# Light-matter interactions in the vacuum of ultra-strongly coupled systems

Daniele De Bernardis1,2, Gian Marcello Andolina2, and Iacopo Carusotto1
1Pitaevskii BEC Center, INO-CNR and Dipartimento di Fisica, Università di
Trento I-38123 Trento, Italy  2JEIP, USR 3573 CNRS, Collège de France, PSL
Research University, 11 Place Marcelin Berthelot, F-75321 Paris, France
1Pitaevskii BEC Center, INO-CNR and Dipartimento di Fisica, Università di
Trento I-38123 Trento, Italy  2JEIP, USR 3573 CNRS, Collège de France, PSL
Research University, 11 Place Marcelin Berthelot, F-75321 Paris, France

###### Abstract

We theoretically study how the peculiar properties of the vacuum state of an
ultra-strongly coupled system can affect basic light-matter interaction
processes. In this unconventional electromagnetic environment, an additional
emitter no longer couples to the bare cavity photons, but rather to the
polariton modes emerging from the ultra-strong coupling, and the effective
light-matter interaction strength is sensitive to the properties of the
distorted vacuum state. Different interpretations of our predictions in terms
of modified quantum fluctuations in the vacuum state and of radiative reaction
in classical electromagnetism are critically discussed. Whereas our discussion
is focused on the experimentally most relevant case of intersubband polaritons
in semiconductor devices, our framework is fully general and applies to
generic material systems.

The non-empty nature of the quantum vacuum is among the most fascinating
effects emerging from quantum mechanics and quantum field theories Milonni
(1994). Fundamental effects of atomic physics such as the Lamb shift Bethe
(1947), the spontaneous emission Milonni (1984), and the vacuum-field Rabi
oscillations Burstein and Weisbuch (2012); Sanchez-Mondragon  _et al._ (1983);
Agarwal (1984); Berman (1994) can be traced back to the quantum fluctuations
of the electromagnetic field in the vacuum state. However, the quantum vacuum
reveals itself also at a more macroscopic scale, for instance through the
Casimir forces Milton (2004); Klimchitskaya  _et al._ (2009); Wang  _et al._
(2021), bringing the idea that this intriguing feature of quantum physics can
be exploited for nano-manipulation and nano-mechanical devices Haroche  _et
al._ (1991); Capasso  _et al._ (2007); Gong  _et al._ (2021). In the last
years, the physics of the quantum vacuum attracted a growing interest also
from the point of view of condensed matter physics, as innovative ways to
manipulate the basic interaction mechanisms and induce new states of quantum
matter have been claimed Garcia-Vidal  _et al._ (2021); Bloch  _et al._
(2022). A crucial ingredient here is the capability to reach the _ultrastrong
coupling_ (USC) regime of light-matter interactions Ciuti  _et al._ (2005);
Frisk Kockum  _et al._ (2019); Forn-Díaz  _et al._ (2019) where the extremely
large value of the coupling strength of polarizable emitters to the
electromagnetic field leads to a significant distortion of the properties of
the electromagnetic vacuum and its quantum fluctuations.

In this Letter, we give a new twist to this research by investigating how the
peculiar properties of the USC vacuum affect basic light-matter interaction
processes involving another emitter used as a probe. Experimentally observable
consequences of this physics are highlighted, such as marked modifications of
the vacuum-field Rabi oscillations and of the spontaneous emission rate. While
these features are naturally understood as an experimentally accessible
evidence of the distorted quantum vacuum state, alternative interpretations
based on classical electromagnetism, fluctuation-dissipation theorems, and
electrostatic forces are also proposed and critically discussed.

Modulo straightforward modifications, our framework applies to generic systems
where the USC can be achieved Frisk Kockum  _et al._ (2019); Forn-Díaz  _et
al._ (2019), from cyclotron excitations in metallic resonators Scalari  _et
al._ (2012, 2013); Paravicini-Bagliani  _et al._ (2019), to Josephson-
junction-based devices coupled to superconducting microwave resonators
Yoshihara  _et al._ (2017); Tomonaga  _et al._ (2023), excitons in 2D
materials Rose  _et al._ (2022); Anantharaman  _et al._ (2023). For the sake
of concreteness, in this Letter, we focus our attention on a semiconductor-
based platform based on intersubband (ISB) transitions in quantum wells (QW),
where USC was first predicted Ciuti  _et al._ (2005) and observed Anappara
_et al._ (2009); Jouy  _et al._ (2011); Todorov  _et al._ (2010); Dietze  _et
al._ (2013); Askenazi  _et al._ (2014); Cortese  _et al._ (2021); Pisani  _et
al._ (2023). Such a platform still remains among the most promising platforms
for the study of quantum vacuum effects Liberato  _et al._ (2007); Mornhinweg
_et al._ (2021); Halbhuber  _et al._ (2020).

We specifically consider the geometry sketched in Fig. 1(a), based on a planar
metallic cavity mode ultrastrongly coupled to an ISB transition in a heavily
doped QW, in the following called the _dresser_. The dressed vacuum of the USC
regime is then used to non-perturbatively influence the coupling to the
electromagnetic field of another QW, called the _emitter_. This latter QW is
taken to be much less doped, so that the coupling to light of its own ISB
transition is far below the ultra-strong coupling regime. Depending on its
strong vs. weak coupling to light, the distortion of the vacuum state in the
USC regime is visible as a strong reinforcement (suppression) of the vacuum-
field Rabi oscillation frequency or of the spontaneous emission rate when the
emitter is resonant with lower (upper) polariton branch.

Figure 1: (a) Illustration of the considered setup. The cavity consists of two
plane-parallel metallic mirrors of surface $S$ at a distance $L_{c}$ enclosing
a pair of two quantum wells (QWs) called dresser and emitter. The electric
field of the TM0-modes cavity modes and the electronic polarization associated
to the intersubband (ISB) transition in the QWs oscillate along a direction
perpendicular to the mirrors plane. (b) Example of typical polariton spectra
emerging from the emitter-cavity-dresser Hamiltonian in Eq. (1). The color
represents the dominant component of each polaritonic branch: blue, red, green
colors respectively refer to the cavity field, dresser, emitter, and black
indicates the regions of maximum hybridization. The left and right panels
correspond to an emitter on resonance with the lower and the upper polariton.
Parameters: $\Omega_{\rm d}/\omega_{\rm d}=1$, $\Omega_{e}/\omega_{\rm
d}=0.1$, $\omega_{e}/\omega_{\rm d}=0.5-2$ (left and right panel).

_The model_ — More in detail, we consider a planar electromagnetic cavity of
surface $S$ and height $L_{c}$, hosting two polarizable QW slabs well
separated in space along the direction $z$ perpendicular to the cavity plane.
On the cavity side, we focus our attention on the so-called TM0-modes Jackson
(1999): these modes are polarized along the $z$-axis and are at the heart of
semiconductor-based cavity QED setups based on quantum well ISB transitions
Todorov and Sirtori (2012, 2014). The two QWs (dresser and emitter) are
coupled to the cavity mode by the same type of ISB dipole transitions Liu and
Capasso (2000). The different QW sizes and doping densities result in
different values of the resonant frequency and of the coupling strength to the
cavity mode.

Throughout this work, we place ourselves in the so-called electric dipole
picture, dubbed $d\cdot E$ in the literature Cohen-Tannoudji  _et al._ (1989);
Craig and Thirunamachandran (1998). In this picture, the total Hamiltonian can
be written in the following form Todorov and Sirtori (2012)

$\displaystyle H\approx$ $\displaystyle
H_{c-\text{d}}+\hbar\omega_{e}\sum_{\textbf{k}}b_{\textbf{k}}^{{\dagger}}b_{\textbf{k}}+$
(1)
$\displaystyle-i\frac{\hbar\Omega_{e}}{2}\sum_{\textbf{k}}\sqrt{\frac{\omega_{\textbf{k}}}{\omega_{e}}}\left(a_{\textbf{k}}-a_{-\textbf{k}}^{{\dagger}}\right)\left(b_{-\textbf{k}}+b_{\textbf{k}}^{{\dagger}}\right),$

where the emitter-cavity interaction is taken at lowest order in its coupling
strength and

$\displaystyle H_{c-\text{d}}$
$\displaystyle=\hbar\omega_{\text{d}}\sum_{\textbf{k}}d_{\textbf{k}}^{{\dagger}}d_{\textbf{k}}+\sum_{\textbf{k}}\hbar\omega_{\textbf{k}}a_{\textbf{k}}^{{\dagger}}a_{\textbf{k}}$
(2)
$\displaystyle\quad-i\frac{\hbar\Omega_{\text{d}}}{2}\sum_{\textbf{k}}\sqrt{\frac{\omega_{\textbf{k}}}{\omega_{\text{d}}}}\left(a_{\textbf{k}}-a_{-\textbf{k}}^{{\dagger}}\right)\left(d_{-\textbf{k}}+d_{\textbf{k}}^{{\dagger}}\right)$
$\displaystyle\quad+\frac{\hbar\Omega_{\text{d}}^{2}}{4\omega_{\text{d}}}\sum_{\textbf{k}}\left(d_{\textbf{k}}+d_{-\textbf{k}}^{{\dagger}}\right)\left(d_{\textbf{k}}+d_{-\textbf{k}}^{{\dagger}}\right),$

describes the (arbitrarily large) coupling of the dresser QW to the cavity.
Here, $a_{\textbf{k}}$ is the annihilation operator of the TM0-cavity photon
mode at in-plane wavevector k, with dispersion relation $\omega_{\textbf{k}}$.
The $b_{\textbf{k}}$ and $d_{\textbf{k}}$ operators are, instead, the
annihilation operators of collective ISB excitations of wavevector k in the
emitter or dresser QW, with a k-independent frequency $\omega_{d,e}$.
Restricting ourselves to a weak excitation regime, the $b_{\textbf{k}}$ and
$d_{\textbf{k}}$ operators can be safely approximated as bosonic Cominotti
_et al._ (2023).

The strength of the light-matter coupling of each QW is quantified by the
dresser and emitter Rabi frequencies $\Omega_{{\rm d},e}$ parameters, which
are determined by the corresponding 2D electron densities $n_{{\rm d},e}$ (and
thus by the doping density) via the relation $\Omega_{{\rm d},e}^{2}=f_{{\rm
d},e}e^{2}n_{{\rm d},e}/(\epsilon_{0}mL_{c})$ Todorov and Sirtori (2012). Here
$e$ is the electron charge and $m$ is the effective electron mass. $f_{{\rm
d},e}$ is the adimensional oscillator strength parameter determined by the
overlap of the electronic wavefunctions in the QW Todorov and Sirtori (2012)
and exactly equal to $1$ in the case of parabolic wells Ciuti  _et al._
(2005). Assuming all other parameters to be constant, the scaling of
$\Omega_{{\rm d},e}$ with the doping level $n_{{\rm d},e}$ allows to
experimentally control the strength of the light-matter coupling in the
dresser and the emitter. The ISB resonance frequencies $\omega_{d,e}$ are then
tuned by the geometry and the depth of the confinement potential in the QWs.

_Emitter-polariton interactions_ — As already mentioned, in this work we focus
on a regime where the emitter is much less doped than the dresser, $n_{e}\ll
n_{\rm d}$, so the coupling of emitter to light $\Omega_{e}$ is much smaller
than all other frequencies and can be taken at lowest order while the highly
doped dresser is in the USC regime,
$\Omega_{e}\ll\\{{\omega_{e},~{}\omega_{\rm
d},~{}\omega_{\textbf{k}}\\}}\simeq\Omega_{\rm d}$. In this regime, the
emitter no longer probes the bare cavity photon modes but rather the cavity-
dresser polariton modes resulting from the hybridization of the cavity photons
and the dresser ISB excitations due to $H_{c-\text{d}}$ in (2).

When the emitter is in the strong emitter-polariton coupling regime with a
Rabi frequency exceeding the polariton linewidths, $\Omega_{e}\gg\gamma_{\rm
lp,up}$, the full polariton spectra arising from the mixing of all three modes
shown in Fig.1(b) display a selective anticrossing of the emitter (green) mode
with the lower (left panel) or the upper (right panel) cavity-dresser
polariton depending on the value of the emitter frequency: in the Figure, for
each polariton branch, the color indicates the dominant cavity (blue), dresser
(red), or emitter (green) nature, and black represents a maximal mixing. As a
most remarkable feature, for the same value of $\Omega_{e}$, we notice that
the anti-crossing with the lower polariton is much wider than the one with the
upper polariton.

