Exact generalized Fokker-Planck equation for arbitrary dissipative quantum systems

Volume 28A, number

PHYSICS

4

2 December

LETTERS

ute to the Kr specific

heat. A detailed discussion of this work will appear in a forthcoming paper.

1968

1. J. S.Brown and G.K.Horton,

Phys. Rev. Letters 18 (1967) 647. 2. G. Jelinek, private communication. 3. S. L.Ruby, Y. Hazoni and M. Pasternak, Phys. Rev.

129 (196$ 826. 4. S. Margulies and J. R. Ehrman, Nucl. Instr. Methods 12 (1961) 131. 5. M. Pasternak, A. Simopoulos, S. Bukshpan and T. Sonnino, Phys. Letters 22 (1966) 52. 6. R.H.Beaumont, H. Chihara and J.A.Morrison, Proc Phys. Sot. (London) 78 (1961) 1462.

We gratefully acknowledge helpful discussions with S. L. Ruby, R. 0. Simmons, B. J. Alder, R. J. Borg, R. Booth, R. Ohlwiler, D. N. Pipkorn and H. L. Sahlin.

*****

EXACT

GENERALIZED ARBITRARY

I. Institut

FOKKER-PLANCK

DISSIPATIVE

QUANTUM

H .HAKEN ftir theoretische Physik Stuttgart, Received

der

FOR

SYSTEMS

Universith’t,

Gemzany

17 October

We derive an explicit and exact partial differential vided the system is described by a density matrix

EQUATION

equation question.

1968

for

a c-number

distribution

function,

pro-

We consider a quantum system which is coupled to reservoirs (or heatbaths). We assume as usual that the heatbath variables may be eliminated by second order perturbation theory. Provided the heatbaths are Markoffian (i.e., they have a short memory), the resulting density matrix equation may be written in the form [l] *:

(1) The first term on the right hand side describes the coherent motion of the system, the sum represents dissipation and fluctuation. In the following we drop the first term, since otherwise the final result (3) would become too lengthy. (It can be easily treated, however). The Pik’s are the usual projection operators from quantum state k to quantum state i with the property Pik Plm = Pim dkl . We associate with each Pfk a classical variable “ik and define a distribution functionf(v) by f(v) = N J exp (- z? vipik)

tr (E

eltp (X;k P&P)

{d&}

.

(2)

N is the normalization

constant. The product is arranged in such a way that (from left to right) i < k, then i = k, and finally i > k. The path of.integrations is to be chosen appropriately. A function of the type (2) had been introduced previously by Haken et al. [2], who derived an exact Fokker-Planck type equation for two-level atoms. As had been shown there 121, such a function allows to calculate all expectation values, even if mul; ti-time averages (with time-ordered operators) are required. The non-trivial task consists in deriving an equation of motion for f(v). Starting from eq. (l), we have derived such an equation in the following form: (3) * Our procedure

286

of deriving

an equation of the type (3) can be easily

extended to non-Markoffian

reservoirs.

Volume 28A, number 4

PHYSICS

LETTERS

2 December 1968

a

Using the rule xik - - we define Mxp as follows: svik Mxc, (x) = ,&A- zkmn A&lk cml,Ip e~hm with AZ,, erwise.

= Oka n (iii, XGi = Xmi ; X&i

- Xm*i6i> ,);

(4)

- *El)

&im is the Kronecker symbol, bi>m = 1 for i > m and = 0 oth-

= Xim. oksn=b*+C

xkll ‘X1112 - 1. Xlln with k < I1 < 12 . . < n for the

upper arrow, provided k 11 > 12 . . > Zj > n for the lower arrow, provided k 2 n . Otherwise 0 vanishes. C[klm is defined by 6i.l (bkm - Xkm bm? k (1 - bib))

for igk,

bk,

for i > k,l > m ,

0

eq(-xii

+ Xkk)(&il -

li 61, i)

lsrn

,

otherwise

Finally, Uli = 1 for 1 > i , oli = 0 for 1 = i , and oli = -1 for 1 < i . There are two constraints in eqs. (1) and (2) to be observed: 1) Be ause we want a real distri tion function, f, we postulate %jk = Vij . 2) Gil account Of C! Pjj = 1 one has to require bE V” 21-- 1* In subsequent publications we will show that eq. (3) has widespread applications, e.g., in the theory of lasers and asers, non-linear quantum optics, spin resonance, spin waves, and phase transitions. Our above tre ment includes the harmonic oscillator and, more generally, the lightfield in its quan5 tized form. Gn the other hand, for this case classical distribution functions can be defined in the form of the Wigner function [3] or by means of the diagonal coherent state representation of Glauber [4] and Sudarshan [5]. ‘It will be demonstrated in a forthcoming paper that these representations can be obtained from the representation (2) by a suitable projection technique. -X~j

vjk

,

=

The author wishes to thank Dipl. Phys. H. Vollmer and Professor W. Weidlich for several useful discussions, and bipl. Phys. P.Reinker for checking some of the results.

References 1. Density matri equations for different systems were derived by a number of authors, in particular by R. K. Wangsne s and F. Bloch, Phys. Rev. 89 (1953) 728. P. N. Argyres, in: Magnetic and electric resonance and relaxation, ed. J. Smidt (North-Holland, Amsterdam, 1963) p, 555. W. Weidlich and F. Haake, Z. Physik 185 (1965) 30; 186 (1965) 203. C. R. Willis, Phys. Rev. 147 (1966) 406. 2. H. Haken, H. Risken and W. Weidlich, Z. Physik 206 (1967) 355. 3. E. P. Wiener. Phvs. Rev. 40 (1932) 749; J. E. Moial, .Proc. Cambridge Phil. Sot. 45 (1949) 99. 4. R. J. Glauber. ‘Phvs. Rev. 130 (1963) 2529; 131 (1963) 2766. 5. E. C. G. Sudar$h&, Phys. Rev.‘Letters 10 (1963) 277.

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