# Photon condensation in magnetic cavity QED

Juan Román-Roche  Instituto de Nanociencia y Materiales de Aragón (INMA),
CSIC-Universidad de Zaragoza, Zaragoza 50009, Spain  Fernando Luis  Instituto
de Nanociencia y Materiales de Aragón (INMA), CSIC-Universidad de Zaragoza,
Zaragoza 50009, Spain  David Zueco  Instituto de Nanociencia y Materiales de
Aragón (INMA), CSIC-Universidad de Zaragoza, Zaragoza 50009, Spain

(August 27, 2024)

###### Abstract

We show that a system of magnetic molecules coupled to microwave cavities (or
$LC$ resonators) undergoes the equilibrium superradiant phase transition when
their mutual interaction exceeds a critical value and that this transition is
experimentally observable. In order to do that, we calculate the equilibrium
thermodynamics of a macroscopic number of spins (interacting or not) coupled
to the quantized magnetic field of a cavity (multimode or not). The effects of
the coupling are first illustrated by the vacuum-induced ferromagnetic
ordering in a quantum Ising model. Then, we compute how the coupling modifies
the magnetic phase diagram of ${\rm Fe_{8}}$ dipolar crystals, which
exemplifies the cooperation between intrinsic and photon induced spin-spin
interactions. To conclude, we demonstrate that a simple transmission
experiment can resolve the superradiant transition, and thus measure the
quantum electrodynamical control of magnetism.

In 1973, Hepp and Lieb showed that $N\to\infty$ polar molecules located inside
a resonant electromagnetic cavity undergo a second order transition from a
normal to a superradiant phase. The $\mathbb{Z}_{2}$ (parity) symmetry is
spontaneously broken leading to a ferroelectric-like state in the ’matter’ and
to a nonzero population of photons in the cavity at equilibrium Hepp and Lieb
(1973a, b); Wang and Hioe (1973). However, $47$ years and a global pandemic
later, this quantum phase transition is yet to be measured Kirton  _et al._
(2018). During this time, the community has enjoyed a winding succession of
proposals on how to achieve the superradiant phase transition (SPT), each
shortly matched by its corresponding no-go theorem Rzażewski  _et al._ (1975);
Nataf and Ciuti (2010); Viehmann  _et al._ (2011); Hagenmüller and Ciuti
(2012); Chirolli  _et al._ (2012); Vukics  _et al._ (2014). Nowadays, the
_non-observation_ of the phase transition is well understood. The crux of the
matter resides in the approximations used to derive the Hamiltonian solved by
Hepp and Lieb, the so-called Dicke model. On one side, there is the treatment
of the $A^{2}$-term (the diamagnetic term) Garziano  _et al._ (2020). On the
other, matter truncations in different gauges must be done consistently as
pointed out by Keeling, showing that the phase transition, if any, is
completely attributable to matter interactions Keeling (2007). Consequently,
the matter phase diagram remains unaltered despite it being immersed in a
cavity and no photonic population develops. The same conclusion has been
recently revisited Stokes and Nazir (2020). A final step for closing the
debate is found in the work by Andolina and collaborators Andolina  _et al._
(2019). In the thermodynamic limit $N\to\infty$, their no-go theorem
illustrates how Gauge invariance inherently prohibits the SPT for electric
dipoles in the long-wavelength limit. The latter implies that matter cannot
respond, i.e. a physical current does not appear in it, to a static and
uniform electromagnetic field. Therefore, to make the SPT observable, either
the nature of the coupling or the spatial field distribution must be, in some
way, revisited. To he best of our knowledge, this strategy has been
theoretically pursued using electron gases that either possess a Rashba spin-
orbit coupling Nataf  _et al._ (2019) or that are subjected to a spatially-
varying electromagnetic field Nataf  _et al._ (2019); Guerci  _et al._ (2020);
Andolina  _et al._ (2020). Other proposals consider systems in the ultrastrong
light-matter coupling regime Bernardis  _et al._ (2018).
Here, we propose an alternative set-up, based on magnetic molecules that
couple to superconducting microwave resonators via the Zeeman interaction
Jenkins  _et al._ (2013). Artificial magnetic molecules Gatteschi  _et al._
(2006); Bartolomé  _et al._ (2016), designed and synthesized by chemical
methods, consist of a high-spin cluster core surrounded and stabilized by a
cloud of organic molecular ligands. The ability to chemically tune their
relevant properties, such as the ground state spin, magnetic anisotropy and
mutual interactions, combined with their stability as isolated molecular
units, confer them a potential interest as magnetic memories in spintronic
devices Bogani and Wernsdorfer (2008) and as qubits for scalable quantum
information schemes Moreno-Pineda  _et al._ (2018); Gaita-Ariño  _et al._
(2019); Atzori and Sessoli (2019). Besides, they tend to organize forming
crystals, which makes them model systems to study pure magnetic dipolar order
and quantum phase transitions Fernández and Alonso (2000); Morello  _et al._
(2003); Luis  _et al._ (2005); Wen  _et al._ (2010); Burzurí  _et al._ (2011).
This work explores the realization of the Dicke model (and generalizations of
it), which undergoes the equilibrium SPT, in a crystal of molecular
nanomagnets coupled to a on-chip microwave cavity. We compute the critical
condition that triggers the SPT at zero and finite temperatures under rather
general conditions, _e.g._ for a spatially-varying magnetic field or in the
presence of direct molecule-molecule interactions, and discuss a feasible
method to detect the transition. Based on this theoretical framework, we also
build the phase diagram for the purely cavity-driven ferromagnetism and study
how the spin-photon coupling enhances the intrinsic ferromagnetic order of a
crystal of Fe8 molecular clusters Burzurí  _et al._ (2011).

