates. Others show preference for one product over another and changes to morphology. , Theoretical descriptions of VSC-modified chemistry suggest various mechanisms including changes to the energy landscape, , altered transition state dynamics, , and modified vibrational energy redistribution pathways, , but a complete picture has yet to emerge. It is unclear what processes are responsible for these observations and to what extent polaritons play a leading role.

1.

Vibrational strong coupling implemented in this study. (a) Schematic diagram of the sample configuration. (b) Energy diagram showing the formation of polaritons (UP and LP) and manifold of dark states (DS). (c) Cavity transmission spectra of the cavity-coupled 1.0 M NH₄SCN in DMF.

One mechanistic proposal is that VSC modifies molecular vibrational dynamics in ways that drive altered chemistry. In pursuit of this idea, time-resolved infrared spectroscopy has been used to monitor vibrational kinetics and spectral changes after exciting the polariton system. − These sorts of nonlinear spectroscopic studies have been successful in showing that VSC can direct photophysical relaxation processes and enable new energy transfer pathways. − However, interpreting transient polariton spectra is complicated by the presence of the large reservoir population. In fact, there is an emerging consensus that transient polariton spectral features, once attributed to excited-state transitions within the polariton manifold, are instead dominated by molecular population dynamics of the optically dark reservoir. − Recent studies have revisited transient polariton spectra within this framework in vibrational systems, ,, while two-dimensional electronic spectroscopy has resolved distinct spectral signatures of polaritons, dark states, and uncoupled molecules in organic exciton-polaritons. Direct 2D-IR measurements on plasmonic antenna lattices have further confirmed that reservoir states exhibit bare-molecular lifetimes. However, these studies have largely been conducted at resonance between the photonic and molecular modes, within a narrow range of coupling strengths, and in systems where molecular inhomogeneous broadening is small relative to the Rabi splitting. How significant molecular disorder and detuning from resonanceconditions common in liquid-phase VSC experimentsalter the reservoir dynamics reflected in transient polariton spectra remains unexplored. We extend this effort using ammonium thiocyanate (NH₄SCN) dissolved in dimethylformamide (DMF) (Figure c), whose high solubility enables Rabi splittings spanning 38 cm^–1 to 95 cm^–1 across multiple detuning conditions. Using ultrafast pump–probe spectroscopy, we characterize both uncoupled and cavity-coupled vibrational dynamics across this parameter space. Applying a second-order kinetics model adapted from Mao and Xiong, we find that the retrieved decay constants are sensitive to detuning conditions. This reveals hidden dynamics of dark states in MVP systems that emerge when probing a broad parameter space.

Measuring steady-state spectra and tracking changes in the transient response of the uncoupled system, summarized in Figure , provide a baseline for understanding the coupled system dynamics. We use NH₄SCN in DMF, which dissociates into ${\mathrm{NH}}_4^{+}$ and SCN^– ions. FTIR spectra (Figure a for 1.0 and 3.5 M, Figure S2 for other concentrations) show a transition centered at ∼2055 cm^–1, corresponding to the ν₃ mode of SCN^–, commonly labeled the CN stretch. While the overall peak center remains nearly constant, the CN stretch frequency is highly sensitive to the local coordination environment, , and the line shape evolves significantly with concentration. Single Lorentzian functions fit well for the 0.45 and 1.0 M samples, indicating a largely homogeneous ensemble of fully solvated, free SCN^– ions. At 2.2 and 3.5 M, however, these fits become inadequate. Savitzky–Golay second-derivative spectroscopy (see the Supporting Information) reveals a concentration-dependent shoulder near 2036 cm^–1 and a feature near 2070 cm^–1 consistent with contact ion pair formation. Peak-normalized difference spectra confirm that the broadening is asymmetric, with greater excess absorption on the red side, ruling out purely symmetric inhomogeneous broadening and indicating the emergence of distinct structural subpopulations above 1.0 M. The line width, measured as full width at half-maximum (fwhm), is ∼19 cm^–1 to 20 cm^–1 in the 0.45 and 1.0 M samples, but increases to 31 cm^–1 and 35 cm^–1 at 2.2 and 3.5 M, respectively. Concentration-dependent parameters for the uncoupled system are summarized in Table .

2.

