# Quantum entangled states of a classically radiating macroscopic spin

Ori Somech  Department of Chemical & Biological Physics, Weizmann Institute of
Science, Rehovot 7610001, Israel  Ephraim Shahmoon  Department of Chemical &
Biological Physics, Weizmann Institute of Science, Rehovot 7610001, Israel

###### Abstract

Entanglement constitutes a main feature that distinguishes quantum from
classical physics and is a key resource of quantum technologies. Here we show,
however, that entanglement may also serve as the essential ingredient for the
emergence of classical behavior in a composite nonlinear radiating system. We
consider the radiation from a macroscopic spin emitter, such as the collective
radiation from an atomic ensemble. We introduce a new class of macroscopic
spin states, the coherently radiating spin states (CRSS), defined as the
asymptotic eigenstates of the SU(2) lowering operator. We find that a spin
emitter in a CRSS radiates classical-like coherent light, although the CRSS
itself is a quantum entangled state exhibiting spin squeezing. We further show
that CRSS are naturally produced in Dicke superradiance and underlie the
dissipative Dicke phase transition. Our CRSS theory thus provides new concepts
for studying the quantum physics of radiation, with applications in current
platforms involving collections of atoms or spins, their consideration in
quantum technologies such as metrology and lasing, and the many-body theory of
spin systems.

The classical limit of radiation, such as that emitted by an antenna or laser,
is typically associated with coherent states of the quantized electromagnetic
field mandel_wolf_1995 ; MW ; SCU . This raises the question of what
constitutes a classical emitter. Namely, considering a dipole emitter
described quantum mechanically, what is the quantum state of this dipole which
radiates a coherent state of the field. In a linear system, where the dipole
is described by an harmonic oscillator, a coherent state of this oscillator
linearly transforms to produce a coherent state also of the field. So, the
classical-like state of the dipole radiates classical-like fields. One may
wonder whether this latter conclusion is general and equally applies to
relevant nonlinear quantum systems.

In particular, consider a dipole described by an SU(2) spin-$j$ wherein
nonlinearity is exhibited by a finite Hilbert space of $2j+1$ states. This
system is of fundamental and practical importance as it is often used to model
the radiation from a collection of atom-like emitters, as in superradiance
GROSS1982301 ; Dicke ; mandel_wolf_1995 ; EMAN : $N$ two-level atoms comprise
a macroscopic dipole with a pseudo-spin $j=N/2\gg 1$. Such collective
radiation effects appear in prominent quantum platforms involving ensembles of
atoms HAR ; TOMs1 ; TOMs2 ; FLD  or artificial emitters MAJ . Hence, they play
a key role in various quantum phenomena and technologies, ranging from
dissipative phase transitions KES ; DRUMMOND1978160  and spin squeezing
Alejandro ; A.M.Rey ; yelin ; REY  to narrowband superradiant lasers HOL ;
TOM1  and the precision of optical lattice clocks REY ; CHAYE ; HEN .

The classical behavior of a macroscopic SU(2) spin is typically associated
with coherent spin states (CSS) atomiccoherentstates ; spinsqueezingreview ;
ASA . These states are characterized by an average spin of length $j$ along a
certain direction and a minimal uncertainty circle perpendicular to it (Fig.
1a) DowlingWigner . Moreover, for a spin-$j$ composed of $N=2j$ two-level
atoms, the CSS contains no entanglement: it is a product state of all
constituent atoms. However, the radiation emitted by a spin $j$ in a CSS is
not that of a classical dipole: in fact, it can exhibit quantum correlations
typical of a nonlinear system, as we show below. So, CSS are classical-like
states of the spin itself but they radiate non-classical light.

Figure 1:  Spin states represented on the Bloch sphere (southern hemisphere).
(a) CSS: the variance contour perpendicular to the mean spin vector (green
arrow) exhibits no spin squeezing (circular contour). However, its projection
onto the $xy$ “dipole plane” _is_ squeezed (elliptical), leading to non-
classical radiation (see _Dipole-projected squeezing_ below). (b) CRSS
exhibits spin squeezing (elliptical contour perpendicular to mean spin) but no
dipole-projected squeezing (circular contour on $xy$ plane), hence it radiates
classical light.

In contrast, here we introduce the _coherently radiating spin state_ (CRSS) as
the state of a spin system which produces classical-like coherent state
radiation (Fig. 1b). We find that such states exist in the macroscopic limit
$j\gg 1$ and are given by the approximate eigenstates of the spin lowering
operator $\hat{J}=\hat{J}_{x}-i\hat{J}_{y}$ with continuous complex
eigenvalues $\alpha$ restricted to $|\alpha|<j$. This is in analogy to bosonic
coherent states defined as the eigenstates of the non-Hermitian boson lowering
operator, and in contrast to CSS which are eigenstates of an Hermitian spin
operator. We show that the CRSS possesses several interesting features: (1)
CRSS is a nonclassical state exhibiting spin squeezing which increases as
$|\alpha|\rightarrow j$. For a spin-$j$ describing $N=2j$ two-level atoms this
means that the constituent atoms are pairwise entangled LEW ; SOR ;
spinsqueezingreview . (2) CRSS is the steady-state of driven-dissipative Dicke
superradiance DRUMMOND1978160 ; CAR ; LAW ; RABk , and is at the origin of the
dissipative Dicke phase transition DRUMMOND1978160 ; CAR ; KES  and associated
spin squeezing Alejandro ; yelin ; A.M.Rey . This allows us to analytically
find the scaling $N^{-\frac{1}{3}}$ of the best achievable spin squeezing. (3)
The radiated light from a CRSS is a coherent state, whose amplitude is
proportional to $\alpha$ and to the coherent drive amplitude, just as in a
linear system. CRSS thus forms a remarkable case of the quantum-classical
border, wherein a composite quantum nonlinear system is perceived by light as
a classical linear scatterer, provided that its constituents are quantum
entangled.

