Zeitschrift ftir Physik 204, 223--239 (1967)

Quantummechanical Solutions of the Laser Masterequation. II W . WEIDLICH, H . RISKEN, a n d H. HAKEN Institut ftir Theoretische Physik der Technischen Hochschule Stuttgart Received May 3, 1967 A Fokker-Planck equation for a distribution function over the macroscopic observables of the laser essentially equivalent to that recently obtained by RISI~N, Scgr~m and WEIDLICH is derived from the fundamental quantummechanical laser masterequation. The general method used is the expansion of the statistical operator in a complete set of projection operators of the atoms and the lightfield. The assumptions leading from the microscopic equation of motion to the macroscopic semictassical Fokker-Planck equation are explicitly introduced and discussed.

Introduction

In the last years essentially three methods have been used to study the influence of quantum fluctuations on the statistics of iaserlight and related problems: a) The use of Heisenberg equations including fluctuating Langevinoperator forces 1-4 for all relevant laser observables. b) A Fokker-Planck equation with quantum mechanically determined dissipation and fluctuation coefficients for a distribution function over the values of the laser observables s-8. c) The masterequation for the statistical operator of the laser

system9 -

11

The methods a) and b) are equivalent in classical systems like the Brownian motion (compare CHA~DRASEKHAR*). They also lead to * In Wax, selected papers on noise and stochastic processes, Dover Publications (1954). 1 HAK~N,H.: Z. Physik 181, 96 (1964); 182, 346 (1965); 190, 327 (1966); -- Phys. Rev. Letters 13, 329 (1964). 2 HAI~N, H., u. W. WErOLtCH: Z. Physik 189, 1 (1966). 3 SAUERMANN, H . : Thesis, Stuttgart 1965; -- Z. Physik 188, 480 (1965); 189, 312 (1966). 4 LAX, M., in: Physic of quantum electronics (ed. by KELLEY,P.L., B. LAX, and P.E. TANNENWALD).New York: McGraw Hill 1966; -- Phys. Rev. 145, 110 (1966). s R~SKeN,H.: Z. Physik 186, 85 (1965); 191, 302 (1966). 6 SCribeD, CH., u. H. RISKEN: Z. Physik 189, 365 (1966). 7 RISKEN,H., CH. SCHMIO, and W. WZIDLICH: Phys. Letters 29, 489 (1966a). -Z. Physik 193, 37 (1966b); 194, 337 (1966c). s LAX, M., and W.H. LOmSELC: QIX Quantum Fokker-Planck solution for Laser noise. Vorabdruck 1966. 9 WEIDLICH,W., u. F. HAAKE: Z. Physik 185, 30 (1965); 186, 203 (1965). 10 SCULLY, M., and W.E. LA~Cmjr.: Phys. Rev. Letters 16, 853 (1966). tl FLECK, J.A.: Phys. Rev. 149, 309 (1966a); 149, 322 (1966b). 16"

224

W. WEIDLICH, H. RISKEN, and H. HAKEN:

equivalent results in the quantummechanical case of the laser system, if certain approximations are made in solving the originally fully quantummechanical Heisenberg equations of motion (compare 1,7). The present paper treats the relation between methods c) and b). The problem of the derivation of a Fokker-Planck equation for the distribution function about the values of macroscopic observables from the fundamental equation of motion of the statistical operator is of general importance in statistical mechanics. Derivations of this sort (see for instance VAN KAMPEN12) necessarily contain certain assumptions on the structure of the system and the macroscopic observables. Therefore it is of some value to substantiate the assumptions involved in a relatively simple though nontrivial system like the laser, where we have only few welldefined macroscopic observables (like the lasing lightmode operators and the operators for the collective level occupation numbers and dipole moments for the active atoms), and where the masterequation for the statistical operator can be solved in different approximations and different representations 9 - 11,13. In a preceding paper 13 (now denoted by I) we have solved the laser-masterequation by using the Glauber representation 14 for the lightmode and a complete set of operators for the active atoms and by making an ansatz for the statistical operator general enough to include the influence of quantum fluctuations on the lightmode. This method leads to a set of coupled equations for some distribution functions, but not to a Fokker-Planck equation. In the present paper we show that by a slight alteration of the ansatz and by some further approximations we essentially obtain the Fokker-Planck equation of method b), which was derived earlier 7 by using quantummechanically determined dissipation and fluctuation coefficients defined in analogy to the corresponding classical coefficients.

