- Email: [email protected]
On the optical stochastic Bloch equations
27 March 1995
PHYSICS
ELSEVIER
Physics Letters A 199 (1995)
LETTERS
A
163-168
On the optical stochastic Bloch equations Pedro J. Colmenares a, J.L. Paz b, Rafael Almeida a aDepartamento de Quimica, Universidad de 10s Andes, L.a Hechicera, Me%& 5101, Venezuela h Departamento
de Q&mica. Vniversidad Simbn Bolivar, Apartado 89000. Caracas 108OA, Venezuela
Received 24 March 1994; revised manuscript received 17 January 1995; accepted for publication Communicated by A.R. Bishop
19 January
1995
Abstract
We present a methodology, based on the use of a theorem due to Novikov, to solve the optical stochastic Bloch equations for a two-level system up to first order in the external field in the presence of a heat reservoir. Two cases are studied. In the first, a white noise bath is considered, while in the second colored noise (Omstein-Uhlenbeck process) is analysed.
1. Introduction
The dynamics of nonlinear optics phenomena in molecular systems is normally analyzed by means of the optical “conventional” Bloch equations (OCBE) . This set of equations is derived from the Liouville-von Neumann equation in which the state of the system is described by the reduced density operator. For a two-level system defined by the states 1a) and 1b), the OCBE read
h,(t) dt
dtdt) -
dt
= ; [(K)ab(OPba(f) -P&At) (~l),(t)l =
-
(im0
dpadt> -=-
dp&)
dt
dt
+
l/TdpbJt)
’
-
i
(K)bAt)dt)
-
$ (PD-p3
9
(l.la)
1
,
(l.lb)
(l.lc)
value in the absence of where pr,( t) = p,,(t) - pbb( t) is the difference in state populations, pb is its equilibrium radiation and T, and T2 are the longitudinal and transversal relaxation times, respectively [ 11. Here, it is assumed that the states )a) and ]b) are coupled through an electric dipole transition moment, /.L,whose nondiagonal matrix elements are I-L,~and /_Q_and its diagonal components pan and /..Q,~(permanent dipole moments) are assumed to be negligible compared to their nondiagonal counterpart and therefore taken as zero. The interaction Hamiltonian that corresponds to the incident wave E( t) is taken as H,(t) = - p.E( t) , In this equation four typical frequencies arise as a result of the different time scales of the processes responsible for them. The first is the Rabis frequency, 0 = phE/h, which is determined by the coupling between the electromagnetic field and the system, the second the transition Bohr frequency w, of the two-level atom and finally, the two frequencies l/T, and 1 /T2 associated to solvent dissipation mechanisms. Elsevier Science B.v. SSDIO375-9601(95)00081-X
164
P.J. Colmenares et al. /Physics
Letters A 199 (1995) 163-168
Most of the work done in nonlinear optics with the OCBE considers the transition frequency w, as a parameter. This assumption is valid in regimes of weak solute-solvent interaction for a homogeneous distribution of the resonant frequencies. However, if one is interested in the development of radiation from a two-level system driven by a monochromatic field and immersed in a condensed medium or in a dense vapor cell, the assumption is no longer valid. Thus, it has been reported that the OCBE cannot describe observed optical signal intensities for systems interacting strongly with the surrounding medium (i.e. an impurity in an ion solid [ 2,3] ). For those cases, molecular collisions induce frequency shifts, and due to the nature of the collisions, they lead to stochastic modulation of the frequency. This shift in the transition frequency becomes a function of time and can be represented by Z(r)=%+rl(r)
9
(1.2)
where q(t) comprises all the stochasticity of z(t) . In this context, a number of works [ 4-81 have been presented recently. Several treatments have been employed to derive modified optical Bloch equations: continued fraction expansion [ 41, statistical analysis of i(t) [ 51, decorrelation and diagrammatic methods [ 6,7], numerical methods [ 81, etc. We have recently presented [ 91 a method to solve the optical stochastic Bloch equations (OSBE), which is based on solving perturbatively in the external field the OSBE and then taking the average over the realizations t(t) of the formal solution for the coherences pba(t) . Even though this procedure proved to be analytically elegant, one may ask oneself if it is not possible to employ a different approach, namely, first to average the system of equations ( l.la)-( 1.lc) and then to solve directly for the average of the coherences. This Letter is devoted to the discussion of this question and the mathematical problems posed when following this approach.
