Decoherence and localization in quantum two-level systems

Decoherence and localization in quantum two-level systems

PHYSICA ELSEVIER Physica A 248 (1998) 393 418 Decoherence and localization in quantum two-level systems Ting Yu * Theoretical Physics Group, Blacket...

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PHYSICA ELSEVIER

Physica A 248 (1998) 393 418

Decoherence and localization in quantum two-level systems Ting Yu * Theoretical Physics Group, Blackett Laboratory, Imperial College, Prince Consort Road, London SW7 2BZ, UK Received 13 May 1997

Abstract

We study and compare the decoherent histories approach, the environment-induced decoherence and the localization properties of the solutions to the stochastic Schr6dinger equation in quantum-jump simulation and quantum-state diffusion approaches, for a quantum two-level system model. We show, in particular, that there is a close connection between the decoherent histories and the quantum-jump simulation, complementing a connection with the quantum-state diffusion approach noted earlier by Di6si, Gisin, Halliwell and Percival. In the case of the decoherent histories analysis, the degree of approximate decoherence is discussed in detail. In addition, the various time scales regarding the decoherence and localization are discussed. By using the yon Neumann entropy, we also discuss the predictability and its relation to the upper bounds of degree of decoherence.

PACS: 03.65.Bz; 03.65.Ca; 42.50. Lc Keyword~: Decoherence; Localization; Quantum jumps

I. Introduction

Two primary paradigms - the environment-induced decoherence approach, proposed by Zurek [1 5], and the consistent histories approach by Griffiths [6] and later by Omnbs [ 7 - 1 2 ] and by Gell-Mann and Hartle [ 13-15] - have been recently developed to solve the fundamental issues in quantum theory, especially, quantum measurement problems and the transition from quantum to classical. The environment-induced decoherence emphasizes the division between the system and its environment. The interaction of the system with its environment is responsible for the decay of the quantum coherence of the system. The decoherent histories approach is designed to provide the most general descriptions for a closed system by using the concept of history -- a * E-mail: [email protected] 0378-4371/98/$19.00 Copyright @ 1998 Elsevier Science B,V. All rights reserved

PHS0378-4371(97)00554-2

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sequence of events at a succession of times. Both approaches are applicable to open quantum systems. Another set of viable theories within the framework of quantum mechanics are the various unravellings of the master equation as stochastic Schrrdinger equations for the single member of the ensemble. Among others, the quantum-state diffusion and the quantum-jump simulation approaches have been extensively studied in recent years (e.g., see Refs. [ 16-22]). As phenomenological theories, these stochastic approaches are not only of theoretical interest but also of practical value. The open quantum system provides a unified framework to exhibit the properties of the various approaches we have mentioned above. The master equation which describes the evolution of the open quantum system plays a central role in the investigations into the decay of quantum coherence due to interaction with a much larger environment. However, it does not tell us how an individual member of an ensemble evolves in a dissipative environment. The unravelling of the master equation as stochastic Schr6dinger equation could provide such a description within its domain of applicability. Corresponding to the decoherence process in the density operator formalism, in stochastic Schrrdinger equation approaches, the solution to the stochastic Schr6dinger equation often possesses a very remarkable property - the solution tends to localize at some special states after a localization time scale. For quantum diffusion approach, this localization property has been justified in many different situations [21-26]. It is worth emphasizing that a key point in these approaches is the mutual influence between the system of interests and its environment. This mutual influence is the common sources of the many different phenomena such as dissipation, fluctuation, decoherence, localization, etc. Analysis of decoherence and localization properties is usually rather involved. The entanglement of the complicated mathematics and the subtle conceptual issues often tends to make a detailed scrutiny of the basic concepts impossible. The attractiveness of two-level system model is that, perhaps it is one of the simplest yet physically meaningful models. The purpose of this paper is to employ a widely used two-level system model as a unified framework to examine the dynamics of the open quantum system by the decoherent histories, the environment-induced decoherence and the stochastic Schr6dinger equations. The main aim of the present paper is to establish the connections between the decoherence process and the localization process. The various time scales concerning these processes are discussed. In particular, we have shown a close connection between the decoherence histories and quantum-jump simulation, complementing a connection with quantum-state diffusion noted earlier by Dirsi et al. [25]. As by-product, by using this simple model, we have compared the decoherent histories and environment-induced decoherence approaches in some details. In the case of the approximate decoherence, we provide a detailed analysis of the degree of decoherence of the two-level models, which is important for a real physical process. The plan of this paper is as follows. In Section 2, we briefly present our two-level system model and its basic properties. We study the consistent histories approach and

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its relation to the environment-induced decoherence, the degree of decoherence and predictability by using von Neumann entropy in Sections 3 and 4, respectively. We study the unravelling of master equation and the localization properties of the solutions to the stochastic Schr6dinger equations in both quantum-jump simulations and quantum-state diffusions in Section 5. We summarize and conclude in Section 6. In the appendix we present a proof of the Theorem given in Section 4.

2. The model

A fundamental building block in quantum theory is the two-level system. We consider a two-level atom system, which is radiatively damped by its interaction with the many modes of a radiation field in thermal equilibrium at temperature T. The upper level and lower level are denoted by 12> and l l>, respectively. This is a typical example of the so-called system-plus-reservoir models in which quantum mechanics is fully implemented. In the case of our model, the system of interest is a single atom, and the reservoir is represented by the quantum radiation. Under some conditions, one may derive the master equation for the two-level atom by tracing out over the radiation variables. This master equation describes a Markovian process and consequently takes the standard Lindblad form (in the Schr6dinger picture) (e.g., see Refs. [19,27,28]): F; = - ~i [ H , p ] +

~(-fi + 1) (2apa t

-

atap

+ 2~(2a~pa - a d p - p a d ) .

-

pata) (2.1

)

Here, the Hamiltonian of the atom is given by H = --or=, 2

(2.2)

where ~2 is the renormalized frequency. The Lindblad operators, which model the effects of the environment in this situation, are L, = ~ l ) a ,

L2 = X f l ~ a ~.

(2.3)

The transition rate from [2) ~ [1) is described by the term proportional to (y/2)(fi+ 1 ) which contains both the simulated transitions and spontaneous transitions rates, and the transition rate from I1) --+ 12) is described by the term proportional to (7/2)fi which gives the rate of absorptive transition caused by taking thermal photons from the radiation. The damping constant 7 is the Einstein A-coefficient, and ~ = ~(og, T) is chosen as -

1 e h°~/k~T- I

(2.4)

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We use ax, ay and az to denote Pauli matrices and a , a t atomic lowering and raising operators, which are defined in the usual way ,

ox=

oy=

0

'

oz=

_

(2.5)

and a = l ( a x - i~Ty),

a t = ½(ax + ia),).

