≤$\leq$

# Vibrational polariton transport in disordered media

Enes Suyabatmaz  Raphael F. Ribeiro<EMAIL_ADDRESS>Department of
Chemistry and Cherry Emerson Center for Scientific Computation, Emory
University, Atlanta, GA, 30322

###### Abstract

Chemical reactions and energy transport phenomena have been experimentally
reported to be significantly affected by strong light-matter interactions and
vibrational polariton formation. These quasiparticles exhibit nontrivial
transport phenomena due to the long-range correlations induced by the photonic
system and elastic and inelastic scattering processes driven by matter
disorder. In this Article, we employ the Ioffe-Regel criterion to obtain
vibrational polariton mobility edges and to identify distinct regimes of
delocalization and transport under variable experimental conditions of light-
matter detuning, disorder, and interaction strength. Correlations between the
obtained trends and recent observations of polariton effects on reactivity are
discussed, and essential differences between transport phenomena in organic
electronic exciton and vibrational polaritons are highlighted. Our transport
diagrams show the rich diversity of transport phenomena under vibrational
strong coupling and indicate that macroscopic delocalization is favored at
negative detuning and large light-matter interaction strength. We also find
the surprising feature that, despite the presence of dephasing-induced
inelastic scattering processes, macroscopic lower polariton delocalization and
wave transport are expected to persist experimentally, even in modes with
small photonic weight.

##  I Introduction

Electromagnetic field confinement enables strong interactions between
molecular excitations and optical microcavity resonances, leading to the
formation of hybrid modes denoted molecular polaritons [1, 2, 3, 4, 5]. Recent
experiments have shown that certain polaritonic materials have desirable
properties for applications in chemical and materials science, such as
controllable charge conductivity [6, 7], energy transfer phenomena [8, 9, 10,
11, 12], and chemical reactivity [13, 14, 15, 16, 17, 18]. However, the
optimal conditions and mechanisms underlying various observations of polariton
effects on chemical systems remain unknown [19, 20, 4]. For instance,
experiments operating under strong light-matter interactions between bright
normal modes of a 3D molecular ensemble and resonant planar infrared (IR)
microcavities have suggested vibrational polariton formation induces cavity-
assisted ultrafast intermolecular energy transfer [11] and chemical reactivity
modulation [13, 14, 15, 16, 17, 18]. Theoretical studies [21, 22, 23, 24] have
suggested the reported vibrational strong coupling (VSC) effects on reactivity
are caused by anomalous intra or intermolecular vibrational energy
redistribution induced by the light-matter coupling, but the key factors
determining the intensity and direction of the VSC-induced change on
reactivity and energy transfer remain largely unknown [19, 20, 4].

Figure 1: Transport phase diagram for vibrational (lower) polaritons emergent
from VSC between an ensemble of $-$C$-$Si of the PFA molecule and an IR
microcavity reported in Ref. [16] with $\hbar\Omega_{R}=15.9~{}\text{meV}$,
$\sigma_{M}=0.967~{}\text{meV}$ and $\gamma_{M}=3.97~{}\text{meV}$ (see
Subsec. II.6). The continuous red and blue curves contain the low and high-
energy mobility edges $q_{\text{min,inel}}^{\text{L}}$ and
$q_{\text{max,inel}}^{\text{L}}$ for a given detuning. The dotted red and blue
curves correspond to the lower and upper boundaries of the delocalization-
localization transition in wave number space according to the Ioffe-Regel
criterion in the absence of inelastic scattering. Table 1 provides a
description of the transport phenomena expected in each region.

Despite the preponderance of questions on the impact of collective (ensemble)
vibrational strong coupling on energy transfer in IR microcavities,
theoretical and experimental investigations of space-time resolved polariton
propagation have been primarily restricted to electronic (organic) exciton-
polaritons [25, 26, 27].

Polariton propagation and energy transport phenomena in disordered organic
electronic exciton-polariton systems was first examined theoretically by
Agranovich et al. [28] (see also [29]). This pioneering work showed that
ballistic, diffusive, and subdiffusive (Anderson localization) propagation are
likely to coexist in polaritonic materials due to the large energetic disorder
in molecular aggregate samples. Corroboration from numerical simulations was
later provided in 1D by Michetti and LaRocca [30] and Agranovich and Gartstein
[31], and experimentally examined in Ref. [32]. Recent experiments have shown
additional features of organic exciton-polariton propagation, including
quality factor-dependent speed [25], tunable energy transport velocity and
effective diffusion constant [27], and superfluid transport under intense
pumping (above the condensation threshold) [33].

Collective IR strong coupling shows significant differences relative to
organic exciton strong coupling. These differences include the much longer IR
microcavity lengths relative [16, 34] to the organic, the one to two orders of
magnitude weaker collective light-matter interaction strengths achieved with
dipole-active high-frequency molecular vibrational transitions relative to the
bright electronic excitations of organic materials, and the much narrower
vibrational line shapes. The stark contrast in light-matter interaction
properties of organic and vibrational excitations suggests the wave functions
of vibrational polaritons and their corresponding energy transport processes
have qualitatively distinct features relative to those of organic exciton-
polaritons. In this work, we present a detailed theoretical analysis of the
properties of transport phenomena in disordered systems under VSC.

We note that several recent studies have investigated polariton and weakly
coupled wave functions [35, 36, 37, 38, 39, 40, 41] and energy transfer
dynamics [42, 43, 44, 45, 46, 47] in disordered 0D (with the optical cavity
treated as a zero-dimensional device represented by either one or two isolated
boson modes with orthogonal field polarizations) and 1D (with a one-
dimensional set of photonic modes) [48, 49, 50, 51] environments. However, as
is well known from wave function localization theory [52, 53, 54, 55], key
qualitative aspects of wave transport are strongly dependent on the number of
spatial dimensions and low-dimensional properties are not generalizable [52,
53, 55] (e.g., the tight-binding model with short-range hopping only has a
matter-insulator disorder induced quantum phase transition in 3D [53, 56]).

In this Article, we adopt the Ioffe-Regel criterion (IRC) to examine
vibrational polariton transport phenomena in a 3D disordered medium. We
unravel optimal conditions for maximization of the density of macroscopically
delocalized polariton modes in a molecular material under VSC with a planar
microcavity and present vibrational polariton transport phase diagrams (see
Fig. 1) identifying different polariton transport phenomena as a function of
cavity-matter detuning, energetic disorder, and molecular dephasing rate for
selected experimental systems. Our findings indicate that the density of
delocalized polaritons modes is larger for strong coupling between redshifted
microcavities and molecular systems with slow dephasing and large collective
light-matter interaction strength (Rabi splitting). We also report that the
density of macroscopically delocalized polaritons is drastically more impacted
by a change in the Rabi splitting than by a change of detuning, and we explore
the connection of these trends to recent experimental observations of
polariton chemistry dependence on collective light-matter interaction
strength.

This Article is organized as follows. Section II reviews the theoretical
framework, including a description of the light-matter Hamiltonian, disorder-
induced scattering processes, and the criteria we employed in the
classification of polariton wave functions and transport regimes. In Sec. III,
we present our main results, including vibrational polariton transport phase
diagrams and the density of macroscopically delocalized polariton modes at
variable parameters of the light-matter system. A comparison between transport
phenomena in organic electronic exciton and vibrational polaritons is provided
in Subsec. III.4. We conclude by presenting a summary of our results and a
discussion of implications to future polariton chemistry research in Sec. IV.

##  II Theory

In this section, we explain the theoretical formalism employed to characterize
vibrational polariton transport phenomena in macroscopic leaky planar
microcavities. Our goal is to construct transport phase diagrams (see Fig. 1
for an example of a detuning$-$wave number diagram) for vibrational polaritons
and to identify optimal conditions for macroscopic delocalization under VSC.
We employ the IRC [57, 54] to obtain nonperturbative estimates of the low and
high-energy boundaries (mobility edges) separating regions in the (in-plane)
wave number space where polaritons show macroscopic delocalization from those
regions containing localized modes. The latter include weakly coupled
molecular states with small photon content as well as low-wave number
polariton modes, which are delocalized on a molecular length scale yet have
finite localization length [58]. The scattering mean free paths required as
input to the IRC are obtained following Litinskaya and Reineker [29], who
applied a method from Lifshitz [59] to obtain elastic and inelastic scattering
polariton rates from the microscopic light-matter Hamiltonian in a disordered
medium.

For the sake of completeness and to introduce our notation, we summarize the
employed methodology below. In Subsec. II.1, we introduce the light-matter
Hamiltonian and the mean-field solutions employed in Subsec. II.2 to obtain
polariton scattering mean free paths originating from the various types of
disorder-induced scattering treated in this work. The IRC for wave function
localization is revisited in Subsec. II.3, and the classification of polariton
transport phases is described in Subsec. II.4. In Subsec. II.5, we present the
method we applied to estimate the density of delocalized LP modes following
the IRC. We conclude the presentation of the theoretical tools used in this
work in Subsec. II.6 where we provide some comments on the choice of
experimental parameters employed in subsequent quantitative analysis,
including a description of how we estimated static disorder parameters for
experimental systems that did not include such data.

###  II.1 Light-Matter Hamiltonian and mean-field solutions

The investigated systems consist of an ideal planar (Fabry-Perot) microcavity
under collective strong coupling with a molecular ensemble. The optical cavity
is formally infinite and periodic along the in-plane directions $x,y$, and it
is finite with length $L_{C}\in~{}O(\mu m)$ along the longitudinal $z$ axis.
The molecular material hosted by the photonic device is isotropic,
homogeneous, and has significant IR oscillator strength at a transition
sampled from a Gaussian probability distribution function.

The microcavity resonances are specified by their longitudinal wave vector
$k_{z}(m)=m\pi/L_{C},~{}m=\\{1,2,...\\}$, continuous in-plane wave vector
$\mathbf{q}=(q_{x},q_{y})$ corresponding to the projection of the wave
momentum along the mirror planes, and polarization $\lambda=\text{TE or TM}$
[60]. The empty cavity Hamiltonian is given by

$\displaystyle
H_{\text{L}}=\sum_{m\mathbf{q}\lambda}E_{C}(q,m)a_{m\mathbf{q}\lambda}^{\dagger}a_{m\mathbf{q}\lambda},$
(1)

where $q=\sqrt{q_{x}^{2}+q_{y}^{2}}$, $E_{C}(q,m)=\hbar\omega_{C}(q,m)$ and
the energy-momentum dispersion of the photonic subsystem is

$\displaystyle E_{C}(q,m)=E_{C}\sqrt{1+\left(\frac{qL_{C}}{m\pi}\right)^{2}},$
(2)

with $E_{C}\equiv E_{C}(0,m)=(\hbar c/\sqrt{\epsilon_{c}})\times m\pi/L_{C}$,
and $\epsilon_{c}$ is the background dielectric constant. This microcavity
model is approximate, for it assumes perfect mirrors and idealized modes with
infinite quality factor. The main effects of cavity decay (induced by coupling
to a continuum of free space EM modes [61, 62, 63]) are incorporated
perturbatively in our estimates of polariton inelastic scattering in Subsec.
II.2.

The molecular system is treated in the low-energy approximation where only the
states (namely, the ground and first excited state) with near-resonant
transitions with the photonic resonator are explicitly treated (the effects of
remaining intramolecular degrees of freedom inducing dephasing and polariton
decay are effectively accounted by inelastic scattering processes that we
discuss later). Intermolecular interactions are ignored since they provide
negligible bandwidth to the spectrum of the molecular system in every studied
case (as evidenced by the dispersionless molecular absorption spectrum outside
an optical microcavity of typical molecular systems in disordered phases). It
follows the bare molecule Hamiltonian can be written as

$\displaystyle
H_{M}=\sum_{i=1}^{N_{M}}\left(\hbar\omega_{M}+\sigma_{i}\right)\ket{1_{i}}\bra{1_{i}},$
(3)

where $\omega_{M}$ is the mean molecular transition frequency, and
$\sigma_{i}$ is a random variable sampled from a Gaussian probability
distribution function with variance $2^{-1/2}\sigma_{M}$ corresponding to
static disorder induced on the molecular normal-modes by slow system-
environment (solvent) correlations on the transport timescales relevant to us.
Note that our work is focused on vibrational polariton transport in the dilute
quasiparticle regime where the polariton density is always small enough that
nonlinearities of the light-matter Hamiltonian can be ignored.

In all investigated scenarios, we examine the polaritons originating from the
strong interaction of a single microcavity band with fixed $k_{z}\equiv
k_{z}(m)$ and the near-resonant molecular ensemble. The effects of other
photonic modes will be ignored as they are either highly off-resonant with the
molecular material, or in cases where the molecular resonance overlaps more
than a single photon band, the corresponding resonance in-plane wave vectors
are far enough from each other that the polaritons resulting from distinct
photonic bands may be treated independently.

