duced in the framework of quantum stochastic differential equations by Gardiner and Collet. I. INTRODUCTION. In quantum mechanics, measurement theory ...
PHYSICAL REVIE%' A
Measurement
VOLUME 34, NUMBER 3
theory and stochastic differential equations in quantum
mechanics
Alberto Barchielli Dirpartimento
di Fisica dell Universita di Milano, Istituto Xazionale di Fisica Xucleare, Sezione di Milano, Via Celoria 16, 20133 Melano, Italy {Received 16 January 1986)
Continuous (in time) measurements can be introduced in quantum mechanics by using operationvalued measures and quantum stochastic calculus. In this paper quantum stochastic calculus is used for showing the connections between measurement theory and open-system theory. In particular, it is shown how continuous measurements are strictly related to the concept of output channels, introduced in the framework of quantum stochastic differential equations by Gardiner and Collet.
I.
wave detectors i
INTRODUCTION
and to the problem of the description bodies. Relations to other approaches are studied in Refs. 34 and 35. Recently, quantum stochastic calculus has also been used in the treatment of continuous measurements and by means of it very general results have been obtained. ' However, up to now quantum stochastic calculus has been used more as a mathematical tool than as something arising from an idealized picture of a physical system; but for certain applications, ' ' a different point of view might be useful. The aim of this paper is to present a simplified version of the theory of continuous measurements starting from QSDE's and taking the point of view that the quantum VA'ener process and related objects describe not only some kind of noise, but also the channel which brings information from the system to the observer. Indeed, in a recent paper on QSDE's (written mostly in view of possible applications in quantum optics) Gardiner and Collet consider the possibility of using in this sense the quantum Wiener process and study the correlation functions for what they call the "out fields. As we shall see, these quantities are strictly linked to the correlations appearing in continuous measurement theory, which offers also a systematic way of coinputing them.
of macroscopic In quantum mechanics, measurement theory and opensystem theory are strictly connected. Usually, by "open system" one means some relevant system interacting with its surrounding (a thermal bath, for instance); but also an observed system can be considered as "open" with respect to the measuring apparatus. This connection is so strict that Davies has treated both topics in a book entitled Quantum Theory of Open Systems. ' In particular, very close relations arise, also at a mathematical level, between the treatment of open systems and continuous (in time) measurements. In the treatment of open systems, essentially two methods have been followed: that of master equations and that of quantum stochastic differential equations (QSDE's). In the first approach the equation for the reduced dynamics of the system is studied, which, under suitable Markov approximations, reduces to a quantum master equation. '6 In the second approach, the Markov approximations are made on the Heisenberg equations of motion for the system operators. In this way, Heisenberg equations take a form very similar to that of classical stochastic differential equations and, for treating them, the new mathematical technique of quantum stochastic calculus has been developed. This approach is equivalent to the introduction of an extremely idealized bath; now, the master equation for the reduced dynamics of the relevant system is an exact consequence of the unitary evolution of the total system. In the case of continuous measurements an axiomatic approach has been followed which is very close to that of master equations in open-system theory. First, counting processes have been introduced by Davies under the name of quantum stochastic processes then, the situation in which some observable of a system is continuously followed in its time evolution has been considered and the concept of operation-valued stochastic process has been introduced. Finally, it has been shown that both cases can be described by the same formalism. ' The dynamics of the system, including the perturbation due to the measurement, turns out to be given by a quantum master equation. Applications have been worked out, related to communication quantum to gravitationalsystems,
"
'
'
"
II.