(2.6)

The master equation, Eq. (2.1), has been widely discussed in many places and is of importance in many quantum optical problems [19,27]. A convenient way to solve the master equation (Eq. (2.1)) is to write down it in the basis 12), I1): /J22 = - 7 ( ~ + 1 )P22 q- 7nPll ,

(2.7)

/)11 = ~ ( n -~- l ) P 2 2 -- 7 n P l l ,

(2.8)

/J21 = -

[~(2~+ l)+ico]p2,,

(2.9)

/912 = -

[ ~ ( 2 " - I - 1) -- i~o] pl2 •

(2.10)

The first two equations are the well-known Einstein rate equations. The general solutions to Eqs. (2.7) and (2.10) are as follows: p22(t) = - - B l

~+1

+ B2e -;'~2n+l)t

(2.11)

p l l ( t ) = B1 - B2e -~'(2~+j)t ,

(2.12)

P21 (t) = B3e -[(~'/2)(2~+1)+i,,)lt,

(2.13)

p l z ( t ) = B4e -[e'/2)(2~+1 )-i,,,k,

(2.14)

w h e r e B i (i = l, 2, 3, 4) are arbitrary constants which can be easily determined once the

initial condition is given. For the initial density matrix with Tr(p0) = 1 we easily get B~

H+I 2~+ 1 '

B2 ~- P22(0)

(2.15) 2~+ 1 '

(2.16)

B3 = P 2 1 ( 0 ) ,

(2.17)

B4 = P 1 2 ( 0 ) •

(2.18)

It immediately follows from the solutions of the master equation that the density operator p tends to the stationary density operator Ps as t ---+ oc: P --~ P" =

0

0

~

"

(2.19)

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It is seen from the above that the off-diagonal elements, which represent the quantum coherence between the excited state and the ground state o f the atom, vanish in the stationary state. This fact implies that, due to the influence o f the random noise, the quantum coherence decays exponentially as time evolves. This is an elementary example o f environment-induced decoherence (e.g., see Refs. [ 1 - 3 ] ) . Diagonalization occurs in the basis [2), [1). it has been shown by Joos and Zeh [29] in their seminal paper that the decoherence processes are typically very effective in a wide variety o f situations. In the case o f our model, from Eqs. (2.9) and (2.10), it follows that the decoherence time scale tD is given by 1 tD ~ 7 ( 2 ~ + 1) "

In particular, when h e ) ~

(2.20) k~T,

h (2.21)

to ~ ",'ksT

While for h~o ~> kB T, i.e., at the low-temperature limit: 1 tL~ ~ - •

(2.22)

For the time being, we are mainly concerned with the two types o f time scales: ~ One is the decoherence time - which is the time scale on which the off-diagonal elements o f the density matrix are suppressed; and the other is the relaxation time which is time scale on which the system approaches thermal equilibrium. It might be useful to note that these two types o f time scales coincide in our two-level model. This is, certainly, a special feature o f this simple model, which is unlikely shared by other models. 2 It has been found that, for example, the decoherence time in quantum Brownian motion model is typically much shorter than the relaxation time [26]. It is easily shown that the master equation (2.1) is invariant under unitary transtbrmations o f the Lindblad operators: a H UaU*,

a t ~-+ U a * U t ,

(2.23)

where U is a unitary matrix. Correspondingly, the density operator p transforms in the same way: p ~

UpU t .

(2.24)

Thus, when t ---* oc p ~

Up, U ~ .

(2.25)

Generally, the density matrix Up.,.U t is no longer diagonal. This indicates that environment-induced decoherence does not occur in other bases. This property is useful i Another two time scales - decoherent history time and localization time are to be discussed in Sections 3 and 4, respectively. 2 It is not difficult to contemplate a two-level model in which those two time scales are different.

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on comparison between the environment-induced decoherenee and decoherent histories approaches.

3. Decoherent histories in the two-level system model

3.1. Decoherent histories The decoherent histories approach [6-15] offers a sensible way to assign probabilities to a sequence of properties of a quantum system without referring to the measurements or to a classical domain. A history is defined in general as a sequence of properties of a closed system occurring at different times, which is denoted as

C~

=

P(n)tt ~,, ~ n ) ~ ,...,

p(l)l, :q ~ t l )~,

(3.1)

where P~i)(ti) are the projection operators in the Heisenberg picture at times t~: p ( i ) ( t i ) = ei/h(t,-to )H p(i) e-i/h(t,-to )H

(3.2)

here, H is the Hamiltonian of the closed system. These projection operators satisfy exhaustive and exclusive conditions:

P(~i)(ti)P~'i)(ti) = 6~,13,P~i)(ti) .

P(~i)(ti) : I,

(3.3)

The superscript (i) labels the set of projections used at time ti and ~i denotes the particular alternative. A natural way to assign the probability to a history is

p(C~) = Tr(C~p(to)C~) - Tr(P(~)(t.) --

...

p(l)'t )p(to)P~')(t,) • '-P~,, (") (tn)) ~ l ~ l

(3.4)

However, one finds that Eq. (3.4) generally does not satisfy the usual probability sum rules. The necessary and sufficient condition to guarantee that the probability sum rules hold is that the real part of the decoherence functional D[~, a t] defined by D[~,_~'] = Tr(C~p(to)C~,)

(3.5)

vanishes for any two different histories C= and Ca,, i.e., ReD[c~,_~'] • 0,

V~ ¢; ~ .

(3.6)

The sets of histories satisfying Eq. (3.6) are said to be consistent (or weakly decoherent). Physical mechanisms causing Eq. (3.6) to be satisfied typically lead also to the stronger condition D[~,_~'] = 0,

V~ ¢ _~',

(3.7)

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which is called medium decoherence [14]. (In this paper, we simply refer it as decoherence). Although the decoherent histories approach was primarily designed for a closed system, the approach is of particular importance for the open system which may be regarded as a subsystem of a large closed system. For the open quantum system, a natural coarse-graining is to focus only on the properties of the distinguished system whilst ignoring the environment. In this case, a natural selection of projections at each time is of the form P~ ® I f', where P~ is a projection onto the distinguished subsystem and I ~' denotes the identity projection on the environment. In the Markovian regime, the decoherence functional could be constructed entirely in terms of the reduced density matrix of the system [30]: D[ Z,Z'] = Tr(P~,,(")Kt,,_, ~ "[p(,, L_~,, 1)•

..K[~[P~')K~'[po]P(',)]...P ( ' ' - I h p~,(,~' ) ~j-~,, ,

,,

(3.8)

where the trace is taken over the distinguished system only. The quantity K[j ,[.] is super-propagator for the reduced density operator: Pt - K~[p0]. In what follows, we shall make a detailed analysis of the decoherent histories in the two-level model described by the master equation (Eq. (2.1)) which depicts a Markovian process. First, let us consider the projection operators represented by PI = II)
and

P2 =

I2)<21



(3.'))

Obviously, {Pi,

i = 1,2} form a set of complete and exclusive projection operators. Physically, P1 may represent that the atom emits a photon whereas P2 may represent that the atom absorbs a photon. Then the decoherence functional at two time points is given by D[Z,Z'] = 0"~ : , Tr(P~2Kt; 2 ~ [P~,I K 0~,[P0]P~,, l ]) •

(3.10)

It is easily shown that, for any 2 × 2 matrix A, the matrix PiApj (i ¢ j) is an upper (or a lower) triangle matrix. From Eqs. (2.9) and (2.10) we know that K/"~[ .] propagates the matrix with zero diagonal elements into the matrix with zero diagonal elements. So, for any initial density matrix P0, the trace in Eq. (3.10) is exactly zero tbr any different pairs of histories (Z ¢ Z') and for any interval t2 - tl. This demonstrates that the set of histories consisting of projectors, Eq. (3.9), are exactly decoherent. The generalization to n time points is straightforward. The exact decoherence for any time interval is slightly surprising. (The density matrix, by contrast, only becomes exactly diagonal as t ~ oc). This exactness is due to the simplicity of the model and we do not expect it to be a generic feature. Next, consider more general projection operators which correspond to the projection to any direction. With any direction denoted by a unit vector n = (sin 0 cos qS, sin 0 sin 4~, cos qS), we associate a vector In) which belongs to the Hilbert space of the two-level system In)=cos OI1)-e

--i¢~ " 0

s,n~12) .