Radiation from a dipole eigenstate
The idea that an eigenstate of the lowering operator $\hat{J}$ should produce
coherent-state radiation is in fact quite generic and can be intuitively
understood as follows. Consider a Hermitian dipole operator in the general
form $\hat{d}_{\mathrm{H}}=\hat{d}+\hat{d}^{{\dagger}}$ coupled to the
Hermitian field $\hat{E}_{\mathrm{H}}=\hat{E}+\hat{E}^{{\dagger}}$ (with
$[\hat{E},\hat{E}^{{\dagger}}]=1$), both taken dimensionless here. The light-
matter interaction Hamiltonian,
$\propto\hat{d}_{\mathrm{H}}\hat{E}_{\mathrm{H}}$, within the rotating-wave
approximation SCU , takes the form

$\hat{H}\propto\hat{d}\hat{E}^{{\dagger}}+\hat{E}\hat{d}^{{\dagger}}.$ (1)

If the system is constantly pumped into an eigenstate $|\alpha\rangle$ of
$\hat{d}$ with eigenvalue $\alpha$ then we may effectively replace
$\hat{d}\rightarrow\alpha$ in Hamiltonian (1) such that the evolution
$e^{-\frac{i}{\hbar}\hat{H}t}$ is the displacement operator of the field,
generating a coherent-state out of an initial vacuum state (Appendix). So, an
eigenstate of $\hat{d}$ produces coherent-state radiation. Indeed, for a
linear dipole, where $\hat{d}$ is a boson lowering operator, $|\alpha\rangle$
is a coherent state of the dipole, which linearly transforms via (1) to
produce a coherent-state field. However, for the nonlinear dipole represented
by a spin $j$, we first consider the possible existence of eigenstates of
$\hat{d}=\hat{J}$ and then study their generation and radiation properties.

Asymptotic eigenstates of $\hat{J}$
Strictly speaking, the operator $\hat{J}$ does not have any eigenstates except
for the eigenstate $|j,-j\rangle$ with an eigenvalue zero, where we use the
common notation $|j,m\rangle$ for the joint eigenstate of the total angular
momentum operator $\hat{J}_{x}^{2}+\hat{J}_{y}^{2}+\hat{J}_{z}^{2}$ and its
$z$-axis projection $\hat{J}_{z}$, with corresponding eigenvalues $j(j+1)$ and
$m$. However, we show below that for a macroscopic spin, $j\gg 1$, there are
states that are approximate eigenstates of $\hat{J}$ up to excellent accuracy.
We call such states CRSS. Speaking loosely, we find that in the limit
$j\rightarrow\infty$ the operator $\hat{J}$ has a set of eigenstates with
complex eigenvalues $\alpha$ satisfying

$\hat{J}|j,\alpha\rangle=\alpha|j,\alpha\rangle,\quad\alpha=jre^{-i\varphi},\quad
0<r<1,\quad r,\varphi\in\mathbb{R}.$ (2)

More formally, we define this condition as,

$\text{$\underset{j\rightarrow\infty}{\text{lim}}$}\epsilon_{j}(r)=0,\quad\epsilon_{j}(r)\equiv\left\|\hat{J}|j,jre^{-i\varphi}\rangle-
jre^{-i\varphi}|j,jre^{-i\varphi}\rangle\right\|.$ (3)

Here $\epsilon_{j}(r)$ is the proximity error of a state $|j,\alpha\rangle$
for being an eigenstate of the operator $\hat{J}$ with an eigenvalue
$\alpha=jre^{-i\varphi}$ ($r<1$), and we request that this quantity approaches
zero for $j\rightarrow\infty$. Importantly, the ratio
$\alpha/j=re^{-i\varphi}$ is kept constant while taking the limit
$j\rightarrow\infty$. This allows to account for excitation amplitudes
$\alpha$ comparable to $j$, where nonlinearity becomes significant (beyond the
Holstein-Primakoff approximation).

Figure 2:  CRSS as approximate eigenstates of the spin-$j$ lowering operator
$\hat{J}$. (a) State coefficients $|a_{m}/a_{-j}|$ from Eq. (5) with $j=25$.
For $r=0.75$ (dashed line) $a_{j}$ approximately vanishes whereas for $r=0.84$
(solid line) it does not. For the latter, the maximum and minimum points
$m_{\mp}$ are marked. In the CRSS ansatz (6), all coefficients $m>m_{+}$ are
set to zero, and the condition (3) is satisfied for all $r<1$. (b) Normalized
coefficients $|a_{m}|$ of the ansatz state (6) for $j=100$. The Gaussian shape
agrees well with the analytical result (11).

We show a simple way to define a state that satisfies Eq. (3). Assuming Eq.
(2) can be solved, we insert to the latter the general state
$\sum_{m=-j}^{j}{a_{m}|j,m\rangle}$ and obtain the recursion relation,

$a_{m+1}=\frac{\alpha}{\sqrt{j(j+1)-m(m+1)}}a_{m},\quad-j\leq m\leq j-1,$ (4)

together with $a_{j}=0$. These two conditions result in a contradiction for
any $|\alpha|>0$ unless $a_{j}$ tends to zero in the required limit
$j\rightarrow\infty$. The solution for the recursion relation (4) is given by

$\displaystyle a_{m}$ $\displaystyle=$ $\displaystyle
a_{-j}e^{f_{m}}e^{-i(m+j)\varphi},$ $\displaystyle f_{m}$ $\displaystyle=$
$\displaystyle(m+j)\ln(rj)-\frac{1}{2}\sum_{k=-j}^{m-1}\ln\left[j(j+1)-k(k+1)\right].$
(5)

In Fig. 2a we plot $|a_{m}/a_{-j}|$ as function of $m$, observing that
depending on $r$, $a_{j}$ does not always tend to zero. We find analytically
and numerically that as $j\rightarrow\infty$, $a_{j}\rightarrow 0$ only for
$r<0.804...$ SM . To satisfy Eq. (3) in the full domain $0<r<1$, we use the
following observation: for a truncated state
$|\psi_{s}\rangle=\sum_{m=-j}^{s}a_{m}|j,m\rangle$ with $s<j$, we obtain that
the proximity error is proportional to the last coefficient
$\left\|\hat{J}|\psi_{s}\rangle-\alpha|\psi_{s}\rangle\right\|=|\alpha
a_{s}|$. Using Eq. (4), we can identify the regions for which $|a_{m}|$ is
increasing or decreasing as a function of $m$, finding the maximum ($m_{-}$)
and minimum ($m_{+}$) points as the roots of the quadratic equation
$|\alpha|^{2}=j(j+1)-m(m+1)$,
$m_{\pm}=\frac{1}{2}\left(-1\pm\sqrt{-4|\alpha|^{2}+4j^{2}+4j+1}\right)$
(rounded to integers), see Fig. 2a. For the minimal proximity error we thus
choose to truncate at $s=m_{+}$, defining our ansatz eigenstate and resulting
error as

$\displaystyle|j,jre^{-i\varphi}\rangle_{\mathrm{ans}}=\sum_{m=-j}^{m_{+}}a_{m}|j,m\rangle,\quad\epsilon_{j}(r)=jr|a_{m_{+}}|.$
(6)

The error can be readily estimated numerically for given $j$ and $r$, using
Eq. (5) for $|a_{m_{+}}|$ and demanding state normalization
$|a_{-j}|=\left[\sum_{m=-j}^{m_{+}}e^{2f_{m}}\right]^{-1/2}$. This is
presented in Fig. 3, observing that for $j\gg 1$, the error decreases
exponentially with $j$ for any $r<1$. We also find that at the limit
$j\rightarrow\infty$ the state (6) minimizes the error $\epsilon_{j}(r)$ for
fixed $j$ and $r$ (Appendix). We thus conclude that the state (6) is an
eigenstate of $\hat{J}$ in the sense of Eq. (3), i.e. it is a CRSS to an
excellent approximation.