Figure 1: Schematic picture of a CPW resonator coupled to a spin ensemble.
$\bm{E}_{\rm mw}$ and $\bm{B}_{\rm mw}$ are the cavity’s microwave electric
and magnetic fields, respectively. $\bm{B}_{\rm ext}$ is the external magnetic
field that induces a Zeeman splitting between the spin energy levels.

## Magnetic cavity QED.-

In the last decade, there has been growing interest in studying the coupling
of electron Ruggenthaler  _et al._ (2018) and, particularly, spin Xiang  _et
al._ (2013); Clerk  _et al._ (2020) ensembles to the field fluctuations
generated by superconducting resonators or cavities. Such hybrid platforms
complement circuit QED, where the "spins" are artificial and made of
superconducting circuits based on Josephson junctions Wallraff  _et al._
(2004); Blais  _et al._ (2020). Different magnetic species have been already
studied in this context, including impurity spins in semiconductors Kubo  _et
al._ (2010); Schuster  _et al._ (2010); Weichselbaumer  _et al._ (2019),
lanthanide ions Bushev  _et al._ (2011); Probst  _et al._ (2014) and magnetic
molecules Bonizzoni  _et al._ (2017); Mergenthaler  _et al._ (2017); Gimeno
_et al._ (2020); Bonizzoni  _et al._ (2020). In connection with the goal of
observing the SPT, set ups hosting molecular crystals offer the crucial
advantage of coupling a macroscopic number of identical and perfectly
organized spins to a single cavity mode (cf Fig. 1). A vast majority of
molecular nanomagnets are both neutral and exhibit a close to zero electric
dipole. Their response to external stimuli is then accurately described by a
simple ’giant’ spin effective Hamiltonian ${\cal H}_{S}$, which includes the
effects of magnetic anisotropy and the couplings to magnetic fields $\bm{B}$
Gatteschi  _et al._ (2006); Jenkins  _et al._ (2013). The latter enter ${\cal
H}_{S}$ via the Zeeman term ${\mathcal{H}}_{\rm
Z}=-\frac{g_{e}\mu_{B}}{\hbar}\bm{S}\cdot\bm{B}$. Here
$[S_{i},S_{j}]=i\hbar\epsilon_{ijk}S_{k}$ are spin operators,
$\mu_{B}=\frac{e\hbar}{2m}$ is the Bohr magneton and $g_{e}$ the Landé factor
($=2$ for an electron spin). The diamagnetic response, which arises mainly
from the molecular ligands surrounding the magnetic core, is much smaller that
the paramagnetic one and can be safely neglected, especially at sufficiently
low temperatures. These results provide the experimental background for the
following discussion.
Let us summarize the main steps to model the molecules-cavity system. We do it
in the single mode case. For the full derivation, also in the multimode case,
see SM . The electromagnetic field is quantized, yielding the cavity
Hamiltonian $\mathcal{H}_{\rm c}=\hbar\Omega a^{\dagger}a$, with $a$
($a^{\dagger}$) the photonic annihilation (creation) operators,
$[a,a^{\dagger}]=1$. The resonance frequency $\Omega$ is determined by the
circuit design, either its dimensions in the case of CPW-cavities or the
inductance and capacitance in the case of $LC$ lumped element resonators, and
ranges typically between $1$ and $10$ GHz. The local quantized magnetic field
generated by the superconducting currents can be written as $\bm{B}_{\rm
mw}({\bm{r}})=\bm{B}_{\rm rms}({\bm{r}})(a^{\dagger}+a)$, with $B_{\rm
rms}^{2}({\bm{r}})=\langle 0|B_{\rm mw}^{2}({\bm{r}})|0\rangle$ its zero-point
fluctuations.