Uncoupled NH₄SCN vibrational dynamics in DMF. (a) FTIR absorption spectra of 1.0 and 3.5 M NH₄SCN in DMF and pure DMF solvent. Dashed lines are fits using Lorentzian line shape functions. (b) Transient transmission spectra of 1.0 and 3.5 M NH₄SCN in DMF probed at 1, 5, and 10 ps after pump excitation. Excited-state absorption (ESA) peaks at ∼2030 cm^–1 (1.0 M) and ∼2026 cm^–1 (3.5 M), while ground-state bleach (GSB) peaks at ∼2060 cm^–1 (1.0 M) and ∼2063 cm^–1 (3.5 M). (c) Time-resolved traces at ESA and GSB peak positions.

1. Concentration-Dependent Parameters for NH₄SCN in DMF Outside Cavity .

				τ (ps)
Concentration (M)	νvib (1 ← 0) (cm–1)	fwhm (cm–1)	Anharmonicity Δν (cm–1)	1 ← 0	2 ← 1	global
0.45	2056	20	27	11.04 ± 0.16	12.06 ± 0.34	11.73 ± 0.12
1.0	2055	19	30	8.07 ± 0.34	9.88 ± 0.03	9.56 ± 0.03
2.2	2055	31	31	7.16 ± 0.52	6.43 ± 0.11	6.51 ± 0.04
3.5	2055	35	38	4.89 ± 0.18	4.56 ± 0.08	4.58 ± 0.01

^aFWHM and v = 1 ← 0 peak positions are from FTIR absorption spectra fit with Lorentzian functions. Anharmonic shifts and time constants are from single exponential fits to transient absorption data.

Vibrational relaxation dynamics of XCN^– (X = O, S, Se) have been studied previously in a variety of solvents, − making it a well-characterized system for comparison. Transient absorption spectra of the CN stretch mode consist of two main contributions, shown in Figure b for 1.0 M at three delay times. Ground-state bleaching (GSB) appears as decreased absorption at v = 1 ← 0 due to depopulation of the ground state, while excited-state absorption (ESA) arises from a hot-band transition at v = 2 ← 1. An isosbestic point can be seen near 2048 cm^–1. We track the transient response for four sample concentrations0.45, 1.0, 2.2, and 3.5 Mand use the 1.0 and 3.5 M samples (Figure b) for comparison with the polariton response below.

Anharmonicity of the CN stretch causes the v = 2 ← 1 transition to be red-shifted relative to the v = 1 ← 0 transition. Overlap between the ESA and GSB features prevents easy characterization of the anharmonic shift and line width of the ESA peak. We attempt to retrieve these by fitting the transient spectra with a sum of two Gaussian functions, keeping the center and width of the GSB fixed to values obtained from FTIR spectra. We find that anharmonicity increases from 27 cm^–1 at 0.45 M to 38 cm^–1 at 3.5 M, as summarized in Table . The anharmonicity in the 0.45 M concentration sample agrees with previous reports at low concentrations. , Broadening of the ESA peak is also observed with increasing concentration. At 1.0 M the line width is ∼55 cm^–1, increasing to ∼68 cm^–1 at 3.5 M.

The amplitudes of the transient absorption signals at the v = 1 ← 0 and v = 2 ← 1 bands were tracked as a function of delay time between incident pump and probe pulses. Global fits of the ESA and GSB time traces return time constants τ_global,1.0M = 9.56 ps and τ_global,3.5M = 4.58 ps for 1.0 and 3.5 M, respectively. Single-frequency kinetic traces for all concentrations are well-fit by single-exponential decay functions from which we extract time constants shown in Table . At the two lowest concentrations studied (0.45 and 1.0 M), the ESA and GSB time constants differ by almost two picoseconds, whereas they converge at higher concentrations. The 1.0 M solution exhibits decay constants τ_ESA,1.0M ≈ 9.88 ps, τ_GSB,1.0M ≈ 8.07 ps for ESA and GSB, respectively. The concentration-dependent broadening is directly linked to the accelerated vibrational dynamics: ion pairing with ${\mathrm{NH}}_4^{+}$ and clustering of SCN^– anions enhance intermolecular interactions, opening additional vibrational energy relaxation pathways that shorten the vibrational lifetime. ,− The dissimilarity of ESA and GSB decay in the first few picoseconds (<5 ps) at low solute concentration could be ascribed to the presence of vibrationally hot ground state that also absorbs around the ESA region.

