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A new wave equation for a continuous nondemolition measurement
Volume 140, number 7,8
PHYSICS LETTERS A
9 October 1989
A N E W WAVE E Q U A T I O N F O R A C O N T I N U O U S N O N D E M O L I T I O N M E A S U R E M E N T V.P. BELAVKIN i
lnstttute of Phystcs, N Copermcus Untverstty, Toruh, Poland Received 12 January 1989; rewsed manuscript recelved 17 May 1989;accepted for pubhcatlon 7 August 1989 Communicated by J.P. Vlgmr
A stochastic model for nondemolmon continuous measurement m a quantum system is given. It ~s shown that the posterior dynamics, including a continuous collapse of the wave function, ~s described by a nonhnear stochastm wave equatmn. For a particle m an electromagnetm field ~t reduces to the Schr6dmger equation w~thextra ~magmarystochastic potentmls.
According to the statistical interpretation of the wave function ~/(t, x), a q u a n t u m measurement o f a simple observable .(" o f a particle reduces it to an eigenfunction o f 2" by a r a n d o m j u m p (collapse, or reduction of the wave packet). Such a j u m p cannot be described by the Schr6dinger equation because the latter corresponds to the unobserved particle. To ignore this fact gives rise to the various quantum paradoxes of the Zeno kind. The aim o f this paper is to derive a dissipative wave equation with an extra stochastic nonlinear term describing the quantum particle under continuous nondemolition observation. This derivation can be obtained in the framework o f quantum stochastic theory o f continuous measurements developed in refs. [ 1-3 ], and the quantum nonlinear filtering method recently announced in ref. [ 4 ]. The derivation can be done in general terms but for simplicity here we consider a q u a n t u m spinless particle in an electromagnetic field. The unperturbed dynamics o f such a particle with mass m is given by the SchrSdinger equation
o f the standard white noise e = (e~, e2, e3) (error),
( e,(r)ek(s) ) =CS,kc~(r--s) , and the particle coordinate operator process X = (X~, X2, )(3 ) in the Heisenberg picture ff(t) = U(t)*XU(t) with amplification x / ~ ( 2 > 0 is the accuracy coeffficient). Indirect measurement o f the position o f the particle described by (2) perturbs its dynamics ( 1 ) in such a way that the vector process ] ' = ( )'l, ~'2, ~'3) is a commutative one (self-nondemolition), [ Y,(r), Yk(s) ] = 0
(3)
and satisfies the nondemolition principle [3,4 ] [ }',(t), Z ( s ) ] = 0 ,
t<~s,
(4)
where U = e A / c and V = e ~ , .4(t, x ) and ~ ( t , x) are the scalar and vector field potentials. A continuous process }" is taken as the sum
with respect to all future Heisenberg operators 2 ( t ) = U ( t ) * Z U ( t ) of the particle. This means that the unitary evolution U(t) can no longer be the resolving operator for eq. (2) but must be defined for an extended quantum system involving an apparatus with field coordinate described by the white noise e. A very nice model o f such an extended unitary evolution is based on the q u a n t u m stochastic Schr6dinger equation
I ' ( t ) = (22)~/2,~(t) + e ( t )
dU+KUdt= (dB+L-L + dB)U,
ihOc,/Ot= [ ( U + i h V ) 2 / 2 m + V]~u ,
( 1)
(2)
(5)
K = L ÷ L / 2 + iH/h, introduced by Hudson and ParPermanent address. M.I.E.M., B. Vusovsky 3/12, Moscow 109028, USSR.
