# Manipulating the Dipolar Interactions and Cooperative Effects in Confined
Geometries

Hadiseh Alaeian  Elmore Family School of Electrical and Computer Engineering,
Department of Physics and Astronomy, Purdue Quantum Science and Engineering
Institute, Purdue University, West Lafayette, IN 47907, US  Artur Skljarow  5
Physikalisches Institut, Center for Integrated Quantum Science and Technology
(IQST), Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
Stefan Scheel  Institut für Physik, Universität Rostock, Albert-Einstein-
Straße 23-24, 18059 Rostock, Germany  Tilman Pfau  5 Physikalisches Institut,
Center for Integrated Quantum Science and Technology (IQST), Universität
Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany  Robert Löw  5
Physikalisches Institut, Center for Integrated Quantum Science and Technology
(IQST), Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
<EMAIL_ADDRESS>
###### Abstract

To facilitate the transition of quantum effects from the controlled laboratory
environment to practical real-world applications, there is a pressing need for
scalable platforms. One promising strategy involves integrating thermal vapors
with nanostructures designed to manipulate atomic interactions. In this
tutorial, we aim to gain deeper insights into this by examining the behavior
of thermal vapors that are confined within nanocavities or waveguides and
exposed to near-resonant light. We explore the interactions between atoms in
confined dense thermal vapors. Our investigation reveals deviations from the
predictions of continuous electrodynamics models, including density-dependent
line shifts and broadening effects. In particular, our results demonstrate
that by carefully controlling the saturation of single atoms and the
interactions among multiple atoms using nanostructures, along with controlling
the geometry of the atomic cloud, it becomes possible to manipulate the
effective optical nonlinearity of the entire atomic ensemble. This capability
renders the hybrid thermal atom-nanophotonic platform a distinctive and
valuable one for manipulating the collective effect and achieving substantial
optical nonlinearities.

  *

Keywords: dipole-dipole interaction, cooperative effects, spin model, mean-
field theory, nonlinear optics

##  1 Introduction

Atoms represent ideal systems for investigating cooperative quantum phenomena
due to their narrow radiative linewidth and consistent reproducibility. When
photons are emitted by two distinct atoms, they often exhibit remarkable
similarity, setting atoms apart from artificial light emitters such as quantum
dots, vacancy centers, or rare-earth ions. Unlike these artificial emitters,
atoms cannot be customized to user specifications but offer predictable
properties with additional interactions involving their environment, like
phonons or nuclear spins.

In certain situations, cold and thermal atomic clouds can be confined very
close to macroscopic surfaces, such as wedged vapor cells [1, 2], nano-fibers
[3, 4], or atom-cladded nanophotonic devices [5, 6, 7, 8, 9, 10, 11, 12, 13].
In these cases, the electromagnetic characteristics of the macroscopic
surroundings influence the interactions between the atomic dipoles. These
effects can be analyzed within the framework of macroscopic quantum
electrodynamics (QED) [14, 15], which describes the quantized electromagnetic
field in various absorptive and dispersive linear media using the classical
electromagnetic Green’s tensor. This can be seen as a generalized mode
expansion where photons occupy the same modes as classical wave counterparts.
The theory accounts for quantum fluctuations in matter and fields, which can
spontaneously polarize neutral atoms, leading to effective interactions with
the surrounding macroscopic objects. This interaction with surfaces results in
an additional radiative line shift, known as the Casimir-Polder shift [16]. It
is one of several electromagnetic dispersion interactions caused by quantum
vacuum fluctuations, depending on variations in atom and body properties with
frequency.

Furthermore, in an ensemble of atoms where many dipoles interact
simultaneously, new properties can emerge. Notable examples are altered
spontaneous emission rates, referred to as sub- or superradiance, initially
predicted by Dicke in 1954 [17] and elaborated on by Lehmberg [18, 19].
Resonant atom-atom interactions can have both positive and negative effects.
For instance, in an array of atomic clocks, undesired interactions reduce
coherence and precision. Conversely, they can be harnessed to engineer light-
matter interactions, creating a perfect mirror using a single atomic or
excitonic layer [20, 21, 22, 23, 24].

In a continuous medium, a polarized atom alters its local environment,
affecting the local field at the position of other atoms, known as the local
field correction. This is described by the density-dependent Lorentz–Lorenz
shift. In a discrete medium of closely spaced individual atoms, superradiance
results in a dispersive effective interaction, sometimes referred to as the
cooperative Lamb shift [25], observed in thin atomic layers [26, 27, 28].
Various theoretical approaches have been proposed [29, 30, 31] to explain some
but not all observed features. In systems with numerous interacting atoms in a
disordered ensemble, the resulting inhomogeneous broadening of atomic
resonance frequencies leads to saturation of the optical response, resulting
in an effective refractive index of the atomic medium that reaches
approximately $n\approx 1.7$ [32].

This work employs a first-principle approach to unify interpretations and
treatments of dipole-dipole interactions (DDI) and cooperative effects in the
literature. While the primary focus is on thermal atomic ensembles
experiencing motional dephasing, the theoretical framework is broadly
applicable. The paper’s structure is as follows: Section 2 uses the most
general formalism of macroscopic quantum electrodynamics to derive the dipole-
dipole interaction, starting from the general quantum master equation. It
discusses the impact of resonant and virtual photon transitions on cooperative
shifts and interactions, shedding light on different approximations commonly
used in literature to describe cooperative effects in various physical
platforms across different limits. To the best of our knowledge, the approach
presented here represents the first comprehensive quantum treatment of this
problem. The section concludes by deriving the well-known spin model often
used to describe cooperative effects in atomic systems.

In Section 3, the spin model is employed to calculate the evolution of the
joint density matrix of the atomic ensemble in free space. For the two-atom
case, the effective non-Hermitian Hamiltonian is directly diagonalized to
determine the energies and decay rates of hybridized states, illustrating
their sub(super)-radiant characteristics. The study is then extended to larger
ensembles with more than two atoms, with a focus on thermal atoms subject to
transit and Doppler broadening and dephasing. While exact diagonalization of
the ensemble effective Hamiltonian becomes impractical in this limit due to
the rapid dephasing of thermal atoms, rendering the mean-field (MF)
approximation a valid approach. A Monte-Carlo approach is used to account for
all motional effects, examining dipolar interactions in various dimensions,
from 1D to 3D clouds, emphasizing the crucial role of geometry and
dimensionality in emerging dipolar effects.

Section 4 summarizes recent experiments that utilize thermal atoms to
investigate and manipulate dipolar interactions. Finally, Section 5 provides a
summary of the results and suggests intriguing directions for exploring
dipolar atomic ensembles at various dimensions.

##  2 Atom-atom interactions from first principles

The interaction of an atomic ensemble with a common light field leads to an
effective light- induced dipole-dipole interaction as well as cooperative
effects [33, 34]. To derive the simplest case of light-induced dipolar
interaction from first principles, we consider $N$ atoms interacting with the
electromagnetic field in free space. After tracing over the electromagnetic
degrees of freedom, one obtains an effective dipole-dipole interaction between
the atoms. In what follows, we will start from the most general description of
an atomic ensemble interacting with quantized electromagnetic fields to derive
a quantum master equation that describes the evolution of the reduced atomic
density matrix. Then, we show how, by employing several approximations such as
the rotating wave, Born-Markov, and mean-field approximations, one obtains
various descriptions of the interacting emitter ensemble including the well-
known classical coupled dipole model. This pedagogical approach presented here
clarifies the range of validity of each approximation and delimits their
applicability in modeling the experimental scenarios and describing their
results as summarized in Sec. 4.

In the most general form, and when expanded in Fourier space, the electric
field operator reads [15]

$~{}\hat{\mathbf{E}}(\mathbf{R})=\int_{0}^{\infty}d\omega\left(\hat{\mathbf{\mathcal{E}}}(\mathbf{R},\omega)+\hat{\mathbf{\mathcal{E}}}^{\dagger}(\mathbf{R},\omega)\right)\,,$
(1)

where $\hat{\mathbf{\mathcal{E}}}(\mathbf{R},\omega)$ is related to
$\hat{\mathbf{f}}(\mathbf{R},\omega)$, the annihilation operator of the
medium-assisted field excitation at position $\mathbf{R}$ and frequency
$\omega$, as [35, 15]

$~{}\hat{\mathbf{\mathcal{E}}}(\mathbf{R},\omega)=i\left(\frac{\omega}{c}\right)^{2}\sqrt{\frac{\hbar}{\pi\epsilon_{0}}}\int
d\mathbf{R^{\prime}}\sqrt{\textrm{Im}(\epsilon(\mathbf{R^{\prime}},\omega))}\,G(\mathbf{R},\mathbf{R^{\prime}},\omega)~{}\hat{\mathbf{f}}(\mathbf{R^{\prime}},\omega)\,,$
(2)

where $\epsilon(\mathbf{R},\omega)$ is the permittivity of the medium and
$G(\mathbf{R},\mathbf{R^{\prime}},\omega)$ is the classical electromagnetic
Green’s tensor determined via the Helmholtz equation [36]

$\left[\nabla\times\nabla\times-\left(\frac{\omega}{c}\right)^{2}\epsilon(\mathbf{R},\omega)\right]G(\mathbf{R},\mathbf{R^{\prime}},\omega)=\delta(\mathbf{R}-\mathbf{R^{\prime}})I\,.$
(3)

For an ensemble of $N$ multi-level atoms with transition frequencies
$\omega_{n}$ interacting with the continuum modes of the electromagnetic
field, the total Hamiltonian reads as [37]

$~{}\hat{\mathcal{H}}=\hat{H}_{A}+\hat{H}_{F}+\hat{H}_{AF}=\hbar\sum_{A,n}\omega_{A,n}\hat{\sigma}_{A,nn}+\hbar\int
d\mathbf{R}\int_{0}^{\infty}d\omega~{}\omega\,\hat{\mathbf{f}}^{\dagger}\hat{\mathbf{f}}-\sum_{A}\hat{\mathbf{d}}_{A}\cdot\hat{\mathbf{E}}(\mathbf{R}_{A})\,,$
(4)

where the first and second terms correspond to the free atom and the
electromagnetic field energies, respectively, and the last term describes the
atom-field interaction to the leading order, i.e. in electric-dipole
approximation.

