N O N - P E R T U R B A T I V E S O L U T I O N O F G M E * )

T. MANCAL

Faculty of Mathematics and Physics, Charles University, Ke Karlovu 3, 121 16 Praha 2, Czech Republic

Received 22 October 1997

A general non-perturbative method is used for the solution of the generalized master equation (GME) for a simple two-level model of the exciton-phonon system coupled to the termodynamic bath. The Argyres-Kelley projection superoperator is used to exclude superfluous information contained in the density matrix. A new partly analytic method of the solution of the generalized master equation is suggested and new analytical expressions for the memory functions are derived.

1 I n t r o d u c t i o n

In this work we extend a special method for the solution of the generalized master equation (GME) suggested in [1]. This method is based on the use of in- teresting properties of some generalized Fourier expansions enabling to convert integro-differential equations into algebraic ones.

We use a simple model system (the same as in [2]) as a suitable basis for the development of the general and accurate method of the solution of GME. The model system consists of three interacting parts: two electronic energy levels and harmonic oscillator represented by the model Hamiltonian and thermodynamic bath creating and annihilating stochastical excitations of the harmonic oscillator. The interaction between the energy levels and the bath is realized via the phonon mode. Our t reatment is completely unperturbative and all approximations are of numerical character only.

2 M o d e l H ~ m i l t o n l a n

The most important part of the system consists of two energy levels, describing the ground and first excited states of an atom or molecule. The excitation on this atom or molecule will be called the exciton. The second part, the harmonic oscil- lator, decribes one optical phonon interacting linearly with the above mentioned energy levels. These two parts together can simulate isolated linear molecule with the ground and first excited energy levels, oscillating in linear approximation. To describe the interaction of this system with surrounding, we introduce the third part of the model system, the thermodynamic bath.

Following [2], we assume that the two level system is excited at t ime t = 0. The two level system can be described by the Pauli ai matrices. The phonon mode is described in the harmonic approximation and the remaining acoustic vibrations

*) Presented at the Czech-Israeli-German Symposium "Dynamical Processes in Condensed Molecular Systems", Prague, Czech Republic, 26-30 May 1997.

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T. ManUal

and fast electronic states are assumed to be much faster than the dynamics of the two level system.

We assume that I1) and 12) are the first excited and ground state electronic levels with the energies h12e and -h12e, respectively. Here, h is the Planck constant and 12 is the single phonon mode frequency. The Hamiltonian of the two-level system and the optical phonon mode is assumed in the form

(b + + b)tDa~ } H = Hex,ph + Hint = h ~ (¢a'z "4- b+b) + v f ~ , + A~r~) , (1)

where b and b + are optical phonon annihilation and creation operators. D and A are the relative coupling constants of the exciton with the optical mode. The exciton-phonon interaction is linear in the phonon displacement.

Now we discuss the interaction with the reservoir not included in Eq. (1). In this work, we use the generalized Haken-Strobl-Reineker model [3]. Henceforth, we assume that L ~ denotes the Liouville superoperator of our exciton-phonon system L' . . . . ( 1 /h ) [H , . . . ] . Similarly, L ~ denotes the Liouville superoperator describing the interaction of the phonon mode with the bath. According to [4], the stochastical contribution of the bath reads

L~ot,n~,p,-c,q6 =

[( i ~ p 6 . q 2 ~ ~ - ~., ~,o - (1 - ~° , )6 .~6~, ~ ( ~ , . + ~ ¢ ) . (2)

Here, the phonon eigenstates are denoted by the Greek letters and the exciton eigenstates by the Latin ones. The 7~t~ coefficients are taken in the form [5]

-y.~ = ~ ( ~ + 1)~ , .÷1 + ~ e x p -k-B--~ e~, . -a - (3)

Here, the coefficient k determines the strength of the interaction between the har- monic oscillator and the bath. We stress that k need not be small. The total Liouville superoperator of the problem equals L = L 1 + L ".