When placed inside of a resonant cavity, the transmission spectra show two peaks (UP and LP) of equal amplitude separated in frequency by the Rabi splitting, Ω _R , as shown in Figure and Figure a for different concentrations. The target cavity mode, ν_cav, can be identified beforehand by checking the transmission spectrum of a cavity filled only with DMF. Rabi splitting magnitudes are 38, 63, 76, and 95 cm^–1 for concentrations 0.45, 1.0, 2.2, and 3.5 M, respectively (Figure , all data shown in Figure S15). All magnitudes exceed both the cavity mode line width (fwhm_cav ∼ 13 cm^–1) and respective molecular absorption line width (19 cm^–1 to 35 cm^–1), confirming that the system fulfills the standard strong coupling criteria. For a collection of N identical molecules coupled to a single cavity mode, the normal-mode splitting follows the well-known $\sqrt{N}$ scaling, , often derived from the Tavis-Cummings Hamiltonian ,, but equally recovered from coupled-oscillator models ,−

$${\mathrm{\Omega }}_R\propto \sqrt{\frac{{\nu }_0|\mu {|}^2}{n_{\mathrm{b}}^2}\left(\frac{N}{V}\right)}$$ 1

where ν₀ is the resonance frequency, μ is the vibrational transition dipole moment, n _b is the background refractive index, N is the number of molecules, and V is the cavity mode volume. Here, the term N/V is effectively represented by the solute concentration, C. The relatively high Ω _R values can be attributed to the strong CN transition dipole moment, , together with favorable solvation of the SCN^– anion in DMF. − The linear dependence of Ω _R on $\sqrt{C}$ is clearly observed in our measurements, as shown in Figure b. We see the same trend when using DMSO as the solvent (Figure b). The slightly different slope values for the two different solvents may be attributed to the difference in the effective μ and n _b between the DMF- and DMSO-dissolved NH₄SCN, originating from different solvation environments.

3.

Concentration-dependent Rabi splitting in NH₄SCN under VSC. (a) Normalized cavity transmission spectra showing polariton peaks for the cavity-coupled 0.45 M (lighter purple line), 1.0 M (purple line), and 3.5 M (darker purple line) NH₄SCN in DMF at zero detuning. (b) Rabi splitting Ω _R vs square root of concentration $\sqrt{C}$ for the cavity-coupled NH₄SCN in DMF (purple dots) and DMSO (violet dots). Dashed lines show linear fits confirming ${\mathrm{\Omega }}_R\propto \sqrt{C}$ scaling.

The transient response of cavity-coupled SC^– exhibits features characteristic of all vibrational polariton systems studied to date. ,,, A derivative-like feature appears near the UP frequency, while a large absorptive feature appears near the LP frequency, as can be seen in Figures a–c for a resonant cavity and detuned cavity modes. Excitation of the reservoir population reduces the number of coupled ground-state molecules (N _eff), contracting the Rabi splitting $({\mathrm{\Omega }}_{\mathrm{R}}\propto \sqrt{N_{\mathrm{eff}}})$ and shifting the polariton peaks inward, which manifests as the derivative-like line shape in the differential transmission spectrum. − This would also be observed near the LP frequency if not for overlap of the transmissive LP peak with the ESA of the reservoir population. The negative portion of the UP feature becomes larger with respect to the absorptive peak near the LP frequency as the cavity mode is tuned from the red to the blue. The positive portion of the derivative also becomes slightly more pronounced.

4.

Pump–probe spectroscopy of NH₄SCN under VSC: (a) negative, (b) zero, and (c) positive detuning transient transmission spectra of the cavity-coupled 1.0 M NH₄SCN in DMF probed at 1, 3, and 5 ps after pump excitation. Corresponding cavity transmission spectra are shown directly above. (d–f) Time-resolved transient transmission spectra for each detuning condition, measured at the positive peak of LP and negative peak of UP.

Rabi splitting contraction can be used to infer the fraction of excited molecules, which is otherwise difficult to measure directly. Here, we study the contraction at four different concentrations of NH₄SCN in DMF and deduce trends. At 1 ps delay between the pump and probe pulses, we fit both the pump-on and pump-off polariton peaks at resonance (zero detuning) to Lorentzian functions and calculate the difference between their center frequencies. Using the above relationship between Ω _R and N, we estimate the fraction of depleted ground state molecules to be 2.1, 0.6, 0.3, and nearly 0% for the 0.45, 1.0, 2.2, and 3.5 M solutions, respectively (see the Supporting Information for details). The two polariton peaks at low concentrations overlap significantly with the v = 1 ← 0 vibration band. This allows the laser pulse to excite a larger fraction of molecules compared to the higher concentrations. Two factors contribute to the lower excitation fraction at higher concentrations. First, the polariton peaks are shifted further away from the absorptive maximum toward the tails. Second, the peak amplitudes decrease with increasing concentration, reducing the overall absorption cross section at the polariton frequencies. The pump energy is held constant at ∼2.4 μJ per pulse across all concentrations to minimize variation in excitation conditions.

