The quantum langevin forces for dynamical systems with linear dissipation and the Lindblad equation

PHYSICA ELSEVIER

Physica A 236 (1997) 335-352

The quantum Langevin forces for dynamical systems with linear dissipation and the Lindblad equation Rouslan L. Stratonovich 1 Mathematics Department, University of Nottingham, NG7 2RD, UK Received 22 May 1996

Abstract The general form of the quantum master equation (the Lindblad-Kossakowski equation) for linear and nonlinear systems with linear dissipation is considered. This equation and the quantum regression theorem are used for finding two-time commutators and symmetrized moments of the Langevin forces. The obtained properties of the forces are unusual for physical systems interacting with the environment systems. In the last section the random Hamiltonian and the corresponding stochastic equations are considered. Keywords: Quantum master equation; Linear dissipation; Operator random forces; Symmetrized moments

I. Introduction In the recent years quantum dynamical dissipative processes described by the quantum master equation

=Lp

(1.1)

(that differs from the von Neumann equation) and by the quantum regression theorem (QRT) are intensively studied. Eq. (1.1) for the linear dissipation will be written in detail later (see (2.1)). In this case, operators Gj entering (1.1) and (2.1) depend on coordinates and momenta linearly. Eq. (1.1) and the related semigroup o f the density-matrix transformations are very interesting objects from the mathematical point o f view. Equations for the dynamicalvariable mean values corresponding to the semigroup were first considered in [1]. Later 10n leave of absence from Physics Department, Moscow State University, 119899 Moscow, Russia. 0378-4371/97/$17.00 Copyright (~) 1997 Elsevier Science B.V. All rights reserved PII S0378-4371 (96)00261-0

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the semigroup and its generator determining the form of master equation were studied by a number of researchers [2-8]. It is well known that many non-quantum dissipative processes are Markov ones and are studied with the help of the Markov master equation, for example, the FokkerPlanck equation. It is quite natural to regard (1.1) and (2.1) as direct generalizations of the non-quantum Markov master equation. But the situation is not so simple. There exist doubts that the theory of quantum Markov processes based on (1.1) and QRT are applicable to the physical dissipative processes (see e.g. [9,10]). Some of them will be pointed out below. In this paper, it will be shown that according to (1.1) and QRT the equations (1 = p / m + ~(t) ,

(1.2)

t

b = - U I (q) - 7P + q(t)

(l.3)

are valid in the one-dimensional linear-dissipation case. Here /_/1' = d U l ( x ) / d x , Ul(x) is a c-function, ¢(t), q(t) are the operator Langevin forces having two-time moments ½([~(tl),~(t2)]+) = V2t~(tl2) ,

½([r/(tl),~/(/2)]+) = U26(t12),

1 ([¢(tl), ~/(t2 )]+) = -- UV cos ~o6(t12 ) (tl2 = tl - t 2 ) . V :

(/)11,/)12 . . . . .

(1.4)

In (1.4) ~p is the angle between vectors U = (Ull,Ul2 . . . . . Unl,Un2), /)nl, ?.)n2); U, V a r e d e f i n e d by

c = IuI =

u

+u2

,

v=lvl--

v~+v2

1_~=1

Besides, the equation UVcos Z = -lh 7

(1.5)

should be valid, Z being the angle between U and V' = ( - v l 2 , vll . . . . . --Vn2, v,1 ). From (1.5) we get V 2 = /h272/(U cos Z) 2 ~> ¼h2y2/U 2 . This means that V 2 cannot be zero if 7 > 0, U < cx~, h > 0. In other words action of the non-zero Langevin force ~(t) in (1.2) is inevitable. This force is difficult to explain for ordinary physical system. In fact, the dissipation is a consequence of interaction of the original system S with some environment E (for instance, the heat bath). The Hamiltonian of the total system S + E is /-/tot = H + HE + Hint •

(1.6)

Usually, the interaction Hamiltonian Hint is only expressed through the coordinates q and Q of S and E. Then [Hint, q] -----0, and we have i i 4 = ~[Htot, q] = ~ [ H , q ] ,

(1.7)

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i.e. q = p/m if H = p 2 / ( 2 m ) + Ul(q). We see that no force ~(t) caused by interaction with E appears on the right-hand side of (1.7). Another difficulty. For thermal equilibrium at temperature T the well-known KuboMartin-Schwinger formula is valid: (1.8)

(C(t)D(t'))o = (D(t')C(t + ih/kT)o .

