# Spin-self-organization in an optical cavity facilitated by inhomogeneous
broadening

Marc Nairn  Institut für Theoretische Physik, Universität Tübingen, Auf der
Morgenstelle 14, 72076 Tübingen, Germany  Luigi Giannelli  Dipartimento di
Fisica e Astronomia “Ettore Majorana”, Università di Catania, Via S. Sofia 64,
95123 Catania, Italy  INFN, Sezione di Catania, 95123, Catania, Italy
Giovanna Morigi  Theoretische Physik, Universität des Saarlandes, Campus E26,
D-66123 Saarbrücken, Germany  Sebastian Slama  Center for Quantum Science and
Physikalisches Institut, Universität Tübingen, Auf der Morgenstelle 14, 72076
Tübingen, Germany  Beatriz Olmos  Institut für Theoretische Physik,
Universität Tübingen, Auf der Morgenstelle 14, 72076 Tübingen, Germany  Simon
B. Jäger  Physics Department and Research Center OPTIMAS, University of
Kaiserslautern-Landau, D-67663, Kaiserslautern, Germany

(September 3, 2024)

###### Abstract

We study the onset of collective spin-self-organization in a thermal ensemble
of driven two-level atoms confined in an optical cavity. The atoms
spontaneously form a spin-pattern above a critical driving strength that sets
a threshold and is determined by the cavity parameters, the initial
temperature, and the transition frequency of the atomic spin. Remarkably, we
find that inhomogeneous Doppler broadening facilitates the onset of spin-self-
organization. In particular, the threshold is non-monotonic when increasing
the spin transition frequency and reaches a minimum when the Doppler
broadening is of similar magnitude. This feature emerges due to Doppler-
induced resonances. Above the threshold, we find cooperative dynamics of spin,
spatial, and momentum degrees of freedom leading to density modulations, fast
reduction of kinetic energy, and the emergence of non-thermal states. More
broadly, our work demonstrates how broadening can facilitate strong light-
matter interactions in many-body systems.

Introduction.– The study of strong light-matter interactions is an active
research field including fundamental studies on out-of-equilibrium phases
Baumann  _et al._ (2010); Klinder  _et al._ (2015); Mivehvar  _et al._ (2019,
2021); Landig  _et al._ (2016); Marsh  _et al._ (2021); Kroeze  _et al._
(2023); Guo  _et al._ (2021); Schmidt  _et al._ (2014); Slama  _et al._ (2007)
and exotic dynamics in open quantum systems Iemini  _et al._ (2018); Gong  _et
al._ (2018); Defenu  _et al._ (2024); Young  _et al._ (2024); Schuster  _et
al._ (2020). Strong light-matter interactions can be realized by placing
atoms, molecules or materials in an optical cavity with practical applications
in new generations of sensors and clocks Meiser  _et al._ (2009); Bohnet  _et
al._ (2012); Ludlow  _et al._ (2015); Pezzè  _et al._ (2018); Luo  _et al._
(2024), cavity-enhanced chemistry Lidzey  _et al._ (1998, 2000); Schwartz  _et
al._ (2011); Herrera and Spano (2016); Hirai  _et al._ (2023), as well as
metamaterials such as light-induced superconductors Koppens  _et al._ (2011);
Cavalleri (2018); de la Torre  _et al._ (2021). A workhorse for describing
features in such systems are the Tavis-Cummings and Dicke models Kirton  _et
al._ (2019); Larson and Mavrogordatos (2021); Thompson  _et al._ (1992);
Hertzog  _et al._ (2019). The latter is famous for the prediction of a phase
transition to a superradiant state that exhibits macroscopic coherence in the
cavity field and the atomic medium above a critical light-matter coupling
strength Dimer  _et al._ (2007); Baumann  _et al._ (2010); Zhiqiang  _et al._
(2017). This coupling strength is determined by the transition frequency of
the cavity-coupled states, while the broadening of this transition is
generally considered detrimental. In particular, for thermal gases that
involve large inhomogeneous broadening the critical coupling strength is
determined by the thermal energy Black  _et al._ (2003); Domokos and Ritsch
(2002); Asbóth  _et al._ (2005); Arnold  _et al._ (2012); Schütz  _et al._
(2016). Similar effects are also relevant in Fermi gases Keeling  _et al._
(2014); Piazza and Strack (2014); Zhang  _et al._ (2021); Helson  _et al._
(2023); Zwettler  _et al._ (2024); Marijanović  _et al._ (2024) where the
Fermi energy can determine the threshold. The bridge between these two regimes
from a single well-resolved transition frequency to a broad distribution of
frequencies is, however, largely unexplored although relevant not only for
thermal atoms but also molecules and materials that are naturally prone of
inhomogeneous broadening.

In this work, we fill this gap based on a model of thermal atoms with two
ground states (spins) whose transition frequency can be tuned and is
inhomogeneously broadened due to the Doppler effect. By external driving one
can engineer strong coupling between the spins and the cavity field. If this
coupling exceeds a threshold we find a superradiant transition which is
accompanied by spatio-temporal spin organization. We call this effect spin-
self-organization. The dynamics recover the Dicke phase transition Dimer  _et
al._ (2007) if the transition frequency exceeds the width of the Doppler-shift
distribution (Doppler width); and it the opposite case it reduces to thermal
self-organization Domokos  _et al._ (2001); Asbóth  _et al._ (2005); Schütz
_et al._ (2015). However, if the Doppler width and transition frequency are
comparable, we find spin-self-organization facilitated by Doppler-resonances.
Due to these resonances the critical coupling strength exhibits a minimum that
can be below the one of the Dicke model and well below the one of thermal
self-organization. We study the emergent patterns above threshold using an
approach based on the truncated Wigner approximation Schachenmayer  _et al._
(2015); Wootters (1987); Walls and Milburn (2008); Polkovnikov (2010). Here,
we find regimes of weak spin-self-organization where patterns survive only on
a very short timescale. In addition we also find strong spin-self-organization
stabilized on long timescales by collective reduction of kinetic energy due to
resonant coupling of specific momentum classes. This work provides a
theoretical description of thresholds and pattern-formation dynamics in
strongly interacting light-matter systems facilitated by moderate
inhomogeneous broadening.

