8 . O p e n Q u a n tu m S y s te m s
8 . 1 C o m b i n e d e v o l u t i o n o f s y s t e m an d b at h
8 . 2 S u p e r o p e r at o r s
8 . 3 T h e K r au s R e p r e s e n t at i o n T h e o r e m
8 . 3 . 1 A m p l i t u d e - d a mp i n g
8 . 3 . 2 P h a s e - d a m p i n g
8 . 3 . 3 D e p o l a r i z i n g p r o c e s s
8 . 4 T h e M as t e r Eq u at i o n
8 . 4 . 1 M a r k o v a p p r o x i ma t i o n
8 . 4 . 2 Li n d b l a d e q u a t i o n
8 . 4 . 3 R e d fi e l d - B o rn t h e o r y o f r e l a x a t i o n
8 . 5 O t h e r d e s c r i p t i o n o f o p e n q u an t u m s y s t e m d y n am i c s
8 . 5 . 1 S t o c h a s t i c Li o u v i l l e e q u a t i o n a n d c u m u l a n t s
8 . 5 . 2 S t o c h a s t i c W a v e f u n c t i o n s
W e n o w p r o c e e d t o t h e n e x t s t e p o f o u r p r o g r a m o f u n d e r s t a n d in g t h e b e h a v io r o f o n e p a r t o f a b ip a r t it e q u a n t u m s y s t e m . W e h a v e s e e n t h a t a p u r e s t a t e o f t h e b ip a r t it e s y s t e m m a y b e h a v e lik e a m ix e d s t a t e w h e n w e o b s e r v e s u b s y s t e m A a lo n e . W h a t if w e w a n t t o k n o w t h e d y n a m ic s o f A o n ly ? C a n w e d e s c r ib e it s e v o lu t io n e v e n if w e d o n ’t h a v e f u ll k n o w le d g e o f B ? ( t h e b a t h ) W e a s s u m e t h a t t h e s t a t e o f t h e b ip a r t it e s y s t e m u n d e r g o e s u n it a r y e v o lu t io n : h o w d o w e d e s c r ib e t h e e v o lu t io n o f A a lo n e ?
8 . 1 Co m b i n e d e v o l u t i o n o f sy s t e m a n d b a t h
W e w ill fi r s t s t a r t in t r o d u c in g t h e e v o lu t io n o f a n o p e n q u a n t u m s y s t e m b y c o n s id e r in g it a s a p a r t o f a la r g e r ( c lo s e d ) s y s t e m u n d e r g o in g t h e u s u a l u n it a r y e v o lu t io n . T h e t o t a l H ilb e r t s p a c e is t h u s H = H S ⊗ H B a n d w e a s s u m e t h e in it ia l s t a t e is r e p r e s e n t e d b y t h e s e p a r a b le d e n s it y m a t r ix ρ = ρ S ⊗ | 0 ) ( 0 | B 2 3 . T h e e v o lu t io n o f t h e t o t a l s y s t e m is
t h e n
ρ ( t ) = U S B ( ρ S ⊗ | 0 ) ( 0 | B ) U S † B
L L
If w e a r e o n ly in t e r e s t e d in t h e e v o lu t io n o f t h e s y s t e m S w e c a n a t t h is p o in t p e r f o r m a p a r t ia l t r a c e o n B
ρ S ( t ) = T r B { ρ ( t ) } = ( k | U S B ( ρ S ⊗ | 0 ) ( 0 | B ) U S † B | k ) = ( k | U S B | 0 ) ρ S ( 0 ) ( 0 | U S † B | k )
k k
k k
{ | ) } H ( | | ) { | ) ( | }
( | | ) | ) | ) H { | ) ( | } ( | | )
w h e r e k is a n o r t h o n o r m a l b a s is f o r B . A s t h e r e s u lt o f k U S B 0 = T r B 0 k U S B w e o b t a in a n o p e r a t o r M k t h a t a c t s o n ly o n t h e S H ilb e r t s p a c e . F o r e x a m p le , in a m a t r ix r e p r e s e n t a t io n t h e e le m e n t s o f M k a r e s im p ly M i , j = i M k j ( w it h i , j d e fi n e d o n S ) ; t h a t is , w e h a v e M i , j = T r j , 0 i , k U S B = i , k U S B j , 0 .
L
N o w w e c a n w r it e t h e e v o lu t io n o f t h e s y s t e m o n ly d e n s it y m a t r ix a s
ρ S ( t ) = M ( ρ S ( t ) ) = M k ρ S ( 0 ) M k †
k
2 3 H e r e w e o n l y a s s u m e t h a t t h e s y s t e m B i s i n a p u r e s t a t e t h a t w e i n d i c a t e a s | 0 ) , w e a r e n o t a s s u mi n g t h a t B i s a T LS .
L
S in c e t h e p r o p a g a t o r U S B is u n it a r y , w e h a v e t h a t
M k † M k = 1 1 S
k
? Q u e s t i o n : P r o v e t h e a b o v e .
k k k S B S B k
I n s e r t i n g t h e d e fi n i t i o n f o r M k w e h a v e
M † M k = L ( 0 | U † | k ) ( k | U S B | 0 ) = ( 0 | U †
( L | k ) ( k | ) U S B | 0 ) = 1 1 S .
L
T h e p r o p e r t ie s o f t h e s y s t e m d e n s it y m a t r ix a r e p r e s e r v e d b y t h e m a p :
L k M k ρ S M k † } = T r { ρ S L k M k M k † } = T r { ρ S 1 1 } )
1 . ρ S ( t ) is h e r m it ia n : ρ A ( t ) † = ( L { k M k ρ S ( 0 ) M k † ) † = L k M k ρ S ( 0 ) † M k † = ρ S ( t ) .
2 . ρ S ( t ) h a s u n it t r a c e . ( s in c e T r
3 . ρ S ( t ) is p o s it iv e .
In t h e s p e c ia l c a s e w h e r e t h e r e is o n ly o n e t e r m in t h e s u m , w e r e v e r t t o t h e u n it a r y e v o lu t io n o f t h e d e n s it y m a t r ix . In t h a t c a s e , a p u r e s t a t e , f o r e x a m p le , w o u ld r e m a in p u r e . If t h a t is n o t t h e c a s e , t h a t is , t h e e v o lu t io n is n o t u n it a r y , it m e a n s t h a t in t h e c o u r s e o f t h e e v o lu t io n t h e s y s t e m S a n d b a t h B b e c a m e e n t a n g l e d , s o t h a t ρ A is in a m ix e d s t a t e a f t e r p a r t ia l t r a c e . B e c a u s e o f t h e lo s s o f u n it a r it y , s u p e r o p e r a t o r s a r e in g e n e r a l n o t in v e r t ib le a n d t h u s t h e r e is a s p e c ifi c a r r o w o f t im e .
A . A n c i l l a r y B a th
In m a n y c a s e s it is n o t p o s s ib le t o f u lly c a lc u la t e t h e e v o lu t io n o f t h e t o t a l s y s t e m ( S + B ) a s e it h e r it is t o o la r g e o r w e h a v e im p e r f e c t k n o w le d g e o f t h e b a t h . H o w e v e r , if w e h a v e a d e s c r ip t io n o f t h e s y s t e m d y n a m ic s in t e r m s o f t h e o p e r a t o r s u m , it is p o s s ib le t o a lw a y s a u g m e n t t h e s y s t e m a n d fi n d a la r g e r , c o m p o s it e s y s t e m t h a t e v o lv e s u n it a r ily a n d y ie ld s t h e o p e r a t o r s u m u p o n p a r t ia l t r a c e . T h e a n c illa r y s y s t e m m ig h t h o w e v e r n o t h a v e a ll t h e c h a r a c t e r is t ic o f t h e ( u n k n o w n ) p h y s ic a l b a t h . W h a t w e a r e lo o k in g f o r is in f a c t a m in im a l d e s c r ip t io n f o r t h e b a t h .
L
{ | ) } | ) H
H
W e c h o o s e a s a n c illa r y H ilb e r t s p a c e B a s p a c e o f d im e n s io n s a t le a s t e q u a l t o t h e n u m b e r o f t e r m s in t h e o p e r a t o r s u m . T h is s p a c e w ill h a v e t h e n a s e t o f o r t h o n o r m a l v e c t o r s k , a n d w e c a n d e fi n e a n o r m a liz e d s t a t e 0 B o n B . T h e n t h e u n it a r y e v o lu t io n o p e r a t o r o f t h e c o m b in e d s y s t e m is d e fi n e d b y im p o s in g t h e r e la t io n s h ip :
U S B ( | ψ ) S ⊗ | 0 ) B ) = ( M k ⊗ 1 1 ) ( | ψ ) S ⊗ | k ) B , ) ∀ | ψ ) S ∈ H S
k
L
T h is e n s u r e s t h a t t h e e v o lu t io n o f t h e r e d u c e d s y s t e m is g iv e n b y t h e K r a u s m a p . T h e t o t a l s y s t e m e v o lu t io n is :
U S B ( ρ S ⊗ | 0 ) ( 0 | B ) U S † B = ( M k ⊗ 1 1 ) | ψ S , k ) ( ψ S , h | ( M h † ⊗ 1 1 )
k , h
a n d u p o n t a k in g t h e p a r t ia l t r a c e :
L L
ρ S ( t ) = T r B { ρ ( t ) } = ( j | ( M k ⊗ 1 1 ) | ψ S , k ) ( ψ S , h | ( M h † ⊗ 1 1 ) | j )
L
j k , h
= ( j | k ) ( h | j ) ( M k † | ψ ) ( ψ | M h )
L
j
ρ S ( t ) = M j † | ψ ) ( ψ | M j
j
A lt h o u g h t h is r e la t io n s h ip d o e s n ’t f u lly d e fi n e t h e o p e r a t o r o n t h e f u ll H ilb e r t s p a c e , w e c a n e x t e n d t h e o p e r a t o r a s d e s ir e d . In p a r t ic u la r w e w a n t it t o b e u n it a r y ( a n d t h is im p o s e s a d d e d c o n s t r a in t s ) . A s t h e o p e r a t o r U S B a s d e fi n e d a b o v e p r e s e r v e s t h e in n e r p r o d u c t o n t h e f u ll H ilb e r t s p a c e , a u n it a r y e x t e n s io n o f it t o t h e f u ll s p a c e d o e s in d e e d e x is t s . F u r t h e r m o r e , w e c a n c h e c k t h a t u p o n t a k in g a p a r t ia l t r a c e o n B w e r e t r ie v e t h e o p e r a t o r s u m a s d e s ir e d , f o r a n in it ia l p u r e s t a t e o n S . B u t a n y d e n s it y m a t r ix c a n b e e x p r e s s e d a s a n e n s e m b le o f p u r e s t a t e s , h e n c e t h is p r o p e r t y is t r u e f o r a n y g e n e r a l s t a t e o n S .
B . N o n - u n i q u e n e s s o f th e s u m r e p r e s e n ta ti o n
T h e o p e r a t o r s u m is o f c o u r s e n o t u n iq u e , s in c e t h e c h o ic e o f t h e s e t { | k ) } w a s q u it e a r b it r a r y a n d n o t u n iq u e . If w e h a d c h o s e n a n o t h e r s e t { | h ) } w e w o u ld h a v e a r r iv e d t o a d iff e r e n t s u m
L
ρ S ( t ) = N h ρ S ( 0 ) N h †
h
? Q u e s t i o n : W h a t i s t h e r e l a t i o n s h i p b e t w e e n t h e o p e r a t o r s M a n d N ?
L
T h e y a r e r e l a t e d b y t h e s i mp l e u n i t a r y t r a n s f o r ma t i o n t h a t c o n n e c t s t h e t w o s e t s o f o r t h o n o r m a l v e c t o r s N h = U h k M k w i t h
| h ) = k U h k | k ) .
