# Polaritons in an electron gas – quasiparticles and Landau effective
interactions

Miguel Angel Bastarrachea-Magnani  Departamento de Física, Universidad
Autónoma Metropolitana-Iztapalapa, San Rafael Atlixco 186, C.P. 09340, CDMX,
México.  Department of Physics and Astronomy, Aarhus University, Ny Munkegade,
DK-8000 Aarhus C, Denmark  Jannie Thomsen  Department of Physics and
Astronomy, Aarhus University, Ny Munkegade, DK-8000 Aarhus C, Denmark  Arturo
Camacho-Guardian  T.C.M. Group, Cavendish Laboratory, University of Cambridge,
JJ Thomson Avenue, Cambridge, CB3 0HE, U.K.  Georg M. Bruun  Department of
Physics and Astronomy, Aarhus University, Ny Munkegade, DK-8000 Aarhus C,
Denmark  Shenzhen Institute for Quantum Science and Engineering and Department
of Physics, Southern University of Science and Technology, Shenzhen 518055

###### Abstract

Two-dimensional semiconductors inside optical microcavities have emerged as a
versatile platform to explore new hybrid light-matter quantum states. The
strong light-matter coupling leads to the formation of exciton-polaritons,
which in turn interact with the surrounding electron gas to form
quasiparticles called polaron-polaritons. Here, we develop a general
microscopic framework to calculate the properties of these quasiparticles such
as their energy and the interactions between them. From this, we give
microscopic expressions for the parameters entering a Landau theory for the
polaron-polaritons, which offers a simple yet powerful way to describe such
interacting light-matter many-body systems. As an example of the application
of our framework, we then use the ladder approximation to explore the
properties of the polaron-polaritons. We furthermore show that they can be
measured in a non-demolition way via the light transmission/reflection
spectrum of the system. Finally, we demonstrate that the Landau effective
interaction mediated by electron-hole excitations is attractive leading to red
shifts of the polaron-polaritons. Our work provides a systematic framework to
study exciton-polaritons in electronically doped two-dimensional materials
such as novel van der Waals heterostructures.

##  I Introduction

Semiconductors in optical microcavities constitute a rich setting for
exploring hybrid light-matter quantum systems with potential optoelectronic
applications Sanvitto and Kéna-Cohen (2016); Kavokin  _et al._ (2017). An
important example is the case of exciton-polaritons, which are quantum
mechanical superpositions of photons and bound electron-hole pairs confined in
a two-dimensional (2D) semiconductor layer inside an optical cavity Hopfield
(1958); Weisbuch  _et al._ (1992). An appealing feature of polaritons is that
they inherit the properties of both their fundamental constituents thereby
providing a tunable way to transfer attributes from matter to light, and
viceversa. Hence, not only can they be selectively excited, controlled and
detected by optical means, but they also possess strong interactions that
introduce novel non-linear optical effects Laussy (2012); Carusotto and Ciuti
(2013). As exciton-polaritons can be considered bosons for extended
temperature and density ranges , they exhibit effects like Bose-Einstein
condensation and superfluidity Kasprzak  _et al._ (2006); Balili  _et al._
(2007); Amo  _et al._ (2009a, b); Kohnle  _et al._ (2011, 2012); Lagoudakis
_et al._ (2008); Sanvitto  _et al._ (2010), although the pump-loss nature of
the experiments leads to a number of important differences compared to the
equilibrium condensates.

Atomically thin transition-metal dichalcogenids (TMDs) Radisavljevic  _et al._
(2011); Mak and Shan (2016); Wang  _et al._ (2018) are among the 2D materials
that have been in the spotlight in recent years. They are composed by two
hexagonal planes of a transition metal atom M (Mo, W) that covalently binds
with chalcogen atoms (S, Se, Te) to form an hexagonal lattice with a trigonal
prismatic arrangement ($\mbox{MX}_{2}$) Bromley  _et al._ (1972); Mak  _et
al._ (2010); Zhu  _et al._ (2011). It has been found that atomically thin
layers of TMDs are thermodynamically stable and that they are direct-gap
semiconductors from the visible to the infrarred Novoselov  _et al._ (2005);
Wang  _et al._ (2018); Mak  _et al._ (2010); Splendiani  _et al._ (2010). The
extrema of the bands are located at the finite momentum $K^{+}$ ($K^{-}$)
points in the hexagonal Brillouin zone and connected by a broken inversion
symmetry. Together with a strong spin-orbit coupling (SOC) this leads to
valley-spin locking, i.e., the coupling between the valley and spin degrees of
freedom Ramasubramaniam (2012); Xiao  _et al._ (2012); Echeverry  _et al._
(2016). As a result, there are valley selective optical rules Cao  _et al._
(2012); Yu  _et al._ (2015); Wang  _et al._ (2018), which, together with
strong light-matter coupling Dufferwiel  _et al._ (2015); Liu  _et al._ (2015)
offer a promising playground for spin optoelectronics and valleytronics Xiao
_et al._ (2012); Xu  _et al._ (2014); Schaibley  _et al._ (2016).

The large binding energy of excitons in TMDs as compared to other microcavity
semiconductors like quantum-wells Chichibu  _et al._ (1996); Qiu  _et al._
(2013); He  _et al._ (2014), combined with the possibility to control the
electron density in the different valleys, opens up exciting new venues to
explore Bose-Fermi mixtures in a hybrid light-matter setting Mak  _et al._
(2012); Emmanuele  _et al._ (2020); Julku  _et al._ (2021). This has
stimulated a number of studies regarding the properties electron-exciton
mixtures and their coupling to light Suris  _et al._ (2001); Suris (2003);
Rapaport  _et al._ (2001); Qarry  _et al._ (2003); Bajoni  _et al._ (2006);
Pimenov  _et al._ (2017); Efimkin and MacDonald (2018); Glazov (2020);
Kyriienko  _et al._ (2020); Rana  _et al._ (2020); Efimkin  _et al._ (2020).
In particular, the emergence of new quasiparticles, the so-called Fermi-
polaron-polaritons have been observed Sidler  _et al._ (2016). They can be
roughly described as a coherent superposition of photons and Fermi polarons,
which are formed by the polaritons interacting with the surrounding electron
gas (2DEG) in analogy with what is observed in atomic gases Schirotzek  _et
al._ (2009); Kohstall  _et al._ (2012); Koschorreck  _et al._ (2012); Cetina
_et al._ (2015, 2016); Scazza  _et al._ (2017); Adlong  _et al._ (2020);
Fritsche  _et al._ (2021).

Two recent experiments have observed large energy shifts of these polaron-
polaritons due to the injection of itinerant electrons in a monolayer TMD
indicating the presence of induced interactions between them Tan  _et al._
(2020); Emmanuele  _et al._ (2020), which opens the door to exploring
interacting quasiparticles in a new hybrid light-matter setting. Landau’s
theory of quasiparticles stands out as a powerful yet simple framework to
describe precisely such interacting many-body systems, including their single
particle and collective properties both in- and out-of-equilibrium Landau
(1957a, b); Baym and Pethick (1991). In light of this, an important question
concerns how to calculate the parameters entering such a Landau theory for
polaron-polaritons.

Inspired by this, we present here a theoretical framework for polaron-
polaritons in a 2DEG in terms of Green’s functions. We moreover show how this
can be used to calculate the parameters of a Landau theory of polaron-
polaritons, which encompasses the strong light-matter coupling. Apart from
assuming that the concentration of the polaron-polaritons is much smaller than
that of the 2DEG and that equilibrium theory can be applied, our theory is
completely general. We then give a concrete example of these results by
employing an approximate many-body theory, the so-called ladder approximation,
which includes strong two-body correlations leading to a bound state between
an exciton and an electron, i.e. a trimer. Using this, we explore the
different polaron-polariton branches and demonstrate how the
transmission/reflection spectrum of the system offer a new experimental way to
determine the energy and residue of the underlying polarons in a non-
demolition way. The energy of the polaron-polaritons is then shown to decrease
with their concentration corresponding to an attractive Landau quasiparticle
interaction mediated by particle-hole excitations in the 2DEG.