thasarathy in ref. [ 5 ]. Here H is the Hamiltonian which in the spinless case is
0 3 7 5 - 9 6 0 1 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physms Publishing Division )
355
Volume 140, number 7,8
PHYSICS LETTERS A
H= [eA(t, x)/c+ihV]Z/2m+e~(t, x),
(6)
L(t) and B(t) are the vector-operators (columns) of the particle and of a Bose field respectively in the Schrtidinger picture, L ÷ = (L~) and B ÷ = (B~) are rows of conjugate operators, L+L=Y.L~Lj, B+L=~B~Lj, L+B=YL'fBj. The Bose field operators Bj(t) are defined in Fock space as annihilations by the canonical commutation relations [B,(r), B~(s) ] = 0 , [/), ( r ) , / ~ (s) ] =~,k6(r-s) for the (generahzed) derivations/~j(t) =dBffdt, and dB in (5) are the forward increments d B ( t ) =
B(t+dt) -B(t).
m~=e(E+XXI~/c) + h ( 2 2 ) '/2 Im J~*,
~=-v~(;¢), B=vxA(~), )/2. Note that the extra stochastic force h
f(t) =~(22) t/2 Im B*= v ( 2 / 2 ) , / 2 ( B . _ B ) I
perturbs the Hamiltonian dynamics of the particle and due to the observation (2) is another white noise of intensity h 2;t/2, which does not commute with the error e:
[e~(t),fk(s) ] = h (22)'/28jk6(t-s) .
Let us represent the error in (2) as the operatorvalued vector-process
e(t) = 2 Re B(t) = B ( t ) + B ( / ) * , having the correlataons of the standard white noise,
( e,(r)e~(s) ) = (/~,(r)J3~(s)) =cS,~6(r-s) ,
s>>.t, j = 1 , 2 , 3 ,
where Q ( t ) = i e(r) dr=2ReB(t)
(8)
1
Due to the openness of the observed partacle as a quantum system in the Bose reservoir, its prior dynamics is a mixing described by the irreversible master equation d
with respect to the vacuum state of the Bose field. One can easily prove that the nondemolition principle ( 3 ), (4) is fulfilled if L (t) = (22) ~/2X where X is the coordinate vector-operator of the particle given m the Schrtidinger representation as multiplication by x=(x~, x2, x3). Indeed, in this case Y ( t ) = f8 ~'(r) dr as an output process Y= ~ with respect to the evoluuon U m the sense of refs. [4,6-8 ]:
O_j(t)=U(s)*Qj(t)U(s),
9 October 1989
dt (Z),+ (ZK+K*Z-L+ZL),=O,
(9)
obtained by averaging (7) with respect to the product (ao= ¢/® I0) of an initial wave function ~ of the particle and the vacuum state 10) of the Bose field. This means that the prior expectations (Z)t= ((at IZ(a,) of the particle observables Z for (at= U(t) (ao cannot be described in terms of a wave function involving only the particle, in spite of the fact that the initial state of the partacle is a pure one. The posterior dynamics of the particle is described by posterior mean values ~(t) = ( Z ) t, which are defined as conditional expectations ( Z ) ' = e ' ( U ( t ) * Z U ( t ) ) = ~t(Z(t) )
(10)
0
is the standard Wiener process in Fock space. Hence ~'j(t) commutes with ~'k(s) and Z(s) for all s>~ t due to the commutativity of e)(t) with ek(s) and Z. The corresponding quantum Langevm equation d 2 + ( 2 R + K ' * 2 - £ +Z£) dt =rib + [Z,/~] + [£+, 2~] dB,
(7)
where I~(t)=U(t)*KU(t), £j(t)=U(t)*LjU(t), gives the quantum stochastic Lorentz equation for 2 = ~ ' . j = 1, 2, 3, 356
of Z=U*ZU with respect to the observables (~t= {Q(r) I r~< t} up to the current time instant t and the inmal state vector (ao.As it was proved in ref. [ 6 ], the nondemolition principle (3), (4) is necessary and sufficaent for the existence of the conditional expectation ~': Z ~ E t ( Z ) with respect to any initial (ao. For a fixed 2, it xs a non=anticipating c-valued function ~: q~z(t, q) = ( Z ) (q') on the observed traJectories q = {q (t) } of the output process Y= ~; the conditional expectation ( must satasfy the positive projection conditions
Volume 140, number 7,8
~'(2"2)>_.0,
PHYSICS LETTERS A
~'(z(t,O))(q)=z(t,q).