8 . 2 S u p e r o p e r a t o r s
M
M
W e w a n t t o d e s c r ib e t h e q u a n t u m e v o lu t io n o f s y s t e m s in t h e m o s t g e n e r a l c a s e , w h e n t h e s y s t e m e v o lv e s n o n - u n it a r ily d u e t o t h e p r e s e n c e o f a n e n v ir o n m e n t 2 4 . A s w e h a v e s e e n , t h e s t a t e s n e e d t o b e d e s c r ib e d b y d e n s it y o p e r a t o r s . T h e r e f o r e , t h e e v o lu t io n is t o b e r e p r e s e n t e d b y a m a p c o n n e c t in g t h e in it ia l d e n s it y m a t r ix t o t h e e v o lv e d o n e ρ ( t ) = [ ρ ( 0 ) ]. T h e m o s t g e n e r a l c h a r a c t e r is t ic s o f t h is m a p w ill b e d e t e r m in e d b y t h e f a c t t h a t t h e p r o p e r t ie s o f t h e d e n s it y m a t r ix s h o u ld b e in g e n e r a l m a in t a in e d ( s u c h a s u n it t r a c e ) . A s t h e m a p is a n o p e r a t o r a c t in g o n o p e r a t o r s , it is c a lle d a s u p e r o p e r a t o r .
M o s t g e n e r a lly , w e c a n d e fi n e a q u a n t u m o p e r a t o r d e s c r ib in g t h e t im e e v o lu t io n la w f o r d e n s it y m a t r ic e s a s a m a p
M : ρ → ρ ′ w it h t h e f o llo w in g p r o p e r t ie s
1 . L in e a r
2 . T r a c e p r e s e r v in g
3 . H e r m it ic it y p r e s e r v in g
4 . P o s it iv e
4 ’ ( C o m p le t e ly p o s it iv e )
A . L i n e a r i t y
—
—
{ }
M
A lt h o u g h a n o n - lin e a r m a p c o u ld a ls o a lw a y s m a p a d e n s it y m a t r ix t o a n o t h e r d e n s it y m a t r ix , if w e im p o s e lin e a r it y w e a r r iv e a t r e s u lt s t h a t a r e m o r e p h y s ic a l. S p e c ifi c a lly , t h e lin e a r it y p r o p e r t y r e t a in s t h e e n s e m b le in t e r p r e t a t io n o f t h e d e n s it y m a t r ix . W h a t w e m e a n is t h e f o llo w in g . S u p p o s e w e c a n w r it e a d e n s it y o p e r a t o r a s a lin e a r s u p e r p o s it io n o f t w o d e n s it ie s , ρ = a ρ 1 + ( 1 a ) ρ 2 . T h e m e a n in g o f t h is e x p r e s s io n is t h a t w it h p r o b a b ilit y a w e h a v e a s y s t e m d e s c r ib e d b y ρ 1 a n d w it h p r o b a b ilit y 1 a b y ρ 2 . If t h e m a p d e s c r ib in g t h e t im e e v o lu t io n la w is lin e a r , t h is p r o b a b ilis t ic in t e r p r e t a t io n is v a lid a ls o f o r t h e e v o lv e d s t a t e . A s s u m e n o w t h a t t h e m a p is n o t lin e a r , f o r e x a m p le it d e p e n d s o n t h e t r a c e o f t h e d e n s it y m a t r ix : ( ρ ) = e i A T r { ρ M } ρ e − i A T r { ρ M } , w h e r e M is a n o p e r a t o r in t h e H ilb e r t s p a c e o f ρ a n d A a n H e r m it ia n o p e r a t o r . W e n o w c o n s id e r a d e n s it y o p e r a t o r ρ 1 s u c h t h a t T r ρ 1 M = 0 . W e a s s u m e t h a t w e d o n o t k n o w e x a c t ly h o w w e p r e p a r e d t h e s y s t e m , b u t w it h 5 0 % p r o b a b ilit y is in ρ 1 . A s s u m e t h e n t h e d e n s it y
m a t r ix ρ = 1 ( ρ 1 + ρ ⊥ ) , s u c h t h a t T r { ρ ⊥ M } = 0 . T h e n , M ( ρ ) = ρ a s t h e t r a c e s a r e z e r o . If w e n o w in s t e a d c o n s id e r
2
2
t h e in it ia l d e n s it y m a t r ix ρ = 1 ( ρ 1 + ρ ) ( t h a t is , s t ill a 5 0 % p r o b a b ilit y o f b e in g in ρ 1 ) , w h e r e T r
{ M ρ }
> 0 w e
o b t a in a n e v o lu t io n f o r ρ 1 . T h a t m e a n s , t h a t in t h e t w o s c e n a r io s , t h e s y s t e m b e h a v e s d iff e r e n t ly , e v e n if w e h a d
p r e p a r e d it in t h e s t a t e ρ 1 ( r e m e m b e r t h e p r o b a b ilis t ic in t e r p r e t a t io n ) , s o t h a t t h e e v o lu t io n o f a p o t e n t ia l s t a t e o f a s y s t e m ρ 1 d e p e n d s o n a n o t h e r p o t e n t ia l s t a t e ( ρ ⊥ o r ρ ) , e v e n if t h is s e c o n d s t a t e n e v e r o c c u r r e d .
2 4 T h i s p r e s e n t a t i o n i n t h i s s e c t i o n a n d t h e f o l l o w i n g e x a mp l e s a r e t a k e n f r o m J . P r e s k i l l ’ s n o t e s a t
h t t p : / / w w w . t h e o r y . c a l t e c h . e d u / p e o p l e / p r e s k i l l / p h 2 2 9 /
B . T h e s u p e r o p e r a to r p r e s e r v e s tr a c e a n d h e r m i ti c i t y
M { }
≤ { } ≤ { }
S in c e t h e d e n s it y m a t r ix t r a c e h a s t h e p r o p e r t y t o d e s c r ib e t h e s u m o f t h e p r o b a b ilit ie s o f a ll p o s s ib le s t a t e s in t h e e n s e m b le , it is im p o r t a n t t h a t t h e t r a c e b e p r e s e r v e d . A n e x c e p t io n c a n b e m a d e f o r o p e r a t o r s t h a t d e s c r ib e m e a s u r e m e n t ( a n d n o t t im e e v o lu t io n ) . In t h a t c a s e 0 T r ρ 1 . In t h is c a s e , T r ρ r e p r e s e n t t h e p r o b a b ilit y t h a t t h e m e a s u r e m e n t o u t c o m e d e s c r ib e d b y t h e m a p h a s o c c u r r e d a n d t h e n o r m a liz e d fi n a l s t a t e is ρ / T r ρ . A s m o r e t h a n o n e o u t c o m e o f t h e m e a s u r e m e n t is p o s s ib le , t h e p r o b a b ilit y o f o b t a in in g ρ m ig h t b e le s s t h a n o n e .
T h e s u p e r o p e r a t o r p r e s e r v e s t h e h e r m it ic it y o f t h e d e n s it y m a t r ix : [ M ( ρ ) ] † = M ( ρ ) if ρ † = ρ
C . P o s i ti v i t y a n d c o m p l e te p o s i ti v i t y
M
M M ⊗
T h e p r o p e r t y o f p o s it iv it y m e a n s t h a t t h e m a p is s u c h t h a t ( ρ ) is n o n - n e g a t iv e if ρ is . A lt h o u g h t h is c o n d it io n is e n o u g h t o o b t a in a v a lid d e n s it y m a t r ix , it le a d s t o a c o n t r a d ic t io n w h e n w e c o n s id e r c o m p o s it e s y s t e m s . L e t ’s t a k e a v a lid m a p 1 o n s y s t e m 1 . T h e n , if w e c o n s id e r a b ip a r t it e s y s t e m a n d w e a p p ly t h e m a p 1 1 1 w e w o u ld lik e t o s t ill o b t a in a d e n s it y m a t r ix o n t h e c o m p o s it e s y s t e m . U n f o r t u n a t e ly , if t h e m a p is s im p ly p o s it iv e , t h is is n o t a lw a y s t h e c a s e . T h u s , w e r e q u ir e it t o b e c o m p l e t e l y p o s it iv e . A m a p is c o m p le t e ly p o s it iv e if M 1 ⊗ 1 1 2 is p o s it iv e f o r a n y e x t e n s io n H 2 o f t h e H ilb e r t s p a c e H 1 .
8 . 3 Th e Kr a u s R e p r e se n t a t i o n Th e o r e m
W e h a v e s e e n in t h e p r e c e d in g s e c t io n s t w o d iff e r e n t w a y s o f d e s c r ib in g t h e e v o lu t io n o f a n o p e n s y s t e m .
T h e fi r s t d e s c r ip t io n s t a r t e d f r o m t h e e v o lu t io n o f a c o m p o s it e s y s t e m ( in c lu d in g t h e s y s t e m o f in t e r e s t a n d a b a t h ) a n d b y t r a c in g o v e r a r r iv e d a t a d e s c r ip t io n o f t h e o p e n e v o lu t io n v ia t h e o p e r a t o r s u m .
T h e s e c o n d d e s c r ip t io n w a s in s t e a d q u it e a b s t r a c t , a n d o n ly d e fi n e d t h e p r o p e r t ie s o f t h e lin e a r m a p d e s c r ib in g t h e e v o lu t io n in o r d e r t o a r r iv e a t a n a c c e p t a b le ( p h y s ic a l) e v o lv e d s t a t e ( t h a t s t ill p o s s e s s t h e c h a r a c t e r is t ic s o f a d e n s it y o p e r a t o r ) . T h e K r a u s r e p r e s e n t a t io n t h e o r e m r e c o n c ile s t h e s e t w o d e s c r ip t io n , b y s t a t in g t h a t t h e y a r e e q u iv a le n t .
S
→
• T h e o r e m: A n y o p e r a t o r ρ S ( ρ ) in a s p a c e o f d im e n s io n s N 2 t h a t o b e y s t h e p r o p e r t ie s 1 - 3 ,4 ’ ( L in e a r it y , T r a c e p r e s e r v a t io n , H e r m it ic it y p r e s e r v a t io n , c o m p le t e p o s it iv it y ) c a n b e w r it t e n in t h e f o r m :
L L
K K
S ( ρ ) = M k ρ M k † , w it h M k † M k = 1 1
k = 1 k = 1
S
≤
w h e r e K N 2 is t h e K r a u s n u m b e r ( w it h N S t h e d im e n s io n o f t h e s y s t e m ) . A s s e e n a b o v e , t h e K r a u s r e p r e s e n t a t io n is n o t u n iq u e 2 5 .
W e c o n s id e r t h r e e im p o r t a n t e x a m p le s o f o p e n q u a n t u m s y s t e m e v o lu t io n t h a t c a n b e d e s c r ib e d b y t h e K r a u s o p e r a t o r s . T o s im p lif y t h e d e s c r ip t io n w e c o n s id e r j u s t a T L S t h a t is c o u p le d t o a b a t h .
8 . 3 . 1 A m p l i t u d e - d a m p i n g
T h e a m p lit u d e - d a m p in g c h a n n e l is a s c h e m a t ic m o d e l o f t h e d e c a y o f a n e x c it e d s t a t e o f a ( t w o - le v e l) a t o m d u e t o s p o n t a n e o u s e m is s io n o f a p h o t o n . B y d e t e c t in g t h e e m it t e d p h o t o n ( “ o b s e r v in g t h e e n v ir o n m e n t ” ) w e c a n g e t in f o r m a t io n a b o u t t h e in it ia l p r e p a r a t io n o f t h e a t o m .
| )
| ) | )
| ) | )
W e d e n o t e t h e a t o m ic g r o u n d s t a t e b y 0 A a n d t h e e x c it e d s t a t e o f in t e r e s t b y 1 A . T h e “ e n v ir o n m e n t ” is t h e e le c t r o m a g n e t ic fi e ld , a s s u m e d in it ia lly t o b e in it s v a c u u m s t a t e 0 E . A f t e r w e w a it a w h ile , t h e r e is a p r o b a b ilit y p t h a t t h e e x c it e d s t a t e h a s d e c a y e d t o t h e g r o u n d s t a t e a n d a p h o t o n h a s b e e n e m it t e d , s o t h a t t h e e n v ir o n m e n t h a s m a d e a t r a n s it io n f r o m t h e s t a t e 0 E ( “ n o p h o t o n ” ) t o t h e s t a t e 1 E ( “ o n e p h o t o n ” ) . T h is e v o lu t io n is d e s c r ib e d b y a u n it a r y t r a n s f o r m a t io n t h a t a c t s o n a t o m a n d e n v ir o n m e n t a c c o r d in g t o
| 0 ) S | 0 ) E → | 0 ) S | 0 ) E
�
| 1 ) S | 0 ) E → 1 − p | 1 ) S | 0 ) E + √ p | 0 ) S | 1 ) E
( O f c o u r s e , if t h e a t o m s t a r t s o u t in it s g r o u n d s t a t e , a n d t h e e n v ir o n m e n t is a t z e r o t e m p e r a t u r e , t h e n t h e r e is n o t r a n s it io n .)
