Abstract
Conventional approaches to probing ultrafast molecular dynamics rely on the use of synchronized laser pulses with a welldefined time delay. Typically, a pump pulse excites a molecular wavepacket. A subsequent probe pulse can then dissociate or ionize the molecule, and measurement of the molecular fragments provides information about where the wavepacket was for each time delay. Here, we propose to exploit the ultrafast nuclearpositiondependent emission obtained due to large light–matter coupling in plasmonic nanocavities to image wavepacket dynamics using only a single pump pulse. We show that the timeresolved emission from the cavity provides information about when the wavepacket passes a given region in nuclear configuration space. This approach can image both cavitymodified dynamics on polaritonic (hybrid light–matter) potentials in the strong light–matter coupling regime and baremolecule dynamics in the intermediate coupling regime of large Purcell enhancements, and provides a route towards ultrafast molecular spectroscopy with plasmonic nanocavities.
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Introduction
The interaction of light and matter is one of the most fundamental ways to unveil the laws of nature and also a very important tool in the control and manipulation of physical systems. When a confined light mode and a quantum emitter interact, the timescale for the energy exchange between both constituents can become faster than their decay or decoherence times, and the system enters the strong coupling regime^{1,2,3}. In this regime, the excitations of the system become hybrid light–matter states, the socalled polaritons, separated by the vacuum Rabi splitting Ω_{R}. Due to their relatively large dipole moments and large excitonbinding energies, strong coupling can be achieved with organic molecules at room temperature down to the few or even singlemolecule level^{4,5,6}. Strong coupling can lead to significant changes in the behavior of the coupled system, affecting properties such as the optical response^{3,4,5,6,7,8,9,10,11,12}, energy transport^{13,14,15,16}, chemical reactivity^{17,18,19,20,21,22,23,24,25}, and intersystem crossing^{22,26,27}. However, up to now these setups did not provide direct information on the molecular dynamics.
A wellknown approach to directly probe molecular dynamics is through the use of ultrashort coherent laser pulses, pioneered in the fields of femtochemistry^{28} and attosecond science^{29}. This allows to observe and control nuclear and electronic dynamics in atoms and molecules at their natural timescale (fs and subfs), and is a fundamental tool towards a better understanding of chemical and electronic processes^{28,29,30,31,32,33,34}. In particular, realtime imaging of molecular dynamics can be achieved in experiments with a pump–probe setup with femtosecond resolution combined with the measurement of photoelectron spectra^{31}. Although similar approaches could, in principle, provide a dynamical picture of molecules under strong light–matter coupling^{35,36,37}, common molecular observables (such as dissociation or ionization yields, or photoelectron spectra) are difficult to access in typical experimental setups, with molecules embedded in a solidstate matrix and confined within nanoscale cavities^{4,5,6}. Another powerful approach is given by transient absorption spectroscopy, where the change of the absorption spectrum of a probe pulse is monitored as a function of time delay after a pump pulse. Although this can provide significant insight about molecular dynamics^{38}, the interpretation of the spectra is nontrivial due to the competition between several distinct effects (such as groundstate bleach, stimulated emission, and excitedstate absorption) in the spectrum^{39}, such that transient absorption spectroscopy only gives an indirect fingerprint of the molecular dynamics.
In this study, we demonstrate that the ultrafast emission induced by strong coupling to plasmonic modes can be used to monitor molecular wavepacket dynamics by measuring the timeresolved light emission of the system after excitation by an ultrashort laser pulse, without the need of a synchronized probe pulse. Our approach exploits the fact that the light–matter hybridization in a molecule is nuclearpositiondependent. Consequently, efficient emission only occurs in regions where the polaritonic potential energy surface (PoPES)^{18,40} on which the nuclear wavepacket moves possesses a significant contribution of the cavity mode, as sketched in Fig. 1. In addition, due to the very low lifetime (or, equivalently, low quality factor) of typical plasmonic nanocavity modes on the order of femtoseconds, emission from the cavity also becomes an ultrafast process. Instead of using a probe pulse to learn where the nuclear wavepacket is at a given time delay, we thus use the nuclearpositiondependent emission to learn when the wavepacket passes a given spatial region. Tracking the timedependent emission from the cavity then gives direct information about the nuclear dynamics by effectively clocking the time it takes the wavepacket to perform a roundtrip in the PoPES through an alloptical measurement. We note that a variety of experimental techniques allow the measurement of timedependent light pulses with fewfemtosecond resolution, e.g., intensity crosscorrelation^{41}, SPIDER^{42}, FROG^{43}, or dscan^{44}.
