- Email: [email protected]
Classical states via decoherence
9 February
1998
PHYSICS
ELSEVIER
Physics Letters A 238
( 1998)
LETTERS
A
219-222
Classical states via decoherence Gh.-S. Paraoanu a,1, H. Scutaru b*2 a Deporrment of Physics, University ofIllinois at Urbana-Champaign, * Department
1110 W Green St.. Urbana, II. 61801. USA of Theoretical Physics, Institute ofAtomic Physics. Bucharest-Magurele, POB MG-6, Romnmu
Received I2 August 1997; revised manuscript received 21 November
1997: accepted for publication 24 November
I997
Communicated by P.R. Holland
Abstract The initial states which minimize the predictability loss for a damped harmonic oscillator are identified as quasi-free states with a symmetry dictated by the environment’s diffusion coefficients. For an isotropic diffusion in phase space, coherent states (or mixtures of coherent states) are selected as the most stable ones. @ 1998 Elsevier Science B.V. PAC.? 03.65.B~; 05.30.-d;
05.4O.+j
The emergence of classical reality from the underlying quantum description of the world is one of the most fascinating unsolved problems of present-day physics. Decoherence was proposed as a mechanism for the selection of classical (preferred) states: due to the interaction with an environment (external degrees of freedom ) classical states are singled out as the most stable ones. Zurek, Habib and Paz [ I] invented a criterion, called “predictability sieve”, for distinguishing the preferred set of states from the rest of the Hilbert space: classical states are characterized by the least increase in entropy. They addressed this problem in the context of the Caldeira-Leggett model (a particle moving in a potential and linearly coupled with a bath of harmonic oscillators [ I-41 ) . They found that coherent states are selected, via a predictability sieve, as the most “classical” ones. ’ On leave from: Department of Theoretical Atomic Physics, Bucharest-Magurele,
Physics, Institute of
PO Box MG-6,
Romania.
In contradistinction to this approach, we will study the problem of classicality in the framework of Lindblad’s theory, in which the structure of the master equation is derived from very general assumptions concerning the mathematical description of the evolution of an open quantum system [.5]. This strategy leads to general results (since Lindblad equations work for a large class of physically interesting systems [ 6]), and also avoids any problems related to the non-preservation of the positivity of the reduced density matrix, which was perceived by some authors [ 71 as an inconvenience of the Caldeira-Leggett type of evolution, We will identify the states with classical behavior as the states which minimize the rate of entropy increase. We will apply this criterion in an identical fashion for both pure and mixed initial states, using a formalism that does not require one to discriminate between them. Lindblad’s result shows that a general form for the generator of a completely positive dynamical semigroup in the Schrijdinger picture is [5 ]
E-mail: [email protected]. ’ E-mail: [email protected].
0375-9601/9X/$19.00
@ 1998 Elsevier Science B.V. All rights reserved.
Pff SO375-9601(97)00925-O
220
Gh.4.
Paraoanu, H. Scutaru/Physics
Letters A 238 (1998) 219-222
where
(1)
at(f)
Z(t) where for the operators vj, we will take [6] vj = Ujp + bjq, where j = 1,2 while aj and bj are complex numbers. For the Hamiltonian H of the system we will consider [ 61 a second-order polynomial operator in coordinate and position written as a sum of a harmonic-oscillator part and a mixed term,
(2) We denote the diffusion
coefficients
by
will denote the dispersion
dstt) dt
(
(3) is
2
A = -ImxaTbj.
(4)
j=l
It is easy to show [6] that the following must be satisfied,
D, D,, -
D& 3
inequality
+ z(t)YT
+ 2D,
-(A-p) --w
-(A?&
> ’
YT is the transposed matrix of Y and D is a 2 x 2 matrix with elements Dqq = m&D,, IDq = Vqp = D, and Vpp = Dpp/mw. For the case of the damped harmonic oscillator, it is known that a certain class of states (quasi-free states [ 81) has the property of invariance under the action of the quantum semigroup. This can be seen by writing the Fokker-Planck equation which corresponds to (6) in the form [ 93
(5)
.