B y e v a lu a t in g t h e p a r t ia l t r a c e o v e r t h e e n v ir o n m e n t , w e fi n d t h e K r a u s o p e r a t o r s
0 S E 1 S E
M = ( 0 | U | 0 ) = ( 1 √ 0 ) , M = ( 1 | U | 0 ) = ( 0 √ p )
0 1 − p 0 0
| ) | )
T h e o p e r a t o r M 1 in d u c e s a “ q u a n t u m j u m p ” , t h e d e c a y f r o m 1 A t o 0 A , a n d M 0 d e s c r ib e s h o w t h e s t a t e e v o lv e s if n o j u m p o c c u r s . T h e d e n s it y m a t r ix e v o lv e s a s
S ( ρ ) = M 0 ρ M 0 † + M 1 ρ M 1 † =
1 − p ρ 1 0 ( 1 − p ) ρ 1 1 0 0
= ( √ ρ 0 0 √ 1 − p ρ 0 1 ) + ( p ρ 1 1 0 )
= ( ρ √ 0 0 + p ρ 1 1
√ 1 − p ρ 0 1 ) .
1 − p ρ 1 0 ( 1 − p ) ρ 1 1
2 5 T h e p r o o f c a n b e f o u n d i n P r o f . P r e s k i l l o n l i n e n o t e s .
→ −
— ≈ → − ≈ → ∞
If w e a p p ly t h e c h a n n e l n t im e s in s u c c e s s io n , t h e ρ 1 1 m a t r ix e le m e n t d e c a y s a s ρ 1 1 ρ 1 1 ( 1 p ) n s o if t h e p r o b a b ilit y o f a t r a n s it io n in t im e in t e r v a l δ t is Γ δ t , t h e n t h e p r o b a b ilit y t h a t t h e e x c it e d s t a t e p e r s is t s f o r t im e t is ( 1 Γ δ t ) t /δ t e − Γ t , t h e e x p e c t e d e x p o n e n t ia l d e c a y la w . A ls o w e h a v e ρ 1 2 ρ 1 2 ( 1 p ) n / 2 ρ 1 2 e − Γ t / 2 . A s t , t h e d e c a y p r o b a b ilit y a p p r o a c h e s u n it y , s o
( )
S ( ρ ) = ρ 0 0 + ρ 1 1 0
0 0
T h e a t o m a lw a y s w in d s u p in it s g r o u n d s t a t e . T h is e x a m p le s h o w s t h a t it is s o m e t im e s p o s s ib le f o r a s u p e r o p e r a t o r t o t a k e a m ix e d in it ia l s t a t e t o a p u r e s t a t e .
In t h e c a s e o f t h e d e c a y o f a n e x c it e d a t o m ic s t a t e v ia p h o t o n e m is s io n , it m a y n o t b e im p r a c t ic a l t o m o n it o r t h e e n v ir o n m e n t w it h a p h o t o n d e t e c t o r . T h e m e a s u r e m e n t o f t h e e n v ir o n m e n t p r e p a r e s a p u r e s t a t e o f t h e a t o m , a n d s o in e ff e c t p r e v e n t s t h e a t o m f r o m d e c o h e r in g . R e t u r n in g t o t h e u n it a r y r e p r e s e n t a t io n o f t h e a m p lit u d e - d a m p in g c h a n n e l, w e s e e t h a t a c o h e r e n t s u p e r p o s it io n o f t h e a t o m ic g r o u n d a n d e x c it e d s t a t e s e v o lv e s a s
( a | 0 ) S + b | 1 ) S ) | 0 ) E → � a | 0 ) S + b √ 1 − p | 1 ) S � | 0 ) E + √ p | 0 ) A | 1 ) E
| )
| )
| ) | )
If w e d e t e c t t h e p h o t o n ( a n d s o p r o j e c t o u t t h e s t a t e 1 E o f t h e e n v ir o n m e n t ) , t h e n w e h a v e p r e p a r e d t h e s t a t e 0 A o f t h e a t o m . In f a c t , w e h a v e p r e p a r e d a s t a t e in w h ic h w e k n o w w it h c e r t a in t y t h a t t h e in it ia l a t o m ic s t a t e w a s t h e e x c it e d s t a t e 1 A a s t h e g r o u n d s t a t e c o u ld n o t h a v e d e c a y e d . O n t h e o t h e r h a n d , if w e d e t e c t n o p h o t o n , a n d o u r p h o t o n d e t e c t o r h a s p e r f e c t e ffi c ie n c y , t h e n w e h a v e p r o j e c t e d o u t t h e s t a t e 0 E o f t h e e n v ir o n m e n t , a n d s o h a v e
p r e p a r e d t h e a t o m ic s t a t e
a | 0 ) S + b √ 1 − p | 1 ) S
2
( o r m o r e p r e c is e ly , if w e n o r m a liz e it : ( a | 0 ) S
+ b √ √ 1 − p | 1 ) S ) / √ 1 − p b 2 ) . T h e n p ( 0 ) = | a | 2 →
| a |
1 − p | b | 2
> | a | 2 .
T h e a t o m ic s t a t e h a s e v o lv e d d u e t o o u r f a ilu r e t o d e t e c t a p h o t o n , it h a s b e c o m e m o r e lik e ly t h a t t h e in it ia l a t o m ic
s t a t e w a s t h e g r o u n d s t a t e !
8 . 3 . 2 P h a s e - d a m p i n g
P h a s e d a m p in g d e s c r ib e s a p r o c e s s w h e r e t h e s y s t e m in t e r a c t s w it h a la r g e e n v ir o n m e n t c o m p o s e d o f m a n y s m a ll s u b s y s t e m s . T h e in t e r a c t io n o f t h e s y s t e m w it h e a c h o f t h e e n v ir o n m e n t s u b s y s t e m s is w e a k ( c o m p a r e d t o t h e s y s t e m e n e r g y , b u t s t r o n g c o m p a r e d t o t h e s u b s y s t e m e n e r g y ) . T h e r e f o r e t h e s y s t e m is u n c h a n g e d , w h ile t h e e n v ir o n m e n t s u b s y s t e m is c h a n g e d . S in c e t h e r e w ill b e m a n y o f t h e s e in t e r a c t io n s w it h t h e e n v ir o n m e n t s u b s y s t e m , t h e ir c o m b in e d a c t io n d o e s h a v e a n e ff e c t o n t h e s y s t e m , h o w e v e r it w ill n o t b e e n o u g h t o c h a n g e it s e n e r g y .
A n e x a m p le is t h e in t e r a c t io n o f a d u s t p a r t ic le w it h p h o t o n s . C o llis io n o f t h e p a r t ic le w it h o n e p h o t o n is n o t g o in g t o c h a n g e t h e p a r t ic le s t a t e . H o w e v e r , if t h e p a r t ic le w a s in t h e g r o u n d o r e x c it e d s t a t e , t h e p h o t o n w ill a c q u ir e m o r e o r le s s e n e r g y in t h e c o llis io n , t h u s b e in g e x c it e d t o it s fi r s t o r s e c o n d e x c it e d s t a t e . W e n o w f o r m a liz e t h is m o d e l. W h e n lo o k in g a t t h e u n it a r y e v o lu t io n o f t h is p r o c e s s , o n ly t h e e n v ir o n m e n t c h a n g e s :
| 0 ) S | 0 ) E → √ 1 − p | 0 ) S | 0 ) E + √ p | 0 ) S | 1 ) E = | 0 ) S ( √ 1 − p | 0 ) E + √ p | 1 ) E )
| 1 ) S | 0 ) E → √ 1 − p | 1 ) S | 0 ) E + √ p | 1 ) S | 2 ) E = | 1 ) S ( √ 1 − p | 0 ) E + √ p | 2 ) E )
√ √
T h u s a p o s s ib le u n it a r y is
1 − p √ p 0 0 0 0
√
—
p 1 p 0 0 0 0
0 0 1
0 0 0
U = 0 0 0 √ 1 − p 0 √ p
0 0 0
√ p 0 1 − p
0 0 0 0 1 √ 0
T h e K r a u s o p e r a t o r a r e f o u n d b y o p e r a t in g t h e p a r t ia l t r a c e o f t h e o p e r a t o r a b o v e :
M 0 = ( 0 | U | 0 ) = √ 1 − p 1 1 M 1 = ( 1 | U | 0 ) = √ p | 0 ) ( 0 | M 2 = ( 2 | U | 0 ) = √ p | 1 ) ( 1 |
T h e s t a t e e v o lu t io n is t h e n
In m a t r ix f o r m :
3
L
S ( ρ ) = M k ρ M k † = ( 1 − p ) ρ + p | 0 ) ( 0 | ρ | 0 ) ( 0 | + p | 1 ) ( 1 | ρ | 1 ) ( 1 |
k = 1
( )
S ( ρ ) = ρ 0 0 ( 1 − p ) ρ 0 1
( 1 − p ) ρ 1 0 ρ 1 1
→ − −
C o n s id e r in g t h e B lo c h v e c t o r : [ n x , n y , n z ] [( 1 p ) n x , ( 1 p ) n y , n z ] ( t h a t is , t h e t r a n s v e r s c o m p o n e n t a r e r e d u c e d . F o r p = 1 t h e s t a t e b e c o m e s d ia g o n a l) . A s s u m e p = p ( ∆ t ) = Γ ∆ t is t h e p r o b a b ilit y o f o n e s u c h s c a t t e r e v e n t s d u r in g t h e t im e ∆ t . T h e n if w e h a v e n s u c h e v e n t s in a t im e t = n ∆ t t h e o ff - d ia g o n a l t e r m s b e c o m e ∝ ( 1 − p ) n =
)
( 1 − Γ ∆ t ) t / ∆ t ≈ e − Γ t :
S ( ρ , t ) = (
ρ 0 0 e − Γ t ρ 1 0
e − Γ t ρ 0 1 ρ 1 1
C o n s id e r f o r e x a m p le a n in it ia l p u r e s t a t e α | 0 ) + β | 1 ) . A t lo n g t im e s , t h is s t a t e r e d u c e s t o :
S ( ρ , t ) =
| β | 2
− →
0 | β | 2
( | α | 2 e − Γ t α β ∗ ) t → ∞ ( | α | 2 0 )
e − Γ t α ∗ β
t h u s a n y p h a s e c o h e r e n c e is lo s t a n d t h e s t a t e r e d u c e s t o a c la s s ic a l, in c o h e r e n t s u p e r p o s it io n o f p o p u la t io n s . B e c a u s e in t h is p r o c e s s p h a s e c o h e r e n c e is lo s t ( b u t t h e e n e r g y / p o p u la t io n is c o n s e r v e d ) t h e p r o c e s s is c a lle d d e p h a s in g a n d t h e t im e c o n s t a n t 1 / Γ is u s u a lly d e n o t e d b y T 2 . T h e n w e h a v e a r e p r e s e n t a t io n o f t h e s u p e r o p e r a t o r , b y e x p r e s s in g ρ a s a lin e a r v e c t o r : S ( ρ , t ) = S ( t ) ρ , w h e r e S = d i a g ( [1 , e − Γ t , e − Γ t , 1 ]) .
8 . 3 . 3 D e p o l a r i z i n g p r o c e s s
{ | ) | ) }
T h e d e p o la r iz in g c h a n n e l is a m o d e l o f a d e c o h e r in g q u b it t h a t h a s p a r t ic u la r ly n ic e s y m m e t r y p r o p e r t ie s . W e c a n d e s c r ib e it b y s a y in g t h a t , w it h p r o b a b ilit y 1 - p t h e q u b it r e m a in s in t a c t , w h ile w it h p r o b a b ilit y p a n “ e r r o r ” o c c u r s . T h e e r r o r c a n b e o f a n y o n e o f t h r e e t y p e s , w h e r e e a c h t y p e o f e r r o r is e q u a lly lik e ly . If 0 1 is a n o r t h o n o r m a l b a s is f o r t h e q u b it , t h e t h r e e t y p e s o f e r r o r s c a n b e c h a r a c t e r iz e d a s :
1 . B it - fl ip e r r o r : | ψ ) → σ x | ψ ) o r | 0 ) → | 1 ) & | 1 ) → | 0 ) .