The remainder of the manuscript is structured as follows. In Section. II, we
introduce the system and discuss the formation of the hybrid light-matter
polaritons. In Section. III, we turn our attention to the effects of
interactions and show how this can be described microscopically. We then
connect this to Landau’s quasiparticle theory providing microscopic
expressions for the quasiparticle energies and their effective interactions.
In Sec. IV, we apply these results to the ladder approximation, and analyse
the predicted properties of the quasiparticles and the interactions between
them. We also propose a new way to measure those via the light
transmission/reflection spectrum. Finally, in Sec. V we present our
conclusions and offer some perspectives.

##  II System

We consider a 2D semiconductor in an optical microcavity. Photons in the
cavity are strongly coupled to excitons in the semiconductor and the excitons
in turn interact with a 2D electron gas (2DEG). The Hamiltonian for the system
is $\hat{H}=\hat{H}_{0}+\hat{H}_{I}$ where

$\displaystyle\hat{H}_{0}=\sum_{\mathbf{k}}\left[\varepsilon_{e\mathbf{k}}\hat{e}_{\mathbf{k}}^{\dagger}\hat{e}_{\mathbf{k}}+\varepsilon_{x\mathbf{k}}\hat{x}_{\mathbf{k}}^{\dagger}\hat{x}_{\mathbf{k}}+\varepsilon_{c\mathbf{k}}\hat{c}_{\mathbf{k}}^{\dagger}\hat{c}_{\mathbf{k}}\right]+\sum_{\mathbf{k}}\Omega\left(\hat{x}_{\mathbf{k}}^{\dagger}\hat{c}_{\mathbf{k}}+\hat{c}_{\mathbf{k}}^{\dagger}\hat{x}_{\mathbf{k}}\right)$
(1)

are the non-interacting and the light-matter coupling terms. Here
$\hat{x}_{\mathbf{k}}^{\dagger}$, $\hat{c}_{\mathbf{k}}^{\dagger},$ and
$\hat{e}_{\mathbf{k}}^{\dagger},$ creates an exciton, photon, and electron
respectively with two-dimensional crystal momentum $\mathbf{k}$. The energy of
these particles is $\varepsilon_{x\mathbf{k}}=\mathbf{k}^{2}/2m_{x},$
$\varepsilon_{c\mathbf{k}}=\mathbf{k}^{2}/2m_{c}+\delta$, and
$\varepsilon_{e\mathbf{k}}=\mathbf{k}^{2}/2m_{e}$, where $m_{x}$, $m_{c}$, and
$m_{e}$ is their mass and $\delta$ is the detuning between the exciton and
photon energies at zero momentum. We set $\hbar=k_{B}=1$ throughout. For
concreteness, we take $m_{c}=10^{-5}m_{x}$, $m_{x}=2m_{e}$ and assume the
light-matter coupling $\Omega$ to be real. The energy offset of the electrons
will be absorbed into their chemical potential. It follows from the optical
and valley selection rules of TMDs Radisavljevic  _et al._ (2011); Mak and
Shan (2016); Wang  _et al._ (2018) that polarised photons couple to excitons
in a specific spin and valley state, which in turn predominantly interacts
with the 2DEG in the opposite valley. Here, we focus on a given spin and
valley and therefore suppress those degrees of freedom in Eq. (1) and onwards.
The excitons are assumed to have a binding energy much larger than any other
relevant energy scale in the system so that they can be considered as point
bosons. For high exciton densities or localised excitons, their composite
nature becomes important and the point boson approximation breaks down,
leading to changes in the effective light-matter interaction and saturation
effects Emmanuele  _et al._ (2020); Kyriienko  _et al._ (2020).

The non-interacting Hamiltonian Eq. (1) is readily diagonalised by means of a
Hopfield transformation Hopfield (1958)

$\displaystyle\begin{bmatrix}\hat{x}_{\mathbf{k}}\\\
\hat{c}_{\mathbf{k}}\end{bmatrix}=\begin{bmatrix}\mathcal{C}_{\mathbf{k}}&-\mathcal{S}_{\mathbf{k}}\\\
\mathcal{S}_{\mathbf{k}}&\mathcal{C}_{\mathbf{k}}\\\
\end{bmatrix}\begin{bmatrix}\hat{L}_{\mathbf{k}}\\\
\hat{U}_{\mathbf{k}}\end{bmatrix}$ (2)

where $L_{\mathbf{k}}^{\dagger}$ ($U_{\mathbf{k}}^{\dagger}$) are the creation
operators of lower and upper polaritons respectively with momentum
$\mathbf{k}$. The corresponding Hopfield coefficients are
$\mathcal{C}_{\mathbf{k}}^{2}=(1+\delta_{\mathbf{k}}/\sqrt{\delta_{\mathbf{k}}^{2}+4\Omega^{2}})/2$
and $\mathcal{S}_{\mathbf{k}}^{2}=1-\mathcal{C}_{\mathbf{k}}^{2}$ with
$\delta_{\mathbf{k}}=\varepsilon_{c\mathbf{k}}-\varepsilon_{x\mathbf{k}}$, and

$\displaystyle\varepsilon_{\sigma\mathbf{k}}=\frac{1}{2}\left(\varepsilon_{c\mathbf{k}}+\varepsilon_{x\mathbf{k}}\pm\sqrt{\delta_{\mathbf{k}}^{2}+4\Omega^{2}}\right),$
(3)

giving the energy of the standard upper $\sigma=\text{U}$ and lower
$\sigma=\text{L}$ exciton-polaritons in absence of the Fermi sea. Interactions
between the excitons and electrons in opposite valleys are described by the
term

$\displaystyle\hat{H}_{I}={\frac{1}{\mathcal{A}}}\sum_{\mathbf{q},\mathbf{k},\mathbf{k}^{\prime}}V_{\mathbf{q}}\hat{e}_{\mathbf{k}+\mathbf{q}}^{\dagger}\hat{x}_{\mathbf{k}^{\prime}-\mathbf{q}}^{\dagger}\hat{x}_{\mathbf{k}^{\prime}}\hat{e}_{\mathbf{k}},$
(4)

where $\mathcal{A}$ is the area of the system. For small Fermi energies and
relevant momenta the electron-exciton interaction can be approximated as a
contact one $V_{\mathbf{q}}\simeq\mathcal{T}_{0}$ Sidler  _et al._ (2016).
This is equivalent to treating the exciton-polaritons as point-like bosons.
Also, we assume that the Coulomb interaction between the electrons are
included by a renormalisation of their dispersion using Fermi liquid theory
Shankar (1994); Ahn and Sarma (2021), and we furthermore neglect the direct
interaction between excitons. For small densities, the latter is rather weak
due to the large binding energy of the excitons, which is typically two orders
of magnitude larger than the rest of energy scales Chichibu  _et al._ (1996);
Qiu  _et al._ (2013); He  _et al._ (2014); Zhu  _et al._ (2015), and it can
easily be included at the mean-field level.

##  III Fermi polaron-polaritons

We now consider the situation where the density of exciton-polaritons is small
compared to the electron density. In this case, the effects of the exciton-
polaritons on the 2DEG can be neglected and the problem reduces to that of
mobile bosonic impurities in an electron gas. The interaction between the
exciton-polaritons and the surrounding electron gas then gives rise to the
formation of quasiparticles denoted Fermi polaron-polaritons or, in short,
polaron-polaritons. Apart from the presence of strong light coupling this has
strong similarities to the formation of Fermi polarons in atomic gases
Massignan  _et al._ (2014). In this section we will describe their generic
properties both from a microscopic point of view as well as using Landau’s
quasiparticle framework. We will furthermore provide precise links between the
two descriptions when appropriate. While these results are general, we will
illustrate them by using a microscopic approximated many-body theory as an
example.

