The decay time of the time-dependent transient stochastic dynamics, in presence of an external force

N.H

15 February 1995

OPTICS COMMUNICATIONS Optics Communications 114( 1995 ) 50 l-508

Full length article

The decay ti,me of the time-dependent transient stochastic dynamics, in presence of an external force J.I. Jimknez-Aquino Departamento de Fikica, Universidad Autdnoma Metropolitana-lztapalapa. 09340 MLxico. Disrrito Federal, MLxico

Received 18 April 1994; revised version received 23 August 1994

Abstract We characterize the transient behavior of the time-dependent linear stochastic systems in presence of an external force, by means of the integration of the nonlinear relaxation time (NLRT), supported by the quasi-deterministic (QD) theory. The formalism is applied to study the transient dynamics of the switch-on times of a single mode semiconductor laser, in presence of an external optical signal, as a prototype modei. The time scale. for this laser model is essentially given in terms of a relevant scaling parameter.

1. Introduction The study of the transient dynamics of switch-on times of a class-A laser, in presence of an external signal has been well characterized in works given in Refs. [ l-51. This class-A laser is one, which can be described by a single equation for the electric field after adiabatic elimination of the atomic variables. Very recently a theoretical analysis of the semiconductor laser model, in terms of the passage times (F’T) distribution, has been done in Ref. [8] and the detection of the presence of a very weak injected signal by measurements of the passage time variance have been reported also in Ref. [ 91. The purpose of this paper is also to characterize the dynamical relaxation of the time-dependent linear stochastic systems, under the action of an external force, these are systems modulated by the linear Langevin-type equation with the time-dependent drift coefficient and submitted to the action of a constant external force, through a time scale called, nonlinear relaxation time (NLRT) [ 51. The semiconductor laser model, classified in the dy-

namical systems in the category of type-B laser [ 6,7], is an optical system, in which such adiabatic elimination of the atomic variables is not possible, and also is a system that can be study in context of our formalism. Here we want to show an alternative and simple method, following the idea of Ref. [ 51, that permit us to characterize the time-dependent transient dynamics of systems, which can evolve toward their corresponding steady states. The evolution of these systems toward such stationary states suggest the existence of a time scale, which measures the dynamical relaxation at the region close to those steady states. As an example of the theoretical formalism, we characterize the transient stochastic dynamics of the switch-on times of a semiconductor laser, in presence of the external optical signal. We then construct this work as follows. In Sec. 2.1 we introduce the time-dependent linear stochastic dynamics with an external force, for a set of xi, i = I,... ,n independent physical variables. In Sec. 2.2 the NLRT associated with the first moment of the square modulus of the vector S = (XI,. . . ,x,), this

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J.I. Jidnez-Aquino

Communications I

is (I) = (S’), is defined and reduced to a manipulable integration with the help of the QD theory. In Sec. 3 the prototype model in terms of the rate equations for the semiconductor laser is presented. Those are the rate equation for the complex electric field E(t) = El ( t) + i E2 ( t) , coupled with the equation for the carrier number N(t) [ 91. In this section some of the preliminaries algebraic steps are the same of those given in Ref. [9]. The NLRT associated with the average of the amplitude of the electric field is calculated, in terms of a relevant scaling parameter. Concluding remarks are given in Sec. 4. Finally in the Appendix we detail the calculation of the results given in Sec. 3.

(1995) 501-508

Here we will assume that the quantity r( 0) = 0, which means that the initial condition is fixed at the unstable state. The connection with QD approach is made taking into account the fluctuations of the initial conditions, around the initial unstable state, say r(0) = h2, with h a random variable such that h2 = hf +. . . + hi. After this stochastic beginning the dynamical evolution is essentially deterministic, and noise plays no important role. Therefore this connection suggest the solution, for the variable r, of the linear deterministic equation associated with Eq. (2.1); it is i = 2a(t)

r,

(2.4)

whose solution reads 2. The NLRT and the QD approach for time-dependent linear stochastic dynamics, presence of an external force

r(t) = h2 ezwCr),

(2.5)

in where t

2.1. The time-dependent linear stochastic dynamics with an external force Here we propose the Langevin model with timedependent drift coefficient, in presence of an external force, in terms of a set of n independent physical variables, which reads ii=a(t)Xi+Ci+ti(t),

izl,...,

n,

(2.1)

where ti ( t) is the noise with zero mean and correlation function (5i(t>5j(t’))=aosijs(t-t’),

j=l,...,n.