The polariton modes appear robust to molecular inhomogeneity. In steady-state spectra, LP and UP line widths remain nearly constant at 25 ± 3 and 21 ± 3 cm^–1, respectively, across the concentration range studied, even as the molecular absorption broadens from 19 to 35 cm^–1. This robustness persists in the transient response. The absorptive feature near the LP frequencyarising from the difference between pump-on and pump-off polariton peaksmaintains a constant fwhm of 20 ± 1 cm^–1 at 1 ps delay across all concentrations, despite the ESA broadening from 55 to 68 cm^–1. The polariton line width remains stable whether the molecules are in the ground or excited state. ,,

Next we examine the nonlinear response of the polaritons. Figures a–c shows the transient transmission spectra of the cavity-coupled 1.0 M SCN^– in DMF solution at three cavity mode detuning conditions and at three pump–probe delay times (1, 3, and 5 ps). Detuning is defined as Δ = ν_cav – ν_vib, where Δ < 0 and Δ > 0 are termed negative and positive detuning (ND and PD), respectively. For the zero-detuned (ZD) sample, Δ ≈ 0, the large absorptive feature near the LP frequency appears near 2027 cm^–1. The derivative-like feature near the UP frequency has a strong negative peak around 2082 cm^–1. We recorded the single-wavelength kinetics at these two frequencies to obtain the traces shown in Figures d–f. We first examine kinetics at zero detuning, where the cavity mode is resonant with the vibrational mode. All traces are well-fit with single exponential decay functions after the instrument response. For 1.0 M at zero detuning, we find decay constants τ_LP = 6.4 ± 0.1 ps and τ_UP = 5.0 ± 0.2 ps, which are faster than the globally fitted noncavity time constant τ_global,1.0M = 9.56 ps (though this apparent acceleration is an artifact of spectral convolution, as discussed below). The concentration dependence mirrors the uncoupled systemfaster decay at higher concentrations (Table ). As noted by others, , interpretation is not straightforward because polariton signals are convoluted with the reservoir response, but the consistency suggests polariton relaxation is influenced by the same intermolecular interactions.

2. Decay Times for Cavity-Coupled Samples .

Concentration (M)	Detuning	Δ (cm–1)	τLP (ps)	τUP (ps)
0.45	ND	–12	9.8 ± 0.2	7.3 ± 0.3
	ZD	–5	10.2 ± 0.7	7.6 ± 0.8
	PD	11	9.1 ± 0.3	6.4 ± 0.2
1.0	ND	–18	5.6 ± 0.1	4.7 ± 1.1
	ZD	–5	6.4 ± 0.1	5.0 ± 0.2
	PD	17	6.5 ± 0.2	5.2 ± 0.3
3.5	ND	–41	2.3 ± 0.1	7.1 ± 3.0
	ZD	–11	2.9 ± 0.4	3.7 ± 1.9
	PD	23	3.1 ± 0.5	3.7 ± 1.8

^aDecay constants are obtained from single exponential fits to time-resolved traces after the instrument response time. Detuning conditions: ND = negative detuning (Δ < 0), ZD = zero detuning (Δ ∼ 0), PD = positive detuning (Δ > 0).

Next we examine detuned samples. By translating the sample laterally, we exploit spatial inhomogeneity in cavity length and small probe beam spot (∼200 μm) to tune the cavity mode frequency ν_cav away from ν_vib. For negative detunings, the LP is more transmissive than the UP as shown in Figure a. Positive detunings, shown in Figure c, result in a more transmissive UP peak. For 1.0 M concentration, as shown in Figure d,f, we find decay constants of τ_LP = 5.6 ± 0.1 ps and τ_UP = 4.7 ± 1.1 ps for negative detuning, and τ_LP = 6.5 ± 0.2 ps and τ_UP = 5.2 ± 0.3 ps for positive detuning. There appears to be a slight lengthening of relaxation times for both LP and UP when going from negative to positive detuning, as summarized in Table . This trend is consistent for both 1.0 and 3.5 M concentrations. However, the changes are small and, for UP measurements, mostly within uncertainty, so we hesitate to draw strong conclusions from this observation alone. Fully analyzing the temporal evolution of transient polariton spectra requires more sophisticated methods, which we present below.

