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Brownian motion of a quantum harmonic oscillator
Vol. 10 (1976)
REPORTS
BROWNIAN
MOTION
ON
MATHEMATICAL
OF A QUANTUM
No. 3
PHYSICS
HARMONIC
OSCILLATOR
G~~RAN LINDBLAD Department
of Theoretical
Physics, Royal Institute (Received
May
of Technology,
S-100 44 Stockholm,
Sweden
4, 1976)
The problem of describing the Brownian motion of a quantum harmonic oscillator or free particle is treated in the formalism of quantum dynamical semigroups. Certain inequalities involving the friction and diffusion coefficients and Planck’s constant are derived. The nature of the quantum Langevin equation is discussed.
1. Introduction The problem of describing the Brownian motion of a quantum harmonic oscillator has been studied repeatedly from several points of view ([l], [5]-[7], [9], [IO], [14]-[16]). The object of this paper is to give a formalism which in a certain sense is as close as possible to the classical Ornstein-Uhlenbeck theory. The starting point is a recent result giving the most general form of the bounded generator of a quantum dynamical semigroup [l 11. This form is here assumed to be valid also for the unbounded generator in the special model we consider. The arbitrary operators contained in the expression for the generator are restricted to a class described by a few parameters. This is done by adopting a condition which is a quantum generalization of the classical formula for the higher order moments of a Gaussian stochastic process in terms of the first and second moments. The main difficulty of any quantum mechanical description of Brownian motion lies in the concept of friction. In classical mechanics the friction force cannot be derived from a Hamiltonian theory. This is obvious from the fact that the friction causes a decrease in the phase space volume, i.e. Liouville’s theorem is not valid. The contraction also contradicts the Heisenberg uncertainty relations: Any finite volume in phase space will in the course of time fall into a volume smaller than h/2. The diffusion term in the Langevin equation increases the volume in phase space of course, the equilibrium state being such that the two tendencies balance. As we will show below, if we can dispose of the parameters in the classical Ornstein-Uhlenbeck theory (for a free particle), then we can always construct a violation of the uncertainty relations. This points to the necessity of some quantum modification of the classical formulae involving the friction and diffusion coefficients. We realize that in quantum theory we cannot have separate friction and diffusion terms, each of which induces a behaviour similar to that given by their classical counterparts. 13931
394
G. LINDBLAD
In the case of an overdamped
quantum
harmonic
oscillator
we find that the friction
coefficient must satisfy an inequality containing Planck’s constant. As another result we find that in general the equilibrium state is not the canonical state associated with the harmonic oscillator Hamiltonian. This condition holds only for a choice of parameters which do not correspond to the classical harmonic oscillator. One consequence of this is that all the equipartition conditions cannot hold. An interesting problem in the theory of Brownian motion is the equivalence of the Fokker-Planck and Langevin approaches. This equivalence is known to hold in the classical case. It seems that the Langevin theory is difficult to formulate for processes which are such that the canonical commutation relations are not conserved. We find a way of describing the time development of the class of systems considered here by a pair of classical Wiener processes. The rule of obtaining the mean values of polynomials in P and Q must, however, take into account the values of the commutators. The usual Langevin formalism adopted in quantum optics involves fluctuating operator-valued forces which do not’ operate in the Hilbert space of the system itself, hence there is a considerable difficulty of interpretation in contradistinction to the classical case ([8], [2]). 2. Classical
Brownian motion
We will recall some formulas from the Ornstein-Uhlenbeck theory tion for a harmonic oscillator [17]. The Langevin equations read dp =
of Brownian
-~pdt-mmn)2qdt+ydW(t), (2.1)
dq = m-‘pdt,
where /I is the friction coefficient, y* = mBkT and W(t) stands for the standard process: the unique stationary Gaussian stochastic process defined by
E[ W(t) W(t’)] for expectation
= min(t,
value
Wiener
W(0) = 0,
E[W(t)l = 0, where E stands
mo-
t, t’ > 0,
t’),
([4]). If we write
Ip = p--EM, we find that for given initial
values
= mkT,
limE[dp:]
,+CC
In the limit cu --f 0 the second
of p and q,
relation
limE[dq:]
*-CO
is replaced
= h-T/mu*.
by
E[z!lq:] z 2Dt, where the diffusion
constant
D is derived
from the equipartition
law to be
D = kT/mp.
If the dispersions /3t s 1:
of p, q in the initial
EL+:1 z mkT.
state are taken
into account,
E [Aqf] z (m@)-2Apg + Aqi + 2Dt.
we find that for
BROWNIAN
MOTION OF A QUANTUM
HARMONIC
OSCILLATOR
395
We now consider the question whether these relations can be valid for a quantum particle. First note that the quantum equipartition values are identical to the classical ones for w = 0. If the uncertainty relations are satisfied for t = 0 we have (mj3)- 2dp; +nq: Choose
the initial
a h/m/9.
state such that the equality
holds.
Then
side is smaller than h2/4. This means If we can choose kT/b M h/pt then the right-hand that if the parameters /I, T are free to take any values, then we can arrange the initial conditions in such a way that the contraction of the phase space due to the friction term dominates over the diffusion during a certain time interval, thereby violating the uncertainty relations. For completeness we give here the equations of motion of the average observables which are linear or quadratic in p and q. The equations are derived from (2.1) by averaging over the Wiener process. &p
= -pp-mo2q,
$
(p’) = - 2t!lp2-2mw2pq+
$
(q2) = 2m-‘pq,
f&q) where we have written 3. Quantum mechanical
2y2,
(2.2)
= m-‘p2-mw2q2-bpq,
p for EEp], etc. formalism
We will use a formalism which is closely related to the classical Fokker-Planck theory but which works in the Heisenberg picture. A more detailed description of the mathematical apparatus with necessary definitions is given in [l 11.The dynamics of a dissipative quantum system living in the Hilbert space &’ is assumed to be described by a semigroup of normal completely positive maps @* of B(S) into itself: >
@, E CPM% Q’s’ @t = W)
@s+t ,
= 1,
C&(X) + X ultra weakly when
t + 0, all X E B(S).
G. LINDBLAD
396 The generator
L is defined
by
W) = for X in some dense domain
in B(S).
$
@t(X)
,I=0
When
L is bounded
we have shown
that it is of
the form L(X) = i[H, X]+ CL,(X), (3.1) Li(X) = Vi+XVi-&
{Vi+ Vi, X},
[ll]. where H s.a. E B(S), Vi E B(X) and ( , } d enotes the anticommutator In the present case of a harmonic oscillator the condition that L is bounded cannot hold. We will assume this form for the generator with Hand I’i unbounded as the simplest way to construct an appropriate model. From the fact that for this model we can give the action of !Dt on the Weyl operators explicitly, it will be shown in the Appendix that CDtand L have the properties demanded. We now want to determine the form of the operators H, Vi. In order to construct a tractable model we must assume that they are “nice” functions of P and Q. This rules out symmetric non-essentially selfadjoint operators and isometric non-unitary operators. In order that the system should describe the Brownian motion of a harmonic oscillator, we must restrict the choice of L guided by the classical equations (2.2). Let fin denote the set of polynomials in P and Q of degree less or equal to n. We may consider the following conditions on L. (a) The equations of motion for elements in 17, should be as close as possible It is evident from 3 2 that they cannot be the same. (b) The equilibrium fluctuations of P and Q should be given by the quantum law. This condition cannot be exactly fulfilled as we will see below. A weaker but more precise version of (a) is (c) L(U,)
=n1,
L(17,)
can actually
(d) Apart from
the harmonic potential
invariance
equipdrtition
= rr.