(3.11)

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Then one can define the following projection operators on the Hilbert space of the system: P+ = In)(nl, where

In')

P _ = In')(n' I ,

(3.12)

is the orthogonal complimentary of In):

In') = e

i~b • 0

0

sm~[1)+cos~12).

(3.13)

We shall show that a set of histories consisting of the projection operators P+ and P _ are approximately decoherent. To this end, first, note that for any 2 x 2 matrix A,

Tr(P+AP_ ) = Tr(P_AP+ )

= 0.

(3.14)

Hence, from Eqs. (2.11 )-(2.14), it can be seen that, after the propagation of Ki''+j [-], all of the diagonal elements of matrix K;'~[p±Kt~ [P0]P~:] contain an exponential damping factor Damping factor

=

(3.15)

e -7(2~+1)t

Thus, we conclude

V_~¢_~'.

O[~,_~'] ,~ 0,

(3.16)

This proves that the set of histories consisting of P+,P_ are approximately decoherent if time interval between tk and tk+~ is larger than the characteristic time scale: 1

tdecoherence

y(2~ + 1 )

(3.17)

This is an expected result. We will give a more detailed estimate of the degree of decoherence in the next section. Note that /decoherence decreases as the coupling 7 is made stronger. From Eq. (2.4), it is easy to see that the decoherence is more effective if the temperature of the bath, T, increases. Conversely, decreasing temperature will make the system spend more time to decohere. The maximum decoherence time for a set of histories is 1/7 which corresponds to zero temperature of bath. In this case, the damping is caused by only spontaneous emission, and then the decoherence process is not very effective. In summary, we find exact decoherence of histories characterized by the projections onto I1) and 12), and approximate decoherence in any other basis. Finally, we examine the probabilities for two times histories consisting of projections Eq. (3.9). These are given by p ( 1 , 2 ) = Tr(Pi p(2, 1 ) =

K[{ [P2Kid [Po ]P2 ])

Tr(P2Kt~ [P, Ki'~[P0]P, ])

~+1 - 2- ~-+ 1 (1 -

6)p22(t~),

(3.18)

-

6)pll(h ),

(3.19)

2H+ 1

(1 -

T Yu/Physica A 248 (1998) 393 418 p(1,

1) =

Tr(P1K[~[P,K~I [ P o ] P 1 ] )

-

p(2,

2)

Tr(P2K[~[P2K[~[ p o ] P 2 ] )

--

=

- 2if+

1 1

- -

2if+ 1

401

[ff + (ff + 1 )6]P22(tl ),

(3.20)

[(if+ 1 ) + ~ 6 ] p l l ( t l ) .

(3.21)

where ff [ P22(tl) - 2ff + ~ + P22(0)

ff ] 6 , 2ff + 1

(3.22)

p ll(tl) -- 2~ ~ ++1~

2ffff+ ~ 1 6 ,

(3.23)

IP22(0)

and 6 = e x p { - 7 ( 2 f f + l ) A t } ( A t = t i - t i i , i = 1,2). As for the n times histories, the calculation for the elementary probabilities will be straightforward. For instance, p(1, 1, • "" 1) = Tr(PIK~',"

l

--(2if+l)"

, [ P 1 K,,,' ....: [Pl

I

[~+(~+

• • • K[~ [P, K[~ [Po]Pi]

1 ),~]"

P22(tl).

• • • P, ]P, ] )

(3.24)

Similarly, one may calculate the transition probabilities, etc. 3.2. Decoherent history vs environment-htduced decoherence

Both decoherent histories and environment-induced decoherence approaches are designed to solve the fundamental problems in quantum theory, in particular, quantum measurement problems and transition from quantum to classical. The comparison on the two approaches is of interest and of importance. Although it is believed that there is a close interrelationship between them, they are by no means equivalent. First of all, they differ in the conceptual aspect [4]. The environment-induced decoherence emphasizes the division between the system of interest and its environment. The mutual interaction is responsible for the decay of the quantum coherence. While the decoherent histories approach to quantum theory permits prediction to be made in genuinely closed systems, such as the whole universe. Apart from this conceptual difference, with this two-level system we will be able to see some other interesting differences. One such difference is that, for any direction In), the sets of histories consisting of the projections P~ In) (nl and P_ = [n')(n' I are decoherent (approximately) for any initial state. However, we have seen from Section 3 that the density matrix tends to become diagonal only in a particular basis I1), 12). Another intriguing difference is the time scales in two formulations. As was shown in the previous section, the histories consisting of the projection onto two levels I1) and 12) are exactly decoherent for any time, moreover, it is independent of the initial density matrices. Whereas environment-induced decoherence in the same basis 11), 12) could only occur after a certain time, which is dependent on the initial state. From the above we see that those two formalisms differ enormously in this two-level model.

402

T. YulPhysica A 248 (1998) 393 418

It should be noted that the differences exhibited here may not be of generality. It is likely that these differences are purely due to the simplicity of the model or due to the approximations employed in the derivation of master equation (such as BornMarkov approximations). However, at any rate, we have shown, within the domain of applicability of the model presented here, the differences between these two formalisms are significant. Moreover, our discussions here could serve as useful hints for further studies on more realistic models.

4. Degree of decoherence and predictability In this section, we will explicitly estimate the degree of decoherence. We show that the degree of decoherence is determined by the largest and the smallest eigenvalues of the projection operators and density matrix at a certain time (see below Section 4.1). By using von Neumann entropy, we also discuss the predictability and its relation to the upper bounds of degree of decoherence.

4.1. Approximate decoherence Physically, one would not expect the decoherence takes place exactly. Therefore, the investigation of the approximate decoherence is of importance. In practical problems, one can, at best, only expect that probability sum rules are satisfied up to order ~, for some constant ~ < 1. Namely, the interference terms do not have to be exactly zero, but small than probabilities by a factor of e. One simple inequality which turns out to be very useful to the study of the degree of decoherence is [31,32]:

ID[ _~.0~t]]2 ~

a2D[_~._~]D[_~'._~'].

(4.1)

We say that a system decoheres to order a if the decoherence functional satisfies Eq. (4.1). As shown in Ref. [31], such a condition implies that the most probability sum rules will then be satisfied to order a. Based on this two-level model we will study the degree of decoherence in some detail. To begin with, we establish the following trace inequality which is useful to our studies of the approximate decoherence. Theorem. Suppose that M and N are two n x n positive definite matrices. Let P and Q be two n × n Hermitian matrices satisfying QP = PQ = O.