2 . P h a s e - fl ip e r r o r : | ψ ) → σ z | ψ ) o r | 0 ) → | 0 ) & | 1 ) → − | 1 ) .
3 . B o t h e r r o r s : | ψ ) → σ y | ψ ) o r | 0 ) → i | 1 ) & | 1 ) → − i | 0 ) .
| ) | ) | ) | )
If a n e r r o r o c c u r s , t h e n ψ e v o lv e s t o a n e n s e m b le o f t h e t h r e e s t a t e s σ x ψ , σ y ψ , σ z ψ .
H H ⊗ H H
T h e d e p o la r iz in g c h a n n e l c a n b e r e p r e s e n t e d b y a u n it a r y o p e r a t o r a c t in g o n S E = S E , w h e r e E h a s d im e n s io n 4 . T h e u n it a r y o p e r a t o r U S E a c t s a s
U S E | ψ ) S ⊗ | 0 ) E → √ 1 − p | ψ ) S ⊗ | 0 ) E +
3
x
S
E
y
S
E
z
S
E
+ � p [ σ | ψ ) ⊗ | 1 ) + σ | ψ ) ⊗ | 2 ) + σ | ψ ) ⊗ | 3 ) ]
{ | ) }
T h e e n v ir o n m e n t e v o lv e s t o o n e o f f o u r m u t u a lly o r t h o g o n a l s t a t e s t h a t “ k e e p a r e c o r d ” o f w h a t t r a n s p ir e d ; if w e c o u ld o n ly m e a s u r e t h e e n v ir o n m e n t in t h e b a s is µ , µ = 0 , 1 , 2 , 3 , w e w o u ld k n o w w h a t k in d o f e r r o r h a d o c c u r r e d ( a n d w e w o u ld b e a b le t o in t e r v e n e a n d r e v e r s e t h e e r r o r ) .
K r a us r e pr e s e n t a t i o n: T o o b t a in a n o p e r a t o r - s u m r e p r e s e n t a t io n o f t h e c h a n n e l, w e e v a lu a t e t h e p a r t ia l t r a c e o v e r t h e e n v ir o n m e n t in t h e { | µ ) E } b a s is . T h e n
x
2
3
y
3
3
z
M µ = ( µ | U S E | 0 ) E
0
1
3
M = √ 1 − p 1 1 , M
= � p σ , M
= � p σ , M
= � p σ
A g e n e r a l in it ia l d e n s it y m a t r ix ρ S o f t h e q u b it e v o lv e s a s
ρ → ρ ′ = ( 1 −
p
p ) ρ + 3 ( σ x ρ σ x + σ y ρ σ y + σ z ρ σ z )
2
2
It is a ls o in s t r u c t iv e t o s e e h o w t h e d e p o la r iz in g c h a n n e l a c t s o n t h e B lo c h s p h e r e . A n a r b it r a r y d e n s it y m a t r ix f o r a s in g le q u b it c a n b e w r it t e n a s ρ = 1 ( 1 1 + i n · i σ ) , w h e r e i n is t h e B lo c h v e c t o r ( w it h P = | i n | t h e p o la r iz a t io n o f t h e s p in ) . S u p p o s e w e r o t a t e o u r a x e s s o t h a t i n = i z a n d ρ = 1 ( 1 1 + P z σ z ) . T h e n s in c e σ z σ z σ z = σ z a n d σ x σ z σ x = σ y σ z σ y = − σ z ,
w e fi n d
ρ ′ = 1 − p + p 1 ( 1 1 − P σ ) + 2 p 1 ( 1 1 − P σ ) = 1 [ 1 1 + ( 1 − 4 p ) P σ ]
3 2
z
z
3 2
z
z
2 3
z
z
3
3
o r P z ′ = ( 1 − 4 p ) P z . F r o m t h e r o t a t io n a l s y m m e t r y , w e s e e t h a t P ′ = ( 1 − 4 p ) ir r e s p e c t iv e o f t h e d ir e c t io n in w h ic h P
p o in t s . H e n c e , t h e B lo c h s p h e r e c o n t r a c t s u n if o r m ly u n d e r t h e a c t io n o f t h e c h a n n e l; t h e s p in p o la r iz a t io n is r e d u c e d
3
—
3
b y t h e f a c t o r ( 1 4 p ) ( w h ic h is w h y w e c a ll it t h e d e p o la r iz in g p r o c e s s ) . T h is r e s u lt w a s t o b e e x p e c t e d in v ie w o f t h e o b s e r v a t io n a b o v e t h a t t h e s p in is t o t a lly “ r a n d o m iz e d ” w it h p r o b a b ilit y 4 p .
W h y d o w e s a y t h a t t h e s u p e r o p e r a t o r is n o t in v e r t ib le ? E v id e n t ly w e c a n r e v e r s e a u n if o r m c o n t r a c t io n o f t h e s p h e r e
≤
w it h a u n if o r m in fl a t io n . B u t t h e t r o u b le is t h a t t h e in fl a t io n o f t h e B lo c h s p h e r e is n o t a s u p e r o p e r a t o r , b e c a u s e it is n o t p o s it iv e . In fl a t io n w ill t a k e v a lu e s o f P 1 t o v a lu e s P > 1 , a n d s o w ill t a k e a d e n s it y o p e r a t o r t o a n o p e r a t o r w it h a n e g a t iv e e ig e n v a lu e . D e c o h e r e n c e c a n s h r in k t h e b a ll, b u t n o p h y s ic a l p r o c e s s c a n b lo w it u p a g a in ! A s u p e r o p e r a t o r r u n n in g b a c k w a r d s in t im e is n o t a s u p e r o p e r a t o r .
8 . 4 Th e M a s t e r E q u a t i o n
8 . 4 . 1 M a r k o v a p p r o x i m a t i o n
In t h e c a s e o f c o h e r e n t e v o lu t io n , w e fi n d it v e r y c o n v e n ie n t t o c h a r a c t e r iz e t h e d y n a m ic s o f a q u a n t u m s y s t e m w it h a H a m ilt o n ia n , w h ic h d e s c r ib e s t h e e v o lu t io n o v e r a n in fi n it e s im a l t im e in t e r v a l. T h e d y n a m ic s is t h e n d e s c r ib e d b y a d iff e r e n t ia l e q u a t io n , t h e S c h r ¨ o d in g e r e q u a t io n , a n d w e m a y c a lc u la t e t h e e v o lu t io n o v e r a fi n it e t im e in t e r v a l b y in t e g r a t in g t h e e q u a t io n , t h a t is , b y p ie c in g t o g e t h e r t h e e v o lu t io n o v e r m a n y in fi n it e s im a l in t e r v a ls . It is o f t e n p o s s ib le t o d e s c r ib e t h e ( n o t n e c e s s a r ily c o h e r e n t ) e v o lu t io n o f a d e n s it y m a t r ix , a t le a s t t o a g o o d a p p r o x im a t io n , b y a d iff e r e n t ia l e q u a t io n . T h is e q u a t io n , t h e m a s t e r e q u a t io n , w ill b e o u r n e x t t o p ic . In f a c t , it is n o t a t a ll o b v io u s t h a t t h e r e n e e d b e a d iff e r e n t ia l e q u a t io n t h a t d e s c r ib e s d e c o h e r e n c e . S u c h a d e s c r ip t io n w ill b e p o s s ib le o n ly if t h e e v o lu t io n o f t h e q u a n t u m s y s t e m is “ M a r k o v ia n ,” o r in o t h e r w o r d s , lo c a l in t im e . If t h e e v o lu t io n o f t h e d e n s it y o p e r a t o r ρ ( t ) is g o v e r n e d b y a ( fi r s t - o r d e r ) d iff e r e n t ia l e q u a t io n in t , t h e n t h a t m e a n s t h a t ρ ( t + d t ) is c o m p le t e ly d e t e r m in e d b y ρ ( t ) .
In g e n e r a l t h e d e n s it y o p e r a t o r ρ A ( t + d t ) c a n d e p e n d n o t o n ly o n ρ A ( t ) , b u t a ls o o n ρ A a t e a r lie r t im e s , b e c a u s e t h e e n v ir o n m e n t ( r e s e r v o ir ) r e t a in s a m e m o r y o f t h is in f o r m a t io n f o r a w h ile , a n d c a n t r a n s f e r it b a c k t o s y s t e m . A n o p e n s y s t e m ( w h e t h e r c la s s ic a l o r q u a n t u m ) is d is s ip a t iv e b e c a u s e in f o r m a t io n c a n fl o w f r o m t h e s y s t e m t o t h e r e s e r v o ir . B u t t h a t m e a n s t h a t in f o r m a t io n c a n a ls o fl o w b a c k f r o m r e s e r v o ir t o s y s t e m , r e s u lt in g in n o n - M a r k o v ia n
fl u c t u a t io n s o f t h e s y s t e m .
≪
≫
S t ill, in m a n y c o n t e x t s , a M a r k o v ia n d e s c r ip t io n is a v e r y g o o d a p p r o x im a t io n . T h e k e y id e a is t h a t t h e r e m a y b e a c le a n s e p a r a t io n b e t w e e n t h e t y p ic a l c o r r e la t io n t im e o f t h e fl u c t u a t io n s a n d t h e t im e s c a le o f t h e e v o lu t io n t h a t w e w a n t t o f o llo w . C r u d e ly s p e a k in g , w e m a y d e n o t e b y δ t E t h e t im e it t a k e s f o r t h e r e s e r v o ir t o “ f o r g e t ” in f o r m a t io n t h a t it a c q u ir e d f r o m t h e s y s t e m . A f t e r t im e δ t E w e c a n r e g a r d t h a t in f o r m a t io n a s f o r e v e r lo s t , a n d n e g le c t t h e p o s s ib ilit y t h a t t h e in f o r m a t io n m a y f e e d b a c k a g a in t o in fl u e n c e t h e s u b s e q u e n t e v o lu t io n o f t h e s y s t e m . O u r d e s c r ip t io n o f t h e e v o lu t io n o f t h e s y s t e m w ill in c o r p o r a t e “ c o a r s e - g r a in in g ” in t im e ; w e p e r c e iv e t h e d y n a m ic s t h r o u g h a fi lt e r t h a t s c r e e n s o u t t h e h ig h f r e q u e n c y c o m p o n e n t s o f t h e m o t io n , w it h ω 1 / δ t c o ar s e . A n a p p r o x im a t e ly M a r k o v ia n d e s c r ip t io n s h o u ld b e p o s s ib le , t h e n , if δ t E δ t c o ar s e ; w e c a n n e g le c t t h e m e m o r y o f t h e r e s e r v o ir , b e c a u s e w e a r e u n a b le t o r e s o lv e it s e ff e c t s . T h is “ M a r k o v ia n a p p r o x im a t io n ” w ill b e u s e f u l if t h e t im e s c a le o f t h e d y n a m ic s t h a t w e w a n t t o o b s e r v e is lo n g c o m p a r e d t o δ t c o ar s e , e .g ., if t h e d a m p in g t im e s c a le δ t d am p s a t is fi e s
δ t d am p ≫ δ t c o ar s e ≫ δ t E
8 . 4 . 2 L i n d b l a d e q u a t i o n
H
L L
— H
O u r g o a l is t o g e n e r a liz e t h e L io u v ille e q u a t io n ρ ˙ = i [ , ρ ] t o t h e c a s e o f M a r k o v ia n b u t n o n - u n it a r y e v o lu t io n , f o r w h ic h w e w ill h a v e ρ ˙ = [ ρ ]. T h e lin e a r o p e r a t o r , w h ic h g e n e r a t e s a fi n it e s u p e r o p e r a t o r in t h e s a m e s e n s e t h a t a H a m ilt o n ia n g e n e r a t e s u n it a r y t im e e v o lu t io n , w ill b e c a lle d t h e L in d b la d ia n .
L
W e c a n d e r iv e t h e L in d b la d e q u a t io n f r o m a n in fi n it e s im a l e v o lu t io n d e s c r ib e d b y t h e K r a u s s u m r e p r e s e n t a t io n , w it h t h e f o llo w in g s t e p s :
1 . F r o m t h e K r a u s s u m w e c a n w r it e t h e e v o lu t io n o f ρ f r o m t t o t + δ t a s : ρ ( t + δ t ) = k M k ( δ t ) ρ ( t ) M k † ( δ t ) .
→
2 . W e n o w t a k e t h e lim it o f in fi n it e s im a l t im e , δ t 0 . W e o n ly k e e p t e r m s u p t o fi r s t o r d e r in δ t , ρ ( t + δ t ) = ρ ( t ) + δ t δ ρ .
