# Quantum correlations in the steady state of light-emitter ensembles from
perturbation theory

Dolf Huybrechts<EMAIL_ADDRESS>Univ Lyon, Ens de Lyon, CNRS, Laboratoire
de Physique, F-69342 Lyon, France  Tommaso Roscilde  tommaso.roscilde@ens-
lyon.fr Univ Lyon, Ens de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon,
France

###### Abstract

The coupling of a quantum system to an environment leads generally to
decoherence, and it is detrimental to quantum correlations within the system
itself. Yet some forms of quantum correlations can be robust to the presence
of an environment – or may even be stabilized by it. Predicting (let alone
understanding) them remains arduous, given that the steady state of an open
quantum system can be very different from an equilibrium thermodynamic state;
and its reconstruction requires generically the numerical solution of the
Lindblad equation, which is extremely costly for numerics. Here we focus on
the highly relevant situation of ensembles of light emitters undergoing
spontaneous decay; and we show that, whenever their Hamiltonian is perturbed
away from a U(1) symmetric form, steady-state quantum correlations can be
reconstructed via pure-state perturbation theory. Our main result is that in
systems of light emitters subject to single-emitter or two-emitter driving,
the steady state perturbed away from the U(1) limit generically exhibits spin
squeezing; and it has minimal uncertainty for the collective-spin components,
revealing that squeezing represents the optimal resource for entanglement-
assisted metrology using this state.

##  1 Introduction

Quantum correlations [1], and in particular many-body entanglement [2], are
the central resource of quantum many-body devices processing quantum
information or simulating complex interacting Hamiltonians [3]. The ability to
engineer arbitrary entangled quantum states of _e.g_ a system of $N$ qubits
amounts in principle to controlling an exponentially large (in $N$) amount of
information – as opposed to linear in $N$ for classical information.
Nonetheless quantum information of $N$ qubits is as astronomically large as it
is fragile to the presence of an environment. Since information is stored in
the relative phases and amplitudes of many-body superposition states, it is
extremely sensitive to decoherence and dissipation, generically reducing
entangled states to statistical mixtures of states. To counter this effect,
the most obvious strategy is to decouple qubits from their environment as much
as possible. Yet a more specialized strategy to preserve entanglement in open
quantum systems consists in finding forms of entanglement which are _robust_
to the presence of an environment, or that are even stabilized by it. Such
robustness to, or assistance from the environment can come from distinct
mechanisms. One relevant example is collective dissipation in driven-
dissipative systems, in which the degrees of freedom are driven individually,
but the elementary dissipation processes can induce entanglement in the system
[4]. Through an appropriate engineering of the coupling of the system to the
environment, one can obtain steady state entanglement and complex non-
equilibrium phases of matter [4, 5, 6, 7]. Another important example is the
competition between dissipation (individual or collective) and entangling
Hamiltonian dynamics [8, 9, 10, 11]. As a result, the stationary state of the
dissipative quantum system can feature entanglement going beyond the paradigm
of unitary evolutions, or beyond that of ground-state physics for interacting
Hamiltonians.

Understanding the kind of entangled states which can emerge as stationary
states of driven-dissipative many-body systems is a formidable theoretical
challenge. Indeed reconstructing the stationary state of generic dissipative
systems is numerically prohibitive already in systems of $\sim O(10)$
elementary degrees of freedom (such as qubit ensembles). And approximation
schemes to tackle the stationary state are often limited in the amount of
entanglement that they can describe, either within the system itself (such as
in mean-field approaches [9]); or of the system with the environment (such as
in approaches based on reduced Hilbert spaces [12]). A significant effort has
been devoted in recent years to the development of methods that can accurately
describe the properties of open quantum systems [9, 12, 13, 14, 15, 16, 17,
18, 19, 20, 21, 10]. However, the understanding of their correlation
properties remains very challenging, and difficult to guess without a
dedicated microscopic calculation.

In this work we offer a framework to understand and predict the correlation
properties of stationary states of an important class of dissipative quantum
systems, i.e. ensembles of quantum emitters coupled to the vacuum of the
electromagnetic field [22]. If the Hamiltonian governing the unitary dynamics
of the system has rotation (U(1)) symmetry along the quantization axis of the
emitters, then the steady state of the system is trivial, and it corresponds
to all the emitters in their ground state. We show that a U(1)-symmetry-
breaking perturbation in the Hamiltonian, if treated within pure-state
perturbation theory, induces quantum correlations in the steady state. This
happens at first order in the perturbation, if the perturbation drives pairs
of emitters; or at second order, if the perturbation drives individual
emitters independently. In particular, the perturbed states possess a very
clear form of quantum correlations, i.e. _spin squeezing_ , and they have
minimal uncertainty for the collective-spin operators, so that squeezing is
their optimal metrological resource [23]. The perturbative calculations can be
fully worked out in the case of collective-spin models with individual or
collective emission; they allow one to understand the steady state as a
perturbed, arbitrarily excited eigenstate of the system’s Hamiltonian; and
they can be tested successfully against exact results. The insight gained from
these reference models serves as a very useful template for the understanding
of more complex situations of models with spatially structured interactions
and/or spatially structured collective emission, of direct relevance to
current experiments on collective light-matter interactions [24, 25].

Our paper is structured as follows: Sec. 2 discusses the general setting of
pure-state perturbation theory for the steady state of the dissipative
dynamics; Sec. 3 discusses the models of dissipation of interest to this work;
Sec. 4 introduces the collective-spin properties used to characterize quantum
correlations; Sec 5 applies the perturbative approach to the case of models
with two-emitter driving, formulating the condition under which squeezing is
expected in the steady state in the form of a "squeezing theorem" for
dissipative steady states; Sec. 6 reconstructs the squeezing properties for
the collective XYZ model and transverse-field Ising model with single-emitter
dissipation; Sec. 7 discusses instead models with single-emitter drive and
collective dissipation, with special focus on the driven Dicke model. A
discussion of the experimental implications and conclusions are offered in
Secs. 8 and 9.

Figure 1: Sketch of the physical situations of interest to this work: (a) the
unperturbed reference is a system of light emitters interacting via a
U(1)-symmetric Hamiltonian, and emitting photons in the environment; (b) a
Hamiltonian perturbation $\lambda H_{1}$ associated with a two-emitter drive,
re-exciting (or de-exciting) the emitters in pairs, can break the U(1)
symmetry; (c) a single-emitter (Rabi) drive with strength $\Omega$.

##  2 Pure-state perturbation theory for the steady state of the Lindblad
equation

###  2.1 General setting

The focus of our work are open quantum systems that interact with a Markovian
environment, and can be described by a Lindblad master equation [26]

$\partial_{t}{\rho}={\cal
L}[\rho]=-i\left[{H},{\rho}\right]+\sum_{j}\mathcal{D}\left(\sqrt{\gamma_{j}}\leavevmode\nobreak\
{L}_{j}\right)[\rho],$ (1)

where ${\cal L}[.]$ denotes the Liouvillian superoperator, ${H}$ is the
system’s Hamiltonian, ${\rho}$ the system’s density matrix, and
$\mathcal{D}\left(\sqrt{\gamma_{j}}\leavevmode\nobreak\ {L}_{j}\right)$
represents the dissipative process associated with the jump operators
$\sqrt{\gamma_{j}}\leavevmode\nobreak\ {L}_{j}$, and it reads

$\mathcal{D}\left(\sqrt{\gamma_{j}}\leavevmode\nobreak\
{L}_{j}\right)[\rho]=\frac{\gamma_{j}}{2}\left(2{L}_{j}{\rho}{L}_{j}^{\dagger}-\left\\{{L}_{j}^{\dagger}{L}_{j},{\rho}\right\\}\right).$
(2)

Our goal is the search of the steady state $\rho_{ss}$ such that ${\cal
L}[\rho_{\rm ss}]=0$. The general setting for perturbation theory of Markovian
open quantum systems [27, 28, 29, 30, 31, 32] assumes that the Liouvillian can
be written as ${\cal L}={\cal L}_{0}+\lambda{\cal L}_{1}$, where ${\cal
L}_{0}[.]=-i[H_{0},(.)]+\sum_{j}\mathcal{D}_{0}\left(\sqrt{\gamma_{j}}\leavevmode\nobreak\
{L}_{j}\right)[.]$ is an unperturbed Liouvillian for which the steady state
($\rho_{\rm ss}=\rho_{0}$) is known; while $\lambda{\cal L}_{1}$ is a
perturbation, parametrized by $\lambda\in\mathbb{R}$. Since our focus is on
elementary mechanisms inducing quantum correlations in the steady state, we
shall specialize our interest to the case in which

  1. 1.

the unperturbed steady state is a unique, _pure state_ ,
$\rho_{0}=|\phi_{0}\rangle\langle\phi_{0}|$, corresponding to an eigenstate of
$H_{0}$, and such that, for all $j$, $L_{j}|\phi_{0}\rangle=0$;

  2. 2.

the perturbation is purely Hamiltonian, and the perturbed Hamiltonian reads
$H=H_{0}+\lambda H_{1}$.

The relevance of the above assumptions to experiments is rather clear. The
steady state of systems of $N$ degrees of freedom coupled to a dissipative
environment is typically one in which each degree of freedom loses all its
energy to the environment, and falls to its single-body ground state, so that
$\sum_{j}\mathcal{D}_{0}\left(\sqrt{\gamma_{j}}\leavevmode\nobreak\
{L}_{j}\right)[\rho_{0}]=0$; and at the same time this state is also a
(generically excited) eigenstate of the many-body Hamiltonian $H_{0}$,
${H}_{0}|\phi_{0}\rangle=E_{0}|\phi_{0}\rangle$ so that $[H_{0},\rho_{0}]=0$.
This is the case _e.g._ of light emitters (real or artificial atoms) which are
not driven, and which therefore decay to their single-body ground state after
having emitted. A most natural way of perturbing this steady state in an
experiment is to perturb the Hamiltonian with a driving term $\lambda H_{1}$
which alters its symmetry, such that $[H_{0}+\lambda H_{1},\rho_{0}]\neq 0$.

We want to stress that the perturbation $\lambda H_{1}$, although involving
the Hamiltonian only, should be still viewed as a perturbation to the
Liouvillian super-operator, i.e. as $\lambda{\cal
L}_{1}[.]=i\lambda[H_{1},(.)]$. And its smallness should not be measured in
comparison with the Hamiltonian term of the Liouvillian only (which may be
vanishing, as we will see in the examples offered in this work); but also in
comparison with the dissipative part.

