Š= k5  ¹ Å 4 ; c" e W _ Ë 8 ý  Ò Åy œ ; c  \ ¥  Ò Å] k ù° Ë Ñ ¹ Åy ¢y ¢5 8 ý P c p7 û ß Ã Å R w ‹

w

Š= k5    ¹ Å 4 ; c" e  W _ Ë 8 ý  Ò Åy œ ; c    \ ¥  Ò Å] k ù° Ë Ñ ¹ Åy ¢y ¢5 8 ý P c p7 û ß Ã Å  R w ‹

T g ` @+ ä  · T a : @® £ · T < 0 å  ·  ™ »€ Ða : @

 â

· ¡ ¤ @ /† < Ɠ § F  g„  s ƒ  ½ ¨z  ´, @ /½ ¨ 702-701

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¡ ‘ žó j u

x 9

€ ª œ@ /† < Ɠ § “ §€ ª œõ & ñ  Ò, x 9 € ª œ 627-702

) o

+ ä * å 

 â

· ¡ ¤ @ /† < Ɠ § „  l „   / B N† < Æ Ò, @ /½ ¨ 702-701

‚ Є ç ¡¦  · L |„ ç ¡% ã < ^∗

 â

· ¡ ¤ @ /† < Ɠ § Ó ü t o † < Æõ , @ /½ ¨ 702-701 (2003¸   4 Z 4 17{ 9  ~ à Î6 £ §)

“

¦^ ‰? /  „   > \ " f F  g„  • ¸• ¸_  y Œ ™û Z“   \  ¦  % ò ƒ  í ß –  ~ ½ ÓZ O `  ¦ ² ú ˜o  # Œ Ä »• ¸ % i  . # Œl " f ½ ¨ ô 

Ç s  y Œ ™û Z“   \   H „   -Ÿ í 7 H  s _  Ø  æ[  t õ & ñ ÷  rë ß –  m   ü @ Ò\ " f Å Ò# Q”   y n C_  ´ òõ \  ¦ Ÿ í† < Ê “ ¦ e ”

 . y Œ •y Œ •_  í ß –ê ø ͓   \  ¦  Ø Ô>  ½ + É â Ä º „   -Ÿ í 7 H ÷  rë ß –  m   „   -„    ¢ ¸  H „   -Ô  ¦í  HÓ ü t \   t  6

£

x6   x ½ + É Ã º e ”  “ ¦ Ò q ty Œ • ) a  . # Œl \ " f ½ ¨ô  Ç F  g„  • ¸• ¸\ " f ' Í   P :  % ò ~ ½ ÓZ O Ü ¼– Ð ½ ¨ô  Ç y Œ ™û Z“     H f ”  ] X

„  s + þ A  s 9 þ t – Ðà ԏ : r / B N" î õ  ° ú  s  \  -t  ç ß –  s  { 9 & ñ ô  Ç  â Ä º\  ô  Ç # Œ & h 6   x 0 p x t ë ß –, ¿ º  P :

~

½

ÓZ O Ü ¼– Ð ½ ¨ô  Ç y Œ ™û Z“     H 0 A_   â Ä º÷  rë ß – m   \  -t  ç ß –  s  { 9 & ñ t  · ú §  H  â Ä º\ " f• ¸ & h 6   x 0 p x

 “ ¦ Ò q ty Œ • ) a  .

PACS numbers: 72

Keywords: F  g„  • ¸• ¸ , „   > ,  % ò ƒ  í ß – , y Œ ™û Z“   

I. " e  ] Ø

þ

j  H “ ¦• ¸_  í ß –\ O  oü <  8Ô  ¦ # Q ì ø ͕ ¸^ ‰? /_  „   [ þ t_ 



1 l x \  › ' aô  Ç ƒ  ½ ¨  8¹ ¡ ¤  Ö ¸µ 1 ÏK t “ ¦ e ”  . ì ø ͕ ¸^ ‰? /_ 



r & h “   Ó ü t o | ¾ Ó[ þ t \  @ /ô  Ç s  : r“ É r € ª œ : Ÿ x > % i † < Æ& h ~ ½ ÓZ O  Ü