The subscript 0 means that the moments are equilibrium ones, C and D being arbitrary operator variables. This formula is a consequence of the equilibrium matrix density Ptot = const x exp[-Htot/kT] (but not of the equation p = c o n s t e x p ( - H / k T ) , which may be invalid). It may be shown that formula (1.8) contradicts the results (1.4), (2.32)-(2.34), (2.38). So the theory of the quantum Markov processes based on (1.1) and QRT may be applied only to physical dissipative processes as some approximation and the question arises about the magnitude of the deviation. The work on investigating this deviation is only beginning. The processes with linear dissipation are very convenient for this investigation because another theory is quite applicable to them, namely the theory based on fluctuation-dissipation theorems. In the non-linear case (non-quadratic potential energy, but linear dissipation) the quantum formulas of the Nyquist type (see [11, Section 5.6]), valid for the linear dissipative elements of the system may be applied. The corresponding results perfectly agree with the results obtained by analysis of the dynamic models of heat bath [12-16]. In this paper, statistical properties of the operator Langevin forces are studied on the basis of the quantum Markov theory. For linear dissipation the forces are Gaussian even if the system is non-linear. Therefore their statistical properties are fully determined by two-time moments and commutators. In the many-component linear-dissipation case the dynamic equations with the Langevin forces are i q~ = ~[H,q~] + ~ , i p~=-~[g,p~]-2~Bq~+~/' ~ (~= 1

.....

r),

where summution over repeated subscripts is understood. If H = 1b~#p~ PI~ + U, U = Ul(q), lib,#I[-1 = ][m~#][, they take the form gI= = b~l~P~ + ~ , i /5~ = - ~ [ U , p~] - 7~eP/~ + r h ,

(1.9) (l.10)

with V~ = 2~;,b;,& The form of non-stochastic terms on the right-hand sides of these equations will be used later.

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2. The Langevin forces for linear one-dimensional system Since the density matrix p(t) of the linear system is Gaussian (for the Gaussian initial matrix) we take the Lindblad-Kossakovsky equation of the form i 1 n /~ = - ~ [ H + A , p ] + ~ Z ( 2 G j p G f

- GfGjp - pG+Gj),

(2.1)

j=l

with Gj = ujq + vjp .

(2.2)

Here Ujl + iuj2, vj = v j l + ivj2 are complex numbers. The reason for adding operator A to the Hamiltonian H on the right-hand side of (2.1) will be seen later. For the linear system we may set H + A = T + U + A = ½[aq2 + c ( q p +

pq)+bp2].

(2.3)

From (2.1) the following equation ensues: N

(b) = i

1 Z([Gf,D]Gj

A,D]) +

_ Gf[G,D])

(2.4)

j=l

for any dynamical operator D. First we take q as D and using (2.2), (2.3) and the ordinary commutation relations for q, p we get from (2.4)

1

(q) = c(q) + b(p) + ~ Z

Im(ujv~ )(q) .

(2.5)

J Analogously setting D = p we obtain

1

([9) = - a ( q ) - c(p) + ~ Z

Im(ujv~)(p) .

(2.6)

J

Comparing (2.5) and (2.6) with averaged equations (1.9) and (1.10) for ~ =/~ = 1, that is, Eqs. (1.2), (1.3), we see that there is no contradiction between them if (41) = (r/l) = 0 ,

(2.7)

blL

= b,

Z

Im(ujv~) = - h e = - ½h711 •

U =

½aq 2 ,

(2.8) (2.9)

J

This means that H = ½(aq 2 + bp 2) = ½(m-' p 2 + aq2). Operator A = ½c(qp + pq) = JTll(qP + Pq) is necessary for the ordinary form of the equations ((t) = ( p ) / m , (p) = - a ( q ) - Vii (P) .

(2.10)

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Now we set D = q2 in (2.4) and obtain 2

(0t(q2)) = b([q, p]+) + 2c(q 2) + ~ Z

Im(ujv~ )(q2) + Z J

= b([q, p]+) + ~

Ivjl2 J

fvjf2 .

(2.11)

J

In an analogous way we get (~t(p2))

=

-a([q, p]+) - 27(p 2} + ~

(2.12)

lUj] 2 ,

J

@t(qP)) = ½(cqt[q, P]+) = -a(q 2) + b(P 2} - 7½([q, P]+ ) - Z Re(ujv])

(2.13)

J

(7 = 711). Equating the time derivatives to zero, we will find the stationary solution of Eqs. (2.11)-(2.13). We mark the stationary mean values with the subscript 0 and denote (q2)o From

=

k°l ,

I½[q,p]+)o-- - k 12, °

(p2) = kO2 "

(2.14)

(2.11)-(2.13) we thus obtain 2bk°2 + Z

-2aN°2 - 27,,k2°2 + ~

[vj[2 = 0,

J

lujl 2 = O, J

bk°2 - ak °, - ),l,k°2 - Z Re(uv]) = 0 .