Further, $\hat{\sigma}_{A,mn}=\ket{m_{A}}\bra{n_{A}}$ are the spin transition
operators between electronic levels $n,m$ of the $A^{\textrm{th}}$ atom, and
$\hat{\mathbf{d}}_{A}$ is the transition dipole moment operator of the
$A^{\textrm{th}}$ atom as,

$~{}\hat{\mathbf{d}}_{A}=-q\sum_{m,n}\braket{m_{A}}{\mathbf{r}_{A}}{n_{A}}\hat{\sigma}_{A,mn}=\mathbf{d}_{A,mn}\hat{\sigma}_{A,mn}\,.$
(5)

Note that, for a two-level atom, the above dipole operator simplifies to
$\mathbf{d}\left(\hat{\sigma}_{A}^{+}+\hat{\sigma}_{A}^{-}\right)$ in terms of
the well-known raising and lowering spin operators $\hat{\sigma}_{A}^{\pm}$.
Although a real atom is not a simple two-level system, later it will become
evident under which circumstances this is a sensible approximation.

The joint atom-field density operator evolves according to the von Neumann
equation as

$~{}\dot{\hat{\rho}}(t)=-\frac{i}{\hbar}\left[\hat{H}_{A}+\hat{H}_{F}+\hat{H}_{AF},\hat{\rho}(t)\right]\,,$
(6)

with the formal solution, valid for a time-independent Hamiltonian,

$~{}\hat{\rho}(t)=\hat{U}(t)\hat{\rho}(0)\hat{U}^{\dagger}(t)~{},~{}\hat{U}(t)=e^{-it\hat{\mathcal{H}}/\hbar}\,.$
(7)

To determine the atom-atom interaction, the electromagnetic degrees of freedom
have to be integrated out to obtain the dynamics in terms of atomic
quantities, only. This will become clearer in the interaction picture after
rotating the joint atom-field density operator to the frame of the free atoms
and photons given by $\hat{\mathcal{H}}_{0}=\hat{H}_{A}+\hat{H}_{F}$ via a
unitary transformation $\hat{U}_{0}(t)=e^{-it\hat{\mathcal{H}}_{0}/\hbar}$

$~{}\hat{\mathcal{O}}_{I}(t)=\hat{U}_{0}^{\dagger}(t)\hat{\mathcal{O}}(t)\hat{U}_{0}(t)\,.$
(8)

It is straightforward to show that the time evolution of the rotated joint
density matrix in the interaction picture is given by

$~{}\dot{\hat{\rho}}_{I}(t)=-\frac{i}{\hbar}\left[\hat{H}_{AF_{I}}(t),\hat{\rho}_{I}(t)\right]\,.$
(9)

The reduced atomic density matrix can be obtained by taking the partial trace
over the electromagnetic field degrees of freedom as

$~{}\hat{\rho}_{I}^{A}(t)=\textrm{Tr}_{F}\left(\hat{\rho}_{I}(t)\right)\,.$
(10)

As the goal is to derive an explicit equation of motion for the reduced atomic
density matrix, we can directly use Eq. (10) in Eq. (9) to find the time
evolution of $\hat{\rho}_{I}^{A}(t)$.

At this point, it is helpful to separate the interaction of the atom with the
electromagnetic field into two parts; 1) the interaction with the driving
field which is typically in a coherent state, and 2) the interaction with
other allowed modes, e.g. via the incoherent spontaneous emission into other
electromagnetic modes in free space. Here the subscripts $L$ and $V$ refer to
the first and the second case, respectively. We formally write the time
evolution of the reduced atomic density matrix as

$~{}\dot{\hat{\rho}}_{I}^{A}(t)=-\frac{i}{\hbar}\textrm{Tr}_{L}\left[\hat{H}_{AF_{I}}(t),\hat{\rho}_{I}(t)\right]-\frac{i}{\hbar}\textrm{Tr}_{V}\left[\hat{H}_{AF_{I}}(t),\hat{\rho}_{I}(t)\right]\,.$
(11)

The incoherently populated fluctuating electromagnetic modes can be described
as spatially and spectrally uncorrelated fields with the following properties
[38]:

$\displaystyle~{}\braket{\hat{\mathbf{E}}(\mathbf{R},\omega)}_{V}$
$\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle\braket{\hat{\mathbf{E}}(\mathbf{R},\omega)\otimes\hat{\mathbf{E}}(\mathbf{R^{\prime}},\omega^{\prime})}_{V}$
$\displaystyle=$ $\displaystyle 0\,,$ (12)
$\displaystyle\braket{\hat{\mathbf{E}}^{\dagger}(\mathbf{R},\omega)\otimes\hat{\mathbf{E}}(\mathbf{R^{\prime}},\omega^{\prime})}_{V}$
$\displaystyle=$
$\displaystyle\frac{\hbar\mu_{0}\omega^{2}}{\pi}n_{T}(\omega)\textrm{Im}(G(\mathbf{R},\mathbf{R^{\prime}},\omega))\delta(\omega-\omega^{\prime})\,,$
$\displaystyle\braket{\hat{\mathbf{E}}(\mathbf{R},\omega)\otimes\hat{\mathbf{E}}^{\dagger}(\mathbf{R^{\prime}},\omega^{\prime})}_{V}$
$\displaystyle=$
$\displaystyle\frac{\hbar\mu_{0}\omega^{2}}{\pi}\left(1+n_{T}(\omega)\right)\textrm{Im}(G(\mathbf{R},\mathbf{R^{\prime}},\omega))\delta(\omega-\omega^{\prime})\,,$

where $n_{T}(\omega)$ is the mean number of thermal photons at temperature $T$
and frequency $\omega$ determined via the Bose-Einstein distribution as

$~{}n_{T}(\omega)=\frac{1}{e^{\frac{\hbar\omega}{k_{B}T}}-1}\,.$ (13)

As can be seen, the electric field vanishes on average, consistent with the
assumption about their population via fluctuations, but the field intensity
does not. Later, it will become clear that these contributions lead to
stimulated emission ($\propto n_{T}(\omega)$) and absorption ($\propto
1+n_{T}(\omega)$) triggered by thermal photons and the vacuum. The latter case
is better known as spontaneous emission.

Equation (9) can be formally solved for the interacting density matrix as

$~{}\hat{\rho}_{I}(t)=\hat{\rho}_{I}(0)-\frac{i}{\hbar}\int_{0}^{t}dt^{\prime}~{}\left[\hat{H}_{AF_{I}}(t^{\prime}),\hat{\rho}_{I}(t^{\prime})\right]\,.$
(14)

As can be seen from Eq. (2), the non-driven, spontaneously occupied modes are
described via their correlations, which implies that the formal solution Eq.
(14) should be re-inserted into the second term of Eq. (11), i.e. fluctuating
field contributions, leading to the equation of motion for the reduced atomic
density matrix as

$~{}\dot{\hat{\rho}}_{I}^{A}(t)=-\frac{i}{\hbar}\textrm{Tr}_{L}\left[\hat{H}_{AF_{I}}(t),\hat{\rho}_{I}(t)\right]-\frac{i}{\hbar}\textrm{Tr}_{V}\left[\hat{H}_{AF_{I}}(t),\hat{\rho}_{I}(0)\right]-\frac{1}{\hbar^{2}}\textrm{Tr}_{V}\int_{0}^{t}dt^{\prime}~{}\left[\hat{H}_{AF_{I}}(t),\left[\hat{H}_{AF_{I}}(t^{\prime}),\hat{\rho}_{I}(t^{\prime})\right]\right]\,.$
(15)

We assume that, at $t=0$, atoms and fields are not correlated, and hence
$\hat{\rho}_{I}(0)=\hat{\rho}_{A}(0)\otimes\hat{\rho}_{F}$ 111This is a strong
assumption that has to be checked in each scenario.. Under this condition the
second term in Eq. (15) vanishes.

Under the weak atom-field couplings, the field remains unchanged, i.e.
$\dot{\hat{\rho}}_{F}(t)=0$. This limit, known as _Born_ approximation allows
one to write the joint density matrix as a direct product of atoms and photon
density matrices at all times, i.e.
$\hat{\rho}_{I}(t)\approx\hat{\rho}_{I}^{A}(t)\otimes\hat{\rho}_{F}$ [39]. In
particular, this means that the joint evolution does not correlate or entangle
atoms and light at any time step.

Furthermore, if the atomic states change slowly enough compared to all other
timescales, their time evolution is a _Markovian_ process implying that
$\hat{\rho}_{I}^{A}(t^{\prime})\approx\hat{\rho}_{I}^{A}(t)$ 222This
approximation is justified since the reduced density is already in the rotated
frame and hence the fast-varying part has been already extracted.. With these
assumptions, Eq. (15) can be simplified to [40]

$~{}\dot{\hat{\rho}}_{I}^{A}(t)=-\frac{i}{\hbar}\textrm{Tr}_{L}\left[\hat{H}_{AF_{I}}(t),\hat{\rho}_{I}^{A}(t)\otimes\hat{\rho}_{F}\right]-\frac{1}{\hbar^{2}}\textrm{Tr}_{V}\int\limits_{0}^{\infty}dt^{\prime}~{}\left[\hat{H}_{AF_{I}}(t),\left[\hat{H}_{AF_{I}}(t-t^{\prime}),\hat{\rho}_{I}^{A}(t)\otimes\hat{\rho}_{F}\right]\right],$
(16)

where in the last integral the limit is extended to $\infty$, justified by the
finite correlation times of the field.

###  2.1 Quantum master equation for the reduced atomic density matrix

Starting from Eq. (16), we first calculate the rotated interacting Hamiltonian
$\hat{H}_{AF_{I}}$ as

$\hat{H}_{AF_{I}}(t)=-\sum_{B}\left(e^{it\hat{H}_{A}/\hbar}\hat{\mathbf{d}}_{B}e^{-it\hat{H}_{A}/\hbar}\right)\cdot\left(e^{it\hat{H}_{F}/\hbar}\hat{\mathbf{E}}(\mathbf{R}_{B})e^{-it\hat{H}_{F}/\hbar}\right)\,,$
(17)

where we used $\left[\hat{H}_{A},\hat{H}_{F}\right]=0$, to separate the
unitary rotation $\hat{U}_{0}(t)$ into atom and field parts.