W(t) = .I 0

a(s) ds.

(2.6)

Here h plays the role of random initial conditions, and W(s) is in general a positive function of time t. It is clear that the process r(t) diverges when f goes to infinity. So that, in order to avoid this explosion, we must stop the function r(t) to a prescribed value rf, such that at the time ti, r( ti) = rf. In NLRT we must take this reference value as the steady state value, this is, i-f = rst. Therefore the whole process r( t) can be written explicitly as r(t) = h2e2W(s) 0(ti -t)

+ rStB(t - ti),

(2.7)

(2.2) a(t) is in general a time-dependent drift coefficient, D is the intensity of the noise and Ci is the external signal, considered as the ith component of the constant force, which is a vector C . 2.2. The NLRT and QD approach

0

P-)0 - (4.t

step function. the substitution of Eq. (2.7) into write the formal expression of the with the moment (r) as

(2.8)

We now define the NLRT associated with the average of the square modulus of the vector S, this is (r) = (S2). This definition is [S]

J

where O(s) is the Therefore with Eq. (2.3) we can NLRT associated

(2.3)

The average (ti) accounts for the relevant contribution of the dynamical relaxation, it is basically the mean first passage time (MFRT), which measures the relaxation time around the unstable state, where the effect of the fluctuations and the external force become the mechanisms responsible of the decay process. The

J.I. Jimbez-Aquino

/Optics

second term is a new contribution to the time scale (2.8)) and it is then a characteristic term of the NLRT. It contributes, for long times, with a time scale at a region of deterministic approximation. This is, once the system leaves of the initial state induced by the cooperative effect between the noise and the external force, it enters in a region where such a cooperative effect, have already no an important role on the dynamical evolution of the system. So that, this new contribution accounts for the time scale at a region, that we can call, the smooth region of the dynamical relaxation. Finally, the statistical properties of the random variable h is obtained through the QD analysis as following. The QD approach assumes the formal solution of Eq. (2.1 1, which can be written as

Communications

503

114 (1995) Sol-508

the function W(t), which is assumed to hc a pvsitivc function of time. Let us then define those limits as I

KO = I’M lim

ds eeW(“‘, I

0

J

KI = f_DC, lim

ds e-2w(S)

.

(2.13)

0

Therefore in this limit of approximation, the stochastic process hi(t) becomes a Gaussian random variable with mean value (hi( CO)) = (hi) = C’iKo, and variance given by c2 = (h?) 1 = 2K, D, I - (h.)’

x;(t)

hi(t)

= hi(t) ew”),

=

.I

e-W(S)

(2.9)

Vi(S) ds,

(2.10)

with ~i( t) = Ci + ti(t)). The task of the QD theory is precisely to shows that the stochastic process h;(t) plays the role, for long times, of an effective initial condition, i.e., hj( t = CO) = hi, with hi a Gaussian random variable. For this purpose, we construct the formal expressions for the mean value and the autocorrelation function of hi(t). So the mean value is given by

(hi(t))

= C; /

e-w(s)

(2.14)

which is a quantity proportional to the noise intensity D. With those characteristics of the first two moments of the random variable hi, the marginal probability density for the modulus h, can be constructed formally with the help of the joint probability density P(hl , . . . . h,) of the n independent random variables, which is a Gaussian distribution

P(ht,. ..,h,)

=A0

(2.15) where a2 = ( 1/2a2). To do this, we construct the joint probability density, in the new space of variables u=(u1,... , u,) using the transformation

(2.11)

&,

P(ht,. ..,h,)dh,...dh,

0

+ P(u ,,...,

U,)dK (2.16)

and the autocorrelation

function

will be such that dV = J(u) du is the volume element in the space u, and J(U) is the Jacobian of the transformation given by dh,/‘du,

+ 20

I 0

ds e-2w(s’ .