It is instructive to estimate the error from Eq. (6) also analytically. To
that end, we write $|a_{m_{+}}|=e^{f_{m_{+}}-f_{m_{-}}}|a_{m_{-}}|$,
approximate $f_{m_{\pm}}$ by converting sum to integral in Eq. (5), and
estimate $|a_{m_{-}}|$ using (11) below, obtaining for $j\gg 1$

$\displaystyle\epsilon_{j}(r)\approx
qe^{-2jg(r)},\>g(r)=\mathrm{arctanh}[\sqrt{1-r^{2}}]-\sqrt{1-r^{2}},$ (7)

with the power-law prefactor
$q(j,r)=j^{\frac{3}{4}}(r^{2}\sqrt{1-r^{2}}/\pi))^{\frac{1}{4}}$. Since
$g(r)>0$ for $r<1$, the error (7) decays essentially exponentially with $j$,
exhibiting excellent agreement with the numerical results of Fig. 3, and
yielding the CRSS existence condition $r<1$. The exponent $g(r)$ is a
monotonically decreasing function of $r$, meaning that for increasing values
of $r$ larger $j$ values are required for the error to still be small, as also
seen in Fig. 3. For a given finite $j\gg 1$, we can thus define the range of
validity $0<r<r_{j}$ in which the state (6) is a CRSS to a good approximation,
by defining the respective maximal value $r_{j}$ via

$\epsilon_{j}(r=r_{j})=\frac{1}{e},$ (8a) |  $r_{j}\sim\sqrt{1-\left(\frac{3}{2j}\right)^{\frac{2}{3}}},$ (8b)
---|---

as marked by the dashed curve in Fig. 3b. The analytical result (8b) for
$r_{j}$ expresses its asymptotic scaling with $j$, which is simply obtained by
demanding $2jg(r_{j})=1$ (ignoring the power-law prefactor $q$ for
$j\rightarrow\infty$).

Figure 3:  Proximity error of the CRSS ansatz state (6) [evaluated numerically
using (5)]. (a) $\epsilon_{j}(r)$ vs. $j$ for $r=0.4,0.7,0.9$ (dashed, dash-
dotted, solid lines). Exponential decay is observed for large $j$ in agreement
with (7). (b) $\epsilon_{j}(r)$ as a function of $j$ and $r$. The dashed curve
marks $r_{j}$, defined in (8a) and calculated using (7). For given $j$ and
$0<r<r_{j}$, the state (6) well approximates a CRSS ($\epsilon\ll 1$), as
predicted analytically.

Steady state of driven superradiance
Consider a dipole $\hat{d}=\hat{J}$ damped by radiation to a continuum of
electromagnetic field modes described by the operator $\hat{E}$. The master
equation for the reduced density matrix of the dipole can be derived from
Hamiltonian (1) as GROSS1982301 ; Alejandro ,

$\displaystyle\frac{d\hat{\rho}}{dt}$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}\left(\hat{H}_{\text{nh
}}\hat{\rho}-\hat{\rho}\hat{H}_{\text{nh
}}^{\dagger}\right)+\gamma\hat{J}\hat{\rho}\hat{J}^{\dagger},$
$\displaystyle\hat{H}_{\text{nh}}$ $\displaystyle=$
$\displaystyle\hbar\left(\Delta-i\frac{\gamma}{2}\right)\hat{J}^{{\dagger}}\hat{J}-\hbar\left(\Omega\hat{J}^{{\dagger}}+\Omega^{\ast}\hat{J}\right).$
(9)

Here $\gamma$ and $\Delta$ are spontaneous emission rate and energy shift,
respectively, both induced by the electromagnetic field reservoir, and
$\Omega$ is the amplitude of a resonant coherent-state incident field. In the
case of superradiance, $\hat{J}=\sum_{n=1}^{N}\hat{\sigma}_{n}$ describes the
collective pseudo-spin of $N$ two-level atoms with lowering operators
$\hat{\sigma}_{n}=|g\rangle_{n}\langle e|$ (levels $|g\rangle$ and
$|e\rangle$), which are all identically coupled to the field $\hat{E}$, as can
be realized in an optical cavity HAR ; TOMs1 . The common field reservoir can
then mediate correlations between the atoms via the collective decay
($\gamma$) and dipole-dipole interaction ($\Delta$). Assuming that all atoms
are initially in the ground state,
$\prod_{n}\otimes|g\rangle_{n}=|N/2,-N/2\rangle$, the spin representation is
fixed to $j=N/2$ throughout the dynamics. The steady state of this Lindblad
master equation becomes a pure state if and only if it is an eigenstate of
both the non-Hermitian Hamiltonian $\hat{H}_{\text{nh}}$ and the jump operator
$\hat{J}$ purestate1 . We showed that the eigenstate (2) of $\hat{J}$ exists
at the limit $N\rightarrow\infty$ (CRSS) and it is straightforward to verify
that it is also an eigenstate of $\hat{H}_{\text{nh}}$ if we take

$\displaystyle\alpha=\frac{\Omega}{\Delta-i\gamma/2},$ (10)

yielding $d\hat{\rho}/dt=0$ for $\hat{\rho}=|j,\alpha\rangle\langle j,\alpha|$
with $j=N/2$. This can also be seen for a finite $N=2j\gg 1$ by considering
the overlap between the exact steady-state solution of (9) and the CRSS ansatz
(6), whose deviation from unity behaves similarly to the proximity error in
Fig. 3 (Appendix).

The above considerations establish that for $N=2j\gg 1$, the steady state of
driven-dissipative superradiance is a CRSS characterized by the amplitude
$\alpha$ from Eq. (10), within the validity range $0<r<r_{j}$ defined in Eq.
(8a) for a finite $j$. This means that the properties of CRSS, a pure state,
underlie dissipative Dicke superradiance phenomena, as we show further below.
For example, in the macroscopic limit $N=2j\rightarrow\infty$, the steady-
state solution of Eq. (9) is known to exhibit a second order phase transition
wherein the order parameter $\langle\hat{J}_{z}\rangle$ vanishes for driving
fields $|\Omega|$ exceeding the critical point
$\Omega_{c}=(N/4)\sqrt{\gamma^{2}+4\Delta^{2}}$ DRUMMOND1978160 ; A.M.Rey ; HL
. But the condition $|\Omega|<\Omega_{c}$ for non-vanishing order parameter is
identical to the condition $|\alpha|/j=r<1$ for the existence of CRSS in Eq.
(2). Therefore, the dissipative Dicke phase transition is originated in the
critical point $r=1$ for the existence of eigenstates of $\hat{J}$ in the
limit $j\rightarrow\infty$.