Using the explicit form of the atom-field interaction in the electric-dipole
approximation limit, the Hamiltonian in the interaction picture reads as

$~{}\hat{H}_{AF_{I}}=-\sum_{B}\underbrace{\sum_{m,n}\mathbf{d}_{B,mn}e^{i\omega_{B,mn}t}\hat{\sigma}_{B,mn}}_{\hat{\mathbf{d}}_{B_{I}}(t)}\cdot\underbrace{\int_{0}^{\infty}d\omega~{}\left(e^{i\omega
t}\hat{\mathcal{E}}^{\dagger}(\mathbf{R}_{B},\omega)+e^{-i\omega
t}\hat{\mathcal{E}}(\mathbf{R}_{B},\omega)\right)}_{\hat{H}_{F_{I}}(t)}\,.$
(18)

Since the interaction Hamiltonian can be decomposed as
$\hat{H}_{AF_{I}}=\hat{H}_{A_{I}}\otimes\hat{H}_{F_{I}}$, one can directly
calculate the contribution of the driving field in Eq. (16) as [41]

$\displaystyle\textrm{T}_{L}$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}\textrm{Tr}_{L}\left[\hat{H}_{AF_{I}}(t),\hat{\rho}_{I}^{A}(t)\otimes\hat{\rho}_{F}\right]$
(19) $\displaystyle=$
$\displaystyle-\frac{i}{2\hbar}\textrm{Tr}_{L}\Big{(}\left[\hat{H}_{A_{I}}(t),\hat{\rho}_{I}^{A}(t)\right]\otimes\left\\{\hat{H}_{F_{I}},\hat{\rho}_{F}\right\\}$
$\displaystyle+\left\\{\hat{H}_{A_{I}}(t),\hat{\rho}_{I}^{A}(t)\right\\}\otimes\left[\hat{H}_{F_{I}},\hat{\rho}_{F}\right]\Big{)}$
$\displaystyle=$
$\displaystyle-\frac{i}{\hbar}\textrm{Tr}_{L}\left(\hat{H}_{F_{I}}\hat{\rho}_{F}\right)\left[\hat{H}_{A_{I}}(t),\hat{\rho}_{I}^{A}(t)\right]\,,$

where we used $\textrm{Tr}_{L}[\hat{H}_{F_{I}},\hat{\rho}_{F}]=0$ and
$\textrm{Tr}_{L}\\{\hat{H}_{F_{I}},\hat{\rho}_{F}\\}=2\textrm{Tr}_{L}(\hat{H}_{F_{I}}\hat{\rho}_{F})$.

Assuming the typical spectroscopy scenario using a coherent field
$\ket{E_{\mathbf{r},\omega}}$, the field density operator can be written as
$\hat{\rho}_{F}=\ket{E_{\mathbf{r},\omega}}\bra{E_{\mathbf{r},\omega}}$ and
hence the partial trace $\textrm{Tr}_{L}$ over the field can be simplified as

$\displaystyle~{}\textrm{Tr}_{L}\left(\hat{H}_{F_{I}}\hat{\rho}_{F}\right)$
$\displaystyle=\textrm{Tr}_{L}\int_{0}^{\infty}d\omega\left(e^{i\omega
t}\hat{\mathcal{E}}^{\dagger}(\mathbf{R}_{B},\omega)+e^{-i\omega
t}\hat{\mathcal{E}}(\mathbf{R}_{B},\omega)\right)\ket{E_{\mathbf{r},\omega}}\bra{E_{\mathbf{r},\omega}}$
(20) $\displaystyle=\int_{0}^{\infty}d\omega~{}\left(e^{i\omega
t}\mathbf{E}_{\mathbf{r},\omega}^{*}+e^{-i\omega
t}\mathbf{E}_{\mathbf{r},\omega}\right)=\mathbf{E}(\mathbf{r},t)\,,$

where in the last line we employed the unity trace property of the density
matrix $\hat{\rho}_{F}$ followed by the inverse Fourier transform to re-
constitute the driving field in the time domain from its frequency components.

This simplifies the driving field contribution to the evolution of the reduced
atomic density matrix to

$~{}\textrm{T}_{L}=\frac{i}{\hbar}\left[\sum_{B}\hat{\mathbf{d}}_{B_{I}}(t)\cdot\mathbf{E}(\mathbf{R}_{B},t),\hat{\rho}_{I}^{A}(t)\right]\,.$
(21)

The final equation is the Rabi Hamiltonian describing the dynamics of atoms
driven by the classical field $\mathbf{E}(\mathbf{R}_{B},t)$. As can be seen,
the coupling of each atom to the driving field is independent of other atoms
in the ensemble which is consistent with the aforementioned Born
approximation.

On the other hand, there will be effective atom-atom interactions that arise
from the second term in Eq. (16). Those modes are not externally driven and
are solely populated by photons emitted by the atoms, whose contributions can
be determined as

$\displaystyle~{}\textrm{T}_{V}=$
$\displaystyle-\frac{1}{\hbar^{2}}\textrm{Tr}_{V}\int_{0}^{\infty}dt^{\prime}~{}\left[\hat{H}_{AF_{I}}(t),\left[\hat{H}_{AF_{I}}(t-t^{\prime}),\hat{\rho}_{I}^{A}(t)\otimes\hat{\rho}_{F}\right]\right]$
$\displaystyle=$
$\displaystyle-\frac{1}{\hbar^{2}}\sum_{C,D}\int_{0}^{\infty}dt^{\prime}~{}\hat{\mathbf{d}}_{C_{I}}(t)\cdot\textrm{Tr}_{V}\\{\hat{\mathbf{E}}_{I}(\mathbf{R}_{C},t)\otimes\hat{\mathbf{E}}_{I}(\mathbf{R}_{D},t-t^{\prime})\hat{\rho}_{F_{I}}\\}\hat{\mathbf{d}}_{D_{I}}(t-t^{\prime})\hat{\rho}_{I}^{A}(t)$
$\displaystyle+\frac{1}{\hbar^{2}}\sum_{C,D}\int_{0}^{\infty}dt^{\prime}~{}\hat{\mathbf{d}}_{C_{I}}(t)\hat{\rho}_{I}^{A}(t)\cdot\textrm{Tr}_{V}\\{\hat{\mathbf{E}}_{I}(\mathbf{R}_{C},t)\otimes\hat{\mathbf{E}}_{I}(\mathbf{R}_{D},t-t^{\prime})\hat{\rho}_{F_{I}}\\}\hat{\mathbf{d}}_{D_{I}}(t-t^{\prime})$
$\displaystyle+\frac{1}{\hbar^{2}}\sum_{C,D}\int_{0}^{\infty}dt^{\prime}~{}\hat{\mathbf{d}}_{D_{I}}(t-t^{\prime})\cdot\textrm{Tr}_{V}\\{\hat{\mathbf{E}}_{I}(\mathbf{R}_{D},t-t^{\prime})\otimes\hat{\mathbf{E}}_{I}(\mathbf{R}_{C},t)\hat{\rho}_{F_{I}}\\}\hat{\rho}_{I}^{A}(t)\hat{\mathbf{d}}_{C_{I}}(t)$
$\displaystyle-\frac{1}{\hbar^{2}}\sum_{C,D}\int_{0}^{\infty}dt^{\prime}\hat{\rho}_{I}^{A}(t)~{}\hat{\mathbf{d}}_{D_{I}}(t-t^{\prime})\cdot\textrm{Tr}_{V}\\{\hat{\mathbf{E}}_{I}(\mathbf{R}_{D},t-t^{\prime})\otimes\hat{\mathbf{E}}_{I}(\mathbf{R}_{C},t)\hat{\rho}_{F_{I}}\\}\hat{\mathbf{d}}_{C_{I}}(t)\,,$

where we used the identity

$\left(\mathbf{v}_{1}\cdot\mathbf{v}_{2}\right)\left(\mathbf{v}_{3}\cdot\mathbf{v}_{4}\right)=\mathbf{v}_{1}\cdot\left(\mathbf{v}_{2}\otimes\mathbf{v}_{3}\right)\mathbf{v}_{4}$
(23)

to re-write the integral kernels as tensor products.

The correlation properties of these fluctuation fields summarized in Eq. 2
should be used to simplify the integral kernels, after employing the Fourier
transformation, as

$\displaystyle\textrm{Tr}_{V}\\{\hat{\mathbf{E}}_{I}(\mathbf{R}_{C},t)\otimes\hat{\mathbf{E}}_{I}(\mathbf{R}_{D},t-t^{\prime})\hat{\rho}_{F_{I}}\\}$
$\displaystyle=\int_{0}^{\infty}d\omega
d\omega^{\prime}\braket{\mathbf{E}(\mathbf{R}_{C},\omega)\mathbf{E}^{\dagger}(\mathbf{R}_{D},\omega^{\prime})}_{V}e^{-i(\omega-\omega^{\prime})t-i\omega^{\prime}t^{\prime}}$
$\displaystyle+\int_{0}^{\infty}d\omega
d\omega^{\prime}\braket{\mathbf{E}^{\dagger}(\mathbf{R}_{C},\omega)\mathbf{E}(\mathbf{R}_{D},\omega^{\prime})}_{V}e^{i(\omega-\omega^{\prime})t+i\omega^{\prime}t^{\prime}}$
$\displaystyle=\frac{\hbar\mu_{0}}{\pi}\int_{0}^{\infty}d\omega\left((1+n(\omega))e^{-i\omega
t^{\prime}}+n(\omega)e^{i\omega
t^{\prime}}\right)\omega^{2}\textrm{Im}(G(\mathbf{R}_{C},\mathbf{R}_{D},\omega))\,.$
(24)

Other terms of Eq. (2.1) can be calculated similarly and expressed in terms of
photon density and Green’s tensor.

Combining the results with Eq. (21) and inserting the final expression into
Eq. (16), we obtain the quantum master equation for the reduced atomic density
matrix $\hat{\rho}_{I}^{A}(t)$ in the interaction picture. The atomic density
operator in the original frame can be obtained using the unitary
transformation
$\hat{\rho}^{A}(t)=e^{-it\hat{H}_{A}/\hbar}\hat{\rho}_{I}^{A}(t)e^{it\hat{H}_{A}/\hbar}$.
Using the explicit form of the dipole operator $\mathbf{d}_{C,D}$ in terms of
the dipole-allowed transitions as in Eq. (5), the evolution of the reduced
atomic density matrix is given by

$\displaystyle~{}\frac{d}{dt}\hat{\rho}^{A}(t)=-\frac{i}{\hbar}\left[\hat{H}_{A}-\sum_{B,m,n}\mathbf{d}_{B,mn}\cdot\mathbf{E}(\mathbf{R}_{B},t)\hat{\sigma}_{B,mn},\hat{\rho}^{A}(t)\right]$
$\displaystyle+\sum_{C,D,m,n,i,j}\mathbf{d}_{C,mn}\cdot\mathbf{H}_{ji}(\mathbf{R}_{C},\mathbf{R}_{D})\cdot\mathbf{d}_{D,ij}\left(\hat{\sigma}_{C,mn}\hat{\sigma}_{D,ij}\hat{\rho}^{A}(t)-\hat{\sigma}_{D,ij}\hat{\rho}^{A}(t)\hat{\sigma}_{C,mn}\right)$
$\displaystyle-\sum_{C,D,m,n,i,j}\mathbf{d}_{C,mn}\cdot\mathbf{H}^{*}_{ji}(\mathbf{R}_{C},\mathbf{R}_{D})\cdot\mathbf{d}_{D,ij}\left(\hat{\sigma}_{C,mn}\hat{\rho}^{A}(t)\hat{\sigma}_{D,ij}-\hat{\rho}^{A}(t)\hat{\sigma}_{D,ij}\hat{\sigma}_{C,mn}\right)\,,$

where $\mathbf{H}_{ij}(\mathbf{R}_{C},\mathbf{R}_{D})$ are the Green’s
function-dependent coefficients for two atoms located at $\mathbf{R}_{C,D}$
for $ji^{\textrm{th}}$ transition, as detailed in [42].