Results
Single molecule
We first illustrate these ideas using a minimal model system: a singlemode nanocavity containing a molecule with two electronic states and a single vibrational degree of freedom, which for simplicity we approximate as a harmonic oscillator (with displacement between the ground and excited state due to exciton–phonon coupling). Our model is then equivalent to the Holstein–Jaynes–Cummings model that has been widely used in the literature to model strongly coupled organic molecules^{19,45,46,47}, with the main difference that we explicitly treat cavity losses and driving by an ultrashort (fewfs) laser pulse, and monitor the timedependent emission. Although this is a strongly reduced model that allows for a straightforward interpretation, we will later show that the results we observe are also obtained in realistic simulations of molecules with a plethora of vibrational modes leading to rapid dephasing^{10}. The system is described by the Hamiltonian (setting ℏ = 1)
where σ^{+} (σ^{−}) is the raising (lowering) operator for the electronic state with excitation energy ω_{e} = 3.5 eV, whereas p and q are the massweighted nuclear momentum and position operators for the vibrational mode with frequency ω_{v} = 0.182 eV and exciton–phonon coupling strength λ_{v} = 0.192 eV (with these parameters, we reproduce the properties of the anthracene molecule; see Methods for further details). The cavity is described through the photon annihilation (creation) operators a (a^{†}), with a photon energy chosen on resonance with the exciton, ω_{c} = ω_{e}. In addition to the coherent dynamics described by the Hamiltonian, the cavity mode decays with rate γ_{c} = 0.1 eV, described by a standard Lindblad decay operator (see Methods for details). The photon–exciton coupling is described through the Rabi splitting at resonance, Ω_{R} = 2E_{1ph}(r_{m}) ⋅ μ_{eg}, where E_{1ph}(r_{m}) is the quantized mode field of the cavity at the molecular position and μ_{eg} is the transition dipole moment of the molecule (in principle, this is qdependent, but is taken constant here for simplicity). Finally, the cavity mode is coupled through its effective dipole moment μ_{c} to an external (classical) laser pulse \(E(t)={E}_{0}\cos ({\omega }_{{\rm{L}}}t)\exp ({\sigma }_{{\rm{L}}}^{2}{t}^{2}/2)\), with central frequency ω_{L}, spectral bandwidth σ_{L}, and a corresponding duration of ≈1.67∕σ_{L} full width at half maximum (FWHM) of intensity. We note that as the cavity mode is driven by the external field, the effective pulse felt by the molecule (in particular in the weak coupling limit) is slightly distorted and not just given by E(t).
