Extracting accurate light-matter couplings from disordered polaritons

The vacuum Rabi splitting (VRS) in molecular polaritons stands as a fundamental measure of collective light-matter coupling. Despite its significance, the impact of molecular disorder on VRS is not fully understood yet. This study delves into the complexities of VRS amidst various distributions and degrees of disorder. Our analysis provides precise analytical expressions for linear absorption, transmission, and reflection spectra, along with a"sum"rule, offering a straightforward protocol for extracting accurate collective light-matter coupling values from experimental data. Importantly, our study cautions against directly translating large VRS to the onset of ultrastrong coupling regime. Furthermore, for rectangular disorder, we witness the emergence of narrow side bands alongside a broad central peak, indicating an extended coherence lifetime even in the presence of substantial disorder. These findings not only enhance our understanding of VRS in disordered molecular systems but also open avenues for achieving prolonged coherence lifetimes between the cavity and molecules via the interplay of collective coupling and disorder.

In this article, we revisit the problem of VRS and disorder, embarking on a comprehensive study of linear absorption , transmission  , and reflection  properties of molecular polaritons, considering various distributions and magnitudes of disorder.Our aim is not only to elucidate the intricate interplay between molecular disorder and VRS, but also to provide a robust method for accurately extracting light-matter coupling parameters.Our findings align with recent reports, demonstrating that VRS tends to increase with disorder, reaches a saturation point, and eventually decreases to zero for a wide range of disorder distributions.Note that , ,  do not in general give the same value of VRS [30].Hence, while some of the results are () ()  () Fig. 1: (A) Linear spectroscopy of molecular polaritons as absorption (), transmission  (), and reflection () (B) For weak disorder, a perturbative approach to understand the role of disorder is useful.The zeroth-order Hamiltonian can be taken to be the disorderless system, comprising of the photon mode interacting with the  degenerate molecules to form the upper (UP) and lower (LP) polaritons, alongside  − 1 dark states ({ 1 , ...,   −1 }).The VRS at resonance in this case happens to be 2 √  , where  is the single molecule light-matter coupling.The molecular disorder then perturbatively couples the polaritons to the manifold of dark states, inducing level repulsion between the polaritons, as discussed in Refs.[33,36], and [37], thus increasing the VRS.This repulsion depends on the variance of the disorder distribution (⟨ 2  ⟩, where   =   −  0 and  0 represents the center of the distribution).Given the perturbative character of this analysis, it does not apply to strong disorder.
already known in the literature, we deem it valuable to collect all the results in a single study.Significantly for experiments, we unveil what seems to be a ubiquitous scenario: the presence of substantial disorder can dramatically enhance the VRS leading to an apparent onset of the ultrastrong coupling regime, despite the underlying collective light-matter coupling firmly residing within the strong coupling regime.Moreover, we introduce a novel sum rule that proves instrumental in extracting precise values of collective light-matter coupling, particularly when both absorption and transmission can be measured in the experimental setup.Crucially, this sum rule demonstrates generality across all types of disorder.In the case of a rectangular distribution (which was briefly discussed in Ref. [34]), we observe the emergence of two narrow polariton peaks in the spectra, reminiscent of the pronounced spectral narrowing witnessed in the context of surface lattice resonances [39] and other phenomena associated with Wood anomalies [40][41][42][43][44].

Model
For concreteness, we consider  two level systems coupled to a single photon mode [in the rotating wave ap-proximation (RWA), Tavis-Cummings model [19]]: where  ℎ and  are the photon frequency and annihilation operator,  , and   = |  ⟩⟨  | are the frequency and annihilation operator for the −th two level system,   is the amplitude of the −th transition dipole, and ℏ = √︁ 20 is the vacuum field amplitude where  0 is the vacuum permittivity and  is the mode volume.In the thermodynamic limit ( → ∞), the linear absorption, transmission, and reflection spectra are given by (see Refs. [45] and [46]) Here  =   +   is the total cavity decay rate, and the respective decay rates into the left and right photon continua are denoted by   and   .The linear molecular susceptibility () is given by Considering the case when ℏ , ≫    and assuming that all  two level systems have the same transition-Tab.1: Real and imaginary parts of the molecular susceptibility

Real or Imaginary
Gaussian a Lorentzian Rectangle ∫︀  0   2 is the Dawson function.
dipole amplitude , the molecular susceptibility becomes where  2 = || 2 is the square of the single molecule light-matter coupling, (  ) is the probability distribution of excitation frequencies, and the real and imaginary parts of the molecular susceptibility are − , where  is the Cauchy principal value, and  ′′ () =  2  ().
To explore the effects of (  ), we consider Lorentzian, Gaussian, and rectangular disorder.Table 1 lists the analytical expressions of  ′ () and  ′′ () for the three different distributions.

