# Single Photon Superradiance and Subradiance as Collective Emission From
Symmetric and Antisymmetric States

Nicola Piovella  Dipartimento di Fisica ”Aldo Pontremoli”, Università degli
Studi di Milano, Via Celoria 16, I-20133 Milano, Italy & INFN Sezione di
Milano, Via Celoria 16, I-20133 Milano, Italy  Stefano Olivares  Dipartimento
di Fisica ”Aldo Pontremoli”, Università degli Studi di Milano, Via Celoria 16,
I-20133 Milano, Italy & INFN Sezione di Milano, Via Celoria 16, I-20133
Milano, Italy

###### Abstract

Recent works have shown that collective single photon spontaneous emission
from an ensemble of $N$ resonant two-level atoms is a rich field of study.
Superradiance describes emission from a completely symmetric state of $N$
atoms, with a single excited atom prepared with a given phase, for instance
imprinted by an external laser. Instead, subradiance is associated with the
emission from the remaining $N-1$ asymmetric states, with a collective decay
rate less than the single-atom value. Here, we discuss the properties of the
orthonormal basis of symmetric and asymmetric states and the entanglement
properties of superradiant and subradiant states. On the one hand, by
separating the symmetric superradiant state from the subradiant ones, we are
able to determine the subradiant fraction induced in the system by the laser.
On the other hand, we show that, as the external laser is switched off and the
atomic excitation decays, entanglement in the atomic ensemble appears when the
superradiant fraction falls below the threshold $1/N$.

##  I Introduction

Cooperative light scattering by a system of $N$ two-level atoms has been a
topic studied since many years [1]. Many studies in the past have been focused
on a diffusive regime dominated by multiple scattering [2], where light
travels over distances much larger than the mean free path. More recently, it
has been shown that light scattering in dilute systems induces a dipole–dipole
interaction between atom pairs, leading to a different regime dominated by
single scattering of photons by many atoms. The transition between single and
multiple scattering is controlled by the optical thickness parameter
$b(\Delta)=b_{0}/(1+4\Delta^{2}/\Gamma^{2})$ [3, 4], where $b_{0}$ is the
resonant optical thickness, $\Delta$ is the detuning of the laser frequency
from the atomic resonance frequency and $\Gamma$ is the single atom decay
rate. A different cooperative emission is provided by superradiance and
subradiance, both originally predicted by Dicke in 1954 [5] in fully inverted
system. Whereas Dicke superradiance originates from constructive interference
between many emitted photons, subradiance is based on destructive interference
effect, leading to the partial trapping of light. Subradiant states are
important in quantum information to protect entanglement against decoherence,
especially in the case of two two-level atoms forming a dark state [6, 7].

Dicke states have been considered for a collection of $N$ two-level systems
[8], realized, e.g., by atoms [9] or quantum dots [10]. In contrast to an
initially fully inverted system with $N$ photons stored by $N$ atoms, states
with at most one single excitation have attracted considerable attention in
the context of quantum information [11, 12, 13], where the accessible Hilbert
space can be restricted to a single excitation by using, e.g., the Rydberg
blockade [14, 15, 16]. A particular kind of single-excitation superradiance
has been proposed by Scully and coworkers [17, 18, 19] in a system of $N$ two-
level atoms prepared by the absorption of a single photon (Timed Dicke state).
A link between this single-photon superradiance and the more classical process
of cooperative scattering of an incident laser by $N$ atoms has been proposed
by a series of theoretical and experimental papers [20, 21, 22]. In such
systems of driven cold atoms, subradiance has been also predicted [23] and
then observed [24]; after that, the laser is abruptly switched off and the
emitted photons detected in a given direction. Subradiance, by itself, has
attracted a large interest for its application in quantum optics as a possible
method to control spontaneous emission, storing the excitation for a
relatively long time. For instance, it has been shown [25] that it is possible
to use subradiance to store a photon in a small volume for many atomic
lifetimes and later switch the subradiant state to a superradiant state which
emits a photon in a time shorter than the single-atom lifetime. Such a process
has potential applications in quantum technology, e.g., quantum memories.
Furthermore, it has been proven that the distribution of the atoms in an
extended ensemble has a substantial effect on cooperative spontaneous emission
[26]. For instance, for atomic distributions with mirror symmetry [27], the
anti-symmetric states are subradiant, even when half of the atoms are randomly
distributed as long as the mirror symmetry is maintained. Periodic
distribution with intrinsic mirror symmetry can be realized in ion traps, and
the subradiant states can be prepared by specially tailored antisymmetric
optical modes. Also, the presence of an optical cavity may enhance the
generation of subradiant states. In ref. [28], it was shown that by tuning the
cavity decay parameter, the nonequilibrium phase transition of cooperative
resonance fluorescence changes drastically, amplifying the subradiant Dicke
states through cavity-assisted coherences. Letting the system relax into the
ground state generates a dark state cascade that can be utilized to store
quantum information.

The aim of this paper is to provide a mathematical description of the single-
excitation states in terms of superradiant and subradiant states, i.e.,
separating the fully symmetric state by the remaining antisymmetric ones.
Symmetric and subradiant excited states are distinguished by their decay
rates, once populated by a classical external laser and observed; after that,
the laser is switched off: the symmetric state has a superradiant decay rate
proportional to $N\Gamma$, where $\Gamma$ is the single-atom decay, whereas
the antisymmetric states have a decay rate slower than $\Gamma$.

A crucial point is to determine if such subradiant states are entangled or
not, in view of a possible application as quantum memories. Once the time
evolution of these states is characterized, we will apply the criteria of the
spin-squeezing inequalities introduced by Tóth [29] to detect entanglement in
the superradiant and subradiant states. We outline that we limit our study to
the linear regime, where the excitation amplitude is proportional to the
driving incident electric field. In this linear regime, we must consider the
entanglement criteria which are independent from the value of the driving
field, i.e., abandoning these criteria which lead to expressions which depend
nonlinearly on the driving field, as will be discussed in the following.

The paper is organized as follows. In Section II, we present the Hamiltonian
describing the dynamics of $N$ two-level atoms interacting with the driving
field and write the equation of motion in the linear regime. Then, we
calculate the decay rate and the transition rates between different elements
of the so-called Timed Dicke basis with its symmetric and antisymmetric
states. Section III introduces the collective spin operator and the formalism
of the spin-squeezing inequalities to assess entanglement. Conclusions are
eventually drawn in Section IV.

