# Collective Lamb Shift and Modified Linewidth of An Interacting Atomic Gas

Hanzhen Ma  Department of Physics, University of Connecticut, Storrs,
Connecticut 06269, USA  Department of Physics, Harvard University, Cambridge,
Massachusetts 02138, USA  Susanne F. Yelin  Department of Physics, Harvard
University, Cambridge, Massachusetts 02138, USA

###### Abstract

Finding a comprehensive and general description of the collective Lamb shift
and cooperative broadening in a radiatively interacting system is a long-
standing open question. Both energy levels and linewidth of individual atoms
are modified by the exchange of real and virtual photons making up the dipole-
dipole interaction. We introduce a method to theoretically study weakly-
driven, low-excited ensembles of two-level atoms, and obtain an analytic
description of the collective Lamb shift and linewidth via a self-consistent
formalism including infinite order of correlations which stem from only two-
body interactions. We predict the dependency of these quantities, as
measurables, on system parameters: the number density of the ensemble, the
detuning of an external probe field, and the geometry of the sample.

Ensembles of radiators can manifest collective effects as a consequence of
dipole-dipole interactions mediated by photons. These effects can alter the
radiative properties, including the buildup of collective modes called
superradiance and subradiance with enhanced or suppressed decay rates [1, 2,
3, 4, 5], as well as the shift of transition frequency [6]. The collective
Lamb shift [7, 8], which results from the exchange of virtual photons between
radiators, has attracted much attention in both theoretical discussions and
experimental studies [9, 10, 11]. In the case of single excitation, the shift
can be found by diagonalizing the equations of motion [12, 13, 14]. Meanwhile,
classical electrodynamics simulations are also used to study spatially
extended systems [15, 16, 17]. Experimental observations of collective Lamb
shift have been reported in solid state samples [18], atomic arrays [19] and
atomic gas [20, 21, 22, 23, 24, 25, 26]. In most discussions to date, the
studies of many-body radiative properties focus either on the buildup of
collective modes [27, 28, 29], or the effects of re-absorption and re-emission
of real and virtual photons. Through collective modes, people have established
a frequently employed pathway for studying super- and subradiance, while the
re-absorption and re-emission of real and virtual photons are commonly
utilized to study radiation trapping and vacuum effects. In this letter, we
focus on the perspective of virtual photon exchange, in order to study the
collective Lamb shift and the modification to linewidth in a dense sample. In
contrast to the usual Lamb shift and natural linewidth arising from the
interaction with the electromagnetic modes of the vacuum, in a dense sample
the changed linewidth and shift results from the virtual interaction of
electromagnetic modes with all the atoms. As the linewidth and the shift of a
collective system are connected via the Kramers-Kronig relations, they need to
be studied simultaneously.

A method often employed is the coupled-dipole model [13, 30], which explores
superradiance and corresponding shift by solving the collective eigenmodes. In
contrast, we want to study how the fundamental properties of individual atomic
transitions, such as the energy levels and linewidths, are modified in the
environment of many atoms with dipole-dipole interaction. For this, a more
comprehensive method to describe virtual photon exchange is needed. A
measurement protocol specific to collective Lamb shift should take into
account the integrated effect of an ensemble, ideally as the effect on a probe
field weak enough such that one can fully neglect excitations and light
intensity which are at least second order in the Rabi frequency. As a
comparison, many other treatments to date identify the collective Lamb shift
as the frequency shift of transmitted light through a sample. However, the
transmitted light measurement from an ensemble includes a combination of
light-intensity-dependent effects (superradiance and corresponding frequency
shift) and light-intensity-independent vacuum effects (spontaneous emission
and Lamb shift). Our method is different from others as we manage to
explicitly distinguish the two types of effects analytically, so the
collective Lamb shift we describe purely stems from vacuum fluctuation and is
irrelevant to the buildup of collective modes. Our pragmatic definition of
collective Lamb shift is, therefore, the limiting case of collective shift
with no excitation. The analytical framework of our method can be found in
Ref. [31], where the light-intensity-dependent and independent effects are
differentiated and then treated on an equal footing.

In this Letter, we develop a self-consistent relation that analytically
describes the collective Lamb shift and the linewidth of an interacting
ensemble. This relation is governed by the local number density of atoms and
the detuning of the external probe field. The scheme to derive such a relation
can be summarized as follows: First, a master equation for any arbitrary
choice of two probe atoms (Fig. 1 (a)) is derived under the Markov
approximation. The modified energy levels and linewidths of the probe atoms
can be obtained by the real and imaginary parts of a dressed Green’s function
that describes the emission and re-absorption of real and virtual photons
between probe and background atoms (Fig. 1 (b)). Second, the dressed Green’s
function is calculated via the Dyson equation [32], and is expressed in terms
of averaged two-point correlators of atomic operators. Third, by virtue of the
quantum regression theorem [33], the averaged two-point atomic correlators are
treated self-consistently and take into account multiple scattering of virtual
photons to infinite order. This leads, finally, to a closed form of the
modified linewidths and energy levels for a steady state. To predict
measurable effects in experiments, one needs to take into account the geometry
of the sample and average over the distribution of probe atoms. More details
of the mathematical framework to this treatment can be found in Refs. [34,
31].

Figure 1: (a) A pictorial demonstration of the atomic gas driven by an
external probe field. The reduced density matrix includes two arbitrarily
chosen probe-atoms, while all other atoms as well as the field are traced out.
(b) The emission and reabsorption of virtual photons result in the
modifications of Lamb shift and spontaneous linewidth.

