1. (1017).

2. (1020).

2.1. , . 2.2. fi

.

3. (1022).

3.1. .

3.2. ( ).

3.3. * 1. 3.4. *

.

4. fl (1030).

4.1. . 4.2. fi-

. 4.3. -

.

5. ' (1034).

(1034).

1.

fl

, , -, - ,-, [1 12]. fi [13 16], - [17], - - ( '). , - , , [2,9]. fl [18, 19], (fl) , , - . - . - e. , fl - , , .

fi .A - fl, . , .. , [12]. `

1998 . 168, 9 ' fl

' fl fi

`

fi.. fi

( ) - (fl). fl , . , . fl - . , , . - .

PACS number: 03.65.Bz

#fi.. fi 1998

fi.. fi. . .fl. fl,117924 fi, . 53, . (095) 132-62-19 (095) 938-22-51E-mail: mensky@sci.lebedev.ru

21 1998 .

1 ' , .

, fl2.

fl , - . , .. .

' fl, , ().

, ( ), -. fl , ( ).

- . . , -, - "" , -. - - , - , . fl , .

, jc1i,jc2i - (, , ) - jci c1jc1i c2jc2i. , jc1i,jc2i , , (, - , ), . -.

, - , , , . (environment) jci jc1i jc2i.

, , jc1j2 jc2j2. - (decoherence). - . , , .

- . - , [20], , - [21, 22]. , , , ().

, [23, 24]. , - , - , .

( ) . , - . , , (.. , ) . . , - ( ). , - e . - .

, . . ` , ( ) - - . , - , - . fi , -, . . , . , , - ( ), , .

, - , . -, -

2 "decoherence". - "-". fi , "" , "interference" "-".

1018 fi.. fifl [fl 1998

, . . , , .

"-" , - . - .

, " ?" " ?" - .

, fl,

c1j1i c2j2i ! j1i; p1 jc1j2 ;

j2i ; p2 jc2j2 :

(

j1i j2i , . - , - p1 p2, . , - , . fi - , j1i, ,, , - j2i.

- Pi, i - . i-, jci jcii Pijci, .. - . , i- , 3

pi kcik2 hcijcii hcjPijci : 1

, : ? . ( . [25, 26, 10, 11]). [12]. ' [27,28], : , . fi

2.1.

, , fl. , ? , , ? .

-, - (, - ) - . . , , ( )' [13 16].

' , , - . fl ? -, . -, , . -, , () -. , - .

, , , , , - . , - . .

( ) . - - . - .

fi , S, ( )M .

S, M ( ).

(), - .

. -

. : - ? -, , . , , -.

-

3 , jcii, , - . - .

. 168, 9] `fl fl ` flfl fl 'fifl 1019

. , , . - , , - . .

2.

"" , , , , , .

2.1. , fi , (, ) - fl [29]. - , () . , - , , - ., , A, a 0, a 00. :

jci c 0ja 0i c 00ja 00i ! ja0i; p1 jc 0j2 ;ja 00i; p2 jc 00j2 :

(2

, , - r0 jcihcj :

r0 jc 0j2 c 0c 00c 0c 00 jc 00j2

!! jc

0j2 00 jc 00j2

! r : 3

r , (2), - .

, (3), - . - . ' -:

ha 0jrja 00i ha 00jrja 0i 0 : 4

ja 0i ja 00i , . , - ( ).

, . -

(- )

, (entanglement) , - . .

fi , , , , . - M, S. , jf0i. , , . ja 0ijf0i ja 0ijf 0i, - ja 00ijf0i ja 00i jf 00i. .

, S M , jC0i jcijf0i, jci - , . :

jC0i jcijf0i c 0ja 0i c 00ja 00ijf0i !

! c 0ja 0ijf 0i c 00ja 00ijf 00i jCi : 5

, (2) (3) (5) - , .. . jCi , S M , - .

jCihCj , - :

r TrfjCihCj jc 0j2ja 0iha 0j jc 00j2ja 00iha 00j hf 00jf 0ic 0c 00ja 0iha 00j hf 0jf 00ic 0c 00ja 00iha 0j : 6

. -, , .. (4) .

() , , , ( jf 0i jf 00i) " ". , .

