###### Abstract

In this chapter we report on the measurements of the overlap distribution of
the replica symmetry breaking solution in complex disordered systems. After a
general introduction to the problem of the experimental validation of the
Parisi order parameter, we focus on the systems where the measurement has been
possible for the first time: random lasers. Starting from first principles of
light-matter interaction we sketch the main steps leading to the construction
of the statistical mechanical model for the dynamics of light modes in a
random laser, a spherical multi-p-spin model with complex spins. A new overlap
is introduced, the intensity fluctuation overlap, whose probability
distribution, under specific assumptions, is equivalent to the Parisi overlap
distribution. The experimental protocol for measuring this overlap is based on
the possibility of experimentally realizing real replicas. After a description
of the first experiment on the random laser made of T5COx grains we review and
discuss various experiments measuring the overlap distribution, as well the
possible connection with Levy-like distribution of the intensity of the light
modes around the laser threshold, the connection with turbulence in fiber
lasers and the role of spatial etherogeneities of light modes in random media.

##  Chapter 23 Replica symmetry breaking in random lasers:
experimental measurement of the overlap distribution

###  1 Introduction

The theory of replica symmetry breaking (RSB) relies on an order parameter
which is not a number or a vector, rather it is a function of a continous
variable.[1, 2] This is the main novelty of the theory of Parisi: to conceive
and construct an order parameter able to identify a thermodynamic phase of a
system with multi-equilibria, a signature for complex disordered systems. The
function in question is the probability distribution $P(q)$ of the overlap $q$
between equilibrium states, or its cumulative

$x(q)=\int^{q}dq^{\prime}\ P(q^{\prime}),$ (1)

or, equivalently, the inverse of the cumulative $q(x)$, see Fig. 1. Numerous
and astonishing are the outcomes of this idea, right in the original replica
formalism, as well as in further reformulations, such as the cavity method [3,
4, 5]. Though $40$ years have passed since the conception of such quantity in
spin-glasses, to experimentally observe the order parameter in its full
functional glory is still a rather difficult and challenging task.

Figure 1: Left: sketch of a full replica symmetry breaking probability
distribution of overlaps between states. The Edwards-Anderson parameter
$q_{\rm EA}$ denotes the modal of the distribution, its meaning being the
overlap of a state with itself. The thick grey bar represents a Dirac delta on
$q=q_{\rm EA}$. Right: the functional order parameter $q(x)$, i.e., the
inverse of the cumulative distribution (1), for a full RSB system. The point
$x_{c}$ of discontinuous derivative is Eq. (1) computed at $q=q_{\rm EA}^{-}$.

In this paper, after exploring the early attempts to experimentally expose the
inner structure of the organization of states predicted by the RSB theory in
complex disordered systems, such as spin-glasses and structural glasses, we
will show how, with some preliminar theoretical work, the distribution of the
values of the overlaps between states can be sampled in random lasers [6, 7,
8, 9, 10, 11, 12].

A first measurement of a peculiar behavior compatible with the - soon incoming
- Parisi picture was performed by Nagata, Keesom and Harrison in 1978[13] on a
spin glass alloy, the cuprate-manganese, CuMn. There, the magnetic
susceptibility was carefully measured in static magnetic fields using two
procedures. In the first one the system is cooled down at nearly zero magnetic
field and, then, at different temperatures the system is perturbed by a
magnetic field and the response is acquired. In the second protocol the CuMn
is cooled down embedded in a constant uniform magnetic field. Then, at each
temperature the field is perturbed and the field cooled susceptibility
measured. Both the zero-field-cooled (ZFC) and the field-cooled (FC)
susceptibility behaviours, respectively $\chi_{\rm FC}$ and $\chi_{\rm ZFC}$,
are qualitatively reproduced by the RSB theory. It holds $\chi_{\rm
ZFC}=\beta(1-q_{\rm EA})$ and $\chi_{\rm FC}=\beta(1-\langle q\rangle)$, where
$q_{\rm EA}$ is the Edwards-Anderson parameter, else called the self-overlap
of a state with itself, and $\langle q\rangle$ denotes the average over the
distribution $P(q)$. This early kind of experiments on spin-glasses provided
modal and average values of the distribution.

Figure 2: Reproduced from Ref. 13 with permission of the American Physical
Society. Magnetic susceptibility of the spin-glass CuMn in the static limit.
Curves (b) and (d) are related to measurements taken after zero-field cooling
and curves (a) and (c) to measurements after cooling in a field of $5.90G$.
The two top curves (a) and (b) are the temperature dependece of the
susceptibility of an alloy with $2.02\%$ of Manganese. The two lower curves
(c) and (d) are measured on a sample with $1.08\%$ of Mn.

How to measure the whole distribution? In principle one needs to measure many
configurations of spins $\\{\sigma\\}$ in time, in a well thermalized spin
glass. This task is very hard, because of the difficulty of measuring several
atomic spins at once and because glassy systems hardly reach equilibrium.

Exploiting exactly the slow relaxation dynamics of glassy systems, an
alternative procedure circumventing the direct measure of spins configurations
was devised in 1998 by Franz, Mezard, Parisi and Peliti [14]. Under the
assumption of stochastic stability they were able to prove that in aging out-
of-equilibrium complex disordered systems, in the long time limit of both the
waiting and the observation time the ratio between the response and the
correlation comes out to be equivalent to the cumulative overlap distribution
at equilibrium. In formulas, being the overlap $q$ the asymptotic limit of the
two-time self-correlation function,

$q=\lim_{t\to\infty}C(t,t^{\prime}),$

the long times limit of the fluctuation-dissipation ratio,

$\tilde{X}(q)\equiv\lim_{t,t^{\prime}\to\infty}\frac{\chi(t,t^{\prime})}{1-C(t,t^{\prime})},$
(2)

results to be equal to the cumulative (1). Furthermore, under those
assumptions the integrated response function depends on $t^{\prime}$ and $t$
exclusively through the correlation function:
$\chi(t,t^{\prime})=\chi(C(t,t^{\prime}))$.

To measure such a fluctuation-dissipation ratio in real experiments, though,
proved harder than expected. Three impressive experiments were carried out in
the last 20 years,[15, 16, 17, 18] yielding the behaviours of response vs
correlation behavior $\chi(C(t,t^{\prime}))$ reproduced in figure 3. We
briefly report and comment the outcomes.

Figure 3: Fluctuation-Dissipation ratio on ageing systems. (A) the ${\rm
CdCr}_{1.7}{\rm In}_{0.3}\rm S_{4}$ spin-glass, reproduced from Ref. 15 with
permission from American Physical Society, (B) Laponite clay, reproduced from
Ref. 17 with permission from American Physical Society, and (C) polivynil-
acetate, reproduced from Ref. 18 with permission from Nature Publishing Group.

The first experiment is on samples of micrometric powder grains of an
insulating spin-glass ${\rm CdCr}_{1.7}{\rm In}_{0.3}\rm S_{4}$[15, 16]. Very
careful SQUID measurements of magnetic fluctuations allowed to provide the
$\chi(C)$ function of the correlation function $C(t,t^{\prime})$ displayed in
fig. 3-A. The experiment is very accurate and deals with extremely weak
thermodynamic fluctuations. Strict preacutions must be taken and kept for the
whole duration of the experiment, but very long measurements are required in
order to acquire enough statistics. One day of measuremenys is required to
provide data curves with long waiting times and, furthermore, the outcome does
not appear clean enough to demostrate the equivalence of (1) and (2) in the
infinite times limit.

In an experiment on the reorientational dynamics of Laponite disks[17], using
the orientational correlation functions measured by depolarized dynamic light
scattering and the corresponding response function via the electric field
induced birefringence, another kind of $\chi(C)$ is obtained. In this case
laponite is shown to undergo a glassy transition in density and not in
temperature. At high packing density it provides the behavior of a (fragile)
structural glass. Therefore, it is expected to be one step RSB-like in the
asymptotic limit, as, actually shown in fig. 3-B, where $\chi(C)$ displays two
slopes. In this experiment, though, as the waiting time $t^{\prime}=t_{w}$
grows the FDR tends to the equilibrium ratio $1/T$ and the out-of-equilibrium
counterpart of the cumulative overlap distribution appears to vanish.

Finally, the nanoscale polarization fluctuations and dielectric responses were
measured in polyvinyl acetate (PVAc), [18] also known to be a fragile glass.
PVAc displays aging at $T\simeq 0.98\ T_{g}$ and at such temperature the
system is expected to be one-step RSB, i. e., $\chi(C)$ is expected to tend to
a function as the one in Fig. 3-C, whose derivative is a step function.
Instead, according to the analysis with the available data a continuously
bending $\chi(C)$ is interpolated, hinting the occurrence of a continuous RSB,
as in the proper spin-glass case, or hinting the occurrence of strong finite
time pre-asymptotic effects.

To sum up, so far, through stochastic stability in off-equilibrium dynamics no
$P(q)$ has been experimentally demonstrated so far. Experimentally, indeed,
things turn out to be different even from the most sophisticated numerical
simulations, see [19] for a recent reference.

This is where photonics comes in: unlike single spins in amorphous magnets, or
local degrees of freedom (e.g., density, orientational, polarization
fluctuations) in structural glasses, single light mode intensities are degrees
of freedom accessible to experimental measurements, at least partially. This
partiality can be dealt with introducing a new overlap, the intensity
fluctuation overlap, whose distribution is equivalent to the Parisi overlap
distribution (under given conditions that will be discussed in section 2.3).

Before showing how a $P(q)$ can be acquired in experiments on random lasers
and other photonic systems we very briefly recall how and to what extent a
random laser is a complex disordered system possibly undergoing a transition
to a RSB phase.

###  2 Random Lasers as Complex Disordered Systems

Random lasers are made of an optically active medium and randomly placed
scatterers[20, 7, 21, 22, 9, 23, 24, 10, 25, 11, 26, 12, 27] (sometimes both
in one [7]). The first provides the gain, the latter provide the high
refraction index and the feedback mechanism needed to lead to amplification by
stimulated emission. As opposed to ordered standard multimode lasers, random
lasers do not require complicated construction and rigid optical alignment,
have omni-directional emission and high operational flexibility. They give
rise to a number of promising applications in the field of speckle-free
imaging [28, 29], granular matter [25, 30], remote sensing [31, 32, 12],
medical diagnostics and biomedical imaging [33, 34, 35, 36, 12], optical
amplification and optoelectronic devices [37, 38, 12].

Random lasers emission spectra above a pump threshold may show multiple sub-
nanometer spectral peaks, as well as single narrow curves with 5-10 nm width.
Depending on the material, and its optical and scattering properties, random
spectral fluctuations between different pumping shots (i. e., different
realizations of the same random laser) may or may not vary significantly. A
wide variety of spectral features is reported [39, 40, 30, 41, 26], depending
on material compounds and experimental setups. Random lasers can be built in
very different ways, can be both solid or liquid, can be 2D or 3D, the
optically active material can be confined or spread all over the volume.
Moreover, random lasers are, usually, open systems where light can propagate
in any direction rather than being confined between well specific boundaries
(mirrors) as in standard lasers and the emission acquisition can be on the
whole solid angle.

