# Superradiance of non-interacting atoms

M. Bojer  Quantum Optics and Quantum Information, Institut für Optik,
Information und Photonik, Friedrich-Alexander-Universität Erlangen-Nürnberg
(FAU), 91058 Erlangen, Germany  J. von Zanthier  Quantum Optics and Quantum
Information, Institut für Optik, Information und Photonik, Friedrich-
Alexander-Universität Erlangen-Nürnberg (FAU), 91058 Erlangen, Germany

###### Abstract

Fully-excited two-level atoms separated by less than the transition wavelength
cooperatively emit light in a short burst, a phenomenon called superradiance
by R. Dicke in 1954. The burst is characterized by a maximum intensity scaling
with the square of the number of atoms $N$ and a temporal width reduced by $N$
compared to the single atom spontaneous decay time. Both effects are usually
attributed to a synchronization of the electric dipole moments of the atoms
occurring during the process of light emission. Contrary to this explanation,
it was recently shown by use of a quantum path description that the peak
intensity results from the quantum correlations among the atoms when occupying
symmetric Dicke states. Here we investigate from this perspective the temporal
evolution of the ensemble, starting in the small sample limit, i.e., when the
atoms have mutual separations much smaller than the transition wavelength
$\lambda$ and pass down the ladder of symmetric Dicke states. In addition, we
explore the temporal evolution for the case of non-interacting atoms with
mutual separations much larger than $\lambda$. We show that in this case a
similar superradiant burst of the emitted radiation is observed if the quantum
correlations of the atoms are generated by conditional photon measurements
retaining the atomic ensemble within or close to the symmetric subspace.

A single two-level atom prepared in the excited state will exponentially decay
towards the ground state and spontaneously emit a photon with a characteristic
rate $\Gamma$. Assuming not only one atom but $N$ non-interacting atoms with
mutual separation much larger than the transition wavelength, i.e., with all
atoms radiating independently of each other, will lead to the same exponential
decay with identical decay rate as in the single atom case. This behavior
changes if the atoms are placed in close proximity to each other, i.e., closer
than the transition wavelength, such that they are subject to the dipole-
dipole interaction. In this case, instead of an exponential decay, the light
is emitted in a short burst, with a maximum intensity scaling with the square
of the number of atoms $N$ and a duration decreasing linearly with $N$ [1], a
phenomenon referred to as superradiance by R. Dicke in 1954 [2]. After this
seminal work, a plethora of theoretical investigations of superradiance and
the collective emission of light of atomic ensembles have been published [3,
4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], some of them confirmed also
experimentally [16, 17, 18, 19, 20, 21, 22, 23, 24]. Commonly, in the
literature, the square law of the peak intensity as well as the reduced decay
time is explained by a phase-locking of the electric dipole moments of the
individual atoms leading to synchronization and a single macroscopic dipole
building up during light emission. However, as already outlined by Dicke in
his original publication [2], the atomic ensemble radiates light while passing
down the ladder of symmetric Dicke states, whereby none of the Dicke states
possesses an electric dipole moment. This fact causes the interpretation of
synchronized atomic dipole moments or a macroscopic dipole moment to be highly
questionable.

In this paper, we recapitulate an alternative interpretation for the
superradiant light burst, valid in the small volume limit, that does not make
use of the synchronization of individual atomic dipole moments. Rather, it is
based on the build-up of atomic dipole-dipole correlations present for all
symmetric Dicke states, leading to a superradiant emission behavior due to
quantum path interferences among the corresponding quantum states [10, 25].
Moreover, we will show that a superradiant light burst can be observed also in
the case of distant non-interacting atoms if appropriate correlations among
the atoms are generated by measuring photons in the far field such that the
detection is unable to distinguish the individual photon source [26, 27, 28].
Here again, no atomic dipole moments are present or become phase-locked, since
the atoms are far distant and do not interact. We will investigate the role of
the measurement process and show that it projects the atomic states towards
the symmetric subspace each time a photon is recorded. To this aim, we will
derive the trace distance to the symmetric subspace before and after the
photon measurement.

The paper is organized as follows. In Sect. I we review Dicke’s model of
superradiance in the small sample limit and apply a quantum path formalism to
interpret the corresponding results. In Sect. II we focus on the case of
distant non-interacting atoms and show that a similar superradiant light burst
as in the small sample limit can be observed if the photons scattered by the
ensemble are consecutively measured in the far field. We will explain this
behavior again via a quantum path formalism of the contributing quantum
states. This approach is particularly apt to follow the dynamical evolution of
the atomic ensemble tailored by consecutive photon measurements. We also study
how the photon measurements project the atomic state towards the symmetric
subspace and in this way enhance the quantum correlations. In Sect. III we
finally conclude.

##  I Small sample superradiance

In this section we recount the phenomenon of superradiance in the small sample
limit as considered by Dicke in his seminal paper in 1954 [2]. We start by
introducing the Hamiltonian of $N$ independent non-interacting two-level atoms

$\hat{H}_{S}=\hbar\omega_{0}\sum_{\mu=1}^{N}\hat{S}_{z}^{(\mu)}\,$ (1)

where
$\hat{S}_{z}^{(\mu)}=\frac{1}{2}(\hat{S}_{+}^{(\mu)}\hat{S}_{-}^{(\mu)}-\hat{S}_{-}^{(\mu)}\hat{S}_{+}^{(\mu)})$
is the $z$-component of the pseudo-spin operator $\hat{\bm{S}}^{(\mu)}$ of the
$\mu$th atom, $\mu\in\\{1,...,N\\}$, and $\hat{S}_{\pm}^{(\mu)}$ are the
corresponding pseudo-spin raising and lowering operators, respectively.
Motivated by the Hamiltonian $\hat{H}_{S}$ we define the collective pseudo-
spin operators as

$\begin{split}\hat{\bm{S}}&=\sum_{\mu=1}^{N}\hat{\bm{S}}^{(\mu)}\;,\;\hat{S}_{z}=\sum_{\mu=1}^{N}\hat{S}_{z}^{(\mu)}\,,\\\
\hat{S}_{+}&=\sum_{\mu=1}^{N}\hat{S}_{+}^{(\mu)}\;,\;\hat{S}_{-}=\sum_{\mu=1}^{N}\hat{S}_{-}^{(\mu)}\,.\end{split}$
(2)

The eigenstates of $\hat{H}_{S}$ are characterized by the three quantum
numbers $J$, $M$ and $\alpha$. In this way, we can write the eigenstates as
$\ket{J,M,\alpha}$ which are also eigenstates of $\hat{S}_{z}$ as well as
$\hat{S}^{2}$, obeying the eigenvalue equations

$\displaystyle\hat{S}_{z}\ket{J,M,\alpha}$ $\displaystyle=\hbar
M\ket{J,M,\alpha}\,,$ (3) $\displaystyle\hat{S}^{2}\ket{J,M,\alpha}$
$\displaystyle=\hbar^{2}J(J+1)\ket{J,M,\alpha}\,,$ (4)

where $\alpha$ accounts for the degeneracy of a given value of $J$. The
degeneracy can be shown to be [2]

$\beta=\frac{N!(2J+1)}{(\frac{N}{2}+J+1)!(\frac{N}{2}-J)!}$ (5)

with $\alpha\in\\{1,2,...,\beta\\}$, whereas the different values of $J$ and
$M$ are given by

$\displaystyle J$ $\displaystyle\in\left\\{0,1,...,\frac{N}{2}\right\\}$
$\displaystyle\text{for }N\text{ even}\,,$ (6) $\displaystyle J$
$\displaystyle\in\left\\{\frac{1}{2},\frac{3}{2},...,\frac{N}{2}\right\\}$
$\displaystyle\text{for }N\text{ uneven}\,,$ (7) $\displaystyle M$
$\displaystyle\in\\{-J,-J+1,...,J\\}\,.$ (8)

The states $\ket{J,M,\alpha}$ are the so-called Dicke states. In the following
we restrict ourselves to the symmetric subspace characterized by $J=N/2$.
Here, we find $\beta=1$, so that we can omit the third index and simply write
$\ket{J,M}$ ($J=N/2$).
In his seminal paper Dicke considered $N$ interacting atoms confined to a
volume smaller than the transition wavelength cubed. As a consequence one can
neglect the phase factors of the electromagnetic field in the interaction
Hamiltonian [2, 29]. However, we note that this symmetry property only holds
for specific configurations of the atomic ensemble [1]. Assuming the
corresponding symmetry to be fulfilled, the total Hamiltonian including the
light-matter interaction in the dipole approximation commutes with
$\hat{S}^{2}$ and $\hat{S}_{z}$. As a consequence, the time evolution does not
mix different $J$ subspaces. Correspondingly, starting with a state in the
symmetric subspace, the time evolution will retain the state in the symmetric
subspace at any later times. The corresponding master equation in the
interaction picture reads [29]

$\partial_{t}\hat{\rho}=-\gamma(\hat{S}_{+}\hat{S}_{-}\hat{\rho}+\hat{\rho}\hat{S}_{+}\hat{S}_{-}-2\hat{S}_{-}\hat{\rho}\hat{S}_{+})\,,$
(9)

where $\gamma=\Gamma/2$ is the half decay rate. By projecting Eq. (9) onto the
basis $\ket{J,M}$ we obtain the following rate equations for the diagonal
elements of the density matrix [1, 29]

$\begin{split}\partial_{t}\rho_{M,M}=&-2\gamma(J+M)(J-M+1)\rho_{M,M}\\\
&+2\gamma(J+M+1)(J-M)\rho_{M+1,M+1}\,,\end{split}$ (10)

where $\rho_{M,M}=\braket{J,M}{\hat{\rho}}{J,M}$. Note that
$2\gamma(J+M)(J-M+1)$ is the rate with which the state $\ket{J,M}\bra{J,M}$
decays to the state $\ket{J,M-1}\bra{J,M-1}$ and $2\gamma(J+M+1)(J-M)$ is the
rate with which it gets populated by the state $\ket{J,M+1}\bra{J,M+1}$.