4
= Yz(t)
afw(nl,x2,r)
A2fi2
-
matrix,
where
j=I
and the friction constant
= Tr(p(t)C’)
With the notations above, Heisenberg’s inequality reads det _‘Z(t) 3 fi2/4. For Z(t) the time-evolution is known [6],
Y= j=l
uct(t)
= Tr(df)C),
69
2 X~$,[xj_fW(~!,~2~~)l c =;j=, I
With the above notation, the master equation which governs the evolution of the system takes the form
Q(t) dt
= +&p(t)1
- &w(l)Pl
t 10)
where fw is the Wigner function of a quasi-free
- [PTP(t)41}
~w(xI,x~,~)
= [(2r)2detx(t)]-“2
x exp[-+XT(t)S:-‘X(t)], - $$[P,
[P*P(t)ll-
+ $$[P,
[q,p(t)ll.
z$%&
[s,P(t)ll
state
(11)
and
(6)
For the correlations between we will use the definition
two operators C and C’
CC’ + C’C -
2
uc(t)ac,tt),
>
(7)
It is now easy to verify that the Gaussian Wigner function ( 1 l), with the time-dependence of the mean values np(t), (am and of dispersions a,,(l), v,,,,(t), a,,q( t) given by Eqs. (3.26)) (3.27) from Ref. [ 61,
Gh.-S. Paraoann, H. Sruturu/Ph_vsicx Letters A 238 (I 998) 219-222
is a solution of Eq. ( 10). Thus, quasi-free states are preserved during the evolution of the system. Starting with such a state, we want to calculate the rate of linear entropy variation. The initial states with the lowest rate of linear entropy increase will be identified as the most stable (i.e. the most classical-like) states. The linear entropy is a convenient measure of the purity of a quantum state and is defined by = 1 - Tr(p*).
s(p) For
a quasi-free
(12)
Let us notice, for the beginning, that the positive 2 x 2 matrix X( t) can be diagonalized, at any particular instant t, in the form (see also Refs. [ 12,13 1) S(t)
= (18)
where N(t) is a real positive number (the parameter), A(t) is the area occupied by in phase space and O(t) is an orthogonal matrix for which we will employ the usual
state we have O(r) =
s(p)
= I - $,
121
(13)
- sin8(t) cos8(r)
cosB(t) ( sine(t)
squeezing the system symplectic form
1 .
A similar formula holds for the D-matrix, where A(t)
is the “area” in phase space, (19)
A(t)
= ;dm. with
The condition for a quasi-free state to be a pure one is A(t) = 1, that is, det X( t) = Ii*/4 (the equality case in the Heisenberg relation). The time-derivative of s(t) can be calculated by making use of the following relation, d[ln detX(f)l
=TJ
( 14)
dt
00 =
z z-‘(r)
1
a,,(t)lmw --(T&f)
= ~ detT(t)
-up,(t) mwg,(t)
> . (15)
Then we obtain from (9) and (14) that
(16) But the rate of the linear entropy increase is given by ds(f1 -_=_ dt
I A(t)’
(20)
Here, A is a parameter controlling the intensity of diffusion, d characterizes the degree of anisotropy and cp is the rotation angle. Now, ( 16) and ( 17) imply ds
where
cos cp - sin p sin cp cos(D > .
,zO
= L[-2A+TJ(X-‘(O)D)]. A(O)
(21)
This result is even more general when a pure state at t = 0 (so that A (0) = that it does not depend on the kind of are starting with (quasi-free states or Ref. [6] it was shown that dTr(p*) ~ dt
2-J
the system is in 1) . in the sense initial states we not). Indeed, in
= 2TrCpUp)) [Tr(pv,pV;)
-Tr(p’V;Vj)l
3 0.
(22)
dA(r) dt
’
(17)
so the rate of linear entropy increase is given entirely by the time derivative (16) of the area A(t) for all states (including pure states). In the following we shall find the squeezing parameter of the initial state for which the rate of increase of A(t) is minimized.