Putting all together, the gauge invariant full cavity-spin Hamiltonian can be
written as follows Jenkins  _et al._ (2013, 2016)

$\displaystyle\mathcal{H}$
$\displaystyle=\mathcal{H}_{S}+\mathcal{H}_{c}+\mathcal{H}_{I}$
$\displaystyle=\mathcal{H}_{S}+\hbar\Omega
a^{\dagger}a+\sum_{j}\frac{\lambda_{j}}{\sqrt{N}}\left(e^{i\theta_{j}}S^{+}_{j}+{\rm
h.c.}\right)(a^{\dagger}+a)\,,$ (1)

where the spin ladder operators $S^{\pm}=S^{x}\pm iS^{y}$ and the coupling
constants

$\frac{\lambda_{j}}{\sqrt{N}}=\frac{g_{e}\mu_{B}}{2\hbar}|B_{\rm
rms}(\bm{r}_{j})|\,.$ (2)

Here, ${\bm{r}_{j}}$ is the position vector of the $j$-th spin. The phases
$\theta_{j}$ in Eq. (1) are defined through $B_{{\rm
rms},x}({\bm{r}_{j}})+iB_{{\rm rms},y}({\bm{r}_{j}})=|B_{\rm
rms}({\bm{r}_{j}})|e^{i\theta_{j}}$. If the molecules are $S=1/2$ non-
interacting spins, like in a sufficiently diluted free radical sample,
$\mathcal{H}_{S}=\hbar\frac{\omega_{z}}{2}\sum_{j}\sigma_{j}^{z}$ and the
Hamiltonian (1) matches exactly the Dicke model. Notice that this model does
not suffer from the “$A^{2}$-issue” because the coupling is of Zeeman kind,
rather than minimal (electric). Besides, the truncation of the electronic
degrees of freedom to a finite dimensional Hilbert space is not an
approximation but follows from the fact that we are dealing with ’real’ spins
obeying the angular momentum commutation relations. Both properties combined
permit to avoid the no-go theorems for the SPT Andolina  _et al._ (2019).

## Exact results at $N\to\infty$.-

In order to study specific spin models and how their properties are affected
by the coupling to light _and vice versa_ , we found it convenient to obtain
an effective spin Hamiltonian where the light degrees of freedom have been
traced out. Following Hepp and Lieb’s original derivation Hepp and Lieb
(1973b), this effective Hamiltonian is defined by the following expression,
exact in the $N\to\infty$ limit,

$\bar{Z}=Tr_{S}\left(\frac{1}{\pi}\int d^{2}\alpha\
e^{-\beta\mathcal{H}(\alpha)}\right)=Tr_{S}\left(e^{-\beta\mathcal{H}_{S,\text{eff}}}\right)\,,$
(3)

where $\mathcal{H}(\alpha)=\expectationvalue{\mathcal{H}}{\alpha}$ and
$\mathcal{H}$ is the total Hamiltonian given by Eq. (1). The resulting
effective Hamiltonian is (See Sec. II of the SM SM )

$\mathcal{H}_{S,\text{eff}}=\mathcal{H}_{S}-\frac{1}{\hbar\Omega}\left[\sum_{j}\frac{\lambda_{j}}{\sqrt{N}}\left(e^{i\theta_{j}}S^{+}_{j}+{\rm
h.c.}\right)\right]^{2}\,.$ (4)

It is apparent that the light-matter coupling translates into an effective
Ising-type ferromagnetic interaction among all spins that drives the quantum
phase transition. This formulation is particularly handy for studying spin
models that are well captured by a mean field approach, as we show below.

The critical point can be obtained by noticing that
$[\mathcal{H}_{I},\mathcal{H}_{c}]\sim[\mathcal{H}_{I},\mathcal{H}_{S}]\sim
1/N$. Then, in the thermodynamic limit, system and cavity factorize. Adapting
Andolina and collaborators’ prescription Andolina  _et al._ (2019) to our
Hamiltonian (1), together with its generalization to finite temperature (fully
developed in Secs. III and IV of the SM SM ), we can express the critical
condition in terms of the static response function $\mathcal{R}(T)$ of the
bare spin model. This critical condition reads

$|\mathcal{R}(T)|\geq\frac{\hbar\Omega}{2}\,,$ (5)

where

$\mathcal{R}(T)=-\frac{\sum_{m,n}e^{-\beta\epsilon_{m}}|\bra{\psi_{m}}\sum_{j}\frac{\lambda_{j}}{\sqrt{N}}\left(e^{i\theta_{j}}S^{+}_{j}+{\rm
h.c.}\right)\ket{\psi_{n}}|^{2}\frac{e^{\beta\Delta_{mn}}-1}{\Delta_{mn}}}{\sum_{m}e^{-\beta\epsilon_{m}}}\,,$
(6)

with $\ket{\psi_{m}}$, $\epsilon_{m}$ the eigenstates and eigenenergies of the
bare spin Hamiltonian $\mathcal{H}_{S}$ and
$\Delta_{mn}=\epsilon_{m}-\epsilon_{n}$. In the case of uniform coupling, the
static response function is proportional to the magnetic susceptibility
$\mathcal{R}=\lambda^{2}\chi_{\perp}$ in the $xy$-plane, perpendicular to the
external dc magnetic field (see Fig. 1).