3 Use of the Laguerre functions

To find the probability of finding the excitation on the energy levels, we have to solve the Liouville equation

-p ( t ) = - i L p ( t ) (4)

with the initial condition

p(0) : I1)(11 ~ pR, (5)

where pR is the phonon density matrix. This initial condition means that we con- sider the system excited at t = 0. Since we are not interested in the complete

464 Czech..J. Phys. 48 (1998)

Non-perturbative solutioa of GME

information described by the density matrix operator we use the so-called Argyres- Kelley projection superoperator [6]

PA - ~ Im)(nl ~ paT~ph(mlAIn). (6)

It takes an average over the phonon states and selects the elements of the den- sity matrix describing the two level system. The master equation for the relevant information Pp(t) follows from the Nakajima-Zwanzig identity

O pp( t ) = - i P L P p ( t ) - - P)p( to) iPLPe-;(1-P)L( t - to) (1

- ~ i PLe-i(1-P)L('-r)(1 -- P ) L P p ( r ) d r . (7)

Equation (7) is usually called generalized master equation (GME). Taking the Argyres-Kelley projection superoperator (6) and the initial conditions (5) the sec- ond term on the right hand side of (7) disappears and we get equations

[ /0' ] -~p. . . ( t) = - ~ ~m.~,p,,(t) + w ~ . , , ( t - r ) p ( ~ ) d ~ . (8)

Pq

Here, we introduce the reduced exciton density matrix operator

p . , . (t) = Trph (nlp(t)[m), (9)

the time local terms um,pq and the so-called memory functions wm,pq(t):

u~.pq = Trph(mlnlp)(ql ® pain), (10)

w,,.,,-,pq(t) = Trph(mlL exp{-i(1 - P ) L t } ( I - P)LIp)(ql ® pain)- (11) Now we use some interesting properties of the Laguerre functions to convert

GME into an algebraic form. The Laguerre functions are defined as

-I.,(t) = L , ( t ) e - " ' , (12)

where a > 0, a 6 R and n = 0, 1,.. . . L,( t) is the Laguerre polynomial of the n-th order L , ( t ) = ( 1 / n ! ) e X ( d " / d z " ) ( e - Z x " ) . This set of functions is orthonormal with respect to the scalar product with the integration

0 ° e-(1-2a) ' . . , dr. (13)

The generalized master equation (7) is the convolution integro-differential equation. For the solution of such an equation we use the equivalences [7]

fo ' L---,(t - r )Lm(r)dr = "L,+m(t) - (t) , -i.+.~+1

Czech. J. Phys. 48 (1998)

(14)

465

T. Man~.al

n - t n-1 0----T"(t) = - E L,n( t )e -a ' -aLn( t )e -" '=- E-Lm(t)-a-L,~(t) . (15)

rn----0 m=0

Supposing that we can expand ppq(t) and Wrnnpq(t) into the power series in L, ,( t) ,

{DO O<3

p,q(t) = E p , q,kTk(t), w,nnpq(t)= Ewrn"'"k-L~(t)' (161 k=0 k=0

where Ppq,k and Wmnpq,k are the expansion coefficients, we can transform Eq. (8) into the algebraic form

2 B~) = E Mmn,,ppq,k. (171

pq

Here we denoted

Mmnpq = Umnpq + Wm,pq,0 + (1 - a)dimp6nq, (18)

k-1 [ 2

s(k2 -~ ~rnt~nl --j~=O pr,,.n,j -- E (Wrnnpq,k_j -- Wmnpq,le_j_l) ppq, (191

Introducing the transformation of the indices i = 2m q- n - 2 we can rewrite the solution of Eq. (17) in the form

4 Pi,k = E ( M ) 5 ' B~ t)- (20)

j= l

Now, for given wm,~q,k, we can calculate pm,.k. To solve our problem completely, we must determine the expansion coefficients of the memory functions w,~npq,k.

4 A p p r o x i m a t i o n of t h e m e m o r y f u n c t i o n s

In this section we find the expansion coefficients of the exponential function, involved in the memory function definition (11), into the Laguerre functions. In this way we derive new analytical expression for the expansion coefficients wm,~pq,k of the memory functions.

Using the scalar product (131 we get

Wrnnpq,k = Trph(mIL exp(-i(1 - P)Lt)'Lk(t)e:~; e-(1-2a)tdt

x (1 - P)LIp)(q] ® pRIn ). (211

We can see from Eq. (211 that we have to calculate the following integral

fo ~° exp(-At)Lk(t)dt . (22 /

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Non-perturbative solution of GME

Using the properties of the Laguerre polynomial Lk(t) [7] and the per partes inte- gration in Eq. (22) we get

fo ° exp( -mt )Lk( t )d t = (A - (23) 1)kA-(k+l).