###  III.1 Microscopic theory

Despite the fact that polariton systems are driven by external lasers, many of
their steady-state properties can be accurately described using equilibrium
theory with a few modifications, such as chemical potentials being determined
by the external laser frequencies Carusotto and Ciuti (2013). We therefore
employ finite temperature quantum field theory to analyse the problem
microscopically Fetter and Walecka (1971). Since the electrons are unaffected
by the excitons, we can focus on the cavity photons and excitons described by
the $2\times 2$ exciton-photon finite-temperature Green’s function
${\mathcal{G}}(\mathbf{k},\tau)=-\langle
T_{\tau}\\{\hat{\Psi}_{\mathbf{k}}(\tau)\hat{\Psi}_{\mathbf{k}}^{\dagger}(0)\\}\rangle$,
where
$\hat{\Psi}_{\mathbf{k}}=[\hat{x}_{\mathbf{k}},\hat{c}_{\mathbf{k}}]^{T}$ and
$T_{\tau}$ denotes the imaginary time ordering. By Fourier transforming, it
can be written in terms of the free propagator $\mathcal{G}_{0}(k)$ and the
proper self-energy $\mathbf{\Sigma}(k)$ as

$\displaystyle{\mathcal{G}}^{-1}(k)={\mathcal{G}_{0}}^{-1}(k)-\mathbf{\Sigma}(k)=\begin{bmatrix}i\omega_{l}-\varepsilon_{x\mathbf{k}}&0\\\
0&i\omega_{l}-\varepsilon_{c\mathbf{k}}\end{bmatrix}-\begin{bmatrix}\Sigma_{xx}(k)&\Omega\\\
\Omega&0\end{bmatrix}.$ (5)

where $k=(\mathbf{k},\omega_{l})$, $\omega_{l}=2\pi lT$ with $l=0,\pm
1,\ldots$ is a bosonic Matsubara frequency, $T$ is the temperature, and
$\Sigma_{xx}(k)$ is the exciton self-energy. As usual, one can obtain the
retarded Green’s function by analytic continuation
$\mathcal{G}(\mathbf{k},\omega)=\left.\mathcal{G}(\mathbf{k},i\omega_{l})\right|_{i\omega_{l}\rightarrow\omega+i0^{+}}$.

In the absence of light, the problem is equivalent to impurity particles
interacting with a Fermi sea, which is known to lead to the formation
quasiparticles called Fermi polarons Chevy (2006); Massignan  _et al._ (2014);
Levinsen and Parish (2015). The coupling to light turns these polarons into
polaron-polaritons, and in analogy with Eq. (3) the energy of these
quasiparticles is given by the self-consistent solutions of

$\displaystyle\varepsilon_{{\sigma}\mathbf{k}}=\frac{1}{2}\left[\varepsilon_{c\mathbf{k}}+\varepsilon_{x\mathbf{k}}+\Sigma_{xx}\left(\mathbf{k},\varepsilon_{\sigma\mathbf{k}}\right)\pm\sqrt{\left[\delta_{\mathbf{k}}-\Sigma_{xx}\left(\mathbf{k},\varepsilon_{\sigma\mathbf{k}}\right)\right]^{2}+4\Omega^{2}}\right].$
(6)

Here, the subindex $\sigma$ denotes the different quasiparticle branches
emerging in the system. Also, a new set of Hopfield coefficients arise giving
the matter and photon components of the polaron-polaritons. As in Eq. (2) they
are

$\displaystyle\mathcal{C}_{\mathbf{k}\sigma}^{2}=\frac{1}{2}+\frac{\varepsilon_{c\mathbf{k}}-\varepsilon_{x\mathbf{k}}-\Sigma_{xx}\left(\mathbf{k},\varepsilon_{\sigma\mathbf{k}}\right)}{2\sqrt{\left[\varepsilon_{c\mathbf{k}}-\varepsilon_{x\mathbf{k}}-\Sigma_{xx}\left(\mathbf{k},\varepsilon_{\sigma\mathbf{k}}\right)\right]^{2}+4\Omega^{2}}}\hskip
14.22636pt\text{ and }\hskip
14.22636pt\mathcal{S}_{\mathbf{k}\sigma}^{2}=1-\mathcal{C}_{\mathbf{k}\sigma}^{2}.$
(7)

###  III.2 Landau theory

Landau’s description of macroscopic systems in terms of quasiparticles is a
highlight in theoretical physics and provides a remarkably simple yet accurate
description of otherwise complex many-body systems Landau (1957a, b). This
includes both their single-particle and collective equilibrium and non-
equilibrium properties, and it is therefore important to understand how it can
be applied to polaron-polaritons. We now address this question and provide
precise links between Landau’s framework and the microscopic theory in the
previous section.

The foundation of Landau’s theory idea is to write the energy $E$ of a system
in powers of its low energy excitations, which have particle like properties,
i.e. the quasiparticles as Baym and Pethick (1991)

$\displaystyle
E=E_{g}+\sum_{\mathbf{q},\sigma}\varepsilon_{\mathbf{k}\sigma}^{0}n_{\mathbf{k}\sigma}+\frac{1}{2{\mathcal{A}}}\sum_{\mathbf{k},\mathbf{k}^{\prime},\sigma,\sigma^{\prime}}\mathsf{f}_{\mathbf{k}\sigma,\mathbf{k}^{\prime}\sigma^{\prime}}n_{\mathbf{k}\sigma}n_{\mathbf{k}^{\prime}\sigma^{\prime}}+...,$
(8)

where $E_{g}$ is the ground state energy of the system and
$\varepsilon_{\mathbf{k}\sigma}^{0}$ is the quasiparticle energy. The
distribution function in a given quasiparticle branch $\sigma$ is given by
$n_{\mathbf{k}\sigma}$, and
$\mathsf{f}_{\mathbf{k}\sigma,\mathbf{k}^{\prime}\sigma^{\prime}}$ is the
interaction between quasiparticles in branches $\sigma$ and $\sigma^{\prime}$
with momenta $\mathbf{k}$ and $\mathbf{k}^{\prime}$. In principle, there are
terms of higher order in $n_{\mathbf{k}\sigma}$ in Eq. (8), which correspond
to three-body interaction terms and higher. However, such terms are usually
not important for realistic densities and it is standard in Landau’s
quasiparticle theory to truncate the series at quadratic order corresponding
to including two-body interactions, as we do here.

In the present case, the quasiparticles are the polaron-polaritons and their
energy $\varepsilon_{\mathbf{k}\sigma}^{0}$ are given by solutions of Eq. (6)
taking the zero impurity limit, i.e. a vanishing quasiparticle distribution
function $n_{\mathbf{k}\sigma}=0$. The ground state of the system is simply
the 2DEG with no polaron-polaritons present with the energy
${\mathcal{A}}n_{e}\varepsilon_{F}/2$ where $n_{e}$ is the density of the 2DEG
with Fermi energy $\varepsilon_{F}$. When the number of quasiparticles is non-
zero, it follows from Eq. (8) that their energy is

$\displaystyle\varepsilon_{\mathbf{k}\sigma}=\varepsilon_{\mathbf{k}\sigma}^{0}+\frac{1}{\mathcal{A}}\sum_{\mathbf{k}^{\prime}\sigma^{\prime}}\mathsf{f}_{\mathbf{k}\sigma,\mathbf{k}^{\prime}\sigma^{\prime}}n_{\mathbf{k}^{\prime}\sigma^{\prime}}.$
(9)

It follows from Eq. (9) that the interaction between the quasiparticles can be
found as Bastarrachea-Magnani  _et al._ (2021)