The perturbed steady state can be written in general as a power series of the
perturbation

$\rho_{\rm ss}=\rho_{0}+\lambda\rho_{1}+\lambda^{2}\rho_{2}+...$ (3)

and this state will be generically a mixed state. Yet, since $\rho_{0}$ is
pure, we can imagine that the corrections $\lambda\rho_{1}$,
$\lambda^{2}\rho_{2}$, etc. alter minimally its purity, so that the perturbed
state can in fact be searched for in the form of a _pure_ state as well. Under
this assumption we write $\rho_{\rm ss}$ as
$|\phi_{0}^{\prime}\rangle\langle\phi_{0}^{\prime}|$ with

$|\phi_{0}^{\prime}\rangle=|\phi_{0}\rangle+\lambda|\psi_{1}\rangle+\lambda^{2}|\psi_{2}\rangle+...$
(4)

so that

$\displaystyle\rho_{ss}$ $\displaystyle\approx\leavevmode\nobreak\
\rho_{0}+\lambda\left(|\phi_{0}\rangle\langle\psi_{1}|+|\psi_{1}\rangle\langle\phi_{0}|\right)$
$\displaystyle+\lambda^{2}\left(|\psi_{1}\rangle\langle\psi_{1}|+|\psi_{2}\rangle\langle\phi_{0}|+|\phi_{0}\rangle\langle\psi_{2}|\right)+O(\lambda^{3})\leavevmode\nobreak\
.$ (5)

Here we shall provide the main results of pure-state perturbation theory
relevant for the study provided below.

Injecting the expansion Eq. (3) into the Lindblad equation, imposing the
condition of stationarity, and equating terms of same order in $\lambda$, one
obtains the following conditions on the first- and second-order corrections to
the steady state:

$\displaystyle\dot{\rho}_{1}$
$\displaystyle=-i[H_{0},\rho_{1}]-i[H_{1},\rho_{0}]$ (6)
$\displaystyle+\sum_{j}\gamma_{j}\left[L_{j}\rho_{1}L_{j}^{\dagger}-\frac{1}{2}\\{L^{\dagger}_{j}L_{j},\rho_{1}\\}\right]=0,$
$\displaystyle\dot{\rho}_{2}$
$\displaystyle=-i[H_{0},\rho_{2}]-i[H_{1},\rho_{1}]$ (7)
$\displaystyle+\sum_{j}\gamma_{j}\left[L_{j}\rho_{2}L_{j}^{\dagger}-\frac{1}{2}\\{L^{\dagger}_{j}L_{j},\rho_{2}\\}\right]=0.$

Given the Hamiltonian eigenbasis $H|\phi_{n}\rangle=E_{n}|\phi_{n}\rangle$,
and assuming the unperturbed steady state $|\phi_{0}\rangle$ to be a _non-
degenerate_ eigenstate of $H$, we can decompose the pure-state perturbations
$|\psi_{1}\rangle$ and $|\psi_{2}\rangle$ in Eq. (4) on the
$\\{|\phi_{n}\rangle\\}$ basis as

$\displaystyle|\psi_{1}\rangle$ $\displaystyle=\sum_{n\neq
0}c_{n}|\phi_{n}\rangle$ $\displaystyle|\psi_{2}\rangle$
$\displaystyle=\left(1-\frac{1}{2}\langle\psi_{1}|\psi_{1}\rangle\right)|\phi_{0}\rangle+\sum_{n\neq
0}d_{n}|\phi_{n}\rangle\leavevmode\nobreak\ .$ (8)

As in ordinary perturbation theory for Hamiltonian eigenstates [33], the
first-order correction can be taken to be orthogonal to the unperturbed state,
while the second-order one needs to have overlap with the unperturbed state in
order to enforce normalization of the perturbed state.

Imposing the conditions of stationarity, Eqs. (6)-(7), on the states
$|\psi_{1}\rangle$ and $|\psi_{2}\rangle$, and projecting them onto the
Hamiltonian eigenstates $\langle\phi_{n}|$ from the left and
$|\phi_{0}\rangle$ from the right, leads to the following coupled equations
for the $c_{n}$ and $d_{n}$ coefficients:

$\displaystyle(E_{n}-E_{0})c_{n}+\langle\phi_{n}|H_{1}|\phi_{0}\rangle$ (9)
$\displaystyle-\frac{i}{2}\sum_{m}c_{m}\langle\phi_{n}|\sum_{j}\gamma_{j}L^{\dagger}_{j}L_{j}|\phi_{m}\rangle=0,\leavevmode\nobreak\
$
$\displaystyle\left(E_{n}-E_{0}\right)d_{n}+\langle\phi_{n}|H_{1}|\psi_{1}\rangle-
c_{n}\langle\phi_{0}|H_{1}|\phi_{0}\rangle$
$\displaystyle+i\sum_{j}\gamma_{j}\langle\phi_{n}|L_{j}|\psi_{1}\rangle\langle\psi_{1}|L_{j}^{\dagger}|\phi_{0}\rangle$
$\displaystyle-\frac{i}{2}\sum_{j}\sum_{m}\gamma_{j}\langle\phi_{n}|L_{j}^{\dagger}L_{j}|\phi_{m}\rangle
d_{m}=0$ (10)

where in the second equation we have kept the $|\psi_{1}\rangle$ symbol for
the sake of brevity.

The above equations clearly provide the formal solution to the perturbation
problem if the matrix elements of the Hamiltonian perturbation $H_{1}$ and of
the jump operators $L_{j},L_{j}^{\dagger}$ on the basis of the unperturbed
Hamiltonian eigenstates are known. We would like to stress that this aspect is
made possible because we are restricting our attention to the steady state of
the Lindblad dynamics, and we are taking it as a pure state. Previous works
[27, 28, 29] have rather tackled the more ambitious problem of reconstructing
the full perturbed Liouvillian spectrum and the corresponding generalized
eigenvectors. This problem is significantly more involved, as the equations
defining the perturbation to the generalized eigenvectors require the pseudo-
inversion of non-Hermitian super-operators.

###  2.2 Special case: $\sum_{j}\gamma_{j}L^{\dagger}_{j}L_{j}$ commutes with
the unperturbed Hamiltonian

A special case – relevant to all the examples that we shall provide in the
following – is the one in which the jump operators satisfy the following
condition

$[\sum_{j}\gamma_{j}L^{\dagger}_{j}L_{j},H_{0}]=0\leavevmode\nobreak\ .$ (11)

For this choice of jump operators, $\sum_{j}\gamma_{j}L^{\dagger}_{j}L_{j}$ is
diagonal on the eigenbasis of $H_{0}$, simplifying significantly the equations
Eqs. (9)-(2.1) above. Indeed this allows for an explicit expression of the
first- and second-order corrections to the pure stationary state:

$\displaystyle|\psi_{1}\rangle=-\sum_{n\neq
0}\frac{\langle\phi_{n}|H_{1}|\phi_{0}\rangle}{\langle
H_{0}^{\rm(NH)}\rangle_{n}-E_{0}}\leavevmode\nobreak\ |\phi_{n}\rangle$ (12)
$\displaystyle|\psi_{2}\rangle=$ $\displaystyle\sum_{n\neq 0,m\neq
0}\frac{\langle\phi_{n}|H_{1}|\phi_{m}\rangle\langle\phi_{m}|H_{1}|\phi_{0}\rangle}{(\langle{H}_{0}^{\rm(NH)}\rangle_{n}-E_{0})(\langle{H}_{0}^{\rm(NH)}\rangle_{m}-E_{0})}\leavevmode\nobreak\
|\phi_{n}\rangle$ $\displaystyle-\sum_{n\neq
0}\frac{\langle\phi_{0}|H_{1}|\phi_{0}\rangle\langle\phi_{n}|H_{1}|\phi_{0}\rangle}{(\langle{H}_{0}^{\rm(NH)}\rangle_{n}-E_{0})^{2}}|\phi_{n}\rangle$
$\displaystyle-\frac{1}{2}\sum_{n\neq
0}\frac{|\langle\phi_{n}|H_{1}|\phi_{0}\rangle|^{2}}{\left|\langle{H}_{0}^{\rm(NH)}\rangle_{n}-E_{0}\right|^{2}}\leavevmode\nobreak\
|\phi_{0}\rangle\leavevmode\nobreak\ $
$\displaystyle-i\sum_{j}\gamma_{j}\sum_{n\neq
0}\frac{\langle\phi_{n}|L_{j}|\psi_{1}\rangle\langle\psi_{1}|L_{j}^{\dagger}|\phi_{0}\rangle}{\langle{H}_{0}^{\rm(NH)}\rangle_{n}-E_{0}}\leavevmode\nobreak\
|\phi_{n}\rangle$ (13)

Here we have introduced the unperturbed, non-Hermitian (NH) Hamiltonian

${H}_{0}^{\rm(NH)}={H}_{0}-i\sum_{j}\frac{\gamma_{j}}{2}{L}_{j}^{\dagger}{L}_{j},$
(14)

such that
$\langle{H}_{0}^{\rm(NH)}\rangle_{n}=\langle\phi_{n}|{H}_{0}^{\rm(NH)}|\phi_{n}\rangle=E_{n}-i\sum_{j}\frac{\gamma_{j}}{2}\langle{L}_{j}^{\dagger}{L}_{j}\rangle_{n}$
are the complex, unperturbed eigenenergies of the NH Hamiltonian, possessing
otherwise the same eigenbasis as $H_{0}$ thanks to the hypothesis Eq. (11).

Comparing with standard non-degenerate perturbation theory [33] it is
immediate to verify that the first-order correction Eq. (12) has the same form
as that of the first-order perturbation to the eigenstates in closed systems
when replacing $H_{0}\to{H}_{0}^{\rm(NH)}$. The same applies as well to the
first three lines of Eq. (13), but the fourth line is instead a new term
purely introduced by dissipation.

##  3 Systems of light emitters

###  3.1 Individual vs. collective spontaneous emission

In the rest of our work we shall specialize our attention to systems of light
emitters – such as real or artificial atoms undergoing spontaneous emission –
described as dissipative ensembles of two-level systems or qubits, with ground
state $|g\rangle$ and excited state $|e\rangle$. They can be described in
terms of spin operators ${\bm{S}}_{i}$ $(i=1,....,N)$, with corresponding spin
states $|g\rangle=|\downarrow\rangle$ and $|e\rangle=\uparrow\rangle$ along
the quantization axis $z$. A useful quantity in the rest of this work will be
the collective spin operator ${\bm{J}}=\sum_{i}{\bm{S}}_{i}$. In the following
we will assume the unperturbed Hamiltonian $H_{0}(\\{\bm{S}_{i}\\})$ to be
U(1) symmetric, namely conserving the collective spin component $J^{z}$,
$[H_{0},J^{z}]=0$, or, equivalently, the number of excited emitters.

The coupling to the environment will be described in terms of spontaneous
emission of photons, in the two limits of 1) individual spontaneous emission
and 2) collective spontaneous emission:

  1. 1.

_individual emission_ : in this case the jump operators are single-spin
lowering operators $L_{j}\to S_{i}^{-}$, describing emitters which are all
coupled to different light modes. If we assume the decay rates to be uniform,
$\gamma_{j}=\gamma$, we have that $\sum_{j}\gamma_{j}L_{j}^{\dagger}L_{j}$
reduces to $\gamma\sum_{i}S_{i}^{+}S_{i}^{-}=\gamma(J^{z}+N/2)$, therefore
commuting with the unperturbed Hamiltonian;

  2. 2.