(4.2)

IT r ( M P N Q )12 <~~2T r ( M P N P ) T r ( M Q N Q ) ,

(4.3)

Then

403

T. YulPhysica A 248 (1998) 393 418

where c = min{~a¢, gX }, here gM = (2ma xM _ Amin~m)/(Ama x~M + /"minZM),'~t."N = (/'max'N _ Xmin'N)/ "N N M M ~N "N (/~ma~ + )'rain)' and are the maximal and the minimal eigenvalues J[max, )~min /'max, Amin of M and N, respectively.

Remark. In fact, the condition that both M and N are the positive-definite matrices could be generalized to that one is positive definite, say M, while the other N is positive semidefinite. In this case, ~: = e M. It is hoped that the above theorem is also useful in some other cases. The theorem is proved in the appendix. For a general initial state represented by P0 (pure or mixed state), the decoherence functional of two time points may be written as D[ Z,_~']

Tr(P±K[~[P K~'[po]P+]).

(4.4)

We now write A = K0'[p0],

(4.5)

B

(4.6)

/~rl [P±].

Then Eq. (4.4) may be rewritten, in the new notation, as

D[ ~_,or'] = Tr(BP_AP+ ) .

(4.7)

Note that /£ in Eq. (4.6) is the super-propagator for the projection operators

P(t) = Ko[P(O)]. -t

(4.8)

The evolution equation for the projection operators is given by

15 ~[H,

+~(fi+l)(2atPa . ataP. Pata)+2fi(2aPa* . .

aatP aatP), (4.9)

where H, a, a f are defined as before. Note that the evolution equation for the projection operator P is different from that for the density operator p, Eq. (2.1). This reflects the difference between the Schr6dinger and Heisenberg pictures in the density operator formalism. The explicit form of Eq. (4.9) may be written as P22 =

'~'(fi

Jr- 1 )(Pll

-

(4.10)

P22),

(4.11)

Pll = "/~(P22 - P I I ) , /521 = - [~(2n )' - + 1 ) -

io)]

P21

P,2 = - [ ' 2 (2fi + 1 ) + i(o] P,2.

,

(4.12) (4.13)

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The general solutions to the above equations are P22(t) = C1 + Cze -7(2~+1)t ,

Pll(t)

,.~ _-~,(2~+l)t _~ _}_~-£t~2~

= CI -

(4.14) (4.15)

P21(t) = C3 e-[;'/2(z~+l)-i~°]t ,

(4.16)

Pl2(t) = C4e -[''/z(2~+l )+io~]t,

(4.17)

where Ci (i = 1,2, 3, 4) are arbitrary constants. For given initial values, these constants can be expressed as ~+1 C1 -- 2ff q _ ~ P 2 2 ( 0 ) Jr- 2 ~ - - ] - P I I ( 0 ) ,

(4.18)

~+1 C2 -- - (P22(0) - P l l ( 0 ) ) , 2~+ 1

(4.19)

C3 = P 2 1 ( O ) ,

(4.20)

C4 = P I 2 ( 0 )

.

(4.21)

From the definitions given by Eqs. (4.5) and (4.6), it is easy to see that in general, both A and B could be positive-definite matrices, and since P _ and P+ are projection operators, so the condition given by Eq. (4.2) is automatically satisfied. Using the theorem above, we immediately arrive at

[Tr( BP+AP_ )]2 ~
(4.22)

That is, ID[z,Z,][2 ~<82D [ Z , z ] D [ z , ,Z , ].

(4.23)

where e = min{c A, e~}, CA =

[2 A

ca _

I;~f - ,~fl

(4.24)

-- )~A[,

2f + ) , f

"

(4.25)

Here 2 A (i = 1,2) and 22 (i = 1,2) are two eigenvalues of A and B, respectively. (Note that 2~ + 22a = 1.) From Eqs. (4.24) and (4.25), it is easily seen that the degree of decoherence may depend on both the projection operators we use and the initial state of the system. This is also an expected result. For the two-level system, EA and e 8 can be calculated exactly. Consider, first, the eigenvalues of A. Since Eq. (2.1) preserves the trace, 8A can be written as gA

= V/1 - 4ZIA2A = X/I -- 4 d e t A .

(4.26)

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The determinant of A can be explicitly evaluated from the general solutions Eqs. (2.11)-(2.14): ~(B+I) d e t A - - ( ~2 ~ ++l ) +

[ ~ B+I I.-~--~Pll(0)+-f~P22(0)

p22(0)

2B+ 1

p~(0)

2~(~+1) ( 2 ~ + 1) 2

2 ~ + lJ

"

] p21(0)p12(0)A ,~ (4.27)

where (5 = exp{-~/(2B + l)tl}. In order that a set of histories are to be decoherent, one expects that ,5 should be small. Similarly, c 8 can be expressed as

cB =

1

4 det B (Tr B)2 .

(4.28)

From Eqs. (4.14)-(4.17), T r B and detB can be easily obtained: 1

T r B = 2Ct + ~ i - C 2 6 1 detB = C f +

k~+

GC2

,

(4.29)

- C3C4

]

(51 - - - C 2 6 1

~+

l

,

(4.30)

where 6t --- e x p { - 7 ( 2 ~ + 1 )(t2 - tl )}. The above discussions show explicitly how the degree of decoherence is related to the projection operators, the initial states and the temperature of bath, as well as the time-spacing interval, in accordance with our general expectations. It may be helpful to consider some special cases in which the simpler expressions for ~:.4 and c e may be obtained. In the long-time limit, the density matrix will tend to the stationary density matrix. Then we may get a much simpler expression for ~:A: 1

C4 ~ - 2~+ 1

(4.3 1 )

As mentioned before, for the decoherent histories, /i and ~l should be small. If we only keep the terms up to the first order of ,51, then t:8 becomes 1/2 ~,B ~-~

C3C461 1 C 2 + ~-~C1C261

(4.32)

Similarly, the expression for e A can be obtained from Eq. (4.27). It is seen from the above expressions that the degree of decoherence improves as the bath temperature increases. We also see that the projections with the smaller off-diagonal elements will give a better degree of decoherence. For a given system with the initial state, then the matter for investigation is to determine which histories, i.e., which string of projections, will lead to the decoherence condition being satisfied. Therefore, we see that eB serves as the main criterion for the degree of decoherence.