Hence for any particle operator Z the posterior process ~ ( t ) = ( Z ) ' is a classical (commutative) stochastic one; its averaging over all observable trajectories q coincides with the prior mean value: ( ~ ( t ) ) = (Z)t. As a linear map Z ~ , ( Z ) ' it is described by the quantum filtering equation
Comparing ( 11 ) and ( 15 ) and taking into account the relation ( ( Z - ~ ) L ) ( Z ( L - [ ) ) , one obtains K=I. +I./2 + ilt/h with
F.=L-Rei+i~/h,
(17)
where ~(t)=(f~, ~2, r3)(/) and g(t) are arbitrary (inessential) real functions of qt and
H = H - h Re [ I m L - P R e L - ~
d ( Z ) ' + ( ZK+ K * Z - L +ZL ) ' dt
= ( 2 L + L + 2 ) ' dO,
9 October 1989
(11)
obtained in ref. [4] for the case considered
(18)
is the Hamiltoman of the particle. Putting ~= 0, g= 0, we obtain the following stochastic dissipative equation, d~b+ (~+I.12+iI71h)(~ dt = / ~ d 0 ,
Y(t) = i 2 Re[L(r) d r + d B l = 0 ( t ) 0
and in ref. [6] for general output nondemolition processes with respect to the initial vacuum Bose state. Here Z ( t ) = Z - ~ ( t ) I is the deviation of Z in the Schr6dinger picture from the posterior mean value ~(t) = ( Z ) t and
(19)
which is nonlinear because/~ a n d / 4 depend on (17). Multiplying the postenor normalized wave function O(t,x) by the stochastic amplitude d(t) which satisfies the Ito equation dO+(Re/)26dt/2=(Ref)(dO,
6(0)=1,
(20)
using Ito's formula one can easily obtain d(6~b) = {Re l d 0 - (Re i) 2 dt/2
0(t)=Q(t)-
i [[(r)+[*(r)] dr o
(12)
= { (Re/+/S) dO
~s the observed innovating Wiener process, i(t) ( q ) = ( L ) (q'). Let us prove that for any initial wave funcnon ~(x) of the open particle the posterior state (9) is pure and is given by
e(t)=
f O(t,x)*ZO(t,x)dx--O(t)+ZO(t), (13)
where the posterior wave function ~b( t, x ) (q) = ~0( q', x) satisfies the stochastic wave equation d~b+/~'0 dt = £~b d 0 ,
(o(O,x)=~(x).
+ L d Q - K dt + £ Re [ dt}6(o
(14)
Indeed, if ~bsatisfies eq. (14) in the Ito form, then ( Z ) = ~b+Z~b satisfies the following equation,
- [(Re i)2/2+K'+/~ Re i] dt}6O = (L d O - K d t ) e O , where the relanon d~tdOk=Olkdt for d O = d Q 2 Re [dt, L= L - Re £ / ~ = K - L Re [+ (Re i) 2 / 2 was taken into account. Hence the nonnormalized posterior wave funcnon 2 (t, x) = ~(t) 0 (t, x) satisfies the linear stochastic equation d2+ (L +L/2 + 1H/h)2 dt=L2 dO.