_collective emission_ : in this case the jump operator is unique, $L_{j}\to
J^{-}=\sum_{i}S_{i}^{-}$, describing emission of all emitters into the same
light mode. In this case
$\sum_{j}\gamma_{j}L_{j}^{\dagger}L_{j}=\frac{\Gamma}{N}J^{+}J^{-}=\frac{\Gamma}{N}\left[{\bm{J}}^{2}-(J^{z})^{2}+J^{z}\right]$.
This operator commutes with the unperturbed Hamiltonian when the latter
conserves not only the total magnetization, but also the collective spin
length, namely if $[{\bm{J}}^{2},H_{0}]=0$.

Figure 2: Sketch of perturbation theory from the unperturbed
$|\phi_{0}\rangle=|\downarrow\downarrow...\downarrow\rangle$ state. We picture
here a generic spectrum of a $U(1)$-symmetric Hamiltonian, whose eigenstates
can be labeled by the quantum number $J^{z}$. The two-emitter drive to first
order, or the one-emitter drive to second order, connect the unperturbed
state, with $J^{z}=-N/2$ and maximum spin length ${\bm{J}}^{2}=J(J+1)$ and
$J=N/2$, with states with $J^{z}=-N/2+2$ and spin length $J\geq N/2-2$. Hence
the perturbed state is a superposition of highly atypical, large-spin
Hamiltonian eigenstates.

For both of the above cases of interest to this work, the pure state
$|\phi_{0}\rangle=|gg...g\rangle=|\downarrow\downarrow...\downarrow\rangle$ is
the unique steady state of the unperturbed dissipative dynamics. Moreover both
cases satisfy the assumption of the previous Sec. 2.2 (adding the extra
condition $[{\bm{J}}^{2},H_{0}]=0$ in case 2), which leads to the explicit
expressions for the state perturbation as in Eqs. (12)-(13). This situation
lends itself to a rather interesting interpretation: the perturbed, pure
steady state of the dissipative evolution can be viewed as the perturbed
version of the Hamiltonian eigenstate $|\phi_{0}\rangle$ induced by the
Hamiltonian $H_{1}$ to first order, and by the $H_{1}$ and the jump operators
$L_{j},L_{j}^{\dagger}$ to second order, namely as the perturbation of a
(generically) excited Hamiltonian eigenstate. The perturbation operators
$H_{1}$ and $L_{j},L_{j}^{\dagger}$ will mainly admix $|\phi_{0}\rangle$ with
other excited states in the same energy range, but with a fundamental
restriction as we will highlight below.

###  3.2 Extremal properties of the unperturbed and perturbed state

A relevant $H_{1}$ perturbation to $U(1)$ symmetric Hamiltonians breaks the
U(1) symmetry, namely the conservation of the number of excited emitters. This
implies that the perturbation corresponds generically to _driving_ , which we
will consider as having two different forms (see sketch in Fig. 1): 1) _two-
emitter drive_ exciting/de-exciting pairs of emitters together, of the kind
$S_{i}^{+}S_{j}^{+}+S_{i}^{-}S_{j}^{-}$ (for $i\neq j$) where
$S_{i}^{\pm}=S_{i}^{x}\pm iS_{i}^{y}$ are raising/lowering spin operators; and
2) _single-emitter drive_ , of the kind $S^{+}_{i}+S^{-}_{i}$. The two-emitter
drive will be discussed in Secs. 5, while the single-emitter drive will be
discussed in Sec. 7.

At this point, it is crucial to understand the special role played by the
unperturbed state $|\phi_{0}\rangle$ within perturbation theory. Indeed this
state is extremal in that it maximizes (in absolute value) the projection
$J^{z}$, and hence the collective spin length ${\bm{J}}^{2}=J(J+1)$ with
$J=N/2$. All the operators associated to single-emitter and two-emitter
drives, as well as the jump operators $L,L^{\dagger}$ related to emission, can
only connect the $|\phi_{0}\rangle$ state to states with $J=N/2-2$ (to first
order in the two-emitter drive operators, and to second order in the single-
emitter drive and jump operators) – see sketch in Fig. 2. This implies that,
to lowest order in perturbation theory, the perturbed state has only support
on high-spin Hamiltonian eigenstates, representing a highly atypical portion
of the spectrum, even when the unperturbed Hamiltonian $H_{0}$ is chaotic.
This aspect suggests that, regardless of the details of the Hamiltonian, the
nature of the perturbed steady state will be close to that obtained for models
with long-range interactions (both in $H_{0}$ as well as in $H_{1}$), whose
eigenstates possess a well defined total spin length ${\bm{J}}^{2}$ with small
or no fluctuations. In the rest of this work we will work with collective-spin
Hamiltonians, whose choice is not only convenient technically (see below), but
also justified by the above remarks.

##  4 Collective-spin properties and spin squeezing

Before we apply pure-state perturbation theory to specific examples, it is
important to introduce the quantities that we shall use to characterize the
quantum correlations induced by the perturbation in the steady state of the
system.

The unperturbed steady state of our interest
$|\phi_{0}\rangle=|\downarrow\downarrow...\downarrow\rangle$ is an example of
a _coherent spin state_ (CSS). The collective spin of this state points along
the $z$ axis, $\langle J^{z}\rangle=-\frac{N}{2}$, and the transverse spin
components have equal uncertainties

${\rm Var}(J^{x})={\rm Var}(J^{y})=N/4\leavevmode\nobreak\ .$ (15)

In fact we can extend the above property by considering a generalized
definition of the collective spin as having transverse, perpendicular
components $J_{1}^{\perp}$ and $J_{2}^{\perp}$, such that
$[J_{1}^{\perp},J_{2}^{\perp}]=iJ^{z}$, in the form

$\displaystyle J_{1}^{\perp}=$
$\displaystyle\sum_{i}\left(\cos\theta_{i}S_{i}^{x}+\sin\theta_{i}S_{i}^{y}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{i}\left(S_{i}^{+}e^{-i\theta_{i}}+S_{i}^{-}e^{i\theta_{i}}\right)\leavevmode\nobreak\
,$ (16)

and

$\displaystyle J_{2}^{\perp}=$
$\displaystyle\sum_{i}\left(-\sin\theta_{i}S_{i}^{x}+\cos\theta_{i}S_{i}^{y}\right)$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\sum_{i}\left(S_{i}^{+}e^{i\theta_{i}}+S_{i}^{-}e^{-i\theta_{i}}\right)\leavevmode\nobreak\
.$ (17)

where $\theta_{i}$ are local angles. For a CSS we have that ${\rm
Var}(J_{1}^{\perp})={\rm Var}(J_{2}^{\perp})=N/4$, and in particular ${\rm
Var}(J_{1}^{\perp}){\rm Var}(J_{2}^{\perp})=|\langle J^{z}\rangle|^{2}/4$,
saturating the Heisenberg uncertainty relation for spin operators. As a
consequence the CSS represents an example of a _minimal uncertainty_ state.

As we shall see, Hamiltonian perturbations breaking its U(1) symmetry have the
immediate effect of redistributing the uncertainties on the transverse
components in the steady state, inducing _squeezing_ of the collective spin.
Squeezing of a transverse collective spin component can be quantified by the
squeezing parameter [34]

$\xi^{2}_{R}=\frac{N\min_{\perp}{\rm Var}(J^{\perp})}{\langle
J^{z}\rangle^{2}}\leavevmode\nobreak\ .$ (18)

where the minimization $\min_{\perp}$ is made over all possible choices of the
local angles $\theta_{i}$. If $\xi_{R}^{2}<1$ the collective spin exhibits
squeezing, which is a fundamental form of quantum correlation. Indeed squeezed
states are entangled [35]; and their entanglement has an immediate
metrological significance, making them more sensitive to rotations more than
any coherent spin state [34].

Because the collective spin components must satisfy the Heisenberg uncertainty
relation ${\rm Var}(J_{1}^{\perp}){\rm Var}(J_{2}^{\perp})\geq|\langle
J^{z}\rangle|^{2}/4$, if one component (e.g. $J_{1}^{\perp}$) acquires the
minimum variance $\min_{\perp}{\rm Var}(J^{\perp})$ and is squeezed (namely
$\xi_{R}^{2}<1$), the other one (e.g. $J_{2}^{\perp}$) must be _anti-_
squeezed (namely $\xi_{R}^{2}>1$ for it), and in particular it has the
property of exhibiting the largest variance among all the collective-spin
components transverse to the average orientation. The anti-squeezed component
captures therefore the strongest form of spin-spin correlations appearing in
the system. To quantify the quantum nature of these correlations, one can
introduce the quantum Fisher information (QFI) associated with
$J_{2}^{\perp}$, which is most generally defined for a state
$\rho=\sum_{k}p_{k}|k\rangle\langle k|$ (with $|k\rangle$ forming an
orthonormal basis) as [36]

${\rm
QFI}(J_{2}^{\perp})=2\sum_{kl}\frac{(p_{k}-p_{l})^{2}}{p_{k}-p_{l}}|\langle
k|J_{2}^{\perp}|l\rangle|^{2}\leavevmode\nobreak\ .$ (19)

In particular the QFI bounds the uncertainty with which one can estimate the
angle $\phi$ of a rotation $e^{-i\phi J^{\perp}_{2}}$, generated by the
operator $J_{2}^{\perp}$, by making arbitrary measurements on the state,
$(\delta\phi)^{2}\geq 1/{\rm QFI}(J_{2}^{\perp})$. The QFI satisfies the
inequality (descending from the above bound and the fact that ${\rm
QFI}(J_{2}^{\perp})\leq 4{\rm Var}(J_{2}^{\perp})$ [23]):

$4{\rm Var}(J_{1}^{\perp}){\rm Var}(J_{2}^{\perp})\geq{\rm
Var}(J_{1}^{\perp}){\rm QFI}(J_{2}^{\perp})\geq|\langle J^{z}\rangle|^{2}$
(20)

implying that

$\xi_{R}^{-2}\leq{\rm QFI}(J_{2}^{\perp})/N\leavevmode\nobreak\ .$ (21)

If the squeezed state is a state of minimal uncertainty, this means that the
inequality chain Eq. (20) collapses to an equality, and in particular that
$\xi_{R}^{-2}={\rm QFI}(J_{2}^{\perp})/N$. For such a state squeezing is the
optimal metrological resource, namely rotations can be best estimated by
simply measuring the average orientation of the collective-spin operator.