Both approaches show a relation between light and matter observables given by

$\expectationvalue{a}=\frac{1}{\hbar\Omega}\expectationvalue{\sum_{j}\frac{\lambda_{j}}{\sqrt{N}}\left(e^{i\theta_{j}}S^{+}_{j}+{\rm
h.c.}\right)}_{S}\,,$ (7)

indicating that the phase transition is marked by the onset of both a
macroscopic population of photons in the cavity and a spontaneous
magnetization in the spin system. These results prove that magnetic materials
with high susceptibilities and low ordering temperatures, conditions that are
fulfilled by some molecular crystals Morello  _et al._ (2003); Luis  _et al._
(2005); Wen  _et al._ (2010); Burzurí  _et al._ (2011), are ideal for
experimentally realizing the phase transition.
In the case of independent spins coupled to a photon field pointing along $x$,
we obtain the critical condition

$\bar{\lambda}^{2}=\frac{\omega_{z}\Omega}{4}\left[(2S+1)\operatorname{cth}\left(\beta\frac{\omega_{z}}{2}(2S+1)\right)-\operatorname{cth}\left(\beta\frac{\omega_{z}}{2}\right)\right]^{-1}$
(8)

of the Dicke model for a spin $S$, which depends on the coupling only through
the collective coupling $\bar{\lambda}\equiv N^{-1}\sum_{j}\lambda_{j}^{2}$,
_i.e._ the root mean square of the spin-dependent couplings

$\bar{\lambda}^{2}=V^{-1}_{\rm sample}\int_{V_{\rm
sample}}\lambda^{2}(\bm{r})dV=\frac{g_{e}^{2}\mu_{B}^{2}\mu_{0}}{8\hbar}\rho\nu\Omega\,,$
(9)

where $V_{\rm sample}$ is the cavity volume occupied by spins and
$\rho=N/V_{\rm sample}$ is the density of spins. In the second equality, we
have used Eq. (2) and the Virial theorem and introduced the filling factor
$\nu=I(V_{\rm sample})/I(V_{\rm total})$ with $I(V)=\int_{V}|B_{\rm
rms}(\bm{r})|^{2}dV$.

Figure 2: Mean field phase diagram of the cavity-spin Hamiltonian for $S=1/2$
spins with Ising interactions. The black region is antiferromagnetic. The grey
region is superradiant and thus ferromagnetic. The white region is not
superradiant nor magnetically ordered. For clarity, right and upper subplots
showcase the standard $\expectationvalue{\sigma_{x}}$ and staggered
$\expectationvalue{\sigma_{x}}_{\rm stag}$ magnetizations, as well as the
number of photons per spin $\expectationvalue{a^{\dagger}a}/N$, along vertical
and horizontal slices of the phase diagram indicated by the arrows. The solid
black curve is the relevant phase boundary for a $1D$ model with PBC obtained
with the superradiance criterion (5) by exact diagonalization using a Lanczos
routine with $14$ sites and a Lanczos basis size of $60$. The parameters used
are $\rho=5.1\cdot 10^{20}\ {\rm cm}^{-3}$, $\Omega=1.4\times 10^{9}\ {\rm
s}^{-1}$ and $\nu=1$.

## Vacuum fluctuations driven ferromagnetism.-

To illustrate the effects of the coupling between the cavity and the magnetic
molecules as well as the interplay between intrinsic and light-induced
ferromagnetic interactions, we showcase two examples using realistic
experimental parameters. From a theoretical perspective, Fig. 2 shows the
alterations to the zero-temperature phases of the paradigmatic quantum Ising
model for effective $S=1/2$ spins, defined by ${\cal
H}_{S}=\frac{\omega_{z}}{2}\sum_{j}\sigma_{j}^{z}-\frac{J}{2}\sum_{i<j}\sigma_{i}^{x}\sigma_{j}^{x}$.
Note that according to Eq. (7) ferromagnetic and superradiant phases are
synonymous, i.e. ferromagnetic ordering is always accompanied by a finite
photon population in the cavity. In particular, regions I, II and III are
superradiant, as evidenced by the finite values of magnetization
$|\langle\sigma_{x}\rangle|$ and photon number $\langle a^{\dagger}a\rangle/N$
shown in the insets of Fig. 2. Remarkably, the light-induced effective
ferromagnetic interaction overpowers the intrinsic antiferromagnetic
interaction extending region I into the $J<0$ sector. In Region II, it is the
synergy between intrinsic and induced ferromagnetism that gives rise to
superradiance. Finally region III is intrinsically ferromagnetic even in the
absence of the cavity and thus becomes superradiant when coupled to one.

Figure 3: $B_{\rm c}-T_{\rm c}$ phase boundary determined with a quantum mean-
field calculation. The solid line corresponds to the bare spin model 10 and is
verified against experimental data (dots) obtained for a crystal of Fe8
molecular clusters Burzurí  _et al._ (2011), whose structure Weighardt  _et
al._ (1984) is shown in the inset. The external field, $\bm{B}$, is applied
perpendicular to the easy magnetization axis $x$:
$\bm{B}=B_{\perp}\left(0,\sin\phi,-\cos\phi\right)$ with $\phi=68º$. The
magnetic anisotropy parameters are $D/k_{B}=0.294$ K and $E/k_{B}=0.046$ K,
and the coupling $J/k_{B}=2.85\times 10^{-3}$ K. The remaining lines are
obtained with the same method taking into account the coupling to a microwave
cavity for a range of filling factors $\nu$. The parameters used are
$\rho=5.1\cdot 10^{20}\ {\rm cm}^{-3}$, taken from the crystal lattice of Fe8
and $\Omega=1.4\times 10^{9}\ {\rm s}^{-1}$.