Interpreting the temporal behavior of the coupled response turns out to be nontrivial. A simple analysis of the decay at single frequencies can be misleading. Time constants can vary significantly with the probing frequency and the spectral response is sensitive to small changes in refractive index of both the medium and cavity mirror materials. , More importantly, the coupled response, particularly near the excited-state absorptive feature, is a complex combination of the polariton transmissive peaks and the reservoir response such that simple subtraction of one from the other does not yield polariton-only excited-state information. − , For these reasons, a more careful analysis is required to extract physically meaningful information from the transient spectra. We adapt a recently introduced second-order kinetics approach, which is validated for W­(CO)₆ at resonance, and apply it here for the first time to detuned cavity conditions and to a system (NH₄SCN in DMF) with substantial inhomogeneous broadening. This approach represents the differential transmission:

$$\mathrm{\Delta }T(\nu ,t)=T_{\mathrm{cav}}(\nu ,A(t))-T_{\mathrm{cav}}(\nu ,A(t=0))$$ 2

as an expanded Taylor series with respect to the transition amplitude A(t) of the molecular excited-state transition:

$$\mathrm{\Delta }T(\nu ,t)={\frac{\partial T_{\mathrm{cav}}}{\partial A}|}_{t=0}A(t)+{\frac{1}{2}\frac{{\partial }^2{T}_{\mathrm{cav}}}{\partial A^2}|}_{t=0}{(A(t))}^2$$ 3

This is used to fit the transient spectra at multiple time delays simultaneously, allowing prediction of the excited-state population kinetics while accounting for nonlinearities in the response. For molecules with multiple excited-state transitions, the expansion generalizes to a multivariate form that can capture intermode correlations resolvable by multidimensional techniques such as 2D-IR. , For NH₄SCN, which exhibits a single excited-state transition (v = 2 ← 1), the expansion reduces to the scalar form given here. Time delays used are 1, 3, and 5 ps, where 5 ps is the latest time delay with sufficient signal-to-noise ratio. Details of the procedure are provided in the Supporting Information.

We perform this analysis on the transient spectra of NH₄SCN in DMF at 0.45, 1.0, and 3.5 M concentrations under three different cavity detuning conditions: ZD (on-resonance), ND, and PD. The extracted kinetics are summarized in Table and spectra for the 1.0 M sample are shown in Figure (corresponding 0.45 and 3.5 M data are in the Supporting Information). At ZD, the model extracts 11.8, 9.6, and 4.3 ps decay time constants for the 0.45, 1.0, and 3.5 M solutions, respectively, which agree with the global relaxation time constants, and the shapes of the spectral responses are captured well. It is clearly seen that the retrieved decay constants do not reflect the decay constant of the kinetic traces at UP and LP transitions given in Table . Figures d–f, shown below the overall spectra, illustrates the contributions from the first- and second-order terms. The first-order basis spectrum ∂T/∂A is reminiscent of a pair of out-of-phase derivatives of Lorentzian functions at the polariton frequencies with amplitudes that vary with perturbative ground state population changes. ,, In all cases the second-order contribution is negligible. This is expected for excitation fractions well below 10%, and most pump–probe experiments operate in this linear regime. A first-order-only fit yields time constants within 5% of the second-order values, and both are independently verified by a full analytical transmission fit (see the Supporting Information).

3. Extracted Population Decay Time Constants from Second-Order Kinetics Model .

		τ (ps)
System	Condition	0.45 M	1.0 M	3.5 M
Noncavity		11.73 ± 0.12	9.56 ± 0.03	4.58 ± 0.01
Cavity	ND	8.5 ± 2.7	5.7 ± 0.6	3.1 ± 0.3
Cavity	ZD	11.8 ± 2.9	9.6 ± 2.0	4.3 ± 0.4
Cavity	PD	7.3 ± 1.1	5.8 ± 1.0	3.2 ± 0.3

^aNon-cavity values are global fit time constants from single exponential fits to uncoupled transient absorption data. Cavity-coupled values are extracted from second-order kinetics model fits to transient transmission spectra at multiple time delays (1, 3, 5 ps). Detuning conditions: ND = negative detuning (Δ < 0), ZD = zero detuning (Δ ∼ 0), PD = positive detuning (Δ > 0).