I. Expansion of the Statistical Operator in Projection Operators Before treating the laser problem explicitly we make some general remarks on the expansion of the statistical operator in projection operators. The statistical operator W is a positive definite hermitian operator with Tr W= 1 and may therefore always be expanded in the form

W=~ ~ojP,j; J

a~j~0;

~ o~j= 1,

(1.1)

J

where P~,~ are projection operators on the eigenstates ~pj of W. Due to the time-dependence of W, ~Pi and P,j are time-dependent. Apart from 12 KAMPEN, N.G. VAN: Physica 23, 707, 816 (1957); 25, 1294 (1959). 13 WEIDLICH, W., H. RISKEN U. H. HAKEN: Z. Physik 201, 396 (1967).

14 GLAUBER,R.J.: Phys. Rev. 130, 2529 (1963a); 131, 2766 (1963b).

Quantummechanical Solutions of the Laser Masterequation. II

225

this wellknown spectral decomposition of W, we may also chose a fixed, timeindependent complete system of projection operators Pox, belonging to states ~ , and expand W in this way:

W---Z g~Po~

(1.2)

E

where g~ are timedependent coefficients. [We remark that e.g. for a n-dimensional space a complete operator set consists of n 2 operators P0~; therefore the ~ cannot all be orthogonal to each other in contrast to the %. in (1.1).] From W= W + and Tr W= 1 there follows g~ = g*;

2 g~ = 1.

(1.3)

If and only if all g~ are > O, we may consider W as representing an ensemble E composed of independent ensembles E~ in the states 0~, where the systems of E~ occur in E with the probability g~ ~ O. On the other hand there always exist W, for which not all g~ are positive. For instance the expansion of

W----P~=Z g~Po~ K

for q~~ has to contain also negative g~. Otherwise we would have a contradiction to NEUMANN'S wellknown theorem 15 that from P~o= 2~ W1+22 W2, with 2j>O, 21+)o2=1, Tr Wj=l there follows WI= W2 =P~, resp. that the ensemble belonging to the pure state P~ is "irreducible" and cannot be decomposed in a nontrivial way into two ensembles W 1 4 WE. We conclude that on the exact microscopic level a probabilistic interpretation of the g~ in (1.2) is not possible, because they may become negative. Let us now go over to the macroscopic level. We assume, that there exist m macroscopic observables A (~), l= 1...m, which, at least approximately, commute and by which the macroscopic behaviour of the system is described. If ]~(1)... ~(m), ~) are the eigenstates of A (~) A(t) [~(1)... ~(,,); ~ ) = ~(~)[~(1)... ~(m); ~),

(1.4)

with ~(*) as eigenvalues of A (~) and ~ as degeneration index, the spectral representation of A (t) is A(~)=

~ ~(1)

15 NEUMANN, J.v.:

Springer 1932.

...

~(1) Q (~(1)... ~(m); ~)

(1.5)

a(m),

Mathematische Grundlagen der Quantenmechanik. Berlin:

226

W. W~IDLICrI,H. RISKEN,and H. HAKEN:

where Q (~(~)... ~o,); ~) is the projection operator on 1~(1)... ~(m); r (For simplicity we assume that all a(o, ~ are discrete.) The probability that in ~h~, after measurement, the values a(1).., a(,o of A(~)...A (m) are found, is given by p~(~(1) ... a(~)) = Tr(Po~ ~ Q(,(,)... ~(~); ~)). (1.6) From (1.6) follows the corresponding probability for the ensemble described by W:

o~(~('... ~("))= :rr(w% Q.(~,('... ~,("); ~)) r =~, g~ p~(a(1)... ~(m)).

(1.7)

Though some g~ may be negative, as we have seen before, co(a(1)... ~('~)) is a positive definite function, as W is a positive definite hermitian operator. We see that the probability distribution of the a(~)...a(m) is partly due to the quantummechanical dispersion in the states ~ and partly due to the fact, that W is a mixture of different ~ . It is now convenient to write the index x more explicitly x = (a (~ 2; ~). a (~ is the expectation value of A (~ in ~ : a(~ rr(Pg,~ A (0) =

~,

a (0 p~(a~

~(")).