2. Method and discussion Let us consider a transition frequency z(r) such that the stochastic function q(t) in Eq. ( 1.2) is characterized by the average (v(t)) = 0 and a correlation function (v(t) q(s) > whose mathematical form will be specified later on. The average of OSBE over the distribution of q(t) will be given by
d(p,(O), dt
dhxz(Q )a dt
_ 2i
= -(l/T,
+iwo)(P6a(t))rl-i(rl(t)Pba(t))?l-
d(padt>), = EM&)), dt
(2.la)
- x [(HI)ab(Pba(t))$-(Pab(t))g(HI)bal
dt
*
$ (K)b0(PD(r))V9
(2.lb) (2.lc)
Here ( ), denotes the average over the distribution of realizations of v(t). It is important to notice that in Eqs. (2.1 b), (2. lc) we have a cross correlation term involving the coherences pbaand the noise v(t) . Thus, if this system is to be solved it is necessary first to deal with this term. Several methods have been proposed to solve similar kinds of multiplicative stochastic differential equations ] 10,111. As was done in Refs. [ 10,111, the cross correlation appearing in (2.lb) and implicitly in (2.1~) will be solved with the help of a theorem due to Novikov [ 121. It states that if q(t) is a Gaussian random function and x[ ~1 is a functional of v(t), then
(2.2) In the functional x[ 91 is identified with the coherence pbll, to apply (2.2) we need to calculate the functional derivative Sp,,( t) /Sq( t’) . In order to do so, Eq. ( 1.lb) can be formally integrated rendering
165
P.J. Colmenares et al. /Physics Letters A 199 (1995) 163-168
f
~ba(t)=pbn(O)- ;
I
I
I ds 1l/T, +itW
ds (~JI)~~(s)PD(s) -
Next, by taking the functional
derivative
ds (HI)ba(~)
.
(2.3)
of (2.3) with respect to q( t’) we get
-
ds [l/T,
-
SPbu(S) -iph(t’) S?7(t’)
+i{((s)]
0
r’ This expression
l~b,z(s)
0
0
satisfies the differential
.
(2.4)
equation
-[l/T, +i&t)]
SPba(O
-
kt(t’)
(2.5)
’
subject to the initial condition (2.6) Integrating
(2.5) and taking the average over the distribution
of realizations
of q(t),
we obtain
(2.7)
With this result, the evaluation of expression (2.2) depends on the form of the correlation function (q(t) v( t’) ). If the rate of frequency fluctuations T is much larger than the width of the distribution of frequency fluctuations produced by the surrounding media, the thermal bath is well described as a white noise. For this case, the correlation function of 7)(t) for a given noise intensity y is (2.8)
(77(r)7)(rf))=2y6(r-t’). Thus, after substituting renders (77(r)P~At))~=
(2.7) and (2.8) into (2.2) and performing
the necessary integrations,
theorem
(2.9)
-iY(PbAO)rl,
whose complex conjugate
Novikov’s
is (2.10)
(77(t)P,b(f))7)=iy(p~b(t>),.