As can be inferred from Eq. (2.1), the first line corresponds to the single-
atom dynamics due to the free atom and the driving field. The second and the
third lines, on the other hand, describe the light-induced atom-atom
interaction, in the forms of spin exchange as
$\hat{\sigma}_{C,mn}\hat{\sigma}_{D,ij}$. In the next subsection, we show how
this general form can be used to derive the commonly-used simplified spin
model master equation [43, 44, 45, 46, 47, 48] as well as the well-known
classical coupled dipole model in some specific limits.

###  2.2 The coupled-spin master equation and classical coupled-dipole limit

Using the Heisenberg equation of motion obtained from Eq. (2.1), one can study
the atomic polarization generated by the atom-light interactions as well as
the light-induced dipole-dipole interaction within the ensemble. Let us assume
that the driving field is a monochromatic coherent field as
$\mathbf{E}_{0}(\mathbf{r})e^{-i\omega_{L}t}+\mathrm{c.c.}$, and it is near-
resonant with only one of the electronic transitions at $\omega_{0}$, such
that $\omega_{L}-\omega_{0}\ll\omega_{L}-\omega_{mn}$ for $\omega_{mn}$ being
the transition frequency of other allowed transitions as
($m\longleftrightarrow n$). This allows one to simplify the equations of
motion noticeably, since such near-resonant exciting laser couples only those
state pairs efficiently, while the far-detuned levels remained almost
unperturbed and hence can be dropped from Eq. (2.1). Further, we assume a
small detuning as $\omega_{L}-\omega_{0}\ll\omega_{L}+\omega_{0}$ which allows
one to employ the _rotating-wave approximation_ to further ignore fast-
rotating, i.e. energy non-conserving, terms in the Rabi Hamiltonian, and only
retain excitation-conserving terms.

Using common notations of $g,e$ for the states of this two-level atom with
transition frequency $\omega_{0}$ and with sub-levels of $\mu,\nu$,
respectively, and in the rotated the frame of the laser
$\hat{\tilde{\sigma}}_{B_{\mu\nu}}^{ge}(t)=\hat{\sigma}_{B_{\mu\nu}}^{ge}(t)e^{i\omega_{L}t}$,
the dipole moment of the $A^{\textrm{th}}$ atom will be given as

$\braket{\hat{\mathbf{d}}_{A,ge}}=\Sigma_{\mu,\nu}\braket{\hat{\sigma}_{B_{\mu\nu}}^{ge}}\mathbf{d}_{A_{\mu\nu}}^{ge}\,.$
(26)

The time evolution of the spin operator is given via

$\displaystyle~{}\frac{d}{dt}\braket{\hat{\tilde{\sigma}}_{A_{\mu\nu}}^{ge}(t)}$
$\displaystyle=i\left(\omega_{L}-\omega_{0}\right)\braket{\hat{\tilde{\sigma}}_{A_{\mu\nu}}^{ge}(t)}$
$\displaystyle-\frac{i}{\hbar}\sum_{\kappa}\left(\braket{\hat{\tilde{\sigma}}_{A_{\kappa\nu}}^{ee}(t)}\mathbf{d}_{A_{\kappa\mu}}^{eg}-\braket{\hat{\tilde{\sigma}}_{A_{\mu\kappa}}^{gg}(t)}\mathbf{d}_{A_{\nu\kappa}}^{eg}\right)\cdot\mathbf{E}_{0}(\mathbf{R}_{A})$
$\displaystyle-i\frac{\mu_{0}\omega_{0}^{2}}{\hbar}\sum_{B\neq
A}\sum_{\delta\epsilon\kappa}\mathbf{d}_{A_{\kappa\mu}}^{eg}\cdot
G(\mathbf{R}_{A},\mathbf{R}_{B},\omega_{0})\cdot\mathbf{d}_{B_{\delta\epsilon}}^{ge}\braket{\hat{\tilde{\sigma}}_{A_{\kappa\nu}}^{ee}(t)\hat{\tilde{\sigma}}_{B_{\delta\epsilon}}^{ge}(t)}$
$\displaystyle+i\frac{\mu_{0}\omega_{0}^{2}}{\hbar}\sum_{B\neq
A}\sum_{\delta\epsilon\kappa}\mathbf{d}_{A_{\nu\kappa}}^{eg}\cdot
G(\mathbf{R}_{A},\mathbf{R}_{B},\omega_{0})\cdot\mathbf{d}_{B_{\delta\epsilon}}^{ge}\braket{\hat{\tilde{\sigma}}_{A_{\mu\kappa}}^{gg}(t)\hat{\tilde{\sigma}}_{B_{\delta\epsilon}}^{ge}(t)}$
$\displaystyle+i\sum_{\epsilon}\left(-\omega_{A_{\nu\epsilon}}^{e^{\textrm{CP}}}+i\frac{\Gamma_{A_{\nu\epsilon}}^{e}}{2}\right)\braket{\hat{\tilde{\sigma}}_{A_{\mu\epsilon}}^{ge}(t)}+\left(\omega_{A_{\mu\epsilon}}^{g^{\textrm{CP}}}+i\frac{\Gamma_{A_{\mu\epsilon}}^{g}}{2}\right)\braket{\hat{\tilde{\sigma}}_{B_{\epsilon\nu}}^{ge}(t)}\,.$

This is the _coupled-spin_ model, whose first line describes the dynamics of a
coherently-driven atom, while the second line is the light-induced dipole-
dipole interaction due to the exchange of a real photon, i.e. a resonant
photon that swaps the spins of the two atoms. In the last line
$\omega^{\textrm{CP}},\Gamma$ corresponds to the collective phase shift and
dissipation due to the other off-resonant electronic states of the atom [42].
Such a shift was first studied by Casimir and Polder and hence it is preferred
to as Casimir–Polder (CP) shift [16], whose detailed form can be found in A.

This treatment also takes into account the modifications of the local density
of the optical states in any environment if the proper Green’s function is
used. Physically, this leads to a modified decay rate, an effect first
predicted and studied by Purcell [49, 50].

In general, the coupled-spin master equation of Eq. (2.2) leads to an infinite
hierarchy of moments for the spins due to the spin correlation terms in the
resonant dipole-dipole interaction. However, if such correlations are ignored,
the _mean-field_ approximation can be employed to simplify them as
$\braket{\hat{\sigma}_{C,mn}(t)\hat{\sigma}_{D,ij}(t)}\approx\braket{\hat{\sigma}_{C,mn}(t)}\braket{\hat{\sigma}_{D,ij}(t)}$.
This approximation converts the equations of motion to a closed system of
coupled nonlinear equations. Finally, in a weakly-driven limit, where the
atoms are not strongly polarized, i.e.
$\braket{\hat{\sigma}_{A}^{ee}(t)-\hat{\sigma}_{A}^{gg}(t)}\approx-1$, the
mean-field version of Eq. (2.2) simplifies to the familiar classical coupled-
dipole model.

In the next section, we start from the quantum master equation (2.2) to study
the canonical problem of light-induced dipole-dipole interaction between two
atoms.

##  3 An interacting ensemble of atoms

Ignoring the fluctuation effects on the lineshift and broadening, and
considering a near-resonant monochromatic coherent drive in the form of
$\mathbf{E}_{0}(\mathbf{R})e^{-i\omega_{L}t}$ (cf. Sec. 2.2), Eq. (2.1) for
the reduced atomic ensemble density operator $\hat{\rho}^{A}(t)$ can be
simplified to the following master equation

$\displaystyle~{}\frac{d}{dt}\hat{\tilde{\rho}}^{A}(t)=$
$\displaystyle-i\left[\sum_{B}\Delta_{B}\hat{\sigma}_{B}^{+}\hat{\sigma}_{B}^{-}-\left(\frac{\mathbf{d}_{B}\cdot\mathbf{E}_{0}(\mathbf{R}_{B})}{\hbar}\hat{\sigma}_{B}^{+}+\textrm{H.c.}\right)-\sum_{C\neq
B}\epsilon_{BC}\hat{\sigma}_{B}^{+}\hat{\sigma}_{C}^{-},\hat{\tilde{\rho}}^{A}(t)\right]$
(28)
$\displaystyle+\sum_{B,C}\kappa_{BC}\left(\hat{\sigma}_{B}^{-}\hat{\tilde{\rho}}^{A}(t)\hat{\sigma}_{C}^{+}-\frac{1}{2}\\{\hat{\sigma}_{B}^{+}\hat{\sigma}_{C}^{-},\hat{\tilde{\rho}}^{A}(t)\\}\right)\,,$

where $\Delta_{B}=\omega_{0}-\omega_{L}$ is the detuning of the
$B^{\textrm{th}}$ atom from the excitation laser,
$\epsilon_{BC}=3\pi\Gamma_{0}\mathbf{d}_{B}\cdot\textrm{Re}(G(\mathbf{R}_{B},\mathbf{R}_{C},\omega_{0}))\cdot\mathbf{d}_{C}$
and
$\kappa_{BC}=3\pi\Gamma_{0}\mathbf{d}_{B}\cdot\textrm{Im}(G(\mathbf{R}_{B},\mathbf{R}_{C},\omega_{0}))\cdot\mathbf{d}_{C}$
correspond to the dispersive and dissipative dipole-dipole interactions,
respectively. Further, $\Gamma_{0}$ is the decay rate of a two-level atom in
free space and is related to the transition frequency $\omega_{0}$ and dipole
matrix element $\mathbf{d}$ as

$~{}\Gamma_{0}=\frac{\omega_{0}^{3}}{3\pi\hbar\epsilon_{0}c^{3}}|\mathbf{d}|^{2}\,.$
(29)

The dyadic Green’s function $G(\mathbf{r},\mathbf{r^{\prime}},\omega)$ depends
on the geometry as well as the optical properties of the environment, and has
a closed-form solution in only a few limited cases.

