# Microscopic theory of cavity-enhanced interactions of dipolaritons

E. R. Christensen  Center for Complex Quantum Systems, Department of Physics
and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark.
A. Camacho-Guardian  Departamento de Física Química, Instituto de Física,
Universidad Nacional Autónoma de México, Apartado Postal 20-364, Ciudad de
México C.P. 01000, Mexico  O. Cotlet  Institute for Quantum Electronics, ETH
Zurich, Zurich, Switzerland  A. Imamoglu  Institute for Quantum Electronics,
ETH Zurich, Zurich, Switzerland  M. Wouters  TQC, Universiteit Antwerpen,
Universiteitsplein 1, B-2610 Antwerp, Belgium  G. M. Bruun  Center for Complex
Quantum Systems, Department of Physics and Astronomy, Aarhus University, Ny
Munkegade 120, DK-8000 Aarhus C, Denmark.  Shenzhen Institute for Quantum
Science and Engineering and Department of Physics, Southern University of
Science and Technology, Shenzhen 518055, China.  I. Carusotto  INO-CNR BEC and
Dipartimento di Fisica Center, Università di Trento, Via Sommarive 14, I-38050
Povo, Trento, Italy

###### Abstract

We develop a theoretical model for the interaction strength between a pair of
exciton-polaritons with finite and parallel static electric-dipole moments. In
stark contrast to simple perturbative arguments predicting the interaction to
be reduced due to the hybridisation with cavity photons, we find that the
strong coupling to a cavity mode leads to a drastic enhancement of the
polariton interactions as compared to bare excitons. This enhancement is
caused by the small polariton mass due to its photonic component, which
increases the kinetic cost associated with minimising the dipole-dipole
repulsion. Microscopic numerical calculations for one-dimensional geometries
show that the interaction strength for the lower polariton branch is maximal
when the exciton and the cavity mode are close to resonance, and can easily
exceed state-of-the-art cavity decay rates for realistic confinement lengths,
thus entering the polariton blockade regime. Analytical extension of our
results to two-dimensional polariton systems highlight dipolar polaritons as a
most promising solid-state platform to realize strongly interacting photonic
systems.

##  I Introduction

The dramatic advances in experimental capabilities regarding hybrid light-
matter quantum systems have opened the door to engineering systems that posses
exotic phases of matter of fundamental interest and could serve as building
blocks for future quantum technology Chang  _et al._ (2014); Kurizki  _et al._
(2015). In this regard, microcavity exciton-polaritons (polaritons), namely
bosonic quasiparticles that emerge from the strong coupling of an optical
excitation of the material with a cavity photon mode in a semiconductor
microcavity, have proven especially appealing. Characterized by an effective
mass four orders of magnitude lower than the electron mass along with
interactions inherited from their matter component, polaritons showcase a
multitude of intrinsically interesting collective quantum phenomena such as
non-equilibrium Bose-Einstein condensation Bloch  _et al._ (2022); Fontaine
_et al._ (2022) and superfluidity Carusotto and Ciuti (2013); Amo  _et al._
(2009), and constitute promising candidates for realizing strongly correlated
fluids of light Verger  _et al._ (2006); Togan  _et al._ (2018); Delteil  _et
al._ (2019); Muñoz-Matutano  _et al._ (2019)

A crucial step towards the realization of such quantum correlated states of
matter is to achieve strong polariton-polariton interactions. This requires an
enhancement of the interaction constant, which in turn calls for a microscopic
understanding of the polariton-polariton scattering process. In the last
decade, several pioneering theoretical and experimental studies have appeared
on polariton interactions and enhanced polariton interactions compared to
exciton ones have been predicted Ciuti  _et al._ (1998); Rochat  _et al._
(2000); Glazov  _et al._ (2009); Wouters (2007); Carusotto  _et al._ (2010);
Takemura  _et al._ (2014); Estrecho  _et al._ (2019); Levinsen  _et al._
(2019); Hu  _et al._ (2020); Camacho-Guardian  _et al._ (2021); Bastarrachea-
Magnani  _et al._ (2021); Camacho-Guardian  _et al._ (2022); Bleu  _et al._
(2020); Li  _et al._ (2021). Yet, most of these studies have focused on
excitons with a short range interaction, which significantly simplifies the
description of the scattering process.

A tempting strategy to reinforce polariton interaction is to use excitons
possessing a static electric dipole. Such dipolar polaritons (dipolaritons)
have been explored experimentally in different settings in quantum wells and
optical wave guides Cristofolini  _et al._ (2012); Byrnes  _et al._ (2014);
Rosenberg  _et al._ (2016); Togan  _et al._ (2018); Rosenberg  _et al._
(2018). A thorough understanding of dipolar excitons calls for more
sophisticated theoretical description due to the strength and range of the
dipolar interaction.

Motivated by these exciting developments, in this article we develop a
microscopic description of two polaritons interacting via their parallel
dipole moments. As a measure of the strength of interactions, we numerically
calculate the interaction-induced energy shift of the lowest two-particle
state in a one-dimensional (1D) wire of finite length. In contrast to a naive
expectation that the hybridisation of the excitons with the photons would
decrease their interaction, a dramatic enhancement of the interaction strength
is found: the reason is that the mass of the polaritons is much smaller than
that of the excitons, which increases the kinetic energy cost associated with
minimising the dipole-dipole repulsion.

In specific, three qualitatively different regimes emerge from our analysis:
From a Tonks-Girardeau fermionization regime for long wires through an
intermediate regime where the energy shift scales with the inverse wire length
to a polariton blockade regime in short wires. The scaling laws exhibited by
the results of our exact diagonalization are quantitatively recovered by a
Born-Oppenheimer type approach, and allow to extend our theory to two
dimensional systems. For traditional semiconductor-based devices, dipolaritons
offer a moderate improvement of the interaction strength compared to standard
contact-interacting polaritons. A sizable reinforcement is however predicted
for tightly-bound excitons with a large oscillator strength, e.g. in 2D
materials, and in multiple-well geometries. This enhancement is of great
promise in view of realizing strongly correlated fluids of light in a solid-
state platform.