We start by analyzing the system response in the strong coupling regime (Ω_{R} = 0.4 eV) after excitation by an ultrashort laser pulse with σ_{L} = 0.1 eV, while scanning the laser frequency ω_{L}. For σ_{L} = 0.1 eV, the duration of the pulse is ≈11 fs. The laser intensity is chosen small enough to remain in the singleexcitation subspace (i.e., within linear response). The instantaneous radiative emission rate from the cavity is given by E_{R} = γ_{c,r}〈a^{†}a〉, where γ_{c,r} is the radiative decay rate of the cavity excitations. As it corresponds to a constant (systemdependent) factor, we set it to unity in the figures shown in the following. Estimates of the achievable photon yields in realistic systems are given in the Discussion section. In Fig. 2, the timedependent radiative emission intensity E_{R} and the exciton population 〈σ^{+}σ^{−}〉 are shown. We observe that when the laser pulse is resonant with the lower polariton region, i.e., for ω_{L} between 3.2 and 3.5 eV, the cavity emission is modulated in time with a period of around 26 fs, whereas no such oscillation is observed when the upper polariton branch is excited for ω_{L} between 3.5 and 3.8 eV. This behavior can be understood with the help of the PoPES, shown in Fig. 1. They are obtained by treating nuclear motion within the Born–Oppenheimer approximation, i.e., with q treated as an adiabatic parameter (see Methods for details). Within the Franck–Condon approximation, shortpulse excitation creates a copy of the vibrational ground state (centered at q = 0) on the relevant polaritonic PES. This vibrational wavepacket will then evolve on the potential surface, performing oscillatory motion, with the character of the wavepacket also oscillating between photondominated and excitondominated depending on nuclear position. However, as radiative emission of the cavity mode is orders of magnitude faster than from the bare molecule (typically, femtoseconds compared with nanoseconds), efficient emission is only possible in regions where the relevant PoPES has a significant photon contribution. Focusing first on the lower polariton, this condition is fulfilled for q < 0 for the parameters chosen here, explaining the observed temporal modulation of the emission intensity, which effectively corresponds to clocking of the nuclear wavepacket motion. Furthermore, the period of this motion is determined by the curvature of the lower polariton PoPES, which is different to the baremolecule oscillation period T_{v} ≈ 22.7 fs. Fitting the lower polariton curve to a harmonic oscillator for the current parameters gives an oscillation period of 25.9 fs, in excellent agreement with the observed modulation frequency of 26 fs. The temporal emission modulation thus also provides a direct fingerprint of the strong couplinginduced modifications of molecular structure. On the other hand, excitation to the upper polariton creates a wavepacket that spends most of its time in the region with efficient emission (q > 0 for the upper PoPES), such that no clear oscillation between photonic and excitonic character, and thus no modulation in the emission intensity, are observed.
Up to now, we have confirmed that molecular dynamics imprints its fingerprint in the timedependent radiative emission of the cavity. We now demonstrate that the timeresolved emission intensity indeed provides a direct quantitative probe of the nuclear wavepacket dynamics. In Fig. 3, we show the nuclear probability density ∣ψ(q)∣^{2} in the singleexcitation subspace under resonant excitation of the lower polariton, Fig. 3a, and upper polariton, Fig. 3b, respectively. For case (a), the wavepacket starts periodic motion around the minimum of the lower polariton curve, \({q}_{\min }\approx 15\, {\rm{a.u.}}\), after the initial excitation at t ≈ 0. In the upper panel, we show E_{R} and the probability to find the nuclei at q ≤ 0, given by \({\int }_{\infty }^{0} \psi (q){ }^{2}{\rm{d}}q\). The observed good agreement demonstrates that it is possible to track the position of the nuclear wavepacket in time through the emission from the cavity. The similarly good agreement found in Fig. 3b, with the integral this case performed for q ≥ 0 corresponding to excitation of the upper polariton branch reinforces this notion. We again observe that a lesspronounced oscillation is observed for excitation of the UP branch. We also note that for case (b), there is a small contribution of the lower polariton to the excitation (as this is energetically still allowed), explaining the slightly worse agreement between the full calculation and the simplified approximation based on integrating the nuclear probability density.