$\displaystyle\frac{\mathsf{f}_{\mathbf{k}\sigma,\mathbf{k}^{\prime}\sigma^{\prime}}}{\mathcal{A}}=\frac{d\varepsilon_{\mathbf{k}\sigma}}{dn_{\mathbf{k}^{\prime}\sigma^{\prime}}}=Z_{\mathbf{k}\sigma}\mathcal{X}_{\mathbf{k}\sigma}^{2}\frac{\partial\Sigma_{xx}(\mathbf{k},\varepsilon_{\mathbf{k}\sigma})}{\partial
n_{{\mathbf{k}}^{\prime}\sigma^{\prime}}},$ (10)

where

$\displaystyle
Z_{\mathbf{k}\sigma}^{-1}=1-\mathcal{X}_{\mathbf{k}\sigma}^{2}\partial_{\omega}\Sigma_{xx}(\mathbf{k},\varepsilon_{\mathbf{k}\sigma})$
(11)

is the residue of a polaron-polariton in branch $\sigma$ with momentum
$\mathbf{k}$ and we have used Eq. (6) in the second equality. Here,
$\mathcal{X}_{\mathbf{k}\sigma}=\mathcal{S}_{\mathbf{k}\sigma}$ when the
quasiparticle energy is determined using the $+\sqrt{\ldots}$ version of the
upper polariton poles in Eq. (6), whereas
$\mathcal{X}_{\mathbf{k}\sigma}=\mathcal{C}_{\mathbf{k}\sigma}$ when the
$-\sqrt{\ldots}$ version of the lower polariton in Eq. (6) is used. Compared
to the usual microscopic many-body formula for Landau’s quasiparticle
interaction Camacho-Guardian and Bruun (2018); Giuliani and Vignale (2005),
Eq. (10) has the additional feature of containing the many-body Hopfield
coefficients. They reflect that it is only the excitonic part of the
quasiparticles which interact with the surrounding 2DEG.

Equations (5)-(10) provide a framework for describing polaron-polaritons in a
2DEG microscopically and moreover show how to connect this to Landau’s
quasiparticle theory. The main assumptions are that the concentration of
polaron-polaritons is much smaller than that of the electrons so that their
effects on the 2DEG can be neglected, and that we can use equilibrium theory
to describe its steady state properties. We now illustrate these results using
an approximate many-body theory.

##  IV The ladder approximation

To give a concrete example of the results in the previous section, we apply
the much used so-called ladder approximation to describe polaritons
interacting with a 2DEG. This theory has turned out to be surprisingly
accurate for mobile impurities in atomic Fermi gases Massignan  _et al._
(2014), which is a problem with many similarities to the one at hand. The
basic idea is to include the two-body scattering physics exactly in a many-
body environment and it is thus particularly suited to describe systems with
strong two-body correlations such as molecule formation or hard core repulsion
Fetter and Walecka (1971). In the present context, the molecules correspond to
bound states of an exciton and an electron, i.e. a trion, which indeed have
been observed in TMDs Lampert (1958); Thilagam (1997); Esser  _et al._ (2001);
Mak  _et al._ (2012); Courtade  _et al._ (2017); Zhu  _et al._ (2014); Ganchev
_et al._ (2015); Sie  _et al._ (2016) motivating the use of this
approximation. In the ladder approximation, the exciton self-energy is given
by

$\displaystyle\Sigma_{xx}(k)=\frac{T}{\mathcal{A}}\sum_{q}\mathcal{G}_{e}(q)\mathcal{T}(k+q),$
(12)

where $k=(\mathbf{k},i\omega_{l})$,
$\mathcal{G}^{-1}_{e}(\mathbf{k},i\omega_{j})=i\omega_{j}-\xi_{\mathbf{k}}^{e}$
is the electron propagator with $i\omega_{j}=(2j+1)\pi T$ a fermionic
Matsubara frequency, and $\sum_{q}$ denotes a sum over both Matsubara
frequencies and 2D momentum. The electron energy is taken with respect to the
Fermi energy of the 2DEG, i.e.,
$\xi_{e\mathbf{k}}=\varepsilon_{e\mathbf{k}}-\varepsilon_{F}$. In Eq. (12), we
have introduced the exciton-electron scattering matrix given by Wouters
(2007); Carusotto  _et al._ (2010); Bastarrachea-Magnani  _et al._ (2019)

$\displaystyle\mathcal{T}(k)=\frac{1}{\text{Re}\Pi_{V}(\mathbf{k}=0,\varepsilon_{T})-\Pi(k)},$
(13)

where $\Pi(k)$ is the in-medium exciton-electron pair-propagator

$\displaystyle\Pi(k)=-\frac{T}{\mathcal{A}}\sum_{q}\mathcal{G}^{(0)}_{xx}(k+q)\mathcal{G}_{e}(-q)=\sum_{\sigma}\int\\!\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\mathcal{X}^{2}_{\sigma\mathbf{k}+\mathbf{q}}\frac{1+n_{B}(\xi_{\mathbf{k}+\mathbf{q}\sigma})-n_{F}(\xi_{e-\mathbf{q}})}{i\omega_{j}-\xi_{\mathbf{k}+\mathbf{q}\sigma}-\xi^{e}_{-\mathbf{q}}}.$
(14)

Here,
$\mathcal{G}^{(0)}_{xx}(k)=\sum_{\sigma}\mathcal{X}^{2}_{\sigma\mathbf{k}}/(i\omega_{l}-\xi_{\mathbf{k}\sigma})$
is the exciton Green’s function in the absence of interactions expressed in
terms of the upper $\sigma=U$ and lower polariton $\sigma=L$ with
$\xi_{\mathbf{k}\sigma}=\varepsilon_{\mathbf{k}\sigma}-\mu_{\sigma}$ where
$\varepsilon_{\mathbf{k}\sigma}$ is given by Eq. (3). In this way, we include
the hybridisation of the exciton and the photon in the scattering matrix. Note
that we have introduced the chemical potentials $\mu_{\sigma}$ to account for
a non-zero concentration of the polaritons described by the Bose-Einstein
distribution $n_{B}(x)=[\exp(\beta x)-1]^{-1}$, whereas $n_{F}(x)=[\exp(\beta
x)+1]^{-1}$ is the Fermi-Dirac distribution for the electrons.

In deriving Eq. (13) we have assumed a momentum independent exciton-electron
interaction, which is accurate for $k_{F}a_{B}^{x}\ll 1$, where $a_{B}^{x}$ is
the Bohr radius giving the typical size of the exciton. Also, the bare
coupling strength has been expressed in terms of the energy $\varepsilon_{T}$
of the trion in the absence of the 2DEG as
$\text{Re}\Pi_{V}(0,\varepsilon_{T})=\mathcal{T}_{0}^{-1}$ Wouters (2007);
Carusotto  _et al._ (2010); Bastarrachea-Magnani  _et al._ (2019). At the
level of a single impurity and zero temperature, the $\mathcal{T}$-matrix
formalism is equivalent to Chevy’s variational ansatz Chevy (2006), which has
recently been employed to explore Fermi polaron-polaritons in TMD monolayers
Sidler  _et al._ (2016). As we shall demonstrate below, our field-theoretical
approach is however readily extended to include the effects of temperature and
a non-zero quasiparticle concentration. Such effects are usually challenging
to incorporate in a variational approach.

###  IV.1 Zero polaron-polariton density

We now discuss the properties of polaron-polaritons in the limit where their
density vanishes, which corresponds to taking
$n_{B}(\xi_{\sigma\mathbf{k}+\mathbf{q}})\rightarrow 0$ in Eq. (14). In this
case, the Matsubara sum in Eq. (12) yields

$\displaystyle\Sigma_{xx}(k)=\int\\!\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}n_{F}(\xi_{e\mathbf{q}})\mathcal{T}(\mathbf{k}+\mathbf{q},i\omega_{\nu}+\xi_{e\mathbf{q}}).$
(15)

In Fig. 1, we show the zero momentum photonic spectral density
$A_{cc}(\omega)=-2\mbox{Im}G_{cc}(\mathbf{k}=0,\omega)$ as a function of the
detuning $\delta$ obtained by inverting Eq. (5). We use the experimentally
realistic values $\Omega=8\mbox{meV}$ and $\varepsilon_{T}=-25\mbox{meV}$ Mak
_et al._ (2012); Wang  _et al._ (2015). In Fig. 1 (a)-(b) we show the spectral
function for increasing electron densities with
$\varepsilon_{F}/\varepsilon_{T}=0.015$ ($n_{e}=8.0\mbox{x}10^{10}$) and 0.19
($n_{e}=1.0\mbox{x}10^{12}$), respectively. For a typical experimental
temperature $T\approx 1K,$ Tan  _et al._ (2020) the thermal energy remains
much smaller than the Rabi coupling $(k_{B}T/\Omega\approx 0.05)$, the trion
binding energy, and the Fermi energy of the system. Temperature effects are
therefore expected to be negligible.