##  5 Two-emitter driving and individual emission: squeezing theorem

###  5.1 Two-emitter driving

The first example of perturbation that we shall consider is a most general,
parity-conserving bilinear Hamiltonian perturbation $H_{1}$ in the form
$H_{1}=H_{1,d}+H_{1,o}$ where

$H_{1,d}=\sum_{lm}\left(J_{lm}S_{l}^{+}S_{m}^{-}+{\rm h.c.}\right)+H_{\rm
diag}$ (22)

is a $J^{z}$-conserving term, and

$H_{1,o}=\sum_{lm}\left(K_{lm}S_{l}^{+}S_{m}^{+}+{\rm
h.c.}\right)\leavevmode\nobreak\ $ (23)

is a $J^{z}$-non-conserving one. Here $H_{\rm diag}$ is an arbitrary
Hamiltonian which is diagonal on the eigenbasis of the $S_{i}^{z}$ operators.
The perturbation $H_{1,o}$ contains two-emitter driving terms, which are
essential to perturb the steady state.

Indeed, given that $[H_{0},J^{z}]=0$, the Hamiltonian eigenstates
$|\phi_{n}\rangle$ can be chosen to be also eigenstates of $J^{z}$. In
particular $|\phi_{0}\rangle$ is the only state in the $J^{z}=-N/2$ sector.
Given that $H_{1,d}$ cannot couple different $J^{z}$ sectors, we have that

$\langle\phi_{n}|H_{1,d}|\phi_{0}\rangle\sim\delta_{n0}\leavevmode\nobreak\ .$
(24)

Hence the state $|\phi_{0}\rangle$ can only be perturbed by the $H_{1,o}$
operator. Within that operator the terms of the kind $S_{l}^{+}S_{m}^{+}$ are
the only ones that do not annihilate the $|\phi_{0}\rangle$ state, and they
map that state onto states belonging to the $J^{z}=-N/2+2$ sector, that we
will denote as ${\cal S}_{-N/2+2}$ in the following.

Since the perturbation $H_{1}$ introduces emitter-emitter couplings, it can by
itself give rise to quantum correlations in the steady state. Hence we will
consider the case of individual (i.e. uncorrelated) spontaneous emission
$L_{j}\to S_{i}^{-}$ with a uniform rate $\gamma$ for all emitters. This
coupling to the environment has clearly the tendency to decorrelate the
emitters. The form of quantum correlations stabilized in the steady state will
therefore be of a kind which is robust to a decorrelating environment.

###  5.2 Squeezing theorem

Applying the formulas of Sec. 2.2, the first-order perturbation to the steady
state takes the form

$|\psi_{1}\rangle=\sum_{n\in{\cal
S}_{-N/2+2}}\frac{\langle\phi_{n}|\sum_{lm}K_{lm}S_{l}^{+}S_{m}^{+}|\phi_{0}\rangle}{E_{0}-E_{n}+i\gamma}|\phi_{n}\rangle,$
(25)

since

$\frac{1}{2}\langle\phi_{n}|\sum_{i}S_{i}^{+}S_{i}^{-}|\phi_{n}\rangle=\frac{1}{2}(-N/2+2+N/2)=1\leavevmode\nobreak\
.$ (26)

As a consequence

$\displaystyle\langle(J^{\perp})^{2}\rangle^{\prime}=\frac{N}{4}+\lambda\left(\langle\phi_{0}|(J^{\perp})^{2}|\psi_{1}\rangle+{\rm
c.c.}\right)+O(\lambda^{2}),$ (27)

where we introduced the short-hand notation for the expectation values on the
perturbed state
$\langle(...)\rangle^{\prime}=\langle\phi_{0}^{\prime}|(...)|\phi_{0}^{\prime}\rangle$.
The matrix element $\langle\phi_{0}|(J^{\perp})^{2}|\psi_{1}\rangle$ can be
written as

$\langle\phi_{0}|(J^{\perp})^{2}|\psi_{1}\rangle=-\frac{1}{4}\sum_{n\in{\cal
S}_{-N/2+2}}\frac{\sum_{lm}\sum_{ij}e^{i(\theta_{i}+\theta_{j})}\langle\phi_{n}|K_{lm}S_{l}^{+}S_{m}^{+}|\phi_{0}\rangle\langle\phi_{0}|S_{i}^{-}S_{j}^{-}|\phi_{n}\rangle}{E_{n}-E_{0}-i\gamma}=-F(\\{\theta_{i}\\})\leavevmode\nobreak\
.$ (28)

Given that

$\langle J^{z}\rangle^{\prime}=-\frac{N}{2}+O(\lambda^{2}),$ (29)

the squeezing parameter to first order takes the value

$\xi_{R}^{2}=1-\frac{8\lambda}{N}\leavevmode\nobreak\ {\rm
Re}[F(\\{\theta_{i}\\})]+O(\lambda^{2})\leavevmode\nobreak\ .$ (30)

The above results implies the following

###### Theorem 1

The steady state of an ensemble of emitters with U(1)-symmetric unperturbed
Hamiltonian subject to individual emission – whose unique steady state is
$|\phi_{0}\rangle=|\downarrow\downarrow...\rangle$ – is perturbed to a spin-
squeezed state (within first-order perturbation theory) by a perturbation of
the form of Eq. (23) iff

$\max_{\\{\theta_{i}\\}}{\rm Re}[F(\\{\theta_{i}\\})]>0\leavevmode\nobreak\ $
(31)

where the function $F$ is defined in Eq. (28).

The angles $\\{\theta_{i}\\}$ maximizing the function ${\rm Re}[F]$ define the
generalized collective-spin component $J^{\perp}_{1}$ which is maximally
squeezed; and, in turn, the angles $\\{\theta_{i}+\pi/2\\}$ define the anti-
squeezed component $J_{2}^{\perp}$, namely the local spin components
$-\sin(\theta_{i})S_{i}^{x}+\cos(\theta_{i})S_{i}^{y}$ which are most strongly
correlated with one another between different sites.

If the condition Eq. (31), defining the squeezed transverse component
$J^{\perp}_{1}$, is satisfied, then the perpendicular component
$J^{\perp}_{2}$ is automatically anti-squeezed since
$\langle\phi_{0}|(J_{1}^{\perp})^{2}|\psi_{1}\rangle=-\langle\phi_{0}|(J_{2}^{\perp})^{2}|\psi_{1}\rangle$,
and therefore

$\langle(J^{\perp}_{2})^{2}\rangle^{\prime}=\frac{N}{4}+2\lambda\max_{\\{\theta_{i}\\}}{\rm
Re}[F({\theta_{i}}]+O(\lambda^{2}).$ (32)

As a consequence, the perturbed state has the property

$\langle(J^{\perp}_{1})^{2}\rangle^{\prime}\langle(J^{\perp}_{2})^{2}\rangle^{\prime}=\frac{N^{2}}{16}+O(\lambda^{2})=\frac{|\langle
J^{z}\rangle^{\prime}|^{2}}{4}+O(\lambda^{2})$ (33)

namely it is a _minimal uncertainty_ state, for which squeezing is the optimal
metrological property.

The above theorem extends to dissipative steady states a simple theorem
recently put forward by one of the authors for Hamiltonian ground states [37].
The present result highlights the importance of spin squeezing as the first
form of quantum correlation that can be stabilized when perturbing the
factorized stead state $|\phi_{0}\rangle$ with a bilinear perturbation (a two-
emitter drive). The assumption of having a bilinear perturbation in the spin
operators is not a fundamental limitation. If we had included in $H_{1}$ an
off-diagonal perturbation which is linear in the spin operators, _e.g._
proportional to $J^{x}$ or $J^{y}$, this would not change the above results
within first-order perturbation theory. Indeed, due to the structure of the
squeezing parameter Eq. (18), linear operators in the spins cannot give
contributions to first order, because they cannot compensate the effect of the
bilinear operator $(J^{\perp})^{2}$ on the unperturbed state (see Eq. (12)).
On the other hand they can perturb the state to second order, as we will
discuss in Sec. 7.

In the following section we will test the predictions of pure-state
perturbation theory and the relevance of spin squeezing in two examples widely
studied in the recent literature on dissipative quantum spin models.

##  6 Application to collective spin Hamiltonians

In the following we will consider collective-spin Hamiltonians of the form

${H}=\sum_{\alpha=x,y,z}\left[\frac{\mathcal{J}_{\alpha}}{N}(J^{\alpha})^{2}+h_{\alpha}J^{\alpha}\right],$
(34)

which can be decomposed into an unperturbed, U(1)-symmetric part, and a
symmetry-breaking perturbation. Specifically, we will discuss the collective
XYZ model [38] and the collective transverse-field Ising model [11] with
dissipation in the form of individual emission with uniform decay rates. These
models are particularly suitable for a test of our perturbative results, since
they admit an efficient numerical solution thanks to the permutational
symmetry of their Hamiltonians [15], offering a valuable benchmark. For these
numerical simulations we resort to the implementation of the permutational
invariant solver in QuTiP [39, 40].

###  6.1 Example I: the dissipative XYZ model

We consider the collective (or infinite-dimensional) XYZ Heisenberg model [38]
with individual spontaneous emission, whose Hamiltonian can be written as

$H=\frac{1}{N}\sum_{i,j}\left({\cal J}_{x}S_{i}^{x}S_{j}^{x}+{\cal
J}_{y}S_{i}^{y}S_{j}^{y}+{\cal J}_{z}S_{i}^{z}S_{j}^{z}\right),$ (35)

where the sums $\sum_{i(j)}$ run over the $N$ emitters. This Hamiltonian –
both in its collective-spin form, as well as when cast on finite-dimensional
lattices – has been the subject of many theoretical studies in the recent past
[8, 9, 41, 42, 43, 38, 10].

Choosing $\mathcal{J}_{x}=\mathcal{J}_{y}$, one recovers a U(1)-symmetric XXZ
model, conserving the magnetization $J^{z}$. The dissipative XXZ model clearly
admits the state $|\phi_{0}\rangle=|\downarrow\downarrow...\downarrow\rangle$
as its unique steady state. It is useful to recast the latter state as a Dicke
state $|\phi_{0}\rangle=|J=N/2,M=-N/2\rangle$ with maximal spin length
${\bm{J}}^{2}=J(J+1)=N/2(N/2+1)$ and $J^{z}=M$.

