Ultrafast vibrational excitation transfer on resonant antenna lattices revealed by two-dimensional infrared spectroscopy

Vibrational polaritons emerge when molecular vibrational transitions strongly interact and hybridize with the confined photonic modes of the infrared cavity.^1–4 Strongly coupled systems are typically discussed within a framework of upper polariton (UP) and lower polariton (LP) states and a large number of reservoir states, which are treated as uncoupled molecules because of their predominantly localized character.^5,6 Vibrational polaritons are envisioned to revolutionize multiple fields of chemistry and physics;^7–10 however, some related aspects are actively debated.^11–14 Recent time- and frequency-resolved pump–probe¹⁵ and two-dimensional infrared (2DIR) spectroscopy¹⁶ experiments conducted with vibrational polaritons in Fabry–Pérot-style photonic cavities revealed the transfer of the excitation from the polaritons to the reservoir,^17,18 where the excitation energy can relax to the ground state via the intramolecular pathways. Because of the differences in the final density of states (DOS), the transfer rate in the opposite direction, namely, from the reservoir to polaritons, is expected to be N times smaller, where N is the number of coupled molecules.^5,6,19,20 The corresponding transfer in vibrational polariton systems was not unambiguously identified experimentally so far,^18,21 which is in contrast with polaritons involving molecular excitons in the visible spectral region, where ultrafast relaxation from the reservoir to polaritons is well known.^22,23

In experiments with infrared Fabry–Pérot cavities, the spectroscopic windows available for signal observation in the spectral region around the polaritonic transitions are limited to transmission through the polariton resonances themselves, which have a typical bandwidth of 10–20 cm⁻¹. In addition, excited state absorption of the reservoir molecules can overlap with the lower polariton-to-ground state transition.¹⁶ Under such experimental conditions, detection of potentially weak signals involving the direct excitation of the reservoir modes at the transition frequency where the system is opaque can be challenging.^18,21 To overcome these limitations, we used an alternative cavity design that utilizes the special optical properties of the two-dimensional arrays of plasmonic nanostructures.^24,25 Such arrays can be designed to possess narrowband lattice resonances, where the incident light is diffracted along the surface at the grazing angle.²⁶ Here, the optical extinction is associated with the resonant excitation, whereas the nearby spectral region remains transparent. The light propagating along the surface interacts with the molecules deposited on the array. Indeed, such nanostructure lattices, referred to as open cavities, have been successfully used to fabricate polaritonic systems with lattice resonances strongly coupled to molecular excitons.²⁷ Recently, we extended the open cavity design to the mid-infrared spectral region and demonstrated strong coupling between the lattice resonances of infrared antenna arrays and the carbonyl stretching (CO) vibration of the (poly)methyl methacrylate (PMMA) polymer molecules spin-coated on the antennas.²⁸ 

In this work, we studied the ultrafast dynamics of vibrational polaritons within an open cavity using 2DIR spectroscopy. We fabricated an antenna array by electron-beam lithography on a CaF₂ window with individual antenna dimensions of 0.9 × 0.2 × 0.07 µm³ (length, width, and height), a longitudinal array period (along the antenna axis) of D_L = 1.4 µm, and a transverse period of D_T = 4.15 µm.^29,30 A scanning electron micro-image of the array is shown in Fig. 1(a), and the corresponding linear spectrum collected at the normal incidence is shown in Fig. 1(b). In order to match the refractive indices of the substrate and environment, the sample was immersed in carbon tetrachloride (CCl₄). In CCl₄, the antenna-lattice resonance (ALR) of the array appears at the frequency $ν̄g,ALR$ = 1653 cm⁻¹ and has a quality factor of Q = 150. Here, the index “g” represents the ground state of the system. A 400 nm-thick layer of PMMA was spin-coated on the array, leading to hybridization of the ALR with the CO vibration of PMMA (⁠$ν̄g,co$ = 1730 cm⁻¹) and the formation of upper (⁠$ν̄g,UP$ = 1767 cm⁻¹) and lower (⁠$ν̄g,LP$ = 1688 cm⁻¹) polaritons, as shown in Fig. 1(c). Note that because a thin polymer layer has an effective refractive index different from that of the bulk CCl₄, the ALR transition undergoes a blue shift compared to that in CCl₄.²⁸ Fitting to a 2 × 2 Jaynes–Cummings model reveals that in PMMA the ALR is slightly detuned from the CO transition with $ν̄g,ALR$ = 1725 cm⁻¹. The Rabi splitting of 79 cm⁻¹ observed in the spectrum significantly exceeds the bandwidth of both the ALR (Δ_ALR = 11 cm⁻¹) and CO (Δ_CO = 22 cm⁻¹) transitions. An additional transition that appears at the $ν̄g,co$ frequency in the spectrum is associated with a molecular reservoir as well as with entirely uncoupled molecules.^21,31