We next investigate the dependence of the effects discussed above on the Rabi splitting Ω_{R}, focusing in particular on the case of smaller Ω_{R}, which would correspond to the weak coupling regime. The corresponding timeresolved radiative emission E_{R} is shown in Fig. 4a on a logarithmic scale. As we have observed the lower polariton branch to display more interesting dynamics, the central laser frequency is chosen such that the lower polariton branch is excited for each Rabi frequency, i.e., ω_{L} = ω_{e} − Ω_{R}/2. Several regimes can be clearly distinguished: for small coupling, Ω_{R} ≲ 0.03 eV, the molecules barely participate in the dynamics and the response is dominated by the excitation and subsequent ringdown (with time constant τ_{c} = ℏ∕γ_{c} ≈ 6.6 fs) of the bare cavity mode (green line in Fig. 4b). In contrast, within the strong coupling regime, Ω_{R} ≳ 0.10 eV, the previously discussed oscillations can be seen, with the modulation frequency increasing concomitantly with Ω_{R} due to the increasingly large modification of the polaritonic PES, and thus the nuclear oscillation period (blue line in Fig. 4b). For intermediate values of Ω_{R}, a slightly different behavior is observed: emission occurs over relatively long times, but is again modulated over time, with a period of around 23 fs, in good agreement with the baremolecule vibrational period T_{v} ≈ 22.7 fs. This can be understood by examining the molecular PES in the case of weak coupling, as shown in Fig. 4c. In that case, the potential energy surfaces are almost unmodified and the initial laser pulse only excites the cavity mode, but the relatively large coupling is sufficient to allow efficient energy transfer to the molecule (exactly in the Franck–Condon region) within the lifetime of the cavity mode, such that the emission is not fully dominated by the cavity response. The molecular wavepacket then again oscillates, now within the bare molecular excitedstate PES. However, for nuclear configurations where the molecular exciton and the cavity mode are resonant (within the cavity bandwidth), the molecular radiative decay is enhanced strongly through the Purcell effect, leading to ultrafast emission exactly when the nuclear wavepacket crosses the resonant configuration (q ≈ 0 for the parameters considered here). In the intermediate coupling regime, it is important to point out that the oscillations will be more clear when the cavity has an ultrafast decay. This can be seen when comparing the radiative emission for two different decay rates, γ_{c} = 0.1 and 0.3 eV (solid red and dashed dark red lines in Fig. 4b), where the oscillations are more prominent for more lossy cavities. We note that the more relaxed requirements for Ω_{R} in this intermediate regime should make it more easily accessible in controlled experimental setups^{6}.
Multiple molecules
Up to now, we have focused on the case of a single molecule under strong coupling. Although this serves to highlight the principal properties of the setup, it is still extremely challenging to achieve in experiment. On the other hand, collective strong coupling can yield significant Rabi splittings in available plasmonic nanocavities even for small numbers of molecules (e.g., 200 meV for three or four molecules^{5}). In this situation, several molecules are coherently coupled to the same photonic mode, with the collective Rabi splitting scaling as \(\sqrt{N}\). In Fig. 5, we demonstrate that the polaritonic molecular clock also works in this situation. We plot the timeresolved radiative emission for N = 1, 2, and 4 molecules while keeping the collective Rabi splitting fixed at Ω_{R} = 0.4 eV for easier comparison. This shows that the coherent wavepacket motion of multiple molecules moving on a collective PoPES can be accessed directly with our setup. We note that this is in strong contrast to standard pump–probe techniques, where only singlemolecule observables are typically interrogated. In contrast, the PoPES in the case of collective strong coupling describe nuclear motion of the polaritonic supermolecule^{48,49} and depend on all molecular coordinates. In the Supplementary Note 2, we use the timedependent variational matrix product state (TDVMPS) approach^{10,50} to show that this approach works even when taking into account all vibrational degrees of freedom and the associated dephasing. In particular, the effect of dephasing is not significantly stronger in the manymolecule case than for a single molecule. Consequently, the proposed setup could provide a route to directly probe multimolecule coherent nuclear wavepacket motion.
Nonharmonic potentials
We next demonstrate that our approach is not restricted to displacedharmonic oscillator models and also gives direct insight into molecular dynamics in more complex potentials. To that end, we treat a molecule described by displaced Morse potentials, corresponding to anharmonic oscillators. The molecular Hamiltonian is given by
where j ∈ {g, e}, with parameters D = 1.0 eV, a = 0.025 a.u., q_{g} = 0, q_{e} = 18.2 a.u., δ_{g} = 0, and δ_{e} = 3.18 eV. Figure 6a shows the uncoupled PES and the corresponding PoPES under strong coupling, whereas Fig. 6b shows the timedependent radiative emission as a function of the driving laser frequency. In contrast with the simple displacedharmonicoscillator model treated before, the oscillation period of the timedependent radiative emission now depends on the laser frequency. The insight this provides into the polaritonic PES becomes clear by comparing the peak times of the cavity emission with the energydependent classical oscillation period within the lower PoPES, \(T({\omega }_{{\rm{L}}})=2{\int }_{{q}_{\min }}^{{q}_{\max }}{\rm{d}}q{[\frac{2}{M}({E}_{{\rm{gs}}}+{\omega }_{{\rm{L}}}{V}_{{\rm{LP}}}(q))]}^{1/2}\), where E_{gs} is the groundstate energy and M is the reduced mass. The green lines in Fig. 6b show that the peak emission happens exactly at t = nT(ω_{L}), with n = 0, 1, 2, …, demonstrating that the polaritonic molecular clock captures the nuclear wavepacket motion accurately and provides a direct picture of the dynamics also in nonharmonic potentials. In the Supplementary Note 1, we furthermore show that our scheme could also be used to study photodissocation dynamics within the weak coupling regime for a model molecule similar to methyl iodide^{33}.