By taking this result into account, Eqs. (2.1) read
d(pDW), dt d(Pdf)
f
[ (f&)ab(pba(f)),j
>t, _
dt d(Pndt)
dt
>T
-
(Pab(r))v(Hl)bal
-
T
‘I 1
-P'D 7
(2.11a)
- - [l/Th+i~ol(pdO),-
i
(fW~,(pdt)),,
(2.1 lb)
= - tl/T;-i%l(Pab(t))V+
;
(HI)nb@D(f))vr
(2.1 lc)
166
P.J. Colmenares et al. /Physics
where we have defined an effective transversal 1 -=T;
Letters A 199 (1995) 163-168
relaxation time 1/T;
by
1 (2.12)
T2 +-I’
which measures the delay in the relaxation of the ensemble polarization due to the heat reservoir. Furthermore, the splitting of the upper level due to the bath is indirectly measured in this approach through the parameter y. It is easy to check that we can recover the classical result, Eqs. ( 1.la)-( 1. lc), for vanishing noise strength. Here, it is worth noticing that the separation of the contribution of the heat reservoir in the 1 /T a expression is a consequence of the fact that white noise has a relaxation time very close to zero. Thus, a time scale separation between the contribution of the bath and all other contributions should be possible. Once at this point, we realize that the system of equations (2.11) and the OCBE are algebraically similar and therefore, can be solved by similar mathematical approaches. In particular, if we apply a perturbation method in the intensity of the laser field it can be proved that the results reported by us [9] for the averages (pba( t)), and (pab( t) ), in the white noise limit can be recovered. If, on the other hand, the relaxation rate r is comparable to the width of the distribution of frequency fluctuations, the thermal bath has to be described as a colored noise. If this case is considered, integration of (2.2) becomes a much more complicated task. This can be illustrated if an Omstein-Uhlenbeck process (OUP) [ 131 is considered. For this process, the correlation function is an exponential decay, i.e. (2.13)
(rl(t)77(t’))=Y7exp(-71t--1’OT
where again y is the strength of the noise, r is its bandwidth and the product -yr is the variance of its distribution. Since an OUP is Gaussian, we could apply Novikov’s theorem to calculate the cross correlation (rev). However, due to mathematical difficulties involving the evaluation of the integrals appearing in (2.2), the theorem cannot be applied so easily as in the white noise case. Despite this, an alternative approximate manner of dealing with this problem is possible if the so-called “quasi-white noise limit” [ 111 is considered. This approximation is based on the assumption that in the small correlation time limit, the correlation function of the OUP decays SO fast that only times t = t’ are important in the evaluation of the integral appearing in (2.2). If this is the case, Sp,,( t) / STJ(t’) can be substituted by its expansion around t= t’ [ 10,111. Thus, to first order this expansion reads
&ha(t) -=Wt’)
Cjpbn(t) Sq(t’)
,t_,
+ d &&3(f) (f-t) dt’ h(f) ,-+,
is given by Eq. (2.6) evaluated at t’ = t, that is
The first term of this expansion CjPba(f)
Mt’)
=
-ipdt)
(2.14)
.
(2.15)
,
,“I
while the second is obtained by taking the derivative of (2.4) with respect to t’. The result of this is
d &m(t) --= dr’ +(t’)
-i-
dpdt’> +[1/T2+i&t’)] dt’
~Pba(S) Wt’) S+,’
I (Who(s)
$$)-
jd.s$([liT,+i&s)] I’
E).
(2.16)
In order to proceed, we need to take the functional derivative of the difference in the state population, po, in the limit s + t’. This is done by following a procedure similar to the one employed to calculate Sph/G~ (Eqs. (2.3)(2.6) ) after which the following result is obtained,
P.J. Colmenares
(H)
-=-
I
et al. /Physics
(2.17)
Aldus.
ob
,I
After evaluating
167
Letters A 199 (1995) 163-168
I’
the last two equations at s = t’ we finally obtain
--d a~bn(f) dt’ Sq(t’)
,,_, =
-i
dPba(t) ____ -iph(t)[l/Tz dr
The cross correlation function into (2.2). This reads
(pba(
t) v(
+i{(t)]
(2.18)
.
t) ) is found then by taking the average of Eq. (2.14)
and substituting
it
I
(Pbntt)dt))r,
=
-
YT
I
(H,)bo(S) n
dsexp(-dt--I)
(PD(s)).(s-O
where we have used Eq. (2.13) for the approximate result, it lacks the simplicity after the result (2.19) is substituted into state populations turn into a complicated
I
(2.19)
)
0
correlation function of the OUP. We notice that despite (2.19) being an displayed by that obtained for the white noise case, Eq. (2.9). Moreover, Eqs. (2.la)-( 2. lc), the equations for the averages of the coherences and system of integro-differential equations.
