Universality class of the glassy random laser Jacopo Niedda1 2Giacomo Gradenigo3 4 2Luca Leuzzi2 1and Giorgio Parisi1 2 5 6 1Dipartimento di Fisica Universit a di Roma Sapienza Piazzale A. Moro 2 I-00185 Roma Italy

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Universality class of the glassy random laser

Jacopo Niedda,1, 2 Giacomo Gradenigo,3, 4, 2 Luca Leuzzi,2, 1, ∗and Giorgio Parisi1, 2, 5, 6

1Dipartimento di Fisica, Universit`a di Roma “Sapienza”, Piazzale A. Moro 2, I-00185, Roma, Italy

2NANOTEC CNR, Soft and Living Matter Lab, Roma, Piazzale A. Moro 2, I-00185, Roma, Italy

3Gran Sasso Science Institute, Viale F. Crispi 7, 67100 L?Aquila, Italy

4INFN-Laboratori Nazionali del Gran Sasso, Via G. Acitelli 22, 67100 Assergi (AQ), Italy

5INFN, Sezione di Roma-1, P.le A. Moro 5, 00185, Rome, Italy

6Accademia Nazionale dei Lincei, Palazzo Corsini - Via della Lungara, 10, I-00165, Roma, Italy

By means of enhanced Monte Carlo numerical simulations parallelized on GPUs we study the

critical properties of the spin-glass-like model for the mode-locked glassy random laser, a 4-spin

model with complex spins with a global spherical constraint and quenched random interactions.

Implementing two diﬀerent boundary conditions for the mode frequencies we identify the critical

points and the critical indices of the random lasing phase transition with ﬁnite size scaling techniques.

The outcome of the scaling analysis is that the mode-locked random laser universality class is

compatible with a mean-ﬁeld one, though diﬀerent from the mean-ﬁeld class of the Random Energy

Model and of the glassy random laser in the narrow band approximation, that is, the fully connected

version of the present model. The low temperature (high pumping) phase is ﬁnally characterized

by means of the overlap distribution and evidence for the onset of replica symmetry breaking in the

lasing regime is provided.

I. INTRODUCTION

When light propagates through a random medium,

scattering reduces information about whatever lies across

the medium and the electromagnetic ﬁeld, composed by

many interfering wave modes, provides a complicated

emission pattern as light undergoes multiple scattering.

If enough power is pumped into the medium multiple

scattering may support the population inversion of atoms

and molecules above some optical gap, yielding a random

laser [1–14]. Random lasers are made of an optically ac-

tive medium and randomly placed scatterers (sometimes

both in one [2]). The ﬁrst provides the gain, the lat-

ter provides 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 con-

struction and rigid optical alignment, have a low cost,

undirectional emissions, high operational ﬂexibility and

give rise to a number of promising applications in the

ﬁeld of speckle-free imaging [15, 16], granular matter

[9, 10], remote sensing [13, 17, 18], medical diagnostics

and biomedical imaging [13, 19–22], optical ampliﬁcation

and optoelectronic devices [13, 23, 24].

Random lasers may show multiple sub-nanometer

spectral peaks above a pump threshold [2], as well as

smoother, though always disordered, emission spectra.

Depending on the material, and its optical and scattering

properties, random spectral ﬂuctuations between diﬀer-

ent 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 [10, 12, 25–27],

depending on material compounds and experimental se-

tups. Random lasers can be built in very diﬀerent ways,

∗luca.leuzzi@cnr.it

can be both solid or liquid, can be 2D or 3D, the optically

active material can be conﬁned or spread all over the vol-

ume. Moreover, random lasers are, usually, open systems

where light can propagate in any direction rather than os-

cillating between well speciﬁc boundaries (mirrors) as in

standard lasers and the emission acquisition can only be

directional rather than on the whole solid angle. Finally,

also the scattering strength and the pumping conditions

may aﬀect the emission.

