Molecular strong coupling: evaluating dipole oscillator and cavity field parameters

In this report we use material parameters to calculate the strength of the expected Rabi splitting for a molecular resonance. As an example we focus on the molecular resonance associated with the C=O bond in a polymer host, specifically the stretch resonance at ∼1730 cm−1. Two related approaches to modelling the anticipated extent of the coupling are examined, and the results compared with data from experiments available in the literature. The approaches adopted here indicate how material parameters may be used to assess the potential of a material to exhibit strong coupling, and also enable other useful parameters to be derived, including the molecular dipole moment and the vacuum cavity field strength.


Introduction
Strong coupling between a molecular resonance and a photonic mode such as that supported by an optical microcavity is a rapidly expanding field.Originally the realm of atom optics cavity quantum electrodynamics (cQED) [1,2], strong coupling is emerging as a powerful new paradigm for molecular materials [3,4].When the rate of the interaction between the molecular resonators and the cavity mode exceeds the damping rates then the molecular and cavity modes lose their separate identity and two new hybrid (polariton) modes emerge, an upper and a lower polariton.These hybrid polariton modes inherent both light and matter characteristics.A hallmark of the strong coupling interaction is a splitting of the polariton modes, i.e. the upper polariton and lower polariton are separated in frequency (energy) by the Rabi frequency, Ω R .Interest is particularly strong in the context of molecular vibrational modes since there are exciting prospects for modifying some chemical reactions with this approach [5].The first demonstrations of strong coupling employing infrared molecular resonances were based on the C=O bond, specifically the stretch resonance at ∼1730 cm −1 .This molecular resonance is common in a wide range of polymeric materials that are convenient for many experiments, primarily because of the ease with which they may be spin cast to form optical micro-cavities; polymers include PVA [6] and PMMA [7,8].
In this report we show how the bulk optical properties of appropriate molecular materials can be used to successfully predict the extent of strong coupling.Although we carry out such an analysis here in the context of the C=O bond, and specifically we compare our analysis with the experiments reported by Shalabney et al and Long et al the approach is very general and may be applied to excitonic as well as vibrational resonances, it should thus find application to materials as diverse as 2D transition metal dichalcogenides [9] and photosynthetic materials [10].Whilst the required formulae are all available in the literature, they are scattered among various sources, and often need care in interpretation.Furthermore, a full understanding of the relevant formulae is shown to be essential if quantitatively useful data are to be extracted.

Background
We consider N molecular resonators that interact with a cavity vacuum electric field, E vac , through their electric dipole moment μ.The interaction energy of one resonator is μ E vac (we assume the dipole moment and the field are aligned).The strength (RMS) of the vacuum field is , where V is the volume of the cavity mode, ω 0 is the resonance frequency of the relevant molecular transition, and ε b is the background permittivity of the molecular material [11].With N molecular resonators coupled to the same cavity mode, the strength of the interaction is scaled up by N [11].It is this scaling with N that can lead to the coupling being sufficient for two new hybrid modes (polaritons) to emerge, the original molecular resonance and cavity mode losing their separate identity [3,4,12].To make contact with the literature it is convenient to write the interaction energy for N molecules in a cavity as ÿΩ R , where Ω R is known as the Rabi frequency [13].We thus have,  so that the material parameters we need if we are to calculate the Rabi splitting are the dipole moment and the frequency associated with the molecular transition, μ, ω 0 , the concentration of the molecular resonators, N V , and the background permittivity of the material in which the molecules are embedded, ε b .It is important to note that we have assumed that the molecules fill the volume of the cavity mode.Our estimate of the Rabi splitting is thus an upper limit since in some situations the molecular resonators do not fill the mode volume, see e.g.[14].In addition we will wish to see whether the coupling strength dominates over the molecular and cavity (de-phasing) decay rates (γ M and γ C respectively), i.e. whether g N > γ C , γ M ; this is the condition that needs to be met if the system is to be in the strong coupling regime.From a practical point of view a more convenient version of this condition is [11], where Γ is the linewidth of the molecular transition and K is the cavity linewidth.Out of curiosity, we may also want to evaluate the mode volume V, the quality factors of the molecular and optical resonances, Q M and Q C respectively, and the vacuum field strength E vac .Looking at equation (1) we can see that to evaluate the interaction strength g N we will need to determine: the dipole moment μ; the resonance frequency ω 0 ; the concentration N/V of molecular resonators, and the refractive index of the host medium, n b .Below we look at two ways of accomplishing this.Let us look at the essentials first, i.e. the number density of C=O bonds, the C=O bond stretch resonance frequency, and the associated dipole moment.From easiest to hardest these are: First, the resonance frequency.This is easily be determined from IR transmission measurements, and is found to be [8] ∼1734 cm −1 , which is equivalent to a wavelength of 5.8 μm, and an angular frequency of 3.26 × 10 14 rad s −1 .We will look at the IR transmission spectrum in more detail below.