####  2.1 The leading model Hamiltonian

Since the original proposal that a random laser might be described by means of
a spin-glass-like Hamiltonian,[42, 43] various derivations have been put
forward [44, 45, 46, 47] of the leading model

$\mathcal{H}[\bm{a}]=-\sum_{k_{1},k_{2}}J^{(2)}_{k_{1}k_{2}}a^{*}_{k_{1}}\
a_{k_{2}}-\sum_{k_{1},k_{2},k_{3}k_{4}}J^{(4)}_{k_{1}k_{2}k_{3}k_{4}}a_{k_{1}}a^{*}_{k_{2}}a_{k_{3}}a^{*}_{k_{4}}+\mbox{c.c.},$
(3)

where $a$’s are complex numbers denoting the complex amplitudes of the light
modes and the $J$’s denote quenched random mode-couplings. The most
fundamental derivation is the construction starting from the light-matter
interaction between the atoms or molecules of the gain medium, thus displaying
an optical gap, and the electromagnetic (e.m.) field of the light.[45] We
briefly skecth the basic steps moving from a quantum description of the
operator dynamics to a classical Hamiltonian theory for the stationary regime
of light amplification by stimulated emission in a disordered medium. The
quantum stochastic differential equations describing the interaction of an
atom with an optical gap $\omega_{o}$ between two levels $|A\rangle$ and
$|B\rangle$, see figure 4, with the electromagnetic field of frequency
$\omega_{\lambda}$, represented by the creation and annihilation operators
$\alpha$ and $\alpha^{\dagger}$, are the Jaynes-Cummings equations[48]

$\displaystyle\dot{\alpha}_{\lambda}$ $\displaystyle=$
$\displaystyle-\imath\omega_{\lambda}a_{\lambda}-\sum_{\mu}\gamma_{\lambda\mu}\alpha_{\mu}+\int
d\bm{r}g^{\dagger}_{\lambda}(\bm{r})\ \sigma_{-}(\bm{r})+F_{\lambda}$ (4)
$\displaystyle\dot{\sigma}_{-}(\bm{r})$ $\displaystyle=$
$\displaystyle-(\gamma_{\bot}+\imath\omega_{o})\sigma_{-}(\bm{r})+2\sum_{\lambda}g_{\lambda}(\bm{r})\
\sigma_{z}(\bm{r})\ \alpha_{\lambda}+F_{-}(\bm{r})$ (5)
$\displaystyle\dot{\sigma}_{z}(\bm{r})$ $\displaystyle=$
$\displaystyle\gamma_{\parallel}\left(S\rho(\bm{r})-\sigma_{z}(\bm{r})\right)-\sum_{\lambda}\left(g^{\dagger}_{\lambda}(\bm{r})\
\alpha^{\dagger}_{\lambda}\sigma_{-}(\bm{r})+\text{h.c.}\right)+F_{z}(\bm{r}),$
(6)

where $\sigma_{z}(\bm{r})\equiv|A\rangle\langle A|-|B\rangle\langle B|$ is the
population inversion operator, $\sigma_{-}(\bm{r})\equiv|B\rangle\langle A|$
is the lowering operator and $\sigma_{+}(\bm{r})\equiv|A\rangle\langle B|$ the
raising operator. The coefficients $g_{\lambda}(\bm{r})$ are the atom-field
coupling constants, $\gamma_{\mu\nu}$ is the damping matrix and it is
associated to the fact that the cavity is open in random lasers and radiative
modes are there, as well. The coefficients $\gamma_{\bot}$,
$\gamma_{\parallel}$, are, respectively, the polarization decay rate and the
population inversion rate
($\gamma_{\parallel}\equiv(\gamma_{A}+\gamma_{B})/2$, see figure 4). The
atomic density is $\rho(\bm{r})$ and $S$ represents the intensity of the
external optical pumping.

Figure 4: Pictorial representation of the light-matter interaction. The atom
here illustrated has five states denoted by letters $|A\rangle$, $|B\rangle$,
$|C\rangle$, $|D\rangle$ and $|G\rangle$, the ground state. $\gamma_{x}$, with
$x=A,B,C,D$ are the decay rate of the excited states to a state of lower
energy. The energy gap between $|A\rangle$ and $|B\rangle$ is
$\hbar\omega_{o}$ where $\omega_{o}$ lies in the optical frequency spectrum
and contributes to the laser in the regime of high external pumping, when many
atoms coherently emit at once.

Noise terms $F_{\lambda}$, $F_{-}(\bm{r})$ and $F_{z}(\bm{r})$ are there, as
well, and they pertain to different stochastic phenomena. The term
$F_{\lambda}$ is due to the presence of radiative modes in open cavities and
the effetive interation between those and the modes inside the cavity. The
atomic noise terms $\mathcal{F}_{-}$ and $\mathcal{F}_{z}$ arise because of
the intercation between the e.m. field and the optically active medium.

Applying perturbation theory the dependence of the atomic operators
$\sigma_{-}$ and $\sigma_{z}$ on the field operator $\alpha_{\lambda}$ can be
worked out, leading to a single equation for coupled $\alpha$’s. Degrading
from quantum operators to complex numbers we are, eventually, left with a
single stochastic differential equations for the electromagnetic field modes
$\alpha$ in the cavity. These modes, at frequencies $\omega_{\lambda}$, are
called “cold cavity” or “passive” modes because they are intrinsic of the
system and not activated by any external pumping. In presence of external
excitation and above the lasing threshold the evolution in the lasing regime
is better expressed in the basis $\\{E_{k}(\bm{r})\\}$ of the so-called slow
amplitude modes. A slow amplitude mode $a_{k}$ is a mode that, for long enough
time, displays a harmonic oscilating behavior at some given angular frequency
$\omega_{k}$, so that the overall electromagnetic field can be written as

$E(\bm{r},t)=\sum_{k=1}^{N}a_{k}(t)\ E_{k}(\bm{r})\ e^{\imath\omega_{k}t}\ \ \
+\ \ \mbox{c.c.}$ (7)

for a system with $N$ modes. The relationship between passive and slow
amplitude modes is not unique and can always be expressed in the form

$\alpha_{\lambda}(t)=\sum_{k}M_{\lambda k}\ a_{k}(t)\ e^{\imath\omega_{k}t}.$
(8)

For a simple case where the relationship between passive and lasing modes can
be worked out exactly at the quantum level one can see, e.g., Refs. [49, 50].
It is called slow amplitude approximation because the dynamics of $a_{k}(t)$
is much slower than the one of the mode oscillation $e^{\imath\omega_{k}t}$,
so that the Fourier transform of $a_{k}(t)\ e^{\imath\omega_{k}t}$ tends to a
Dirac delta $\delta(\omega-\omega_{k})$.

Carrying out this transformation eventually leads to a stochastic differential
equation of the potential kind for the slow amplitudes:

$\displaystyle\dot{a}_{k}(t)=-\frac{\partial\mathcal{H}[\bm{a}]}{\partial
a^{*}_{k}}+F_{k}(t)$ (9)

where the noise $F_{k}(t)$ can be taken as white noise and the Hamiltonian
$\mathcal{H}[\bm{a}]$ turns out to be

$\mathcal{H}[\bm{a}]=-\sum_{\bm{k}\ |\ {\rm
FMC(\bm{k})}}J^{(2)}_{k_{1}k_{2}}a^{*}_{k_{1}}\ a_{k_{2}}-\sum_{\bm{k}\ |\
{\rm
FMC(\bm{k})}}J^{(4)}_{k_{1}k_{2}k_{3}k_{4}}a_{k_{1}}a^{*}_{k_{2}}a_{k_{3}}a^{*}_{k_{4}}+\mbox{c.c.}.$
(10)

In the above Hamiltonian definition a constraint is imposed on the frequencies
induced by the slow amplitude condition, that we call the Frequency Matching
Condition (FMC). For two and four modes it reads, respectively,

$\displaystyle\left|\omega_{k_{1}}-\omega_{k_{2}}\right|\lesssim\gamma$ (11)
$\displaystyle\left|\omega_{k_{1}}-\omega_{k_{2}}+\omega_{k_{3}}-\omega_{k_{4}}\right|\lesssim\gamma,$
(12)

where $\gamma$ is the finite linewidth of the modes. This can only be derived
in a quantum theoretical approach,[49] whereas, for what concerns the
classical approach, we can include $\gamma$ as a parameter coherent with
experimental observations. In equation (10) two effective coupling terms
appear: a two- and a four-mode coupling, whose expressions are derived as the
Hamiltonian is built. In the slow amplitude mode basis they depend on the
spatial intersection of the eigenfunctions $E_{k}(\bm{r})$ of the modes
modulated by the spatial profile of, respectively, the linear $\chi^{(1)}$ and
the nonlinear $\chi^{(3)}$ susceptibility of the random medium:

$\displaystyle J_{k_{1}k_{2}}$ $\displaystyle\propto$ $\displaystyle\int
d\bm{r}\ E_{k_{1}}(\bm{r})\ E_{k_{2}}(\bm{r})\
\chi^{(1)}(\bm{r}|\omega_{k_{1}},\omega_{k_{2}})$ (13) $\displaystyle
J_{k_{1}k_{2}k_{3}k_{4}}$ $\displaystyle\propto$ $\displaystyle\int d\bm{r}\
E_{k_{1}}(\bm{r})\ E_{k_{2}}(\bm{r})\ E_{k_{3}}(\bm{r})\ E_{k_{4}}(\bm{r})\
\chi^{(3)}(\bm{r}|\omega_{k_{1}},\omega_{k_{2}},\omega_{k_{3}},\omega_{k_{4}})$
(14)

One last fundamental ingredient of modes dynamics in a lasing material to be
combined in the formulation of the Hamiltonian dynamics is gain saturation.