3. Final comments In summary, we have presented a methodology for solving the OSBE for systems interacting, to any degree, with a bath reservoir. This is accomplished by employing a theorem (due to Novikov) [ 121 for stochastic Gaussian processes, that has been used in the past but in different contexts [ 10,111. Two cases have been studied. In the first one a white noise bath is considered. This leads to OSBE algebraically similar to OCBE. This result turns out to be important, since it makes it possible to solve the OSBE by means of the extensive number of techniques available for the OCBE. It is worth noticing, that for this case, the result is identical to an equivalent one obtained by us [ 91 when the Bloch equations are solved by means of a perturbative treatment and the result is averaged over the distribution of the random variable. Furthermore, the equations show an effective transversal relaxation time, where the contribution of the heat reservoir appears explicitly. This contribution can be associated with the rate constant of spectral diffusion within the upper state. Unfortunately, when colored noise is considered (OUP), the solution of the problem becomes more complicated. However, if the “quasi white-noise” limit is taken we are able to transform the Bloch equations into a set of integrodifferential equations, which to our knowledge, can only be solved by numerical techniques.
Acknowledgement This research was supported in part by Consejo de Desarrollo Cientifico, Humanistico versidad de 10s Andes (CDCHT-ULA, Grant C-671-94B), Decanato de Investigaciones Bolivar and Consejo Nactional de Investigaciones Cientfficas y Tecnologicas (CONICIT).
References 1 I ] K. Blum, Density matrix: theory and applications
(Plenum, New York, 1981).
y Tecnologico of Uniof Universidad Simon
168
P.J. Colmenares et al. /Physics
L,etters A I99 (1995) 163468
[2] R.G. DeVoe and R.G. Brewer, Phys. Rev. Lett. 50 (1983) 1269.
[3] P.R. Berman and R.G. Brewer, Phys. Rev. A 32 (1985) 2784. [4] G.S. AgarwaJ and N. Nayak, Phys. Rev. A 33 (1986) 391. [5] P.R. Berman, J. Opt. Sot. Am. B 3 (1986) 572; 3 (1986) 564. [6] H. Tsunetsugu and E. Hanamura, Solid State Commun. 55 (1985) 397. [ 71 T. Kushida and S. Kinoshita, Phys. Rev. A 41 ( 1990) 6042. [S] G. Vemury and G.S. Agarwal, Phys. Rev. A 48 (1993) 2386. [9] P.J. Colmenares, R. Almeida and J.L. Paz, Stochastic effects of the solvent on the absorptive and dispersive processes of the two-level system up to first order in the external field, submitted for publication. [ IO] J.M. Sancho, M. San Miguel, S.L. Katz and J.D. Gunton, Phys. Rev. A 26 (1982) 1589. [ 111 P.J. Colmenares, J. Theor. Biol. 161 (1993) 175. [ 121 EA. Novikov, Sov. Phys. JBTP 20 ( 1965) 1290. [ 13) R.F. Fox, Phys. Rep. 48 (1978) 181.
PHYSICS
ELSEVIER
Physics Letters A 199 (1995)
LETTERS
A
163-168
On the optical stochastic Bloch equations Pedro J. Colmenares a, J.L. Paz b, Rafael Almeida a aDepartamento de Quimica, Universidad de 10s Andes, L.a Hechicera, Me%& 5101, Venezuela h Departamento
de Q&mica. Vniversidad Simbn Bolivar, Apartado 89000. Caracas 108OA, Venezuela
Received 24 March 1994; revised manuscript received 17 January 1995; accepted for publication Communicated by A.R. Bishop
19 January
1995
Abstract
We present a methodology, based on the use of a theorem due to Novikov, to solve the optical stochastic Bloch equations for a two-level system up to first order in the external field in the presence of a heat reservoir. Two cases are studied. In the first, a white noise bath is considered, while in the second colored noise (Omstein-Uhlenbeck process) is analysed.