(2.12)

J

Now well, the QD approach assumes the convergence of the integral terms of (2.11) and (2.12) for long times. This is so, because of the characteristics of

J(u)

=

; ah,,/&,

... ..( .

dh,/Ju, ;

.

(2.17)

ah,,/&,

Therefore, the space of variables that we are interested , u,) , so the formal expression in,willbeu=(h,Ua.. of the joint probability density, using Eq. (2.15), reads

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J.I. Jimhez-Aquino

/Optics Communications 114 (1995) 501-508

n = N/N,,, z = F/I,,,

P(h,u2,...,u,)dV =Aoexp{-a2[h2+K~C2-2Ka(h.U)]}dV

&

where the dot means the scalar product. Finally the marginal probability density of the modulus h can be calculated, by integrating over the rest of the variables LQ,...,~,, and by calculating the Jacobian of the transformation.

In this section we will study a semiconductor laser model submitted to the action of an external signal, which is a prototype model that can be formulated in the context of the formalism of Sec. 2. The rate equations for the semiconductor laser model are essentially given in terms of two quantities, those are the amplitude of the electric field E( t’) and the carrier number N( t’) [ 91. These equations are

+ keEe + (P’N) “2[(

1 + s(E)*

t’) ,

=

l&l*

=

F [ g(‘” -,““I

- ‘1,

(3.5)

and (3.6)

N,, = Nth = No + Y/g,

are the steady state values of the intensity and carrier number for an injection current C = Ye& above its threshold value C* = yeN* (for s = 0), and F is the amplitude of the electric field. The rate equations become

3. The semiconductor laser model with an external signal and the NLRT

1 + s/El*

(3.4)

where (2.18)

g(N - NO) + iag(N - No)

t = (yye)1/2t’,

i=g[(.-l)+ia(n-l+$)]Z +

se+ fiS(r>,

(3.7)

h=-MfE(A.l),

(3.8)

where we have defined the following parameters

dimensionless

1 E

(3.1) y2

=

E Y’

(3.2) rl=

P’ &(yy*)‘/2’

gN,, P =Y ’

5 = cu-p,.

(3.9)

where Ee = Fe exp( yzt/2) is the external field whose amplitude and frequency are F, and y2/2 respectively. The parameters k,, g, y, ye, p’, a, s and C = ye& with No the carrier number at transparency, are all the appropriate parameters of the model [ 8,9]. The spontaneous emission noise is modeled by a complex Gaussian white noise 5( t’) with zero mean and correlation

The spontaneous emission noise has the correlation given by (3.3). The Eqs. (3.7) and (3.8) are a set of two coupled equations that can be formulated in the context of the JZq. (2.9), by integrating (3.8) and substituting it into Eq. (3.7). So the solution of (3.8) gives

(&t’> l*(P))

n(t)

= 2&r’-

t”).

(3.3)

In the linear regime we can neglect the nonlinear gain parameter of saturation (s = 0) in the Eq. (3.1) and the nonlinear coupling term in the Eq. (3.2). Next, both Eqs. (3.1) and (3.2) are better analyzed in a frame of reference in which the electric field E rotates with the oscillation frequency of the laser, corresponding to the transparency value of the carrier number [ 91. In this new frame of reference and for dimensionless variables [ 8,9]

=n(O)e+‘+(A+

l)(l

-eP’),

(3.10)

where n(0) is the initial condition. Therefore we obtain the time-dependent linear stochastic dynamics with an external field, for the laser model, by writing (3.7) as follows (3.11) where r(t) given by

is a time-dependent

complex

coefficient

J.I. JimPnez-Aquino /Optics Communications 114 (199s) Sol-508

r(f) = (n(t) - 1) +icu

n(t)

- 1 + -?alL -)

.

(3.12)

In order to progress further, we define the time I at which N crosses its threshold value Nst, this is n(r) = 1 IS], where

iJ

1 + A -n(O)

(

In

E

A

= 1 +A[1

n(t)

n(s) - 1 M ks

-e-C(r-t7].