We consider an ensemble of identical two-level atoms that interact with the
electric field, with the full Hamiltonian

$H=\sum_{i=1}^{N}\hbar\omega_{0}\sigma^{\dagger}_{i}\sigma_{i}+\sum_{\vec{k},\lambda}\hbar\omega_{l}(\vec{k})a^{\dagger}_{\vec{k},\lambda}a_{\vec{k},\lambda}-\sum_{i=1}^{N}\vec{p}_{i}\cdot(\vec{\cal
E}_{i}+\vec{E}_{i})$ (1)

where $\sigma^{\dagger}_{i}=\left|e_{i}\right\rangle\\!\\!\left\langle
g_{i}\right|$ and $\sigma_{i}=\left|g_{i}\right\rangle\\!\\!\left\langle
e_{i}\right|$ are the raising and lowering operators of the $i$-th atom,
$a^{\dagger}$ and $a$ are the creation and annihilation operators of the
photons. We assume that the atoms mostly couple to the field modes with
frequencies $\omega_{l}$ that are close to the atomic resonant frequency
$\omega_{0}$.
$\vec{p}_{i}=\hat{\epsilon}_{i}\wp(\sigma_{i}+\sigma_{i}^{\dagger})$ is the
dipole operator of the $i$-th atom with real dipole matrix element $\wp$.
$\vec{\cal E}_{i}$ is the external classical driving field, while
$\vec{E}_{i}$ is the quantized field operator, at the position of the $i$-th
atom. Here, we focus on the dynamics of two arbitrarily chosen probe atoms,
and formally trace out the quantized field and the remaining $N-2$ atoms [33,
35]. A Lindblad form master equation with rotating-wave approximation in the
rotating frame is obtained for the two probe-atoms [31]

$\displaystyle\dot{\rho}$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}[\tilde{H}_{0},\rho]+i\Omega\sum_{i=1,2}[\sigma_{i}+\sigma_{i}^{\dagger},\rho]-i\\!\sum_{i,j=1,2}\\!\delta_{ij}[\sigma^{\dagger}_{j}\sigma_{i},\rho]$
(2)
$\displaystyle-\sum_{i,j=1,2}\frac{\gamma_{ij}}{2}(\sigma_{j}^{\dagger}\sigma_{i}\rho-2\sigma_{i}\rho\sigma_{j}^{\dagger}+\rho\sigma_{j}^{\dagger}\sigma_{i})$

where
$\tilde{H}_{0}=\hbar(\omega_{0}-\omega_{c})\sum_{i}\sigma^{\dagger}_{i}\sigma_{i}=\hbar\Delta_{0}\sum_{i}\sigma^{\dagger}_{i}\sigma_{i}$
is the two-atom free Hamiltonian in the rotating frame, with a classical
driving frequency $\omega_{c}$. $\Omega=|\wp{\cal E}_{0}/\hbar|$ is the Rabi
frequency, with the driving amplitude ${\cal E}_{0}$. Note that a positive
$\Delta_{0}$ stands for a red-detuned driving field. The two sets of
parameters, $\gamma_{ij}$ and $\delta_{ij}$, describe the modified spontaneous
linewidth and the collective Lamb shift respectively.

It is commonly accepted that for atoms placed in free-space, $\gamma_{ij}$ and
$\delta_{ij}$ are associated with the free-space Green’s tensor that results
from the dipole-dipole interaction. However, this is no longer correct in a
dense system, since one must consider the effect of the medium, namely, the
multiple scattering of virtual photons between the particles (Fig. 1 (b)). We
define the dressed Green’s function in the medium as

$D_{ij}(\tau,t)=D(\vec{r}_{i},t+\tau,\vec{r}_{j},t)=\theta(\tau)\langle[E^{-}_{i}(t+\tau),E^{+}_{j}(t)]\rangle$
(3)

and its Fourier transform with respect to $\tau$,
$\tilde{D}_{ij}(\omega,t)=\int_{-\infty}^{+\infty}D_{ij}(\tau,t)e^{-i\omega\tau}d\tau$,
where $E^{\pm}_{i(j)}(t)$ is the positive (negative) component of the field
operator in the Heisenberg picture, and $\theta(\tau)$ is the Heaviside step
function. The modified spontaneous linewidth and the collective Lamb shift are
then written in terms of the dressed Green’s function [34, 31]

$\displaystyle\gamma_{ij}(t)=$
$\displaystyle\frac{\wp^{2}}{\hbar^{2}}\int_{-\infty}^{+\infty}\langle[E_{j}^{+}(t),E_{i}^{-}(t+\tau)]\rangle
e^{-i(\omega_{0}-\omega_{l})\tau}d\tau$ (4a) $\displaystyle\delta_{ij}(t)=$
$\displaystyle-\frac{i\wp^{2}}{2\hbar^{2}}\int_{0}^{+\infty}\langle[E_{j}^{+}(t+\tau),E_{i}^{-}(t)]\rangle
e^{i(\omega_{0}-\omega_{l})\tau}d\tau$
$\displaystyle+\frac{i\wp^{2}}{2\hbar^{2}}\int_{0}^{+\infty}\langle[E_{j}^{+}(t),E_{i}^{-}(t+\tau)]\rangle
e^{-i(\omega_{0}-\omega_{l})\tau}d\tau$ (4b)