The outline of the paper is as follows. In Sec. II, we present the model, and
in Sec. III we give results for the ground state energy from exact
diagonalization and discuss the different regimes determined by the wire
length. We introduce the Born-Oppenheimer approach in Sec. IV, and Sec. V
discusses the role of light coupling. In Sec. VI we use the Born-Oppenheimer
approach to gain insight into different non-trivial scalings, and Sec. VII
contains experimental considerations as well as comparisons between
dipolaritons and non-dipolar polaritons. Finally, we provide concluding
remarks in Sec. VIII. In the Appendices, we provide additional details
regarding the Born-Oppenheimer model and on the peculiar dependence of the
interaction energy on the exciton-photon detuning. We also provide additional
numerical results for a slightly more sophisticated model of coupled direct-
and indirect excitons, and for a mathematically similar quasi-1D coupled wire
system.

##  II Model

As illustrated in Fig. 1, we start by considering two dipolar excitons
confined to a quantum wire and coupled to a cavity mode in an effectively 1D
geometry. The dipole moment of the excitons is perpendicular to the wire
giving rise to a strong repulsion, and we therefore expect that the short-
range details of the excitons do not matter, which allows us to describe them
as point-like particles Yu and Yao (2021).

The 1D dynamics along the quantum wire is described by the Hamiltonian

$\displaystyle\hat{H}=\sum_{k=1,2}\int
dx_{k}\sum_{i,j=\\{C,X\\}}\hat{\Psi}_{i}^{\dagger}(x_{k})h^{0}_{ij}(x_{k})\hat{\Psi}_{j}(x_{k})$
$\displaystyle+\frac{1}{2}\int
dx_{1}dx_{2}\big{[}\hat{\Psi}^{\dagger}_{X}(x_{1})\hat{\Psi}^{\dagger}_{X}(x_{2})$
$\displaystyle\times
V_{\text{dd}}(|x_{1}-x_{2}|)\hat{\Psi}_{X}(x_{1})\hat{\Psi}_{X}(x_{2})\big{]}.$
(1)

Here, $h^{0}_{ij}$ are the entries of the $h^{0}$ matrix

$\displaystyle
h^{0}(x_{k})=\begin{pmatrix}-\frac{\hbar^{2}}{2m_{C}}\frac{\partial^{2}}{\partial
x_{k}^{2}}+\Delta+\delta_{C}&\Omega\\\
\Omega&-\frac{\hbar^{2}}{2m_{X}}\frac{\partial^{2}}{\partial
x_{k}^{2}}\end{pmatrix},$ (2)

and we impose hard-wall boundary conditions at the edges $x_{1},x_{2}=0,L$, of
the wire. The exciton and cavity-photon masses are taken as $m_{X}=2m_{e}$,
i.e. twice the electron mass, and $m_{C}=10^{-4}\,m_{X}$, respectively. The
corresponding field operators are $\hat{\Psi}_{X/C}$. The Rabi frequency
$\Omega$ quantifies the strength of the exciton-photon coupling. In order to
set the energy of the lowest photon mode on resonance with the exciton one, a
detuning
$\Delta=\left({m_{X}}^{-1}-{m_{C}}^{-1}\right){\hbar^{2}\pi^{2}}/{(2L^{2})}$,
is added to the photon energy. Any further detuning is contained in the
parameter $\delta_{C}$.

The dipolar interaction potential has the usual form

$V_{\text{dd}}(x_{1},x_{2})=\frac{e^{2}d^{2}}{4\pi\epsilon|x_{1}-x_{2}|^{3}}\equiv\frac{C_{3}}{|x_{1}-x_{2}|^{3}}$
(3)

where $e$ is the electron charge, $\epsilon$ is the dielectric constant of the
material, and $d$ is the spatial separation between electrons and holes in the
exciton. In addition to coupled Cristofolini  _et al._ (2012) and wide quantum
wells Rosenberg  _et al._ (2016) as well as TMD bilayers Datta  _et al._
(2022), such dipolar excitons have recently been realized in
electrostatically-defined quasi-1D geometries Thureja  _et al._ (2022). If
further embedded in a laterally confined, thus effectively one-dimensional
microcavity, such a set-up could constitute a highly tunable platform to study
1D dipolaritons.

Figure 1: Sketch of the system under consideration. Two dipolar excitons in a
quantum wire (horizontal middle line) strongly coupled to a cavity photon mode
(yellow wave). The dipole moment of the excitons is perpendicular to the wire
leading to a strong repulsion.  Figure 2: Black solid line: Interaction energy
of a pair of dipolaritons as a function of the confinement length $L$. For
large $L$-values (right, red region), the interaction energy tends to the
Tonks-Girardeau (TG) limit of two impenetrable bosons of mass $2m_{C}$ (red
solid). For intermediate $L$-values (middle, green region), the energy shift
is well-described by the perturbative Born-Oppenheimer (BO) model (green
dashed). For small $L$-values (left, blue region) the energy shift approaches
the Jaynes-Cummings (JC) regime of a zero-dimensional quantum dot (blue dashed
horizontal). For intermediate to large confinement length (green and red
regions), the interaction shift for the polaritons is larger than the one for
bare dipolar excitons (lower orange full). A qualitatively similar behaviour
is observed for contact-interacting polaritons (lower red dashed) with energy
shift $\Delta E_{non-dip}\propto\Omega a_{\rm exc}/L$, with $a_{\rm exc}$ the
exciton Bohr radius set to $5\,$nm.