##  II The Model

We consider $N\gg 1$ two-level atoms with the same atomic transition frequency
$\omega_{a}$, linewidth $\Gamma$ and dipole $d$ (polarization effects are
neglected). The atoms are driven by a monochromatic plane wave with electric
field $E_{0}$, frequency $\omega_{0}$ and wave vector $\mathbf{k}_{0}$,
detuned from the atomic transition by $\Delta_{0}=\omega_{0}-\omega_{a}$ (see
Figure 1); $|g_{j}\rangle$ and $|e_{j}\rangle$ are the ground and excited
states, respectively, of the $j$-th atom, $j=1,\ldots,N$, which is placed at
position $\mathbf{r}_{j}$.

We consider the single excitation effective Hamiltonian [30, 31] in the scalar
approximation. This approximation is appropriate in the dilute limit, where
inter-atomic distances are larger than the optical wavelength, making near-
field terms decaying as $1/r^{3}$ negligible. If we assume that only one
photon is present, when tracing over the radiation degrees of freedom, the
dynamics of the atomic system can be described by the non-Hermitian
Hamiltonian $\hat{H}=\hat{H}_{0}-i\hat{H}_{\rm eff}$, where [30, 32]

$\displaystyle\hat{H}_{0}=-\hbar\Delta_{0}\sum_{j=1}^{N}{\hat{\sigma}_{j}}^{\dagger}{\hat{\sigma}_{j}}+\frac{\hbar\Omega_{0}}{2}\sum_{j=1}^{N}\left(\hat{\sigma}_{j}e^{-i\mathbf{k}_{0}\cdot\mathbf{r}_{j}}+{\hat{\sigma}_{j}}^{\dagger}e^{i\mathbf{k}_{0}\cdot\mathbf{r}_{j}}\right)$
(1) $\displaystyle\hat{H}_{\rm
eff}=\frac{\hbar\Gamma}{2}\sum_{j,m}G_{jm}\,\hat{\sigma}_{j}^{\dagger}\hat{\sigma}_{m}.$
(2)

Figure 1: Scheme of the system and parameters.

Here, $\Omega_{0}=dE_{0}/\hbar$ is the Rabi frequency,
$\hat{\sigma}_{j}=|g_{j}\rangle\langle e_{j}|$ and
$\hat{\sigma}_{j}^{\dagger}=|e_{j}\rangle\langle g_{j}|$ are the lowering and
raising operators, with commutations rules
$[\hat{\sigma}_{j},\hat{\sigma}_{m}^{\dagger}]=-\delta_{jm}\hat{\sigma}_{zj}$,
where $\hat{\sigma}_{zj}=|e_{j}\rangle\langle e_{j}|-|g_{j}\rangle\langle
g_{j}|$ are the population difference operators, and

$G_{jm}=\left\\{\begin{array}[]{ll}\Gamma_{jm}-i\,\Omega_{jm}&\mbox{if}~{}j\neq
m,\\\\[4.30554pt] 1&\mbox{if}~{}j=m,\end{array}\right.$ (3)

where

$\Gamma_{jm}=\frac{\sin(k_{0}r_{jm})}{k_{0}r_{jm}}\quad\mbox{and}\quad\Omega_{jm}=\frac{\cos(k_{0}r_{jm})}{k_{0}r_{jm}}.$
(4)

Note that $\hat{H}_{\rm eff}$ contains both real and imaginary parts, which
takes into account that the excitation is not conserved since it can leave the
system by emission. By writing the Heisenberg’s equations for the operators
$\hat{\sigma}_{j}$ and $\hat{\sigma}_{zj}$, one can solve them by iteration in
powers of $\Omega_{0}$. One can see that assuming weak excitation
($\Omega_{0}\ll\Gamma$), we can approximate (in the Heisenberg’s equation for
$\hat{\sigma}_{j}$) $\hat{\sigma}_{zj}-\approx\hat{I}_{j}$, where
$\hat{I}_{j}$ is the identity operator for the $j$th atom [33]. Following the
approach reported in refs. [30, 31, 32], one can show that this approximation
amounts to the linear regime in which all the processes generating more than
one photon at the same time are neglected [33]. Thereafter, only one atom
among the $N$ atoms can be found in the excited state, whereas all others are
in their ground state. Then, the generic state of the atomic system belongs to
an $(N+1)$\- dimensional Hilbert space and can be written as [17, 25]

$|\Psi\rangle=\alpha|g\rangle+\sum_{j=1}^{N}\beta_{j}e^{i\mathbf{k}_{0}\cdot\mathbf{r}_{j}}|j\rangle$
(5)

where $|g\rangle=|g_{1},\dots,g_{N}\rangle$ and
$|j\rangle=|g_{1},\ldots,e_{j},\ldots,g_{N}\rangle$, $j=1,\ldots,N$. From the
Schrödinger equation $i\hbar(\partial/\partial
t)|\Psi\rangle=\hat{H}|\Psi\rangle$ and assuming $\alpha\approx 1$ in the
weak-excitation approximation, we obtain the following equations for the
coefficients $\beta_{j}$ of the state (5):

$\displaystyle\dot{\beta}_{j}$
$\displaystyle=\left(i\Delta_{0}-\frac{\Gamma}{2}\right)\beta_{j}-\frac{i\Omega_{0}}{2}-\frac{\Gamma}{2}\sum_{m\neq
j}\widetilde{G}_{jm}\,\beta_{m}(t),\qquad(j=1,\ldots,N)$ (6)

where $\dot{\beta}_{j}$ refers to the time derivative of the coefficient
$\beta_{j}$ and
$\widetilde{G}_{jm}=G_{jm}e^{-i\mathbf{k}_{0}\cdot(\mathbf{r}_{j}-\mathbf{r}_{m})}$.
We observe that a more fundamental Master equation approach, describing the
atomic system dynamics within the same approximations (scalar approximation
and single-excitation approximation), leads to the same Equation (6) [32, 22].