For a pure state p2 = p and pop self-adjoint operator 0. Then dTr(p’) ___ dt
= ; x[ , i
= TJ( p0) p for any
ITr(pV,)1* - Tr(pb’j*V,)
1 2 0.
(23)
Gh.-S. Paraoanu, H. Scutaru/Physics
222
We have Tr(pt$)
selects states with the same degree of anisotropy. When the initial state is pure,
= ajap + bjaq. Hence we get
= -2A
-‘Whl
$1
Letters A 238 (1998) 219-222
1=0 + &&Q,(O)
t=o + D,,a,(O)
- =',up,q(O)l
= -2A+Tr(X’(O)D), which has the same form as the rate of linear entropy increase calculated before (see (2 1) ) for pure (A(0) = 1) initial quasi-free states. We want to finding the states which produce the least increase of the phase-space area at the initial moment t = 0. By minimizing the expression -2A + Tr( X-‘(0)D) + ~{cos2[e(o) + sin*[e(O)
.
= -2h -cp] [N2(o)d-2+N-2(0)d21
- p] [N2(0)d2
+ N-2(O)d-2]},
with respect to N (0) and 0 (0)) one gets ds
mm [ df
II
,=0
= 2A - A( A(0)2
corresponding
to
N*(O) = d,
0’(O) = q.
.
(24)
(25)
So, in the general case of an anisotropic diffusion, the minimum variation of the area in phase space is obtained when the squeezing parameter of the state equals d, the degree of anisotropy of the diffusion, and the characteristic rotation angle of the correlation and diffusion matrices are equal. For an isotropic diffusion, DVp = Dqy, Dp4 = 0, i.e. d = 1 (and the rotation angle vanishes trivially from all the relations, since now the system has rotation symmetry in phase space) ; we obtain N* (0) = 1. This case corresponds to many models of dissipation, especially from quantum optics (see Ref. [ 111) ; the same result was obtained by Zurek, Habib and Paz [ I] in the context of the Caldeira-Leggett environment model. In other words, a phase-space isotropic environment favors a symmetric state, while an anisotropic environment
=2(4-A)
20,
where the last inequality comes from (5) and expresses the fact that the linear entropy of pure states in an environment always increases, because the pure initial states become more and more mixed. Our result shows that the values of N* (0) and 0* (0) are independent of the overall magnitude A of the diffusion. They are also independent of A( 0). Thus, the same degree of squeezing is singled out, irrespective of the purity of the initial state, thus confirming previous insights [ 1,4] regarding the structure of the mixed preferred states: they can be seen as the thermalization of the selected pure states. We conclude by emphasizing the main results of this Letter. In general, the pure or mixed state which produces the minimum rate of increase in the area occupied by the system in phase-space is a quasi-free state which has the same symmetry as that induced on the evolution in phase-space by the diffusion coefficients. For isotropic phase-space diffusion, the selected pure states are the coherent states.
References W.H. Zurek, S. Habib, J.P. Paz, Phys. Rev. Lett. 70 ( 1993) 1187. 121A 121 A.O. Caldeira, A.J. Leggett, Physica (Amsterdam) (1983) 587. 131 W.G.Unruh, W.H. Zurek, Phys. Rev. D 40 (1989) 1071. r41 W.H. Zurek, Progr. Theor. Phys. 89 ( 1993) 281. 151 G. Lindblad, Commun. Math. Phys. 48 ( 1976) 119. 161 A. Sandulescu. H. Scutaru, Ann. Phys. (N.Y.) 173 ( 1987) 277. 171 V. Ambegaokar, Phys. Today 46 (4) ( 1993) 82. 181 H. Scutaru, Phys. Leo. A 141 (1989) 223. [91 A. 1%~. Helv. Phys. Acta., 67 (1994) 436. [lOI G.S. Agarwal, Phys. Rev. A 3 (1971) 828. Ann. Phys. (N.Y.) 89 [III R.S. Ingarden, A. Kossakowski, (1975) 451. 1121 H. Scutaru, Phys. Lett. A 200 ( 1995) 91. [I31 R. Balian, C. De Dominicis, C. Itzykson, Nucl. Phys. 67 (1965) 609.