QUANTUM STOCHASTIC
DIFFERENTIAL EQUATIONS The rigorous formulation of QSDE's is based on a now well-developed theory of stochastic integration with respect to suitable noncommuting "differentials" or "integrators. Here I do not want to review this theory, but I recall only few results and practical rules, using a nonrigorous language; moreover, I limit myself to the case of boson quantum stochastic calculus. Let us introduce the Bose fields aj(t), a~(t) satisfying the canonical coinmutation relations (CCR's)
"
s), [aj(t), a;(s)] =0; [oj(t),o; (s)] =5J5(t — in the context of QSDE's the pair of operators
(2. 1) A/,
(A/),
defined by
= A/—
t
f o;(s)ds,
(A/)t=
1986
t
f a, (s)
sd,
(2.2)
The American Physical Society
MEASUREMENT THEORY AND STOCHASTIC DII'E'ERENTIAL. . . is called quantum Wiener process. Then one must choose a suitable representation of the commutation rules (2. 1) and an initial state for the fields. We take a Gaussian initial state, without any coherent part (which, however, could be easily introduced), defined by the moments
&A') =&(A')') &
&
where
dA~Jd
A,') =5&nJ min(t,
s),
A/(A, ') ) =5J(nJ+1) min(t,
nJ)0,
~cJ
(2.3)
dU,
=
g
s),
&nJ(nJ+1);
~
4
where the differentials 4
(2.6}
and M(t), EJ(t), FJ(t), and G(t) are "adapted" operatorvalued processes, which means that they depend on the Wiener process A,', (A,')t only for times s & t; by Eqs. (2. 1) and (2.6) adapted processes commute with the fundamental differentials, i.e.,
[E,(t), dA,') =0, [EJ(t},d(A/)t]=0,
.. . .
(2.7)
If N(t) is another process satisfying an equation analogous to Eq. (2.5), then the differential of the product M(t)N(t) is given by d(M(t)N(t))
= (dM(t))N(t)+M(t}(dN(t)) + (dM (t) )(dN (t) ),
(2.8) where the "Ito correction" (dM(t))(dN(t)) can be computed by means of the following multiplication rules: dX(h) =h
[H (t), X(t) ]dt +
A/+RId(A~J) RJ d—
I
+cJRJ
t U„UO ——I +cJ'RJ ]dt j iH d—
(2. 10)
where the RJ's and H are operators in A and H =H . The solution U, of Eq. (2. 10) is a family of unitary operators on AA, which represents the evolution operator for the total system (system S and Bose fields), in the interaction picture with respect to the free dynamics of the fields It is instructive to write down the formal solution of Eq. (2. 10},from which unitarity is apparent. By using the multiplication rules (2.9}, one can see that Eq. (2. 10) is formally equivalent to
dU, =exp
iH
dt+g—[
RJd(A/)t] RJ dA~J+—
U, ,
(2.11)
are intended to "point into the fu-
A/-
(2.9)
(2.5)
ture" (Ito prescription), i.e., dA( =A)'+ui
5— ,Jc&'dt,
=5,J(n& +. 1)dt, =d(AJ) dt =(dt)2=0.
(2.4)
= g [EI(t)dA J+FJ(t)d (A/)t]+G(t)dt,
=
——,' f (nJ+1)RJ RJ+nJRJRJ
higher-order moments reduce to the previous ones because of the Gaussian character of the state. The case n, =0, cJ =0, corresponds to the use of a Fock representation of the commutation rules (2. 1) and the other cases to certain ' non-Pock representations. The non-Pock cases can be reduced to the Pock one by doubling the fields. In the physical literature the field operator aJ(t) =dA~Jldt and its formal adjoint are called quantum white noise; in the case of c, +0 one speaks of "squeezed" white noise. M Now let A be the Hilbert space of the system of interest the Hilbert space on which the chosen (system S) and representation of the CCR's (2. 1) acts. One can give a of rigorous meaning to equations for operators in AA the type
dM(t)
d(A,')
Consider now the stochastic Schrodinger equation
) denotes the quantum expectation values and
&
d(A~J)
(A,')
dAJdt
)]'= —5; c min(t s),
'= — 5,JcJdt,
d (A,~)tdA, '=5,JnJ. dt,
=0
AJA,') =[&(A,') (AJ)
&(A~J)
dA~JdA,
1643
because in the series expansion of the exponential only the terms up to second order survive and give rise exactly to Eq. (2. 10). Then, the formal solution of Eq. (2. 11) is U, =T exp
—R~
—iH s+
AJ+RJ.