T. YulPhysica A 248 (1998) 393-418

406

4.2. yon Neumann entropy and predictability It is also of interest to compute the von Neumann entropy of p(t) [4,5,33]. We will discuss how the initial density matrix and the yon Neumann entropy are related to the upper bounds of the degree of decoherence. We will also discuss the preferred states by using the yon Neumann entropy rather than linear entropy, in this two-level model. In the case of system-plus-reservoir model, the pure states of the system, due to the interaction with environment, will typically deteriorate into the mixtures with the different rates. The rate at which pure initial states evolve into the mixtures reflects the stability of those pure states which are continuously monitored by the environment. The von Neumann entropy provides a convenient measure of the loss of predictability: S = - T r ( p In p ) ,

(4.33)

By definition, the more predictable state (pure state) may have less increase of the entropy in a fixed time period. This characterization process of predictability is called the predictability sieve (coined by Zurek [4,5]) which has been studied recently in quantum Brownian motion model by using the linear entropy [4,5,33,34]. We will see that the two-level system serves as a very nice toy model to employ this "predictability sieve" by directly using the von Neumann entropy. For the purpose of the evaluating the entropy, we choose a special basis in which p is diagonal. Let 21 and 22 be the eigenvalues of p, then Eq. (4.33) reduces to 2

S = - Z

)~i In 2 i

(4.34)

.

i=1

Obviously, 21 and 22 can be expressed as 1 + ~A

21-

~ ,

1 - ~:A

22 =

2

(4.35)

Hence, Eq. (4.34) can be rewritten as S=

-

[ ( 1 2 ~ ) l n ( l +) ~ ~ A -

-

+

~(1--eA)ln( 1 ~ A ) ] 2

A t r i v i a l o b s e r v a t i o n shows that the y o n N e u m a n n e n t r o p y

"

S(eA) is

(4.36) a m o n o t o n i c a l l y de-

creasing function of e A. Here, we find an interesting relation between the predictability of initial state and ~;A, which is an upper bound of the degree of decoherence. Namely, the von Neumann entropy provides a restriction on the upper bound of the degree of decoherence. Precisely, the initial density matrix which leads to larger entropy production may give smaller cA. This relation between the predictability and the degree of the decoherence is a physically expected result. To obtain a higher degree of the decoherence one would expect that the environment has stronger influence on the system of interest, such as increasing the temperature of the bath. Then the predictability of the state, correspondingly, decreases.

T. Yu/Physica A 248 (1998) 393 418

407

I would like to point out here that the actual degree of decoherence could be much smaller than the upper bound eA, since it is often typically undercut by the lesser upper bound e,8. Moreover, the matter for investigation in histories approach is to determine which histories will satisfy the given degree of decoherence. In contrast, our goal here is merely to see how the initial states are related to the upper bounds of the degree of decoherence, hence, we do not take any particular set of histories into account. Next, by using the yon Neumann entropy we shall find the most predictable states, those states will, by definition, generate the least entropy for a given time interval. Since the entropy S (Eq. (4.36)) is the monotonically decreasing function of ~:~1 it is equivalent to find the states which give rise to the largest ~;A. From Eqs. (4.26) and (4.27) it is easily seen that I~) = e V ~

I1) :k

12)

(4.37)

minimize the yon Neumann entropy, and therefore are the preferred states. This is slightly surprising from both decoherent histories and environment-induced decoherence points of view. At first sight, one might expect that I 1) and 12) would be the preferred states, since this basis plays a very special role in both formalisms. However, from Eqs. (2.9) and (2.10) it is easy to see that the pure states ]1) and 12) will immediately deteriorate into the mixed states. Namely, those two states are most vulnerable to the influence of the environment. This explains why the basis [1) and [2) are not the preferred states. Also, we see from the above discussions that the states, which diagonalize the density matrix, are not necessarily same as the preferred states that are sorted by the Zurek's predictability sieve.

5. Unravelling of the master equation The master equation provides an ensemble description of a quantum system. The unravelling of master equation as the stochastic Schr6dinger equation for the state vector has provided many insights into the foundation of quantum theory, especially in quantum measurement and the useful tools to study various practical problems in the quantum optics (e.g., see Refs. [23,24]). In this section, we will study the localization in the two different unravellings of the master equation quantum-jump simulation and quantum-state diffusion approaches. The former uses the discrete random variables whereas the latter uses the continuous random variables. 5,1. Quantum-jump simulation

In the measurement schemes, such as direct photodetection, the master equation, which models the measurement process, in some sense describes the lack of information of the systems. Namely, it describes the measurement process in which the results of

T. YulPhysica

408

measurement

are not extracted.

A 248 (1998)

The quantum-jump

393-418

simulations,

by contrast, mimic that

which may be observed in a single run of the experiment. The state of the system in this situation is represented by a wave function. The whole physical process under consideration

is the combinations

are characterized the individual stochastic

of continuous

by the discrete

system

umavellings

random

is usually

governed

evolutions

variables.

by a stochastic

are said to be equivalent

and abrupt jumps

Therefore,

which

the wave function

differential

equation.

to the master equations

of The

if the former,

after the stochastic average, could reproduce the latter. The alternative description by a single wave function is not confined in the measurement processes. In general, any master equation with Lindblad form could be unravelled into the stochastic Schriidinger equation. For the master following form:

equation,

Id+) = -;H

Eq. (2. l), the stochastic

Schrodinger

equation

takes the

,$) dt +

(5.1) are the Lindblad operators representing the inHere LI = dma, L2 = fiat fluence of the environment and Ni = LtL; (i = 1,2). (N;) = ($lNil$) represents quantum average and M represents the ensemble average. The real random variables dWi(i = 1,2) satisfy dW*dWj

= JijdWi,

M(dWi)

= (Ni) dt

(5.2) (i = 1,2).

(5.3)

Under condition, Eq. (5.2) it is easy to see that dWi only take two values: 0 and 1. The master equation (Eq. (2.1)) can be recovered from the stochastic Schrodinger equation (Eq. (5.1)) in the sense that if I$) is the solution to Eq. (5.1) then p = Ml$)($l satisfies the master equation (Eq. (2.1)). In what follows,

we shall discuss

the “localization”

properties

of the single-jump

trajectories. Here, by “localization” we mean that the quantum-state vector generated by the stochastic Schriidinger equation will converge to some fixed states in the mean square. More precisely, let A be an operator (not necessarily quantum mean-square deviation as a(A,A) If the solution

Hermitian),

then we define the

= (AtA) - (‘4+)(A). of the stochastic

Schrodinger

(5.4) equation

(Eq. (5.1))

satisfies (5.5)

namely, the dispersion of the operator A tends to decrease as time evolves. Then we say that the solution localizes at the eigenstates of the operator A (A is sometimes called the collapse operator).

T. Yu/Physica A 248 (1998) 393 418

409

For the stochastic Schr6dinger equation for the quantum-jump simulation in a twolevel system, the collapse operator is a:. Then quantum mean-square deviation in this case is (Ao~) 2 = 1

15.6)

(o~) 2 .

In order to prove the localization, we should first derive the evolution equation of the expectation value of o: by using the following tbrmula: d (m) = (~,]AldO) + (dtp]A]~9) + (dtPlAldt)) ,

(5.7)

where A is an operator. From Eq. (5.1), it is straightforward to arrive at the following equation: d (o:) = (1 - @ ~ ) ) d W l

- (1 + (O-z))dW2 + [@:) (N, + N2} + (NL - N 2 ) ] d t .

(5.8) Notice that d( A g z ) 2 ~- - 2 (~Tz) d (ffz) - (d (o'z)) 2 .