$

1020 fi.. fifl [fl 1998

( 1023). , - () . -, , . ,

hf 0jf 00i hf 00jf 0i 0 : 7 , -

r TrfjCihCj jc 0j2ja 0iha 0j jc 00j2ja 00iha 00j 8

(4). . (., , [30 32]). ' : , (environment-induced superselection) [28].

fi - , (5). (. [12] ) , (.. ) ha 00jrtja 0i2 expka 00 a 02t : 9 , - . (9), td - -:

td a 00 a 02 : 10

, , , , , . , , , .

2.2. fi , , - , 2.1. fi S, M . , .

(., , [33, 34, 30, 31, 35,28, 36 38, 4, 3, 39, 7, 40, 12]). fl , , , - . -

[36]. - , , :

(Caldeira & Leggett, 1983)

S, M. (, , , ). - , , - 4

_r ihH; r 1

2kr r r ; 11

k 2ZkBTh2

12

T Z, , :

mr Z_r V 0r Ft : 13

, - [40, 41]. , - , , . :

(Konetchnyi, Mensky & Namiot, 1993)

4 [36] . fi , - .

. 168, 9] `fl fl ` flfl fl 'fifl 1021

(11),

k 2l2t

14

, l t . ( - ) .

, (11), - ( ). 3, , - , - .

3.

2, . , ,

- (fl) - .

3.1. fi fl - , - fl. () . t0 t tN:ct UtN; tN1PiN1UtN1; tN2 . . .

. . .Ut3; t2Pi2Ut2; t1Pi1Ut1; t0ct0 : 15

- fi1; i2; . . . ; iN1g. , , .. . , - fl, . - .

, - , fl (., -

, [42, 2, 43]). - . fl - , .

-, , . - .

1. S

_r ihH; r 1

2kA; A; r ; 16

H , A , -, .. "" , k -. (16) - [44], . fi 2.2, - . [44] - - . (16) , - , - , - (. . 2).

2. [9] S

j _ci i

hH kA at2jci : 17

at A t, -. , -, , -: , , .. ( , ). - , , , , . - (16) (17), at. 3.2.

3. -

djci i

hH kA hAi2jci dt kp A hAijci dw

18

[45] (17). w

$

1022 fi.. fifl [fl 1998

( ), . , , dw2 dt, ( ). - [46 48], (16). , . (18) , - (17), . [49] - , , - , .

- fl .

fl - fl. . - . (, , ). - at, A. - w.

fl , , . - - , . ( ) -

(.,, [50]). ( ) () - -. , - (. 3.5).

3.2. ( ) - () . - ( ) , () - . - . (.. ) - .

3.2.1. . - , . , . - - . , .

- ( )

UTq 00; q 0 dq exp

i

hSq

d p dq exp

i

h

T0

p _qHp; q; t : 19

' Sq , :

Sq T0

dt Lq; _q; t ; 20

H . (19) - ( ), -. , . - .

fl + +

:Feynman 1948

&

# :Mensky 1979

!:Feynman & Vernon 1963

#:Golubtsova & Mensky 1989;Mensky, Onotrio & Presilla 1991

& #

# :Diosi 1989;Gisin 1989Belavkin 1989Onofrio, Presilla & Tarmbini 1996

! :Lindblad 1976

. 168, 9] `fl fl ` flfl fl 'fifl 1023

Ut (19), ..

hq 00jUtjq 0i Utq 00; q 0 ; 21

, .. -

jcti Utjc0i ; r 0 Ut r0U yt : 22

, jcti , jc0i. H , . Ut ,

Uyt Ut 1 ; 23

jcti (, , jc0i ).

, , - (19) . - , , - , - , ( ). - , , () , (19).

, - , . , , . , , .. - -. , . fl, , - , , . , - , . , -.

(). - [51]. [52, 53, 2,9] (. [54, 55, 42, 56]). fi , , -: 1) , , 2) , , (, , ).

, . , - - ( ). t , .. at - . - a fatj 04 t4Tg. , - a ., , t at. Da. a , , , a, Da(. 1). , , , .

, - a , q Ia,

U aTq 00; q 0 Ia

dq expi

hSq

: 24

, a , p; q - Ja ,

U aTq 00; q 0 Ja

dp dq expi

h

T0

p _qHp; q; t: 25

fi . [6], Ia Ja .

Ia Ja. , , , . fl,

a

0 T t

. 1., .