AJ
~
J
(2. 12)
T means the time-ordered product. The rigorous proof of existence, uniqueness, and unitarity of the solution of Eq. (2. 10) is given in the quoted literature (when the operators RJ and H are bounded). Let now X be any system operator and consider the Heisenberg picture where
X(t) = U, XU„RJ(t) = U, RJ U„H(t) = U, HU, . (2. 13) (2.9), one easily By using Eq. (2.10) and the rules (2.7}— obtains for X(t) the following QSDE:
yJ [ [RJ (t),X(t)]dA,'+ [X(t),RJ (t)]d (A,')t — (n, + —, ) [[R,.(t), [R, (t),X(t))]+[R, (t), [R, (t),X(t)]]jdh —,
——,'(IRJ~(t), [RJ(t),X(t)] j —IRJ(t), [RJ~(t),X(t)] j )dt t)—]]dhj, t) —]]dk , c,'[R, (t), [R, (t), X(— , c, [R, (h), [R; (t),X(— '
(2. 14)
34
ALBERTO BARCHIELLI where
I
A, B}=AB+BA.
Note that in Eqs. (2. 10) and (2. 14} the parameters nj and eJ appear; they characterize the initial state of the fields or, better, in a mathematical language, the representation of the CCR's (2. 1). This is due to the use of an Ito-type calculus. Formally, also a Stratonovich-type calculus can be introduced; in this case, the initial data do not appear in the evolution equations. Similarly one can introduce the Heisenberg picture for the fields
. -=Ut ~JU — t t
t
(2. 15)
~
these fields correspond to the "output modes" of Gardiner and Collet, The while the AtJ are the "input modes. reason for this terminology is that the fields do not play only the role of noise, but they can also represent the channel which carries signals in and out of the system S.
Obviously, the fields we have introduced give a very rough picture of a physical system; in Ref. 30 some ideas are given about the approximations needed for obtaining this idealized picture starting from a physically more "realistic" one. By using Bqs. (2.7) (2. 10), one easily obtains the following QSDE for the out fields:
—
dA,""'=dA J+R~(t)dt
.
(2. 16)
Finally, taking the expectation of Eq. (2. 14) with respect to the initial state (2.3) of the fields, one obtains for the reduced dynamics of the relevant system S the following master equation:
"
dp(t)/dt
=Lp(t),
where the generator given by
(2. 17) (or Liouvillian)
L turns out to be
l
Lp=
i [H p—]
'
—,
g
I
' — (nj+ —, )([RJ,[RJ,p]]+[RJ,[Rq, p)])
+ —,'([R, tR, ,p})
,p}])+c. [R,[Rt p]]+c'[R,[R,p]]} . [RJ, IR—
(2. 18)
I
The fact that no approximation is needed f'or obtaining a Markovian master equation is due to the extremely "singular" coupling between system and fields. Moreover, it is possible to show that by using quantum stochastic calculus the most general master equation can be obtained.
III. CONTINUOUS MEASUREMENTS Let us consider Eq. (2. 10) with a modified initial condit &s &0. Now, the solution of this equation tion U, is a unitary operator U„, which depends on the fields aj(r), aj (r) only for times r such that t & r &s and, thereJ (AJ) J with u &s. Moreover, we fore, commutes with AJ,
=I,
have
Ut=U„U„ t&s&O;
U,
— = U„.