Let us first focus on the limit $\delta\gg|\Omega|$ where the photon is
decoupled from the excitons and electrons. In addition to the photon, there
are two quasiparticle branches in this limit: The so-called attractive polaron
corresponding to the exciton attracting the electrons around it giving a
quasiparticle energy below the trion energy, and the repulsive polaron
corresponding to the electron repelling the electrons around it giving an
energy above zero. We see that the repulsive polaron has most spectral weight
for low electron density with $\varepsilon_{F}/\varepsilon_{T}=0.015$, whereas
the attractive branch starts to gain most spectral weight for high electron
density with $\varepsilon_{F}/\varepsilon_{T}=0.19$. This is consistent with
what it is found for polarons in atomic gases, since a small electron density
with $\varepsilon_{F}\ll\varepsilon_{T}$ corresponds to the so-called BEC
limit and a large electron density $\varepsilon_{F}\gg\varepsilon_{T}$
corresponds to the BCS limit. For atomic gases, one indeed has that the
residue of the attractive polaron approaches unity in the BCS limit whereas
that of the repulsive polaron vanishes and vice versa in the BEC limit Schmidt
_et al._ (2012); Massignan  _et al._ (2014).

---
Figure 1: Photon spectral distribution $A_{cc}(\mathbf{k}=0,\omega)$ for
$n_{e}=8.0\mbox{x}10^{10}$ ($\varepsilon_{F}/\varepsilon_{T}=0.015$) (a) and
$1.0\mbox{x}10^{12}$ ($\varepsilon_{F}/\varepsilon_{T}=0.19$) (b). We observe
three quasiparticle branches L, M and U of exciton-polaron-polaritons (red
curves). The yellow solid curves correspond to the uncoupled photon and
exciton energies, while the cyan lines give the polariton branches in absence
of electron-exciton interactions. The horizontal green solid line indicates
the bare binding energy of the trion $\varepsilon_{T}$ and the dashed yellow
the binding energy in the presence of many-body correlations. (c) Size of the
Rabi coupling for the L-M branches (attractive polaron) $\Omega_{\text{LM}}$
(blue) and the M-U branches (repulsive polaron) $\Omega_{\text{MU}}$ (black)
as a function of the ratio $\varepsilon_{F}/\varepsilon_{T}$. (d) Value of the
detuning where the avoided crossings between the polaron-polariton branches
occur with the same color coding as in (c). The background colors show the 2D
polaron spectral function in the absence of light. For the calculations we
employ an additional artificial broadening $\eta/2\Omega=0.01$.

When $\delta/|\Omega|$ decreases, the photon starts to couple to the
attractive and repulsive polarons resulting in three hybrid light-matter
quasiparticle branches, which we have denoted as the upper U, middle M, and
lower L polaron-polaritons. There are two prominent avoided crossings between
these branches as it can be seen in Fig. 1 (a)-(b), and their size and
position can be understood as follows. In absence of any light-matter
coupling, the impurity forms an attractive (repulsive) polaron with energy
$\omega^{a(r)}_{\mathbf{k}}$ and residue $Z^{a(r)}_{\mathbf{k}}$ Schmidt  _et
al._ (2012); Massignan  _et al._ (2014); Efimkin and MacDonald (2018). The
coupling of these polarons to the photon can be described by the photon
Green’s function

$\displaystyle
G^{-1}_{cc}(\mathbf{k},\omega)\approx\omega-\varepsilon_{c\mathbf{k}}-\Omega^{2}\left[\frac{Z^{a}_{\mathbf{k}}}{\omega-\omega^{a}_{\mathbf{k}}}+\frac{Z^{r}_{\mathbf{k}}}{\omega-\omega^{r}_{\mathbf{k}}}\right],$
(16)

which is illustrated in Fig. 2. It describes the repeated transitions between
the photon and the polarons by the Rabi coupling as the polaron-polariton
propagates through the medium. Equation (16) includes only the quasiparticle
peaks of the exciton propagator and ignores any many-body continuum of states
in the spirit of Landau theory. From Eq. (16), we see that the matrix element
giving the size of the avoided crossing of the photon branch with the
repulsive and attractive polarons is

$\displaystyle\Omega_{\text{UM}}=\Omega\sqrt{Z^{r}_{\mathbf{k}}}\hskip
28.45274pt\text{ and }\hskip
28.45274pt\Omega_{\text{LM}}=\Omega\sqrt{Z^{a}_{\mathbf{k}}},$ (17)

respectively. This explains why the avoided crossing for the
repulsive/attractive polaron is large/small for small electron density
$\varepsilon_{F}/\varepsilon_{T}=0.015$ in Fig. 1(a), since this corresponds
to the BEC limit where the residue of the repulsive polaron approaches unity
Schmidt  _et al._ (2012); Massignan  _et al._ (2014). In the same fashion, the
avoided crossing of the repulsive/attractive polaron is small/large for large
electron density in Fig. 1(b), since this corresponds to the BCS limit where
the attractive polaron has a residue close to unity and the residue of the
repulsive polaron vanishes.

---
Figure 2: Feynman diagram for the coupling of the photon propagator (black,
wavy line) to the exciton (red line). The dotted lines represent the Rabi
coupling.

To explore this further, we plot in Fig. 1(c) the size of the two avoided
crossings extracted as the minimum energy difference between the polaron-
polariton branches as a function of the electron density. This clearly shows
how $\Omega_{\text{UM}}$ decreases with increasing electron density reflecting
the decreasing weight of the repulsive polaron. As the BCS limit is
approached, the repulsive polaron becomes ill-defined and we cannot determine
$\Omega_{\text{UM}}$. Mirroring this, $\Omega_{\text{LM}}$ increases with
increasing electron density since the residue of the attractive polaron
increases as the BCS limit is approached. Since the avoided crossing of the
photon with the exciton in the absence of electrons is given by $\Omega$, we
conclude from this that the residues of the repulsive and attractive polarons
can be extracted by measuring the size of their avoided crossings.

Furthermore, from Eq. (17) we see that the position of the avoided crossings
is determined by when the energies of the attractive and repulsive polarons
cross the photon branch. To illustrate this, we plot in Fig. 1(d) the value of
the detuning where the avoided crossings occur as a function of the electron
density. We also plot the spectral function of the polaron in a 2D Fermi gas
in the absence of light coupling determined from Eq. (5) setting $\Omega=0$
Schmidt  _et al._ (2012). The good agreement between the peaks of this
spectral function giving the energies of the attractive and repulsive polarons
in a Fermi gas and the positions of the two avoided crossings confirms that
the underlying physics indeed is driven by the coupling of polarons to light.

In conclusion, these results unfold a new experimental way to determine the
energy and residue of the polaron in a non-demolition way by detecting the
light transmission/reflection spectrum of the system. This method represents
an important alternative to earlier approaches based on Rabi-oscillations in
radio-frequency (RF) spectroscopy Nascimbène  _et al._ (2009); Kohstall  _et
al._ (2012); Scazza  _et al._ (2017); Camacho-Guardian  _et al._ (2020). We
note that these avoided crossings have already been observed experimentally
Sidler  _et al._ (2016); Efimkin and MacDonald (2017); Tan  _et al._ (2020).

###  IV.2 Non-zero polaron-polariton density

We now consider the case of a non-zero polaron-polariton density focusing on
how this affects their energy. From this, we will derive a microscopic
expression for Landau’s quasiparticle interaction within the ladder
approximation.