1024 fi.. fifl [fl 1998

, 5.

, , . - :

U aTq 00; q 0 dqwaq exp

i

hSq

26

, ,

U aTq 00; q 0 dp dqwa p; q exp

i

h

T0

p _qHp; q; t:

27

, .fl, q a a, waq q, a , .

' 1. wa p; q . , , - , - . , ( ). . , - .

U aTq 00; q 0 U aT , (21), ( , -)

jcaTi U aTjc0i ; raT U aT r0U aTy : 28

- U aT, . jcaTi, , , (, ). fi raT , - , . , - .,

Pa TrraT TrU aT r0U aTy

29

, a. - da,

daPa 1 : 30

( ) a, :

rT da raT

daU aT r0U aTy : 31

, - . fi rT r0, :

da U aTyU aT 1 : 32

(30), - .

3.2.2. fi . fl , - , . , , :A Ap; q; t.

a fatj 04 t4Tg : 33

, A - t at. , A,

At Apt; qt; t ; 34 a. " " . - -

A a2T 1T

T0

At at2 dt : 35

(27)

wa p; q expkA a2

T

expkT0

At at2 dt: 36

- , ( A). fi -

5 , q, _q, , p m _q (. [57, 9]).

7 fl, . 168, 9

. 168, 9] `fl fl ` flfl fl 'fifl 1025

, .

fi -, . , . -, , A. , - , .

(36) . , , - , ( -) , .. - , ., , - - (. 1 . 1025).

k (36) "", . ,

k 1TDa2T

: 37

k (36), , DaT , T. , - (36) , , At, at DaT.

k , DaT :

Da2T 1

T; 38

.. . fi , -. , - (9).

(36) , (27)

UaT q 00; q 0

d p dq exp

i

h

T0

p _qHp; q; tdt

kT0

Ap; q; t at2 dt : 39

() (19), - H

Ha p; q; t Hp; q; t ikhAp; q; t at2 : 40

, , (28),

- (40):

j _ci ihHajci

i

hH kA at2jci : 41

, .. a. -, a - (31), - [50] (16).

' 2. , A ( -). , , at, , - , - , , . , 3.2.1, [41]. , -, " -". .

3.2.3. * fl .fl - (27), , . ,, . , , , ( ) -, , , - . . , - a .

, [58 60], , - , a.

, a , p; q, -, ..

At at ; 6

At Apt; qt; t : p Dp; q Dq, -

6 fi , , p; q, a, .

1026 fi.. fifl [fl 1998

Dp;Dq ( Dp Dpt, Dq Dqt ). At DAt. , DAt , Dat. , p Dp; q Dq Da a. , a, .. - .

fl p; q

S p; q T0

p _qHp; q; t dt 42

, DS p Dp; q Dq. - , , , - a. , a , DS h , a 7.

, , - a ,

maxjDAtj4Dat

Xi

T0

dt

Dpi

_qi qHqpi

Dqi

_pi qHqqi

9h :43

, , - " ". a , p; q

_qi qHqpi dFtqAp; q; t

qpi; 44

_pi qHqqi dFtqAp; q; t

qqi45

" " dFt, -T

0

dtdFtDat9h : 46

, (43) - , - p; q p Dp; q Dq (. 2):

maxjDAtj4Dat

Xi

T0

dt dsi

9h ; 47

dsit Dpitdqit Dqitdpit ;

dqi dqi 1m

pi dt ; dpi dpi Fi dt : 48

fl (47) , . , - ( ). Dp dq Dq dp h : 49 , Dp dq Dq dp h : 50

Dp Dt. , Dq , (50) . dp , (47)., ,

Dpjdqj Dpq 00 q 0 pm Dt

h ; 51 q 0, q 00 . ', , q 00 q 0=Dt - p=m h=DpDt. - [61]

Dpjv 00 v 0jDt h ; 52

. - (47) .

3.2.4. , . - , -

7 . , (39) -, . a, - .

p

SD; d

D Dp;Dq

d dp; dq

q

p

q

d

D

. 2. , - () (), p; q p Dp; q Dq, a. h , a c .

7*

. 168, 9] `fl fl ` flfl fl 'fifl 1027

. , , , - , . , , - (. 4.3). - -. -.

, - . - () - . - , , . : , ( ). -, , - [62].

, (39) - - (41) , - , - . . () . - .