This equation is obeyed even if A is a positive definite matrix. To obtain the expan- sion coefficients of the memory function (21) we must take A = i ( E - P ) L + ( 1 - a ) E , where E is the unit superoperator and a is the exponent in Laguerre function (12). Then, we get

wmnpq,k = Trph(mlL[i(E - P )L - aE]k[i(E - P )L + (1 - a)E] -k-1

x (E - P)LIp)(ql ® pain ). (24)

At this point we have the w,nnpq,k coefficients, which are necessary for the eval- uation of the p,n,~,k coefficients. The last step is to calculate the matrix elements of L and (E - P )L in the basis of the eigenfunctions of the two level energy system and the harmonic oscillator.

The above mentioned method converts the problem of the solution of the Li- ouville equation into the problem of the solution of the algebraic equations for the reduced density matrix elements (9). The use of the complete set of the La- guerre functions enables to derive new analytical expressions (24) for the memory functions. We shall see in the following section that expression (24) increases sub- stantially the accuracy of the results.

5 Numer ica l results

To test the method we calculated the probability Pii(t) of finding the system on the first excited level for several values of the parameters e, k, D, A, and/~ = (1/kBT), where k is the Boltzmann constant and T is the temperature of the bath. We put I2 = 1, which is equivalent to the choice of the time unit I2 - i . We denote the maximum (cutt-off) phonon number as N and take N = 8 as a suitable value [2].

The most important result of this paper is that, in contrast to [2], our method makes it possible to perform calculations even for long times (about 10 times greater than in [2]) when the probabilities pll(t) and p~( t ) aproach their equilibrium val- ues. In the short time region we completely confirmed the results of [2]. In the case when the system interacts strongly (k = 1) with the bath we can see that no strong oscillations of the probability pll( t ) occur. The short time region shows that the time derivative of pll( t) at t = 0 is zero, in contrast to the results following from the Pauli master equation. However, there are small oscillations for short times. For long times the system relaxes to the equilibrium state and no oscillations occur. The probability pl 1 (t) for different temperatures shows that increased temperature leads to a steeper decay of Pll(t) and increases the equilibrium value of pll( t ) at long times.

Czech. J. Phys. 48 (1998) 467

T. ManUal: Non-perturbative solution of GME

In contrast to the above mentioned situation the missing interaction with the bath (/c = 0) leads to a steeper decay of p11(t) in the begining, however, the system does not relax to any equilibrium state and strong oscillations of the probability occur in the long time region.

We also investigated the case of the weak coupling to the bath (it = 0.1). We found that very similar oscillations as in the case of switched-off interaction with the bath occur for short times, but in long time region the system relaxes to the equilibrium state. It shows that even the weak interaction with the infinite bath leads to the relaxation.

The dependence of Pll(t) on D and A was discussed in detail in [2] and will not be dealt with here.

6 Conclusions

We suggested a new general method of the solution of the generalized master equation and used it for investigation of the time behavior of the model system. The new complete set of orthogonal functions enabled to derive new analytical expressions (24) for the memory function expansion coefficients and increase the accuracy of the computation.

The method suggested in this paper is non-perturbative and is applicable to any Hamiltonian and Liouville superoperator. This property of the method makes it suitable for the solution of any physical problem, where GME formalism is used.

References

[1] L. Sk£1a, O. Bflek: Phys. Status Solidi B 114 (1982) K51.

[2] M. Men~ik: J. Phys.: Condens. Matter 7 (1995) 7349.

[3] V.M. Kenkre and P. Reineker: Exciton Dynamics in Molecular Crystals and Aggregates, Springer Tracts in Modern Physics, Springer-Verlag, Berlin-Heidelberg, 1982.

[4] V. ~£pek and V. SzScs: Phys. Status Solidi B 131 (1985) 667.

[5] L. Landau and E. Teller: Phys. Z. Sowjetunion 10 (1936) 34.

[6] P.N. Argyres and P.L. Kelley: Phys. Rev. A 134 (1964) 98.

[7] M. Abramowitz and I.A. Stegun: Handbook of Mathematical Functions, Dover, New York, 1972.

468 Czech. J. Phys. 48 (1998)