Figure 2: (a) Colorplot of the polariton-vacuum Rabi frequencies as a function
of the photon mode frequency $\omega_{\textbf{k}}$ and of the dresser Rabi
frequency $\Omega_{\rm d}$, for an emitter resonant with the lower
($\omega_{e}=\omega_{{\rm lp},\textbf{k}}$, left panel) or upper
($\omega_{e}=\omega_{{\rm up},\textbf{k}}$, right panel) polariton. (b)
Colorplot of the vacuum fluctuations of the displacement field in the TM0
mode.

In formal terms, we can rewrite the total Hamiltonian of Eq. (1) in the
polariton basis as

$\displaystyle H$
$\displaystyle\approx\hbar\omega_{e}\sum_{\textbf{k}}b_{\textbf{k}}^{{\dagger}}b_{\textbf{k}}+$
(3)
$\displaystyle\quad+\sum_{\textbf{k}}\hbar\omega_{\text{up},\textbf{k}}p_{\text{up},\textbf{k}}^{{\dagger}}p_{\text{up},\textbf{k}}+\sum_{\textbf{k}}\hbar\omega_{\text{lp},\textbf{k}}p_{\text{lp},\textbf{k}}^{{\dagger}}p_{\text{lp},\textbf{k}}+$
$\displaystyle\quad+i\sum_{\textbf{k}}\left(\frac{\hbar\Omega_{\text{up},\textbf{k}}}{2}p_{\text{up},\textbf{k}}b_{\textbf{k}}^{{\dagger}}+\frac{\hbar\Omega_{\text{lp},\textbf{k}}}{2}p_{\text{lp},\textbf{k}}b_{\textbf{k}}^{{\dagger}}\right)+\text{h.c.},$

where $p_{\,{\rm up},\textbf{k}},p_{\,{\rm lp},\textbf{k}}$ are the
annihilation operators of a cavity-dresser polariton in the upper or lower
polariton branches of (2) with eigenfrequencies

$\omega_{{\rm
up/lp},\textbf{k}}^{2}=\frac{\omega^{2}_{\textbf{k}}+\bar{\omega}_{\rm
d}^{2}}{2}\pm\sqrt{\frac{\left(\bar{\omega}_{\rm
d}^{2}-\omega^{2}_{\textbf{k}}\right)^{2}}{4}+\Omega_{\rm
d}^{2}\omega^{2}_{\textbf{k}}}.$ (4)

where $\bar{\omega}_{\rm d}=\sqrt{\omega^{2}_{\rm d}+\Omega^{2}_{\rm d}}$
includes the depolarization shift of the dresser frequency Załużny (1982);
Ando  _et al._ (1982); Cominotti  _et al._ (2023). The effective polariton-
vacuum Rabi frequencies quantifying the coupling strength between the emitter
and the lower/upper cavity-dresser polaritons read (Secs.III and IV of the SM)

$\frac{\Omega_{{\rm
up},\textbf{k}}}{\Omega_{e}}=\sqrt{\frac{\omega^{2}_{\textbf{k}}}{\omega_{e}\omega_{{\rm
up},\textbf{k}}}}\sin\theta_{\textbf{k}}\textrm{~{},~{}~{}}\frac{\Omega_{{\rm
lp},\textbf{k}}}{\Omega_{e}}=\sqrt{\frac{\omega^{2}_{\textbf{k}}}{\omega_{e}\omega_{{\rm
lp},\textbf{k}}}}\cos\theta_{\textbf{k}}\,,$ (5)

with

$\cos^{2}\theta_{\textbf{k}}=\frac{\omega_{{\rm
up},\textbf{k}}^{2}-\omega_{\textbf{k}}^{2}}{\omega_{{\rm
up},\textbf{k}}^{2}-\omega_{{\rm
lp},\textbf{k}}^{2}}\textrm{~{},~{}~{}}\sin^{2}\theta_{\textbf{k}}=\frac{\omega_{\textbf{k}}^{2}-\omega_{{\rm
lp},\textbf{k}}^{2}}{\omega_{{\rm up},\textbf{k}}^{2}-\omega_{{\rm
lp},\textbf{k}}^{2}}.$ (6)

Interestingly, all the information regarding the hybridization between the
cavity photon and the dresser excitations due to the USC is contained in the
hybridization angle $\theta_{\textbf{k}}$, which summarizes into a single
parameter the Hopfield coefficients expressing the polariton operators
$p_{\text{lp},\textbf{k}}$ and $p_{\text{up},\textbf{k}}$ in terms of the
cavity photon $a_{\textbf{k}}$ and dresser $d_{\textbf{k}}$ operators and
their hermitian conjugates Ciuti  _et al._ (2005); De Liberato (2017).

The reformulation in terms of the polariton Hamiltonian (3) provides a
physical understanding of the peculiar features displayed by the emitter-
polariton coupling that have have observed in Fig.1(b). In Fig. 2(a) we show a
color plot of the polariton-vacuum Rabi frequencies $\Omega_{{\rm
lp},\textbf{k}}$ and $\Omega_{{\rm up},\textbf{k}}$ as a function of the
wavenumber k and of the dresser Rabi frequency $\Omega_{\rm d}$, when the
emitter is resonant with some state on the lower $\omega_{e}=\omega_{{\rm
lp},\textbf{k}}$ (left panel) or the upper $\omega_{e}=\omega_{{\rm
up},\textbf{k}}$ (right panel) polariton branch. In the full polariton
spectrum of Fig.1(b), $\Omega_{{\rm lp/up},\textbf{k}}$ quantifies the
magnitude of the Rabi splitting.

For a weak dresser Rabi frequency $\Omega_{\rm d}\ll\omega_{\rm d}$, the
resonant lower and upper polariton-vacuum Rabi frequencies display similar
behavior, with $\Omega_{{\rm lp(up)},\textbf{k}}\simeq\Omega_{e}$ when the
polariton has a fully photonic nature and $\Omega_{{\rm
up(lp)},\textbf{k}}\simeq 0$ when the polariton has a fully excitonic nature.
In general, while the photonic weight is redistributed between the upper and
lower polariton, the total weight is conserved, $\Omega_{{\rm
lp},\textbf{k}}^{2}+\Omega_{{\rm up},\textbf{k}}^{2}\approx\Omega_{e}^{2}$, as
a consequence of the weakly dressed regime where $\omega_{{\rm
up/lp},\textbf{k}}\simeq\omega_{\textbf{k}}$.

The physics drastically changes when the dresser enters the USC regime for
$\Omega_{\rm d}\simeq\omega_{\rm d}$. Here $\omega_{{\rm
up/lp},\textbf{k}}\neq\omega_{\textbf{k}}$, and the square-root prefactors in
(5) start to matter: the coupling with the lower polariton $\Omega_{{\rm
lp},\textbf{k}}$ is reinforced and remains significant up to higher
wavenumbers, while the upper polariton’s coupling $\Omega_{{\rm
up},\textbf{k}}$ is significantly reduced. As an immediate consequence the
conservation of the photonic weight is strongly violated, $\Omega_{{\rm
lp},\textbf{k}}^{2}+\Omega_{{\rm up},\textbf{k}}^{2}\neq\Omega_{e}^{2}$. This
quite remarkable behavior is due to the mixing of creation and annihilation
operators in the Bogoliubov transformation to polariton operators Ciuti  _et
al._ (2005), so that the normal and anomalous terms constructively
(destructively) interfere in determining the coupling of the lower (upper)
polariton to the emitter.

This marked asymmetry of the Rabi splitting of the lower and upper polariton
branches is straightforwardly observed in a cavity transmission/reflection
spectroscopy experiment. Fig.3 displays examples of simulated transmission
spectra well in the strong emitter-polariton coupling regime $\Omega_{\rm
lp/up,\textbf{k}}\gg\gamma_{\rm lp/up}$ where all polariton branches are well
separated.

_Quantum fluctuations in the USC vacuum_ — In order to obtain a deeper
understanding of the relation between the modified emitter light-matter
coupling strength and the properties of the USC dressed vacuum, we evaluate
the fluctuations of the cavity electric displacement field
$D_{\textbf{k}}=i\sqrt{\epsilon_{0}\hbar\omega_{\textbf{k}}/(2SL_{c})}(a_{\textbf{k}}-a_{-\textbf{k}}^{{\dagger}})$.
Within our dipole representation, this field -rather than the electric field-
represents in fact the correct electromagnetic degree of freedom to describe
light-matter interactions Cohen-Tannoudji  _et al._ (1989); Craig and
Thirunamachandran (1998); Stengel  _et al._ (2009). By using the hybridization
angle $\theta_{\textbf{k}}$ defined above, we can express this quantity as
(Sec.V of the SM)

$\frac{\langle{\rm vac}|D^{2}_{\textbf{k}}|{\rm
vac}\rangle}{(\epsilon_{0}\mathcal{E}_{\textbf{k}})^{2}}=\frac{\omega_{\textbf{k}}}{\omega_{\rm
up,\textbf{k}}}\sin^{2}\theta_{\textbf{k}}+\frac{\omega_{\textbf{k}}}{\omega_{\rm
lp,\textbf{k}}}\cos^{2}\theta_{\textbf{k}}\\\
=\frac{\omega_{e}}{\omega_{\textbf{k}}}\left(\frac{\Omega_{\rm
lp,\textbf{k}}^{2}}{\Omega_{e}^{2}}+\frac{\Omega_{\rm
up,\textbf{k}}^{2}}{\Omega_{e}^{2}}\right),$ (7)

where
$\mathcal{E}_{\textbf{k}}^{2}=(\hbar\omega_{\textbf{k}})/(2\epsilon_{0}SL_{c})$
are the quantum fluctuations of the electric (or, in this case equivalently,
of the displacement) field in a bare cavity. The peculiarity of the USC is
then visible in Fig. 2(b), where we display a color plot of the total electric
displacement fluctuations in the different k modes as a function of the
strength of the cavity dresser coupling $\Omega_{d}$.

On the one hand, for weak or moderate $\Omega_{d}$, the prefactors
$\omega_{\textbf{k}}/\omega_{\\{\textrm{up,lp}\\},\textbf{k}}$ on the first
line in Eq. (7) are close to one and thus play a minor role; thanks to the
trigonometric identity
$\sin^{2}\theta_{\textbf{k}}+\cos^{2}\theta_{\textbf{k}}=1$ associated to the
conservation of the photonic weight, the two contributions then sum up to the
standard bare vacuum fluctuations. On the other hand, the total fluctuations
get substantially increased in the USC regime, in connection with the
increased value of the lower polariton-vacuum Rabi frequencies. More in
specific, the second line of (7) shows that the contributions of the lower and
upper polariton frequencies to the vacuum fluctuations have an amplitude
proportional to the polariton-emitter Rabi frequencies $\Omega_{\rm
lp/up,\textbf{k}}$, a result that extends to the USC vacuum case the
traditional concept of vacuum-field Rabi splitting Sanchez-Mondragon  _et al._
(1983); Agarwal (1984).

Figure 3: Colorplot of the transmission spectrum as a function of the incident
wavevector k and frequency $\omega$. The white dashed line marks the
wavevector location of the emitter minimal anticrossing, the green dashed line
marks the emitter frequency $\omega_{e}$. The two yellow circles mark the
locations where the polariton maximally hybridizes with the emitter. As in the
previous figures, the left/right panels correspond to an emitter on resonance
with the lower/upper polariton. Parameters: $\Omega_{e}/\omega_{\rm d}=0.2$,
$\Omega_{\rm d}/\omega_{\rm d}=1$, $\omega_{e}/\omega_{\rm d}=0.7$ (left
panel) and $1.6$ (right panel). The photonic, dresser and emitter decay rates
are $\gamma/\omega_{\rm d}=0.01$, $\kappa_{\rm d}=\kappa_{e}=0.05\,\omega_{\rm
d}$.