This condition Translation
to (2.2).
means
be demanded
for the separate
pieces in (3.1).
term L should be translation
invariant.
that P,
w31 = -wp, a.
We also want to find a way of implementing the idea that the process is Gaussian, i.e. that if s.a. elements in fl, have a normal distribution at t = 0, then this is true for all t > 0. We remember the classical formula for the third and fourth moments in terms of the first and second moments for a set of jointly normally distributed variables: E[z&C,dXzdX,]
= 0,
E [OX, &‘, OX, dX,]
= E [AX, AX,] E [AX, AX,] + +E[dX,dX,]E[dX,dX,]+E[dX,dX,]E[nx,nx,]
BROWNIAN
MOTION
OF A QUANTUM
HARMONIC
for any choice of (Xi} in the set and where dX = X-E[X]. The corresponding relation will be formulated using a set of dissipation functions [ll] D(X,, m-1
X2) = U~,~J-~(~,)~,
quantum
-~1Ux,),
>x2, X3) = WXlJf2 9X3) -x1 q-G
m-,,~,,~,,a
397
OSCILLATOR
9X3) 3
= ~(X,,~,~,,~,)+~,D(~,,~,)~,-~,~(x,,~,,~,)--D(x,,x,,x,)x,.
Note that D(X, Y) is chosen to be bilinear here instead of sesquilinear as in [ll]. The dissipation functions are all zero when L is a derivation. A simple computation gives from (3.1) that they have the positivity properties [vi, X]+[vj,
o(x+,
x, = C
D(X+,
Y+, Y, X) = C
D(x+, y+, Y,x> =
Xl 2 03
[Vi, X]+Y+Y[Vi,
X]~
C[[viyxl,Y]+[[vyX],
0,
Y]
(3.2)
> 0.
In order to obtain the quantum counterpart of the moment relations above we develop the expressions into sums of products of expectations, replace the classical variables by the quantum observables in 17, and E by Qt. The order of the operators is important of course. A clever choice of ordering and differentiation at t = 0 gives the quantum moment conditions (e) D(Xr, X2, Xd -X2 D(X,, D(X,yX~,X~,X~)
= 0,
X4 = 0, Ull (Xi> C II,.
We can see from the solution below that condition (e) implies normal distribution for elements in fl, is conserved in time. From the relation all X, YEn, D(x+, y+, x, Y) = 0,
that the property
of
and (3.2) follows that [[Vi, x], Y] = 0, i.e. that [Vi, X] ~fl,, and Vi EIT, for all i. It then follows that both conditions in (e) are satisfied and that D(X, Y) en0 for all X, Y ~17,. We note that in the classical case the diffusion comes into (2.2) only through the term 2y2 which is of zero order in p and q. The dissipative part of the generator L defined by the Vi in (3.1) satisfies condition (c) if all Vi ~p7,. Consequently, we must demand this also of the Hamiltonian part, i.e. H E II2 . The term in H belonging to 17, is chosen to be zero for simplicity and we write H = i [m-‘P2
+ mozQ2 + ;i(PQ + QP)]
.
Write Vi = UiP+biQ. The constant term is omitted as its contribution to the generator to a term in the Hamiltonian linear in P and Q. It is easily seen that for every map of the form L is equivalent
X+CV,+XV,=
c
[lail’PXP+a*biPXQ+b*UiQXP+
jbi\‘QXQ]
G. LINDBLAD
398
it is possible to choose the Vi such that the sum reduces to two terms, i = 1,2. We also see that the quantities
determine the map uniquely (compare [3], 0 1.4) and hence the non-Hamiltonian of the generator L in (3.1). Introduce the vectors b=
a = (a1,6),
(b19bZ)9
part
ai9biEC,
and the real parameters x, ,u through (a,b)s Ca,*bi = X+ip. Then the parameters pn, o, I, la], lbl, x,,ucompletely specify the generator L. of the form (3.1) with the Hamiltonian given above. We can easily calculate the action of L on P and Q: L(P) = (p-A)P-mdQ, L(Q) = m-lP+(,u+I)Q.
A comparison with (2.2) gives p = -;3,
21=
/!?.
(3.3)
It is easily checked that ,D = - 1 is precisely the relation which must be satisfied if L is to be translation invariant apart from the harmonic potential term (condition (d)). We will continue to use the general values of L and ,B in the next section. The action of L on L!* is calculated to be: L(Pz)
= ~(,u-~)P~-~~co~R+~~~~,
L(Q’)
= 2Cj~+il)Q~+2rn-‘R+lal~,
L(R) = m-‘P2-mo2Q2+2,uR-x,
where R = + (PQ i-QP). We note that only the presence of the terms Ial2 and x distinguishes this from (2.2). Note that if a = 0 then ,u = 0, hence we cannot obtain a set of equations identical to (2.2). 4. Solution of the quantum equations of motion
We introduce a new basis in 17, instead of P and Q: X = P+m(;l+v)Q,
Y = P+m(A-v)Q.
where y2 = w2 - A2. We find that [X, Y] = 2imv,
L(X) = cp+v)X,
L(X) = (p-,lJ)X.
BROWNIAN
To X, Y correspond
MOTION
parameter
OF A QUANTUM
vectors
HARMONIC
399
OSCILLATOR
A, B.
Vi = AiX+BiY, A = (2mv)-‘[-m(A-v)a+b], B = (2mv)-‘[m(il+v)a-b]. Introduce
the Weyl operators
ewbW+qY)l
JV(E, V) =
=
exp(itX)exp(inY)exp(imvtq)
= exp(iqY)exp(iEX)exp(
- imv&j).
Obviously, any polynomial in X and Y can be obtained There are also the useful relations
from
W(6, 7) by differentiation.
[X, IV] = -2mvrjW,
{X, W} = -2i-$W,
[Y, W] = 2mvtW,
(Y, W> = -2i-$
W.
We want to find the semigroup C$ generated by L.As is shown in the Appendix to find the action of ot on the Weyl operators. Qt(W([, 7)) is the unique -$
(Gt W) = L(Qt W). The solution @@IV,
where 5,) q,, ft are determined +((W’expX) L(W,) find
is easily
= [+i(x,
expressed
turns
out to be of the form
7)) = WG,
= wexpf;,
W~}~~+~i{Y,w,}i.l,+W,j,lexpf; and
i* = CU+v)Et,
The solutions
Qexpff
by the equation
in commutators
j
it is enough solution of
= -2(mv)2/q1A
anticommutators.
= L(W,)expJ;. By identification
we
;7r = (/J--V)%, -5tB12.
are elementary ,$, = eCc+VE 3
qt = &-w
77 (4.1)
ft([, From them e.g.
r) = -2(mv)2
we can obviously
,i Ie(P--Y)T~A-e(~+Y’r5B12dz.
find out how Qt operates
QCX = e’/‘+““X ,
on any polynomial
@( Y = e(~-“)‘Y
in x and Y,
400
G. LINDBLAD
We can write
$1.
@,(W(E, r7)) = expti(~~lX+“Il~tY)+fi(~, The calculation of the following combinations is real, i.e. that we have the underdamped case.