(3.14)

= h(t) eAcr),

(3.15)

here A ( t ) reads

+ 0(&s)2,

s = I - i.

tar

( 3.20~

Which means that n(s) is a linear process in time. and the real part of A(t) is also approximated by

(3.13)

So that, the formal solution of (3.11) for times later than i leads to an equation similar to (2.9), which is written as z(t)

So, the solution (3.14) can be approximated times of interest such that E( t - F) << 1. cl<

,5

>

A simpler and more physical approach in this situation is to consider a deterministic evolution from t = t with the initial condition n(t = 0 = 1 and r_(t = i) = h(co) [ 81. This leads to

50,

Al(s) =

L 2E

[n(d) - 1] ds’ E & .$, 4

s

(3.21)

0

So in the spirit of the formalism of Sec. 2 we can observe from (3.19) that CT’(?) converges to a finite value for long times, such that (&CL/~) I/“( t - r) >> 1. Therefore the process h(t) can be replaced by a time independent Gaussian random variable h( t = cx) = h, with mean value and variance given respectively by (see Appendix 1 c’J4h

SeJ;; (h) = -e 2Ja3J

[I -@(c/2Jt)l~

(3.22)

and (3.16)

(T2=25J( 2E//LLU &$i&

z(O)

The marginal probability density for the modulus h, such that h = hl + ih:! and h2 = h: + h$ can be constructed in this particular case, using Eq. (2.18) for n = 2, and it reads

(3.17)

P(h)=-$lo($)exp(

and

s i I

h(t)

=

[se + Jn(‘)rl((t’)l

eeA(“)df’ +

The initial value z (0) is the initial value of the field E(O), which is assumed to be zero for fixed initial conditions. Therefore the process h(t) is Gaussian with mean value (jl(r))

,,ie-Act’~ dr’,

=

(3.18)

i and autocorrelation a?(t)

= (Ih(f)

= 477 J i

/

n( t’) e --AI

dt’,

where A I( t’) is the real part of the A(t) .

-&Ch’+d’)), (3.24)

where d2 = l(h)\’ and la(x) is the modified Bessel function of order zero [ lo]. On the other hand, taking into account the time differences s = t - i, together with the hypothesis for the initial condition z(i) = h, the deterministic equation of Eq. (3.11) is i =

- I(

(3.23)

ET(r)

2.

The solution of this equation of s as

(3.25) can be written in terms

(3.19) lz(s)12 = h2ezAlcs). which is similar to (2.5).

(3.26)

506

J.I. Jimknez-Aquino /Optics Communications 114 (1995) 501-508

Because of stochastic character of (3.26) we will have a stochastic time fi or well si = ti - 7 for which )z )* will reach the steady state value jst, this is jst = jh]2e(AP/2)Sf Under these circumstances written as S=T-t=

the NLRT (2.8)

(si) - i(F(vsi)),

(3.27)

small, and the effect of the external signal is not already takes into account. Then in that region the contribution of average of the Dawson function, for long times, such that VSi >> 1, in terms of the scaling parameter r, can be reduced to (see Appendix)

can be

(F( VS.)) I M l/J;

(3.28)

Finally, in the limit of r >> 1, the NLRT (3.28), for a single mode semiconductor laser in presence of an external signal is written as

where (si) measures the MFPT, F(x) is the Dawson function [ IO], and Y = m. In order to evaluate the average (si), we propose the appropriate change of variables R2 = 2h2/X2 and p2 = 2d2/_Z2, such that J? = 2a2. The calculation of this average follows from the Eqs. (3.27) and (3.24) with the help of the following scaling parameter

+ln(P2/2)

-P(l)

+ l] + 0(rP3’*).