T h is im p lie s t h a t t h e K r a u s o p e r a t o r s h o u ld b e e x p a n d e d a s M k = M ( 0 ) + √ δ tM ( 1 ) + δ tM ( 2 ) + . . . .
T h e n t h e r e is o n e K r a u s o p e r a t o r s u c h t h a t M
= 1 1 + δ t ( − i H
k 2 k k
0 √ + K ) + O ( δ t ) w it h K h e r m it ia n ( s o t h a t ρ ( t + δ t )
L
is h e r m it ia n ) , w h ile a ll o t h e r s h a v e t h e f o r m : M k = δ tL k + O ( δ t ) , s o t h a t w e e n s u r e ρ ( t + δ t ) = ρ ( t ) + δ ρ δ t :
ρ ( t + δ t ) = M 0 ρ ( t ) M 0 † + M k ρ M k †
L
k > 0
= [ 1 1 + δ t ( − i H + K ) ] ρ [ 1 1 + δ t ( i H + K ) ] + δ t L k ρ L † k
L
k
= ρ − i δ t [ H , ρ ] + δ t ( K ρ + ρ K ) + δ t L k ρ L † k
k
3 . K a n d t h e o t h e r o p e r a t o r s L k a r e r e la t e d t o e a c h o t h e r , s in c e t h e y h a v e t o r e s p e c t t h e K r a u s s u m n o r m a liz a t io n c o n d it io n ,
2
k
k
K = − 1 L L † L .
k > 0
→
4 . F in a lly w e s u b s t it u t e K in t h e e q u a t io n a b o v e a n d t a k e t h e lim it δ 0 : ρ ( t + d t ) = ρ ( t ) + d t ρ ˙ . W e t h u s o b t a in t h e L in d b la d e q u a t io n
M
ρ ˙ ( t ) = L [ ρ ] = − i [ H , ρ ( t ) ] +
L k ρ ( t ) L † k − 2 L † k L k ρ ( t ) − 2 ρ ( t ) L † k L k
L ( 1 1 )
k = 1
H
L
T h e fi r s t t e r m in [ ρ ] is t h e u s u a l S c h r ¨ o d in g e r t e r m t h a t g e n e r a t e s u n it a r y e v o lu t io n ( t h u s w e id e n t if y t h e h e r m it ia n o p e r a t o r w it h t h e u s u a l H a m ilt o n ia n ) . T h e o t h e r t e r m s d e s c r ib e t h e p o s s ib le t r a n s it io n s t h a t t h e s y s t e m m a y u n d e r g o d u e t o in t e r a c t io n s w it h t h e r e s e r v o ir . T h e o p e r a t o r s L k a r e c a lle d L in d b la d o p e r a t o r s o r q u a n t u m j u m p
o p e r a t o r s . E a c h L k ρ L † k t e r m in d u c e s o n e o f t h e p o s s ib le q u a n t u m j u m p s , w h ile t h e − 1 L † k L k ρ − 1 ρ L † k L k t e r m s a r e
2 2
n e e d e d t o n o r m a liz e p r o p e r ly t h e c a s e in w h ic h n o j u m p s o c c u r .
| ) | )
If w e r e c a ll t h e c o n n e c t io n b e t w e e n t h e K r a u s r e p r e s e n t a t io n a n d t h e u n it a r y r e p r e s e n t a t io n o f a s u p e r o p e r a t o r , w e c la r if y t h e in t e r p r e t a t io n o f t h e m a s t e r e q u a t io n . W e m a y im a g in e t h a t w e a r e c o n t in u o u s ly m o n it o r in g t h e r e s e r v o ir , p r o j e c t in g it in e a c h in s t a n t o f t im e o n t o t h e | µ ) E b a s is . W it h p r o b a b ilit y 1 − O ( δ t ) , t h e r e s e r v o ir r e m a in s in t h e s t a t e 0 E , b u t w it h p r o b a b ilit y o f o r d e r δ t , t h e r e s e r v o ir m a k e s a q u a n t u m j u m p t o o n e o f t h e s t a t e s µ E . W h e n w e s a y
t h a t t h e r e s e r v o ir h a s “ f o r g o t t e n ” t h e in f o r m a t io n it a c q u ir e d f r o m t h e s y s t e m ( s o t h a t t h e M a r k o v ia n a p p r o x im a t io n
a p p lie s ) , w e m e a n t h a t t h e s e t r a n s it io n s o c c u r w it h p r o b a b ilit ie s t h a t in c r e a s e lin e a r ly w it h t im e . T h is is e q u a t io n is a ls o c a lle d t h e K o s s a k o w s k i- L in d b la d e q u a t io n 2 6 .
T h e L in d b la d e q u a t io n a b o v e is e x p r e s s e d in t h e S c h r ¨ o d in g e r p ic t u r e . It is p o s s ib le t o d e r iv e t h e H e is e n b e r g p ic t u r e L in d b la d e q u a t io n , w h ic h h a s t h e f o r m :
d x = i [ H , x ] + L L † x L − 1
L † L x + x L † L ,
d t
w h e r e x is t h e o b s e r v a b le u n d e r s t u d y .
k k 2 k k k k
k
A n o t h e r w a y t o e x p r e s s t h e L in d b la d e q u a t io n is f o r a ” v e c t o r iz e d ” d e n s it y m a t r ix : ρ ˙ = ( H + G ) ρ , w it h t h e g e n e r a t o r
G : M
m
m
2
m
m
2
m
m
G = L L ¯ ⊗ L − 1 1 1 ⊗ ( L † L ) − 1 ( L ¯ † L ¯ ) ⊗ 1 1
m = 0
— H ⊗ − ⊗ H
a n d t h e H a m ilt o n ia n p a r t w ill b e g iv e n b y H = i ( 1 1 1 1 ) . In t h is f o r m , t h e L in d b la d e q u a t io n b e c o m e s a lin e a r e q u a t io n ( a m a t r ix m u lt ip ly in g a v e c t o r , if w e a r e c o n s id e r in g e .g . d is c r e t e s y s t e m s ) . T h u s it is “ e a s y ” t o s o lv e t h e d iff e r e n t ia l e q u a t io n , fi n d in g :
ρ ( t ) = e x p [( H + G ) t ] ρ ( 0 ) ,
G
w h e r e w e id e n t if y t h e s u p e r o p e r a t o r S = e x p [ ( H + ) t ] . M o r e d e t a ils o n h o w t o c o n v e r t f r o m K r a u s s u m , t o L in d b la d t o s u p e r o p e r a t o r d e s c r ip t io n o f t h e o p e n q u a n t u m s y s t e m d y n a m ic s c a n b e f o u n d in T . F . H a v e l, R o b u s t p r o c e d u r e s f o r c o n v e r t i n g a m o n g L i n d b l a d , K r a u s a n d m a t ri x r e p r e s e n t a t i o n s o f q u a n t u m d y n a m i c a l s e m i g r o u p s , J . M a t h . P h y s . 4 4 , 5 3 4 ( 2 0 0 3 ) .
A . E x a m p l e : s p i n - 1 / 2 d e p h a s i n g
D e p h a s in g , o r t r a n s v e r s e r e la x a t io n , is t h e p h e n o m e n o n a s s o c ia t e d w it h t h e d e c a y o f t h e c o h e r e n c e t e r m s ( o ff d ia g o n a ls ) in t h e d e n s it y m a t r ix . In N M R , s in c e t h e s ig n a l is d u e t o t h e e n s e m b le o f s p in s , a c o h e r e n c e t e r m w h ic h la s t s f o r e v e r w o u ld r e q u ir e a ll t h e s a m e s p in s o f t h e d iff e r e n t m o le c u le s t o p r e c e s s a b o u t t h e m a g n e t ic fi e ld a t e x a c t ly t h e s a m e r a t e . A s p r e v io u s ly m e n t io n e d , t h e f r e q u e n c y o f a s in g le s p in d e p e n d s o n t h e lo c a l m a g n e t ic fi e ld , w h ic h d e p e n d s o n t h e e x t e r n a l fi e ld , a n d o n t h e fi e ld c r e a t e d b y t h e s u r r o u n d in g s p in s . D u e t o r a p id t u m b lin g , t h e a v e r a g e
2 6 A n d r z e j K o s s a k o w s k i O n q u a n t u m s t a t i s t i c a l m e c h a n i c s o f n o n - H a m i l t o n i a n s y s t e m s , R e p . M a t h . P h y s . 3 2 4 7 ( 1 9 7 2 ) G ¨ o r a n Li n d b l a d O n t h e g e n e r a t o r s o f q u a n t u m d y n a m i c a l s e m i g r o u p s , C o m m u n . M a t h . P h y s . 4 8 1 1 9 ( 1 9 7 6 ) ) .
fi e ld o v e r t im e is t h e s a m e , b u t d o e s v a r y a c r o s s t h e s a m p le a t a p a r t ic u la r g iv e n t im e . T h is in s t a n t a n e o u s v a r ia t io n c a u s e s t h e id e n t ic a l s p in s o f a ll t h e m o le c u le s t o s lo w ly d e s y n c h r o n iz e a n d t h e r e f o r e lo s e c o h e r e n c e a c r o s s t h e s a m p le . A n o t h e r e x a m p le o f d e p h a s in g w a s a lr e a d y p r e s e n t e d w h e n w e d e s c r ib e d p h a s e - d a m p in g f o r a d u s t p a r t ic le in t e r a c t in g w it h m a n y p h o t o n s .
T h e d e p h a s in g n o is e c a n b e t h o u g h t a s a r is in g f r o m r a n d o m z r o t a t io n a c r o s s t h e s a m p le , s o t h a t t h e s t a t e o f t h e s y s t e m c a n b e d e s c r ib e d b y a s t a t is t ic a l a v e r a g e o v e r a d is t r ib u t io n o f r o t a t io n a n g le s q ( ϑ ) :
C o n s id e r a n in it ia l d e n s it y o p e r a t o r
S d ( ρ ) = J
d ϑ q ( ϑ ) e − i ϑ / 2 σ z ρ ( 0 ) e i ϑ / 2 σ z
( )
ρ ( 0 ) = ρ 0 0 ρ 0 1
ρ 1 0 ρ 1 1
T h e e v o lu t io n u n d e r a p r o p a g a t o r U ϑ = e − i ϑ / 2 σ z g iv e s
( ρ 0 0 ρ 0 1 e − i ϑ )
ρ ( ϑ ) = ρ 1 0 e i ϑ ρ 1 1
T a k in g t h e in t e g r a l o v e r t h e a n g le d is t r ib u t io n w e fi n d
( ) J − ( ) ( )
ρ 1 0 Γ
ρ 1 1
ρ ( ϑ ) = (
ρ 0 0
∗
ρ 0 1 Γ ) ,
2
w h e r e Γ = e − i ϑ = q ( ϑ ) e − i ϑ d ϑ . If q ( ϑ ) = q ( ϑ ) ( a s g iv e n b y a n is o t r o p ic e n v ir o n m e n t ) w e o b t a in e − i ϑ = c o s ϑ . F o r a n o n - M a r k o v ia n e n v ir o n m e n t w h e r e m e m o r y e ff e c t s a r e p r e s e n t , w e c a n d e s c r ib e t h e d is t r ib u t io n q ( ϑ ) a s a G a u s s ia n s t o c h a s t ic p r o c e s s , s o t h a t Γ = ( c o s ϑ ) ≈ e − ( ϑ 2 ) / 2 = e − t 2 /T 2 . F o r a M a r k o v ia n p r o c e s s in s t e a d w e h a v e a n
e x p o n e n t ia l d e c a y Γ = e − t /T 2 .
W e c a n a ls o e x p lic it ly e v a lu a t e S d :
S d ( ρ ) = J d ϑ q ( ϑ ) [c o s ( ϑ / 2 ) 1 1 − i s in ( ϑ / 2 ) σ z ] ρ ( 0 ) [c o s ( ϑ / 2 ) 1 1 + i s in ( ϑ / 2 ) σ z ] =
= J d ϑ q ( ϑ ) [c o s 2 ( ϑ / 2 ) ρ ( 0 ) + s in 2 ( ϑ / 2 ) σ z ρ ( 0 ) σ z − i c o s ( ϑ / 2 ) s in ( ϑ / 2 ) ( σ z ρ ( 0 ) − ρ ( 0 ) σ z ) ]
B y e v a lu a t in g t h e in t e g r a l, a n d a s s u m in g a g a in a s y m m e t r ic d is t r ib u t io n , w e h a v e :
S d ( ρ ) = ( 1 − p ) ρ ( 0 ) + p σ z ρ ( 0 ) σ z
2
w h e r e p = J d ϑ q ( ϑ ) s in 2 ( ϑ / 2 ) . B y c o m p a r is o n w it h t h e p r e v io u s r e s u lt w e fi n d p = 1 − Γ .