The light-matter interaction is treated in the Coulomb gauge within the
rotating-wave-approximation

$\displaystyle
H_{\text{LM}}=\sum_{m\mathbf{q}\lambda}\sum_{n=1}^{N_{M}}g_{m\mathbf{q}\lambda}(\mathbf{R}_{n})a_{m\mathbf{q}\lambda}\sigma_{n}^{+}+g_{m\mathbf{q}\lambda}^{*}(\mathbf{R}_{n})a_{m\mathbf{q}\lambda}^{\dagger}\sigma_{n}^{-},$
(4)

where $\sigma_{n}^{+}=\ket{1_{n}}\bra{G}$ and
$\sigma_{n}^{-}=\ket{G}\bra{1_{n}}$ and $\ket{G}$ is the vacuum state of the
light-matter system where all microcavity modes and (relevant) molecular
degrees of freedom are in their ground-state. Assuming the electromagnetic
environment is generated by a perfect planar microcavity, the interaction
strength $g_{m\mathbf{q}\lambda}(\mathbf{R}_{n})$ between molecule $n$
(treated as a point-dipole) located at $\mathbf{R}_{n}=(x_{n},y_{n},z_{n})$
and the $(m\mathbf{q}\lambda)$ mode of the EM field is given in the Coulomb
gauge by

$\displaystyle
g_{m\mathbf{q}\text{TE}}(\mathbf{R}_{n})=-i\frac{E_{M}}{E_{C}(q)}\sqrt{\frac{E_{C}(q)}{2\epsilon}}\frac{e^{i\mathbf{q}\cdot\mathbf{R}_{n}}}{\sqrt{S}}\mu_{n}\cdot\mathbf{f}_{m\mathbf{q}\text{TE}}(\mathbf{R}_{n}),$
(5) $\displaystyle
g_{m\mathbf{q}\text{TM}}(\mathbf{R}_{n})=-i\frac{E_{M}}{E_{C}(q)}\sqrt{\frac{E_{C}(q)}{2\epsilon}}\frac{e^{i\mathbf{q}\cdot\mathbf{R}_{n}}}{\sqrt{S}}\mu_{n}\cdot\mathbf{f}_{m\mathbf{q}\text{TM}}(\mathbf{R}_{n}),$
(6)

with mode profile vectors $\mathbf{f}_{m\mathbf{q}\text{TE}}(\mathbf{R})$ and
$\mathbf{f}_{m\mathbf{q}\text{TM}}(\mathbf{R})$ given by [64, 65]

$\displaystyle\mathbf{f}_{m\mathbf{q}\text{TE}}(\mathbf{R})=\sqrt{\frac{2}{L_{C}}}\text{sin}\left(k_{z}z\right)\mathbf{e}_{z}\times\mathbf{e}_{\mathbf{q}},$
(7)
$\displaystyle\mathbf{f}_{m\mathbf{q}\text{TM}}(\mathbf{R})=\sqrt{\frac{2}{L_{C}}}\frac{k^{2}}{k_{z}^{2}}\left[\text{sin}\left(k_{z}z\right)\mathbf{e}_{\mathbf{q}}-\frac{iq}{k_{z}z}\text{cos}(k_{z}z)\mathbf{e}_{z}\right],$
(8)

where $k^{2}=k_{z}^{2}+q^{2}$, $\mathbf{e}_{z}$ is the unit vector along the
$z$ axis, $\mathbf{e}_{\mathbf{q}}=(q_{x},q_{y})/q$, the molecular transition-
dipole moments $\mu_{n}$ are uncorrelated random vectors sampled from a
uniform distribution on a circle along the $xy$ plane (for simplicity since
this assumption is inconsequential), and the molecular positions
$\mathbf{R}_{n}$ are uniformly distributed with a mean distance between
neighboring molecules equal to $a\equiv\rho^{-1/3}$ where $\rho$ is the
molecular density.

The disorder makes it challenging to extract the exact spectrum of the total
light-matter Hamiltonian

$\displaystyle H=H_{\text{M}}+H_{\text{L}}+H_{\text{LM}},$ (9)

due to the large number of coupled light and matter degrees of freedom.
However, approximate solutions can be obtained in the weak disorder limit
where elastic and inelastic scattering processes are irrelevant, leading to
the mean-field lower polariton (LP) and upper polariton (UP) energies
$E_{\text{L}}^{(0)}(q)$ and $E_{\text{U}}^{(0)}(q)$ [66, 67]

$\displaystyle
E_{\text{U,L}}^{(0)}(q)=\frac{E_{M}+E_{C}(q,m)}{2}\pm\frac{1}{2}\sqrt{\left(\hbar\Omega_{R}\right)^{2}+\left[E_{M}-E_{C}(q,m)\right]^{2}},$
(10)

where $\Omega_{R}$ is the Rabi splitting frequency (collective light-matter
interaction strength). The photonic and molecular weights of the mean-field LP
and UP solutions can be straightforwardly obtained from the Schrodinger
equation using the same assumptions employed to obtain Eq. 10 [29, 68]. While
the polariton energies obtained from Eq. 10 resemble those given by the Tavis-
Cummings model [69, 70], this occurs here solely because, in the mean-field
limit, Eq. 9 can be decomposed into a direct sum of $2\times 2$ Hamiltonians
describing the interaction of the cavity mode $(m,\mathbf{q},\lambda)$ with a
collective matter polarization mode with same symmetry [71, 29]. Nevertheless,
our treatment of vibrational polariton propagation below relies fundamentally
on the interaction of a common set of molecules with multiple EM modes. The
mean-field polariton eigenvalues and eigenfunctions provide the starting point
for the treatment of polariton transport in disordered media described in the
next subsections.

###  II.2 Scattering mechanisms and mean free paths

Molecular energetic, dipolar, and translational disorder and system-
environment interactions may cause polariton scattering, randomization of
their in-plane wave vector, and incoherent transitions between polaritons and
reservoir modes [28, 29, 58]. Elastic scattering induced by nonvanishing
static energetic disorder and structural medium fluctuations preserve the wave
vector magnitude $q$ but randomizes its direction, whereas inelastic
scattering (photon leakage or via molecular system-bath interactions) also
dissipates energy from the polariton system and therefore does not conserve
$q$.

The average distance between elastic [inelastic] scattering events for a wave
packet with mean momentum $q$ in the weak scattering limit is given by the
mean free path $l_{\text{el}}(q)$ [$l_{\text{inel}}(q)$]. As discussed below,
the mean free paths associated with distinct scattering mechanisms may be
employed to characterize [54, 55, 58, 72, 73] the transport properties of
polariton quasiparticles.

The mean polariton free path $l_{s}^{\text{P}}(q)$ ($\text{P}=\text{LP}$ or
UP), associated with the scattering mechanisms $s=$ inelastic, resonance
(elastic), and induced by fluctuations of molecular orientation and positions
(elastic), can be obtained from the corresponding scattering rates
$\Gamma_{s}^{\text{P}}(q)/\hbar$ and the (mean-field) polariton group velocity
$v_{\text{P}}(q)$ via

$\displaystyle l_{s}^{\text{P}}(q)=\frac{\hbar
v_{\text{P}}(q)}{\Gamma_{s}^{\text{P}}(q)},$ (11) $\displaystyle
v_{\text{P}}(q)=\frac{1}{\hbar}\frac{\partial E_{\text{P}}^{(0)}(q)}{\partial
q}=\left(\frac{L_{C}}{m\pi}\right)^{2}\frac{qE_{C}^{2}}{\hbar
E_{C}(q,m)}P_{C}^{\text{P}}(q),$ (12)

where $P_{C}^{\text{P}}(q)$ is the photon weight in the corresponding (mean-
field) polariton mode. The wave-vector broadening induced by the scattering
process $s$ on the polariton mode $\text{P}(q)$ is given by $\delta
q_{s}^{\text{P}}(q)=\left[l_{s}^{\text{P}}(q)\right]^{-1}$ [55].

Elastic scattering can be induced by static (diagonal) energetic disorder (Eq.
3) and structural fluctuations of the environment (off-diagonal disorder, Eqs.
4). As we show below, elastic scattering produced by the inhomogeneous
distribution of molecular transition energies is dominant. This process,
typically denoted by resonance scattering [74, 29] occurs when a polariton
with wave vector $\mathbf{q}$ is converted into another with wave vector
$\mathbf{q}^{\prime}$ with $q^{\prime}=q$, via an intermediate nearly resonant
weakly coupled molecular excited-state. Resonance scattering is often
considered to be weak relative to absorptive processes (e.g., “decay into dark
modes”). However, as shown by Agranovich et al. [75], and Litinskaya et al.
[76] in their earlier studies of inorganic microcavities, resonance scattering
can dominate over inelastic processes under experimentally achievable
scenarios where the energy scale of polariton inelastic scattering is smaller
than the energetic disorder.

When the density of states of the bare molecular system is significant at
$E_{\text{P}}^{(0)}(q)$, resonance scattering is expected to be strong, and a
perturbative treatment is invalid. However, a nonperturbative estimate of the
scattering rate $\Gamma_{\text{res}}^{\text{P}}(q)$ may be obtained as the
imaginary part of the complex polariton energies $E_{\text{LP}}(q)$ or
$E_{\text{UP}}(q)$ arising as solutions to the complex eigenvalue equation
(originating from Eq. 9 after assuming scattering induced by structural
fluctuations can be ignored [29])

$\displaystyle
E(q)-E_{C}(q,m)=\frac{\left(\hbar\Omega_{R}\right)^{2}}{4}\int_{-\infty}^{\infty}dE_{j}\frac{\rho(E_{j})}{E(q)-E_{j}}$
$\displaystyle=\frac{\left(\hbar\Omega_{R}\right)^{2}\sqrt{\pi}}{4i\sigma}e^{-[E(q)-E_{M}]^{2}/\sigma^{2}}\text{erfc}[-i[E(q)-E_{M}]/\sigma],$
(13)

where
$\rho(E_{j})=(1/\sigma_{M}\sqrt{\pi})\text{exp}\left[-(E_{j}-E_{M})^{2}/\sigma_{M}^{2}\right]$,
and erfc is the complementary complex error function. This equation can be
solved numerically to find $\Gamma_{\text{res}}^{\text{P}}(q)$ as
$\text{Im}\left[E_{\text{P}}(q)\right]=\Gamma_{\text{res}}^{\text{P}}(q)$ for
application in Eq. 11 providing the resonant scattering mean free path for
polariton mode $P$ with wave number $q$.

Fourth-order perturbation theory in structural deviations from the mean [29]
provides an estimate of the mean free path for scattering induced by medium
fluctuations from the corresponding elastic scattering rate

$\Gamma_{\text{fluc}}^{\text{P}}(q)\approx\frac{3m^{2}\pi^{2}}{2}\left(\frac{a}{L_{C}}\right)^{3}\frac{\left(\hbar\Omega_{R}\right)^{2}}{4E_{C}}P_{M}^{\text{P}}(q),$
(14)

where $P_{M}^{\text{P}}(q)$ denotes the total molecular probability in the
polariton mode $\text{P}(q)$. Note that in VSC of small and medium-sized
molecules with Fabry-Perot cavities, the intermolecular distance $a$ is almost
always on the order of a few to tens of nanometers, whereas
$L_{C}=O\left(10^{3}-10^{4}~{}\text{nm}\right)$, and $\hbar\Omega_{R}\ll
E_{C}$. Therefore, $\Gamma_{\text{fluc}}$ will almost always be negligible
relative to resonance elastic scattering and inelastic decay, especially at
$q\gg 0$.

In the absence of strong interactions between the normal-modes strongly
coupled to the microcavity, inelastic scattering processes induce polariton
decay via at least two mechanisms: (i) local interactions between the strongly
coupled molecular states and intra or intermolecular (e.g., low-frequency
solute-solvent) bath modes which induce polariton dephasing (via decay into
weakly coupled or dark modes)[58, 77], and (ii) photon escape via leaky cavity
mirrors [60]. Both processes extinguish polaritons while transferring energy
irreversibly into a thermal reservoir (molecular and photonic, respectively)
with a significantly larger density of states. The rates of both types of
events are given by $\gamma_{M}/\hbar$ and $\gamma_{C}/\hbar$, respectively.
The cavity leakage rate $\gamma_{C}$ can be obtained from the bare cavity mode
linewidth or quality factor [60, 78] $Q=\hbar\omega_{C}/\gamma_{C}$, whereas
$\gamma_{M}$ can be obtained from the molecular homogeneous absorption
linewidth. In this work, we obtain an approximation for $\gamma_{M}$ from the
polariton linewidth at resonance [79] (see Subsec. II.6). Recently, Li et al.
reported an additional contribution to polariton inelastic scattering via
direct interactions between nearest neighbor molecular modes contributing to
polaritons [80]. In Appendix B, we provide an approximate upper bound to this
intermolecular dephasing mechanism for systems where the solute of a mixture
undergoes VSC (like those we examined in this work, see Table
LABEL:tab:parameters), numerically compare it to Eq. 15, and show that the
processes (i) and (ii) listed above provide the dominant inelastic scattering
pathways (we also note that, in scenarios where strong coupling occurs with
neat liquids or molecular crystals, the dephasing pathway introduced in Ref.
[80] likely plays a more important role).

Assuming $\gamma_{M}$ is dominated by dephasing induced by interactions of
each molecular normal-mode with a low-frequency bath, the total inelastic
scattering rate for a polariton with in-plane momentum $q$ can be estimated by
(see Appendix A for a derivation)

$\Gamma_{\text{inel}}^{\text{P}}(q)=P_{M}^{\text{P}}(q)\gamma_{M}+P_{C}^{\text{P}}(q)\gamma_{C}.$
(15)

Note that, in principle, $\Gamma_{\text{inel}}^{\text{P}}(q)$ includes both
processes that conserve the polariton-number [e.g.,
$\text{P}(q)\rightarrow\text{P}^{\prime}(q^{\prime})$ with $q^{\prime}\neq q$
and P and $\text{P}^{\prime}$ refer to either LP or UP], and processes where a
polariton decays into a weakly coupled molecular state or decays via emission
of a photon bringing the system into the ground-state. The processes
annihilating polaritons are expected to be dominant in collective strong
coupling measurements due to the large density of weakly coupled molecular
modes and ultrafast photon leakage via highly leaky mirrors. Regardless of
their outcomes, inelastic processes are induced by interaction with an
incoherent reservoir. Therefore, the inelastic scattering mean free path may
also be viewed as the polariton phase-breaking length since it provides the
average distance traveled by a wavepacket with mean wave number $q$ before the
phase relations between its components are randomized.