5.

Second-order kinetic analysis of 1.0 M NH₄SCN in DMF at different detunings. (a–c) Transient transmission spectra at 1 ps delay (purple dots) for negative, zero, and positive detuning, respectively. Solid magenta lines show second-order kinetic model fits. (d–f) First-order (∂T/∂A) and second-order (∂ ² T/∂A ²) basis spectra for each detuning condition.

The time constants found for cases with the cavity detuned away from the molecular resonance are consistently shorter than the uncoupled values (Table ). Detuned lifetimes range from 7.3 to 8.5 ps, 5.7 to 5.8 ps, and 3.1 to 3.2 ps for concentrations of 0.45, 1.0, and 3.5 M, respectively. These values can be compared with the lifetimes of 11.7, 9.6, and 4.6 ps observed in non-cavity samples at the same concentrations. This observation is somewhat unexpected because we expect the late-time VSC response to exhibit similar kinetics to the bare molecule. , However, this may be understood as an effect of optical filtering or newly opened decay channels.

The central question motivating this work is how reservoir dynamics manifest in transient vibrational polariton spectra across a range of molecular and polaritonic conditions. Single-wavelength kinetics would initially seem to suggest that polariton relaxation is faster at higher concentrations, but this assessment is complicated and potentially misleading on three counts. First, the spectral response contains both early time polariton dynamics and the evolving molecular response filtered through the coupled mode structure. Population relaxation, spectral diffusion, , and the resulting Rabi splitting evolution all reshape the line shape on overlapping time scales, making it difficult to isolate individual contributions at any given frequency. Second, the probing frequency influences the apparent decay time constant because the time-dependent evolution of the line shape means that measurements at different frequencies sample different parts of the evolving spectrum. , Third, the standard logarithmic conversion from transmission to absorption assumes a Beer–Lambert exponential, but cavity transmission is an absorption-modulated Airy function; the mapping is not one-to-one, so direct comparison of coupled and uncoupled ΔA can be misleading. In order to extract meaningful kinetics from the coupled system, one must therefore account for the cavity optics and apply a model that captures the nonlinear response of the whole system, as done here.

The second-order kinetics model retrieves reservoir lifetimes at zero detuning that match the uncoupled molecular values across three concentrations spanning the homogeneous-to-inhomogeneous transition: 0.45, 1.0, and 3.5 M. Three independent approaches converge on these extracted lifetimes: the second-order kinetics framework, first-order kinetics only, and full Fabry–Perot transfer-matrix fitting all agree within 5%, and synthetic single-lifetime spectra confirm the method introduces no systematic bias at any detuning (see the Supporting Information). This validation extends the model, originally demonstrated for W­(CO)₆ in the homogeneously broadened limit, to a liquid-phase system with pronounced concentration-dependent inhomogeneous broadening. Having established that the model can reliably extract dark-state decay rates, we turn to a finding not accessible in prior work: polariton line width robustness across a range of molecular conditions. Across all four concentrations tested (including 2.2 M in zero-detuning studies), both steady-state peaks and the transient absorptive feature near LP maintain constant line widths of 20 ± 1 cm^–1 even as the molecular absorption and ESA broaden significantly with concentration. That the nonlinear absorptive feature is stable against molecular disorder indicates that the LP after broadband excitation is insensitive to inhomogeneous broadening of the v = 2 ← 1 transition despite a quite broad ESA peak in the molecular nonlinear spectrum. The line shape is also robust to detuning, but the apparent decay kinetics shift with off-resonance conditions, which we discuss below.

The detuned regime reveals a more complex picture. Both PD and ND yield faster apparent decay than uncoupled lifetimes. One possibility is spectral weighting by the cavity transmission function, the “optical filter” scenario analyzed by Schwennicke et al. The second-order kinetics model extracts an effective population amplitudeA(t) weighted by the cavity’s spectral transmittance, so that off resonance the extracted kinetics are dominated by molecular subpopulations near the absorption wings, which may have different inherent relaxation rates. , However, spectral weighting alone does not naturally account for the symmetric acceleration under positive and negative detuning. Subpopulations on opposite wings of the absorption band need not relax identically, yet the nearly equal lifetimes at ±Δ suggest a mechanism that depends on |Δ| rather than on its sign.