_Spontaneous emission_ — While the Rabi splitting in the strong emitter-
polariton coupling regime provides a most direct experimental signature of the
modified Rabi couplings (5), a related effect is visible also in the
spontaneous emission rate in the weak emitter-polariton coupling regime
$\gamma_{\rm lp/up}\gg\Omega_{\rm lp/up,\textbf{k}}$ where the Rabi
oscillations are replaced by an irreversible emission process. In our system,
the spontaneous emission is mediated by the resonant polariton mode and its
rate reads Purcell  _et al._ (1946) $\Gamma_{\rm
lp/up,\textbf{k}}\approx{\Omega_{\rm lp/up,\textbf{k}}^{2}}/{\gamma_{\rm
lp,up}}$: in analogy with the corresponding modification of the Rabi splitting
in the strong coupling regime, the effect of the USC vacuum state in the weak
emitter-polariton coupling regime will be to reinforce (suppress) the
spontaneous emission into the lower (upper) polariton by an amount related to
the modified quantum fluctuations of the electric displacement field (7). In
physical terms, this prediction can be understood as a novel form of the
Purcell effect Purcell  _et al._ (1946); Englund  _et al._ (2005), where one
acts on the modification of the overall intensity of the field fluctuations in
the USC vacuum rather than on just their frequency redistribution.
Interestingly, our result is in line with the theory of spontaneous emission
in dielectric media, and has the key advantage of a clear disentanglement of
local field effects Glauber and Lewenstein (1991); Knoester and Mukamel
(1989); Scheel  _et al._ (1999); Barnett  _et al._ (1992); Berman and Milonni
(2004); de Vries and Lagendijk (1998)

_Discussion_ — Our previous derivations were carried out within a quantum
language, so the modifications of the vacuum-field Rabi splitting and of the
spontaneous emission rate were naturally related to the ones of the zero-point
fluctuations (7). In connection to the intense debate that took place on the
physical origin of spontaneous emission in terms of quantum fluctuation and/or
radiative reaction effects Senitzky (1973); Milonni  _et al._ (1973); Dalibard
_et al._ (1982), it is interesting to assess whether our predictions can be
equivalently expressed in classical terms of radiative reaction by the dressed
electromagnetic field in the USC device.

As a first step in this direction, it is straightforward to notice that our
theory can be directly reformulated in terms of motion equations for the
cavity $a_{\textbf{k}}$ and the emitter $b_{\textbf{k}}$ and dresser
$d_{\textbf{k}}$ polarization fields stemming from the Hamiltonian (1) and re-
interpreted as classical variables. This set of equations can be summarized in
a dispersion relation that can equivalently derived directly from Maxwell
equations (Secs.VI-VIII of the SM) of the form
$\omega^{2}\,\epsilon_{r}(\omega)=c^{2}k^{2}$, with the effective dielectric
permittivity

$\epsilon_{r}(\omega)=\frac{1}{1-\frac{\Omega_{d}^{2}}{\bar{\omega}_{d}^{2}-\omega^{2}-i\kappa_{\rm
d}\omega}-\frac{\Omega_{e}^{2}}{\bar{\omega}_{e}^{2}-\omega^{2}-i\kappa_{e}\omega}}\,,$
(8)

that is an analog of the _Clausius-Mossotti_ formula in a multiple slab
geometry Hopfield (1958); Böttcher  _et al._ (1974) (here $\kappa_{\rm
d},\kappa_{e}$ represent the dresser and emitter radiative losses, as already
introduced in Fig. 3). Interestingly, the deviation of this expression from a
naive simple sum of the individual slabs’ permittivities,
$\epsilon_{r}(\omega)\approx 1+\Omega_{\rm d}^{2}/(\bar{\omega}_{\rm
d}^{2}-\omega^{2}-i\kappa_{\rm
d}\omega)+\Omega_{e}^{2}/(\bar{\omega}_{e}^{2}-\omega^{2}-i\kappa_{e}\omega)$
provides a signature of the key role played by the electrostatic interaction
between the emitter and the dresser noticed in De Bernardis  _et al._ (2018);
Amelio  _et al._ (2021); Sáez-Blázquez  _et al._ (2023); Pantazopoulos  _et
al._ (2023), whose strength is indeed proportional to the product of the
electron densities $n_{d}n_{e}\sim\Omega^{2}_{d}\Omega_{e}^{2}$. Once again,
$\bar{\omega}^{2}_{d,e}={\omega}^{2}_{d,e}+\Omega_{d,e}^{2}$ are the
depolarization-shifted values of the dresser and emitter frequencies Załużny
(1982); Ando  _et al._ (1982); Cominotti  _et al._ (2023).

This formulation in terms of classical equations of motion is the strong
emitter-polariton coupling generalization of the radiative reaction equations
Senitzky (1973); Dalibard  _et al._ (1982) in the case of a resonantly peaked
density of states of the electromagnetic modes. The analogy to the radiative
reaction equations is even closer in the weak emitter-polariton coupling
regime: close to resonance to the lower (upper) polariton mode, integrating
out the polariton field leads to a radiative damping term of the usual form
(Sec. VIII of the SM)
$\dot{b}_{\textbf{k}}=-i\omega_{e}b_{\textbf{k}}-\Omega_{\rm
lp(up),\textbf{k}}^{2}/(2\gamma_{\rm lp(up)})\,b_{\textbf{k}}\,$. Physically,
this close correspondence between the radiative reaction and the quantum
fluctuations is a direct manifestation of the fluctuation-dissipation theorem
relating the quantum fluctuations of the dressed polariton field in its vacuum
state and the susceptibility of vacuum to the polarization of the emitter
Barnett  _et al._ (1992); Dalibard  _et al._ (1982). All together, these
arguments show how the whole vacuum phenomenology described so far can be
equivalently re-interpreted in terms of electrodynamics in dense dielectric
media.

_Conclusions and outlook_ — In this work, we have theoretically investigated
how basic light-matter interaction processes are modified in the distorted
vacuum state of a semiconductor-based cavity QED system in the ultra-strong
coupling (USC) regime. In particular, we have focused our attention on
quantities of direct experimental access such as the vacuum-field Rabi
splitting of an additional emitter strongly coupled to USC polaritons, and its
spontaneous emission in the weak emitter-polariton coupling limit. In both
cases, a signature of the distorted vacuum state of the USC regime is visible
as a marked asymmetry between the polariton branches.

Even though our discussion is focused on a specific material system of major
experimental interest, the predicted effects generally apply to any optical
system in the USC regime. As such, our conclusions are of interest for a broad
community of researchers, from circuit-QED devices, to semiconductor
optoelectronics and terahertz optics and validate the picture that engineering
the QED vacuum is indeed a powerful tool to control optical processes and, on
the longer run, possibly also manipulate the properties of materials Garcia-
Vidal  _et al._ (2021); Bloch  _et al._ (2022); Andolina  _et al._ (2022).

From a conceptual standpoint, we have shown that our predictions can be
equivalently understood in terms of classical radiative reaction in dielectric
materials or in terms of quantum fluctuations in the distorted USC vacuum
state, the two pictures being connected by the fluctuation-dissipation
theorem. On one hand, the connection to classical radiative reaction provides
a new point of view on vacuum effects as a tool to control materials, a
concept that is attracting a growing interest, but is typically investigated
within a quantum language Ashida  _et al._ (2020, 2023); Arwas and Ciuti
(2023); Curtis  _et al._ (2023); Garcia-Vidal  _et al._ (2021); Bloch  _et
al._ (2022). On the other hand, the quantum picture provides a transparent
framework to go beyond a Fermi golden rule analysis of spontaneous emission in
dielectric materials and describe the interplay of ultra-strong coupling with
more sophisticated light-matter interaction phenomena such as parametric
downconversion, which directly involve the amplification of quantum
fluctuations.

###### Acknowledgements.

We are grateful to Raffaele Colombelli, Francesco Pisani, Yanko Todorov, Marco
Schirò, Simone De Liberato, Peter Rabl and Giuseppe La Rocca for useful
discussions. We acknowledge funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(Grant agreement No. 101002955 – CONQUER), from the Provincia Autonoma di
Trento, the Q@TN initiative, and the PNRR MUR project PE0000023-NQSTI.

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Supplementary material for:
Modification of light-matter interactions in the vacuum of ultrastrongly
coupled systems

Supplementary material for:
Modification of light-matter interactions in the vacuum of ultrastrongly
coupled systems

Daniele De Bernardis1,2, Gian Marcello Andolina2, and Iacopo Carusotto1

##  I Intersubband polaritons in the dipole representation

We review here the details of the derivation of the intersubband (ISB)
polariton Hamiltonian valid in the ultrastrong coupling (USC) regime, based on
Ref. Todorov and Sirtori (2012). Initially we focus only on a single quantum
well (QW) and in the next section we generalize it to a multi-well
configuration.

The polarization density can be written explicitly in terms of
raising/lowering operators of the respective intersubband transition Todorov
and Sirtori (2012)

$\textbf{P}(\textbf{r})=\frac{e}{S}\sum_{\mu>\nu,\textbf{k}}\zeta_{\mu\nu}(z)e^{i\textbf{k}\cdot\textbf{r}_{\parallel}}\left(B_{\mu\nu\textbf{k}}+B_{\mu\nu-\textbf{k}}^{{\dagger}}\right)\textbf{u}_{z},$
(S1)

Here $e$ is the electron charge, $S$ is the surface area of the slab (assumed
to be equal to the cavity) and $\textbf{u}_{z}$ is a unit vector along the
$z$-axis. The in-plane position is $\textbf{r}_{\parallel}=(x,y,0)$ and
$\textbf{k}=(k_{x},k_{y},0)$ is the in-plane wavector. The dipole density is
given by

$\zeta_{\mu\nu}(z)=z\psi_{\mu}^{*}(z)\psi_{\nu}(z).$ (S2)

Here $\psi_{\mu}(z)$ is the wavefunction of the $\mu$-th level of the QW
Todorov and Sirtori (2012); Todorov (2015, 2014), defined in the interval
$[z_{\rm qw}-L_{\rm qw}/2,z_{\rm qw}+L_{\rm qw}/2]$, where $L_{\rm qw}$ is the
size of the quantum well along the $z$-axis and $z_{\rm qw}$ is its central
position. The ISB transition operators are given by

$B_{\mu\nu\textbf{k}}=\sum_{\textbf{q}}\tilde{\Psi}^{{\dagger}}_{\nu,\textbf{q}-\textbf{k}}\tilde{\Psi}_{\mu,\textbf{q}},$
(S3)

where $\tilde{\Psi}_{\mu,\textbf{k}}$ is the Fermionic operator that
annihilates an electron with in-plane momentum k in the $\mu$-th subband. It
is convenient to rewrite this operator as

$B_{\mu\nu\textbf{k}}=\sqrt{N_{\mu\nu}}b_{\mu\nu\textbf{k}}.$ (S4)

where $N_{\mu\nu}=\braket{\hat{N}_{\nu}}-\braket{\hat{N}_{\mu}}$ is the
electron population imbalance between two subbands. Considering only low
energy transitions which take place just from the ground state (Fermi sea) to
an upper band and assuming $N_{\mu\nu}\gg 1$ (heavy doping), we have that

$[b_{\mu\nu\textbf{k}},b_{\mu^{\prime}\nu^{\prime}\textbf{k}\,^{\prime}}^{{\dagger}}]\simeq\delta_{\mu\mu^{\prime}}\delta_{\nu\nu^{\prime}}\delta_{\textbf{k}\,\textbf{k}\,^{\prime}}$
(S5)

and the transition operators behave as Bosonic creation/annihilation operators
for the collective excitation. Considering only transitions from the lowest
subband, in the low excitation limit, we have that $N_{\mu\nu=0}\approx N$,
where $N$ is the total number of electrons. From now on we always consider
only transitions from the lowest subband, suppressing the index $\nu$. The QW
Hamiltonian is then just given by a collection of harmonic oscillators

$H_{\rm qw}=\hbar\omega_{\rm
qw}\sum_{\textbf{k}}b_{\textbf{k}}^{{\dagger}}b_{\textbf{k}}.$ (S6)

The light-matter Hamiltonian for the QW coupled to the cavity field can be
derived by considering the total energy of the system, which is given by the
matter’s energy plus the electromagnetic energy Jackson (1999)

$\begin{split}H&=H_{\rm qw}+H_{\rm em}\\\ &=H_{\rm qw}+\int
d^{3}r\left[\frac{\textbf{E}^{2}(\textbf{r})}{2\epsilon_{0}}+\frac{\epsilon_{0}c^{2}}{2}\textbf{B}^{2}(\textbf{r})\right].\end{split}$
(S7)

The coupling between matter and the electromagnetic field is provided by the
fact that the charged matter generates and changes the electric and magnetic
field. In the so-called dipole gauge this is realized by the following minimal
coupling substitution in the cavity electric field energy density Jackson
(1999)

$\epsilon_{0}\textbf{E}(\textbf{r})=\textbf{D}(\textbf{r})-\textbf{P}(\textbf{r}).$
(S8)

Notice that here we take only the displacement field to be transverse Cohen-
Tannoudji  _et al._ (1989), such that
$\vec{\nabla}\cdot\textbf{D}(\textbf{r})=0$. In this way we correctly recover
Maxwell equations with a polarizable medium, where
$\vec{\nabla}\cdot\textbf{E}(\textbf{r})=-\vec{\nabla}\cdot\textbf{P}(\textbf{r})/\epsilon_{0}=\rho_{b}(\textbf{r})/\epsilon_{0}$,
where $\rho_{b}(\textbf{r})$ is the so-called bound charge density Jackson
(1999).