###  II.3 Ioffe-Regel criterion for localization

Using the scattering mean free paths associated with inelastic, resonant, and
fluctuation-induced elastic scattering obtained from the expressions above,
the regions in the $q$ space where vibrational polariton wave functions are
macroscopically delocalized (can transport energy over arbitrarily large
distances) can be determined via the Ioffe-Regel criterion [57, 54].

According to the IRC, polariton localization occurs when the (in-plane)
wavelength approaches its (shortest) mean free path. Specifically, the wave
number space boundaries between macroscopically delocalized and localized
polariton modes are obtained from the condition

$\displaystyle ql^{\text{P}}(q)=\frac{2\pi l^{\text{P}}(q)}{\lambda(q)}\approx
1,$ (16)

where $l^{\text{P}}(q)$ is the mean free path corresponding to the strongest
scattering process perturbing $\text{P}=$ LP or UP with wave vector
$\mathbf{q}$ and $\lambda(q)$ is the in-plane wavelength.

For a given set of parameters characterizing the light-matter system (detuning
$u=E_{C}-E_{M}$, Rabi splitting $\Omega_{R}$, static disorder $\sigma_{M}$,
cavity leakage rate, $\gamma_{C}$, and molecular homogeneous line width
$\gamma_{M}$), the wave number boundaries $q_{\text{min}}$ and
$q_{\text{max}}$ separating localized polariton modes ($q>q_{\text{max}}$ and
$q<q_{\text{min}}$) from delocalized ($q_{\text{min}}<q<q_{\text{max}}$)
determine the mobility edges for polariton transport. The IRC for wave
localization can also be equivalently written as $q/\delta q<1$ in terms of
the mean in-plane wave number fluctuation $\delta q=1/l^{\text{P}}(q)$.

Notably, while the IRC is understood to provide only an approximate estimate
for the mobility edges of a disordered system, it has numerical support from
multiple works [76, 81, 72] and may be justified formally [74, 82] using the
scaling theory of localization [53, 55] and the self-consistent theory of
localization [83, 55].

To find the LP and UP mobility edges for a given light-matter system, we first
(numerically) solve the independent IRC equations associated with each
scattering mechanism $s$

$\displaystyle ql^{\text{P}}_{s}(q)=1,~{}~{}s\in\\{\text{fluc, res, inel}\\}.$
(17)

The set of solutions of this equation may have 0, 1, or 2 elements depending
on $s$ and the parameters of the light-matter system. For instance, the
resonance scattering LP mean free path is, in general, a nonmonotonic function
of $q$ at zero or negative detuning. Therefore, the generic set of solutions
of Eq. 17 for LP resonance scattering is given by
$Q_{\text{res}}^{\text{LP}}\equiv\\{q_{\text{min,res}}^{\text{L}},q_{\text{max,res}}^{\text{L}}\\}$,
where min and max refer to the corresponding minimum and maximum solutions.
This implies that in the absence of inelastic and fluctuation scattering, only
LP modes with $q_{\text{min,res}}^{\text{L}}<q<q_{\text{max,res}}^{\text{L}}$
would show macroscopic delocalization.

In comparison to resonance scattering, inelastic processes have qualitatively
distinct effects on polariton wave packet evolution that we have anticipated
above and will further discuss below. Nevertheless, inelastic mean free paths
also show, in general, nonmonotonic behavior with $q$ (Fig. 3). We denote the
associated typical solution set of Eq. 17 by
$Q_{\text{inel}}^{\text{L}}\equiv\\{q_{\text{min,inel}}^{\text{L}},q_{\text{max,inel}}^{\text{L}}\\}$.
In principle, these solutions carry the same interpretation as the resonant
case discussed above: only LP modes with
$q_{\text{min,inel}}^{\text{L}}<q<q_{\text{max,inel}}^{\text{L}}$ are
delocalized over macroscopic distances according to the IRC.

Conversely, for the systems examined in this article, the LP fluctuation mean
free path is always macroscopic (Fig. 3), and the generic solution set of Eq.
17 for $s=\text{fluc}$ has, a single element,
$Q_{\text{fluc}}^{\text{L}}\equiv\\{q_{\text{fluc}}^{\text{L}}\\}$. This
implies that in the absence of energetic disorder and inelastic scattering,
all LP modes with $q>q_{\text{fluc}}^{\text{L}}$ are delocalized.

The sets $Q_{\text{fluc}}^{\text{L}},Q_{\text{res}}^{\text{L}}$ and
$Q_{\text{inel}}^{\text{L}}$ contain all information required to construct the
LP transport phase diagrams. Strictly speaking, long-lived propagating
polariton modes must satisfy $ql_{s}^{\text{P}}(q)>1$ for each type of
scattering $s$. Therefore, the minimum $q$ (low-energy mobility edge) is
determined by the scattering process $s$ with a maximum value of
$q_{\text{min},s}$, since $ql_{s}^{\text{L}}(q)<1$ for all
$q<q_{\text{min},s}$ implying localized polariton modes according to the IRC.
On the other hand, the high-energy mobility edge is set by the scattering
process with a minimum value of $\\{q_{\text{max},s}\\}$, since strong
localization of the polariton wave function happens for $q>q_{\text{max}}$
regardless of the effectiveness of other scattering mechanisms. To summarize,
the minimum and maximum in-plane wave numbers (low and high-energy mobility
edges) for macroscopically delocalized LP modes are given in general by

$\displaystyle
q_{\text{min}}^{\text{L}}=\text{max}\left\\{q_{\text{min,res}}^{\text{L}},q_{\text{min,fluc}}^{\text{L}},q_{\text{min,inel}}^{\text{L}}\right\\},$
(18) $\displaystyle
q_{\text{max}}^{\text{L}}=\text{min}\left\\{q_{\text{max,res}}^{\text{L}},q_{\text{max,inel}}^{\text{L}}\right\\}.$
(19)

A similar analysis can be performed for UP, but in that case, we only find a
low-energy mobility edge as Eq. 17 admits, in general, a single solution
$q_{\text{min}}^{\text{U}}$. All modes with $q<q_{\text{min}}^{\text{U}}$ are
localized, while UP states with $q>q_{\text{min}}^{\text{U}}$ display
macroscopic delocalization. This occurs even in redshifted microcavities since
only a small fraction of UP modes have significant molecular content. Note
that for sufficiently large $q$, UP modes may be regarded as photons weakly
perturbed by off-resonant molecular scattering, see, e.g., Fig. 2). Therefore,
from now on, we will focus our analysis of transport via the LP modes emergent
from VSC.

Table 1: Summary and description of polariton classification criteria adopted in this work. Below and elsewhere in the text, $l_{\text{el}}^{\text{L}}$ refers to the shortest elastic mean free path among resonance and fluctuation-induced elastic scattering. Likewise, $q_{\text{min,el}}^{\text{L}}$ and $q_{\text{max, el}}^{\text{L}}$ refer to the solutions of Eqs. 18 and 19 in the absence of inelastic scattering (i.e., when $q_{\text{min,inel}}=0$ and $q_{\text{max,inel}}\rightarrow\infty$). Mode character |  Criterion |  Description
---|---|---
delocalized |  $\lambda<2\pi l_{s}^{\text{L}}$ |  Delocalized modes are the optimal energy transport carriers. In other words, wave packets consisting of a narrow superposition of polariton modes with wave vector centered around $\mathbf{q}$ with $ql_{s}^{\text{L}}(q)>1$ travel ballistically with a group velocity $|\nabla_{\mathbf{q}}\omega_{P}(\mathbf{q})|$ prior to elastic and inelastic scattering times [84, 55]. However, delocalized modes can behave in significantly different fashions depending on their main scattering mechanism. When elastic scattering prevails over inelastic collisions, interference effects lead to weak localization, i.e., a reduced diffusion constant and enhanced coherent backscattering [55]. The classical diffusive regime (with decay) for energy transport arises only when incoherent inelastic scattering that conserves the polariton number occurs on a faster timescale than elastic ($l_{\text{inel}}<l_{\text{el}}$).
strongly damped propagating |  $2\pi l_{\text{el}}^{\text{L}}>\lambda>2\pi l_{\text{inel}}^{\text{L}}$ |  Assuming that the dominant effect of inelastic scattering is to induce polariton light emission or molecular absorption via dephasing, strongly damped propagating modes are expected to transport energy with weak elastic scattering within their lifetimes.
localized LPs |  $\lambda>2\pi l_{\text{el}}^{\text{L}}$ and $q<q_{\text{min,el}}^{\text{L}}$ |  Localized LPs satisfy the IRC for localization and have small enough $q$ that their photon content is significant (typically the case for redshifted or near-resonant cavities). Conventional kinetic theory fails for these modes. If a localized wave packet is launched with mean in-plane wave number $q$ satisfying the criterion given here, it is expected to show transient diffusive behavior and localization. If inelastic scattering dominates, then it is likely that the wave packet will decay before showing signatures of strong localization.
(localized) weakly coupled excitons |  $\lambda>2\pi l_{\text{el}}^{\text{L}}$ and $q>q_{\text{max}}^{\text{L}}$ |  The majority of these states has less than 1$\%$ photonic content. Note that there is no sharp boundary between localized LP and weakly coupled molecular modes (Fig. 1). However, we make a distinction between these two types of localized states because LP modes with $q\rightarrow 0$ are expected to show a delocalization length that is large on the molecular length scale (as verified by numerical simulations [31, 30]), whereas weakly coupled molecular modes are expected to localize over much smaller distances [50, 49, 51].
|  |

###  II.4 Polariton transport phases

The IRC allows us to distinguish macroscopically delocalized polaritons from
strongly localized modes. This classification may be further refined by
recognizing two essential aspects of polariton kinetics.

First, while modes with $q<q_{\text{min}}^{\text{L}}$ and
$q>q_{\text{max}}^{\text{L}}$ are both strongly localized according to the
IRC, their localization length can differ by many orders of magnitude [30, 31,
58]. In particular, LP modes with $q<q_{\text{min}}^{\text{L}}$ have
significant photon content and are expected to be localized over length scales
that are orders of magnitude larger than LP modes with insignificant photonic
weight and $q\gg q_{\text{max}}^{\text{L}}$. Therefore, in the phase diagrams
we construct here, we make a distinction between localized LP modes and weakly
coupled excitons. Both types of modes are strongly localized in the
macroscopic sense and belong to the LP branch of the polariton dispersion, but
the vast majority of the weakly coupled excitons have almost insignificant
photonic content, whereas the inverse is true for localized LP modes. Note
this distinction is also suggested by Agranovich [68] and Litinskaya [58] and
has also been highlighted in recent experimental work [85].

Second, we denote by “overdamped propagating modes”, polariton states which
are only considered to be localized according to IRC due to strong inelastic
scattering. The corresponding modes would be delocalized in the absence of
molecular dephasing ($\gamma_{M}=0$) and photon leakage ($\gamma_{C}=0$).
Given that polariton decay by cavity leakage or via dephasing into the weakly
coupled reservoir are the dominant inelastic scattering processes, we expect
that overdamped propagating modes can transport energy efficiently during
their lifetime. In other words, polariton wave function strong localization is
unlikely to be a generic feature of overdamped propagating modes, although
more work is needed to rigorously characterize the transport properties of
these states. A summary of our classification scheme is given in Table 1.

###  II.5 Density of delocalized LP modes

For the various light-matter systems investigated in this paper, we also
analyze under various conditions the total density of delocalized LP modes
$\rho_{L}^{\text{deloc}}$. This quantity provides a simple global measure of
delocalization in a polaritonic system. Given $q_{\text{min}}^{\text{L}}$ and
$q_{\text{max}}^{\text{L}}$, we find $\rho^{\text{deloc}}_{L}$ from

$\rho^{\text{deloc}}_{L}\equiv\frac{1}{2\pi}\int_{q_{\text{min}}^{L}}^{q_{\text{max}}^{L}}\mathrm{d}q~{}q=\frac{\left(q_{\text{max}}^{\text{L}}\right)^{2}-\left(q_{\text{min}}^{\text{L}}\right)^{2}}{4\pi}.$
(20)

Note that we only report the density of delocalized LP modes since the
corresponding density of delocalized UP modes would be dependent on the choice
of high-energy cutoff momentum. In any case, the fraction of delocalized UP
modes is expected to be close to one since only a small fraction of UPs (with
generally near zero in-plane wave number) have significant molecular content
and may become localized (see, e.g., Fig. 2).

###  II.6 Light-matter systems and parameters

The set of parameters employed to obtain the vibrational polariton phase
diagrams presented in this Article are listed in Table LABEL:tab:parameters.
All input parameters were either obtained directly or derived from
experimental data in the cited references. The molecular energetic disorder
parameters $\sigma_{M}$ for the systems in Refs. [16, 33] were estimated in
the following way: the relevant bare molecule excitations were assumed to have
a Voigt line shape [86] with Lorentzian FWHM $\gamma_{M}$ and Gaussian
inhomogeneous broadening $\sigma_{M}$. The parameter $\gamma_{M}$ was obtained
from the (approximate) LP line width
$\gamma_{\text{LP}}=(\gamma_{M}+\gamma_{C})/2$ at resonance [79]. Using the
experimental value for the Voigt line width $f_{V}$ and the approximate form
[87]

$\displaystyle
f_{V}\approx\frac{\gamma_{M}}{2}+\sqrt{\frac{\gamma_{M}^{2}}{4}+4\sigma_{M}^{2}\text{ln}(2)},$
(21)

we solved for $\sigma_{M}$ in each case to obtain the values reported in Table
LABEL:tab:parameters.