Our starting point is Eq. (12) for the exciton self-energy. For a non-zero
density of excitons, evaluating the Matsubara sum yields Bastarrachea-Magnani
_et al._ (2021)

$\displaystyle\Sigma_{xx}(\mathbf{k},i\omega_{\nu})=$
$\displaystyle\int\\!\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\Big{[}n_{F}(\xi_{e\mathbf{q}})\mathcal{T}(\mathbf{k}+\mathbf{q},i\omega_{\nu}+\xi_{e\mathbf{q}})$
$\displaystyle+\int_{-\infty}^{\infty}\\!\frac{d\omega^{\prime}}{\pi}\,\frac{n_{F}(\omega^{\prime})\mbox{Im}\mathcal{T}(\mathbf{k}+\mathbf{q},\omega^{\prime}+i0^{+})}{i\omega_{\nu}-\omega^{\prime}+\xi_{e\mathbf{q}}}-\frac{n_{F}(\omega^{\text{tr}}_{\mathbf{k}+\mathbf{q}})Z^{\text{tr}}_{\mathbf{k}+\mathbf{q}}}{i\omega_{\nu}-\omega^{\text{tr}}_{\mathbf{k}+\mathbf{q}}+\xi_{e\mathbf{q}}}\Big{]}.$
(18)

Compared to Eq. (15), the finite exciton density gives rise to the two new
terms in the second line of Eq. (18). The last term is a contribution coming
from a non-zero population of the trion state, which appears as a pole in the
many-body scattering matrix at the energy $\omega^{\text{tr}}_{\mathbf{k}}$
with residue $Z^{\text{tr}}_{\mathbf{k}}$. This results in an interaction
between the trions and the excitons mediated by the exchange of an electron
Bastarrachea-Magnani  _et al._ (2021), which has been observed to give rise to
large optical non-linearities. We neglect this term in the following assuming
a zero population of trions and refer the reader to Ref. Emmanuele  _et al._
(2020) for an analysis of the interesting interaction between excitons and
trions mediated by electron exchange.

A non-zero exciton density enters the self-energy explicitly via the second
term in Eq. (18), which comes from the branch-cut of the exciton-electron
scattering matrix. Physically, it corresponds to the propagation of an
electron and an exciton with population $n_{F}(\omega)$. The exciton density
also enters the scattering matrix $\mathcal{T}$ via the exciton-electron pair
propagator given by Eq. (14). In Fig. 4, we plot the energy shift of the
lowest polaron-polariton branch
$\Delta\varepsilon_{{\mathbf{q}}\text{L}}=\varepsilon_{{\mathbf{q}}\text{L}}-\varepsilon_{{\mathbf{q}}\text{L}}^{0}$
for ${\mathbf{q}}=0$ as a function of its density
$n_{L}={\mathcal{A}}^{-1}\sum_{\mathbf{q}}n_{B}(\xi_{{\mathbf{q}}L})$ for
several values of the cavity detuning. Here,
$\varepsilon_{{\mathbf{q}}\text{L}}^{0}$ denotes the energy of the lower
polaron-polariton branch in the limit of vanishing density consistent with the
notation in section III.2. The energy shift is obtained by solving Eq. (6) for
a varying chemical potential of the polaritons. We see that the energy shift
is _negative_ and depends approximately linearly with density $n_{L}$. From
Landau theory, this negative shift corresponds to an _attractive interaction_
between the quasiparticles as can be seen explicitly from Eq. (9).

To derive a microscopic expression for the interaction between the polaron-
polaritons, it follows from Eq. (10) that we must evaluate the derivative of
the exciton self-energy with respect to their distribution
$n_{{\mathbf{q}}\sigma}=n_{B}(\xi_{{\mathbf{q}}\sigma})$. We thus expand Eq.
(18) as
$\Sigma_{xx}(\mathbf{k},\omega)=\Sigma_{n_{\sigma}=0}(\mathbf{k},\omega)+\delta\Sigma(\mathbf{k},\omega)+\mathcal{O}(n_{\sigma}^{2}),$
and evaluating this on-shell with $\omega=\xi_{\mathbf{k}\sigma}$ one obtains
Bastarrachea-Magnani  _et al._ (2021)

$\displaystyle\frac{\partial\Sigma_{xx}(\mathbf{k},\xi_{\mathbf{k}\sigma})}{\partial
n_{{\mathbf{k}}^{\prime}\sigma^{\prime}}}=$
$\displaystyle\mathcal{X}_{\mathbf{\mathbf{k}^{\prime}}\sigma^{\prime}}^{2}\int\\!\frac{d^{2}\mathbf{p}}{(2\pi)^{2}}\frac{1}{\xi_{\mathbf{k}\sigma}-\xi_{\mathbf{k}^{\prime}\sigma^{\prime}}+\xi_{\mathbf{p}e}-\xi_{\mathbf{k}-\mathbf{k}^{\prime}+\mathbf{p}e}}\times$
$\displaystyle\left[n_{F}(\xi^{e}_{\mathbf{p}})\mathcal{T}^{2}(\mathbf{k}^{\prime}-\mathbf{p},\xi_{\mathbf{k}\sigma}+\xi_{\mathbf{p}}^{e})-n_{F}(\xi^{e}_{\mathbf{k}-\mathbf{k}^{\prime}+\mathbf{p}})\mathcal{T}^{2}(\mathbf{k}^{\prime}-\mathbf{p},\xi_{\mathbf{k}-\mathbf{k}^{\prime}+\mathbf{p}}^{e}+\xi_{\mathbf{k^{\prime}}\sigma^{\prime}})\right].$
(19)

Here it is understood that all energies $\xi_{\mathbf{k}\sigma}$ as well as
the $\mathcal{T}$ matrix are evaluated for vanishing quasiparticle density.
This expression can be generalised to a non-zero density by using the full
density-dependent $\mathcal{T}$-matrix as shown in Appendix A. Note that since
we are using a non self-consistent approximation, it is the density of the
bare upper and polaritons that enter inside the exciton self-energy. To derive
Eq. (IV.2), we have identified these densities with those of the polaron-
polaritons, which corresponds to the first step in a self-consistent
calculation.

---
Figure 3: Feynman diagram of the interaction between quasiparticles $\sigma$
(red lines) and $\sigma^{\prime}$ (cyan lines) mediated by the 2DEG. The
wiggly line corresponds to the induced interaction which translates to a
$\mathcal{T}$-matrix repeated scattering mediated by an electron-hole pair
(black lines) in the 2DEG.

The effective interaction between polaron-polaritons in branches $\sigma$ and
$\sigma^{\prime}$ with momenta ${\mathbf{k}}$ and ${\mathbf{k}}^{\prime}$ can
now be obtained by inserting Eq. (IV.2) in Eq. (10). Equation (IV.2) is
illustrated diagrammatically in Fig. 3, which shows that it corresponds to an
induced interaction between two polaron-polaritons mediated by particle-hole
excitations of the electron gas. Indeed, when the polaron-polariton energy is
detuned far from the trion energy one can approximate the scattering matrices
in Eq. (IV.2) by the constant
$\mathcal{T}\simeq\mathcal{T}(\mathbf{0},\xi_{\mathbf{k}\sigma})$, and the
interaction becomes proportional to the 2D Lindhard function Bastarrachea-
Magnani  _et al._ (2021), which is characteristic of a particle-hole mediated
interaction Giuliani and Vignale (2005). For stronger interaction between the
excitons and the electrons, one must retain the full energy and momentum
dependence of the scattering matrix in Eq. (IV.2).