Assuming the system to be initially in the fully excited state
$\ket{N/2,N/2}$, the rate equations can be solved analytically. One finds that
the system descends the ladder of symmetric Dicke states $\ket{N/2,M}$, with
$N/2\geq M\geq-N/2$, with a radiated intensity not following the usual
exponential decay as in the case of a single atom but displaying a short burst
with a peak intensity proportional to $N^{2}$ and a width inversely
proportional to $N$ (see Fig. 1).

Figure 1: Intensity $I(t)$ against $\gamma t$ for an initially fully excited
system $\ket{N/2,N/2}$ in case of a small sample dynamics for three different
total numbers of atoms $N$. Due to the collective emission behavior one
obtains a short burst instead of an exponential decay.

The usual explanation for this behavior is the build up of a macroscopic
dipole moment emitting the light coherently with increasing amplitude. As can
be seen from Eq. (10), the time evolution of the atomic ensemble involves only
the population of symmetric Dicke states. Since each Dicke state $\ket{J,M}$
does not carry any dipole moment, i.e., $\braket{\hat{S}_{\pm}}=0$, this
explanation appears highly questionable. In what follows, we will show that
the outlined evolution of the radiated intensity can be attributed to the
build-up of dipole-dipole correlations among the atoms resulting from the
entanglement of the atoms in the symmetric Dicke states $\ket{J,M}$ [25]. To
this aim, we give a short explanation of the production of dipole-dipole
correlations and the resulting intensity evolution in terms of a quantum path
interpretation [10].

Neglecting proportionality constants, the radiated intensity is given by [10]

$\begin{split}I=&\braket{\hat{S}_{+}\hat{S}_{-}}=\sum_{\mu,\nu}\braket{\hat{S}_{+}^{(\mu)}\hat{S}_{-}^{(\nu)}}\\\
=&\sum_{\mu}\braket{\hat{S}_{+}^{(\mu)}\hat{S}_{-}^{(\mu)}}+\sum_{\begin{subarray}{c}\mu,\nu\\\
\mu\neq\nu\end{subarray}}\braket{\hat{S}_{+}^{(\mu)}}\braket{\hat{S}_{-}^{(\nu)}}\\\
&+\sum_{\begin{subarray}{c}\mu,\nu\\\
\mu\neq\nu\end{subarray}}\left(\braket{\hat{S}_{+}^{(\mu)}\hat{S}_{-}^{(\nu)}}-\braket{\hat{S}_{+}^{(\mu)}}\braket{\hat{S}_{-}^{(\nu)}}\right)\,,\end{split}$
(11)

where the first part describes the incoherent contribution of each atom to the
intensity, the second part the coherent contribution from the non-vanishing
dipole moments and the third part characterizes the quantum correlations. In
the case of Dicke states Eq. (11) reduces to [10]

$I=\sum_{\mu}\braket{\hat{S}_{+}^{(\mu)}\hat{S}_{-}^{(\mu)}}+\sum_{\begin{subarray}{c}\mu,\nu\\\
\mu\neq\nu\end{subarray}}\braket{\hat{S}_{+}^{(\mu)}\hat{S}_{-}^{(\nu)}}\,,$
(12)

since $\braket{\hat{S}_{\pm}^{(\mu)}}=0$. This shows that there is no coherent
contribution to the radiated intensity which can be assigned to a dipole
moment of the atoms, but only an incoherent contribution stemming from the
dipole-dipole correlations.

As an illustration of the quantum path formalism we start by looking at the
symmetric Dicke state $\ket{1,0}$ for $N=2$ atoms, i.e.,
$\ket{1,0}=\frac{1}{\sqrt{2}}(\ket{e,g}+\ket{g,e})$, where $\ket{e}$ and
$\ket{g}$ denote the excited and ground state of the individual atoms,
respectively. Plugging this state into Eq. (12), we find

$I=1+1=2\,.$ (13)

Compared to the separable states $\ket{e,g}$ or $\ket{g,e}$ alone, i.e.,
states with the same number of atoms in the excited state but no entanglement,
the intensity of the symmetric Dicke state is enhanced. This results from the
quantum correlations among the atoms due to the entanglement of the state
$\ket{1,0}$ [25]. More physical insight into the contribution of the
individual terms is gained by a quantum path interpretation [10] (see Fig. 2):

Figure 2: Quantum paths that lead to a detection event. Filled out dots
correspond to atoms being in the excited state, empty dots correspond to atoms
being in the ground state.

In order to record a photon, the photon must be either emitted by the first
atom or the second atom leading to the two quantum paths illustrated in Fig.
2. After the detection of the photon has taken place the state of the atomic
ensemble is the total ground state $\ket{g,g}$. Since the two quantum paths
drawn in the figure lead to the same final state and are added coherently in
the initial state $\ket{1,0}$, the two quantum paths interfere with each
other. Setting the electric field amplitudes to $1$ [see Eq. (11)], we can now
find the radiated intensity: The state $\ket{1,0}$ is the sum of the two
product states $\ket{e,g}$ and $\ket{g,e}$, each having a single excitation; a
detection event leads only to one final state $\ket{g,g}$; therefore, the
number of quantum paths per final state is

$\begin{split}&\frac{\text{\\# product states}\cdot\text{\\# quantum paths per
product state}}{\text{\\# different final states}}\\\ &=\frac{2\cdot
1}{1}=2\,.\end{split}$ (14)

When calculating the total intensity, i.e., taking the modulus square of the
sum of the transition amplitudes, each of the quantum paths is multiplied with
itself (incoherent contribution) and in addition interferes with the other
path (interference term), leading in total to $2^{2}$ quantum path pairs.
Taking the normalization of the state $\ket{1,0}$ into account, the radiated
intensity thus calculates to $(1/\sqrt{2})^{2}\cdot 2^{2}\cdot 1=2$, where
$2^{2}$ is the number of quantum path pairs and $1$ is the number of different
final states. In this way, by just counting the number of quantum path pairs
we obtain the same value as by a straightforward mathematical calculation
leading to Eq. (13).

We now apply the quantum path formalism to an arbitrary symmetric Dicke state
$\ket{J,M}$ and use the results obtained above to explain the phenomenon of
superradiance in the small sample limit. We start by noting that
$M=J-n_{g}=N/2-n_{g}$, where $n_{g}$ denotes the number of atoms in the ground
state of the state $\ket{J,M}$. The intensity radiated by an atomic ensemble
in the state $\ket{J,M}$ can then be calculated to [10]

$I=[\mathcal{N}\cdot\mathcal{P}_{\text{no.}}^{\text{pair}}\cdot
f_{\text{no.}}]_{\ket{J,M}}\,,$ (15)

where $\mathcal{N}$ denotes the squared normalization of the state
$\ket{J,M}$, $\mathcal{P}_{\text{no.}}^{\text{pair}}$ is the number of quantum
path pairs leading to the same final state and $f_{\text{no.}}$ is the number
of different final states. In the case of a symmetric Dicke state $\ket{J,M}$
we find that $\mathcal{N}=\binom{N}{n_{g}}^{-1}$ and
$f_{\text{no.}}=\binom{N}{n_{g}+1}$. The number of quantum path pairs
$\mathcal{P}_{\text{no.}}^{\text{pair}}$ needs more explanation. The state
$\ket{J,M}$ contains $\binom{N}{n_{g}}$ product states, where for each of
these product states one has $N-n_{g}$ different quantum paths leading to a
detection event. Thus, one finds for Eq. (14) [10]

$\binom{N}{n_{g}}\cdot(N-n_{g})/f_{\text{no.}}=\binom{N}{n_{g}}\cdot(N-n_{g})/\binom{N}{n_{g}+1}=n_{g}+1$
(16)

single quantum paths per final state. Each of these paths is multiplied with
itself and in addition interferes with all the other quantum paths leading in
total to $\mathcal{P}_{\text{no.}}^{\text{pair}}=(n_{g}+1)^{2}$ quantum path
pairs. We thus find for the intensity [10]

$\begin{split}I&=\binom{N}{n_{g}}^{-1}\cdot(n_{g}+1)^{2}\cdot\binom{N}{n_{g}+1}\\\
&=(N-n_{g})(n_{g}+1)=(J+M)(J-M+1)\;.\end{split}$ (17)

Note that this expression, found by counting the different interfering quantum
paths leading to a detection event, gives the same result as derived from the
quantum mechanical calculation
$\braket{\hat{S}_{+}\hat{S}_{-}}=\braket{J,M}{\hat{S}_{+}\hat{S}_{-}}{J,M}=(J+M)(J-M+1)$,
presented by Dicke in his seminal paper [2]. However, the quantum path
interference interpretation gives a more transparent explanation apt to
display the physical origin of the observed superradiant intensity.

The quantum path interpretation can be applied also to compute the time
evolution of the intensity in the small sample limit. Since the master
equation (9) causes the atomic ensemble to descend the ladder of symmetric
Dicke states, the time evolved state can be written as

$\hat{\rho}(t)=\sum_{M=-J}^{J}\rho_{M,M}(t)\ket{J,M}\bra{J,M}\,.$ (18)

The temporal evolution of the intensity can thus be expressed in mathematical
terms as

$I(t)=\operatorname{Tr}[\hat{S}_{+}\hat{S}_{-}\hat{\rho}(t)]=\sum_{M=-J}^{J}\rho_{M,M}(t)\braket{J,M}{\hat{S}_{+}\hat{S}_{-}}{J,M}\,.$
(19)

Applying the quantum path interpretation to each state in the sum of Eq. (19)
the intensity $I(t)$ at any time $t$ can be computed. Thus, the quantum path
formalism is able to explain the entire superradiant emission dynamics of the
atomic ensemble in the small sample limit. In the next section we show that
one can get a similar behavior in the case of distant non-interacting atoms,
where the build up of quantum correlations is not caused by the time evolution
of the atomic ensemble, but by consecutive photon measurements.