Figure 1: Cavity setup and spin-self-organization. (a): Schematic of the atom-
cavity configuration and the initial state of the dynamics. (b): Emergence of
the spin-self-organization phase transition as spin-position correlations
contained in order parameter $\langle\hat{X}\rangle$ grow once the coupling
strength $g$ is tuned beyond the critical $g_{c}$. The symmetry-broken
$\langle\hat{X}\rangle\neq 0$ phase splits into two real-space spin-aligned
configurations. (c): Phase diagram for a thermal Doppler width
$\delta_{D}=10\omega_{R}$, with the critical coupling solid line $\Omega_{c}$
marking the onset of (weak) spin-self-organization obtained from the stability
analysis (4). The dashed line represents
$\Omega_{c}^{\mathrm{Dicke}}=\omega_{0}$. The dotted line marks the boundary
between the strongly and weakly spin-self-organized phases.

Physical setup.– The system under consideration consists of a gas of $N$ atoms
at finite temperature inside a lossy cavity. The atoms are tightly trapped
along the transverse directions, such that their motion is restricted to the
longitudinal axis of the cavity [see Fig. 1(a)]. Considering each atom as an
external laser- and cavity-driven few-level system in a double Lambda
configuration Dimer  _et al._ (2007); Baden  _et al._ (2014); Zhang  _et al._
(2018), the higher-lying levels can be adiabatically eliminated, such that
only two levels coupled via the cavity remain, $|{g}\rangle$ (ground state)
and $|{e}\rangle$ (excited state), reducing each atom to a spin-$1/2$
particle. An appropriate choice of the laser parameters [see Supplementary
Material (SM) sup  and references therein] leads to dynamics described by the
Lindblad equation

$\partial_{t}\hat{\rho}=-\frac{\mathrm{i}}{\hbar}\left[\hat{H},\hat{\rho}\right]+2\kappa\left(\hat{a}\hat{\rho}\hat{a}^{\dagger}-\frac{1}{2}\\{\hat{a}^{\dagger}\hat{a},\hat{\rho}\\}\right),$
(1)

where $\kappa$ and $\hat{a}$ are the decay rate and the mode annihilation
operator of the cavity field. The Hamiltonian for the internal (spin and
cavity photons) and external (motional) degrees of freedom reads

$\begin{split}\hat{H}=&-\hbar\Delta\hat{a}^{\dagger}\hat{a}+\hat{H}_{\mathrm{at}}+\hbar
Ng\hat{X}\left(\hat{a}^{\dagger}+\hat{a}\right),\end{split}$ (2)

where the cavity detuning is given by $\Delta$. The Hamiltonian
$\hat{H}_{\mathrm{at}}=\sum_{j}\left[\hat{p}_{j}^{2}/(2m)+\hbar\omega_{0}\hat{\sigma}_{j}^{z}/2\right]$
consists of the atomic kinetic and internal energies, where $\omega_{0}$ is
the atomic inversion frequency and
$\hat{\sigma}_{j}^{z}=\hat{\sigma}_{j}^{+}\hat{\sigma}_{j}^{-}-\hat{\sigma}_{j}^{-}\hat{\sigma}_{j}^{+}$,
with $\hat{\sigma}_{j}^{-}=|{g}\rangle_{j}\langle{e}|$,
$\hat{\sigma}_{j}^{+}=|{e}\rangle_{j}\langle{g}|$. The cavity field is coupled
collectively with strength $Ng$ to the operator

$\hat{X}=\frac{1}{N}\sum_{j=1}^{N}\hat{\sigma}^{x}_{j}\cos{(k\hat{x}_{j})},$
(3)

whose expectation value is the _spin-self-organization_ order parameter.
Importantly, it includes, besides the internal spin coherences
$\sim\hat{\sigma}_{j}^{x}=\hat{\sigma}_{j}^{+}+\hat{\sigma}_{j}^{-}$, the
information of the cavity mode function with wave vector $k$. Note that by
ignoring $\cos(k\hat{x}_{j})$ or $\hat{\sigma}_{j}^{x}$ we recover the Dicke
Baumann  _et al._ (2010); Kirton  _et al._ (2019); Klinder  _et al._ (2015);
Hepp and Lieb (1973) or spatial self-organization models Domokos and Ritsch
(2002); Asbóth  _et al._ (2005); Niedenzu  _et al._ (2011); Arnold  _et al._
(2012), respectively. In our work, however, it is crucial that spin and
motional dynamics are treated on equal footing. The build-up of
$\langle\hat{X}\rangle$ is accompanied by emergent spin-position correlations
[see Fig. 1(b)] and results in superradiant emission into the cavity.

Stability analysis and spin organization.– We start by analyzing the dynamics
of the system at very short timescales in order to identify the parameter
regime where the system will undergo spin-self-organization. For this, we
assume that initially the cavity is empty, i.e.
$\langle\hat{a}^{\dagger}\hat{a}\rangle=0$, and the atoms are in the ground
state $|{g}\rangle$ and in a disordered state, with the initial position of
the $j$-th spin $x_{j}$ distributed uniformly along the cavity axis. The
initial momentum distribution is thermal with (Doppler) width
$\delta_{D}=k\sqrt{\langle\hat{p}^{2}\rangle}/m=k/\sqrt{m\beta}$ where
$\beta=1/(k_{B}T)$ is the inverse temperature of the atoms [see Fig. 1(a)].