(1.8)

~(1) ... ~(rn)

The 2 are further indices by which the function p~(a~ characterized, so that we may write

~(")) is to be

p~(~(~)... ~(~)) = p,~,, ~(~(~)--. ~(~)); finally, the ~/ distinguish between those ~ , which lead to the same probability distribution function Pa"), ~(~0)... aO")) of the macroscopic variables. Then we may write instead of (1.7): co(a(~).., a("))=

=

~

a ( t ) , ),, r/

a(l)j ).

g(a (~ 2; ~1) Pa(,), ~(,o)... a(m))

G(a(l) 2) Pa(~), )3,,~(I) ...

~(m)) ,

(1.9)

and
~

~(0 co(ao)... a(m))

a(1) ,.. a(m)

= ~. G(a (~), 2) a (~

(1.10)

a(1), ),

with

G(a( ~ 2) = ~ g(a (~, 2, r/). r/

(1.11)

Quantummechanical Solutions of the Laser Masterequation. I[ The time development of a)(~~

227

(m)) is determined by that of all

g(a (~ 2; ~/), which is known in principle from the microscopic equations of motion. On the other hand, the Pare, z(e(a).., e(m)) are constant functions in time, because the Po~ are time independent. If the macroscopic observables A (~ are appropriately chosen, the function G(a (~ 2), which by (1.9) already determines co(e(~ e(m)), will be positive definite in general, as in the sum (1.11) eventual negative terms g(a (t), 2, q) usually will be overcompensated by positive terms. The formulas of this chapter are exact relations. But only by an approximation it is possible to derive an equation of motion for G(a (~ ;t) alone out of the microscopic equations for all g(a (~ 2; t/). If it is compatible with these equations, as in the case of our masterequation, the simplest approximation is, to put equal all g(a (~ 2, t/) with different ,/. This means that we average the amplitudes g(a (~ 2, t/) over a "cell" of states P~,(~(,),z, ~), which have the same properties with respect to the macroscopic variables A (~ II. Derivation of the Fokker-Planek-Equation from the Masterequation for the Laser System We now specialize the general consideration of wI to the case of a laser with A active two-level atoms coupled to one lightmode in resonance with the homogeneously broadened atomic transition frequency. Our notation is the same as in 11 a. There we have derived the fully quantum theoretical equation of motion for the statistical operator of the laser system [comp. I; (1.9)]. -7-

lfl) o(fl,/3* .... )(ill,

(2.1)

where eo(fl, fl*, ...) is a c-number function with respect to fl, fl* but an operator in all creation and annihilation operators ,,(.~)+ _j , a}~) of thej-th level of the v-th atom (v= I...A). The equation reads [comp. I; (1.13a)].

d~=(-i)g= dt =1

[-H V2 FHZ (~)' o7a + u (v) q,, 0)7a--~-L

( 8o.) a(1,,)+a(o,,) a(o,,)+a~,,)~7) +\Or1 (2.2)

-i

-~7~,,

+~=1

aa')+a(o")+~,, -~v2]f+ A~ \ ct /F

228

W . WEIDLICH, H . RISKEN, a n d H . HAKEN:

with

fl= 89

fl*=89

(2.3)

[comp. I; (1.12)], n(~)_r.(~)+ .~(~) .&)+ a(~));

n(~)-t.(~) ..t@ --k~l

+ a(o~)+a(o~)+ a(l~)); (2.4)

,,(v)+;,(v)~

H ( V ) _ ;/.,(v)+ .(v)

[comp. I; (1.15)]

\ clt ]a,,--~- {[a('~

a(o~)]+

(.o; a(1")+ a(')]} +

~1

L'-~O

+~/ol T {[ai')+ a(J); r176 a(')* a(l")] + r'&)+L',,,-(')c9; a(o"4+ a(J)]} q2 ~ra(~)+ a(,). (L 0 0 ,

~ " 1.(,)+ .(~)l+[a?)+ "1 J

(2.5)

a?)co; a(o~)+ a(o')]}

[comp. I; (1.3)] [ (~2 (D

3209~

(2.6)

[comp. I; (1.14a)]. The operators H~,~), H (~),/7(z~) for the inversion and the components of the dipole moment of the v-th atom, respectively, have eigenstates ~r "~(*) Z~ ) [comp. I; (1.16)] H~*)g'(_g) = -v'+ -,t,(*).,

//(~) q)~)=-t-p~);