---
Figure 4: Energy shift of the L polaron-polariton branch as a function of
their concentration for representative values of the cavity detuning from
$\delta/2\Omega=-3.0$ to $2.0$. The color coding is indicated in the figure.
We employ a finite but small temperature $\beta\varepsilon_{F}=0.1$

We now return to Fig. 4 where the energy shift of the lowest polaritonic
branch ($\sigma^{\prime}=\text{L}$) is shown as a function of the same lowest
polariton concentration ($\sigma=\text{L}$). We can understand it in terms of
the effective interaction between the lowest polaron-polaritons. The
interaction is attractive since the energy shift is negative, and it increases
in strength with the detuning $\delta$. The reason for this is two-fold.
First, it is the excitonic component that interacts with the electrons and
this component increases with the detuning for the lowest polaron-polariton.
Second, the energy of the lowest polaron-polariton approaches the trion energy
with increasing $\delta$, which gives rise to strong resonant effects in the
electron-exciton scattering. As a result, we see from Fig. 4 that there can be
a sizeable negative energy shift of the polaron-polariton due the attractive
interaction mediated by particle-hole excitations in the 2DEG. So far, one has
instead observed a temporary positive energy shift corresponding to a
repulsive interaction, which has been attributed to a non-equilibrium phase
filling effect Tan  _et al._ (2020). It would thus be very interesting to
investigate this further experimentally as the effective interaction between
quasiparticles is a key component of Landau’s quasiparticle theory and because
it may give rise to strong non-linear optical effects Bastarrachea-Magnani
_et al._ (2021); Camacho-Guardian  _et al._ (2021).

##  V Conclusions

We presented a theoretical framework for describing polaron-polaritons in 2D
semiconductors inside optical microcavities. Microscopic expressions for the
parameters entering a Landau quasiparticle theory were given, which provides a
simple yet accurate way to describe this new system of interacting hybrid
light-matter quasiparticles. Our framework is general apart from assuming that
the concentration of the quasiparticles is much smaller than the surrounding
electron gas and that equilibrium theory can be applied. To illustrate the
results, the ladder approximation was then used to explore the system. We also
proposed a new non-demolition scheme to probe the energy and residue of the
polaron-polaritons via the Rabi splittings in the light
transmission/reflection spectrum. Finally, we showed that the Landau effective
interaction between the polaron-polaritons mediated by particle-hole
excitations in the electron gas, is attractive.

Our theoretical framework provides a systematic way to analyse current
experiments exploring exciton-polaritons in monolayer TMDs Sidler  _et al._
(2016); Tan  _et al._ (2020). It can moreover be extended to study a new class
of exciton-polaritons in van der Waals heterostructures with interlayer
Feshbach resonances Schwartz  _et al._ (2021); Kuhlenkamp  _et al._ (2021),
hybridised inter- and inter-layer excitons Alexeev  _et al._ (2019),
dipolaritons Togan  _et al._ (2018), and spatially localised excitons Zhang
_et al._ (2021); Camacho-Guardian and Cooper (2021). The rich features
predicted in these systems Shimazaki  _et al._ (2020); Kennes  _et al._ (2021)
open the door to using polaritons as quantum probes in strongly correlated
electronic states Shimazaki  _et al._ (2021), and to realise and control
strongly interacting photons. An exciting perspective is to explore the regime
of higher polaron-polariton concentrations, where many intriguing phases such
as a Bose-Einstein condensate of polaron-polaritons Julku  _et al._ (2021),
superconductivity, and supersolidity Cotleţ  _et al._ (2016) have been
predicted.

##  Appendix A Strong coupling polariton interactions

We take the self-energy as calculated in Eq. 18, but without considering the
$\mathcal{T}$-matrix real pole,

$\displaystyle\Sigma_{xx}(\mathbf{k},i\omega_{\nu})=\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}n_{F}(\xi_{e\mathbf{q}})\mathcal{T}(\mathbf{k}+\mathbf{q},i\omega_{\nu}+\xi_{e\mathbf{q}})$
(20)
$\displaystyle+\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\int_{-\infty}^{\infty}\,\frac{d\omega^{\prime}}{\pi}\,\frac{n_{F}(\omega^{\prime})\mbox{Im}\mathcal{T}(\mathbf{k}+\mathbf{q},\omega^{\prime}+i0^{+})}{i\omega_{\nu}-\omega^{\prime}+\xi_{e\mathbf{q}}},$

Next, we employ the following relationships

$\displaystyle\mbox{Im}\mathcal{T}=\left[\left(\mbox{Re}\mathcal{T}\right)^{2}+\left(\mbox{Im}\mathcal{T}\right)^{2}\right]\,\mbox{Im}\Pi\left[\left(\mathcal{T}-i\mbox{Im}\mathcal{T}\right)^{2}+\left(\mbox{Im}\mathcal{T}\right)^{2}\right]\,\mbox{Im}\Pi=$
(21)
$\displaystyle\left[\mathcal{T}^{2}-2i\mathcal{T}\mbox{Im}\mathcal{T}-\left(\mbox{Im}\mathcal{T}\right)^{2}+\left(\mbox{Im}\mathcal{T}\right)^{2}\right]\,\mbox{Im}\Pi=\left[\mathcal{T}^{2}-2i\mathcal{T}\mbox{Im}\mathcal{T}\right]\,\mbox{Im}\Pi.$

This becomes a series over the imaginary part of the pair propagator. We
separate the principal and imaginary parts of the pair propagator in Eq. 14 as

$\displaystyle\Pi(\mathbf{q},\omega)=\sum_{\sigma}\int\frac{d^{2}\mathbf{p}}{(2\pi)^{2}}\frac{}{}\mathcal{X}^{2}_{\sigma\mathbf{q}+\mathbf{p}}\left[1-n_{F}(\xi_{e-\mathbf{p}})+n_{B}(\xi_{\sigma\mathbf{q}+\mathbf{p}})\right]\mbox{x}$
(22)
$\displaystyle\left[\mathcal{P}\frac{1}{\omega-\xi_{e-\mathbf{p}}-\xi_{\sigma\mathbf{q}+\mathbf{p}}}-i\pi\delta(\omega-\xi_{e-\mathbf{p}}-\xi_{\sigma\mathbf{q}+\mathbf{p}})\right],$

inserting it in Eq. 21 we obtain

$\displaystyle\mbox{Im}\mathcal{T}(\mathbf{q},\omega)=-\pi\left[\mathcal{T}^{2}-2i\mathcal{T}\mbox{Im}\mathcal{T}\right]\mbox{x}$
(23)
$\displaystyle\sum_{\sigma}\int\frac{d^{2}\mathbf{p}}{(2\pi)^{2}}\mathcal{X}^{2}_{\sigma\mathbf{q}+\mathbf{p}}\left[1-n_{F}(\xi_{e-\mathbf{p}})+n_{B}(\xi_{\sigma\mathbf{q}+\mathbf{p}})\right]\delta(\omega-\xi_{e-\mathbf{p}}-\xi_{\sigma\mathbf{q}+\mathbf{p}}).$

Substituting this result in the second term of Eq. 20 and using that
$n_{F}(x+y)(1-n_{F}(x)+n_{B}(y))=n_{F}(x)n_{B}(y)$, therefore the self-energy
reads

$\displaystyle\Sigma_{xx}(\mathbf{k},\omega)=\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\left\\{\frac{}{}n_{F}(\xi_{e\mathbf{q}})\mathcal{T}(\mathbf{k}+\mathbf{q},\omega+\xi_{e\mathbf{q}})\right.$
(24)
$\displaystyle\left.-\sum_{\sigma}\int\frac{d^{2}\mathbf{p}}{(2\pi)^{2}}\frac{\mathcal{X}^{2}_{\sigma\mathbf{k}+\mathbf{q}+\mathbf{p}}n_{B}(\xi_{\sigma\mathbf{k}+\mathbf{q}+\mathbf{p}})n_{F}(\xi_{e-\mathbf{p}})}{\omega-\xi_{\sigma\mathbf{k}+\mathbf{q}+\mathbf{p}}+\xi_{e\mathbf{q}}-\xi_{e-\mathbf{p}}+i0^{+}}\right.\mbox{x}$
$\displaystyle\left.\left[\mathcal{T}^{2}(\mathbf{k}+\mathbf{q},\xi_{e-\mathbf{p}}+\xi_{\sigma\mathbf{k}+\mathbf{q}+\mathbf{p}}+i0^{+})\right.\left.-2i\mathcal{T}(\mathbf{k}+\mathbf{q},\xi_{e-\mathbf{p}}+\xi_{\sigma\mathbf{k}+\mathbf{q}+\mathbf{p}}+i0^{+})\mbox{Im}\mathcal{T}(\mathbf{k}+\mathbf{q},\xi_{e-\mathbf{p}}+\xi_{\sigma\mathbf{k}+\mathbf{q}+\mathbf{p}}+i0^{+})\right]\right\\}.$