##  III Results and Discussion

We provide transport phase diagrams and the density of delocalized polariton
modes for several systems under vibrational strong coupling in this section.
Our focus is on IR polariton energy transport, but we also provide a
comparison to a recently studied electronic (organic) exciton-polariton system
below. Table LABEL:tab:parameters includes the input parameters, LP mobility
edges, and density of delocalized LP modes for selected systems.

In Subsec. III.1, we begin our analysis of vibrational polariton transport by
investigating $q-u~{}(\text{wave number}$-detuning) transport phase diagrams
obtained with parameters representing the vibrational strong coupling between
the $m=2$ band of an IR microcavity with a normal-mode involving a significant
contribution from the $\text{C}-\text{Si}$ bond of
1-phenyl-2-trimethylsilylacetylene (PTA) studied recently by Thomas et al.
[16]. This study established that an earlier reported VSC effect on chemical
reactivity [13] can be controlled by varying the light-matter interaction
strength via manipulation of the molecular density.

Subsequently, we analyze how the previously obtained detuning-dependent
transport phase diagram is modified in the limit where molecular dynamical
disorder (here represented by the bare molecule dephasing rate $\gamma_{M}$;
Inh. lim. case in Table LABEL:tab:parameters) vanishes and the bare molecule
line width is entirely due to inhomogeneous broadening, where $\sigma_{M}$
takes its maximal value (assuming a Voigt line shape as explained in Sec.
II.6). This discussion is followed with an exploration of the transport phase
diagram obtained in the homogeneous broadening limit where $\sigma_{M}=0$ and
$\gamma_{M}$ is maximized (Hom. lim. in Table LABEL:tab:parameters).

Subsection III.2 examines the effects of variable disorder on polariton
transport phenomena. For a fixed (zero) detuning, we establish the universal
forms of the $q-\sigma_{M}$ and $q-\gamma_{M}$ transport phase diagrams
providing the critical disorder and dephasing rates beyond which no
macroscopic delocalized LP mode exists according to the IRC criterion.

In Subsection III.3, we present an analysis of the effect of the collective
light-matter interaction strength on the mobility edges and density of
delocalized polariton modes at zero detuning and $\sigma_{M}$ and $\gamma_{M}$
fixed according to the experimental data of Ref. [16].

We conclude this section with a comparison of the main features of organic
exciton and vibrational polariton by comparing their (detuning dependent)
transport phase diagrams (Subsec. III.4).

Table 2: Input parameters, low and high-energy LP mobility edges (minimum and maximum boundaries, $q_{\text{min}}$ and $q_{\text{max}}$ respectively), and density of delocalized LP modes $\rho_{L}^{\text{deloc}}\left(\text{cm}^{-2}\right)$ for the infrared and organic polariton systems examined in this work. Energies are given in meV, and wave vector magnitudes are given in cm-1. Inh. and Hom. refer respectively to inhomogeneous and homogeneous limits of the bare molecule line shape employed for theoretical analysis of systems with all parameters corresponding to a particular experiment except for the energetic disorder $\sigma_{M}$ and dephasing rate $\gamma_{M}/\hbar$ which take the values listed on the Table and explained in Sec. III.1. N/A means not available. | $E_{C}$ | $E_{M}$ | $\hbar\Omega_{R}$ | $\sigma_{M}/\hbar\Omega_{R}$ | $a~{}(\text{nm})$ |  $L_{C}~{}(\mu$m) | $\hbar\gamma_{M}$ | $\hbar\gamma_{C}$ | $q_{\text{min}}^{\text{U}}$ | $q_{\text{min}}^{\text{L}}$ | $q_{\text{max}}^{\text{L}}$ | $P_{C}\left(q_{\text{max}}^{\text{L}}\right)$ | $\rho_{L}^{\text{deloc}}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$-$C$-$Si (PFA) [16] | 106.6 | 106.6 | 15.87 | 0.06 | 0.69 | 6.0 | 3.97 | 3.47 | 2551 | 3595 | 6843 | 0.103 | $2.7\times 10^{6}$
$-$C$-$Si (PFA) (Inh.) | 106.6 | 106.6 | 15.87 | 0.18 | 0.69 | 6.0 | 0.00 | 3.47 | 1904 | 1904 | 11592 | 0.021 | $1.0\times 10^{7}$
$-$C$-$Si (PFA) (Hom.) | 106.6 | 106.6 | 15.87 | 0.00 | 0.69 | 6.0 | 4.84 | 3.47 | 2654 | $-$ | $-$ | $-$ | 0.00
$-$NCO stretch (HMDI) [34] | 269 | 281 | 13.2 | 0.16 | N/A | 1.6 | 3.31 | 6.2 | 4437 | 3237 | 11257 | 0.045 | $9.3\times 10^{6}$
TDAF exciton [33] | 3045 | 3500 | 600 | 0.52 | 1.13 | 0.13 | 18 | 5.0 | 93276 | 13501 | 234471 | 0.111 | $4.36\times 10^{9}$

###  III.1 Detuning dependent vibrational polariton transport phase diagrams

In order to obtain a typical transport phase diagram for a material under VSC,
we first examined the system studied experimentally in Ref. [16] consisting of
VSC between $-$C$-$Si (PFA) with an IR microcavity at zero detuning (other
relevant parameters are listed on the first row of Table
LABEL:tab:parameters). Fig. 2 shows the LP and UP polariton dispersion curves
for this system, along with their mobility edges $q_{\text{min}}^{\text{L}}$,
$q_{\text{max}}^{\text{L}}$ and $q_{\text{min}}^{\text{U}}$ separating the
macroscopically delocalized LP and UP modes from their localized counterparts
($q<q_{\text{min}}^{\text{L}}$ and $q>q_{\text{max}}^{\text{L}}$ for LP and
$q<q_{\text{min}}^{\text{U}}$ for UP). In this system, inelastic scattering is
stronger than elastic for all delocalized LP modes which satisfy $\lambda<2\pi
l_{\text{el}}$ and $\lambda<2\pi l_{\text{inel}}$ simultaneously. Therefore,
the localization-delocalization boundaries, according to the IRC, are all
determined by inelastic scattering. All UP modes with
$q<q_{\text{min}}^{\text{U}}=2551~{}\text{cm}^{-1}$ are localized (in the
thermodynamic limit), but their localization length is expected to be
macroscopic since their photonic content is greater than $50\%$ and $q\approx
0$. Note the absence of high-energy delocalization-localization UP boundary as
follows from the fact that at large $q$ the UP is dominated by its photonic
character and is only weakly affected by the disorder in the molecular system.

Figure 2: Top: Dispersion of plane wave-like states (continuous curves) and
corresponding localization-delocalization boundaries $q_{\text{min}}$,
$q_{\text{max}}$ for vibrational polaritons experimentally probed in Ref. [16]
(see the first row of table LABEL:tab:parameters). The low and high-energy
mobility edges $q_{\text{min}}^{\text{P}}$ ($\text{P}=\text{L}$ or U for LP
and UP, respectively) and $q_{\text{max}}^{\text{L}}$ separate delocalized
modes from the localized states satisfying the IRC with dominant inelastic
scattering $2\pi l_{\text{inel}}(q)<\lambda(q)$. Bottom: Photon content of the
corresponding UP and LP bands.

Conversely, LP has both low- and high-energy mobility edges at
$q_{\text{min}}^{\text{L}}=3595~{}\text{cm}^{-1}$ and
$q_{\text{max}}^{\text{L}}=6843~{}\text{cm}^{-1}$. Both are determined by
inelastic scattering since
$l_{\text{inel}}^{\text{L}}(q)<l_{\text{res}}^{\text{L}}(q)$ and
$l_{\text{fluc}}^{\text{L}}(q)$ for all relevant $q$ satisfying either
$ql_{\text{inel}}^{\text{L}}(q)>1$ (delocalized modes in the presence of both
inelastic and elastic scattering) or $ql_{\text{el}}^{\text{L}}(q)>1$
(delocalized modes according to the IRC only in the absence of inelastic
scattering).

In Fig. 3, we provide $1/q=\lambda(q)/2\pi$ and the mean free paths associated
with fluctuation, resonance, and inelastic scattering for the studied system.
The dominance of decay processes induced by inelastic over resonance (elastic)
scattering here observed is a consequence of the small inhomogeneous
broadening ($\sigma_{M}\approx 1~{}\text{meV}$) of the molecular system
relative to the photon leakage rate ($\gamma_{C}\approx 3.5~{}\text{meV}$) and
the molecular dephasing rate $(\gamma_{M}\approx 4~{}\text{meV})$. Note that
the resonance scattering mean free path eventually becomes shorter than the
inelastic mean free path at the strong localization region where $q>2.5\times
10^{4}~{}\text{cm}^{-1}$ and both elastic and inelastic mean free paths are
much smaller than the corresponding in-plane LP wavelength $2\pi/q$.

Figure 3: Comparison of mean free paths associated to scattering induced by
structural fluctuations (left) $l_{\text{fluc}}^{\text{L}}(q)$, and resonance
$l_{\text{res}}^{\text{L}}(q)$, and inelastic $l_{\text{inel}}^{(L)}(q)$
(right) processes for the system experimentally studied in Ref. [16] with
largest Rabi splitting (see first row of Table LABEL:tab:parameters). The
figures show that elastic scattering (resonance and medium fluctuation-
induced) is relatively weak compared to inelastic processes
[$l_{\text{inel}}^{\text{L}}(q)<l_{\text{res}}^{\text{L}}(q)$ and
$l_{\text{fluc}}^{\text{L}}(q)$] for all $q$ where LP is delocalized according
to the IRC.

The detuning-dependent transport phase diagram for the system investigated in
this section [16] is shown in Fig. 1 for variable detuning
$u\in[-26,63]~{}\text{meV}$. Here, we find, in agreement with earlier studies
[29, 58, 50], that a more negatively detuned optical cavity favors
delocalization by increasing the density of macroscopically delocalized LP
modes. In addition, we find that for detuning $u$ greater than approximately
0.2 meV, the LP phase-breaking length $l_{\text{inel}}^{\text{LP}}(q)$ is
smaller than the in-plane wavelength for all wave numbers, and no macroscopic
delocalized modes are likely to exist. In any case, there remains a finite
density of strongly damped propagating modes with
$q_{\text{min,res}}^{\text{L}}<q<q_{\text{max,res}}^{\text{L}}$ for all
$u\in(0.2~{}\text{meV},63~{}\text{meV})$ (Fig. 1 and Table 1). At detunings
$u$ greater than 63 meV, only localized LP and weakly coupled molecular modes
exist according to the employed classification scheme.

In Fig. 4, we utilize the mobility edges obtained in Fig. 1 to quantify the
density of delocalized LP modes per unit area for variable detuning. We also
report $\rho_{\text{L}}^{\text{deloc}}(u)$ obtained in the absence of
inelastic scattering (“elastic” curve). While the number of short and long-
lived delocalized LP modes per unit area increases as the detuning becomes
more negative, the effect of detuning on the density of delocalized LP modes
is small, especially in the presence of inelastic scattering, which
drastically reduces $\rho_{\text{L}}^{\text{deloc}}(u)$ for $u\leq 0$.

Figure 4: Detuning dependence of the density of delocalized LP modes for the
system in Ref. [16] with strongest Rabi splitting (see the first row of Table
LABEL:tab:parameters) in the presence (continuous curve) and absence of
inelastic scattering (dotted curve).

To assess the impact of static and dynamical molecular disorder on detuning
dependent transport phase diagrams, we present in Figs. 5 and 6 $q-u$
transport diagrams for the following limiting situations of molecular
disorder. In Fig. 5, we take the inhomogeneous limit of the molecular line
width where $\gamma_{M}=0$ (and $\sigma_{M}$ is maximal since all the
molecular line width would be due to static disorder$-$ see second row of
Table LABEL:tab:parameters), whereas Fig. 6 examines the vibrational
polaritons emergent in the homogeneous limit of the molecular line width where
$\sigma_{M}=0$ (and $\gamma_{M}$ is maximal since all the molecular line width
would be due to homogeneous dephasing $-$ see third row of Table
LABEL:tab:parameters).

In the inhomogeneous limit (Fig. 5), the low-energy mobility edge for LP
transport is determined by inelastic scattering (produced in this case only by
photonic leakage) for all $u$ less than approximately 14 meV, and by resonant
scattering for all $u>14$ meV. Conversely, the high-energy delocalization-
localization boundary is always determined by resonance scattering, since in
the limit where $\gamma_{M}=0$, we find that for each detuning below the
critical, there exists $q^{*}$ such that for $q>q^{*}$, elastic scattering
dominates over inelastic, i.e., $l_{\text{inel}}(q)>l_{\text{el}}(q)$ (this
region is to the right of the dashed line in Fig. 5). In this case, we only
observe overdamped LP modes in a narrow region
$q_{\text{min,el}}^{\text{L}}<q<q_{\text{min,inel}}^{\text{L}}$ where
$l_{\text{inel}}(q)<1/q<l_{\text{el}}(q)$ and the photonic weight is
sufficiently large that cavity leakage leads to efficient inelastic scattering
despite the absence of molecular dephasing. Note that while $\gamma_{M}=0$ is
an unrealistic condition, molecular dephasing can be slowed down by lowering
the temperature.