Second, the number density.Shalabney et al use the polymer PVA C , for which the density is [15] 1.19 g cm −3 .Thus, 1 cm 3 contains 1.19 g of the polymer.The molecular mass of PVA C is 86 (the repeat unit contains 4 carbon atoms, 2 oxygen atoms and 6 hydrogen atoms), so that 1 cm 3 contains 1.19/86 moles.Since each repeat unit contains one C=O bond, the number of C=O bonds per cm 3 is thus 8.33 × 10 21 , so that the density of bonds is 8.33 × 10 27 per m 3 .We assume these numbers also apply to the spun films used in the strong coupling experiments.
Third, the dipole moment.We can calculate the dipole moment in two ways, from the Lorentz oscillator model, and from molar absorption.We look at the Lorentz oscillator first.
The first approach to evaluate the dipole moment we use here is to fit a Lorentz oscillator (LO) model for the permittivity of the PVA C to the measured transmission spectrum.The LO model is incorporated into a Fresnel-based calculation of the transmittance, the sample consisting of the spun polymer film on top of a substrate, see figure 1.A convenient 'know nothing in advance' formulation of the Lorentz oscillator permittivity is, where ε(ω) is the frequency dependent permittivity, ε b is a constant across the frequency range of interest (it represents the contribution to the permittivity of resonances in other spectral regions).The frequencydependent refractive index, n(ω), and permittivity ε(ω) are related through ε(ω) = n 2 (ω).The molecular damping rate is γ MD , and corresponds to the full-width at half-maximum of the resonance in the LO model, note that the de-phasing rate and the damping rate are related through, γ M = γ MD /2.Note also that, with reference to equation (2), γ MD ∼ Γ.The reduced oscillator strength of the transition is f ¢.For the Fresnel calculation of the transmittance we also need the thickness of the polymer layer, and the refractive index of the substrate.The dipole moment μ is related to the oscillator strength, f, by [16], so that if we can find f then we can use (4) to find the dipole moment.The difference between f ¢ and f is discussed below.
The experimental information we have to work with is the IR transmission spectrum.Shalabney et al provide such a spectrum for a film of PVA C on a Germanium substrate (for which we take the permittivity to be ε Ge = 16.0, the superstrate is air).In assessing the transmittance data we need to take account of the reflection that occurs at the substrate/air interface, so that to simulate the experimental data we need to multiply our Fresnel-calculated data by 0.64 1 .In the experiment the overall transmittance of the sample was measured.Figure 1 shows the transcribed experimental data from Shalabney et al (red data points) together with the result of a Fresnel-based calculation for the transmittance, where the parameters of the Lorentz oscillator model for the C=O bond have been varied to provide a reasonable 'by eye' match to the experimental data.The inset shows the spectra in the neighbourhood of the C=O stretch, the main part of the figure covers a wider energy range.In the Figure 1.IR transmittance spectrum of PVA C .Shown as red points are data transcribed from the report of Shalabney et al [6].Shown as a black line is the result of a Fresnel-based match to the experimental data.The inset lower right shows the spectral region around the C=O resonance in more detail.The schematic upper left shows the configuration of the transmittance measurement, the PVA C film is on a Ge substrate.See table 1 for details of the parameters used, see also footnote 2.
Table 1.Table of Lorentz oscillator parameters for the C=O bond for the different models considered.The oscillator strengths are dimensionless, the units for the other parameters are as given at the head of each column. 1 The transmission of the Ge/air interface, with is T , where n Ge = 4.
main figure there is a gentle periodic modulation of transmittance, arising from interference due to the two surfaces of the polymer film, which are separated by d = 1.70 μm, see figure 1a in [6].The parameters used in the model are given in table 1 below. 2lthough equation (3) is convenient, it is not the most appropriate equation to use in finding the dipole moment because f ¢ does not convey the physical origin of the Lorentz oscillator model.Instead we will use, where now the parameters are given in terms of wavenumber, (cm −1 ).