Figure 3: Colorplots of the different components of the two-particle ground
state wavefunction for different confinement lengths $L=10^{3}\ \mu m$ (top
red panel), $L=1\ \mu m$ (middle green panel), and $L=10^{-2}\ \mu m$ (bottom
blue panel). Horizontal and vertical axes show $x_{1}$ and $x_{2}$
respectively, as shown for upper left panel. Left, center, right panels
correspond to respectively the $CC$, $XC=CX$, $XX$ components.

Starting from the Hamiltonian (1), we numerically solve the two-body problem
to obtain the exact ground state energy and wave function of two interacting
dipolar polaritons. The ground state wave function is symmetric under particle
exchange and has four components

$\Psi(x_{1},x_{2})=\sum_{i,j\in\\{C,X\\}}\psi_{i,j}(x_{1},x_{2})$ (4)

giving the excitonic and photonic amplitudes of the two polaritons. Since we
are interested in varying the confinement length, $L$, across orders of
magnitude and the dipolar potential varies strongly for small separation
lengths, we diagonalise the Hamiltonian in Eq. (1) numerically by transforming
to relative- and center of mass coordinates and discretising the spatial axes
in a non-uniform grid Tan  _et al._ (1990). Because of the large size of the
Hamiltonian matrix, we only look for the lowest eigenvalue, $E_{LP-LP}$, with
a Lanczos-type algorithm Lanczos (1950); Stewart (2002). We define the
interaction energy as the difference between the interacting and non-
interacting ground states of the two-particle system, i.e.

$\Delta E=E_{LP-LP}-2E_{LP}.$ (5)

If not otherwise specified, in all figures we set $d=5\,$nm,
$\epsilon=10\epsilon_{0}$, $\Omega=1\,$meV, and $\delta_{C}=0$.

##  III Numerical results

In Fig. 2, the interaction energy $\Delta E$ is plotted as a function of the
wire length $L$ for the resonant case $\delta_{C}=0$. Three regimes can be
identified as a function of $L$.

###  III.1 Long wire regime

For a long wire, i.e. large $L$ (right red region in Fig. 2), the interaction
energy (black solid line) shows a $1/L^{2}$ scaling. This can be interpreted
as the Tonks-Girardeau (TG) limit of two impenetrable hard-core bosonic
polaritons Tonks (1936); Girardeau (1960): the dipolar interactions are strong
enough to dominate over the kinetic energy and the polaritons get effectively
fermionized. The physics is then fully determined by the impenetrability
condition and does not depend on the specific shape and range of the
interaction potential. Quantitatively, the ground state energy
$5\hbar^{2}\pi^{2}/[2(2m_{C})L^{2}]$ is equal to that of two non-interacting
_fermions_ with _polariton mass_ $m_{LP}=2m_{C}$, which occupy the lowest
($n=1$) and second lowest ($n=2)$ energy levels of the infinite square well of
width $L$ Lieb and Liniger (1963). The interaction energy is obtained by
subtracting twice the energy of the $n=1$ state, which gives the result

$\Delta E_{TG}=\frac{3\hbar^{2}\pi^{2}}{2(2m_{C})L^{2}}$ (6)

plotted as a red solid line in Fig. 2. To further corroborate this
interpretation, the top row of Fig. 3 shows the absolute square of the
normalized wavefunction components. This shows that in this large $L$ regime,
the entanglement of the photon and the exciton makes all four components have
comparable weights and all vanish along the $x_{1}\simeq x_{2}$ contact line,
including the purely photonic one $\psi_{CC}$.

###  III.2 Short wire regime

In the opposite limit of a short wire shown by the blue region in Fig. 2, we
observe a saturation of the interaction energy to a finite value. This can be
understood as a Jaynes-Cummings (JC) regime Jaynes and Cummings (1963); Tian
and Carmichael (1992), where the electronic system behaves as a zero
dimensional quantum dot and can only host a single matter excitation. On the
small $L$ panels on the bottom row of Fig. 3, this is visible as a strong
suppression of the two-exciton component $\psi_{XX}$ by the blockade effect
(right panel). Setting this component to zero, the Hamiltonian is readily
diagonalized giving the ground state energy $-\sqrt{2}\Omega$ for the two-
particle state. Since the energy of a single lower polariton is $-\Omega$, it
follows that the interaction energy is

$\Delta E_{JC}=(2-\sqrt{2})\Omega.$ (7)

Equation (7) is plotted as a horizontal blue dashed line in Fig. 2.

###  III.3 Intermediate wire length regime

The physics is richest for intermediate wire lengths $L$ shown by the central
green panel in Fig. 2. As one can hint on the figure, here the interaction
energy displays a $1/L$ scaling typical of weakly interacting bosons Lieb and
Liniger (1963), to be contrasted to the $1/L^{2}$ scaling of the impenetrable
boson case discussed above for the large $L$ regime.

The $1/L$ scaling can be qualitatively understood in terms of the effective
potential that the polaritons inherit from the dipole-dipole interaction
between their excitonic components. Following the JC argument above, we can
approximate this potential as a barrier of height $(2-\sqrt{2})\Omega$ and
range set by the distance where the dipolar interaction energy equals the Rabi
coupling, namely $R=(C_{3}/\Omega)^{1/3}$. At a perturbative Born-
approximation level, this simple argument results in an estimate for the
interaction energy

$\Delta E_{interm}\propto\frac{\Omega^{2/3}C_{3}^{1/3}}{L}$ (8)

which recovers the functional trend found in the numerics. A more rigorous and
quantitative derivation of this key result, and a detailed analysis of its
parameter dependence will be presented in the next section.

Using Eqs. (6)-(8), we can estimate the cross-over between the regime of long
and intermediate wire lengths to occur at
$L\sim(m\Omega^{2/3}C_{3}^{1/3})^{-1}$, whereas the cross-over between the
short and intermediate regimes is at $L\sim C_{3}^{1/3}/\Omega^{1/3}$.