[ll
1998
PHYSICS
ELSEVIER
Physics Letters A 238
( 1998)
LETTERS
A
219-222
Classical states via decoherence Gh.-S. Paraoanu a,1, H. Scutaru b*2 a Deporrment of Physics, University ofIllinois at Urbana-Champaign, * Department
1110 W Green St.. Urbana, II. 61801. USA of Theoretical Physics, Institute ofAtomic Physics. Bucharest-Magurele, POB MG-6, Romnmu
Received I2 August 1997; revised manuscript received 21 November
1997: accepted for publication 24 November
I997
Communicated by P.R. Holland
Abstract The initial states which minimize the predictability loss for a damped harmonic oscillator are identified as quasi-free states with a symmetry dictated by the environment’s diffusion coefficients. For an isotropic diffusion in phase space, coherent states (or mixtures of coherent states) are selected as the most stable ones. @ 1998 Elsevier Science B.V. PAC.? 03.65.B~; 05.30.-d;
05.4O.+j
The emergence of classical reality from the underlying quantum description of the world is one of the most fascinating unsolved problems of present-day physics. Decoherence was proposed as a mechanism for the selection of classical (preferred) states: due to the interaction with an environment (external degrees of freedom ) classical states are singled out as the most stable ones. Zurek, Habib and Paz [ I] invented a criterion, called “predictability sieve”, for distinguishing the preferred set of states from the rest of the Hilbert space: classical states are characterized by the least increase in entropy. They addressed this problem in the context of the Caldeira-Leggett model (a particle moving in a potential and linearly coupled with a bath of harmonic oscillators [ I-41 ) . They found that coherent states are selected, via a predictability sieve, as the most “classical” ones. ’ On leave from: Department of Theoretical Atomic Physics, Bucharest-Magurele,
Physics, Institute of
PO Box MG-6,
Romania.
In contradistinction to this approach, we will study the problem of classicality in the framework of Lindblad’s theory, in which the structure of the master equation is derived from very general assumptions concerning the mathematical description of the evolution of an open quantum system [.5]. This strategy leads to general results (since Lindblad equations work for a large class of physically interesting systems [ 6]), and also avoids any problems related to the non-preservation of the positivity of the reduced density matrix, which was perceived by some authors [ 71 as an inconvenience of the Caldeira-Leggett type of evolution, We will identify the states with classical behavior as the states which minimize the rate of entropy increase. We will apply this criterion in an identical fashion for both pure and mixed initial states, using a formalism that does not require one to discriminate between them. Lindblad’s result shows that a general form for the generator of a completely positive dynamical semigroup in the Schrijdinger picture is [5 ]
E-mail: [email protected]. ’ E-mail: [email protected].
0375-9601/9X/$19.00
@ 1998 Elsevier Science B.V. All rights reserved.
Pff SO375-9601(97)00925-O
220
Gh.4.
Paraoanu, H. Scutaru/Physics
Letters A 238 (1998) 219-222
where
(1)
at(f)
Z(t) where for the operators vj, we will take [6] vj = Ujp + bjq, where j = 1,2 while aj and bj are complex numbers. For the Hamiltonian H of the system we will consider [ 61 a second-order polynomial operator in coordinate and position written as a sum of a harmonic-oscillator part and a mixed term,
(2) We denote the diffusion
coefficients
by
will denote the dispersion
dstt) dt
(
(3) is
2
A = -ImxaTbj.
(4)
j=l
It is easy to show [6] that the following must be satisfied,
D, D,, -
D& 3
inequality
+ z(t)YT
+ 2D,
-(A-p) --w
-(A?&
> ’
YT is the transposed matrix of Y and D is a 2 x 2 matrix with elements Dqq = m&D,, IDq = Vqp = D, and Vpp = Dpp/mw. For the case of the damped harmonic oscillator, it is known that a certain class of states (quasi-free states [ 81) has the property of invariance under the action of the quantum semigroup. This can be seen by writing the Fokker-Planck equation which corresponds to (6) in the form [ 93
(5)
.