Our 2DIR setup involves three infrared femtosecond laser pulses focused on the sample in the square-like boxcar geometry, making the polar angle θ and the azimuthal angle φ = 45° with respect to the normal to the plane of the array, as shown in Fig. 2(a). It is useful to discuss such experiments within the concept of dynamic gratings.³² First two laser pulses excite two polariton waves, which interfere and create a grating, whereas the third beam diffracts off the grating, generating nonlinear emission into the phase-matched direction.^33,34 Because at such an experimental configuration the polariton dispersion plays an important role, we collected linear spectra of the coupled system at various θs, which is equivalent to varying the size of the boxcar’s square. The results of these measurements, shown in Fig. 2(b) as a color map, illustrate the dispersion of the polariton states.

In order to better understand polariton dispersion, we first calculated the ALR dispersion using the so-called dipole sum approach, which is extensively employed to calculate the spectra of plasmonic arrays (see the supplementary material for details).^4,24,27 The resulting ALR transition frequencies, $ν̄g,ALR0,±1θ$⁠, are shown by white dotted lines in Fig. 2(b). Using the obtained values of $ν̄g,ALR0,±1θ$⁠, we calculated the polariton dispersion numerically with the Tavis–Cummings-like model Hamiltonian³⁵ (see the supplementary material for details), where we accounted for both the inhomogeneous distribution of the molecular transition frequencies (approximated as Δ_CO) and the distribution of the coupling strength constants obtained using the method described in Refs. 28 and 36. Because the inhomogeneity of the molecular transitions in the disordered polymer lifts the dark character of the reservoir modes,³⁷ they appear around the $ν̄g,CO$ frequency as a dispersionless band. The calculated frequencies of the polariton states, which are in good agreement with the experimental results, are denoted as black circles and squares in Fig. 2(b).

As expected, when the degeneracy of the ALR modes is lifted for the off-normal incidence, four polariton states are obtained. As shown in Fig. 2(b), the four polaritons behave differently: one of the two states emerging from LP (θ = 0) has a strong transition dipole moment and its frequency asymptotically approaches the $ν̄g,ALR0,−1$ dispersion curve for large θs, whereas the transition dipole moment of the second state vanishes and its frequency converges to that of the reservoir modes. We refer to these states as bright LP (LPb) and dark LP (LPd). Analogously, the frequency of one of the two states emerging from UP(θ = 0) asymptotically approaches the $ν̄g,ALR0,+1$ dispersion curve (UPb), whereas the frequency of the second state converges to the reservoir frequency (UPd). However, unlike the LPd state, the UPd state has a non-vanishing transition dipole moment for small θs, as shown in Fig. 2(c) for θ = 4°.

Inspection of the composition of the eigenstates, whose calculated frequencies are presented in Fig. 2(b), shows that a higher degree of hybridization occurs for small θs. For larger θ values, the ALR and the $ν̄g,co$ frequencies detune and the admixtures of the molecular component into polaritons are reduced. In our 2DIR setup, θ ∼ 4° [see Fig. 2(c)], which results in Hopfield coefficients that are summarized in Table I (see the supplementary material for details). We found that LPb is composed of a mixture of the $ν̄g,ALR0,−1$ and $ν̄g,CO$ modes, whereas LPd has a predominantly molecular character. Reservoir modes involve various mixtures of ALR modes and $ν̄g,CO$ modes. The UPd state has a molecular character, whereas the UPb state is a mixture of the $ν̄g,CO$ and $ν̄g,ALR0,+1$ modes.

2DIR spectroscopy³⁸ was conducted with sequences of three ∼70 fs laser pulses having a central wavelength of 5.8 μm; the pulses were focused on the sample with a spot size of ∼150 μm. More details about our setup are given in Refs. 39 and 40. We systematically scanned the so-called coherence time interval between the first two excitation pulses and probed the excited molecules with the third pulse, which arrived after the waiting time interval T. The emitted third-order signal is heterodyned with the local oscillator pulse and is spectrally resolved on the array detector. The latter measurement generates the detection frequency axis of the 2D spectrum, whereas the excitation frequency is obtained by the Fourier transformation of the collected dataset with respect to the coherence time interval. The absolute-value absorptive spectrum⁴¹ of the coupled system collected with the excitation laser pulses incident at θ ∼ 4° and at waiting time T = 0.3 ps is shown in Fig. 3 along with the corresponding absolute-value rephasing and non-rephasing spectra.⁴² The data feature three diagonal peaks at the transition frequencies of $ν̄g,UPd=1745$ cm⁻¹, $ν̄g,CO=1730$ cm⁻¹, and $ν̄g,LPb=1680$ cm⁻¹. A strong cross-peak with the excitation frequency of $ν̄g,CO$ and the detection frequency of $ν̄g,LPb$ is seen in the spectrum.