¼– Ð & ñ w n ÷ &  H  כ s  ˜ м # & h s  . s M : Å Ò  ) a @ / © œs  ÷ &



 H  כ “ É r „  l „  • ¸\  ¦ Ÿ í† < Ê   H „   [ þ t_  à º5 Å x‰ & ³ © œs “ ¦ Ã

º5 Å x \  % ò † ¾ Ó`  ¦ Šҍ  H f  ­ ä ¼a Ë >B j m 7 £ §(scattering mecha- nism) s  . s  Qô  Ç ƒ  ½ ¨×  æ \  @ /³ ð ÷ &  H  כ “ É r ‚  + þ A6 £ x² ú š s

 : r s   [1–13]. s  כ “ É r ì ø ͕ ¸^ ‰\ " f_  y n C_  f  ¨ à ºü < ~ ½ Ó Ø

 ¦1 p x`  ¦ Ÿ í† < Ê   H r ç ß –& h Ü ¼– Ð ”  1 l x   H €  •ô  Ç „  l  © œ\  _  ô 

Ç | × ¼a Ë >(perturbation)\  @ /K " f $ í / B N& h Ü ¼– Ð & h 6   x ÷ &“ ¦ e ”

 . r ç ß –& h Ü ¼– Ð ”  1 l x   H „  l  © œ\  _ K " f Ò q tl   H „  

•

¸• ¸\  @ /ô  Ç ‚  + þ A6 £ x² ú šs  : r“ É r € ª œ : Ÿ x > % i † < Æ_   „ ½ Ó0 A\ 

#

Œ Q t   ^ ‰> í ß –l Z O `  ¦ Ÿ í† < Ê “ ¦ e ” Ü ¼ 9  % ò l Z O • ¸ Õ

ª ×  æ 
 . s  l Z O \ " f  H  % ò  ( Vq » ¡ §ƒ  í ß –  : pro- jector)\  ¦ # Qb  G>  ‚  × þ ˜   H \    " f   É r „  > h+ þ Ad ” `  ¦

∗

E-mail: [email protected]

°

ú >   ) a  . ‘ : r ƒ  ½ ¨”  “ É r Y > t _   % ò  \  ¦  6   x # Œ Y > 

>

h_  s  : r`  ¦ • ¸Ø  ¦ô  Ç & h s  e ”  . [4,12,14,15]  % ò    H é ß – { 9

„    % ò  ,  ^ ‰ % ò  – Ð ½ ¨ì  r ÷ &“ ¦, s [ þ t y Œ •y Œ •“ É r   r

  © œI _ ” > r  % ò  (state-dependent projector),  © œI 1 l q w n

 % ò  (state-independent projector)1 p x Ü ¼– Ð ì  r À Ó  ) a  .



© œI _ ” > r  % ò  \  ¦ & h 6   x½ + É  â Ä º (L d J _k = const · J z )  ÷ &

#

Q & ñ  l  © œs    § 4 `  ¦  â Ä º 7 £ ¤ f ” ] X „  s + þ A  s 9 þ t – Ðà ԏ : r /

B

N" î õ  ° ú  s  \  -t  ç ß –  s  { 9 & ñ ô  Ç  â Ä º\  ô  Ç # Œ  6   x

½ +

É Ã º e ”   [16–18]. Õ ª Q
  © œI 1 l qw n  % ò  \  ¦ & h 6   x €   (L d J _k ^∗ = 0)  ÷ &Ù ¼– Ð 0 A_   â Ä º ÷  rë ß – m   \  -t  ï  r 0 A  s _  ç ß –  s  { 9 & ñ t  · ú §“ É r  â Ä º\ • ¸  | à Ðf ”   

“

¦ Šҁ © œ÷ &“ ¦ e ”   [19]. ¢ ¸,  © œI _ ” > r  % ò  \  ¦ & ñ _    H

~

½ ÓZ O “ É r  € ª œ    H  z  ´`  ¦ · ú ˜>  ÷ &% 3  . [20] ‘ : r ƒ  ½ ¨”  