Spin observables and squeezing
Having established the existence of CRSS and its relation to superradiance, we
now examine the properties of spin variables in a CRSS. Using Eq. (2), the
average dipole $\langle\hat{J}\rangle$ is trivially given by the amplitude
$\alpha$, agreeing with mean-field results of superradiance DRUMMOND1978160 ;
yelin ; HL . Notably, in the case of superradiance, Eq. (10) implies that this
dipole amplitude is linear with the incident field $\Omega$, a result expected
for a harmonic oscillator but less intuitive for our nonlinear spin scatterer.
We note that this is valid for very strong fields $|\alpha|\sim j\gg 1$ well
beyond the typical linear limit $|\Omega|\ll\Omega_{c}$ ($|\alpha|\ll j$).
Such linear-scatter behavior will lead to scattered light which remains in a
coherent state as anticipated above for CRSS and further discussed below.

Nonlinearity of the spin dipole is revealed by considering the observable
$\hat{J}_{z}$. The moments of $\hat{J}_{z}$ can be estimated directly from the
distribution $|a_{m}|^{2}\propto e^{2f_{m}}$ from Eq. (5) and Fig. 2b. To this
end, we expand the integral approximation of $f_{m}$ around its peak at
$m=m_{-}$, and find for $j\gg 1$ that $|a_{m}|^{2}$ is well-described by a
Gaussian,

$\displaystyle|a_{m}|^{2}\approx\frac{1}{\sqrt{2\pi
w^{2}}}e^{-\frac{(m-m_{-})^{2}}{2w^{2}}},\quad m_{-}\approx-j\sqrt{1-r^{2}},$
(11)

of width $w^{2}=jr^{2}/(2\sqrt{1-r^{2}})$ (Appendix). This yields
$\langle\hat{J_{z}}\rangle\approx m_{-}\approx-j\sqrt{1-r^{2}}$ in agreement
with the mean-field solution of superradiance DRUMMOND1978160 ; yelin ; HL ,
and exhibiting the second order phase transition at $r=1$ as discussed above.
Together with
$\langle\hat{J}_{x}-i\hat{J}_{y}\rangle=\langle\hat{J}\rangle=\alpha=jre^{-i\varphi}$
and writing $r=\sin\theta$, this yields a spin vector
$\hat{\mathbf{J}}=(\hat{J}_{x},\hat{J}_{y},\hat{J}_{z})$ whose mean of length
$j=N/2$ is directed at an angle $\theta$ away from the south pole of the Bloch
sphere with an angle $\varphi$ along the $xy$ plane (Fig. 1b). This mean spin
is equivalent to that of a CSS (Fig. 1a), however, the quantum fluctuations
around it are different, as can be characterized by spin squeezing.

The spin squeezing parameter is defined by
$\xi^{2}=(2j/|\langle\hat{\mathbf{J}}\rangle|^{2})\min_{\phi}\mathrm{Var}[\hat{J}^{\bot}_{\phi}]$,
where $\hat{J}^{\bot}_{\phi}$ is the projection of the spin vector onto an
angle $\phi$ on the plane perpendicular to the direction of the mean spin
$\langle\hat{\mathbf{J}}\rangle$ spinsqueezingreview ; spinsqueezingparameter1
; KIT . $\xi^{2}<1$ signifies a non-classical spin-squeezed state wherein the
constituent $N=2j$ spins-$1/2$ are pairwise entangled, and the best (minimum)
achievable squeezing is bounded by the Heisenberg limit $\xi^{2}=1/N$ SOR ;
LEW ; spinsqueezingreview . Using property (2) of the CRSS, together with
$\langle\hat{J}_{z}\rangle=m_{-}$, $\mathrm{Var}[\hat{J}_{z}]=w^{2}$ from
(11), we find $\xi^{2}=\sqrt{1-r^{2}}$ (Appendix). This result agrees with
that obtained for steady-state superradiance using a mean-field approach (with
linearized fluctuations) yelin ; HL , and as such, it is strictly exact only
for $j\rightarrow\infty$. However, our CRSS theory allows to extend the
analysis also for a finite $j\gg 1$. For a given $j$ we recall the validity
region $0<r<r_{j}$ defined in Eq. (8a). Since minimal $\xi^{2}=\sqrt{1-r^{2}}$
is achieved for maximal $r$, then for a given $j$ the predictable optimal
squeezing for a CRSS is given by

$\xi^{2}_{\min}(j)=\sqrt{1-r^{2}_{j}},$ (12a) |  $\xi^{2}_{\min}(j)\sim\left(\frac{3}{2j}\right)^{\frac{1}{3}}\propto N^{-\frac{1}{3}}.$ (12b)
---|---

Here (12b) is obtained using $r_{j}$ from (8b) and expresses the asymptotic
scaling of the CRSS optimal squeezing with $j=N/2\rightarrow\infty$. The
predicted scaling $N^{-\frac{1}{3}}$ does not reach the Heisenberg limit
$N^{-1}$.

Figure 4:  Spin squeezing $\xi^{2}$ in superradiance. (a) $\xi^{2}$ as a
function of $r=|\Omega|/\Omega_{c}$ for the steady state of Eq. (9) with
$j=N/2=25,100$ (dots, squares). The optimal (minimum) value $\xi_{\min}^{2}$
is seen around $r=r_{j}$ from Eq. (8a) [calculated using (7)]. Agreement with
CRSS theory (solid line) is observed within the validity region $r<r_{j}$. (b)
$\xi_{\min}^{2}$ defined in (a) plotted as a function of $j$ in log-log scale
(dots). Agreement with CRSS theory (12a) (solid line) is observed. Inset:
increasing $j$, (12a) scales as $j^{-0.32}$, in close agreement with the
asymptotics $j^{-\frac{1}{3}}$ from (12b) (solid and dashed lines,
respectively).