(2.15)

J

Now we use the quantum regression theorem (see, for example, [17]). According to it the formulas

Ot,(x,(h)Xv(t2)) =-d,o(xo(h)x,.(h) ) at tl > t2. Or, (xv(t2)xu(h)) = -d,p(Xv(t2)xp(t,))

at t, > t2

are valid if the equations {2u(t)} = -d,p(xo(t)) denoted Xl = q, x2 = p and

(2.16)

(i.e. (2.10)) hold. Here we have

(2.17) Formulas (2.16) imply the equations

cg~k~v(O = -d~pkpv(~) 6qrfcl~v('c)

=-d~pfcpv('r)

at r > 0 ,

(2.18)

at ~ > 0 ,

(2.19)

with

k~v(z) = ½([xa(t + r)xv(t)])+,

/ ~ = ([x~(t + z),xv(t)]) .

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The values (2.14) play the part of the "initial" values for Eq. (2.1 8) (i.e. k~(0) = k°~). The "initial" conditions for (2.19) are obvious ~m,(0) : ( 0[p,q] [ q o ] ) = i ~ ( _

~ ~) .

(2.20)

Further, we will denote this matrix as . Eqs. (2.18) and (2.19) are easily solved with the help of the Laplace transforms. Denoting v17

~,

0 O~

(2.2 1)

K~v(s) = f e-S~k~v(Z) d z , o

we get -

I~(s)

=

=

s+7

s + ~

k

(2.22)

'

(2.23)

.

Results (2.22) and (2.23) help to find the matrix spectral density Sg)(09) = f

ei~°(t~-'2)k,~(tl - t2)d(tl - t2)

--00 OQ

OO

(2.24)

= f ei°~k~v(z)dz + f e-i°~k~(-z)dz , 0

0

Hence using (2.21) and the equation k~v(-z) = kw(z) valid for the stationary process we get (2.25)

S~X)(09) = K~v(i09) + Kv~(-i¢o) .

If we remove averaging in (2.10) we get equations with Langevin forces (2.26)

2~ = -duvxv + (~ ,

where ~1 = ~1, (z = ql. In the spectral language this means (2.27)

~ = (i096u~ + d~v)xv .

Using the last formula we can get the relation between (2.24) and matrix spectral density of Langevin forces [ S((v)(09) I : ( i09 - b ) a i09 + 7'

)(09) (-i09 StuX -b

a ) . -i09 + 7

(2.28)

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R.L. StratonovichIPhysica A 236 (1997) 335 352

Substituting (2.22) into (2.25) and using (2.28) we obtain S(u~)(co) = / ~ o ( - / o 9 -b

a

)+(io9

-io) + 7

-b a

)/~o

ioJ + y

(2.29)

"

Hence multiplying matrices and taking into account (2.15) we find Sl[ ( ( O ) :

~lv,[ 2, J

sI~'(¢o) -- - Z Re(ujv;), J

S~)(fD) ~---~

lUjl 2 •

J (2.30)

Using (2.9) and (2.30) it can be proved that tSl~)(~o)[ < {SI1 (~)(0))$22 (~')(09) - (h7/2) 2} 1/2

(2.31 )

Note that according to (2.31) ~tl , ) cannot be equal to zero in the non-trivial case when 7~=0, h ~ 0 . From (2.30) after the reverse Fourier transformation we have two-time symmetrized moments of the Langevin forces ½([~1(/1), ~l(t2)]+) = ~ ]vj[2¢~(/12), J

(2.32)

½([¢, (t,), ¢2(t2)]+) = - Z Re(vyu7 )6(q2), J

(2.33)

½([¢2(tl), ¢2(t2)]+) = ~ luj[2cS(tl2) J

(2.34)

(t12 = tl -- t2).

With the same method, but using equation /Tm.(-r ) = -/7,,~(~),

(2.35)

which is valid when the stationariness condition is met, and formulas ~(x), , m' 1,o9) = k~,v(i¢o) -- Rvu(--i~o), a

s+7

--I

(2.36) (2.37)

((2.37) being analogous to (2.22)), we get

Of course the two-time mean commutators and symmetrized moments of q and p may also be written easily. Due to Gaussian properties of Langevin's forces the triple and higher commutators are equal to zero. Higher moments can be expressed through the two-time ones. Formulas of the type (2.32)-(2.34) and (2.38) are also valid in the non-linear case when potential energy U = Ul(q) is arbitrary. Then another method (see the next section) should be applied for their derivation.