The total light-matter Hamiltonian is then given by Cohen-Tannoudji  _et al._
(1989)

$\begin{split}H=H_{\rm qw}+\int
d^{3}r\left[\frac{\left(\textbf{D}(\textbf{r})-\textbf{P}(\textbf{r})\right)^{2}}{2\epsilon_{0}}-\frac{\epsilon_{0}c^{2}}{2}\textbf{A}(\textbf{r})\cdot\nabla^{2}\textbf{A}(\textbf{r})\right].\end{split}$
(S9)

The electric displacement in the cavity made of two perfect parallel mirrors
can be written as Jackson (1999); Sáez-Blázquez  _et al._ (2023)

$\begin{split}\textbf{D}(\textbf{r})=i\sum_{n,\textbf{k},\sigma}\sqrt{\frac{\epsilon_{0}\hbar\omega_{n,\textbf{k}}}{2}}\Bigg{[}\textbf{w}_{n,\textbf{k}}^{\sigma}(\textbf{r}_{\parallel},z)a_{n,\textbf{k}}-\left(\textbf{w}_{n,\textbf{k}}^{\sigma}(\textbf{r}_{\parallel},z)\right)^{*}a_{n,-\textbf{k}}^{{\dagger}}\Bigg{]}.\end{split}$
(S10)

Here the cavity mode functions are solutions of the Poisson equation and can
be written as Sáez-Blázquez  _et al._ (2023)

$\textbf{w}_{n,\textbf{k}}^{\sigma}(\textbf{r}_{\parallel},z)=\frac{e^{i\textbf{k}\cdot\textbf{r}_{\parallel}}}{\sqrt{S}}\sqrt{\frac{2}{L_{c}\left(1+\delta_{n,0}\right)}}\begin{pmatrix}i\varepsilon^{(x)}_{n,\textbf{k},\sigma}\sin(k_{n}z)\\\
i\varepsilon^{(y)}_{n,\textbf{k},\sigma}\sin(k_{n}z)\\\
\varepsilon^{(z)}_{n,\textbf{k},\sigma}\cos(k_{n}z)\\\ \end{pmatrix},$ (S11)

where $S=L_{x}L_{y}$ is the surface area of the rectangular cavity $L_{c}$ is
the cavity height, and $k_{n}=\pi/L_{c}n$, with $n=0,1,2\ldots$ is the
wavevector along the cavity axis, where metallic boundary conditions are
assumed. The index $\sigma=1,2$ is the polarization index.

Considering only ISB transitions from the lowest subband, the interaction
light-matter Hamiltonian for a single QW placed at $z=z_{\rm qw}$ is

$\begin{split}H_{I}&=\int
d^{3}r\frac{1}{\epsilon_{0}}\left[-\textbf{D}\cdot\textbf{P}+\frac{P^{2}}{2}\right]\\\
&=-i\frac{\hbar\omega_{P}}{2}\sum_{n,\textbf{k},\sigma,\mu}\varepsilon^{(z)}_{n,\textbf{k},\sigma}\sqrt{f^{n}_{\mu
0}\frac{\omega_{n,\textbf{k}}}{\omega_{\mu
0}}}(a_{n,\textbf{k}}-a_{n,-\textbf{k}}^{{\dagger}})\left(b_{\mu-\textbf{k}}+b_{\mu\textbf{k}}^{{\dagger}}\right)\\\
&+\frac{\hbar\omega_{P}^{2}}{4}\sum_{\textbf{k}}\sum_{\mu,\mu^{\prime}}\frac{I_{\mu\mu^{\prime}}}{\sqrt{\omega_{\mu
0}\omega_{\mu^{\prime}0}}}\left(b_{\mu\textbf{k}}+b_{\mu-\textbf{k}}^{{\dagger}}\right)\left(b_{\mu^{\prime}\textbf{k}}+b_{\mu^{\prime}-\textbf{k}}^{{\dagger}}\right)\end{split}$
(S12)

Here the main coupling parameter is _the plasma frequency_

$\omega_{P}^{2}=\frac{Ne^{2}}{\epsilon_{0}mSL_{c}},$ (S13)

and $m$ is the electron effective mass. The generalized oscillator strength is
defined as

$\begin{split}f^{n}_{\mu 0}=\frac{2m\omega_{\mu
0}}{\hbar}\frac{2}{1+\delta_{n0}}\left[\int_{-\infty}^{\infty}dz\zeta_{\mu
0}(z)\cos\left(k_{n}z\right)\right]^{2}.\end{split}$ (S14)

For $n=0$ it coincides with the usual oscillator strength of the $\mu$-th
dipole transition $f_{\mu 0}=2m\omega_{\mu 0}z_{\mu 0}^{2}/\hbar$, where

$z_{\mu 0}=\braket{\mu}{z}{0}=\int_{-\infty}^{\infty}dz\zeta_{\mu 0}(z)$ (S15)

is the ISB dipole matrix element. At higher $n>0$, it also contains all the
multipoles matrix elements, as it has to be by interacting with a non-
homogeneous electric field. The higher the $n$-th mode is the more multipolar
will be the ISB excitation.

In the last term of Eq. (S12) we introduced the $P^{2}$-interaction tensor
strength

$\begin{split}I_{\mu\mu^{\prime}}&=\frac{4m^{2}L_{c}}{\hbar^{2}}\omega_{\mu
0}\omega_{\mu^{\prime}0}\int_{-\infty}^{\infty}dz\zeta_{\mu
0}(z)\zeta_{\mu^{\prime}0}(z)\\\ &=\frac{4m^{2}L_{c}}{\hbar^{2}}\omega_{\mu
0}\omega_{\mu^{\prime}0}\int_{-\infty}^{\infty}dzdz^{\prime}\delta(z-z^{\prime})\zeta_{\mu
0}(z)\zeta_{\mu^{\prime}0}(z^{\prime})\\\
&=\frac{4m^{2}L_{c}}{\hbar^{2}}\omega_{\mu
0}\omega_{\mu^{\prime}0}\sum_{n=0}\int_{-\infty}^{\infty}dz\zeta_{\mu
0}(z)\chi_{n}(z)\times\int_{-\infty}^{\infty}dz\zeta_{\mu^{\prime}0}(z)\chi_{n}(z),\end{split}$
(S16)

where in the last equality we introduced a complete basis such that
$\sum_{n=0}\chi_{n}(z)\chi_{n}(z^{\prime})=\delta(z-z^{\prime})$.

Truncating the ISB transitions to $\mu=1$ only, we suppress also the index
$\mu$ having $b_{\mu,\textbf{k}}\mapsto b_{\textbf{k}}$ and defining
$\omega_{\rm qw}=\omega_{10}$ and $f^{n}_{\rm qw}=f^{n}_{10}$. The interaction
Hamiltonian becomes

$\begin{split}H_{I}\approx-i\frac{\hbar\omega_{P}}{2}\sum_{n,\textbf{k},\sigma}\varepsilon^{(z)}_{n,\textbf{k},\sigma}\sqrt{f^{n}_{\rm
qw}\frac{\omega_{n,\textbf{k}}}{\omega_{\rm
qw}}}(a_{n,\textbf{k}}-a_{n,-\textbf{k}}^{{\dagger}})\left(b_{-\textbf{k}}+b_{\textbf{k}}^{{\dagger}}\right)+\frac{\hbar\omega_{P}^{2}}{4\omega_{\rm
qw}}\sum_{\textbf{k},n}f_{\rm
qw}^{n}\left(b_{\textbf{k}}+b_{-\textbf{k}}^{{\dagger}}\right)\left(b_{\textbf{k}}+b_{-\textbf{k}}^{{\dagger}}\right)\end{split}$
(S17)

where we have used $\chi_{n}(z)=\sqrt{2/(L_{c}(1+\delta_{n,0}))}\cos(k_{n}z)$
Todorov and Sirtori (2012).

The ISB transition couple mostly with the so-called TM0 mode Todorov (2015),
which corresponds to the $n=0$ mode, entirely polarized along the cavity axis
($z$-axis), with $\varepsilon^{(z)}_{0,\textbf{k},\sigma}=\delta_{z,\sigma}$.
The light-matter Hamiltonian can be further simplified to

$\begin{split}&H\approx\hbar\omega_{\rm
qw}\sum_{\textbf{k}}b_{\textbf{k}}^{{\dagger}}b_{\textbf{k}}+\sum_{\textbf{k}}\hbar\omega_{\textbf{k}}a_{\textbf{k}}^{{\dagger}}a_{\textbf{k}}-i\frac{\hbar\omega_{P}}{2}\sum_{\textbf{k}}\sqrt{f_{\rm
qw}\frac{\omega_{\textbf{k}}}{\omega_{\rm
qw}}}(a_{\textbf{k}}-a_{-\textbf{k}}^{{\dagger}})\left(b_{-\textbf{k}}+b_{\textbf{k}}^{{\dagger}}\right)+\frac{\hbar\omega_{P}^{2}}{4\omega_{\rm
qw}}\sum_{\textbf{k}}f_{\rm
qw}\left(b_{\textbf{k}}+b_{-\textbf{k}}^{{\dagger}}\right)\left(b_{\textbf{k}}+b_{-\textbf{k}}^{{\dagger}}\right),\end{split}$
(S18)

where we have suppressed the $n$ index everywhere. After all these passages we
are left with an effective light-matter Hamiltonian describing the interaction
with only the TM0 cavity mode Todorov and Sirtori (2012), which can be
rewritten as

$\begin{split}H_{{\rm TM}_{0}}=H_{\rm qw}+L_{c}\int
d^{2}r\left[\frac{\left(D(\textbf{r}_{\parallel})-P(\textbf{r}_{\parallel})\right)^{2}}{2\epsilon_{0}}-\frac{\epsilon_{0}c^{2}}{2}A(\textbf{r}_{\parallel})\nabla^{2}A(\textbf{r}_{\parallel})\right].\end{split}$
(S19)

Here the displacement and the vector potential fields are scalar variables,
meant to describe the $z$-component of the TM0 modes, which are independent of
the position along the $z$-axis (perpendicular to the cavity plane) and thus
depends only from the in-plane position $\textbf{r}_{\parallel}$

$D(\textbf{r}_{\parallel})=i\sum_{\textbf{k}}\sqrt{\frac{\epsilon_{0}\hbar\omega_{\textbf{k}}}{2SL_{c}}}e^{i\textbf{k}\cdot\textbf{r}_{\parallel}}(a_{\textbf{k}}-a_{-\textbf{k}}^{{\dagger}})$
(S20)
$A(\textbf{r}_{\parallel})=\sum_{\textbf{k}}\sqrt{\frac{\epsilon_{0}\hbar}{2SL_{c}\omega_{\textbf{k}}}}e^{i\textbf{k}\cdot\textbf{r}_{\parallel}}(a_{\textbf{k}}+a_{-\textbf{k}}^{{\dagger}}),$
(S21)

with
$[a_{\textbf{k}},a_{\textbf{k}\,^{\prime}}^{{\dagger}}]=\delta_{\textbf{k},\textbf{k}\,^{\prime}}$.
The effective polarization density coupled to the TM0 modes is also a scalar
quantity, and is given by

$P(\textbf{r}_{\parallel})=\frac{ez_{10}\sqrt{N}}{SL_{c}}\sum_{\textbf{k}}e^{i\textbf{k}\cdot\textbf{r}_{\parallel}}\left(b_{\textbf{k}}+b_{-\textbf{k}}^{{\dagger}}\right),$
(S22)

It is worth noticing that nor the electromagnetic field densities nor the
polarization densities of this TM0-effective-description depends from the
$z$-coordinate and so the overall physics does not depend from the position of
the QW inside the cavity or its distance with respect to one or the other
metallic plate. This is in principle in contradiction with basics
electromagnetism, where any charge configuration between metallic plates would
generates image charges which will affect its behaviour as a function of the
distance from the plate De Bernardis  _et al._ (2018). This contradiction
emerges as a consequence of the truncation to the TM0 mode only, for which we
have discarded all the information regarding the presence of image charges and
eventual Coulombic corrections. However we argue that this corrections are
most often negligible for our specific aims.