(3.1)
tion channels and on them observations can be made. For simplicity, I consider measurements only on one field, say on ai"'(t). There are many ways for treating these measurements. In Refs. 12 and 13 only measurements of the self-adjoint part of the fields were considered, while in this paper I want to treat a joint measurement of both the self-adjoint and anti-self-adjoint parts of field 1. In order to see how to introduce this measurement, let us start from a one-dimensional system (an oscillator) characterized by the creation and annihilation operators a t, a. In the most general formulation of quantum mechanics observables are represented by effect-valued (or positive-operator-valued) measures (EVM's). This generalization is essential when one has to describe a joint measurement of noncommuting quantities. A simple joint measurement of a and a t is represented by the following
'
EvM:39
Using the fact that A J and UT, commute, we can write AtJ*
"' = U, AtJ Ut = U,
UT
J =UTAtJUT
VT&t,
f
(3.2)
(z) =
d z
—exp(za
f (z), —z'a) g}( P ~
where g} is any normalized squeezed state defined by
and, therefore, the out fields
~
a'"'(t) =dA 1'"'/dt
[a'"'(t)] =d(A J'"')t/dt
F(B)= J
t AtJ UT t U,
«a+&a
(3.3)
satisfy the same CCR's (2. 1) as the in fields. Another way to obtain this result is to use Eq. (2. 16) of Sec. II and Eqs. (2.23) and (2.24} of Ref. 30. Because fields referring to different times commute, measurements of them at different times do not interfere and one can easily treat a continuous measurement too. The fields we have introduced can play different physical roles: soine of them can represent pure noise, so that any information flowing into them is lost for the experimenter; but other ones can play the role of communica-
)
I
~
exp(z
"a —za t),
state; we choose for
@&=o,
~
(3.4)
f} a (3.5)
then we have
(3.6)
If p
}—
is a statistical operator, then the quantity
P (8
~
p):Tr
I
F (B)p
(3.7)
is interpreted as the probability of finding the result z CB, when the system is in the state p. Equation (3.7) defines a true probability distribution (positivity, normalization,
MEASUREMENT THEORY AND STOCHASTIC DIFFERENTIAI. . . . and o-additivity hold). The parameters a and p characterize the measuring procedure. It is simpler to work not with the EVM F(8) itself, but with its Fourier transform (the analog of the characteristic function of a probability measure), which is defined by
:f
d z exp[i(kz'+k'z)]f (z) .
V(k)—
operator, choose Imk by taking the hmit V
"
For computWe call V(k) the "characteristic operator. (z) in Eq. (3.4) can ing it, note that the operator density be rewritten as
f
V, [k]
(3.9)
~
f [ik (s)d (A,')t+ik'(s)dA,
= exp
——,' ~ak(s) —Pk'(s)
z & is a coherent state for the annihilation operator aa +pat. Then, by using the properties of coherent states, one obtains
where
(3. 11)
which is the Fourier transform of the projection-valued measure associated with the self-adjoint operator a + at. Equation (3.10) is easily generalized to the infinitedimensional case. For the moment we do not consider system S; only the fields are present. Let us introduce the following operator:
(3.8)
f (z) = —za+z'p& &za+z'p ~,
=0, p& 0, and a=( 1+p )'~; then,
p~+ oo, we obtain (k}=exp[ik(a+a )],
~
'
(3. 12)
ds]
~
+ik'a ——,' ak —pk'
V(k)=exp(ika
~
)
~
k(s) is a complex-valued test function. The operator V, [k] is the "functional Fourier transform" of an EVM representing the joint measurement of ai(s), ai(s} for all times s, such that Ops ~t. The quantum expectation value of V, [k] is the characteristic functional of a stochastic process. In general, from a generalized characteristic functional all probabilities can be obtained by taking the anti-Fourier transform; moreover, the moments can be directly computed by functional differentiation of the characteristic functional, i.e., where
(3.10}
.
The characteristic operator (3.10) describes a measurement of the quantity associated with the operator a. For obtaining a characteristic operator describing the measurement of the self-adjoint part of the operator a alone, it is sufficient to take Imk =0. In this way we describe an imprecise measurement of a+a . For obtaining the usual prescription for the measurement of a self-adjoint
« z(ti)
i) 5 —V[k]&/5k'(ti)
)» =(
z "(t
z(t„)z'(t„+i)
5k'(t„)5k(t„+i)
&
5k(t
« »
~
where we have denoted by z(t) the stochastic process and by the mean with respect to the probability associated with the characteristic functional & V, [k] &. In our case, by using Eqs. (2.3) and (3.12), we obtain &
~i[k] =expI &
— [(2&i+1+ a '+ ~
I
I
p ') k(t) I
I
i
)k(t)' —«'p+ci)k'(t}']I
(3.14)
(3.15a) ~
S is
' —(ap'+c
p+ci )5(t —s),
«z(t)z'(s)»=-, '(2n, +1+ ]a['+ p[')5(t When the system
I
distribution
of a complex white noise z(t); z (t) is a Gaussian process with zero mean and corre-
which is the characteristic functional lations
« z(t)z(s) » = —(a
I
(3.13)
) k
—
s) .
(3.15b)
present, we have to take into account the evolution of the total system.