The last equation can be transformed to a nonstochastlc linear equation for #(t) = exp[ - LO(t)]2(0:
ihO~/(t)/Ot=H(O_(t))~/(t), d ( Z ) = ( Z / £ + / ( " 2 - / - . +Z/~) dt
= ( z £ + £ + z } dO,
~9(0) = ~ ,
(22)
where
(15)
obtained by using Ito's formula d(0+Z{b) = d 0 + Z(b+ 0 +Z d~+d(b+ Z d0
(21)
iH(Q(t) ) = h exp[ - L O ( t ) ] (K+L2/2) exp[LQ(t) ]
(16)
and the Ito multiplication table dO_kdOt=~kldt.
is a perturbed Hamiltonian W(t)HW*(t), W(t)= e x p [ - L ~ ( t ) ] ( W * = W - ' , if L * = - L ) . Indeed, 357
Volume 140, number 7,8
PHYSICS LETTERS A
with the help of Ito's formula, we obtain
c(q') 2= j exp[ (22)l/2xq(t) ] I ~u(q', x) 12 d x .
d(v= e - LO d~ + de - LO~ _ L 2e - LO~ dt
(25)
=e-LO-( L d O - K dt )~ - (L d O - L 2 d t / 2 ) e - LO)~--L2~ dt
= - e - L O ( K + L 2 / 2 ) e LOtp dt 1
- thH(~(t))(udt. Eq. (22) for any observed trajectory ¢(t) can be viewed as the Schrrdinger equataon (1) with complex potentials U and V. In the case o f position observation, L = (2/2)~/2x, these potentials have the form
U(t, x) = e A ( t , x ) / c + i h ( 2 / 2 ) l / 2 q ( t ) , V(t, x) = e ~ ( t , x) - i h [ (2/2)~/2x]2,
9 October 1989
(23)
and we get
Note that the indirect nondemolition measurement considered is complete in the sense that the posterior state of the particle is pure if the initial state is pure. Hence the posterior dynamics o f such indirectly completely observed particle is pure contrary to the prior dynamics which is always mixed (for 2 = 0 ) even for v a c u u m q u a n t u m noise. In the case of noncomplete measurement, ff for instance, an extra bath is added, or if the noise has a nonzero temperature, as ts supposed m ref. [ 3 ], this fact is no more true. The author is grateful to Professor R.S. Ingarden and Dr. P. Staszewski for their hospitality and useful discussions during his stay at the Institute o f Physics, N. Copernicus University, Torufi, where this work was completed.
H ( q ) = e x p [ - (2/2)WExq] × ( H - ½ih2x 2) e x p [ x q ( 2 / 2 ) t/2 ]
References
1
- 2m { ( e / c ) A + [ V - (2/2)1/2q] 2} h)~ 2
+e~+ ~x
.
The solution ~(t, x) (q) = ~u(q', x) o f the wave equation ( 2 ) for an observed trajectory q = {q (t) } defines the posterior normalized wave function ~(q', x) = exp[ (2/2 )l/2xq(t) - I n c(qt) ]~(qt, x) , (24) where In 6 ( t ) = i (2/2)~/2"~2d(~-Lfc2dt/2 O
can be obtained from the normahzation condition
358
[ 1] v P Belavkm, Radtotekh. Elektron 25 (1980) 1445 [ 2] A. Barchielh and G. Lupterl, J. Math. Phys. 26 ( 1985) 2222. [3] V.P. Belavkm, m: Information complexity and control m quantum physics, Proc. CISM Conf. Udine 1985, eds. A Blaqmere, S. Diner and G. Lochak (Springer, Berhn, 1987) p. 311 [4] V.P Belavkm, m: Stochasuc methods m mathematics and physics, Proc. XXIV Winter School on Theoretical Physics, eds R. Gtelerak and W Karwowskx (World Scientific, Singapore, 1989) p. 310. [ 5 ] R S. Hudson and K.R. Parthasarathy, Commun. Math Phys 93 (1984) 301 [6] V.P Belavkm, Quantum stochasuc calculus and quantum nonlinear filtenng, Preprint Centro Matemat~coV. Volterra, Umverslta dl Roma II, to be pubhshed. [7] V.P. Belavkm and P. Staszewskl, Phys. Lett. A 140 (1989) 359. [ 8] A. Barchmlh, in: Quantum probablhty and apphcaUons III, eds. L. Accardl and W. yon Waldenfels (Spnnger, Berlin, 1988) p. 37.