We will now find f and γ MD and will also find f k and γ MDk , so that we may make a comparison between our results and those of Shalabney et al.Comparing equations (3) and (5) we have, From the 'fit' shown in figure 1 we note that we used ω 0 = 3.26 × 10 14 rad s −1 and f 0.018 ¢ = , and from this we can calculate ω p via (6) using the bond density discussed earlier, i.e.N/V = 8.0 × 10 27 m −3 .Doing so we find ω p = 5.1 × 10 15 , so that f = 0.7 × 10 −4 .The background permittivity was taken to be ε b = 1.99 [6].Next, noting that to convert rad s −1 to cm −1 we divide by 1.88 × 10 11 , we find, f k = 54 × 10 3 , k o = 1734 cm −1 , and γ MDk = 16 cm −1 .In table 1 we bring all of these data together, including the data from Shalabney et al i.e. their results from fitting their own data.Table 1 thus  Cm].Whilst the exact value of the dipole moment will depend on the specific host, the value found here is in the range of values given by Grechko and Zanni for various materials [17], note that Shalabney et al state a value for the dipole moment of 1D.An alternative to using the Lorentz oscillator model is to make more direct use of extinction (transmission) measurements as a convenient and simple way to characterise a molecular material by adopting a different approach based on the molar absorption coefficient,  .For a film or solution of the molecules of interest, of concentration C m , and of path length l, the molar absorption coefficient 3 is given by [19,20], where I 0 and I t are the incident and transmitted intensities.Note that the commonly used units for the path length, l, are cm, whilst the units for the molecular concentration, C m , are moles per dm −3 , i.e. moles per litre; the molar absorption coefficient thus has units of dm 3 mol −1 cm −1 .Now, Turro (see [19] where n is the wavenumber in cm −1 .This can often be approximated as, see [19]  ´- where dn (∼Γ) is the width (FWHM) of the extinction feature in wavenumbers (cm −1 ). 5 We note that the oscillator strength found using equations (11) and ( 12) is less than the oscillator strength employed in the Lorentz oscillator model, by a factor of n b .This is associated with the change in energy density of the light in the molecular material (c.f.air), see equation 9.29 in [23].As a result the oscillator strength derived from extinctiontype measurements should instead be written as, For the C=O data of Shalabney et al (see figure 1), max  corresponds to T = 0.045/0.45= 1/10 (see footnote 6 ), and the path length is l = 1.7 × 10 −4 cm.For the molecular concentration, C m , 1 cm 3 contains 1.19/86 moles, so that there are 13.8mol dm −3 , i.e.C m = 13.8 mol dm −3 .Bringing these factors together we can use (10) to evaluate max  and find 426 max =  .Finally we can estimate the width dn from the molecular absorption data in figure 2 to be dn » 21 cm −1 ; care is needed in making this estimate, see below.We can thus calculate f via equation (13), and find f = 5.5 × 10 −5 .This compares with the × 10 −5 we obtained earlier, see table 1, surprisingly similar given the rather crude approximations we have made.The associated dipole moment is μ = 0.26 D.
We can now look at the coupling strength and compare it to the decay rates to see if the strong coupling regime applies.From equation (1) we can calculate the value of g N for the C=O bond in PVA C , doing so we find g N = 79 cm −1 if we use parameters from the LO model, and g N = 71 cm −1 if we use parameters directly from the extinction data.Note that in using equation (1) we need to divide the right-hand side by 3 , see footnote. 7igure 2. IR transmittance spectra (experimental and calculated), molar absorption coefficient, and imaginary part of the permittivity for PVA C .Transmittance: shown as black circles are data transcribed from the report of Shalabney et al [6]; shown as a black line is the result of a Fresnel-based match to the experimental data (note that the transcribed data have been scaled, as described in footnote 2).The molar absorption coefficient  (r.h.axis) is calculated from the transmittance data according to equation (10) (note that for this calculation I 0 is 0.43, i.e. the value the transmission would have been in the absence of the molecular resonance (see insert to figure 1).Also shown is the imaginary part of the PVA permittivity (as calculated from the Lorentz oscillator model), scaled for ease of comparison with the molar absorption coefficient.