In the inverse scenario where the molecular line shape is assumed to be in the
homogeneous limit with $\sigma_{M}=0$ and the molecular FWHM is determined
entirely by fast dephasing processes, the LP mobility edges are strictly
determined from inelastic scattering, since resonance scattering is absent and
fluctuation scattering is too weak (see, e.g., Fig. 3). Fig. 6 shows the $q-u$
phase diagram obtained under these conditions. Notably, the critical detuning
(close to zero meV) is drastically smaller relative to that in the
inhomogeneous limit where $\sigma_{M}$ is large and $\gamma_{M}$ vanishes
(Fig. 5), but only slightly smaller relative to the experimental system which
operates near its critical detuning (Fig. 1). Overall, a much richer diversity
of polariton transport phenomena is expected for a system with weak inelastic
and strong elastic scattering (Fig. 5) in comparison to the opposite case with
strong inelastic and weak elastic scattering processes (Fig. 6).

Figure 5: Transport phase diagram for the inhomogeneous limit (see Table
LABEL:tab:parameters) of a vibrational polariton system examined in Ref. [16].
All parameters are from [16] except for the variable detuning and the
molecular homogeneous line width $\gamma_{M}$, which is set to zero in order
to study the transport diagram in the limit where the molecular line shape is
dominated by inhomogeneous broadening ($\sigma_{M}=2.91~{}\text{meV}$ and
$\gamma_{M}=0$). Note that inelastic scattering remains present since
polariton decay can occur via photon leakage ($\gamma_{C}=3.47$ meV). The
green dashed line represents the boundary between delocalized LP modes where
elastic and inelastic scattering prevails. The weakening of inelastic
processes relative to the system examined in Fig. 1 leads to a new region
($q>q^{*}$, where $q^{*}$ is given by the green dashed line) where resonant
elastic scattering dominates over inelastic, and weak wave localization and
coherent backscattering phenomena are expected. Figure 6: Transport phase
diagram for the homogeneous limit (see Table LABEL:tab:parameters) of the
vibrational polariton system of Ref. [16] with all parameters as in Figs. 1
and 5, except that here $\sigma_{M}=0~{}\text{meV}$ and
$\gamma_{M}=4.84~{}\text{meV}$ (obtained on the assumption that the bare
molecule line width is entirely due to fast dephasing processes).

###  III.2 Disorder - wave number transport phase diagrams at zero detuning

In this subsection, we examine the impacts of static and dynamic disorder on
the localization of molecular vibrational LP modes at fixed zero detuning. In
analogy with studies of metal-insulator Anderson transitions [55, 56], we
present mobility edges, identify regions of the quasiparticle spectrum where
(coherent) diffusion is absent and estimate critical values of disorder above
which all polaritons are strongly localized according to the IRC.

Figure 7 shows the $q-\sigma_{M}$ and $q-\gamma_{M}$ transport phase diagrams
and corresponding critical values for the system with the largest collective
light-matter interaction strength ($\hbar\Omega_{R}$ = 15.9 meV) among those
examined by Thomas et al. in Ref. [16]. We find the critical energetic
disorder beyond which all LP modes are localized according to the IRC is given
by $\sigma_{M}^{\text{c}}\approx 0.55\times\hbar\Omega_{R}$, whereas the
critical molecular dephasing rate $\gamma_{M}^{c}$ is characterized by the
weaker requirement $\gamma_{M}^{c}\approx 0.3\times\hbar\Omega_{R}$. The
condition on $\gamma_{M}$ for strong damping is significantly milder than that
for strong Anderson localization of the analogous closed system, including
only static energetic disorder. However, it is important to note that both
$\gamma_{M}^{\text{c}}$ and $\sigma_{M}^{\text{c}}$ increase nontrivially with
increasing $\Omega_{R}$ and decreasing detuning.

Notably, while $\gamma_{M}$ and $\sigma_{M}$ are related to the line widths of
the bare molecule transition, they contribute to inelastic and elastic
scattering in qualitatively distinct ways. Specifically, $\gamma_{M}$ is
directly proportional to the molecular contribution to the LP inelastic
scattering rate (Eq. 15), whereas the same is not necessarily true for
$\sigma_{M}$, as this quantity controls the magnitude of the LP resonance
scattering rate by modulating the bare molecule density of states at the LP
energy (see Eq. 13). A final caveat about a direct comparison between
$\sigma_{M}^{c}$ and $\gamma_{M}^{c}$ is that, as is well known for the
Anderson model, critical points for the wave function delocalization-
localization transition depend on the disorder probability distribution
function, and thus, it is possible that other distributions such as the
uniform may give weaker requirements for a localization-delocalization
polariton transition.

The diagrams presented in Fig. 7 are similar to those obtained in studies of
Anderson localization. This suggests that $\sigma_{M}^{\text{c}}$ is the
critical point of a continuous phase transition.

Figure 7: Left: $(\sigma_{M},q)$ phase diagram for system in Ref. [16] in the
inhomogeneous limit of the molecular system (with $\gamma_{M}=0$, see Table
LABEL:tab:parameters). Right: $(\gamma_{M},q)$ phase diagram for the system in
Ref. [16] in the homogeneous limit of the molecular system (with
$\sigma_{M}=0$, see Table LABEL:tab:parameters).

###  III.3 Effects of light-matter interaction strength on LP delocalization
at zero detuning

To identify the effect of variable light-matter interaction strength on the
localization-delocalization boundaries of vibrational polaritons, we examine
the systems experimentally probed in Ref. [16] with $\hbar\Omega_{R}(\text{in
meV})=15.9,12.0,11.4,9.7,8.2,$ and 6.8, and intermolecular distance
$a=0.69~{}\text{nm}$ for $\hbar\Omega_{R}=15.9~{}\text{meV}$ (the values of
$a$ for the other $\Omega_{R}$ are approximated from
$\Omega_{R}\propto\sqrt{N_{M}/V}\approx a^{-3/2}$). Disorder and microcavity
parameters are equal for these systems (given in the first row of Table
LABEL:tab:parameters). The behavior of the high-energy LP and UP mobility
edges and the density of delocalized LP modes as a function of $\Omega_{R}$
are presented in Fig. 8 (we also computed localization-delocalization
boundaries for additional interaction strengths not examined experimentally in
Ref. [16]).

Figure 8: Top: Zero-detuning low and high-energy mobility edges for the
delocalization-localization transition of LP according to the IRC as a
function of collective light-matter interaction strengths probed in Ref. [16].
Bottom: Density of macroscopically delocalized LP modes for the same systems
as in the top figure.

Figure 8 shows that in the presence of inelastic scattering, the Rabi
splitting has almost no effect on the density of delocalized LP modes.
Furthermore, the IRC (Eq. 16) implies macroscopically delocalized LP modes
only exist at the largest probed collective light-matter interaction
strengths, namely 14 and 15.9 meV.

The situation is significantly more interesting when inelastic processes are
ignored. In particular, while the low-energy LP localization-delocalization
boundary set by elastic (resonance) scattering is essentially independent of
$\Omega_{R}$, Fig. 8 shows that the high-energy mobility edge $q_{\text{max,
el}}^{\text{L}}$ and the density of delocalized LP modes
$\rho_{\text{L}}^{\text{deloc}}$ increases exponentially with $\Omega_{R}$.
This behavior is ascribed to $q_{\text{min}}^{\text{L}}$ being close to zero
for all $\Omega_{R}$, while the $q_{\text{max},\text{el}}^{\text{L}}$ are
determined by resonance scattering that gets largely suppressed at a fixed LP
wave number $q$ with increasing $\Omega_{R}$. The substantial effect of
$\Omega_{R}$ on the resonance scattering mean free path (for modes with
moderate to large $q$) occurs as a result of larger $\Omega_{R}$ leading to
greater energetic separation between the LP and the center of the molecular
reservoir density of vibrational modes.

Our finding that the density of (strongly damped) propagating LP modes
increases with $\Omega_{R}$ resembles the key observation of Ref. [16] that
the cavity effect on reactivity also increases with $\Omega_{R}$.

To conclude, we contrast the detuning and Rabi splitting dependence of
$\rho^{\text{deloc}}_{\text{L}}$ presented in Figs. 4 and 8 (bottom),
respectively. These figures show that increasing light-matter interaction
strength is much more effective at inducing the formation of propagating LP
modes than reducing the light-matter detuning. This issue is relevant because
experimental work suggests that zero-detuning is optimal for maximizing
polariton effects on chemistry. While our findings suggest otherwise that
larger values of $\rho_{\text{L}}^{\text{deloc}}$ occur at negative detuning,
the difference with respect to zero-detuning (“on-resonance” case) is small.

###  III.4 Comparison between electronic exciton and vibrational polaritons

In this final subsection, we make a comparison between electronic exciton and
vibrational polaritons. Table 1 presents the main parameters required to
obtain the polariton dispersion and mobility edges from the application of the
IRC to the organic microcavity system of Ref. [33]. Our choice to examine this
system was motivated by the large Rabi splitting and the superfluid phase
transition achieved under intense pumping.

To provide evidence of the generality of our prior analysis of vibrational
polariton phases, we compare the obtained organic electronic exciton-polariton
transport diagram in Fig. 9 with the detuning dependent vibrational polariton
transport diagram (Fig. 10) obtained for the LP modes arising from the strong
coupling of $-$NCO stretch modes with an IR microcavity experimentally studied
by Simpkins and coworkers [34] (the experimental parameters describing this
light-matter system are listed in Table LABEL:tab:parameters).

Figure 9: LP transport phase diagram for the electronic TDAF exciton-polariton
system in Ref. [33] (see Table LABEL:tab:parameters). Figure 10: Transport
phase diagram for LP modes emergent from vibrational strong coupling between
$-\text{NCO}$ normal-modes and an infrared microcavity [34] with parameters
listed in Table LABEL:tab:parameters

A key feature of the presented electronic (organic) exciton-polariton
transport phase diagram is the dramatically broader interval of wave numbers
supporting macroscopic delocalization in comparison to the vibrational
polariton system of Fig. 10. At the experimentally investigated detunings
($u=-455~{}\text{meV}$ for the electronic (organic) exciton-polariton system
and $u=-12~{}\text{meV}$ for the vibropolariton system), the prediction of the
IRC for the density of delocalized LP modes of the electronic exciton-
polariton is larger by three orders of magnitude relative to the IR system.

It is notable that the electronic exciton-polaritons show a much smaller
density of overdamped modes in comparison to the IR polaritons. This behavior
is due to the greater energetic disorder of the electronic excitons, which
leads to resonance scattering dominance over inelastic processes even at
relatively small $q$ (Fig. 9). A similar feature only arose for the
vibrational polaritons studied in this paper in the inhomogeneous limit
examined in Fig. 5, when we analyzed the excitations arising in the limit
where the entire line width of the bare molecule ensemble is due to static
inhomogeneity of the vibrational transition energies and $\gamma_{M}$ was
assumed to vanish. Roughly, we observe, in general, that transport phase
diagrams with significant regions of the wave number space including
delocalized LP modes with $l_{\text{inel}}>l_{\text{el}}$ (e.g., Figs. 9 and
5) occur only when $\sigma_{M}/\gamma_{M}\gg 1$.

The observed suppressed scattering and large density of delocalized electronic
exciton LP modes in Fig. 9 are likely key features enabling the observation of
quasi-long-range off-diagonal order and superfluidity in this system [33].
Typical vibrational polariton systems (e.g., Figs. 1 and 10) will likely
exhibit challenges reaching a similar quasi-Bose Einstein Condensate (quasi-
BEC) regime in large part due to the significantly stronger inelastic
scattering and reduced density of vibrational LP modes which lead to generic
vibrational polariton phase diagrams resembling Figs. 1 and 10 which have a
small density of delocalized LP modes according to the IRC.

##  IV Conclusions

We applied the Ioffe-Regel criterion to construct transport phase diagrams and
identify the main features of transport phenomena of vibrational polaritons
emergent from collective strong interactions of molecular vibrational modes
and planar IR microcavities. We unraveled optimal conditions of negative
detuning, low disorder, and large Rabi splitting for the maximization of
delocalization in vibrational polaritons.

Lower polariton modes were classified as (macroscopically propagating)
delocalized, (propagating) strongly damped, localized LPs, and as localized
weakly coupled excitons. Each of these categories corresponds to a
qualitatively distinct regime of polariton delocalization and transport that
was characterized in various experimentally studied systems.

We found that despite the reduced density of delocalized vibrational LP modes
relative to the weakly coupled, a finite and significant density of
macroscopic delocalized vibrational LPs persists in general, even at zero
detuning. Macroscopic delocalization was found to be maintained under typical
vibrational strong coupling conditions even for polariton states with nearly
insignificant photon content (e.g., less than ten percent). Our results
provide significant new insight into transport phenomena under VSC and are
expected to be particularly relevant in informing future investigations of
polariton-assisted processes that may largely benefit from wave function
delocalization. In addition, while we make no statements about the properties
of the strongly localized molecular reservoir modes (“weakly coupled
excitons”), we expect that polariton-induced perturbations of local dynamical
processes (e.g., reactivity and vibrational energy relaxation) involving dark
modes will be greater under conditions where a larger number of polaritons are
macroscopically delocalized and the vibrational polariton high-energy mobility
edges occur at higher $q$. Therefore, our analysis of polariton transport
phenomena is expected to be relevant in the future rationalization of
polariton chemistry trends and for the establishment of optimal conditions for
the control of chemical kinetics with photonic materials.