//(x~)X~) = -t-Z(~)_ (2.7)

with the corresponding projection operators P = p2 [eomp. I; (1.17)] + ----a(~)+ a~'); P0(v) --4--2

~l

P0- =a(o~)+ a(o~) q~0

u0

~\~l

u, 0 T

P(~ -z-' 1."1F'~(~) + "1"(~)J"~(~)+-"o "o"(*),(,~(*)+,__~,.,.1 a_ a(o~)-a(o~)+a?))] (2.8) rt(~) _ o ( 9) _ p(~) 9 riO,) _ p(~) _ p ( ~ ; ~xrt(~) - ~ pO') z + - - ~ x -(~) P Let us now introduce the following projection operators in the space of all A atoms d+

P(d+ d_,r+ r_,s+ s_; p)=-P[drs, DRS; p]=p [I P~,~ x V=I

D v=d++l

O+~+ v=D+I

D+R v=D+r++l

,+R+~+

D+R+S=A

v=D+R+I

v = D + R + s + +1

(2.9)

Quantummechanical Solutions of the Laser Masterequation. II

229

with

D=(d+ +d_);

d=(d+ -d_);

R=(r+ +r_);

r=(r+-r_);

S=(s+ +s_);

s=(s+-s_);

D+R+S=A

(2.10)

p denotes a permutation of the A atom indices v. Clearly we have A! d+! d_! r+! r_[ s+[ s_! different projection operators (2.9), each of them projecting on a state where (d+, d_, r+, r_, s+, s_) atoms are in the states (0+, 0 - , ~0+, rp_, )~+, Z-), respectively. As the P(d+ d_, r+ r_, s+ s_ ;p) form an even overcomplete system of operators for the atom space (six operators Po• Px• Pe-+ for the twodimensional space of every atom) it is possible and completely general, to expand W in the following way, which is the explicit form of (2.1) with respect to the atomic variables:

W=[. ~ -

2 g(flfi*;drs, DRS;p)[fl)(filP[drs, DRS;p].

(2.1t)

DRS; d r~; p

Corresponding to (1.3) the trace condition reads: g(flfl*; drs, DRS; p ) = I .

Tr W=

(2.12)

DRS; drs; p

In (2.11), (2.12) we have used the variables (2.10). We now introduce as macroscopic observables of the atomic system the operators D, R, S of the collective inversion and the components of the collective dipole moment, respectively: A

D = z~'0V r1(v)., V=I

A

R = }-' H(ev);

A

S = ~ H~v)..

V=I

(2.13)

v=l

Because of (2.14) [comp. I; (1.18)] we have a (o= Tr (P [d 1"s, D R S; p] A(~ with a (~ (d, r, s); A(~ (D, R, S) respectively.

(2.i5)

230

W, WEIDLICH, H. RISKEN, a n d H. HAKEN:

It is also easy to calculate the probability distribution function p~ [comp. (1.6)] for the states P [dr s; D R S; p]. Let us firstly consider the observable D. In the state P [drs, DRS; p] there are D=d+ + d_ atoms in eigenstates if+ resp. ~_ of the corresponding components -t-t hey) ~0 of D, while the other r + + r _ + s + + s _ = R + S atoms are in states 9+, q~-, Z+, X-. In everyone of these states, independently for each atom, the corresponding component/-/~v) of D takes the value + 1 or ( - 1 ) with probability 89 if measured. Therefore, if 6 is one of the eigenvalues A , A - 2, A - 4,... ( - A) of D, the probability zc~(fi) to measure the value 6 of D in P[drs, DRS; p] by simple combinations follows to be

with ( d - (R + S)) < 6 < (d+ (R + S)). Similarly, the probabilities foR(p), rCs(a) to measure the value p of R, respectively a of S in P[drs, DRS, p] read

~.(p) = ~/._(~+~)"

\2]

(D + S)!

( D+S+p-r.),2 " (D+S2(P-r))'

(2.16")

( r - ( D + S))< p<(r +(D+ S)), (2.16"')

(s-(D + R))__<~____(r+(O + R)). Neglecting the fact that D, R, S do not exactly commute the probability

pa,~(6pa) for the values 6, p, a of D, R, S in the state P[drs, DRS;p] is DRS

Pa r ~(3 p a) = no(6 ) nR(p) ns(a) .