PHYSICS LETTERS A
9 October 1989
A N E W WAVE E Q U A T I O N F O R A C O N T I N U O U S N O N D E M O L I T I O N M E A S U R E M E N T V.P. BELAVKIN i
lnstttute of Phystcs, N Copermcus Untverstty, Toruh, Poland Received 12 January 1989; rewsed manuscript recelved 17 May 1989;accepted for pubhcatlon 7 August 1989 Communicated by J.P. Vlgmr
A stochastic model for nondemolmon continuous measurement m a quantum system is given. It ~s shown that the posterior dynamics, including a continuous collapse of the wave function, ~s described by a nonhnear stochastm wave equatmn. For a particle m an electromagnetm field ~t reduces to the Schr6dmger equation w~thextra ~magmarystochastic potentmls.
According to the statistical interpretation of the wave function ~/(t, x), a q u a n t u m measurement o f a simple observable .(" o f a particle reduces it to an eigenfunction o f 2" by a r a n d o m j u m p (collapse, or reduction of the wave packet). Such a j u m p cannot be described by the Schr6dinger equation because the latter corresponds to the unobserved particle. To ignore this fact gives rise to the various quantum paradoxes of the Zeno kind. The aim o f this paper is to derive a dissipative wave equation with an extra stochastic nonlinear term describing the quantum particle under continuous nondemolition observation. This derivation can be obtained in the framework o f quantum stochastic theory o f continuous measurements developed in refs. [ 1-3 ], and the quantum nonlinear filtering method recently announced in ref. [ 4 ]. The derivation can be done in general terms but for simplicity here we consider a q u a n t u m spinless particle in an electromagnetic field. The unperturbed dynamics o f such a particle with mass m is given by the SchrSdinger equation
o f the standard white noise e = (e~, e2, e3) (error),
( e,(r)ek(s) ) =CS,kc~(r--s) , and the particle coordinate operator process X = (X~, X2, )(3 ) in the Heisenberg picture ff(t) = U(t)*XU(t) with amplification x / ~ ( 2 > 0 is the accuracy coeffficient). Indirect measurement o f the position o f the particle described by (2) perturbs its dynamics ( 1 ) in such a way that the vector process ] ' = ( )'l, ~'2, ~'3) is a commutative one (self-nondemolition), [ Y,(r), Yk(s) ] = 0
(3)
and satisfies the nondemolition principle [3,4 ] [ }',(t), Z ( s ) ] = 0 ,
t<~s,
(4)
where U = e A / c and V = e ~ , .4(t, x ) and ~ ( t , x) are the scalar and vector field potentials. A continuous process }" is taken as the sum
with respect to all future Heisenberg operators 2 ( t ) = U ( t ) * Z U ( t ) of the particle. This means that the unitary evolution U(t) can no longer be the resolving operator for eq. (2) but must be defined for an extended quantum system involving an apparatus with field coordinate described by the white noise e. A very nice model o f such an extended unitary evolution is based on the q u a n t u m stochastic Schr6dinger equation
I ' ( t ) = (22)~/2,~(t) + e ( t )
dU+KUdt= (dB+L-L + dB)U,
ihOc,/Ot= [ ( U + i h V ) 2 / 2 m + V]~u ,
( 1)
(2)
(5)
K = L ÷ L / 2 + iH/h, introduced by Hudson and ParPermanent address. M.I.E.M., B. Vusovsky 3/12, Moscow 109028, USSR.