##  II Superradiance of far distant independent atoms

In this section we investigate the radiated intensity in the case of $N$ two-
level atoms with mutual separations much larger than the transition
wavelength, such that dipole-dipole interactions can be neglected. In this
limit, the atomic ensemble, e.g., starting from the fully excited state, does
not stay in the subspace of symmetric Dicke states. Rather, in the course of
photon emissions, the system will leave the symmetric subspace populating also
non-symmetric Dicke states. However, by performing photon measurements at
particular positions the atomic system can be forced back towards the
symmetric subspace [11, 28, 25]. Measuring the photons consecutively in time
the temporal evolution of the photon emission displays again a similar peaked
intensity profile as in the case of the small atomic sample. The physical
explanation for this behavior can once again be obtained employing the quantum
path formalism.

Note that a measurement of a photon does not create a dipole moment in the
atomic ensemble. Neither does a macroscopic dipole moment build up in the
course of the temporal evolution of the system due to atom-atom interactions,
since the atoms are far distant from each other. The observed superradiant
emission behavior thus results again only from the quantum correlations and
the entanglement of the atomic ensemble, in this case generated by photon
detection. This occurs even for atoms with separations much larger than the
transition wavelength known under the name of entanglement at a distance [30,
31, 32, 33, 34, 35, 36, 37, 38].

###  II.1 Master equation and differential equations for the density matrix
elements

Since we consider atoms with mutual separations much larger than the
transition wavelength any interactions between the particles can be neglected
. The master equation in the interaction picture then contains only the
spontaneous decay term of the individual atoms. For $N$ atoms it reads [29]

$\partial_{t}\hat{\rho}=-\gamma\sum_{\mu=1}^{N}(\hat{S}_{+}^{(\mu)}\hat{S}_{-}^{(\mu)}\hat{\rho}+\hat{\rho}\hat{S}_{+}^{(\mu)}\hat{S}_{-}^{(\mu)}-2\hat{S}_{-}^{(\mu)}\hat{\rho}\hat{S}_{+}^{(\mu)})\,.$
(20)

Next, we project the master equation onto an orthonormal basis, in this case
the bare basis given by the tensor product of the single atom states. Let
$\ket{\alpha}$ and $\ket{\beta}$ denote two states of this orthonormal basis.
The projection then reads (see App. A)

$\begin{split}\partial_{t}\rho_{\alpha,\beta}(t)=&\;2\gamma\sum_{\mu=1}^{N}\rho_{\eta^{(\mu)},\delta^{(\mu)}}(t)\\\
&-\gamma\sum_{\mu=1}^{N}\rho_{S_{+}^{(\mu)}\xi,\beta}(t)-\gamma\sum_{\mu=1}^{N}\rho_{\alpha,S_{+}^{(\mu)}\zeta}(t)\,,\end{split}$
(21)

where the indices of the density matrix elements on the right hand side of Eq.
(21) indicate the following conditions

$\begin{split}\text{1.
term:}\quad&\hat{S}_{-}^{(\mu)}\ket{\eta}=\ket{\alpha}\,,\,\hat{S}_{-}^{(\mu)}\ket{\delta}=\ket{\beta}\,,\\\
\text{2. term:}\quad&\hat{S}_{+}^{(\mu)}\ket{\xi}=\ket{\alpha}\,,\\\ \text{3.
term:}\quad&\hat{S}_{+}^{(\mu)}\ket{\zeta}=\ket{\beta}\;.\end{split}$ (22)

Note that only if the conditions of Eq. (22) are fulfilled a non-zero
contribution from the individual terms of the right hand side of Eq. (21) is
obtained.

In what follows we denote the states of the bare basis of the atoms with
respect to the number of atoms being in the ground state. I.e., we define the
fully excited state by $\ket{e}\coloneqq\ket{e,e,...,e}$, whereas the states
where atom $j$ is in the ground state and all the other atoms are in the
excited state is denoted by
$\ket{g_{j}^{(1)}}\coloneqq\hat{S}_{-}^{(j)}\ket{e}$. States with an arbitrary
number of atoms in the ground state are defined and denoted analogously.

As an example for the application of Eqs. (21) and (22), we calculate the rate
equation for the matrix element $\rho_{e,e}(t)$. Since it is a diagonal
element, the two conditions for the first term reduce to one condition
$\hat{S}_{-}^{(\mu)}\ket{\eta}=\hat{S}_{-}^{(\mu)}\ket{\delta}=\ket{e}$;
however, since the state $\ket{e}$ is the fully excited state there exist no
states $\ket{\eta},\ket{\delta}$ which fulfill this condition, so this term
vanishes. The second and third condition merge to the same condition
$\hat{S}_{+}^{(\mu)}\ket{\xi}=\hat{S}_{+}^{(\mu)}\ket{\zeta}=\ket{e}$. Here,
it is obvious that there exists only one state which fulfils this condition,
namely $\ket{g_{\mu}^{(1)}}$. Moreover, the summation over all atoms gives a
factor of $N$ for each of the two terms. Alternatively, to follow more closely
the physical process, one can interpret the equation
$\hat{S}_{+}^{(\mu)}\ket{\xi}=\hat{S}_{+}^{(\mu)}\ket{\zeta}=\ket{e}$ with
$\mu\in\\{1,2,...,N\\}$ as a dynamical variable. In this case, one finds
immediately the $N$ states which fulfill this condition, namely
$\ket{g_{\mu}^{(1)}}$. In both cases, the rate equation for the matrix element
$\rho_{e,e}(t)$ reads

$\partial_{t}\rho_{e,e}(t)=-2\gamma N\rho_{e,e}(t)\,,$ (23)

which, since every atom decays independently, just leads to the well-known
exponential decay from the state $\rho_{e,e}(t)$ with a rate $2\gamma N$.

Applying this formalism to all the other states characterized by the number of
ground state atoms on finds a set of first-order differential equations for
the density matrix elements. Assuming a number of reasonable restrictions on
the initial conditions (see App. A), one can reduce the number of differential
equations, which can then be solved analytically to get the full consecutive
time evolution of the density matrix upon conditional measurements of photons
in the forward direction intersecting the free time evolutions.

###  II.2 Intensity correlation functions

Instead of solving the time evolution of the density matrix, let us
investigate multi-photon measurements at particular locations and at
consecutive times as outlined in the introduction to Sect. II above. Such
conditional multi-modal photon measurements are described by higher-order
intensity correlation functions, where the $m$th-order intensity correlation
function is defined as [39]

$G_{\hat{\rho}}^{(m)}(\bm{r}_{1},t_{1},...,\bm{r}_{m},t_{m})=\braket{\hat{E}^{(-)}(\bm{r}_{1},t_{1})...\hat{E}^{(-)}(\bm{r}_{m},t_{m})\hat{E}^{(+)}(\bm{r}_{m},t_{m})...\hat{E}^{(+)}(\bm{r}_{1},t_{1})}_{\hat{\rho}}\,.$
(24)

Here, $\braket{\cdot}_{\hat{\rho}}$ denotes the expectation value with respect
to the quantum state $\hat{\rho}$ and

$\hat{E}^{(+)}(\bm{r}_{s},t_{s})=\sum_{\mu=1}^{N}e^{-ik_{0}\hat{\bm{r}}_{s}\bm{R}_{\mu}}\hat{S}_{-}^{(\mu)}(t_{s})$
(25)

is the positive frequency part of the electric field operator evaluated at
position $\bm{r}_{s}$ and time $t_{s}$. Note that in Eq. (25), we set the
electric field amplitude to unity and used the scalar notation since we assume
that the transition dipole matrix elements of the atoms are perpendicular to
the detection plane. Further, $k_{0}$ is the wave number, $\hat{\bm{r}}_{s}$
is the unit vector in the direction of observation, and $\bm{R}_{\mu}$ and
$\hat{S}_{-}^{(\mu)}(t_{s})$ are the position and the pseudo-spin lowering
operator of the $\mu$th atom in the Heisenberg picture, respectively. Since
Eq. (24) represents a conditional multi-modal measurement, it can also be
written in the form

$G_{\hat{\rho}}^{(m)}(\bm{r}_{1},t_{1},...,\bm{r}_{m},t_{m})=G_{\hat{\rho}_{m-1}}^{(1)}(\bm{r}_{m},t_{m})\cdot
G_{\hat{\rho}}^{(m-1)}(\bm{r}_{1},t_{1},...,\bm{r}_{m-1},t_{m-1})$ (26)

where [26, 11, 40]

$\hat{\rho}_{m-1}=\frac{\hat{E}^{(+)}(\bm{r}_{m-1},t_{m-1})...\hat{E}^{(+)}(\bm{r}_{1},t_{1})\hat{\rho}\hat{E}^{(-)}(\bm{r}_{1},t_{1})...\hat{E}^{(-)}(\bm{r}_{m-1},t_{m-1})}{\mathrm{Tr}[\hat{E}^{(+)}(\bm{r}_{m-1},t_{m-1})...\hat{E}^{(+)}(\bm{r}_{1},t_{1})\hat{\rho}\hat{E}^{(-)}(\bm{r}_{1},t_{1})...\hat{E}^{(-)}(\bm{r}_{m-1},t_{m-1})]}$
(27)

is the state after $m-1$ photon measurements at positions
$\bm{r}_{1},\ldots,\bm{r}_{m-1}$ and times $t_{1},\ldots,t_{m-1}$, with
$\hat{\rho}_{0}=\hat{\rho}$. Note that $G_{\hat{\rho}_{m-1}}^{(1)}$ depends on
all former $m-1$ photon detection positions and detection times, since
$\hat{\rho}_{m-1}$ depends on all former $m-1$ photon detection events.