Gain saturation is fundamental to have a stationary solution at all in the
systems under study, that might be represented as a potential, equilibrium-
like, solution to the stochastic equations. The most “energetic” phenomenon
that can occur in a system of atoms in a laser is when they all are in their
excited optical level and they all emit a photon at once. As such an event
were to occur, the atoms emission would be soon afterwards depleted because
they have to invert their population and that takes time. Therefore, the gain
- that is the capability of the material of amplifying light - saturates at a
certain level $E_{\rm sat}$. Its behavior as a function of the total energy
$\mathcal{E}$ pumped into the system, is usually modeled as [51, 46]

$g(\mathcal{E})=\frac{g_{0}}{1+\frac{\mathcal{E}}{E_{\rm sat}}}.$

Even though the external pumping continues the amount of lasing will not
increase indefinitely, no matter how strong the pumping is , but it will reach
a stationary regime. The overall energy $\mathcal{E}$ shared by the modes in
the cavity slowly fluctuates between an upper and a lower bound, but its
change is much slower than the one of the single mode intensities. As a
further approximation, then, in the spin-glass theory for random lasers we
assume that the dynamics of the total energy of $N$ modes is so slow to be
considered as a constant with respect to the dynamics of the single photon
emissions at any time $t$, i.e.,

$\mathcal{E}=\sum_{k}|a_{k}(t)|^{2}=\epsilon\ N=\mbox{const}.$ (15)

In figure 5 we pictorially sketch this approximation in a two-mode case. From
the point of view of the phasors $a_{k}(t)$, whose dynamics is governed by
(9), Eq. (15) is a global contraint for the dynamics of a configuration to be
on the $2N$ hyper-sphere of radius $\sqrt{\epsilon N}$. This is the reason why
this kind of spins are refereed to as “spherical” in statistical mechanics
literature [52]. We will, therefore, call the leading model (10) a complex
spherical $2$+$4$-spin spin-glass model.

With such a global constraint the stationary regime can be described as if the
system were at equilibrium at an effective temperature (a “photonic”
temperature $T_{\rm ph}$) related to the ratio $\mathcal{P}$ between the
external pumping rate and the spontaneous emission rate. The latter is
proportional to the real heat bath temperature $T$, whereas the first one is
proportional to the energy (15) stored into the photonic system. In a formula,
the role of the temperature driving the lasing transition is played by the
rescaled temperature

$T_{\rm
ph}=\frac{T}{\epsilon^{2}}=\frac{1}{\mathcal{P}^{2}}=\frac{1}{\beta_{\rm
ph}}.$ (16)

Figure 5: A sketch of the spherical approximation for the gain saturation.

####  2.2 An analytic solution in the narrowband approximation

The spin-glass model (10) in the limit of narrow bandwidth of the random laser
spectrum tends to the generalization of the fully connected spherical
$2$+$4$-spin spin-glass model[53, 54, 55, 56] to complex spins. In the
narrowband approximation[57, 51] all resonances have frequencies so close to
each other that their difference is of the order of the linewidth $\gamma$, so
that the conditions (11) and (12) are always satisfied. Under the further
assumption that all modes have a spatial extension of the order of the whole
volume of the lasing material, Eqs. (13) and (14) imply that the interaction
network can be taken as fully connected.

The fully connected limit has been extensively studied in the thermodynamic
limit in Refs. [58, 45, 46, 47] where the couplings (13) and (14) are chosen
to be independent identically distributed gaussian variables of given mean
$\bar{J}/N^{\delta}$ and variance $\sigma_{J}^{2}/N^{\delta}$. The exponent is
$\delta=1$ in the two-body term and $\delta=3$ in the four body term. In this
way the magnitude of each coupling decreseas as the number of coupling of a
single mode and the Hamiltonian (3) on the fully connected graph of
interaction if always extensive.

The ratio $R_{J}\equiv\sigma_{J}/\bar{J}$ between mean square displacement and
mean value represents the degree of disorder of the system.222The model can be
implemented with two different degrees of randomness for the $2$\- and the
$4$-body interactions [45] but for the sake of the presentation we consider
them equal to each other. Glassy random lasers, i.e., random lasers displaying
anomalous shot-to-shot fluctuations of the emission spectra, will generally
have a large $R_{J}$ (even infinite if we take $\bar{J}=0$). In Figure 6 we
reproduce a typical phase diagram in the pumping rate and degree of disorder.
Four thermodynamic phases can occur depending on the degree of randomness
$R_{J}$. As the disorder is small (or none) the model (3) displays an
incoherent wave regime at low pumping and a Standard Mode-Locking laser regime
at high pumping $\mathcal{P}$. This regime also represents Random Lasers with
no glassy features, that is, lasers with random resonances in the spectrum but
no anomalous flutucations from shot to shot (the resonances are random but
they are always the same). As randomness in the mode-coupling increases a
Phase-Locking Wave regime is predicted to occur, that is a regime where the
phases of the modes are locked while no resonance occurs in the mode
intensities. This phase vanishes as $R_{J}\to\infty$. Eventually, for large
enough randomness a Glassy Random Laser occurs above a certain pumping
threshold. This is the regime where the spin-glass theory of multistate
systems is necessary in order to deal with the complexity of the spectral
behavior.

In the highly (quenched) disordered region the order parameter identifying the
random lasing threshold is the probability distribution of the overlap

$q_{\rm{ab}}=\Re\left[\frac{1}{N}\sum_{k=1}^{N}\bar{a}^{(\rm a)}_{k}a^{(\rm
b)}_{k}\right].$ (17)

between any two replicas a and b.

Some instances of RSB solution in different points of the phase diagram, and
the relative $P(q)$’s are reported in Figure 6 (the Parisi distributions are
displayed in the rightmost column). For low pumping the incoherent wave regime
is described by a replica symmetric solution. Increasing the pumping the
system undergoes a transition to a Glassy Random Laser with infinite breakings
of the replica symmetry (Full RSB). Increasing further $\mathcal{P}$ the laser
moves to a 1-Full RSB phase and, eventually, for very large pumping, where the
pairwise (i.e., linear) contribution to (3) plays no role anymore, a one step
RSB phase correctly describes the thermodynamic behavior fo the Glassy Random
Laser.

Figure 6: An instance of a phase diagram of a random laser model (3) in the
narrowband approximation (left). Four thermodynamic phases can occur depending
on the degree of randomness $R_{J}$: Incoherent Wave (IW) at low $\mathcal{P}$
and low disorder $R_{J}$, Standard Mode locking (SLM) at low $R_{J}$ and high
pumping $\mathcal{P}$, Phase-Locking Wave (PLW) in a range of disorder $R_{J}$
large but not extremely large, Glassy Random Laser (GRL) above a certain
pumping threshold for large $R_{J}$. The latter is the regime of broken
replica symmetry and needs to be analyzed by means of Parisi theory for
complex disordered systems. The dotted lines are first order transition lines,
that is, above and below those dotted lines a region of phase separation is
expected (spinodal lines, and how to computed them, are reported in Ref. 45).
On the right the IFO and Parisi overlap distributions $P(\mathcal{C})$ and
$P(q)$ are displayed in four points of the phase diagram along a line of
increasing pumping for a given degree of disorder. For low pumping the IW
regime is described by a replica symmetric solution. Increasing $\mathcal{P}$
the system becomes a GRL with a Full RSB. Increasing further $\mathcal{P}$ the
laser moves to a 1-Full RSB phase and, eventually, for very large pumping, the
behavior of the GRL is described by a one step RSB phase. Repoduced from Ref.
59 with permission from Nature Publishing Group.

To measure such an overlap one has to access real and imaginary parts of the
complex amplitude of each light mode $a_{k}$ in each replica. Else said the
intensity and the phase:

$\displaystyle I_{k}^{(\rm a)}$ $\displaystyle=$
$\displaystyle\left|a_{k}^{(\rm a)}\right|^{2}$ (18)
$\displaystyle\phi_{k}^{(\rm a)}$ $\displaystyle=$
$\displaystyle\mbox{arg}\left(a_{k}^{(\rm a)}\right).$ (19)

As will be discussed in a while, even though the intensities can easily be
measured from the emission spectra, the phases in random lasers are hardly
accessible experimentally. To overcome such a hindrance it is possible to
define an overlap between replica involving intensities alone. Or, better,
involving their fluctuations.

####  2.3 The intensity fluctuation overlap (IFO)

Let us define the fluctuation between the intensity $I_{k}^{(\rm a)}$ of a
single resonance (a single mode $k$) in the spectrum of a single replica
$($a$)$ and its average at equilibrium $\langle I_{k}^{(\rm a)}\rangle$:

$\Delta_{k}^{(\rm a)}=\frac{I_{k}^{(\rm a)}-\langle I_{k}^{(\rm
a)}\rangle}{2\sqrt{2}\ \epsilon}=\frac{I_{k}^{(\rm a)}-\langle I_{k}^{(\rm
a)}\rangle}{2\sqrt{2T}},$ (20)

where the normalization factor $\sqrt{8}$ is a pedantry and simply depends on
the coefficients in the definition of (3).

We can, then, introduce the overlap between the intensity fluctuations of two
replicas $\rm a$ and $\rm b$:

$\mathcal{C}_{\rm ab}\equiv\frac{1}{N}\sum_{k=1}^{N}\Delta_{k}^{(\rm
a)}\Delta_{k}^{(\rm b)}.$ (21)

In the fully connected model with large disorder the distribution of the
Intensity Fluctuation Overlap (IFO) is proved to be equivalent to the one of
the Parisi overlap (17) squared. Indeed, for each couple of replicas the
identity holds:

$\mathcal{C}_{\rm ab}=q_{\rm ab}^{2}\qquad,\qquad\mbox{a}\neq\mbox{b}$ (22)

In Figure 6 we show some examples of both distributions as the system
undergoes a phase transition from an incoherent wave regime to a lasing regime
that is random and glassy.

####  2.4 Mode-locked random laser theory and numerical simulations

Moving to more realistic random lasers one should relax the narrow-band
assumption. Indeed, the optically active materials composing random lasers
have, usually, a wide emission spectrum in which many resonances take place
above the lasing threshold. That is, a more realistic system is modeled by
equation (10), implementing the conditions (11), (12). Such conditions imply a
dilution of the interaction network of $\mathcal{O}(N)$ [50, 60] and induce
some kind of metrics in the space of the frequencies. 333 A model with
$4$-body interaction, that represents the fundamental non-linear ingredient to
yield lasing at high pumping (low temperature) will, therefore, display
$\mathcal{O}(N^{3})$ mode-coupling terms in the Hamiltonian (if the modes are
spatially extended to a finite fraction of the volume of the lasing material).
This also implies that the $\delta$ exponent introduced in section 2.2 in the
average and variance of the random coupling distribution will be descreased by
one. The $2$-body term will have a sparse structure ($\delta=0$: each mode has
a finite number of connection, whatever $N$), whereas it will be $\delta=2$
for the $4$-mode couplings.

Even though the $2$-mode terms are important for the occurrence of a RSB
solution of the full kind, in this section we will focus on the $4$-mode
contribution alone, that is responsible for the onset of a glassy lasing
regime, though as a one step RSB solution. This means that in the lasing
regime one should expect a $P(q)$, or $P(C)$, like in figure 6-d and 6-a: a
single central peak at high temperature that, as the lasing threshold is
overcome, also displays two more side peaks at low $T$.