We perform a linear stability analysis of this unorganized atomic
configuration, which is reported in the SM sup . We find that spin-self-
organization requires the coupling strength to overcome a threshold $g_{c}$
[cf. Fig. 1(b)]. Defining a rescaled coupling strength as
$\Omega=-2Ng^{2}\Delta/(\kappa^{2}+\Delta^{2})$, this threshold is given by

$\displaystyle\Omega_{c}=\left[\int_{0}^{\infty}dt\,e^{-(\delta_{D}t)^{2}/2}\sin([\omega_{0}+\omega_{R}]t)\right]^{-1},$
(4)

where $\omega_{R}=\hbar k^{2}/(2m)$ is the recoil frequency. At very low
temperatures, i.e. $\delta_{D}\ll\omega_{R}$, we recover previously known
results from spinor self-ordering Mivehvar  _et al._ (2017); Kroeze  _et al._
(2018, 2019); Chiacchio and Nunnenkamp (2019), where
$\Omega_{c}^{\text{Spinor}}\approx\omega_{0}+\omega_{R}$: the sum of the usual
Dicke threshold Kirton  _et al._ (2019) and self-organization at zero
temperature Nagy  _et al._ (2008) determined by $\omega_{0}$ and $\omega_{R}$,
respectively. The focus of this work is, however, on the regime
$\delta_{D}\gg\omega_{R}$ which is fundamentally different: the threshold in
spinor self-ordering is determined by the gap of only two states Mivehvar  _et
al._ (2017); Kroeze  _et al._ (2018, 2019); Chiacchio and Nunnenkamp (2019)
[e.g. $|{g,p=0}\rangle\leftrightarrow(|{e,\hbar k}\rangle+|{e,-\hbar
k}\rangle)/\sqrt{2}$], in our case, the cavity field will couple many states.

Now, we may distinguish two extreme regimes: (i) When
$\omega_{0}\gg\delta_{D}$, the internal transition frequency $\omega_{0}$ is
well resolved. Here, the coupling strength needs to overcome $\omega_{0}$ to
enable the formation of a coherent spin state and we find essentially the
threshold of the Dicke model Kirton  _et al._ (2019)
$\Omega_{c}^{\text{Dicke}}=\omega_{0}$. (ii) When $\delta_{D}\gg\omega_{0}$,
the internal structure of the atoms cannot be resolved. In this regime we find
the threshold of spatial self-organization Asbóth  _et al._ (2005); Niedenzu
_et al._ (2011); Schütz  _et al._ (2015), namely, $\Omega_{c}^{\text{Self-
org.}}=2/(\hbar\beta)$. In the intermediate regime $\Omega_{c}$ is shown in
Fig. 1(c) as function of $\omega_{0}$ for a fixed value of
$\delta_{D}=10\omega_{R}$. Starting from $\omega_{0}=0$ where
$\Omega_{c}=\Omega_{c}^{\text{Self-org.}}=100\omega_{R}$ we find that any
finite value of $\omega_{0}$ results in a drastic reduction of $\Omega_{c}$.
For the extreme case $\omega_{0}\gg\delta_{D}$, we find
$\Omega_{c}\approx\Omega_{c}^{\text{Dicke}}$ but, more remarkably, for finite
but large $\omega_{0}$ we find $\Omega_{c}<\Omega_{c}^{\text{Dicke}}$. This
shows that small but finite values of $\delta_{D}$ facilitate spin-self-
organization. Moreover, the non-monotonic behavior of $\Omega_{c}$ is a
consequence of working in the regime $\delta_{D}\gg\omega_{R}$ and is not
present in spinor self-ordering. In general, the threshold $\Omega_{c}$ is not
the sum of $\Omega_{c}^{\text{Dicke}}$ and $\Omega_{c}^{\text{Self-org.}}$ in
strong contrast to spinor self-ordering. Instead we find a minimum of
$\Omega_{c}$ reached for $\delta_{D}\sim\omega_{0}$, where both motion and
spin evolve on equal timescales. In this regime we find rich dynamics, as we
will discuss in the following.

Figure 2: Spin-self-organization and bunching. Dynamics for 100 trajectories
with the parameters: $\kappa=100\omega_{R},\,\Delta=-100\omega_{R},$ and
$\delta_{D}=10\omega_{R}$. (a): Peak value $\langle
X_{p}^{2}\rangle_{\mathrm{ens}}$ of the numerical simulation for $N=100$ atoms
without cavity shot-noise. The solid black line represents the theoretically
predicted threshold $\Omega_{c}$ from the stability analysis (4). (b):
Dynamics for $N=100,\ 800$ and $1600$ in different regions of the phase
diagram corresponding to strong spin-self-organized (triangle marker, upper
panel) and weak spin-self-organized (star marker, lower panel) behaviours. The
results are benchmarked against the mean-field results obtained for $N=100$
without cavity shot-noise. The shaded regions bound 90% of the trajectories.
(c): Final value $\langle X_{f}^{2}\rangle_{\mathrm{ens}}$ at
$t_{f}\omega_{R}=10$, showcasing that only a portion of the phase diagram can
host strong spin-self-organization, i.e. one that is robust in time. The
dotted line is drawn to guide the eye. (d): Bunching parameter
$\langle\mathcal{B}\rangle_{\mathrm{ens}}$ at $t_{f}\omega_{R}=10$. Regions
with $\langle\mathcal{B}\rangle_{\mathrm{ens}}=0.5$ correspond to an
homogeneous spin density across the cavity. The emergence of an increasingly
bunched pattern at the mode-function antinodes as $\Omega$ is increased is
showcased in the cut at $\omega_{0}/\omega_{R}=30$, where
$0.5<\langle\mathcal{B}\rangle_{\mathrm{ens}}\leq 1$ (only one of two possible
patterns is shown here, with 1000 trajectories for improved resolution).