(4.2)
are then very simple.
We assume
that Y
(4.3)
A,(X, Y) = imv(1 -e2/ct)-
(2mv)2Re(B,
A) 1 e’/“&. 0
5. Equilibrium
states
We now want to determine librium. If ,U+Y < co, then
the parameters
limQ+(VE,
A, B from the equipartition
7)) = expk(5,
t-as
$1, (5.1)
fm(E, q) = -(mv)2[/~~++v~-‘~B~2E2+~~~-~~1-1jA~2~2-2J~~-’Re(A, This means
that if we define the dual semigroup @:e(w)
then it follows
as indicated
in the Appendix
where the equilibrium
fluctuations (AX’)
dl
= coo defined
= 0,(X,X)
Y) = 2(m~)~j/~-~~]-‘lA~~,
= d,(X,
Y) = i~~~~-2(n~~~)2~~l-1Re(B, A),
L!lx = X-(X), of A, B and a simple
As DtX --f 0, c#+Y -+ 0,
= ~(~v)~I,u+Y/-~~B~~,
where
The definition
by
= exp if- (5 I 41.
in X, Y are given by foe, of course.
on the state space through
that
state Q, is uniquely
e, [ Wt, The equilibrium we find that
@F operating
B)tq].
= n(@t W),
lim@c *+cC (in trace norm)
law at equi-
iX) calculation
lm(A,
= n,(X),
etc.
gives the relation
B) = -,uu/2mv.
(5.2)
BROWNIAN
In order that
MOTION
OF A QUANTUM
(5.2) shall be consistent
the Schwartz
(Re(A , B))‘+ must be satisfied.
In terms
(WA,
When Y = 0 this is the ordinary
401
inequality
fluctuations
we find the relation
-(;(dXdY+dYdx))2
uncertainty
OSCILLATOR
B))* 6 IA121B12
of the equilibrium
(1 -Y2/~2)(‘4X2)(dY2)
HARMONIC
relation
We can compare fm(E, 7) with the expectation state corresponding to the Hamiltonian
B (mv)“.
(5.3)
(h = 1). of W(E, r) in the Gibbs
canonical
HO = t (m-’ P2 + mw2Q2). T
and the temperature
eT = Z,lexp(
- H,/kT),
ZT = Tr [exp( - H,/kT)]
.
We know that ([12]) e=(exp i(uQ + UP)) = exp[ - (4m2w)-l(u2 + (mwv2))coth(w/2kT)]. In order to obtain
eT(B’(5,
7)) we put
24= m((i+v)6+(1+)7j), If e,
is equal to eT we can identify
the coefficients
ZI = 5+q. in (5.1) to obtain
jAJ2 = [2mv2(l+y)]-r1w\p-_ylcoth(o/2kT), lBi2 = (2mv2w)-‘;l(A+v)I,u+vIcoth(w/2kT),
(5.4)
Re(A , B) = - (2mv2)-‘ol,u[ coth(w/2kT). These relations tion
are equivalent
to the equipartition
law. (5.3) now reads after simplifica-
,LL(,u~ - AZ)-+ < coth(co/2kT)
(5.5)
(note that 1 > o > 0). (5.5) is a necessary and sufficient condition for the existence of a solution of (5.4) for A and B. Because the left-hand side is greater than unity, the inequality holds only for sufficiently large values of T. Especially, if (3.3) holds as indicated by classical analogy or translation invariance, then there is never a solution. If (5.5) does not hold we can at most satisfy two of the equipartition conditions, say those for AX2 and dY2 which define ]A] and IBI. Alternatively we can fulfil the equipartition conditions for dP2 and 0Q2. In order that it shall be possible to satisfy the equipartition conditions for dX2 and dY2 it is a necessary and sufficient condition that the weaker inequality (Im(A, is satisfied.
B))2 < lA121B12
(5.4) gives after simplification: I/&J~-‘(,u~ -y2)-’
< coth(w/2kT).
(5.6)
402
G. LINDBLAD
In the case ,u = -;Z we obtain
the relation Y < ocoth(o/2kT).
(5.7)
If inequality (5.6) is not satisfied we can at most satisfy one equipartition condition. An analogous calculation for the underdamped case (a~ imaginary) gives the same condition (5.5) for the existence of a Gibbs equilibrium state. The inequality is fulfilled for all temperatures if and only if il = 0 (note that R < o in this case). This is the choice made in most papers on the subject, but then the equations of motion for P and Q are not the classical ones. 6. The free Brownian particle If we use (3.3) and let o --+ 0 we find that Y=
X = P+m/?Q, From
P.
(4.3) follows that &(X, x) = (mB)21B12t, limd,(Y,
f-+LX
The equipartition
conditions
Y) = $ m2&412.
on dX 2, dY2 in (5.4) now give in the limit w -+ 0: IAl2 = JB12 = 2kT/m/?.
For fit large we have
(6.1) Identification
gives
which is the classical constant reintroduced,
result.
The compatibility
condition
(5.7) now reads,
with Planck’s
h/9 < 4kT. Equivalently
we get a condition
on D: 4mD k h.
It is easy to see that this inequality implies that the reasoning of $ 2 leading to a contradiction with the uncertainty relations cannot be carried through. If one chooses to fulfil only the equipartition condition on dY2 (= dP2 in this case) and defines D from (6.1) as D = ; JB12, we obtain
from
(5.6), with A introduced D > h2/3/16mkT.
Hence,
for given 13, T we can choose
the parameters
such that
D = max(kT/m/?, h2/I/16mkT).
BROWNIAN
7. Langevin
OF A QUANTUM
MOTION
HARMONIC
403
OSCILLATOR
equations (4.2) can be represented in the following processes with the correlation
It is easily seen that the solution Wx(t), WY(t) be normalized Wiener E[Wx(t)
Wr(t)l
= - [Re(A, B)/IAI * 1Bl]min(t,
way. Let
t’).
Write Fx(t)
= 2mvlBI S etP+Y)(f-T)dWx(z), 0
F,(t)
= 2mvlAl ~e(P-‘)C’+fW,(t),
(7.1)
0
F(t) = W,(t) + TW), u,(W(E, q)) = expi(S~tX+r~tY+F(t)). A simple calculation
gives @,t(W) =
E[@Ql.
(7.2)
We also def$ne
u,(w),&q=o = @t-Y+J,x(t).
u,(X) = -if Derivation
gives da&Q
which is the Langevin
equation G(~‘)
= L(cxJ’)dr+2mv(B(dWx(t) for the observable
d2 = - Jp
X.
at(W) e_11=o = (GY,
E[cc,(X2)] = @Q-)2+E[F,(t)2] Similarly
we can calculate
(7.3)
clr and Qt working
%dr> = f(M),
= @,(X2).
on any function @,(f(x))
=
of X only (or Y only)
H.f~~J)l
just as in the classical Langevin theory. When dealing with functions we must take into account the fact that the commutation relations in time. cr,(XY) = - - d2 dWy E[a,(XY)]
f uw
iZTIZO
+imv
by
of both X and Y are not conserved
= cc,(X)a,(Y)+imv(l-e2”‘),
= Gt(XY) = @,(.X)O~(Y)+EIFx(t)Fr(t)]+imv(l-e2”’).