In order to understand the character of the CO/LPb cross-peak shown in Fig. 3, we numerically diagonalized the Hamiltonian accounting for the second tier states (see the supplementary material)^35,43 and examined possible pathways of the excited state absorption from the reservoir. We obtained that signals involving states with energy approaching that of the doubly excited reservoir³⁵ (⁠$ν̄CO,2CO$ = 1707 cm⁻¹) do not overlap the spectral region of the CO/LPb cross-peak, as shown in Fig. S1 of the supplementary material. In addition, in agreement with Ref. 43, we obtained new possible pathways of the excited state absorption from the reservoir to two doubly excited states located at ca. $ν̄g,CO+ν̄g,LPb$ and ca. $ν̄g,CO+ν̄g,UPd$ above the ground state such that the corresponding signals are expected to appear with the excitation frequency of $ν̄g,CO$ and the detection frequencies of $∼ν̄g,LPb$ and $∼ν̄g,UPd$⁠, respectively. Because in cases where these pathways lead to observable signals, both CO/LPb and CO/UPd peaks are expected to appear in the spectrum, whereas only a former peak is seen in Fig. 3, we ruled out the corresponding excited state absorption as a possible origin of the sole CO/LPd cross-peak.

Therefore, we assigned the CO/LPd cross-peak to the stimulated emission from the LPb state, which follows the excitation of the reservoir, as well as to absorption into the doubly excited LPb state with $ν̄LPb,2LPb$ = 1677 cm⁻¹. The corresponding third-order signals appear out of phase and interfere destructively. However, because of the anharmonicity of LPb provided by the molecular components and the difference in the transition dipole moments of the $ν̄g,LPb$ and $ν̄LPb,2LPb$ transitions, the overall cross-peak signal does not vanish.^44,45 For larger detuning between the ALR and CO transitions, molecular contributions to the LPb decrease. This leads to smaller anharmonicity of the LPb state and weaker cross-peak signals, as illustrated in Fig. S2 of the supplementary material.

There are two possible scenarios for the appearance of the CO/LPb cross-peak: (1) direct coupling between the reservoir and LPb modes and (2) transfer of the excitation energy between the two.^46–48 In the former case, however, one expects that both CO/LPb and LPb/CO cross-peak signals will be present. Therefore, because the LPb/CO cross-peak does not appear in our data (Fig. 3), we attribute the CO/LPb cross-peak to the unidirectional excitation energy transfer from the reservoir to LPb. Such a transfer can be described by the kinetic scheme shown in Fig. 3(d), which illustrates possible excitation and relaxation pathways of the system. Here, for the sake of simplicity, the manifold of the fundamental reservoir excitations is represented by a single state $R$⁠.

In order to obtain more information on the coupled system, we examined the time evolution of the diagonal and cross-peak amplitudes obtained from a series of the rephasing 2DIR spectra collected at different waiting times. In these experiments, relatively weak excitation laser pulses (pulse energy of 0.2 μJ) were used. Fitting the peak amplitude data to exponential decay functions revealed that the diagonal peaks corresponding to $ν̄g,UPd$⁠, $ν̄g,CO$⁠, and $ν̄g,LPb$ are well-described by a single exponential decay with the time constants of τ_UPd = 0.86 ± 0.11 ps, τ_CO = 0.78 ± 0.06 ps, and τ_LPb = 0.87 ± 0.13 ps, respectively, as shown in Fig. 4. As expected, reservoir states decay with the same time constant as the uncoupled CO modes in the bare polymer film (⁠$τCObare$ = 0.79 ± 0.04 ps). A decay time constant of UPd is similar to that of the reservoir because its composition is dominated by the molecular modes, as discussed above. A decay time constant of LPb matches the bandwidth of the corresponding resonance. We found that the time the wave packet takes to escape the region of the sample monitored in the experiment is longer since the group velocity is $vg=dω/dkθ=4°≈$ 35 µm/ps, which we determined by fitting the experimental data of the dispersion relations in Fig. 2(b) with a parabolic function.⁴⁹ Therefore, the decay of the LPb diagonal peak is determined by the dissipation of the excitation and not by the wave-packet transport.