Recall that for strong coupling we require the strong coupling criterion, equation (2) to be met, i.e. 2g N > (Γ + K )/2 where Γ is the width of the C=O resonance, which we found above to be 21 cm −1 , and K is the width of the cavity resonance.Shalabney et al give the the FWHM of their cavity resonance, as measured in transmission, to be 140 cm −1 (17 meV), so that (Γ + K )/2 = 80 cm −1 ; the criterion for strong coupling is thus easily satisfied.It is interesting to note that the measured Rabi splitting in the experiment of Shalabney et al is ∼170 cm −1 , implying a value for g N of ∼85 cm −1 .
An element of confusion can arise when carrying out some of the analysis described here.The problem concerns the width of the molecular resonance.Clearly the width that goes into the Lorentz oscillator model is based on assuming a Lorentzian profile, but in practice this is unlikely to be more than an approximation to the response of the real material.Although one could determine the width directly from the transmittance data, doing so is in general not appropriate since the width of the transmittance feature changes with concentration (especially at the high concentrations often needed to achieve strong coupling).Instead one should use a lineshape that is concentration independent, and the molar absorption coefficient we have used here is one such parameter.Note also that it is important not to use wavelength as the variable in such spectra, frequency is more appropriate [25].In figure 2 we present a closer zoom-in of the transmittance data shown in figure 1.In addition, the molar absorption coefficient (derived from the transmittance data using equation (10)) and the imaginary part of the PVA permittivity are shown.The imaginary part of the PVA permittivity has a FWHM of 16 cm −1 , as expected (see table 1), whilst the transmittance dip has a FWHM, estimated from figure 2, of nearly 30 cm −1 .The molar absorption coefficient however has a width of 21 cm −1 and it is this width that is needed when making use of the extinction data to evaluate the oscillator strength 8 .
For completeness, let us calculate the relevant Q factors.For the molecular resonance this is It may also be useful to look at the vacuum field strength.A reasonable approximation can be found by estimating the mode volume V, and using the well-known expression for the vacuum field strength [11], For the mode volume, to a reasonable approximation this is d r eff 2 p where d is the cavity thickness, and r eff is the radius of the mode, this radius is in turn is determined by the width of the mode in terms of in-plane wavevector [27,28], and is given by, where R is the reflectance (intensity reflection coefficient) of the cavity mirrors.A Fresnel multi-layer calculation can be used to work out R for the upper and lower mirrors.The mirrors are 10 nm thick gold, and the permittivity of gold at this wavelength was estimated to be ε Au = −1000 + 200i, based on data in [29].For the upper mirror the three media are PVA/Au/air, for the lower mirror they are PVA/Au/Ge.Values of R upper = 0.85 and R lower = 0.8 were found.Taking an average value of ∼0.8 we find r eff = 7.3 μm so that the mode volume is V d r eff 2 p = = 300 μm 3 .We can now calculate the vacuum field strength with the help of equation (14), we find the field strength to be E RMS = 1.7 ×10 3 Vm −1 .(Note that Shalabney et al took their mode volume to be V n b 3 ( ) l = = 70 μm 3 ).Finally we can estimate the number of C=O bonds involved from the density (8.3×10 27 m −3 ) and the mode volume (300 ×10 −18 m 3 ) as N = 2 × 10 12 .It is now clear that the single molecule coupling strength, g N N = 1 ×10 −4 cm −1 is much much smaller than the de-phasing/decay rates, strong coupling here really is a collective effect.

Conclusion
We have shown how two relatively simple models can be used to understand and predict the extent of the Rabisplitting observed in vibrational strong coupling experiments.The analysis has been presented with the aim of providing a clear link between the extent of the Rabi-splitting and material parameters.We have used this worked example (the case of the C=O bond) to show how a number of related parameters may be evaluated, notably the dipole moment, the oscillator strength, the vacuum field strength and the cavity mode volume of planar Fabry-Perot-type cavities.Whilst the formulae used here are already available in the literature, how they should be implemented, what pitfalls might arise, and what assumptions are being made in their use have been presented here in a coherent way in the hope that in doing so others may use them to speed up the evaluation of molecular materials for strong coupling in their own research.