We next consider a related, albeit slightly more realistic model. It
corresponds to an specific molecular material, a crystal of Fe8 clusters with
$S=10$, which shows a ferromagnetic phase transition purely induced by dipolar
interactions below a critical temperature $T_{\rm c}(B_{\perp}=0)\cong 0.6$ K
and a zero-temperature critical magnetic field $B_{\rm c}\simeq 2.65$ T
Burzurí  _et al._ (2011). The magnetic phase diagram measured for a magnetic
field perpendicular to the magnetic anisotropy axis, here chosen as the $x$
axis, is shown in Fig. 3. As expected for a system dominated by long-range
dipolar interactions, it agrees very well with the predictions of a mean-field
Hamiltonian

$\displaystyle\mathcal{H}_{S}=$ $\displaystyle-
DS_{x}^{2}+E\left(S_{z}^{2}-S_{y}^{2}\right)-g_{e}\mu_{B}\vec{B}\cdot\vec{S}$
$\displaystyle-2J\expectationvalue{S_{x}}S_{x}+J\expectationvalue{S_{x}}^{2}\,$
(10)

with parameters given in Fig. 3. The combination of high spin, thus high
susceptibility, and negligible exchange interactions, which lead to a quite
low $T_{\rm c}$ even for a densely concentrated spin lattice, makes this
material well suited to obtain and measure the SPT.

This expectation is borne out by calculations that include the coupling to a
microwave cavity, whose results are also included in Fig. 3. They show that
light-matter interaction has a remarkable effect on the equilibrium phase
diagram, enhancing both $B_{\rm c}$ and $T_{\rm c}$, the latter by almost as
much as a factor six, depending on the filling of the cavity. This enhancement
evidences that the cavity induces quite strong ferromagnetic correlations, a
characteristic signature of the SPT, which in this case co-operate with the
intrinsic interactions between the spins in the crystal. Achieving filling
factors well above $0.1$ seems quite reasonable provided that one achieves a
sufficiently good interface between the chip and the magnetic material. The
microwave magnetic field extends only a few tens of microns above the circuit
surface while high quality crystals with sizes of a few mm can be synthesized.
It can also be seen from these results that, even after the introduction of
direct spin-spin interactions, the transition depends only on the filling
factor, as shown in SM . To finish, we want to emphasize that the two previous
examples show how with magnetic coupling the light-induced and intrinsic
interactions add up, modifying the bare matter phase diagram. This is _the_
signature of photon condensation and, thus, of the occurence of the SPT
Keeling (2007); Andolina  _et al._ (2019).

## Transmission experiment for resolving the transition.-

Finally, we discuss how to measure the phase transition. For this, we consider
a conventional transmission experiment. In such experiment, the cavity
(resonator) is coupled to a microwave transmission line. Then, a signal is
sent through the latter. This signal interacts with the cavity that is coupled
to the spins and the transmitted (and/or the reflected) signal is recorded. We
calculate this transmission using the input-output formalism, as discussed in
SM ,

$t=-\frac{i\kappa}{(\omega-\Omega)+i\kappa+4\lambda^{2}\chi_{\perp}(\omega)}\,.$
(11)

Here, $\chi_{\perp}(\omega)$ is the transverse AC-susceptibility for the spins
driven through the cavity field

$\chi_{\perp}(\omega)=\frac{\omega_{z}\expectationvalue{\sigma_{z}}_{0}}{(\omega-i\gamma)^{2}-\omega_{z}^{2}-\frac{16\lambda^{4}\expectationvalue{\sigma_{x}}_{0}^{2}}{\Omega^{2}}}\,.$
(12)

The key feature in (11) is the spin-dependence through its dynamical
susceptibility, which gives hope that the phase transition is resolved. Figure
4 confirms this. The phase transition reveals itself in the form of a
resonance in the low-$\omega$ regime. In Fig. 4, we show 2D transmission plots
for temperatures larger or smaller than the zero-field critical temperature
$T_{\rm c}(0)$. Note that according to Eq. 8 the critical temperature at zero
field has a quadratic dependence on $S$, hence the discrepancy between the
critical temperature in Figs. 3 and 4. In the former limit, the system remains
subradiant, and paramagnetic, regardless of $\omega_{z}$ and shows only the
well known avoided crossing at $\omega=\Omega=\omega_{z}$. As temperature is
decreased below $T_{\rm c}(0)$ the phase becomes dependent on $\omega_{z}$ and
a new resonance, i.e. a new transmission channel, appears at
$\omega_{z}=\omega_{\rm z,c}$. This behaviour constitutes a clear signature of
the SPT and shows that this transition can be detected in a standard
temperature-dependent transmission experiment, provided that the spin-photon
coupling, or the spin density, is large enough.