To analytically solve a diluted (that is not sparse, nor fully connected)
system with a deterministic prescription is still an unsolved problem in
replica theory but one can resort to Monte Carlo numerical simulations. 444We
also stress that to use a sparse approximation in this case would not allow to
use the cavity method because the variables are not locally confined, the
“spins” are not $|s|^{2}=1$ as the Ising, the XY [61, 62] or the Heisenberg
spins. Instead light mode amplitudes only have an overall constraint (15) and
locally one or a few of them might condensate the whole power at disposal,
$|a|^{2}=\mathcal{O}(N)$, leaving nothing for all the others. This is a case
where the equivalence of the canonical and the microcanonical ensembles break
down and cannot be studied in the present form.

In figure 7 we report instances of average overlap distributions at various
temperatures $T$ for a random laser model of $N=66$ on a mode-locked graph
obtained simulating different replicas with the Exchange Monte Carlo, or
“Parallel Tempering” algorithm. This exploits the properties of detailed
balance in a Markov chain dynamics of the system of spins through different
parallel thermal baths in order to drastically reduce the thermalization time
in the spin-glass phase. The implementation of the dynamics, moreover, is
carried out on GPU’s so that both the just mentioned parallel dynamics and the
computation of every single energy update can be spread simultaneously on
several threads. This is a technical but crucial point in order to have
reasonable simulation times in a system that is Non-Deterministic Polynomial
Complete, has continuous variables and a number of interactions growing like
$N^{3}$. We notice that the IFO distributions, that in the fully connected
case are equivalent to the Parisi distributions, on the mode-locked graph
appear to have much stronger finite size effects, as analyzed in Ref. 63. On
the other hand their signal of a transition to a multi-state phase of
nontrivial correlations is much more evident than with the $P(q)$, that one
has to look in the log scale to appreciate the occurrence of side peaks.

We stress that, even if diluted with the FMC, Eq. (12), the thermodynamic
solution is expected to be a mean-field one, as in the case of the fully
connected model, as confirmed by recent accurate numerical results [63].

Figure 7: Left: average Parisi overlap distribution $P(q)$ from numerical
simulation of the dynamics of the $4$-spin complex spherical spin model with
$\bar{J}=0$ (“infinite” degree of randomness $R_{J}$), $N=66$ modes at four
different temperatures across the critical point, $T_{c}=0.61(3)$[63]. Right:
average distribution $P(\mathcal{C})$ of the Intensity fluctuation overlap,
Eq. (21), for the same system. In both plots, at each temperature,
distributions are averaged over $100$ disordered samples. All samples are
thermalized, that is the configurations are all at equilibrium and the time
average coincides with the ensemble average.

When moving from analytical and numerical results to real experiments
different aspects have to be taken into account. As in the numerical
simulations, also in experiments the number of modes is finite. However, in
experiments on glassy random lasers it is rather difficult to scale the system
studying a controlled trend in $N$ and perform finite size scaling as in
numerical simulations [64]. Moreover, also the mode resolution is finite and
we do not know much about the spatial extension of the modes and, therefore,
about the interaction graph, nor about the magnitude and sign of the
couplings. As anticipated, when the IFO are introduced in experiments only the
mode intensity is acquired and not the mode phase. Moreover, one does not have
access to instantaneous emission intensitities but only to the integrated
intensity emission spectra. It is, thus, rather difficult to directly control
equilibration. Notwithstanding it has been possible to obtain clear signatures
of RSB across the lasing threshold in experiments on a particular subset of
random lasers, having a quenched scattering structure and displaying large
shot-to-shot spectral fluctuations.

###  3 Experimental measurements of the Parisi order parameter

In this section we will describe and discuss the first experimental
demonstration of RSB reported in ref. 65 and give an overview of the numerous
interesting results that have been published after this work. A subsection
will be dedicated to material requirements for reproducing real replicas in
random lasers.

####  3.1 Experimental procedure

In Ref. 65 the authors measure shot to shot intensity of the emitted light
from a suitable random laser, calculate the fluctuation overlap $\mathcal{C}$
of these experimental observables and build its distribution at various input
pumping. The so obtained $P(\mathcal{C})$ profiles unambigously reproduce the
predictions of RSB theory made by Giorgio Parisi.

The authors use a thiophene (T5OCx) dye chemically treated to have a thick
amorphous solid material. When this material is pumped by an external source,
above a certain threshold the fluorescence is amplified and a stimulated
emission is obtained [66, 67, 68]. The amplification is sustained by the
multiple light scattering inside the disordered system. A representative
confocal image of the sample is depicted in fig 8a. For the investigation of
emission fluctuations the sample is pumped by an external pulsed laser and at
each pulse the emission is collected. A sketch of the pumping and collecting
geometry is given in the inset of fig. 8b.

Figure 8:  a) 3D-reconstruction of confocal microscopy Z-stack images of a
T5OCx solid sample. The right and the bottom panels report the yz- and the xz-
sections, respectively. Scale bar: 20$\mu$m. b) High resolution single shot
spectra taken in the same conditions, 10mJ pump energy. Inset: sketch of the
experiment. Reproduced from Ref. 65 with permission from Nature Publishing
Group.

Fig. 8b shows single shot RL emission spectra, taken at identical experimental
conditions. The distinct peaks are the activated modes or resonances of the
disordered laser. Their configuration changes from shot to shot evidencing
that each time the system is pumped the numerous passive modes, characteristic
of the material, compete for the available gain, giving rise to several
different compositions of the activated spectral peaks [69]. This is also
evidenced by the direct visualisation of the sample during pumping, as
reported in fig. 9, where four different fluorescence images taken at four
single shots exhibit different emission patterns and spectra. From shot to
shot the size and the brightness of the luminous spots change showing that
also the spatial structure of the modes changes. Such behaviour is strongly
dependent on the input energy and it occurs above a pumping threshold that is
the RL energy threshold. Below this pumping energy only spontaneous emission
(fluorescence) is observed, above this threshold the system becomes a RL with
strongly variable emission.

Figure 9:  Snapshots of RL emissions. Single shot optical images and
corresponding emission spectra (insets) during the pumping of the sample in
the same experimental conditions. The input energy is 10mJ. Reproduced from
Ref. 65 with permission from Nature Publishing Group.

In fig. 10 emission spectra of subsequent 100 shots at two different pump
energies are shown. At low energy (fig. 10a) only noisy variations of the
spontaneous emission are observed, while at high energy (fig. 10b) the spectra
fluctuate randomly from pulse to pulse. These experimental results can be
analyzed in the framework of replica theory. Indeed, the solid property of the
material guarantees the reproducibility of the same disordered network of mode
interactions from shot to shot. Bond-disorder is quenched and real replicas of
the same random laser sample are reproduced at each illumination shot. Each
spectrum can, thus, be considered as the intensity configuration of a
different state of the same thermodynamic glassy phase. Though for the theory,
cf. Eqs. (3), (10), the spins are complex mode amplitudes, the experimental
accessible observables are their intensities (18), $I_{k}^{\rm a}$, at
wavelength $\lambda_{k}$ of the shot $\rm a$: $k$ is the spin index and a is
the replica index.

Figure 10:  a-b) Emission spectra at low energy 1mJ (a) and high energy 12mJ
with evident fluctuations (b). The experimentally accessible variable, coarse
graining the behavior of single modes, is the intensity $I(k)$ at a given
wavelength $\lambda_{k}$. In figure b the resonances $I(k)$ for two and for
different replicas (spectra), denoted by $\alpha$ (red arrows) and $\beta$
(blue arrows), are pointed out at.

It is important to recall that the effective statistical mechanics Hamiltonian
variables are the complex amplitudes $a_{k}$, [70, 44, 45] not accessible in
the experiments. While, as anticipated in section 2.2, their square moduli,
the emitted intensities, are easily measurable. In section 2.3 we reported
that such coarse-graining is refined enough to validate the possible breaking
of replica symmetry in random lasers and their glassy-like behaviour by
introducing the IFO parameter $\mathcal{C}$, Eq. (21), between pulse-to-pulse
intensity fluctuations in different replicas. Now we are considering
experimental replicas and the experimental IFO is naturally defined as

$\mathcal{C}_{\rm ab}=\frac{1}{\mathcal{N}_{\rm
ab}}\sum_{k=1}^{N}\Delta_{k}^{({\rm a})}\Delta_{k}^{({\rm b)}},$ (23)

where

$\Delta_{k}^{({\rm a})}\equiv I_{k}^{(\rm a)}-\bar{I}(k)$ (24)

with $\bar{I}(k)$ the average over $N_{s}$ replicas (emission spectra) of each
mode intensity

$\bar{I}_{k}=\frac{1}{N_{s}}\sum_{\rm a=1}^{N_{s}}I_{k}^{(\rm a)}$ (25)

and where the normalization factor is

$\mathcal{N}_{\rm ab}\equiv\sqrt{\sum_{k=1}^{N}\left(\Delta_{k}^{({\rm
a})}\right)^{2}}\sqrt{\sum_{k=1}^{N}\left(\Delta_{k}^{({\rm b})}\right)^{2}}$
(26)

From the $N_{s}$ measured spectra the set of all $N_{s}(N_{s}-1)/2$ values of
$\mathcal{C}$ for each different input energy is calculated and their
distributions $P(\mathcal{C})$ are depicted in fig. 11 (top).

Figure 11:  Top line: distribution function of the experimental overlap
$P(\mathcal{C})$, Eq. (23), showing replica symmetry breaking by increasing
pump energy. Bottom line: distribution of the theoretical IFO (21) by
increasing pumping. Reproduced from Ref. 65 and 59 with permission from Nature
Publishing Group.

Figure 12:  The value $q_{\rm max}$ corresponding to the position of the
maximum of $P(|\mathcal{C}|)$ versus pumping. A sharp transition from $0$ to a
large value is evidenced. Reproduced from Ref. 65 with permission from Nature
Publishing Group.

Six examples are reported in fig. 11, top row of panels, for increasing pump
energy. At low energy (fig. 11 left-most plot), all overlaps are centred
around the zero value, meaning that the electromagnetic modes (spins) are
independent and not interacting, they are in the paramagnetic regime. By
increasing energy, modes are coupled by the nonlinearity and this corresponds
to a non-trivial overlap distribution. In the high energy glassy phase, with
all modes highly interacting and frustrated by the disorder, $\mathcal{C}$
assumes all possible values in the range $[-1,1]$. Such behaviour of the
$P(\mathcal{C})$ evidences the fact that the correlation between intensity
fluctuations in any two replicas depends on the replicas selected. The variety
of possible correlations extends to the whole range of values. This is a
manifestation of the breaking of the replica symmetry. In the bottom row the
analytical results of the fully connected model of sections 2.2-2.3 are
reproduced for a heuristic comparison.