We now show how this theory applies also for spin squeezing in superradiance.
In Fig. 4a we plot $\xi^{2}$ as a function of $r=|\Omega|/\Omega_{c}$ for the
exact steady-state solution of Eq. (9) with $j=25,100$ (see SM  for
calculation details, in analogy to Alejandro ; yelin ). We observe excellent
agreement with the theoretical CRSS result $\sqrt{1-r^{2}}$ within the
expected validity region $r<r_{j}$ defined in (8a). Moreover, for $r>r_{j}$,
$\xi^{2}$ begins to grow, setting its minimum at around $r=r_{j}$. This
suggests that the CRSS prediction $\xi^{2}_{\mathrm{min}}$ from Eq. (12a) also
predicts the best achievable (minimal) spin squeezing in superradiance. This
is indeed what is observed in Fig. 4b, where the minimal $\xi^{2}$ of
superradiance, obtained with respect to $r$ for different $j$ values as in
Fig. 4a, is plotted as a function of $j$. Excellent agreement with the CRSS
prediction from (12a) is exhibited. Within this range of $j$ values, a linear
fit to the log-log plot yields the scaling $j^{-0.28}$. Extending the range of
$j$ values as in the inset of Fig. 4b, $\xi^{2}_{\min}$ from Eq. (12a) scales
as $j^{-0.32}$, already in close agreement with the predicted asymptotic
scaling $j^{-\frac{1}{3}}$ from Eq. (12b).

Radiated light
After discussing the spin system and its observables, we now turn our
attention to the light it radiates. We begin by showing the counter-intuitive
result, that the steady-state light emission in Dicke superradiance is a
classical coherent state whose amplitude is linear with the incident driving
field. This is identical to scattering by a linear dipole although the
spin-$j$ dipole is highly nonlinear, and nonlinear optical systems typically
generate photon interactions and quantum correlated light QNLOr . The crucial
point however is that we showed that in steady-state the superradiant spin is
pumped into a CRSS, which is expected to radiate classically as argued above.
To illustrate this, consider a generic relation between the field $\hat{E}$ in
some detection mode and the radiating dipole $\hat{J}$,

$\displaystyle\hat{E}(t)=\hat{E}_{0}(t)+G\hat{J}(t),$ (13)

written in the Heisenberg picture, similar to an input-output relation
derivable from (1) MW ; WEL . Here $\hat{E}_{0}(t)$ is the freely evolving
field, excluding interaction with the dipole, and the c-number $G$ is related
to $\gamma$ (see Appendix, e.g. for superradiance in a cavity). In the
Schrödinger picture, the solution of Eq. (9) at a steady state time $t$ is a
pure CRSS state of the dipole, $|j,\alpha\rangle_{d}$, and since the total
dipole+field system is closed, then the total state is separable,
$|\psi(t)\rangle=|j,\alpha\rangle_{d}\otimes|\chi\rangle_{f}$ with some field
state $|\chi\rangle_{f}$. We now show that this state is a coherent state,
i.e. an eigenstate of the (Schrödinger picture) field operator $\hat{E}(0)$.
Assuming a vacuum (or coherent) state for the field mode $\hat{E}(0)$ at
initial time $t=0$, and using Eqs. (13) and (2), we find
$\hat{E}(0)|\psi(t)\rangle=G\hat{J}(0)|\psi(t)\rangle=G\alpha|\psi(t)\rangle$
(Appendix). This proves that the steady state radiation is a coherent state
with an amplitude $G\alpha\propto\Omega$ linear in the incident field. So the
nonlinear spin system driven to an entangled many-body state (CRSS), scatters
light as a classical linear system. A similar relation (13) also holds for the
instantaneous emission at a given time $t$ without residing to steady-state
conditions (Appendix), leading to the general conclusion that a CRSS radiates
coherent-state light.

Dipole-projected squeezing
Although the CRSS is a non-classical spin squeezed state, we found that it
does not radiate quantum states such as squeezed light. This is also
intuitively understood as follows. From Hamiltonian (1) we observe that the
field perceives the spin directly only through the dipole lowering operator
$\hat{J}=\hat{J}_{x}-i\hat{J}_{y}$ and hence only through its projection onto
the $xy$ “dipole plane”. This suggests that non-classical squeezed light is
produced only if a similar squeezing exists in the projected $xy$ spin, hence
motivating the distinction between spin squeezing of the total spin and that
of the dipole-projected spin. To this end, we define the dipole quadrature
$\hat{J}_{\phi}=(e^{i\phi}\hat{J}+e^{-i\phi}\hat{J}^{{\dagger}})/2$, in
analogy to the field quadrature
$\hat{E}_{\phi}=e^{i\phi}\hat{E}+e^{-i\phi}\hat{E}^{{\dagger}}$. Using Eq.
(13) with $[\hat{E},\hat{E}^{{\dagger}}]=1$ and
$[\hat{J},\hat{J}^{{\dagger}}]=-2\hat{J}_{z}$, the relation between the
variances of these quadratures is found

$\displaystyle\mathrm{Var}[\hat{E}_{\phi}]=1+4G^{2}\left(\mathrm{Var}[\hat{J}_{\phi}]+\frac{1}{2}\langle\hat{J}_{z}\rangle\right),$
(14)

where $G$ is taken real without loss of generality. Considering the Heisenberg
uncertainty
$\mathrm{Var}[\hat{J}_{\phi}]\mathrm{Var}[\hat{J}_{\phi+\frac{\pi}{2}}]\geq|\langle\hat{J}_{z}\rangle|^{2}/4$,
we define the _dipole-projected squeezing_ for a quadrature $\phi$ as
$\mathrm{Var}[\hat{J}_{\phi}]<|\langle\hat{J}_{z}\rangle|/2$. Then, from Eq.
(14) we conclude that for $\langle\hat{J}_{z}\rangle<0$ the emitted light
becomes squeezed, i.e. $\mathrm{Var}[\hat{E}_{\phi}]<1$, provided that the
dipole-projected spin is also squeezed.

For CRSS, using the definition (2) we obtain
$\mathrm{Var}[\hat{J}_{\phi}]=-\langle\hat{J}_{z}\rangle/2$, so that no dipole
squeezing exists and vacuum noise-level is obtained for the light,
$\mathrm{Var}[\hat{E}_{\phi}]=1$, as expected for a coherent state. This is
nicely seen by the geometrical picture of Fig. 1b: the noise contour
perpendicular to the mean spin direction $\theta,\varphi$ presents the spin
squeezing of CRSS with the $\varphi$ and $\varphi+\frac{\pi}{2}$ axes
exhibiting increased and reduced variances $(j/2)/\cos\theta$ and
$(j/2)\cos\theta$, respectively (Appendix). However, upon projection of the
noise contours onto the $xy$ dipole plane, the $\varphi$ axis variance is
multiplied by a factor $\cos^{2}\theta$ so that both axes have identical
noises and no projected squeezing exists.