The Hamiltonian Eq. (35) can be rewritten in the form

${H}=H_{0}+\lambda H_{1},$ (36)

with

$H_{0}=\frac{\mathcal{J}}{N}\left[(J^{x})^{2}+(J^{y})^{2}\right]+\frac{\mathcal{J}_{z}}{N}(J^{z})^{2},$
(37)

and

$\displaystyle\lambda H_{1}=\frac{\delta{\cal
J}}{2N}\left[(J^{+})^{2}+(J^{-})^{2}\right]$ (38)

where we have introduced the symbols

${\cal J}=\frac{{\cal J}_{x}+{\cal J}_{y}}{2}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ {\delta\cal J}=\frac{{\cal
J}_{x}-{\cal J}_{y}}{2}\leavevmode\nobreak\ .$ (39)

Given the permutation invariance of the Hamiltonian (including the
perturbation), we can choose uniform angles $\theta_{i}=\theta$ in the
definition of the transverse spin components Eqs. (16) and (17) without loss
of generality. In particular the permutationally invariant perturbation will
couple the Dicke state $|\phi_{0}\rangle=|J=N/2,M=-N/2\rangle$ to another
Dicke state only, namely $|\phi_{2}\rangle=|J=N/2,M=-N/2+2\rangle$ via a
double spin flip. This means that the sum of Eq. (28) collapses to only one
state. Moreover Dicke states are all eigenstates of the all-to-all XXZ
Hamiltonian with eigenvalues

$\displaystyle E(J,M)=$ $\displaystyle\langle J,M|H_{0}|J,M\rangle,$ (40)
$\displaystyle=$ $\displaystyle\frac{\cal J}{N}J(J+1)+\frac{1}{N}({\cal
J}_{z}-{\cal J})M^{2},$ (41)

so that the only relevant transition energy is $E_{2}-E_{0}=2\left({\cal
J}-{\cal J}_{z}\right)\left(N-2\right)/N$.

Moreover we get

$\displaystyle\langle\phi_{2}|H_{1}|\phi_{0}\rangle$
$\displaystyle=\frac{\delta\cal
J}{2N}\left\langle\phi_{2}\right|(J^{+})^{2}\left|\phi_{0}\right\rangle,$ (42)
$\displaystyle=\frac{\delta\cal J}{2}\sqrt{2\left(N-1\right)/N}.$

As a consequence the function $F\left(\\{\theta\\}\right)$ from theorem 1
takes the form

$\displaystyle F(\\{\theta\\})=$ $\displaystyle\frac{1}{8}\frac{\delta{\cal
J}(N-1)e^{i2\theta}}{\left({\cal J}-{\cal
J}_{z}\right)\left(N-2\right)/N-i\gamma},$ (43) $\displaystyle=$
$\displaystyle\frac{\alpha e^{i2\theta}}{\beta-i\gamma}.$ (44)

where we introduced the symbols

$\displaystyle\alpha$ $\displaystyle=\frac{N-1}{4}\leavevmode\nobreak\
\delta{\cal J}$ (45) $\displaystyle\beta$
$\displaystyle=\frac{2(N-2)}{N}\left({\cal J}-{\cal
J}_{z}\right)\leavevmode\nobreak\ .$ (46)

In terms of these symbols the real part of $F$ reads

${\rm Re}[F(\\{\theta\\})]=\frac{\alpha\left(\beta\cos 2\theta-\gamma\sin
2\theta\right)}{\beta^{2}+\gamma^{2}}.$ (47)

Figure 3: (a) Minimal spin squeezing parameter $\xi_{\rm R}^{2}$ as a function
of $\delta\mathcal{J}$ for the exact solution (full black lines) and the
perturbative solution (dashed colored lines). This corresponds to a
perturbation perpendicular to the $\mathcal{J}$ axis (see also Fig. 5).
Parameters used are $N=20$, $\mathcal{J}_{z}=\gamma$ , and
$\mathcal{J}=-1.8\gamma,-0.8\gamma,-0.2\gamma$ (resp. blue, dark red, and
orange, as indicated on the figure). (b) The angle $\phi_{min}$ for which the
spin squeezing parameter is minimal (see panel (a)). Same parameters as panel
(a) (from top to bottom; $\mathcal{J}=-0.2\gamma,-0.8\gamma,-1.8\gamma$).

####  6.1.1 Onset of squeezing in the perturbed state

The presence of squeezing in the steady state is then governed by condition
(31), and depends on the angle $\theta$. The angle $\theta$ extremizing ${\rm
Re}[F(\\{\theta_{i}\\})]$ is given by (see appendix A)

$\theta_{ex.}=\frac{1}{2}\tan^{-1}\left(-\frac{\gamma}{\beta}\right)+k\frac{\pi}{2}\quad
k\in\mathbb{Z}$ (48)

and it exhibits a subtle dependence on both the Hamiltonian parameters and the
dissipation strength that we will discuss in Sec. 6.1.2.

For the extremal angle $\theta_{\rm ex}$ the real part of the $F$ function
takes the form (see appendix A)

${\rm Re}\left[F(\left\\{\theta_{\rm
ex}\right\\})\right]=\frac{\alpha\cos(k\pi)}{\sqrt{\beta^{2}+\gamma^{2}}}.$
(49)

The sign of this function (determining the existence of squeezing) depends on
the integer $k$ and on the sign of $\alpha$, but it does not depend on the
sign of $\beta$. Yet, whenever the perturbation breaks the U(1) symmetry,
namely $\alpha\neq 0$, one can always find an extremizing angle for which
${\rm Re}(F)>0$: this is achieved by choosing $k$ even (odd) for $\alpha>0$
($\alpha<0$). This implies that _the U(1)-symmetry-breaking perturbation
always induces squeezing in the steady state_ , as per the above theorem. The
above result is clearly exhibited in Fig. 3, where we compare the squeezing
parameter predicted by perturbation theory with the exact result for $N=20$
spins as an example. There we see that, as predicted by pure-state
perturbation theory, the squeezing parameter $\xi_{R}^{2}$ decreases linearly
from its unperturbed unit value. The deviation from the perturbative
predictions can clearly be attributed to beyond-linear effects, as well as to
the fact that the steady-state develops some finite entropy (albeit
subextensive [44]), not accounted for by pure-state perturbation theory.

The prediction of perturbation theory for the squeezing parameter agrees with
the exact result over a larger interval of perturbation values when
$\mathcal{J}\ll\mathcal{J}_{z}$ and $\mathcal{J}\gg\mathcal{J}_{z}$, namely
far away from the SU(2)-symmetric point (for the Hamiltonian) ${\cal
J}_{x}={\cal J}_{y}={\cal J}_{z}$. To understand this result it is important
to remind that, for a critical value of the perturbation $\delta{\cal J}$
(dependent on the value of ${\cal J}/\gamma$), the steady-state phase diagram
features a phase transition [38] from a paramagnetic phase – continuously
connected with the unperturbed U(1)-symmetric steady state; to a ferromagnetic
phase, which develops long-range correlations for one spin component in the
$xy$ plane (see the sketch in Fig. 5, and Sec. 6.1.2 for further discussion).
The critical value of $\delta{\cal J}$ is minimal (in absolute value) for
${\cal J}={\cal J}_{z}$ and it grows upon increasing the difference between
${\cal J}$ and ${\cal J}_{z}$. The predictions of perturbation theory break
down upon approaching the phase transition, and this occurs for larger
$\delta{\cal J}$ values the larger the anisotropy between ${\cal J}$ and
${\cal J}_{z}$. Nonetheless, even when perturbing the steady state away from
the SU(2) symmetric point, perturbation theory correctly captures the onset of
squeezing, showing that this is a universal feature of the paramagnetic phase
close to the U(1)-symmetric line ${\cal J}_{x}={\cal J}_{y}$. As we will
discuss in the next section, pure-state perturbation theory also predicts
correctly which collective-spin component is squeezed and which one is anti-
squeezed; and in this sense, it provides insight into important features of
the steady-state phase diagram, even going beyond its strict range of
applicability.

Figure 4: Optimal angle $\theta$ in the $(x,y)$-plain that minimises the spin
squeezing parameter in the XYZ model as predicted by (a) the perturbative
approach and (b) the exact solution for a system with $N=20$ spins and
$\mathcal{J}_{z}=\gamma$. Black dashed lines show the mean-field phase
transition between the paramagnetic phase and ferromagnetic phase –
corresponding to the exact solution for the all-to-all connected model in the
thermodynamic limit [38, 45].

####  6.1.2 Angle of squeezing vs. anti-squeezing; insight into the
ferromagnetic phases

Eq. (48) from perturbation theory gives an explicit prediction of the extremal
angle for squeezing and anti-squeezing as a function of the Hamiltonian
parameter and the dissipation strength. The extremal squeezing angle
$\theta_{\rm ex}$ is plotted in Fig. 4 across the steady-state phase diagram,
as a function of the ratios ${\cal J}_{x}/\gamma$ and ${\cal J}_{y}/\gamma$ at
fixed ${\cal J}_{z}/\gamma=1$, comparing pure-state perturbation theory with
the exact result from Ref. [38]. We observe that the subtle dependence of the
angle on both the Hamiltonian parameters and the dissipation strength is
correctly captured by perturbation theory within its strict regime of
applicability, namely within the paramagnetic phase continuously connected
with the unperturbed limit ${\cal J}_{x}={\cal J}_{y}$.

In order to make sense of the angle map of Fig. 4, it is instructive to take
the limite of a negligible dissipation, i.e. to consider
$|\mathcal{J-J}_{z}|\gg\gamma$. In this limit the ratio $\gamma/\beta$ becomes
negligible, and therefore one obtains ${\rm
Re}[F(\theta)]\approx\alpha\cos(2\theta)/\beta$, which for $\alpha/\beta>0$ is
maximized by $\theta=0\leavevmode\nobreak\ ({\rm mod}\leavevmode\nobreak\
\pi)$ and for $\alpha/\beta<0$ it is maximized by
$\theta=\pi/2\leavevmode\nobreak\ ({\rm mod}\leavevmode\nobreak\ \pi)$. The
two conditions, $\alpha/\beta>0$ and $\alpha/\beta<0$, divide the $({\cal
J}_{x},{\cal J}_{y})$ plane into four quadrants, which are oriented with the
$\pi/4$-rotated axes $\delta{\cal J}$ and ${\cal J}-{\cal J}_{z}$. The
condition $\alpha/\beta>0$ is met when $\delta{\cal J}>0$ and ${\cal J}>{\cal
J}_{z}$, defining the quadrant I; or $\delta{\cal J}<0$ and ${\cal J}<{\cal
J}_{z}$, defining the quadrant III. In these two quadrants the maximum
squeezing angle is therefore $\theta=0\leavevmode\nobreak\ ({\rm
mod}\leavevmode\nobreak\ \pi)$. The angle $\theta=\pi/2\leavevmode\nobreak\
({\rm mod}\leavevmode\nobreak\ \pi)$ is instead the maximum squeezing angle in
the quadrant II: $\delta{\cal J}>0$ and ${\cal J}<{\cal J}_{z}$; and quadrant
IV: $\delta{\cal J}<0$ and ${\cal J}>{\cal J}_{z}$. The four quadrants with
$\theta=0\leavevmode\nobreak\ ({\rm mod}\leavevmode\nobreak\ \pi)$ and
$\theta=\pi/2\leavevmode\nobreak\ ({\rm mod}\leavevmode\nobreak\ \pi)$ are
clearly visible in Fig. 4(a); and they are repeated in Fig. 5 for clarity.
Introducing a finite dissipation $\gamma$ has then the main effect of rounding
off the transition between the two values of $\theta=0\leavevmode\nobreak\
({\rm mod}\leavevmode\nobreak\ \pi)$ or $\pi/2\leavevmode\nobreak\ ({\rm
mod}\leavevmode\nobreak\ \pi)$ when moving above or below the ${\cal
J}_{x}={\cal J}_{y}$ line.