1 (SJ = sn - ( 2rAIU)

p; =

-+

lnW2/2)1,

(3.30)

(3.34)

where

l/2

--lu(l)

(3.33)

The parameter p2/2 = 1(h)12/2a2 in (3.33) can be written, with the help of Eq. (A.6) of the Appendix, as p2/2=P;G(5),

which is a large quantity of order In A’i2/q for l2A << 1 [ 81. Making an expansion in 7-l we then get

LwP2/2)

1 S = so - (2&&r)%) tE,(P2/2)

(3.32)

(3.29)

7 = ln(j,t/P),

x

+ 0( 7-3’2).

TS,” zz&J5X

(3.35)

corresponds to the value of p2/2 when d = 0, and G( 6) is a function of frequency mismatch y2 given by

such that G(b)

1)

3

= em2’ 11 - @(&J_,12.

(3.36)

(3.31)

>

where Ei (x) is the integral exponential function [ lo] and ?P ( 1) is the digamma function [ lo]. SO is the value of (3.30) in absence of external field (j? = 0). Therefore Eq. (3.30), which is the same as that of Ref. [ 93, measures the MFPT of the dynamical relaxation around the unstable state, induced by the cooperative effect between the noise and the external signal. So that, it is also the relevant contribution of the decay process, because of, the universal logarithmic term, for the noise and the external signal, characteristic of the dynamical relaxation of the unstable states. The second term of (3.28) is the new contribution to the dynamical relaxation of the laser semiconductor model. As we said in Sec. 2.2, the contribution of this time scale is basically at the region of deterministic approximation, there, where the noise effect is very

We could also approximated the function G(&) for very small values of the Error function, this is, Ifid<< 1. So to first order in @J(X) we get

G(c) M ee2’1exp[ (WE/J;;) d-1

12. (3.37)

So, within the limits of our approximations, for the laser model, we show the analytical behavior of the result (3.33) in the caption of Fig. 1. The black curve corresponds to the time scale (3.33) with the corresponding analytical result (3.37) for G( Cu). Here the optimum frequency mismatch for detection occurs at ~5= -1, where the curve reaches its minima, and we note a local symmetry in the interval -2 < (Y < 0. On the other hand, the dashed line corresponds to the

J.I. Jim&e=-Aquino

/Optics Communications

:io7

I14 (1995) 501-508

Acknowledgments Financial support from CONACYT knowledged.

( MCxico ) is X-

Appendix A With the approximation given in Eq. (3.20) the mean of h(t) given in Eq.(3.18) can be written as

I -2

0

2

ig. I. Behavior of the NLRT (3.33) versus ~2 = a - YZ/Y jr the semiconductor laser. The solid line is obtained with the xactly Error function. The dashed line corresponds to an aproximation of the Error function. The laser parameters used I the calculations are the same as those given in Ref. [91: = 5.6 x 10-4s--‘, y = 4 x IO” s-‘, ye = 5 x lOas-‘, ‘= 14xIO’~s-L,~‘=~.Ix104s-‘,No=6.8x107,C~=0.905, ‘,‘, = 3.4 x 10’hs-‘.

cy = 5, and .sf = 5 x 10V3.

ame time scale (3.33), but now with the approximaon (3.38). In this case the minima is shifted at & z -8/5, where the optimum frequency mismatch for de:ction takes place. It is clear here, that the dashed urve shows a better symmetry than the black one. [owever both curves show a monotonously decreaslg behavior. Concluding

remarks

We have characterized the time-dependent linear ochastic dynamics, in presence of an external force, )r a set of independent physical variables, by mean f the NLRT connected with the QD theory. This time :ale for such dynamical systems, contains not only le MFPT (first term of (2.8) ), which accounts for le relevant contribution to the dynamical relaxation, ut also a new contribution (second term of (2.8)), lhich accounts, in the limit of long times, for the time :ale at a region of deterministic approximation. The calculation of the new contribution to the time :ale, is not directly integrable, because of the charac:ristics of the function W(t). However a systematic lethod. in terms of the Dawson function, for the laser lode1 or any other model can be formulated. Finally the steady state value rStcomes, in a natural lay, from any time-dependent nonlinear transient dyamics, as it is does in the case of the semiconductor iser system.