In what follows, we assume for simplicity that the atoms are arranged along
the $x$-axis and that all detectors are located in the far field at the same
position on the $y$-axis. In addition, we suppose the atomic dipole moments
pointing towards the $z$-axis, i.e., perpendicular to the detection plane
($xy$-plane). The intensity correlation functions then only depend on the
different detection times $t_{1},\ldots,t_{m-1}$, but not on the particular
detection positions $\bm{r}_{1},\ldots,\bm{r}_{m-1}$, since the positive
electric field operator in the far field in this case is simply given by
$\hat{E}^{(+)}(\bm{r}_{s},t_{s})=\hat{S}_{-}(t_{s})$ [see Eq. (25)]. In this
way, neither geometric phases nor atomic dipole patterns need to be accounted
for in the calculation of
$G_{\hat{\rho}}^{(m)}(\bm{r}_{1},t_{1},...,\bm{r}_{m},t_{m})$. As a
consequence, we can calculate simply the multi-time correlations of the
collective pseudo-spin raising and lowering operators. This allows in
particular to compare the result with the small sample case discussed in Sect.
I.

Rewriting Eq. (26) as

$G_{\hat{\rho}_{m-1}}^{(1)}(t_{m})=\frac{G_{\hat{\rho}}^{(m)}(t_{1},...,t_{m})}{G_{\hat{\rho}}^{(m-1)}(t_{1},...,t_{m-1})}\,,$
(28)

where we have already neglected the position dependency, and applying Eq. (62)
of App. B involving the quantum regression theorem [29, 40] leads to

$\begin{split}&G_{\rho_{m-1}}^{(1)}(t_{m})=\frac{\prod_{i=1}^{m}e^{-2\gamma
t_{i}}\braket{\overbrace{\hat{S}_{+}(0)...\hat{S}_{+}(0)}^{m\text{
times}}\overbrace{\hat{S}_{-}(0)...\hat{S}_{+}(0)}^{m\text{
times}}}}{\prod_{i=1}^{m-1}e^{-2\gamma
t_{i}}\braket{\underbrace{\hat{S}_{+}(0)...\hat{S}_{+}(0)}_{m-1\text{
times}}\underbrace{\hat{S}_{-}(0)...\hat{S}_{+}(0)}_{m-1\text{ times}}}}\\\
&=e^{-2\gamma
t_{m}}\braket{\hat{S}_{+}(0)\hat{S}_{-}(0)}_{\hat{\rho}_{m-1}}=e^{-2\gamma
t_{m}}G_{\hat{\rho}_{m-1}}^{(1)}(0)\,.\end{split}$ (29)

Therefore, to compute $G_{\rho_{m-1}}^{(1)}(t_{m})$ we only have to calculate
the correlation function $G_{\hat{\rho}_{m-1}}^{(1)}(0)$ at time $t=0$.

Assuming an initially fully excited system, the correlation function at time
$t=0$ is simple to derive. This is due to the fact that for
$\hat{\rho}=\ket{e}\bra{e}$ the state after $m$ photon detections at $t=0$ is
given by $\hat{\rho}_{m}=\ket{J,M}\bra{J,M}$, where $J=N/2$ and $M=N/2-m$,
i.e., the measurement causes the system to descent down the ladder of
symmetric Dicke states. In this way, we obtain for
$G_{\hat{\rho}_{m-1}}^{(1)}(0)$

$\begin{split}&\braket{\hat{S}_{+}(0)\hat{S}_{-}(0)}_{\hat{\rho}_{m-1}}=\braket{J,M}{\hat{S}_{+}\hat{S}_{-}}{J,M}\\\
&=(J+M)(J-M+1)=m\cdot(N-m+1)\,.\end{split}$ (30)

In Fig. 3, the conditional intensities $G_{\hat{\rho}_{m-1}}^{(1)}(t_{m})$ are
plotted against $\gamma t$ for the initially fully excited system. The
different intersections of the conditional intensity correspond to the
consecutive times at which a spontaneously emitted photon is detected.

Figure 3: Conditional intensities $G_{\rho_{m-1}}^{(1)}(t_{m})$ against
$\gamma t$ for an initially fully excited system $\ket{e}$ with $m=1$ to $m=N$
in consecutive steps for three different $N$.

We see that the measurement process leads to steps in the conditional
intensity, resulting from the quantum correlations among the atoms produced
each time a photon is recorded. For instance, for $N=5$, the five consecutive
conditional intensities $G_{\hat{\rho}_{m-1}}^{(1)}(t_{m})$ are given by [see
Eq. (30)]

$\begin{split}G_{\hat{\rho}_{0}}^{(1)}(t_{1})&=5\cdot e^{-2\gamma t_{1}}\,,\\\
G_{\hat{\rho}_{1}}^{(1)}(t_{2})&=8\cdot e^{-2\gamma t_{2}}\,,\\\
G_{\hat{\rho}_{2}}^{(1)}(t_{3})&=9\cdot e^{-2\gamma t_{3}}\,,\\\
G_{\hat{\rho}_{3}}^{(1)}(t_{4})&=8\cdot e^{-2\gamma t_{4}}\,,\\\
G_{\hat{\rho}_{4}}^{(1)}(t_{5})&=5\cdot e^{-2\gamma t_{5}}\,.\end{split}$ (31)

From Eq. (31), we see that after every intersection the intensity follows the
usual single-atom decay at a rate $2\gamma$ resulting from the fact that the
atoms are non-interacting (see Fig. 3). However, due to the increase of the
corresponding coefficients, as calculated in Eq. (30), a superradiant burst
like the one in the small sample limit is observed, even for distant atoms, if
conditional photon measurements are considered. In the next section we show
that this superradiant behavior can be explained again by the quantum path
formalism.

###  II.3 Quantum path interference interpretation

Starting from the differential equations for the density matrix elements in
the case of distant atoms [see Eq. (21) and App. A] we can analytically solve
the time evolution of the density operator and in this way compute the
temporal evolution of the intensity in the forward direction as considered in
Sect. II.2. Alternatively, the latter can be derived employing the quantum
path formalism. This allows to give a transparent physical explanation of the
time-dependent intensity, in particular if conditional measurements are
involved.

We assume again that the initial state is the fully excited state $\ket{e}$.
For this state we obviously have $\mathcal{N}=1$,
$\mathcal{P}_{\text{no.}}^{\text{pair}}=1$ and $f_{\text{no.}}=N$, so that the
initial coefficient equals $N$ [see Eq. (15)]. Further, from the differential
equations for the density matrix elements we obtain for the density operator
$\hat{\rho}_{0}(t_{1})$ at time $t_{1}$

$\hat{\rho}_{0}(t_{1})=p_{e}(t_{1})\ket{e}\bra{e}+p_{g^{(1)}}(t_{1})\sum_{j}\ket{g_{j}^{(1)}}\bra{g_{j}^{(1)}}+p_{g^{(2)}}(t_{1})\sum_{j<k}\ket{g_{jk}^{(2)}}\bra{g_{jk}^{(2)}}+\dots\,,$
(32)

where the probabilities, obtained by solving the differential equations in
App. A, are given by

$\begin{split}p_{e}(t_{1})&=e^{-2N\gamma t_{1}}\\\
p_{g^{(1)}}(t_{1})&=e^{-2N\gamma t_{1}}\cdot(e^{2\gamma t_{1}}-1)\\\
p_{g^{(2)}}(t_{1})&=e^{-2N\gamma t_{1}}\cdot(e^{2\gamma t_{1}}-1)^{2}\\\
&\phantom{h}\vdots\\\ p_{g^{(s)}}(t_{1})&=e^{-2N\gamma t_{1}}\cdot(e^{2\gamma
t_{1}}-1)^{s}\,,\end{split}$ (33)

with $\ket{g^{(0)}}=\ket{e}$. We note that due to the particular initial
conditions (see App. A) and the photon measurement in forward direction all
states with the same number of ground state atoms evolve with the same time
behavior; in this case, we can write the probabilities listed in Eq. (33)
simply in front of the individual sums appearing in Eq. (32).

Next, we derive the intensity radiated in the forward direction at time
$t_{1}$ using in Eq. (32) the quantum path interference interpretation for
each term with the same number of ground state atoms. For each of these terms
we have $\mathcal{N}=1$. Thus, we only need to compute for each of these terms
the number of interfering quantum path pairs and the number of final states.
For a single product state with $n_{g}$ atoms in the ground state we have
$N-n_{g}$ excited states, i.e., $N-n_{g}$ possible single quantum paths.
Further, for each product state we have exactly $\binom{N-n_{g}}{1}$ different
final states, so that $\mathcal{P}_{\text{no.}}^{\text{pair}}=1^{2}$.
Consequently, the coefficient in front of each block of the density matrix
with the same number of ground state atoms becomes according to Eq. (15)

$\binom{N}{n_{g}}\cdot\mathcal{N}\cdot\mathcal{P}_{\text{no.}}^{\text{pair}}\cdot
f_{\text{no.}}=\binom{N}{n_{g}}\cdot 1\cdot 1^{2}\cdot\binom{N-n_{g}}{1}\,,$
(34)

where $\binom{N}{n_{g}}$ is the number of terms in each block with the same
number of ground state atoms. The correlation function for the first
measurement thus reads

$\displaystyle G_{\hat{\rho}_{0}}^{(1)}(t_{1})$
$\displaystyle=\sum_{s=0}^{N-1}e^{-2N\gamma t_{1}}\cdot(e^{2\gamma
t_{1}}-1)^{s}\cdot\binom{N}{s}\cdot 1\cdot 1^{2}\cdot\binom{N-s}{1}$
$\displaystyle=1\cdot(N-1+1)e^{-2\gamma t_{1}}=Ne^{-2\gamma t_{1}}\,,$ (35)

where we wrote the second-last expression in Eq. (35) as in Eq. (30).