thus $D_{ij}\\!\sim\\!\gamma_{ij}/2+i\delta_{ij}$. These expressions can be
formally derived by tracing out the $N-2$ background atoms and the quantized
field [36]. For two arbitrarily chosen probe atoms, Eq. (2) is effectively
averaged over all possible choices of atomic pairs, thus a permutational
symmetry is imposed such that $\gamma_{11}=\gamma_{22}$,
$\gamma_{12}=\gamma_{21}$, and similarly for $\delta_{ij}$. A Comprehensive
derivation of Eqs. (2) and (4) for a dense atomic gas can be found in Ref.
[31], where the authors also take into account the contribution to the
broadening and frequency shift which scales with the number of photons. Here,
we consider a low-excited gas and neglect those effects. One can reproduce the
free-space Green’s tensor by using the free-field operators in Eq. (3) [34],
resulting in a function of distance between two spatial points [37, 27, 28,
29]

$\displaystyle\tilde{D}^{(0)}_{\alpha\beta}(r,\omega)$ $\displaystyle=$
$\displaystyle-\frac{i\hbar e^{-i\omega
r/c}}{4\pi\epsilon_{0}r}\bigg{[}\delta_{\alpha\beta}\bigg{(}\frac{\omega^{2}}{c^{2}}-i\frac{\omega}{c}\frac{1}{r}-\frac{1}{r^{2}}\bigg{)}$
(5)
$\displaystyle+\frac{x_{\alpha}x_{\beta}}{r^{2}}\bigg{(}-\frac{\omega^{2}}{c^{2}}+3i\frac{\omega}{c}\frac{1}{r}+\frac{3}{r^{2}}\bigg{)}\bigg{]}$

and the scalar free-space Green’s function 111If one looks at different
polarizations, the Dyson equation here can be easily rewritten in the tensor
form. But here we assume two-level atoms with transitions of only one
polarization by averaging over random polarization and replacing
$x_{\alpha}x_{\beta}/r^{2}\\!\\!\to\\!\\!\delta_{\alpha\beta}/3$

$\tilde{D}^{(0)}(r,\omega)=-\frac{i\hbar\omega^{2}}{6\pi\epsilon_{0}c^{2}r}e^{-i\omega
r/c}$ (6)

In a dense sample, the dressed Green’s function $D_{ij}$ obeys a general form
of Dyson equation [32, 34]

$\displaystyle
D(\vec{r}_{i},t_{i},\vec{r}_{j},t_{j})=D^{(0)}(\vec{r}_{i},t_{i},\vec{r}_{j},t_{j})-\int\\!dt_{k}\\!\int\\!dt_{l}\\!\int\\!d^{3}\\!r_{k}$
$\displaystyle\times
D^{(0)}(\vec{r}_{i},t_{i},\vec{r}_{k},t_{k})P(\vec{r}_{k},t_{k},t_{l})D(\vec{r}_{k},t_{l},\vec{r}_{j},t_{j})$

which consists of the free-space term from Eq. (6) as well as all orders of
corrections that result from the multiple scattering of virtual photons.

The source function $P$ has the following form in a continuum approximation

$P(\vec{r}_{i},t_{1},t_{2})=\frac{\wp^{2}{\cal
N}}{\hbar^{2}}\theta(t_{1}-t_{2})\langle[\sigma^{\dagger}_{i}(t_{1}),\sigma_{i}(t_{2})]\rangle$
(8)

where ${\cal N}$ is the number density of the atoms. By iteration, the second
term in Eq. (Collective Lamb Shift and Modified Linewidth of An Interacting
Atomic Gas) includes a sum of all orders in $P$, describing multiple
scattering of virtual photons to all orders. The correlation function on the
right hand side of Eq. (8) can be further expressed in terms of the elements
of the density matrix by using the quantum regression theorem [36].

In Ref. [34, 31], we solve Eq. (Collective Lamb Shift and Modified Linewidth
of An Interacting Atomic Gas) in Fourier space for $\tilde{D}_{ij}$ for a gas
of randomly polarized radiators:

$\tilde{D}_{ij}(\omega,t)=-\frac{i\hbar\omega_{0}^{2}}{6\pi\epsilon_{0}c^{2}r}{\rm
exp}\bigg{(}-i\frac{\omega_{0}r}{c}\sqrt{1+\frac{2i\hbar}{3\epsilon_{0}}\tilde{P}(\omega,t)}\bigg{)}$
(9)

where $r=|\vec{r}_{i}-\vec{r}_{j}|$ is the distance between the radiators.
Here, the source function $\tilde{P}$ takes a spatial average and is
independent of the location. Note that by setting $\tilde{P}=0$, Eq. (9)
reduces to Eq. (6) in the randomly polarized condition, so the Green’s
function is indeed dressed by the source function $\tilde{P}$.

We further assume a weak-driving condition in the sample. Since the number of
excitations scales as $O(\Omega^{2})$ and we keep only up to $O(\Omega)$
terms, we will neglect all effects that depend on the number of excitations.
This approximation is valid, for example, in systems described in Ref. [21,
26]. We will approach the limit $\Omega\to 0$ in the expression of
$\tilde{P}$. The Fourier variable in Eq. (9) takes the value
$\omega=\omega_{0}-\omega_{l}\approx\delta_{11}$. The source function is then
simplified to [36]

$\tilde{P}(\omega\approx\delta_{11},t)=\frac{\wp^{2}}{\hbar^{2}}{\cal
N}\frac{-2}{\gamma_{11}-2i\Delta_{0}}$ (10)

which depends on the density of atoms ($\cal N$), the modified single-atom
linewidth ($\gamma_{11}$) and the detuning of the driving field
($\Delta_{0}$).