We next discuss the requirements that must be fulfilled for the phenomena described above to be observed. First, the molecule needs to have sufficiently strong exciton–phonon coupling (i.e., a sufficiently large change in the qdependent excitation frequency) to lead to significant spatial modulation of the cavity and exciton components of the PoPES. Furthermore, the slope of the (polaritonic) PES in the Franck–Condon region has to be large enough for the nuclear wavepacket to leave the initial position before it has time to decay completely (although this problem could be mitigated by, e.g., choosing the cavity to be resonant in another region of nuclear configuration space instead of at the equilibrium configuration). For the Holsteintype molecular model studied here, these conditions are satisfied if λ_{v} is comparable to the vibrational frequency ω_{v}, and both are comparable to the cavity decay rate γ_{c}. These properties are fulfilled for several organic molecules that have been used in strong coupling experiments, such as anthracene^{51} or the rylene dye [N,N0Bis(2,6diisopropylphenyl)1,7 and 1,6bis(2,6diisopropylphenoxy)perylene3,4:9,10tetracarboximide]^{52}. In addition, to be able to observe coherent wavepacket motion, internal vibrational relaxation and dephasing, which typically occurs on the scale of tens to hundreds of femtoseconds in solidstate environments, must be slow enough compared with the dynamics of interest. In the Supplementary Note 2, we demonstrate that this is the case for the anthracene molecule by comparing the Holstein–Jaynes–Cummings model calculation with largescale quantum dynamics simulations including all vibrational modes of the molecule, performed using the TDVMPS approach^{10,50}.
To summarize, we have proposed a scheme to probe and image molecular dynamics by measuring the timedependent radiative emission obtained after shortpulse excitation of a system containing few molecules and a nanocavity with large light–matter coupling, close to or within the strong coupling regime. We show that this approach enables to retrieve a direct mapping of nuclear wavepacket motion in the time domain, also in the fewmolecule case, where this scheme provides a direct fingerprint of coherent multimolecular nuclear dynamics. In the strong coupling regime, this gives access to the cavitymodified molecular dynamics occurring on the PoPES, whereas in the weak coupling regime it allows probing of the baremolecule excitedstate dynamics. By exploiting the ultrafast emission dynamics in typical highly lossy plasmonic nanocavity, we obtain the timeresolved dynamics without the need for a pump–probe setup with synchronized femtosecond pulses. In addition, in contrast to the common approaches of femtochemistry, our proposed scheme does not require direct access to molecular observables such as photoelectron spectra or fragmentation yields, which are difficult to obtain for typical experimental geometries. Instead, it only relies on optical access to the nanocavity mode. In addition, the scheme only depends on the properties of the first few electronic states of the molecules, and is not affected by, e.g., the multitude of ionization channels that have to be taken into account in photoionization^{34}. As only a single excitation is imparted to the molecules and the dynamics are probed through the photons emitted upon relaxation to the ground state, the molecules are left intact after the pulse. At the same time, this implies that the absolute photon numbers to be measured are small. This could be mitigated by using highrepetitionrate sources (readily available for the low laser intensities required), as well as collecting the response from an array of identical nanocavities, taking advantage of highly reproducible setups available nowadays, e.g., through DNA origami^{6,53}. Finally, we mention that although the cavity decay rate γ_{c} in a plasmonic cavity is typically large and leads to fewfemtosecond lifetimes as required for the discussed approach, this rate is often dominated by nonradiative contributions that do not lead to farfield emission. However, fortunately the same plasmonic nanocavity architectures that provide the current largest coupling strengths, such as nanoparticleonmirror geometries, also provide a significant radiative quantum yield of close to 50%^{54,55,56}.