4
= Yz(t)
afw(nl,x2,r)
A2fi2
-
matrix,
where
j=I
and the friction constant
= Tr(p(t)C’)
With the notations above, Heisenberg’s inequality reads det _‘Z(t) 3 fi2/4. For Z(t) the time-evolution is known [6],
Y= j=l
uct(t)
= Tr(df)C),
69
2 X~$,[xj_fW(~!,~2~~)l c =;j=, I
With the above notation, the master equation which governs the evolution of the system takes the form
Q(t) dt
= +&p(t)1
- &w(l)Pl
t 10)
where fw is the Wigner function of a quasi-free
- [PTP(t)41}
~w(xI,x~,~)
= [(2r)2detx(t)]-“2
x exp[-+XT(t)S:-‘X(t)], - $$[P,
[P*P(t)ll-
+ $$[P,
[q,p(t)ll.
z$%&
[s,P(t)ll
state
(11)
and
(6)
For the correlations between we will use the definition
two operators C and C’
CC’ + C’C -
2
uc(t)ac,tt),
>
(7)
It is now easy to verify that the Gaussian Wigner function ( 1 l), with the time-dependence of the mean values np(t), (am and of dispersions a,,(l), v,,,,(t), a,,q( t) given by Eqs. (3.26)) (3.27) from Ref. [ 61,
Gh.-S. Paraoann, H. Sruturu/Ph_vsicx Letters A 238 (I 998) 219-222
is a solution of Eq. ( 10). Thus, quasi-free states are preserved during the evolution of the system. Starting with such a state, we want to calculate the rate of linear entropy variation. The initial states with the lowest rate of linear entropy increase will be identified as the most stable (i.e. the most classical-like) states. The linear entropy is a convenient measure of the purity of a quantum state and is defined by = 1 - Tr(p*).
s(p) For
a quasi-free
(12)
Let us notice, for the beginning, that the positive 2 x 2 matrix X( t) can be diagonalized, at any particular instant t, in the form (see also Refs. [ 12,13 1) S(t)
= (18)
where N(t) is a real positive number (the parameter), A(t) is the area occupied by in phase space and O(t) is an orthogonal matrix for which we will employ the usual
state we have O(r) =
s(p)
= I - $,
121
(13)
- sin8(t) cos8(r)
cosB(t) ( sine(t)
squeezing the system symplectic form
1 .
A similar formula holds for the D-matrix, where A(t)
is the “area” in phase space, (19)
A(t)
= ;dm. with
The condition for a quasi-free state to be a pure one is A(t) = 1, that is, det X( t) = Ii*/4 (the equality case in the Heisenberg relation). The time-derivative of s(t) can be calculated by making use of the following relation, d[ln detX(f)l
=TJ
( 14)
dt
00 =
z z-‘(r)
1
a,,(t)lmw --(T&f)
= ~ detT(t)
-up,(t) mwg,(t)
> . (15)
Then we obtain from (9) and (14) that
(16) But the rate of the linear entropy increase is given by ds(f1 -_=_ dt
I A(t)’
(20)
Here, A is a parameter controlling the intensity of diffusion, d characterizes the degree of anisotropy and cp is the rotation angle. Now, ( 16) and ( 17) imply ds
where
cos cp - sin p sin cp cos(D > .
,zO
= L[-2A+TJ(X-‘(O)D)]. A(O)
(21)
This result is even more general when a pure state at t = 0 (so that A (0) = that it does not depend on the kind of are starting with (quasi-free states or Ref. [6] it was shown that dTr(p*) ~ dt
2-J
the system is in 1) . in the sense initial states we not). Indeed, in
= 2TrCpUp)) [Tr(pv,pV;)
-Tr(p’V;Vj)l
3 0.
(22)
dA(r) dt
’
(17)
so the rate of linear entropy increase is given entirely by the time derivative (16) of the area A(t) for all states (including pure states). In the following we shall find the squeezing parameter of the initial state for which the rate of increase of A(t) is minimized.