(2.17)

DRS

Treating 6, p, a as continuous variables, we approximately derive from (2.17) for the probability ~,s(6, p, a) d6 dp d~r to measure in state

vRs P[drs, DRS;p] values 6, p, o" of D, R, S within the intervals d6, dp, da

Quantummechanical Solutions of the Laser Masterequation. II

231

the formula Pd~ ~(0, P, ~) d3 dp da DRS

d~5 dp d a ]/8n3(R +S)(D+ S)(D+ R)

(a-d)~

e

2 (R+S)

(,_~)2

~_~)~

2 (D+S) - - 2 (D+R)

(2.17')

Corresponding to formulas (1.9), (1.10), (1.11) we now obtain:

~o(/~/~*,6p~)= 2 g(~p*,d,'s, DRS; p)p~r~(6p~) DRS; drs; p

DRS

= ~G(flfl*,drs, DRS)pd~(~pa) d r s DRS

(2.18)

DRS

as positive definite probability distribution function and the expectation values

2

(2.19)

=~ cl~fi ~ G(fi fl*,drs, D R S ) a (0 7C d r s DRS

with A(0 = (b, b+,D,R, S),

a (0 = (fl, fl*, 6, p, ~r),

a (t) = (fi, fl*, d, r, s),

respectively and

G(flfl*,drs, D R S ) = ~ , g(fifi*,drs, D R S , p).

(2.20)

P

Clearly we have to identify the indices (a(~ ,t; ~) of w 1 with (tiff* drs;

D R S ; p), respectively. Now we shall show that it is possible by some approximations to derive an equation of motion for G(flfi*, drs, D R S ) alone from the microscopic equations for all g(flfl*, drs, D R S ; p). Inserting the statistical operator (2.11) into the equation of motion (2.2), we will have to calculate expressions like

L--e

,p

drs,,

p l;

Ot

)A

(P[drs, V~ S; P3 ~?)+ a~)-a~o"§ a~" P[d,-s, V R S; p])...

232

W . WEIDLICH, H . RISKEN, a n d H . HAKEN:

I

+

§ ,-~[el

I

,--~ [ t'-I

for which this table, following from the definitions (2.4), (2.5), (2.8) of all operators, will be useful (index v omitted here). Using this table and the equation of motion, one easily sees that in general we have to expect also negative amplitudes g(flfl*, drs, DRS; p) as already pointed out in w1. For instance also negative g(...) occur at time to+6t, if we start at to with a pure state P(do ro So, Do Ro So;Po) of the atom system corresponding to an amplitude

go(fl fl*,drs, D RS; P) =f(fi, fl*) 6d ao6,,o 6,,o &n0o 6~ Ro&SSo6p po I

+

>0. The decisive approximation, which will lead to an equation of Fokker-Planck type for G(fifl*, drs, DRS) is the assumption, that at t = to the amplitudes

g(fl fl*, dr s, D RS; p)=g(fl fl*, dr s, D RS) in (2.11) are equal for all permutations p resp. for all A! =

~

~

+

d+!d_! r+! r_!s+! s_! different projection operators

P[drs, D RS; p].

e

l+

Because of the symmetry of (2.2) in all atoms, we then will have this property for all later times t>to. It is now not necessary but convenient to use^ instead of (2.11)a statistical operator W, which is equivalent to W under the above assumption. 6(fl fl*, a ," s, D R S) x

Y.

Y.

2,2 2 2

DRS drs

xlfl)(fli P[drs, DRS]

(2.21)

Quantummechanical Solutions of the Laser Masterequation. II

233

with

,drs, m Rs)=

*,drs, m RS, p) P

A] g([l fi*, d r s, D R S) d+!d_!r+!r_!s+!s_!

(2.L~)

and P[drs, DRS] being one arbitrary representative of the P[drs, DRS;p]. Applying the table and putting all resulting P[drs, DRS;p] with different p equal to one P [drs, D R S], which is allowed in our symmetrical statistical operator, we e.g. obtain A

Y

P(d+ d_, r+ r_, s+

V=I

=d+(P(d+ - 1, d_, r+, r_, s+ + 1, s _ ) - P ( d + - 1, d_, r+, r_, s+, s_ + 1))- d_(P(d+, d_ - 1, r+, r_, s+ + 1, s _ ) - P ( d + , d_ - 1, r+, r_, s+, s_ + 1))-

(2.23)