thasarathy in ref. [ 5 ]. Here H is the Hamiltonian which in the spinless case is
0 3 7 5 - 9 6 0 1 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physms Publishing Division )
355
Volume 140, number 7,8
PHYSICS LETTERS A
H= [eA(t, x)/c+ihV]Z/2m+e~(t, x),
(6)
L(t) and B(t) are the vector-operators (columns) of the particle and of a Bose field respectively in the Schrtidinger picture, L ÷ = (L~) and B ÷ = (B~) are rows of conjugate operators, L+L=Y.L~Lj, B+L=~B~Lj, L+B=YL'fBj. The Bose field operators Bj(t) are defined in Fock space as annihilations by the canonical commutation relations [B,(r), B~(s) ] = 0 , [/), ( r ) , / ~ (s) ] =~,k6(r-s) for the (generahzed) derivations/~j(t) =dBffdt, and dB in (5) are the forward increments d B ( t ) =
B(t+dt) -B(t).
m~=e(E+XXI~/c) + h ( 2 2 ) '/2 Im J~*,
~=-v~(;¢), B=vxA(~), )/2. Note that the extra stochastic force h
f(t) =~(22) t/2 Im B*= v ( 2 / 2 ) , / 2 ( B . _ B ) I
perturbs the Hamiltonian dynamics of the particle and due to the observation (2) is another white noise of intensity h 2;t/2, which does not commute with the error e:
[e~(t),fk(s) ] = h (22)'/28jk6(t-s) .
Let us represent the error in (2) as the operatorvalued vector-process
e(t) = 2 Re B(t) = B ( t ) + B ( / ) * , having the correlataons of the standard white noise,
( e,(r)e~(s) ) = (/~,(r)J3~(s)) =cS,~6(r-s) ,
s>>.t, j = 1 , 2 , 3 ,
where Q ( t ) = i e(r) dr=2ReB(t)
(8)
1
Due to the openness of the observed partacle as a quantum system in the Bose reservoir, its prior dynamics is a mixing described by the irreversible master equation d
with respect to the vacuum state of the Bose field. One can easily prove that the nondemolition principle ( 3 ), (4) is fulfilled if L (t) = (22) ~/2X where X is the coordinate vector-operator of the particle given m the Schrtidinger representation as multiplication by x=(x~, x2, x3). Indeed, in this case Y ( t ) = f8 ~'(r) dr as an output process Y= ~ with respect to the evoluuon U m the sense of refs. [4,6-8 ]:
O_j(t)=U(s)*Qj(t)U(s),
9 October 1989
dt (Z),+ (ZK+K*Z-L+ZL),=O,
(9)
obtained by averaging (7) with respect to the product (ao= ¢/® I0) of an initial wave function ~ of the particle and the vacuum state 10) of the Bose field. This means that the prior expectations (Z)t= ((at IZ(a,) of the particle observables Z for (at= U(t) (ao cannot be described in terms of a wave function involving only the particle, in spite of the fact that the initial state of the partacle is a pure one. The posterior dynamics of the particle is described by posterior mean values ~(t) = ( Z ) t, which are defined as conditional expectations ( Z ) ' = e ' ( U ( t ) * Z U ( t ) ) = ~t(Z(t) )
(10)
0
is the standard Wiener process in Fock space. Hence ~'j(t) commutes with ~'k(s) and Z(s) for all s>~ t due to the commutativity of e)(t) with ek(s) and Z. The corresponding quantum Langevm equation d 2 + ( 2 R + K ' * 2 - £ +Z£) dt =rib + [Z,/~] + [£+, 2~] dB,
(7)
where I~(t)=U(t)*KU(t), £j(t)=U(t)*LjU(t), gives the quantum stochastic Lorentz equation for 2 = ~ ' . j = 1, 2, 3, 356
of Z=U*ZU with respect to the observables (~t= {Q(r) I r~< t} up to the current time instant t and the inmal state vector (ao.As it was proved in ref. [ 6 ], the nondemolition principle (3), (4) is necessary and sufficaent for the existence of the conditional expectation ~': Z ~ E t ( Z ) with respect to any initial (ao. For a fixed 2, it xs a non=anticipating c-valued function ~: q~z(t, q) = ( Z ) (q') on the observed traJectories q = {q (t) } of the output process Y= ~; the conditional expectation ( must satasfy the positive projection conditions
Volume 140, number 7,8
~'(2"2)>_.0,
PHYSICS LETTERS A
~'(z(t,O))(q)=z(t,q).