1. Introduction
The dynamics of nonlinear optics phenomena in molecular systems is normally analyzed by means of the optical “conventional” Bloch equations (OCBE) . This set of equations is derived from the Liouville-von Neumann equation in which the state of the system is described by the reduced density operator. For a two-level system defined by the states 1a) and 1b), the OCBE read
h,(t) dt
dtdt) -
dt
= ; [(K)ab(OPba(f) -P&At) (~l),(t)l =
-
(im0
dpadt> -=-
dp&)
dt
dt
+
l/TdpbJt)
’
-
i
(K)bAt)dt)
-
$ (PD-p3
9
(l.la)
1
,
(l.lb)
(l.lc)
value in the absence of where pr,( t) = p,,(t) - pbb( t) is the difference in state populations, pb is its equilibrium radiation and T, and T2 are the longitudinal and transversal relaxation times, respectively [ 11. Here, it is assumed that the states )a) and ]b) are coupled through an electric dipole transition moment, /.L,whose nondiagonal matrix elements are I-L,~and /_Q_and its diagonal components pan and /..Q,~(permanent dipole moments) are assumed to be negligible compared to their nondiagonal counterpart and therefore taken as zero. The interaction Hamiltonian that corresponds to the incident wave E( t) is taken as H,(t) = - p.E( t) , In this equation four typical frequencies arise as a result of the different time scales of the processes responsible for them. The first is the Rabis frequency, 0 = phE/h, which is determined by the coupling between the electromagnetic field and the system, the second the transition Bohr frequency w, of the two-level atom and finally, the two frequencies l/T, and 1 /T2 associated to solvent dissipation mechanisms. Elsevier Science B.v. SSDIO375-9601(95)00081-X
164
P.J. Colmenares et al. /Physics
Letters A 199 (1995) 163-168
Most of the work done in nonlinear optics with the OCBE considers the transition frequency w, as a parameter. This assumption is valid in regimes of weak solute-solvent interaction for a homogeneous distribution of the resonant frequencies. However, if one is interested in the development of radiation from a two-level system driven by a monochromatic field and immersed in a condensed medium or in a dense vapor cell, the assumption is no longer valid. Thus, it has been reported that the OCBE cannot describe observed optical signal intensities for systems interacting strongly with the surrounding medium (i.e. an impurity in an ion solid [ 2,3] ). For those cases, molecular collisions induce frequency shifts, and due to the nature of the collisions, they lead to stochastic modulation of the frequency. This shift in the transition frequency becomes a function of time and can be represented by Z(r)=%+rl(r)
9
(1.2)
where q(t) comprises all the stochasticity of z(t) . In this context, a number of works [ 4-81 have been presented recently. Several treatments have been employed to derive modified optical Bloch equations: continued fraction expansion [ 41, statistical analysis of i(t) [ 51, decorrelation and diagrammatic methods [ 6,7], numerical methods [ 81, etc. We have recently presented [ 91 a method to solve the optical stochastic Bloch equations (OSBE), which is based on solving perturbatively in the external field the OSBE and then taking the average over the realizations t(t) of the formal solution for the coherences pba(t) . Even though this procedure proved to be analytically elegant, one may ask oneself if it is not possible to employ a different approach, namely, first to average the system of equations ( l.la)-( 1.lc) and then to solve directly for the average of the coherences. This Letter is devoted to the discussion of this question and the mathematical problems posed when following this approach.
2. Method and discussion Let us consider a transition frequency z(r) such that the stochastic function q(t) in Eq. ( 1.2) is characterized by the average (v(t)) = 0 and a correlation function (v(t) q(s) > whose mathematical form will be specified later on. The average of OSBE over the distribution of q(t) will be given by
d(p,(O), dt
dhxz(Q )a dt
_ 2i
= -(l/T,
+iwo)(P6a(t))rl-i(rl(t)Pba(t))?l-
d(padt>), = EM&)), dt
(2.la)
- x [(HI)ab(Pba(t))$-(Pab(t))g(HI)bal
dt
*
$ (K)b0(PD(r))V9
(2.lb) (2.lc)
Here ( ), denotes the average over the distribution of realizations of v(t). It is important to notice that in Eqs. (2.1 b), (2. lc) we have a cross correlation term involving the coherences pbaand the noise v(t) . Thus, if this system is to be solved it is necessary first to deal with this term. Several methods have been proposed to solve similar kinds of multiplicative stochastic differential equations ] 10,111. As was done in Refs. [ 10,111, the cross correlation appearing in (2.lb) and implicitly in (2.1~) will be solved with the help of a theorem due to Novikov [ 121. It states that if q(t) is a Gaussian random function and x[ ~1 is a functional of v(t), then
(2.2) In the functional x[ 91 is identified with the coherence pbll, to apply (2.2) we need to calculate the functional derivative Sp,,( t) /Sq( t’) . In order to do so, Eq. ( 1.lb) can be formally integrated rendering
165
P.J. Colmenares et al. /Physics Letters A 199 (1995) 163-168
f
~ba(t)=pbn(O)- ;
I
I
I ds 1l/T, +itW
ds (~JI)~~(s)PD(s) -
Next, by taking the functional
derivative
ds (HI)ba(~)
.