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3. Symmetrized moments of Langevin's forces in the multi-component case

We denote the transformation from the density matrix to the Wigner distribution by W:

w(q,p)=W[p]--(2.J)-r fexp(-i -lp.y.).(q+ ½y,q-½Y) dy,

(3.1)

where p(q',q") = (qtlPlq") is the density matrix in the coordinate representation, q = ( q i ..... q,.), Y = (Yl,..., Yr). Using (3.1) we easily obtain the following relations: i]~ t3 \ W[q~M] = (q~, + --f-~p~)W[M],

W[Mq~,]=-(q:,

ih W[p~M] = (p~ - - - ~

h o+@ --f ~) W[Mp~] = ( ip~,

~ ~3~-)W[M],

W[M], (3.2)

with arbitrary operator M. We will use the Lindblad-Kossakowski equation of the form (2.1) where now

Gj = uj~q~ + vj~p~,

A = ½c~[q~, p~]+ .

(3.3)

Applying the transformation W to both sides of Eq. (2.1) and using (3.3) and formulas (3.2) give the equation for the Wigner distribution w(q, p):

r~(q, p) = Cow + 7~p-~p (p~w)

1

[

a2w

2

*

~2w

* a2w ]

J

(3.4) Here we have assumed that

E Im ( uj~ v~lt) = -- ½h ~/i = - hc ~ , J

E Im(u#u~fl) = O, J

Z Im(t,j~v~) = 0.

(3.5)

J

The superoperator [0 is caused by H:

LoW = W{(ih)-l[H, p]} for H = T + U. This expression may be represented in the Moyal form [18] 2 h ~ ~3 ~ O LoW = ~ s i n { ~ [(~q~)n(~'~p~)w -(-~p~)z4(-~q~)w]) H ( q ' p ) w ( q ' p ) '

(3.6)

with H(q, p) = (2nh)rW[H]. In (3.6) subscripts H and w indicate to which function the corresponding partial derivatives refer. Supposing that T = ½b~¢p~p¢, and that

R.L. Stratonovich/Physica A 236 (1997) 335 352

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Ul = (2~h)rw[u] does not depend on momenta, we get from (3.6) o~

1

c~w b~llpl~q+l~l )! /~0W -- 63Ul(q) Oq= 63p= = (2l+1 I(~)Ut

(~)w

--

63

63

(@)21

21+1

(3.7)

The operator

(3.8)

A = ¼7~/3[q~,Pe]+

appearing on the right-hand side of (2.1) is selected in such a way that the dissipative parts of the drift vectors in (3.4) and (3.7), or to be precise in the expression

~U1 63w Ow c~ 63q~ 63P~ b~pl~q ~ + ? ~ - ~ p (p~w), correspond to the appropriate terms in Eqs. (1.9) and (1.10). Transition to the Wigner distribution w(q,p) means that symmetrization ordering of operators q and p is chosen. After choosing operator ordering, concepts of the non-quantum probability theory may be applied with the only reservation that probabilities and probability densities are now not necessarily semipositive. According to non-quantum statistical theory existence of the terms with second derivatives signifies that the delta-correlated Langevin forces are present. Due to symmetrization operator ordering not other moments than the symmetrical ones should be taken as having a delta-type form. Thus we obtain from (3.4)

l 5([~(/1 ), ~fl(t2)]+) = Z

VjeV;f;(~(tl2),

(3.9)

J

Re(rj=Uj*ll)6(h2)

' ([~z(tl), q/~(t2)]+) = - Z

(3.10)

J

uj~uj~a(t,2 ).

½([rh(t,), r//~(t2)]+ } = ~

(3.11 )

J

These formulas are the generalization of the results (2.32)-(2.34) of the previous section. We will denote for short Z J

= v e,

= J

- ~ Re(vj~u]~) = W~I~.

(3.12)

Note that the Langevin forces are Gaussian even in the case of the non-linear Eqs. (1.9) and (1.10) if dissipation is linear.

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4. Two-time commutators of Langevin forces in the general case of dissipation

linear

Eqs. (1.9) and (1.10) may be written in a more compact form:

:~,(t) = fu(x(t)) + ~ ,

(4.1)

where (xl,... ,X2r) = ( q l . . . . . qr, Pl,..., Pr), ( ( 1 , . . . , ( 2 r ) = (~1 . . . . . ~r, ?]l . . . . . ~]r)- From (4.1) we have / ,

T

xu(z) - x,(0) = / f,(x(t))dt + / ~,(t)dt. 0

(4.2)

0

Any instant to may be taken instead of 0 (we are writing it 0 for short). We may write the following approximate expression on the right-hand side of (4.2): / .

T

x~(z) - x~(O) = / ~u(t)dt + f~(x(O))z + O(23/2).