Numerical simulation of the dynamics.– To study the dynamics in the regime
where $\delta_{D}\gg\omega_{R}$, we combine two established methods Giannelli
(2020): a discrete truncated Wigner simulation for the spins Schachenmayer
_et al._ (2015); Wootters (1987), and truncated Wigner simulation for the
semiclassical dynamics of continuous variable systems Polkovnikov (2010);
Walls and Milburn (2008). We simulate trajectories of the semiclassical fields
$a_{x}\sim(\hat{a}+\hat{a}^{\dagger})/2$
$a_{p}\sim\mathrm{i}(\hat{a}^{\dagger}-\hat{a})/2$, positions $x_{j}$, momenta
$p_{j}$ including opto-mechanical forces, and Pauli matrices
$s_{j}^{\alpha}\sim\hat{\sigma}_{j}^{\alpha}$ with $\alpha=x,y,z$. Their
dynamics are coupled by stochastic differential equations, and expectation
values can be calculated by performing an ensemble average
$\langle\,.\,\rangle_{\mathrm{ens}}$ over many realizations of the initial
state (for equations of motion and further information see SM sup ). To
describe the initial state, we set $s_{j}^{z}=-1$ and sample
$s_{j}^{\alpha}=\pm 1$ ($\alpha=x,y$) independently. The atomic positions are
chosen randomly and uniformly from $[0,2\pi/k)$ and the momenta are sampled
from a Gaussian with zero mean and variance $\langle
p^{2}\rangle_{\mathrm{ens}}=m/\beta$. To incorporate the initial vacuum
fluctuations in the cavity field, we sample $a_{x}$ and $a_{p}$ from a
Gaussian with zero mean and $\langle
a_{x}^{2}\rangle_{\mathrm{ens}}=1/4=\langle a_{p}^{2}\rangle_{\mathrm{ens}}$.

Using this simulation method we calculate the time evolution of the order
parameter $\langle X^{2}(t)\rangle_{\mathrm{ens}}$ for a fixed value of
$\delta_{D}=10\omega_{R}$ varying the atomic frequency $\omega_{0}$ and the
coupling $\Omega$. For low enough values of $\Omega$, the system remains for
all times in the unorganized phase, i.e. $\langle
X^{2}(t)\rangle_{\mathrm{ens}}\approx 0$. As the coupling is increased, this
unorganized phase is no longer stable and the system _jumps_ at very short
times $t_{p}$ into the spin-self-organized phase, confirmed by a value
$\langle X^{2}_{p}\rangle_{\mathrm{ens}}\equiv\langle
X^{2}(t_{p})\rangle_{\mathrm{ens}}>0$. This timescale $t_{p}$ is determined by
the collective coupling strength of the atoms to the cavity which is for our
parameters much shorter than $1/\omega_{R}$ [see Fig. 2(b)]. In Fig. 2(a) we
have plotted $\langle X^{2}_{p}\rangle_{\mathrm{ens}}$ for a range of values
of $\omega_{0}$ and $\Omega$, and observe an almost perfect coincidence of the
numerically obtained transition from spin unorganized to self-organized phase
with the line predicted by our stability analysis.

Our stability analysis does not predict, though, whether the spin-self-
organized phase reached at short times is the true stationary state of the
system. Actually, as depicted in Fig. 2(b), the numerical evolution of the
system for a finite amount of atoms $N$ can decay back into an unorganized
state. In order to determine whether spin-self-organization is present in the
thermodynamic limit, we perform a finite-size scaling analysis and compare our
results with the simulation of the dynamics in the absence of shot-noise [see
Fig. 2(b)]. In the thermodynamic limit, we observe $\langle
X^{2}\rangle_{\mathrm{ens}}\to 0$ only for low values of $\omega_{0}$, as
$\omega_{R}t\gg 1$. Here, the internal energy of the spins is insufficient to
sustain organization on longer times and dissipation channels become
commensurate with the Hamiltonian dynamics, resulting in a relaxation back to
the unorganized state. For large enough values of $\Omega$ and $\omega_{0}$,
however, the spins remain organized. To highlight these different regimes we
have depicted in Fig. 2(c) the observable $\langle
X^{2}_{f}\rangle_{\mathrm{ens}}\equiv\langle
X^{2}(t_{f})\rangle_{\mathrm{ens}}$ at $t_{f}=10\omega_{R}^{-1}$. The
timescales $t_{f}$ is chosen to be of several $1/\omega_{R}$ such that the
spins have sufficient time to redistribute themselves over several
wavelengths. Spin-self-organization on these very different timescales,
$t_{p}$ and $t_{f}$, are referred to as weak and strong spin-self-organized
phases in this work [see Fig. 1(c)].