(2.3)
of (2.3) with respect to q( t’) we get
-
ds [l/T,
-
SPbu(S) -iph(t’) S?7(t’)
+i{((s)]
0
r’ This expression
l~b,z(s)
0
0
satisfies the differential
.
(2.4)
equation
-[l/T, +i&t)]
SPba(O
-
kt(t’)
(2.5)
’
subject to the initial condition (2.6) Integrating
(2.5) and taking the average over the distribution
of realizations
of q(t),
we obtain
(2.7)
With this result, the evaluation of expression (2.2) depends on the form of the correlation function (q(t) v( t’) ). If the rate of frequency fluctuations T is much larger than the width of the distribution of frequency fluctuations produced by the surrounding media, the thermal bath is well described as a white noise. For this case, the correlation function of 7)(t) for a given noise intensity y is (2.8)
(77(r)7)(rf))=2y6(r-t’). Thus, after substituting renders (77(r)P~At))~=
(2.7) and (2.8) into (2.2) and performing
the necessary integrations,
theorem
(2.9)
-iY(PbAO)rl,
whose complex conjugate
Novikov’s
is (2.10)
(77(t)P,b(f))7)=iy(p~b(t>),.
By taking this result into account, Eqs. (2.1) read
d(pDW), dt d(Pdf)
f
[ (f&)ab(pba(f)),j
>t, _
dt d(Pndt)
dt
>T
-
(Pab(r))v(Hl)bal
-
T
‘I 1
-P'D 7
(2.11a)
- - [l/Th+i~ol(pdO),-
i
(fW~,(pdt)),,
(2.1 lb)
= - tl/T;-i%l(Pab(t))V+
;
(HI)nb@D(f))vr
(2.1 lc)
166
P.J. Colmenares et al. /Physics
where we have defined an effective transversal 1 -=T;
Letters A 199 (1995) 163-168
relaxation time 1/T;
by
1 (2.12)
T2 +-I’
which measures the delay in the relaxation of the ensemble polarization due to the heat reservoir. Furthermore, the splitting of the upper level due to the bath is indirectly measured in this approach through the parameter y. It is easy to check that we can recover the classical result, Eqs. ( 1.la)-( 1. lc), for vanishing noise strength. Here, it is worth noticing that the separation of the contribution of the heat reservoir in the 1 /T a expression is a consequence of the fact that white noise has a relaxation time very close to zero. Thus, a time scale separation between the contribution of the bath and all other contributions should be possible. Once at this point, we realize that the system of equations (2.11) and the OCBE are algebraically similar and therefore, can be solved by similar mathematical approaches. In particular, if we apply a perturbation method in the intensity of the laser field it can be proved that the results reported by us [9] for the averages (pba( t)), and (pab( t) ), in the white noise limit can be recovered. If, on the other hand, the relaxation rate r is comparable to the width of the distribution of frequency fluctuations, the thermal bath has to be described as a colored noise. If this case is considered, integration of (2.2) becomes a much more complicated task. This can be illustrated if an Omstein-Uhlenbeck process (OUP) [ 131 is considered. For this process, the correlation function is an exponential decay, i.e. (2.13)
(rl(t)77(t’))=Y7exp(-71t--1’OT
where again y is the strength of the noise, r is its bandwidth and the product -yr is the variance of its distribution. Since an OUP is Gaussian, we could apply Novikov’s theorem to calculate the cross correlation (rev). However, due to mathematical difficulties involving the evaluation of the integrals appearing in (2.2), the theorem cannot be applied so easily as in the white noise case. Despite this, an alternative approximate manner of dealing with this problem is possible if the so-called “quasi-white noise limit” [ 111 is considered. This approximation is based on the assumption that in the small correlation time limit, the correlation function of the OUP decays SO fast that only times t = t’ are important in the evaluation of the integral appearing in (2.2). If this is the case, Sp,,( t) / STJ(t’) can be substituted by its expansion around t= t’ [ 10,111. Thus, to first order this expansion reads
&ha(t) -=Wt’)
Cjpbn(t) Sq(t’)
,t_,
+ d &&3(f) (f-t) dt’ h(f) ,-+,
is given by Eq. (2.6) evaluated at t’ = t, that is
The first term of this expansion CjPba(f)
Mt’)
=
-ipdt)
(2.14)
.