##  IV Born-Oppenheimer energy landscapes

In order to put our interpretation of the intermediate regime on a solid
quantitative basis, we perform a Born-Oppenheimer (BO) like approximation
inspired by molecular physics and consider an effective potential landscape
determined by the energy of the ground state of two particles located at a
fixed distance $r=|x_{1}-x_{2}|$. This can be calculated by diagonalizing the
$4\times 4$ matrix that describes the two-particle problem in the
$CC,XC,CX,XX$ basis, where interactions are included as an energy shift of the
$XX$ state by the dipolar potential $V_{\text{dd}}(r)$ while the kinetic
energy is neglected, see App. A for details. This approach is expected to be
valid as long as the kinetic energy is smaller than the splitting of the BO
energy landscapes.

The energies of the three symmetric eigenstates under particle-exchange are
plotted in Fig. 4 as a function of $r$. For large $r$ values the eigenenergies
recover the ones of the three combinations of non-interacting lower (LP) and
upper (UP) polaritons. For smaller $r$, the dipolar potential (green dashed)
becomes important and the eigenenergies are shifted up in energy. On the one
hand, the highest energy state (red solid) tends to a pair of repulsively
interacting excitons and thus diverges as the dipolar potential. On the other
hand, the shift of the lowest energy state (black solid) saturates to a finite
value $(2-\sqrt{2})\Omega$ determined by the JC energy shift: for such small
distances, the dipolar potential is large enough to suppress the coupling to
the $XX$ state. As expected, the size of this saturation region compares
qualitatively well with the value of the dipolar radius $R$ introduced in the
previous Section.

A prediction for the interaction energy of a pair of dipolaritons can now be
extracted by evaluating the expectation value of the BO energy in the non-
interacting ground state wavefunction. Such a perturbative estimate is
expected to be accurate when the wire is short so that the energy gap to the
next confined state well exceeds the interaction energy. The result, plotted
as dashed green line in Fig. 2, captures in a quantitatively accurate way the
full numerical calculation (black solid) except in the large $L$ region. It is
important to note that this result stems from the application of perturbation
theory to the non-perturbative BO energy landscapes. In contrast to most
previous works on polariton-polariton interactions Ciuti  _et al._ (1998);
Rochat  _et al._ (2000); Glazov  _et al._ (2009); Estrecho  _et al._ (2019),
the result is therefore fully non-perturbative in the exciton interaction
potential. The slight deviation at relatively small $L$ values around the
blue/green interface stems from corrections to the BO model due to kinetic
energy effects.

For even smaller $L$ (blue region), the simple BO model perfectly recovers the
JC energy shift in the quantum dot limit. This interpretation is further
confirmed in the middle row of Fig. 3 for an intermediate $L=1\ \mu$m: as
expected from the BO energy landscape, the ground state closely resembles the
product of free-particle sine-shaped wavefunctions with a narrow depression
along the contact line in the $XX$-component only. Physically, this reflects
the fact that when the polaritons are too close, one of them is forced to
break into a pure cavity photon.

Figure 4: Born-Oppenheimer energy landscapes as a function of the separation
$r$ between the polaritons. The green dashed line indicates the dipolar
potential between bare excitons weighted by the Hopfield coefficient.

##  V Role of strong coupling with light

At the perturbative level, one would expect the light coupling to reduce the
interaction between the polaritons by the factor $|u_{X}|^{4}$ where $u_{X}$
is the excitonic Hopfield coefficient, as compared to that between bare
dipolar excitons. Our numerics find a completely different behaviour, with
much stronger interactions between polaritons than between bare excitons. The
reason of this exciting behaviour is that hybridisation with light makes
polariton mass much smaller that the exciton one. This in turn increases the
kinetic energy cost of localisation to minimise the interaction energy. This
key result is easily appreciated by comparing in Fig. 2 the interaction energy
of a pair of dipolaritons with that of two bare excitons interacting via the
same dipolar potential (yellow lower solid).

The enhancement of the interaction strength is most visible in the large $L$
limit governed by TG physics (red region), where the lighter mass of
dipolaritons provides a $m_{X}/m_{C}\sim 10^{4}$ enhancement factor of the
kinetic energy. A marked enhancement is also visible in the intermediate
regime (green middle region), again due to the kinetic energy of the photonic
component. Only for unrealistically short wires, the coupling to light leads
to a reduction of the effective strength of interaction by allowing excitons
to avoid overlapping by turning one of the polaritons into a photon.

Figure 5: Numerical calculation (solid blue) and power-law analytical fit (red
dashed) for the energy shift at fixed $L=1\,\mu$m as function of Rabi coupling
$\Omega$ (top left), of dipolar strength $C_{3}$ (top right) and as function
of detuning $\delta_{C}$ in linear (bottom left) and log-log (bottom right)
scales. Here, larger $L$-values are also shown and the solid red and green
dashed lines indicate the BO approximation and the analytical fit
respectively.

##  VI Dependence on parameters

Further insight into the physics of our model can be obtained by means of a
closer look at the scaling of the interaction energy with the system
parameters.

In Fig. 5 we plot the interaction energy as a function of $\Omega$ (top left)
and of the dipolar strength $C_{3}$ (top right) for $L=1\,\mu$m corresponding
to the most relevant intermediate wire length regime. The two panels confirm
the $\Omega^{2/3}$ and $C_{3}^{1/3}$ behaviors predicted by the simple
argument underlying Eq. (8) and highlight the crucial role of the Rabi
coupling $\Omega$ in the interaction process.

To assess how the exciton-photon detuning $\delta_{C}$ influences the
interaction energy, we plot in the bottom left panel of Fig. 5 $\Delta E$ as a
function of detuning for $L=1\ \mu$m and $\Omega=1$ meV. The most apparent
feature is that the interaction energy reaches its maximum in the vicinity of
the $\delta_{C}=0$ resonance condition. This result is to be contrasted with
the naive argument of standard perturbation theory that interactions scale
with the excitonic Hopfield coefficient as $\Delta E\sim|u_{X}|^{4}$ and are
maximum on the far excitonic side at large and positive $\delta_{C}$. Once
again, this confirms the key role played by the kinetic energy of the photonic
component in reinforcing the interaction energy.