To find the state after the first measurement, we have to apply the collective
spin lowering and raising operators onto the time-evolved state Eq. (32). The
individual states in Eq. (32) then transform to

$\begin{split}\ket{g^{(0)}}&\rightarrow\ket{g^{(0)}_{S_{-}}}\coloneqq
S_{-}\ket{g^{(0)}}=\sum_{j=1}^{N}S_{-}^{(j)}\ket{e}=\sum_{j=1}^{N}\ket{g_{j}^{(1)}}\\\
\ket{g_{j}^{(1)}}&\rightarrow\ket{g_{j,S_{-}}^{(1)}}\coloneqq
S_{-}\ket{g_{j}^{(1)}}=\sum_{k=1}^{N}S_{-}^{(k)}\ket{g_{j}^{(1)}}=\sum_{\begin{subarray}{c}k=1\\\
k\neq j\end{subarray}}^{N}\ket{g_{jk}^{(2)}}\\\
\ket{g_{jk}^{(2)}}&\rightarrow\ket{g_{jk,S_{-}}^{(2)}}\coloneqq
S_{-}\ket{g_{jk}^{(2)}}=\sum_{l=1}^{N}S_{-}^{(l)}\ket{g_{jk}^{(2)}}=\sum_{\begin{subarray}{c}l=1\\\
l\neq j,l\neq k\end{subarray}}^{N}\ket{g_{jkl}^{(3)}}\\\
&\phantom{h}\vdots\end{split}$ (36)

Solving the corresponding differential equations we can write the time-evolved
state after the first photon measurement as

$\hat{\rho}_{1}(t_{2})=\frac{1}{N}\left\\{p_{g^{(0)}_{S_{-}}}(t_{2})\ket{g^{(0)}_{S_{-}}}\bra{g^{(0)}_{S_{-}}}+p_{g_{S_{-}}^{(1)}}(t_{2})\sum_{j}\ket{g_{j,S_{-}}^{(1)}}\bra{g_{j,S_{-}}^{(1)}}+p_{g_{S_{-}}^{(2)}}(t_{2})\sum_{j,k}\ket{g_{jk,S_{-}}^{(2)}}\bra{g_{jk,S_{-}}^{(2)}}+\dots\right\\}\,,$
(37)

where the probabilities are independent of the index and given by

$p_{g^{(s)}_{S_{-}}}(t_{2})=e^{-2(N-1)\gamma t_{2}}(e^{2\gamma
t_{2}}-1)^{s}\,.$ (38)

Using Eq. (37), we can again apply the quantum path interference formalism to
calculate the intensity after measurement of the first photon. The
normalization is $\mathcal{N}=1/N$. Furthermore, a state originating from a
previous state with $N-n_{g}+1$ excitations is the sum of $N-n_{g}+1$
different states giving rise to interference. Note that here $n_{g}$ denotes
the number of atoms being in the ground state after the first measurement, not
the initial number of ground state atoms, i.e., $1\leq n_{g}\leq N-1$. Since
we consider the first-order intensity correlation function after the
measurement of the first photon, i.e., the conditional second-order
correlation function [see Eq. (28)] we have to distribute two ground states
among all initially excited atoms, leading to $\binom{N-n_{g}+1}{2}$ different
final states. Again, we have $N-n_{g}$ quantum paths for each different
product state with $n_{g}$ atoms in the ground state (see above). Thus, we
find $\mathcal{P}_{\text{no.}}^{\text{pair}}=2^{2}=4$ interfering quantum path
pairs so that the coefficient of each block reads

$\binom{N}{n_{g}-1}\cdot\frac{1}{N}\cdot 2^{2}\cdot\binom{N-n_{g}+1}{2}\,.$
(39)

Finally, to calculate the conditional second-order intensity correlation
function, i.e., the first-order intensity correlation function after the
measurement of the first photon, we have to sum over all contributions in Eq.
(37). Shifting the summation index down by one, we find

$G_{\hat{\rho}_{1}}^{(1)}(t_{2})=\sum_{s=0}^{N-2}e^{-2(N-1)\gamma
t_{2}}\cdot(e^{2\gamma t_{2}}-1)^{s}\cdot\binom{N}{s}\cdot\frac{1}{N}\cdot
2^{2}\cdot\binom{N-s}{2}=2(N-2+1)e^{-2\gamma t_{2}}=2(N-1)e^{-2\gamma
t_{2}}\,,$ (40)

where again we wrote the second-last expression in Eq. (40) as in Eq. (30).
The presented method to find the conditional intensity correlation function
after measurement of the first photon can be applied to an arbitrary order
$m$. In this case the general probabilities are found to be

$p_{s}(t_{m})=e^{-2(N-m+1)\gamma t_{m}}(e^{2\gamma t_{m}}-1)^{s}\,,$ (41)

where we characterize the probability by the initial number of ground state
atoms $s$ of the time-evolved state Eq. (32) before the first measurement. The
normalization for an arbitrary correlation order $m$ is given by

$\mathcal{N}=\frac{1}{[(m-1)!]^{2}}\cdot\binom{N}{m-1}^{-1}\,.$ (42)

Considering subsequent measurement processes, we find for states with
$N-n_{g}$ excitations $\binom{N-n_{g}+(m-1)}{m}$ different final states, where
the number of ground state atoms $n_{g}$ refers to the number of ground state
atoms before the last measurement. The number of interfering terms can then be
counted to

$\underbrace{(m-1)!}_{\text{multiple
counting}}\underbrace{\binom{N-n_{g}+(m-1)}{m-1}}_{\text{interfering quantum
states}}\,,$ (43)

where we split up the expression into a factor that accounts for the multiple
counting of the same quantum state and the number of different interfering
quantum states. Note that the multiple counting factor squared cancels exactly
with a factor coming from the normalization. The number of single quantum
paths per final state thus calculates to

$\frac{\binom{N-n_{g}+(m-1)}{m-1}\cdot(N-n_{g})}{\binom{N-n_{g}+(m-1)}{m}}=m\,,$
(44)

so that the number of interfering quantum path pairs is given by $m^{2}$. The
$m$th-order intensity correlation function can thus finally be written as

$G_{\hat{\rho}_{m-1}}^{(1)}(t_{m})=\sum_{s=0}^{N-m}e^{-2(N-m+1)\gamma
t_{m}}(e^{2\gamma t_{m}}-1)^{s}\binom{N}{s}\cdot\binom{N}{m-1}^{-1}\cdot
m^{2}\cdot\binom{N-s}{m}=m(N-m+1)\cdot e^{-2\gamma t_{m}}\,,$ (45)

which is the same result as the one calculated by use of the quantum
regression theorem [see Eqs. (29) and (30) above]. However, in the derivation
of Eq. (45), we explicitly solved the first-order differential equations for
the density matrix elements and then applied the quantum path formalism, what
allows for a transparent physical interpretation. In this way, we are able to
explain the occurring superradiant behavior of distant atoms if consecutive
measurements of the radiated photons are considered. In the next section we
analyze the role of the measurement process more quantitatively and show that
it projects the time-evolved states towards the symmetric superradiant
subspace. The latter implies the appearance of quantum correlations leading to
the observed coherent emission of spontaneous radiation.

Figure 4: Conditional intensities $G_{\rho_{m-1}}^{(1)}(t_{m})$ against
$\gamma t$ for an initially fully excited system $\ket{e}$ for $N=5$ and $m=1$
to $m=5$ in consecutive steps (left axis) together with the trace distance Eq.
(48) (right axis). The measurement projects the state closer to the symmetric
subspace leading to a similar course of the conditional intensities as in the
small sample case.

###  II.4 Role of measurement process - distance to symmetric subspace

In this subsection we investigate the distance of the time-evolved conditioned
states obtained after consecutive measurement of $m$ photons to the symmetric
subspace spanned by the symmetric Dicke states $\ket{J,M}$, with $J=N/2$ and
$M=N/2-n_{g}$. To that aim, let us calculate the trace distance

$T(\hat{\rho}_{m},\hat{\rho}_{m,\text{proj}})=\frac{1}{2}\operatorname{Tr}\left[\sqrt{(\hat{\rho}_{m}-\hat{\rho}_{m,\text{proj}})^{\dagger}(\hat{\rho}_{m}-\hat{\rho}_{m,\text{proj}})}\right]$
(46)

between the time-evolve state $\hat{\rho}_{m}$ obtained after the measurement
of $m$ photons and its projection onto the symmetric subspace

$\hat{\rho}_{m,\text{proj}}=\sum_{M,M^{\prime}}\ket{J,M}\braket{J,M}{\hat{\rho}_{m}}{J,M^{\prime}}\bra{J,M^{\prime}}\,.$
(47)

In this way we obtain the final result for the trace distance (see App. C)

$T(\hat{\rho}_{m},\hat{\rho}_{m,\text{proj}})=\frac{1}{2}\sum_{n_{g}=0}^{N}\binom{N}{n_{g}}\text{Time}(m,n_{g})[\text{Diag}(m,n_{g})-\text{Diag}_{\text{proj}}(m,n_{g})]\,,$
(48)

where the different functions are defined in App. C. In Figure 4 we show for
$N=5$ the trace distance between the time-evolved states after the measurement
of $m$ photons, $m\in\\{1,...,N\\}$, and the corresponding states projected
onto the symmetric subspace, together with the conditional intensities Eq.
(45). We see that each photon measurement projects the state closer to the
symmetric subspace resulting in a temporal behavior of the conditional
intensities similar to the one observed in the small sample case.