In the last years experiments on a certain class of ran-

dom lasers provided evidence of particularly non-trivial

correlations between the shot-to-shot ﬂuctuations of the

emission spectra. We will refer to those as glassy ran-

dom lasers [13, 28–32]. These special correlations are

predicted by a theory based on statistical mechanics of

complex disordered systems [33–35]. Indeed, it has been

shown that these ﬂuctuations are compatible with an or-

ganization of mode conﬁgurations in clusters of states,

similar to the one occurring for complex disordered sys-

tems displaying multiequilibria, as the spin glasses. Such

a correspondence has been analytically explained prov-

ing the equivalence between the distribution of the Inten-

sity Fluctuation Overlaps (IFO) and the distribution of

the overlap between states, the so-called Parisi overlap,

the order parameter of the glass transition [36]. Though

the analytical proof assumes narrow-band spectra, such

that all modes - within their line widths - can be consid-

ered at the same frequency [33, 37, 38], numerical simu-

lations have provided evidence that the onset of nontriv-

ial distributions of IFO and Parisi overlap distributions

occur at the same (critical) temperature also in realis-

tic models for random multimode lasers [39]. In these

models, the four-waves non-linear mixing between elec-

tromagnetic ﬁeld modes is controlled by a deterministic

selection rule depending on modes frequencies, termed

mode-locking. In mode-locked lasers interactions are pos-

sible only for the quadruplets of modes whose frequencies

arXiv:2210.04362v3 [cond-mat.dis-nn] 24 Feb 2023

ωksatisfy the condition

|ωk1−ωk2+ωk3−ωk4|< γ, (1)

with γbeing the typical line-width of the modes. We

will refer to Eq. (1) as Frequency Matching Condition

(FMC). In standard mode-locked lasers such selection

rule is implemented by ad hoc nonlinear devices (e.g., sat-

urable absorbers for passive mode-locking [40]) that are

not there in random lasers. As hypothesized in [41] and

recently experimentally demonstrated in [12], though, in

random lasers mode-locking occurs as a self-starting phe-

nomenon. We call an interaction network built on the

mode-locking selection rule in Eq. (1) a Mode-Locked

(ML) graph.

From the point of view of statistical mechanics of com-

plex disordered systems, random lasers represent, so far,

the only physical system where the relevant degrees of

freedom, namely the complex amplitudes of the light

modes, naturally form a dense interaction network of

the kind for which replica symmetry breaking mean-ﬁeld

theory [42] is proved to work, as in high dimension spin-

glasses or structural glasses made of hard spheres [43].

It is not by chance that random lasers are, so far, the

only complex disordered system providing experimental

evidence of a continuous replica symmetry-breaking pat-

tern [13, 28–32]. Actually, mean-ﬁeld theory for an inﬁ-

nite number of replica symmetry breakings has rigorously

been derived [44, 45] only for fully connected systems, in-

cluding the random laser model in the narrow-band ap-

proximation [33, 38]. Using the cavity method it is, then,

possible to compute a replica symmetry breaking (RSB)

phase also in systems with sparse interactions[46] (the

Viana-Bray model, for instance [47–49]). Still, the cor-

rect mean-ﬁeld theory which describes ML random lasers

has yet to be found, due to some peculiarities of the in-

teraction between light modes that will be detailed in the

following.

In this work we resort to Monte Carlo numerical simu-

lations of the dynamics of a leading model for multimode

random lasers, the Mode-Locked (ML) 4-phasor model

[33, 34, 38, 50].

Even if the phenomenology of the model is quite rich

already in the narrow bandwidth approximation, going

beyond the fully-connected case is necessary to achieve

a realistic description of random lasers in the spin-glass

theoretical framework. If Nis the number of modes, the

FMC leads to O(N) dilution in the interaction graph: the

total number of interactions, which is of order O(N4) in

the complete graph, is, thus, reduced to O(N3) in the

diluted graph [51].

Therefore, as far as the the interaction graph is con-

cerned, the ML 4-phasor model places itself in an inter-

mediate position between the complete and the sparse

graph, the latter being the case where the number of

couplings per variable does not scale with Nin the ther-

modynamic limit. The analytical solution of a spin-glass

model in such an intermediate regime of dilution is a

very hard problem to address, since standard mean-ﬁeld

techniques such as RSB theory, [52, 53], do not straight-

forwardly apply and the cavity method for sparse [48] or

diluted dense networks [54] does not allow to devise close

equations for global order parameters and provide a fully

explicit solution. Eventually, to the best of our knowl-

edge, no spin-glass model has been solved exactly out of

the fully connected or the sparse case. Hence, one needs

to perform numerical simulations in order to investigate

the physics of the model.