The angle $\theta$ shown in the figure determines the maximally squeezed
component of the collective spin, but correspondingly the angle
$\theta+\pi/2\leavevmode\nobreak\ ({\rm mod}\leavevmode\nobreak\ \pi)$
determines the anti-squeezed spin component, developing the strongest
ferromagnetic correlations. These correlations become long-ranged across the
paramagnetic-to-ferromagnetic transition, whose description is clearly beyond
the scopes of perturbation theory. Yet the nature of the ferromagnetic
correlations can be understood to a large extent from perturbation theory.

First of all, it is remarkable to see that the dominant correlations are in
the $xy$ plane, and they are ferromagnetic _regardless_ of the signs of the
Hamiltonian couplings ${\cal J}_{x}$ and ${\cal J}_{y}$. Moreover the spin
component exhibiting the strongest correlations does not correspond
necessarily to the spin component with the largest Hamiltonian coupling.
Perturbation theory predicts that the strongest ferromagnetic correlations are
exhibited by the $x$ spin components in quadrants I and III, and by the $y$
spin component in quadrants II and IV; while the strongest coupling in the
Hamiltonian is ${\cal J}_{x}$ in quadrants II and III, and ${\cal J}_{y}$ in
quadrants I and IV. Clearly the onset of ferromagnetism in the steady state
cannot be easily understood in terms of energy extremization.

Figure 5: Sketch of the steady-state phase diagram of the dissipative XYZ
model, indicating the squeezed/anti-squeezed collective-spin components
predicted by perturbation theory.

On the other hand, the pure-state perturbation picture predicts that the
perturbed state has the simple form of a superposition of two Dicke states:

$\displaystyle|\phi_{0}^{\prime}\rangle\approx$
$\displaystyle|J=N/2,M=-N/2\rangle$
$\displaystyle+\zeta|J=N/2,M=-N/2+2\rangle$ (50)

such that

$\displaystyle\langle\phi_{0}^{\prime}|S_{i}^{x}S_{j}^{x}|\phi_{0}^{\prime}\rangle$
$\displaystyle=\zeta^{\prime}$
$\displaystyle\langle\phi_{0}^{\prime}|S_{i}^{y}S_{j}^{y}|\phi_{0}^{\prime}\rangle$
$\displaystyle=-\zeta^{\prime}$ (51)

where $\zeta^{\prime}=\zeta/\sqrt{2N(N-1)}$ . Given that

$\zeta=-\frac{\sqrt{2N(N-1)}}{4(N-2)}\frac{\delta{\cal J}}{{\cal J}-{\cal
J}^{z}}$ (52)

we obtain positive, ferromagnetic correlations (and anti-squeezing) for the
$x$ spin component when ${\delta{\cal J}}<0,{{\cal J}-{\cal J}^{z}}>0$ or
${\delta{\cal J}}>0,{{\cal J}-{\cal J}^{z}}<0$ (quadrants II and IV
respectively), while negative correlations (ans squeezing) appear for the $y$
spin component. On the other hand, for ${\delta{\cal J}}>0,{{\cal J}-{\cal
J}^{z}}>0$ and ${\delta{\cal J}}<0,{{\cal J}-{\cal J}^{z}}<0$ (quadrants I and
III respectively) ferromagnetic correlations appear for the $y$ spin component
etc. This is in agreement with the previous analysis of the extremizing angle.
The picture of the correlation structure is summarized in Fig. 5.

Figure 6: Minimal spin squeezing parameter $\xi_{\rm R}^{2}$ in the
dissipative TFI model as a function of $\mathcal{J}_{x}$ for the exact
solution (full black line) and the perturbative solution (dashed orange line).
Inset: the angle $\phi_{min}$ for which the spin squeezing parameter is
minimal. Parameters used are $N=20$,$\Delta=-6\gamma$ (as indicated on the
figure).

###  6.2 Example II: the dissipative transverse field Ising model

In the following we discuss a second model with two-emitter driving, namely
the transverse-field Ising (TFI) model with collective (or infinite-
dimensional) interactions and spontaneous local emission along the field axis
[11], whose Hamiltonian is given by

$H=\frac{{\cal
J}_{x}}{N}\sum_{i,j}S_{i}^{x}S_{j}^{x}+\Delta\sum_{i}S_{i}^{z}\leavevmode\nobreak\
.$ (53)

The dissipation is the same as for the previous model, namely we have single-
emitter jump operators $L_{j}\to S_{i}^{-}$ with uniform single-emitter decay
rate $\gamma$.

The Hamiltonian $H$ can be decomposed as a U(1)-symmetric unperturbed term and
a symmetry-breaking perturbation ${H}=H_{0}+\lambda H_{1}$, with

$H_{0}=\Delta J^{z},$ (54)

and

$\displaystyle\lambda H_{1}$ $\displaystyle=\frac{{\cal
J}_{x}}{N}\sum_{i,j}S_{i}^{x}S_{j}^{x}$ (55) $\displaystyle=\frac{{\cal
J}_{x}}{4N}\left[(J^{+})^{2}+(J^{-})^{2}+J^{+}J^{-}+J^{-}J^{+}\right]\leavevmode\nobreak\
.$

The unique steady state $|\phi_{0}\rangle$ in the unperturbed limit
$\lambda=0$ is, as before, the coherent spin state aligned with the $-z$ axis.
And, as in the previous example, it is connected by the perturbation $H_{1}$
to only one eigenstate of the unperturbed Hamiltonian, namely the Dicke state
$|\phi_{2}\rangle=|J=N/2,M=-N/2+2\rangle$. Since Dicke states have the
unperturbed energies $E(J=N/2,M)=\Delta M$, the energy difference entering in
the denominator of the first-order correction to the steady state, Eq. (12),
is simply $E_{2}-E_{0}=2\Delta$.

Figure 7: Optimal angle $\theta$ in the $(x,y)$-plain that minimises the spin
squeezing parameter in the TFI model as predicted by (a) the perturbative
approach and (b) the exact solution for a system with $N=20$. Black dashed
lines show the mean-field phase transition between the paramagnetic phase and
ferromagnetic phase.  Figure 8: Sketch of the steady-state phase diagram of
the dissipative TFI model, indicating the squeezed/anti-squeezed collective-
spin components predicted by perturbation theory.

The matrix element in the numerator of the first-order correction to the
steady state reads instead

$\displaystyle\langle\phi_{2}|H_{1}|\phi_{0}\rangle$
$\displaystyle=\frac{{\cal
J}_{x}}{4N}\left\langle\phi_{2}\right|(J^{+})^{2}\left|\phi_{0}\right\rangle,$
(56) $\displaystyle=\frac{{\cal
J}_{x}}{4}\sqrt{\frac{2\left(N-1\right)}{N}}\leavevmode\nobreak\ .$ (57)

As a consequence the $F$ function entering in the squeezing parameter reads

$\displaystyle F(\\{\theta_{i}\\})=\frac{e^{i2\theta}}{8}\frac{{\cal
J}_{x}(N-1)}{2\Delta-i\gamma}=\frac{\alpha e^{i2\theta}}{\beta-i\gamma},$ (58)

where $\alpha$ and $\beta$ are given by

$\displaystyle\alpha$ $\displaystyle=\frac{1}{8}{\cal J}_{x}(N-1)$ (59)
$\displaystyle\beta$ $\displaystyle=2\Delta\leavevmode\nobreak\ .$ (60)

The $F$ function has the same form as for the XYZ model, Eq. (43), and
therefore the angle $\theta$ extremizing it and the corresponding extremal
value have the same expression as those in Eqs. (48) and (49). As a
consequence, for any (finite) value of the perturbation ${\cal J}_{x}$ and of
the field $\Delta$ one can always find an angle $\theta$ for which squeezing
appears. Fig. 6 shows an example of the optimal squeezing parameter and
optimal angle for a system with $N=20$ spins, and for a field with strength
$\Delta=-6\gamma$. The pure-state perturbation results capture the linear
onset of squeezing upon adding a finite ${\cal J}_{x}$ perturbation. The
agreement with the exact result – obtained again with the technique developed
in Ref. [15] – degrades upon increasing the perturbation because the steady
state develops some finite entropy, and most importantly because the system
can experience a phase transition to ferromagnetism [11] (see Fig. 8 for a
sketch of the phase diagram).

The full dependence of the squeezing angle on the Hamiltonian parameters
${\cal J}_{x}$ and $\Delta$, and on the dissipation strength $\gamma$, is
shown in Fig. 7. To understand the main features of the angle map across the
phase diagram, we can extract the optimal squeezing angle simply by taking the
limit of negligible dissipation ($|\gamma/\beta|\ll 1$) in Eq. (47), which
gives ${\rm Re}F\approx\alpha\cos(2\theta)/\beta$. This implies that
$\theta=0\leavevmode\nobreak\ ({\rm mod}\leavevmode\nobreak\ \pi)$ is the
optimal squeezing angle if $\alpha/\beta>0$, i.e. if $\Delta>0$ and ${\cal
J}_{x}>0$, or $\Delta<0$ and ${\cal J}_{x}<0$ (quadrants I and III in the
$({\cal J}_{x},\Delta)$ plane – see Fig. 8); while
$\theta=\pi/2\leavevmode\nobreak\ ({\rm mod}\leavevmode\nobreak\ \pi)$ is the
optimal squeezing angle in the two complementary quadrants $\Delta>0$ and
${\cal J}_{x}<0$, or $\Delta<0$ and ${\cal J}_{x}>0$ (quadrants II and IV
respectively). The presence of a finite dissipation $\gamma$ introduces a
smooth crossover in the squeezing angle between quadrant I and quadrant II,
and between quadrant III and quadrant IV. Fig. 8 summarizes the structure of
correlations (i.e. the squeezed vs. anti-squeezed collective-spin component)
across the phase diagram of the system.

##  7 Single-emitter drive and second order perturbation theory

In this section we consider the most elementary form of drive which perturbs a
dissipative system of emitters out of the state with all emitters in the
ground state – namely single-emitter (or Rabi) drive. This generically
corresponds to a perturbation in the form

$\lambda{H}_{1}=-\frac{1}{2}\sum_{i}\Omega_{i}\left(e^{i\phi_{i}}S_{i}^{+}+e^{-i\phi_{i}}S_{i}^{-}\right)$
(61)

where $\Omega_{i}$ is the local amplitude of the Rabi drive at site $i$, and
$\phi_{i}$ the local phase.