For $\delta_{C}>0$, the interaction energy decays towards the much smaller
value for two bare excitons in the TG regime,
$3\hbar^{2}\pi^{2}/(2m_{X}L^{2})$. As discussed in App. B.1, the first
correction to this asymptotic value comes from the mixing of excitons with the
much lighter photons, which gives a contribution to the effective mass
$\propto 1/\delta_{C}^{2}$.

For large negative detunings $\delta_{C}<0$, the system is mostly photonic and
the energy shift again drops to zero. To pin down the $\delta_{C}$-dependence
on the negative side, we display in the bottom right panel of Fig. 5 a log-log
plot of the interaction energy as function of (negative) $\delta_{C}$ for
different values of $L$. For large and negative $\delta_{C}$, the dependence
displays a very non-trivial $|\delta_{C}|^{-10/3}$ law (green dashed lines)
which differs from all previous models. A justification of this unexpected
scaling is given in App. B.2 in terms of the BO model.

We emphasize how this $|\delta_{C}|^{-10/3}$ dependence is different from the
one that one would expect from a naive Born-approximation calculation for
which the interaction potential for polaritons is proportional to the
$|u_{X}|^{4}$ Hopfield coefficient and, therefore, to
$(\Omega/\delta_{C})^{4}$ Carusotto and Ciuti (2013). Within our theory, this
dependence is also peculiar to the dipolariton case: for polaritons emerging
from contact-interacting excitons, the radius of the potential peak is in fact
fixed by the exciton radius, so the interaction energy scales in a slightly
slower way as $|\delta_{C}|^{-3}$. Interestingly, this result recovers the
prediction of the standard model for the nonlinearity stemming from saturation
of the excitonic oscillator strength, proportional to the $u_{X}^{3}u_{C}$
Hopfield coefficients Carusotto and Ciuti (2013).

##  VII Experimental considerations

We conclude our work with a discussion of actual experimental observability of
our predictions and of their relevance for the on-going quest for realising
strongly correlated polariton fluids.

From Fig. 2, it appears that the TG regime is hardly observable in state-of-
the-art wire devices whose linewidth $\gamma$ is typically on the $\mu$eV
range at the very best Sun  _et al._ (2017). On the contrary, effective
polariton blockade $\Delta E>\gamma$ is expected to be within reach in
confined wire geometries of $\mu$m-range length with realistically narrow
linewidths, where the findings of Fig. 2 suggest that dipolaritons offer a
significant advantage compared to standard contact-interacting polaritons
(dashed red curve).

###  VII.1 Two-dimensional systems

To go beyond 1D systems and get insight on experimentally most relevant 2D
geometries, we take advantage of the analytical understanding extracted from
the 1D calculations and extend the BO theory to higher dimensions. Figure 2
shows how our full numerics are well recovered by the perturbation theory
based on the BO potential, while the calculations in App. D show how the BO
model is quantitatively accurate also for a coupled-wire geometry. All this
supports that the BO model is valid independently of the geometry and the
dimensionality.

For a generic $D$-dimensional model, the BO model gives a dipolariton
interaction energy that scales as

$\Delta E_{D}\propto\Omega R^{D}/L^{D}=\Omega^{1-D/3}C_{3}^{D/3}/L^{D}.$ (9)

As a most salient feature, this scaling implies that for any $D\leq 2$ the
interaction energy grows with $\Omega$, which implies that, for fixed dipolar
length $d$ and dipolar interaction strength $C_{3}$, more tightly-bound
excitons with a stronger oscillator strength are favourable in view of
achieving strong interactions. As opposed to the one-dimensional case, we note
that for $D=2$ there is no clear division between interaction- and kinetic
energy dominated regimes as function of confinement length since both shifts
scale as $1/L^{2}$. However, this crossover can be accessed by tuning the
parameters $\Omega$, $C_{3}$ and $\delta_{C}$.

The dipolariton interaction-induced scaling is to be contrasted with non-
dipolar polaritons where interactions are stronger for Wannier excitons with a
large Bohr radius $a_{\rm exc}$. In this case, $a_{\rm exc}$ (and not the
length $R$) sets the radius of the effective hard-core potential shown in Fig.
4. It follows that the two-polariton interaction energy scales as $\Delta
E_{non-dip}\propto\Omega a_{\rm exc}^{D}/L^{D}$. Since the Rabi coupling
scales as $\Omega\sim a_{\rm exc}^{-D/2}$, the resulting interaction energy
scales as $\Delta E_{non-dip}\propto a_{\rm exc}^{D/2}\propto 1/\Omega$ which
manifestly favors Wannier excitons with a large Bohr radius.

###  VII.2 Quantitative remarks

For typical parameter values of GaAs-based devices used in this paper with
$\Omega=1\,$meV, a dipolar length $d=5\,$nm and an exciton radius $a_{\rm
exc}=5\,$nm, we have $R\simeq$ $15\,$nm and the interaction energy of dipolar
polaritons is a factor of $\sim 3$ (in 1D) and $\sim 9$ (in 2D) larger than
the non-dipolar one and is on the edge of photon blockade already for
realistic pillars of $1\,\mu$m area.