The use of the bare basis $\\{|g\rangle,|j\rangle\\}$ has the advantage that
Equation (6) can be easily solved numerically. However, it does not
distinguish between symmetric and anti-symmetric states that play a relevant
role in our investigation. For this reason we introduce the Timed Dicke (TD)
basis with a single excitation [18, 34]:

$\left\\{|g\rangle,|+\rangle_{\mathbf{k}_{0}},|1\rangle_{\mathbf{k}_{0}},\ldots,|N-1\rangle_{\mathbf{k}_{0}}\right\\},$
(7)

where $|g\rangle$ has been introduced above, whereas

$|+\rangle_{\mathbf{k}_{0}}=\frac{1}{\sqrt{N}}\sum_{j=1}^{N}e^{i\mathbf{k}_{0}\cdot\mathbf{r}_{j}}|j\rangle$
(8)

is the symmetric Timed Dicke (STD) state and

$|s\rangle_{\mathbf{k}_{0}}=\frac{1}{\sqrt{s(s+1)}}\left\\{\sum_{j=1}^{s}e^{i\mathbf{k}_{0}\cdot\mathbf{r}_{j}}|j\rangle-s\,e^{i\mathbf{k}_{0}\cdot\mathbf{r}_{s+1}}|s+1\rangle\right\\},\qquad(s=1,\ldots,N-1)$
(9)

are the anti-symmetric ones. The advantage of using the TD basis is that
$|s\rangle_{\mathbf{k}_{0}}$ are collective states involving $s+1$ atoms.
Considering (9) and the definition of $|+\rangle_{\mathbf{k}_{0}}$, it is
straightforward to verify that, within the considered Hilbert space, the TD
basis is complete, namely:

$|g\rangle\langle
g|+|+\rangle_{\mathbf{k}_{0}}\langle+|+\sum_{s=1}^{N-1}|s\rangle_{\mathbf{k}_{0}}\langle
s|=\hat{I}$ (10)

and orthonormal, since $\langle g|+\rangle_{\mathbf{k}_{0}}=\langle
g|s\rangle_{\mathbf{k}_{0}}=\,_{\mathbf{k}_{0}}\langle+|s\rangle_{\mathbf{k}_{0}}=0$,
and $\,{}_{\mathbf{k}_{0}}\langle
s|s^{\prime}\rangle_{\mathbf{k}_{0}}=\delta_{s,s^{\prime}}$.

To highlight the physical meaning of the TD states, it is useful to evaluate
the following transition rates between the basis elements. On the one hand, we
find

$\,{}_{\mathbf{k}_{0}}\langle+|\hat{H}_{0}|+\rangle_{\mathbf{k}_{0}}=-\hbar\Delta_{0}\qquad\mbox{and}\qquad\,_{\mathbf{k}_{0}}\langle+|\hat{H}_{\rm
eff}|+\rangle_{\mathbf{k}_{0}}=\frac{\hbar\Gamma_{+}}{2}-i\hbar\,\Omega_{+}$
(11)

with $\hat{H}_{0}$ and $\hat{H}_{\rm eff}$ given in Equations (1) and (2),
respectively, and

$\displaystyle\Gamma_{+}$
$\displaystyle=\Gamma\left(1+\frac{1}{N}\sum_{j=1}^{N}\sum_{m=1\atop(m\neq
j)}^{N}\widetilde{\Gamma}_{jm}\right),$ (12) $\displaystyle\Omega_{+}$
$\displaystyle=\frac{\Gamma}{N}\sum_{j=1}^{N}\sum_{m=1\atop(m\neq
j)}^{N}\widetilde{\Omega}_{jm}$ (13)

where

$\widetilde{\Gamma}_{jm}=\Gamma_{jm}\cos[\mathbf{k}_{0}\cdot(\mathbf{r}_{j}-\mathbf{r}_{m})]\qquad\mbox{and}\qquad\widetilde{\Omega}_{jm}=\Omega_{jm}\cos[\mathbf{k}_{0}\cdot(\mathbf{r}_{j}-\mathbf{r}_{m})]$
(14)

and $\Gamma_{jm}$ and $\Omega_{jm}$ have been introduced in Equation (4). For
a cloud of cold atoms with a Gaussian distribution with parameter
$\sigma_{r}$, one can prove that $\Gamma_{+}\approx\Gamma(1+b_{0}/12)$ [21,
33], where $b_{0}=3N/\sigma^{2}$ is the resonant optical thickness, with
$\sigma=k_{0}\sigma_{r}$ . Thus, we can conclude that
$|+\rangle_{\mathbf{k}_{0}}$ is a superradiant state for very large $b_{0}$
[18, 19].

On the other hand, the transition rates for the states
$|s\rangle_{\mathbf{k}_{0}}$, $s=1,\dots,N-1$, read

$\,{}_{\mathbf{k}_{0}}\langle
s|\hat{H}_{0}|s\rangle_{\mathbf{k}_{0}}=-\hbar\Delta_{0}\qquad\mbox{and}\qquad\,_{\mathbf{k}_{0}}\langle
s|\hat{H}_{\rm
eff}|s\rangle_{\mathbf{k}_{0}}=\frac{\hbar\Gamma_{s}}{2}-i\hbar\,\Omega_{s}$
(15)

with

$\displaystyle\Gamma_{s}=\Gamma\left[1+\frac{1}{s+1}\left(\frac{1}{s}\sum_{j=1}^{s}\sum_{m=1\atop(m\neq
j)}^{s}\tilde{\Gamma}_{jm}-2\sum_{j=1}^{s}\tilde{\Gamma}_{j,s+1}\right)\right],$
(16)
$\displaystyle\Omega_{s}=\frac{\Gamma}{2(s+1)}\left(\frac{1}{s}\sum_{j\neq
m=1}^{s}\tilde{\Omega}_{jm}-2\sum_{j=1}^{s}\tilde{\Omega}_{j,s+1}\right).$
(17)

In this case, the decay rates $\Gamma_{s}$ are less than the single-atom decay
rate $\Gamma$, and the states $|s\rangle_{\mathbf{k}_{0}}$ turn out to be
subradiant [25].