Finally, our work has provided a new perspective on challenges expected in the
way of reaching quasi-BEC and superfluid behavior in the VSC regime. By
comparing the delocalization in electronic (organic) exciton and vibrational
polariton systems, we found the former tend to have a density of macroscopic
delocalized modes several orders of magnitude larger than the latter. This
suggests that the pumping threshold for the achievement of an LP
nonequilibrium BEC phase under VSC will be much larger relative to that
required for electronic (organic) exciton condensation since the condensate is
formed from the establishment of long-range off-diagonal order, which is
facilitated by wave function delocalization.

ACKNOWLEDGMENTS

We acknowledge generous start-up funds from the Emory University Department of
Chemistry.

DATA AVAILABILITY

The data that supports the findings of this study are available from the
corresponding author upon reasonable request.

Appendix A: Rate of inelastic scattering

In this Appendix, we provide a more detailed discussion of the inelastic
scattering processes considered in the main manuscript and a derivation of Eq.
15. We assume the inelastic scattering events are uncorrelated and may be
ascribed to three qualitatively distinct processes corresponding to cavity
leakage, molecular population relaxation, and molecular dephasing. For
concreteness, the light-matter system is supposed to be in the vibrational
strong coupling regime. We begin with the general Hamiltonian below, including
intermolecular interactions and system-bath coupling,

$\displaystyle
H=H_{\text{L}}+H_{\text{M}}+H_{\text{LM}}+H_{\text{B}}+V_{\text{MM}}+V_{\text{SB}},$
(22)

where $H_{\text{L}}$, $H_{\text{M}}$, and $H_{\text{LM}}$ are the same as in
the main text, $V_{\text{MM}}$ contains quadratic nearest-neighbor
electrostatic dipole-dipole interactions (see Appendix B),
$H_{\text{B}}=H_{\text{B}}^{C}+H_{\text{B}}^{M}$, where $H_{\text{B}}^{C}$ and
$H_{\text{B}}^{M}$ are Hamiltonians for the bath weakly coupled to the cavity
and molecular subsystems, and
$V_{\text{SB}}=V_{\text{SB}}^{C}+V_{\text{SB}}^{M}$ represents all light and
matter interactions with their corresponding baths ($V_{\text{SB}}^{C}$ and
$V_{\text{SB}}^{M}$, respectively). Each cavity mode with in-plane momentum
$\mathbf{q}$ and polarization $\lambda$ (we omit the index $m$ since we
consider a single cavity band to be relevant) is assumed to interact
bilinearly with an independent set of free space harmonic modes with creation
and annihilation operators $c_{\mathbf{q}\lambda\omega}^{s,\dagger}$ and
$c_{\mathbf{q}\lambda\omega}^{s}$, with $\omega>0$ and $s=L~{}\text{or}~{}R$
corresponding to free space modes to the left and right of the cavity,
respectively [88]

$\displaystyle
V_{\text{SB}}^{C}=\hbar\sqrt{\frac{\gamma_{C}}{4\pi}}\sum_{s=L,R}\sum_{\mathbf{q}\lambda}\int_{0}^{\infty}\mathrm{d}\omega~{}\left[a_{\mathbf{q}\lambda}^{\dagger}c_{\mathbf{q}\lambda\omega}^{s}+c_{\mathbf{q}\lambda\omega}^{s,\dagger}a_{\mathbf{q}\lambda}\right],$
(23)

where we assume without loss of generality, the cavity decay rate $\gamma_{C}$
is frequency and polarization independent.

The molecular system-bath interactions are described as follows. Each molecule
$i$ interacts independently with a set of low and high-frequency bath modes
$\\{\alpha\\}$, with creation and annihilation operators
$b_{i\alpha}^{\dagger},b_{i\alpha}$ and $B_{i\beta}^{\dagger}$ and
$B_{i\beta}$, respectively. The low-frequency bath modes correspond to, e.g.,
librations with frequencies that are much lower (and thus are highly off-
resonant) than $E_{M}/\hbar$. These modes form a quasicontinuum that induces
dephasing of the molecular polarization and typically provides the dominant
contribution to the molecular absorption lineshape in free space [89, 90]. In
solution, solute-solvent interactions are stronger than solute-solute, and the
bare molecule homogeneous broadening linewidth $\gamma_{M}=1/T_{2}$ is often
entirely determined by the interaction of the low-frequency bath modes with
the matter polarization. High-frequency environment modes have a much lower
density of states than low-frequency, but the former are nevertheless involved
with vibrational population relaxation via excitation energy transfer
involving one or multiple vibrational quanta of the system. Nevertheless,
vibrational population relaxation, in general, occurs on a much longer
timescale ($T_{1}\gg T_{2}$) than dephasing induced by low-frequency modes
[89, 91]. For this reason, we will ignore the contribution of population
relaxation to the polaritonic inelastic scattering. This suggests the
following well-known representation of the matter-bath interactions [90, 92]

$\displaystyle
V_{\text{SB}}^{M}=\sum_{i=1}^{N_{M}}\sum_{\alpha}\hbar\omega_{\alpha}\lambda_{\alpha}\sigma_{i}^{+}\sigma_{i}^{-}\left(b_{i\alpha}+b_{i\alpha}^{\dagger}\right).$
(24)

We reiterate our assumption that each molecule has an independent set of bath
modes and note that del Pino et al. [77] have shown that this scenario
(compared to those where the bath modes interact simultaneously with multiple
strongly coupled molecules) provides the fastest transition rates for
polariton $\rightarrow$ weakly coupled mode in the absence of strong direct
forces between the molecules involved in strong coupling (e.g., when
$V_{\text{MM}}$ has negligible effects on polariton dynamics).

In the mean-field limit [29], the normal-modes of
$H_{\text{M}}+H_{\text{LM}}+V_{\text{MM}}$ consist of degenerate TE and TM
polaritons and longitudinal matter polarization modes (“LO modes”) which are
decoupled from light [71, 93]. We will assume the LO modes are dispersionless
and the energy shift induced by $V_{\text{MM}}$ relative to $E_{M}$ is
negligible compared to system-bath interactions. These assumptions are typical
for effective Hamiltonians representing solute dynamics.

Dynamical fluctuations of the intermolecular distance, orientation, and
molecular excitation energies also enable inelastic scattering, i.e.,
polariton decay into weakly coupled modes[80], corresponding in our model to
strongly localized linear combinations of longitudinal modes and polaritons
with nearly negligible photon content (e.g., less than $0.1\%$). In this
Appendix, we focus on the inelastic scattering induced by intramolecular
system-bath coupling and cavity decay. Polariton decay into weakly coupled
modes via fluctuations of the direct intermolecular interaction strength is
discussed in Appendix B.

We assume the initial state of the inelastic scattering process is a direct
product of a polariton pure state and the bath thermal density operator and
compute the rate of polariton decay using Fermi’s golden rule. This requires
the evaluation of

$\displaystyle
R_{FI}^{\text{SB},M}\equiv\sum_{n_{bI},n_{bF}}p_{n_{bI}}|\braket{F,n_{bF}}{V_{\text{SB}}^{M}}{I,n_{bI}}|^{2},$
(25) $\displaystyle
R_{FI}^{\text{SB},C}\equiv\sum_{n_{bI},n_{bF}}p_{n_{bI}}|\braket{F,n_{bF}}{V_{\text{SB}}^{C}}{I,n_{bI}}|^{2},$
(26)

where $I$ is a polariton state, $F$ is the final state corresponding either to
a weakly coupled molecular mode or to the cavity-matter global ground state,
$n_{bI}$ and $n_{bF}$ are initial and final bath Fock states, and $p_{n_{bI}}$
is the probability for a particular initial state of the light-matter bath.
The terms $R_{FI}^{\text{SB},M}$ with $I$ corresponding to a lower polariton
mode $P(\mathbf{q})$ and $F$ corresponding to weakly coupled molecular modes
$D_{\mu}$ with $\mu=1,2,...,N_{M}-N_{\text{P}}$ (where $N_{\text{P}}$ is the
number of lower polariton modes as defined above, and the set
$\\{\ket{D_{\mu}}\\}$ contains LP modes with very small photonic content, such
that they may be viewed as weakly-coupled excitons according to Table 1) with
$E_{D_{\mu}}>E_{P(\mathbf{q})}$ can be readily computed in the mean-field
limit leading to the following rate of LP inelastic decay into the dense
manifold of weakly coupled states

$\displaystyle k_{D\leftarrow P(\mathbf{q})}^{\text{SB},M}$
$\displaystyle=\frac{2\pi}{\hbar}\sum_{\mu}\sum_{mn}c_{mP(\mathbf{q})}^{*}c_{nP(\mathbf{q})}c_{mD_{\mu}}c_{nD_{\mu}}^{*}\sum_{\alpha}(\hbar\omega_{\alpha}\lambda_{\alpha})^{2}\braket{n_{\alpha}}\delta\left[E_{\alpha}-(E_{D_{\mu}}-E_{P(\mathbf{q})})\right]$
$\displaystyle\approx\frac{2\pi}{\hbar}\sum_{\mu}P_{M}^{P}(\mathbf{q})P_{M}^{D_{\mu}}\sum_{mn}\frac{\delta_{mn}}{N_{M}^{2}}\sum_{\alpha}(\hbar\omega_{\alpha}\lambda_{\alpha})^{2}\braket{n_{\alpha}}\delta\left[E_{\alpha}-(E_{D_{\mu}}-E_{P(\mathbf{q})}))\right]$
$\displaystyle\approx\frac{2\pi}{\hbar}P_{M}^{P}(\mathbf{q})\frac{N_{M}-N_{\text{P}}}{N_{M}}\sum_{\alpha}(\hbar\omega_{\alpha}\lambda_{\alpha})^{2}\braket{n_{\alpha}}\delta\left[E_{\alpha}-(E_{M}-E_{P(\mathbf{q})})\right]$
$\displaystyle\approx\frac{1}{\hbar}P_{M}^{P}(\mathbf{q})J(E_{M}-E_{P(\mathbf{q})})\braket{n(E_{M}-E_{P(\mathbf{q})})}.$
(27)

where $\braket{n_{\alpha}}$ is the mean occupation number of the molecular
bath mode $\alpha$ and
$J(\omega)=\sum_{\alpha}(\hbar\omega_{\alpha}\lambda_{\alpha})^{2}\delta(E-E_{\alpha})$
is the bath spectral density. Using the high-temperature limit of
$\braket{n(\omega)}=k_{\text{B}}T/\hbar\omega$ (note that for vibrational
polaritons, $E_{M}-E_{P(\mathbf{q})}$ is almost always much smaller than room
temperature) and the Ohmic spectral density $J(\omega)=\eta\omega
e^{-\omega^{2}/\omega^{2}_{\text{cut}}}$ [94] with
$\omega_{\text{cut}}\gg\omega_{M}-\omega_{P(\mathbf{q})}$ and $\eta>0$, it
follows that

$\displaystyle k_{D\leftarrow P(\mathbf{q})}^{\text{SB},M}$
$\displaystyle\approx\frac{\eta k_{\text{B}}T}{\hbar}P_{M}^{P}(\mathbf{q})$
$\displaystyle\approx\gamma_{M}P_{M}^{P}(\mathbf{q}),$ (28)

where we used $\eta k_{\text{B}}T/\hbar=\gamma_{M}$, thus obtaining the matter
contribution to Eq. 15 of the main text. This result agrees with that obtained
by del Pino et al. [77], who examined the effects of system-bath interactions
on vibrational strong coupling between a cavity photon mode and a molecular
ensemble [77].

The computation of the photonic contribution to the inelastic scattering rate
via Fermi’s Golden rule follows the same ideas as above, except the system
final state $F$ is the light-matter ground state, and the polariton decay
process is induced by the photonic bath

$\displaystyle k_{G\leftarrow
P(\mathbf{q})}^{\text{SB},C}=P_{C}^{P}(\mathbf{q})\gamma_{C}.$ (29)

In the absence of sufficiently strong interactions between the molecular modes
contributing to polariton formation, the total rate of polariton inelastic
scattering (Eq. 15) follows from Eqs. 28 and 29. Molecular population
relaxation was also ignored since matter dephasing induced by coupling to a
low-frequency bath tends to be much more effective at inducing polariton
transitions than vibrational population relaxation processes, which typically
take much longer than polarization dephasing [77].

Appendix B: Polariton inelastic scattering via intermolecular dephasing
pathway

Fluctuations in the intermolecular interactions between molecular modes
contributing to collective strong light-matter coupling may also lead to
polariton dephasing [80]. Assuming this mechanism is incoherent, mediated by
dipole-dipole interactions, and the molecular dipole vectors are independently
distributed over the unit sphere, the polariton inelastic scattering induced
by this process has rate [80]

$\displaystyle k_{D\leftarrow
P(\mathbf{q})}^{\text{dd},M}=2\pi\Delta^{2}P_{M}^{P}(\mathbf{q})S_{P(\mathbf{q})},$
(30)

where $S_{P(\mathbf{q})}$ is the spectral overlap integral between the
polariton $P(\mathbf{q})$ and the weakly coupled mode reservoir

$\displaystyle
S_{P(\mathbf{q})}=\int_{0}^{\infty}\mathrm{d}\omega~{}\rho_{D}(\omega)\rho_{P(\mathbf{q})}(\omega),$
(31)

$\Delta^{2}=N_{nn}\braket{U_{nn}^{2}}$, $N_{nn}$ is the number of nearest
neighbor strongly coupled molecules, and $\braket{U_{nn}^{2}}$ is the mean
squared fluctuation of the intermolecular transition-matrix element
$[V_{\text{MM}}]_{ij}$ (where $i$ and $j$ are nearest-neighbors). Note that
our definition of $\Delta^{2}$ differs slightly from Ref. [80], as we perform
an incoherent sum over the possible final states. This follows from our
assumption that the molecular dipole vectors are randomly distributed and
uncorrelated (so there is no interference between $D\leftarrow P(\mathbf{q})$
pathways mediated by transition-matrix elements $[V_{\text{MM}}]_{ij}$ with
fixed molecule $i$ and its nearest neighbors $j$).