In this section, we focus on the physics of such dipolar interactions by
limiting ourselves to the simplest case, i.e. an atomic ensemble in free
space, where a closed-form Green’s tensor with the following form is available
[51]

$~{}G=\left(I+\frac{1}{k_{0}^{2}}\nabla\nabla\right)\frac{e^{ik_{0}R}}{4\pi
k_{0}R}\,,$ (30)

where $R=|\mathbf{R}_{2}-\mathbf{R}_{1}|$ and $k_{0}=\omega_{0}/c$.

Using this Green’s function, the atom-field interaction energy is given by

$~{}\mathcal{G}=\frac{k_{0}^{3}e^{ik_{0}R}}{4\pi\epsilon_{0}k_{0}R}\left[\left(\mathbf{r}\times\mathbf{d}_{1}\right)\times\mathbf{r}+\left[3\mathbf{r}\left(\mathbf{r}\cdot\mathbf{d}_{1}\right)-\mathbf{d}_{1}\right]\left(\frac{1}{(k_{0}R)^{2}}-\frac{i}{k_{0}R}\right)\right]\cdot\mathbf{d}_{2}\,,$
(31)

where $\mathbf{r}=(\mathbf{R}_{2}-\mathbf{R}_{1})/R$ is the unit vector
between the two emitters. The dispersive and dissipative interaction effects
are related to $\mathcal{G}$ via $\epsilon_{12}=\textrm{Re}(\mathcal{G})$ and
$\kappa_{12}=\textrm{Im}(\mathcal{G})$.

###  3.1 Canonical problem: two interacting atoms

To develop a better understanding of the dipole-dipole interaction, we
approximate the master equation (28) with a non-Hermitian Hamiltonian as

$~{}\frac{d}{dt}\hat{\tilde{\rho}}^{A}(t)=-i\hat{\mathcal{H}}_{\textrm{eff}}\hat{\tilde{\rho}}^{A}(t)+i\hat{\tilde{\rho}}^{A}(t)\hat{\mathcal{H}}_{\textrm{eff}}^{\dagger}\,,$
(32)

where the effective non-Hermitian operator is

$\hat{\mathcal{H}}_{\textrm{eff}}=\hat{H}-i\left(\frac{\Gamma_{0}}{2}\left(\hat{\sigma}_{1}^{+}\hat{\sigma}_{1}^{-}+\hat{\sigma}_{2}^{+}\hat{\sigma}_{2}^{-}\right)+\kappa_{12}\left(\hat{\sigma}_{1}^{+}\hat{\sigma}_{2}^{-}+\hat{\sigma}_{2}^{+}\hat{\sigma}_{1}^{-}\right)\right)\,.$
(33)

We start by writing $\hat{\mathcal{H}}_{\textrm{eff}}$ in the two-spin basis
of
$\ket{1}=\ket{g_{1}g_{2}},\ket{2}=\ket{e_{1}g_{2}},\ket{3}=\ket{g_{1}e_{2}},\ket{4}=\ket{e_{1}e_{2}}$
as

$~{}\mathcal{H}_{\textrm{eff}}=\begin{bmatrix}0&-\Omega_{1}&-\Omega_{2}&0\\\
-\Omega_{1}^{*}&\left(\Delta-i\frac{\Gamma_{0}}{2}\right)&-\left(\epsilon_{12}+i\kappa_{12}\right)&-\Omega_{2}\\\
-\Omega_{2}^{*}&-\left(\epsilon_{12}+i\kappa_{12}\right)&\left(\Delta-i\frac{\Gamma_{0}}{2}\right)&-\Omega_{1}\\\
0&-\Omega_{2}^{*}&-\Omega_{1}^{*}&\left(2\Delta-i\Gamma_{0}\right)\end{bmatrix}\,.$
(34)

The effect of dipole-dipole interaction can be clarified in the weak-probe
limit, i.e. $\Omega_{1,2}\approx 0$, where the following new states can be
determined

$\displaystyle~{}\omega_{1}$ $\displaystyle=0\longrightarrow\ket{1}\,,$
$\displaystyle\omega_{2}$
$\displaystyle=\left(\Delta-\epsilon_{12}\right)-\frac{i}{2}\left(\Gamma_{0}+2\kappa_{12}\right)\longrightarrow\ket{+}=\frac{\ket{2}+\ket{3}}{\sqrt{2}}\,,$
$\displaystyle\omega_{3}$
$\displaystyle=\left(\Delta+\epsilon_{12}\right)-\frac{i}{2}\left(\Gamma_{0}-2\kappa_{12}\right)\longrightarrow\ket{-}=\frac{\ket{2}-\ket{3}}{\sqrt{2}}\,,$
$\displaystyle\omega_{4}$
$\displaystyle=2\Delta-i\Gamma_{0}\longrightarrow\ket{4}\,,$ (35)

where the real and imaginary parts of $\omega_{i}$ are the frequency of the
normal mode and its corresponding decoherence rate, respectively.

As can be seen, the interaction removes the degeneracy of the two entangled
states $\ket{\pm}$, splitting them to $\Delta\mp\epsilon_{12}$ decaying at
$\Gamma_{0}\pm 2\kappa_{12}$, respectively. Typically, the mode with the
faster (slower) decay rate compared to an isolated atom is called the super
(sub)-radiant state.

In general, $\mathcal{G}$ describes a non-local and retarded interaction
however by limiting oneself to the near-field only, i.e. $k_{0}R\ll 1$,
$\mathcal{G}$ simplifies to

$~{}\mathcal{G}\approx\frac{3\mathbf{r}\left(\mathbf{r}\cdot\mathbf{d}_{1}\right)-\mathbf{d}_{1}}{4\pi\epsilon_{0}R^{3}}\cdot\mathbf{d}_{2}\,.$
(36)

This limit is typically referred to as the _coherent_ interaction, where the
interaction has a dispersive-only contribution ($\epsilon_{12}\neq 0$) without
any dissipative part ($\kappa_{12}=0$).

It is straightforward to expand Green’s function for small inter-atomic
separations $k_{0}R$ to find

$~{}\mathbf{d}_{1}\cdot\textrm{Im(G)}\mathbf{d}_{2}\simeq\frac{k_{0}}{6\pi}\mathbf{d}_{1}\cdot\mathbf{d}_{2}-\frac{k_{0}^{3}R^{2}}{60\pi}\left[2\mathbf{d}_{1}\cdot\mathbf{d}_{2}-(\mathbf{d}_{1}\cdot\mathbf{r})(\mathbf{d}_{2}\cdot\mathbf{r})\right]\,,$
(37)

where the first term is responsible for the decay rate $\Gamma_{0}$, and the
second term accounts for a finite propagation time between the two emitters.

From the near-field form of Eq. (36) it is evident that in an atomic ensemble,
the strongest interaction arises from two dominant arrangements of the atoms
being arranged either side-by-side (blue-colored spheres) or head-to-tail
(red-colored spheres). In the first case, $\mathbf{r}\cdot\mathbf{d}_{1}=0$,
the interaction energy is positive and hence repulsive which leads to an
increase in the energy and a blue-shift of the two-atom spectrum. On the other
hand, for the head-to-tail configuration, the interaction energy is negative,
meaning an attractive potential and hence a red-shifted two-atom spectrum.
Besides, as the near-field interaction scales as $1/R^{3}$ (cf. Eq. (36)), the
energy shift $\epsilon_{12}$ becomes proportional to the ensemble density.

Figure 1: Free-space dipole-dipole interaction of two polarized atoms. (a) The
DDI-induced energy shift, vs. the distance between two emitters $r$ in units
of the reduced wavelength $\bar{\lambda}$. (b) The DDI-induced broadening, for
the same dipole orientations as in (a). (c) The energy of the two-atom case
for two different orientations. In each case, configurations are depicted with
arrows and higher levels indicate larger energies.

Beyond the near-field regime, however, Green’s function oscillates with
distance $R$ as a result of the retardation, hence the DDI-induced energy
shift and the decay rates become strongly space-dependent. Figure 1(a),(b)
shows the real and imaginary parts of the light-induced interaction between
two atoms as a function of their separation, respectively. In each panel, the
side-by-side configuration case is shown in blue while the orange line shows
the behavior of the head-to-tail configuration. Figure 1(c) summarizes the
energies of the two interacting atoms when arranged head-to-tail or side-by-
side compared to the individual atom energy.

Figure 2(a)-(d) shows the excited state population of the two fixed atoms
$\braket{\hat{\sigma}_{ee}}$ vs. detuning at different normalized inter-atomic
spacing $r$, and four possible dipole orientations shown with black arrows.
The corresponding lineshift, shown in Fig. 2 (e)-(h), is extracted by fitting
a Lorentzian to each spectrum, and the resonance is plotted as a function of
the normalized inter-atomic spacing $(kr)^{-3}$. As one can see, the induced
lineshift scales linearly with the density (in 3D) due to the near-field
$R^{-3}$ scaling in Eq. (36).

Figure 2: DDI vs. detuning and inter-atomic separation. (a)-(d) Waterfall plot
of the excited state population $\braket{\hat{\sigma}_{ee}}$ of two stationary
atoms as a function of probe detuning $\Delta$ at different inter-atomic
spacings. (e)-(h) The lineshift $s$ and (i)-(l) linewidth $\Gamma^{\prime}$ of
the spectrum vs. normalized distance $(kr)^{-3}$ extracted from the
corresponding spectrum in (a)-(d). All plots are color-coded with the colormap
in the top left corner of the panel (a) that shows the inter-atomic distance
$r$ in units of the reduced wavelength $\bar{\lambda}$. The rows are separated
in head-to-tail (a,e,i), parallel side-by-side (b,f,j), anti-parallel side-by-
side (c,g,k), and anti-parallel head-to-tail (d,h,l) configurations as
depicted with arrows in each row at the right edge. Dashed lines in (e)-(h)
mark simplified linear predictions $\propto r^{-3}$ for the lineshift. The
driving field strength is $\Omega=0.01\Gamma$ within the weak-probe limit.