In fig. 12 a useful parameter $q_{\rm max}$ corresponding to the position of
the maximum of $P(|\mathcal{C}|)$ versus pumping is shown: it changes
drastically from $0$ to a very large valure next to $1$ signaling a phase
transition at about 3mJ, this is the RL energy threshold. As shown in figures
(6) and (7), the onset of the transition to the glassy phase occurs as soon as
the $P(\mathcal{C})$ develops non-gaussian tails. Taking as a reference the
Full Width Half Maximum (FWHM) of the low pumping gaussian distribution one
might, therefore, improve the identification of the random lasing threshold as
the pumping energy at which the area outside the range of $2$ FWHM is
sensitively larger than $2\%$.

These findings besides being a first experimental demonstration of the RSB
theory, are also relevant in photonics. The overlap distributions and $q_{\rm
max}$ are now used in many works to measure the energy threshold of RLs and
are considered a powerful tool to identify the RL behaviour and distinguish it
from other emission mechanisms.

####  3.2 Material requirements for reproducing real replicas.

Before reporting the main results of RSB in other kinds of RL, it is important
to stress the concept that not all random lasers show strong emitted intensity
fluctuations and consequent breaking of replica symmetry. In ref. 65, for
instance, it is demonstrated the lack of RSB in a RL made of a dyed colloidal
dispersion. Indeed, in a fluid sample the particles tend to move from shot to
shot during a single experiment and, therefore, the set of possible multiple
scattering trajectories experienced by the light pumped into the system
change. This implies that the optical susceptibility, as well as the normal
mode profiles in (13) and (14) change giving rise to a different quenched
disordered sample every shot. Under this conditions a RL sample is not
replicated but a new RL, with a new random coupling configuration, is realized
at every shot, unless the liquid is not stable enough (from the point of view
of photonic time-scales) to allow the realization of replicas. Within this
discussion, we report that RSB has been evidenced in liquid phase RL samples
for instance in Ref. 71 and Ref. 72.

In Ref. 71 a dyed sol-gel colloidal suspension with modified amorphous TiO2
scattering particles, made on purpose to strongly hinder photodegradation and
precipitation, shows clear glassy random laser behaviour. The spectra are
analysed by following the procedure reported in section 3.1 and RSB is
evidenced. Although the authors do not deepen on the fact that in liquid
materials the structural composition changes in time due to the brownian
motion of the particles, they directly test the robustness of their samples
against tens of thousand of shots and give evidence that emission spectra from
single pump shots can be properly considered as replicas. As pumping increases
from the incoherent wave regime across the lasing threshold the
$P(\mathcal{C})$ behaviour is, indeed, qualitatively similar to the one
reported in figure 11: a low pumping $P(\mathcal{C})$ with a single gaussian
peak in zero develops long tails around the lasing threshold that become side
peaks at larger and larger IFO values upon increasing the pumping.

The opposite approach, pumping energy on a fluid RL whose microscopic
scatterers position certainly changes from shot to shot, is followed in Ref.
72. Experimental evidence is provided of the motion of the scattering
particles from shot to shot and, looking at the IFO distribution, the
robustness of RSB theory also in systems with annealed disorder is claimed. If
the statistical mechanical description of glassy random lasers as spin-glass
models is to hold, however, one would expect the occurrence of a behavior
corresponding to a replica symmetric solution [73]. Our point of view is that
in solid systems single shot spectra are considered as replicas because all
experimental parameters are quenched and only the strong interaction between
modes (spins) at high pumping (low temperature) causes the breaking of their
symmetry. In fluid materials whose typical diffusion time-scale is shorter
than the experimental time (even though possibly longer then the single shot
duration) the microscopic matter composition changes from shot to shot. Even
though still many modes are randomly activated and interact with quenched
disordered couplings at any shot and the system is a random laser it is not
possible to look for replica symmetry breaking because real replicas (i.e.,
same realization of the random coupling network) are not there.

Looking at the IFO distributions reported in Ref. 72, indeed, thery turn out
to have a sharp change from a gaussian to a purely bimodal distribution across
the lasing threshold. A more likely interpretation of this phenomenon might,
thus, be the occurrence of bistability, a known phenomenon in standard lasers.
Because of randomness (at each shot a different realization of disordered mode
couplings is yielded) some samples will be lasing after an illuminating shot,
while some other might still be in the fluorescent regime. This implies that
the average spectrum $\bar{I}_{k}$ over all emissions will never be similar to
any of the single emission spectra: it will be far away from both the
fluorescent spectra and the lasing emission spectra. Therefore, no
$\Delta_{k}$ in Eq. (24) will be small and the only values of the IFO
available will be large (positive or negative), inducing a bimodal
$P(\mathcal{C})$.

This is also what probably occurs in some other experiments on ordered lasers.

#####  3.2.1 RSB in ordered cavity

We mention two very bright works where RSB has been claimed to occur in
standard ordered cavities. Basak et al. in 2016 report on strong intensity
fluctuations in both liquid and solid dye lasers with Fabry-Perot cavities
obtained by the cuvette walls in the former and by the interface of polymeric
thin slab with air in the latter [74]. The samples are pumped with pulsed
lasers and by increasing the pumping energy at laser threshold strong
intensity fluctuations are observed. These are analyzed by means of the IFO
distribution and RSB is put forward as an explanation of their behaviour.

Actually, the very detailed analysis performed by the authors demostrates the
existence of a critical interval of pumping energy where, from shot to shot, a
fluorescent or a laser emission take place. This is a clear indication of
bistability, in laser language, corresponding to phase separation in a first
order phase transition, in statistical mechanics. It is not unexpected, as the
statistical physics model for the ordered (or for the not too disordered)
multi-mode laser in a closed cavity corresponds to Eq. (3) with only the four-
mode coupling part and with small enough $R_{J}$ for the $J$’s distribution,
cf. figure 6.

Another example is the work of Moura et al. in Ref. 75 where the IFO
distribution is measured in the spontaneous mode-locking regime of a multimode
Q-switched Nd:YAG laser. The authors rightly assert that the observed
phenomenon is quite distinct from what investigated in random lasers
characterized by incoherently oscillating modes. As they display the
$P(\mathcal{C})$, calculated via (23), (24) and (26), vs pumping the lasing
threshold is identified as the sharp transition from a phase of a single
gaussian peaked $P(\mathcal{C})$ to a clean bimodal $P(\mathcal{C})$.

Since in Ref 75 the ensemble of spectra from which the $P(\mathcal{C})$ is
computed is very instructively displayed, it is, therefore, possible to see
that above threshold only two types of spectra are present: a broad, flat
fluorescent one and a narrow, spiky lasing one. Once again, we face
bistability. This is expected in the theory transition between model-locking
lasers (ordered or slightly random) as reported, for an instance, by the
dotted line in figure 6.

From this latter study we also report a interesting phenomenon. As pumping is
large enough so that all shots are lasing, strong fluctuations appear to be
there in the (narrow) spectra and the $P(\mathcal{C})$ appears to have a more
complicated shape, though always basically bimodal, in which overlaps between
the side peaks have a finite probability to occur. It would be interesting to
understand whether this is due to experimental noise correlated with the
strong pumping or it is the onset of a glassy phase as, for instance, in Fig.
6 for $0.5<R_{J}<1$.

####  3.3 Evidence of RSB in other different RLs and nonlinear waves

A completely different class of RLs showing RSB is furnished by erbium-based
random fiber lasers (Er-RFL) and widely investigated by A. Gomes and
coworkers. A first demonstration in given in ref. 76, where the authors employ
a 30-cm-long Er-RFL, pumped by a continous wave source. The authors discuss
and show that this kind of RFL is a multimode laser with more than 200
longitudinal modes. They analyse emission intensity fluctuation over 1500
acquisition spectra (replicas) for different pumping power and measure the IFO
from Eq. (23). In fig. 13 the IFO distribution $P(q)$ is depicted together
with the parameter $q_{\rm max}$, at which $P(q)$ assumes its global maximum.
555We warn the reader that the symbol $q$, rather than $\mathcal{C}$, is used
here and in the following to denote the intensity fluctuation overlap. It is
demonstrated in fig. 13-e that both sharp line narrowing and $q_{\rm max}$
growth from $0$ to $1$ occur at the same identical input power interval, that
is the laser threshold. This confirms the findings in Ref. 65.

Figure 13:  a-d) IFO distribution $P(q)$ obtained from experimental data at
different pump powers (normalized with respect to the threshold power $P_{\rm
th}$). e) $q_{\rm max}$ (circles) versus normalized pump power, together with
the FWHM (triangles) for comparison. The inset shows the results for pump
powers up to 12$P_{\rm th}$. Reproduced from Ref. 76 with permission from
American Physical Society.

#####  3.3.1 RSB and Lévy flight

Transition from paramagnetic to glassy states in random lasers has been also
analyzed in correspondence to the transition from diffusive to super diffusive
(or Lévy flight) regimes of light propagation. Strong emission fluctuations in
RLs have been described in the past by using the diffusive model of light and
by successfully identifying different statistical regimes for disordered
lasers [77, 78]. Recently, the strong emission fluctuations in RLs have been
analyzed both in the frameworks of Lévy statistics and spin glass theory.
Interesting similarities and differences have been demonstrated.

A first comparison between the two statistical physics properties is reported
by Gomes et al. in Ref. 79. In this work the physical origin of the possible
correspondence between the Lévy flight statistics of emission intensity and
the photonic RSB glassy transition in RLs is both theoretically and
experimentally investigated. The RL consists of crystalline powders of
Nd3+-doped YBO3 (Nd:YBO) and pumped by a pulsed laser. The authors for the
first time explain the comparison between the RSB transition to the photonic
glassy phase and the changes in the statistics of intensity fluctuations in
RLs within the same theoretical framework based on the Langevin equations
describing the evolution of the mode amplitudes. While such theoretical
approach has been introduced and widely used by the spin-glass community [42,
43, 70, 44], only recently it has been considered for the statistical
distribution of RL intensities [80]. Starting from the Langevin equation the
probability density function of emission intensity is obtained. The steady
state solution of such equation gives the Lévy-like distribution of
intensities. The authors show that, for a given disorder strength, by
increasing pumping rate the statistics of emission intensities shifts
progressively from an initial Gaussian to a Lévy-like and, then, again to a
Gaussian regime, as predicted by the diffusive model of light propagation in
random media. The IFO distribution calculated on the same experimental data
show a transition from the spontaneous emission-paramagnetic to the RL glassy
behaviour and a recurrence to the paramagnetic regime at higher pumping,
similarly to the Lévy statistics results. Once again, this critical behavior
is interpreted as a RSB transition, whereas, looking both at the strict
bimodal shape of the IFO distribution and at the series of spectra displayed
in the work, the transition is probably a first order phase transition with
coexistence of fluorescence and (random) laser phases. That is, bistability.