Combining Eq. (3) and Eqs. (4):

$\displaystyle\gamma_{ij}$ $\displaystyle=-2\frac{\wp^{2}}{\hbar^{2}}{\rm
Re}\big{(}\tilde{D}_{ij}\big{)}$ (11a) $\displaystyle\delta_{ij}$
$\displaystyle=\frac{\wp^{2}}{\hbar^{2}}{\rm Im}\big{(}\tilde{D}_{ij}\big{)}$
(11b)

For simplicity, we define the effective wave number $q_{0}$

$q_{0}=q_{0}^{\prime}+iq_{0}^{\prime\prime}=\frac{\omega_{0}}{c}\sqrt{1+\frac{2i\hbar}{3\epsilon_{0}}\tilde{P}}$
(12)

By taking the limit $r\to 0$, the single-atom spontaneous linewidth is
associated with the real part of the effective wave number $q_{0}$:

$\gamma_{ii}=\gamma_{ij}(r\to 0)=-2\frac{\wp^{2}}{\hbar^{2}}\lim_{r\to 0}{\rm
Re}\big{(}\tilde{D}_{ij}\big{)}=\gamma_{0}\frac{\lambda}{2\pi}q_{0}^{\prime}$
(13)

where $\lambda=2\pi c/\omega_{0}$ is the wavelength of the transition, and
$\gamma_{0}=\wp^{2}\omega^{3}_{0}/3\pi\hbar\epsilon_{0}c^{3}$ is the free-
space spontaneous linewidth. For the Lamb shift, we renormalize to the free-
space value:

$\displaystyle\delta_{ii}^{\prime}$ $\displaystyle=$
$\displaystyle\delta_{ij}(r\to 0)-\delta_{ij}^{(0)}(r\to 0)$ (14)
$\displaystyle=$ $\displaystyle\frac{\wp^{2}}{\hbar^{2}}\lim_{r\to 0}{\rm
Im}\big{(}\tilde{D}_{ij}-\tilde{D}^{(0)}_{ij}\big{)}=-\gamma_{0}\frac{\lambda}{4\pi}q_{0}^{\prime\prime}$

By substituting Eq. (13) and Eq. (14) to Eq. (12) and making use of the form
in Eq. (10), one can obtain a self-consistent relation for the single-atom
terms $\gamma_{ii}$ and $\delta_{ii}^{\prime}$

$1+\frac{2C}{2\frac{\Delta_{0}}{\gamma_{0}}+i\frac{\gamma_{ii}}{\gamma_{0}}}=\bigg{(}\frac{\gamma_{ii}}{\gamma_{0}}-i\frac{2\delta_{ii}^{\prime}}{\gamma_{0}}\bigg{)}^{2}$
(15)

where we have defined the cooperativity parameter $C=\frac{\lambda^{3}{\cal
N}}{4\pi^{2}}$ which is proportional to the number density of atoms. Eq. (15)
is the first main result of this paper. In an ensemble of interacting atoms,
both the spontaneous linewidth $\gamma_{ii}$ and the Lamb shift
$\delta_{ii}^{\prime}$ of an individual atom are modified due to the
collective nature of the dynamics. Under the weak-driving and low-excitation
conditions, both quantities together satisfy Eq. (15) which is governed by the
density of the radiators ($C$) and the detuning of the driving ($\Delta_{0}$).

Figure 2: (a) The single-atom spontaneous linewidth modified by the collective
effect, in units of the free-space linewidth $\gamma_{0}$. The value changes
with respect to the detuning of the external driving field ($\Delta_{0}$), and
the number density of emitters ($C$). (b) The single-atom Lamb shift modified
by the collective effect, as a function of detuning. (c) The single-atom
spontaneous linewidth as a function of particle density, with three different
detunings. (d) The single-atom Lamb shift as a function of particle density.

Fig. 2 shows the solution of Eq. (15), which has a modified Lorentzian
profile. By varying the detuning of the external driving field and the density
of particles, one can tune both the modified linewidth and the Lamb shift. At
low-density or in far-detuned regions, both quantities recover the free-space
values, namely, $\gamma_{11}\to\gamma_{0}$ and $\delta_{11}^{\prime}\to 0$. As
the particle density gets higher, the collective effect predominates and leads
to modification of both quantities. It is clear to see from Fig. 2(a) that a
red-detuned driving field leads to an enhancement of spontaneous emission,
while a blue-detuned driving field results in a suppression. A positive
$\delta_{11}^{\prime}$ stands for a red shift of radiated light in our
definition.

In Fig. 2(c,d) we plot the linewidth and the Lamb shift as functions of
particle density, with different external detunings. Here, the atomic density
$C=1$, for example, corresponds to $\sim\\!40$ particles contained in a cubed
wavelength, which is $\sim\\!8\times 10^{13}/cm^{3}$ for Rb 780 nm transition.
The minimum of $\gamma_{11}$ around $C=0.8$ corresponds to the valley at the
blue-detuned region in Fig. 2(a). In the low density region ($C<0.5$), the
Lamb shift increases linearly, while for high density, it exhibits sub-linear
behavior.