, fl, , . (. - (15)).

- . , - - . 1) 2) , () .

, , - , - . , - . , , , , . , , . ,

.

4 . ' , .

[52, 53, 2], -- [63, 60, 64]. - , .

1. - [66].

2. - [66].

3. - [67].

4. [68].

5. [69].

3.3. * (41) [45]. at ct

a c x2kp ; Ct exp

1

2

t0

dt x2ct ; 53

ct . (41)

j _Ci i

hH kA c2

2kp

A cxjCi 54

, w dw x dt,

djCi i

hH kA c2

jCi dt

2kp

A cjCi dw :55

, jCi ,hCj hdCjjCi jdCi hCjCi ; 56

dw2 dt ; c hCjAjCi : 57

(55) (57) . w (), . :

Pw hcjci exp 12

t0

_w2t dt: 58

-. w

1028 fi.. fifl [fl 1998

Dt -

PDw 12pDtp exp

12

Dw2

Dt

; 59

, , - :

Dw2 PDwDw2 dDw Dt : 60

dw2 dt, [70].

(55) , , , - - . - [46, 71, 47]. , , - , - (16). (55) , [44].

3.4. * fl - [72 75], (consistent histories). , , .. .

, , , . , ( ), . - , -fi [74].

a fi1; i2; . . . ; iN1g [74] , - .

Ua UtN; tN1PiN1UtN1; tN2 . . .. . .Ut3; t2Pi2Ut2; t1Pi1Ut1; t0 ; 61

, (15) . , , . Ua , .

U Xa

Ua 62

, .. 8.

-fi , . , , .

-fi ""

Pa Tr Ua rU ya : 63 , , - .

fl fag , , / ( ). ( ). - - :

b Xa2b

a ; Ub Xa2b

Ua : 64

""

Pb Tr Ub rU yb 65

"" .

, " " , fag . - , , - . fag , - , - . (63) , .

fl fag - , fbg , , , :

Pb Xa2b

Pa : 66

8 , (61) , (15), - , .

. 168, 9] `fl fl ` flfl fl 'fifl 1029

, fag , - fbg (64), - (66). , (66) ,

Caa 0 Tr Ua rU ya 0 67

Caa 0 C aa 0 0 pri a 6 a 0 : 68

, (68) , fag .

, - " ", .. , , , - - (. 2.1). - , , - , [76]. , , -, .

, , . , , ( ). fag, , . fl - .

4. fl

- . , . , fl - . fi fl - - , . - , - , - .

[77, 78]. ,

, o , .. ( '). - . , - Et, .

fl - , , . E, , . ( (41)) , E, - PE. ( ) , E . , , E, . , . fi , .

[18], fl - . [19] , , - fl .

4.1. H H0 V, H0 "" , V , . , H0. 3.2 -

wE p; q expkT0

H0pt; qt; t Et2 dt ; 69

.. , E.

HE H0 V }ikhH0 Et

2; 70

a

qqtjcti

i

hH kH0 Et2jcti : 71

, jcti . (29) , -

1030 fi.. fifl [fl 1998

E:PE kcTk2 : 72

jnt exp iEnthjni ; 73

- :

_Cn kEn Et

2Cn i

h

Xn 0hjnjVjjn 0 iCn 0 74

- :

PE Xn

CnT2 : 75 V 0

CnT Cn0 expkT0

dtEn Et

2 Cn0 exp

TTlr

En E2TDE 2

; 76

Tlr 1kDE 2 77

, .. , . " " DE , .

(76) , T4Tlr (.. - Tlr) , ( ) . , E ,

Et En;c n; E En jCn0j2.

, E ( , . (35)) En, Cn ( t T) , , . (75), jCn0j2.

E ,.. , - Cn T . - - .

, , , -. , En, jCn0j2. - , jCn0j2 (

). fl , , - .

T5Tlr, , :

Emax Emin E DETlr=T

p,

, DE;CnT Cn0 Emin Emax;CnT Emin;Emax. , T

DETlr=T

p. T

, T Tlr, , .. - En. E En 9. , n, jCn0j2.

fi , T5Tlr - , . , Tlr, , . Tlr - .

, , , - - "" - [18]. . fl , - , ( Cn0) En, n- .fl , , , fl. fl , .. , .