The novelty of these findings lies at higher excitation pulse energy well
above the threshold, where fluctuations inside the single spectrum decline
considerably with the consequent restoration of the Gaussian diffusive regime
for the intensities, but the RL becomes more and more disordered from the
point of view of mode coupling and the IFO distribution appears to become non-
trivial, hinting the possible onset of a glassy random laser. In this article
the authors do not give any explanation of the existence of a strict causal
link between the self-averaged Gaussian regime of intensity fluctuations above
the threshold and the observed suppression of the glassy phase, leaving the
subject to further studies. The link between the onset of the Lévy statistical
regime of intensity fluctuations and the emergence of the RL regime has been
reported also in a one dimensional Er-RFL [81].

A different result is shown in ref. 72, where the authors conclude that the RL
transition and the Lévy regime onset may not have a clear causal relation.
They claim that the former is strongly related to the threshold and less on
the magnitude of the fluctuations, while the latter emerges for large
fluctuations and only under strict conditions of the ratio between gain and
scattering properties of the material. As a demonstration they report the
clear experimental result where the Lévy regime is suppressed while a RSB
transition is present. A further study on the matter is probably required,
also including the role of bistability in non-glassy random lasers.

#####  3.3.2 RSB and Turbulence

In 2018 Gonzales et al. employ an erbium random fiber laser to demonstrate for
the first time the coexistence of turbulence-like and spin-glass-like behavior
from the same set of measurements [82].

For the theoretical analysis they introduce the photonic Pearson correlation
coefficient that can be considered as a generalized time dependent expression
of the spin-glass overlap parameter. If $\tau$ is the time of data
acquisition, the time dependent IFO between replicas $\alpha$ and $\beta$ is
defined as

$Q_{\alpha\beta}(\tau)=\frac{\sum_{k}\Delta_{k}^{(\alpha)}(\tau)\Delta_{k}^{(\beta)}(\tau)}{\sqrt{\sum_{k}\left(\Delta_{k}^{(\alpha)}(\tau)\right)^{2}}\sqrt{\sum_{k}\left(\Delta_{k}^{(\beta)}(\tau)\right)^{2}}},$
(27)

with $k$ being the wavelength index in the emission spectra, $\alpha$ and
$\beta$ single shot spectra indexes (replicas). Besides the definition (24),
for $\tau=0$, i.e.,

$\Delta_{k}^{(\alpha)}(0)\equiv I_{k}^{(\rm a)}(0)-\bar{I}_{k}(0),$

when $\tau>0$ the time-dependent spectral fluctuation is defined as

$\Delta_{k}^{(\alpha)}(\tau)\equiv
I_{k}^{(\alpha)}(\tau)-I_{k}^{(\alpha)}(0)-\left[\bar{I}_{k}(\tau)-\bar{I}_{k}(0)\right].$

With these notations Eq.(27) quantifies the emission intensity fluctuations in
time and it is sensitive to both liquid-glass transition and fluid dynamics
phenomena such as turbulence. It is demonstrated that above threshold the
distribution of the Pearson coefficient for short times is centered around
$Q\sim 0$, leading to the unimodal behaviour of $P(Q)$ in the turbulent-like
state. While always above threshold but at large time scales it assumes a
distribution resembling the profile typical of a RSB IFO distributions. In
this regime the intermittency vanishes and a crossover to the non-turbulent
behavior takes place.

The authors state that, since the overlap parameter considers all separation
times between spectra, the statistical weight of the replica overlaps with
long times dominates over the short time series. As a consequence,
$P(Q(\tau))$ actually appears qualitatively similar to the overlap
distribution that characterizes the RSB spin glass phase.

#####  3.3.3 RSB maps

Very recently RSB theory has been applied to obtain real time maps of the
laser activity in a heterogeneous RL [83]. The random lasers are made of
polymeric ribbon-like and highly porous fibers with evident RL action from
separated micrometric domains that alternatively switch on and off by tuning
the pumping light intensity. This novel effect is visualised by building for
the first time replica symmetry breaking maps of the emitting fibers with
micrometric spatial resolution. The overlap parameter is calculated directly
from the images. 100 single shot frames are recorded at fixed experimental
condition and each one is a replica. Four examples are illustrated in fig. 14,
where the variation of the spatial emission from shot to shot is evident. In
this work the measurable quantities corresponding to spins are the intensities
of the pixels recorded by the camera. The authors calculate the overlap
between these observables that are the spatial (transverse) modes while
previously only temporal (longitudinal) modes were considered [65].

Figure 14:  Four fluorescence images taken in identical experimental
situations and regarded as replicas. Reproduced from Ref. 83 with permission
from ACS Publications.

Each frame of 1024$\times$1376 pixels is divided into squares of 16$\times$16,
named macropixels, and the overlap $q^{j}_{\alpha,\beta}$ for each macropixel
$j$ is calculated as:

$q^{(j)}_{\alpha\beta}=\frac{\sum_{x_{k},y_{k}}\Delta^{(\alpha,j)}(x_{k},y_{k})\
\Delta^{(\beta,j)}(x_{k},y_{k})}{\sqrt{\sum_{x_{k},y_{k}}\left(\Delta^{(\alpha,j)}(x_{k},y_{k})\right)^{2}}\sqrt{\sum_{x_{k},y_{k}}\left(\Delta^{(\beta,j)}(x_{k},y_{k})\right)^{2}}}.$
(28)

where $\alpha$ and $\beta$ are replica indexes, $j$ is the macropixel index,
and $x_{k}$ and $y_{k}$ are pixel indexes inside one macropixel running from 1
to 16. In eq. (28)

$\Delta^{(\alpha,j)}(x_{k},y_{k})\equiv
I^{(\alpha,j)}(x_{k},y_{k})-{\overline{I^{(j)}}}(x_{k},y_{k}),$

with $I^{(\alpha,j)}(x_{k},y_{k})$ being the intensity at the pixel with
coordinate $(x_{k},y_{k})$ of the $j^{\rm th}$ macropixel for the replica
$\alpha$ and where

${\overline{I^{(j)}}}(x_{k},y_{k})\equiv\frac{1}{N_{s}}\sum_{\alpha=1}^{N_{s}}I^{(\alpha,j)}(x_{k},y_{k})$

is the average intensity over $N_{s}$ shots for each pixel. With this
procedure, from the images one has access to the sets of all
$N_{s}(N_{s}-1)/2$ values $q^{(j)}$ of $q^{(j)}_{\alpha\beta}$ and they
determine the distributions $P(q^{(j)})$ for all 64$\times$86 macropixels $j$.
This is done for different pumping energy. Following the previous work, the
authors calculate $q_{\rm max}$ for all macropixels and obtain replica
symmetry breaking maps at various pumping. The results are shown in fig. 15.
Domains with $q_{\rm max}>0.5$ are laser-ON and those with $q_{\rm max}<0.5$
are laser-OFF.

Figure 15:  Replica symmetry breaking maps. Maps of the $q_{\rm max}$values
calculated for each macropixel at different input energies. ON and OFF
indicate the activation status of the RL emission.

In this work the mapping of $q_{\rm max}$ allows for the first time the
visualisation of heterogeneous RL with switching and variable activity. This
procedure is proposed as a robust tool to identify the presence and the
spatial extension of RL activity from fluorescence images.

#####  3.3.4 RSB in nonlinear waves

In 2017 Pierangeli et al. [84] report the observation of the breaking of
replica symmetry in nonlinear optical propagation. They investigate intensity
profiles of a laser beam transmitted from a photorefractive disordered slab
waveguide. The nonlinearity of the material is tuned and strong fluctuations
of light above a certain threshold are evidenced, as in the intensity map of
fig. 16. Differently from previous works, the replica overlap is calculated
over the spatial autocorrelation of the intensities $I(x)$, and its
distribution undergoes a clear transition into a non-trivial distribution as
the nonlinearity exceeds a threshold value. Results are shown in fig. 16.
These findings demonstrate that nonlinear propagation can manifest features
typical of spin-glasses thanks to the coexistence and interplay of the two
main ingredients: disorder and nonlinearity.

Figure 16:  Transmitted intensity profiles along one direction $I(x)$ by
increasing nonlinearity. At high nonlinearity RSB regime of strong
fluctuations is evidenced. The corresponding distribution $P(q)$ in the
paramagnetic (left) and glassy (right) regimes are depicted.

###  4 Conclusions and Outlook

In this chapter we have considered the problem of the experimental measure of
the functional order parameter for a multiequilibria phase in complex
disordered systems, representing the transition to a replica symmetry broken
thermodynamic phase. After a flash introduction of the Parisi overlap
distribution we briefly discussed how the outcomes of the RSB theory have soon
cast light on peculiar results in early experiments on spin-glass
susceptibility. We have, then, been very rapidly reviewing the attempt to
reconstruct the equilibrium overlap distribution through measurements of the
off-equilibrium fluctuation-dissipation ratio in systems satisfying stochastic
stability and we did our best to give a state-of-the-art report of the
experiments tried so far along that approach. Finally, we have diffusively
discussed, both theoretically, numerically and experimentally, the approach to
the problem by means of complex random photonic systems, the random lasers,
and reported the first measurements of the Parisi order parameter through the
measurements of light intensity spectral fluctuations and the acquisition of
overlaps between fluctuations of different real replicas of the systems.

The study of nonlinear photonics systems by means of statistical mechanics of
disordered systems is an inspiring example of a constructive sinergy leading
to advances in both fields. The idea that random lasers might be described as
disordered systems and treated with the tools of statistical physics in order
to fully unveil and understand their complex behavior [42, 43, 70, 44] has
brought to the development of a spin-glass model[58, 45] whose RSB order
parameter could finally be accessible to experimental measurements.[59, 65,
76, 79, 12]

Furthermore, such sinergy has stimulated several new studies on different open
issues in photonics in random media. To mention a few, the statistical physics
approach has been applied in relation to the onset of a power-law (Lévy-like)
distribution of the emission intensity in random laser,[77, 78, 80, 79, 72,
81] to the characterization of turbulence in photonic systems,[82] to the
spatial distribution of interacting modes [26, 83] and the self-substained
nature of mode-locking in random lasers [44, 26] and has lead to the study of
glassiness in nonlinear photonic systems other than random lasers [84].
Several rather interesting issues remain open in this line of research
combining photonics and statistical physics, including the study of a first
order critical behavior with phase coexistence and its relationship to the
possible occurrence of bistability in random lasers, or the measurement and
control of mode phases whose impact would have important consequences in the
making and the technological use of random lasers.

### Acknowledgements

The authors thank Fabrizio Antenucci, Miguel Ibañez-Berganza, Giacomo
Gradenigo, Claudio Maggi, Alessia Marruzzo, Jacopo Niedda and Giorgio Parisi
for very many useful scientific exchanges.