(4.3)

o

This approximation is sufficient for us. Substituting (4.3) into formula

Yt,v(z) d___ef([X~(Z) -- x~,(O),Xv(Z) -- Xv(0)])

(4.4)

gives

[~(tl),~v(t2)])dtldt2 +O(z2),

Yuv('c)= 0

v > O.

(4.5)

o

On the other hand, opening the braces in (4.4), we get r , v -- ( [ x , ( r ) . x ~ ( ~ ) ] ) + ([x~(0).xv(O)]) - ([x~(~).x~(0)]) - ( [ x ~ ( 0 ) . x v ( r ) ] )

(4.6) For the stationary process ([x~(z),Xv(Z)]) = ([x~(0),xv(0)]). Then (4.6) gives Y#v(z) = -([x~(z) - x#(0),xv(0)]) + ([xv(Q - xv(0),x#(0)]) .

(4.7)

According to the quantum regression theorem, equations (~(t)xp(0)) = (f~(x(t))xp(O)), (x,(O)~,(~))

(4.8)

(xp(O)L(x(t)))

=

are valid when 0?~(t)) = (f~(x(t))) is valid for the additive random forces. The latter equations follow from (4.1) for (~) = 0. Eqs. (4.8) imply ([~(t).xAO)])

=

([L(x(t)).xAO)])

or

([6u(t),xp(0)]) = ([f~(x(0) + 6u(t)),xp(O)]),

(4.9)

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345

with 6~(t) = x~(t) -xu(O). Eqs. (4.9) may be solved with various degrees of accuracy. A very rough approximation ([3,(t),xp(0)]) = ([L(x(0)),xp(0)]) + O(t) is sufficient for us. It gives after integration from 0 to ([xp(r) - xp(0),xp(0)]) = ([f~(x(0),xp(0)])z + O(~2).

(4.10)

Substituting (4.10) into (4.7), we get Y,,,(r) = - ([f,(x(O)),x,(O)]}r + ([fi(x(O)),x,(O)])~ + O(~ 2).

(4.1 1 )

Comparing (4.5) and (4.11) we conclude that ([~(tl),~v(t2)]) = {-([L(x(O)),xv(O)]) + ([L(x(O)),x,(O)])}6(t12) .

(4,12)

Here any argument may be taken instead of 0. Now we take into consideration the actual form of functions f,(x). From (1.9), (1.10) and (4.1) we have

f~(x) --= b~.~.p~. ,

(4.13)

i

fr+~(X) = - ~ [ U ( q ) , p~] - 7~6P~ at

~ =

1 , . . . , r . For /~ = ct~
(4.14) in (4.12), we get with the help of (4.13)

( [ ~ ( t l ) , ~/¢(t2)]) = { - b ~ ([p~, q/~]) + b~, ([p;,, q~]) }6(tl 2 ). Hence ([~(tl), ~/3(t2)]) = 0 . Now we set # = r + ~

(4.15)

> r, v - - - - r + f l > r and due to (4.14) we obtain

i ([~r+~(tl), ~r+/~(t2)]) = { -- -h ([[U, p~], p~] - [[U,p~],p~])j6(tl2).

(4.16)

Using the identity

[[A,B], C] + [[B, C],A] + [[C,A],B] = 0

(4.17)

with A =- p~., B = U , C = p ~ we see that expression in angle brackets on the righthand side of (4.16) vanishes. Therefore ([r/~(tl), t//~(t2)]) = 0 . Finally, we set in get

(4.12)

(4.18)

# = ~
([~(tl),q~(t2)]) = ~ -k

i

v =

r+fl

> r and owing to (4.13) and (4.14) we

if[U, p~], q~]) - 7~6[P6, q~] }b(tl2).

(4.19)

R.L. Stratonovich l Physica d 236 (1997) 335-352

346

According to (4.17) the commutator with U may be expressed by means of [[p/~, q~], U] and [[q~, U], p/~]. Both of them are equal to zero because [q~, U(q)] --- 0, and because are the c-numbers. Consequently, (4.18) gives

[p#,q~]

(4.20)

([¢~(tl), q#(t2)]) = ihT/~6(tl2).

Formulas (4.15), (4.18) and (4.20) are the generalizations of the previous result (2.38).

5. On possibility of the discontinuous two-time commutators In the previous considerations two-time commutators were supposed to be continuous, i.e. having the same property as the symmetrized moments. It should be noted, however, that the possibility of discontinuous commutators exists due to the fact that the second term on the right-hand side of (2.36) appears with the minus sign unlike in the case of Eq. (2.25). Owing to this fact we may set

f~v(O+O)=ih( ~-I

~+l)fl ,

(5.1)

where ct, fl, e are arbitrary. Then according to (2.35) we have

fC~v(O-O)=ih(_~-ct_1

-~+l)_fl .