For a CSS, the opposite situation occurs, as seen in Fig. 1a. Whereas the
noise contour at the plane perpendicular to the main spin direction is a
circle of variance $j/2$ so that no spin squeezing exists spinsqueezingreview
, the projection onto the $xy$ dipole plane reduces the noise at the $\varphi$
direction to $(j/2)\cos^{2}\theta$, smaller than
$|\langle\hat{J}_{z}\rangle|/2=(j/2)\cos\theta$. This means that a CSS at the
southern Bloch hemisphere, $\langle\hat{J}_{z}\rangle<0$, radiates squeezed
light. The latter is also verified by a direct calculation of
$\mathrm{Var}[\hat{E}_{\phi}]$ using a CSS SM .

The above results can be summarized in the following counter-intuitive manner:
CSS, which is a non-correlated many-body spin, emits quantum-correlated
squeezed light; whereas CRSS, which is a quantum-correlated (spin-squeezed)
many-body spin, emits non-correlated coherent light. The fact that
correlations are necessary in a spin system to produce classical non-
correlated light is nicely seen from a microscopical picture. For dipole-
projected squeezing, and hence light squeezing to exist, the phase-dependent
variance
$\mathrm{Var}[\hat{J}]=\langle\hat{J}^{2}\rangle-\langle\hat{J}\rangle^{2}$
also has to exist. Writing this quantity using constituent spin operators
$\hat{J}=\sum_{n=1}^{N}\hat{\sigma}_{n}$ and recalling
$\hat{\sigma}_{n}^{2}=0$, we have

$\displaystyle\mathrm{Var}[\hat{J}]=-\sum_{n}\langle\hat{\sigma}_{n}\rangle^{2}+\sum_{n}\sum_{n\neq
m}\left(\langle\hat{\sigma}_{n}\hat{\sigma}_{m}\rangle-\langle\hat{\sigma}_{n}\rangle\langle\hat{\sigma}_{m}\rangle\right).$
(15)

Assuming a coherent drive $\Omega$, where $\langle\hat{\sigma}_{n}\rangle\neq
0$, and for statistically independent atoms as in a CSS, the second term
vanishes such that $\mathrm{Var}[\hat{J}]\neq 0$ and light squeezing exists.
This typical situation occurs in resonance fluorescence from an ensemble of
individual atoms, where the nonlinearity of each two-level atom generates
squeezed light singleatom ; Natom . For squeezing to vanish, we must have a
non-vanishing second term in Eq. (15), namely, non-vanishing correlations
between atoms. For a pure state, these correlations imply entanglement, which
is exactly the situation in a CRSS, where the pairwise entanglement due to
spin-squeezing leads to the exact cancellation of the two terms in (15).

Outlook
This study introduces CRSS as a new class of macroscopic spin states which
possess rather unusual properties: they are many-body entangled states which
behave as macroscopic classical emitters.

These new insights open several important directions. First, since CRSS theory
was shown here to describe Dicke superradiance, the above predictions on both
photon and collective-spin observables could be evaluated experimentally in
platforms such as cavities HAR ; TOMs1 , waveguides Alejandro ; KIMsr ; CHArmp
, and superconducting resonators MAJ . Moreover, CRSS theory may be
conceptually and technically useful for the study of various related
collective radiation phenomena. For example, it is interesting to explore the
possible role of CRSS in superradiant lasing HOL ; TOM1  and whether it can be
generalized to account for collective radiation beyond the permutation-
symmetric Dicke case GROSS1982301 ; YEL . This could have important
consequences for understanding and exploiting superradiance both in ordered
and disordered ensembles coop ; RUI ; ANA ; RUO3 ; KSR1 ; BRW2 ; YEb  and for
quantum metrology CHAYE ; HEN ; REY ; YEr .

Another direction is the exploration of the very nature of CRSS and its
relation to many-body entangled spin states. The optimal spin squeezing
contained in a CRSS exhibits a different scaling, $N^{-\frac{1}{3}}$, than
that of spin-squeezed states generated by one- and two-axis twisting
Hamiltonians spinsqueezingreview ; KIT , with $N^{-\frac{2}{3}}$ and $N^{-1}$,
respectively. It is interesting to study the existence of unitary
transformations that generate this different class of spin squeezing.
Moreover, although CRSS underlies the dissipative Dicke phase transition, it
is in fact a pure state. Hence, it might be possible to relate it to quantum
phase transitions in Hamiltonian many-body spin systems.

More generally, relying on the finding that a macroscopic spin is perceived as
a classical emitter provided that its constituents are quantum entangled, it
should be considered if and how this idea extends to other composite quantum
systems.

## APPENDIX

Radiation from a lowering-operator eigenstate
Here we show that for a generic dipole+field system, $\hat{H}=\hbar
g(\hat{E}\hat{d}^{{\dagger}}+\hat{E}^{{\dagger}}\hat{d})$ (coupling constant
$g$), that is pumped into an eigenstate of $\hat{d}$ with an eigenvalue
$\alpha$, one obtains a coherent state of the field $\hat{E}$, which is
equivalent to that obtained by the replacement $\hat{d}\rightarrow\alpha$ in
$\hat{H}$. We assume that the system is pumped to a state $|\psi(t)\rangle$
that satisfies at any given time $t$,
$\hat{d}(0)|\psi(t)\rangle=\alpha|\psi(t)\rangle$, where $\hat{d}(0)$ is the
Scrödinger picture operator. Such a pumping mechanism can be supplied by an
external reservoir as in the steady state of Eq. (9) with
$\hat{J}\leftrightarrow\hat{d}$ and as discussed above for superradiance. We
note

$\displaystyle\hat{E}(0)|\psi(t)\rangle=\hat{U}(t)\hat{E}(t)|\psi(0)\rangle,\quad\hat{U}(t)=e^{-\frac{i}{\hbar}\hat{H}t},$
(16)

where $|\psi(t)\rangle=\hat{U}(t)|\psi(0)\rangle$ and
$\hat{E}(t)=\hat{U}^{{\dagger}}(t)\hat{E}(0)\hat{U}(t)$ is the Heisenberg
picture operator, formally obtained from the Heisenberg equations as

$\displaystyle\hat{E}(t)=\hat{E}(0)-ig\int_{0}^{t}dt^{\prime}\hat{d}(t^{\prime}).$
(17)

Inserting Eq. (17) into Eq. (16), assuming an initial vacuum state of the
field, $\hat{E}(0)|\psi(0)\rangle=0$, and using
$\hat{d}(t^{\prime})=\hat{U}^{{\dagger}}(t^{\prime})\hat{d}(0)\hat{U}(t^{\prime})$,
we obtain

$\displaystyle\hat{E}(0)|\psi(t)\rangle=-ig\hat{U}(t)\int_{0}^{t}dt^{\prime}\hat{U}^{{\dagger}}(t^{\prime})\hat{d}(0)|\psi(t^{\prime})\rangle$
$\displaystyle=-ig\alpha\hat{U}(t)\int_{0}^{t}dt^{\prime}\hat{U}^{{\dagger}}(t^{\prime})|\psi(t^{\prime})\rangle=-ig\alpha
t|\psi(t)\rangle,$ (18)

where the pumping assumption,
$\hat{d}(0)|\psi(t^{\prime})\rangle=\alpha|\psi(t^{\prime})\rangle$, was used
in the second equality. Equation (18) then shows that $|\psi(t)\rangle$ is a
coherent state with an amplitude $-ig\alpha t$ just as one would get from the
displacement operator $\hat{U}(t)=e^{-\frac{i}{\hbar}\hat{H}t}$ formed by
setting $\hat{d}\rightarrow\alpha$ in $\hat{H}$.