As discussed before, an attractive interaction leads to a negative shift and a
repulsive interaction to a positive one. Further, it is straightforward to
check that for parallel spins, i.e. $\mathbf{d}_{1}\cdot\mathbf{d}_{2}\geq 0$,
the dipole-dipole interaction in the side-by-side arrangement increases the
energy by $3/4\Gamma_{0}$, while the head-to-tail configuration reduces the
energy by $3/2\Gamma_{0}$. The estimated lineshift from the near-field
approximation is plotted as a dashed line in Fig. 2(e)-(h), which is close to
the exact solution obtained from the full Green’s function, shown in circles.
In both cases, the attractive shift of the head-to-tail configuration is twice
as strong as the repulsive shift of the side-by-side one. This simple near-
field interaction picture will be useful when we investigate dipolar
interactions in different dimensions in the next section.

Figure 2 (i)-(l) illustrates the dissipative effect of the dipole-dipole
interaction in modifying the decay rates of the collective states. For both
optically-active cases in Fig. 2(i) and (j), the effective decay rate of the
entangled states $\Gamma^{\prime}$ is larger than the free space lifetime,
hence it is a super-radiant mode. On the other hand, for the optically dark
states shown in Fig. 2(k) and (l), the effective decay rates of the entangled
states are smaller than the isolated atom implying a longer-lived and hence a
sub-radiant state.

###  3.2 Dipole-dipole interaction in a thermal cloud in lower dimensions

To model the experiments involving thermal atoms subject to motional dephasing
and the Doppler effect, as well as studies on cooperative effects in solid-
state systems with spectral wandering, a detailed investigation of the
inhomogeneous effects is required. In such cases, it is no longer sufficient
to consider the fixed atoms as done so far. Instead, one has to take into
account the random velocities and trajectories of individual atoms as well as
a time- varying mutual interaction. The resulting Doppler and transit time
effects play crucial roles in the optical properties of the ensemble and in
modifying the static picture. To that end, we developed a Monte-Carlo
algorithm to evolve the density matrix of the atomic ensemble in time, using
the quantum master equation derived in Eq. (11). We take into account the
atoms in the ensemble with random velocities and trajectories that can be
updated at each time step to fully capture the motional effects such as
Doppler and finite-time atom-field interactions.

Figure 3: DDI between two flying atoms. Dynamic dipole-dipole interaction of
two flying atoms. The difference in the excited state population
$\Delta\sigma$ between two interacting atoms and two non-interacting ones is
plotted as a function of time-of-flight. The orange and purple curves
correspond to individual atoms’ quantities. Four different probe laser (red
arrow) configurations are considered: (a) counter-propagating atoms with
(anti-) parallel probe, (b) counter-propagating atoms and orthogonal probe,
(c) co-propagating atoms with a parallel probe, and (d) co-propagating atoms
with an anti-parallel probe. The laser is linearly polarized with an out-of-
plane field. The absolute velocities of all atoms are equal in each scenario.
The driving field is weak $\Omega=0.01\Gamma$ and the probe is resonant, i.e.
$\Delta=0$.

Figure 3 shows some important possible dipolar collisions with moving atoms in
a typical thermal cloud. In all panels, we plot the difference in the excited
state population between the interacting atoms
$\braket{\hat{\sigma}_{ee}^{\textrm{int}}}$ and the non-interacting case
$\braket{\hat{\sigma}_{ee}^{\textrm{0}}}$ with
$\Delta\sigma=\braket{\hat{\sigma}_{ee}^{\textrm{int}}}-\braket{\hat{\sigma}_{ee}^{0}}$.
This way, we isolate the dynamics of the atomic states in the driving field
from the interaction effects.

In Fig. 3 (a), two atoms start at $t=0$ at a distance $R$ and fly towards each
other with identical velocities $v_{1}=-v_{2}$. The atoms are shown as orange
and purple circles, while the direction of motion of the atoms and the laser
direction are depicted as black and red arrows, respectively. At $t\approx
2.2~{}\mu s$, the atoms meet, an event that leads to a change in
$\Delta\sigma$ as a result of the interaction. Further, as the dipolar forces
are weak, they have negligible effects on the velocity of the fast-moving
thermal atoms and hence the motion of atoms can be treated classically. The
resulting oscillations of the excited state population quickly vanish as the
distance between both atoms increases after the collision. Note that, the
evolution is not identical for both atoms as they experience different Doppler
shifts concerning the probe laser.

When the probe direction is perpendicular to the atoms’ trajectories as shown
in Fig. 3(b), the scenario for both atoms is symmetric, and hence both atoms
evolve equally due to the driving laser as well as the mutual dipolar
interactions. Unlike case (a) here, as soon as both atoms interact, they
become detuned to the probe laser which leads to an overall smaller effect.

In Fig. 3 (c) both atoms move in the side-by-side configuration and start at a
distance of $R=\lambda/20$ with the same velocity $v_{1}=v_{2}$. Due to their
initial configuration, they interact from the start $t=0$ and evolve towards a
new steady state based on their identical detuning relative to the probe. As
both atoms are in the near-field of one another at all times, they experience
a strong repulsive interaction and hence a dipolar-induced blueshift. However,
due to the relative motion of the atoms and the laser propagation, the driving
field is Doppler shifted to lower energies, i.e. red-shifted. As a result,
both atoms are shifted out of resonance which reduces their interaction. This
is observable as a reduced oscillation. As their interaction decreases, the
interaction-induced blueshift is reduced as well and hence their detuning
decreases which consequently leads to their stronger polarization. This
behavior is noticeable as a revived population in Fig. 3(c) at $t\approx
0.35$. A similar scenario is depicted in Fig. 3(d), however, as both atoms
travel toward the laser, the blue-detuned laser will be in resonance with the
blueshifted atoms. Due to this resonant driving field and hence increased
polarization, both atoms interact strongly with one another as can be seen in
the amplitude of the oscillations compared to Fig. 3(c). All other
interactions in an ensemble resulting from arbitrary trajectories of two atoms
can be thought of in these four cases summarized in Fig. 3 with the only
differences being the randomized orientations of the atoms and a larger
coordination number.

Figure 4: Tunability of DDI with the ensemble dimensionality. Continuous
transition of dipolar interactions from 2D to 1D without Purcell enhancement.
(a) Waterfall plot of the excited state spectrum for different aspect ratios
between transversal and longitudinal dimensions. The ratio
$L_{z}/L_{x}=L_{z}/L_{y}$ increases from bottom to top. Dashed lines show
Voigt fits into the spectrum. (b) Lineshift and linewidth extracted from the
Voigt fit as a function of transverse to longitudinal aspect ratio. Region I
and Region II mark the regime where attractive and repulsive interactions
dominate the spectrum, respectively. The density is kept constant for each
aspect ratio. The transversal dimensions are $L_{x}=L_{y}=0.3\lambda$. (c)
Lineshift as a function of density for a 2D (I) and a 1D (II) probe volume
referring to the markers in (b). Points represent simulated lineshifts, dashed
lines are linear fits and solid lines show the expected lineshift for a simple
picture of a static dipole ensemble in the near field with the associated
dipole orientation. The driving field strength is $\Omega=0.01\Gamma$.

The previous results imply that the effect of dipolar interactions can be
modified noticeably by changing the atomic orientations or the coordination
numbers, so it is interesting to explore if the well-known cooperative effects
can be modified via the ensemble _geometry or dimensionality_. From the two-
atom discussions, it is evident that the ensemble can be confined such that
certain dipole configurations are favored, e.g. the head-to-tail configuration
dominates the ensemble when confined to 2D. To study this more systematically,
we start with a 2D ensemble with $L_{x}=L_{y}=0.3\lambda$ and
$L_{z}=0.05\lambda$ and continuously increase the volume by increasing $L_{z}$
such that the initial 2D geometry is changed to a pseudo-1D rectangular
”tube”.

The corresponding excited state population vs. detuning for various lengths
and the energy shift extracted from these spectra as a function of the tube
length transitioning from 2D to 1D is depicted in Fig. 4(a) and (b),
respectively. Note that the linewidth is a result of the Doppler and transit
time broadening, due to the small transversal dimensions $L_{x,y}$, as well as
the frequent boundary collisions. The Doppler detuning emerges from the
velocity component along the z-axis (parallel to the laser) which did not
exist in the pure 2D case. The Doppler broadening continuously vanishes with
decreasing $L_{z}$. The atoms with a large $v_{z}$ along the laser immediately
collide with the wall and only scatter a small amount of the probe light,
while atoms with small $v_{z}$ interact longer with the probe field and hence
contribute to the signal more. This gradual contribution of the dephasing with
increasing $L_{z}$ leads to the broadening depicted in Fig. 4(b). For
$L_{z}\geq\lambda$ the broadening remains almost unchanged as the volume is
large enough to fully resolve the dephasing along the $z$-axis and an
equilibrium condition for the transversal collisions, along the $x$\- and
$y$-axis, is reached. The lineshift depicted in panel (b) is extracted from a
Voigt fit to the simulated data in Fig. 4(a), shown as dashed lines in the
same panel.

One can see from the orange line in Fig. 4(b) that the interaction varies from
an attractive to a repulsive corresponding to red and blue shifts, for
$L_{z}\leq L_{x}$ and $L_{z}\geq L_{x}$, respectively. The former is a
quasi-2D scenario where the head-to-tail configurations of the dipoles are
dominant which leads to a redshift of the averaged spectrum. In contrast, the
latter corresponds to an elongated pseudo-1D cloud and accommodates more side-
by-side atom pairs with repulsive interactions which leads to a blueshift,
ultimately. At $L_{x}=L_{y}=L_{z}$, denoted by a vertical dashed line in Fig.
4(b), the near-field dipole interactions from various arrangements cancel out
each other and the collective shift vanishes. Hence, a rearrangement of
dipoles using different confinements can switch the collective behavior from
attractive to repulsive without manipulating each atom. Note that both, red-
shift as well as blue-shift, saturate at $L_{z}/L_{x}\leq 0.1$ and
$L_{x}/L_{z}\geq 10$, respectively, implying that at least an aspect ratio of
$\approx 10$ between transversal and longitudinal confinement is required to
render the system to 2D or 1D.

Figure 4(c) and (d) show the lineshift for different densities at simulation
conditions marked with ”I” and ”II” in panel (b). Both shifts scale linearly
with the normalized density $(k_{0}R)^{-3}$ as shown with linear fits (dashed
line) to the simulated data (points). However, when compared to the near-field
approximations (solid lines in Fig. 4(c),(d)), the collective effect is
noticeably weaker. This can be attributed to the perturbed repulsive
interactions resulting in a blueshift in (II) by attractive interactions as
the transversal dimensions are still finite such that occasionally atom pairs
can be found in the head-to-tail configuration.