Figure 3: The two-atom terms of (a) modified linewidth and (b) Lamb shift as
functions of distance between the two particles ($r$) in units of transition
wavelength, with different particle density ($C$). In both calculations the
driving is on-resonant with $\Delta_{0}=0$.

The two-atom terms $\gamma_{12}$ and $\delta_{12}$ can be calculated from Eq.
(11) as functions of inter-atom distance, and are shown in Fig. 3. An
oscillatory behavior is shown, which results from the constructive/destructive
contributions from the paired atom. In the small $r$ limit, both quantities
reduce to the single-atom values, while in the large $r$ limit both vanish.

So far, we have been looking at the linewidth and energy shift for any single
atom. But what are the measurable effects for the whole atomic gas? After
choosing two probe atoms, both the dynamics and the steady states of the
system can be solved via Eq. (2), which depend on the positions of the probe
atoms. There is no obvious way to average over $\gamma_{ij}$ or $\delta_{ij}$,
but one can average over single-atom density matrices. Since the diagonal
elements of the density matrices are trivially 1 and 0 for the ground and
excited state populations in the weak excitation limit, the only important
quantity is the off-diagonal element $\langle e|\rho|g\rangle$ which is
related to the susceptibility. Starting from the ensemble average of the two-
atom density matrix over all possible choices of probe atom pairs, $\rho^{\rm
eff}=\frac{1}{n(n-1)}\sum_{\alpha}\rho^{(\alpha)}$ where $n$ is the total
particle number and $\alpha$ denotes choices of probe atoms, the effective
susceptibility of the whole ensemble is related to the averaged density matrix
by

$\displaystyle\chi$ $\displaystyle\propto$ $\displaystyle{\cal N}\frac{\langle
e_{1}|\rho^{\rm eff}|g_{1}\rangle+\langle e_{2}|\rho^{\rm
eff}|g_{2}\rangle}{2}$ (16) $\displaystyle=$
$\displaystyle\frac{1}{n(n-1)}\sum_{\alpha}{\cal N}\frac{\langle
e_{1}|\rho^{(\alpha)}|g_{1}\rangle+\langle
e_{2}|\rho^{(\alpha)}|g_{2}\rangle}{2}$

namely, one can first calculate the matrix elements $\langle
e_{i}|\rho^{(\alpha)}|g_{i}\rangle$ for each probe atom pair, and then average
over the ensemble.

For a weak driving, the $\Omega$ term in Eq. (2) can be treated as a
perturbation. Eq. (2) then leads to the steady-state equations of the two-atom
density matrix in the first order of $\Omega$, under a particular choice of
probe atom pair $\alpha$:

$\displaystyle 0\\!=$
$\displaystyle-\bigg{[}\frac{\gamma_{11}}{2}+i(\delta_{11}+\Delta_{0})\bigg{]}\rho_{eg}\\!+\\!\big{(}\frac{\gamma_{12}}{2}+i\delta_{12}\big{)}m_{eg}+i\Omega$
(17a) $\displaystyle 0\\!=$
$\displaystyle-\bigg{[}\frac{3}{2}\gamma_{11}\\!+\\!\gamma_{12}\\!+\\!i(\delta_{11}\\!+\\!\Delta_{0})\bigg{]}m_{eg}$
$\displaystyle-\\!\big{(}\gamma_{11}\\!+\\!\frac{\gamma_{12}}{2}\\!-\\!i\delta_{12}\big{)}\rho_{eg}-i\Omega$
(17b)

where
$\rho_{eg}=\frac{1}{2}(\langle\sigma_{1}\rangle+\langle\sigma_{2}\rangle)$ is
the average single-atom coherence, and $m_{eg}$ is defined as
$m_{eg}=\frac{1}{2}(\langle\sigma_{1}\sigma_{z2}\rangle+\langle\sigma_{z1}\sigma_{2}\rangle)$,
with $\sigma_{i}=\left|g_{i}\right\rangle\\!\\!\left\langle e_{i}\right|$ and
$\sigma_{zi}=\left|e_{i}\right\rangle\\!\\!\left\langle
e_{i}\right|-\left|g_{i}\right\rangle\\!\\!\left\langle g_{i}\right|$. The
solution to Eq. (17) is

$\displaystyle\rho_{eg}$
$\displaystyle=\frac{\Omega}{\delta_{11}+\delta_{12}+\Delta_{0}-i\frac{\gamma_{11}+\gamma_{12}}{2}}$
(18a) $\displaystyle m_{eg}$ $\displaystyle=-\rho_{eg}$ (18b)

By substituting Eq. (18a) to Eq. (16), one can calculate the effective
susceptibility of the atomic gas. Here, we employ a continuous approximation
and take the average via an integration over the sample.

$\displaystyle\rho^{\rm eff}_{eg}$ $\displaystyle=$
$\displaystyle\int\\!\\!\\!\\!\int
d^{3}r_{1}d^{3}r_{2}f(\vec{r}_{1})f(\vec{r}_{2})$ (19)
$\displaystyle\times\frac{\Omega}{\delta_{11}+\delta_{12}(r)+\Delta_{0}-i\frac{\gamma_{11}+\gamma_{12}(r)}{2}}$

where $f(\vec{r})$ is the particle distribution function and
$r=|\vec{r}_{1}-\vec{r}_{2}|$.