-s+(P(d+ + 1, d_, r+, r_, s+ - 1, s _ ) -P(d+,d_-l,r+,r_,s+-l,s_))+ +s_(P(d+ + l , d _ , r + , r _ , s + , s _ - l ) - P ( d + , d_ + 1, r+, r_, s+, s_ - 1)). The insertion of (2.21) into the equation of motion (2.2), the explicit use of formulas like (2.23) which are derived from the table and the fact, that P(d+ d_, r+ r_, s+ s_), P(d'+ d;, r'+ r ' , s'+s ' ) are independent operators for (d+d_,r+r_,s+s_)#(d'+d; , r+ ' r " , s'+s'_) now lead by straight forward calculation to the following difference equation for G(flfi*, drs; DRS):

dG(fi fl*,drs; D RS) dt =g {-~-[(D+l-(d-1))2 4 (D+l+d+l) 2

(G(d-l,r,s+l)-G(d-l,r,s-1))+

(G(d+l,r,s-1)-G(d+l,r,s+l))+

(S+1 +(s+l)) ( G ( d + a , r , s + l ) - C ( d - l , , . , s + l ) ) + 2

234

W. WEIDLICH,H. RISKEN,and H. HAKEN:

4 (S+l-(s-1)) 2

(G(d-i,r,s-1)-G(d+l,r,s-1))]-

v2 [.(D+ 1 - ( d - 1)) (G (d- l, r - 1, s)- G (d- 1, r + 1, s)) + 2 2 (D+l +(d+ 1)) 2

(G(d+l,r+l,s)-G(d+l,r-l,s))+

4 (R+l+(r+l)) 2

(G(d-l,r+l,s)-G(d+l,r+l,s))+

4 ( R + l - (2r - 1 ) )

(G(d+l,r-l,s)-G(d-l,r-l,s))] +

[(D+ 1 +(d+l)) 2

(G(d+l,r,s+l)-G(d+l,r,s-1))-

1 (R+t+(r+l)) 2 2

(G(d,r+l,s-1)-G(d,r+l,s+l))-

1 (R+l-(r-1)) 2 2

(G(d,r-l,s-1)-G(d, r-l,s+l))-

- l ((S2S~) G(d,r,s) (S+(s+2))G(d,r,s+2))+ 2 2 -~ (S-(s--1)) G(d+l,r,s-1) (S+(s+l)) G(d+l,r,s+l)+ 2

2

+~

G(d,r,s)

a [(D+I+(d+I)) c~v2 2 1 (S+l+(s+l)) -t2 2

2

o,:d,,,,s_2:,)]_

(G(d+l,r-l,s)-G(d+l,r+l,s))+ (2.24)

(G(d,r-l,s+l)-G(g,r+l,s+O)+

1 (S + 1 - ( s - 1)) (G (d, r - 1, s - 1 ) - G(d, r + i, s - 1))+

-t2

2

+~1 @~f-~ G(d,r,s)

(R + (r2 + 2)) G (d, r + 2, s)) +

(R+l+(r+l))G(d+l,r+l,s) 2

+~

~

G(d,r-2,s)

(R+l-(r-1))G(d+l,r_l,s)+ 2

2

QuantummechanicalSolutionsof the Laser Masterequation.II

( \ ~v~

c3(vzG))q_2tr[~2G + 632G~} ) + (D -2 d) G (d, r, s)) -F

+{7ol ((D-(~ -2)) G(d-2, r,s)

(D+d) G(d, r, s)]\ + /

+71o (-(D+(~-+2)) G(d+2, r,s) + ~

235

2

[(R+12-(r+l))(G(d-l,r+l,s)-G(d+l,r+l,s))+

q ( R + l - ( r - 1 ) ) (G(d-l,r-l,s)-G(d+l,r-l,s))+ 2 q ( S + l + ( s + l ) ) (G(d-l,r,s+l)-G(d+l,r,s+l))+ 2

(s + 1 (s 2

1))

q 3

(G(d- 1, r,s- 1) - G (d + 1, r , s - 1))| +

+_~_ [(R+(2+2)) G(d,r+2,s) -(R+r)2 G(d,r,s)+ ( R - ( r - 2 ) ) G(d,r-2, s)

2

(R-r) G(d,r,s)+ 2

q (S+(s+2))2 G(d,r,s+2)

(S+s)2 G(d,r,s)+

+

(S-s)

2

G(d,r,s-2)

2

I'} G(d,r,s)]~ j) "