(5.2)

We see from (5.1) and (5.2) that functions/~uv(z) are discontinuous at z = 0 if at least one quantity of ~, fl, e is not equal to zero. It is natural to set

~,v(o) = ½[£-~(0 + o) + ,~,~(0 - 0)]

(5.3)

in accordance with the Dirichlet conditions. It is obvious that (5.3) coincides with (2.20) (due to (5.1) and (5.2)) as it should be. In the case (5.1) the formula

I£(s)=ih( sa s + ~ - b ) - ' (

~_1

~+l)fl

(5.4)

should hold instead of (2.37). Using (2.36) and (5.4) we then obtain the following commutators of Langevin forces: ([~(t]), ~(tz)])

= ih

( - 6-2~8t, 6-2~0t,) ~(tl - 2~c3t, -2fl0t,

t2),

(5.5)

with 6 -- a~ + b/~ + (~ + 1)~. This result replaces (2.38). the symmetric r x r In the linear multi-dimensional case, when U = l matrices ~,/~, g should be taken into consideration instead of a,/~, y. Then commutators of Langevin forces obtained by the same method are

ia~q~q~,

([~(tl), ~/~(t2)]) =

ih(bg- gb- 2~Ot, )6(t12),

(5.6)

R.L. StratonovichlPhysica A 236 (1997) 335-352

347

= ih(7 T + ~T _}_~ .jr. t ~ -- 2~,, )6(t12),

(5.7)

([~a(tl), q/3(t2)])

([q~(tl), r/e(t2)]) = ih(ga - ag +/}5 r - ~fi - 2/}c~t,)6(t,2) .

(5.8)

This generalizes (5.5). Lindblad's equation and the quantum regression theorem are unable to determine ~, fi, ~ if discontinuance of ([q=(tl),qe(t2)]), ([q~(tl),pfl(t2)]), ([p~(tl),pfl(t2)])is allowed. We may try to determine d, fi, g with the help of the condition of vanishing of undesirable forces {l(t),...,{r(t), which are difficult to account for by physical reasoning. Letting commutators (5.6) and (5.7) to be zero we get d = 0, g = 0, fi = --/~-19T. The last equation means fl~.~,= -2=> Then (5.8) takes the form ([q~(tl), ~//~(t2)]) = 2ih)~=l~Ot, a(q2 ).

(5.9)

This result may be corroborated by the fluctuation-dissipation theorems. It should be noted, however, that formula (5.9) contradicts the consequence (3.1 1) of the Lindblad equation. Let us show this by considering only one variable ~/1 -= q when )ql ¢ 0. We introduce two operators

Yl = 2a f

e -oltl cosooltq(t)dt,

--oo

Y2 =

2a f

e -
(5.10)

--(2,O

Applying (5.9) we get the commutator ([YI'Y2]) =4°'2 7

7 e x p ( - o - ] t ] - o-]fl])cos~olt sinoolfl[q(t),q(fl)]dtdt'

--oo --o~

= 8ih211~r2

e-

~rt

d at' cos ~ol t ~-7[esinoolt]dt

(or > 0). This gives {[Yl, Y2]) = 4ih211aCOl .

(5.11)

In the analogous way we can obtain from (3.1 1)

(y~) = 2,h Z uj, J (y2) = 2crh Z J

2 (-02 +

20.2

--~-+7 '

]Ujl ]2 ~o2 + ~r2 .

(5.12)

(5.13)

R.L. Stratonovich/PhysicaA 236 (1997) 335 352

348

From (5.11)-(5.13) where 211 and )-]j luj~l 2 are independent of ~ol it is easily seen that the well-known inequality

Al <[y,, y2])12 ~<(y~)(y2) (valid for any pair of the self-adjoint operators Yl,Y2) is violated for sufficiently large For absence of the above contradiction the correlators ([q~(tl),q~(t2)]+) of the Langevin forces should be more singular functions than (3.11) if (5.9) is valid. Such is the case for the Langevin forces obtained in [12-14] and with the help of the quantum Nyquist formula. Note that the infinite mean square values of velocities and momenta follow from the quantum Nyquist formula and therefore the state of the system cannot be described by the density matrix p in this case.

6. The random Hamiltonian depending on the Langevin forces and return to the Lindblad equation: stochastic equations If we introduce the random Hamiltonian Hr(t) =

½b~l~p~p~+U+A+p~(t)-q~q~(t)=H+A+p~-q~q~

(6.1)

(A = ½c~[q~, p~]+), the time-evolution unitary operator can be written in the form Utto = T e x p

-~

i

Hr(t')dt' t0

=Texp

{; -~

i

[(H

+A)dt' + p~d~(t')- q~d~(t')]t,

to

}

.