Furthermore, in contrast to non-dipolar polaritons, the scaling in Eq. 9
predicts a wide margin for further improvement in the dipolar case by using,
e.g., dipolar excitons in two-dimensional transition metal dichalcogenide
(TMD) materials or Frenkel excitons in organic materials where excitons are
much more tightly bound. Mueller and Malic (2018); Knupfer (2003). Of course,
this conclusion holds provided a similar value of the dipolar length $d$ can
be obtained also for stronger bound excitons. This could be realised, e.g., in
a two-layer set-up with a layer separation larger the exciton Bohr radius and,
at the same time, a strong tunnel coupling of electrons or holes between two
layers. In this case, the hybridized intra-layer and inter-layer excitons
could ensure concurrently a large $\Omega$ (due to intra-layer component of
the hybrid exciton) while maintaining a large dipole moment (due to inter-
layer component) Cristofolini  _et al._ (2012); Agranovich  _et al._ (1998).
Evidence of excitons with a sizable oscillator strength and, at the same time,
a large spatial size has been reported in Thureja  _et al._ (2022).

Thanks to the spatially long-range shape of the dipolar interactions, a
further increase of $\Omega$ can be obtained using $N$ quantum wells, or TMD
monolayers separated by few layers of hBN. In standard non-dipolar polaritons,
an increase in $\Omega$ would be in fact compensated by the reduced
interactions between delocalized excitons in the $N$ wells/layers. For
dipolaritons with typical dipolar interaction range $R$ in the $10$ nm range,
several electronically disconnected layers could be placed such that the
vertical distance between the first and last layer is smaller than $R$. Here,
inter-layer and intra-layer dipolar exciton interactions would be comparable,
and dipolar interactions would be able to block excitons in every pair of
layers. This results in a factor of $\sqrt{N}$ increase of $\Omega$ which from
Eq. 9 gives an overall $N^{(1-D/3)/2}$ increase of the interaction energy. All
together, these improvements should be sufficient to bring dipolariton systems
Togan  _et al._ (2018) into the regime of strong interactions.

##  VIII Conclusions and outlook

In this article we have developed a microscopic model of the interactions
between dipolar polaritons in a 1D geometry. Depending on the wire length,
different regimes are identified, from a single-mode quantum dot-like
behaviour in short wires, to impenetrable polaritons in long ones. The
physically most relevant regime lies in between and can be quantitatively
described in terms of perturbation theory using an effective Born-Oppenheimer
potential. In particular, our work highlights the crucial role of the light-
matter coupling in increasing interactions between polaritons over bare
exciton ones for physically relevant wire lengths. Further theoretical work
will address the extension of our theory to Rydberg polaritons propagating in
atomic gases and semiconductor microcavities which have been recently
demonstrated Peyronel  _et al._ (2012); Gorshkov  _et al._ (2011); Firstenberg
_et al._ (2013); Walther  _et al._ (2018); Orfanakis  _et al._ (2022) as a
promising alternative platform for strongly interacting polaritons, albeit in
a conceptually different propagating geometry for atomic gases Larré and
Carusotto (2015).

In addition to its theoretical interest as the first microscopic model
polariton interactions, our results have the strong experimental interest of
suggesting dipolaritons emerging from tightly bound excitons in TMDs and/or in
multiple quantum well geometries as a promising candidate for achieving strong
interactions. Such a result would open a new era in the study of strongly
correlated fluids of light using state-of-the-art semiconductor optics
technology.

## Acknowledgments

We acknowledge financial support from the Danish National Research Foundation
through the Center of Excellence “CCQ” (Grant agreement no.: DNRF156) and the
U.S. Army CCDC Atlantic Basic and Applied Re- search via Grant No.
W911NF-19-1-0403. The work at ETH Zurich was supported by the Swiss National
Science Foundation (SNSF) under Grant Number 200020_207520. I. Carusotto
acknowledges financial support from the European Union H2020-FETFLAG-2018-2020
project “PhoQuS” (n.820392), from the Provincia Autonoma di Trento, and from
the Q@TN initiative. We also thank Deepnakur Thureja, Puneet Murty, Daniele De
Bernardis and Thomas Pohl for many discussions.

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##  Appendix A Details on the Born-Oppenheimer approximation

Here we give additional technical details on the Born-Oppenheimer
approximation discussed in the main text. To this purpose, we consider the
4-by-4 matrix:

$\displaystyle
h_{BO}=\begin{pmatrix}2(\epsilon_{C,1}+\Delta+\delta_{C})&\Omega&\Omega&0\\\
\Omega&\epsilon_{C,1}+\Delta+\delta_{C}+\epsilon_{X,1}&0&\Omega\\\
\Omega&0&\epsilon_{C,1}+\Delta+\delta_{C}+\epsilon_{X,1}&\Omega\\\
0&\Omega&\Omega&2\epsilon_{X,1}+V_{dd}(r),\end{pmatrix}$ (10)

where $r$ is the separation between the two particles, and
$\Delta=\epsilon_{X,1}-\epsilon_{C,1}$, meaning that $\delta_{C}=0$ is the
resonance.

The Born-Oppenheimer energy landscapes $V_{BO,i}$ shown in Fig. 4 of the main
text are obtained by calculating the eigenvalues of $h_{BO}$ as a function of
the distance $r$. As we are dealing with bosonic particles, we can restrict to
the symmetric subspace under exchange of the two particles, which reduces the
basis to the states $CC$, $XX$ and $XC+CX$ states. In particular, the lowest
eigenvalue describes the effective interaction potential $V_{BO}$ between a
pair of lower polaritons. To calculate the energy shift of the ground state in
the confinement potential, perturbation theory prescribes that we calculate

$\Delta
E=\int_{0}^{L}\int_{0}^{L}dx_{1}dx_{2}V_{BO}(x_{1}-x_{2})|\psi_{LP}(x_{1})|^{2}|\psi_{LP}(x_{2})|^{2},$
(11)

where $|\psi_{LP}(x_{1})|^{2}$ is the absolute square of the non-interacting
lower-polariton wavefunction in the box.

Most remarkably, even though this calculation is perturbative in the effective
Born-Oppenheimer interaction potential $V_{BO}$, it is not perturbative in the
underlying exciton-exciton interaction potential.