We interpret these observations through a unified picture in which the cavity modes open additional decay channels for the dark-state reservoir (Figure S17). , The total dark-state decay rate can be decomposed as

$${\mathrm{\Gamma }}_{\mathrm{DS}}={\mathrm{\Gamma }}_{\mathrm{DS}\to \mathrm{vib}}+{\mathrm{\Gamma }}_{\mathrm{DS}\to \mathrm{ph}}$$ 4

where Γ_DS→vib is the intrinsic vibrational relaxation rate (detuning-independent) and Γ_DS→ph is the cavity-enabled photonic decay channel. This photonic channel is opened by molecular disorder: inhomogeneous broadening partially mixes dark and bright character, giving nominally dark states a finite amplitude on the polariton modes. The photonic channel proceeds through both polariton branches, Γ_DS→ph = η_UPΓ_DS→UP + η_LPΓ_DS→LP, where the scattering rate from the dark-state manifold into each polariton branch scales as Γ_DS→UP/LP ∝ σ²/(ΔE)² with σ the inhomogeneous line width and ΔE the energy gap, and the photon leak rate scales as η_UP/LP ∝ |C _UP/LP|² the photonic Hopfield coefficient. The scattering rate and photonic weight are reciprocal: the polariton branch nearest the dark states has the largest scattering rate (ΔE small) but the smallest photonic weight (matter-like), while the distant branch has a smaller scattering rate but a larger photon leak rate (photon-like). At zero detuning, both branches are symmetrically offset from the reservoir and are moderately photonic, minimizing the net photonic channel. Under detuning, the interplay of enhanced scattering into the near branch and enhanced leakage through the far branch amplifies Γ_DS→ph. Because σ²/(ΔE)² and |C _UP/LP|² depend on |Δ| rather than its sign, the acceleration is approximately symmetric. See Figure S17 for the schematic of this symmetric decay behavior.

The observed symmetry is consistent with our area-based Hopfield analysis (see the Supporting Information), which shows that the experimental detunings are highly symmetric for the 0.45 and 1.0 M samples (Table and Table S1). The 3.5 M sample still shows acceleration despite its more inhomogeneous line shape and less symmetric detunings (Δ = −41 versus +23 cm^–1); the shorter uncoupled lifetime (∼4.6 ps) compresses the kinetic headroom, consistent with a dynamic-range effect. At high concentrations, 2D-IR spectroscopy can resolve distinct subensemblesfree ions and contact ion pairs ,, whose different relaxation dynamics may further modulate the detuning dependence. A definitive test requires frequency-resolved lifetime measurements such as 2D-IR spectroscopy under vibrational strong coupling; temperature-dependent measurements and molecular dynamics simulations could further elucidate the relative contributions of spectral weighting and cavity-enabled decay channels.

In conclusion, we systematically mapped transient dynamics of the SCN^– anion in DMF solution under vibrational strong coupling across Rabi splittings from 38 cm^–1 to 95 cm^–1 and multiple detuning conditions. By extending the second-order kinetics framework beyond the resonant, homogeneously broadened regime in which it was introduced, we show that retrieved reservoir lifetimes match uncoupled values at zero detuning even in a strongly inhomogeneously broadened system, but that off-resonance conditions yield consistently faster apparent decay. This detuning-dependent accelerationnot previously accessible because prior studies operated exclusively at resonanceis consistent with cavity-enabled photonic decay channels whose net rate is governed by a reciprocal interplay between dark-state scattering rates and polariton photonic weights, augmented by spectrally biased sampling of the molecular response by the detuned cavity transmission function. The absence of coherent Rabi oscillations is consistent with the system being near the disorder-limited delocalization threshold. These results demonstrate that detuning and molecular disorder are essential control parameters for interpreting transient polariton spectra and provide a systematic baseline for future ultrafast studies of vibrational polariton dynamics.

Methods

NH₄SCN crystalline powder is dissolved in N,N-dimethylformamide (DMF) (FUJIFILM Wako) or dimethyl sulfoxide (DMSO) (FUJIFILM Wako). A small drop is pipetted onto one side of a calcium fluoride (CaF₂) window and then the other window is placed on top to ensure no air bubbles form. Cavity samples are prepared similarly. Custom dichroic mirrors on Al₂O₃ substrate are designed to be reflective in the mid-infrared region of interest (ν = 2000 cm^–1 to 2400 cm^–1). Laser-cut Kapton spacers (12 μm thickness) set the nominal cavity length.