Figure 4: Left panel: Phase boundary of the spin $1/2$ Dicke model in the
$\omega_{z,{\rm c}}-T_{\rm c}$ plane, the vertical dashed lines represent the
$\omega_{z}$ sweeps to be performed in the transmission experiment and their
relative location in terms of $T_{\rm c}(0)$. Middle and right panels: Plots
of the transmission for different temperatures as a function of the ratios
between $\omega$, $\omega_{z}$ and $\Omega$ for a spin $1/2$ Dicke model. A
dotted black line marks the transition to the superradiant phase at
$\omega_{\rm z,c}$. The parameters used are $\kappa=\gamma=0.025\Omega$,
$\rho=5.1\times 10^{20}\,{\rm cm}^{-3}$, $\Omega=1.4\times 10^{9}\ {\rm
s}^{-1}$ and $\nu=0.25$.

## Conclusions.-

In this work, we have shown that the coupling of a macroscopic number of
spins, in particular crystals of molecular nanomagnets, to the quantum vacuum
fluctuations of cavities or $LC$ resonators generates ferromagnetic spin-spin
interactions that lead to an equilibrium superradiant phase transition. Using
realistic parameters, we find that it also gives rise to detectable signatures
in the transmission of microwaves through such a hybrid set-up, thus providing
a feasible solution to the long-standing problem of measuring the SPT. Our
results also present these systems as ideal for exploring the quantum
electrodynamical control of matter, baptized as _cavity QED materials_ Rokaj
_et al._ (2020). Recent studies have shown that quantum light fluctuations can
modify properties such as excitonic transport Feist and Garcia-Vidal (2015);
Schachenmayer  _et al._ (2015); Orgiu  _et al._ (2015), chemical reactivity
Thomas  _et al._ (2016, 2019a), superconductivity Sentef  _et al._ (2018);
Schlawin  _et al._ (2019); Curtis  _et al._ (2019); Thomas  _et al._ (2019b)
and the ferroelectric phase in quantum paramagnetic materials Pilar  _et al._
(2020); Schuler  _et al._ (2020); Ashida  _et al._ (2020). Here, we show that
they can also generate, modify and control long-range ordered magnetic phases.

## Acknowledgments.-

The authors acknowledge funding from the EU (COST Action 15128 MOLSPIN,
QUANTERA SUMO and FET-OPEN Grant 862893 FATMOLS), the Spanish MICINN
(MAT2017-88358-C3-1-R, RTI2018-096075-B-C21, PCI2018-093116, EUR2019-103823),
the Gobierno de Aragón (Grant E09-17R Q-MAD) and the BBVA foundation (Leonardo
Grants 2018).