It is often claimed that in the quantum equations necessary in order to restore the canonical commutation under the dynamical semigroup a’, ([2], [8], [9]).
of motion relations
the Langevin forces are which are not conserved
404
G. LINDBLAD
There seems to be some difficulty in the interpretation of the Langevin equations actually produced to remedy this defect. In fact the fluctuating operator valued forces are not operators in the Hilbert space of the system considered (the harmonic oscillator in our case). Consequently, if we introduce sample paths for the observables of the system, these must include operators belonging to the heat bath. Thus it seems that the influence of the heat bath has not been reduced out of the description to the same extent as in the classical Langevin theory. There is a simple argument that this is no accident: we cannot in general reproduce the semigroup @‘rby adding fluctuating operator valued forces to the equations of motion for X and Y in such a way that averages of functions of X and Y can be calculated in the simple way that we could use for functions of X or Y alone. Assume the existence of such forces F,, Fy and define c(* on X, Y through da,(X) = L(a,X)dt+dF,(t), da,(Y) = _L(a,Y)dr+dF,(t). The solution
is still given
by a,(X) = Q,(X) + i e(“+‘)(‘-“)dF,(z),
E with respect to the stochastic proetc. We assume that there is defined an expectation cesses FX, Fy and a space of sample paths. Furthermore, we demand that the fluctuating forces restore the commutation relations
and that
c(~is extended
to all functions cc&x,
of X and
Y through
Y) = f(a, X, at Y).
For a given path of (F,, Fy) we know from the theory relations that there is a unitary equivalence a,(X) = Ut+XUt,
of the canonical
commutation
a,(Y) = U;‘YU,,
where U, depends on the path. Then it follows that c4(f(X,
Y)) = U:f(x,
Y) U,,
@,(f(X, V) = E[a,(f(X, V)] = E[U:f(X, r) U,l. This holds for all observables in B(S) especially for trace class operators. It is obvious that for these the trace is preserved. One can easily check in (3.1) that Qt is trace-preserving if and only if ,LL= 0. In this case the commutation relations are conserved and we can work with the scalar fluctuating forces defined in (7.1). We remark that (7.1)-(7.3) can also be used when W, and W, are Gaussian stochastic processes other than the Wiener process. In this way we can also construct a non-Markovian dynamics which may be relevant in the quantum case [7].
BROWNIAN
MOTION
OF A QUANTUM
HARMONIC
OSCILLATOR
In order that the map @, defined by (7.2) shall be completely must satisfy condition (A.2) given in the Appendix, i.e.:
E[F~(t)lE[F:(t)l-E[F,(t)F,(t)l’
positive
405 for all t, we
> IW, ~llz(~l -e2pW)2.
Appendix In order to verify that the map di, defined by (4.2) defines a semigroup of completely positive maps in the sense of [ll] we must first prove that it is completely positive. Let the linear map @ be given on the Weyl operators (as defined in § 4) by @(IV,
v)) = II%‘, v’)
expf(f , r)
where the map (E, 7) + (F, q’) is given by a linear
(A-1)
transformation
K
(F, r’) = VE, r), and j(E, r) is a quadratic
form in 5 and 7: -f(E,
r) = &+28&7+y~2.
Let & be the C*-algebra spanned by the Weyl operators, and M,(d) the IZx n matrix algebra over &, with matrix indices i,j = 1, . . . , n. Note that d is actually the norm closed linear span of the W(E, r). @ is defined on d by linearity and continuity. By definition @ is completely positive on & if for every n Aij
E
M”(d)+
*
By linearity and density arguments for all elements of the type
=
it follows
Aij
A simple calculation
A:j
=
w(Fi,
@(Ai/)
Mu+
E
that it is enough
Ti)+
w(fj,
. to demand
qj).
gives @(Aij) Lij
=.f(ti-tj;.,
=
w(f:,
qi)+
w(Ci,
T/J)exp&j,
~i-rj)+imv(l-d)(E*rj-ri~j), A
=
det
K.
Obviously, @(Aij) is positive in M,(d) if and only if exp&j matrix. By Lemma 1.7 of [13] we know that a sufficient condition is the positivity of
is a positive scalar valued for the positivity of expLij
Lij = Lij+LOO-Lie-Loj
for every choice of 0 in the index set. Note that Lii = 0, Lij = Lj*i. A simple calculation L:j = 2rT’ L’
gives
‘rj,
where
this condition
(Ei-ECJ,
Ti-rlO),
L’=
[
c1 p+is
@-id y
1 )
s = &rv(l-d).
406
G. LINDBLAD
Obviously,
Llj is positive
iff L’ is positive, a > 0,
y 3 0,
i.e. iff “y-/Y
> 82.
(A-2)
These conditions imply that f(6, r) is negative definite. If (A.2) is not fulfilled, then L’ has at least one negative eigenvalue. By choosing cl to be the corresponding eigenvector we find that eXp(Lij), i, j = 0, 1, is not positive. From this follows that exP(Lj)
= exP(L~i)exP(L:j)exp(L,j)
is not positive. Consequently, (A.2) is a necessary and sufficient condition for the positivity of ~(Aij). Hence @ is completely positive on JZZ’if and only if (A.2) is satisfied. From Proposition 4.2 of [5] it follows that @ can be continued to an ultraweakly continuous map @ E CP(Z),. Now we can check that (4.2) defines a semigroup of the desired type. The semigroup property on ~2 follows from direct computation. (A.2) is satisfied, hence Qt is completely positive on .RZ. From Propositions 4.2 and 4.3 of [5] follows that Qt can be extended to a semigroup of completely positive maps such that Qt(X) --t X ultraweakly, t + 0, all X, as outlined at the end of [5]. Note that in [5] the continuity property is formulated in an equivalent way in terms of the dual semigroup on the set of normal states: @Fo + e,
t + 0 in trace norm.
In the same way we can show that in 9 5 we have, in the trace norm, @Q-Q,,
t-too.
REFERENCES [l] [2] [3] [4] [5] [6] [7] [S] [9] [lo] [II] [12] [13] [14] [15] [16] [17]
Agarwal, G. S.: Phys. Rev. A4 (1971), 739. -: Springer Tracts Mod. Phys. 70, 1973. Arveson, W.: Acta Math. 123 (1969), 141. Breiman, L. : Probability, Addison-Wesley, Reading, 1968. Davies, E. B.: Commun. math. Phys. 27 (1972), 309. -: ibid. 33 (1973), 171. Ford, G. W., M. Kac, P. Mazur: J. Math. Phys. 6 (1964), 504. Haken, H.: Handb. Phys. 25, 2c, Springer, Berlin, 1970. Lax, M.: Phys. Rev. 145 (1966), 1IO. Lewis, J. T., L. C. Thomas: Ann. Inst. H. PoincarB, A22 (1975), 241. Lindblad, G.: On the gerrerators of yuanfunz &nanzira/ semigroups, to appear in Commun. Phys. Messiah, A.: Mechutzique quaniique I, Ch. 12, Dunod, Paris, 1962. Parthasarathy, K. R., K. Schmidt: Springer Lecture Notes in Mathematics 272 (1972). Schwinger, J.: J. Math. Phys. 2 (1961), 407. Senitzky, I. R.: Phys. Rev. 119 (1960), 670. Ullersma, P.: Physica 32 (1966), 27. Wax, N.: Selected pupers on noise and stochastic processes, Dover, New York, 1959.
math.