“ É

r s [ þ t_   % ò ~ ½ ÓZ O `  ¦ + ‹" f Y > Y >  ì ø ͕ ¸^ ‰? /_   s 9 þ t – Ð à

ԏ : r „  s \ " f_  f  ¨ à º‚  ; Ÿ ¤`  ¦ > í ß –Ù þ ¡ . Õ ª Q
 s [ þ t`  ¦

½

¨› ¸& h Ü ¼– Ð q “ §   H { 9 \   H ™ èf . Ë y  K  M ® o  . ‘ : r  7 Hë  H \ 

"

f  H ¿ º> h_   % ò  \  ¦ & ñ _  “ ¦ s [ þ t`  ¦ + ‹" f „  • ¸• ¸ J $ ™

"

fü < ‚  ; Ÿ ¤† < Êà º(linewidth function) x 9  s  ¢ - aÖ  ¦(relaxation

-100-

~

½ ÓZ O `  ¦ ™ è> hô  Ç .
  s  ¿ º ~ ½ ÓZ O s  # Qb  G>   Ø Ô 9

#

Q‹ "   â Ä º\  ˜ Ð   | à Ðf ”  >   6   x| ¨ c à º e ”   H t \  @ /K 

"

f ž Ð_  ô  Ç . # Œl \ " f  H o Ä ºq _  ~ ½ Ó& ñ d ” \ " f Ø  ¦µ 1 Ï 

#

Œ x 9 • ¸ƒ  í ß – \  ¦ ½ ¨ “ ¦ „   l  _  ”  1 l x+ þ A_  „  l  © œ\ 

@

/ô  Ç „  À Ó_  ¨ î ç  H`  ¦ ½ ¨ # Œ ‚  + þ A  H  \ " f_  6 Ÿ §(Ohm)_ 

^

‰> \  ¦ ‚  × þ ˜   H { 9 ì ø Í& h “   ‚  + þ A6 £ x² ú šs  : r~ ½ Ód ” `  ¦ G × þ ˜ô  Ç



. s M :
 
  H „  • ¸• ¸J $ ™" fü < Õ ª כ s  Ÿ í† < Ê   H y Œ ™û Z

“

  \  Å Ò3 l q # Œ ž Ð_  l – Ð ô  Ç . ] j 2  © œ\ " f  H F  g„  

•

¸• ¸ J $ ™" fü < ¿ º  % ò  \  ¦ + ‹" f y Œ ™û Z“   \  ¦ + þ Ad ”  o “ ¦ q

“ §ô  Ç . ] j 3 © œ\ " f  H    : r`  ¦ ë “ B  H  .

II. ° Ë Ñ ¹ Åy ¢y ¢ • Ö " eÑ ÷ P c p7 û ß Ã Å 8 ý ] k ùÅ k Ä× D

F 

gf  ¨ à º\  @ /ô  Ç ‚  + þ A6 £ x² ú šs  : r \ " f_  ¿ º  © œI _ ” > r  % ò

~

½ ÓZ O `  ¦ q “ § l  0 A # Œ | × ¼a Ë >s  \ O `  ¦ M :_   „   >  _  K x 9 ž Ðm î ß –`  ¦

H _eq = H _d + V = X

α

E _α a ^† _α a _α + V (1)



 “ ¦ s  כ \  @ /6 £ x   H o Ä ºq  ƒ  í ß – (Liouville oper- ator)\  ¦

L eq = L d + L v (2)



  .

E α   H  © œI  |α >\ " f_  é ß –{ 9  „   _  \  -t s “ ¦ a ^† _α (a α )  H  © œI  α\  @ /ô  Ç Ò q t$ í (™ èY > )ƒ  í ß – s  . s M :_  x 9

• ¸ƒ  í ß – \  ¦ ρ eq – Ð ¿ ºl – Ð ô  Ç . # Œl \  „   l  ü < ° ú  

“ É

r r ç ß –_ ” > r | × ¼a Ë >`  ¦  Œ •6   x r v €   >   H q ¨ î + þ As   ) a  .

Š

© œF G     H  (dipole approximation)\ " f_  | × ¼a Ë >_  K x 9 

ž Ðm î ß –“ É r

H _int = e X

l=1,2,3

X

αβ

(x _l ) _αβ a ^† _α a _β E _l exp(iωt) + c.c (3)

–

Ð Ñ ü t à º e ”  . e  H „   _  „  l | ¾ Ó_  ] X @ /u , x l “ É r é ß –{ 9 

„ 

 _  0 Au ý a³ ð– Ð" f x ¹ = x, x 2 = y, x 3 = z s “ ¦, ω  H

„ 

l  © œ_  y Œ •”  1 l x à º, c.c  H 4 Ÿ ¤ ™ è/ B NÓ  os  . e ” _ _  ƒ  í ß –  Y \  @ /K " f Y αβ =< α |Y |β >s  .  „   > _  „  À Ӄ   í