F r o m t h e s u p e r o p e r a t o r , w e c a n fi n d t h e c o r r e s p o n d in g K r a u s s u m d e c o m p o s it io n :
√
M 0 = 1 − p 1 1 , M 1 = √ p σ z
W e w a n t n o w t o d e s c r ib e t h is s a m e e v o lu t io n u n d e r a d e p h a s in g e n v ir o n m e n t b y a L in d b la d e q u a t io n . N o t ic e t h a t t h is is g o in g t o b e p o s s ib le o n ly if w e h a v e a M a r k o v ia n e n v ir o n m e n t , Γ = e − t /T 2 .
2 2
C o n s id e r t h e a c t io n o f t h e s u p e r o p e r a t o r S d ( ρ ) = 1 + Γ ρ ( 0 ) + 1 − Γ σ z ρ ( 0 ) σ z . If w e c o n s id e r a s m a ll t im e Γ = e − δ t /T 2 ≈
1 − δ t/ T 2 a n d w e o b t a in :
γ δ t γ δ t
S d ( ρ , δ t ) = ρ − 2 ρ + 2 σ z ρ σ z
w h e r e γ = 1 / T 2 . T h e n , t a k in g t h e d iff e r e n c e ρ ( δ t ) − ρ ( 0 ) in t h e lim it δ t → 0 w e h a v e
∂ ρ γ γ 1
√
∂ t = 2 ( σ z ρ σ z − ρ ) = 2 ( σ z ρ σ z − 2 { σ z σ z , ρ } )
w h e r e w e u s e d t h e f a c t σ 2 = 1 1 . T h u s γ σ z is t h e L in d b la d o p e r a t o r f o r d e p h a s in g .
z 2
2
A s s u m e n o w t h a t w e h a d c o n s id e r e d a n o n - M a r k o v ia n e n v ir o n m e n t , f o r w h ic h Γ = e − ( t /T 2 ) . T h e n if w e t r ie d t o fi n d
— ∝
t h e in fi n it e s im a l t im e e v o lu t io n , w e c a n n o t d e fi n e a d iff e r e n t ia l e q u a t io n , s in c e ρ ( δ t ) ρ ( 0 ) is n o t δ t . F o r t h is t y p e o f e n v ir o n m e n t , t h e L in d b la d e q u a t io n c a n n o t b e d e fi n e d .
8 . 4 . 3 R e d fi e l d - B o r n t h e o r y o f r e l a x a t i o n
C o n s id e r a s y s t e m S c o u p le d t o a n e n v ir o n m e n t E ( t h e h e a t b a t h ) s u c h t h a t
H = H 0 + V = H S + H E + V ,
L ⊗
a n d V d e s c r ib e s t h e in t e r a c t io n b e t w e e n t h e s y s t e m a n d t h e e n v ir o n m e n t . M o s t g e n e r a lly it w ill t a k e t h e f o r m
V = k A k B k ( t ) , w it h A a c t in g o n t h e s y s t e m a n d B o n t h e e n v ir o n m e n t ( a n d w e h a v e e v e n a llo w e d f o r a t im e - d e p e n d e n c e o f t h e r a n d o m e n v ir o n m e n t fi e ld ) . In t h e S c h r ¨ o d in g e r p ic t u r e , t h e t im e e v o lu t io n o f t h e d e n s it y
d t
m a t r ix is g iv e n b y t h e L io u v ille e q u a t io n , i l d ρ ( t ) = [ H , ρ ( t ) ].
D e fi n e t h e in t e r a c t io n p ic t u r e d e n s it y m a t r ix
I
k
k
ρ ( t ) ≡ e i ( H S + H E ) t ρ ( t ) e − i ( H S + H E ) t ,
a n d s im ila r ly t h e in t e r a c t io n - p ic t u r e s y s t e m - e n v ir o n m e n t in t e r a c t io n
I
k
k
V ( t ) ≡ e i ( H S + H E ) t V e − i ( H S + H E ) t .
T h e n t h e e v o lu t io n in t h e in t e r a c t io n p ic t u r e is g iv e n b y
d t
k
0
k
k
k
I
I
i l d ρ I ( t ) = e i H 0 t ( [ H , ρ ( t ) ] − [ H , ρ ( t ) ]) e − i H 0 t = e i H 0 t [ V , ρ ( t ) ] e − i H 0 t = [ V ( t ) , ρ ( t ) ] .
T h is h a s t h e f o r m a l s o lu t io n
J
1 t
i
ρ I ( t ) = ρ I ( 0 ) + l
0
d t 1 [ V I ( t 1 ) , ρ I ( t 1 ) ]
( N o t e t h a t t h is is t h e s a m e e q u a t io n a s a b o v e , e x c e p t in in t e g r a l f o r m ) . E x p a n d in g o n c e ( b y in s e r t in g t h e s a m e e q u a t io n a t t h e p la c e o f ρ I ( t ) ) w e o b t a in ,
J
1 t
ρ I ( t ) = ρ I ( 0 ) + i l
1 t t 1
J J
d t 1 [ V I ( t 1 ) , ρ I ( 0 ) ] + ( i l ) 2
d t 1 d t 2 [ V I ( t 1 ) , [ V I ( t 2 ) , ρ I ( t 2 ) ]]
0 0 0
W e c o u ld r e p e a t t h is p r o c e s s t o o b t a in a n in fi n it e s e r ie s ( t h e D y s o n s e r ie s w e a lr e a d y s a w ) .
{ }
L e t u s c o n c e n t r a t e in s t e a d o n t h e e v o lu t io n o f t h e ( in t e r a c t io n p ic t u r e ) r e d u c e d d e n s it y m a t r ix ρ S = T r E ρ I , o b t a in e d b y t r a c in g o v e r t h e e n v ir o n m e n t . T o o b t a in t h e a v e r a g e d e n s it y o p e r a t o r , w e a ls o n e e d t o t a k e a n e n s e m b le a v e r a g e o v e r t h e r a n d o m fl u c t u a t in g e n v ir o n m e n t :
J
1 t
ρ S ( t ) = ρ S ( 0 ) + i l
1 t t 1
J J
d t 1 ( T r E { [ V I ( t 1 ) , ρ I ( 0 ) ] } ) + ( i l ) 2
d t 1 d t 2 ( T r E { [ V I ( t 1 ) , [ V I ( t 2 ) , ρ I ( t 2 ) ]] } ) .
0 0 0
W e w a n t t o fi n d a n e x p lic it e x p r e s s io n f o r t h e s y s t e m e v o lu t io n o n ly ( in t h e f o r m o f a d iff e r e n t ia l e q u a t io n ) . T o d o t h is , w e w ill m a k e a n u m b e r o f a p p r o x im a t io n s .
A . S i m p l i fi c a ti o n : S e p a r a b i l i t y a n d e n e r g y s h i f t
W e fi r s t a s s u m e t h a t a t t im e t = 0 t h e s y s t e m a n d e n v ir o n m e n t a r e in a s e p a r a b le s t a t e :
ρ ( 0 ) = ρ S ( 0 ) ⊗ ρ E ( 0 ) .
( t h is c a n a lw a y s b e o b t a in e d b y c h o o s in g t = 0 a p p r o p r ia t e ly ) .
L
T h is c o n d it io n h e lp s s im p lif y in g t h e s e c o n d t e r m in t h e L H S o f t h e e x p r e s s io n a b o v e . W e h a v e
T r E { [ V I ( t 1 ) , ρ I ( 0 ) ] } = [ A I ( t 1 ) , ρ S ( 0 ) ]T r E { B k ( t ) ρ E } ,
k
t h a t is , w e c o n s id e r t h e e x p e c t a t io n v a lu e o f t h e o p e r a t o r s B k . In g e n e r a l w e w ill a ls o n e e d t o t a k e a n e n s e m b le a v e r a g e o v e r t h e r a n d o m fl u c t u a t in g fi e ld ( B k ( t ) ) , a s w e lo o k a t e x p e c t a t io n v a lu e s f o r t h e d e n s it y o p e r a t o r .
W e c a n n o w m a k e t h e a s s u m p t io n t h a t ( B E ) E = 0 , w h ic h im p lie s ( V I ( t ) ) E ∼ T r E { V I ( t ) ρ E ( 0 ) } = 0 . T h is is n o t r e s t r ic t iv e , s in c e , if V is o f t h e f o r m V = A S ⊗ B E w it h ( B E ) E / = 0 w e c a n r e p la c e V w it h V = A S ⊗ ( B E − ( B E ) E ) ,
L
( ) E H ( ) E
a n d s im u lt a n e o u s ly a d d A S B E t o S . W it h t h is c o n d it io n , V = 0 a n d s in c e ρ E ( 0 ) h a s t h e s a m e f o r m in b o t h S c h r ¨ o d in g e r a n d in t e r a c t io n p ic t u r e s , t h e r e s u lt h o ld s in t h e in t e r a c t io n p ic t u r e a ls o . T h e s a m e a r g u m e n t c a n b e m a d e if V = α A S , α ⊗ B E , α . T h e n t h e s e c o n d t e r m in t h e e q u a t io n a b o v e v a n is h e s a n d w e h a v e
ρ S ( t ) = ρ S ( 0 ) + ( i l ) 2
d t 1 d t 2 ( T r E { [ V I ( t 1 ) , [ V I ( t 2 ) , ρ I ( t 2 ) ] ] } ) .
1 J t J t 1
0 0
B . A s s u m p ti o n 1 : B o r n a p p r o x i m a ti o n
⊗
W e c a n a lw a y s w r it e ( in a n y p ic t u r e ) ρ ( t ) = ρ S ( t ) ρ E ( t ) + ρ c o r r e l a t i o n ( t ) , w h ic h m a y b e t a k e n a s a d e fi n it io n o f ρ c o r r e l a t i o n . L e t u s a s s u m e ( a s d o n e in t h e p r e v io u s s e c t io n ) t h a t t h e in t e r a c t io n is t u r n e d o n a t t im e t = 0 , a n d t h a t p r io r t o t h a t t h e s y s t e m a n d e n v ir o n m e n t a r e n o t c o r r e la t e d ( ρ c o r r e l a t i o n ( 0 ) = 0 ) . T h is a s s u m p t io n is n o t v e r y r e s t r ic t iv e , s in c e w e c a n a lw a y s fi n d a t im e p r io r t o w h ic h t h e s y s t e m a n d e n v ir o n m e n t d id n o t in t e r a c t . N o w h o w e v e r w e m a k e a s t r o n g e r a s s u m p t io n .
≈ ≫
W e w ill a s s u m e t h a t t h e c o u p lin g b e t w e e n t h e s y s t e m a n d t h e e n v ir o n m e n t is w e a k , s o t h a t ρ ( t ) ≈ ρ S ( t ) ⊗ ρ E ( t ) , f o r t im e s c a le s o v e r w h ic h p e r t u r b a t io n t h e o r y r e m a in s v a lid . F u r t h e r m o r e , w e w ill a s s u m e t h a t t h e c o r r e la t io n t im e τ E ( a n d t h u s t h e r e la x a t io n t im e ) o f t h e e n v ir o n m e n t is s u ffi c ie n t ly s m a ll t h a t ρ E ( t ) ρ E ( 0 ) if t τ E .
N o t e t h a t s in c e w e a s s u m e t h a t t h e e n v ir o n m e n t is in a t h e r m a l e q u ilib r iu m , it h a s a t h e r m a l d e n s it y m a t r ix
ρ E ( 0 ) = 1 L e k T | n ) ( n | ,
—
E n E
Z E
n B E E
H
w h ic h is a s t a t io n a r y s t a t e , i.e ., [ ρ E ( 0 ) , E ] = 0 , s o t h a t ρ E ( 0 ) h a s t h e s a m e f o r m in b o t h t h e in t e r a c t io n p ic t u r e a n d S c h r ¨ o d in g e r p ic t u r e . T h e n
J J
1 t t 1
0 0
ρ S ( t ) = ρ S ( 0 ) + ( i l ) 2
d t 1 d t 2 ( T r E { [ V I ( t 1 ) , [ V I ( t 2 ) , ρ S ( t 2 ) ⊗ ρ E ( 0 ) ]] } ) .