REPORTS
BROWNIAN
MOTION
ON
MATHEMATICAL
OF A QUANTUM
No. 3
PHYSICS
HARMONIC
OSCILLATOR
G~~RAN LINDBLAD Department
of Theoretical
Physics, Royal Institute (Received
May
of Technology,
S-100 44 Stockholm,
Sweden
4, 1976)
The problem of describing the Brownian motion of a quantum harmonic oscillator or free particle is treated in the formalism of quantum dynamical semigroups. Certain inequalities involving the friction and diffusion coefficients and Planck’s constant are derived. The nature of the quantum Langevin equation is discussed.
1. Introduction The problem of describing the Brownian motion of a quantum harmonic oscillator has been studied repeatedly from several points of view ([l], [5]-[7], [9], [IO], [14]-[16]). The object of this paper is to give a formalism which in a certain sense is as close as possible to the classical Ornstein-Uhlenbeck theory. The starting point is a recent result giving the most general form of the bounded generator of a quantum dynamical semigroup [l 11. This form is here assumed to be valid also for the unbounded generator in the special model we consider. The arbitrary operators contained in the expression for the generator are restricted to a class described by a few parameters. This is done by adopting a condition which is a quantum generalization of the classical formula for the higher order moments of a Gaussian stochastic process in terms of the first and second moments. The main difficulty of any quantum mechanical description of Brownian motion lies in the concept of friction. In classical mechanics the friction force cannot be derived from a Hamiltonian theory. This is obvious from the fact that the friction causes a decrease in the phase space volume, i.e. Liouville’s theorem is not valid. The contraction also contradicts the Heisenberg uncertainty relations: Any finite volume in phase space will in the course of time fall into a volume smaller than h/2. The diffusion term in the Langevin equation increases the volume in phase space of course, the equilibrium state being such that the two tendencies balance. As we will show below, if we can dispose of the parameters in the classical Ornstein-Uhlenbeck theory (for a free particle), then we can always construct a violation of the uncertainty relations. This points to the necessity of some quantum modification of the classical formulae involving the friction and diffusion coefficients. We realize that in quantum theory we cannot have separate friction and diffusion terms, each of which induces a behaviour similar to that given by their classical counterparts. 13931
394
G. LINDBLAD
In the case of an overdamped
quantum
harmonic
oscillator
we find that the friction
coefficient must satisfy an inequality containing Planck’s constant. As another result we find that in general the equilibrium state is not the canonical state associated with the harmonic oscillator Hamiltonian. This condition holds only for a choice of parameters which do not correspond to the classical harmonic oscillator. One consequence of this is that all the equipartition conditions cannot hold. An interesting problem in the theory of Brownian motion is the equivalence of the Fokker-Planck and Langevin approaches. This equivalence is known to hold in the classical case. It seems that the Langevin theory is difficult to formulate for processes which are such that the canonical commutation relations are not conserved. We find a way of describing the time development of the class of systems considered here by a pair of classical Wiener processes. The rule of obtaining the mean values of polynomials in P and Q must, however, take into account the values of the commutators. The usual Langevin formalism adopted in quantum optics involves fluctuating operator-valued forces which do not’ operate in the Hilbert space of the system itself, hence there is a considerable difficulty of interpretation in contradistinction to the classical case ([8], [2]). 2. Classical
Brownian motion
We will recall some formulas from the Ornstein-Uhlenbeck theory tion for a harmonic oscillator [17]. The Langevin equations read dp =
of Brownian
-~pdt-mmn)2qdt+ydW(t), (2.1)
dq = m-‘pdt,
where /I is the friction coefficient, y* = mBkT and W(t) stands for the standard process: the unique stationary Gaussian stochastic process defined by
E[ W(t) W(t’)] for expectation
= min(t,
value
Wiener
W(0) = 0,
E[W(t)l = 0, where E stands
mo-
t, t’ > 0,
t’),
([4]). If we write
Ip = p--EM, we find that for given initial
values
= mkT,
limE[dp:]
,+CC
In the limit cu --f 0 the second
of p and q,
relation
limE[dq:]
*-CO
is replaced
= h-T/mu*.
by
E[z!lq:] z 2Dt, where the diffusion
constant
D is derived
from the equipartition
law to be
D = kT/mp.
If the dispersions /3t s 1:
of p, q in the initial
EL+:1 z mkT.
state are taken
into account,
E [Aqf] z (m@)-2Apg + Aqi + 2Dt.
we find that for
BROWNIAN
MOTION OF A QUANTUM
HARMONIC
OSCILLATOR
395
We now consider the question whether these relations can be valid for a quantum particle. First note that the quantum equipartition values are identical to the classical ones for w = 0. If the uncertainty relations are satisfied for t = 0 we have (mj3)- 2dp; +nq: Choose
the initial
a h/m/9.
state such that the equality
holds.
Then
side is smaller than h2/4. This means If we can choose kT/b M h/pt then the right-hand that if the parameters /I, T are free to take any values, then we can arrange the initial conditions in such a way that the contraction of the phase space due to the friction term dominates over the diffusion during a certain time interval, thereby violating the uncertainty relations. For completeness we give here the equations of motion of the average observables which are linear or quadratic in p and q. The equations are derived from (2.1) by averaging over the Wiener process. &p
= -pp-mo2q,
$
(p’) = - 2t!lp2-2mw2pq+
$
(q2) = 2m-‘pq,
f&q) where we have written 3. Quantum mechanical
2y2,
(2.2)
= m-‘p2-mw2q2-bpq,
p for EEp], etc. formalism
We will use a formalism which is closely related to the classical Fokker-Planck theory but which works in the Heisenberg picture. A more detailed description of the mathematical apparatus with necessary definitions is given in [l 11.The dynamics of a dissipative quantum system living in the Hilbert space &’ is assumed to be described by a semigroup of normal completely positive maps @* of B(S) into itself: >
@, E CPM% Q’s’ @t = W)
@s+t ,
= 1,
C&(X) + X ultra weakly when
t + 0, all X E B(S).
G. LINDBLAD
396 The generator
L is defined
by
W) = for X in some dense domain
in B(S).
$
@t(X)
,I=0
When
L is bounded
we have shown
that it is of
the form L(X) = i[H, X]+ CL,(X), (3.1) Li(X) = Vi+XVi-&
{Vi+ Vi, X},
[ll]. where H s.a. E B(S), Vi E B(X) and ( , } d enotes the anticommutator In the present case of a harmonic oscillator the condition that L is bounded cannot hold. We will assume this form for the generator with Hand I’i unbounded as the simplest way to construct an appropriate model. From the fact that for this model we can give the action of !Dt on the Weyl operators explicitly, it will be shown in the Appendix that CDtand L have the properties demanded. We now want to determine the form of the operators H, Vi. In order to construct a tractable model we must assume that they are “nice” functions of P and Q. This rules out symmetric non-essentially selfadjoint operators and isometric non-unitary operators. In order that the system should describe the Brownian motion of a harmonic oscillator, we must restrict the choice of L guided by the classical equations (2.2). Let fin denote the set of polynomials in P and Q of degree less or equal to n. We may consider the following conditions on L. (a) The equations of motion for elements in 17, should be as close as possible It is evident from 3 2 that they cannot be the same. (b) The equilibrium fluctuations of P and Q should be given by the quantum law. This condition cannot be exactly fulfilled as we will see below. A weaker but more precise version of (a) is (c) L(U,)
=n1,
L(17,)
can actually
(d) Apart from
the harmonic potential
invariance
equipdrtition
= rr.