We want to emphasize that even the strong spin-self-organization visible in
Fig. 2(c) occurs well below the usual threshold for spatial self-organization
$\Omega_{c}^{\text{Self-org.}}=100\omega_{R}$. Let us remark that in this
regime the initial temperature is too high to allow for conventional atomic
self-organization into a spatial density pattern. From this point of view the
light-forces are insufficient to trap the atoms at anti-nodes of the cavity
mode. This insight together with $\langle X^{2}_{f}\rangle_{\mathrm{ens}}>0$
suggests the spin exhibits a position dependent alignment
$s_{j}^{x}\propto\cos(kx_{j})$ and the density is homogeneous in space.
Remarkably and against this intuition, we find that the density also exhibits
a spatial pattern. To demonstrate this ordering, we calculate the bunching
parameter $\mathcal{B}=\sum_{j=1}^{N}\cos^{2}(kx_{j})/N$ Ritsch  _et al._
(2013) at time $t_{f}$ and show the result in Fig. 2(d), where a density
grating is characterized by values
$\langle\mathcal{B}\rangle_{\mathrm{ens}}>1/2$. We find that the strong spin-
self-organized phase is indeed accompanied by a density modulation. The
resulting state is one where both cavity-modulated spin-alignment and real-
space density waves are present, as can be observed in the inset of Fig. 2(d).
It should be noted that the timescales for the evolution and saturation of
$\langle\mathcal{B}\rangle_{\mathrm{ens}}$ are different than those of
$\langle X^{2}\rangle_{\mathrm{ens}}$: whereas the spin-self-organization
order parameter builds up on a very short timescale $t_{p}$, bunching
manifestly requires longer times $t_{f}$ as the initially homogeneous atomic
gas has to assemble itself at the cavity field antinodes (see SM sup ). Our
findings show that density gratings and spin-self-organization are possible
for temperatures that are a priori too high for ordinary self-organization. To
understand this behavior further, we go on to study the kinetic energy and
atomic momentum distribution.

Figure 3: Final momentum state statistics. (a): Kinetic energy at time
$t_{f}=10/\omega_{R}$ in units of $\hbar\omega_{R}$, for an initial kinetic
energy of $E_{\text{kin}}=25\hbar\omega_{R}$ ($\delta_{D}=10\omega_{R}$).
Spin-self-organization is clearly accompanied by cooling. (b): Kurtosis
$\mathcal{K}$ of the final momentum distribution. Regions with
$\mathcal{K}>3$, which have tails that asymptotically approach zero more
slowly than a Gaussian, are clearly identified in the transition region
$\Omega=\omega_{0}\gg\delta_{D}$ and where $\Omega>\omega_{0}\sim\delta_{D}$.
(c): Sketches of the energy-momentum distribution and favoured transitions
(black arrows) for a fixed momentum distribution of width $\delta_{D}$ at the
non-gaussian region $\omega_{0}\sim\delta_{D}$. Transitions from the ground to
the excited state at the tails of the momentum distribution are strongly
favoured, leading to an effective reduction of the energy gap the system needs
to overcome to spin-self-organize.

Momentum characteristics.– In Fig. 3(a) we show the mean kinetic energy
$E_{\mathrm{kin}}=\langle\sum_{j}p_{j}^{2}/(2m)\rangle_{\mathrm{ens}}/N$ after
the time $t_{f}$. We find a significant decrease of the kinetic energy with
respect to the initial value $E_{\mathrm{kin}}(t=0)=25\hbar\omega_{R}$ in the
spin-self-organized region. A simple model to understand this feature can be
derived by adiabatically eliminating the cavity degrees of freedom relying on
$\omega_{0},\delta_{D}\ll\Delta,\kappa$. In this regime, we find that the
atomic dynamics are described by the Hamiltonian

$\hat{H}_{\mathrm{ad}}=\hat{H}_{\mathrm{at}}-\frac{\hbar\Omega
N}{2}\hat{X}^{2}.$ (5)

For energy to be conserved, thus, an increase of $\langle\hat{X}^{2}\rangle$
is always accompanied by an increase of $\langle\hat{H}_{\mathrm{at}}\rangle$.
Since the atoms start in the internal ground state, their internal energy can
only increase on short timescales. However, since $\hat{X}$ couples internal
and external degrees of freedom, it provides also a channel for energy
exchange. An atom with momentum $p>0$ can now exhibit a recoil which decreases
or increases its kinetic energy. The corresponding frequency gaps are given by
$\Delta E_{-}(p)\approx\omega_{0}-kp/m$ and $\Delta
E_{+}(p)\approx\omega_{0}+kp/m$, where we dropped the contribution of the
recoil energy. Since the process that results in reduction of kinetic energy
is closer to resonance $|\Delta E_{-}|<|\Delta E_{+}|$ the atoms will more
likely couple to lower momentum states. This is confirmed by our simulations,
where we find that $E_{\mathrm{kin}}$ reduces on very short timescales $\sim
t_{p}$. We remark that this process is very distinct from cavity cooling
Vuletić and Chu (2000); Horak  _et al._ (1997); Maunz  _et al._ (2004);
Hosseini  _et al._ (2017). The process that we describe here is a collective
transfer of energy from external to internal degrees of freedom. Furthermore,
it is striking that the phase diagram and some dynamical features share
several similarities with the dynamics observed when the atoms are
incoherently pumped and the cavity-atom coupling is described by the rotating
wave approximation Jäger  _et al._ (2019, 2020). Differing from that model,
the dynamics here analysed is determined by Doppler resonances and the
counter-rotating terms turn out to be essential to trigger these dynamics.
This opens the possibility to tune the response of the system by controlling
the ratio of $\delta_{D}$ and $\omega_{0}$.

