Generalised non-linear optical master equations

J. Phys. B: At. Mol. Phys. 19 (1986) 3457-3475. Printed in Great Britain

Generalised non-linear optical master equations taking into account the correlation time of relaxational perturbations S Ya Kilin and A P Nizovtsev Institute of Physics, BSSR Academy of Sciences, Minsk 220602, USSR

Received 4 April 1985, in final form 25 November 1985

Abstract. Generalised master equations in two equivalent forms, namely differential master equations ( D M E ) and integro-differential master equations ( I D M E ) , are obtained. Both differ from the Bloch ones in that they take into account the density matrix dynamics during the correlation time 7c of the relaxational perturbations. On the basis of these equations, the non-linear optical problems of high-power radiation absorption, the resonance fluorescence spectrum and the free induction decay signal are discussed.

1. Introduction

In describing non-linear optical phenomena, relaxation is usually taken into account by introducing phenomenological decay rates into the equations of motion for the density matrix (master equations ( M E ) ) . For resonance phenomena these equations are the Bloch ones, in which relaxation is described by two characteristic times, those of longitudinal and transverse relaxation. At the present time, however, there are a number of optical experiments whose results are at variance with the Bloch theory: (i) the non-Lorentzian driving field frequency dependence of the resonance fluorescence integral intensity (Carlsten et al 1977); (ii) the unusual power dependence of absorption saturation due to the driving field effect on the relaxation process (the bleaching effect on the spectral line wings (Zuev et a1 1976, Bonch-Bruevich et a1 1980)); and (iii) the ‘anomalous’ power dependence of the free induction decay rate (De Voe and Brewer 1983). In order to describe these optical phenomena accurately, it is necessary to remove the main assumption of the relaxation theories that result in the Bloch ME-the assumption that the correlation time T~ of perturbations responsible for relaxation is negligible. In practice, however, the relaxational perturbations induced by a heat bath are characterised by a finite correlation time T~ (collision duration in gases, inverse spectrum width of electron-phonon interactions in crystals etc). Thus there exists a timescale of the order of T~ upon which one can speculate about the dynamics of relaxation or of the bath motion. In general, it is clear that the assumption T~ + 0 is justified only when the quantum system under consideration (atom, molecule, impurity in crystal etc) may be considered to be unchanging during time intervals of about T ~ . In optics, this assumption is violated when the characteristic time ti=SZ-’ of an atom-field interaction is less than T ~ i.e. , when SZ.rCa 1, where SZ = ( E ~ + ~ V is~ the ) ” ~ 0022-3700/86/213457

+ 19$02.50

@ 1986 The Institute of Physics

3457

3458

S Ya Kilin and A P Nizovtsev

generalised Rabi frequency, E is the resonance detuning and V is the Rabi frequency. In this case, the atom has time to be modified by the field during T ~ and , as a result the atomic relaxation becomes field dependent. Earlier attempts to take into account the effects of field on relaxation were made in order to describe radiofrequency rather than optical phenomena, namely the NMR saturation (Kubo and Tomita 1954, Redfield 1955, Bloch 1956, 1957, Tomita 1958, Hubbard 1961, Abragam 1961, Argyres and Kelley 1964). Among these authors, only Argyres and Kelley incorporated the correlation time rc strictly into their formalism, which was based on the projection operator method (Zwanzig 1960). Unfortunately, the general M E obtained by them has not found a wide applicability until quite recently. In linear optics, a dependence of relaxation on field is well known and manifests itself as the field-frequency dependence of relaxation rates following from a number of dynamical and stochastic theories of spectral line broadening (see, e.g., Mukamel 1979, 1982a, Burnett et a1 1980, Burnett and Cooper 1980, Boulet et a1 1980, Allard and Kielkopf 1982, Mukamel and Grinbert 1982) which describe in a natural way the non-Lorentzian character and asymmetry of spectral line shapes. There is not, at present, very much theoretical work in non-linear optics which reflects in its formalism the dynamics of relaxation over times of about rc and takes into account powerful radiation-field effects on relaxation. Takagahara et a1 (1977a, b) have developed the so called 'stochastic theory of the intermediate-state interaction' and have discussed second-order optical processes on the basis of this theory, adopting several different stochastic models for relaxational perturbations. Mukamel and coworkers have utilised the tetradic scattering formalism (Fano 1963, Ben-Reuven and Mukamel 1975) to describe dephasing effects on multiphoton processes (Mukamel 1982b, 1983), as well as the absorption line broadening (Rabin et a1 1982) and resonance fluorescence spectrum (Deng and Mukamel 1984) in the medium-field-strength case y t ’ > 0. According to (3), p ( t ) = d ( t , O ) p ( O ) = 9 ( t , t ’ ) p ( t ’ ) = 9 ( t , t’)9(t’,O)p(O)