Now we focus on the state (5), which, in the TD basis, reads

$|\Psi\rangle=\alpha|g\rangle+\beta_{+}|+\rangle_{\mathbf{k}_{0}}+\sum_{s=1}^{N-1}\gamma_{s}|s\rangle_{\mathbf{k}_{0}},$
(18)

with

$\displaystyle\beta_{+}$
$\displaystyle=\,_{\mathbf{k}_{0}}\langle+|\Psi\rangle=\frac{1}{\sqrt{N}}\sum_{j=1}^{N}\beta_{j},$
(19) $\displaystyle\gamma_{s}$ $\displaystyle=\,_{\mathbf{k}_{0}}\langle
s|\Psi\rangle=\frac{1}{\sqrt{s(s+1)}}\left(\sum_{j=1}^{s}\beta_{j}-s\,\beta_{s+1}\right).$
(20)

Thanks to the considerations made above about the properties of the TD states,
we can easily find the probability to find our state in a superradiant, i.e.
STD, and subradiant state, that is:

$P_{+}=|\beta_{+}|^{2}$ (21)

and

$P_{\rm sub}=\sum_{s=1}^{N-1}|\gamma_{s}|^{2},$ (22)

respectively. Moreover, from the normalization of the state $|\Psi\rangle$, it
follows that

$\displaystyle\sum_{j=1}^{N}|\beta_{j}|^{2}$
$\displaystyle=|\beta_{+}|^{2}+\sum_{s=1}^{N-1}|\gamma_{s}|^{2}=P_{+}+P_{\rm
sub}.$ (23)

Finally, the superradiant fraction of excited atoms is given by

$f_{\rm SR}=\frac{P_{+}}{P_{+}+P_{\rm
sub}}=\frac{\left|\overline{\beta}\right|^{2}}{~{}\overline{|\beta|^{2}}~{}}$
(24)

where we introduced the mean quantities:

$\displaystyle\overline{|\beta|^{2}}=\frac{1}{N}\sum_{j=1}^{N}|\beta_{j}|^{2}$
(25)
$\displaystyle\left|\overline{\beta}\right|=\frac{1}{N}\left|\sum_{j=1}^{N}\beta_{j}\right|.$
(26)

In turn, the subradiant fraction is $f_{\rm sub}=1-f_{\rm SR}$.

##  III Entanglement Properties of the Superradiant and Subradiant Collective
States

As the system we are investigating in this paper consists of a large number of
atoms, $N\gg 1$, we cannot asses its entanglement properties by individually
addressing the single particles. Nevertheless, it is known that spin-squeezing
can be used to create large-scale entanglement [35]. A two-level atom can be
considered a spin-$1/2$ system, described in terms of Pauli’s operators of the
angular moments. Therefore, here we consider suitable generalized spin-
squeezing inequalities [29] based only on collective quantities that are
accessible and can be measured experimentally.

Given $N$ two-level atoms, we start defining the following collective angular
momentum operators:

$\hat{J}_{k}=\frac{1}{2}\sum_{j=1}^{N}\hat{\sigma}_{j}^{(k)},\qquad\qquad(k=x,y,z)$
(27)

where $\hat{\sigma}_{j}^{(k)}$ are the Pauli matrices associated with the
$j$-th atom. If we assume the state given by Equation (5), we can calculate
the expectation values of the first and second moments of $\hat{J}_{k}$ using
the result of the previous section (see Appendix A for details):

$\displaystyle\langle\hat{J}_{x}\rangle$ $\displaystyle=\Re{\rm
e}[\alpha^{*}\sum_{j=1}^{N}\beta_{j}],$ (28)
$\displaystyle\langle\hat{J}_{y}\rangle$ $\displaystyle=-\Im{\rm
m}[\alpha^{*}\sum_{j=1}^{N}\beta_{j}],$ (29)
$\displaystyle\langle\hat{J}_{z}\rangle$
$\displaystyle=-N\left(\displaystyle\frac{1}{2}-\overline{|\beta|^{2}}\right),$
(30)

and

$\displaystyle\langle\hat{J}_{x}^{2}\rangle=\langle\hat{J}_{y}^{2}\rangle=\frac{N}{2}\left(\frac{1}{2}+N\,\left|\overline{\beta}\right|^{2}-\overline{|\beta|^{2}}\right),$
(31)
$\displaystyle\langle\hat{J}_{z}^{2}\rangle=N\left[\frac{N}{4}-(N-1)\,\overline{|\beta|^{2}}\right],$
(32)

$\beta_{j}$ being the solutions of Equation (6) and $\overline{|\beta|^{2}}$
and $\left|\overline{\beta}\right|$ are given in Equations (25) and (26),
respectively. One can also straightforwardly evaluate the corresponding
variances
$(\Delta\hat{J}_{k})^{2}=\langle\hat{J}_{k}^{2}\rangle-\langle\hat{J}_{k}\rangle^{2}$,
$k=x,y,z$.

In ref. [29], G. Tóth and co-workers proved that a sufficient condition to
have entanglement is the violation of suitable inequalities involving the
first and second moments of the $\hat{J}_{k}$ operators evaluated above.
Though they proposed four inequalities, in our case, only one of them is
relevant to our system, namely:

$(\Delta\hat{J}_{x})^{2}+(\Delta\hat{J}_{y})^{2}+(\Delta\hat{J}_{z})^{2}\geq\frac{N}{2},$
(33)

since the other three Tóth’s inequalities in ref. [29] are:

$\displaystyle\langle\hat{J}^{2}_{x}\rangle+\langle\hat{J}^{2}_{y}\rangle+\langle\hat{J}^{2}_{z}\rangle$
$\displaystyle\leq\frac{N(N+2)}{4},$ (34a)
$\displaystyle\langle\hat{J}^{2}_{k}\rangle+\langle\hat{J}^{2}_{l}\rangle-\frac{N}{2}$
$\displaystyle\leq(N-1)(\Delta\hat{J}_{m})^{2},$ (34b)
$\displaystyle(N-1)\left[(\Delta\hat{J}_{k})^{2}+(\Delta\hat{J}_{l})^{2}\right]$
$\displaystyle\geq\langle\hat{J}^{2}_{m}\rangle+\frac{N(N-2)}{4}$ (34c)

(where $k,l,m$ take all the possible permutations of $x,y,z$) are not useful,
as the first and the third are never violated, whereas the second one leads to
a condition on the $\beta_{j}$ going beyond the linear regime assumed to solve
Eq. (6), as shown in Appendix B.

The inequality (33) is more interesting. Its l.h.s. can be rewritten as a
function of the superradiant fraction (24), that is:

$\displaystyle(\Delta\hat{J}_{x})^{2}+(\Delta\hat{J}_{y})^{2}+(\Delta\hat{J}_{z})^{2}$
$\displaystyle=\frac{N}{2}+N^{2}\overline{|\beta|^{2}}\left(N\,|\overline{\beta}|^{2}-\overline{|\beta|^{2}}\right)$
(35)
$\displaystyle=\frac{N}{2}+N^{2}\left(\overline{|\beta|^{2}}\right)^{2}\left(N\,f_{\rm
SR}-1\right),$ (36)

which can be solved within the linear regime assumed throughout the paper.
Hence, the inequality (33) is violated when the superradiant fraction is
$f_{SR}<1/N$, thus highlighting the entanglement of the collective atomic
state. This will occur when the driving laser is cut and the superradiant
component decays faster than the subradiant one, until the subradiant fraction
becomes larger than $1-1/N$.