(6.2)

Here 7- indicates that the operators in brackets are ordered chronologically according to the values of the subscript t I. The differentials of the integral processes t =

t

(6.3)

=

J 0

J 0

enter into the right-hand side of (6.2). Let us write the differential of the operator (6.2):

d Utto : exp { - i [(H + A)dt + p~d~(t) - q~dO~(t)] }U,o =(1-h[(H+A)dt+

p~d~(t)-q~dq~(t)]- ~-~-~(p~d~l

~ _q~dq~)2 ) (6.4)

R.L. Stratonovich/Physica A 236 (1997) 335-352

349

Here the stochastic differential expression is understood in the Ito sense, and therefore the terms quadratic in d ~ , dO~ have been written. According to (3.9)-(3.12), (4.15), (4.18) and (4.20) we have

{d~(t)d~fl(t)) = (d~l~d~) = V~l~dt, ih

(dq~dq~) = (aql~dO~) = U~l~dt,

(6.5)

(dq~d~13) = (W~I~ - ~7~l~)dt. ih

(6.6)

Besides that, (~) = 0, (q~) = 0. The operators H, A, p~, q~ in (6.2), (6.4) are the operators of the Schr6dinger picture. The operators of the Heisenberg picture are defined by the formula DH(t) = UtotDUtto

(6.7)

= Utot). See, for example, formula (5.2.37) from [11]. Using (6.4) and (6.7) we get

DH(I + d t ) = DH(t) + d D H = Ut0t{ 1 + ~[(Hi + A ) d t + p~d~a - q~dVl~] - ~(pal

d ~~ - q~d0~) 2 }

xDH(t)(I~ -- ~[(Hi +A)dt + p ~ d ~ - q~d?l~] - -~-~(p~d~l

~ - q~dqa)2}Uuo

I

(6.8) Let us average both sides of this equation with respect to increments d~(t), dq~(t), which are independent of the preceding increments. These increments commute with DH(t) due to statistical independence. Owing to (6.5) and (6.6) we have

ih ((p~d~ - q~dq~)2) = [p~p~V~ + q~q~U~ - p~q~(W~ + ~ 7 ~ ) -

-q~p~(W~/~

ih ,~] - -fY~l~)J dt.

(6.9)

According to (6.5), (6.6), (6.8), (6.9) after averaging we get from (6.8)

\

Z

-(W~-

d(dq = - 2~2 2 V~flp~D H H(t)pflH + U~flGD H HqfiH

ih2,/~)q~pl~'~ H H DHTj+}+ h [ H . + AH,D H] = [r(t)DU(t) .

(6.10)

If we multiply both sides of (6.10) by p and take trace, we get

dDH(t) dt / = (LT (t)DH(t)) '

(6.11)

R.L. Stratonovich/ Physica A 236 (1997) 335-352

350 which is equivalent to

dp(t) _ Lp(t) dt

(6.12)

where

• +A,p] + -~1 {2[V~p~pp~ + U~q~pq~- (W~ + lihT~)q~pp~ Lp = -~[/4

-(W~-½ihT~)q~p~,p]+}.

(6.13)

This is the reduced form of Lindblad's equation equivalent to (2.1) for our case. The arbitrary Lindblad equation can be obtained by using the same method (see Appendix). Now we will consider stochastic differential equations. It is easy to prove that N

aim Z

N

A~(tj)A~(tj) = V~llZ,

!i~moZ A~l~(tj)Aq~ (tj) = U ~ z ,

~--+0

~

j=O

j=O

N

lim Z A ~ ( t j ) A ~ ( t j ) = 6---+0 j=0

(W~ + ½i~/~)'c ,

(6.14)

where A~(tj) = ~=(tj+l)- {~(tj), AO#(tj) = O~(tj+l)- ~l~(tj), to = t, tu = t + Z, e = maxj(tj+l --tj). Convergence in (6.14) takes place in the sense of the norm [IBII = (BBr) 1/2. Due to (6.14) we may write equations

d~d~e = d~ed~ ~ = V~#dt,

dO,dO# = aoaao = U~#dt ,

d~,d71~ : (W,, + ½ihY~#) dt ,

(6.15)

which do not contain averaging and are stronger than (6.5) and (6.6). Using them we obtain from (6.8) i n H ] d ~~ ( t ) dDH(t) = £T (t)DH(t)dt + ~[p~,D

i H H ]dt/~(t), ~ -~[q~,D

(6.16)

which is the equation in the Ito sense• Hence, letting D n = q~ and D n = p n, we derive Eqs. (1•9) and (1.10). In the Schr6dinger picture the Ito-type stochastic equation i

-

i

dp(t) = Lp(t) dt - ~ [p~, p(t)] d ~ ( t ) + ~ [q~, p(t)] d~l~(t)

(6.17)

corresponds to (6.16).

