Resonant activation in bistable semiconductor lasers
Stefano Lepri1, ∗ and Giovanni Giacomelli1
Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche,
via Madonna del Piano 10, I-50019 Sesto Fiorentino, Italy
(Dated: November 4, 2018)
We theoretically investigate the possibility of observing resonant activation in the hopping dynam-
ics of two-mode semiconductor lasers. We present a series of simulations of a rate-equations model
under random and periodic modulation of the bias current. In both cases, for an optimal choice of
the modulation time-scale, the hopping times between the stable lasing modes attain a minimum.
The simulation data are understood by means of an effective one-dimensional Langevin equation
with multiplicative fluctuations. Our conclusions apply to both Edge Emitting and Vertical Cavity
Lasers, thus opening the way to several experimental tests in such optical systems.
PACS numbers: 42.55.Px, 05.40.-a, 42.65.Sf
I. INTRODUCTION
It is currently established that stochastic fluctuations
may have a constructive role in enhancing the response
of nonlinear systems to an external coherent stimulus.
Relevant examples are the enhancement of the decay
time from a metastable state (noise–enhanced stability)
[1, 2], the synchronization with a weak periodic input sig-
nal (stochastic resonance) [3] or the regularizaton of the
response at an optimal noise intensity (coherence reso-
nance) [4].
Another instance is the phenomenon of resonant acti-
vation that was discovered by Doering and Gadoua [5].
They showed that the escape of an overdamped Brown-
ian particle over a fluctuating barrier can be enhanced
by suitably choosing the correlation time of barrier fluc-
tuations themselves. In other words, the escape time
from the potential well attains a minimum for an optimal
choice of such correlation time. Since its discovery, the
phenomenon received a considerable attention from the-
orists (see e.g. Refs. [6, 7, 8, 9, 10]). Detailed studies by
means of analog simulations have also been reported for
both Gaussian and dichotomous fluctuations [11]. More
recently, the phenomenon has been shown to occur also
for the case in which the barrier oscillates periodically
[12, 13].
To our knowledge, experimental evidences of resonant
activation were only given for a bistable electronic cir-
cuit [14] and, very recently, for a colloidal particle sub-
ject to a periodically–modulated optical potential [15].
It is therefore important to look for other setups where
the effect could be studied in detail. As a matter of
fact, multimode laser systems are good candidates to
investigate noise–activated dynamics like the switching
among modes induced by quantum fluctuations (sponta-
neous emission) [16]. In particular, semiconductor lasers
proved to be particularly versatile for detailed experi-
∗Electronic address: stefano.lepri@isc.cnr.it
mental investigations of modulation and noise-induced
phenomena like stochastic resonance [17, 18] and noise–
induced phase synchronization [19]. In those previous
studies, the resonance regimes are attained by a suit-
able random modulation of the bias current which can
be tuned in a well-controlled way. It is thus natural to
argue about the possibility of observing resonant activa-
tion with the same type of experimental setup.
In this paper, we theoretically demonstrate the phe-
nomenon of resonant activation in a generic rate–
equations model for a two-mode semiconductor laser un-
der modulation of the bias current. The basic ingredients
that act in the theoretical descriptions are a fluctuating
potential barrier and some activating noise. In the laser
system, the latter is basically provided by spontaneous
emission while current fluctuations, that appear addi-
tively into the rate equations, effectively act multiplica-
tively if a suitable separation of time scales holds [20]. In
a previous paper [21], we have explicitely demonstrated
such multiplicative–noise effects on the mode–hopping
dynamics. This was shown by a reduction to a bistable
one–dimensional potential system with both multiplica-
tive and additive stochastic forces. Several predictions
drawn from such a simplified model are in good agree-
ment with the experimental observations carried out for
a bulk, Edge-Emitting Laser (EEL) [21]. In the present
context, we will show that this reduced description is of
great help in the interpretation of simulation data.
The outline of the paper is the following. In Sec. II
we recall the model for a two-mode semiconductor laser.
In Sec. III we present the numerical simulation for two
physically distinct cases displaying resonant activation.
These results are discussed and interpreted by compar-
ing with the reduced one–dimensional Langevin model
mentioned above (Sec. IV). We draw our conclusions in
Sec. V.
II. RATE EQUATIONS
Our starting point is a stochastic rate-equation model
for a semiconductor laser that may operate in two longi-
http://arxiv.org/abs/0704.0206v1
mailto:stefano.lepri@isc.cnr.it
tudinal modes whose complex amplitudes are denoted by
E±. Both of them interact with a single carrier density
N that provides the necessary amplification. The two
modes have very similar linear gains, provided that their
wavelengths are almost equal and they are close to the
gain peak. Let J(t) denote the bias (injection) current,
the model can be written as [21]
Ė+ =
(1 + iα)g+ − 1
2DspN ξ+ (1a)
Ė− =
(1 + iα)g− − 1
2DspN ξ− (1b)
Ṅ = γ
J(t)−N − g+|E+|2 − g−|E−|2
where γ is carrier density relaxation rate, α is the
linewidth enhancement factor [22]. The modal gains read
N ± ε(N −Nc)
1 + s|E±|2 + c|E∓|2
, (2)
where ε determines the difference in differential gain
among the two modes while Nc defines the carrier
density where the unsaturated modal gains are equal.
The parameters s and c are respectively the self- and
cross-saturation coefficients. The ξ± are two indepen-
dent, complex white noise processes with zero mean
[〈ξ±(t)〉 = 0] and unit variance [〈ξi(t)ξ∗j (t′)〉 = δijδ(t−t′)]
that model spontaneous emission. The noise terms in
Eqs. (1a) and (1b) are gauged by the spontaneous emis-
sion coefficient Dsp.
All quantities are expressed in suitable dimensionless
units. In particular, time is normalized to the photons’
lifetime, which for semiconductor laser is typically of the
order of a few picoseconds or less (see e.g. [22, 23, 24])
A detailed analysis of the stationary solutions of
Eqs. (1) is reported in Ref. [25]. For a constant bias cur-
rent J(t) = J0 and Dsp = 0, Eqs. (1) admit four different
steady state solutions: the trivial one E± = 0, two single-
mode solutions — E+ 6= 0, E− = 0 and viceversa — and
a solution where both modes are lasing, E± 6= 0. For
Nc > 1, and c > s, there exist a finite interval of J0 val-
ues for which the two single–mode solutions coexist and
are stable while the E± 6= 0 is unstable (bistable region).
Here, for Dsp > 0 the laser performs stochastic mode-
hopping, with the total emitted intensity remains almost
constant while each mode switches on and off alternately
at random times. We point out that the emission in each
mode is nonvanishing even in the “off” state, as the av-
erage power spontaneously emitted in each mode at any
time is given by 4DspN [recall that Eqs. (1) are usually
interpreted in Itô sense [23]]. Observation of this be-
haviour has been reported in several experimental works
on EELs [26, 27, 28].
We remark that while Eqs. (1) aim at modeling EELs,
the results presented henceforth would apply also to po-
larization switching in Vertical Cavity Surface Emitting
Lasers (VCSELs). Indeed, experimental data [29] show
strong similarities between this phenomenon and the lon-
gitudinal mode dynamics. On the theoretical side, this
analogy is supported by the fact that the polarization dy-
namics in VCSELs is described by models that are math-
ematically similar to the one discussed here [30, 31, 32].
In the following, we will focus on the effect of the ex-
ternally imposed fluctuation/modulation of the injected
current. This situation is modeled by letting
J(t) = J0 + δJ(t) . (3)
The DC value J0 sets the working point and will be al-
ways chosen to be in the bistability region. We focus
on the case in which δJ is a Ornstein-Uhlenbeck process
with zero average 〈δJ(t)〉 = 0 and correlation time τ :
˙δJ = −
ξJ (4)
that means
〈δJ(t)δJ(0)〉 = DJ exp(−|t|/τ) . (5)
This choice is suitable to model a finite-bandwith noise
generator. Notice that τ and the variance of fluctuations
DJ = 〈δJ2〉 can be fixed independentely.
Another case of experimental interest that we will con-
sider is using the current modulation
δJ = A sinΩt (6)
To assess the nature of the stochastic process at hand,
it is important to introduce the relevant time scales. We
define first of all the switching or relaxation time TR as
the typical time for the emission to change from one mode
to the other. The main quantities we are interested in are
the Kramers or residence times T± defined as the average
times for which the emission occurs in each mode. In
semiconductor lasers T± are generally much larger than
TR. Typically, TR ∼ 1− 10ns while residence times may
range between 0.1 and 100 µs [29, 33]. The third time-
scale is of course given by the characteristic time of the
external driving, namely, τ and 2π/Ω respectively.
In the following, we will study how the hopping dy-
namics changes upon varying these latter parameters as
well as the strength of the perturbation.
III. NUMERICAL SIMULATIONS
In this Section we present the outcomes of a series of
numerical simulation of Eqs. (1). In Ref. [21] it was
observed that the sensitivity of each of the T± on the
imposed current fluctuations may be notably different
depending on the parameters’ choice. This is a typi-
cal signature of the multiplicative nature of the stochas-
tic process. In particular, one can argue [21] that such
“simmetry-breaking” effects mostly depend on the ratio
εσ/δ where
, δ =
. (7)
The parameter σ represents the gain saturation induced
by the total power in the laser, while δ describes the
reduction in gain saturation due to partitioning of the
power between the two modes.
The possibility of obtaining qualitatively different re-
sponses depending on the actual parameters corresponds
to the different experimental observations reported for
both EELs [21, 28, 33] and VCSELs [17, 29]. Those two
classes of lasers were indeed found to display markedly
different simmetry-breaking effects under current modu-
lation. To account for those features, we consider two
different sets of phenomenological parameters. For defi-
niteness, in both cases we fix ε = 0.1, s = 1.0, Nc = 1.1,
γ = 0.01 and change the values of c and Dsp (see Table
I). The first set (δ = 0.05) corresponds to the case in
which added modulation changes the hopping time scale
in an almost symmetric way. On the contrary, in the
second case (δ = 0.15) the asymmetry effect of the noise
is stronger [21]. We can thus consider the two as rep-
resentative of the VCSELs and EELs case respectively.
The value of J0 has been empirically adjusted to yield
T+ ≃ T− ≡ Ts and an almost symmetric distribution of
intensities in absence of modulation. The actual values
are about 10% above the laser threshold. The sponta-
neous emission coefficient Dsp has been chosen to yield a
value of the residence times of the same order of magni-
tude of the experimental ones.
In the following, we decide to set α = 0 which is ap-
propriate for our EEL model where the phase dynamics
is not relevant [21]. This choice may however not be fully
justified for the VCSEL case. In this respect, the sim-
ulations presented below are representative of the VC-
SEL dynamics only in a qualitative sense. Nonetheless,
it should be pointed out that a 1D Langevin model in-
dependent of α describes also the VCSEL case [30, 31].
Since resonant activation is mainly due to the multiplica-
tive noise effect described by such equations [see Eq. (9)
below] we consider this as an indirect proof that phe-
nomenology we will report below should be observable
also in the VCSEL case.
The largest part of the simulations were performed
with Euler method with time steps 0.01-0.05 for times
in the range 107 − 108 time units depending on the val-
ues of τ and Ω. For comparison, some checks with Heun
method [34] have also been carried on. Within the sta-
tistical accuracy, the results are found to be insensitive
the the choice of the algorithm.
A. Stochastic modulation
Let us start illustrating the results in the case of
stochastic current modulation (Eq. (4)). In Fig. 1 and
TABLE I: The parameter values used in the two series of
simulations of Eqs. (1), the other values are given in the text.
c Dsp J0 δ σ
1.1 0.7 × 10−5 1.197 0.05 1.05
1.3 1.5 × 10−5 1.194 0.15 1.15
2 we report the measured dependence of the residence
times T± on the correlation time τ for the two parameter
sets given in Table I and different values of the noise vari-
ance DJ . In all cases, the curves display well-pronounced
minima at an optimal value of τ . This is the typical sig-
nature of resonant activation. The minima are almost
located between the relaxation time TR and the hopping
time Ts (marked by the vertical dashed lines). The val-
ues of TR reported in the figures have been estimated
from the reduced model discussed in the next Section,
see Eq. (14) below.
The effect manifest in a different way for the second
parameter set. In the case of Fig. 1 both times attain a
minimum, albeit with different values. On the contrary
the data of Fig. 2 show that one of the two times is hardly
affected from the external perturbation regardless of the
value of τ . In other terms, we can tune the current corre-
lation in such a way that emission along only one of the
two modes is strongly reduced (about a factor 10 in the
simulation discussed here).
B. Periodic modulation
Let us now turn to the case of sinusoidal current mod-
ulation (Eq. 6). In Fig. 3 and 4 we report the measured
dependence of the residence times T± on the frequency Ω
for the two parameter sets given in Table I and different
values of the amplitude A. For comparison with the pre-
vious case we choose A such that the RMS value of (6)
is roughly equal to the variance of (4), i.e. A ≃
2DJ .
As in the previous case, the curves display resonant
activation at an optimal value of Ω. For the second set
of parameters, one of the two hopping times is more re-
duced than the other (compare Fig. 4 with Fig. 2). It
should be also noticed that the data in Fig. 2 display
some statistical fluctuations while the curves for the pe-
riodic modulation are smoother.
IV. INSIGHTS FROM A REDUCED MODEL
In order to better understand the activation phe-
nomenon it is useful to reduce the five–dimensional dy-
namical system (1) to an effective one-dimensional sys-
tem. This has been accomplished in Ref. [21]. For com-
pleteness, we only recall here some basic steps of the
derivation. In the first place, we introduce the change of
Correlation time τ
=4  10
=4  10
=4  10
FIG. 1: (Color online) Simulations of the rate equations with
Ornstein-Uhlenbeck current fluctuations, parameter set with
c = 1.1 (see text and Table I): residence times T+ (squares)
and T− (circles) for increasing values of the current variance
DJ . The values of the relaxation time TR and the hopping
time Ts (in absence of modulation) are marked by the vertical
dashed lines.
coordinates
E+ = r cosφ exp iψ+, E− = r sinφ exp iψ− . (8)
In these new variables, r2 is the total power emitted by
the laser, and φ determines how this power is partitioned
among the two modes. The values φ = 0, π/2 correspond
to pure emission in mode + and − respectively. The
phases ψ± do not influence the evolution of the modal
amplitudes and carrier density and can be ignored.
In order to simplify the analysis, we assume that (i)
The difference between modal gains is very small, i.e.,
Nc >∼ 1, ε≪ 1, c >∼ s; (ii) the laser operates close enough
to threshold, so that r2 ≪ 1 and the saturation term is
small: in this limit, r and N decouple to leading order
from φ; (iii) r and N can be adiabatically eliminated and
(iv) only their fluctuations around the equilibrium val-
ues due to J are retained. This last assumption holds for
weak spontaneous noise and amounts to say that r and
Correlation time τ
=1  10
=5  10
=1  10
FIG. 2: (Color online) Simulations of the rate equations with
Ornstein-Uhlenbeck current fluctuations, parameter set with
c = 1.3 (see text and Table I): residence times T+ (squares)
and T− (circles) for increasing values of the current variance
N are stochastic processes given by nonlinear transfor-
mations of J (see Eqs. (16) in Ref. [21]). This requires
that J does not change too fast. For example, in the
case of the Orstein–Uhlenbeck process, Eq. (4), τ should
be larger than the relaxation time of the total intensity.
The validity of the above reduction has been carefully
checked against simulations of the complete model [21].
For the scope of the present work, we performed a fur-
ther check by comparing the spectrum of fluctuations of
r2 with the imposed one, Eq. (4). Indeed, the behaviour
is the same for τ > TR while for shorter τ some differ-
ences are detected. This means that the reduced descrip-
tion discussed below becomes less and less accurate. On
the other hand, in this regime spontaneous fluctuation
should dominate and this limitation become less relevant
for our purposes.
Altogether, the hopping dynamics is effectively one-
dimensional and is described by the slow variable φ. Its
Modulation period 2π/Ω
A=0.020
A=0.090
A=0.009
FIG. 3: (Color online) Simulations of the rate equations with
sinusoidal modulation of the current, parameter set with c =
1.1 (see text and Table I): residence times T+ (squares) and
T− (circles) for increasing values of modulation amplitude A.
evolution is ruled by the effective Langevin equation
φ̇ = −1
a cos 2φ+ b
sin 2φ +
tan 2φ
2Dφ ξφ (9)
where, together with (7) we have defined the new set of
parameters
(1 + σ)Nc − 1
1 + σ
(J − 1) (11)
1 + σ
(J − Js) (12)
(1 + σJ)2
(1 + σ)(J − 1)
Dsp . (13)
We remind in passing that the same equation (9) has
been derived by Willemsen et al. [30, 31] to describe po-
larization switches in VCSELs (see also Ref. [35] for a
similar reduction). The starting point of their derivation
Modulation period  2π/Ω
6 A=0.014
A=0.030
A=0.055
FIG. 4: (Color online) Simulations of the rate equations with
sinusoidal modulation of the current, parameter set with c =
1.3 (see text and Table I): residence times T+ (squares) and
T− (circles) for increasing values of modulation amplitude A.
is the San Miguel-Feng-Moloney model [36]. The physi-
cal meaning of the variable φ is different from here as it
represents the polarization angle of emitted light. This
supports the above claim that, upon a suitable reinterpre-
tation of variables and parameters, many of the results
presented henceforth may apply also to the dynamics of
VCSELs.
In absence of modulation (δJ = 0), Eq. (9) is bistable
in an interval of current values where it admits two stable
stationary solutions φ± and an unstable one φ0 (double-
well). This regime correspond to the bistability region of
model (1). Notice that for J0 = Js, b = 0 the hopping
between the two modes occurs at the same rate. The
above definitions allows an estimate of relaxation time TR
defined above. This is is the inverse of the curvature of
the potential in φ0. For J0 = Js this is straightforwardly
evaluated to be
(1 + σ)
δ(Js − 1)
For the two parameter sets given in Table I one finds
TR = 210 TR = 77.0, respectively. These are the values
emploied to draw the leftmost vertical lines in Figs. 1-4.
The effect of a time-dependent current is to make the
coefficients a, b and Dφ fluctuating. It can be shown [21]
that the effect onDφ can be recasted as a renormalization
of the intensity of the spontaneous-emission noise. How-
ever, for the parameters employed in the present work it
turns out that this correction is pretty small and will be
neglected henceforth by simply considering Dφ as con-
stant [38]. For simplicity, we also disregard the depen-
dence of Dφ on δJ in the drift term of Eq. (9). Under
those further simplifications the Langevin equation can
be rewritten as
φ̇ = −U ′(φ)− V ′(φ) δJ +
2Dφ ξφ (15)
where we have express the force term as derivatives of
the “potentials”
U(φ) = −
δ(J0 − 1)
16(1 + σ)
cos 4φ−
εσ(J0 − Js)
4(1 + σ)
cos 2φ
−Dφ ln sin 2φ (16)
V (φ) = − δ
16(1 + σ)
cos 4φ− εσ
4(1 + σ)
cos 2φ. (17)
Langevin equations of the form (15) with (4) have been
thoroughly studied in the literature (see e.g. [7, 8, 9, 10,
11] and references therein) as prototypical examples of
the phenomenon of activated escape over a fluctuating
barrier. In view of their non-Markovian nature, their
full analytical solution for arbitrary τ is not generally
feasible. Several approximate results can be provided in
some limits.
For an arbitrary choice of the parameters, V has a
different symmetry with respect to U meaning that the
effective amplitude of multiplicative noise is different
within the two potential wells. If this difference is large
enough, current fluctuation will remove the degeneracy
between the two stationary solutions. This is best seen
by computing the istantaneous potential barriers ∆U±(t)
close to the symmetry point J0 = Js . For weak noise
and δJ ≪ (Js − 1), they are given to first-order in δJ(t)
∆U±(t) ≃
8(1 + σ)
(Js − 1) +
δ ± 2εσ
8(1 + σ)
δJ(t) . (18)
Obviously, this last expression makes sense only when
the fluctuating term is sub-threshold i.e. whenever the
system is bistable. In the case of periodic modulation,
formula (18) allows estimating the range of amplitude
values for a sub-threshold driving
δ(Js − 1)
δ ± 2εσ
. (19)
Using this condition, along with the parameter values at
hand, we deduce that the cases displayed in lower panels
of Figs. 3 and 4 correspond to superthreshold driving.
However, while the minima are much more pronounced
than in the other panels, there is no qualitative difference
in the system response. In the case of stochastic modu-
lation, the same remark applies in a probabilistic sense
for the last panels of Figs. 1 and 2.
Altogether, the mode switching can be seen as an acti-
vated escape over fluctuating barriers given by Eq. (18).
The statistical properties of the latter process is con-
trolled by the current fluctuations. We now discuss the
properties of various regimes. For simplicity, we refer
to the case of stochastic modulations. Most of the re-
marks and formulas reported in the following Subsection
should apply also to the periodic case by replacing τ and
DJ with 2π/Ω and A
2/2 whenever appropriate.
A. Fast barrier fluctuations: τ < TR ≪ T±
As we already pointed out, in this regime the reduc-
tion to Eq. (15) is not justified. We may thus only expect
some qualitative insight on the behaviour of the rate-
equations. From a mathematical point of view, some
analytical approximations for equations like (15) are fea-
sible in this limit (see e.g. Ref. [8] for the stochastic
case). For our purposes, it is sufficient to note that in
this regime the effect of δJ is hardly detected for both
types of driving (see again Figs. 1-4). Note also that
working at DJ fixed means that for τ → 0 the fluctua-
tion become negligible.
B. Resonant activation: TR < τ ≪ T±
If TR < τ we are in the colored noise case. The prob-
lem is amenable of a kinetic description which amounts
to neglect intrawell motion and reduce to a rate model
describing the statistical transitions in terms of transition
rates. If we consider τ as a time scale of the external driv-
ing we can follow the terminology of Ref. [39] and refer
to this situation as the “semiadiabatic” limit of Eq. (15).
In this regime, the residence time is basically the short-
est escape time, which in turn correspond to the lowest
value of the barrier (the noise is approximatively con-
stant in the current range considered henceforth). For
the case of interest, δ < 2εσ we can use (18) to infer
that the minimal values of ∆U± should be attained for
δJ ∝ ∓
DJ respectively. This yields
T± ≃ Ts exp
−K 2εσ ± δ
1 + σ
where K is a suitable numerical constant. Notice that δ
controls the asymmetry level: if δ ≪ 2εσ the two resi-
dence times decrease at approximatively the same rate.
This prediction is verified in the simulations and also in
the experiment [21].
As a further argument in support of the above rea-
soning, we also evaluated the probability distributions
of the residence times obtained from the simulation of
the rate equations. In Fig. 5, we show two representa-
tive cumulative distributions. The data are well fitted
by a Poissonian P (T ) = 1 − exp(−T/T±) for both the
stochastic and periodic modulation cases. This confirms
that hopping occours preferentially when a given (mini-
mal) barrier occurs.
0 0.5 1 1.5
0 1 2 3 4
FIG. 5: (Color online) Cumulative distributions of the resi-
dence times in the resonant activation region, parameter set
with c = 1.3 (see text and Table I). Left panel: stochastic
modulation with DJ = 5×10
−4, τ = 1.638×103 . Right panel:
periodic modulation with A = 0.03 and period 1.286 × 104.
We report only the histograms for the times whose averages
are denoted by T+ in the text. Solid line is the cumulative
Poissonian distribution with the same average.
C. Slow barrier, frequent hops: TR ≪ T± ≪ τ
This corresponds to the adiabatic limit in which the
time scale of the external driving is slower than the intrin-
sic dynamics of the system [39]. To a first approximation
we can here treat current fluctuations in a parametric
way. Correction terms may be evaluated by means of
a suitable perturbation expansion in the small parame-
ter 1/τ [10]. If δJ is small enough for the expression
(18) to make sense, the escape time can be estimated as
the average of escape times over the distribution of bar-
rier fluctuations, i.e. 〈T±〉δJ . For the case of Eq. (4),
the variable δJ is Gaussian and we can use the identity
〈expβz〉 = exp(β2〈z2〉/2) to obtain [11]
T± ≃ Ts exp
[2(δ ± 2εσ)2
(1 + σ)2D2
. (21)
This reasoning implies that for large τ the residence times
should approach two different constant values. A closer
inspection of the graphs (in linear scale) reveals that this
is not fully compatible with the data of Fig.1 even for
the smallest value of DJ . In several cases, T± continue to
increase with τ and no convincing evidence of saturation
is observed. We note that the same type of behaviour
was already observed in the analog simulations data of
Ref. [11]. There, an increase of hopping times duration at
large τ was found. The Authors of Ref. [11] explained this
as an effect of a too large value of the noise fluctuation
forcing the system to jump roughly every τ . We argue
that the same explanation holds for our case. This is also
consistent with the fact that the exponential factors in
Eq. (21) evaluated with the simulation parameters turn
out to be much larger than unity.
V. CONCLUSIONS
In this paper, we have explored numerically and ana-
lytically the effects of external current fluctuations on the
mode-hopping dynamics in a model of a bistable semicon-
ductor laser. To the best of our knowledge, this setup
provides the first theoretical evidence of resonant activa-
tion in a laser system. As the phenomenon has hardly
received any experimental confirmation in optics, we be-
lieve that our study may open the way to future research
in this subfield.
The model we investigated is based on a rate-equation
description, where the bias current enters parametrically
into the evolution of the modal amplitudes. We consid-
ered, two kinds of current flutuations, namely, a stochas-
tic process ruled by an Orstein-Uhlenbeck statistics, and
a coherent, sinusoidal modulation. These choices are mo-
tivated by the aim of proposing a suitable setup for an
experimental verification of our results. Upon varying the
characteristic time-scale of the imposed fluctuations, we
have shown that the residence times attain a minimum
for a well-defined value, which is the typical signature
of resonant activation. The magnitude of the effect can
be different depending on the parameters of the model.
Moreover, the response of the system appears very much
similar for both periodic and random modulations.
The reduction of the rate equations to a one-
dimensional Langevin equation allowed us to recast the
problem as an activated escape over a fluctuating bar-
rier. To first approximation, the fluctuating barrier (mul-
tiplicative term) is mainly controlled by current modu-
lations while the spontaneous noise act as an additive
source. This simplified description has allowed us to
draw some predictions (e.g. the dependence of residence
times on noise strength) and to better understand the
role of the physical parameters. Given the generality of
the description, our results should apply to a broad class
of multimode lasers, including both Edge Emitting and
Vertical Cavity Lasers.
From an experimental point of view, driving the laser
in a orders-of-magnitude wide range of time-scales is
more feasible in the case of a sinusoidal modulation than
for a colored, high frequency noise. However, given the
evidence of a resonant activation phenomenon for such
modulation, our results indicate that it occurs almost for
the same parameters in the case of colored noise, provided
that the RMS of the modulations equals the amplitude
of the added noise. Thus, the phenomenon could be fully
exploited along those lines. Since the reported experi-
mental evidences of the phenomenon are so far scarce,
we hope that the present work could suggest a detailed
characterization in optical systems that allows for both
very precise measurements and careful control of param-
eters.
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  We theoretically investigate the possibility of observing resonant activation
in the hopping dynamics of two-mode semiconductor lasers. We present a series
of simulations of a rate-equations model under random and periodic modulation
of the bias current. In both cases, for an optimal choice of the modulation
time-scale, the hopping times between the stable lasing modes attain a minimum.
The simulation data are understood by means of an effective one-dimensional
Langevin equation with multiplicative fluctuations. Our conclusions apply to
both Edge Emitting and Vertical Cavity Lasers, thus opening the way to several
experimental tests in such optical systems.

Resonant activation in bistable semiconductor lasers
Stefano Lepri1, ∗ and Giovanni Giacomelli1
Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche,
via Madonna del Piano 10, I-50019 Sesto Fiorentino, Italy
(Dated: November 4, 2018)
We theoretically investigate the possibility of observing resonant activation in the hopping dynam-
ics of two-mode semiconductor lasers. We present a series of simulations of a rate-equations model
under random and periodic modulation of the bias current. In both cases, for an optimal choice of
the modulation time-scale, the hopping times between the stable lasing modes attain a minimum.
The simulation data are understood by means of an effective one-dimensional Langevin equation
with multiplicative fluctuations. Our conclusions apply to both Edge Emitting and Vertical Cavity
Lasers, thus opening the way to several experimental tests in such optical systems.
PACS numbers: 42.55.Px, 05.40.-a, 42.65.Sf
I. INTRODUCTION
It is currently established that stochastic fluctuations
may have a constructive role in enhancing the response
of nonlinear systems to an external coherent stimulus.
Relevant examples are the enhancement of the decay
time from a metastable state (noise–enhanced stability)
[1, 2], the synchronization with a weak periodic input sig-
nal (stochastic resonance) [3] or the regularizaton of the
response at an optimal noise intensity (coherence reso-
nance) [4].
Another instance is the phenomenon of resonant acti-
vation that was discovered by Doering and Gadoua [5].
They showed that the escape of an overdamped Brown-
ian particle over a fluctuating barrier can be enhanced
by suitably choosing the correlation time of barrier fluc-
tuations themselves. In other words, the escape time
from the potential well attains a minimum for an optimal
choice of such correlation time. Since its discovery, the
phenomenon received a considerable attention from the-
orists (see e.g. Refs. [6, 7, 8, 9, 10]). Detailed studies by
means of analog simulations have also been reported for
both Gaussian and dichotomous fluctuations [11]. More
recently, the phenomenon has been shown to occur also
for the case in which the barrier oscillates periodically
[12, 13].
To our knowledge, experimental evidences of resonant
activation were only given for a bistable electronic cir-
cuit [14] and, very recently, for a colloidal particle sub-
ject to a periodically–modulated optical potential [15].
It is therefore important to look for other setups where
the effect could be studied in detail. As a matter of
fact, multimode laser systems are good candidates to
investigate noise–activated dynamics like the switching
among modes induced by quantum fluctuations (sponta-
neous emission) [16]. In particular, semiconductor lasers
proved to be particularly versatile for detailed experi-
∗Electronic address: stefano.lepri@isc.cnr.it
mental investigations of modulation and noise-induced
phenomena like stochastic resonance [17, 18] and noise–
induced phase synchronization [19]. In those previous
studies, the resonance regimes are attained by a suit-
able random modulation of the bias current which can
be tuned in a well-controlled way. It is thus natural to
argue about the possibility of observing resonant activa-
tion with the same type of experimental setup.
In this paper, we theoretically demonstrate the phe-
nomenon of resonant activation in a generic rate–
equations model for a two-mode semiconductor laser un-
der modulation of the bias current. The basic ingredients
that act in the theoretical descriptions are a fluctuating
potential barrier and some activating noise. In the laser
system, the latter is basically provided by spontaneous
emission while current fluctuations, that appear addi-
tively into the rate equations, effectively act multiplica-
tively if a suitable separation of time scales holds [20]. In
a previous paper [21], we have explicitely demonstrated
such multiplicative–noise effects on the mode–hopping
dynamics. This was shown by a reduction to a bistable
one–dimensional potential system with both multiplica-
tive and additive stochastic forces. Several predictions
drawn from such a simplified model are in good agree-
ment with the experimental observations carried out for
a bulk, Edge-Emitting Laser (EEL) [21]. In the present
context, we will show that this reduced description is of
great help in the interpretation of simulation data.
The outline of the paper is the following. In Sec. II
we recall the model for a two-mode semiconductor laser.
In Sec. III we present the numerical simulation for two
physically distinct cases displaying resonant activation.
These results are discussed and interpreted by compar-
ing with the reduced one–dimensional Langevin model
mentioned above (Sec. IV). We draw our conclusions in
Sec. V.
II. RATE EQUATIONS
Our starting point is a stochastic rate-equation model
for a semiconductor laser that may operate in two longi-
http://arxiv.org/abs/0704.0206v1
mailto:stefano.lepri@isc.cnr.it
tudinal modes whose complex amplitudes are denoted by
E±. Both of them interact with a single carrier density
N that provides the necessary amplification. The two
modes have very similar linear gains, provided that their
wavelengths are almost equal and they are close to the
gain peak. Let J(t) denote the bias (injection) current,
the model can be written as [21]
Ė+ =
(1 + iα)g+ − 1
2DspN ξ+ (1a)
Ė− =
(1 + iα)g− − 1
2DspN ξ− (1b)
Ṅ = γ
J(t)−N − g+|E+|2 − g−|E−|2
where γ is carrier density relaxation rate, α is the
linewidth enhancement factor [22]. The modal gains read
N ± ε(N −Nc)
1 + s|E±|2 + c|E∓|2
, (2)
where ε determines the difference in differential gain
among the two modes while Nc defines the carrier
density where the unsaturated modal gains are equal.
The parameters s and c are respectively the self- and
cross-saturation coefficients. The ξ± are two indepen-
dent, complex white noise processes with zero mean
[〈ξ±(t)〉 = 0] and unit variance [〈ξi(t)ξ∗j (t′)〉 = δijδ(t−t′)]
that model spontaneous emission. The noise terms in
Eqs. (1a) and (1b) are gauged by the spontaneous emis-
sion coefficient Dsp.
All quantities are expressed in suitable dimensionless
units. In particular, time is normalized to the photons’
lifetime, which for semiconductor laser is typically of the
order of a few picoseconds or less (see e.g. [22, 23, 24])
A detailed analysis of the stationary solutions of
Eqs. (1) is reported in Ref. [25]. For a constant bias cur-
rent J(t) = J0 and Dsp = 0, Eqs. (1) admit four different
steady state solutions: the trivial one E± = 0, two single-
mode solutions — E+ 6= 0, E− = 0 and viceversa — and
a solution where both modes are lasing, E± 6= 0. For
Nc > 1, and c > s, there exist a finite interval of J0 val-
ues for which the two single–mode solutions coexist and
are stable while the E± 6= 0 is unstable (bistable region).
Here, for Dsp > 0 the laser performs stochastic mode-
hopping, with the total emitted intensity remains almost
constant while each mode switches on and off alternately
at random times. We point out that the emission in each
mode is nonvanishing even in the “off” state, as the av-
erage power spontaneously emitted in each mode at any
time is given by 4DspN [recall that Eqs. (1) are usually
interpreted in Itô sense [23]]. Observation of this be-
haviour has been reported in several experimental works
on EELs [26, 27, 28].
We remark that while Eqs. (1) aim at modeling EELs,
the results presented henceforth would apply also to po-
larization switching in Vertical Cavity Surface Emitting
Lasers (VCSELs). Indeed, experimental data [29] show
strong similarities between this phenomenon and the lon-
gitudinal mode dynamics. On the theoretical side, this
analogy is supported by the fact that the polarization dy-
namics in VCSELs is described by models that are math-
ematically similar to the one discussed here [30, 31, 32].
In the following, we will focus on the effect of the ex-
ternally imposed fluctuation/modulation of the injected
current. This situation is modeled by letting
J(t) = J0 + δJ(t) . (3)
The DC value J0 sets the working point and will be al-
ways chosen to be in the bistability region. We focus
on the case in which δJ is a Ornstein-Uhlenbeck process
with zero average 〈δJ(t)〉 = 0 and correlation time τ :
˙δJ = −
ξJ (4)
that means
〈δJ(t)δJ(0)〉 = DJ exp(−|t|/τ) . (5)
This choice is suitable to model a finite-bandwith noise
generator. Notice that τ and the variance of fluctuations
DJ = 〈δJ2〉 can be fixed independentely.
Another case of experimental interest that we will con-
sider is using the current modulation
δJ = A sinΩt (6)
To assess the nature of the stochastic process at hand,
it is important to introduce the relevant time scales. We
define first of all the switching or relaxation time TR as
the typical time for the emission to change from one mode
to the other. The main quantities we are interested in are
the Kramers or residence times T± defined as the average
times for which the emission occurs in each mode. In
semiconductor lasers T± are generally much larger than
TR. Typically, TR ∼ 1− 10ns while residence times may
range between 0.1 and 100 µs [29, 33]. The third time-
scale is of course given by the characteristic time of the
external driving, namely, τ and 2π/Ω respectively.
In the following, we will study how the hopping dy-
namics changes upon varying these latter parameters as
well as the strength of the perturbation.
III. NUMERICAL SIMULATIONS
In this Section we present the outcomes of a series of
numerical simulation of Eqs. (1). In Ref. [21] it was
observed that the sensitivity of each of the T± on the
imposed current fluctuations may be notably different
depending on the parameters’ choice. This is a typi-
cal signature of the multiplicative nature of the stochas-
tic process. In particular, one can argue [21] that such
“simmetry-breaking” effects mostly depend on the ratio
εσ/δ where
, δ =
. (7)
The parameter σ represents the gain saturation induced
by the total power in the laser, while δ describes the
reduction in gain saturation due to partitioning of the
power between the two modes.
The possibility of obtaining qualitatively different re-
sponses depending on the actual parameters corresponds
to the different experimental observations reported for
both EELs [21, 28, 33] and VCSELs [17, 29]. Those two
classes of lasers were indeed found to display markedly
different simmetry-breaking effects under current modu-
lation. To account for those features, we consider two
different sets of phenomenological parameters. For defi-
niteness, in both cases we fix ε = 0.1, s = 1.0, Nc = 1.1,
γ = 0.01 and change the values of c and Dsp (see Table
I). The first set (δ = 0.05) corresponds to the case in
which added modulation changes the hopping time scale
in an almost symmetric way. On the contrary, in the
second case (δ = 0.15) the asymmetry effect of the noise
is stronger [21]. We can thus consider the two as rep-
resentative of the VCSELs and EELs case respectively.
The value of J0 has been empirically adjusted to yield
T+ ≃ T− ≡ Ts and an almost symmetric distribution of
intensities in absence of modulation. The actual values
are about 10% above the laser threshold. The sponta-
neous emission coefficient Dsp has been chosen to yield a
value of the residence times of the same order of magni-
tude of the experimental ones.
In the following, we decide to set α = 0 which is ap-
propriate for our EEL model where the phase dynamics
is not relevant [21]. This choice may however not be fully
justified for the VCSEL case. In this respect, the sim-
ulations presented below are representative of the VC-
SEL dynamics only in a qualitative sense. Nonetheless,
it should be pointed out that a 1D Langevin model in-
dependent of α describes also the VCSEL case [30, 31].
Since resonant activation is mainly due to the multiplica-
tive noise effect described by such equations [see Eq. (9)
below] we consider this as an indirect proof that phe-
nomenology we will report below should be observable
also in the VCSEL case.
The largest part of the simulations were performed
with Euler method with time steps 0.01-0.05 for times
in the range 107 − 108 time units depending on the val-
ues of τ and Ω. For comparison, some checks with Heun
method [34] have also been carried on. Within the sta-
tistical accuracy, the results are found to be insensitive
the the choice of the algorithm.
A. Stochastic modulation
Let us start illustrating the results in the case of
stochastic current modulation (Eq. (4)). In Fig. 1 and
TABLE I: The parameter values used in the two series of
simulations of Eqs. (1), the other values are given in the text.
c Dsp J0 δ σ
1.1 0.7 × 10−5 1.197 0.05 1.05
1.3 1.5 × 10−5 1.194 0.15 1.15
2 we report the measured dependence of the residence
times T± on the correlation time τ for the two parameter
sets given in Table I and different values of the noise vari-
ance DJ . In all cases, the curves display well-pronounced
minima at an optimal value of τ . This is the typical sig-
nature of resonant activation. The minima are almost
located between the relaxation time TR and the hopping
time Ts (marked by the vertical dashed lines). The val-
ues of TR reported in the figures have been estimated
from the reduced model discussed in the next Section,
see Eq. (14) below.
The effect manifest in a different way for the second
parameter set. In the case of Fig. 1 both times attain a
minimum, albeit with different values. On the contrary
the data of Fig. 2 show that one of the two times is hardly
affected from the external perturbation regardless of the
value of τ . In other terms, we can tune the current corre-
lation in such a way that emission along only one of the
two modes is strongly reduced (about a factor 10 in the
simulation discussed here).
B. Periodic modulation
Let us now turn to the case of sinusoidal current mod-
ulation (Eq. 6). In Fig. 3 and 4 we report the measured
dependence of the residence times T± on the frequency Ω
for the two parameter sets given in Table I and different
values of the amplitude A. For comparison with the pre-
vious case we choose A such that the RMS value of (6)
is roughly equal to the variance of (4), i.e. A ≃
2DJ .
As in the previous case, the curves display resonant
activation at an optimal value of Ω. For the second set
of parameters, one of the two hopping times is more re-
duced than the other (compare Fig. 4 with Fig. 2). It
should be also noticed that the data in Fig. 2 display
some statistical fluctuations while the curves for the pe-
riodic modulation are smoother.
IV. INSIGHTS FROM A REDUCED MODEL
In order to better understand the activation phe-
nomenon it is useful to reduce the five–dimensional dy-
namical system (1) to an effective one-dimensional sys-
tem. This has been accomplished in Ref. [21]. For com-
pleteness, we only recall here some basic steps of the
derivation. In the first place, we introduce the change of
Correlation time τ
=4  10
=4  10
=4  10
FIG. 1: (Color online) Simulations of the rate equations with
Ornstein-Uhlenbeck current fluctuations, parameter set with
c = 1.1 (see text and Table I): residence times T+ (squares)
and T− (circles) for increasing values of the current variance
DJ . The values of the relaxation time TR and the hopping
time Ts (in absence of modulation) are marked by the vertical
dashed lines.
coordinates
E+ = r cosφ exp iψ+, E− = r sinφ exp iψ− . (8)
In these new variables, r2 is the total power emitted by
the laser, and φ determines how this power is partitioned
among the two modes. The values φ = 0, π/2 correspond
to pure emission in mode + and − respectively. The
phases ψ± do not influence the evolution of the modal
amplitudes and carrier density and can be ignored.
In order to simplify the analysis, we assume that (i)
The difference between modal gains is very small, i.e.,
Nc >∼ 1, ε≪ 1, c >∼ s; (ii) the laser operates close enough
to threshold, so that r2 ≪ 1 and the saturation term is
small: in this limit, r and N decouple to leading order
from φ; (iii) r and N can be adiabatically eliminated and
(iv) only their fluctuations around the equilibrium val-
ues due to J are retained. This last assumption holds for
weak spontaneous noise and amounts to say that r and
Correlation time τ
=1  10
=5  10
=1  10
FIG. 2: (Color online) Simulations of the rate equations with
Ornstein-Uhlenbeck current fluctuations, parameter set with
c = 1.3 (see text and Table I): residence times T+ (squares)
and T− (circles) for increasing values of the current variance
N are stochastic processes given by nonlinear transfor-
mations of J (see Eqs. (16) in Ref. [21]). This requires
that J does not change too fast. For example, in the
case of the Orstein–Uhlenbeck process, Eq. (4), τ should
be larger than the relaxation time of the total intensity.
The validity of the above reduction has been carefully
checked against simulations of the complete model [21].
For the scope of the present work, we performed a fur-
ther check by comparing the spectrum of fluctuations of
r2 with the imposed one, Eq. (4). Indeed, the behaviour
is the same for τ > TR while for shorter τ some differ-
ences are detected. This means that the reduced descrip-
tion discussed below becomes less and less accurate. On
the other hand, in this regime spontaneous fluctuation
should dominate and this limitation become less relevant
for our purposes.
Altogether, the hopping dynamics is effectively one-
dimensional and is described by the slow variable φ. Its
Modulation period 2π/Ω
A=0.020
A=0.090
A=0.009
FIG. 3: (Color online) Simulations of the rate equations with
sinusoidal modulation of the current, parameter set with c =
1.1 (see text and Table I): residence times T+ (squares) and
T− (circles) for increasing values of modulation amplitude A.
evolution is ruled by the effective Langevin equation
φ̇ = −1
a cos 2φ+ b
sin 2φ +
tan 2φ
2Dφ ξφ (9)
where, together with (7) we have defined the new set of
parameters
(1 + σ)Nc − 1
1 + σ
(J − 1) (11)
1 + σ
(J − Js) (12)
(1 + σJ)2
(1 + σ)(J − 1)
Dsp . (13)
We remind in passing that the same equation (9) has
been derived by Willemsen et al. [30, 31] to describe po-
larization switches in VCSELs (see also Ref. [35] for a
similar reduction). The starting point of their derivation
Modulation period  2π/Ω
6 A=0.014
A=0.030
A=0.055
FIG. 4: (Color online) Simulations of the rate equations with
sinusoidal modulation of the current, parameter set with c =
1.3 (see text and Table I): residence times T+ (squares) and
T− (circles) for increasing values of modulation amplitude A.
is the San Miguel-Feng-Moloney model [36]. The physi-
cal meaning of the variable φ is different from here as it
represents the polarization angle of emitted light. This
supports the above claim that, upon a suitable reinterpre-
tation of variables and parameters, many of the results
presented henceforth may apply also to the dynamics of
VCSELs.
In absence of modulation (δJ = 0), Eq. (9) is bistable
in an interval of current values where it admits two stable
stationary solutions φ± and an unstable one φ0 (double-
well). This regime correspond to the bistability region of
model (1). Notice that for J0 = Js, b = 0 the hopping
between the two modes occurs at the same rate. The
above definitions allows an estimate of relaxation time TR
defined above. This is is the inverse of the curvature of
the potential in φ0. For J0 = Js this is straightforwardly
evaluated to be
(1 + σ)
δ(Js − 1)
For the two parameter sets given in Table I one finds
TR = 210 TR = 77.0, respectively. These are the values
emploied to draw the leftmost vertical lines in Figs. 1-4.
The effect of a time-dependent current is to make the
coefficients a, b and Dφ fluctuating. It can be shown [21]
that the effect onDφ can be recasted as a renormalization
of the intensity of the spontaneous-emission noise. How-
ever, for the parameters employed in the present work it
turns out that this correction is pretty small and will be
neglected henceforth by simply considering Dφ as con-
stant [38]. For simplicity, we also disregard the depen-
dence of Dφ on δJ in the drift term of Eq. (9). Under
those further simplifications the Langevin equation can
be rewritten as
φ̇ = −U ′(φ)− V ′(φ) δJ +
2Dφ ξφ (15)
where we have express the force term as derivatives of
the “potentials”
U(φ) = −
δ(J0 − 1)
16(1 + σ)
cos 4φ−
εσ(J0 − Js)
4(1 + σ)
cos 2φ
−Dφ ln sin 2φ (16)
V (φ) = − δ
16(1 + σ)
cos 4φ− εσ
4(1 + σ)
cos 2φ. (17)
Langevin equations of the form (15) with (4) have been
thoroughly studied in the literature (see e.g. [7, 8, 9, 10,
11] and references therein) as prototypical examples of
the phenomenon of activated escape over a fluctuating
barrier. In view of their non-Markovian nature, their
full analytical solution for arbitrary τ is not generally
feasible. Several approximate results can be provided in
some limits.
For an arbitrary choice of the parameters, V has a
different symmetry with respect to U meaning that the
effective amplitude of multiplicative noise is different
within the two potential wells. If this difference is large
enough, current fluctuation will remove the degeneracy
between the two stationary solutions. This is best seen
by computing the istantaneous potential barriers ∆U±(t)
close to the symmetry point J0 = Js . For weak noise
and δJ ≪ (Js − 1), they are given to first-order in δJ(t)
∆U±(t) ≃
8(1 + σ)
(Js − 1) +
δ ± 2εσ
8(1 + σ)
δJ(t) . (18)
Obviously, this last expression makes sense only when
the fluctuating term is sub-threshold i.e. whenever the
system is bistable. In the case of periodic modulation,
formula (18) allows estimating the range of amplitude
values for a sub-threshold driving
δ(Js − 1)
δ ± 2εσ
. (19)
Using this condition, along with the parameter values at
hand, we deduce that the cases displayed in lower panels
of Figs. 3 and 4 correspond to superthreshold driving.
However, while the minima are much more pronounced
than in the other panels, there is no qualitative difference
in the system response. In the case of stochastic modu-
lation, the same remark applies in a probabilistic sense
for the last panels of Figs. 1 and 2.
Altogether, the mode switching can be seen as an acti-
vated escape over fluctuating barriers given by Eq. (18).
The statistical properties of the latter process is con-
trolled by the current fluctuations. We now discuss the
properties of various regimes. For simplicity, we refer
to the case of stochastic modulations. Most of the re-
marks and formulas reported in the following Subsection
should apply also to the periodic case by replacing τ and
DJ with 2π/Ω and A
2/2 whenever appropriate.
A. Fast barrier fluctuations: τ < TR ≪ T±
As we already pointed out, in this regime the reduc-
tion to Eq. (15) is not justified. We may thus only expect
some qualitative insight on the behaviour of the rate-
equations. From a mathematical point of view, some
analytical approximations for equations like (15) are fea-
sible in this limit (see e.g. Ref. [8] for the stochastic
case). For our purposes, it is sufficient to note that in
this regime the effect of δJ is hardly detected for both
types of driving (see again Figs. 1-4). Note also that
working at DJ fixed means that for τ → 0 the fluctua-
tion become negligible.
B. Resonant activation: TR < τ ≪ T±
If TR < τ we are in the colored noise case. The prob-
lem is amenable of a kinetic description which amounts
to neglect intrawell motion and reduce to a rate model
describing the statistical transitions in terms of transition
rates. If we consider τ as a time scale of the external driv-
ing we can follow the terminology of Ref. [39] and refer
to this situation as the “semiadiabatic” limit of Eq. (15).
In this regime, the residence time is basically the short-
est escape time, which in turn correspond to the lowest
value of the barrier (the noise is approximatively con-
stant in the current range considered henceforth). For
the case of interest, δ < 2εσ we can use (18) to infer
that the minimal values of ∆U± should be attained for
δJ ∝ ∓
DJ respectively. This yields
T± ≃ Ts exp
−K 2εσ ± δ
1 + σ
where K is a suitable numerical constant. Notice that δ
controls the asymmetry level: if δ ≪ 2εσ the two resi-
dence times decrease at approximatively the same rate.
This prediction is verified in the simulations and also in
the experiment [21].
As a further argument in support of the above rea-
soning, we also evaluated the probability distributions
of the residence times obtained from the simulation of
the rate equations. In Fig. 5, we show two representa-
tive cumulative distributions. The data are well fitted
by a Poissonian P (T ) = 1 − exp(−T/T±) for both the
stochastic and periodic modulation cases. This confirms
that hopping occours preferentially when a given (mini-
mal) barrier occurs.
0 0.5 1 1.5
0 1 2 3 4
FIG. 5: (Color online) Cumulative distributions of the resi-
dence times in the resonant activation region, parameter set
with c = 1.3 (see text and Table I). Left panel: stochastic
modulation with DJ = 5×10
−4, τ = 1.638×103 . Right panel:
periodic modulation with A = 0.03 and period 1.286 × 104.
We report only the histograms for the times whose averages
are denoted by T+ in the text. Solid line is the cumulative
Poissonian distribution with the same average.
C. Slow barrier, frequent hops: TR ≪ T± ≪ τ
This corresponds to the adiabatic limit in which the
time scale of the external driving is slower than the intrin-
sic dynamics of the system [39]. To a first approximation
we can here treat current fluctuations in a parametric
way. Correction terms may be evaluated by means of
a suitable perturbation expansion in the small parame-
ter 1/τ [10]. If δJ is small enough for the expression
(18) to make sense, the escape time can be estimated as
the average of escape times over the distribution of bar-
rier fluctuations, i.e. 〈T±〉δJ . For the case of Eq. (4),
the variable δJ is Gaussian and we can use the identity
〈expβz〉 = exp(β2〈z2〉/2) to obtain [11]
T± ≃ Ts exp
[2(δ ± 2εσ)2
(1 + σ)2D2
. (21)
This reasoning implies that for large τ the residence times
should approach two different constant values. A closer
inspection of the graphs (in linear scale) reveals that this
is not fully compatible with the data of Fig.1 even for
the smallest value of DJ . In several cases, T± continue to
increase with τ and no convincing evidence of saturation
is observed. We note that the same type of behaviour
was already observed in the analog simulations data of
Ref. [11]. There, an increase of hopping times duration at
large τ was found. The Authors of Ref. [11] explained this
as an effect of a too large value of the noise fluctuation
forcing the system to jump roughly every τ . We argue
that the same explanation holds for our case. This is also
consistent with the fact that the exponential factors in
Eq. (21) evaluated with the simulation parameters turn
out to be much larger than unity.
V. CONCLUSIONS
In this paper, we have explored numerically and ana-
lytically the effects of external current fluctuations on the
mode-hopping dynamics in a model of a bistable semicon-
ductor laser. To the best of our knowledge, this setup
provides the first theoretical evidence of resonant activa-
tion in a laser system. As the phenomenon has hardly
received any experimental confirmation in optics, we be-
lieve that our study may open the way to future research
in this subfield.
The model we investigated is based on a rate-equation
description, where the bias current enters parametrically
into the evolution of the modal amplitudes. We consid-
ered, two kinds of current flutuations, namely, a stochas-
tic process ruled by an Orstein-Uhlenbeck statistics, and
a coherent, sinusoidal modulation. These choices are mo-
tivated by the aim of proposing a suitable setup for an
experimental verification of our results. Upon varying the
characteristic time-scale of the imposed fluctuations, we
have shown that the residence times attain a minimum
for a well-defined value, which is the typical signature
of resonant activation. The magnitude of the effect can
be different depending on the parameters of the model.
Moreover, the response of the system appears very much
similar for both periodic and random modulations.
The reduction of the rate equations to a one-
dimensional Langevin equation allowed us to recast the
problem as an activated escape over a fluctuating bar-
rier. To first approximation, the fluctuating barrier (mul-
tiplicative term) is mainly controlled by current modu-
lations while the spontaneous noise act as an additive
source. This simplified description has allowed us to
draw some predictions (e.g. the dependence of residence
times on noise strength) and to better understand the
role of the physical parameters. Given the generality of
the description, our results should apply to a broad class
of multimode lasers, including both Edge Emitting and
Vertical Cavity Lasers.
From an experimental point of view, driving the laser
in a orders-of-magnitude wide range of time-scales is
more feasible in the case of a sinusoidal modulation than
for a colored, high frequency noise. However, given the
evidence of a resonant activation phenomenon for such
modulation, our results indicate that it occurs almost for
the same parameters in the case of colored noise, provided
that the RMS of the modulations equals the amplitude
of the added noise. Thus, the phenomenon could be fully
exploited along those lines. Since the reported experi-
mental evidences of the phenomenon are so far scarce,
we hope that the present work could suggest a detailed
characterization in optical systems that allows for both
very precise measurements and careful control of param-
eters.
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