Furthermore, one can “reverse-engineer” the effective linewidth $\gamma^{\rm
eff}$ and frequency shift $\delta^{\rm eff}$ by the following relation:

$\rho^{\rm eff}_{eg}=\frac{\delta^{\rm eff}+i\frac{\gamma^{\rm
eff}}{2}}{(\delta^{\rm eff})^{2}+(\frac{\gamma^{\rm eff}}{2})^{2}}\Omega$ (20)

That is, after calculating the susceptibility via Eq. (19) for a given atomic
distribution $f(\vec{r})$, one can immediately separate its real and imaginary
parts and obtain the effective linewidth and frequency shift behind the
susceptibility. $\gamma^{\rm eff}$ and $\delta^{\rm eff}$ are the quantities
that can be directly measured in experiments.

Figure 4: (a) A pictorial demonstration of a spherical gas of radiators with
radius $R$. The propagation path of the probe light is drawn in red shadow.
(b) The effective linewidth of a spherical gas, as a function of the radius,
plotted with different particle densities. (c) The effective Lamb shift of a
spherical cloud. (d) A pictorial demonstration of a cylindrical gas of
radiators, with varying length $L$ and fixed radius $r=0.2\lambda$. (e) The
effective linewidth of a cylindrical gas, as a function of the length, plotted
with different particle densities. (f) The effective Lamb shift of a
cylindrical gas.

As an example, we consider a spherical gas of atoms with uniform distribution
$f(\vec{r})=\frac{1}{V}$ and radius $R$ (Fig. 4 (a)). Here, the first
integration in Eq. (19) is done over the whole sample, as each atom is
influenced by every other atoms. The second integration is done along the
propagation path of the probe light with a small waist compared to the
transverse size of the sample, as these atoms along the path have major
influence on the measured quantities. The result is shown in Fig. 4 (b,c),
where the effective, collective linewidth $\gamma^{\rm eff}$ and the
collective Lamb shift $\delta^{\rm eff}$ of the sample are plotted as
functions of the sphere’s radius, with different cooperativity parameter $C$.
By and large, both $\gamma^{\rm eff}$ and $\delta^{\rm eff}$ decrease with
larger radius of the atomic gas. As $R\to\infty$, both quantities
asymptotically approach the values of the single-atom terms $\gamma_{11}$ and
$\delta_{11}$, which are the homogeneous limits for an infinite size ensemble.
With sufficiently high particle density, the collective Lamb shift shows an
oscillatory behavior in small $R$. The minimum of $\delta^{\rm eff}$ (least
red shift) is found when the radius is near half of the transition wavelength.

Similarly, a numerical result for a cylinder-shaped gas is shown in Fig. 4
(e,f). This geometry resembles an elongated gas realized in recent experiments
such as Ref. [39, 40, 41]. The radius of the cylinder is fixed to
$0.2\lambda$, and the modified linewidth and Lamb shift are plotted as
functions of the length $L$. Both quantities also approach the homogeneous
limits as the bulk volume increases. The oscillatory behavior as the bulk
volume changes in both geometries is in qualitative agreement with some
experiments [21, 26], although they have slab-shaped gases instead. Exact
matching in values is not achieved, because the transmitted light measured in
experiments includes additional effects beyond vacuum fluctuation, such as the
etalon effect from the parallel surfaces, which plays an important role in the
measurement of light shift.

In conclusion, we have developed a theoretical framework to describe the
collective radiation of dense ensembles. In particular, a weakly driven, low-
excited condition is imposed, and the collective modification of Lamb shift
and spontaneous linewidth are studied both analytically and numerically. We
find that these quantities are connected via a dressed Green’s function that
describes the exchange and multiple scattering of virtual photons in the
medium. The dressed Green’s function, together with the master equation, lead
to a self-consistent relation of linewidth and Lamb shift that shows their
dependencies on the density of radiators and the detuning of the probe field.
The linewidth and collective Lamb shift have a modified Lorentzian profile
with respect to the detuning of the probe field, and the modification scales
with the particle density. We find both enhancement and suppression of the
decay rate, and an overall red-shift of the collective Lamb shift. We
numerically predict the effective linewidth and Lamb shift that can be
directly measured by experiments, and find oscillatory behavior as the size of
the sample changes. These effects are at the order of the vacuum linewidth
$\gamma_{0}$, and are already noticeable for a density low as $\sim\\!8\times
10^{13}/cm^{3}$ for Rb 780nm transition. Our work provides insights into the
collective effects of radiative systems and is instructive to future
experiments.

We thank Stefan Ostermann and Oriol Rubies-Bigorda for helpful discussions. We
would like to thank the NSF for funding through PHY-1912607 and PHY-2207972
and the AFOSR through FA9550-19-1- 0233.

## References

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##  I Supplemental Material

In this supplemental material, we provide the derivation of the two-atom
master equation for a driven dense atomic gas that is used in the main text.
The expressions of modified broadening terms $\gamma_{ij}$ and shift terms
$\delta_{ij}$ are directly obtained from the master equation. To calculate the
two-point correlators in the source function $P$, we make use of the quantum
regression theorem and solve for the correlators in the Fourier space.