1-i (h(t))

X- >

/exp

( -5’2-iaS’2-i&.G)

ds’,

0

(A.11 where a = ( +/4). Now in the limit of ( A,u/~) ‘I2 ( tt) >> 1, the integral in (A.1) is transformed in

(h) =

27

exp( -bx2 + cx) dx,

(A.21

0

where b = ( 1+ ia) and c = (iEY/2E&). This integral is evaluated according to Ref. [ 101, and it reads (h) = -e seJ;; 26

+[l

-@(c/2&)],

(A.3)

where Q(x) is the Error function. The factor in the exponential is a complex number given by c2/4b = -D + iaD, where we have defined D as ii=

D=

(A.4)

4Ghj_&( 1 + a2).

Because b is a complex number, we can write (A.3) as

(h)= &(I x

sefi

exp[-D+i(Da-

B)]

+~2)1/4

[I-@(v%/G)J,

where B = ( l/2) tan-’ (cu). So the quantity obtained from (AS) as

(AS) 1(h) 1’ is

508

J.I. Jidnez-Aquino

/Optics Communications 114 (1995) 501-508

The variance (3.23) is calculated from (3.19) using the same limit of approximation for (h). So that

F(q)

& I =i [ln(j,/h2)]

.=5

= .$[ ln(jr/Z2) a+)

n( t’) e -2A,(r’) dt’,

=477

(A.7)

s

= +[T-

However for the change variable s’ = t’ - f, we have s

&t>

= 4?7

J

n(d) e-2AlCs’)ds’,

(A.81

ln( R2/2> 1 -‘j2

-

ln(R2/2)]-‘/2

1

i

-li2

--!--

In(R2/2)

+ ....

=mf4J;5

(A.1 1)

where we have used again the expansion in r-‘. The second term of (A.1 1) is obtained with the help of Eq. (3.24). It has the following form (see Ref. [ IO] )

0

ln(R2/2)

which is the same as

= EI (P2/2)

= 477

I(

[n(d) -11 e-

2A,(s’)

+

e-2A,(s’)

&’ >

0

s =

_ % d(_2Ai(s’))

477

+ P( l>, (A.12)

s

a(t)

+ y + ln(P2/2)

e-2Al(s’)

with P ( 1) = -y and y the Euler constant. This is the coefficient of r-3/2 in (A.1 1) that we have neglected in (3.32), but has a relevant contribution in the calculation of (Si) given in (3.30).

.I( 0

+e

-2A,(s’)

ds’

References

> = 4F[

1 _ e-2A~W~

++

ie-2h(.i)

[I1 G. Vemuri, and R. Roy, Phys. Rev. A 39 (1989) 2539. r21 S. Balle, F. De Pascuale and M. San Miguel, Phys. Rev. A

&‘,

41 (1990)

0

(A.91 So, in the limit of our approximations we use (3.21) to see that u2( 00) = a2, which is a constant given by

=277(26//z+

J_).

(A.lO)

Eq. (3.32) is evaluated using again the limit V( t F) >> 1 wherev= m. In this case the Dawson function is approximated by F(n) M 1/2x, so

5012;

r31 J.I. Jimeiiez-Aquino

and J.M. Sancho, Phys. Rev. A 43 (1991) 589. 141 P. Yung. G. Vemuri and R. Roy, Optics Comm. 78 ( 1990) 58. and J.M. Sancho, Phys. Rev. E 47 [51 1.1. Jimknez-Aquino (1993) 1558. [61 A. Mecozzi, S. Piazzola, A. D’Ottavi and P Spano, Phys. Rev. A 38 ( 1988) 3136; Appl. Phys. Lett. 55 ( 1989) 769. 171 ET. Arecchi, W. Gadomski, R. Meucci and J.A. Roversi, Phys. Rev. A 39 ( 1989) 4004. [81 S. Balle, P. Colet and M. San Miguel, Phys. Rev. A 43 (1991) 498. [91 M.C. Torrent, S. Balle, M. San Miguel and J.M. Sancho, Phys. Rev. A 47 (1993) 3390. 1101 Handbook of Mathematical Functions, eds. M. Abramowitz and LA. Stegun (Dover, New York, 1972).