(5.9)

Then, inserting Eq. (5.8) into the above equation, taking the ensemble means and remembering Eq. (5.3), we obtain M d ( A a ~ ) 2 = - ~ 1° / ( n + 1)(1

-

(O':))2(1

÷

(O'z))

--

7~(1 + (o:))2(1 - ( a : ) ) . (5.10)

The right-hand side of Eq. (5.10) is non-positive, and that it vanishes if and only if 10) is 12) or I1). Hence, we conclude that the solution to the stochastic Schr6dingcr equation (Eq. (5.1)) will localize at 12) or I1) after a certain time. That is, any initial state (which will be a superposition of 11) and 12)) will tend to a solution in which the atom undergoes stochastic jumps between I1 ) and 12). Let us now estimate this localization time. From Eq. (5.10), a few manipulations directly give M d ( A a z ) 2 ~ - 7(2n + 1 ) ( A ~ ) 2 • So the localization rate tlocalization ~

/localization

(5.11)

is

1

7(2~ + 1) '

(

5.12 )

which agrees with the decoherence time scale Eq. (2.20). Note that this is the minimum localization time. The actual time for localization might be larger than this time. In some sense, that the localization in quantum-jump simulation chooses the basis tl ), 12) appears to be natural, since it correspond to the trajectories that would actually observed in an individual experiment. As expected, the set of histories consisting of projection onto the basis give the best degree of decoherence. In addition, we have seen

T. Yu/PhysicaA 248 (1998) 393-418

410

that density matrix become diagonal in this basis. Here, we have demonstrated a close connection between the different approaches. This is one of the main results in the paper. The connection we have established here bridges the two different approaches - decoherent histories and quantum-jump simulations. The former is regarded as a fundamental theory with a wide range of applicability, whilst the latter is mainly seen as a tool with great practical values, in particular, in the computational aspects.

5.2. Quantum-state diffusion In this subsection, we will illustrate the localization process in another unravelling of the master equation - the quantum-state diffusion approach, which was introduced by Gisin and Percival [16] to describe the quantum open system by using a stochastic Schrrdinger equation (which is often called the Langevin-Ito stochastic differential equation) for the normalized pure-state vector of an individual system of the ensemble. Similar to the quantum-jump simulation, a solution of the Langevin-Ito equation for the diffusion of a pure quantum state in state-space represents a single member of an ensemble whose density operator satisfies the corresponding master equation. Generally, if the master equation takes the standard Lindblad form: /~ = - ~ i[ H , p ] + ~

(LipL~ - ~Li I t Lip - ~pL~Li )

(5.13)

Then, correspondingly, the Langevin-Ito stochastic equation can be written as

Id~l) z -~n I ~l) dt -]- ~i ( (Z] lZi - ~Z]Z i - ~ IZ] l (Zi) ) I~l)dt + Z ( L i - (Li)) ]~) d~i,

(5.14)

i

where H is a Hamiltonian (of the open system) and Li a r e Lindblad operators, as before, (Li) = (~'lLilq~). The complex Wiener processes d~i satisfy

M(d~i) = 0,

M(d~id~j) = 0,

M(d~TdCj) = 6ijdt,

(5.15)

where M denotes a mean over the ensemble. Quantum-state diffusion reproduces the master equation in the mean: p = M I~') (ffl,

(5.16)

where [ff) satisfy the quantum-state diffusion equation (Eq. (5.14)), then it can be shown that p satisfies the master equation (Eq. (5.13)). In order to show the localization properties of the Langevin-Ito equation, we now consider the simplest case which is assumed that the bath temperature is zero (~i = 0). In this case the master equation Eq. (2.1) reduces to

i f~ = -~[n,p] + ~(2apa t - atap - pata) .

(5.17)

T. Yu/Physica A 248 (1998) 393 418

411

Then the corresponding Langevin-lto equation is given by I~') dt + Z(2 2 (at) a - ata - (a t) (a))Itk)dt

i Id,/,) = - ~ H

+xfT(a- (a))[email protected],

(5.18)

where d~ is the complex Wiener process satisfying

M(d~) = O,

M(d~d~) = O, M(d~*d~) = dt,

(5.19)

where M denotes a mean over probability distribution. The evolution of the quantum average of operators can be calculated by using the following formula:

d(G) = ~ i( [ H , GI) d t - ~ Z1

(L~[Li, \ " G]+[G, Lt]Li)dt i

+ Z(O-(G t, Li) d~i + O-(L~,G) d~,* ),

(5.20)

i

where

a(A,B) = (AtB) - (A t ) (B).

(5.21)

Using Eq. (5.20), it is straightforward to get the following equations: d(a,)=

-

(a,,)-~-(ax

dt

+ 2~;[l + (o-~,)- (at) 2 +i(o-x)(ay)]d~ +~[1

[email protected])=

+ (o-:) -

-~(ax)+

[email protected])

(5.22)

dt

[-i(l+(a:))+i(o-v)2-(av}@v)]d~

+~[i(1

d (a:)

i (O'r)(Gv)]d~* ,

(o-x) 2 -

+ (o--)) - i (o-),)2

(o-,) (a,.)] d~*,

(5.23)

-[(a:) 7 + 7]dt °'~/(1 + (o-:))((ar) - i(av))d¢

~'~/(1 +

@~) )( (ax)

+

i (av) )d~* .

(5.24)

Moreover, we need to calculate the higher-order moments. For any Hermitian operator A we have from Eq. (5.4),

d(AA) 2 = d((A 2) - (A) x) -- d (A 2) - 2 (A)d (A) - (d

(A)) 2 .

(5.25)

72 Yu/Physica A 248 (1998) 393 418

412 Then, we easily obtain

Md(A~x)

2 = 2 ~ (O'x)(O'y)+ 7 (O'x) 2 - ~(Ao'x) 4 - 7(Ao'x) 2 (O'z) 3' 2 (O'z)2

27 ((O'x) (O'v))2 ,

(5.26)

~ )4 - 7(AO-y) 2 (O'z) M d (Aoy) 2 = - 2 ( o (crx) (Cry) -/- g (O'y) 2 -- ~(AO-y 2 (O'z)2 -- i ((O'x) (0"3))2 "

(5.27)

Now, we are in the position to consider the localization of solutions to Eq. (5.18). Using the master equation, it is very easy to see that the atom will soon collapse into the lower state I1) and keeps there forever. Here, we shall demonstrate that any solution to the Langevin-Ito equation given by Eq. (5.18) will localize at the lower state after a localization time. The collapse operator in this case is (5.28)

A = o~ + i~h,. Then by using Eq. (5.4) we get

a(A,A) = (Ao'x) 2 + (Affr) 2 + 2 (az).

(5.29)

Hence, we have

M d a ( A , A ) = 7 (ax)2 + 7 (ffy)2-- ~(A~Yx)4- ~7(A~yy) 4 --)'(AO'x) 2 (~Yz) -- 7(AtYy) 2 (O'z) -- 7 (O'z) 2 - 2 7 ( @ z ) + 1) - 7((ax)

(O'y))2.

(5.30)

In order to prove that the left-hand side of Eq. (5.30) as non-positive, let us denote

(AO'x) 2 = 1 - - Y ,

(5.31)

(Aoy) 2 = I + Y ,

(5.32)

(~) = -1 +Z.

(5.33)

Substituting Eqs. (5.31-(5.33) into Eq. (5.30), we have

M

a(At,A)

[

7 -R2-X

1

Y-2Z-~(Y-Z)

,

2-5(X-Z)2

],

(5.34)

where R = (ox)(a),). Note that

X + Y + 2Z = a(A,A)>~O.