###  I.1 I. Derivation of $\gamma_{ij}$ and $\delta_{ij}$

Here, We outline the derivation of the two-atom master equation and read off
the expressions of $\gamma_{ij}$ and $\delta_{ij}$. Details of the derivation
is given in Ref. [31]. The full Hamiltonian can be separated into two parts

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}\hbar\omega_{0}\sigma^{\dagger}_{i}\sigma_{i}+\hbar\omega_{l}a^{\dagger}a-\sum_{i\neq
1,2}\vec{p}_{i}\cdot(\vec{\cal
E}_{i}+\vec{E}_{i})-\sum_{i=1,2}\vec{p}_{i}\cdot(\vec{\cal
E}_{i}+\vec{E}_{i})$ (21) $\displaystyle=$ $\displaystyle H_{0}+V$ (22)

where $V=-\sum_{i=1,2}\vec{p}_{i}\cdot(\vec{\cal E}_{i}+\vec{E}_{i})$ only
contains two probe atoms. We implement the Keldysh formalism and define the
effective two-atom time-evolution operator along the Keldysh contour

${\cal S}^{\rm eff}_{\cal C}\equiv\langle{\cal S}_{\cal
C}\rangle_{E}=\left\langle{\cal T}_{\cal C}{\rm
exp}\left\\{-\frac{i}{\hbar}\int_{\cal
C}d\check{\tau}V^{I}(\check{\tau})\right\\}\right\rangle_{E}$ (23)

where $\langle...\rangle_{E}={\rm Tr}_{E}(...\rho_{E})$ denotes an average
over the environmental degrees of freedom, namely, the $N-2$ atoms and the
quantized field.
$V^{I}(t)=e^{iH_{0}(t-t_{0})/\hbar}Ve^{-iH_{0}(t-t_{0})/\hbar}$ is the $V$
term in the interaction picture. $\cal C$ represents an integration path from
$-\infty$ to $t$, then from $t$ to $-\infty$, with ${\cal T}_{\cal C}$ being
the normal time-ordering for the first part and the inverse time-ordering for
the second. The matrix element of the density matrix can be written as the
expectation value of the corresponding projection operator.

$\rho_{\alpha\beta}(t)=\langle{\cal T}_{\cal C}{\cal S}^{\rm eff}_{\cal
C}P^{I}_{\beta\alpha}(t)\rangle_{0}$ (24)

where
$P^{I}_{\beta\alpha}(t)=e^{iH_{0}(t-t_{0})/\hbar}\left|\beta\right\rangle\\!\\!\left\langle\alpha\right|e^{-iH_{0}(t-t_{0})/\hbar}$
and $\langle...\rangle_{0}={\rm Tr}(...\rho(t_{0}))$. Keeping the first and
second order terms in Eq. (23) and taking the time derivative of Eq. (24)
gives rise to the two-atom master equation up to second-order correlations.
The related terms in the master equation are

$\displaystyle\dot{\rho}$
$\displaystyle=.....-\frac{\wp^{2}}{2\hbar^{2}}\\!\\!\sum_{i,j=1,2}(\sigma_{j}^{\dagger}\sigma_{i}\rho-2\sigma_{i}\rho\sigma_{j}^{\dagger}+\rho\sigma_{j}^{\dagger}\sigma_{i})\int_{-\infty}^{+\infty}\langle[E_{j}^{+}(t),E_{i}^{-}(t+\tau)]\rangle
e^{-i(\omega_{0}-\omega_{l})\tau}d\tau$
$\displaystyle+\frac{\wp^{2}}{2\hbar^{2}}\\!\\!\sum_{i,j=1,2}[\sigma^{\dagger}_{j}\sigma_{i},\rho]\int_{0}^{+\infty}\\!\\!\bigg{(}\\!\langle[E_{j}^{+}(t),E_{i}^{-}(t+\tau)]\rangle
e^{-i(\omega_{0}-\omega_{l})\tau}\\!-\\!\langle[E_{j}^{+}(t+\tau),E_{i}^{-}(t)]\rangle
e^{i(\omega_{0}-\omega_{l})\tau}\\!\bigg{)}d\tau$

The first term is the Lindblad term, with its coefficient defined as
$\gamma_{ij}/2$, and the second term resembles the Hamiltonian time-evolution,
with a coefficient defined as $-i\delta_{ij}$. Therefore, we obtain the
expressions for $\gamma_{ij}$ and $\delta_{ij}$ in the main context.

###  I.2 II. Derivation of the source function $P(\vec{r}_{i},t+\tau,t)$ and
its Fourier transform $\tilde{P}(\vec{r}_{i},\omega,t)$

In order to obtain the source function, one need to calculated the related
two-point correlators of the atomic operators. To do this, we define the
average excited population $\langle\sigma_{ee}\rangle$ and average single-atom
coherence $\langle\sigma\rangle$ as

$\displaystyle\langle\sigma_{ee}\rangle$ $\displaystyle=$
$\displaystyle(\langle\sigma_{ee1}+\sigma_{ee2}\rangle)/2$ (25)
$\displaystyle\langle\sigma\rangle$ $\displaystyle=$
$\displaystyle(\langle\sigma_{1}\rangle+\langle\sigma_{2}\rangle)/2$ (26)

where $\sigma_{eei}=\left|e_{i}\right\rangle\\!\\!\left\langle e_{i}\right|$
and $\sigma_{i}=\left|g_{i}\right\rangle\\!\\!\left\langle e_{i}\right|$. The
master equation directly leads to the Maxwell-Bloch equations of motion