##  II Polariton Hamiltonian for two stacked wells

Since the polarization of the matter inside the cavity is given by the sum
over all its individual components, i.e. as a sum over all different quantum
wells

$\textbf{P}(\textbf{r})=\sum_{i=1}^{N_{\rm qw}}\textbf{P}_{i}(\textbf{r}),$
(S23)

and each polarization do not overlap with the others $\int
d^{3}r\textbf{P}_{i}(\textbf{r})\cdot\textbf{P}_{j}(\textbf{r})=0$ if $i\neq
j$, we can easily generalize the Hamiltonian in Eq. (S19) to a multi-wells
setup. Notice that this condition of non-overlapping polarization densities is
exactly the condition of localized dipoles used in the standard applications
of the dipole picture Cohen-Tannoudji  _et al._ (1989). As a consequence the
direct Coulomb interaction between different QWs disappears in favour of a
fully local mediated interaction through the dynamical cavity field.

In the simplest case of two QWs, the dresser and emitter, as in the main text,
the resulting Hamiltonian is

$\begin{split}H=H_{\rm d}+H_{e}+L_{c}\int
d^{2}r\left[\frac{D^{2}}{2\epsilon_{0}}-\frac{\epsilon_{0}c^{2}}{2}A\nabla^{2}A\right]+L_{c}\int
d^{2}r\frac{1}{\epsilon_{0}}\left[-DP_{\rm d}+\frac{P_{\rm
d}^{2}}{2}\right]+L_{c}\int
d^{2}r\frac{1}{\epsilon_{0}}\left[-DP_{e}+\frac{P_{e}^{2}}{2}\right].\end{split}$
(S24)

Since the two QWs have different electron number $N_{\rm d},N_{e}$, they also
have different plasma frequencies. These can be cast into the two Rabi
frequencies

$\begin{split}&\Omega_{\rm d}^{2}=f_{\rm d}\frac{N_{\rm
d}e^{2}}{\epsilon_{0}mSL_{c}},\\\
&\Omega_{e}^{2}=f_{e}\frac{N_{e}e^{2}}{\epsilon_{0}mSL_{c}}.\end{split}$ (S25)

Notice that here we are using all the definitions of Sec. I replacing all the
subscripts ${\rm qw}\mapsto{\rm d},e$.

Introducing the creation-annihilation operators like in Sec. I, the total
cavity QED Hamiltonian for the dresser and emitter QWs is thus given by

$\begin{split}H=&\omega_{\rm
d}\sum_{\textbf{k}}d_{\textbf{k}}^{{\dagger}}d_{\textbf{k}}+\omega_{e}\sum_{\textbf{k}}b_{\textbf{k}}^{{\dagger}}b_{\textbf{k}}+\sum_{\textbf{k}}\omega_{\textbf{k}}a^{{\dagger}}_{\textbf{k}}a_{\textbf{k}}\\\
&-\frac{i\hbar\Omega_{\rm
d}}{2}\sum_{\textbf{k}}\sqrt{\frac{\omega_{\textbf{k}}}{\omega_{\rm
d}}}\left(a_{\textbf{k}}-a_{-\textbf{k}}^{{\dagger}}\right)\left(d_{-\textbf{k}}+d_{\textbf{k}}^{{\dagger}}\right)-\frac{i\hbar\Omega_{e}}{2}\sum_{\textbf{k}}\sqrt{\frac{\omega_{\textbf{k}}}{\omega_{e}}}\left(a_{\textbf{k}}-a_{-\textbf{k}}^{{\dagger}}\right)\left(b_{-\textbf{k}}+b_{\textbf{k}}^{{\dagger}}\right)\\\
&+\frac{\hbar\omega_{{\rm d}}^{2}}{4\omega_{\rm
d}}\sum_{\textbf{k}}\left(d_{-\textbf{k}}+d_{\textbf{k}}^{{\dagger}}\right)\left(d_{-\textbf{k}}+d_{\textbf{k}}^{{\dagger}}\right)+\frac{\hbar\omega_{{e}}^{2}}{4\omega_{e}}\sum_{\textbf{k}}\left(b_{-\textbf{k}}+b_{\textbf{k}}^{{\dagger}}\right)\left(b_{-\textbf{k}}+b_{\textbf{k}}^{{\dagger}}\right).\end{split}$
(S26)

Assuming that the emitter polarization is always small, such that it never
goes in the ultrastrong coupling regime we can neglect the $\sim
P_{e}^{2}$-term of the emitter, which is the last term in Eq. (S26). In this
way we recover the cavity-dresser-emitter Hamiltonian in the main text.

##  III Polaritons from canonical transformations

Here we consider the cavity-dresser Hamiltonian only defined in Eq. (S18) In
this section we will suppress the vector notation for the in-plane wavevector
$\textbf{k}\mapsto k$ in order to simplify the notation.

We then reintroduce the real canonical coordinates defining the quadrature
operators

$\begin{split}D_{k}&=-i\sqrt{\frac{\omega_{k}}{2}}\left(a_{k}-a^{{\dagger}}_{-k}\right),\\\
A_{k}&=\frac{1}{\sqrt{2\omega_{k}}}\left(a_{k}+a^{{\dagger}}_{-k}\right),\\\
\Pi_{k}&=-i\sqrt{\frac{\omega_{\rm
d}}{2}}\left(d_{k}-d^{{\dagger}}_{-k}\right),\\\
X_{k}&=\frac{1}{\sqrt{2\omega_{\rm
d}}}\left(d_{k}+d^{{\dagger}}_{-k}\right).\\\ \end{split}$ (S27)

The cavity-dresser Hamiltonian becomes

$\begin{split}&H_{c-{\rm
d}}=\sum_{k}\left[\frac{\Pi_{k}^{2}}{2}+\frac{\omega_{\rm d}^{2}+\Omega_{\rm
d}^{2}}{2}X_{k}^{2}\right]+\\\
&+\sum_{k}\left[\frac{D_{k}^{2}}{2}+\frac{\omega_{k}^{2}}{2}A_{k}^{2}\right]+\sum_{k}\Omega_{\rm
d}D_{k}X_{k}.\end{split}$ (S28)

We make a canonical transformation (just a re-labelling)

$\begin{split}\tilde{D}_{k}&=-\omega_{k}A_{k},\\\
\tilde{A}_{k}&=\frac{1}{\omega_{k}}D_{k},\\\ \end{split}$ (S29)

We then have

$\begin{split}&H_{c-{\rm
d}}=\sum_{k}\left[\frac{\Pi_{k}^{2}}{2}+\frac{\omega_{\rm d}^{2}+\Omega_{\rm
d}^{2}}{2}X_{k}^{2}\right]+\\\
&+\sum_{k}\left[\frac{\tilde{D}_{k}^{2}}{2}+\frac{\omega_{k}^{2}}{2}\tilde{A}_{k}^{2}\right]+\sum_{k}\Omega_{\rm
d}\omega_{k}\tilde{A}_{k}X_{k}.\end{split}$ (S30)

The Hamiltonian in this form can be cast to a matrix, considering $H_{c-{\rm
d}}=1/2v^{T}Mv$, where $v^{T}=(X_{k},\tilde{A}_{k},\Pi_{k},\tilde{D}_{k})$,
and

$M=\begin{pmatrix}\omega_{\rm d}^{2}+\Omega_{\rm d}^{2}&\Omega_{\rm
d}\omega_{k}&0&0\\\ \Omega_{\rm d}\omega_{k}&\omega_{k}^{2}&0&0\\\ 0&0&1&0\\\
0&0&0&1\\\ \end{pmatrix}$ (S31)

Diagonalising $M$ is now equivalent to diagonalise the Hamiltonian. We can
achieve this by considering the following unitary transformation

$U=\begin{pmatrix}R&0_{2\times 2}\\\ 0_{2\times 2}&R\\\ \end{pmatrix}$ (S32)

where $R$ is a $2\times 2$ rotation

$R=\begin{pmatrix}\cos\theta&\sin\theta\\\ -\sin\theta&\cos\theta\\\
\end{pmatrix}$ (S33)

In order to further simplify all the expressions we make implicit the
dependence from the wavenumber $q$. Moreover we define

$\begin{split}&\Omega_{X}=\omega_{\rm d}^{2}+\Omega_{\rm d}^{2}\\\
&G=\Omega_{\rm d}\omega_{k}\\\ &\Omega_{A}=\omega_{k}^{2}\\\ \end{split}$
(S34)

so

$M=\begin{pmatrix}\Omega_{X}&G&0&0\\\ G&\Omega_{A}&0&0\\\ 0&0&1&0\\\
0&0&0&1\\\ \end{pmatrix}$ (S35)

We then find

$\begin{split}\cos\theta&=\sqrt{\frac{1}{2}\left(1+\frac{\Omega_{X}-\Omega_{A}}{\sqrt{\left(\Omega_{X}-\Omega_{A}\right)^{2}+4G^{2}}}\right)}=\sqrt{\frac{\omega_{\rm
up}^{2}-\omega_{k}^{2}}{\omega_{\rm up}^{2}-\omega_{\rm lp}^{2}}}\\\
\sin\theta&=\sqrt{\frac{1}{2}\left(1-\frac{\Omega_{X}-\Omega_{A}}{\sqrt{\left(\Omega_{X}-\Omega_{A}\right)^{2}+4G^{2}}}\right)}=\sqrt{\frac{\omega_{k}^{2}-\omega_{\rm
lp}^{2}}{\omega_{\rm up}^{2}-\omega_{\rm lp}^{2}}}\\\ \end{split}$ (S36)

with the diagonal matrix

$M^{\prime}=\begin{pmatrix}\omega_{\rm up}^{2}&0&0&0\\\ 0&\omega_{\rm
lp}^{2}&0&0\\\ 0&0&1&0\\\ 0&0&0&1\\\ \end{pmatrix}$ (S37)

where

$\omega_{\rm
up/lp}^{2}=\frac{\Omega_{X}+\Omega_{A}}{2}\pm\frac{1}{2}\sqrt{\left(\Omega_{X}-\Omega_{A}\right)^{2}+4G^{2}}$
(S38)

Working in the canonical representation has the convenience that, together
with the eigenfrequencies, we have full access on the hybridization of the
degrees of freedom due to the USC regime. Differently from previous works
Frisk Kockum  _et al._ (2019); Forn-Díaz  _et al._ (2019), we can condense the
full knowledge of the four Hopfield coefficients Hopfield (1958); Ciuti  _et
al._ (2005), in a single mixing angle $\theta_{k}$.

Figure S1: Canonical Hopfield coefficient
$\sin\theta_{\textbf{k}},~{}\cos\theta_{\textbf{k}}$ as a function of the in-
plane momentum and of the Rabi dressing $\Omega_{\rm d}$.

In Fig. S1 we show the behaviour of the canonical Hopfield coefficients as a
function of the in-plane wavenumber $k$ and the dresser Rabi frequency
$\Omega_{\rm d}$. When the dresser is only weakly doped, and $\Omega_{\rm
d}\ll\omega_{\rm d}$ we observe a very small hybridization ($\cos\sim 1$ and
$\sin\sim 0$ or vice versa), which become substantial only when the TM0 is
resonant with the dresser frequency at $ck=\omega_{\rm d}$ (and
$\cos\sim\sin\sim 1/2$). On contrary, when the dresser is in the USC regime
$\Omega_{\rm d}\simeq\omega_{\rm d}$, the light-matter hybridization becomes
important on a large range of wavenumbers.