The ordered version of the ML 4-phasor model has

been extensively studied through numerical simulations

in [55, 56], where the essential consequences of the FMC

on the topology of the interaction graph have been inves-

tigated. In particular, the dilution induced by the FMC

Eq. (1) has been compared with a random dilution of

the same order, revealing important diﬀerences between

the two cases. The random diluted graph has a homo-

geneous topology and its phenomenology is compatible

with the mean-ﬁeld solution [37, 57]. On the other hand

the inhomogeneities induced by the FMC lead to a graph

characterized by a correlated topology and its behaviour

signiﬁcantly diﬀers from the homogeneous mean-ﬁeld so-

lution because of the onset of phase waves, at least at all

simulated N. Already in the ordered case, thus, the ML

model might display very strong ﬁnite size eﬀects. The

more so when quenched disordered couplings are consid-

ered.

Large sizes are hard to simulate because the mode vari-

ables are continuous (complex) numbers and because the

total number of interactions grows like N3with the num-

ber Nof modes. Because of these eﬀects it has not been

possible so far to identify the universality class of the

modes. By looking at the speciﬁc heat behaviour, it has

been observed [39] that for a O(N) dilution having a ran-

dom homogeneneous or a deterministic topology for the

same model makes a great diﬀerence in terms of inter-

polation of the critical properties in the thermodynamic

limit. A random homogeneous O(N) dilution of the fully

connected network allows to see, already at relatively

small sizes, a glass transition of the mean-ﬁeld kind in

the same universality class of the Random Energy Model

(REM), which is the reference mean-ﬁeld model for disor-

dered systems with non-linear interactions. On the other

hand the deterministic dilution yields apparently a dif-

ferent result.

To unravel such possible diﬀerence here we carefully in-

vestigate the universality class of the ML 4-phasor model,

providing simulations of systems of large enough sizes,

large statistics and, above all, introducing a trick to dras-

tically reduce ﬁnite size eﬀects.

After a description of the model in Section II, in Sec-

tion III we explain the strategy used to reduce the ﬁnite

size eﬀects due to the heterogeneous FMC dilution. In

Section IV we present a simple argument to get the expo-

nent for the ﬁnite-size scaling (FSS) regime of the speciﬁc

heat in the REM and generalize it deriving boundaries for

the critical exponents of a generic mean-ﬁeld universality

class. We, then, compare this prediction to the speciﬁc

heat behaviour in the equilibrium numerical simulations

and, through FSS analysis, we assess that the scaling of

the speciﬁc heat near the glass transition temperature is

compatible with a mean-ﬁeld theory, which is the main

outcome of the present work. Eventually, in Section V

we present the behaviour of the overlap probability distri-

bution upon lowering the temperature across the random

lasing transition, that turns out to be a glass transition.

The trick used to reduce ﬁnite-size eﬀects turns out to be

useful in identifying more clearly signatures of glassiness.

II. THE MODE-LOCKED 4-PHASOR MODEL

The ML 4-phasor model has its roots in the quantum

theory of the electromagnetic ﬁeld and matter interaction

in an open system. A full account of the derivation of the

classical stochastic dynamics from the quantum many-

body dynamics of light coupled with matter can be found

in [34].

The main point is that by considering laser media

where the characteristic time of atomic pump and loss are

much shorter than the lifetimes of the resonator modes,

the atomic variables, i.e., matter ﬁelds, can be removed

obtaining non-linear equations for the electromagnetic

ﬁeld alone.

The stochastic diﬀerential equation for the time evolu-

tion of the modes akreads as

dak1

dt =X

k|FMC(k)

g(2)

k1k2ak2

k|FMC(k)

g(4)

k1k2k3k4ak2ak3ak4+ηk1(t),(2)

where the expression of the sum over the indices ksatisfy-

ing a FMC, like (1), will be soon clariﬁed in Eq. (4). The

dynamic variables are ak(t) = Ak(t)eiφk(t), the complex

amplitudes of the light modes comprised by the discrete

spectrum of the electromagnetic ﬁeld

E(r, t) =

k=1

ak(t)eiωktEk(r) + c.c. (3)