This condition Translation
to (2.2).
means
be demanded
for the separate
pieces in (3.1).
term L should be translation
invariant.
that P,
w31 = -wp, a.
We also want to find a way of implementing the idea that the process is Gaussian, i.e. that if s.a. elements in fl, have a normal distribution at t = 0, then this is true for all t > 0. We remember the classical formula for the third and fourth moments in terms of the first and second moments for a set of jointly normally distributed variables: E[z&C,dXzdX,]
= 0,
E [OX, &‘, OX, dX,]
= E [AX, AX,] E [AX, AX,] + +E[dX,dX,]E[dX,dX,]+E[dX,dX,]E[nx,nx,]
BROWNIAN
MOTION
OF A QUANTUM
HARMONIC
for any choice of (Xi} in the set and where dX = X-E[X]. The corresponding relation will be formulated using a set of dissipation functions [ll] D(X,, m-1
X2) = U~,~J-~(~,)~,
quantum
-~1Ux,),
>x2, X3) = WXlJf2 9X3) -x1 q-G
m-,,~,,~,,a
397
OSCILLATOR
9X3) 3
= ~(X,,~,~,,~,)+~,D(~,,~,)~,-~,~(x,,~,,~,)--D(x,,x,,x,)x,.
Note that D(X, Y) is chosen to be bilinear here instead of sesquilinear as in [ll]. The dissipation functions are all zero when L is a derivation. A simple computation gives from (3.1) that they have the positivity properties [vi, X]+[vj,
o(x+,
x, = C
D(X+,
Y+, Y, X) = C
D(x+, y+, Y,x> =
Xl 2 03
[Vi, X]+Y+Y[Vi,
X]~
C[[viyxl,Y]+[[vyX],
0,
Y]
(3.2)
> 0.
In order to obtain the quantum counterpart of the moment relations above we develop the expressions into sums of products of expectations, replace the classical variables by the quantum observables in 17, and E by Qt. The order of the operators is important of course. A clever choice of ordering and differentiation at t = 0 gives the quantum moment conditions (e) D(Xr, X2, Xd -X2 D(X,, D(X,yX~,X~,X~)
= 0,
X4 = 0, Ull (Xi> C II,.
We can see from the solution below that condition (e) implies normal distribution for elements in fl, is conserved in time. From the relation all X, YEn, D(x+, y+, x, Y) = 0,
that the property
of
and (3.2) follows that [[Vi, x], Y] = 0, i.e. that [Vi, X] ~fl,, and Vi EIT, for all i. It then follows that both conditions in (e) are satisfied and that D(X, Y) en0 for all X, Y ~17,. We note that in the classical case the diffusion comes into (2.2) only through the term 2y2 which is of zero order in p and q. The dissipative part of the generator L defined by the Vi in (3.1) satisfies condition (c) if all Vi ~p7,. Consequently, we must demand this also of the Hamiltonian part, i.e. H E II2 . The term in H belonging to 17, is chosen to be zero for simplicity and we write H = i [m-‘P2
+ mozQ2 + ;i(PQ + QP)]
.
Write Vi = UiP+biQ. The constant term is omitted as its contribution to the generator to a term in the Hamiltonian linear in P and Q. It is easily seen that for every map of the form L is equivalent
X+CV,+XV,=
c
[lail’PXP+a*biPXQ+b*UiQXP+
jbi\‘QXQ]
G. LINDBLAD
398
it is possible to choose the Vi such that the sum reduces to two terms, i = 1,2. We also see that the quantities
determine the map uniquely (compare [3], 0 1.4) and hence the non-Hamiltonian of the generator L in (3.1). Introduce the vectors b=
a = (a1,6),
(b19bZ)9
part
ai9biEC,
and the real parameters x, ,u through (a,b)s Ca,*bi = X+ip. Then the parameters pn, o, I, la], lbl, x,,ucompletely specify the generator L. of the form (3.1) with the Hamiltonian given above. We can easily calculate the action of L on P and Q: L(P) = (p-A)P-mdQ, L(Q) = m-lP+(,u+I)Q.
A comparison with (2.2) gives p = -;3,
21=
/!?.
(3.3)
It is easily checked that ,D = - 1 is precisely the relation which must be satisfied if L is to be translation invariant apart from the harmonic potential term (condition (d)). We will continue to use the general values of L and ,B in the next section. The action of L on L!* is calculated to be: L(Pz)
= ~(,u-~)P~-~~co~R+~~~~,
L(Q’)
= 2Cj~+il)Q~+2rn-‘R+lal~,
L(R) = m-‘P2-mo2Q2+2,uR-x,
where R = + (PQ i-QP). We note that only the presence of the terms Ial2 and x distinguishes this from (2.2). Note that if a = 0 then ,u = 0, hence we cannot obtain a set of equations identical to (2.2). 4. Solution of the quantum equations of motion
We introduce a new basis in 17, instead of P and Q: X = P+m(;l+v)Q,
Y = P+m(A-v)Q.
where y2 = w2 - A2. We find that [X, Y] = 2imv,
L(X) = cp+v)X,
L(X) = (p-,lJ)X.
BROWNIAN
To X, Y correspond
MOTION
parameter
OF A QUANTUM
vectors
HARMONIC
399
OSCILLATOR
A, B.
Vi = AiX+BiY, A = (2mv)-‘[-m(A-v)a+b], B = (2mv)-‘[m(il+v)a-b]. Introduce
the Weyl operators
ewbW+qY)l
JV(E, V) =
=
exp(itX)exp(inY)exp(imvtq)
= exp(iqY)exp(iEX)exp(
- imv&j).
Obviously, any polynomial in X and Y can be obtained There are also the useful relations
from
W(6, 7) by differentiation.
[X, IV] = -2mvrjW,
{X, W} = -2i-$W,
[Y, W] = 2mvtW,
(Y, W> = -2i-$
W.
We want to find the semigroup C$ generated by L.As is shown in the Appendix to find the action of ot on the Weyl operators. Qt(W([, 7)) is the unique -$
(Gt W) = L(Qt W). The solution @@IV,
where 5,) q,, ft are determined +((W’expX) L(W,) find
is easily
= [+i(x,
expressed
turns
out to be of the form
7)) = WG,
= wexpf;,
W~}~~+~i{Y,w,}i.l,+W,j,lexpf; and
i* = CU+v)Et,
The solutions
Qexpff
by the equation
in commutators
j
it is enough solution of
= -2(mv)2/q1A
anticommutators.
= L(W,)expJ;. By identification
we
;7r = (/J--V)%, -5tB12.
are elementary ,$, = eCc+VE 3
qt = &-w
77 (4.1)
ft([, From them e.g.
r) = -2(mv)2
we can obviously
,i Ie(P--Y)T~A-e(~+Y’r5B12dz.
find out how Qt operates
QCX = e’/‘+““X ,
on any polynomial
@( Y = e(~-“)‘Y
in x and Y,
400
G. LINDBLAD
We can write
$1.
@,(W(E, r7)) = expti(~~lX+“Il~tY)+fi(~, The calculation of the following combinations is real, i.e. that we have the underdamped case.