Figures 2–4 present a typical result. We numerically evaluate the values of
$\beta_{j}$ solving the linear equations (6) for $N=10^{3}$ and a Gaussian
distribution with $\sigma=k_{0}\sigma_{r}=10$. The laser is cut off after
$\Gamma t=20$. Figure 2 shows $P=(1/N)\sum_{j}|\beta_{j}|^{2}$ for
$\Delta_{0}=10\Gamma$ vs. time. Figure 3 shows the left-hand side of Equation
(33) normalized to $N$, i.e.,
$C=[(\Delta\hat{J}_{x})^{2}+(\Delta\hat{J}_{y})^{2}+(\Delta\hat{J}_{z})^{2}]/N$
vs. time, for the same parameters of Figure 2 and three different values of
detuning, $\Delta_{0}=8\Gamma$ (red line), $\Delta_{0}=9\Gamma$ (blue line)
and $\Delta_{0}=10\Gamma$ (black line). The inset shows the region where the
inequality (33) is violated, namely when $C<1/2$. Figure 4 shows the
superradiant and subradiant fractions, $f_{SR}$ (continuous blue line) and
$f_{\mathrm{sub}}$ (dashed red line), as defined by Equation (24), for the
same parameters of Figure 2. The dotted black line is the value $1/N$. We
observe that when the laser is on, the subradiant fraction is only about the
$3\%$ of the total excitation. As soon as the laser is cut off, the
superradiant fraction decays fast and the subradiant fraction increases,
becoming dominating for $\Gamma t>20.5$. The atoms become entangled when
$f_{SR}<1/N$, at time larger than $\Gamma t>24$.

Figure 2: Plot of $P=(1/N)\sum_{j}|\beta_{j}|^{2}$ vs. time, obtained by
solving Equation (6) for $N=10^{3}$, $\Delta_{0}=10\Gamma$ and a Gaussian
distribution with $\sigma=k_{0}\sigma_{R}=10$. The laser is cut off after
$\Gamma t=20$.

Figure 3:
$C=[(\Delta\hat{J}_{x})^{2}+(\Delta\hat{J}_{y})^{2}+(\Delta\hat{J}_{z})^{2}]/N$
vs. time for the same parameters of Figure 2 and three different values of
detuning, $\Delta_{0}=8\Gamma$ (red line), $\Delta_{0}=9\Gamma$ (blue line)
and $\Delta_{0}=10\Gamma$ (black line). The inset magnifies the plot around
$C=1/2$, showing the violation of the spin-squeezing inequality (33).

Figure 4: Superradiant and subradiant fractions, $f_{SR}$ (continuous blue
line) and $f_{sub}$ (dashed red line), for the same parameters of Figure 2.
The dotted black line is the value $1/N$.

##  IV Conclusions

In this work, we addressed the open question concerning the relation between
the entanglement of an atomic ensemble and its superradiant and subradiant
fraction. To this aim, we considered the state of $N$ two-level atoms with
positions $\mathbf{r}_{j}$ (with $j=1,\dots,N$), where only one is excited
among them. Then, we discussed their symmetry properties: the single-
excitation Hilbert space can be spanned by a completely symmetric state (the
“Symmetric Timed Dicke state”) and $N-1$ asymmetric ones, where the first has
a superradiant decay rate proportional to $N$, while the others have
subradiant decay rates less than the single-atom value. Remarkably, the
projection on the symmetric and asymmetric states allows us to calculate the
superradiant and subradiant fractions of the ensemble.

To address the problem of the birth of the entanglement, we studied the
relevant case of an ensemble of $N$ atoms driven by an external, uniform
laser. In the framework of the linear regime, the probability amplitude of
excitation is proportional to the incident electric field, and the
superradiant fraction is largely dominant, with only a small fraction of atoms
in the subradiant states. However, when the laser is cut off, the superradiant
fraction rapidly decays to zero, leaving only the subradiant one, as it has
been observed in the experiments of ref. [24].

In order to investigate the entanglement properties of the atomic ensemble, we
exploited suitable spin squeezing inequalities, based on the first and the
second order moments of collective spin operators. We have found that one of
these inequalities is violated when the superradiant fraction decreases below
the value $1/N$. Therefore, to have entanglement the probability of finding
$N$ atoms in the superradiant state must be less than the average probability
per atom to be in the excited state. Conversely, when the superradiant
fraction is larger than $1/N$, no entanglement can be detected by measuring
the moments of the collective spin operators.

##  Appendix A Calculation of the First and Second Moments of $\hat{J}_{k}$

In this appendix, we explicitly calculate the first and second moments of the
collective angular momentum operators used in the text, namely:

$\hat{J}_{k}=\frac{1}{2}\sum_{j=1}^{N}\hat{\sigma}_{j}^{(k)},\qquad\qquad(k=x,y,z)$
(37)

where $\hat{\sigma}_{j}^{(k)}$ are the Pauli matrices associated with the
$j$-th atom. It is useful to introduce the identities
$\hat{\sigma}_{j}^{\dagger}=\left(\hat{\sigma}_{j}^{(x)}+i\hat{\sigma}_{j}^{(y)}\right)/2$
and
$\hat{\sigma}_{j}=\left(\hat{\sigma}_{j}^{(x)}-i\hat{\sigma}_{j}^{(y)}\right)/2$,
which are the raising and lowering operators appearing in Equations (1) and
(2), and $\sigma_{j}^{(z)}=|e_{j}\rangle\langle e_{j}|-|g_{j}\rangle\langle
g_{j}|$. We define

$\displaystyle\tilde{\sigma}_{j}^{\dagger}=e^{i\mathbf{k}_{0}\cdot\mathbf{r}_{j}}\hat{\sigma}_{j}^{\dagger}\,,\qquad\tilde{\sigma}_{j}=e^{-i\mathbf{k}_{0}\cdot\mathbf{r}_{j}}\hat{\sigma}_{j}$
(38)

and the collective operators:

$\displaystyle\hat{J}_{-}=\sum_{j=1}^{N}\tilde{\sigma}_{j}\,,\quad\hat{J}_{+}=\sum_{j=1}^{N}\tilde{\sigma}_{j}^{\dagger}\quad\mbox{and}\quad\hat{J}_{z}=\frac{1}{2}\sum_{j=1}^{N}\hat{\sigma}_{j}^{(z)}$
(39)

with commutation relations:

$\displaystyle[\hat{J}_{+},\hat{J}_{-}]=2\hat{J}_{z},\qquad[\hat{J}_{z},\hat{J}_{\pm}]=\pm
2\hat{J}_{\pm}.$ (40)

We can now apply the formalism of ref. [29], calculating the first- and
second-order moments of the collective operators
$\hat{J}_{x}=(\hat{J}_{+}+\hat{J}_{-})/2$ and
$\hat{J}_{y}=(\hat{J}_{+}-\hat{J}_{-})/(2i)$. Explicitly,

$\displaystyle\langle\hat{J}_{x}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left\\{\langle\hat{J}_{+}\rangle+\langle\hat{J}_{-}\rangle\right\\},$
(41) $\displaystyle\langle\hat{J}_{y}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2i}\left\\{\langle\hat{J}_{+}\rangle-\langle\hat{J}_{-}\rangle\right\\},$
(42) $\displaystyle\langle\hat{J}_{x}^{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left\\{\langle\hat{J}_{+}\hat{J}_{-}\rangle+\langle\hat{J}_{-}\hat{J}_{+}\rangle+\langle\hat{J}_{+}^{2}\rangle+\langle\hat{J}_{-}^{2}\rangle\right\\},$
(43) $\displaystyle\langle\hat{J}_{y}^{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left\\{\langle\hat{J}_{+}\hat{J}_{-}\rangle+\langle\hat{J}_{-}\hat{J}_{+}\rangle-\langle\hat{J}_{+}^{2}\rangle-\langle\hat{J}_{-}^{2}\rangle\right\\},$
(44)

where the expectations are evaluated using the $\beta_{j}$ as evaluated from
Equation (6) (see Section II). Since

$\displaystyle\tilde{\sigma}_{j}|\Psi\rangle$
$\displaystyle=\beta_{j}|g\rangle,$ (45)
$\displaystyle\tilde{\sigma}_{j}^{\dagger}|\Psi\rangle$ $\displaystyle=\alpha
e^{i\mathbf{k}_{0}\cdot\mathbf{r}_{j}}|j\rangle,$ (46)
$\displaystyle\hat{\sigma}_{j}^{(z)}|\Psi\rangle$
$\displaystyle=-\alpha|g\rangle+\beta_{j}e^{i\mathbf{k}_{0}\cdot\mathbf{r}_{j}}|j\rangle-\sum_{m=1}^{N}(1-\delta_{jm})\beta_{m}e^{i\mathbf{k}_{0}\cdot\mathbf{r}_{m}}|m\rangle$
$\displaystyle=-|\Psi\rangle+2\beta_{j}e^{i\mathbf{k}_{0}\cdot\mathbf{r}_{j}}|j\rangle,$
(47)

we derive

$\displaystyle\langle\hat{J}_{-}\rangle$
$\displaystyle=\sum_{j=1}^{N}\langle\tilde{\sigma}_{j}\rangle=\alpha^{*}\sum_{j=1}^{N}\beta_{j},$
(48) $\displaystyle\langle\hat{J}_{+}\rangle$
$\displaystyle=\sum_{j=1}^{N}\langle\tilde{\sigma}_{j}^{\dagger}\rangle=\alpha\sum_{j=1}^{N}\beta_{j}^{*},$
(49) $\displaystyle\langle\hat{J}_{z}\rangle$
$\displaystyle=\frac{1}{2}\sum_{j}\langle\hat{\sigma}_{j}^{(z)}\rangle=-\frac{N}{2}+\sum_{j}|\beta_{j}|^{2},$
(50) $\displaystyle\langle\hat{J}_{x}\rangle$
$\displaystyle=\Re\textrm{e}\left[\alpha^{*}\sum_{j=1}^{N}\beta_{j}\right],$
(51) $\displaystyle\langle\hat{J}_{y}\rangle$
$\displaystyle=-\Im\textrm{m}\left[\alpha^{*}\sum_{j=1}^{N}\beta_{j}\right].$
(52)

Recalling that the system has a single excitation, we have
$\tilde{\sigma}_{j}\tilde{\sigma}_{m}^{\dagger}|\Psi\rangle=0$ and
$\tilde{\sigma}_{j}^{\dagger}\tilde{\sigma}_{m}^{\dagger}|\Psi\rangle=0$, and
we obtain:

$\langle\hat{J}_{-}^{2}\rangle=\langle\hat{J}_{+}^{2}\rangle=0.$ (53)

Furthermore,

$\displaystyle\langle\hat{J}_{+}\hat{J}_{-}\rangle$
$\displaystyle=\sum_{j,m}\langle\tilde{\sigma}_{j}^{\dagger}\hat{\sigma}_{j}\rangle$
$\displaystyle=\sum_{j,m}\beta_{j}^{*}\beta_{m}=\left|\sum_{j=1}^{N}\beta_{j}\right|^{2}$
(54)

and, using the commutation rule (40) and Equations (50) and (54), we find

$\displaystyle\langle\hat{J}_{-}\hat{J}_{+}\rangle$
$\displaystyle=\langle\hat{J}_{+}\hat{J}_{-}\rangle-2\langle\hat{J}_{z}\rangle$
$\displaystyle=N+\left|\sum_{j=1}^{N}\beta_{j}\right|^{2}-2\sum_{j=1}^{N}|\beta_{j}|^{2}.$
(55)

From these results, we can calculate the expectations of
$\langle\hat{J}_{k}^{2}\rangle$, $k=x,y,z$, namely:

$\displaystyle\langle\hat{J}_{x}^{2}\rangle$
$\displaystyle=\langle\hat{J}_{y}^{2}\rangle=\frac{N}{4}+\frac{1}{2}\left|\sum_{j=1}^{N}\beta_{j}\right|^{2}-\frac{1}{2}\sum_{j=1}^{N}|\beta_{j}|^{2},$
(56) $\displaystyle\langle\hat{J}_{z}^{2}\rangle$
$\displaystyle=\frac{N^{2}}{4}-(N-1)\sum_{j=1}^{N}|\beta_{j}|^{2}.$ (57)

It is remarkable that the second-order moments do not depend on $\alpha$,
which appears only in the expression for $\langle\hat{J}_{x,y}\rangle$.