We acknowledge the support from the European Research Council (ERC) under the
European Union’s Horizon 2020 Research and Innovation Program, Project
LoTGlasSy (Grant Agreement No. 694925), the support of LazioInnova - Regione
Lazio under the program Gruppi di ricerca 2020 \- POR FESR Lazio 2014-2020,
Project NanoProbe (Application code A0375-2020-36761).

## References

  * 1. G. Parisi, A sequence of approximated solutions to the S-K model for spin glasses, _J. Phys. A: Math. Gen._ 13, L115, (1980).
  * 2. G. Parisi, The order parameter for spin glasses: a function on the interval 0-1, _J. Phys. A: Math. Gen._ 13, 1101, (1980).
  * 3. M. Mézard, G. Parisi, and M. A. Virasoro, SK model: The replica solution without replicas, _Europhysics Letters (EPL)_. 1(2), 77–82 (jan, 1986).  10.1209/0295-5075/1/2/006.  URL https://doi.org/10.1209/0295-5075/1/2/006.
  * 4. M. Mézard, G. Parisi, and M. Virasoro, _Spin Glass Theory and Beyond_. (World Scientific (Singapore), 1987).
  * 5. M. Mézard and G. Parisi, The Bethe lattice spin glass revisited, _Eur. Phys. J. B_. 20, 217, (2001).
  * 6. D. S. Wiersma, M. P. Vanalbada, and A. Lagendijk, Random laser, _Nature_. 373, 203, (1995).
  * 7. H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, Random laser action in semiconductor powder, _Phys. Rev. Lett._ 82(11), 2278, (1999).
  * 8. H. Cao, Lasing in random media, _Waves in Random Media and Complex Media_. 13(3), R1–R39, (2003).
  * 9. D. S. Wiersma, The physics and applications of random lasers, _Nature Physics_. 4, 359, (2008).
  * 10. J. Andreasen, A. A. Asatryan, L. C. Botten, B. A. Byrne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. E. Türeci, and C. Vanneste, Modes of random lasers, _Adv. Optics and Photonics_. 3, 88–127, (2011).
  * 11. I. Viola, L. Leuzzi, C. Conti, and N. Ghofraniha, _Basic Physics and Recent Developments of Organic Random Lasers_ , In _Organic Lasers_.  Pan Stanford, (2018).
  * 12. A. S. Gomes, A. L. Moura, C. B. de Araújo, and E. P. Raposo, Recent advances and applications of random lasers and random fiber lasers, _Progress in Quantum Electronics_. 78, 100343, (2021).  ISSN 0079-6727.  https://doi.org/10.1016/j.pquantelec.2021.100343.  URL https://www.sciencedirect.com/science/article/pii/S0079672721000288.
  * 13. S. Nagata, P. Keesom, and H. Harrison, Low-dc-field susceptibility of cumn spin glass, _Phys. Rev. B_. 19, 1633, (1979).
  * 14. S. Franz, M. Mézard, and G. P. L. Parisi, Measuring Equilibrium Properties in Aging Systems, _Phys. Rev. Lett._ 81, 1758, (1998).
  * 15. D. Hérisson and M. Ocio, Fluctuation-dissipation ratio of a spin glass in the aging regime, _Phys. Rev. Lett._ 88, 257202 (Jun, 2002).  10.1103/PhysRevLett.88.257202.  URL https://link.aps.org/doi/10.1103/PhysRevLett.88.257202.
  * 16. D. Hérisson and M. Ocio, Off-equilibrium fluctuation-dissipation relation in a spin glass, _Eur. Phys. J. B_. 40, 283–294, (2004).  URL https://doi.org/10.1140/epjb/e2004-00278-6.
  * 17. C. Maggi, R. Di Leonardo, J. C. Dyre, and G. Ruocco, Generalized fluctuation-dissipation relation and effective temperature in off-equilibrium colloids, _Phys. Rev. B_. 81, 104201, (2010).  10.1103/PhysRevB.81.104201.  URL https://link.aps.org/doi/10.1103/PhysRevB.81.104201.
  * 18. H. Oukris and N. Israeloff, Nanoscale non-equilibrium dynamics and the fluctuation–dissipation relation in an ageing polymer glass, _Nature Phys_. 6, 135–138, (2010).  URL https://doi.org/10.1038/nphys1482.
  * 19. M. Baity-Jesi, E. Calore, A. Cruz, L. A. Fernandez, J. M. Gil-NarviÃ³n, A. Gordillo-Guerrero, D. IÃ±iguez, A. Maiorano, E. Marinari, V. Martin-Mayor, J. Monforte-Garcia, A. M. Sudupe, D. Navarro, G. Parisi, S. Perez-Gaviro, F. Ricci-Tersenghi, J. J. Ruiz-Lorenzo, S. F. Schifano, B. Seoane, A. TarancÃ³n, R. Tripiccione, and D. Yllanes, A statics-dynamics equivalence through the fluctuation–dissipation ratio provides a window into the spin-glass phase from nonequilibrium measurements, _Proceedings of the National Academy of Sciences_. 114(8), 1838–1843, (2017).  10.1073/pnas.1621242114.  URL https://www.pnas.org/doi/abs/10.1073/pnas.1621242114.
  * 20. H. Cao, Y. G. Zhao, H. C. Ong, S. T. Ho, J. Y. Dai, J. Y. Wu, and R. P. H. Chang, Ultraviolet lasing in resonators formed by scattering in semiconductor polycrystalline films, _Appl. Phys. Lett._ 73(25), 3656, (1998).
  * 21. H. Cao, J. Y. Xu, S. H. Chang, and S. T. Ho, Transition from amplified spontaneous emission to laser action in strongly scattering media, _Phys. Rev. E_. 61, 1985, (2000).
  * 22. M. Anni, S. Lattante, T. Stomeo, R. Cingolani, G. Gigli, G. Barbarella, and L. Favaretto, Modes interaction and light transport in bidimensional organic random lasers in the weak scattering limit, _Phys. Rev. B_. 70, 195216, (2004).
  * 23. K. L. van der Molen, R. W. Tjerkstra, A. P. Mosk, and A. Lagendijk, Spatial extent of random laser modes, _Phys. Rev. Lett._ 98, 143901, (2007).
  * 24. A. Tulek, R. C. Polson, and Z. V. Vardeny, Naturally occurring resonators in random lasing of $\pi$-conjugated polymer films, _Nature Phys._ 6, 303 – 310, (2010).
  * 25. V. Folli, A. Puglisi, L. Leuzzi, and C. Conti, Shaken granular lasers, _Phys. Rev. Lett._ 108, 248002, (2012).
  * 26. F. Antenucci, G. Lerario, B. S. Fernandéz, L. De Marco, M. De Giorgi, D. Ballarini, D. Sanvitto, and L. Leuzzi, Demonstration of self-starting nonlinear mode locking in random lasers, _Phys. Rev. Lett._ 126, 173901 (Apr, 2021).  10.1103/PhysRevLett.126.173901.  URL https://link.aps.org/doi/10.1103/PhysRevLett.126.173901.
  * 27. Y. Eliezer, S. Mahler, A. A. Friesem, H. Cao, and N. Davidson, Controlling nonlinear interaction in a many-mode laser by tuning disorder, _Phys. Rev. Lett._ 128, 143901 (Apr, 2022).  10.1103/PhysRevLett.128.143901.  URL https://link.aps.org/doi/10.1103/PhysRevLett.128.143901.
  * 28. B. Redding, A. A. Choma, and H. Cao, Speckle-free laser imaging using random laser illumination, _Nature Photon._ 6, 355–359, (2012).
  * 29. M. Barredo-Zuriarrain, I. Iparraguirre, J. Fernández, J. Azkargorta, and R. Balda, Speckle-free near-infrared imaging using a nd3+ random laser, _Laser Physics Letters_. 14(10), 106201 (sep, 2017).  10.1088/1612-202x/aa7874.  URL https://doi.org/10.1088\%2F1612-202x\%2Faa7874.
  * 30. V. Folli, N. Ghofraniha, A. Puglisi, L. Leuzzi, and C. Conti, Time-resolved dynamics of granular matter by random laser emission, _Scientific Rep._ 3, 2251, (2013).
  * 31. E. Ignesti, F. Tommasi, L. Fini, F. Martelli, N. Azzali, and S. Cavalieri, A new class of optical sensors: a random laser based device., _Sci. Rep._ p. 35225, (2016).
  * 32. Y. Xu, L. Zhang, S. Gao, P. Lu, S. Mihailov, and X. Bao, Highly sensitive fiber random-grating-based random laser sensor for ultrasound detection, _Opt. Lett._ 42(7), 1353–1356 (Apr, 2017).  10.1364/OL.42.001353.  URL http://ol.osa.org/abstract.cfm?URI=ol-42-7-1353.
  * 33. R. C. Polson and Z. V. Vardeny, Random lasing in human tissues, _Appl. Phys. Lett._ 85, 1289–1291, (2004).
  * 34. Q. Song, S. Xiao, Z. Xu, J. Liu, X. Sun, V. Drachev, V. M. Shalaev, O. Akkus, and Y. L. Kim, Random lasing in bone tissue, _Opt. Lett._ 35(9), 1425–1427 (May, 2010).  10.1364/OL.35.001425.  URL http://ol.osa.org/abstract.cfm?URI=ol-35-9-1425.
  * 35. F. Lahoz, I. R. Martín, M. Urgellés, J. Marrero-Alonso, R. Marín, C. J. Saavedra, A. Boto, and M. Díaz, Random laser in biological tissues impregnated with a fluorescent anticancer drug, _Laser Physics Letters_. 12(4), 045805 (mar, 2015).  10.1088/1612-2011/12/4/045805.  URL https://doi.org/10.1088%2F1612-2011%2F12%2F4%2F045805.
  * 36. Y. Wang, Z. Duan, Z. Qiu, P. Zhang, J. Wu, D. Zhang, and T. Xiang, Random lasing in human tissues embedded with organic dyes for cancer diagnosis, _Sci. Rep._ 7, 8385, (2017).
  * 37. D. T. W. Lin, Y. C. Hu, and C. Cheng, The optimization of the thermal response on the zno flexible pyroelectric film temperature sensor, _IEEE Sensors Journal_. 12(2), 397–403 (Feb, 2012).  ISSN 2379-9153.  10.1109/JSEN.2011.2161666.
  * 38. Y.-M. Liao, Y.-C. Lai, P. Perumal, W.-C. Liao, C.-Y. Chang, C.-S. Liao, S.-Y. Lin, and Y.-F. Chen, Highly stretchable label-like random laser on universal substrates, _Adv. Mat. Technol._ 1, 1600068, (2016).
  * 39. S. K. Turitsyn, S. A. Babin, A. E. El-Taher, P. Harper, D. V. Churkin, S. I. Kablukov, J. D. Ania-Castanon, V. Karalekas, and E. V. Podivilov, Random distributed feedback fibre laser, _Nat Photon_. 4, 231–235, (2010).