(4.2)
are then very simple.
We assume
that Y
(4.3)
A,(X, Y) = imv(1 -e2/ct)-
(2mv)2Re(B,
A) 1 e’/“&. 0
5. Equilibrium
states
We now want to determine librium. If ,U+Y < co, then
the parameters
limQ+(VE,
A, B from the equipartition
7)) = expk(5,
t-as
$1, (5.1)
fm(E, q) = -(mv)2[/~~++v~-‘~B~2E2+~~~-~~1-1jA~2~2-2J~~-’Re(A, This means
that if we define the dual semigroup @:e(w)
then it follows
as indicated
in the Appendix
where the equilibrium
fluctuations (AX’)
dl
= coo defined
= 0,(X,X)
Y) = 2(m~)~j/~-~~]-‘lA~~,
= d,(X,
Y) = i~~~~-2(n~~~)2~~l-1Re(B, A),
L!lx = X-(X), of A, B and a simple
As DtX --f 0, c#+Y -+ 0,
= ~(~v)~I,u+Y/-~~B~~,
where
The definition
by
= exp if- (5 I 41.
in X, Y are given by foe, of course.
on the state space through
that
state Q, is uniquely
e, [ Wt, The equilibrium we find that
@F operating
B)tq].
= n(@t W),
lim@c *+cC (in trace norm)
law at equi-
iX) calculation
lm(A,
= n,(X),
etc.
gives the relation
B) = -,uu/2mv.
(5.2)
BROWNIAN
In order that
MOTION
OF A QUANTUM
(5.2) shall be consistent
the Schwartz
(Re(A , B))‘+ must be satisfied.
In terms
(WA,
When Y = 0 this is the ordinary
401
inequality
fluctuations
we find the relation
-(;(dXdY+dYdx))2
uncertainty
OSCILLATOR
B))* 6 IA121B12
of the equilibrium
(1 -Y2/~2)(‘4X2)(dY2)
HARMONIC
relation
We can compare fm(E, 7) with the expectation state corresponding to the Hamiltonian
B (mv)“.
(5.3)
(h = 1). of W(E, r) in the Gibbs
canonical
HO = t (m-’ P2 + mw2Q2). T
and the temperature
eT = Z,lexp(
- H,/kT),
ZT = Tr [exp( - H,/kT)]
.
We know that ([12]) e=(exp i(uQ + UP)) = exp[ - (4m2w)-l(u2 + (mwv2))coth(w/2kT)]. In order to obtain
eT(B’(5,
7)) we put
24= m((i+v)6+(1+)7j), If e,
is equal to eT we can identify
the coefficients
ZI = 5+q. in (5.1) to obtain
jAJ2 = [2mv2(l+y)]-r1w\p-_ylcoth(o/2kT), lBi2 = (2mv2w)-‘;l(A+v)I,u+vIcoth(w/2kT),
(5.4)
Re(A , B) = - (2mv2)-‘ol,u[ coth(w/2kT). These relations tion
are equivalent
to the equipartition
law. (5.3) now reads after simplifica-
,LL(,u~ - AZ)-+ < coth(co/2kT)
(5.5)
(note that 1 > o > 0). (5.5) is a necessary and sufficient condition for the existence of a solution of (5.4) for A and B. Because the left-hand side is greater than unity, the inequality holds only for sufficiently large values of T. Especially, if (3.3) holds as indicated by classical analogy or translation invariance, then there is never a solution. If (5.5) does not hold we can at most satisfy two of the equipartition conditions, say those for AX2 and dY2 which define ]A] and IBI. Alternatively we can fulfil the equipartition conditions for dP2 and 0Q2. In order that it shall be possible to satisfy the equipartition conditions for dX2 and dY2 it is a necessary and sufficient condition that the weaker inequality (Im(A, is satisfied.
B))2 < lA121B12
(5.4) gives after simplification: I/&J~-‘(,u~ -y2)-’
< coth(w/2kT).
(5.6)
402
G. LINDBLAD
In the case ,u = -;Z we obtain
the relation Y < ocoth(o/2kT).
(5.7)
If inequality (5.6) is not satisfied we can at most satisfy one equipartition condition. An analogous calculation for the underdamped case (a~ imaginary) gives the same condition (5.5) for the existence of a Gibbs equilibrium state. The inequality is fulfilled for all temperatures if and only if il = 0 (note that R < o in this case). This is the choice made in most papers on the subject, but then the equations of motion for P and Q are not the classical ones. 6. The free Brownian particle If we use (3.3) and let o --+ 0 we find that Y=
X = P+m/?Q, From
P.
(4.3) follows that &(X, x) = (mB)21B12t, limd,(Y,
f-+LX
The equipartition
conditions
Y) = $ m2&412.
on dX 2, dY2 in (5.4) now give in the limit w -+ 0: IAl2 = JB12 = 2kT/m/?.
For fit large we have
(6.1) Identification
gives
which is the classical constant reintroduced,
result.
The compatibility
condition
(5.7) now reads,
with Planck’s
h/9 < 4kT. Equivalently
we get a condition
on D: 4mD k h.
It is easy to see that this inequality implies that the reasoning of $ 2 leading to a contradiction with the uncertainty relations cannot be carried through. If one chooses to fulfil only the equipartition condition on dY2 (= dP2 in this case) and defines D from (6.1) as D = ; JB12, we obtain
from
(5.6), with A introduced D > h2/3/16mkT.
Hence,
for given 13, T we can choose
the parameters
such that
D = max(kT/m/?, h2/I/16mkT).
BROWNIAN
7. Langevin
OF A QUANTUM
MOTION
HARMONIC
403
OSCILLATOR
equations (4.2) can be represented in the following processes with the correlation
It is easily seen that the solution Wx(t), WY(t) be normalized Wiener E[Wx(t)
Wr(t)l
= - [Re(A, B)/IAI * 1Bl]min(t,
way. Let
t’).
Write Fx(t)
= 2mvlBI S etP+Y)(f-T)dWx(z), 0
F,(t)
= 2mvlAl ~e(P-‘)C’+fW,(t),
(7.1)
0
F(t) = W,(t) + TW), u,(W(E, q)) = expi(S~tX+r~tY+F(t)). A simple calculation
gives @,t(W) =
E[@Ql.
(7.2)
We also def$ne
u,(w),&q=o = @t-Y+J,x(t).
u,(X) = -if Derivation
gives da&Q
which is the Langevin
equation G(~‘)
= L(cxJ’)dr+2mv(B(dWx(t) for the observable
d2 = - Jp
X.
at(W) e_11=o = (GY,
E[cc,(X2)] = @Q-)2+E[F,(t)2] Similarly
we can calculate
(7.3)
clr and Qt working
%dr
= @,(X2).
on any function @,(f(x))
=
of X only (or Y only)
H.f~~J)l
just as in the classical Langevin theory. When dealing with functions we must take into account the fact that the commutation relations in time. cr,(XY) = - - d2 dWy E[a,(XY)]
f uw
iZTIZO
+imv
by
of both X and Y are not conserved
= cc,(X)a,(Y)+imv(l-e2”‘),
= Gt(XY) = @,(.X)O~(Y)+EIFx(t)Fr(t)]+imv(l-e2”’).