We are now ready to also evaluate the variances in the collective angular
momentum operators. Defining
$(\Delta\hat{A})^{2}=\langle\hat{A}^{2}\rangle-\langle\hat{A}\rangle^{2}$, we
have:

$\displaystyle(\Delta\hat{J}_{x})^{2}$
$\displaystyle=\frac{N}{4}+\frac{1}{2}\left|\sum_{j=1}^{N}\beta_{j}\right|^{2}-\frac{1}{2}\sum_{j=1}^{N}|\beta_{j}|^{2}-\left\\{\Re\textrm{e}\left[\alpha^{*}\sum_{j=1}^{N}\beta_{j}\right]\right\\}^{2}$
(58) $\displaystyle(\Delta\hat{J}_{y})^{2}$
$\displaystyle=\frac{N}{4}+\frac{1}{2}\left|\sum_{j=1}^{N}\beta_{j}\right|^{2}-\frac{1}{2}\sum_{j=1}^{N}|\beta_{j}|^{2}-\left\\{\Im\textrm{m}\left[\alpha^{*}\sum_{j=1}^{N}\beta_{j}\right]\right\\}^{2}$
(59) $\displaystyle(\Delta\hat{J}_{z})^{2}$
$\displaystyle=\left(\sum_{j=1}^{N}|\beta_{j}|^{2}\right)\left(1-\sum_{j=1}^{N}|\beta_{j}|^{2}\right).$
(60)

Notice that

$\displaystyle(\Delta\hat{J}_{x})^{2}+(\Delta\hat{J}_{y})^{2}=\frac{N}{2}-\left(\sum_{j=1}^{N}|\beta_{j}|^{2}\right)\left(1-\left|\sum_{j=1}^{N}\beta_{j}\right|^{2}\right)$
(61)

which is independent of $\alpha$.

All the previous results can be simplified by introducing the mean quantities
$\overline{|\beta|^{2}}$ and $\left|\overline{\beta}\right|$ given in
Equations (25) and (26), respectively. The corresponding expressions are
directly reported in Section III.

##  Appendix B Considerations on Equations (34b) and (34c)

In this appendix, we show that, in our case, Equations (34a), (34b) and (34c)
cannot lead to useful conclusions.

In our case, the l.h.s. of Equation (34a) reads:

$\langle\hat{J}_{x}^{2}\rangle+\langle\hat{J}_{y}^{2}\rangle+\langle\hat{J}_{z}^{2}\rangle=\frac{N(N+2)}{4}-N^{2}\sigma_{\beta}^{2}$
(62)

where we defined the particle variance as

$\displaystyle\sigma_{\beta}^{2}$
$\displaystyle=\overline{|\beta|^{2}}-\left|\overline{\beta}\right|^{2},$ (63)
$\displaystyle=\frac{1}{N}\sum_{s=1}^{N-1}|\gamma_{s}|^{2}.$ (64)

As one may expect, Equation (34a) is true, and the equality sign holds for all
the symmetric states, with $\sigma_{\beta}=0$ or, equivalently, $f_{\rm
SR}=1$.

Concerning Equation (34b), if we study the following expression (similar
results can be obtained for the other combination of the involved
expectations):

$\langle\hat{J}_{x}^{2}\rangle+\langle\hat{J}_{y}^{2}\rangle-\frac{N}{2}\leq(N-1)(\Delta\hat{J}_{z})^{2},$
(65)

we derive:

$\displaystyle\left|\overline{\beta}\right|^{2}$
$\displaystyle\leq\overline{|\beta|^{2}}\left[1-(N-1)\overline{|\beta|^{2}}\right]$
(66)

or, equivalently:

$\sigma_{\beta}\geq\sqrt{N-1}\left(|\overline{\beta}|^{2}+\sigma_{\beta}^{2}\right).$
(67)

We can clearly see that these inequalities are nonlinear in
$\overline{|\beta|^{2}}$. Recalling Equations (25) and (26), we conclude that
they cannot lead to useful conditions, since the solution of (66) or (67)
depends on the $\beta_{j}^{2}$ and thus on $\Omega_{0}^{2}$, whereas the
$\beta_{j}$ have been obtained in the framework of the linear regime with
respect $\Omega_{0}$, see Equation (6).

Equation (34c) is never violated for our system. In fact, for instance, we
have:

$\displaystyle(N-1)[(\Delta\hat{J}_{x})^{2}+(\Delta\hat{J}_{y})^{2}]\geq\langle\hat{J}^{2}_{z}\rangle+\frac{N(N-2)}{4}$
(68)

which, using Equations (61) and (57), yields

$\displaystyle
N^{2}\overline{|\beta|^{2}}\left|\overline{\beta}\right|^{2}>0.$ (69)