$\displaystyle\frac{d}{d\tau}\langle\sigma_{ee}\rangle$ $\displaystyle=$
$\displaystyle-\gamma_{11}\langle\sigma_{ee}\rangle-i\Omega(\langle\sigma\rangle-\langle\sigma^{\dagger}\rangle)$
(27) $\displaystyle\frac{d}{d\tau}\langle\sigma\rangle$ $\displaystyle=$
$\displaystyle-\big{[}\gamma_{11}/2+i(\delta_{11}+\Delta_{0})\big{]}\langle\sigma\rangle-i\Omega(2\langle\sigma_{ee}\rangle-1)$
(28) $\displaystyle\frac{d}{d\tau}\langle\sigma^{\dagger}\rangle$
$\displaystyle=$
$\displaystyle-\big{[}\gamma_{11}/2-i(\delta_{11}+\Delta_{0})\big{]}\langle\sigma^{\dagger}\rangle+i\Omega(2\langle\sigma_{ee}\rangle-1)$
(29)

The quantum regression theorem guarantees that two-time correlators follow the
same equations of motion of single-operator expectation values [33]:

$\displaystyle\frac{d}{d\tau}\langle\sigma_{ee1}(t+\tau)\sigma_{1}(t)\rangle$
$\displaystyle\\!=\\!$
$\displaystyle-\gamma_{11}\langle\sigma_{ee1}(t+\tau)\sigma_{1}(t)\rangle-i\Omega\big{[}\langle\sigma_{1}(t+\tau)\sigma_{1}(t)\rangle-\langle\sigma^{\dagger}_{1}(t+\tau)\sigma_{1}(t)\rangle\big{]}$
$\displaystyle\frac{d}{d\tau}\langle\sigma_{1}(t+\tau)\sigma_{1}(t)\rangle$
$\displaystyle\\!=\\!$
$\displaystyle-\big{[}\gamma_{11}/2+i(\delta_{11}+\Delta_{0})\big{]}\langle\sigma_{1}(t+\tau)\sigma_{1}(t)\rangle-i\Omega(2\langle\sigma_{ee1}(t+\tau)\sigma_{1}(t)\rangle-\langle\sigma_{1}(t)\rangle)$
$\displaystyle\frac{d}{d\tau}\langle\sigma_{1}^{\dagger}(t+\tau)\sigma_{1}(t)\rangle$
$\displaystyle\\!=\\!$
$\displaystyle-\big{[}\gamma_{11}/2-i(\delta_{11}+\Delta_{0})\big{]}\langle\sigma_{1}^{\dagger}(t+\tau)\sigma_{1}(t)\rangle+i\Omega\big{[}2\langle\sigma_{ee1}(t+\tau)\sigma_{1}(t)\rangle-\langle\sigma_{1}(t)\rangle\big{]}$

and similarly for $\langle\sigma_{1}(t)\sigma_{ee1}(t+\tau)\rangle$,
$\langle\sigma_{1}(t)\sigma_{1}(t+\tau)\rangle$ and
$\langle\sigma_{1}(t)\sigma_{1}^{\dagger}(t+\tau)\rangle$. Note that $t$ is a
fixed parameter and the initial conditions are such that $\tau=0$. Doing a
Laplace transform with respect to $\tau$ helps to solve the differential
equations easily, given initial conditions such as
$\langle\sigma_{1}^{\dagger}(t)\sigma_{1}(t)\rangle=\langle\sigma_{ee}(t)\rangle$,
$\langle\sigma_{1}(t)\sigma_{1}(t)\rangle=0$ and
$\langle\sigma_{1}(t)\sigma_{ee1}(t)\rangle=\langle\sigma_{1}(t)\rangle$. To
obtain the Fourier transform of the two-point correlators, it suffices to
replace the Laplace variable by $i\omega$. Finally by substituting to Eq. (8)
in the main text, the Fourier transform of $P(\vec{r}_{i},t+\tau,t)$ has the
following form:

$\displaystyle\tilde{P}(\vec{r}_{i},\omega,t)\\!=\\!\frac{\wp^{2}}{\hbar^{2}}{\cal
N}\frac{(2\langle\sigma_{ee}\rangle\\!-\\!1)\big{\\{}\big{[}\gamma_{11}/2\\!+\\!i(\delta_{11}\\!+\\!\Delta_{0}\\!+\\!\omega)\big{]}(\gamma_{11}\\!+\\!i\omega)\\!+\\!2\Omega^{2}\big{\\}}\\!+\\!2\Omega\langle\sigma\rangle(-i\gamma_{11}/2\\!+\\!\delta_{11}\\!+\\!\Delta_{0}\\!+\\!\omega)}{((\delta_{11}+\Delta_{0})^{2}+(\gamma_{11}/2+i\omega)^{2})(\gamma_{11}+i\omega)+4\Omega^{2}(\gamma_{11}/2+i\omega)}$
(30)

With the weak-driving and low-excitation approximations we set
$\langle\sigma_{ee}\rangle=\langle\sigma\rangle=\Omega=0$, and by definition
$\omega=\omega_{0}-\omega_{l}\approx\delta_{11}$. Expectation values are
averaged over all possible probe atoms, and $\tilde{P}$ is independent of the
locations at the end. It reduces to the following simple form

$\tilde{P}(\omega\approx\delta_{11},t)=\frac{\wp^{2}}{\hbar^{2}}{\cal
N}\frac{-2}{\gamma_{11}-2i\Delta_{0}}$ (31)