In contrast to the diagonal peaks, the CO/LPb cross-peak signal has a bi-exponential decay with a fast time constant of ∼100 fs and a slow time constant of ∼0.8 ps. In order to rationalize this observation, we assumed that an impulsive excitation of the reservoir, $R$⁠, is followed by two different relaxation channels: the first channel involves a direct decay to the ground state $g$ with the characteristic time constant τ_CO, whereas the second channel involves the transfer of the excitation to $LPb$ with the time constant τ_CO/LPb. LPb decays with the time constant τ_LPb. The corresponding kinetic scheme is shown in Fig. 3(d). The reversible exchange of the excitation energy between the $LPb$ and $LPd$ states is also accounted for, with the time constants τ_LPb/LPd ≈ τ_LPd/LPb (see Fig. 5). Solving the associated system of rate equations⁴⁶ under the assumption that τ_CO/LPb, τ_LPb/LPd ≪ τ_LPb ≈ τ_LPd, which appears to hold in view of our experimental observations, we obtained the following expression for the waiting time dependence of the CO/LPb cross-peak amplitude:

Fitting the experimental data to the functional form of $SCO/LPbT$ results in the value of τ_LPb/LPd = 0.22 ± 0.11 ps with a₁ = 0.62 ± 0.31 and τ_LPb = 0.80 ± 0.12 ps with a₂ = 0.38 ± 0.20, as shown in Fig. 4(c). The τ_LPb value obtained from this fit matches τ_LPb = 0.87 ± 0.13 ps that we determined from the diagonal peaks’ dynamics (see the above discussion).

Interestingly, we found that increasing the excitation pulse energy affects the ultrafast dynamics of the coupled system. First, we observed that unlike the diagonal peak whose amplitude scales linearly with the excitation pulse energy within the range accessible in our experiments (up to ∼0.8 μJ), the amplitude of the CO/LP cross-peak saturates for pulse energies above ∼0.3 μJ, as shown in Fig. 4(d). Note that our attempts to obtain a 2DIR signal from the ALR in the absence of molecules were unsuccessful. We attribute this to the harmonic character of the ALR as it is well-known that harmonic potentials do not generate third-order 2DIR signals.³⁸ Therefore, the saturation of the cross-peak amplitude is consistent with the limited contribution of the molecular modes to LPb such that the associated transition is saturated much earlier than that of the diagonal CO peak, which, in addition to the reservoir states of the coupled system, also involves transitions of the uncoupled molecules.^21,28,31 We consistently observed that the diagonal LPb peak saturates at pulse energies similar to those of the CO/LPb cross-peak, whereas the diagonal UPd peak, which has a predominantly molecular character, does not saturate at the pulse energies used in our experiments. Furthermore, we found that, for increased energies of the excitation pulses, the waiting time evolution of the cross-peak amplitude changed, compared with that obtained for excitation with weak pulses, as shown in Fig. 4(c) for 0.7 μJ. This is in contrast to the relaxation dynamics of the diagonal peaks of CO and both polaritons, which decay at the same rates as for the weak pulse excitation. The waiting time dependence of the diagonal CO peak for the excitation with 0.7 μJ pulses, shown in Fig. 4(a), is indistinguishable from that of excitation with a 0.2 μJ pulse.

An increase in the excitation pulse energy leads to the sequential population of the higher excited states of the quantum system, referred to as ladder climbing,^50–53 as illustrated in Fig. 5. This process occurs only in molecules strongly coupled to the ALR modes, as they experience enhanced near-fields around infrared antennas.^53,54 In arrays used in the present experiments, we estimated a tenfold effective near-field enhancement.²⁸ Such non-perturbative excitation, which is followed by relaxation of the excited population down the vibrational ladder, can delay the transfer of the excess energy from the reservoir to LPb, leading to a change in the relaxation dynamics observed in the experiment. In order to assess the efficiency of such a process, we resorted to quantum mechanical simulations of the multilevel system, which included an anharmonic vibrational ladder of the CO excitations up to the n = 10 level and LP modes. The dynamics of the open system was evaluated by numerically solving the Lindblad equation,⁵³ as described in detail in the supplementary material.