##  III Conclusion

In conclusion, we have shown that Dicke superradiance consists of two
different aspects. One is the magnitude of the radiated intensity, one is the
duration of the radiated pulse. Starting from the small sample limit, we
explicitly showed that the magnitude of the radiated intensity does not arise
from a synchronization of the individual dipole moments of the atoms, as the
expectation value of the dipole moment is zero for all Dicke states. It rather
stems from the quantum correlations of the atoms which build up in the course
of the dynamic evolution of the system. The superradiant behavior, i.e., the
light burst and its peaked intensity, can eventually be attributed to the
interference of the indistinguishable quantum paths resulting from the
entanglement of the symmetric Dicke states. The shortened time period of the
light burst can be explained by the altered transition rates [see Eqs. (10)
and (18)].

In a next step, we applied the quantum path interference explanation to the
case of far distant non-interacting atoms. While quantum correlations
naturally arise via the coupled dynamics of the atoms in the small sample
case, in the configuration of distant atoms they can be generated by
consecutive conditional photon measurements. The latter lead again to
entanglement between the individual atoms and thus to quantum coherences.
These coherences then give rise to interference effects which can again be
explained using the quantum path interference formalism. The superradiant
burst of the radiated intensity can thus be adequately described also in the
case of distant non-interacting atoms by the interference of indistinguishable
quantum paths. However, in contrast to the small sample case, the period of
the superradiant burst obtained via conditional measurements depends uniquely
on the respective measurement times.

## Acknowledgements

M.B. and J.v.Z. gratefully acknowledge funding and support by the
International Max Planck Research School - Physics of Light. This work was
funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) – Project-ID 429529648 – TRR 306 QuCoLiMa (”Quantum Cooperativity
of Light and Matter”).

##  Appendix A Differential equations for the density matrix elements

If we project the master equation Eq. (20) onto the orthonormal basis given by
the tensor product states, we find

$\begin{split}\partial_{t}\braket{\alpha}{\hat{\rho}}{\beta}=&-\gamma\sum_{\mu=1}^{N}(\braket{\alpha}{\hat{S}_{+}^{(\mu)}\hat{S}_{-}^{(\mu)}\hat{\rho}}{\beta}+\braket{\alpha}{\hat{\rho}\hat{S}_{+}^{(\mu)}\hat{S}_{-}^{(\mu)}}{\beta}-2\braket{\alpha}{\hat{S}_{-}^{(\mu)}\hat{\rho}\hat{S}_{+}^{(\mu)}}{\beta})\\\
=&-\gamma\sum_{\mu=1}^{N}\left(\sum_{\xi}\braket{\alpha}{\hat{S}_{+}^{(\mu)}\ket{\xi}\bra{\xi}\hat{S}_{-}^{(\mu)}\hat{\rho}}{\beta}+\sum_{\zeta}\braket{\alpha}{\hat{\rho}\hat{S}_{+}^{(\mu)}\ket{\zeta}\bra{\zeta}\hat{S}_{-}^{(\mu)}}{\beta}-2\sum_{\eta,\delta}\braket{\alpha}{\hat{S}_{-}^{(\mu)}\ket{\eta}\bra{\eta}\hat{\rho}\ket{\delta}\bra{\delta}\hat{S}_{+}^{(\mu)}}{\beta}\right)\\\
\partial_{t}\rho_{\alpha,\beta}(t)\stackrel{{\scriptstyle\mathclap{(\ast)}}}{{=}}&\;2\gamma\sum_{\mu=1}^{N}\rho_{\eta^{(\mu)},\delta^{(\mu)}}(t)-\gamma\sum_{\mu=1}^{N}\rho_{S_{+}^{(\mu)}\xi,\beta}(t)-\gamma\sum_{\mu=1}^{N}\rho_{\alpha,S_{+}^{(\mu)}\zeta}(t)\,,\end{split}$
(49)

where $\ket{\alpha}$ and $\ket{\beta}$ are two tensor product states and the
equality $(\ast)$ is restricted by the different conditions of Eq. (22) that
have to be fulfilled to get a non-zero contribution from the corresponding
terms on the right hand side of the last line of Eq. (49).
In the following, we restrict ourselves to initial density matrices which do
not have coherences between states with different number of ground state
atoms, i.e., we restrict ourselves to block-diagonal density matrices with
respect to the number of atoms being in the ground state. Moreover, we only
consider initial density matrices, which provide for density matrix elements
following the same kind of differential equation the same initial conditions.
We note that by starting with an initial state that fulfills these two
conditions, also the states after time evolution and consecutive photon
measurements in forward direction fulfill these conditions. This allows us to
reduce the number of real first-order differential equations from $2^{2N}-1$
to $N^{2}$, i.e., we only need to find the time evolution of particular
elements of the blocks of the density matrix.
Using as basis the tensor product basis, a straightforward calculation of the
differential equations of the density matrix elements leads to the following
first-order differential equation for the diagonal elements

$\partial_{t}\begin{pmatrix}\rho_{e,e}(t)\\\
\rho_{g_{j}^{(1)},g_{j}^{(1)}}(t)\\\ \rho_{g_{jk}^{(2)},g_{jk}^{(2)}}(t)\\\
\vdots\end{pmatrix}=\begin{pmatrix}-2\gamma N&0&0&\dots\\\
2\gamma&-2\gamma(N-1)&0&\\\ 0&4\gamma&-2\gamma(N-2)&\\\
\vdots&&&\ddots\end{pmatrix}\begin{pmatrix}\rho_{e,e}(t)\\\
\rho_{g_{j}^{(1)},g_{j}^{(1)}}(t)\\\ \rho_{g_{jk}^{(2)},g_{jk}^{(2)}}(t)\\\
\vdots\end{pmatrix}\,.$ (50)

In an analogous procedure one can find the differential equation for the off-
diagonal elements, which is given by

$\partial_{t}\begin{pmatrix}\rho_{g_{j}^{(1)},g_{k}^{(1)}}(t)\\\
\rho_{g_{jk}^{(2)},g_{jl}^{(2)}}(t)\\\ \rho_{g_{jk}^{(2)},g_{lm}^{(2)}}(t)\\\
\rho_{g_{jkl}^{(3)},g_{jkm}^{(3)}}(t)\\\
\rho_{g_{jkl}^{(3)},g_{jmn}^{(3)}}(t)\\\
\rho_{g_{jkl}^{(3)},g_{mno}^{(3)}}(t)\\\
\vdots\end{pmatrix}=\begin{pmatrix}-2\gamma(N-1)&0&0&0&0&0&\dots\\\
2\gamma&-2\gamma(N-2)&0&0&0&0&\\\ 0&0&-2\gamma(N-2)&0&0&0&\\\
0&4\gamma&0&-2\gamma(N-3)&0&0&\\\ 0&0&2\gamma&0&-2\gamma(N-3)&0&\\\
0&0&0&0&0&-2\gamma(N-3)&\\\
\vdots&&&&&&\ddots\end{pmatrix}\begin{pmatrix}\rho_{g_{j}^{(1)},g_{k}^{(1)}}(t)\\\
\rho_{g_{jk}^{(2)},g_{jl}^{(2)}}(t)\\\ \rho_{g_{jk}^{(2)},g_{lm}^{(2)}}(t)\\\
\rho_{g_{jkl}^{(3)},g_{jkm}^{(3)}}(t)\\\
\rho_{g_{jkl}^{(3)},g_{jmn}^{(3)}}(t)\\\
\rho_{g_{jkl}^{(3)},g_{mno}^{(3)}}(t)\\\ \vdots\end{pmatrix}\,.$ (51)

Solving this reduced number of first-order differential equations by either
taking the matrix exponential or a successive application of the Duhamel-
principle allows us to fully determine the time evolution of an ensemble of
independent atoms for the above described initial states analytically.

##  Appendix B Multi-time correlations of raising and lowering pseudo-spin
operators

We first investigate the $m$th-order single-time correlation function
$\braket{\hat{S}_{+}\hat{S}_{+}...\hat{S}_{+}\hat{S}_{-}...\hat{S}_{-}}_{\hat{\rho}(t)}$
in a state $\hat{\rho}(t)$ that is block-diagonal with respect to the number
of atoms in the ground state, i.e., each block has a well-defined number of
ground state atoms. Writing the expectation value explicitly

$\braket{\hat{S}_{+}\hat{S}_{+}...\hat{S}_{+}\hat{S}_{-}...\hat{S}_{-}}_{\hat{\rho}(t)}=\sum_{i_{1}=1}^{N}...\sum_{i_{m}=1}^{N}\sum_{j_{1}=1}^{N}...\sum_{j_{m}=1}^{N}\braket{\hat{S}_{+}^{(i_{1})}...\hat{S}_{+}^{(i_{m})}\hat{S}_{-}^{(j_{1})}...\hat{S}_{-}^{(j_{m})}}_{\hat{\rho}(t)}$
(52)

leads us to the evaluation of expectation values of the form
$\braket{\hat{S}_{+}^{(i_{1})}...\hat{S}_{+}^{(i_{m})}\hat{S}_{-}^{(j_{1})}...\hat{S}_{-}^{(j_{m})}}_{\hat{\rho}(t)}$.
If either $i_{s}=i_{p}$ or $j_{s}=j_{p}$ for some $s$ and $p$ the expectation
value vanishes, since $[\hat{S}_{-}^{(\mu)}]^{2}=[\hat{S}_{+}^{(\mu)}]^{2}=0$
$\forall\mu\in\\{1,...,N\\}$. In addition, if any $i_{s}\neq j_{p}\;\forall p$
or $j_{p}\neq i_{s}\;\forall s$ the expectation value also vanishes, since we
assume a block-diagonal state with respect to the number of ground state atoms
implying that there are no coherences between states with different number of
atoms in the ground state. Therefore, we can restrict ourselves to expectation
values of the form
$\left\langle\prod_{s=1}^{m}\hat{S}_{+}^{(i_{s})}\prod_{p=1}^{m}\hat{S}_{-}^{(i_{p})}\right\rangle_{\rho(t)}$
with $i_{s}\neq i_{p}\;\forall s\neq p$. To calculate these $m$th-order
correlation functions we first investigate the simple case of the first-order
expectation value
$\braket{\hat{S}_{+}^{(i)}\hat{S}_{-}^{(i)}}_{\hat{\rho}(t)}$. We calculate
the time derivative