Methods
The time dynamics is described by the following Lindblad master equation
where \({{\mathcal{L}}}_{a}[\rho (t)]=a\rho (t){a}^{\dagger }\frac{1}{2}[\rho (t){a}^{\dagger }a+{a}^{\dagger }a\rho (t)]\) is a standard Lindblad decay term modeling the incoherent decay of the cavity mode due to material and radiative losses. The master equation numerical results were obtained using the QuTiP package^{57,58}. The PoPESs used for the interpretation and analysis of the results are obtained by diagonalizing the (undriven) Hamiltonian within the Born–Oppenheimer approximation, i.e., diagonalizing H(t) − p^{2}∕2 for E_{0} = 0 and fixed q^{18}. In Fig. 1, we show the PoPES within the singleexcitation subspace, spanned by the uncoupled states \(\lefte,0\right\rangle\) and \(\leftg,1\right\rangle\), where \(\leftg\right\rangle\) (\(\lefte\right\rangle\)) is the electronic ground (excited) state and \(\leftn=0,1,\ldots \right\rangle\) is the cavity mode Fock state with n photons. The Hamiltonian in this subspace can be written as
and diagonalizing it gives the PoPES plotted in Figs. 1 and 4b.
The parameter values chosen for modeling the molecule were based on abinitio calculations for the anthracene molecule at the TDAB3LYP level of theory using Gaussian 09^{59}. Fitting the PES obtained in these calculations to a displacedharmonic oscillator model using the Duschinsky linear transformation^{60},
yields the parameters {ω_{k}, λ_{k}}, or equivalently the spectral density \({J}_{{\rm{v}}}(\omega )={\sum }_{k}{\lambda }_{k}^{2}\delta (\omega {\omega }_{k})\), determining the vibrational spectrum of the molecule. The decoherence due to the coupling of the vibrational molecular modes with the surrounding bath is taken into account empirically by replacing the discrete peaks in the spectral density by a Lorentzian with 0.3 meV of width; however, the results are not affected by this. The single vibrational mode in Eq. (1) is then taken as the corresponding reaction coordinate, with \({\lambda }_{\mathrm{{v}}}=\sqrt{{\sum }_{k}{\lambda }_{k}^{2}}\) and \({\omega }_{\mathrm{{v}}}={\sum }_{k}{\omega }_{k}{\lambda }_{k}^{2}/{\lambda }_{v}^{2}\)^{10,61}.
In Fig. 7, we show the vibrational spectral density of anthracene (convoluted with a Lorentzian to represent broadening due to interactions with a solidstate environment). It can be seen that ω_{v} is very close to the frequency of the dominant vibrational mode in J_{v}(ω). We have additionally checked the validity of the singlemode approximation by comparing the model calculations above with TDVMPS) calculations^{10,50} in which the full phononic spectral density, describing all vibrational modes of the molecule and surroundings, is taken into account (see Supplementary Note 1 for details).
Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
We thank Alex W. Chin and Florian A. Y. N. Schröder for their help with the TDVMPS calculations, and Clàudia Climent for her help with calculating the displacedharmonic oscillator model and with the Duschinsky linear transformation. This work has been funded by the European Research Council grant ERC2016STG714870 and the Spanish Ministry for Science, Innovation, and Universities—AEI grants RTI2018099737BI00, PCI2018093145 (through the QuantERA program of the European Commission), and CEX2018000805M (through the María de Maeztu program for Units of Excellence in R&D).
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R.E.F. S. and J.F. developed the idea. R.E.F.S. performed the numerical calculations. R.E.F.S., J.P., F.J.G.V., and J.F. contributed to analysis of the results. R.E.F.S. and J.F. wrote the main part of the manuscript. The manuscript was discussed by R.E.F.S., J.P., F.J.G.V., and J.F.
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Silva, R.E.F., Pino, J.d., GarcíaVidal, F.J. et al. Polaritonic molecular clock for alloptical ultrafast imaging of wavepacket dynamics without probe pulses. Nat Commun 11, 1423 (2020). https://doi.org/10.1038/s4146702015196x
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DOI: https://doi.org/10.1038/s4146702015196x
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