On the right side of (2.24) we have omitted the arguments D, R, S in G(fifl*, drs, D RS). Expanding G(fifl*, d+ Ad, r + AR, s + As; D+ AD, R + A R, S+ A S) and the other factors in (2.24) in the usual way:

6(~ ~*, d+a d,... S+A S)

=exp (Ad ~-~+Ar ~-~+As O-~-+AD~-~+AR o-~+AS ~--S) (2.25) x a(P~*,

drs, DR S)

and taking into account only derivatives up to the second order in v~v2, drs, DRS, we immediately obtain a Fokker-Planck equation for

236

W. WEIDLICH,H. RISKEN, and H. HAKEN:

G(fl fl*, drs, D RS): d G(fl fi*, d r s, D R S) dt _~v10 (Qcvt_gs)G)+ ~2 ((tcv2_gr)G) + +--~-d ((gEvl s+v2 r]+Yll [ d - A a~

G)+

0 + ~ ((~ r - g v2 d) C) + ~ ((~ s - g Vxd) a) + 02

62

02

(2.26)

92

+ ~vtz (2x nt, G ) + - ~ v~ (2xnt, G)+--~-7 (],• G)+~s ~ (y• S G)+ 02

+~

62

(~ II[ D - a o d ] G ) -

+ ~02

02

Od 0 r (a~ ~ II r G) - ~

(ao 7 II s G) +

02

(g[D+R+d] G)

OvlOd (gs G)+

02 92 + ~20r (g[D+S+d ] G) OV2~d (grG). The probability distribution function (2.18) written in continuous variables 6, p, o- follows by folding the solution G(pfl*, drs, DRS) of (2.26) with the dispersion function (2.17'):

co(tiff*, • p a) d2fl da d p da =d2fldfdpda

~

~G(fifl*,drs, DRS)x

D,R,S D+R+S=A (~-d) 2

x

,

(2.27)

(p--r)2 . (a--s)2 )

V,8 na (R + S) (D + S) (D + R)

dddrds.

Let us now discuss Eq. (2.26). There are no terms with derivatives 0

~

~

8D ' ~R ' 8S '

02

8D 2'

~2

~2

(~S 2 '

t~R 2 '

~2

~2

02

aD ~R ' aD 8S ' 8R 8S

in it. The fact that no terms with first and second derivatives of D, R, S appear in (2.26), means, that the form of the distribution function G with respect to D, R, S is not determined by (2.26). But this is due to the

Quantummechanical Solutions of the Laser Masterequation. II

237

inherent nonuniqueness of an expansion of the form (2.11), because of the overcompleteness of the system P[drs; DRS;p]. Namely, because of P~,~+ P ~ = P ~ +P(~=P(x~ +P(x~=I (v) (2.28) the coefficients g(fifl*, drs, DRS; p) and G(flfl*, drs, DRS) are not uniquely determined, which shows up also in the Fokker-Planck equation (2.26). Of course, different but equivalent G(flfl*, drs, DRS) should lead to the same function co(tiff*, 6po) in (2.27). (We remark, that this nonuniqueness was avoided in 1 l a by expanding W in a complete, but not overcomplete set of operators. This approach on the other hand leads to equations not of the Fokker-Planck type.) We now may solve (2.26) with D=Do, R=Ro, S=So as parameters in it, which means putting G(flfl*, drs, D RS)= GDoRoSo(flfl*' drs) (~DDo (~RRo (~SSo" The final G(flfl*, drs, DRS) then is a linear combination

G(fl f*, d r s, ORS) = ~

Do Ro So

c(DoRoSo) GDoRoSo(ffl*,drs)6DDo6RRo6SSo

(2.29)

= c(DRS) GDRS(fifl*, d r s) with coefficients c(DRS) corresponding to the initial condition for W. If GD~s(ff*, drs) is normalized to

s d2f dddrds GDRS(flf*, drs)=l, 7"C

we have with (2.22) and the trace condition (2.12): c (D, R, S)= 1.