(5.35)

Then, we show that

Mda(At,A)<~O

(5.36)

T. Yu/Physica A 248 (1998) 393 418

413

and the equality holds if and only if X-- Y=Z=0.

(5.37)

That is, the average in the left-hand sides of Eqs. (5.31)-(5.33) is taken over the ground state I1). This proves that the solution to Eq. (5.18) will localize at the ground state when the evolution time is larger than the localization time. Finally, let us estimate the localization rate of the quantum state evolution. Using Eqs. (5.34) and (5.35), we immediately obtain d M~i~a(A,A)<~

-- 7 ( t T ( A , A ) )

So the localization rate 1 tlocalization ~'~ --

7

tlocalization

2 .

(5 38)

is

(5.39)

In summarizing this section, we have shown the localization process in both quantumjump simulation and quantum-state diffusion. Those localizations have been extensively discussed in the quantum-state diffusion approaches. Here, we have seen that a similar localization process could also occur in the quantum-jump simulation. It should be noted that, in the case of the zero-temperature of our two-level model, for any initial state of the system, the atom will eventually localize at the ground state. Therefore, the system will always evolve from a pure state into the pure state. In this sense, we say that the decoherence and localization are basically trivial in this case. However, the above demonstration of localization can still be regarded as a useful example for showing that quantum-state diffusion picture provides a consistent description with the density matrix formalism and decoherence approach. It is interesting to compare the master equation formulation with their stochastic unravellings. Clearly, the master equations provide a fundamental description of the quantum open systems. But numerical simulation of the many-freedom problems seems rather awkward as it requires a large memory. Moreover, it cannot provide a description for an individual system. The quantum trajectories approach - the unravelling of master equation as the stochastic Schr6dinger equation - could do this job and have advantages over the master equation in computational aspect [16,35]. For this two-level model, the merit of stochastic unravellings is mainly in the conceptual aspects. Generally, the localization process is very difficult to show analytically, if not impossible. Obviously, the unravelling of the master equation is not unique, Quantum-jump simulations and quantum-state diffusions are only two well-known examples, which lie in our interests in this paper. These stochastic unravellings are often connected with certain measurement schemes. For instance, the quantum-jump simulation can be associated with the direct photodetection, and quantum-state diffusion corresponds to the heterodyne detection. In a quantum-jump process, quantum-jump simulation may be a natural candidate for description of the process. The quantum-state diffusion by nature is a continuous diffusion process. However, if the transition is so fast that the "diffusion"

414

T. Yu/Physica A 248 (1998) 393 418

from one level to the other level of atom can be regarded as an instantaneous process, then quantum-state diffusion could also give rise to the "jump" process [20,23]. It should be noted that the applicability of quantum-jump simulation and quantum-state diffusion are different. The preference of these stochastic approaches are largely dependent on the physical models employed and the problems to be solved. In general, the relation between those two approaches is by no means obvious. Undoubtedly, the researching into this relation would be of importance and of interest [36].

6. Summary and conclusions In this paper, based on the two-level system models, we have studied and compared in detail the decoherent histories approach, environment-induced decoherence approach, quantum-jump simulation and quantum-state diffusion. We have demonstrated the localization in both quantum-jump simulations and quantum-state diffusion approaches. Here, we conclude with a summary and a few remarks. We have shown that there are a number of sets of decoherent histories in this two-level model. Clearly, these decoherent histories are not equally important from the physical point of view. Among those, the most natural one is that which consist of the projections onto 11) and 12). We have proven that this set of histories give the best degree of decoherence. Note that the density matrix in the basis 11) and 12) will become diagonal after a typically short time. Moreover, we have shown that the solutions to the stochastic SchrSdinger equation in the quantum-jump simulation will localize at 11) or 12) after certain time which is basically same as the decoherence time. Also, We have shown the localization process in quantum-state diffusion in the case of zero temperature. In addition, we have found that the environment-induced decoherence, decoherent histories and the localization process are more effective as the bath temperature increases. Physically, this is an expected result as the bath at a higher temperature would have stronger influence on the system. These results are in agreement with former studies on the quantum Brownian models [26] as well as on the quantum-optical models [37]. Despite the similarity and agreement mentioned above, it is important to note that there also exists the significant difference. Among others, as was shown in this paper, time scale for decoherent histories could be quite different from that for environmentinduced decoherence, and also different from the time scale for localization. It seems that the investigation into the relationship between those two rival formalisms is a delicate task, and the similarity and difference between the two approaches should be studied on the case-by-case basis. There is no reason to believe that in more realistic physical problems the two approaches would agree each other in every aspect. In connection with the stochastic evolutions, we have shown a close relationship between the decoherent histories and quantum-jump simulations. The solution generated by stochastic Schr6dinger equation (Eq. (5.1)) will randomly jump between the two

72 YuIPhysica A 248 (1998) 393 418

415

levels I1) and 12). However, due to the influence of the bath, the localization process occurs in this basis after a certain time. Meanwhile, as was shown, the set of histories consisting of the projection onto these two levels are perfectly decoherent. Here, we have seen that quantum-jump simulation is entirely compatible with the history point of view. This is a very nice result. Similar results in the quantum-state diffusion have been discussed before [25,26]. The approximate decoherence is of basic importance in practical physical process. By using this two-level system model we can clearly see what determines the degree of decoherence. For a given set of histories, the only adjustable parameters are the temperature of bath, the time-spacing interval and the initial state of the system, in accordance with our general expectations. We have studied the predictability of the pure states in this two-level model. The von Neumann entropy in this situation serves as predictability sieve to sort out the preferred states which yield the smallest entropy production. As by-product, we also see an interesting relation between the upper bound of degree of decoherence and the initial density operators through the yon Neumann entropy. It is important to notice that, as phenomenological theories, both quantum-state diffusion and quantum-jump simulation must be used under some conditions (e.g. see Ref. [20]). The comparison between different approaches therefore must be made in caution, since the correspondence between them is by no means mathematically one-to-one correspondence. Rather, we emphasize that, underlying the quantum open system, the mutual influence between the system and its environment is the common theoretical base of all of those approaches and both decoherence and localization are nothing more than the different manifestations of a single entity. Finally, there are several special features of our model which are worth pointing out explicitly. First, the Hamiltonian of the system is diagonal in the basis I1), 12',,. We see that the evolution equations for the diagonal elements and the off-diagonal elements of the density matrix are decoupled in this situation. One consequence of this is the exactness of decoherence histories of the projections onto I1), 12). An immediate generalization of our model is to consider the Hamiltonian that does not enjoy this property. One such example is that, in addition to a thermal radiation field, the two-level atom is applied a coherent driving field. Notice that, in this situation, the evolution equations for the diagonal and off-diagonal elements of density matrix are no longer decoupled. The possible effect of making this change is that, instead of decoherence, the quantum coherence could be generated due to the influence of the coherent driving field [19]. At last, let us note that the coarse-graining in our two-level system is made by using projection operators on the system whilst ignoring the environment. It would be interesting to consider the general n-dimensional model in which the effect of a further coarse-graining on the degree of decoherence can be discussed. Work towards to this aspect is in progress. The environment-induced decoherence, decoherent histories as well as various stochastic Schr6dinger equations have provided many important insights into the understanding of fundamental problems in quantum theory. The investigation into the similarity and

416

T Yu/Physwa A 248 (1998) 393 418

difference between the different approaches is of importance. More thorough studies in this aspect would be useful. Note added. After completion of the paper, I become aware the related work concerning decoherent histories and quantum jumps carried out independently by T. Brun [38]. His results are in tune with ours in Section 5.