After the initial dynamics, the system may exhibit either weak or strong spin-
self-organization. While this transition is not clearly visible in the kinetic
energy, it is encoded in the momentum distribution. We observe weak spin-self-
organization approximately in the regime $\omega_{0}\lesssim\delta_{D}$. In
this case, there are many atoms that undergo transitions with Doppler
resonances $|\Delta E_{-}(p)|\approx 0$. These are strongly favoured because
of a small energy gap and facilitate the spin-self-organization, in a similar
manner to Umklapp processes in a degenerate fermionic gas Piazza and Strack
(2014) [see Fig. 3(c) left-hand side]. The atoms with the corresponding
momenta quickly reduce their momentum, which leads to a non-Gaussian momentum
distribution with a pronounced central peak [see Fig. 3(c) right-hand side].
This is measured in Fig. 3(b) with the mean kurtosis $\mathcal{K}=\langle
P^{4}\rangle_{\mathrm{ens}}/\langle P^{2}\rangle_{\mathrm{ens}}^{2}$, with
$\langle
P^{n}\rangle_{\mathrm{ens}}=\langle\sum_{j}p_{j}^{n}\rangle_{\mathrm{ens}}/N$.
This quantity is $\mathcal{K}=3$ for a Gaussian, and the value $\mathcal{K}>3$
found in our case indicates a momentum distribution with a steeper peak. For
larger $\omega_{0}$, we find strong spin-self-organization and
$\mathcal{K}\approx 3$. In this regime the system forms a density pattern,
which makes the previous considerations only focusing on internal and kinetic
energy invalid. We remark, however, that the reduction of kinetic energy
reduces the critical coupling strength $\Omega_{c}$ and is therefore a main
reason why the threshold for a density pattern is not determined by the
initial temperature as for the usual self-organization transition.

Conclusions and Outlook.– This work bridges the gap between strong light-
matter interactions in spins with a well-resolved transition frequency to
those dominated by inhomogeneous broadening. We find a new intermediate regime
of spin-self-organization facilitated by Doppler resonances resulting in an
interplay of spin and motional degrees of freedom. This interplay is also
responsible for dramatic changes in the momentum statistics, such as an almost
instantaneous reduction of kinetic energy of the gas. Beyond the fundamental
interest of our study, the regime that we explore is currently experimentally
accessible in several groups, including the experiment of one of the authors
Suarez  _et al._ (2023) (see SM sup  for experimental details). Future works
can explore the long-time behavior of this model where we expect a pronounced
role of dissipation, e.g. in the form of collective cavity cooling. Moreover,
we believe that the theoretical results in this work will also be applicable
to very different physical setups including free fermions with two different
spin species where we expect to see a similar dynamical and threshold
behavior. Another example are insulators where the cavity induces excitations
of different momentum modes in the empty band. More general, our work
demonstrates how moderate inhomogeneous broadening can enhance strong light-
matter interactions in many-body systems.

###### Acknowledgements.

The authors acknowledge support by the state of Baden-Württemberg through
bwHPC and the German Research Foundation (DFG) through grant no INST 40/575-1
FUGG (JUSTUS 2 cluster). The research leading to these results has received
funding from the Deutsche Forschungsgemeinsschaft (DFG, German Research
Foundation) under the Research Unit FOR 5413/1, Grant No. 465199066. S.B.J.
acknowledges support from Research Centers of the Deutsche
Forschungsgemeinschaft (DFG): Projects A4 and A5 in SFB/Transregio 185: OSCAR.
This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) – Project-ID 429529648 – TRR 306 QuCoLiMa (“Quantum
Cooperativity of Light and Matter”), the DFG Forschergruppe WEAVE ”Quantum
many-body dynamics of matter and light in cavity QED” - Project ID 525057097,
and the QuantERA project ”QNet: Quantum transport, metastability, and
neuromorphic applications in Quantum Networks” - Project ID 532771420. G.M.
thanks Helmut Ritsch for helpful comments. LG acknowledges support from the
PNRR MUR project PE0000023-NQSTI.