Hence for any particle operator Z the posterior process ~ ( t ) = ( Z ) ' is a classical (commutative) stochastic one; its averaging over all observable trajectories q coincides with the prior mean value: ( ~ ( t ) ) = (Z)t. As a linear map Z ~ , ( Z ) ' it is described by the quantum filtering equation
Comparing ( 11 ) and ( 15 ) and taking into account the relation ( ( Z - ~ ) L ) ( Z ( L - [ ) ) , one obtains K=I. +I./2 + ilt/h with
F.=L-Rei+i~/h,
(17)
where ~(t)=(f~, ~2, r3)(/) and g(t) are arbitrary (inessential) real functions of qt and
H = H - h Re [ I m L - P R e L - ~
d ( Z ) ' + ( ZK+ K * Z - L +ZL ) ' dt
= ( 2 L + L + 2 ) ' dO,
9 October 1989
(11)
obtained in ref. [4] for the case considered
(18)
is the Hamiltoman of the particle. Putting ~= 0, g= 0, we obtain the following stochastic dissipative equation, d~b+ (~+I.12+iI71h)(~ dt = / ~ d 0 ,
Y(t) = i 2 Re[L(r) d r + d B l = 0 ( t ) 0
and in ref. [6] for general output nondemolition processes with respect to the initial vacuum Bose state. Here Z ( t ) = Z - ~ ( t ) I is the deviation of Z in the Schr6dinger picture from the posterior mean value ~(t) = ( Z ) t and
(19)
which is nonlinear because/~ a n d / 4 depend on (17). Multiplying the postenor normalized wave function O(t,x) by the stochastic amplitude d(t) which satisfies the Ito equation dO+(Re/)26dt/2=(Ref)(dO,
6(0)=1,
(20)
using Ito's formula one can easily obtain d(6~b) = {Re l d 0 - (Re i) 2 dt/2
0(t)=Q(t)-
i [[(r)+[*(r)] dr o
(12)
= { (Re/+/S) dO
~s the observed innovating Wiener process, i(t) ( q ) = ( L ) (q'). Let us prove that for any initial wave funcnon ~(x) of the open particle the posterior state (9) is pure and is given by
e(t)=
f O(t,x)*ZO(t,x)dx--O(t)+ZO(t), (13)
where the posterior wave function ~b( t, x ) (q) = ~0( q', x) satisfies the stochastic wave equation d~b+/~'0 dt = £~b d 0 ,
(o(O,x)=~(x).
+ L d Q - K dt + £ Re [ dt}6(o
(14)
Indeed, if ~bsatisfies eq. (14) in the Ito form, then ( Z ) = ~b+Z~b satisfies the following equation,
- [(Re i)2/2+K'+/~ Re i] dt}6O = (L d O - K d t ) e O , where the relanon d~tdOk=Olkdt for d O = d Q 2 Re [dt, L= L - Re £ / ~ = K - L Re [+ (Re i) 2 / 2 was taken into account. Hence the nonnormalized posterior wave funcnon 2 (t, x) = ~(t) 0 (t, x) satisfies the linear stochastic equation d2+ (L +L/2 + 1H/h)2 dt=L2 dO.