W e c a n a ls o g o f u r t h e r a n d e x p lic it ly w r it e t h e p a r t ia l t r a c e :
k
h
( T r E { [ V I ( t 1 ) , [ V I ( t 2 ) , ρ S ( t 2 ) ⊗ ρ E ( 0 ) ]] } ) = L ( B k ( t 1 ) B h ( t 2 ) ) [ A I ( t 1 ) , [ A I ( t 2 ) , ρ S ( t 2 ) ]]
k , h
( )
J
w h e r e B k ( t 1 ) B h ( t 2 ) = G k , h ( t 1 , t 2 ) is t h e c o r r e la t io n f u n c t io n f o r t h e e n v ir o n m e n t . D iff e r e n t ia t in g , w e g e t
d 1 t
0
d t ρ S ( t ) = ( i l ) 2
o r
d s ( T r E { [ V I ( t ) , [ V I ( s ) , ρ S ( s ) ⊗ ρ E ( 0 ) ]] } ) .
d t ρ S ( t ) = ( i l ) 2
d s
0 k , h
d 1 J t L
[ I [ I ] ]
( B k ( t ) B h ( s ) )
A k ( t ) ,
A h ( s ) , ρ S ( s )
.
T h is s h o u ld p r o p e r ly b e c o n s id e r e d a d iff e r e n c e e q u a t io n , s in c e w e h a v e a s s u m e d t h a t t ≫ τ E .
C . A s s u m p ti o n 2 : M a r k o v a p p r o x i m a ti o n
≈
W e w ill a ls o a s s u m e t h a t w e a r e w o r k in g o v e r t im e s c a le s t h a t a r e s h o r t e r t h a n t h e g r o s s t im e s c a le o v e r w h ic h t h e s y s t e m e v o lv e s , s o t h a t ρ S ( s ) ρ S ( t ) . T h u s w e c a n r e p la c e ρ S ( s ) in t h e in t e g r a l w it h ρ S ( t ) . W e fi n a lly g e t t h e R e d fi e ld e q u a t io n :
J
d 1 t
0
d t ρ S ( t ) = ( i l ) 2
d s T r E { [ V I ( t ) , [ V I ( s ) , ρ S ( t ) ⊗ ρ E ( 0 ) ]] }
J
o r
d t ρ S ( t ) = ( i l ) 2
d s
0 k , h
( B k ( t ) B h ( s ) )
d 1 t L
[ I [ I ] ]
A k ( t ) ,
A h ( s ) , ρ S ( t )
.
W e c a n c h a n g e v a r ia b le s f r o m s → s ′ = t − s ( s o t h a t w e c h a n g e t h e in t e g r a ls a s : J t d s → J 0 d ( t − s ′ ) = − J 0 d s ′ =
0
J t d s ′ ) . T h e n
d t ρ S ( t ) = ( i l ) 2
d s
0 k , h
( B k ( t ) B h ( t − s ) )
d 1 J t L
0 t t
A k ( t ) ,
A h ( t − s ) , ρ S ( t )
.
[ I [ I ] ]
T h e c o r r e la t io n t im e o f t h e t h e r m a l b a t h E is a s s u m e d t o b e v e r y s h o r t , s o t h a t t h e c o r r e la t io n f u n c t io n ( B k ( t 1 − t 2 ) B h ( 0 ) ) E d iff e r s o n ly s ig n ifi c a n t ly f r o m z e r o w h e n t 1 ≈ t 2 . W e c a n t h e r e f o r e e x t e n d t h e lim it o f in t e g r a t io n t o ∞ ( a n d c a ll
t − s = τ ) :
d t
S
( i l ) 2
k
h
k
h
S
d ρ ( t ) = 1 J ∞ d τ L ( B ( t ) B ( τ ) ) [ A I ( t ) , [ A I ( τ ) , ρ ( t ) ] ] .
0 k , h
D . S p e c tr a l d e n s i ti e s
T h e n e x t s t e p in t h e s im p lifi c a t io n p r o g r a m is t o t a k e t h e e x p e c t a t io n v a lu e s w it h r e s p e c t t o t h e e ig e n s t a t e s o f t h e
s y s t e m a n d t h e n F o u r ie r t r a n s f o r m . W e w ill w r it e A k ( t ) = L
A p e − i ω p t :
d
d t
S
ρ ( t ) =
L L J ∞ d τ G
k , h p , q
0
p k
( i l ) 2
k h
k
h
S
( τ ) � A p e − i ω p t , � A q e − i ω q ( t − τ ) , ρ
( t ) � � .
H e r e w e u s e d t h e f a c t t h a t G ( t, τ ) is s t a t io n a r y , a n d t h u s d e p e n d o n ly o n t h e d iff e r e n c e t − τ , G ( t, τ ) = G ( t − τ ) . W e t h e n c h a n g e d v a r ia b le s f r o m τ → t − τ . W e c a n r e w r it e t h e e q u a t io n a s
J
d t
S
( i l ) 2
k
h
S
k h
d ρ ( t ) = L L [ A p , [ A q , ρ
k , h p , q
( t ) ] ] e − i ( ω p + ω q ) t J ∞ d τ G
0
( τ ) e i ω q τ .
0
T h u s w e h a v e t h e in t e g r a l ∞ e i ω τ G ( τ ) = J ( ω ) , w h e r e t h e F o u r ie r t r a n s f o r m o f t h e c o r r e la t io n f u n c t io n G ( τ ) is t h e t h e s p e c t r a l f u n c t io n J ( ω ) . W it h s o m e s im p lifi c a t io n s ( d u e t o s t a t is t ic a l p r o p e r t ie s o f t h e b a t h o p e r a t o r s a n d t o t h e f a c t t h a t w e o n ly t a k e t e r m s r e s u lt in g in a n H e r m it ia n o p e r a t o r ) , w e fi n a lly a r r iv e a t t h e m a s t e r e q u a t io n :
d t
S
k
p − k
d ρ ( t ) = − L L J ( ω ) [ A p
k p
, [ A p , ρ
k
S
( t ) ]]
W e c a n a ls o w r it e t h e m a s t e r e q u a t io n a s t h e R e d fi e ld e q u a t io n ( s u b s c r ip t s in d ic a t e m a t r ix e le m e n t s ) :
L
d
d t ρ a, a ′ = R aa ′ , b b ′ ρ b , b ′
b , b ′ / b − b ′ = a − a ′
8 . 5 O t h e r d e s c r i p t i o n o f o p e n q u a n t u m s y st e m d y n a m i c s
8 . 5 . 1 S t o c h a s t i c L i o u v i l l e e q u a t i o n a n d c u m u l a n t s
S t o c h a s t ic L io u v ille t h e o r y is b a s e d o n a s e m ic la s s ic a l m o d e l o f d e c o h e r e n c e , in w h ic h t h e H a m ilt o n ia n a t a n y in s t a n t in t im e c o n s is t s o f a d e t e r m in is t ic a n d a s t o c h a s t ic p a r t , w h ic h r e p r e s e n t s t h e e ff e c t s o f a r a n d o m n o is e . In t h e s im p le s t c a s e o f N M R T 2 r e la x a t io n , t h is t y p ic a lly t a k e s t h e f o r m
H t o t ( t ) = H d e t ( t ) + H s t ( t ) = H d e t ( t ) + ω ( t ) H N ,
w h e r e H d e t is t h e s t a t ic d e t e r m in is t ic H a m ilt o n ia n , a n d w e s e p a r a t e d t h e s t o c h a s t ic t im e d e p e n d e n c e d e s c r ib e d b y t h e c o e ffi c ie n t ω ( t ) f r o m t h e n o is e g e n e r a t o r H N . ω ( t ) is a r a n d o m v a r ia b le d u e t o s t o c h a s t ic , t im e - d e p e n d e n t fl u c t u a t in g fi e ld s a n d H N is a n o p e r a t o r w h ic h d e s c r ib e s h o w t h e s e c la s s ic a l fi e ld s a r e c o u p le d t o t h e q u a n t u m s y s t e m .
W e n o w in t r o d u c e a s u p e r o p e r a t o r L ( t ) d e fi n e d o n L io u v ille ( o p e r a t o r ) s p a c e v ia
L ( t ) = H t ∗ o t ( t ) ⊗ 1 1 − 1 1 ⊗ H t o t ( t ) = L d e t ( t ) + ω ( t ) L N
w h e r e L N = H N ∗ ⊗ 1 1 − 1 1 ⊗ H N . T h is s u p e r o p e r a t o r is t h e g e n e r a t o r o f m o t io n f o r d e n s it y o p e r a t o r ρ ˆ , m e a n in g
( J t ′ ′ )
ρ ( t ) = U ρ ˆ ( 0 ) = T e x p − i d t L ( t ) ρ ˆ ( 0 )
0
t o o b t a in ( ρ ˆ ( t ) ) =
U
ρ ˆ ( 0 ) . T h e p r o b le m o f c a lc u la t in g t h e a v e r a g e o f t h e e x p o n e n t ia l o f a s t o c h a s t ic o p e r a t o r h a s
w h e r e T is t h e u s u a l D y s o n t im e o r d e r in g o p e r a t o r . S in c e w h a t is a c t u a lly o b s e r v e d in a n e x p e r im e n t is t h e s t a t is t ic a l a v e r a g e o v e r t h e m i ( c r o ) s c o p ic t r a j e c t o r ie s o f t h e s y s t e m ( ρ ˆ ( t ) ) , w e h a v e t o t a k e t h e e n s e m b le a v e r a g e s u p e r p r o p a g a t o r
b e e n s o lv e d b y K u b o 2 7 u s in g t h e c u m u la n t e x p a n s io n .
0
F ir s t , e x p a n d t h e t im e - o r d e r e d a v e r a g e e x p o n e n t ia l S = ( T e x p ( − i J t d t ′ H ( t ′ ) ) ) v ia t h e D y s o n s e r ie s :
0 2 ! 0
1
0
2
1
2
S = 1 1 − i J t d t ′ ( H ( t ′ ) ) + ( − i ) 2 T J t d t J t d t ( H ( t ) H ( t ) ) + · · ·
n ! 0
0
1
n
+ ( − i ) n T J t d t
· · · J t d t
( H ( t 1 ) · · · H ( t n
) ) + · · ·
( H · · · H )
T h e t e r m ( t 1 ) ( t n ) is c a lle d t h e n - t h m o m e n t o f t h e d is t r ib u t io n . W e w a n t n o w t o e x p r e s s t h is s a m e p r o p a g a t o r in t e r m s o f t h e c u m u la n t f u n c t io n K ( t ) , d e fi n e d b y :
S = e K ( t )
T h e c u m u la n t f u n c t io n it s e lf c a n m o s t g e n e r a lly b e e x p r e s s e d a s a p o w e r s e r ie s in t im e :
L
∞ n 2
n
K ( t ) = ( − i t ) K
n !
= − i tK 1
+ ( − i t ) K
2
2 !