## References

  * Baumann  _et al._ (2010) K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger, Nature 464, 1301 (2010).
  * Klinder  _et al._ (2015) J. Klinder, H. Keßler, M. Wolke, L. Mathey, and A. Hemmerich, Proceedings of the National Academy of Sciences 112, 3290 (2015).
  * Mivehvar  _et al._ (2019) F. Mivehvar, H. Ritsch, and F. Piazza, Phys. Rev. Lett. 123, 210604 (2019).
  * Mivehvar  _et al._ (2021) F. Mivehvar, F. Piazza, T. Donner, and H. Ritsch, Advances in Physics 70, 1 (2021).
  * Landig  _et al._ (2016) R. Landig, L. Hruby, N. Dogra, M. Landini, R. Mottl, T. Donner, and T. Esslinger, Nature 532, 476 (2016).
  * Marsh  _et al._ (2021) B. P. Marsh, Y. Guo, R. M. Kroeze, S. Gopalakrishnan, S. Ganguli, J. Keeling, and B. L. Lev, Phys. Rev. X 11, 021048 (2021).
  * Kroeze  _et al._ (2023) R. M. Kroeze, B. P. Marsh, D. A. Schuller, H. S. Hunt, S. Gopalakrishnan, J. Keeling, and B. L. Lev, arXiv:2311.04216 (2023).
  * Guo  _et al._ (2021) Y. Guo, R. M. Kroeze, B. P. Marsh, S. Gopalakrishnan, J. Keeling, and B. L. Lev, Nature 599, 211 (2021).
  * Schmidt  _et al._ (2014) D. Schmidt, H. Tomczyk, S. Slama, and C. Zimmermann, Phys. Rev. Lett. 112, 115302 (2014).
  * Slama  _et al._ (2007) S. Slama, S. Bux, G. Krenz, C. Zimmermann, and P. W. Courteille, Phys. Rev. Lett. 98, 053603 (2007).
  * Iemini  _et al._ (2018) F. Iemini, A. Russomanno, J. Keeling, M. Schirò, M. Dalmonte, and R. Fazio, Phys. Rev. Lett. 121, 035301 (2018).
  * Gong  _et al._ (2018) Z. Gong, R. Hamazaki, and M. Ueda, Phys. Rev. Lett. 120, 040404 (2018).
  * Defenu  _et al._ (2024) N. Defenu, A. Lerose, and S. Pappalardi, Physics Reports 1074, 1 (2024), out-of-equilibrium dynamics of quantum many-body systems with long-range interactions.
  * Young  _et al._ (2024) D. J. Young, A. Chu, E. Y. Song, D. Barberena, D. Wellnitz, Z. Niu, V. M. Schäfer, R. J. Lewis-Swan, A. M. Rey, and J. K. Thompson, Nature 625, 679 (2024).
  * Schuster  _et al._ (2020) S. C. Schuster, P. Wolf, S. Ostermann, S. Slama, and C. Zimmermann, Phys. Rev. Lett. 124, 143602 (2020).
  * Meiser  _et al._ (2009) D. Meiser, J. Ye, D. R. Carlson, and M. J. Holland, Phys. Rev. Lett. 102, 163601 (2009).
  * Bohnet  _et al._ (2012) J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, Nature 484, 78 (2012).
  * Ludlow  _et al._ (2015) A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, Rev. Mod. Phys. 87, 637 (2015).
  * Pezzè  _et al._ (2018) L. Pezzè, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, Rev. Mod. Phys. 90, 035005 (2018).
  * Luo  _et al._ (2024) C. Luo, H. Zhang, V. P. W. Koh, J. D. Wilson, A. Chu, M. J. Holland, A. M. Rey, and J. K. Thompson, Science 384, 551 (2024).
  * Lidzey  _et al._ (1998) D. G. Lidzey, D. D. C. Bradley, M. S. Skolnick, T. Virgili, S. Walker, and D. M. Whittaker, Nature 395, 53 (1998).
  * Lidzey  _et al._ (2000) D. G. Lidzey, D. D. C. Bradley, A. Armitage, S. Walker, and M. S. Skolnick, Science 288, 1620 (2000).
  * Schwartz  _et al._ (2011) T. Schwartz, J. A. Hutchison, C. Genet, and T. W. Ebbesen, Phys. Rev. Lett. 106, 196405 (2011).
  * Herrera and Spano (2016) F. Herrera and F. C. Spano, Phys. Rev. Lett. 116, 238301 (2016).
  * Hirai  _et al._ (2023) K. Hirai, J. A. Hutchison, and H. Uji-i, Chemical Reviews 123, 8099 (2023), pMID: 37390295, https://doi.org/10.1021/acs.chemrev.2c00748 .
  * Koppens  _et al._ (2011) F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, Nano Letters 11, 3370 (2011).
  * Cavalleri (2018) A. Cavalleri, Contemporary Physics 59, 31 (2018).
  * de la Torre  _et al._ (2021) A. de la Torre, D. M. Kennes, M. Claassen, S. Gerber, J. W. McIver, and M. A. Sentef, Rev. Mod. Phys. 93, 041002 (2021).
  * Kirton  _et al._ (2019) P. Kirton, M. M. Roses, J. Keeling, and E. G. Dalla Torre, Advanced Quantum Technologies 2, 1800043 (2019).
  * Larson and Mavrogordatos (2021) J. Larson and T. Mavrogordatos,  _The Jaynes–Cummings Model and Its Descendants_, 2053-2563 (IOP Publishing, 2021).
  * Thompson  _et al._ (1992) R. J. Thompson, G. Rempe, and H. J. Kimble, Phys. Rev. Lett. 68, 1132 (1992).
  * Hertzog  _et al._ (2019) M. Hertzog, M. Wang, J. Mony, and K. Börjesson, Chem. Soc. Rev. 48, 937 (2019).
  * Dimer  _et al._ (2007) F. Dimer, B. Estienne, A. S. Parkins, and H. J. Carmichael, Phys. Rev. A 75, 013804 (2007).
  * Zhiqiang  _et al._ (2017) Z. Zhiqiang, C. H. Lee, R. Kumar, K. J. Arnold, S. J. Masson, A. S. Parkins, and M. D. Barrett, Optica 4, 424 (2017).
  * Black  _et al._ (2003) A. T. Black, H. W. Chan, and V. Vuletić, Phys. Rev. Lett. 91, 203001 (2003).
  * Domokos and Ritsch (2002) P. Domokos and H. Ritsch, Phys. Rev. Lett. 89, 253003 (2002).
  * Asbóth  _et al._ (2005) J. K. Asbóth, P. Domokos, H. Ritsch, and A. Vukics, Phys. Rev. A 72, 053417 (2005).