## References

  * Hepp and Lieb (1973a) K. Hepp and E. H. Lieb, Ann. Phys 76, 360 (1973a).
  * Hepp and Lieb (1973b) K. Hepp and E. H. Lieb, Phys. Rev. A 8, 2517 (1973b).
  * Wang and Hioe (1973) Y. K. Wang and F. T. Hioe, Phys. Rev. A 7, 831 (1973).
  * Kirton  _et al._ (2018) P. Kirton, M. M. Roses, J. Keeling, and E. G. D. Torre, Adv. Quantum. Techn 2, 1800043 (2018).
  * Rzażewski  _et al._ (1975) K. Rzażewski, K. Wódkiewicz, and W. Żakowicz, Phys. Rev. Lett 35, 432 (1975).
  * Nataf and Ciuti (2010) P. Nataf and C. Ciuti, Nat. Commun. 1, 72 (2010).
  * Viehmann  _et al._ (2011) O. Viehmann, J. von Delft, and F. Marquardt, Phys. Rev. Lett 107, 113602 (2011).
  * Hagenmüller and Ciuti (2012) D. Hagenmüller and C. Ciuti, Phys. Rev. Lett 109, 267403 (2012).
  * Chirolli  _et al._ (2012) L. Chirolli, M. Polini, V. Giovannetti, and A. H. MacDonald, Phys. Rev. Lett 109, 267404 (2012).
  * Vukics  _et al._ (2014) A. Vukics, T. Grießer, and P. Domokos, Phys. Rev. Lett 112, 073601 (2014).
  * Garziano  _et al._ (2020) L. Garziano, A. Settineri, O. D. Stefano, S. Savasta, and F. Nori, Phys. Rev. A 102, 023718 (2020).
  * Keeling (2007) J. Keeling, J. Phys. Condens. Matter 19, 295213 (2007).
  * Stokes and Nazir (2020) A. Stokes and A. Nazir, Phys. Rev. Lett 125, 143603 (2020).
  * Andolina  _et al._ (2019) G. M. Andolina, F. M. D. Pellegrino, V. Giovannetti, A. H. MacDonald, and M. Polini, Phys. Rev. B 100, 121109(R) (2019).
  * Nataf  _et al._ (2019) P. Nataf, T. Champel, G. Blatter, and D. M. Basko, Phys. Rev. Lett 123, 207402 (2019).
  * Guerci  _et al._ (2020) D. Guerci, P. Simon, and C. Mora, “Superradiant phase transition in electronic systems and emergent topological phases,” (2020), arXiv:2005.08994 [cond-mat.mes-hall] .
  * Andolina  _et al._ (2020) G. M. Andolina, F. M. D. Pellegrino, V. Giovannetti, A. H. MacDonald, and M. Polini, “Theory of photon condensation in a spatially-varying electromagnetic field,” (2020), arXiv:2005.09088 [cond-mat.mes-hall] .
  * Bernardis  _et al._ (2018) D. D. Bernardis, T. Jaako, and P. Rabl, Phys. Rev. A 97, 043820 (2018).
  * Jenkins  _et al._ (2013) M. Jenkins, T. Hümmer, M. J. Martínez-Pérez, J. García-Ripoll, D. Zueco, and F. Luis, New. J. Phys 15, 095007 (2013).
  * Gatteschi  _et al._ (2006) D. Gatteschi, R. Sessoli, and J. Villain,  _Molecular nanomagnets_, Vol. 5 (Oxford University Press, New York, 2006).
  * Bartolomé  _et al._ (2016) J. Bartolomé, F. Luis, and J. F. Fernández,  _Molecular Magnets: Physics and Applications_ (Springer, Berlin, Heildelberg, 2016).
  * Bogani and Wernsdorfer (2008) L. Bogani and W. Wernsdorfer, Nat. Mater. 7, 179–186 (2008).
  * Moreno-Pineda  _et al._ (2018) E. Moreno-Pineda, C. Godfrin, F. Balestro, W. Wernsdorfer, and M. Ruben, Chem. Soc. Rev. 47, 501 (2018).
  * Gaita-Ariño  _et al._ (2019) A. Gaita-Ariño, F. Luis, S. Hill, and E. Coronado, Nat. Chem. 11, 301 (2019).
  * Atzori and Sessoli (2019) M. Atzori and R. Sessoli, J. Am. Chem. Soc 141, 11339–11352 (2019).
  * Fernández and Alonso (2000) J. F. Fernández and J. J. Alonso, Phys. Rev. B 62, 53 (2000).
  * Morello  _et al._ (2003) A. Morello, F. L. Mettes, F. Luis, J. F. Fernández, J. Krzystek, G. Aromí, G. Christou, and L. J. de Jongh, Phys. Rev. Lett 90, 017206 (2003).
  * Luis  _et al._ (2005) F. Luis, J. Campo, J. Gómez, G. J. McIntyre, J. Luzón, and D. Ruiz-Molina, Phys. Rev. Lett 95, 227202 (2005).
  * Wen  _et al._ (2010) B. Wen, P. Subedi, L. Bo, Y. Yeshurun, M. P. Sarachik, A. D. Kent, A. J. Millis, C. Lampropoulos, and G. Christou, Phys. Rev. B 82, 014406 (2010).
  * Burzurí  _et al._ (2011) E. Burzurí, F. Luis, B. Barbara, R. Ballou, E. Ressouche, O. Montero, J. Campo, and S. Maegawa, Phys. Rev. Lett 107, 097203 (2011).
  * Ruggenthaler  _et al._ (2018) M. Ruggenthaler, N. Tancogne-Dejean, J. Flick, H. Appel, and A. Rubio, Nat. Rev. Chem. 2, 0118 (2018).
  * Xiang  _et al._ (2013) Z.-L. Xiang, S. Ashhab, J. Q. You, and F. Nori, Rev. Mod. Phys. 85, 623 (2013).
  * Clerk  _et al._ (2020) A. A. Clerk, K. W. Lehnert, P. Bertet, J. R. Petta, and Y. Nakamura, Nat. Phys. 16, 257 (2020).
  * Wallraff  _et al._ (2004) A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, Nature 431, 162 (2004).
  * Blais  _et al._ (2020) A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, “Circuit quantum electrodynamics,” (2020), arXiv:2005.12667 [quant-ph] .