As explained in the main text, the quasiparticle interactions are given by the
functional derivative of Eqs. 10 with respect to the quasiparticle
distribution Baym and Pethick (1991); Camacho-Guardian and Bruun (2018)

$\displaystyle\frac{\mathsf{f}_{\sigma\mathbf{k},\sigma^{\prime}\mathbf{k}^{\prime}}}{\mathcal{A}}=Z_{\sigma^{\prime}\mathbf{k}^{\prime}}\frac{\partial\xi_{\sigma\mathbf{k}}}{\partial
n_{\sigma^{\prime}{\mathbf{k}^{\prime}}}}=Z_{\sigma^{\prime}\mathbf{q}^{\prime}}\mathcal{X}_{\sigma^{\prime}\mathbf{k}^{\prime}}^{2}\frac{\partial\Sigma(\mathbf{k},\xi_{\sigma\mathbf{k}})}{\partial
n_{\sigma^{\prime}{\mathbf{k}^{\prime}}}},$ (25)

this entails the calculation of the derivative of the second part of the self-
energy

$\displaystyle\frac{\delta}{\delta
n_{B}(\xi_{\sigma^{\prime}\mathbf{k}^{\prime}})}\left[n_{B}(\xi_{\sigma\mathbf{k}})\left(\mathcal{T}^{2}(\mathbf{k},\omega)-2i\mathcal{T}(\mathbf{k},\omega)\mbox{Im}\mathcal{T}(\mathbf{k},\omega)\right)\right]=$
(26) $\displaystyle\frac{\delta n_{B}(\xi_{\sigma\mathbf{k}})}{\delta
n_{B}(\xi_{\sigma^{\prime}\mathbf{k}^{\prime}})}\left\\{\mathcal{T}^{2}(\mathbf{k},\omega)-2i\mathcal{T}(\mathbf{k},\omega)\mbox{Im}\mathcal{T}(\mathbf{k},\omega)+\right.$
$\displaystyle\left.n_{B}(\xi_{\sigma\mathbf{k}})\left[2\mathcal{T}(\mathbf{k},\omega)-2i\mbox{Im}\mathcal{T}(\mathbf{k},\omega)\right]\frac{\partial\mathcal{T}(\mathbf{k},\omega)}{\delta
n_{B}(\xi_{\sigma^{\prime}\mathbf{k}^{\prime}})}-n_{B}(\xi_{\sigma\mathbf{k}})\mathcal{T}(\mathbf{k},\omega)\left(\frac{\partial\mathcal{T}(\mathbf{k},\omega)}{\delta
n_{B}(\xi_{\sigma^{\prime}\mathbf{k}^{\prime}})}-\frac{\partial\mathcal{T}^{*}(\mathbf{k},\omega)}{\delta
n_{B}(\xi_{\sigma^{\prime}\mathbf{k}^{\prime}})}\right)\right\\}.$

The functional derivative of the $\mathcal{T}$-matrix is given by

$\displaystyle\frac{\delta}{\delta
n_{B}(\xi_{\sigma^{\prime}\mathbf{k}^{\prime}})}\mathcal{T}(\mathbf{k},\omega)=\frac{\mathcal{T}_{0}^{2}}{\left[1-\mathcal{T}_{0}\Pi(\mathbf{k},\omega)\right]^{2}}\frac{\delta\Pi(\mathbf{k},\omega)}{\delta
n_{B}(\xi_{\sigma^{\prime}\mathbf{k}^{\prime}})}=$ (27)
$\displaystyle\mathcal{T}^{2}(\mathbf{k},\omega)\sum_{\sigma}\int\frac{d^{2}\mathbf{p}}{(2\pi)^{2}}\frac{\mathcal{X}_{\sigma\mathbf{k}+\mathbf{p}}^{2}}{\omega-\xi_{e-\mathbf{p}}-\xi_{\sigma\mathbf{k}+\mathbf{p}}}\frac{\delta
n_{B}(\xi_{\sigma\mathbf{k}+\mathbf{p}})}{\delta
n_{B}(\xi_{\sigma^{\prime}\mathbf{k}^{\prime}})}=$
$\displaystyle=\mathcal{T}^{2}(\mathbf{k},\omega)\int\frac{d^{2}\mathbf{p}}{(2\pi)^{2}}\mathcal{X}_{\sigma\mathbf{k}+\mathbf{p}}^{2}\frac{\delta_{\sigma,\sigma^{\prime}}\delta(\mathbf{k}^{\prime}-(\mathbf{k}+\mathbf{p}))}{\omega-\xi_{e-\mathbf{p}}-\xi_{\sigma\mathbf{k}+\mathbf{p}}+i0^{+}}=\frac{\mathcal{X}_{\sigma^{\prime}\mathbf{k}^{\prime}}^{2}\mathcal{T}^{2}(\mathbf{k},\omega)}{\omega-\xi_{e\mathbf{k}-\mathbf{k}^{\prime}}-\xi_{\sigma\mathbf{k}^{\prime}}+i0^{+}}.$

Because the derivative is of the order $\mathcal{T}^{2}$, if we keep only
terms associated to second-order diagrammatic contributions we can approximate

$\displaystyle\frac{\delta}{\delta
n_{B}(\xi_{\sigma^{\prime}\mathbf{k}^{\prime}})}\left[n_{B}(\xi_{\sigma\mathbf{k}})\left(\mathcal{T}^{2}(\mathbf{k},\omega)-2i\mathcal{T}(\mathbf{k},\omega)\mbox{Im}\mathcal{T}(\mathbf{k},\omega)\right)\right]\simeq\mathcal{T}^{2}(\mathbf{k},\omega)\delta(\mathbf{k}-\mathbf{k}^{\prime})\delta_{\sigma,\sigma^{\prime}}.$
(28)

In this way, after substituting the derivative of the $\mathcal{T}$-matrix
into the derivative of the self-energy, the mediated potential on-shell, up to
second order diagrams, reads

$\displaystyle\frac{\partial\Sigma(\mathbf{k},\xi_{\sigma\mathbf{k}})}{\partial
n_{\sigma^{\prime}{\mathbf{k}^{\prime}}}}=\mathcal{X}_{\sigma^{\prime}\mathbf{k}^{\prime}}^{2}\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\frac{1}{\xi_{\sigma\mathbf{k}}-\xi_{\sigma^{\prime}\mathbf{k}^{\prime}}+\xi_{e\mathbf{q}}-\xi_{e\mathbf{k}-\mathbf{k}^{\prime}+\mathbf{q}}+i0^{+}}\mbox{x}$
(29)
$\displaystyle\left[n_{F}(\xi_{e\mathbf{q}})\mathcal{T}^{2}(\mathbf{k}^{\prime}-\mathbf{q},\xi_{\sigma\mathbf{k}}+\xi_{e\mathbf{q}}+i0^{+})-n_{F}(\xi_{e\mathbf{k}-\mathbf{k}^{\prime}+\mathbf{q}})\mathcal{T}^{2}(\mathbf{k}^{\prime}-\mathbf{q},\xi_{e\mathbf{k}-\mathbf{k}^{\prime}+\mathbf{q}}+\xi_{\sigma^{\prime}\mathbf{k}^{\prime}}+i0^{+})\right],$

which is identical to Eq. IV.2 from the main text.

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