The joint measurement
of the output fields (3.3) is represented by the characteristic operator
—U, V, [k]U, = exp '"'[k]=
V,
&&
exp
f k(s)d(A, ""') — f [ k (s) t
i
'
—,
ds
~
exp i
~
+
~
Let us denote by p the initial state of system S and by pr„iq the initial state of the fields, defined by Eqs. (2.3). The quantity &
~~"'[k]
&
= Trss~I ~— i"'[k]t S
r„ia l
(3.17)
is now the characteristic functional of a stochastic process z (t), which has the meaning of "output signal" (it represents the result of the measurement of the output fields). As before, the moments can be computed from Eq. (3.13) (with V, V, "'}. For instance, using Eq. (3.16), we obtain
~
f k'(s)dA, ""'
ak (s) — Pk '(s)
~
]
(3.16)
—&a', "'(t) &, « z(t) » = — &A,""'&= dt
(3.18)
« z(t)z'(s) » = Blas &(A,""')tA,""'& + —,(1+ a '+ p ')5(t —s), (
)
)
« z(t)z(s}» = OtBs &A,""'A,""'&
(
(3.19a)
a*p5(t — s) .— (3.19b)
34
ALBERTO BARCHIELLI Note that these moments are strictly related to the correlation functions (5.1}of Ref. 30. Due to the system-field interaction, the output fields contain information on system S, as one can see from Eq. (2. 16) where the system operator Rz appears. The problem is now to extract this information, so that the whole construction can be interpreted as an indirect continuous measurement on S. This can be obtained by elilninating the fields from the description. Define a linear operator G, [k] from the space T(A) of trace-class operators on A (the space spanned by the statistical operators on A) into itself by
T(A),
G, [k]p=Tr~{ V, [k]Upepg;, idU, ] VpF where
Tr~ denotes
(3.20)
the partial trace over the Hilbert space
of the fields. Equation (3.20) defines the action of G, [k] on p. Then, we have d V,'"'[k] = {ik (t)d (A,""')t+ik'(t)dA,
——,' [(2n, +1+ a '+ ~
where dA,
""'is given
by
~
~
p
&X(t)V,'"'[k]) =T-r„. {XV,[k]U,pep„„,Utj
= Trg{XG, [k]p],
(3.21)
From the definition (3.20) we have, for t =0,
Go[k]=I
(3.22)
and, by taking X = in Eq. (3.21), we see that the characteristic functional (3.17) can be reexpressed as
I
(
'"'[k]) = Tr„{G, [kjpI .
(3.23)
V,
Now it is possible to obtain a closed differential equation for the operator G, [k], in which the fields do not ap(2.9), compear any more. First, by using the rules (2.7) — pute the differential of X(t)V,'"'[k] W.e need the differential of X(t), which is given by Eq. (2. 14), and the differential of V,'"'[k], which turns out to be given by
""' {
') k(t) ' —(ap'+c i )k(t)' (a"p—+c, )k'(t)']dh ~
~
I
'"'[k],
V,
(3.24)
Eq. (2.16). The result of this computation is
d(X(t) V,'"'[k) ) = (dX(t)) V,'"'[k]+X(t)(dV,'""[k])+ik(t){(n, +1)[R,(t) X(t)]+c;[R, (t) X(t)] I V,'"'[k]dt
—ik'(t) {n, [R, (t),X(t)]+c,[R, (t),X(t)] j V,'"'[k]dh
.
(3.25)
Then, take the quantum expectation value of Eq. (3.25) with respect to pph, id and divide by dt (2.3) the terms containing dA J, d (A/) vanish; using the cyclic property of the trace, we obtain
dt
By the f.irst of Eqs.
—— (X(— t)V,'"'[k]) = Tr~{XG,[k]pj dt = Tr„{X[L,+K(k(t) }]G,[k]p],
where
L is the operator
K(k)p=ik(
'
—,
in T(A) defined by
{Ri,pJ+(ni+
'
—,
)[p R i)+ci [p
——,[(2ni+1+ a + ~
~
Eq. (2. 18) and K(k) is given by
{
P~ ) { k
~
Ri]}+ik'(—,' {Ri,pj+(ni+ —, )[Ri,p]+c&[R i,pj)
—(aP +ci )k —(a'P+ci)k"
]p .