ß – J l (l = 1, 2, 3)“ É r J _l = X

λ

_l

,λ

₂

J _l ^∗ = X

λ

₁

,λ

₂

(j _l ) _λ

₁

_λ

₂

a ^† _λ

1

a _λ

₂

(4)

s

“ ¦ j l “ É r é ß –{ 9 „   _  „  À Ӄ  í ß – s  . ‚  + þ A  H  \ 

"

f_  s   „   > _  „  À Ó_  l @ /u (€ © œ © œ^  ¦¨ î ç  H)  H 6

Ÿ §(Ohm)_  + þ Ad ” Ü ¼– Ð j þ t à º e ”  . 7 £ ¤

< J k >= X

l

σ kl (ω)E l (ω) (5)



. ] j 1+ þ Ad ” \ " f  H

σ kl (ω) = −e X

α,β

(x l ) αβ A ^αβ _k (¯ ω)

= −e X

α,β

(x _l ) _αβ < (~¯ ω − L eq ) ⁻¹ J _k > _αβ (6)

s

“ ¦ ] j 2+ þ Ad ” \ " f  H

σ _kl (ω) = −e X

α,β

X

γ,δ

(x _l ) _αβ (j _k ) _γδ A ˜ ^γδ _αβ (¯ ω)

= −e X

α,β

X

γ,δ

(x _l ) _αβ (j _k ) _γδ < (~¯ ω − L eq ) ⁻¹ a ^† _γ a _δ > _αβ (7)

– Ð   H  .

#

Œl \ " f¯ ω ≡ ω − ia(a → 0 ^† ) s “ ¦

< X > _αβ = T _R ρ eq [X, a ^† _α α _β ]

(8)

s

“ ¦ T R   H  ^ ‰à ÔY Us Û ¼(many body trace) s “ ¦ [A, B] ≡ AB − BAs  .

1. < g 1] k ùÅ k Ä

]

j 1 + þ Ad ” \ " f_  ì  r K  “     H  6 £ § õ  ° ú   .

A ^αβ _k (¯ ω) =< (~¯ ω − L eq ) ⁻¹ J k > αβ (9)

s

 כ `  ¦ > í ß – l  0 A # Œ  6 £ § õ  ° ú  “ É r  % ò



(projector)P, Q\  ¦ & ñ _ ô  Ç .

P _k ^αβ X = < X > αβ

< J _k > _αβ J _k (10)

Q ^αβ _k = 1 − P _k ^αβ (11)

s

  % ò ƒ  í ß – \  ¦ d ”  (9)\  Ÿ í† < Ê÷ &# Q e ”   H o Ä ºq  ƒ   



_  š ¸ É rA á ¤ \  L ^eq = L eq (P _k ^αβ + Q ^αβ _k ) ü < ° ú  s  & h 6   x “ ¦ (A − B) ⁻¹ = A ⁻¹ + A ⁻¹ B(A − B) ⁻¹ \  ¦ + ‹" f ì  r K “   \  ¦

„ 

> h €    6 £ § õ  ° ú  s  S X ‰ © œ ) a  .

1

~¯ ω − L eq

J _k = 1

~ ω ¯ − L eq Q ^αβ _k − L eq P _k ^αβ J _k

= 1

~ ω ¯ − L eq Q ^αβ _k J _k + 1

~ ω ¯ − L eq Q ^αβ _k L _eq P _k ^αβ 1

~ ω ¯ − L eq

J _k

= 1

~ ω ¯ J _k + 1

~ ω ¯ − L eq Q ^αβ _k L _eq J _k A ^αβ _k (~¯ ω)

< J k > αβ

(12)

s

 כ `  ¦ & ñ o  €    6 £ § õ  ° ú  s   ) a  .

A ^αβ _k (¯ ω) = < J _k > _αβ

~ ω ¯ − Ω αβ − iΓ αβ (¯ ω) (13)

#

Œl \ " fΩ ^α⠍  H ‚  s 1 l x† < Êà º (lineshift function)s  9, iΓ αβ (ω)  H y Œ ™û Z“   (damping factor)s  . # Œl \ " f P _k ^αβ J k = J k ü < Q ^αβ _k J k = 0 `  ¦ s 6   x €   ‚   s 1 l x † < Êà º“   Ω _α⠍  H  6 £ § õ  ° ú  s  Å Ò# Q”   .