Solving the Lindblad equation results in dynamics that reproduce well the experimental observations. The calculated population of the LPb state shows the effect of saturation with increasing pulse energy, as denoted by a red line in Fig. 4(d). The calculated waiting time evolution of the LPb state population, after excitation with weak and strong laser pulses, is shown in Fig. 5. In both cases, the relaxation of the LPb state is bi-exponential with a fast time constant of 0.1 ps and a slow time constant of 0.8 ps, which reflect the relaxation rates used in the model. However, the contributions of the exponential components to the obtained decay curves are different in two cases: a₁ = 0.55 and a₂ = 0.45 for the fast and slow components, respectively, with weak excitation pulses, whereas a₁ = 0.45 and a₂ = 0.55 with strong excitation pulses. Such a change in the relaxation dynamics obtained with the theoretical calculations qualitatively agrees with the experimental results in Fig. 4(c), where the contribution of the exponential components a₁ = 0.62 and a₂ = 0.38 for the 0.2 μJ excitation pulses changed to a₁ = 0.33 and a₂ = 0.67 for the 0.7 μJ excitation pulses.

The agreement between the experimental data and the calculations validates the proposed kinetic model and allows us to discuss the corresponding results. First, we noted that, within such a model, observing the bi-exponential relaxation dynamics of the LPb state requires that there is no direct excitation transfer from the reservoir to LPd. This suggests that the reservoir to the LPb transfer mechanism involves a dipole–dipole interaction, facilitated by the non-dark character of the reservoir. In addition, analysis of the simulation results reveals that because we assumed that unidirectional transfer from the reservoir to the LPb state occurs on the ultrafast time scale, after the strong pulse excitation, the population of the reservoir integrated over all the levels included in the model is relatively low, ∼0.15, whereas the combined population of LPs is ∼0.35, which places the present experimental conditions very close to the regime of population inversion.⁵³ 

From considerations regarding the final DOS, the ratio of rate constants for the reservoir-to-polariton and polariton-to-reservoir population transfer is expected to scale with the number of coupled molecules: assuming k_R/P ∼1, one expects that k_P/R ∼ N.^5,6,20 In our experiments, we estimated that N ∼ 1 · 10⁹ per antenna with ∼1000 antennas being simultaneously measured. The volume occupied by the coupled molecules obtained from electromagnetic numerical simulations is shown in Fig. S4 of the supplementary material. The sole observation of the R/LPb cross-peak suggests that k_R/LPb is significantly larger than expected, e.g., k_R/LPb ∼ N², such that the reservoir-to-polariton rate overcomes the 1/N scaling of the DOS and outperforms the polariton-to-reservoir transfer. We hypothesize that our results can be rationalized by recalling that, in the strongly coupled systems, ultrafast excitation of the nearly degenerate manifold can prepare correlated quantum states, which feature collective dynamics.⁵⁵ The reservoir-to-polariton transfer with a rate suggesting such a collective behavior appears to be somewhat analogous to Dicke superradiance,⁵⁶ where preparation of a correlated state of N emitters leads to a spontaneous emission rate scaling as ∼N². In this case of the collective radiationless transfer, one may consider the coherence-to-population transfer process,⁵⁷ where N² coherences can be available, when excited within the nearly degenerate manifold of the correlated reservoir states. More work is being conducted to validate this hypothesis.

To summarize, we employed 2DIR spectroscopy to investigate the dynamics of the vibrational polaritons and reservoir states in an infrared open cavity. The cross-peak, associated with the excitation of the reservoir and detection of the LP, indicates a unidirectional transfer of excitation between these states. In addition, the ultrafast energy exchange between two non-degenerate bright and dark LPs was identified with a characteristic time constant of 200 fs. Increasing the intensity of the excitation laser pulses leads to saturation of the cross-peak signal, indicating strong excitation non-linearity. The quantum-mechanical modeling of the experimental results reveals vibrational ladder climbing in the anharmonic levels of the reservoir and reproduces all the observations, including the saturation of the LP population and a change in the time dependence of the relaxation dynamics. We envision that the efficient unidirectional process of the reservoir-to-polariton transfer will play an important role in the future development of room temperature quantum optical devices in the mid-infrared region.

See the supplementary material for additional experimental data and a detailed description of the theoretical models.

This research was supported by the Israel Science Foundation (Grant No. 293/20), the United States–Israel Binational Science Foundation (Grant No. 2020009), and The Helen Diller Quantum Center, Technion. Infrared antenna arrays were prepared at the Micro- and Nano-Fabrication Unit with support from the Russell Berrie Nanotechnology Institute, Technion. L.C. acknowledges enlightening discussions with Professors Uri Peskin (Technion) and Wei Xiong (UCSD).