(3.27)
I
Because Eq. (3.26) holds for any system operator X and any state p, we have that G, [k] satisfies the differential equation
dG, [k] jdh
=[L+K(k(t)}]G,[k] .
(3.28)
The formal solution of this equation, with the initial condition (3.22), can be written as G, [k]=Texp
f ds[L+K(k(s))] .
(3.29)
The operator L +K(k (t) ) on T(A) can be called the generator of G, [k]. If we had introduced a joint measurement on more than one field, we would have obtained for
K (k) a sum of many contributions of the kind of Eq. (3.27). Equations (3.17), (2. 18), (3.27), and (3.22) allow us to reconstruct G, [k] in terms of system operators, so that, by Eq. (3.23), also the characteristic functional describing
our measurement is reexpressed in terms of quantities related to system S alone. We have seen [Eq. (3.13)] that the moments can be obtained by functional differentiation of the characteristic functional. In our case the first moments were given by Eqs. (3. 18) and (3.19). But, using Eqs. (3.23) and (3.29), the same moments can be reexpressed in terms of system operators alone. For the first moments we have
MEASUREMENT THEORY AND STOCHASTIC DIFFERENTIAL. . .
BK(k(t)) Trq, Bk*(t)
i ((z(t) }) = —
((z(t)z*(s)) ) = —Try
B ~
exp(Lt)p
K(k(t})
,
ak(t)ak'(t),
exp(Lt)p 5(t
exp[L (t
—s)]
. BK(k (s))
exp[L (s
—
—8(s —t)Trq
ak(s)
(3.30)
—s)
—s)Try M(k(t)) 6(t— Bk'(t)
=
= Trq{R,p(t) j,
BIC(k (s) }
t)],
BE(k (t)) Bk'(t)
—s) +6(t —s) Trg{Ri exp[L (t —s)]( —{R p(s) j+(n + +6(s t)Trg{R exp[L (s —t)]( ' {Ri, p(t) j +(n i + ' '(2n
—,
+1+ ~~ ~'+
—,
1
—8(t
a'E(k (t) ) ak "(t)'
„2
s)Tr~
exp(Lt)p
~P~')5(t
—,
((z(t)z(s)))= —Tr~
exp(Ls)p
Bk(s)
exp(Lt)p 5(t
M(k (t)) — ak'(t)
)[p(s), R, )+c; [p(s), R, ])j
)[R i,p(t)]+c [R i, p(t)]) j
—,
~
—s}
s)],
exp[L (t —
BZ(k(s))
exp(Ls)p
ak'(s)
—6(s
—s)Tr~{R i exp[L (t —s)]( —,' {R~,p(s) j+(ni+ = —(a'P+ci)5(t s)+6(t— +8(s —t)Tr„{s t j, where
8(t) = 1 for t p 0, 8(0) = —,', 8(t) =0 for t g 0, and
p(t)=exp(Lt)p .
(3.31a)
(3.32)
By Eq. (3.30}, which is the standard quantum expectation of R i at time t, the measurement we have constructed can be interpreted as a joint continuous measurement of the self-adjoint and anti-self-adjoint parts of R i. Note that we have constructed this continuous measurement by introducing a system-field coupling and a measurement on the fields; then, the fields are eliminated from the description. However, the same results can be obtained by introducing in an axiomatic way the concept of a continuous measurement on a quantum system. This is possible in the framework of the operational approach to quantum mechanics (sec, for instance, Ref. 1). Operations (and operation-valued measures) are the mathematical objects which allow us to treat both probabilities and the change of state of a system due to a measurement. We can say measures give a generalization of that operation-valued the concept of an observable and of the von Neumann reduction postulate. In Ref. 7 the physical idea of a continuous observation has been formalized by introducing the notion of an The Fourier operation-valued stochastic process. transform of this object turns out to be exactly the operator G, [k] introduced in this paper. For this Fourier transform the same equation as (3.28) has been obtained, with a generator of the form (2. 18) and (3.27). Then, in Ref. 12, a more general class of operation-valued stochas-
'
—,
t)Trg{s~~— tj
)[R ~,p(s)]+ci[R i, p(s)]) j (3.31b)
tic processes has been constructed, which includes both the case of generators of the type of (2. 18) and (3.27) and the so-called counting processes. ' IV. COUNTING PROCESSES In this section we want to show how counting processes can be obtained by using a generalization of the quantum stochastic calculus introduced in Sec. II. Let us choose for field 1 the Fock representation —0, ci —0); then, it is possible to introduce the so(n i called gauge (or number) process A„~ formally defined by
A,
— =
ai(s)ai(s)ds .