We will obtain an upper bound estimate for $k_{D\leftarrow
P(\mathbf{q})}^{\text{dd},M}$ assuming $\rho_{P(\mathbf{q})}$ and
$\rho_{D}(\omega)$ are centered at $E_{M}$ and have equal linewidth
$\gamma_{M}$. This gives the upper bound
$S_{P(\mathbf{q})}^{0}>S_{P(\mathbf{q})}$. We will employ lineshapes for the
weakly coupled modes given by
$\rho_{D}(\omega)=N(\gamma_{M},\omega_{M})(\gamma_{M}/\pi)\times
1/[(\omega-\omega_{M})^{2}+\gamma_{M}^{2}]$, where $N(\gamma_{M},\omega_{M})$
is a normalization constant and $\omega>0$. To estimate $\Delta^{2}$ in the
absence of knowledge of the magnitude of the dipole-dipole coupling
fluctuations, we assume $U_{nn}$ is equal to the isotropically averaged
magnitude of (free space) dipole-dipole interactions for a system with
$N_{nn}=6$ with mean distance $a$ and dipole magnitude $\mu$

$\displaystyle\braket{\Delta^{2}}\approx\frac{2}{3}N_{nn}\frac{\mu^{4}}{16\pi^{2}\epsilon^{2}a^{6}\hbar^{2}},$
(32)

where $\epsilon=n^{2}\epsilon_{0}$ and $n$ is the (optical) index of
refraction of the medium. Using our estimate for $U_{nn}$, the upper bound
$S_{P(\mathbf{q})}^{0}$ and $P_{P}^{\text{M}}(q)\approx 1$ we find that

$\displaystyle k_{D\leftarrow
P(\mathbf{q})}^{\text{dd},M}<\frac{\mu^{4}}{2\pi\epsilon^{2}a^{6}\hbar^{2}}S_{P(\mathbf{q})}^{0}$
(33)

For an intermolecular distance of $a=1~{}\text{nm}$, $\mu=1~{}\text{D}$,
$n=1.33$ (corresponding to methanol, the solvent employed in Refs. [13, 14]),
and $1/\gamma_{M}=1~{}\text{ps}$ (approximately 4.1 meV/$\hbar$), we find
$1/k_{D\leftarrow P(\mathbf{q})}^{\text{dd},M}>16~{}\text{ps}$. Under similar
conditions where $P_{M}^{P}(\mathbf{q})\rightarrow 1$, we find
$1/k_{D\leftarrow P(\mathbf{q})}^{\text{SB},M}\approx 1~{}\text{ps}$.

The given estimates support the view that inelastic scattering induced by
fluctuations of the dipole-dipole interaction (between molecular normal modes
involved in polariton formation) provides, in general, a negligible change to
the rates obtained from Eq. 15. The case where the intermolecular dephasing
pathway would be expected to be most relevant is that with the shortest mean
intersolute distance $a=0.7~{}\text{nm}$ (corresponding to the atypically
large 5 mol/L concentration). Nevertheless, even in this case, when we take
into account that $\gamma_{M}=3.97~{}\text{meV}/\hbar$,
$\gamma_{C}=3.47~{}\text{meV}/\hbar$ and $E_{M}-E_{P(\mathbf{q})}\approx
5~{}\text{meV}$ at LP mode with $q=q_{\text{max}}^{\text{L}}$ (which has
89.7$\%$ molecular weight and 10.3$\%$ photonic weight), we find the
contribution of $k_{D\leftarrow P(\mathbf{q})}^{\text{dd},M}$ to the inelastic
scattering rate is weaker by a factor of O(10) relative to the rate given by
Eq. 15 (assuming $\mu=1~{}\text{D}$ and with spectral overlap obtained from
analogous distributions for the weakly coupled and polariton modes with
linewidths $\gamma_{M}$ and
$P_{M}^{P}(\mathbf{q})\gamma_{M}+P_{C}^{P}(\mathbf{q})\gamma_{C}$,
respectively).

Importantly, in most molecular systems where a solute interacts strongly with
the infrared EM field, $a$ tends to be greater than 1 nm (i.e., concentration
less than $\approx 1.65~{}\text{mol/L}$), and the transition dipole moments
are also smaller than employed in our estimates, i.e., $\mu<1~{}\text{D}$. The
situation could very well be different in close-packed systems with short mean
intermolecular distances $a\ll 1~{}\text{nm}$ and large transition dipoles
$\mu\gg 1~{}\text{D}$. The former scenario is more common in pure solvent
strong vibrational strong coupling or organic electronic exciton strong
coupling in molecular crystals [95]. However, even in these systems, the Rabi
splitting tends to be significantly larger than considered in our analysis due
to the high molecular density, and thus, the spectral overlap between
polaritons with small photonic content and weakly coupled molecular modes is
also likely smaller than in the upper bound computed above.