After averaging, the excited state population over all atoms and time steps
the effect of confinements along the $x$\- and $y$-axis are noticeable as a
reduction of the expected blueshift in a 1D ensemble. The same is true for the
attractive interactions in (I). Overall, Fig. 4 shows that moving atoms
confined in different free-space geometries do not behave similarly to the
simple static two-atom cases explained in the previous section and one cannot
extend the simple near-field interpretation as described with Eq. (36) to the
free-space ensemble behavior. Also, the aspect ratio regarding transversal and
longitudinal components plays a crucial role in the energy shift of the
spectrum.

##  4 Experiments

Although collective light shifts caused by light-induced dipole-dipole
interaction were predicted 50 years ago in a seminal paper by R. Friedberg et
al. [25], it took a long time until the first experimental evidence of such a
shift was provided. The actual shift depends on various parameters including
the shape of the sample, the density, the level structures, the type of
quantum emitters, and whether hot or ultracold atoms are used. As a side
remark, the dipole-dipole shift is often termed the cooperative Lamb shift.
However, this effect is not related to the fluctuations of the electromagnetic
field, i.e. the origin of the Lamb shift derives from QED (cf. A). Although R.
Friedberg et al., who initially coined this term, revisited their wording
later [52], the term ’Lamb shift’ is still meaningful in a historical context
[53].

The first attempt to measure the collective response was conducted in an
optically-excited thermal xenon gas in 1990 [54]. There, a three-photon
excitation scheme in a macroscopic cell filled with xenon at high densities
and large laser beams compared to wavelengths were used. The excitation scheme
was chosen to avoid radiation trapping and incoherent multiple scattering,
which can also lead to line broadenings and shifts [55, 56].

Further, laser beams were aligned at multiple angles to create 3D excitation
volumes. Predominantly, the Lorentz-Lorenz shift, _aka_ the Clausius–Mossotti
shift, is anticipated to be the primary contributor to the observed line
shift, typically manifesting as a redshift. Additionally, there are minor
corrections attributable to collective effects, which are highly sensitive to
the geometry of the excitation volume. However, these corrections are presumed
to be minimal in a 3D sample, as the net effect of all dipole-dipole
interactions (DDIs) is expected to average out to zero. Nonetheless, the
obtained data shows an overall blueshift for most of the configurations.
Compared to later experiments carried out in rubidium vapors as will be
discussed in what follows, the shifts in xenon are roughly an order of
magnitude larger.

In [57], the authors have analyzed the shifts in various geometrical
arrangements, including 3D configurations, identifying three key components of
the lineshift: 1) the Coulomb term, 2) the cooperative Lamb shift (CLS), and
3) the resonant collision shift.

For larger volumes, the Coulomb term also referred to as the Lorentz-Lorenz
shift, is quantified as
$\Delta_{\textrm{LL}}=-\frac{Nd^{2}}{3\epsilon_{0}\hbar}$, where $N$
represents the ensemble density and $d$ the transition dipole moment. The CLS
conceptualized as a virtual photon exchange between two atoms, introduces a
correction in 3D, expressed as
$\Delta_{\textrm{CLS}}=\frac{\Delta_{\textrm{LL}}}{4}$. The final component,
the resonant collision shift, arises from transient correlations in dipole
orientations during close encounters between atoms, involving a permanent
exchange of excitation.

From the available information, it is hard to delimit the effect of multiple
scattering. Even if such parasitic effects can be neglected, other effects
such as excitation above saturation, transient interaction, Doppler selection,
optical pumping, polarization, and laser linewidth may have influenced the
experimental outcomes. Therefore, a re-investigation of these measurements in
a more controlled experiment using today’s technology and modern narrow-band
CW lasers will be beneficial.

It is worth emphasizing that, our approach and the numerical results
incorporate all these factors, with the limitation that correlations beyond
two-body interactions were not considered. Our simulations also indicate a
shift towards 3D characteristics (cf. Fig. 4). However, due to the limited
size of the systems that we can simulate, it remains unclear how significant
the impact of finite-size effects is in this context.

In another experiment, X-rays were used to excite a solid iron sample inside a
low-Q cavity [58]. They use a 14.4 keV excitation source, i.e. a wavelength of
86 pm, which is smaller than the iron lattice constant of 287 pm. The
advantage of this experiment is the absence of atom-atom interactions, which
simplifies the description. The collective response of the system was studied
in terms of Dicke superradiance accompanied by a collective shift [59] and
measured in the spectra of the superradiant burst. Further, the fluorescence
lineshift showed a scaling with the number of iron atoms. This is not the
typical density scaling as the density of the iron samples is constant. The
larger number of atoms in a bigger iron sample $N$ increased the number of
atoms participating in a Dicke state, leading to an $N$-dependent lineshift
and the superradiant time scale. The observed shifts are about 5 orders of
magnitude smaller than those observed in rubidium (when compared via particle
density), but of course, the x-ray wavelength is about 4 orders of magnitude
smaller. This experiment is also hard to compare to Friedberg’s original
proposal [25] since the collective effects are not due to light-induced DDIs.
The distance between the quantum emitters is larger than the involved
wavelength, a situation closer to the canonical Dicke model. Besides this, the
experiment has been carried out in a cavity and in a transient regime, both of
which are not present in the original Friedberg paper, either. Nonetheless,
the theoretical model applied by the authors describes the observations rather
well.

As geometry plays a crucial role in modifying the collective effects, it is
desirable to obtain good control over the actual arrangement of the quantum
emitters and the excitation volume. The emitters should be as simple as
possible, ideally, two-level systems, to simplify the description of the
excitation dynamics and to reduce the complexity of any theoretical or
numerical simulation. Alkali atoms with their lowest lying S to P transitions
in the VIS/NIR domain are ideal candidates for this. The strength of the
light-induced dipole-dipole interaction becomes significant when the distance
between the atoms gets smaller than the involved wavelengths. This can easily
be achieved with thermal gases of alkali atoms, _e.g._ potassium, rubidium,
and cesium, by heating the metal to 500-700 K. The drawback of such dense
gases is their large optical depth and hence the very low signal-to-noise
ratio. To study light-induced dipole- dipole interactions it is necessary to
stay below the saturation intensity, i.e. a few $mW/cm^{2}$. With densities
similar to a BEC (e.g. at around 150 Celsius in a rubidium vapor) one obtains
optical depths above 1 at a $1\mu m$ thickness [60]. The solution is to use
vapor cells with an extremely narrow spacing, ideally with a wedge-shaped
spacing to adjust the thickness of the sample to thicknesses below $1~{}\mu
m$.

With such wedged cells, two experiments have been performed in rubidium [61]
and potassium [62]. The atoms have been excited in cell lengths ranging from
$(10~{}nm-\mu m)$ with laser beam diameters larger than the cell thickness and
the excitation wavelength. The result is a 2D pancake-shaped atomic cloud
where the expected cooperative collective shift is [25]

$\Delta_{\textrm{CLS}}=\Delta_{\textrm{LL}}-\frac{3}{4}\Delta_{\textrm{LL}}\left(1-\frac{\sin
2kL}{2kL}\right)\,,$ (38)

where $L$ is the length of the vapor cell.

For a very short vapor slab, the system is effectively two-dimensional. For
high enough densities, the interaction is dominated by the non-retarded light-
induced DDI in a plane, where the polarization lies in the plane of the atoms,
and the attractive head-to-tail contribution of the dipole-dipole interaction
dominates over the repulsive side-by-side contribution resulting in a net red-
shift (cf. Sec. 3 for detailed discussions). When extending the sample to
$L=\lambda/4$, the retardation of the propagating laser leads to a $\pi$ phase
shift. The dipoles are now oriented in the opposite direction and the
interaction results in a blueshift. Therefore, one expects an oscillation of
$\Delta_{\textrm{CLS}}$ as a function of the cell thickness which is
consistent with the experimental observations. Further, the theory developed
in [25] matches well with the experimental data although it does not consider
the full quantum correlations between the atoms. A remaining oscillation of
the data as a function of cell length can be explained by the effective
refractive index of the atomic slab itself. In addition, there is a cavity
effect induced by the cell walls as theoretically described in [31].

To avoid the undesired cavity effect of the wedged cells, one could employ the
Light-Induced Atomic Desorption (LIAD) effect to desorb atoms from the cell
surface via a pulsed laser. Using a short pulse, compared to the motion of the
released atoms, for a very short time a 2D ensemble of atoms can be created.
At high laser intensities, the density of this 2D sample can be large and the
light-induced DDI effects can be studied. This has been recently observed in
[2], where the scaling with the transition strength has been studied, by
confirming the quadratic scaling of the lineshift between the D1 (795 nm) and
D2 (780 nm) transitions in rubidium.

Another potential candidate to measure collective lamb shifts in a 2D geometry
is based on the total reflection of light from the glass wall of a vapor cell,
where a very short-range evanescent field can be established inside the atomic
vapor. This technique was used in a pioneering experiment in 1991 to observe
the Lorentz-Lorenz shift in potassium vapor [63]. However, a clear signature
of cooperative Lamb shifts has not been reported in similar experiments, so
far.

Finally, there exists one experimental result on a cooperative line shift in a
1D atomic ensemble [13]. The geometry, in this case, is set by the light mode
confined in a slot waveguide. Since the slot is only 50 nm which is much
smaller compared to the 1529 nm, the wavelength of the 5P-4D transition in
rubidium used in that work, the atoms entering the light mode inside the
waveguide are effectively arranged in one dimension. The polarization is
perpendicular to the atomic chain, and hence the induced dipoles are parallel
to each other. For close-by atoms, the dipole-dipole interaction is repulsive
and the line shift goes predominantly to the blue. An interesting aspect of
having an optical structure close to the atoms is an enhanced Purcell factor
favoring scattering into the waveguide mode [64]. This not only enhances the
DDI along the waveguide but also changes the 1/r3 scaling in free space to an
all-to-all interaction. To describe the observed line shift, the states of the
atoms in a non-uniform field were determined using Bloch equations while the
full motion of the atoms was included. In addition, the Casimir-Polder shifts
of the atomic resonances due to the waveguide as well as a pairwise light-
induced DDI have been included. This method, as detailed in Sec. 3, led to an
excellent agreement with the experimental data [13].