For a pure state p2 = p and pop self-adjoint operator 0. Then dTr(p’) ___ dt
= ; x[ , i
= TJ( p0) p for any
ITr(pV,)1* - Tr(pb’j*V,)
1 2 0.
(23)
Gh.-S. Paraoanu, H. Scutaru/Physics
222
We have Tr(pt$)
selects states with the same degree of anisotropy. When the initial state is pure,
= ajap + bjaq. Hence we get
= -2A
-‘Whl
$1
Letters A 238 (1998) 219-222
1=0 + &&Q,(O)
t=o + D,,a,(O)
- =',up,q(O)l
= -2A+Tr(X’(O)D), which has the same form as the rate of linear entropy increase calculated before (see (2 1) ) for pure (A(0) = 1) initial quasi-free states. We want to finding the states which produce the least increase of the phase-space area at the initial moment t = 0. By minimizing the expression -2A + Tr( X-‘(0)D) + ~{cos2[e(o) + sin*[e(O)
.
= -2h -cp] [N2(o)d-2+N-2(0)d21
- p] [N2(0)d2
+ N-2(O)d-2]},
with respect to N (0) and 0 (0)) one gets ds
mm [ df
II
,=0
= 2A - A( A(0)2
corresponding
to
N*(O) = d,
0’(O) = q.
.
(24)
(25)
So, in the general case of an anisotropic diffusion, the minimum variation of the area in phase space is obtained when the squeezing parameter of the state equals d, the degree of anisotropy of the diffusion, and the characteristic rotation angle of the correlation and diffusion matrices are equal. For an isotropic diffusion, DVp = Dqy, Dp4 = 0, i.e. d = 1 (and the rotation angle vanishes trivially from all the relations, since now the system has rotation symmetry in phase space) ; we obtain N* (0) = 1. This case corresponds to many models of dissipation, especially from quantum optics (see Ref. [ 111) ; the same result was obtained by Zurek, Habib and Paz [ I] in the context of the Caldeira-Leggett environment model. In other words, a phase-space isotropic environment favors a symmetric state, while an anisotropic environment
=2(4-A)
20,
where the last inequality comes from (5) and expresses the fact that the linear entropy of pure states in an environment always increases, because the pure initial states become more and more mixed. Our result shows that the values of N* (0) and 0* (0) are independent of the overall magnitude A of the diffusion. They are also independent of A( 0). Thus, the same degree of squeezing is singled out, irrespective of the purity of the initial state, thus confirming previous insights [ 1,4] regarding the structure of the mixed preferred states: they can be seen as the thermalization of the selected pure states. We conclude by emphasizing the main results of this Letter. In general, the pure or mixed state which produces the minimum rate of increase in the area occupied by the system in phase-space is a quasi-free state which has the same symmetry as that induced on the evolution in phase-space by the diffusion coefficients. For isotropic phase-space diffusion, the selected pure states are the coherent states.
References W.H. Zurek, S. Habib, J.P. Paz, Phys. Rev. Lett. 70 ( 1993) 1187. 121A 121 A.O. Caldeira, A.J. Leggett, Physica (Amsterdam) (1983) 587. 131 W.G.Unruh, W.H. Zurek, Phys. Rev. D 40 (1989) 1071. r41 W.H. Zurek, Progr. Theor. Phys. 89 ( 1993) 281. 151 G. Lindblad, Commun. Math. Phys. 48 ( 1976) 119. 161 A. Sandulescu. H. Scutaru, Ann. Phys. (N.Y.) 173 ( 1987) 277. 171 V. Ambegaokar, Phys. Today 46 (4) ( 1993) 82. 181 H. Scutaru, Phys. Leo. A 141 (1989) 223. [91 A. 1%~. Helv. Phys. Acta., 67 (1994) 436. [lOI G.S. Agarwal, Phys. Rev. A 3 (1971) 828. Ann. Phys. (N.Y.) 89 [III R.S. Ingarden, A. Kossakowski, (1975) 451. 1121 H. Scutaru, Phys. Lett. A 200 ( 1995) 91. [I31 R. Balian, C. De Dominicis, C. Itzykson, Nucl. Phys. 67 (1965) 609.
[ll