The last equation can be transformed to a nonstochastlc linear equation for #(t) = exp[ - LO(t)]2(0:
ihO~/(t)/Ot=H(O_(t))~/(t), d ( Z ) = ( Z / £ + / ( " 2 - / - . +Z/~) dt
= ( z £ + £ + z } dO,
~9(0) = ~ ,
(22)
where
(15)
obtained by using Ito's formula d(0+Z{b) = d 0 + Z(b+ 0 +Z d~+d(b+ Z d0
(21)
iH(Q(t) ) = h exp[ - L O ( t ) ] (K+L2/2) exp[LQ(t) ]
(16)
and the Ito multiplication table dO_kdOt=~kldt.
is a perturbed Hamiltonian W(t)HW*(t), W(t)= e x p [ - L ~ ( t ) ] ( W * = W - ' , if L * = - L ) . Indeed, 357
Volume 140, number 7,8
PHYSICS LETTERS A
with the help of Ito's formula, we obtain
c(q') 2= j exp[ (22)l/2xq(t) ] I ~u(q', x) 12 d x .
d(v= e - LO d~ + de - LO~ _ L 2e - LO~ dt
(25)
=e-LO-( L d O - K dt )~ - (L d O - L 2 d t / 2 ) e - LO)~--L2~ dt
= - e - L O ( K + L 2 / 2 ) e LOtp dt 1
- thH(~(t))(udt. Eq. (22) for any observed trajectory ¢(t) can be viewed as the Schrrdinger equataon (1) with complex potentials U and V. In the case o f position observation, L = (2/2)~/2x, these potentials have the form
U(t, x) = e A ( t , x ) / c + i h ( 2 / 2 ) l / 2 q ( t ) , V(t, x) = e ~ ( t , x) - i h [ (2/2)~/2x]2,
9 October 1989
(23)
and we get
Note that the indirect nondemolition measurement considered is complete in the sense that the posterior state of the particle is pure if the initial state is pure. Hence the posterior dynamics o f such indirectly completely observed particle is pure contrary to the prior dynamics which is always mixed (for 2 = 0 ) even for v a c u u m q u a n t u m noise. In the case of noncomplete measurement, ff for instance, an extra bath is added, or if the noise has a nonzero temperature, as ts supposed m ref. [ 3 ], this fact is no more true. The author is grateful to Professor R.S. Ingarden and Dr. P. Staszewski for their hospitality and useful discussions during his stay at the Institute o f Physics, N. Copernicus University, Torufi, where this work was completed.
H ( q ) = e x p [ - (2/2)WExq] × ( H - ½ih2x 2) e x p [ x q ( 2 / 2 ) t/2 ]
References
1
- 2m { ( e / c ) A + [ V - (2/2)1/2q] 2} h)~ 2
+e~+ ~x
.
The solution ~(t, x) (q) = ~u(q', x) o f the wave equation ( 2 ) for an observed trajectory q = {q (t) } defines the posterior normalized wave function ~(q', x) = exp[ (2/2 )l/2xq(t) - I n c(qt) ]~(qt, x) , (24) where In 6 ( t ) = i (2/2)~/2"~2d(~-Lfc2dt/2 O
can be obtained from the normahzation condition
358
[ 1] v P Belavkm, Radtotekh. Elektron 25 (1980) 1445 [ 2] A. Barchielh and G. Lupterl, J. Math. Phys. 26 ( 1985) 2222. [3] V.P. Belavkm, m: Information complexity and control m quantum physics, Proc. CISM Conf. Udine 1985, eds. A Blaqmere, S. Diner and G. Lochak (Springer, Berhn, 1987) p. 311 [4] V.P Belavkm, m: Stochasuc methods m mathematics and physics, Proc. XXIV Winter School on Theoretical Physics, eds R. Gtelerak and W Karwowskx (World Scientific, Singapore, 1989) p. 310. [ 5 ] R S. Hudson and K.R. Parthasarathy, Commun. Math Phys 93 (1984) 301 [6] V.P Belavkm, Quantum stochasuc calculus and quantum nonlinear filtenng, Preprint Centro Matemat~coV. Volterra, Umverslta dl Roma II, to be pubhshed. [7] V.P. Belavkm and P. Staszewskl, Phys. Lett. A 140 (1989) 359. [ 8] A. Barchmlh, in: Quantum probablhty and apphcaUons III, eds. L. Accardl and W. yon Waldenfels (Spnnger, Berlin, 1988) p. 37.