(4. 1)
A quantum stochastic calculus involving also the "difand it turns out that ferential" dA, can be developed d A, satisfies the multiplication rules
(dA,
)
=dA„dA, d(A, ') =d(A, ')t,
dA, 'dA, =dA,
';
(4.2) all the other products involving d A, vanish. By proceed"' ing as in Sec. II, one can introduce the process A', by
A'"': —U
A U
"' [see Eq. (2. 15}];A', satisfies the QSDE dA',
"'= dA, +R i(t)dA, '+R i(t)d(A, ') +R i (t)R i (t)dt,
(4.4)
34
ALBERTO BARCHIELLI where U, and Rt(t) are given by Eqs. (2. 10) and (2. 13). Moreover, as for At'"'„one can prove that A',
"'= UTA, UT
VT&t
"' '"' [see Eq. (3.2)]. The operators A', and A, commute for every t and s. of Now, let us consider a continuous measurement d A,'"', the analog of Eq. (3.16) is
The moments of the (real) output signal z(t) can be computed by using Eqs. (3.13) and (3.29}; the first ones turn out to be given by &A', "'& = Tr~IR tR~p(t) I, « z(t) » = — dt « (t) ( )» AolltAollt
BtBs V,'"'[k]=U, exp
=exp i
i
J
J t
k(s)dA,
U, 4
k(s)dA,'"'
(4.6)
k(s) is a
real-valued test function. The operator Fourier transform of a is the functional projection-valued measure describing the measurement of d A,'"', 0 t, and it satisfies the QSDE where
V,'"'[k]
(s (
d V,'"'[k] = ( I exp[ik (t)] —1 I d A', "') V,'"'[k]
.
(4.7)
By proceeding as before, it is possible to eliminate the fields; then Eqs. (3.23) and (3.28} are obtained, where the operator E(k } is now given by
K(k)p= [exp(ik)
1]R— tpR t .
(4.8)
'E. B. Davies,
Quantum Theory of Open Systems (Academic, London, 1976). 2E. B. Davies, Commun. Math. Phys. 15, 277 (1969); 19, 83 (1970); 22, 51 (1971). 3M. D. Srinivas and E. B. Dsvies, Opt. Acta 28, 981 (1981); 29, 235 (1982). M. D. Srinivas, in Quantum Probability and Applications to the Quantum Theory of Irreuersible Processes, Vol. 1055 of I.ecture in Mathematics, edited by L. Accardi, A. Frigerio, and V. Gorini (Springer, Berlin, 1984), p. 356. 5A. Barchielli„L. Lanz, and G. M. Prosperi, Nuovo Cimento
¹tes
72B, 79 (1982). 6A. Barchielh, Nuovo Cimento 748, 113 (1983). 7A. Barchielli, L. Lanz, and G. M. Prosperi, Found. Phys. 13,
779 (1983). SG. Lupieri, J. Math. Phys. 24, 2329 (1983). 9G. M. Prosperi, in Ref. 4, p. 301. toA. Barchielli, L. Lanz, and G. M. Prosperi, in Proceedings of the International on Foundations of Quantum Symposium Mechanics, edited by S. Kamefuchi et al. (Physical Society of Japan, Tokyo, 1984), p. 165. A. Barchielli, L. Lanz, and G. Prosperi, in Chaotic Behavior in Quantum Systems, edited by G. Casati (Plenum, New York, 1985), p. 321. ~2A. Barchielli and G. Lupieri, J. Math. Phys. 26, 2222 (1985). ~3A. Barchielh and G. Lupieri, in Quantum Probability ond Ap edit&n Mathematics, plications II, Vol. 1136 of Leetgre ed by L. Accardi and W. von %'aldenfels (Springer, Berlin, 1985), p. 57. t4A. Barchielli, in Stochastic Processes in Classical and Quan turn Systems, Proceedings of the Ascono Corno InternationalConference, June 1985, Ascona, Switzerland, edited by S. Albeverio and D. Merlini (Springer, Berlin, in pxess).