Over the last 33 years, there have been so far only 5 experimental results on
the light-induced cooperative phenomenon in atoms. The sixth result is the
thin iron foil experiment in a low-Q cavity, as described above. In thermal
gases, the short coherence time allows a simplified theoretical treatment that
neglects all higher-order quantum correlations, except the pair correlations
between two atoms. For ultracold atoms, however, the behavior cannot be fully
captured unless higher-order correlations are included.

First experiments have been carried out in one-dimensional systems [65, 66],
but more experiments are on the way. The potential to arrange ultracold atoms
in optical tweezers in arbitrary configurations has inspired a lot of
theoretical studies during the last five years. One candidate to implement
many of these ideas relies on the low-lying clock transitions in the alkaline
earth elements [67]. They are especially suited to study coherent collective
and cooperative effects in ultracold atoms due to their long wavelength and
the possibility of their side-band cooling to the ground state inside the
tweezers.

In parallel, there are attempts to use ensembles of solid-state quantum
emitters to study collective effects [68, 69]. Solid-state emitters have
several advantages compared to atoms as they are fixed in space, they can be
closely packed to high densities, their excitation and emission wavelength are
flexible, and they can be easily integrated with nano-photonic structures.
Their main drawback is the dispersion of absorption/emission wavelengths among
_identical emitters_ , which makes it hard to observe collective effects.
Also, the coupling to the environment, e.g. phonons, spin noise, and charge
noise, leads to additional dephasing and shorter coherence times. We are not
aware of any experiment observing light-induced interactions with many solid-
state emitters thus far, except for a work observing a large collective Lamb
shift of two coupled superconducting “atoms” [70]. The key is the tunability
of the emitters by external electric fields or via local strain of the
supporting solid-state matrix. This has been achieved for optically active
molecules frozen onto a surface at liquid helium temperature. Here two
molecules were brought into resonance by electric field tuning [71]. Also,
three strain-tuned quantum dots coupled via a photonic waveguide have been
enabled to superradiate collectively [72]. How these methods can lead to a
scalable approach has to be investigated in the future.

What are the open questions an experiment can address? There is still only
limited knowledge regarding the influence of geometry and dimensionality on
the collective line shifts. Also, absorption and fluorescence have different
properties, and the connection between super-/subradiance with the various
sources of line shifts is not entirely understood. The role of coherence and
quantum correlations is not very significant in thermal gases, but with longer
coherence times in ultracold atoms, and ensembles of fixed solid-state
emitters they become more relevant. Finally, one has to distinguish between
the steady-state features and the transient responses. Both regimes are
relevant to exploit dense ensembles of quantum emitters for quantum
technologies, e.g. quantum sensors, single-photon storage, or light-driven
quantum computers and simulators.

##  5 Conclusion

Our theoretical framework for examining thermal vapors within macroscopic
environments marks an initial step towards exploiting the considerable
adjustability afforded by this platform. However, it raises several pressing
queries that require further exploration.

Firstly, there is a need to extend our coupled dipole model to accommodate
large atomic densities analogous to those encountered in experimental setups.
Moreover, it is imperative to include the temporal dynamics, accounting for
atomic motion. This temporal consideration becomes pivotal in addressing
nonlocal effects, variations in Doppler shifts, atomic collisions, and other
dynamic phenomena. However, the principal impediment to this endeavor is the
substantial numerical complexity involved in seeking exact solutions to the
coupled dipole model. Recent advances, exemplified by the renormalization
group approach, offer auspicious avenues to approximate solutions and
potentially alleviate the computational demands [73]. This becomes
particularly relevant if avenues for deriving (semi-)analytical solutions can
be explored. An initial step entails investigating the feasibility of
extending this approach to encompass the precise three-dimensional Green’s
tensor associated with macroscopic geometries and stochastically include the
Doppler effects.

Secondly, our developed coupled-dipole model, originally applied to thermal,
exhibits potential for broader applicability. It can be extended to scrutinize
atom-atom interactions across a wider spectrum of macroscopic geometries.
While extensive investigations into atom-atom interactions have occurred in
one- and two-dimensional photonic crystals, the interaction between hot or
cold atoms and diverse structures, such as microresonators, nanofibers,
hollow-core fibers, superconducting chips, and atomic cladding waveguides,
remain unexplored.

Lastly, it is important to extend the analysis to Rydberg atoms. In numerous
experiments, atom-atom interactions were of minimal concern due to low atomic
densities. However, this scenario undergoes a transformation when atoms are
excited to higher energy levels, particularly in Rydberg states. Transitions
between Rydberg levels feature significantly smaller wavenumbers,
necessitating fewer atoms to reach the critical density of $1/k^{3}$.
Consequently, the coupled-dipole model needs to be extended in two critical
aspects. Firstly, it should account for coherences between multiple atomic
levels, given that Rydberg atoms are typically produced through a two-step
excitation process. Secondly, the treatment of atom-wall and atom-atom
interactions should adopt a self-consistent approach. Rydberg atoms are
characterized by potent interactions and minute energy gaps between adjacent
states, suggesting that the impact of interactions may no longer be
perturbatively small. While many existing studies have addressed these aspects
separately, a unified model capable of addressing dense thermal vapors within
arbitrary geometries remains elusive.

Ultimately, once we have comprehensively described all pertinent aspects of a
system, the macroscopic quantum electrodynamics (mQED) formalism can showcase
its full potential. A macroscopic environment, represented by the Green’s
tensor, can be tailored to induce specific system behaviors. Over the long
term, this inverse design approach holds the potential to enhance existing
applications, including vapor-based single-photon sources and quantum
memories, while also opening the door to entirely novel yet-to-come
applications.

## Acknowledgments

The authors would like to acknowledge stimulating and insightful discussions
with H. Dobbertin and C. S. Adams. HA acknowledges financial support from the
Eliteprogramm of Baden-Württemberg Stiftung, DFG (GRK2642 ”Photonic Quantum
Engineers”) through the Mercator Fellowship, the Purdue University Startup
fund, the financial support from the Industry-University Cooperative Research
Center Program at the US National Science Foundation under Grant No. 2224960,
and the Air Force Office of Scientific Research under award number
FA9550-23-1-0489. We gratefully acknowledge Deutsche Forschungsgemeinschaft
(DFG) through the Priority Programme 1929 ’Giant Interaction in Rydberg
systems (GiRyd)’.

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##  Appendix A Casimir-Polder and Lamb Shift

In Eq. 2.2 of the main text we showed how one can obtain the well-known
interacting spin model after several assumptions about the relative
frequencies including the coherent drive frequency as well as the resonance
frequency of the emitters. In particular, when the two-level approximation is
valid the coupling of far-detuned transitions to the laser can be ignored in
the dipole-dipole interaction. However, the presence of these off-resonant
excitations and the photon exchange leads to additional collective corrections
in terms of a frequency shift and a broadening.

In this Appendix, we present the detailed expressions for the collective off-
resonant frequency shift, i.e. $\omega^{m^{\textrm{CP}}}$, and the collective
decay due to the off-resonant transitions, i.e. $\Gamma^{m}$. Following the
general expression of the atomic reduced density matrix and summing over all
other transitions, we will get a state-dependent frequency shift as

$\displaystyle~{}\omega_{A_{\nu\delta}}^{m^{\textrm{CP}}}=$
$\displaystyle-\frac{\mu_{0}\omega^{2}}{\hbar}\sum_{k,\kappa}\mathbf{d}_{A_{\nu\kappa}}^{mk}\cdot\\{[\Theta(\omega_{A_{,}mk})\left(n(\omega_{A,mk})+1)\right)-\Theta(\omega_{A,km})n(\omega_{A,km})]$
$\displaystyle\textrm{Re}(G(\mathbf{R}_{A},\mathbf{R}_{A},\omega_{A,mk}))-\frac{2k_{B}T}{\hbar}\sum_{j=0}^{\infty}\frac{\omega_{A,mk}G(\mathbf{R}_{A},\mathbf{R}_{A},i\xi_{j})}{\xi_{j}^{2}+\omega_{A,mk}^{2}}\\}\cdot\mathbf{d}_{A_{\kappa\delta}}^{km}\,,$

as well as an additional broadening as

$\displaystyle~{}\Gamma_{A_{\nu\delta}}^{m}=$
$\displaystyle\frac{2\mu_{0}\omega^{2}}{\hbar}\sum_{k,\kappa}\mathbf{d}_{A_{\nu\kappa}}^{mk}\cdot\\{[\Theta(\omega_{A_{,}mk})\left(n(\omega_{A,mk})+1)\right)+\Theta(\omega_{A,km})n(\omega_{A,km})]$
$\displaystyle\textrm{Im}(G(\mathbf{R}_{A},\mathbf{R}_{A},\omega_{A,mk}))\\}\cdot\mathbf{d}_{A_{\kappa\delta}}^{km}\,.$

Here, $\xi_{j}=\frac{2\pi k_{B}T}{\hbar}j$ denotes the Matsubara frequencies,
originating from the poles of the Bose-Einstein distribution on the imaginary
axis [15]. Remarkably, the finite temperature only changes the shift and
broadening through the density of the fluctuating photons, but does not
contribute to the resonant dipole-dipole interaction. As a limiting case, the
broadening contains the free-space contribution at zero temperature.
Furthermore, it includes the modifications of the local density of states due
to the presence of any objects through the corresponding Green’s function $G$.
This consequently leads to a modified decay rate as predicted by Purcell [50,
74].

To interpret the results of the collective shift and decay let us focus on the
energy levels $m,k$. The expression
$\mathbf{d}_{A_{nk}}\cdot\textrm{Re}(G(\mathbf{R}_{A},\mathbf{R}_{A},\omega_{A,mk})\cdot\mathbf{d}_{A_{nk}}$
describes the following phenomenon: atom $A$ undergoes the $m\rightarrow k$
transition using a photon with frequency $\omega_{A_{mk}}$ that travels from
the atoms to the environment and comes back to the atom where it is
reabsorbed. The transitions are either triggered by the stimulated emission
and absorption of a thermal photon ($n(\omega_{A_{mk}})$) or by the stimulated
emission due to vacuum fluctuations (term 1).

The free-space part of Green’s tensor formally gives an infinite result. But
it can be renormalized and then gives the Lamb shift [38], which is
complemented by an AC-Stark shift at the finite temperature. Such a shift in
the presence of some macroscopic objects and not in the free space is the so-
called Casimir-Polder shift. The free space Lamb shift is naturally
independent of the macroscopic environment and the position of the atoms.