i.e. the Green’s matrix of the (A+ B) system satisfies the semi-group relation 9(t,0) = 9(t,t ’ ) 9 ( t ’ , 0).

(31)

This feature is also possessed by the Green’s matrix of M E for the reduced density matrix cr(t) in the autonomous regime. In general, we have a(t)=D(t,O)o(O)=SpB 9 ( t , O)p(O)=D(t,t’)a(t’)+Sp,

9(t,t ’ ) ( Z - P ) p ( t ’ ) .

(32)

This suggests that the Green’s matrix D ( t, t ’ ) of the A system satisfies the semi-group condition defining the Markovian processes (Stratonovich 1963):

D ( t, 0) = D ( t, t’)D(t’, 0)

3464

S Y a Kilin and A P Nizovtsev

or, equivalently, in some basis: kf

only in the case where Sp&(t, t ’ ) ( I - P ) p ( t ’ ) = D ( t O)a(O)-D(t, , t’)a(t’)=o.

(34)

The latter is fulfilled when the following three conditions hold: (a) the intermediate time t’ is separated from the initial time t = 0 (which may be t = --CD as well) by a time greater than the time T , of A-system relaxation to the steady state; ( b ) the difference t - t ’ is greater than the correlation time T~ (see (22) and (23)); and (c) the initial density matrix a(0)is the steady-state density matrix. Thus the time behaviour of the A system in regions I and 111 (figure 1) will be non-Markovian, whereas in region I1 it will be Markovian. Obviously, if T , >> T~ the region I11 has no pronounced effect on the system behaviour; the system evolution will, therefore, be Markovian and both DME (1 1) and IDME (15) become DME with the time-independent relaxation operator R. In the case of IDME (15) this follows immediately from equation (16). Otherwise, for T ~ T~ S it is necessary to use the generalised ME (11) and (15). I 0

I1 Tr

111 f-TC

f

f’

Figure 1. Regions of non-Markovian (I, 111) and Markovian (11) time behaviour of the reduced-system evolution.

(v) If the A-system-bath coupling is taken into account in the generalised DME (11) and IDME (15) to the same order (to second order, say) then the results following from these two types of equations will be different (Mukamel 1979). This is due to the fact that the use of perturbation theory in these equations actually means the use of different statistical properties of the bath. Thus the retention of the A-system bath coupling u p to second order in DME (11) corresponds to the neglect of the cumulants mn(T17 , 2 r. . . , 7,) with n 3 3 in expansion (20).In I D M E (19, second-order perturbation theory (i.e. the Born approximation) corresponds, according to (25) and (30), to the neglect of the time ordered cumulants 0,( 7 , , T ~ .,. . , T ~ =) Tm, ( 7 , , T ~ . ., . , 7,) with n23. (vi) The following question may arise: why construct the M E with the kernel rC and relaxation operator R in terms of the Green’s matrices of reduced system evolution, which just determine the dynamics of all physical quantities of the A system? The answer is that for the case where the A system is coupled to a number of other bath-like systems (Bi systems) besides the bath (B system) itself, these new couplings may be taken into account in M E obtained by considering additive relaxational terms. This will be true when correlation times T~~ of relaxational perturbations from the additional B, systems are much smaller than the characteristic times of the A system. (vii) The above M E can also be used as the basis of the stochastic theory of relaxation, where the system-bath coupling is thought of as a random process with specified properties. The averaging over the bath states in the above expressions should in this case be replaced by averaging over random process realisations. Based on this approach, absorption and spectral redistribution in high-power resonant radiation scattering, as well as free induction decay, are considered in D 3.