  * Kubo  _et al._ (2010) Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J.-F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, Phys. Rev. Lett 105, 140502 (2010).
  * Schuster  _et al._ (2010) D. I. Schuster, A. P. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. J. L. Morton, H. Wu, G. A. D. Briggs, B. B. Buckley, D. D. Awschalom, and R. J. Schoelkopf, Phys. Rev. Lett 105, 140501 (2010).
  * Weichselbaumer  _et al._ (2019) S. Weichselbaumer, P. Natzkin, C. W. Zollitsch, M. Weiler, R. Gross, and H. Huebl, Phys. Rev. Appl. 12, 024021 (2019).
  * Bushev  _et al._ (2011) P. Bushev, A. K. Feofanov, H. Rotzinger, I. Protopopov, J. H. Cole, C. M. Wilson, G. Fischer, A. Lukashenko, and A. V. Ustinov, Phys. Rev. B 84, 060501(R) (2011).
  * Probst  _et al._ (2014) S. Probst, A. Tkalčec, H. Rotzinger, D. Rieger, J.-M. L. Floch, M. Goryachev, M. E. Tobar, A. V. Ustinov, and P. A. Bushev, Phys. Rev. B 90, 100404(R) (2014).
  * Bonizzoni  _et al._ (2017) C. Bonizzoni, A. Ghirri, M. Atzori, L. Sorace, R. Sessoli, and M. Affronte, Sci. Rep. 7, 13096 (2017).
  * Mergenthaler  _et al._ (2017) M. Mergenthaler, J. Liu, J. J. L. Roy, N. Ares, A. L. Thompson, L. Bogani, F. Luis, S. J. Blundell, T. Lancaster, A. Ardavan, G. A. D. Briggs, P. J. Leek, and E. A. Laird, Phys. Rev. Lett 119, 147701 (2017).
  * Gimeno  _et al._ (2020) I. Gimeno, W. Kersten, M. C. Pallarés, P. Hermosilla, M. J. Martínez-Pérez, M. D. Jenkins, A. Angerer, C. Sánchez-Azqueta, D. Zueco, J. Majer, A. Lostao, and F. Luis, ACS Nano , 8707–8715 (2020).
  * Bonizzoni  _et al._ (2020) C. Bonizzoni, A. Ghirri, F. Santanni, M. Atzori, L. Sorace, R. Sessoli, and M. Affronte, npj Quantum Inf. 6, 68 (2020).
  * (45) Supplemental Material at [URL will be inserted by publisher] for additional information on the modelling of the interaction between light and molecules, the effective spin Hamiltonian, the critical condition in terms of the spin static response function, the finite temperature stiffness theorem, the dependence of the effective spin-spin interaction on $\bar{\lambda}$ and the transmission experiment.
  * Jenkins  _et al._ (2016) M. D. Jenkins, D. Zueco, O. Roubeau, G. Aromí, J. Majer, and F. Luis, Dalton Trans. 45, 16682 (2016).
  * Weighardt  _et al._ (1984) K. Weighardt, K. Pohl, I. Jibril, and G. Huttner, Angew. Chem. Int. Ed. Engl. 23, 77 (1984).
  * Rokaj  _et al._ (2020) V. Rokaj, M. Ruggenthaler, F. G. Eich, and A. Rubio, “The free electron gas in cavity quantum electrodynamics,” (2020), arXiv:2006.09236 [quant-ph] .
  * Feist and Garcia-Vidal (2015) J. Feist and F. J. Garcia-Vidal, Phys. Rev. Lett 114, 196402 (2015).
  * Schachenmayer  _et al._ (2015) J. Schachenmayer, C. Genes, E. Tignone, and G. Pupillo, Phys. Rev. Lett 114, 196403 (2015).
  * Orgiu  _et al._ (2015) E. Orgiu, J. George, J. A. Hutchison, E. Devaux, J. F. Dayen, B. Doudin, F. Stellacci, C. Genet, J. Schachenmayer, C. Genes, G. Pupillo, P. Samorì, and T. W. Ebbesen, Nat. Mater. 14, 1123 (2015).
  * Thomas  _et al._ (2016) A. Thomas, J. George, A. Shalabney, M. Dryzhakov, S. J. Varma, J. Moran, T. Chervy, X. Zhong, E. Devaux, C. Genet, J. A. Hutchison, and T. W. Ebbesen, Angew. Chem. Int. Ed. 55, 11462 (2016).
  * Thomas  _et al._ (2019a) A. Thomas, L. Lethuillier-Karl, K. Nagarajan, R. M. A. Vergauwe, J. George, T. Chervy, A. Shalabney, E. Devaux, C. Genet, J. Moran, and T. W. Ebbesen, Science 363, 615 (2019a).
  * Sentef  _et al._ (2018) M. A. Sentef, M. Ruggenthaler, and A. Rubio, Sci. Adv. 4, eaau6969 (2018).
  * Schlawin  _et al._ (2019) F. Schlawin, A. Cavalleri, and D. Jaksch, Phys. Rev. Lett 122, 133602 (2019).
  * Curtis  _et al._ (2019) J. B. Curtis, Z. M. Raines, A. A. Allocca, M. Hafezi, and V. M. Galitski, Phys. Rev. Lett 122, 167002 (2019).
  * Thomas  _et al._ (2019b) A. Thomas, E. Devaux, K. Nagarajan, T. Chervy, M. Seidel, D. Hagenmüller, S. Schütz, J. Schachenmayer, C. Genet, G. Pupillo, and T. W. Ebbesen, “Exploring superconductivity under strong coupling with the vacuum electromagnetic field,” (2019b), arXiv:1911.01459 [cond-mat.supr-con] .
  * Pilar  _et al._ (2020) P. Pilar, D. D. Bernardis, and P. Rabl, Quantum 4, 335 (2020).
  * Schuler  _et al._ (2020) M. Schuler, D. D. Bernardis, A. M. Läuchli, and P. Rabl, SciPost Phys. 9, 66 (2020).
  * Ashida  _et al._ (2020) Y. Ashida, A. Imamoglu, J. Faist, D. Jaksch, A. Cavalleri, and E. Demler, “Quantum electrodynamic control of matter: Cavity-enhanced ferroelectric phase transition,” (2020), arXiv:2003.13695 [cond-mat.mes-hall] .