Acknowledgements The author thanks V.P. Belavkin, E.B. Davies and R.F. Streater for interesting discussions and the Department of Mathematics, University of Nottingham for its hospitality.

R.L. Stratonovich / Physica A 236 (1997) 335-352

35 l

The author also wants to express his gratitude to N.G. van Kampen for his interest in this work and important remarks. The financial support from Kapitza Fellowships and from E.P.S.R.C. (project GR/K08024) is greatly appreciated.

Appendix. The general form of Lindblad's equation and the random Hamiltonian If we set

Hr(t)dt = Hldt + Z ( G ] d A j J

+ GjdAJ) ,

(A. 1)

we obtain t

V,,o -- ~-exp { - ~- lI--I d,' + G7 d,'i~(,':) + C~ dAJ(,')l}.

(A.2)

to

Here Aj(t) are the Gaussian processes with properties

[dAj(t),dA~(t)] = 6jk dt .

(A.3)

It is convenient to set dA~(t)dAj(t) = 0 as in [19]. Then due to (A.3) one has

dAj( t ) dA~( t ) = fijk dt .

(A.4)

If operators Gj, G~ in (A.1) depend on t, the stochastic expression in (A.1) is understood in the Ito sense. From (A.2) we have

dUff,) = {I - ~[Hii d t + G ] d A j ( t ) + G j d A J ( t ) ]

- - ~12 ( G j,d A j + G j

dAJ)2]}Uuo " (A.5)

Now we have

( GJ dAj + Gj dAJ ) 2 = G) Gi dt

(A.6)

owing to (A.4) and the equation dA~(t)dAj(t) Section 6 we get the master equation

b = - ~i[ H ~,p]+~1 ~ 2 c j p G J

= 0. With the same method as in

- GJcjp - p c ~ c j )

---Lp

J Instead of (6.17) we now have the Ito-type stochastic equation i

t

i

dp = £p - ~ [G), p] dAj - ~[Gj, p] dA~ ,

R.L. Stratonovich/Physica A 236 (1997) 335-352

352

while stochastic equation of the Heisenberg picture is of the form

dDn(t)

=

i

H ,D H jd,

+

Z 2(G;

t n D H G;H

-

J

+~[(G])H,Dn]dAj(t) + ~[G~ i H ,D H ]dAj(t). t So the operator processes quantum master equation.

Aj(t)

are

sufficient for deriving the general form of the

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

F. Bloch and R.K. Wangsness, Phys. Rev. 89 (1953) 728. E.B. Davies, Commun. Math. Phys. 15 (1969) 119; 19 (1970) 83; 22 (1971) 51. A. Kossakowski, Rep. Math. Phys. 3 (1972) 247. A. Kossakowski, Bull. Acad. Pol. Sci. S6r. Math. Astr. Phys. 20 (1972) 1021; 21 (1973) 649. E.B. Davies, Commun. Math. Phys. 39 (1974) 91. E.B. Davies, Quantum theory of Open Systems (Academic Press, London, 1976). G. Lindblad, Commun. Math. Phys. 48 (1976) 119. V. Gomi, A. Kossakowski and E.C. Sudarshan, J. Math. Phys. 17 (1976) 821. P. Talkner, Ann. Phys. 167 (1986) 390. R.L. Stratonovich and O.A. Chichigina, J. Exp. Theor. Phys. 71 (1994) 56. R.L. Stratonovich, Nonlinear Nonequilibrium Thermodynamics I (Springer-Verlag, Berlin-HeidelbergNew York-Tokyo, 1992). P. Ullersma, Physica 32 (1966) 27, 56, 74, 90. A. L6pez, Z. Phys. 192 (1966) 63. J.T. Lewis and L.C. Thomas, in: Functional Integration and its Applications, ed, A.M. Arther (Clarenden, Oxford, 1975) pp. 97-123. H. Maasen, Phys. Lett, A 91 (1982) 107. R.F. Streater, J. Phys. A 15 (1982) 1477. C.W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences (Springer, Berlin-Heidelberg-New York-Tokyo, 1994). R.P. Moyal, Proc. Cambridge Phil. Soc. 45 (1949) 99. R,L. Hudson and K.R. Parthasarathy, Commun. Math. Phys. 93 (1984) 301.