3465

Generalised non-linear optical master equations

3. Stochastic theory of adiabatic relaxation Let monochromatic radiation of frequency coo and amplitude Eo act on the resonance transition la)-lb) of the quantum system (atom, molecule, impurity in a crystal etc) subject to relaxational perturbations from the bath (atom-perturber collisions in gases, electron-phonon interaction in solids etc). For simplicity, we restrict ourselves to adiabatic perturbations, i.e. those which cause only stochastic modulation of the transition frequency iJ&: f&b( t ) = + 80&( t ) , where 8w&( t ) is the random process. Then the equations for the elements of an unaveraged density matrix p in the basis rotating at the field frequency wo are r = -i(

+ 2uYX)r

E$‘,

(35)

-pabEo/h is the la)-lb) transition-field Eo interaction matrix element, E , = W , b ( t ) wo is the random frequency detuning of the resonance and Yy( v = x, y, z ) are spin matrices for spin Y = 1 in the Cartesian basis representation (Warshalovich et a1 1975):

V =

(1 1 1)

Y x = 0 0 -i

Yy=(!i

; ;)

Yz=(;

;;)

(37)

satisfying the standard commutation relations [YX,Y,]= -iYy [ Y y Y,] , = -iY,. (38) According to relations (7)-( 16), the main quantity determining the relaxation terms of M E is the Green’s matrix [ Y x , Yyl = iY2

(39) The calculation of D( t, 0) can be performed in several ways, which are discussed below. ( a ) Disentanglement of time ordered exponents. Using the Feynman theorem (Feynman 1951, Popov 1958), the matrix D ( t , 0) can be written as ~ ( t0), = (e-””+ e-@”. e-?”-) (40)

or, explicitly for the basis vector In) = ( o h b , z, O b u ) , (eP)* -2( y eP)* -( y2 e@)*) m t , O ) = (r/ePI) -1+2(lePI) (r*lePI) -( y 2 e@) -2( y e@) (eP>

i

where the functions a ( t ) ,P ( t ) and y ( t ) are determined by the relations ici=tm 2 +

E , ~ - v

(.(0)=0)

P = i \ o ‘ d ~(E7+2ua(7)) y = iu

Iof

d7 exp( - p (

7)).

3466


Since the function U (t ) = exp( iu

jot

d r a(.))

satisfies the second-order linear differential equation ij+iEt ir

+ J u l z=~o

(42)

equations (40)-(42) link the problem directly to the widely developed theory of dynamic systems with fluctuating parameters (Kliatskin 1977). According to (14), IDME corresponding to (40a) and incorporating spontaneous transitions have the form b:b=-$A(+Lb-jof

+1

dTjL;‘U;b+i

jot

dT d;-‘(

(+;a

lof

dTG~~‘;‘((+~,-ULb)

-UlbT)

where A is the spontaneous emission rate and the functions are defined by their Laplace transforms

j L b , GLb, U:, / 3 f b

and d:

?pi= [( - l / s + 2(lePl)r”1)(eP)[s1 + 2(y ep)[”](y*le@I)[”]]/A(s)- s

Note that in the case of collisional relaxation, the random process Swab(t ) is the Poisson pulse process Swab(t) = x ~ N = ~g(t - t k , &), where g(t - t k , &) is the single-perturber frequency shift determined by the random parameters t k (the time of closest approach) and & (impact parameter, relative velocity etc). In the usual binary-collision approximation, a one-perturber phaseshift g( t, 6) is considered instead of the all-perturber phaseshift Z kg(t - t k , 6).Using this function, equations (41) are solved (as a rule in an adiabatic approximation) and the functions ep, lePI, y e@,y(ePland y 2 ep are defined; these are then averaged over the random parameters 5. It is in this manner that binary collisions in the strong field are considered by Pestov (1984). It is shown below that the analysis of the radiation effect on the collisional relaxation can easily be carried out for N-perturber collisions as well by using the theory of Poisson random processes (Feynman and Hibbs 1965).

3467


( b ) Cumulant expansion. The expansion (20) is in principle a technique for calculating D ( t, 0) to any degree of accuracy. Let us go over in (39) to the ‘dressed’-state representation performed by the unitary transformation d‘=e x p ( - i W Y ) d exp(iWy)

9;= cos 0spz +sin tWx

(45)

Y : = -sin W, + cos 0,Yx where COS

e = Eo/o

a=

sin 8 = 2u/R

Then D(t, 0) = ( T exp( -i

I + 4(uI2/

Eo

=(Et).