##  Appendix B Dependence of the interaction energy on the exciton-photon
detuning

###  B.1 Positive detuning

Figure 6: Energy shift as function of positive detuning $\delta_{C}$ for
different confinement lengths.

In Fig. 6, we show the numerically calculated dependence on positive detuning
of the system of cavity photon and dipolar exciton in a strictly 1D geometry
as discussed in the main text. For this configuration, we have seen in Fig. 5
of the main manuscript that for large positive detunings $\delta_{C}$, the
energy tends to the energy of two interacting bare excitons. This behaviour is
shown in greater detail in Fig. 6 for different values of $L$.

It is seen that for a large $L$ in the TG-regime, the interaction energy
indeed decreases following the dependence of the effective lower polariton
mass, $m_{LP}^{-1}\simeq
m_{X}^{-1}+m_{C}^{-1}\frac{\Omega^{2}}{\delta_{C}^{2}}$, with a
$1/\delta_{C}^{2}$ slope followed by an asymptotically constant value. For the
smallest $L=1\,\mu$m, the picture is more complicated but in all cases the
energy shift saturates to an energy given by the sum of the TG energy for bare
excitons and the corresponding average value of the dipole-dipole potential.
All these scaling considerations directly extend to the case of polaritons
emerging from contact-interacting excitons.

###  B.2 Negative detuning

As mentioned in the main text, the dependence for negative detuning as
$|\delta_{c}|^{10/3}$ shown in the bottom right panel of Fig. 5 is even more
intriguing. To understand it within the BO picture, we first need to estimate
the height of the potential peak around $r=0$ on the green curve in Fig. 4. To
this purpose, we need to compare the asymptotic energy of two LPs at infinity
with the $r=0$ value. These are obtained as the lowest eigenvalue of the
3-by-3 matrix describing the projection of the BO Hamiltonian on the symmetric
subspace,

$\begin{pmatrix}2\delta_{C}&\sqrt{2}\Omega&0\\\
\sqrt{2}\Omega&\delta_{C}&\sqrt{2}\Omega\\\ 0&\sqrt{2}\Omega&V_{\rm
dip}(r)\end{pmatrix}$ (12)

At $r=0$, the dipolar potential is infinite, so we can restrict to the first
and second rows and columns, obtaining, in the large and negative $\delta_{C}$
limit

$E_{r=0}=2\delta_{C}+\frac{2\Omega^{2}}{\delta_{C}}-\frac{4\Omega^{4}}{\delta_{C}^{3}}$
(13)

At $r=\infty$, particles are non-interacting, so the asymptotic energy is
equal to twice the single LP energy, $E_{r=\infty}=2E_{LP}$ with

$E_{LP}=\delta_{C}+\frac{\Omega^{2}}{\delta_{C}}-\frac{\Omega^{4}}{\delta_{C}^{3}}$
(14)

Taking the difference of the two quantities gives a dependence as

$E(r=0)-E(r=\infty)\simeq\frac{2\Omega^{4}}{|\delta_{C}^{3}|}\,.$ (15)

An estimate of the expectation value of the interaction energy on the ground
state is obtained by multiplying the peak potential times the effective radius
$R_{\delta_{C}}$. Since the energy scale on the diagonal of the matrix (12) is
$\delta_{C}$, the characteristic radius can be estimated as
$R_{\delta_{C}}\sim(C_{3}/\delta_{C})^{1/3}$. Multiplied to the dependence in
(15), this provides a physical explanation to the overall
$|\delta_{C}|^{-10/3}$ observed in the bottom right panel of Fig. 5.

Figure 7: Numerical calculation for the interaction energy for a system with
both direct- and indirect excitons and different values of the inter-exciton
coupling strength $J$.

##  Appendix C Model with direct and indirect excitons

The validity of simplest model considered in the main text is confirmed by a
calculation for a physically realistic model that is inspired from recent
literature on coupled quantum wells Cristofolini  _et al._ (2012) and
explicitly involves two different types of excitons, direct- and indirect. In
this model, the two excitonic species are coupled by a tunnel-like coupling
term and only the indirect exciton has a permanent dipole moment which gives
rise to the dipole-dipole repulsion in the two-particle problem.

The Hamiltonian has the form

$\displaystyle\hat{H}$ $\displaystyle=\sum_{k=1,2}\int
d{x_{k}}\sum_{i,j=\\{C,X,IX\\}}\hat{\Psi}_{i}^{\dagger}({x_{k}})h^{0}_{ij}({x_{k}})\hat{\Psi}_{j}({x_{k}})+\frac{1}{2}\int
d{x_{1}}d{x_{2}}\big{[}\hat{\Psi}^{\dagger}_{IX}({x_{1}})\hat{\Psi}^{\dagger}_{IX}({x_{2}})V_{\text{dd}}(|{x_{1}-x_{2}}|)\hat{\Psi}_{IX}({x_{1}})\hat{\Psi}_{IX}({x_{2}})\big{]},$
(16)

where IX stands for indirect exciton and X simply for the direct exciton. Here
$h^{0}_{ij}$ are elements of the matrix $h^{0}$ given as

$\displaystyle
h^{0}({x_{k}})=\begin{pmatrix}-\frac{\hbar^{2}}{2m_{C}}\frac{\partial^{2}}{\partial
x_{k}^{2}}+\Delta+\delta_{C}&\Omega&0\\\
\Omega&-\frac{\hbar^{2}}{2m_{X}}\frac{\partial^{2}}{\partial x_{k}^{2}}&-J\\\
0&-J&-\frac{\hbar^{2}}{2m_{IX}}\frac{\partial^{2}}{\partial
x_{k}^{2}}\end{pmatrix},$ (17)

with equal masses for the two types of excitons. The detunings $\Delta$ and
$\delta_{C}$ follow the same definitions as in the main text.