The state perturbed by $H_{1}$ to first order experiences a net rotation of
the collective spin away from the $z$ axis, but it remains a coherent spin
state without any form of correlations. Indeed the first-order perturbed state
is a superposition between the unperturbed state $|\phi_{0}\rangle\in{\cal
S}_{-N/2}$ and a state in the magnetization sector with $M=-N/2+1$,
$|\phi_{1}\rangle\in{\cal S}_{-N/2+1}$, so that
$\langle\phi_{1}|S_{i}^{+}S_{j}^{-}|\phi_{0}\rangle=\langle\phi_{1}|S_{i}^{+}S_{j}^{+}|\phi_{0}\rangle=0$
for any $i\neq j$. To observe correlations one needs the superposition of
$|\phi_{0}\rangle$ with a state $|\phi_{2}\rangle$ in the magnetization sector
${\cal S}_{-M/2+2}$, which can only be achieved at second order in the
perturbation. Moreover the appearance of correlations requires also an
interacting unperturbed Hamiltonian and/or collective emission, given that
otherwise the state of the emitters remains perfectly factorized as in the
unperturbed case.

In the following we apply the results of Sec. 2 for second-order perturbation
theory to the relevant example of the driven Dicke model.

###  7.1 Example: the driven Dicke model

The driven Dicke model [46, 47] describes a symmetric coupling between the
emitters and their environment, with a jump operator $\sqrt{\Gamma/N}J^{-}$.
The emitters are driven away from their ground state by a uniform Rabi field
along _e.g._ the $x$-direction, namely the unperturbed Hamiltonian is
vanishing, $H_{0}=0$, while the perturbing Hamiltonian simply reads:

$\lambda H_{1}=-\Omega J^{x}\leavevmode\nobreak\ $ (62)

namely it is of the form of Eq. (61) with uniform $\Omega_{i}=\Omega$ and
$\phi_{i}=0$.

As already pointed out above, within first-order perturbation theory only the
average collective spin can rotate, but correlations cannot appear; while they
appear instead to second order. Applying Eqs. (12) and (13) we obtain for the
perturbed state the form:

$\displaystyle|\phi_{0}^{\prime}\rangle=\left(1-\frac{N\Omega^{2}}{2\Gamma^{2}}\right)\left|-\frac{N}{2}\right\rangle+\frac{i\Omega\sqrt{N}}{\Gamma}\left|-\frac{N}{2}+1\right\rangle$
$\displaystyle-\frac{\Omega^{2}N^{2}}{\Gamma^{2}\sqrt{2N(N-1)}}\left|-\frac{N}{2}+2\right\rangle+\mathcal{O}\left(\frac{\Omega^{3}}{\Gamma^{3}}\right)$
(63)

where we have adopted the shorthand notation $|J=N/2,M\rangle\to|M\rangle$ for
Dicke states. Expressions for various expectation values with respect to
$|\phi^{\prime}_{0}\rangle$ can be found in App. B. We would like to point out
that the collective emission plays a crucial role in keeping the perturbed
state in the symmetric Dicke manifold, thereby allowing for the appearance of
correlations, as the Dicke states are generically entangled except for the
fully polarized ones. If the emission were not fully collective, the last term
in Eq. (13) would admix the unperturbed state with states going out of the
symmetric manifold.

The perturbed state develops a rotated net collective spin

$\langle{\bm{J}}\rangle=\frac{N}{2}(0,\cos\phi,\sin\phi)=\frac{N}{2}\bm{e}_{\phi}$
(64)

where

$\displaystyle\cos\phi$
$\displaystyle=-\frac{2\Omega}{\Gamma}+\mathcal{O}\left(\frac{\Omega^{3}}{\Gamma^{3}}\right)$
$\displaystyle\sin\phi$
$\displaystyle=-\left(1-\frac{2\Omega^{2}}{\Gamma^{2}}\right)+\mathcal{O}\left(\frac{\Omega^{3}}{\Gamma^{3}}\right)\leavevmode\nobreak\
.$ (65)

The squeezed component of the collective spin must therefore be searched for
in the plane transverse to this rotated collective spin, namely as

$J^{\theta}=\cos\theta\leavevmode\nobreak\ J^{x}+\sin\theta
J^{\phi^{\prime}}\leavevmode\nobreak\ $ (66)

where $J^{\phi^{\prime}}=-\sin\phi\leavevmode\nobreak\
J^{y}+\cos\phi\leavevmode\nobreak\ J^{z}$ is the collective spin component
orthogonal to $\bm{e}_{\phi}$ in the $yz$ plane.

By construction $J^{\theta}$ is orthogonal to the net orientation of the
collective spin, and therefore $\langle J^{\theta}\rangle=\langle
J^{x}\rangle=\langle J^{\phi^{\prime}}\rangle=0$. On the other hand

$\displaystyle\langle(J^{\theta})^{2}\rangle$
$\displaystyle=\cos^{2}\theta\leavevmode\nobreak\
\langle(J^{x})^{2}\rangle+\sin^{2}\theta\leavevmode\nobreak\
\langle(J^{\phi^{\prime}})^{2}\rangle$
$\displaystyle+\cos\theta\sin\theta\leavevmode\nobreak\ \langle
J^{x}J^{\phi^{\prime}}+J^{\phi^{\prime}}J^{x}\rangle\leavevmode\nobreak\ .$
(67)

One finds that $\langle
J^{x}J^{\phi^{\prime}}+J^{\phi^{\prime}}J^{x}\rangle=0$ for the perturbed
state in Eq. (63). To evaluate the rest of the above expression, after a
lengthy but straightforward calculation one can show that

$\langle(J^{x})^{2}\rangle=\frac{N}{4}-\frac{N\Omega^{2}}{2\Gamma^{2}}+\mathcal{O}\left(\frac{\Omega^{3}}{\Gamma^{3}}\right)$
(68)

and

$\langle(J^{\phi^{\prime}})^{2}\rangle=\frac{N}{4}+\frac{N\Omega^{2}}{2\Gamma^{2}}+\mathcal{O}\left(\frac{\Omega^{3}}{\Gamma^{3}}\right)\leavevmode\nobreak\
.$ (69)

Hence we find that the $J^{x}$ fluctuations are squeezed below the projection
noise $N/4$, while those of $J^{\phi^{\prime}}$ are anti-squeezed. In fact
these collective spin components correspond to those maximizing (resp.
minimizing) the squeezing parameter, since

$\displaystyle\xi_{R}^{2}$ $\displaystyle=\frac{N\leavevmode\nobreak\ {\rm
Var}(J^{\theta})}{\langle\bm{J}\rangle^{2}}$
$\displaystyle=1+\frac{2\Omega^{2}}{\Gamma^{2}}\left(\sin^{2}\theta-\cos^{2}\theta\right)$
(70)

which is clearly minimized for $\theta=0$, namely the maximally squeezed spin
component corresponds to $J^{x}$. Moreover, we observe that

$\langle(J^{x})^{2}\rangle\langle(J^{\phi^{\prime}})^{2}\rangle=\frac{|\langle\bm{J}\rangle|^{2}}{4}+\mathcal{O}\left(\frac{\Omega^{3}}{\Gamma^{3}}\right)$
(71)

namely the squeezed state is again of minimal uncertainty.

The results for the minimal spin squeezing parameter ($\theta=0$) from second-
order perturbation theory are compared to the exact results in Fig. 9. The
second-order approach reproduces the quadratic onset of squeezing as a
function of the drive $\Omega$, and follows its quadratic development for a
rather broad parameter range – roughly equaling half of the extension of the
paramagnetic phase in the steady state. Second order perturbation theory
clearly breaks down on approach to the super-radiant transition [46, 47],
occurring for $\Omega/\Gamma=1/2$.

Figure 9: Minimal spin squeezing parameter $\xi_{\rm R}^{2}$ in the
dissipative TFI model as a function of $\frac{2\Omega}{\Gamma}$. The black
lines show the exact solutions for $N=50,100,200,300$ (decreasing minimum as
$N$ increases). The orange dashed line shows the result of the second order
perturbative approach.

##  8 Discussion

Throughout this work we have shown that spin squeezing is a form of quantum
correlations which is promoted by the interplay between Hamiltonian driving
and dissipation, either in the form of individual emission or of collective
emission. A few previous works already identified squeezing as a
characteristic feature of steady states in driven-dissipative quantum systems.
In particular Ref. [8] found squeezing in the paramagnetic phase of the
dissipative XYZ model on two- and three-dimensional lattices and nearest-
neighbor interactions by evaluating Gaussian quantum fluctuations around the
mean-field steady state. This result has been confirmed in Ref. [10] for the
two-dimensional dissipative XYZ model, by making use of a truncated cumulant
expansion of quantum fluctuations for each pure state within a quantum-
trajectory approach to the driven dissipative dynamics. Spin squeezing in the
paramagnetic steady state of the driven Dicke model has been previously
reported in Ref. [47]. Our work encompasses all these results for weak
driving, by defining the general conditions under which squeezing is expected
in the paramagnetic phase, close to the trivial (i.e. undriven) steady state.

All of the applications we discussed in this work focus on collective-spin
models (namely on Hamiltonians $H_{0}$ and $H_{1}$ only depending on the
collective spin operator ${\bm{J}}$); yet the results we presented arguably
apply to a much broader class of models. As discussed in Sec. 3.2, first- and
second-order perturbation theory starting from the coherent spin state
$|\phi_{0}\rangle$ can only produce states with $J\geq N/2-2$. Hence both the
unperturbed Hamiltonian $H_{0}$ as well as its perturbation $H_{1}$ act on the
states of interest as projected onto this large-spin sector of Hilbert space

$H_{0(1)}\to H^{\rm(eff)}_{0(1)}={\cal P}_{J\geq N/2-2}\leavevmode\nobreak\
H_{0(1)}\leavevmode\nobreak\ {\cal P}_{J\geq N/2-2}$ (72)

where ${\cal P}_{J\geq N/2-2}$ is the projector on the sector with $J\geq
N/2-2$. Regardless of the range of interactions in $H_{0(1)}$, the projected
Hamiltonians $H^{\rm(eff)}_{0(1)}$ have effectively long-range interactions.
As an example, the Hamiltonian projected onto the sector with maximal $J=N/2$
has only infinite-range interactions (namely it depends only on the ${\bm{J}}$
operator). Therefore the physics of the collective-spin models explored in
this work applies in fact also to their short-range-interacting versions, as
long as one focuses on perturbative effects close to the $U(1)$ symmetric
limit.

Single-emitter driving as in the driven Dicke model is obviously very relevant
to experiments, as it is obtained by coupling the emitters to a Rabi field.
The physics of the driven Dicke model has been recently realized
experimentally in Ref. [25], and one can therefore expect squeezing to be a
feature of the steady state at weak driving, and possibly up to the super-
radiant transition. On the other hand, the experimental relevance of two-
emitter driving is less obvious. Nonetheless, as discussed in Ref. [8], two-
emitter driving can be realized by coupling pairs of emitters to pairs of
lasers that are strongly detuned from single-emitter transitions, but are
instead resonant for two-photon two-emitter transitions.