Acknowledgements The author would like to express his sincere thanks to Jonathan Halliwell for suggesting this project, for encouragements, and for many suggestions which are critically important for the ideas in this paper. He is grateful to Todd Brun for useful conversations and to Lajos Di6si for many useful conversations and encouragements. He is also grateful to Bernhard Meister and Andreas Zoupas for interesting discussions. This work was supported by the SBFSS scholarship from the British Council.

Appendix A. Proof of theorem In this appendix, we shall give a proof of Theorem in Section 3. Since both M and N are positive-definite matrices, one of them, say, N can be decomposed as

N =StS,

(A.1)

where S is an n × n matrix. After an arrangement, the right-hand side of Eq. (4.3) becomes

ITr(MPSt SQ )I = ITr( SQMPSt )[ .

(A.2)

Suppose Xm are an orthonormal basis in n-dimensional space V. Then

ITr(MPNQ)I = ~m Xrm(SQMPSt)xm ,

(m.3)

r is the transpose of Xm. Now, we set where x m Ym = Q g t x m ,

(a.4)

PS?xm •

(A.5)

Zm :

Then the trace in Eq. (A.3) may be rewritten as

Tr(MPNQ) = Z

T ymMzm "

(A.6)

m

Since ym,Zm are orthogonal vectors and M is a positive-definite matrix, then it is not difficult to arrive at the following inequality (see Ref. [39]):

[(yrmMzm)l ~ 3 M( Y mT M y m ) 1/2 (zmMZm) r 1/2 ,

(A.7)

T. Yu/Physica A 248 (1998) 393-418

417

where e M : ()~max m "M ' M x Jr-2min),Amax M ",~/ ~M n are the largest and the smallest --Amin)/(Ama and Ami eigenvalues of M, respectively. Combining Eq. (A.7) with Cauchy's inequality

ambm \

m

2 ant

~ /

t~

b zm "

(A.8)

m

then Eq. (A.3) becomes

r ITr(MPNQ)[ = ~ Xrm(SQMPSt)xm ~ < ~ lymMz.,I ttl

~.M

Ym

Ym

(A.9)

It is easy to identify that

Tr(MPNP) = Z ymMym, r

(A. lO)

m

Tr(MQNQ) = Z

rMz m . zm

(A.I1)

ttl

This proves that

ITr(MPNQ)I <~eM[Tr(MPNP)]I/2[Tr(MQNQ)]I/2.

(A. 12 )

Since M and N are in the completely symmetric position, so a similar result is true also for ~,N. Then it completes the proof of the theorem. []

References [1] [2] [3] [4] [5]

W.H. Zurek, Phys. Rev. D 24 (1981) 1516. W.H. Zurek, Phys. Rev. D 26 (1982) 1862. W.H. Zurek, Phys. Today 44 (10) (1991) 36. W.H. Zurek, Prog. Theor. Phys. 89 (1993) 281 and reference therein. W.H Zurek, in: by J.J. Halliwell, J. Perez-Mercader, W. Zurek (Eds.), Physical Origin of Time Asymmetry, Cambridge University Press, Cambridge, 1994. [6] R.B. Griffiths, J. Stat. Phys. 36 (1984) 219. [7] R. Omn~s, J. Stat. Phys. 53 (1988) 893. [8] R. Oran,s, J. Stat. Phys. 53 (1988) 933. [91] R. Omnbs, J. Stat. Phys. 53 (1988) 957. [10] R. Omn6s, J. Star. Phys. 57 (1989) 357. [11] R. Omnes, Rev. Mod. Phys. 64 (1992) 339. [12] R. Omnbs, The Interpretation of Quantum Mechanics, Princeton University Press, Princeton, New Jersey, 1994. [13] M. Gell-Mann, J. Hartle, in: W. Zurek (Ed.), Complexity, Entropy, and the Physics of Information, SFI Studies in the Science of Complexity, vol. VIII, Addison-Wesley, Reading, USA, 1990. [14] M. Gell-Mann, J. Hartle, Phys. Rev. D 47 (1993) 3345. [15] J. Hartle, in: S. Coleman, J. Hartle, T. Piran, S. Weinberg (Eds.), Quantum Cosmology and Baby Universe, World Scientific, Singapore, 1991. [16] N. Gisin, I.C. Percival, J. Phys. A 25 (I992) 5677.

418

72 Yu/Physica A 248 (1998) 393 418

[17] N. Gisin, 1.C. Percival, J. Phys. A 26 (1993) 2233. [18] N. Gisin, I.C. Percival, J. Phys. A 26 (1993) 5677. [19] H.J. Carmichael, An Open Systems Approach to Quantum Optics, Lecture Notes in Physics, Springer, Berlin, 1994. [20] H.M. Wiseman, G.J. Milburn, Phys. Rev. A 47 (1993) 1652. [21] L. Di6si, J. Phys. A 21 (1988) 2885. [22] I. Percival, J. Phys. A 27 (1994) 1003. [23] N. Gisin, P.L. Knight, I.C. Percival, R.C. Thompson, D.C. Wilson, J. Mod. Opt. 40 (1993) 1663. [24] B. Garraway, P.L. Knight, Phys. Rev. A 49 (1994) 1266. [25] L. Di6si, N. Gisin, J. Halliwell, I. Percival, Phys. Rev. Lett. 74 (1995) 203. [26] J. Halliwell, A. Zoupas, Phys. Rev. D 52 (1995) 7294. [27] C.G. Gardiner, Quantum Noise, Springer, Berlin, 1994. [28] D.F. Walls, G.J. Milbum, Quantum Optics, Springer, Berlin, 1994. [29] E. Joos, H.D. Zeh, Z. Phys. B 59 (1985) 223. [30] J.B. Paz, W.H. Zurek, Phys. Rev. D 48 (1992) 2728. [31] H.F. Dowker, J.J. Halliwell, Phys. Rev. D 46 (1992) 1580. [32] J.N. McElwaine, Phys. Rev. A 53 (1996) 2021. [33] W.H. Zurek, S. Habib, J.P. Paz, Phys. Rev. Lett. 70 (1993) 1187. [34] M. Gallis, Phys. Rev. A 53 (1996) 655. [35] R. Schack, T. Brun, I. Percival, Quantum State Diffusion, Localization and Computation. QMW preprint Th-95-19, unpublished. [36] T.Brun, P. O'Mahony, M. Rigo, From Quantum Trajectories to Classical Orbits, QMW preprint 1996, quant-ph/9608038, unpublished. [37] J. Twamley, Phys. Rev. D 48 (1993) 5730. [38] T.A. Brun, Quantum Jumps as Decoherent Histories, QMW preprint 1996, quant-ph/9606025, unpublished. [39] B. Meister, Ph.D. Thesis, Imperial College, 1996, unpublished.