In Fig. 7 we show the numerical predictions for the energy shifts as a
function of $L$ for $\Omega=5$ meV and two values of the inter-exciton
coupling $J=5$ and 6 meV. For large $L$, the system is in the Tonks-Girardeau
(TG) regime, and it is seen that for a higher $J$, the interaction energy gets
lowered. At the single-particle level, increasing $J$ gives a larger excitonic
component in the lower polariton which in turn increases the effective mass
and lowers the TG energy shift at large $L$. As the confinement length is
decreased, the system enters the intermediate regime, which is sensitive to
the underlying interaction, and increasing $J$ results in a larger energy
shift for a fixed Rabi coupling. In the small $L$ limit, the system decouples
from the two-particle basis state consisting of two indirect excitons, and
Jaynes-Cummings physics is realized. In this regime, the interaction energy is
given as $2f_{1}(\Omega,J)-f_{2}(\Omega,J)$, where $f_{1,2}$ are the
generalized Rabi couplings with subscript indicating either single- or two-
particle system, and it is a increasing function of $J$.

These additional plots confirm the generality of our conclusions, by showing
that the behaviour of the simplest model extends to more complex
configurations including several coupled excitonic species.

##  Appendix D Coupled wire model

Figure 8: Numerical calculation for the interaction energy in a quasi-1D
system for different values of tunneling strength $J$. The Rabi coupling is
set to $\Omega=10$ meV. Other parameters as in the figures of the main text.

In this final section, we consider a mathematically similar model including a
pair of wires. In particular, we assume two kinds of excitons that are
distinguished by living on parallel wires and have the possibility of
tunneling between them and, in this way, of passing next one another at a not
excessive energy cost.

The Hamiltonian is

$\displaystyle\hat{H}$ $\displaystyle=\sum_{k=1,2}\int
d{x_{k}}\sum_{i,j=\\{C,X_{1},X_{2}\\}}\hat{\Psi}_{i}^{\dagger}({x_{k}})h^{0}_{ij}({x_{k}})\hat{\Psi}_{j}({x_{k}})+\sum_{i=1,2}\frac{1}{2}\int
d{x_{1}}d{x_{2}}\big{[}\hat{\Psi}^{\dagger}_{X_{i}}({x_{1}})\hat{\Psi}^{\dagger}_{X_{i}}({x_{2}})V_{\text{dd}}(|{x_{1}-x_{2}}|)\hat{\Psi}_{X_{i}}({x_{1}})\hat{\Psi}_{X_{i}}({x_{2}})\big{]}$
$\displaystyle+\left\\{\frac{1}{2}\int
d{x_{1}}d{x_{2}}\big{[}\hat{\Psi}^{\dagger}_{X_{1}}({x_{1}})\hat{\Psi}^{\dagger}_{X_{2}}({x_{2}})V_{\text{dd}}\left(\sqrt{|{x_{1}-x_{2}}|^{2}+(\Delta
z)^{2}}\right)\hat{\Psi}_{X_{1}}({x_{1}})\hat{\Psi}_{X_{2}}({x_{2}})\big{]}+h.c.\right\\},$
(18)

where h.c. means the hermitian conjugate of the term just before, and
$h^{0}_{ij}$ are elements of the matrix $h^{0}$ given as

$\displaystyle
h^{0}({x_{k}})=\begin{pmatrix}-\frac{\hbar^{2}}{2m_{C}}\frac{\partial^{2}}{\partial
x_{k}^{2}}+\Delta+\delta_{C}&\Omega&\Omega\\\
\Omega&-\frac{\hbar^{2}}{2m_{X_{1}}}\frac{\partial^{2}}{\partial
x_{k}^{2}}&-J\\\
\Omega&-J&-\frac{\hbar^{2}}{2m_{X_{2}}}\frac{\partial^{2}}{\partial
x_{k}^{2}}\end{pmatrix},$ (19)

where the masses of the two excitons are taken to be equal. Note that $\Delta$
should be set to the symmetric state of the reduced system

$\displaystyle\begin{pmatrix}-\frac{\hbar^{2}}{2m_{X_{1}}}\frac{\partial^{2}}{\partial
x_{k}^{2}}&-J\\\ -J&-\frac{\hbar^{2}}{2m_{X_{2}}}\frac{\partial^{2}}{\partial
x_{k}^{2}}\end{pmatrix}$ (20)

that is shifted by an additional $-J$ from the bare excitonic energy level,
i.e. $\Delta=\epsilon_{X,1}-J-\epsilon_{C,1}$. $\epsilon_{X,1}$ and
$\epsilon_{C,1}$ are the lowest excitonic- and photonic single particle states
in the confinement potential.

In Fig. 8 we show the numerical results for the interaction energy as a
function of the box size $L$ for two values of the tunneling amplitude $J$ and
a fixed lateral spacing of $\Delta z=5$ nm. It is seen that for large $L$ the
interaction energy is approximately equal in the two cases. As $L$ is
decreased, one sees a slight dependence on $J$, a smaller $J$ yielding a
slightly larger energy shift. This is explained by the fact that at small
values of $J$, the system does not decouple completely from the mixed states
$|X_{1}X_{2}\rangle$ and $|X_{2}X_{1}\rangle$ where the two particles are
excitons of different kinds. For large values of $J$, the system completely
decouples from these states. This slight dependence on $J$ is more pronounced
when $\Delta z$ is increased. It shows the importance of choosing a large
enough $J$ to obtain a realistic quasi-1D model. The energy shift is again
$J$-independent in the quantum dot limit where light is decoupled completely
from the states that feature dipolar repulsion.

It is interesting to note that the BO energy shift calculated as discussed in
the main text and in the following of the SM is able to accurately capture
(green dashed line) the essential physics for intermediate $L$-values and the
quantum dot limit, as for the strict 1D case. Once again, these plots confirm
that the qualitative behavior and the order of magnitude of the interaction
energy are similar to ones of the purely 1D case discussed in the main text
and support our application of the BO formalism to generic geometries beyond
1D.