The USC Hamiltonian in the new diagonal variables looks like two uncoupled
harmonic oscillators, representing the upper/lower polaritons

$H_{c-{\rm d}}=\frac{\Lambda_{\rm up}^{2}}{2}+\frac{\omega_{\rm
up}^{2}}{2}\xi_{\rm up}^{2}+\frac{\Lambda_{\rm lp}^{2}}{2}+\frac{\omega_{\rm
lp}^{2}}{2}\xi_{\rm lp}^{2}$ (S39)

where

$\begin{pmatrix}\xi_{\rm up}\\\ \xi_{\rm lp}\\\ \Lambda_{\rm up}\\\
\Lambda_{\rm lp}\\\ \end{pmatrix}=U\begin{pmatrix}X\\\ \tilde{A}\\\ \Pi\\\
\tilde{D}\\\ \end{pmatrix}$ (S40)

Evidently the eigenstates of Ham. (S39) are given by the polaritonic operators

$p_{\rm up/lp}=\frac{1}{\sqrt{2\omega_{\rm up/lp}}}\Lambda_{\rm
up/lp}-i\sqrt{\frac{\omega_{\rm up/lp}}{2}}\xi_{\rm up/lp}$ (S41)

##  IV The polaritonic photon

For a moment we focus only on the cavity-dresser Hamiltonian. Using the
canonical transformation diagonalization method explained in Appendix III we
can rewrite the cavity-dresser plasma Hamiltonian as

$\begin{split}&H_{c-{\rm d}}=\sum_{\textbf{k}}\left[\,\hbar\omega_{\rm
lp,\textbf{k}}\,p_{\,{\rm lp},\textbf{k}}^{{\dagger}}p_{\,{\rm
lp},\textbf{k}}\,+\,\hbar\omega_{\rm up,\textbf{k}}\,p_{\,{\rm
up},\textbf{k}}^{{\dagger}}p_{\,{\rm up},\textbf{k}}\right]~{}.\end{split}$
(S42)

Here $p_{\,{\rm lp},\textbf{k}}$ and $p_{\,{\rm up},\textbf{k}}$ represent the
annihilation operators of the lower and upper polaritons, and their
frequencies are given by (see Sec. III of the SM )

$\omega_{\rm up/lp,\textbf{k}}^{2}=\frac{\omega^{2}_{\textbf{k}}+\omega_{\rm
d}^{2}+\Omega_{\rm d}^{2}}{2}\pm\frac{1}{2}\sqrt{\left(\omega_{\rm
d}^{2}+\Omega_{\rm d}^{2}-\omega^{2}_{\textbf{k}}\right)^{2}+4\Omega_{\rm
d}^{2}\omega^{2}_{\textbf{k}}}.$ (S43)

The new polaritonic variables represent the correct degrees of freedom to
describe the cavity-dresser system, and consequently all the physical
quantities must be rewritten in this basis. Since that the coupling between
the cavity and the emitter is given by

$H_{{\rm int},e-c}=-\frac{1}{\epsilon_{0}}\int
d^{3}r\textbf{D}(\textbf{r})\cdot\textbf{P}_{e}(\textbf{r}),$ (S44)

for our aims we mainly need to transform the cavity electric displacement
field, which can be rewritten following Sec. III of the SM and using the
transformation in Eq. (S40)

$\begin{split}D(\textbf{r})=i\sqrt{\frac{\epsilon_{0}\hbar}{2SL_{c}}}\sum_{\textbf{k}}\omega_{\textbf{k}}\,e^{i\textbf{k}\cdot\textbf{r}_{\parallel}}\left(\frac{\sin\theta_{\textbf{k}}}{\sqrt{\omega_{\rm
up,\textbf{k}}}}p_{\,{\rm
up},\textbf{k}}+\frac{\cos\theta_{\textbf{k}}}{\sqrt{\omega_{\rm
lp,\textbf{k}}}}p_{\,{\rm lp},\textbf{k}}\right)+{\rm h.c.}.\end{split}$ (S45)

Using the expression for the emitter polarization given in the main text, and
assuming that the emitter is weakly doped, we can perform the rotating-wave
approximation (RWA) in the emitter-cavity interaction in Eq. (1). It is worth
noticing that the RWA cannot be implemented in Eq. (1) by only discarding the
terms
$\sim\left(a_{\textbf{k}}b_{\textbf{k}}\right),~{}\left(a^{{\dagger}}_{-\textbf{k}}b^{{\dagger}}_{-\textbf{k}}\right)$,
but it requires to switch on the polaritonic picture, and considering the
electric field displacement given by Eq. (S45). This is very similar to what
happens in the open driven/dissipative description of USC, where one has to
switch to the polaritonic picture in order to identify the positive/negative
frequencies operators that, coupling to the external bath, form the correct
jump operator of the system De Liberato (2014); Frisk Kockum  _et al._ (2019);
De Bernardis (2023).

For the sake of completeness is also worth to calculate the dresser
polarization in the polariton basis, that will be useful in the next sections.
It reads

$\begin{split}P_{\rm
d}(\textbf{r})=\sqrt{\frac{\epsilon_{0}\hbar}{2SL_{c}}\frac{\Omega_{\rm
d}^{2}}{\omega_{\rm
d}}}\sum_{\textbf{k}}\,e^{i\textbf{k}\cdot\textbf{r}_{\parallel}}\left(\sqrt{\frac{\omega_{\rm
up,\textbf{k}}}{\omega_{\rm d}}}\cos\theta_{\textbf{k}}p_{\,{\rm
up},\textbf{k}}+\sqrt{\frac{\omega_{\rm lp,\textbf{k}}}{\omega_{\rm
d}}}\sin\theta_{\textbf{k}}p_{\,{\rm lp},\textbf{k}}\right)+{\rm
h.c.}\end{split}$ (S46)

##  V Hybridization angle and vacuum observables

The interest in the hybridization angle $\theta_{k}$ is not only limited in
understanding the cavity-dresser components of the polariton excitations, but
it is also linked to the understanding on how the vacuum of quantum
electrodynamics is modified by the presence of matter. Indeed, by using again
the canonical formalism in Sec. III of the SM, we can compute the expectation
value of any observable over the USC polaritonic vacuum $|\rm vac\rangle$,
defined by

$p_{{\rm up/lp},\textbf{k}}|\rm vac\rangle=0.$ (S47)

Using Eqs. (S45)-(S46) we can calculate the vacuum fluctuations of the cavity
electric field considering that

$E_{\rm TM_{0}}(\textbf{r})=\frac{D(\textbf{r})-P_{\rm
d}(\textbf{r})}{\epsilon_{0}}.$ (S48)

We thus have that

$\begin{split}\langle{\rm vac}|E_{\rm TM_{0},\textbf{k}}^{2}|{\rm
vac}\rangle&=\frac{1}{\epsilon_{0}}\langle{\rm vac}|D^{2}_{\textbf{k}}|{\rm
vac}\rangle+\langle{\rm vac}|P_{\rm d,\textbf{k}}^{2}|{\rm vac}\rangle\\\
&=\mathcal{E}_{\textbf{k}}^{2}\Bigg{[}\frac{\omega_{\textbf{k}}}{\omega_{\rm
up,\textbf{k}}}\sin^{2}\theta_{\textbf{k}}+\frac{\omega_{\textbf{k}}}{\omega_{\rm
lp,\textbf{k}}}\cos^{2}\theta_{\textbf{k}}+\frac{\Omega_{\rm
d}^{2}}{\omega_{\rm d}^{2}}\left(\frac{\omega_{\rm
lp,\textbf{k}}}{\omega_{\textbf{k}}}\sin^{2}\theta_{\textbf{k}}+\frac{\omega_{\rm
up,\textbf{k}}}{\omega_{\textbf{k}}}\cos^{2}\theta_{\textbf{k}}\right)\Bigg{]},\end{split}$
(S49)

where
$\mathcal{E}_{\textbf{k}}^{2}=(\hbar\omega_{\textbf{k}})/(2\epsilon_{0}SL_{c})$.
We immediately notice from the last line of this formula that the electric
field fluctuations take a large contribution from the fluctuations of the
dresser polarization, which are given by the intrinsic fluctuations of matter.
Moreover, after a few algebraic steps, we have that

$\langle{\rm vac}|P_{\rm d,\textbf{k}}^{2}|{\rm vac}\rangle=\frac{\Omega_{\rm
d}^{2}}{\omega_{\rm d}\omega_{\textbf{k}}}\langle{\rm
vac}|D^{2}_{\textbf{k}}|{\rm vac}\rangle,$ (S50)

from which we arrive to

$\langle{\rm vac}|E_{\rm TM_{0},\textbf{k}}^{2}|{\rm
vac}\rangle=\frac{1}{\epsilon_{0}}\left[1+\frac{\Omega_{\rm
d}^{2}}{\omega_{\rm d}\omega_{\textbf{k}}}\right]\langle{\rm
vac}|D^{2}_{\textbf{k}}|{\rm vac}\rangle.$ (S51)

It is worth noticing that - being a gauge non-invariant quantity - the
physical significance of the electric displacement is sometimes considered
obscure and confusing Purcell (1965). However, we can see here that in the
dipole picture, the electric displacement is directly related to the TM0
electric field fluctuations and it thus realizes a good proxy to explore the
USC modifications of the electric field fluctuations.

Another interesting example that we report for completeness is the cavity
virtual photon population

$\begin{split}N_{\rm ph,\textbf{k}}=\braket{{\rm
vac}}{\,a_{\textbf{k}}^{{\dagger}}a_{\textbf{k}}\,}{{\rm
vac}}=\frac{\sin^{2}\theta_{\textbf{k}}}{4\,\omega_{\textbf{k}}}\frac{\omega_{\rm
up,\textbf{k}}^{2}+\omega_{\textbf{k}}^{2}}{\omega_{\rm
up,\textbf{k}}}+\frac{\cos^{2}\theta_{\textbf{k}}}{4\,\omega_{\textbf{k}}}\frac{\omega_{\rm
lp,\textbf{k}}^{2}+\omega_{\textbf{k}}^{2}}{\omega_{\rm
lp,\textbf{k}}}-\frac{1}{2},\end{split}$ (S52)

and the bare-dresser virtual excitation population

$\begin{split}N_{\rm d,\textbf{k}}=\braket{{\rm
vac}}{\,d_{\textbf{k}}^{{\dagger}}d_{\textbf{k}}\,}{{\rm
vac}}=\frac{\sin^{2}\theta_{\textbf{k}}}{4\,\omega_{\rm d}}\frac{\omega_{\rm
lp,\textbf{k}}^{2}+\omega_{\rm d}^{2}}{\omega_{\rm
lp,\textbf{k}}}+\frac{\cos^{2}\theta_{\textbf{k}}}{4\,\omega_{\rm
d}}\frac{\omega_{\rm up,\textbf{k}}^{2}+\omega_{\rm d}^{2}}{\omega_{\rm
up,\textbf{k}}}-\frac{1}{2}.\end{split}$ (S53)

For many years, these quantities were at the center of the discussions around
polaritonic vacuum observables Ciuti  _et al._ (2005); Liberato  _et al._
(2007); De Liberato (2017). However, their individual relevance is now
considered marginal, since their physical meaning explicitly depends from the
chosen representation Pilar  _et al._ (2020). They are important only when
correlated with physical gauge invariant quantities. An example of gauge-
invariant quantities is the differential zero point frequency of the system
$\Delta\omega_{\rm ZP}$ Ciuti  _et al._ (2005). This is obtained subtracting
the bare total zero point frequency for vanishing light-matter coupling from
the interacting one. Taking the vacuum expectation value of the cavity-dresser
Hamiltonian in Eq. (2) we have that

$\begin{split}\Delta\omega_{\rm ZP}=\frac{\omega_{\rm
up,\textbf{k}}+\omega_{\rm
lp,\textbf{k}}}{2}-\frac{\omega_{\textbf{k}}+\omega_{\rm
d}}{2}=\omega_{\textbf{k}}N_{\rm ph,\textbf{k}}+\omega_{\rm d}N_{\rm
d,\textbf{k}}+\Omega_{\rm d}N_{\rm int,\textbf{k}}.\end{split}$ (S54)

Here the last term represents the interaction energy, defined by

$\begin{split}N_{\rm
int,\textbf{k}}=\quad\frac{i}{2}\sqrt{\frac{\omega_{\textbf{k}}}{\omega_{\text{d}}}}\langle{\rm
vac}|\left(a_{\textbf{k}}^{{\dagger}}-a_{-\textbf{k}}^{\vphantom{{\dagger}}}\right)\left(d_{-\textbf{k}}^{\vphantom{{\dagger}}}+d_{\textbf{k}}^{{\dagger}}\right)|{\rm
vac}\rangle+\langle{\rm
vac}|\frac{\Omega_{\text{d}}}{4\omega_{\text{d}}}\left(d_{\textbf{k}}^{\vphantom{{\dagger}}}+d_{-\textbf{k}}^{{\dagger}}\right)\left(d_{\textbf{k}}^{\vphantom{{\dagger}}}+d_{-\textbf{k}}^{{\dagger}}\right)|{\rm
vac}\rangle.\end{split}$ (S55)

It is worth noticing that this term contains both the cavity-dresser
interaction and the dresser self interaction, which is notoriously known as
the $P^{2}$-term De Bernardis  _et al._ (2018); Todorov and Sirtori (2012);
Grießer  _et al._ (2016); Bamba and Ogawa (2014); Schäfer  _et al._ (2020),
responsible of the so-called polariton gap.

Interestingly, for the resonant wavevector $\textbf{k}_{\rm res}$, such that
$\omega_{\textbf{k}_{\rm res}}=\omega_{\rm d}$, the interaction energy exactly
vanishes

$N_{\rm int,\textbf{k}_{\rm res}}=0,$ (S56)

because the positive dresser self interaction ($P^{2}$-term) exactly
compensates the negative cavity-dresser contribution in Eq. (S55). As a
consequence the differential zero point frequency is completely determined by
the cavity and dresser virtual excitation, taking the simple expression

$N_{\rm ph,\textbf{k}_{\rm res}}=N_{\rm d,\textbf{k}_{\rm
res}}=\frac{1}{2}\left(\sqrt{1+\frac{\Omega_{\rm d}^{2}}{4\omega_{\rm
d}^{2}}}-1\right).$ (S57)

In this case the virtual photon number $N_{\rm ph,\textbf{k}_{\rm res}}$
represents the electromagnetic energy that can be released by an instantaneous
suppression of the cavity-dresser coupling Ciuti  _et al._ (2005).