##  9 Conclusions

In this work we have introduced pure-state perturbation theory for the steady
state of open quantum systems, valid when the system Hamiltonian is perturbed
away from a known limit admitting a pure state as steady state. The assumption
of a pure perturbed state allows for closed expressions of the coefficient of
the perturbed state on the unperturbed Hamiltonian eigenbasis, and
particularly so when the non-Hermitian Hamiltonian of the open quantum system
is diagonalized by the eigenbasis of the unperturbed Hamiltonian.

Pure-state perturbation theory offers a significant insight into the physical
properties of the perturbed steady state, in particular for what concerns
quantum correlations. In this work we have focused our attention on ensembles
of light emitters – i.e. qubits or (artificial) atoms – and we characterized
quantum correlations by considering collective-spin properties, in particular
the uncertainty of its components perpendicular to the average orientation.
For ensembles of light emitters which emit photons individually, we could
formulate a theorem for the onset of spin squeezing under the effect of two-
emitter driving, based on first-order perturbation theory. The theorem applies
to relevant models of collective interactions among emitters, i.e. the
dissipative XYZ model [38] and the dissipative transverse-field Ising model
[11] with all-to-all couplings. Furthermore we showed that single-emitter
drive, in the form of Rabi coupling, leads to spin squeezing when the emitters
are collectively coupled to the environment, as modeled by the driven Dicke
model. Perturbation theory allows for explicit expressions of the perturbed
states, and it shows that the spin-squeezed states are of minimal uncertainty,
namely spin squeezing is their optimal resource for metrology, since the
inverse spin-squeezing parameter coincides with the quantum Fisher
information.

Our pure-state perturbation results are tested against exact results, and they
exhibit a rather large domain of accuracy away from the unperturbed limit,
validating the pure-state assumption. The near purity, or low entropy, of the
perturbed state allows the Hamiltonian perturbation to induce _quantum_
correlations in the steady state. It is important to point out that the low
entropy of the steady state is actually a condition valid when the dissipation
is much stronger than the drive, keeping the system close to the (pure) ground
state of each emitter. The fast dissipation prevents the emitters from being
highly excited and therefore from radiating significantly into the environment
– something which would entail instead a significant entropy in the steady
state. Hence the form of quantum correlations described in this work are the
result of a very specific interplay between dissipation and drive, resulting
in the so-called normal phase within the steady-state phase diagram. This can
be contrasted with the super-radiant phase, developing long-range correlations
among the emitters, but also a large entropy due to the important emission of
radiation into the environment.

The perturbative approach allows for analytical predictions of the steady
state and its quantum correlation properties when the system is weakly driven.
But, more generally, the predictions of perturbation theory provide a
significant insight into the mechanisms leading to the appearance of the
specific correlation patterns of the steady state, which can characterize the
phase diagram beyond the strict limit of applicability of the perturbative
approach. This is crucial in the absence of a simple variational principle –
such as the minimization of the free energy – which dictates the form of the
steady state, and which could allow for an intuitive (_e.g._ semiclassical)
prediction of its properties. A relevant example provided in this work is
offered by the appearance of ferromagnetic correlations in the steady state of
the dissipative XYZ model, in spite of the Hamiltonian interactions being
antiferromagnetic; and the fact that the most correlated spin components are
not necessarily those which are most strongly interacting. These predictions
go clearly beyond what one can reconstruct via mean-field theory, which
pictures the whole paramagnetic phase as being identical to the U(1)-symmetric
steady state [38]. The insight acquired via perturbation theory can be used as
a precious guidance in the understanding and the design of many-body quantum
states resulting from the competition between strong dissipation and a weak
drive; and it can be applied rather broadly to dissipative quantum many-body
systems. In this work we focused on ensembles of light emitters, relevant to a
very broad spectrum of physical systems ranging from atomic physics (e.g.
Rydberg atoms [8], atomic ensembles close to the super-radiant transition [47,
25], etc.) to solid-state physics [24]. Yet it is clear that one can apply the
same scheme to different degrees of freedom – _e.g._ to coupled bosonic modes
leaking into the environment, and close to their vacuum state.

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##  Appendix A Calculation of ${\rm Re}\left[F(\left\\{\theta_{\rm
ex}\right\\})\right]$

In this section we detail the calculations of the function ${\rm
Re}\left[F(\left\\{\theta_{\rm ex}\right\\})\right]$ appearing in theorem 1,
assuming that its real part can be written in the form of Eq. (47). That is,
one can write

${\rm Re}\left[F(\left\\{\theta_{\rm
ex}\right\\})\right]=\frac{\alpha\left(\beta\cos 2\theta_{\rm ex}-\gamma\sin
2\theta_{\rm ex}\right)}{\beta^{2}+\gamma^{2}}.$ (73)

The angle $\theta_{\rm ex}$ extremizing the function leads to a vanishing
derivative of Eq. (73) with respect to $\theta$ to zero. The result of Eq.
(48) is then obtained as follows

$\displaystyle\frac{d}{d\theta}{\rm Re}[F(\\{\theta\\})]=0\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\Leftrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \tan 2\theta=-\frac{\gamma}{\beta}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\Leftrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\theta_{ex}=\frac{1}{2}\tan^{-1}\left(-\frac{\gamma}{\beta}\right)+k\frac{\pi}{2}\quad
k\in\mathbb{Z}.$ (74)

We can now substitute this result in Eq. (73), and calculate for which angle
${\rm Re}\left[F(\left\\{\theta_{\rm ex}\right\\})\right]>0$, i.e. spin
squeezing occurs. This yields

${\rm Re}\left[F(\left\\{\theta_{\rm
ex}\right\\})\right]=\frac{\alpha\left[\beta\cos\left(\tan^{-1}\left(-\frac{\gamma}{\beta}\right)+k\pi\right)-\gamma\sin\left(\tan^{-1}\left(-\frac{\gamma}{\beta}\right)+k\pi\right)\right]}{\beta^{2}+\gamma^{2}}\leavevmode\nobreak\
.$ (75)

We can simplify this expression via standard trigonometric formulas, and using
the fact that $\cos(\tan^{-1}(x))=\frac{1}{\sqrt{1+x^{2}}}$ and
$\sin(\tan^{-1}(x))=\frac{x}{\sqrt{1+x^{2}}}$. Hence,

$\displaystyle\cos\left(\tan^{-1}\left(-\frac{\gamma}{\beta}\right)+k\pi\right)=\frac{\cos(k\pi)}{\sqrt{1+\frac{\gamma^{2}}{\beta^{2}}}},$
(76)

and

$\displaystyle\sin\left(\tan^{-1}\left(-\frac{\gamma}{\beta}\right)+k\pi\right)=\frac{-\frac{\gamma}{\beta}}{\sqrt{1+\frac{\gamma^{2}}{\beta^{2}}}}\cos\left(k\pi\right).$
(77)

Substituting this result in Eq. (75) finally allows us to write

$\displaystyle{\rm Re}\left[F(\left\\{\theta_{\rm
ex}\right\\})\right]=\frac{\alpha\left[\beta^{2}\frac{\cos(k\pi)}{\sqrt{\beta^{2}+\gamma^{2}}}+\gamma^{2}\frac{\cos\left(k\pi\right)}{\sqrt{\beta^{2}+\gamma^{2}}}\right]}{\beta^{2}+\gamma^{2}}=\frac{\alpha\cos(k\pi)}{\sqrt{\beta^{2}+\gamma^{2}}}.$
(78)

##  Appendix B Perturbation-theory predictions for the driven Dicke model

In this section we provide the reader with a list of the results for the
expectation values of collective spin operators, and products thereof, for the
driven Dicke model discussed in Section 7.1, as obtained for the perturbed
state $|\phi_{0}^{\prime}\rangle$ given in Eq. 63. After a straightforward
calculation, one obtains up to second order in $\Omega/\Gamma$:

$\displaystyle\langle\phi_{0}^{\prime}|J^{x}|\phi_{0}^{\prime}\rangle$
$\displaystyle=0\leavevmode\nobreak\ ,$ (79)
$\displaystyle\langle\phi_{0}^{\prime}|J^{y}|\phi_{0}^{\prime}\rangle$
$\displaystyle=-\frac{\Omega N}{\Gamma}\leavevmode\nobreak\ ,$ (80)
$\displaystyle\langle\phi_{0}^{\prime}|J^{z}|\phi_{0}^{\prime}\rangle$
$\displaystyle=-\frac{N}{2}+\frac{\Omega^{2}N}{\Gamma^{2}}\leavevmode\nobreak\
,$ (81)
$\displaystyle\langle\phi_{0}^{\prime}|J^{z}|\phi_{0}^{\prime}\rangle^{2}$
$\displaystyle=\frac{N^{2}}{4}-\frac{\Omega^{2}N^{2}}{\Gamma^{2}}\leavevmode\nobreak\
,$ (82)
$\displaystyle\langle\phi_{0}^{\prime}|\left(J^{x}\right)^{2}|\phi_{0}^{\prime}\rangle$
$\displaystyle=\frac{N}{4}-\frac{\Omega^{2}N}{2\Gamma^{2}}\leavevmode\nobreak\
,$ (83)
$\displaystyle\langle\phi_{0}^{\prime}|\left(J^{y}\right)^{2}|\phi_{0}^{\prime}\rangle$
$\displaystyle=\frac{N}{4}+\frac{1}{2}\frac{\Omega^{2}}{\Gamma^{2}}N(2N-1)\leavevmode\nobreak\
,$ (84)
$\displaystyle\langle\phi_{0}^{\prime}|\left(J^{z}\right)^{2}|\phi_{0}^{\prime}\rangle$
$\displaystyle=\frac{N^{2}}{4}-\frac{\Omega^{2}}{\Gamma^{2}}N\left(N-1\right)\leavevmode\nobreak\
.$ (85)

For the expressions of the form
$\langle\phi_{0}^{\prime}|J^{\alpha}J^{\beta}|\phi_{0}^{\prime}\rangle+\langle\phi_{0}^{\prime}|J^{\beta}J^{\alpha}|\phi_{0}^{\prime}\rangle$
one finds

$\displaystyle\langle\phi_{0}^{\prime}|J^{x}J^{y}|\phi_{0}^{\prime}\rangle+\langle\phi_{0}^{\prime}|J^{y}J^{x}|\phi_{0}^{\prime}\rangle$
$\displaystyle=0\leavevmode\nobreak\ ,$ (86)
$\displaystyle\langle\phi_{0}^{\prime}|J^{x}J^{z}|\phi_{0}^{\prime}\rangle+\langle\phi_{0}^{\prime}|J^{z}J^{x}|\phi_{0}^{\prime}\rangle$
$\displaystyle=0\leavevmode\nobreak\ ,$ (87)
$\displaystyle\langle\phi_{0}^{\prime}|J^{y}J^{z}|\phi_{0}^{\prime}\rangle+\langle\phi_{0}^{\prime}|J^{z}J^{y}|\phi_{0}^{\prime}\rangle$
$\displaystyle=\frac{\Omega}{\Gamma}N\left(N-1\right)\leavevmode\nobreak\ .$
(88)