, -. H0 , V -

hj1jVjj2i hj2jVjj1i V0 : 78

(73) jni, V - o DE=h, .. V . (74) -

_C1 ivC2 kE1 Et

2C1 ;

_C2 ivC1 kE2 Et

2C2 ; 79

v V0=h.

9 , .

. 168, 9] `fl fl ` flfl fl 'fifl 1031

(k 0), , .. C1t R1t, C2t R2t,

R1t C10 cos vt iC20 sin vt ; 80R2t C20 cos vt iC10 sin vt : 81

: T, Tlr TR p=v. [18].

' (Tlr5TR5T) Et E1 Et E2. , Et En, jCn0j2. E En,

n. . (TR5T5Tlr) .E DET DE

Tlr=T

p.

(TR Tlr T) .E , . E . ,

( ) .fi 4.2.

4.2. fi [19] (79) , -

, T1;T2 TR=2 p=2v (p-). (78), v (79) . 1 1 2. fi , - , .

(79) Et. Et C1t C2t - PE jC1Tj2 jC2Tj2 E. , .

P2t c2t2 C2t2C1t2 C2t2 :

p-. - - E.

fl 3 (79) . - E ( ) P2 ( )

E2

P

E1

fl

1

0

T1

T2 t T1 T2 t T1 T2 t

. 3.fi .

1032 fi.. fifl [fl 1998

, . , E , . - , , TR=2. - , -, E , , ,.

fl . 3 . , , , . . , - E P2, .. , ().

.

E 80% , - .

1=2.

4.3. [19] , . , p- ( ), , - - . - , , , , . , . , , , , , (. - 4.2).

- [79]. - (. 4). -, , , - . , - , , , - - (79).

Et, - , - .

- : , , , - . (, , ) N . N- - n N=N - ( ) . - N-. , E. E. N-, Et, (79).

, - , , .. , (79). , , - , , [79].

, - , (70) . - , k ( Tlr), .

E2

E1

jFijFi

jF0i

D1

D2

. 4. , - : F0 , F, F ,D1, D2 .

. 168, 9] `fl fl ` flfl fl 'fifl 1033

, , , -. , , -, .

5. '

- , . fl , , -, (. 4.3) , (. 2.2). , - () .

, , (. 4.1). -, - fl. . fl , , -.

fl - - .fi : , - - (. 3). , , , -. , [63].

fl - ( ). fi , . - , . ', .. , - (. 4.1).

fl -, , . , - ( 4.2).fl , -

, (4.3). .

-. . fl , . , ( , ) , (. - [12]). , , -, .

, -. . fi , - , . , , , . , - . - , . , [80]. , , .

- , . H.-D. Zeh, J. Audretsch .. fl.

-, 98-01-00161.

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10. fl 164 449 (1994); fl 165 967 (1995)11. , fi fl 166 651 (1996);

(fi.: " ", 1997)

1034 fi.. fifl [fl 1998

12. Giulini D, Joos E, Kiefei C, Kupsch J, Stamatescu I-O, Zeh H DDecoherence and the Appearance of a Classical World in Quantum

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26. (fi.: fl, 1987)

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(fi.: fl, 1964)30. Zeh H D Found. Phys. 1 69 (1970)31. Zeh H D Found. Phys. 3 109 (1973)32. Joos E, Zeh H D Z. Phys. B 59 223 (1985)33. 53 1434 (1967)34.

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Decoherence and the theory of continuous quantum measurements

M.B. MenskyP.N. Lebedev Physical Institute, Russian Academy of Sciences,

Leninski| prosp. 53, 117924 Moscow, RussiaTel. (7-095) 132-62 19

Fax (7-095) 938-22 51

E-mail: mensky@sci.lebedev.ru

The decoherence of a quantum system, i.e., its becoming partly classical, results from its interaction with environment and is welldescribed in terms of continuous quantummeasurement theory. In the present paper, various approaches to CQM theory are reviewed,of which that using effective complex Hamiltonians is discussed in detail. The effective complex Hamiltonian is obtained from therestricted path integral, the latter highlighting the role of information in the dynamics of the measured system, and is applied to theenergy measurement on a two-level system. For such a measurement, quantum transition monitoring is shown to be possible and itsbackaction on the transition probability analyzed. Performing such a measurement using a long series of soft observations is described.

PACS number: 03.65.Bz

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