Minimization of the proximity error
We showed that the error $\epsilon_{j}(r)$ of state (6) goes to zero
exponentially. Here we show that it also does so at the highest rate. We
define the state $|\alpha,j\rangle_{\mathrm{min}}$ that minimizes
$\epsilon_{j}(r)$ in Eq. (3) while keeping $j$ and $\alpha$ fixed. Writing the
error for a state $|\psi\rangle$ as
$\epsilon_{\psi}=\left\|\hat{K}|\psi\rangle\right\|$ with
$\hat{K}=\hat{J}-\alpha$, $|\alpha,j\rangle_{\mathrm{min}}$ is given by the
state $|\psi\rangle$ which minimizes
$\epsilon^{2}_{\psi}=\langle\psi|\hat{K}^{{\dagger}}\hat{K}|\psi\rangle$.
Since the lower bound of this quantity is the smallest eigenvalue of
$\hat{K}^{{\dagger}}\hat{K}$, its corresponding eigenstate is in fact the
state $|\alpha,j\rangle_{\mathrm{min}}$, and can be readily evaluated
numerically for given $j$ and $\alpha$. Figure 5a compares the two definitions
$|j,\alpha\rangle_{\mathrm{ans}}$ and $|j,\alpha\rangle_{\mathrm{min}}$, by
displaying the infidelity $1-|_{\mathrm{ans}}\langle
j,\alpha|j,\alpha\rangle_{\mathrm{min}}|$. It is seen that this infidelity
decreases exponentially with $j$, indicating that the two definitions coalesce
in the relevant limit $j\rightarrow\infty$.

Superradiance at finite $j$
For a finite $j\gg 1$, the state $|j,\alpha\rangle_{\mathrm{ans}}$ from (6) is
a CRSS to an excellent approximation in the range $r<r_{j}$ with $r_{j}$ from
(8a). We thus expect that within this range $|j,\alpha\rangle_{\mathrm{ans}}$
also well approximates the steady-state of the superradiance master equation
(9). For the case $\Delta=0$, the latter is given by
$\hat{\rho}_{s}\propto[(\hat{J}-\alpha)^{{\dagger}}(\hat{J}-\alpha)]^{-1}$ CAR
; LAW  and can be readily evaluated numerically for given $j$ and
$r=|\Omega|/\Omega_{c}$. In Fig. 5b we plot the infidelity between the state
(6) and the exact steady-state $\hat{\rho}_{s}$, $jr(1-_{\mathrm{ans}}\langle
j,\alpha|\hat{\rho}_{s}|j,\alpha\rangle_{\mathrm{ans}})$, observing a similar
dependence on $j$ and $r$ as that of the CRSS proximity error in Fig. 3b. In
particular, the curve $r_{j}$ (dashed line) indeed overlaps with an infidelity
contour beyond which the state $|j,\alpha\rangle_{\mathrm{ans}}$ well
approximates $\hat{\rho_{s}}$.

Figure 5:  (a) The infidelity between the state $|j,\alpha\rangle_{\min}$ that
minimizes the proximity error and the CRSS ansatz
$|j,\alpha\rangle_{\mathrm{ans}}$ decays exponentially with $j$, so that the
states coalesce for $j\rightarrow\infty$ ($r=0.4,0.7,0.9$ in dashed, dash-
dotted, and solid lines). (b) The infidelity between the exact steady-state
$\hat{\rho}_{s}$ of (9) and the CRSS ansatz behave similar to the proximity
error in Fig. 3b, as expected. Therefore,
$\hat{\rho}_{s}\approx|j,\alpha\rangle_{\mathrm{ans}}\langle j,\alpha|$ forms
an excellent approximation at any finite $j\gg 1$ for $0<r<r_{j}$ (dashed
curve marks $r_{j}$, white parts represent values too small for our numerical
evaluation).

The normalization factor $jr$ in the above infidelity definition is analogous
to that in the proximity error in Eqs. (3) and (6). This error (or infidelity)
definition means that an error of order $<1$ leads to values of typical spin
correlators (e.g. $\langle\hat{J}^{2}\rangle$) which agree with their CRSS
values ($\langle\hat{J}\rangle^{2}$) up to a correction smaller by a factor
$\sim 1/|\alpha|=1/(jr)$ (similar to a mean-field assumption).

Expansion of the distribution $|a_{m}|^{2}$
From Eq. (5) we have $|a_{m}|^{2}\propto e^{2f_{m}}$. Approximating the sum in
$f_{m}$ by an integral SM , we obtain an analytical expression for
$f_{m}=f(m)$ and expand it around the peak $m=m_{-}$,

$\displaystyle f(m)$ $\displaystyle\approx$ $\displaystyle
A_{0}-A_{2}(m-m_{-})^{2}$ (19) $\displaystyle+$ $\displaystyle
A_{3}(m-m_{-})^{3}+A_{4}(m-m_{-})^{4},$

with the expansion coefficients for $j\gg 1$,

$\displaystyle A_{2}(j,r)$ $\displaystyle\approx$
$\displaystyle\frac{\sqrt{1-r^{2}}}{2r^{2}}\frac{1}{j},$ $\displaystyle
A_{3}(j,r)$ $\displaystyle\approx$
$\displaystyle\left(\frac{1}{3r^{4}}-\frac{1}{6r^{2}}\right)\frac{1}{j^{2}},$
$\displaystyle A_{4}(j,r)$ $\displaystyle\approx$
$\displaystyle\frac{\sqrt{1-r^{2}}}{3r^{4}}\left(\frac{1}{4}-\frac{1}{r^{2}}\right)\frac{1}{j^{3}},$
(20)

and where $A_{0}$ is of no importance here. For a Gaussian approximation to be
valid, only the $A_{2}$ term should be significant. This term implies a
distribution width of order
$w=\left[jr^{2}/(2\sqrt{1-r^{2}})\right]^{\frac{1}{2}}\propto\sqrt{j}$, so
that the terms $A_{n}\propto j^{-n+1}$ for $n>2$ are negligible if
$A_{n}w^{n}\ll 1$. It can be verified that this is indeed the case if we
demand $\sqrt{1-r^{2}}j^{\frac{1}{3}}\gg 1$. This is equivalent to the CRSS
validity condition $r<r_{j}$ with the asymptotic $r_{j}$ from Eq. (8b). In
this case, we obtain a Gaussian distribution and, demanding normalization, we
arrive at Eq. (11).