Io1

dT (asp:+E 3 ’ , ) ) )

= exp(-iag:t)(

where Ft = E , -

EO(

T exp( -i

lot

dTE$f,(T)))

and

Spz(7)= exp(iCWiT)9’z exp(-iflY:T)

= cos esp:

-+ sin e[sp: exp(iaT)+ 9’exp(-iR~)].

(47)

Restricting ourselves to the second-order term in the cumulant expansion, we obtain

Thus the relaxation operator of the

R = do-’= -iaV: -

DME

becomes

dT (

~ r ~ T ) ~ z ~ t).z ( T -

Because of its importance for applications, we write such with (TA6 = exp(-isot)uab(t):

DME

(49)

in an explicit form,

(51b) Here, A is the spontaneous emission rate and K ( t - T ) = ( E,&) is the correlation function of stationary perturbations. It is seen from (50) that the effect of adiabatic perturbations

3468


in this approximation amounts to a renormalisation of the decay parameters Yab and of the matrix element of the system-field interaction in the equations for non-diagonal elements of the density matrix. In the autonomous regime, for t >> T ~ the , functions r i b and V i b become constants: Tab Gab

+

+ Re( K sin2 8 = vu1 - a-'{Im( KLin') + i cos e[ K C o-1 Re( K [ ' " ' ) ] } ~ = f A KIO1cos2 8

(52)

where K'" = :j dT K ( T )exp(-sT). Equations (50), with the time-independent parameters (52), can be used to determine the steady-state absorption coefficients for the strong field, as well as the linear response to the additional probe field at frequencies o satisfying the conditions / U - W ~ I T ~ < < 1 and Io -oo*521~c> ~1 (which ~ is true for both high field intensities 2lul.r~>> 1 and line wings l s o l ~>>c l ) , the probability of induced transitions tends to a value determined by spontaneous relaxation alone: dab

=21u/'($A)/[ai+ ($A)2].

Consequently, there occurs a narrowing of the absorption profile Wabs(.so)(as compared with the case a.rC 1. The spectrum of resonance jluorescence excited by high-power laser radiation has been studied in great detail both theoretically (see, for example, Apanasevich and Kilin 1979) and experimentally (Schuda et a1 1974, Wu et a1 1977). As a rule, theoretical analyses were performed using phenomenological relaxation rates. However, the dependence of relaxation parameters on the strong field was observed by Carlsten et a1 (1977). As mentioned above, this dependence arises as a result of the finite correlation time T~ and can be explained by the generalised ME. The steady-state resonance fluorescence spectrum is proportional to (Apanasevich and Kilin 1979) roo

G ( w ) = l i m Re t+m

J

dTexp(iw(t-T))(Y+(t)Y-(T))

= v l u & / 2 8 ( w -wo)+Re[DL:[”Uaa

+(DL~rsl-UbaS-l)~ab]s=i(w-wo)

(63)

where D y is the Green’s matrix of the generalised M E and uij are steady-state values of density matrix elements. Using (43), (44) and ( 6 2 a ) , we obtain, in the low-field case, 1+[2 Re(f~’)/A]+(g@-f!b‘)/s)] G ( w )= v8(w-wo)+Re 1sA f 21’ 4A f[bsd s =i(w-wo)

+

[

+

where

It follows from (64) that in the low-field case the resonance scattering spectrum consists of a non-shifted delta-like component of coherent (Rayleigh) scattering and a broadened resonance fluorescence component. The ratio of the integral intensities of these components

IF/IR= 2 Re (fiy)A-’ is associated with the Fourier transform of the characteristic functional (exp(i {h d7 ET)) of the random process if that causes the system’s relaxation. Consequently, measurement of the ratio IF/IR as a function of the frequency detuning provides complete information about the properties of stochastic perturbations. In particular, using this ratio it is possible to study interatomic potentials for collisional relaxation. As the driving field power increases, there appears a line in the scattering spectrum at a so called three-photon frequency, and the spectrum assumes the characteristic form of a triplet. In the case of well separated triplet components, the scattered light spectrum may be written as

S Ya Kilin and A P Nizoutseu

3472 where n,

= f+z’

are the populations of the ‘dressed’ states and -1

.