##  VI Classical theory of transmission spectra

Here we derive the linear response theory following from our cavity-dresser-
emitter system. We start by considering the total Hamiltonian given in Eq.
(S26) and rewriting it using the quadrature canonical representation for the
cavity, the dresser and the emitter degrees of freedom as given in Sec. III of
the SM

$H=\frac{1}{2}v^{T}M_{\rm tot}v,$ (S58)

where $v^{T}=(X_{\rm d}(k),X_{e}(k),\tilde{A}_{k},\Pi_{\rm
d}(k),\Pi_{e}(k),\tilde{D}_{k})$ is the array containing the canonical
variables obtained generalising the definition in Eq. (S27) to the emitter,
while the Hamiltonian matrix is given by

$M_{\rm tot}=\begin{pmatrix}\omega_{\rm d}^{2}+\Omega_{\rm
d}^{2}&0&\Omega_{\rm d}\omega_{k}&0&0&0\\\
0&\omega_{e}^{2}+\Omega_{e}^{2}&\Omega_{e}\omega_{k}&0&0&0\\\ \Omega_{\rm
d}\omega_{k}&\Omega_{e}\omega_{k}&\omega_{k}^{2}&0&0&0\\\ 0&0&0&1&0&0\\\
0&0&0&0&1&0\\\ 0&0&0&0&0&1\\\ \end{pmatrix}$ (S59)

From here we can write down the equation of motion of the system (using the
Hamilton equations), which match the standard dielectric description from
classical electromagnetism. In the frequency domain, the equation of motion
read

$\mathcal{M}_{k}(\omega)\cdot\begin{pmatrix}A_{k}\\\ X_{\rm d}(k)\\\
X_{e}(k)\\\ \end{pmatrix}=0$ (S60)

where the dynamical matrix is given by

$\begin{split}&\mathcal{M}_{k}(\omega)=\\\
&\begin{pmatrix}\omega_{k}^{2}-(\omega+i\gamma/2)^{2}&i\omega\Omega_{\rm
d}&i\omega\Omega_{e}\\\ -i\omega\Omega_{\rm d}&\omega_{\rm
d}^{2}-(\omega+i\kappa_{\rm d}/2)^{2}&-\Omega_{\rm d}\Omega_{e}\\\
-i\omega\Omega_{e}&-\Omega_{\rm
d}\Omega_{e}&\omega_{e}^{2}-(\omega+i\kappa_{e}/2)^{2}\\\
\end{pmatrix}\end{split}$ (S61)

Notice that here we introduce the cavity, dresser and emitter losses
$\gamma,\kappa_{\rm d},\kappa_{e}$ in a phenomenological way, just inserting a
viscous damping in the equations. The cavity transmission reads Ciuti and
Carusotto (2006); Liberato  _et al._ (2007)

$T_{c}(k,\omega)=\gamma\omega_{k}\left[\mathcal{M}^{-1}_{k}\right]_{00}.$
(S62)

##  VII Details on the hybridization angle measurement protocol

Here we give a detailed description of the protocol to measure the cavity-
dresser hybridization angle from the emitter-cavity-dresser spectrum.

We call $\bar{\omega}_{e+},\bar{\omega}_{e-}$ the frequencies measured from
the cavity transmission at the minimal anticrossing between the emitter and
the cavity-dresser polaritons, while $\bar{k}_{\rm x}$ is the wavevector
realizing the minimal anticrossing (we will keep using the bar $\bar{\cdot}$
only for quantities which are directly measured from an eventual experiment,
and distinguish them from quantities derived from the theory). We can measure
$\bar{k}_{\rm x},\bar{\omega}_{e+},\bar{\omega}_{e-}$ directly from a
transmission (reflection) experiment by only detecting the two peaks around
the emitter frequency. Since $\bar{\omega}_{e\pm}\simeq\omega_{\rm
up/lp}\pm\Omega_{\rm up/lp}$, the upper/lower polariton frequency resonant
with the emitter is given by

$\bar{\omega}_{\rm up/lp}=\frac{\bar{\omega}_{e+}+\bar{\omega}_{e-}}{2}.$
(S63)

(Notice that the distinction between upper and lower polariton is also
experimentally well defined since the two polaritons are separated by a gap,
making them clearly distinguishable). The measured emitter-polariton Rabi
splitting is then given by

$\bar{\Omega}_{\rm up/lp}=\frac{\bar{\omega}_{e+}-\bar{\omega}_{e-}}{2}.$
(S64)

Figure S2: Reconstruction of the mixing angle tangent $\tan\theta_{k}$ using
Eq. (S65) (dots) and analytical prediction using Eq. (S36) from Sec. III of
the SM (solid lines). The emitter frequency is swept through the two
polaritons branches and every time we recorded the splitted frequencies at the
minimal anticrossing in each polariton branch. Parameters: (all)
$\Omega_{e}/\omega_{\rm d}=0.2$, $\gamma=0.01\omega_{\rm d}$, $\kappa_{\rm
d}=\kappa_{e}=0.05\omega_{\rm d}$.

Following Eq. (5), we have that the cavity-emitter anticrossing becomes a
probe of the hybridization angle $\theta_{k}$, through the relation

$\frac{\left[\bar{\omega}_{\rm up}\bar{\Omega}_{\rm
up}\right]_{k}}{\left[\bar{\omega}_{\rm lp}\bar{\Omega}_{\rm
lp}\right]_{k}}=\tan\theta_{k},$ (S65)

where $[\cdot]_{k}$ indicates that the data inside the square brackets are
measured from the minimal anticrossing happening at the wavevector
$\bar{k}_{\rm x}=k$. Repeating this protocol many times while sweeping
$\omega_{e}$ and collecting all the data for each emitter resonance, we can
reproduce the tangent of the mixing angle using Eq. (S65), effectively
realizing a full tomography of the USC cavity-dresser polaritons.

This USC tomographic approach is naturally limited by the quality factors of
the dresser, emitter and cavity $Q_{{\rm d}/e}=\omega_{{\rm d}/e}/\kappa_{{\rm
d}/e}$, $Q_{k}=\omega_{k}/\gamma$, for which if the coupling between the
emitter and the upper polariton at low wavevector is too small to be resolved,
$\bar{\Omega}_{\rm up/lp}\ll\gamma,\kappa_{{\rm d}/e}$, it is impossible to
correctly identify the peaks in the transmission.

In Fig. S2 we show an example of this reconstruction mechanism. Even if here
we show a specific example regarding ISB transition in a TM0 cavity with
linear dispersion, it is important to highlight that our formalism and the
resulting tomography protocol are independent of the specific cavity QED
implementation. Not only does our description apply to any different cavity
dispersions, by only inserting the specific $\omega_{\textbf{k}}$ in all the
equations, but it applies also to any device that couples to the cavity
electric field through a dipole transition. A more detailed discussion about
the generality of our analysis will be contained in Ref. De Bernardis and
Andolina (2023).

In the small wavevector region of Fig. S2 the detection is limited by the
vanishing coupling to the upper polariton, as described in Fig. 2(right
panel). In such circumstances our algorithm to reconstruct the Rabi splitting
gives artificially larger data, predicting a wrong hybridization angle. Also
in the other regions our data in Fig. S2 are affected by numerical noise from
the error committed in the peak detection due to the broadening of the
transmission peaks given by the linewidths $\gamma,\kappa_{\rm d},\kappa_{e}$.
Despite that we could have completely avoided this noise by increasing our
simulation’s numerical accuracy, we decided to keep it in order to simulate
how realistic data could be treated in an experiment and to show that our
protocol works even in the presence of noisy data.

##  VIII Dipolar cQED with slabs

In this appendix we re-derive the whole theory in the main text starting only
from Maxwell equations.

$\vec{\nabla}\cdot\textbf{E}=\frac{\rho}{\epsilon_{0}}$ (S66)
$\vec{\nabla}\times\textbf{B}=\frac{1}{c^{2}}\left(\frac{\textbf{J}}{\epsilon_{0}}+\frac{\partial}{\partial
t}\textbf{E}\right)$ (S67)

$\vec{\nabla}\cdot\textbf{B}=0$ (S68)
$\vec{\nabla}\times\textbf{E}=-\frac{\partial}{\partial t}\textbf{B}$ (S69)

Since we have only dipolar matter we have that

$\rho=-\vec{\nabla}\cdot\textbf{P}_{\rm tot},$ (S70)

where $\textbf{P}_{\rm tot}=\sum_{a}\textbf{P}_{a}$ is the total polarization
vector of all the matter in the system, and the current is given by

$\textbf{J}=\partial_{t}\textbf{P}_{\rm tot}$ (S71)

Taking the rotor of Eq. (S69) and combining with Eq. (S67) we have that

$-\nabla^{2}\textbf{E}+\vec{\nabla}(\vec{\nabla}\cdot\textbf{E})=-\frac{1}{c^{2}}\partial_{t}^{2}\textbf{E}-\frac{1}{c^{2}\epsilon_{0}}\partial_{t}^{2}\textbf{P}_{\rm
tot}.$ (S72)

To this point, we cannot solve this equation, since it is still coupled with
the Gauss law

$\vec{\nabla}\cdot\textbf{E}=-\frac{\vec{\nabla}\cdot\textbf{P}_{\rm
tot}}{\epsilon_{0}}.$ (S73)

We then make an arbitrary split in the electric field, defining a longitudinal
and transverse part

$\textbf{E}=\textbf{E}^{\parallel}+\textbf{E}^{\perp},$ (S74)

where

$\textbf{E}^{\parallel}=\vec{\nabla}(G\star\vec{\nabla}\cdot\textbf{P}_{\rm
tot})/\epsilon_{0}$ (S75)

and

$\textbf{E}^{\perp}=\partial_{t}\textbf{A}.$ (S76)

Here $G$ is the Green’s function of the Poisson equation
$-\nabla^{2}G(\textbf{r},\textbf{r}^{\prime})=\delta(\textbf{r}-\textbf{r}^{\prime})$,
and $\star$ denotes the convolution operator. Evidently
$\vec{\nabla}\times\textbf{E}^{\parallel}=0$ and the vector is longitudinal by
definition, even in the standard sense Cohen-Tannoudji  _et al._ (1989). We
then take A as a transverse vector, having the property
$\vec{\nabla}\cdot\textbf{A}=0$. We highlight that these definitions are
general and true in a cavity setup or in any other confined geometry.

Using these definitions for the electric field we arrive at rewriting the
Maxwell equations in only one equation

$-\nabla^{2}\textbf{E}^{\perp}+\frac{1}{c^{2}}\partial_{t}^{2}\textbf{E}^{\perp}=-\frac{1}{c^{2}\epsilon_{0}}\partial_{t}^{2}\left[\textbf{P}_{\rm
tot}+\vec{\nabla}(G\star\vec{\nabla}\cdot\textbf{P}_{\rm tot})\right].$ (S77)

The part in square brackets on the left-hand side is the transverse projected
polarization

$\textbf{P}_{\rm tot}^{\perp}=\textbf{P}_{\rm
tot}+\vec{\nabla}(G\star\vec{\nabla}\cdot\textbf{P}_{\rm tot}),$ (S78)

with the property

$\vec{\nabla}\cdot\textbf{P}_{\rm tot}^{\perp}=0.$ (S79)

While the longitudinal projection is

$\textbf{P}_{\rm
tot}^{\parallel}=-\vec{\nabla}(G\star\vec{\nabla}\cdot\textbf{P}_{\rm tot})$
(S80)

In order to understand the properties of the resulting electric field, we need
to specify the dynamics of our matter system. Following the standard
literature Jackson (1999); Hopfield (1958) we consider an equation of motion
for the polarization of each constituent of the system

$\partial_{t}^{2}\textbf{P}_{a}+L_{a}(\textbf{P}_{a})=\Omega_{a}^{2}\epsilon_{0}\textbf{E}(\textbf{r}_{a}).$
(S81)

Here $L_{a}(\cdot)$ is a linear differential operator defining the dynamics of
the polarization of each constituent, $\Omega_{a}$ is its Rabi frequency (or,
equivalently, plasma frequency).