  * Arnold  _et al._ (2012) K. J. Arnold, M. P. Baden, and M. D. Barrett, Phys. Rev. Lett. 109, 153002 (2012).
  * Schütz  _et al._ (2016) S. Schütz, S. B. Jäger, and G. Morigi, Phys. Rev. Lett. 117, 083001 (2016).
  * Keeling  _et al._ (2014) J. Keeling, M. J. Bhaseen, and B. D. Simons, Phys. Rev. Lett. 112, 143002 (2014).
  * Piazza and Strack (2014) F. Piazza and P. Strack, Phys. Rev. Lett. 112, 143003 (2014).
  * Zhang  _et al._ (2021) X. Zhang, Y. Chen, Z. Wu, J. Wang, J. Fan, S. Deng, and H. Wu, Science 373, 1359 (2021).
  * Helson  _et al._ (2023) V. Helson, T. Zwettler, F. Mivehvar, E. Colella, K. Roux, H. Konishi, H. Ritsch, and J.-P. Brantut, Nature 618, 716 (2023).
  * Zwettler  _et al._ (2024) T. Zwettler, G. del Pace, F. Marijanovic, S. Chattopadhyay, T. Bühler, C.-M. Halati, L. Skolc, L. Tolle, V. Helson, G. Bolognini, A. Fabre, S. Uchino, T. Giamarchi, E. Demler, and J.-P. Brantut, arXiv:2405.18204 (2024).
  * Marijanović  _et al._ (2024) F. Marijanović, S. Chattopadhyay, L. Skolc, T. Zwettler, C.-M. Halati, S. B. Jäger, T. Giamarchi, J.-P. Brantut, and E. Demler, arXiv:2406.13548 (2024).
  * Domokos  _et al._ (2001) P. Domokos, P. Horak, and H. Ritsch, Journal of Physics B: Atomic, Molecular and Optical Physics 34, 187 (2001).
  * Schütz  _et al._ (2015) S. Schütz, S. B. Jäger, and G. Morigi, Phys. Rev. A 92, 063808 (2015).
  * Schachenmayer  _et al._ (2015) J. Schachenmayer, A. Pikovski, and A. M. Rey, Phys. Rev. X 5, 011022 (2015).
  * Wootters (1987) W. K. Wootters, Annals of Physics 176, 1 (1987).
  * Walls and Milburn (2008) D. Walls and G. J. Milburn,  _Quantum Optics_ (Springer Berlin, Heidelberg, 2008).
  * Polkovnikov (2010) A. Polkovnikov, Annals of Physics 325, 1790 (2010).
  * Baden  _et al._ (2014) M. P. Baden, K. J. Arnold, A. L. Grimsmo, S. Parkins, and M. D. Barrett, Phys. Rev. Lett. 113, 020408 (2014).
  * Zhang  _et al._ (2018) Z. Zhang, C. H. Lee, R. Kumar, K. J. Arnold, S. J. Masson, A. L. Grimsmo, A. S. Parkins, and M. D. Barrett, Phys. Rev. A 97, 043858 (2018).
  * (54) See Supplemental Material for further details about the theory, numerical implementation, and experimental specifications. The Supplemental includes Refs. Dimer  _et al._ (2007); Zhang  _et al._ (2018); Baden  _et al._ (2014); Niedenzu  _et al._ (2011); Jäger  _et al._ (2016); Mivehvar  _et al._ (2017); Kroeze  _et al._ (2018); Kirton  _et al._ (2019); Schachenmayer  _et al._ (2015); Walls and Milburn (2008); Domokos  _et al._ (2001); Stenholm (1986); Rackauckas and Nie (2017, 2020); Suarez  _et al._ (2023).
  * Hepp and Lieb (1973) K. Hepp and E. H. Lieb, Annals of Physics 76, 360 (1973).
  * Niedenzu  _et al._ (2011) W. Niedenzu, T. Grießer, and H. Ritsch, Europhysics Letters 96, 43001 (2011).
  * Mivehvar  _et al._ (2017) F. Mivehvar, F. Piazza, and H. Ritsch, Phys. Rev. Lett. 119, 063602 (2017).
  * Kroeze  _et al._ (2018) R. M. Kroeze, Y. Guo, V. D. Vaidya, J. Keeling, and B. L. Lev, Phys. Rev. Lett. 121, 163601 (2018).
  * Kroeze  _et al._ (2019) R. M. Kroeze, Y. Guo, and B. L. Lev, Phys. Rev. Lett. 123, 160404 (2019).
  * Chiacchio and Nunnenkamp (2019) E. I. R. Chiacchio and A. Nunnenkamp, Phys. Rev. Lett. 122, 193605 (2019).
  * Nagy  _et al._ (2008) D. Nagy, G. Szirmai, and P. Domokos, The European Physical Journal D 48, 127 (2008).
  * Giannelli (2020) L. Giannelli, “Quantum opto-mechanical systems for quantum technologies,”  (2020).
  * Ritsch  _et al._ (2013) H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, Rev. Mod. Phys. 85, 553 (2013).
  * Vuletić and Chu (2000) V. Vuletić and S. Chu, Phys. Rev. Lett. 84, 3787 (2000).
  * Horak  _et al._ (1997) P. Horak, G. Hechenblaikner, K. M. Gheri, H. Stecher, and H. Ritsch, Phys. Rev. Lett. 79, 4974 (1997).
  * Maunz  _et al._ (2004) P. Maunz, T. Puppe, I. Schuster, N. Syassen, P. W. H. Pinkse, and G. Rempe, Nature 428, 50 (2004).
  * Hosseini  _et al._ (2017) M. Hosseini, Y. Duan, K. M. Beck, Y.-T. Chen, and V. Vuletić, Phys. Rev. Lett. 118, 183601 (2017).
  * Jäger  _et al._ (2019) S. B. Jäger, J. Cooper, M. J. Holland, and G. Morigi, Phys. Rev. Lett. 123, 053601 (2019).
  * Jäger  _et al._ (2020) S. B. Jäger, M. J. Holland, and G. Morigi, Phys. Rev. A 101, 023616 (2020).
  * Suarez  _et al._ (2023) E. Suarez, F. Carollo, I. Lesanovsky, B. Olmos, P. W. Courteille, and S. Slama, Phys. Rev. A 107, 023714 (2023).
  * Jäger  _et al._ (2016) S. B. Jäger, S. Schütz, and G. Morigi, Phys. Rev. A 94, 023807 (2016).
  * Stenholm (1986) S. Stenholm, Rev. Mod. Phys. 58, 699 (1986).
  * Rackauckas and Nie (2017) C. Rackauckas and Q. Nie, Discrete and Continuous Dynamical Systems - B 22, 2731 (2017).
  * Rackauckas and Nie (2020) C. Rackauckas and Q. Nie, 2020 IEEE High Performance Extreme Computing Conference (HPEC) , 1 (2020).