?s1=;(;A)(2+sin2 0 ) + ( I o l d t exp(-st)*(f,))

Here, x(E I ) = (exp(i cos 0 d.r &)) is the characteristic functional of the random process. The imaginary and real parts of calculated at relevant frequencies, yield a shift and a width of side componects which depend on the scattered photon frequency w. Such calculations can be performed for the Gaussian and Poisson random processes based on (56) and (57). In the impact regions for side components of the triplet, i.e. in the regions where / w - wo+ RI .rc. Then

f G= KIolcos2 0 +:(fA)(2+sin2 e )

(66)

for the case of the Gaussian process and

f p =(2r[cos

o ( c , / u ) B ( ; ;f ( n - I ) ] ~ @ - ~ )W I -( 1 -

n)l

x e x p [ i r / ( l - n)])u+$(iAj(2+sin20 ) for the case of the Poisson process, describing collisions of particles with the potential change power law g ( t , U, b ) = C n / ( b 2 + u 2 t 2 ) n / 2

where b is the impact parameter, U is the particle velocity, B ( x , y ) and T(x) are beta and gamma functions. It follows from equations (66) and (67) that as the field power increases, the width of the triplet side components decreases to that determined by spontaneous emission alone. The dependence of the side-component narrowing on the field power is determined by the nature of the random process, and, in particular, for collisional relaxation by the power n of the potential f , (cos O ) z / ( f l - l ) , which may be the basis for its experimental determination. Free induction decay is one of the transient coherent optical phenomena for which the finiteness of the correlation time .rc may be essential, especially in solids. In F I D experiments one examines the beating of the field E’( t ) = E l exp( -iwht) + cc at a non-resonant frequency U ; , to which the laser frequency wo is abruptly switched at time t = 0, and the field induced by macroscopic polarisation of the sample, Eab(t ) u a b ( E O ) exp( -hob -;A - KO).Here, gab(E ~ is) the steady-state value of aibfor t < 0, when the system interacts with the resonant laser field E ( t ) , and Ko=I:d.rK(r), K ( t - T ) = ( E ~ E , ) . The measured beat intensity is actually a result of averaging over the inhomogeneous distribution of transition frequencies w,b in the sample:

-

-

Isignal(t)

=

I

de0 EO) Re(Eab(t)E’(t)).

In order to calculate the latter, we have used the Born approximation I D M E (54). Our choice was stimulated by the F I D experiment of De Voe and Brewer (1983) on the Pr3+-LaF3 system for which the homogeneous line broadening is due to fluctuating F-F nuclear spin flips which randomly modulate the Pr3+ optical transition frequency via the Pr-F nuclear dipolar interaction. This modulation can be well described by the random telegraphic noise for which the Born approximation M E are exact (Agarwal


3473

1979). Thus we have taken the correlation function K ( t - 7) in the form K ( t - 7)= ( K o / ~ cexp(-lt ) - T ~ / T ~ We ) . have also assumed the widths of the poles of vaab(eO) to be smaller than those for the distribution P ( E ~ As ) . a result we obtain

-

Isigna,(f) cos[(oo-w&)t] cos(qf+S)[exp(-T-t)+ g exp(-r+t)].

(68)

A detailed analysis of (68) was carried out in a previous paper (Apanasevich et al 1984), where a comparative analysis of similar work on this subject was also given. In the present paper we discuss only the main features of Isignal(t). There are four specific regions of the parameters 4Ko/A = 2( TI/ T2- 1) and I V U ) T ~ = U where the signal behaviour is essentially different. These regions are separated for the case AT^ TI; U >> 2( for T2< T I ) .The FID rate in this case is r- = rR= T;’ + f i D is the narrow intermediate region adjacent to the curve D = 0 and the line U* = 5. A characteristic feature of this region is the bi-exponential FID with the rates Ti= &A+ KO+( p D1’2)1’2 and with the weight of the second exponent of equation (68),

/VI.

*

g - 1.

By virtue of the above subdivision into four regions, the intensity dependence of the FID rate will be different for three types of systems (see figure 2): for TI < T2, the system goes from the Bloch regime immediately into the Redfield one (type a);for T I > T , > ~ T the ~ , system goes first from the Bloch regime to that with changed frequency 7 and phase 5 of beats, where there is an interval of values of v in which T - ( u ) = (1 +V!?)U, and then to the Redfield regime (type p ) ; for TI > 4.rC> T2, the system has no Bloch regime even with U