It is often claimed that in the quantum equations necessary in order to restore the canonical commutation under the dynamical semigroup a’, ([2], [8], [9]).
of motion relations
the Langevin forces are which are not conserved
404
G. LINDBLAD
There seems to be some difficulty in the interpretation of the Langevin equations actually produced to remedy this defect. In fact the fluctuating operator valued forces are not operators in the Hilbert space of the system considered (the harmonic oscillator in our case). Consequently, if we introduce sample paths for the observables of the system, these must include operators belonging to the heat bath. Thus it seems that the influence of the heat bath has not been reduced out of the description to the same extent as in the classical Langevin theory. There is a simple argument that this is no accident: we cannot in general reproduce the semigroup @‘rby adding fluctuating operator valued forces to the equations of motion for X and Y in such a way that averages of functions of X and Y can be calculated in the simple way that we could use for functions of X or Y alone. Assume the existence of such forces F,, Fy and define c(* on X, Y through da,(X) = L(a,X)dt+dF,(t), da,(Y) = _L(a,Y)dr+dF,(t). The solution
is still given
by a,(X) = Q,(X) + i e(“+‘)(‘-“)dF,(z),
E with respect to the stochastic proetc. We assume that there is defined an expectation cesses FX, Fy and a space of sample paths. Furthermore, we demand that the fluctuating forces restore the commutation relations
and that
c(~is extended
to all functions cc&x,
of X and
Y through
Y) = f(a, X, at Y).
For a given path of (F,, Fy) we know from the theory relations that there is a unitary equivalence a,(X) = Ut+XUt,
of the canonical
commutation
a,(Y) = U;‘YU,,
where U, depends on the path. Then it follows that c4(f(X,
Y)) = U:f(x,
Y) U,,
@,(f(X, V) = E[a,(f(X, V)] = E[U:f(X, r) U,l. This holds for all observables in B(S) especially for trace class operators. It is obvious that for these the trace is preserved. One can easily check in (3.1) that Qt is trace-preserving if and only if ,LL= 0. In this case the commutation relations are conserved and we can work with the scalar fluctuating forces defined in (7.1). We remark that (7.1)-(7.3) can also be used when W, and W, are Gaussian stochastic processes other than the Wiener process. In this way we can also construct a non-Markovian dynamics which may be relevant in the quantum case [7].
BROWNIAN
MOTION
OF A QUANTUM
HARMONIC
OSCILLATOR
In order that the map @, defined by (7.2) shall be completely must satisfy condition (A.2) given in the Appendix, i.e.:
E[F~(t)lE[F:(t)l-E[F,(t)F,(t)l’
positive
405 for all t, we
> IW, ~llz(~l -e2pW)2.
Appendix In order to verify that the map di, defined by (4.2) defines a semigroup of completely positive maps in the sense of [ll] we must first prove that it is completely positive. Let the linear map @ be given on the Weyl operators (as defined in § 4) by @(IV,
v)) = II%‘, v’)
expf(f , r)
where the map (E, 7) + (F, q’) is given by a linear
(A-1)
transformation
K
(F, r’) = VE, r), and j(E, r) is a quadratic
form in 5 and 7: -f(E,
r) = &+28&7+y~2.
Let & be the C*-algebra spanned by the Weyl operators, and M,(d) the IZx n matrix algebra over &, with matrix indices i,j = 1, . . . , n. Note that d is actually the norm closed linear span of the W(E, r). @ is defined on d by linearity and continuity. By definition @ is completely positive on & if for every n Aij
E
M”(d)+
*
By linearity and density arguments for all elements of the type
=
it follows
Aij
A simple calculation
A:j
=
w(Fi,
@(Ai/)
Mu+
E
that it is enough
Ti)+
w(fj,
. to demand
qj).
gives @(Aij) Lij
=.f(ti-tj;.,
=
w(f:,
qi)+
w(Ci,
T/J)exp&j,
~i-rj)+imv(l-d)(E*rj-ri~j), A
=
det
K.
Obviously, @(Aij) is positive in M,(d) if and only if exp&j matrix. By Lemma 1.7 of [13] we know that a sufficient condition is the positivity of
is a positive scalar valued for the positivity of expLij
Lij = Lij+LOO-Lie-Loj
for every choice of 0 in the index set. Note that Lii = 0, Lij = Lj*i. A simple calculation L:j = 2rT’ L’
gives
‘rj,
where
this condition
(Ei-ECJ,
Ti-rlO),
L’=
[
c1 p+is
@-id y
1 )
s = &rv(l-d).
406
G. LINDBLAD
Obviously,
Llj is positive
iff L’ is positive, a > 0,
y 3 0,
i.e. iff “y-/Y
> 82.
(A-2)
These conditions imply that f(6, r) is negative definite. If (A.2) is not fulfilled, then L’ has at least one negative eigenvalue. By choosing cl to be the corresponding eigenvector we find that eXp(Lij), i, j = 0, 1, is not positive. From this follows that exP(Lj)
= exP(L~i)exP(L:j)exp(L,j)
is not positive. Consequently, (A.2) is a necessary and sufficient condition for the positivity of ~(Aij). Hence @ is completely positive on JZZ’if and only if (A.2) is satisfied. From Proposition 4.2 of [5] it follows that @ can be continued to an ultraweakly continuous map @ E CP(Z),. Now we can check that (4.2) defines a semigroup of the desired type. The semigroup property on ~2 follows from direct computation. (A.2) is satisfied, hence Qt is completely positive on .RZ. From Propositions 4.2 and 4.3 of [5] follows that Qt can be extended to a semigroup of completely positive maps such that Qt(X) --t X ultraweakly, t + 0, all X, as outlined at the end of [5]. Note that in [5] the continuity property is formulated in an equivalent way in terms of the dual semigroup on the set of normal states: @Fo + e,
t + 0 in trace norm.
In the same way we can show that in 9 5 we have, in the trace norm, @Q-Q,,
t-too.
REFERENCES [l] [2] [3] [4] [5] [6] [7] [S] [9] [lo] [II] [12] [13] [14] [15] [16] [17]
Agarwal, G. S.: Phys. Rev. A4 (1971), 739. -: Springer Tracts Mod. Phys. 70, 1973. Arveson, W.: Acta Math. 123 (1969), 141. Breiman, L. : Probability, Addison-Wesley, Reading, 1968. Davies, E. B.: Commun. math. Phys. 27 (1972), 309. -: ibid. 33 (1973), 171. Ford, G. W., M. Kac, P. Mazur: J. Math. Phys. 6 (1964), 504. Haken, H.: Handb. Phys. 25, 2c, Springer, Berlin, 1970. Lax, M.: Phys. Rev. 145 (1966), 1IO. Lewis, J. T., L. C. Thomas: Ann. Inst. H. PoincarB, A22 (1975), 241. Lindblad, G.: On the gerrerators of yuanfunz &nanzira/ semigroups, to appear in Commun. Phys. Messiah, A.: Mechutzique quaniique I, Ch. 12, Dunod, Paris, 1962. Parthasarathy, K. R., K. Schmidt: Springer Lecture Notes in Mathematics 272 (1972). Schwinger, J.: J. Math. Phys. 2 (1961), 407. Senitzky, I. R.: Phys. Rev. 119 (1960), 670. Ullersma, P.: Physica 32 (1966), 27. Wax, N.: Selected pupers on noise and stochastic processes, Dover, New York, 1959.
math.