+ · · ·
n = 1
E x p a n d in g n o w t h e e x p o n e n t ia l u s in g t h e e x p r e s s io n a b o v e w e h a v e :
1 2 ( − i t ) 2 2
S = 1 1 + K ( t ) + 2 ! ( K ( t ) ) + · · · = 1 1 − i tK 1 + 2 ! ( K 2 + K 1 ) + · · ·
B y e q u a t in g t e r m s o f t h e s a m e o r d e r in t h e t w o e x p a n s io n s w e o b t a in t h e c u m u la n t s K n in t e r m s o f t h e m o m e n t s o f o r d e r a t m o s t n . F o r e x a m p le :
1 J t ( )
K 1 =
K 2 =
d t ′
t 0
t T
2 d t 1 d t 2
0 0
1 J t
H ( t ′ )
H ( t 1 ) H ( t 2 )
— K 1
J t ( ) 2
T h e p r o p a g a t o r c a n t h e r e f o r e b e e x p r e s s e d in t e r m s o f t h e c u m u la n t a v e r a g e s :
( ( H ( t ′ ) ) c )
= ( H ( ( t ′ ) )
) ( ) ( )
H ( t 1 ) H ( t 2 ) c
T h e p r o p a g a t o r c a n t h e r e f o r e b e w r it t e n a s :
= T H ( t 1 ) H ( t 2 )
— H ( t 1 )
H ( t 2 )
S = e x p
— i
d t ′
H ( t ′ ) c −
( J t (
) J t
J t ( ) )
d t 1
d t 2
H ( t 1 ) H ( t 2 ) c + · · ·
0 0 0
2 7 R . K u b o , G e n e r a l i z e d C u m u l a n t Ex p a n s i o n M e t h o d , J o u r n a l o f t h e P h y s i c a l S o c i e t y o f J a p a n , 1 7 , 1 1 0 0 - 1 1 2 0 ( 1 9 6 2 )
c 0
H ( H ) J H
N o t e t h a t if is a d e t e r m in is t ic f u n c t io n o f t im e , t h e e n s e m b le a v e r a g e s c a n b e d r o p p e d a n d ( t ) = t d t ′ ( t ′ ) b e c o m e s t h e t im e - a v e r a g e H a m ilt o n ia n , w h ic h is t h e fi r s t t e r m in t h e Ma g n u s e x p a n s io n . T h e s e c o n d t e r m in t h e c u m u la n t e x p a n s io n , o n t h e o t h e r h a n d , b e c o m e s
T
d t 1
J t J t
( J t ) 2
0 0 0
1 d t 2 H ( t 1 ) H ( t 2 ) − d t ′ H ( t ′ )
0
1
0
2
1
2
0
1
0
2
1
2
= 2 J t d t J t 1 d t H ( t ) H ( t ) − J t d t J t d t H ( t ) H ( t )
= J d t 1 J d t 2 [ H ( t 1 ) , H ( t 2 ) ] ,
0 0 0 t 1
= J t d t 1 J t 1 d t 2 H ( t 1 ) H ( t 2 ) − J t d t 1 J t d t 2 H ( t 1 ) H ( t 2 )
t t 1
0 0
T
w h e r e w e h a v e u s e d t h e f a c t t h a t t h e t im e - o r d e r in g o p e r a t o r s y m m e t r iz e s it s a r g u m e n t w it h r e s p e c t t o p e r m u t a t io n o f t h e t im e p o in t s . T h is is t h e s e c o n d t e r m in t h e M a g n u s e x p a n s io n f o r t h e “ a v e r a g e ” ( e ff e c t iv e ) H a m ilt o n ia n . P r o c e e d in g in t h is f a s h io n o n e c a n in p r in c ip le d e r iv e a v e r a g e H a m ilt o n ia n t h e o r y 2 8 f r o m t h e D y s o n a n d c u m u la n t e x p a n s io n s .
In t e r m s o f t h e s o - c a lle d “ c u m u la n t a v e r a g e s ” ( · · · ) c , t h e s u p e r p r o p a g a t o r is g iv e n b y :
J
J
( ) (
J t ′ ′ t t )
1
U = e x p − i d t ( L ( t ) ) c − 2 T d t 1 d t 2 ( L ( t 1 ) L ( t 2 ) ) c + · · ·
0 0 0
0
P r o v id e d J t d t ′ L ( t ′ ) ≪ 1 f o r a ll t > 0 , w e c a n s a f e ly n e g le c t h ig h o r d e r t e r m s in t h e e x p o n e n t ia l’s a r g u m e n t .
8 . 5 . 2 S t o c h a s t i c W a v e f u n c t i o n s
~
T h e M o n t e C a r lo w a v e f u n c t io n w a s d e r iv e d s im u lt a n e o u s ly in t h e 1 9 9 0 s b y t w o g r o u p s in t e r e s t e d in v e r y d iff e r e n t q u e s t io n s . A g r o u p o f s c ie n t is t s in F r a n c e , D a lib a r d , C a s t in , a n d M ø lm e r , w a n t e d t o s im u la t e la s e r c o o lin g o f a t o m s q u a n t u m m e c h a n ic a lly in t h r e e d im e n s io n s . T h e ir n u m e r ic a l s o lu t io n r e q u ir e d d is c r e t iz in g s p a c e in t o a g r id o f 4 0 x 4 0 x 4 0 p o s it io n s ; t o im p le m e n t t h e m a s t e r e q u a t io n o n s u c h a s p a c e w o u ld h a v e r e q u ir e d a d e n s it y m a t r ix w it h O ( 4 0 6 ) 1 0 9 e n t r ie s s u c h c a lc u la t io n s a r e b e y o n d t h e s c o p e o f e v e n m o d e r n c o m p u t e r s . H o w e v e r , s im u la t in g a w a v e f u n c t io n w it h O ( 4 0 3 ) e n t r ie s is q u it e f e a s ib le . C o n s e q u e n t ly t h e g r o u p s o u g h t t o c o n v e r t t h e m a s t e r e q u a t io n t o s o m e t h in g m o r e lik e t h e S c h r ¨ o d in g e r e q u a t io n 2 9 .
A t t h e s a m e t im e , C a r m ic h a e l w a s in t e r e s t e d in t h e e ff e c t s t h a t c o n t in u o u s m o n it o r in g w o u ld h a v e o n a s y s t e m 3 0 . F o r e x a m p le , a t w o - le v e l a t o m p r e p a r e d in a n e q u a l s u p e r p o s it io n o f s t a t e s c a n d e c a y b y e m it t in g a p h o t o n ; if t h a t p h o t o n is d e t e c t e d , t h e e x p e r im e n t e r k n o w s w it h c e r t a in t y t h a t t h e a t o m is in it s g r o u n d s t a t e . B u t w h a t h a p p e n s 5 0 % o f t h e t im e w h e n a p h o t o n is n o t d e t e c t e d ? C e r t a in ly , a f t e r a lo n g t im e h a s p a s s e d , t h e a t o m m u s t b e in it s g r o u n d s t a t e , b u t h o w d o e s t h a t h a p p e n ? T o s t u d y t h e s e a n d s im ila r q u e s t io n s , C a r m ic h a e l w a n t e d t o in c o r p o r a t e t h e e ff e c t s o f c o n t in u o s m o n it o r in g , a n d u n d e r s t a n d h o w a m e a s u r e m e n t c a n c a u s e t h e s y s t e m s t a t e t o s u d d e n ly j u m p in t o a d iff e r e n t s t a t e .
T h e d e s c r ip t io n o n w h ic h b o t h g r o u p s c o n v e r g e d b e g in s w it h t h e m o s t g e n e r a l f o r m o f t h e m a s t e r e q u a t io n ,
d ρ
d t
= − i [ H , ρ ] + L ( ρ ) ,
L
w it h t h e L in d b la d ia n γ k
L ( ρ ) = −
k
2 ( L † k L k ρ + ρ L k † L k − 2 L k ρ L k † ) .
2 8 S e e f o r e x a m p l e H a e b e r l e n , H i g h R e s o l u t i o n N M R i n S o l i d s : S e l e c t i v e A v e r a g i n g , A c a d e m i c P r e s s I n c . , N e w Y o r k ( 1 9 7 6 )
2 9 J e a n D a l i b a r d , Y v a n C a s t i n a n d K l a u s M l me r W a v e - f u n c t i o n a p p r o a c h t o d i s s i p a t i v e p r o c e s s e s i n q u a n t u m o p t i c s , P h y s . R e v . Le t t . 6 8 , 5 8 0 5 8 3 ( 1 9 9 2 )
3 0 H . J . C a r mi c h a e l Q u a n t u m t r a j e c t o r y t h e o r y f o r c a s c a d e d o p e n s y s t e m s P h y s . R e v . Le t t . 7 0 , 2 2 7 3 2 2 7 6 ( 1 9 9 3 )
U s in g t h is e x p lic it e x p r e s s io n a n d r e a r r a n g in g t h e t e r m s w e h a v e
L
d t 2
k
k
2
k
k
k
k
k
d ρ = − i ( H − i L γ k L † L ) ρ − ρ ( H + L γ k L † L ) + L γ L ρ L †
k k k
= − i H e ff ρ − ρ H e † ff
w h e r e w e h a v e d e fi n e d a n e ff e c t iv e H a m i l t o n i a n
+ γ k L k ρ L † k ,
k
e ff
2
k
k
H = H − i L γ k L † L
k
L
( n o t ic e t h a t t h is is n o t a v a lid H a m ilt o n ia n in t h e u s u a l s e n s e , s in c e it is no t H e r m it ia n , s o it s e ig e n v a lu e s a r e n o t t h e e n e r g y , s in c e t h e y c o u ld b e im a g in a r y n u m b e r s ) .
E x p a n d in g t h e d e n s it y m a t r ix in t e r m s o f a n e n s e m b le o f p u r e s t a t e s , ρ = j p j | ψ j ) ( ψ j | , w e c a n r e w r it e t h e m a s t e r e q u a t io n in a s u g g e s t iv e f o r m :
d t
d ρ = L p
j
e ff
− i ( H
| ψ ) ( ψ | − | ψ ) ( ψ | H †
) + L γ L
k
j
j
k
| ψ ) ( ψ | L †
j
j
j
j
e ff
k
j k
N o w w e c a n in t e r p r e t t h e fi r s t t w o t e r m s o f t h is e q u a t io n a s a S c h r ¨ o d in g e r e v o lu t io n f o r e a c h o f t h e p u r e s t a t e s in t h e d e n s it y m a t r ix e x p a n s io n :
d
d t | ψ j ) = − i H e ff | ψ j )
| ) | ) | )
w h ile w e in t e r p r e t t h e la s t t e r m a s a q u a n t u m j u m p o p e r a t o r t h a t c h a n g e s ψ j in t o ϕ j , k = L k ψ j w it h s o m e p r o b a b ilit y .
W e c a n t h e n h a v e a p r o b a b ilis t ic p ic t u r e o f t h e p u r e s t a t e e v o lu t io n . A f t e r a n in fi n it e s im a l t im e , in t h e a b s e n c e o f
j u m p s , t h e s t a t e w ill h a v e e v o lv e d t o
| ψ j ( t + δ t ) ) = ( 1 − i H e ff ) | ψ j ) / 1 − δ p j ,
√
L L
w h e r e w e h a v e in t r o d u c e d a n o r m a liz a t io n f a c t o r , w h ic h is n e e d e d s in c e t h e H a m ilt o n ia n is n o t h e r m it ia n :
δ p j = δ p j , k = δ t γ k ( ψ j | L † k L k | ψ j )
k k
) �
If in s t e a d a j u m p h a s o c c u r r e d , t h e s t a t e w o u ld h a v e e v o lv e d t o
| ϕ j , k
= γ k δ t L δ p j , k k
| ψ j )
T h u s t h e e v o lu t io n o f t h e d e n s it y m a t r ix is g iv e n b y
ρ ( t + δ t ) = L p j ( 1 − δ p j ) | ψ j ( t + δ t ) ) ( ψ j ( t + δ t ) | + L δ p j , k | ϕ j , k ) ( ϕ j , k |
j k
T h is e x p r e s s io n le a d s u s t o t h e f o llo w in g in t e r p r e t a t io n : t h e s y s t e m u n d e r g o e s a d y n a m ic s t h a t y ie ld s t w o p o s s ib le o u t c o m e s :
— | ) H
1 . w it h p r o b a b ilit y 1 δ p j t h e s y s t e m e v o lv e s t o t h e s t a t e ψ j ( t + δ t ) , a c c o r d in g t o t h e o p e r a t o r e ff w it h a n a p p r o p r ia t e n o r m a liz a t io n
2 . w it h p r o b a b ilit y δ p j t h e s y s t e m j u m p s t o a n o t h e r s t a t e . T h e r e a r e m a n y p o s s ib le s t a t e s t h e s y s t e m c a n j u m p t o , e a c h o n e w it h a p r o b a b ilit y δ p j , k .
T h is p r o b a b ilis t ic p ic t u r e is o f c o u r s e a c o a r s e g r a in in g o f t h e c o n t in u o u s t im e e v o lu t io n . H o w e v e r , b y d is c r e t iz in g t im e it b e c o m e s e a s ie r t o d e v is e a s im u la t io n p r o c e d u r e t o r e p r o d u c e t h e d e s ir e d d y n a m ic s , w it h a w a v e f u n c t io n M o n t e c a r lo p r o c e d u r e .
MIT OpenCourseWare http://ocw.mit.edu
22.51 Quantum Theory of Radiation Interactions
Fall 201 2
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .