Noise induced re-entrant transitions in exciton bistable system
We study a re-entrant transition in an exciton bistable system under the influence of multiplicative noise. The re-entrant behavior is predictable for specific values of the system control parameter when increasing the multiplicative noise intensity. The system of Wannier-Mott excitons exhibits bist...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
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irk-123456789-1215812017-06-15T03:03:43Z Noise induced re-entrant transitions in exciton bistable system Gudyma, Yu. Ivans'kii, B. Semenko, O. We study a re-entrant transition in an exciton bistable system under the influence of multiplicative noise. The re-entrant behavior is predictable for specific values of the system control parameter when increasing the multiplicative noise intensity. The system of Wannier-Mott excitons exhibits bistable ↔ monostable transitions in a window of intermediate amplitudes of the multiplicative noise intensity demonstrating its destructive and constructive role. 2006 Article Noise induced re-entrant transitions in exciton bistable system / Yu. Gudyma, B. Ivans'kii, O. Semenko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2006. — Т. 9, № 1. — С. 88-92. — Бібліогр.: 15 назв. — англ. 1560-8034 PACS 05.40.-a, 78.40.Fy, 42.65.Pc http://dspace.nbuv.gov.ua/handle/123456789/121581 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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We study a re-entrant transition in an exciton bistable system under the influence of multiplicative noise. The re-entrant behavior is predictable for specific values of the system control parameter when increasing the multiplicative noise intensity. The system of Wannier-Mott excitons exhibits bistable ↔ monostable transitions in a window of intermediate amplitudes of the multiplicative noise intensity demonstrating its destructive and constructive role. |
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Gudyma, Yu. Ivans'kii, B. Semenko, O. |
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Gudyma, Yu. Ivans'kii, B. Semenko, O. Noise induced re-entrant transitions in exciton bistable system Semiconductor Physics Quantum Electronics & Optoelectronics |
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Gudyma, Yu. Ivans'kii, B. Semenko, O. |
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Noise induced re-entrant transitions in exciton bistable system |
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Noise induced re-entrant transitions in exciton bistable system |
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Noise induced re-entrant transitions in exciton bistable system |
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Noise induced re-entrant transitions in exciton bistable system |
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Noise induced re-entrant transitions in exciton bistable system |
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noise induced re-entrant transitions in exciton bistable system |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Noise induced re-entrant transitions in exciton bistable system / Yu. Gudyma, B. Ivans'kii, O. Semenko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2006. — Т. 9, № 1. — С. 88-92. — Бібліогр.: 15 назв. — англ. |
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Semiconductor Physics Quantum Electronics & Optoelectronics |
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AT gudymayu noiseinducedreentranttransitionsinexcitonbistablesystem AT ivanskiib noiseinducedreentranttransitionsinexcitonbistablesystem AT semenkoo noiseinducedreentranttransitionsinexcitonbistablesystem |
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2025-07-08T20:09:38Z |
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Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 1. P. 88-92.
© 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
88
PACS 05.40.-a, 78.40.Fy, 42.65.Pc
Noise induced re-entrant transitions in exciton bistable system
Yu. Gudyma*, B. Ivans’kii, O. Semenko
Yu. Fed’kovich Chernivtsi National University
2, Kotsyubinsky str., 58012 Chernivtsi, Ukraine
*E-mail: yurig@ukr.net
Abstract. We study a re-entrant transition in an exciton bistable system under the
influence of multiplicative noise. The re-entrant behavior is predictable for specific
values of the system control parameter when increasing the multiplicative noise intensity.
The system of Wannier-Mott excitons exhibits bistable ↔ monostable transitions in a
window of intermediate amplitudes of the multiplicative noise intensity demonstrating its
destructive and constructive role.
Keywords: exciton, resonance absorption, multiplicative noise, transition.
Manuscript received 10.11.05; accepted for publication 15.12.05.
1. Introduction
Systems with a definite number of competing states of
local stability play a leading role in the contemporary
analysis of various physical phenomena and their
possible applications. In a growing number of cases such
situations occur in open systems, i.e., systems that
require continual fluxes of the energy or matter. Our
focus will be on the dynamics of solid-state Wannier-
Mott excitons with bistable behavior. Due to an
excitonic resonance, an intrinsic absorption optical
bistability may occur in a narrow spectral range beyond
the critical light intensity. An external feedback in the
form of optical resonator is not necessary for such
bistability.
It has been found experimentally that resonatorless
optical bistability near the exciton resonance takes place
at rather small exciton concentrations ( 315
ex cm10~ −n )
and the intensity of incident radiation 1 kW/cm2 at room
temperatures [1, 2]. It is important that optical nonlinear
effects occur during resonance formation of excitons at a
significantly lower incident-light intensity as compared
with excitation of isolated atoms. The exciton
nonlinearities leading to this bistability has been
investigated theoretically in early papers [3, 4]. It has
been found that the concentration of excitons induced by
resonance light of a sufficiently large intensity can be an
ambiguous function of that intensity, if the width and
position of the absorption line depends on the
concentration. In fact, the increase of external radiation
leads to growth of the density of exciton states and
raising exciton-exciton collision role. In this case, the
response of the system depends on the number of
excitons induced by external radiation. Thus, an increase
of the light intensity induces the change of optical
characteristics in semiconductors [5]. The optical device
based on such semiconductors has not been already
designed, but the latter phenomenon has a fundamental
meaning in the study of non-equilibrium systems.
Hence, for the correct description of the position and
linewidth of the exciton, it is impossible to neglect
many-body effects [6]. The exciton-density dependence
of the exciton resonance position has been intensely
investigated by spectrally resolved four-wave mixing [7,
8]. Furthermore, a sophisticated case of the energy
renormalization of the biexciton resonance in ZnSe
quantum wells takes place [9]. These examples indicate
the importance of taking into consideration the
dependence of the resonance frequency on the
concentration of excitations in various systems.
The processes of creation and transformation of an
ordered structure far from equilibrium are similar to
phase transitions in an equilibrium system. Noise
induced phenomena associated with kinetic transitions
have special place in non-equilibrium processes [10] and
have been actively discussed [11-13]. It has become
apparent that even small noise can induce qualitative
changes in the system far from thermal equilibrium [14].
In this work, the optical bistable system near the
excitonic resonance is studied in the presence of the
multiplicative noise. We predict the appearance of noise
induced re-entrant transitions in the narrow region of
light intensities. The bifurcation diagrams showing that
region were obtained by numerically calculating the
extremum of the probability for the stationary density of
excitons in bistable semiconductor.
The paper is organized as follows. In the next
Section, we briefly discuss the basic equations for the
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 1. P. 88-92.
© 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
89
exciton bistable system under some multiplicative noise.
The analysis of its stochastic behavior and re-entrant
transitions are presented in the third Section. Finally,
several conclusions have been given in Sec.IV.
2. Basic equations
Transport equations for the laser radiation with the
intensity I(z) and quasi-particle concentration n(z,t) can
be written in the form
( ) ( ) , = zIn
dz
dI ωα− , (1)
( ) ( )
τ
ωα
∂
∂
∂
∂ nzIn
z
nD
t
n
, +
=
2
2
− , (2)
where ( )n,ωα and D are the light absorption and
quasi-particle diffusion coefficients, respectively; τ is the
quasi-particle lifetime. The surface 0=z of a
semiconductor plate is illuminated with a light beam of
the intensity 0I . In equation (2), we will confine
ourselves to a linear function for the recombination
concentration of excitons.
The frequency dependence of the light-absorption
coefficient ( )ωα for the resonance light absorption is
given by
( )
12
0
–1 = ,
−
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
⎟
⎠
⎞
⎜
⎝
⎛
Δ
′
+
ω
ωωαωα rn . (3)
Collective interactions (interactions of excitons with
lattice or with each other, exchange interactions, etc.) in
the exciton system at high excitation levels cause the
resonance frequency rω ′ to become a function of the
excitation concentration that decreases with n:
( ) = ann rr −′ ωω . This behavior is due to effective
attraction of excitons. The magnitude of the light-
absorption coefficient (3) at fixed frequencies depends
essentially on the resonance frequency. In the case of
excitons, ωΔ and an are the width and shift (in the
linear approximation of the quasi-particle concentration)
of the exciton level rω . The concentration-induced blue
shift and the broadening of the exciton absorption line
increase absorption in the relation (3).
Assuming that the diffusion length exceeds the plate
thickness and introducing the averaged light intensity,
we obtain a nonlinear concentration balance equation for
a system of Wannier-Mott excitons
{ } .),(exp[1 11
0
−− −−−= τωα nlnlI
dt
dn (4)
The macroscopic equation (4) in the frequency
region rωω ≤ has from one to three solutions
depending on the value of the control parameter. In the
last case, this leads to a hysteresis in the quasi-particle
distribution that is determined by the laser-radiation
intensity. This results from a bell shape (pseudo-
Lorentzian form) of the absorption factor as a function
of the exciton concentration in a certain frequency
region. For this reason, the phenomena of resonatorless
optical bistability takes place in the exciton frequency
region. An internal feedback in the system provides a
balance condition between the processes of exciton
generation and recombination. Toyozawa [3] examined
similar nonlinear situation for the system with threshold
behavior of absorption.
Having a fixed incident light frequency, we can
change over to dimensionless variables in Eq. (4):
( )( ) )(
–1+1
– exp–1 = 2 ηη
η
λβ
θ
η f
d
d
≡−
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
⎥
⎦
⎤
⎢
⎣
⎡
Ω
(5)
with τθ / = t , l = 0αλ , ( )( )2r /= ωωω Δ−Ω ,
( )ωωη −ran/= , ( ) lIa ωωτβ −r0/ = . By mechanical
analogy Eq. (5) describes an over-damped oscillatory
system, while the magnitude )(ηf corresponds to
external forces.
3. Re-entrant transitions
Let us examine the changes in the states of the system
induced by fluctuations of the light intensity. For the
system of excitons, the light intensity is an external
parameter that appears in Eq. (5) as a multiplicative
factor. We shall describe the random breakdown of light
coherence by the following stochastic process
( ) ( ),tt σξββ += where the external noise )(tξ is
characterized by very rapid fluctuations (Gaussian white
noise) to compare with the characteristic evolution time
of system. This means that the one-dimensional variable
η undergoes the influence of a random external
perturbation. The intrinsic fluctuations of the system are
not considered, since they do not induce transitions [10].
These fluctuations (both additive and multiplicative)
only shift the region of multiple quasi-particle
distribution towards a major value of the control
parameter.
In terms of generalized functions, the Gaussian white
noise is a derivative of the Wiener process. Therefore,
Eq. (5) is a stochastic differential equation in the sense
of Stratonovich associated with the Fokker-Plank
equation that defines the evolution of the transition
probability ( )θηθη ′′, / ,p [10]
( ) ( ) ( )[ ]
( ) ( )[ ]ηθηθη
∂η
∂
ηθηθη
∂η
∂
∂θ
ηθη∂
′+
′
′
|,,
2
1
+ | ,,–=|,
2
2
pB
pAp
(6)
where
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 1. P. 88-92.
© 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
90
0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0
- 0 . 6
- 0 . 4
- 0 . 2
0 . 0
-dV
/d
η
Phase coordinates
Fig. 1. “Stochastic force” plotted versus phase coordinates for
different values of noise amplitude σ. The line with circles
corresponds to σ2 = 0, and another one (with squares) to σ2 =
= 0.09.
( ) ( )( )[ ]{ }
( )
( )( )[ ]×−Ω+−
−Ω+
−
Ω+
−−Ω+−−
2
2
2
2
11/exp
11
1
2
+ 11/exp1=,
ηλ
η
ηλσ
ηηλβθηA
( )( )[ ]{ } 11/exp1 2ηλ −Ω+−−× ,
( ) ( )( )[ ]{ }222 11/exp1 = , ηλσθη −Ω+−−B .
The states of system can be adequately described by
the stationary probability density p(η), if not to consider
the microscopic transients of a nonlinear system with
respect to an external noise. The stationary solution of
the homogeneous Fokker-Plank equation (6)
( ) ( )( )[ ]{ }
( )( )[ ]{ } /]–11/exp1[2exp
11/exp1
2
0
2
12
UU
Nps
−Ω+−−⎜⎜
⎝
⎛
−×
×−Ω+−−=
∫
−
λβ
σ
ηλη
η
( )( )[ ][ ] ⎟
⎠
⎞−Ω+−− dUU
2211/exp1/ λ (7)
corresponds to the diffusive stochastic process. Here, N
is a constant to be determined from the the normalization
condition
( )∫ =
b
s dp
0
1 ηη . (8)
The limits of integration are given by physical
restrictions of the problem, i.e., by variation of the
exciton concentration from zero to that at which the
Bose condensation takes place.
The solution (7) can be written by analogy with the
well-known canonical distribution of equilibrium
thermodynamics
)/)(2exp()( 2σηη VNps −= . (9)
Here )(ηV is the effective potential. The random
diffusion process )(θη is ergodic, since the probability
density (9) is normalized. It follows from the fact that
the product ηη dps )( is the time over which the
arbitrary trajectory of the diffusion process approaches
the point η at infinity. Thus, the maxima of the
probability density correspond to stable stationary states
in which the system dwells relatively long, while the
minima correspond to unstable stationary states that are
passed very quickly. The macroscopic stationary states
of the system can be found from extremum conditions of
the stationary probability density [10]. Noise induced
transitions appear when the density function changes
from unimodal into bimodal distribution. Thus, for
examination of long-time system behavior, it is
sufficient to study the extremes of the effective potential
( )ηV , since the neighborhood of potential minima of the
2σ order gives the basic contribution in the stationary
probability density. A numerical analysis is carried out
for the model of illuminated CdS-like semiconductor
with the following parameters: =τ 10−3 s,
=a 2.5·10−7 cm3/s, =Ω 100, =λ 0.055. The level of the
light flux intensity is 1020 photon/cm2·s.
Let us consider the behavior of the system for
supercritical values of the control parameter 64.18=cβ .
In the absence of external multiplicative noise, the
system is characterized by one stationary state near the
zero value of η (Fig. 1, curve marked as circles), which
corresponds to the shifted resonance frequency. The
inset shows the function near the zero line. But the
growth of fluctuations of the incident light, which are
characterized by the noise amplitude σ or dispersion
=2σ 0.09, leads to the appearance of bistability in the
concentration dependence (Fig. 1, curve marked as
squares). Hence, the given qualitative reconfiguration of
the dynamical system regime with resonance absorption
of light should be treated as an external noise induced
bistability of exciton states below the critical intensity of
the incident radiation. By the analogy with the motion of
particle in the classic potential )(ηV , this mean that
small external noises transform an infinite motion of
particle into finite motion in a certain interval. On the
other hand, for supercritical values of the controlling
parameter β , the external noise shifts the region of
bistability towards the higher values of the incident light
intensity (suppression of bistability by noise). As an
example, we illustrate this behavior in Fig. 2.
Hence, external white multiplicative noise has a
contradictory role in the open nonlinear system. It leads
to enlargement of the system bistability region and its
shift. Noise may lead to a bistable behavior in systems
where this phenomenon cannot take place without noise.
On the other hand, it can suppress the states that are
already present in the deterministic system. Such
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 1. P. 88-92.
© 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
91
0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0
- 0 . 4
- 0 . 3
- 0 . 2
- 0 . 1
0 . 0
0 . 1
f(
η)
, f
s (
η)
η
h1 h2 h3
h
s
1
h
s
2
h
s
3
Fig. 2. Comparison of deterministic f(η) and stochastic f s(η)
“forces” for β = 20, Ω = 25. The amplitude of the
multiplicative noise intensity is σ = 7.
0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5
1 8 . 5 0
1 8 . 5 5
1 8 . 6 0
1 8 . 6 5
1 8 . 7 0
β
σ
Fig. 3. Phase diagram of the system (dependence of the critical
value of the nondimensional laser radiation intensity β versus the
amplitude of the multiplicative noise intensity σ for various
values of Ω. The value Ω = 50 corresponds to the lines with
circles, Ω = 100 to the lines with triangles and Ω = 150 to the
lines with squares.
combination of destructive and constructive nature of
external multiplicative noise may lead to unusual
behavior of the system at the re-entrant transition. In this
case, the noise-induced transition brings the system to
bimodality; while the re-entrant transition takes places at
a higher value of the multiplicative noise intensity,
making the system to be unimodal again. The system
exhibits a suppression bistability, followed by the re-
entrant transition back to bistability, both induced by
multiplicative noise. In other words, the system
undergoes a suppression of bistability in a narrow
window of intermediate intensities. It will be shown that
our prediction takes place. For this purpose, we will
estimate the critical value of the light intensity by
changing the amplitude of the multiplicative noise
intensity. In order to illustrate the behavior of an optical
bistable system being under excitonic resonance, we
have numerically examined the extremum of the
effective potential (9).
As one can see from Fig. 3, the cβ (critical value of
incident radiation intensity) depends on the parameter Ω
which is the ratio between the value of detuning at the
excitonic resonance and excitonic line width. In the
deterministic case ( 0=σ ), an increase of the parameter
Ω requires higher intensities of light β for bistability
to take place in the exciton system. Probability
distributions are bimodal for points above the curves and
unimodal below. The growing multiplicative noise
increases the critical value of the laser radiation intensity
at first, but than decreases it abruptly at larger values of
σ . At the higher then the critical value of the light
intensity, a crossover takes place from the bistable to
monostable region and back to bistable while increasing
the noise amplitude. The latter re-entrant phenomenon is
accompanied with the shift of the transition line.
Note that the described phenomena resemble the
appearance of stochastic resonance since constructive
manifestation of noise is optimal for the intermediate
value of the multiplicative noise amplitude. Previous
order resumes for a larger value of the noise intensity
amplitude showing its destructive role. It can be easily
seen from Fig. 4 that the latter behavior takes place for a
narrow window of values of the control parameter β .
The stochastic system exhibits re-entrant behavior as a
function of multiplicative Gaussian white noise. Besides,
an increase of noise induces bistability for smaller values
of laser irradiation than in the deterministic case. As
discussed in paper [15], nonlinear systems can undergo
noise induced re-entrant transitions also under the
influence of colored noises.
0 5 10 15 20
15
18
bistable
bistable
β
σ
monostable
Fig. 4. Phase diagram of the system here is the dependence of
the critical value for the nondimensional laser radiation
intensity β versus the amplitude of the multiplicative noise
intensity σ. The dotted lines marks the region of the re-entrant
transition.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 1. P. 88-92.
© 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
92
Quasi-white noise approach uses assumption that the
width of the noise spectrum (~σ ) is by far larger than
the inverse time of passing to a steady regime. Another
simplification used in this approach is neglecting the
internal structure of a laser beam. Qualitative results of
the research doesn’t change, if take into account the
described structure.
4. Conclusions
In this paper, the influence of multiplicative noise on a
system of Wannier-Mott excitons have been examined.
We have presented the evidence of the re-entrant
transition in the bistable model on excitonic resonance.
The re-entrant transition is induced by parametric noise
of the laser radiation intensity, which demonstrates the
dual role of noise amplitude as a control parameter. It
has been found that even a sufficiently small external
multiplicative noise shifts the region of bistability
towards the higher values of the incident light intensity,
in other words, noise suppresses the bistability. But the
further increase of noise induces the bistability of the
stochastic system for smaller values of laser irradiation
than in the deterministic case.
We have numerically constructed the transition line
diagrams, i.e., the boundaries of the bimodal-unimodal
transitions. The form and the position of the lines on the
plane { σβ , } depends on the ratio between the value of
detuning at the excitonic resonance and excitonic line
width. The state diagrams show that the system passes
the monostable region in a window of intermediate
amplitudes of the multiplicative noise intensity for the
specific value of β . The transition is re-entrant, i.e.,
bistability is resumed at a higher noise intensity. An
immediate cause of re-entrant behavior is the resonance
origin of the excitonic absorption mechanism in
semiconductors. Thus, one can expect the appearance of
the latter phenomenon in other similar systems with the
resonance absorption.
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