(2.30)

D, R, S

D+R+S=A The Eq. (2.26) has the same structure as the Fokker-Planck equation derived for on(tiff*, drs) by RISKEN, SCHMID, and WEIDLICH7r [comp. 1.c. formula (3.3)] by calculating quantum mechanically dissipation and fluctuation coefficients for this equation, which are defined in analogy to the classical coefficients. While the dissipation coefficients of both equations agree completely, we see a difference in the fluctuation coefficients: There are coefficients

Qvl~=g[D+R+d]; Qv, a = - g s ; Q~2r=g[D+S+d]; Qv2s=-gr(2.31) 17 Z, Physik. Bd. 204

W. WEIDLICH, H. RISKEN, and H. HAKEN:

238

proportional to the (small compared to 711, ~• x) coupling constant g in (2.26), which disappear in the former equation 7c. Furtheron we have in Eq. (2.26) Qrr=27•

Qdd=271t(D--ao d)~2711D

Q~s=27•

(2.32)

instead of (~r~=(~s~=2yz A; Qaa=2Vll(A-ao d)g2~ll A

(2.32')

in the former equation 7c The latter difference is due to the fact that (2.26) is an equation for G(fifl*,drs, DRS) resp. GoRs(fi*fl, drs) while (3.3), 1.c. 7~ refers to co(fiB*, drs). The relation between both equations is most easily seen, if we compare them for g = O, neglecting the field variables fi, fi*. The solution of

dGoRs(d r s; t) dt 0 ~ O a = Or (7• r GDRS)+ - ~ (7• S GDRS)+--~-~ (7 II(d - A ao) GDRS) + 02

+~

~

02

(7• R GDRS)+ ~

.

(2.26')

~2

(7• S GDm) + ~d ~ (7 II D GDRS)

with initial condition Go Rs (dr s, O) = ~ (d- d') 6 (r- r') 6 (s- s') reads ( d - d ' (0) 2

GD~s(d r s, t) =

e c ~0Y

+

( r - r ' (0) 2 ( s - s ' (0) 2 "J 2R(t) + W Y

[/8~ 3 D(t) R(I) S(t)

(2.33)

with

D(t)=D(1-e-2~.t);

R(t)=R(1-e-2~•

S(t)=S(1-e-Z~r- t)

d=d-Aao; d'=d'-Aao; cl'(t)=d'e-~"t; r,(t)=r,e-r~_t; s,(t)=s, e-~•

(2.34)

Inserting (2.33) with (2.29) in (2.27) and calculating explicitly the integral over dddr ds, we obtain for the probability distribution function co(a, p, a; t): co(a, p, a; t)=

Z

c(DRS)•

DRS D+R+S=A

x

(~'-a"~ (t))~ (p_~, (o)~ (,~_ ~, o))~ e " { 2 (D (t) + R + S) + 2 (D + R (t) + S) t 2 (D + R + S (t)) }

]//8-~ (D (t) + R + S) (D + R (t) + S) (D + R + S (t))

(2.35)

Quantummechanical Solutions of the Laser Masterequation. II

239

On the other hand, the former Fokker-Planck equation (3.3), 1.c. 7c for co(5, p, a; t) (in the same case g=O) reads

dco(6, p,~; t) dt

(9 Op (~lP r176 +-~

(~• A r

(~,• r

(~,11(5-A(ro) r

(2.36) ~-~ (~1 A r

~--~ (y iI Ao))

and allows a solution with initial condition

p, o; o)= oJ(,5, p, ~r; t)=

d')

s')

f O-a" (t))z -} (P-"(O)~ ~ (~-s' e t 2 Ajj ('))~ (t) ~ 2 A• (t) 2 A Z (t)

(2.37)

V87c 3 A2(t) All (t) where ~=5-Aao;

Ail(t)=(1-e-2~"OA;

A•

(2.38)

While both functions (2.35) and (2.37) agree for t-~oo the difference of these probability functions for finite t arises, because the more exact treatment, starting from the masterequation, takes into account that the observables D, R, S cannot have sharp values simultaneously in any state, especially not in the states P[drs, D R S , p], as they do not commute. Correspondingly, the Fokker-Planck equation (2.26) for GDRs(flfl*, drs) contains only this part of the fluctuation, which is not already included in the dispersion of D,R, S in the states P[drs, D R S , p] used to expand W. The main objective of the present paper was to establish the connection between the microscopic variables in the density matrix equation and the Fokkcr-Planck equation which governs the motion of the macroscopic observables. As we will show in a subsequent paper, III, it is also possible to introduce a distribution function for macroscopic observables in such a way, that for it a completely exact equation of motion is obtained. Its leading terms agree with the Fokker-Planck equation of RISKEN, SCHMID and WEIDLICH7.

17"