  * 40. M. Leonetti, C. Conti, and C. Lopez, The mode-locking transition of random lasers, _Nat. Photon._ 5, 615, (2015).
  * 41. Q. Baudouin, N. Mercadier, V. Guarrera, W. Guerin, and R. Kaiser, A cold-atom random laser, _Nat. Phys._ 9, 357–360, (2013).
  * 42. L. Angelani, C. Conti, G. Ruocco, and F. Zamponi, Glassy behavior of light, _Phys. Rev. Lett._ 96, 065702, (2006).
  * 43. L. Angelani, C. Conti, G. Ruocco, and F. Zamponi, Glassy behavior of light in random lasers, _Phys. Rev. B_. 74, 104207, (2006).
  * 44. C. Conti and L. Leuzzi, Complexity of waves in nonlinear disordered media, _Phys. Rev. B_. 83, 134204, (2011).
  * 45. F. Antenucci, A. Crisanti, and L. Leuzzi, Complex spherical 2+4 spin glass: A model for nonlinear optics in random media, _Phys. Rev. A_. 91, 053816, (2015).
  * 46. F. Antenucci, _Statistical physics of wave interactions_. (Springer, 2016).
  * 47. F. Antenucci, A. Crisanti, M. Ibáñez-Berganza, A. Marruzzo, and L. Leuzzi, Statistical mechanics models for multimode lasers and random lasers, _Philosophical Magazine_. 96(7-9), 704–731, (2016).  10.1080/14786435.2016.1145359.
  * 48. E. T. Jaynes and F. W. Cummings, Comparison of quantum and semiclassical radiation theories with application to the beam maser, _Proceedings of the IEEE_. 51, 89, (1963).
  * 49. V. Eremeev, S. E. Skipetrov, and M. Orszag, Quantum theory of a two-mode open-cavity laser, _Phys. Rev. A_. 84, 023816 (Aug, 2011).  10.1103/PhysRevA.84.023816.  URL https://link.aps.org/doi/10.1103/PhysRevA.84.023816.
  * 50. A. Marruzzo.  _STATISTICAL MECHANICS OF CONTINUOUS SPIN MODELS AND APPLICATIONS TO NONLINEAR OPTICS IN DISORDERED MEDIA_.  PhD thesis, Scuola di Dottorato Vito Volterra, Dottorato di Ricerca in Fisica, Università Sapienza di Roma, Piazzale Aldo Mor 5, 00185, Rome, Italy, (2015).
  * 51. A. Gordon and B. Fischer, Phase transition theory of many-mode ordering and pulse formation in lasers, _Phys. Rev. Lett._ 89, 103901 (Aug, 2002).  10.1103/PhysRevLett.89.103901.  URL https://link.aps.org/doi/10.1103/PhysRevLett.89.103901.
  * 52. A. Crisanti and H. Sommers, The spherical $p$-spin interaction spin-glass model - the statics, _Z. Phys. B_. 87, 341, (1992).
  * 53. A. Crisanti and L. Leuzzi, Spherical $2+p$ spin-glass model: An exactly solvable model for glass to spin-glass transition, _Phys. Rev. Lett._ 93, 217203, (2004).
  * 54. A. Crisanti and L. Leuzzi, Spherical 2+p spin-glass model: An analytically solvable model with a glass-to-glass transition, _Phys. Rev. B_. 73, 014412, (2006).
  * 55. A. Crisanti and L. Leuzzi, Equilibrium dynamics of spin-glass systems, _Phys. Rev. B_. 75, 144301, (2007).
  * 56. A. Crisanti and L. Leuzzi, Exactly solvable spin-glass models with ferromagnetic couplings: The spherical multi-p-spin model in a self-induced field, _Nucl. Phys. B_. 870, 176–204, (2013).
  * 57. C. J. Chen, P. K. Wai, and C. R. Menyuk, Stability of passively mode-locked fiber lasers with fast saturable absorption, _Opt. Lett._ 19, 198, (1994).
  * 58. F. Antenucci, C. Conti, A. Crisanti, and L. Leuzzi, General phase diagram of multimodal ordered and disordered lasers in closed and open cavities, _Phys. Rev. Lett._ 114, 043901, (2015).
  * 59. F. Antenucci, A. Crisanti, and L. Leuzzi, The glassy random laser: replica symmetry breaking in the intensity fluctuations of emission spectra, _Sci. Rep._ 5, 16792, (2015).
  * 60. A. Marruzzo, P. Tyagi, F. Antenucci, A. Pagnani, and L. Leuzzi, Improved pseudolikelihood regularization and decimation methods on non-linearly interacting systems with continuous variables, _SciPost Phys._ 5, (2018).
  * 61. A. Marruzzo and L. Leuzzi, Nonlinear xy and p-clock models on sparse random graphs: Mode-locking transition of localized waves, _Phys. Rev. B_. 91, 054201, (2015).
  * 62. A. Marruzzo and L. Leuzzi, Multi-body quenched disordered $xy$ and $p$-clock models on random graphs, _Phys. Rev. B_. 93, 094206, (2016).
  * 63. G. Gradenigo, L. Leuzzi, J. Niedda, and G. Parisi.  Universality class of the glassy random laser.  in preparation, (2022).
  * 64. G. Gradenigo, F. Antenucci, and L. Leuzzi, Glassiness and lack of equipartition in random lasers: The common roots of ergodicity breaking in disordered and nonlinear systems, _Phys. Rev. Research_. 2, 023399 (Jun, 2020).  10.1103/PhysRevResearch.2.023399.  URL https://link.aps.org/doi/10.1103/PhysRevResearch.2.023399.
  * 65. N. Ghofraniha, I. Viola, F. Dimaria, G. Barbarella, G. Gigli, L. Leuzzi, and C. Conti, Experimental evidence of replica symmetry breaking in random lasers, _Nat. Comm._ 6, 6058, (2015).
  * 66. N. Ghofraniha, I. Viola, F. D. Maria, G. Barbarella, G. Gigli, and C. Conti, Random laser from engineered nanostructures obtained by surface tension driven lithograph, _Laser and Photon. Rev._ 7(3), 432, (2013).
  * 67. M. Anni, S. Lattante, R. Cingolani, G. Gigli, G. Barbarella, and L. Favaretto, Far-field emission and feedback origin of random lasing in oligothiopene dioxide neat films, _Appl. Phys. Lett._ 83, 2754, (2003).
  * 68. D. Pisignano, M. Anni, G. Gigli, R. Cingolani, M. Zavelani-Rossi, G. Lanzani, G. Barbarella, and L. Favaretto, Amplified spontaneous emission and efficient tunable laser emission from a substituted thiophene-based oligomer, _Appl. Phys. Lett._ 81(19), 3534–3536, (2002).
  * 69. S. Mujumdar, V. Türck, R. Torre, and D. S. Wiersma, Chaotic behavior of a random laser with static disorder, _Phys. Rev. A_. 76, 033807, (2007).
  * 70. L. Leuzzi, C. Conti, V. Folli, L. Angelani, and G. Ruocco, Phase diagram and complexity of mode-locked lasers: From order to disorder, _Phys. Rev. Lett._ 102, 083901, (2009).
  * 71. P. I. R. Pincheira, A. F. Silva, S. I. Fewo, S. J. M. C. no, A. L. Moura, E. P. Raposo, A. S. L. Gomes, and C. B. de Araújo, Observation of photonic paramagnetic to spin-glass transition in a specially designed tio2 particle-based dye-colloidal random laser, _Opt. Lett._ 41(15), 3459–3462, (2016).
  * 72. F. Tommasi, E. Ignesti, S. Lepri, and S. Cavalieri, Robustness of replica symmetry breaking phenomenology in random laser, _Sci. Rep._ 6, 37113, (2016).
  * 73. W.-K. Chen, H.-W. Hsieh, C.-R. Hwang, and Y.-C. Sheu, Disorder chaos in the spherical mean-field model, _J. Stat. Phys._ 160, 417–429, (2015).
  * 74. S. Basak, A. Blanco, and C. López, Large fluctuations at the lasing threshold of solid- and liquid-state dye lasers, _Sci. Rep._ 6, 32134, (2016).
  * 75. A. L. Moura, P. I. R. Pincheira, A. S. Reyna, E. P. Raposo, A. S. L. Gomes, and C. B. de Araújo, Replica symmetry breaking in the photonic ferromagneticlike spontaneous mode-locking phase of a multimode nd:yag laser, _Phys. Rev. Lett._ 119, 163902, (2017).
  * 76. A. S. L. Gomes, B. C. Lima, P. I. R. Pincheira, A. L. Moura, M. Gagné, E. P. Raposo, C. B. de Araújo, and R. Kashyap, Glassy behavior in a one-dimensional continuous-wave erbium-doped random fiber laser, _Phys. Rev. A_. 94, 011801, (2016).
  * 77. S. Lepri, S. Cavalieri, G. Oppo, and D. Wiersma, Statistical regimes of random laser fluctuations, _Phys. Rev. A_. 75, 063820, (2007).
  * 78. S. Lepri, Fluctuations in a diffusive medium with gain, _Phys. Rev. Lett._ 110, 230603, (2013).
  * 79. A. S. L. Gomes, E. P. Raposo, A. L. Moura, S. I. Fewo, P. I. R. Pincheira, V. Jerez, L. J. Q. Maia, and C. B. de Araújo, Observation of lévy distribution and replica symmetry breaking in random lasers from a single set of measurements, _Sci. Rep._ 6, 27987, (2016).
  * 80. E. P. Raposo and A. S. L. Gomes, Analytical solution for the lèvy-like steady-state distribution of intensities in random lasers, _Phys. Rev. A_. 91, 043827, (2015).
  * 81. B. C. Lima, A. S. L. Gomes, P. I. R. Pincheira, A. L. Moura, M. Gagné, E. P. Raposo, C. B. de Araújo, and R. Kashyap, Observation of lévy statistics in one-dimensional erbium-based random fiber laser, _J. Opt. Soc. Am. B_. 34(2), 293–299, (2017).
  * 82. I. R. R. González, E. P. Raposo, A. M. Macêdo, L. de S. Menezes, , and A. S. Gomes, Coexistence of turbulence-like and glassy behaviours in a photonic system, _Sci. Rep._ 8, 17046, (2018).
  * 83. L. M. Massaro, S. Gentilini, A. Portone, A. Camposeo, D. Pisignano, C. Conti, and N. Ghofraniha, Heterogeneous random laser with switching activity visualized by replica symmetry breaking maps, _ACS Photonics_. 8(1), 376–383, (2021).
  * 84. D. Pierangeli, A. Tavani, F. D. Mei, A. J. Agranat, C. Conti, and E. DelRe, Observation of replica symmetry breaking in disordered nonlinear wave propagation, _Nat. Comm._ 8, 1501, (2017).

## Index