A Ti:sapphire regenerative amplifier (Legend Elite, Coherent) operating at 1 kHz repetition rate produces pulses near 800 nm to pump an optical parametric amplifier (OPERA, Coherent). The latter outputs near IR signal (∼1.35 μm) and idler (∼1.88 μm) pulses. MIR pulses with center wavelength ∼4.8 μm are produced by difference frequency generation between signal and idler pulses in a type II AgGaS₂ crystal. A low-pass filter (<3 μm) after the crystal blocks the outgoing signal and idler pulses. The MIR pulses are separated into pump (∼90%) and probe (∼10%) paths using a CaF₂ window coated with a thin film of gold. The pump is sent to a delay line (stage: KX0725-LG5; controller: D250, Suruga Seiki) and chopped (3501, New Focus) at 500 Hz. A λ/2 waveplate and polarizer combination is used to modulate the pump intensity and polarization with respect to the probe for magic angle measurements (54.7°). The pump and probe pulses are focused using f = 80 mm and 50 mm CaF₂ lenses, respectively, and spatially overlapped in the sample. The sample is held in a demountable flow cell (Harrick Scientific) at room temperature, mounted on a rotated stage so that pump and probe beams share the same incident angle and thus probe the same cavity mode. The transmitted signal is then collimated by an off-axis parabolic (OAP) mirror and then refocused by another OAP mirror before it enters the spectrometer (MS3501i, Solar TII). Inside the spectrometer, the signal is dispersed by a grating (100 L/mm, blaze = 4.16 μm) and then detected with a mercury cadmium telluride (HgCdTe) detector (FTIR-13-1.0, Infrared Associates) at liquid nitrogen temperature. A Fourier-transform infrared (FTIR) spectrometer (FT/IR-4600, JASCO) is used to perform steady-state infrared spectroscopy.

Steady-state FTIR spectra of NH₄SCN solutions were corrected by subtracting a DMF solvent reference spectrum, interpolated linearly to match the sample wavenumber grid. A linear baseline correction was applied using anchor points at 1900 and 2200 cm^–1 to remove residual background contributions. The CN stretch band was fit to a Lorentzian function to extract peak position and full-width at half-maximum (fwhm).

For global analysis, traces were fit simultaneously with a shared lifetime τ while allowing independent amplitudes and offsets for each channel. Uncoupled transient spectra were averaged over multiple scans, and reported time constants reflect experimental reproducibility across repeated measurements.

Rabi splitting was determined from pump–probe transmission spectra by fitting the LP and UP peaks to Lorentzian functions and calculating the frequency difference. Full-width at half-maximum of polariton transient features was determined by Savitzky-Golay smoothing followed by linear interpolation to identify half-maximum crossing points.

Nonlinear least-squares fitting was performed in Julia using the SciML ecosystem (CurveFit.jl, NonlinearSolve.jl). Parameter uncertainties were estimated as standard errors derived from the covariance matrix of the least-squares fit. For the second-order kinetics analysis of cavity-coupled samples, uncertainties are fitting errors rather than experimental reproducibility, as reproducing exact detuning conditions across experiments is difficult. Visualization uses Makie.jl.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpclett.6c00381.

• Additional details of the second-order kinetics model, characterization of the target cavity mode, thiocyanate (SCN^–) absorption and derivative spectroscopy analysis in DMF across different concentrations, global kinetic fits of the uncoupled NH₄SCN in DMF, anharmonicity vs solute concentration, angle-dependence FTIR measurement of NH₄SCN in DMF, photonic-matter fractions and cavity mode estimation, cavity-coupled transient and time-resolved spectra of 2.2 M NH₄SCN in DMF under zero-detuning, cavity-coupled spectra of 0.45 and 3.5 M NH₄SCN in DMF across different detuning conditions, second-order kinetics of the cavity-coupled 0.45 and 3.5 M NH₄SCN in DMF, second-order kinetic model validation via first-order and full analytical transmission fits, calculation of ground-state population reduction across different concentrations, LP transient line width analysis, detuning-dependent dark-state relaxation pathways, and validation of kinetics extraction via synthetic data (PDF)

The authors declare no competing financial interest.