(2.15)
,
,“I
while the second is obtained by taking the derivative of (2.4) with respect to t’. The result of this is
d &m(t) --= dr’ +(t’)
-i-
dpdt’> +[1/T2+i&t’)] dt’
~Pba(S) Wt’) S+,’
I (Who(s)
$$)-
jd.s$([liT,+i&s)] I’
E).
(2.16)
In order to proceed, we need to take the functional derivative of the difference in the state population, po, in the limit s + t’. This is done by following a procedure similar to the one employed to calculate Sph/G~ (Eqs. (2.3)(2.6) ) after which the following result is obtained,
P.J. Colmenares
(H)
-=-
I
et al. /Physics
(2.17)
Aldus.
ob
,I
After evaluating
167
Letters A 199 (1995) 163-168
I’
the last two equations at s = t’ we finally obtain
--d a~bn(f) dt’ Sq(t’)
,,_, =
-i
dPba(t) ____ -iph(t)[l/Tz dr
The cross correlation function into (2.2). This reads
(pba(
t) v(
+i{(t)]
(2.18)
.
t) ) is found then by taking the average of Eq. (2.14)
and substituting
it
I
(Pbntt)dt))r,
=
-
YT
I
(H,)bo(S) n
dsexp(-dt--I)
(PD(s)).(s-O
where we have used Eq. (2.13) for the approximate result, it lacks the simplicity after the result (2.19) is substituted into state populations turn into a complicated
I
(2.19)
)
0
correlation function of the OUP. We notice that despite (2.19) being an displayed by that obtained for the white noise case, Eq. (2.9). Moreover, Eqs. (2.la)-( 2. lc), the equations for the averages of the coherences and system of integro-differential equations.
3. Final comments In summary, we have presented a methodology for solving the OSBE for systems interacting, to any degree, with a bath reservoir. This is accomplished by employing a theorem (due to Novikov) [ 121 for stochastic Gaussian processes, that has been used in the past but in different contexts [ 10,111. Two cases have been studied. In the first one a white noise bath is considered. This leads to OSBE algebraically similar to OCBE. This result turns out to be important, since it makes it possible to solve the OSBE by means of the extensive number of techniques available for the OCBE. It is worth noticing, that for this case, the result is identical to an equivalent one obtained by us [ 91 when the Bloch equations are solved by means of a perturbative treatment and the result is averaged over the distribution of the random variable. Furthermore, the equations show an effective transversal relaxation time, where the contribution of the heat reservoir appears explicitly. This contribution can be associated with the rate constant of spectral diffusion within the upper state. Unfortunately, when colored noise is considered (OUP), the solution of the problem becomes more complicated. However, if the “quasi white-noise” limit is taken we are able to transform the Bloch equations into a set of integrodifferential equations, which to our knowledge, can only be solved by numerical techniques.
Acknowledgement This research was supported in part by Consejo de Desarrollo Cientifico, Humanistico versidad de 10s Andes (CDCHT-ULA, Grant C-671-94B), Decanato de Investigaciones Bolivar and Consejo Nactional de Investigaciones Cientfficas y Tecnologicas (CONICIT).
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