Lorentzian disorder
For Lorentzian disorder, the real and imaginary parts of the susceptibility are From Eqs. 2 and 3, the absorption, transmission, and reflection spectra are given by Figure 2 presents the numerically calculated spectra.
To find the extrema of each spectra, we solve for    () = 0,   () = 0, and   () = 0 separately.Lorentzian disorder affords exact analytical expressions for all regimes of .In the case where the photon mode is resonant with the center of the Lorentzian distribution, i.e.,  ℎ =  0 , we find that the upper (  + ) and lower (  − ) polariton peaks in the absorption are located at the frequencies Similarly, the upper and lower polariton peaks in the transmission spectrum are located at and the peaks in the reflection spectrum are at With prior information on the molecular disorder () and cavity linewidth (), one can use the above equations to extract the correct value for the collective coupling  √  from experimentally obtained spectra.For the transmission spectra, we observe similar behavior to that was shown in Refs.[33][34][35][36][37] for Gaussian disorder: the VRS initially increases with increasing disorder; as disorder increases further, the VRS saturates, and then decreases to zero.It is interesting that this trend is not observed for absorption, as VRS only decreases with disorder.Table 2 summarizes the intricate behavior of VRS across spectra for different disorder distributions.Qualitatively, the increase of VRS with weak disorder is a manifestation of level repulsion (see Figure 1B).It should be noted that Eqs.13-14 are the same as those presented in Ref. [30].Furthermore, the expressions for  = 0, Eq. 16 looks similar to the analytical expressions derived by Refs.[8,34] and [38], but the decrease in VRS for weak disorder differs by a factor of a half compared to our result.This is due to the fact that the real parts of the poles for Eqs.16-18 do not correspond to the true extrema along the real-value frequency axis.

Gaussian and rectangular disorder
For Gaussian and Rectangular disorder, we can still extract semi-analytical results when  ≪  √  (weak disorder) and | −  0 | ≈  √  (about the polariton peaks).For the Gaussian distribution, we employ the asymptotic expansion of the Dawson function [47], similar to Ref. [34], up to [(  −0 ) 3 ] to obtain an approximate expression for the real part of the susceptibility, Meanwhile, for the rectangular distribution, we get, Note that for both distributions  ′′ () ≈ 0 at the polaritonic windows, so no VRS is predicted in the absorption spectrum for both Gaussian and rectangular distributions.The lack of VRS in the absorption spectrum for these disorder distributions is due to the minimal overlap between the molecular absorption spectrum and the polariton transmission, since the tails of the Gaussian die quickly away from  0 , while the rectangular distribution has no tails.Contrast this observation with the analogous one for the Lorentzian distribution, which does present VRS in its absorption spectrum owing to the long tails of the Lorentzian.Using these approximations, we find that for  ℎ =  0 , the transmission and reflection spectra for Gaussian distribution are approximately  3) with the polariton frequencies for both spectra approximately located at Similarly, we find that the transmission and reflection spectra for the rectangular distribution are approximately In this case the polariton frequencies are approximately located at which is similar to Eq. 23.
Figure 3 shows the numerically calculated spectra for Gaussian disorder.The trend in VRS in the transmission and reflection spectra as a function of  is qualitatively the same as that for the Lorentzian.Eq. 23 is in good agreement with the numerical results for  ≪  √  , and upon Taylor expanding around  = 0 to second order, reduces to the analytical results of Refs.[34] and [37].The analytical results of Refs.[33] and [36] qualitatively capture the behavior of the transmission spectrum; however, quantitatively they fits better with the Lorentzian disorder.The transformation of Gaussian into Lorentzian disorder in these references appears to be due to the usage of the Markovian approximation. Figure 3 also highlights the dangers for taking VRS at face value.We observe that the largest VRS in the transmission and reflection spectra is approximately 1.5 × 2 √  .This implies that one must be careful using VRS to determine the strength of the collective light-matter coupling.For example, if VRS/2 0 ≈ 0.1 one may mistakenly claim to be in the ultrastrong coupling regime, while in reality the collective light-matter coupling  √  ≈ 0.07 is still within the strong-coupling regime.√  .The white dashed lines indicate our analytical results for the polariton frequencies, showing strong agreement with the calculated spectra for weak disorder.As done in Ref. [33], the white solid lines in (B) represent individual spectra for weak and strong disorder systems, highlighting the transition from two distinct peaks to a broad central peak as disorder increases.Remarkably, for large disorder, two peaks that are narrower than the line width of the cavity and molecular disorder emerge on either side of the broad central peak.
Figure 4 displays the numerically calculated spectra for the rectangular disorder.Such a scenario was briefly considered in Ref [34]; here, we provide further analysis.At low disorder, the rectangular distribution exhibits similar behavior to the Gaussian distribution, as predicted by Eq. 26.As disorder increases, a broad central peak forms, consistent with both the Lorentzian and Gaussian distributions.Intriguingly, as disorder increases, two sharp sidebands for the polaritons also emerge, each narrower than the cavity linewidth and the width of the rectangular distribution, respectively.This is reminiscent to the pronounced spectral narrowing witnessed in plasmonic surface lattice resonances [39].The unique characteristics of the rectangular disorder, including singularities in its real part of the susceptibility  ′ () near these polariton peaks (see Table 1), are responsible for this phenomenon.The reduced linewidth of these peaks indicates a higher degree of coherence lifetime within the system, even in the presence of significant energetic disorder.Consequently, this observation presents a promising avenue for applications in molecular polariton systems requiring prolonged coherences between cavity and molecules.In a realistic experiment, this phenomenon will only occur if the underlying members of the disorder distribution have a small Lorentzian homogeneous linewidth (which has not been treated in this work); a large such linewidth will smoothen the singularity and reduce the problem to the Lorentzian case.