where Ek(r) is the space-dependent wavefunction of the

mode with frequency ωk. The noise is taken as a white

noise hηk(t)i= 0, hηj(t)ηk(t0)i= 2T δjkδ(t−t0), as we

will later discuss. The amplitudes ak(t) are the remnant

of the original creation and annihilation operators of the

electromagnetic ﬁeld quantization, which have been de-

graded to complex numbers in the semiclassical approx-

imation. By slow amplitude mode it is meant that the

time scale of the amplitude dynamics is larger than the

time scale deﬁned by the frequency of the mode, i.e.,

ω−1

k. Therefore, in the slow amplitude approximation

the phases eiωktcan be averaged out, which, in Fourier

space, taking the Fourier transform of Eq. (3), implies

that ak(t)'ak(t, ω)δ(ω−ωk). Lasing modes are slow

amplitude modes by deﬁnition, since they are character-

ized by a very narrow linewidth γaround their frequency

ωk. The time average of the fast oscillations eiωktleads

to the sum termed FMC in the equation (2). The general

expression for 2n-body interactions reads as

FMC(k) : |ωk1−ωk2+··· +ωk2n−1−ωk2n|.γ, (4)

of which Eq. (1) is the case n= 2. The FMC acts as a

selection rule on the modes participating in the interac-

tions.

The linear terms in Eq. (2) yield diﬀerent contribu-

tions possibly depending on cavity gain and losses and

atom-ﬁeld interaction inside the disordered medium. The

latter expression is the most relevant one in the dynam-

ics:

g(2)

k1k2∝√ωk1ωk2

{x,y,z}

αβ ZV

drαβ(r)Eα

k1(r)Eβ

k2(r),(5)

where (r) is the dielectric permittivity tensor and the

integral is extended over the entire volume Vof the

medium. In particular, the diagonal elements of g(2)

k1k2

represent the net gain curve of the medium (i.e., the gain

reduced by the losses), which plays an important role

mainly below the lasing threshold.

The non-linear couplings g(4)

k1k2k3k4are given by the spa-

tial overlap of the electromagnetic mode wavefunctions

modulated by a non-linear optical susceptibility χ(3)

g(4)

k1k2k3k4∝

j=1

√ωkj

{x,y,z}

αβγδ ZV

drχ(3)

αβγδ ({ωk};r)

×Eα

k1(r)Eβ

k2(r)Eγ

k3(r)Eδ

k4(r),(6)

where, again, the integral is over the whole volume of the

medium. In general, both the linear and the non-linear

couplings are complex numbers and can be written as

[50]

g(2)

k1k2=Gk1k2+iDk1k2,(7)

g(4)

k1k2k3k4= Γk1k2k3k4+i∆k1k2k3k4.(8)

In the standard laser case the linear couplings are di-

agonal and the non-linear ones can be safely considered

as constant: in this case, Dkis the group velocity dis-

persion coeﬃcient and ∆ is the self-phase modulation

coeﬃcient, responsible for the Kerr eﬀect [40]. In the

purely dissipative limit [34, 37], i.e. Dk1k2Gk1k2and

∆k1k2k3k4Γk1k2k3k4, which in standard laser theory

corresponds to neglect the group velocity dispersion and

the Kerr eﬀect, the dynamics of Eq. (2) becomes a po-

tential diﬀerential equation

dak1

dt =−∂H[a]

∂ak1(t)+ηk1(t),

with a Hamiltonian function given by

H=−X

k|FMC(k)

Gk1k2ak1ak2

−X

k|FMC(k)

Γk1k2k3k4ak1ak2ak3ak4+ c.c..(9)

In principle, the noise is correlated, i.e. hηk1ηk2i 6=

δk1k2. However, it can be diagonalized by changing ba-

sis of dynamic variables: the decomposition of resonator

modes into a slow amplitude basis is not unique [58] and

one can use this freedom to build a basis in which the

noise has no correlations. The diagonalization of the

noise can be done at the cost of having non-diagonal lin-

ear interactions, which is not a real complication in the

random laser case, since linear couplings already have

oﬀ-diagonal contributions accounting for the openness of

the cavity.

The laser dynamics is brought to stationarity by gain

saturation, a phenomenon connected to the fact that,

as the power is kept constant, the emitting atoms pe-

riodically decade in lower states saturating the gain of

the laser. In the same way the dynamics induced by

the Hamiltonian Eq. (9) eventually reaches a stationary

regime, when a constraint on the total energy contained

in the system is added. This argument was ﬁrst pro-

posed for standard multimode lasers in Refs [37, 57].