Using all the definitions of the electric field introduced before we arrive to

$\partial_{t}^{2}\textbf{P}_{a}+L_{a}(\textbf{P}_{a})=-\Omega_{a}^{2}\sum_{b}\textbf{P}_{b}^{\parallel}(\textbf{r}_{a})+\Omega_{a}^{2}\epsilon_{0}\textbf{E}^{\perp}(\textbf{r}_{a})$
(S82)
$-\nabla^{2}\textbf{E}^{\perp}+\frac{1}{c^{2}}\partial_{t}^{2}\textbf{E}^{\perp}=-\frac{1}{c^{2}\varepsilon_{0}}\partial_{t}^{2}\sum_{a}\textbf{P}_{a}^{\perp}(\textbf{r}).$
(S83)

We now specialize in a cavity system, truncating the description to the only
TM0 modes Todorov and Sirtori (2012); Todorov (2015, 2014). One can directly
check that the transverse projector (or delta transverse) is given by

$\left[\delta^{\perp}_{ij}(\textbf{r},\textbf{r}\,^{\prime})\right]_{{\rm
TM}_{0}}=\frac{1}{L_{c}}\delta_{zj}\delta(\textbf{r}_{\parallel}-\textbf{r}_{\parallel}\,^{\prime}),$
(S84)

here $\delta_{zj}$ is the Kronecker delta that selects the polarization
direction only along the $z$-direction, which, in our convention, is the
direction perpendicular to the parallel cavity plates. As a consequence, the
longitudinal delta in the TM0 mode is given by

$\left[\delta^{\parallel}_{ij}(\textbf{r},\textbf{r}\,^{\prime})\right]_{{\rm
TM}_{0}}=\delta_{zj}\delta(\textbf{r}_{\parallel}-\textbf{r}_{\parallel}\,^{\prime})\delta(z-z^{\prime})-\frac{1}{L_{c}}\delta_{zj}\delta(\textbf{r}_{\parallel}-\textbf{r}_{\parallel}\,^{\prime}).$
(S85)

Notice that, when $z\neq z^{\prime}$

$\left[\delta^{\parallel}_{ij}(\textbf{r},\textbf{r}\,^{\prime})\right]_{{\rm
TM}_{0}}=-\left[\delta^{\perp}_{ij}(\textbf{r},\textbf{r}\,^{\prime})\right]_{{\rm
TM}_{0}}.$ (S86)

Here we further specialize in the case of multiple infinite slabs, localized
at different $z$-positions. Assuming harmonic dynamics,
$L_{a}(\textbf{P}_{a})=\omega_{a}^{2}\textbf{P}_{a}$, the equations in the TM0
subspace in $k$-space (in-plane) and frequency domain reduce to

$(\omega_{a}^{2}-\omega^{2})P_{a,k}=\Omega_{a}^{2}\sum_{b\neq
a}P_{b,k}+\Omega_{a}^{2}\epsilon_{0}E_{\rm TM_{0},k}$ (S87)
$(c^{2}k^{2}-\omega^{2})E_{\rm
TM_{0},k}=\frac{\omega^{2}}{\epsilon_{0}}\sum_{a}P_{a,k}.$ (S88)

For simplicity from here on we suppress the vectorial notation on $k$.

###  VIII.1 Effective dielectric permittivity

To solve the equations (S87)-(S88) we take

$P_{a,k}=\epsilon_{0}E_{\rm TM_{0},k}\sum_{b}\Pi_{ab}\Omega_{b}^{2}.$ (S89)

The dipole susceptibility is given by $\Pi_{ab}=\mathcal{M}_{ab}^{-1}$, where
the dynamical matrix is

$\mathcal{M}=\begin{pmatrix}\omega_{1}^{2}-\omega^{2}&&-\Omega_{1}^{2}&&-\Omega_{1}^{2}&&...\\\
-\Omega_{2}^{2}&&\omega_{2}^{2}-\omega^{2}&&-\Omega_{2}^{2}&&...\\\
-\Omega_{3}^{2}&&-\Omega_{3}^{2}&&\omega_{3}^{2}-\omega^{2}&&...\\\
...&&...&&...&&...\\\ \end{pmatrix}$ (S90)

From these definitions, we find the relative electric susceptibility of the
ISB multi-slabs setup as

$\epsilon_{r}(\omega)=1+\sum_{a,b}\Pi_{ab}\Omega_{b}^{2}.$ (S91)

The cavity transmission in Eq. (S62) is equivalently given by

$T_{c}(k,\omega)=\frac{\gamma\omega_{k}}{c^{2}k^{2}-\omega^{2}\epsilon_{r}(\omega)-i\gamma\omega+\gamma^{2}/4}.$
(S92)

For example, in the specific case of two slabs (emitter $e$ and dresser ${\rm
d}$) we have

$\mathcal{M}=\begin{pmatrix}\omega_{\rm d}^{2}-\omega^{2}&&-\Omega_{\rm
d}^{2}\\\ -\Omega_{e}^{2}&&\omega_{e}^{2}-\omega^{2}\\\ \end{pmatrix}$ (S93)

and the dipole susceptibility is given by

$\begin{split}\Pi=&\frac{1}{(\omega_{\rm
d}^{2}-\omega^{2})(\omega_{e}^{2}-\omega^{2})-\Omega_{\rm
d}^{2}\Omega_{e}^{2}}\times\\\
&\times\begin{pmatrix}\omega_{e}^{2}-\omega^{2}&&\Omega_{\rm d}^{2}\\\
\Omega_{e}^{2}&&\omega_{\rm d}^{2}-\omega^{2}\\\ \end{pmatrix}\end{split}$
(S94)

As a consequence, the relative permittivity is given by

$\epsilon_{r}(\omega)=1+\frac{\Omega_{\rm
d}^{2}(\omega_{e}^{2}-\omega^{2})+\Omega_{e}^{2}(\omega_{\rm
d}^{2}-\omega^{2})+2\Omega_{\rm d}^{2}\Omega_{e}^{2}}{(\omega_{\rm
d}^{2}-\omega^{2})(\omega_{e}^{2}-\omega^{2})-\Omega_{\rm
d}^{2}\Omega_{e}^{2}}$ (S95)

It is worth noticing that when we take the limit of small Rabi (plasma)
frequencies for the slabs, $\Omega_{{\rm d},e}\ll\omega_{{\rm d},e}$ we
recover the usual relative permittivity for a couple of independent emitters

$\epsilon_{r}(\omega)\approx 1+\frac{\Omega_{\rm d}^{2}}{\omega_{\rm
d}^{2}-\omega^{2}}+\frac{\Omega_{e}^{2}}{\omega_{e}^{2}-\omega^{2}}.$ (S96)

###  VIII.2 Spontaneous emission as classical electromagnetic damping

Here we draw a connection between the modified Purcell spontaneous emission
described in the maintext and the electromagnetic damping arsing in the
classical equations (S87)-(S88).

We start by considering the equations for the dresser and the TM0 electric
field mode

$\begin{split}&\left(\omega_{\rm d}^{2}-\omega^{2}\right)P_{\rm d}-\Omega_{\rm
d}^{2}\epsilon_{0}E_{\rm TM_{0}}=\Omega_{\rm d}^{2}P_{e},\\\
&\left(c^{2}k^{2}-\omega^{2}\right)E_{\rm
TM_{0}}-\frac{\omega^{2}}{\epsilon_{0}}P_{\rm
d}=\frac{\omega^{2}}{\epsilon_{0}}P_{e},\end{split}$ (S97)

coupled together to the emitter by

$\left(\omega_{e}^{2}-\omega^{2}\right)P_{e}=\Omega_{e}^{2}\left(\epsilon_{0}E_{\rm
TM_{0}}+P_{\rm d}\right).$ (S98)

In this last equation we immediately recognise $D=\epsilon_{0}E_{\rm
TM_{0}}+P_{\rm d}$, which is the only relevant degree of freedom that couples
to the emitter. In this section, to simplify the notation, we completely
suppress the index $k$ unless necessary.

By solving Eq. (S97) we find that

$\epsilon_{0}E_{\rm
TM_{0}}=\chi_{e|E}(\omega)P_{e}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}P_{\rm
d}=\chi_{e|{\rm d}}(\omega)P_{e}$ (S99)

where the two susceptibilities are defined by

$\begin{split}&\chi_{e|E}(\omega)=\frac{\omega^{2}\left(\bar{\omega}_{\rm
d}^{2}-\omega^{2}\right)}{\left(c^{2}k^{2}-\omega^{2}\right)\left(\omega_{\rm
d}^{2}-\omega^{2}\right)-\omega^{2}\Omega_{\rm
d}^{2}}=\omega^{2}\frac{\omega_{\rm up}^{2}+\omega_{\rm
lp}^{2}-c^{2}k^{2}-\omega^{2}}{\left(\omega_{\rm
up}^{2}-\omega^{2}\right)\left(\omega_{\rm lp}^{2}-\omega^{2}\right)},\\\
&\chi_{e|{\rm d}}(\omega)=\frac{\Omega_{\rm
d}^{2}c^{2}k^{2}}{\left(c^{2}k^{2}-\omega^{2}\right)\left(\omega_{\rm
d}^{2}-\omega^{2}\right)-\omega^{2}\Omega_{\rm d}^{2}}=\frac{\Omega_{\rm
d}^{2}c^{2}k^{2}}{\left(\omega_{\rm up}^{2}-\omega^{2}\right)\left(\omega_{\rm
lp}^{2}-\omega^{2}\right)},\end{split}$ (S100)

where $\bar{\omega}_{\rm d}^{2}=\omega_{\rm d}^{2}+\Omega_{\rm d}^{2}$. The
emitter dynamics can be rewritten exactly as

$\left(\omega_{e}^{2}-\omega^{2}\right)P_{e}=\Omega_{e}^{2}\left[\chi_{e|E}(\omega)+\chi_{e|{\rm
d}}(\omega)\right]P_{e}$ (S101)

Now we consider that the poles of $\chi_{e|E}(\omega),\,\chi_{e|{\rm
d}}(\omega)$ are lifted by the respective polariton linewidths $\gamma_{\rm
up,lp}$. This is implemented by replacing
$\omega^{2}\longmapsto\omega^{2}+i\gamma_{\rm up,lp}\omega$ in the denominator
of the two susceptibilities. In this way we can specialize on the _Purcell
regime_ where $\gamma_{\rm up,lp}\gg\Omega_{e}$. Joining this condition with
the assumption that the emitter is almost resonant with one of the polaritons,
for instance $\omega_{e}\simeq\omega_{\rm up}$, we can expand the emitter
dynamics and the susceptibilities around this pole, obtaining

$\begin{split}&\chi_{e|E}(\omega)\approx\frac{i}{\gamma_{\rm up}}\omega_{\rm
up}\frac{\omega_{\rm up,lp}^{2}-c^{2}k^{2}}{\omega_{\rm up}^{2}-\omega_{\rm
lp}^{2}}\\\ &\chi_{e|{\rm d}}(\omega)\approx\frac{i}{\gamma_{\rm
up}}\frac{\Omega_{\rm d}^{2}c^{2}k^{2}}{\omega_{\rm up}}\frac{1}{\omega_{\rm
up}^{2}-\omega_{\rm lp}^{2}},\end{split}$ (S102)

from which

$\left(\omega_{e}^{2}-\omega^{2}\right)P_{e}^{+}\approx
2\omega_{e}(\omega_{e}-\omega)P_{e}^{+}=\Omega_{e}^{2}\left[\chi_{e|E}(\omega)+\chi_{e|{\rm
d}}(\omega)\right]P_{e}^{+}\approx i\frac{\Omega_{e}^{2}}{\gamma_{\rm
up}}\frac{\omega_{k}^{2}}{\omega_{\rm up}}\frac{\omega_{\rm
lp}^{2}-\omega_{k}^{2}}{\omega_{\rm lp}^{2}-\omega_{\rm up}^{2}}P_{e}^{+},$
(S103)

where we adopt the notation $P_{e}^{+}$ to indicate that this equation
describes only the dynamics of $P_{e}$ around the pole
$\omega\simeq+\omega_{e}\simeq+\omega_{\rm up}$. Rearranging the terms and
taking the inverse Fourier transform, we finally finds

$i\partial_{t}P_{e}^{+}=\omega_{e}P_{e}^{+}-i\frac{\Omega_{\rm
up}^{2}}{2\gamma_{\rm up}}P_{e}^{+}$ (S104)

One can repeat the reasoning for all the poles $\omega\simeq-\omega_{\rm up}$,
$\omega\simeq\pm\omega_{\rm lp}$, obtaining the same type of result.

Reinterpreting $P_{e}^{+}$ as $b_{\textbf{k}}$ we obtain the equation shown in
the maintext.