Sum rule
From our extensive study of disorder effects on the polariton linear spectrum, it is evident from the three disorder distributions studied in this work, that disorder significantly impacts the value of VRS, seemingly posing a challenge in extracting the precise value of the collective light-matter coupling.However, a simple sum rule can be utilized when both the absorption and transmission polariton spectra are accessible.By integrating the ratio of these two signals (), which is proportional to the imaginary part of the linear molecular susceptibility  ′′ () =  2  (), a robust method for determining the collective light-matter coupling is revealed (see Eqs. 2 and 3): Importantly, this expression is general for any form and strength of disorder.Note that Eq. 27 is presented in the low temperature limit.For finite temperature effects, see Appendix.Therefore, experimental setups that enable the measurement of both absorption and transmission spectra are deemed ideal for extracting accurate collective-light-matter coupling values.
For larger values of , Eqs.16-18 should be used to extract the correct value for  √  .

Conclusions
Our comprehensive investigation has unraveled the intricate relationship between molecular disorder and VRS in molecular polariton.We have derived precise analytical expressions for absorption, transmission, and reflection spectra across various disorder distributions.Furthermore, our study introduces a generalized sum rule for determining the collective lightmatter coupling under any form of disorder.These findings do not only clarify the nuanced behavior of VRS amidst disorder but also establish a reliable framework for extracting light-matter coupling parameters with high accuracy from experimental data.In practical terms, when the experimental set-up allows access to both transmission and absorption signals, the sum rule can be readily applied.In situations where accessing both signals is not possible, researchers can leverage Eq. 2-4, coupled with the analytical forms of the molecular susceptibility outlined in Table 1, to effectively fit their experimental results.Both these approaches ensure the extraction of the correct value of  √  in the presence of disorder of any magnitude.Additionally, for scenarios involving mild disorder, the simplified expressions provided in Table 3 offer a convenient solution for data fitting.Furthermore, our study has unveiled a fascinating phenomenon associated with rectangular disorder -the emergence of narrow sidebands alongside a broad central peak.This intriguing line narrowing, observed in the presence of significant disorder, suggests a higher degree of coherence lifetime within the system.Such behavior is especially promising for applications requiring long-lived coherences between the cavity and molecules, providing an exciting avenue for future research in the realm of molecular polaritons.
A Sum rule for finite temperature

Fig. 2 :
Fig. 2: Numerically calculated (A) absorption, (B) transmission, and (C) reflection spectra for a Lorentzian distribution of excitation energies () centered at  =  0 ;  ℎ =  0 and   =   = 1 2 √  .The white dashed lines indicate our analytical results for the polariton frequencies, showing strong agreement with the calculated spectra overall magnitudes of disorder.As done in Ref.[33], the white solid lines in (B) represent individual spectra for weak and strong disorder systems, highlighting the transition from two distinct peaks to a broad central peak as disorder increases.

Fig. 3 :
Fig. 3: Numerically calculated (A) absorption, (B) transmission, and (C) reflection spectra for a Gaussian distribution of excitation energies () centered at  =  0 ;  ℎ =  0 and   =   = 1 2 √  .The white dashed lines indicate our analytical results for the polariton frequencies, showing strong agreement with the calculated spectra for weak disorder.As done in Ref.[33], the white solid lines in (B) represent individual spectra for weak and strong disorder systems, highlighting the transition from two distinct peaks to a broad central peak as disorder increases.

Fig. 4 :
Fig. 4: Numerically calculated (A) absorption, (B) transmission, and (C) reflection spectra for a rectangular distribution of excitation energies () centered at  =  0 ;  ℎ =  0 and   =   = 1 2 Tab. 3: Vacuum Rabi splitting expressions for when  ≪  √ From Eqs. 2 and 3, we have that Note that if we evaluate the integral counter clockwise over a semicircle  + with an infinite radius in the upper half of the complex plane, we have that Substituting these values into Eq.A2, we arrive at the generalized sum rule for arbitrary  :∫︁   (  ) tanh (︁ ℏ  2   )︁ |(  )| 2 (A5)is the square of the effective single molecule lightmatter coupling.Here (  ) is the probability distribution of excitation frequencies.Note in the limit that ℏ , ≫    and (  ) → , we have that  2   →  2 , recovering the version of the sum rule presented in the main text.