In fact, lasers are strongly out of equilibrium: energy

is constantly pumped into the system in order to keep

population inversion and stimulated emission, and in the

case of cavityless systems also compensate the leakages.

However, a stationary regime can be described as if the

system is at equilibrium with an eﬀective thermal bath,

whose eﬀective temperature (a “photonic” temperature)

accounts both for the amount of energy E=N stored

into the system because of the external pumping and for

the spontaneous emission rate. The latter is proportional

to the kinetic energy of the atoms, e. g., to the heat bath

temperature T. Eventually, the external parameter driv-

ing the lasing transition turns out to be [33, 34, 50]

Tphotonic =T

2.(10)

One can also introduce the pumping rate P[41] as the

inverse of the square root of this ratio:

P2=2

T=1

Tphotonic.

In order to mathematically model the gain saturation,

an overall spherical constraint can be imposed on the

amplitudes ﬁxing the total optical intensity in the system

k=1 |ak|2=N. (11)

The precise value of the couplings in the Hamiltonian

Eq. (9) requires the knowledge of the spatial wavefunc-

tions of the modes, see Eqs. (5) and (6), which is not

available in random lasers, since they are characterized

by a complicated spatial structure of the modes. If, as

it apparently occurs in glassy random lasers, modes are

spatially extended to wide regions of the optically ac-

tive compound, each mode is nonlinearly interacting with

very many others. We will implement such an “extended

modes approximation” [59] in our model, where the only

relevant factor in the mode-coupling is the FMC, rather

than spatial conﬁnement of light modes. In this case,

because of thermodynamic convergence, each coupling

coeﬃcient will be smaller and smaller as the number of

modes increases.

In principle, all couplings involving the same mode

will be correlated. However, because of the spatially

etherogeneous optical nonlinear susceptibility in (6) and

the fact that each coupling coeﬃcient vanishes as Nin-

creases, the role of correlation will be qualitatively neg-

ligible as far as the system displays enough modes. For

this reason, the couplings will be taken as independent

Gaussian random variables in the present work:

P(Jk1···kp) = 1

q2πσ2

exp (−J2

k1···kp

2σ2

p),(12)

with p= 2,4 and σ2

p∼N2−pto ensure the extensivity of

the Hamiltonian and where some rescaling of the modes

and coeﬃcients (g→J) has been performed [34, 41].

Eventually, the stationary properties of the system can

be described by a model whose Hamiltonian is

H[a] = H2[a] + H4[a],(13)

where

H2[a] = −X

k|FMC(k)

Jk1k2ak1ak2+ c.c.

H4[a] = −X

k|FMC(k)

Jk1k2k3k4ak1ak2ak3ak4+ c.c. (14)

As mentioned in section II when introducing the dissi-

pative limit we will consider the J’s as real parameters,

without loss of generality. The eﬀective distribution for

the phasor conﬁguration a={a1, . . . , aN}will, eventu-

ally, be

P[a]∝e−βH[a]δ N −

k=1 |ak|2!,(15)

where βis the inverse temperature.

III. FREQUENCY MATCHING CONDITION

WITHOUT EDGE-BAND MODES

The FMC Eq. (4) is the most peculiar aspect of the

ML 4-phasor model, since it deﬁnes the topology of the

interaction network. The full inclusion of the FMC in

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UniversalityclassoftheglassyrandomlaserJacopoNiedda,1,2GiacomoGradenigo,3,4,2LucaLeuzzi,2,1,andGiorgioParisi1,2,5,61DipartimentodiFisica,UniversitadiRoma\Sapienza",PiazzaleA.Moro2,I-00185,Roma,Italy2NANOTECCNR,SoftandLivingMatterLab,Roma,PiazzaleA.Moro2,I-00185,Roma,Italy3GranSassoScienceInstitute...

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Universality class of the glassy random laser Jacopo Niedda1 2Giacomo Gradenigo3 4 2Luca Leuzzi2 1and Giorgio Parisi1 2 5 6 1Dipartimento di Fisica Universit a di Roma Sapienza Piazzale A. Moro 2 I-00185 Roma Italy

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