## References

  * Ebbesen [2016] T. W. Ebbesen, Hybrid Light–Matter States in a Molecular and Material Science Perspective, Accounts of Chemical Research 49, 2403 (2016).
  * Herrera and Owrutsky [2020] F. Herrera and J. Owrutsky, Molecular polaritons for controlling chemistry with quantum optics, The Journal of Chemical Physics 152, 100902 (2020).
  * Li  _et al._ [2022a] T. E. Li, B. Cui, J. E. Subotnik, and A. Nitzan, Molecular polaritonics: chemical dynamics under strong light-matter coupling, Annual review of physical chemistry 73, 43 (2022a).
  * Ribeiro and Yuen-Zhou [2022] R. F. Ribeiro and J. Yuen-Zhou, Introduction to vibropolaritons: Spectroscopy, relaxation and chemical reactions, in  _Vibrational Dynamics of Molecules_ (World Scientific, 2022) pp. 517–574.
  * Dunkelberger  _et al._ [2022] A. D. Dunkelberger, B. S. Simpkins, I. Vurgaftman, and J. C. Owrutsky, Vibration-cavity polariton chemistry and dynamics, Annual Review of Physical Chemistry 73, 429 (2022).
  * Orgiu  _et al._ [2015] E. Orgiu, J. George, J. A. Hutchison, E. Devaux, J. F. Dayen, B. Doudin, F. Stellacci, C. Genet, J. Schachenmayer, C. Genes, G. Pupillo, P. Samorì, and T. W. Ebbesen, Conductivity in organic semiconductors hybridized with the vacuum field, Nature Materials 14, 1123 (2015).
  * Nagarajan  _et al._ [2020] K. Nagarajan, J. George, A. Thomas, E. Devaux, T. Chervy, S. Azzini, K. Joseph, A. Jouaiti, M. W. Hosseini, A. Kumar, C. Genet, N. Bartolo, C. Ciuti, and T. W. Ebbesen, Conductivity and Photoconductivity of a p-Type Organic Semiconductor under Ultrastrong Coupling, ACS Nano 14, 10219 (2020).
  * Coles  _et al._ [2014] D. M. Coles, N. Somaschi, P. Michetti, C. Clark, P. G. Lagoudakis, P. G. Savvidis, and D. G. Lidzey, Polariton-mediated energy transfer between organic dyes in a strongly coupled optical microcavity, Nature materials 13, 712 (2014).
  * Zhong  _et al._ [2017] X. Zhong, T. Chervy, L. Zhang, A. Thomas, J. George, C. Genet, J. A. Hutchison, and T. W. Ebbesen, Energy Transfer between Spatially Separated Entangled Molecules., Angewandte Chemie 56, 9034 (2017).
  * Georgiou  _et al._ [2018] K. Georgiou, P. Michetti, L. Gai, M. Cavazzini, Z. Shen, and D. G. Lidzey, Control over energy transfer between fluorescent bodipy dyes in a strongly coupled microcavity, ACS Photonics 5, 258 (2018).
  * Xiang  _et al._ [2020] B. Xiang, R. F. Ribeiro, M. Du, L. Chen, Z. Yang, J. Wang, J. Yuen-Zhou, and W. Xiong, Intermolecular vibrational energy transfer enabled by microcavity strong light–matter coupling, Science (New York, N.Y.) 368, 665 (2020).
  * Son  _et al._ [2022] M. Son, Z. T. Armstrong, R. T. Allen, A. Dhavamani, M. S. Arnold, and M. T. Zanni, Energy cascades in donor-acceptor exciton-polaritons observed by ultrafast two-dimensional white-light spectroscopy, Nature Communications 13, 7305 (2022).
  * Thomas  _et al._ [2016] A. Thomas, J. George, A. Shalabney, M. Dryzhakov, S. J. Varma, J. Moran, T. Chervy, X. Zhong, E. Devaux, C. Genet, J. A. Hutchison, and T. W. Ebbesen, Ground-State Chemical Reactivity under Vibrational Coupling to the Vacuum Electromagnetic Field, Angewandte Chemie International Edition 55, 11462 (2016).
  * Thomas  _et al._ [2019] A. Thomas, L. Lethuillier-Karl, K. Nagarajan, R. M. Vergauwe, J. George, T. Chervy, A. Shalabney, E. Devaux, C. Genet, J. Moran, _et al._ , Tilting a ground-state reactivity landscape by vibrational strong coupling, Science 363, 615 (2019).
  * Lather  _et al._ [2019] J. Lather, P. Bhatt, A. Thomas, T. W. Ebbesen, and J. George, Cavity Catalysis by Cooperative Vibrational Strong Coupling of Reactant and Solvent Molecules, Angewandte Chemie International Edition 58, 10635 (2019).
  * Thomas  _et al._ [2020] A. Thomas, A. Jayachandran, L. Lethuillier-Karl, R. M. Vergauwe, K. Nagarajan, E. Devaux, C. Genet, J. Moran, and T. W. Ebbesen, Ground state chemistry under vibrational strong coupling: dependence of thermodynamic parameters on the rabi splitting energy, Nanophotonics 9, 249 (2020).
  * Hirai  _et al._ [2020] K. Hirai, R. Takeda, J. A. Hutchison, and H. Uji-i, Modulation of Prins cyclization by vibrational strong coupling, Angewandte Chemie 132, 5370 (2020).
  * Ahn  _et al._ [2023] W. Ahn, F. Herrera, and B. Simpkins, Modification of ground-state chemical reactivity via light-matter coherence in infrared cavities, Science 380, 1165 (2023).
  * Nagarajan  _et al._ [2021] K. Nagarajan, A. Thomas, and T. W. Ebbesen, Chemistry under vibrational strong coupling, Journal of the American Chemical Society 143, 16877 (2021).
  * Simpkins  _et al._ [2021] B. S. Simpkins, A. D. Dunkelberger, and J. C. Owrutsky, Mode-specific chemistry through vibrational strong coupling (or a wish come true), The Journal of Physical Chemistry C 125, 19081 (2021).
  * Li  _et al._ [2021a] X. Li, A. Mandal, and P. Huo, Cavity frequency-dependent theory for vibrational polariton chemistry, Nature Communications 12, 1315 (2021a).
  * Du and Yuen-Zhou [2022] M. Du and J. Yuen-Zhou, Catalysis by dark states in vibropolaritonic chemistry, Physical Review Letters 128, 096001 (2022).
  * Wang  _et al._ [2022] D. S. Wang, J. Flick, and S. F. Yelin, Chemical reactivity under collective vibrational strong coupling, The Journal of Chemical Physics 157, 224304 (2022).
  * Lindoy  _et al._ [2023] L. P. Lindoy, A. Mandal, and D. R. Reichman, Quantum dynamics of vibrational polariton chemistry, Nat Commun 14, 2733 (2023).
  * Pandya  _et al._ [2022] R. Pandya, A. Ashoka, K. Georgiou, J. Sung, R. Jayaprakash, S. Renken, L. Gai, Z. Shen, A. Rao, and A. J. Musser, Tuning the coherent propagation of organic exciton-polaritons through dark state delocalization, Advanced Science , 2105569 (2022).
  * Xu  _et al._ [2022] D. Xu, A. Mandal, J. M. Baxter, S.-W. Cheng, I. Lee, H. Su, S. Liu, D. R. Reichman, and M. Delor, Ultrafast imaging of coherent polariton propagation and interactions, arXiv preprint arXiv:2205.01176 (2022).
  * Balasubrahmaniyam  _et al._ [2023] M. Balasubrahmaniyam, A. Simkhovich, A. Golombek, G. Sandik, G. Ankonina, and T. Schwartz, From enhanced diffusion to ultrafast ballistic motion of hybrid light–matter excitations, Nature Materials , 1 (2023).
  * Agranovich  _et al._ [2003] V. M. Agranovich, M. Litinskaia, and D. G. Lidzey, Cavity polaritons in microcavities containing disordered organic semiconductors, Physical Review B 67, 085311 (2003).
  * Litinskaya and Reineker [2006] M. Litinskaya and P. Reineker, Loss of coherence of exciton polaritons in inhomogeneous organic microcavities, Physical Review B 74, 165320 (2006).
  * Michetti and La Rocca [2005] P. Michetti and G. La Rocca, Polariton states in disordered organic microcavities, Physical Review B 71, 115320 (2005).
  * Agranovich and Gartstein [2007] V. Agranovich and Y. N. Gartstein, Nature and dynamics of low-energy exciton polaritons in semiconductor microcavities, Physical Review B 75, 075302 (2007).
  * Guebrou  _et al._ [2012] S. A. Guebrou, C. Symonds, E. Homeyer, J. Plenet, Y. N. Gartstein, V. M. Agranovich, and J. Bellessa, Coherent emission from a disordered organic semiconductor induced by strong coupling with surface plasmons, Physical Review Letters 108, 066401 (2012).
  * Lerario  _et al._ [2017] G. Lerario, A. Fieramosca, F. Barachati, D. Ballarini, K. S. Daskalakis, L. Dominici, M. De Giorgi, S. A. Maier, G. Gigli, S. Kéna-Cohen, _et al._ , Room-temperature superfluidity in a polariton condensate, Nature Physics 13, 837 (2017).
  * Simpkins  _et al._ [2015] B. S. Simpkins, K. P. Fears, W. J. Dressick, B. T. Spann, A. D. Dunkelberger, and J. C. Owrutsky, Spanning Strong to Weak Normal Mode Coupling between Vibrational and Fabry–Pérot Cavity Modes through Tuning of Vibrational Absorption Strength, ACS Photonics (2015).
  * Scholes [2020] G. D. Scholes, Polaritons and excitons: Hamiltonian design for enhanced coherence, Proceedings of the Royal Society A 476, 20200278 (2020).
  * Botzung  _et al._ [2020] T. Botzung, D. Hagenmüller, S. Schütz, J. Dubail, G. Pupillo, and J. Schachenmayer, Dark state semilocalization of quantum emitters in a cavity, Physical Review B 102, 144202 (2020).
  * Chávez  _et al._ [2021] N. C. Chávez, F. Mattiotti, J. Méndez-Bermúdez, F. Borgonovi, and G. L. Celardo, Disorder-enhanced and disorder-independent transport with long-range hopping: Application to molecular chains in optical cavities, Physical Review Letters 126, 153201 (2021).
  * Cohn  _et al._ [2022] B. Cohn, S. Sufrin, A. Basu, and L. Chuntonov, Vibrational polaritons in disordered molecular ensembles, The Journal of Physical Chemistry Letters 13, 8369 (2022).
  * Gera and Sebastian [2022a] T. Gera and K. Sebastian, Exact results for the tavis-cummings and huckel hamiltonians with diagonal disorder, The Journal of Physical Chemistry A 126, 5449 (2022a).
  * Gera and Sebastian [2022b] T. Gera and K. Sebastian, Effects of disorder on polaritonic and dark states in a cavity using the disordered tavis–cummings model, The Journal of Chemical Physics 156, 194304 (2022b).
  * Cederbaum [2022] L. S. Cederbaum, Cooperative molecular structure in polaritonic and dark states, The Journal of Chemical Physics 156, 184102 (2022).
  * Li  _et al._ [2021b] T. E. Li, A. Nitzan, and J. E. Subotnik, Collective vibrational strong coupling effects on molecular vibrational relaxation and energy transfer: Numerical insights via cavity molecular dynamics simulations, Angewandte Chemie 133, 15661 (2021b).
  * Zhou  _et al._ [2022] Z. Zhou, H.-T. Chen, J. E. Subotnik, and A. Nitzan, The interplay between disorder, local relaxation and collective behaviors for an ensemble of emitters outside vs inside cavity, arXiv preprint arXiv:2211.16325 (2022).
  * Sun  _et al._ [2022] K. Sun, C. Dou, M. F. Gelin, and Y. Zhao, Dynamics of disordered tavis-cummings and holstein-tavis-cummings models, The Journal of Chemical Physics 156, 024102 (2022).
  * Cui and Nizan [2022] B. Cui and A. Nizan, Collective response in light–matter interactions: The interplay between strong coupling and local dynamics, The Journal of Chemical Physics 157, 114108 (2022).
  * Gómez and Vendrell [2023] J. A. Gómez and O. Vendrell, Vibrational energy redistribution and polaritonic fermi resonances in the strong coupling regime, The Journal of Physical Chemistry A 127, 1598 (2023).
  * Pérez-Sánchez  _et al._ [2023] J. B. Pérez-Sánchez, A. Koner, N. P. Stern, and J. Yuen-Zhou, Simulating molecular polaritons in the collective regime using few-molecule models, Proceedings of the National Academy of Sciences 120, e2219223120 (2023).
  * Tichauer  _et al._ [2021] R. H. Tichauer, J. Feist, and G. Groenhof, Multi-scale dynamics simulations of molecular polaritons: The effect of multiple cavity modes on polariton relaxation, The Journal of Chemical Physics 154, 104112 (2021).
  * Allard and Weick [2022] T. F. Allard and G. Weick, Disorder-enhanced transport in a chain of lossy dipoles strongly coupled to cavity photons, Physical Review B 106, 245424 (2022).
  * Ribeiro [2022] R. F. Ribeiro, Multimode polariton effects on molecular energy transport and spectral fluctuations, Communications Chemistry 5, 1 (2022).
  * Engelhardt and Cao [2022] G. Engelhardt and J. Cao, Polarition localization and spectroscopic properties of disordered quantum emitters in spatially-extended microcavities, arXiv preprint arXiv:2209.02909 (2022).
  * Anderson [1958] P. W. Anderson, Absence of diffusion in certain random lattices, Physical review 109, 1492 (1958).
  * Abrahams  _et al._ [1979] E. Abrahams, P. Anderson, D. Licciardello, and T. Ramakrishnan, Scaling theory of localization: Absence of quantum diffusion in two dimensions, Physical Review Letters 42, 673 (1979).
  * Van Tiggelen [1999] B. A. Van Tiggelen, Localization of waves, Diffuse waves in complex media , 1 (1999).
  * Sheng [2006] P. Sheng,  _Introduction to wave scattering, localization and mesoscopic phenomena_ , Vol. 88 (Springer Science & Business Media, 2006).
  * Evers and Mirlin [2008] F. Evers and A. D. Mirlin, Anderson transitions, Reviews of Modern Physics 80, 1355 (2008).
  * Ioffe and Regel [1960] A. Ioffe and A. Regel, Non-crystalline, amorphous, and liquid electronic semiconductors, in  _Progress in semiconductors_ (1960) pp. 237–291.
  * Litinskaya [2008] M. Litinskaya, Propagation and localization of polaritons in disordered organic microcavities, Physics Letters A 372, 3898 (2008).
  * Lifshitz  _et al._ [1987] I. Lifshitz, M. Kaganov, and V. Tzukernic,  _Selected Works of IM Lifshitz_ (Nauka Moscow, 1987).
  * Kavokin  _et al._ [2017] A. V. Kavokin, J. J. Baumberg, G. Malpuech, and F. P. Laussy,  _Microcavities_ , Vol. 21 (Oxford University Press, 2017).
  * Dalton  _et al._ [1996] B. Dalton, E. Guerra, and P. Knight, Field quantization in dielectric media and the generalized multipolar hamiltonian, Physical Review A 54, 2292 (1996).
  * Viviescas and Hackenbroich [2003] C. Viviescas and G. Hackenbroich, Field quantization for open optical cavities, Physical Review A 67, 013805 (2003).
  * Scholes [2021] G. D. Scholes, Emergence of collective coherent states from strong-light coupling of disordered systems, The Journal of Physical Chemistry A 125, 6739 (2021).
  * Jackson [2021] J. D. Jackson,  _Classical electrodynamics_ (John Wiley & Sons, 2021).
  * Zoubi and La Rocca [2005] H. Zoubi and G. La Rocca, Microscopic theory of anisotropic organic cavity exciton polaritons, Physical Review B 71, 235316 (2005).
  * Litinskaya and Agranovich [2009] M. Litinskaya and V. Agranovich, In-gap polaritons in uniformly filled microcavities, Journal of Physics: Condensed Matter 21, 415301 (2009).
  * Agranovich  _et al._ [2011] V. Agranovich, Y. N. Gartstein, and M. Litinskaya, Hybrid resonant organic–inorganic nanostructures for optoelectronic applications, Chemical reviews 111, 5179 (2011).
  * Agranovich [2009] V. M. Agranovich,  _Excitations in Organic Solids_ , Vol. 142 (OUP Oxford, 2009).
  * Tavis and Cummings [1968] M. Tavis and F. W. Cummings, Exact Solution for an N-Molecule-Radiation-Field Hamiltonian, Physical Review 170, 379 (1968).
  * Tavis and Cummings [1969] M. Tavis and F. W. Cummings, Approximate Solutions for an N-Molecule-Radiation-Field Hamiltonian, Physical Review 188, 692 (1969).
  * Vinogradov [1992] E. A. Vinogradov, Vibrational polaritons in semiconductor films on metal surfaces, Physics reports 217, 159 (1992).
  * Skipetrov and Sokolov [2018] S. E. Skipetrov and I. Sokolov, Ioffe-regel criterion for anderson localization in the model of resonant point scatterers, Physical Review B 98, 064207 (2018).
  * Poduval and Sarma [2023] P. P. Poduval and S. D. Sarma, Anderson localization in doped semiconductors, arXiv preprint arXiv:2302.00680 (2023).
  * Van Tiggelen  _et al._ [1991] B. Van Tiggelen, A. Lagendijk, A. Tip, and G. Reiter, Effect of resonant scattering on localization of waves, Europhysics Letters 15, 535 (1991).
  * Agranovich  _et al._ [1987] V. Agranovich, V. Kravtsov, and I. Lerner, On the possibility of photon localization in doped semiconductors near excitonic resonances, Physics Letters A 125, 435 (1987).
  * Litinskaia  _et al._ [2002] M. Litinskaia, V. Agranovich, and G. La Rocca, Elastic scattering of polaritons in high quality microcavities and weak localization, physica status solidi (a) 190, 395 (2002).
  * del Pino  _et al._ [2015] J. del Pino, J. Feist, and F. J. Garcia-Vidal, Quantum theory of collective strong coupling of molecular vibrations with a microcavity mode, New Journal of Physics (2015).
  * Steck [2017] D. A. Steck,  _Classical and Modern Optics_ , revision 1.7.4 ed. (available online at http://steck.us/teaching, 2017).
  * Houdré  _et al._ [1996] R. Houdré, R. P. Stanley, R. P. Stanley, and M. Ilegems, Vacuum-field Rabi splitting in the presence of inhomogeneous broadening: Resolution of a homogeneous linewidth in an inhomogeneously broadened system, Physical Review A (1996).
  * Li  _et al._ [2022b] T. E. Li, A. Nitzan, and J. E. Subotnik, Polariton relaxation under vibrational strong coupling: Comparing cavity molecular dynamics simulations against fermi’s golden rule rate, The Journal of Chemical Physics 156, 134106 (2022b).
  * Skipetrov and Sokolov [2014] S. E. Skipetrov and I. M. Sokolov, Absence of anderson localization of light in a random ensemble of point scatterers, Physical review letters 112, 023905 (2014).
  * Economou  _et al._ [1984] E. Economou, C. Soukoulis, and A. Zdetsis, Localized states in disordered systems as bound states in potential wells, Physical Review B 30, 1686 (1984).
  * Vollhardt and Wolfle [1980] D. Vollhardt and P. Wolfle, Diagrammatic, self-consistent treatment of the anderson localization problem in d≤ 2 dimensions, Physical Review B 22, 4666 (1980).
  * Allen and Kelner [1998] P. B. Allen and J. Kelner, Evolution of a vibrational wave packet on a disordered chain, American Journal of Physics 66, 497 (1998).
  * Myers  _et al._ [2018] D. Myers, S. Mukherjee, J. Beaumariage, D. Snoke, M. Steger, L. Pfeiffer, and K. West, Polariton-enhanced exciton transport, Physical Review B 98, 235302 (2018).
  * Wertheim  _et al._ [1974] G. Wertheim, M. Butler, K. West, , and D. Buchanan, Determination of the gaussian and lorentzian content of experimental line shapes, Review of Scientific Instruments 45, 1369 (1974).
  * Whiting [1968] E. E. Whiting, An empirical approximation to the voigt profile, Journal of Quantitative Spectroscopy and Radiative Transfer 8, 1379 (1968).
  * Gardiner and Collett [1985] C. W. Gardiner and M. J. Collett, Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation, Physical Review A 31, 3761 (1985).
  * Kenkre  _et al._ [1994] V. Kenkre, A. Tokmakoff, and M. Fayer, Theory of vibrational relaxation of polyatomic molecules in liquids, The Journal of chemical physics 101, 10618 (1994).
  * Mukamel [1999] S. Mukamel,  _Principles of Nonlinear Optical Spectroscopy_ (Oxford University Press on Demand, 1999).
  * Tokmakoff  _et al._ [1994] A. Tokmakoff, B. Sauter, A. Kwok, and M. Fayer, Phonon-induced scattering between vibrations and multiphoton vibrational up-pumping in liquid solution, Chemical physics letters 221, 412 (1994).
  * May and Kühn [2008] V. May and O. Kühn,  _Charge and energy transfer dynamics in molecular systems_ (John Wiley & Sons, 2008).
  * Kittel [2005] C. Kittel,  _Introduction to solid state physics_ (John Wiley & sons, inc, 2005).
  * Weiss [2012] U. Weiss,  _Quantum dissipative systems_ (World Scientific, 2012).
  * Kéna-Cohen  _et al._ [2008] S. Kéna-Cohen, M. Davanço, and S. R. Forrest, Strong Exciton-Photon Coupling in an Organic Single Crystal Microcavity, Physical Review Letters 101, 116401 (2008).