CRSS spin squeezing calculation
For the spin squeezing, one has to calculate the variance of the spin operator
$\hat{J}^{\bot}_{\phi}$ that is projected to the plane perpendicular to the
mean spin vector (MSV). The latter defines the angles $\theta$ and $\varphi$
from the main text, and hence the corresponding rotated coordinate system in
which the MSV points to a rotated $z^{\prime}$ axis. Within this rotated
coordinate system, the lowering spin operator
$\hat{J}^{\prime}=\hat{J}^{\prime}_{x}-i\hat{J}^{\prime}_{y}$ (with
$\hat{J}^{\prime}_{i}$ the spin projection onto the rotated
$i\in\\{x^{\prime},y^{\prime}\\}$ axis) is given in terms of the original spin
operators via the transformation,

$\displaystyle\hat{J}^{\prime}=e^{i\varphi}\frac{\cos{\theta}+1}{2}\hat{J}+e^{-i\varphi}\frac{\cos{\theta}-1}{2}\hat{J}^{{\dagger}}-\sin\theta\hat{J}_{z},$
(21)

recalling $\sin\theta=r$. A general Hermitian spin operator on the plane
perpendicular to the MSV is then given by
$\hat{J}^{\bot}_{\phi}=(e^{i\phi}\hat{J}^{\prime}+e^{-i\phi}\hat{J}^{\prime{\dagger}})/2$.
Calculating the variance of this operator involves various first and second
order moments of $\hat{J}$, $\hat{J}^{{\dagger}}$ and $\hat{J}_{z}$, all of
which can be evaluated using the CRSS property (2), the moments of
$\hat{J}_{z}$ from Eq. (11) and the SU(2) commutation relations
$[\hat{J},\hat{J}^{{\dagger}}]=-2\hat{J}_{z}$ and
$[\hat{J},\hat{J}_{z}]=\hat{J}$. Recalling the mean spin length
$|\langle\hat{\mathbf{J}}\rangle|=j$ in a CRSS, we finally have

$\displaystyle\xi^{2}_{\phi}=\frac{2j\mathrm{Var}[\hat{J}^{\bot}_{\phi}]}{|\langle\hat{\mathbf{J}}\rangle|^{2}}=\cos[2(\phi-\varphi)]\frac{1-c^{2}}{2c}+\frac{1+c^{2}}{2c},$
(22)

with $c=\sqrt{1-r^{2}}$. The minimal noise is obtained for
$\phi=\varphi+\pi/2$, yielding the spin squeezing
$\xi^{2}=\xi^{2}_{\varphi+\pi/2}=c=\sqrt{1-r^{2}}=\cos\theta<1$, whereas the
conjugate quadrature $\phi=\varphi$ exhibits anti-squeezing,
$\xi^{2}_{\varphi}=1/\cos\theta>1$. This is illustrated in Fig. 1b.

The scattered field
The relation (13) between the output field and the radiating dipole can be
generally obtained in the Heisenberg picture. One typical case is that of a
dipole damped by a continuum of photon modes described by an operator
$\hat{E}$ in (1), in equivalence to the master equation (9) and in analogy to
input-output theory MW  (where $G$ is often related to the photon Green’s
function WEL ; CARb ). For example, consider superradiance in a leaky cavity,
where the atoms are situated at positions where they are identically coupled
to the cavity mode $\hat{a}$ and driven by a laser $\Omega$. The Hamiltonian
is given by

$\displaystyle\hat{H}=-\hbar\delta_{a}\hat{J}_{z}-\hbar\delta_{c}\hat{a}^{{\dagger}}\hat{a}+\hbar\left[\left(g^{\ast}\hat{a}^{{\dagger}}+\Omega^{\ast}\right)\hat{J}+\mathrm{h.c.}\right],$
(23)

where $\delta_{a}$ and $\delta_{c}$ are the detunings of the dipole and the
cavity from the driving-laser frequency. Considering the damping of the cavity
to the outside modes thorough its leaky mirrors at a rate $\kappa$, and
exploiting the separation of time scales $\kappa\gg g$, we formally solve the
Heisenberg-Langevin equation for $\hat{a}$ at $t\gg 1/\kappa$, obtaining

$\displaystyle
g\hat{a}(t)=\left(\Delta-i\frac{\gamma}{2}\right)\hat{J}(t)+\hat{f}(t),\quad\Delta-i\frac{\gamma}{2}=\frac{|g|^{2}}{\delta_{c}+i\kappa/2}.$
(24)

Here $\hat{f}(t)$ is a Langevin noise due to the freely evolving (vacuum)
field of the modes to which the cavity is damped, satisfying
$\hat{f}(t)|\psi(0)\rangle=0$, where the initial state $|\psi(0)\rangle$ is in
the field’s vacuum. We identify that Eq. (24) is equivalent to Eq. (13) with
the “detection mode” $\hat{E}$ being the cavity mode. A similar relation can
be obtained between the dipole and the outside field modes. Once the relation
(13) is established, and using $\hat{E}_{0}(t)|\psi(0)\rangle=0$ for the
freely evolving vacuum, a similar algebra to that of Eq. (18) leads to
$\hat{E}(0)|\psi(t)\rangle=G\hat{U}(t)\hat{J}(t)|\psi(0)\rangle=G\hat{J}(0)|\psi(t)\rangle=G\alpha|\psi(t)\rangle$.

A simpler case wherein a relation of the type (13) holds is that of the
instantaneous field emitted at a given time $t\equiv 0$. The general solution
from Eq. (17), for a dipole $\hat{d}=\hat{J}$ and a short evolution time
$\tau$ after $t=0$, is given by
$\hat{E}(\tau)\approx\hat{E}(0)-ig\tau\hat{J}(0)$. Here again it is simple to
show that $\hat{E}(0)|\psi(\tau)\rangle=G\alpha|\psi(\tau)\rangle$ with
$G=-ig\tau$.

###### Acknowledgements.

We acknowledge financial support from the Israel Science Foundation (ISF)
grant No. 2258/20, the ISF and the Directorate for Defense Research and
Development (DDR&D) grant No. 3491/21, the Center for New Scientists at the
Weizmann Institute of Science, the Council for Higher Education (Israel), and
QUANTERA (PACE-IN). This research is made possible in part by the historic
generosity of the Harold Perlman Family.

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