## References

  * Lehmberg [1970] R. H. Lehmberg, Radiation from an $N$-atom system. I. General formalism, Physical Review A 2, 883 (1970).
  * van der Mark  _et al._ [1988] M. B. van der Mark, M. P. van Albada, and A. Lagendijk, Light scattering in strongly scattering media: Multiple scattering and weak localization, Phys. Rev. B 37, 3575 (1988).
  * Labeyrie  _et al._ [2003] G. Labeyrie, E. Vaujour, C. A. Mueller, D. Delande, C. Miniatura, D. Wilkowski, and R. Kaiser, Slow diffusion of light in a cold atomic cloud, Physical review letters 91, 223904 (2003).
  * Guerin  _et al._ [2017] W. Guerin, M. Rouabah, and R. Kaiser, Light interacting with atomic ensembles: collective, cooperative and mesoscopic effects, Journal of Modern Optics 64, 895 (2017).
  * Dicke [1954] R. H. Dicke, Coherence in spontaneous radiation processes, Physical Reveiw 93, 99 (1954).
  * Nourmandipour  _et al._ [2021] A. Nourmandipour, A. Vafafard, A. Mortezapour, and R. Franzosi, Entanglement protection of classically driven qubits in a lossy cavity, Scientific Reports 11, 16259 (2021).
  * Taghipour  _et al._ [2022] J. Taghipour, B. Mojaveri, and A. Dehghani, Witnessing entanglement between two two-level atoms coupled to a leaky cavity via two-photon relaxation, The European Physical Journal Plus 137, 772 (2022).
  * Mandel and Wolf [1995] L. Mandel and E. Wolf,  _Optical coherence and quantum optics_ (Cambridge university press, 1995).
  * Gross and Haroche [1982] M. Gross and S. Haroche, Superradiance: An essay on the theory of collective spontaneous emission, Physics Reports 93, 301 (1982).
  * Lodahl  _et al._ [2004] P. Lodahl, A. Floris van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals, Nature 430, 654 (2004).
  * Brandes [2005] T. Brandes, Coherent and collective quantum optical effects in mesoscopic systems, physics reports 408, 315 (2005).
  * Karasik  _et al._ [2007] R. I. Karasik, K.-P. Marzlin, B. C. Sanders, and K. B. Whaley, Multiparticle decoherence-free subspaces in extended systems, Physical Review A 76, 012331 (2007).
  * Pedersen and Mølmer [2009] L. H. Pedersen and K. Mølmer, Few qubit atom-light interfaces with collective encoding, Physical Review A 79, 012320 (2009).
  * Tong  _et al._ [2004] D. Tong, S. Farooqi, J. Stanojevic, S. Krishnan, Y. Zhang, R. Côté, E. Eyler, and P. Gould, Local blockade of rydberg excitation in an ultracold gas, Physical Review Letters 93, 063001 (2004).
  * Urban  _et al._ [2009] E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. Yavuz, T. Walker, and M. Saffman, Observation of rydberg blockade between two atoms, Nature Physics 5, 110 (2009).
  * Gaëtan  _et al._ [2009] A. Gaëtan, Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, Observation of collective excitation of two individual atoms in the rydberg blockade regime, Nature Physics 5, 115 (2009).
  * Scully  _et al._ [2006] M. Scully, E. Fry, C. Ooi, and K. Wodkiewicz, Directed spontaneous emission from an extended ensemble of $n$ atoms: Timing is everything, Physical Review Letters 96, 010501 (2006).
  * Svidzinsky  _et al._ [2008] A. A. Svidzinsky, J.-T. Chang, and M. O. Scully, Dynamical evolution of correlated spontaneous emission of a single photon from a uniformly excited cloud of $n$ atoms, Physical Review Letters 100, 160504 (2008).
  * Svidzinsky  _et al._ [2010] A. A. Svidzinsky, J.-T. Chang, and M. O. Scully, Cooperative spontaneous emission of $n$ atoms: Many-body eigenstates, the effect of virtual lamb shift processes, and analogy with radiation of $n$ classical oscillators, Physical Review A 81, 053821 (2010).
  * Courteille  _et al._ [2010] P. W. Courteille, S. Bux, E. Lucioni, K. Lauber, T. Bienaimé, R. Kaiser, and N. Piovella, Modification of radiation pressure due to cooperative scattering of light, he European Physical Journal D 58, 69 (2010).
  * Bienaimé  _et al._ [2010] T. Bienaimé, S. Bux, E. Lucioni, P. Courteille, N. Piovella, and R. Kaiser, Observation of a cooperative radiation force in the presence of disorder, Physical Review Letters 104, 183602 (2010).
  * Bienaimé  _et al._ [2013] T. Bienaimé, R. Bachelard, N. Piovella, and R. Kaiser, Cooperativity in light scattering by cold atoms, Fortschritte der Physik 61, 377 (2013).
  * Bienaimé  _et al._ [2012] T. Bienaimé, N. Piovella, and R. Kaiser, Controlled dicke subradiance from a large cloud of two-level systems, Physical Review Letters 108, 123602 (2012).
  * Guerin  _et al._ [2016] W. Guerin, M. O. Araújo, and R. Kaiser, Subradiance in a large cloud of cold atoms, Physical Review Letters 116, 083601 (2016).
  * Scully [2015] M. O. Scully, Single photon subradiance: quantum control of spontaneous emission and ultrafast readout, Physical review letters 115, 243602 (2015).
  * Svidzinsky  _et al._ [2015] A. A. Svidzinsky, X. Zhang, and M. O. Scully, Quantum versus semiclassical description of light interaction with atomic ensembles: Revision of the maxwell-bloch equations and single-photon superradiance, Physical Review A 92, 013801 (2015).
  * Cai  _et al._ [2016] H. Cai, D.-W. Wang, A. A. Svidzinsky, S.-Y. Zhu, and M. O. Scully, Symmetry-protected single-photon subradiance, Physical Review A 93, 053804 (2016).
  * Gegg  _et al._ [2018] M. Gegg, A. Carmele, A. Knorr, and M. Richter, Superradiant to subradiant phase transition in the open system dicke model: Dark state cascades, New Journal of Physics 20, 013006 (2018).
  * Tóth  _et al._ [2007] G. Tóth, C. Knapp, O. Gühne, and H. J. Briegel, Optimal spin squeezing inequalities detect bound entanglement in spin models, Physical review letters 99, 250405 (2007).
  * Akkermans  _et al._ [2008] E. Akkermans, A. Gero, and R. Kaiser, Photon localization and dicke superradiance in atomic gases, Physical review letters 101, 103602 (2008).
  * Bellando  _et al._ [2014] L. Bellando, A. Gero, E. Akkermans, and R. Kaiser, Cooperative effects and disorder: A scaling analysis of the spectrum of the effective atomic hamiltonian, Physical Review A 90, 063822 (2014).
  * Ellinger  _et al._ [1994] K. Ellinger, J. Cooper, and P. Zoller, Light-pressure force in n-atom systems, Physical Review A 49, 3909 (1994).
  * Bienaimé  _et al._ [2011] T. Bienaimé, M. Petruzzo, D. Bigerni, N. Piovella, and R. Kaiser, Atom and photon measurement in cooperative scattering by cold atoms, Journal of Modern Optics 58, 1942 (2011).
  * Scully and Svidzinsky [2010] M. O. Scully and A. A. Svidzinsky, The lamb shift—yesterday, today, and tomorrow, Science 328, 1239 (2010), https://science.sciencemag.org/content/328/5983/1239.full.pdf .
  * Kitagawa and Ueda [1993] M. Kitagawa and M. Ueda, Squeezed spin states, Phys. Rev. A 47, 5138 (1993).