$\begin{split}\frac{d}{dt}\braket{\hat{S}_{+}^{(i)}\hat{S}_{-}^{(i)}}_{\hat{\rho}(t)}=&\,\,\frac{d}{dt}\operatorname{Tr}[\hat{S}_{+}^{(i)}\hat{S}_{-}^{(i)}\hat{\rho}(t)]=\operatorname{Tr}[\hat{S}_{+}^{(i)}\hat{S}_{-}^{(i)}\dot{\hat{\rho}}(t)]\\\
=&-\gamma\sum_{\mu=1}^{N}\left\\{\operatorname{Tr}[\hat{S}_{+}^{(i)}\hat{S}_{-}^{(i)}\hat{S}_{+}^{(\mu)}\hat{S}_{-}^{(\mu)}\hat{\rho}(t)]+\operatorname{Tr}[\hat{S}_{+}^{(i)}\hat{S}_{-}^{(i)}\hat{\rho}(t)\hat{S}_{+}^{(\mu)}\hat{S}_{-}^{(\mu)}]-2\operatorname{Tr}[\hat{S}_{+}^{(i)}\hat{S}_{-}^{(i)}\hat{S}_{-}^{(\mu)}\hat{\rho}(t)\hat{S}_{+}^{(\mu)}]\right\\}\,,\end{split}$
(53)

where we used the master equation Eq. (20). To find the differential equation
the expectation value fulfils we can do the following simplifications

$\displaystyle\begin{rcases}\operatorname{Tr}[\hat{S}_{+}^{(i)}\hat{S}_{-}^{(i)}\hat{S}_{+}^{(\mu)}\hat{S}_{-}^{(\mu)}\hat{\rho}(t)]=\operatorname{Tr}[\hat{S}_{+}^{(i)}\hat{S}_{-}^{(i)}\hat{\rho}(t)\hat{S}_{+}^{(\mu)}\hat{S}_{-}^{(\mu)}]\;\hphantom{=\operatorname{Tr}[\hat{S}_{+}^{(i)}\hat{S}_{-}^{(i)}\hat{\rho}(t)]}\\\
\operatorname{Tr}[\hat{S}_{+}^{(i)}\hat{S}_{-}^{(i)}\hat{S}_{-}^{(\mu)}\hat{\rho}(t)\hat{S}_{+}^{(\mu)}]=\operatorname{Tr}[\hat{S}_{+}^{(i)}\hat{S}_{-}^{(i)}\hat{\rho}(t)\hat{S}_{+}^{(\mu)}\hat{S}_{-}^{(\mu)}]\end{rcases}\text{for}\;i\neq\mu\,,$
(54)
$\displaystyle\begin{rcases}\operatorname{Tr}[\hat{S}_{+}^{(i)}\hat{S}_{-}^{(i)}\hat{S}_{+}^{(\mu)}\hat{S}_{-}^{(\mu)}\hat{\rho}(t)]=\operatorname{Tr}[\hat{S}_{+}^{(i)}\hat{S}_{-}^{(i)}\hat{\rho}(t)\hat{S}_{+}^{(\mu)}\hat{S}_{-}^{(\mu)}]=\operatorname{Tr}[\hat{S}_{+}^{(i)}\hat{S}_{-}^{(i)}\hat{\rho}(t)]\\\
\operatorname{Tr}[\hat{S}_{+}^{(i)}\hat{S}_{-}^{(i)}\hat{S}_{-}^{(\mu)}\hat{\rho}(t)\hat{S}_{+}^{(\mu)}]=0\end{rcases}\text{for}\;i=\mu\,,$
(55)

such that for $i\neq\mu$ all terms in the sum cancel each other, whereas for
$i=\mu$ we get $2\braket{\hat{S}_{+}^{(i)}\hat{S}_{-}^{(i)}}_{\hat{\rho}(t)}$.
Thus, we find the differential equation

$\frac{d}{dt}\braket{\hat{S}_{+}^{(i)}\hat{S}_{-}^{(i)}}_{\hat{\rho}(t)}=-2\gamma\braket{\hat{S}_{+}^{(i)}\hat{S}_{-}^{(i)}}_{\hat{\rho}(t)}\,,$
(56)

which is solved by
$\braket{\hat{S}_{+}^{(i)}\hat{S}_{-}^{(i)}}_{\hat{\rho}(t)}=e^{-2\gamma
t}\braket{\hat{S}_{+}^{(i)}\hat{S}_{-}^{(i)}}_{\hat{\rho}(0)}$. Now we can
generalize this result to get the higher-order equal-time correlation
functions. Therefore, we first consider the second-order correlation function
$\braket{\hat{S}_{+}^{(i_{1})}\hat{S}_{+}^{(i_{2})}\hat{S}_{-}^{(i_{2})}\hat{S}_{-}^{(i_{1})}}_{\hat{\rho}(t)}$.
Since $i_{1}\neq i_{2}$ we have the same situation as before, with the
difference that now we get two contributions from $i_{1}=\mu$ and $i_{2}=\mu$,
such that we immediately find

$\frac{d}{dt}\braket{\hat{S}_{+}^{(i_{1})}\hat{S}_{+}^{(i_{2})}\hat{S}_{-}^{(i_{2})}\hat{S}_{-}^{(i_{1})}}_{\hat{\rho}(t)}=-4\gamma\braket{\hat{S}_{+}^{(i_{1})}\hat{S}_{+}^{(i_{2})}\hat{S}_{-}^{(i_{2})}\hat{S}_{-}^{(i_{1})}}_{\hat{\rho}(t)}\,.$
(57)

In general, we get [41]

$\displaystyle\frac{d}{dt}\left\langle\prod_{s=1}^{m}\hat{S}_{+}^{(i_{s})}\prod_{p=1}^{m}\hat{S}_{-}^{(i_{p})}\right\rangle_{\hat{\rho}(t)}=-2m\gamma\left\langle\prod_{s=1}^{m}\hat{S}_{+}^{(i_{s})}\prod_{p=1}^{m}\hat{S}_{-}^{(i_{p})}\right\rangle_{\hat{\rho}(t)}$
(58)

for which the solution is

$\displaystyle\left\langle\prod_{s=1}^{m}\hat{S}_{+}^{(i_{s})}\prod_{p=1}^{m}\hat{S}_{-}^{(i_{p})}\right\rangle_{\hat{\rho}(t)}=e^{-2m\gamma
t}\left\langle\prod_{s=1}^{m}\hat{S}_{+}^{(i_{s})}\prod_{p=1}^{m}\hat{S}_{-}^{(i_{p})}\right\rangle_{\hat{\rho}(0)}\,.$
(59)

With this result we can now calculate the multi-time correlation functions by
use of the quantum regression theorem, which relates the multi-time
expectation values to single-time expectation values [29, 40]. We find,

$\frac{d}{d\tau}\braket{\hat{S}_{+}^{(i_{1})}(t)\hat{S}_{+}^{(i_{2})}(t+\tau)\hat{S}_{-}^{(i_{2})}(t+\tau)\hat{S}_{-}^{(i_{1})}(t)}=-2\gamma\braket{\hat{S}_{+}^{(i_{1})}(t)\hat{S}_{+}^{(i_{2})}(t+\tau)\hat{S}_{-}^{(i_{2})}(t+\tau)\hat{S}_{-}^{(i_{1})}(t)}\,,$
(60)

such that

$\braket{\hat{S}_{+}^{(i_{1})}(t_{1})\hat{S}_{+}^{(i_{2})}(t_{2})\hat{S}_{-}^{(i_{2})}(t_{2})\hat{S}_{-}^{(i_{1})}(t_{1})}=e^{-2\gamma(t_{2}-t_{1})}\braket{\hat{S}_{+}^{(i_{1})}(t_{1})\hat{S}_{+}^{(i_{2})}(t_{1})\hat{S}_{-}^{(i_{2})}(t_{1})\hat{S}_{-}^{(i_{1})}(t_{1})}$
(61)

with $t=t_{1}$ and $\tau=t_{2}-t_{1}$ and we are left with a single-time
expectation value, which we solved beforehand. A successive application of the
quantum regression theorem to the $m$th-order multi-time correlation functions
then gives [41]

$\begin{split}\braket{\hat{S}_{+}^{(i_{1})}(t_{1})...\hat{S}_{+}^{(i_{m})}(t_{m})\hat{S}_{-}^{(i_{m})}(t_{m})...\hat{S}_{-}^{(i_{1})}(t_{1})}&=e^{-2\gamma(t_{m}-t_{m-1})}e^{-4\gamma(t_{m-1}-t_{m-2})}...e^{-2m\gamma
t_{1}}\cdot\braket{\hat{S}_{+}^{(i_{1})}(0)...\hat{S}_{+}^{(i_{m})}(0)\hat{S}_{-}^{(i_{m})}(0)...\hat{S}_{-}^{(i_{1})}(0)}\\\
&=\prod_{l=1}^{m}e^{-2\gamma
t_{l}}\braket{\hat{S}_{+}^{(i_{1})}(0)...\hat{S}_{+}^{(i_{m})}(0)\hat{S}_{-}^{(i_{m})}(0)...\hat{S}_{-}^{(i_{1})}(0)}\,.\end{split}$
(62)