¹tes
(4.9)
&
&
= « z(t) » 5(t —s) s)][R ~p(s)R ~~] I + B(t s)— Tr~I R &R exp[L (t — (4. 10) + e(s t)Tr— sI t~ I, 1
where p(t) is given by Eq. (3.32}. In Ref. 12 a generator of the form (4.8) is called "Poiswhile a generator of the form (3.27} is called sonian, "Gaussian. Indeed, in the classical case (when operators are replaced by c-numbers), they describe, respectively, Poissonian and Gaussian stochastic processes. Moreover, it is possible to show' that the case of generators of Poisson type corresponds to counting processes in the sense of Refs. 1 — 4. A typical situation, which can be treated by means of this class of processes, is that of one or more photon counters which are continuously operating.
"
"
'5K. R. Parthasarsthy, Boll. Mat. (to be published). ' G. S. Agarwal, Phys. Rev. 178, 2025 (1969). '~F. Haake, in Quantum Statistics in Optics and Solid StatePhysics, Vol. 66 of Springer Tracts in Modern Physics (Springer, Berlin, 1973), p. 98. 18V. Gorini, A. Frigerio, M. Verri, A. Kossakowski, and E. C. G. Sudsrshan, Rep. Math. Phys. 13, 149 (1978). i9H. Spohn, Rev. Mod. Phys. 53, 569 (1980). ~ M. Lsx, Phys. Rev. 145, 110 (1966). 2~W. H. Luisell, Quantum Statistical Properties of Radiation (%'iley, New York, 1973). 22M. Ssrgent III, M. O. Scully, and %'. E. Lamb, Jr. , Laser Physics (Addison-%'esley, Reading, Mass. , 1974). R. L. Hudson and R. F. Streater, Phys. Lett. 86A, 277 (1981). 24R. L. Hudson and K, R. Parthasarathy, in Ref. 4, p. 171. R. L. Hudson and K. R. Parthasarathy, Commun. Math. Phys. 93, 301 (1984). 26R. L Hudson and K. R. Parthasarathy, Acta Appl. Math. 2, 353 (19&4). 27R. L. Hudson and J. M. Lindsay, J. Funct. Anal. 61, 202 (1985). R. L. Hudson and J. M. Lindssy, jn Ref. 13, p. 276. 29A. Frigerio, Publ. RIMS Kyoto Univ. 21, 657 (1985). 30C. %'. Gardiner and M. J. Collet, Phys. Rev. A 31, 3761 (1985). 3~E. B. Davies, IEEE Trans. Info. Theory 23, 530 (1977). 32A. S. Holevo, Izv. Vuz. Mat. 26, 3 (1982) [Sov. Math. 26, 1 (1982)]. 33A. Barchielli, Phys. Rev. D 32„347 (1985). 34A. S. Holevo, Tear. Mat. Fiz. 57, 424 (1983) [Theor. Math. Phys. 57, 1238 (1984)]. 35A. S. Holevo, in Advances in Statistical Signal Processing (JAI, Greenwich, Connecticut, in press).
34 6L. Lanz, O. Melsheimer,
MEASUREMENT THEORY AND STOCHASTIC DIFFERENTIAL. . .
and S. Penati, physics (to be published). 7A. Rimini, in Proceedings of the Theoretica! Physics Meeting A— malfi, l983 (ESI, Napoli, 1984), p. 275. G. C. Ghirardi, A. Rimini, and T. %'eber, in Ref. 13, p. 223.
1649
S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (North-Holland, Amsterdam, 1982). ~I. M. Gel'fand and N. Ya. Vilenkin, Generalized Functions, Vol. 4 of Applications of Harmonic Analysis (Academic, New 39A.
York, 1964).