##  Appendix C Trace distance between post-measurement states and symmetric
subspace

In this Appendix, we explicitly calculate the trace distance between the time-
dependent post-measurement states and the symmetric subspace. Therefore, we
note that due to the restrictions on the initial state the post-measurement
states $\hat{\rho}_{m}$ are block-diagonal with respect to the number of
ground state atoms, i.e., we can write

$\hat{\rho}_{m}=\bigoplus\limits_{n_{g}}\hat{\rho}_{m,n_{g}}\,.$ (63)

Since $M,M^{\prime}$ correspond to a specific number of ground state atoms via
$M=N/2-n_{g}$ the states $\ket{J,M},\ket{J,M^{\prime}}$ only act on the
subspace spanned by the tensor product states with the same number of ground
state atoms. Thus, Eq. (47) reduces to

$\hat{\rho}_{m,\text{proj}}=\sum_{n_{g}}\braket{J,M}{\hat{\rho}_{m,n_{g}}}{J,M}\ket{J,M}\bra{J,M}=\bigoplus\limits_{n_{g}}\braket{J,M}{\hat{\rho}_{m,n_{g}}}{J,M}\ket{J,M}\bra{J,M}\,.$
(64)

In addition, again due to the restrictions on the initial state the sum of the
elements of a given row of $\hat{\rho}_{m,n_{g}}$ gives the same value for
each row. Therefore, the states $\ket{J,M}$ with corresponding number of
ground state atoms $n_{g}$ are eigenstates of the different block matrices
$\hat{\rho}_{m,n_{g}}$. Thus, we can write

$\hat{\rho}_{m,n_{g}}=\braket{J,M}{\hat{\rho}_{m,n_{g}}}{J,M}\ket{J,M}\bra{J,M}\oplus\hat{\tilde{\rho}}_{m,n_{g}}\,.$
(65)

The post-measurement states can then be written as

$\hat{\rho}_{m}=\bigoplus\limits_{n_{g}}\braket{J,M}{\hat{\rho}_{m,n_{g}}}{J,M}\ket{J,M}\bra{J,M}\oplus\hat{\tilde{\rho}}_{m,n_{g}}$
(66)

and using Eq. (64)

$\displaystyle\hat{\tilde{\rho}}_{m}\coloneqq\hat{\rho}_{m}-\hat{\rho}_{m,\text{proj}}=\bigoplus\limits_{n_{g}}\hat{\tilde{\rho}}_{m,n_{g}}\,.$
(67)

Since $\hat{\rho}_{m}$ and $\hat{\rho}_{m,\text{proj}}$ are Hermitian
$\hat{\tilde{\rho}}$ is also Hermitian and we can rewrite Eq. (46) to

$T(\hat{\rho}_{m},\hat{\rho}_{m,\text{proj}})=\frac{1}{2}\operatorname{Tr}\left[\sqrt{\hat{\tilde{\rho}}_{m}^{2}}\right]\,.$
(68)

Furthermore, $\hat{\tilde{\rho}}$ can be diagonalized via some unitary matrix
$U$, such that

$\hat{\tilde{\rho}}_{m,D}=U^{\dagger}\hat{\tilde{\rho}}_{m}U$ (69)

is diagonal. Then,

$\sqrt{\hat{\tilde{\rho}}_{m}^{2}}=U\sqrt{\hat{\tilde{\rho}}_{m,D}^{2}}U^{\dagger}$
(70)

and

$\operatorname{Tr}\left[\sqrt{\hat{\tilde{\rho}}_{m}^{2}}\right]=\operatorname{Tr}\left[U\sqrt{\hat{\tilde{\rho}}_{m,D}^{2}}U^{\dagger}\right]=\operatorname{Tr}\left[\sqrt{\hat{\tilde{\rho}}_{m,D}^{2}}\right]=\sum_{i}|\lambda_{m,i}|\,,$
(71)

where $\lambda_{m,i}$ are the eigenvalues of $\hat{\tilde{\rho}}_{m}$.
Since $\hat{\rho}_{m}$ is a density matrix, all eigenvalues of
$\hat{\rho}_{m}$ are greater than or equal to zero. Thus, also all eigenvalues
of the block-matrices $\hat{\rho}_{m,n_{g}}$ and
$\hat{\tilde{\rho}}_{m,n_{g}}$ are greater than or equal to zero, such that
the same holds true for $\hat{\tilde{\rho}}_{m}$. Then, we find

$\begin{split}T(\hat{\rho}_{m},\hat{\rho}_{m,\text{proj}})&=\frac{1}{2}\operatorname{Tr}\left[\sqrt{\hat{\tilde{\rho}}_{m}^{2}}\right]=\frac{1}{2}\sum_{i}|\lambda_{m,i}|\\\
&=\frac{1}{2}\sum_{i}\lambda_{m,i}=\frac{1}{2}\operatorname{Tr}[\hat{\tilde{\rho}}_{m}]\,,\end{split}$
(72)

such that we only need to compute the trace of the difference matrix Eq. (67).
Therefore, we separately identify the diagonal entries of $\hat{\rho}_{m}$ and
$\hat{\rho}_{m,\text{proj}}$ combinatorially. Both have the same time
dependence coming from the evolution of the density matrix via the master
equation. We define this factor as

$\text{Time}(m,n_{g})\coloneqq e^{-2(N-m)\gamma t_{m+1}}(e^{2\gamma
t_{m+1}}-1)^{n_{g}-m}\,.$ (73)

Note that the difference with respect to Eq. (41) is that $m=1$ corresponds
already to the first post-measurement state, whereas in Eq. (41) $m=1$
corresponds to the initial time-evolved state without a measurement done. In
addition, $s$ in Eq. (41) denotes the initial number of ground state atoms.
Here, $n_{g}$ denotes the current number of ground state atoms, such that a
subtraction of $m$ is needed. Now we are left with counting the occurrence of
diagonal states. In Eq. (43) we have already identified a multiple counting
factor of quantum states coming from the successive application of
measurements. Therefore, we find

$\binom{n_{g}}{n_{g}-m}(m!)^{2}=\binom{n_{g}}{m}(m!)^{2}$ (74)

many occurrences of the same diagonal entry, where $\binom{n_{g}}{n_{g}-m}$
accounts for the number of initial states leading to the considered final
diagonal state and $m!$ comes from the multiple counting. Note that the square
of $m!$ enters, since one gets $m!$ from the ”ket”-vector and $m!$ from the
”bra”-vector. Accounting for the post-measurement normalization we identify
the prefactor of the diagonal entries of $\hat{\rho}_{m}$ to be

$\text{Diag}(m,n_{g})\coloneqq\frac{\binom{n_{g}}{m}(m!)^{2}}{\binom{N}{m}(m!)^{2}}=\frac{\binom{n_{g}}{m}}{\binom{N}{m}}\,.$
(75)

For $\hat{\rho}_{m,\text{proj}}$ the diagonal entries are given by
$\braket{J,M}{\hat{\rho}_{m,n_{g}}}{J,M}\cdot\mathcal{N}$, where
$\mathcal{N}=\binom{N}{n_{g}}^{-1}$ is the squared normalization of the state
$\ket{J,M}$. For the scalar product we find

$\braket{J,M}{\hat{\rho}_{m,n_{g}}}{J,M}=(\\#\text{inc. terms})\cdot
p_{\text{single}}\cdot(\\#\text{coh.
terms})^{2}\cdot\text{Time}(m,n_{g})\cdot\frac{1}{(m!)^{2}}\binom{N}{m}^{-1}\,,$
(76)

where $\text{Time}(m,n_{g})$ denotes the time evolution factor Eq. (73) and
$\frac{1}{(m!)^{2}}\binom{N}{m}^{-1}$ is the post-measurement normalization.
In addition, the number of initial incoherent terms is

$(\\#\text{inc. terms})=\binom{N}{n_{g}-m}$ (77)

and the number of coherent terms is given by Eq. (43)

$(\\#\text{coh.
terms})=\prod_{i=1}^{m}[N-(n_{g}-i)]=m!\binom{N-n_{g}+m}{m}\,.$ (78)

Note that the square accounts again for the ”ket” as well as the ”bra”
contribution. Last, $p_{\text{single}}$ denotes the probability of finding a
single tensor product state within the state $\ket{J,M}$, which can easily be
computed to

$p_{\text{single}}=\mathcal{N}=\binom{N}{n_{g}}^{-1}\,.$ (79)

With the above considerations we define

$\text{Diag}_{\text{proj}}(m,n_{g})\coloneqq\binom{N}{n_{g}-m}\binom{N}{n_{g}}^{-2}\binom{N-n_{g}+m}{m}^{2}\binom{N}{m}^{-1}\,,$
(80)

which gives the prefactor of the diagonal entries for the projected state
$\hat{\rho}_{m,\text{proj}}$. Now, putting everything together we arrive at
the final result for the trace distance

$T(\hat{\rho}_{m},\hat{\rho}_{m,\text{proj}})=\frac{1}{2}\sum_{n_{g}=0}^{N}\binom{N}{n_{g}}\text{Time}(m,n_{g})[\text{Diag}(m,n_{g})-\text{Diag}_{\text{proj}}(m,n_{g})]\,.$
(81)

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