Ω αβ ≡ T R {ρ eq [L eq J k , a ^† _α a β ] }/ < J k > αβ (14)

 À

» d ” \ " f L eq 5 Å q \  Ÿ í† < Ê ÷ &# Q e ”   H L d _   © œ ± ú “ É r

 † ½ Ó t    H   # Œ > í ß – “ ¦, L ^d a ^† _α a β = E αβ a ^† _α a β `  ¦ s  6

  x €  ,

Ω _αβ ∼ = −E αβ (15)

  ) a  . # Œl \ " f E ^αβ = E α − E β s  .

d ”

 (13) \  Ÿ í† < Ê ÷ &# Q e ”   H y Œ ™û Z “     H  6 £ § õ  ° ú   .

iΓ ^αβ _k (¯ ω) < J k > αβ =< L eq Q ^αβ _k (~¯ ω −L eq Q ^αβ _k ) ⁻¹ L eq J k > αβ

(16) s

 כ `  ¦ > í ß – €  ,

iΓ αβ (ω) < J k > αβ = −E αβ < (~¯ ω − L eq Q ^αβ _k ) ⁻¹ L eq J k > αβ +T R {ρ eq [L v a ^† _α a β , (~¯ ω − L eq Q ^αβ _k ) ⁻¹ L eq J k ] }

− T R {ρ eq [L _eq a ^† _α a _β , J _k ] } < (~¯ ω − L eq Q ^αβ _k ) ⁻¹ L _eq J _k > _αβ / < J _k > _αβ (17)

s

 ÷ & 9, L ^d J k = ~ωJ k = constJ K s “ ¦,  © œ  ñ  Œ •6   x s  y

© œ t  · ú §“ É r  â Ä º\  [ Ò2 Ÿ ¤ A] \      6 £ § õ  ° ú  s    H  

 ) a  .

iΓ ^αβ _k (ω) < J _k > _αβ ≈ T R {ρ eq [L _v a ^† _α a _β , (~¯ ω − L d ) ⁻¹ L _v J _k ] } (18)

#

Œl \ " f

< J k > αβ = (f β − f α )(j k ) βα (19) s

 . # Œl \ " f f ^α ≡ T R {ρ eq [a ^† _α a α ] }– Ð  © œI  α\  @ /ô  Ç

„ 

 _  ` …Ø Ôp  n | Ã Ì ì  r Ÿ í† < Êà ºs  .   " f d ”  (13) \  d ”

(15) ü < d ”  (19)\  ¦ V , # Q" f ì  r K “   \  ¦  r  & h # Q˜ Ѐ  



6 £ § õ  ° ú  s   ) a  .

A _αβ (¯ ω) = (j _k ) _βα (f _β − f α )

~¯ ω − E βα − iΓ ^αβ _k (¯ ω) (20)

iΓ ^αβ _k (ω) = T R {ρ eq [L v a ^† _α a β , (~¯ ω −L d ) ⁻¹ L v J k ] }/ < J k > αβ

(21) L d ü < L _v  Å Ò# Qt €   A _αβ (¯ ω) _  ½ ¨^ ‰& h “   + þ AI \  ¦ ½ ¨

½ +

É Ã º e ”  .

2. < g 2] k ùÅ k Ä

]

j 2 + þ Ad ” _  ì  r K  “     H  6 £ § õ  ° ú   .

A ˜ ^γδ _αβ (¯ ω) =< (~¯ ω − L eq ) ⁻¹ a ^† _γ a δ > αβ (22)

_

ô  Ç .

P ˜ _γδ ^αβ X = < X > αβ

< a ^† γ a δ > αβ

a ^† _γ a δ (23)

Q ˜ ^αβ _γδ = 1 − ˜ P _γδ ^αβ (24)

s

 . s  כ `  ¦ + þ Ad ”  1\ " fü < ° ú  “ É r ~ ½ ÓZ O Ü ¼– Ð S X ‰ © œ €  

1

~ ω ¯ − L eq

a ^† _γ a δ = 1

~ ω ¯ − L eq Q ˜ ^αβ _γδ − L eq P ˜ _γδ ^αβ a ^† _γ a δ

= 1

~ ω ¯ − L eq Q ˜ ^αβ _γδ a ^† _γ a _δ + 1

~ ω ¯ − L eq Q ˜ ^αβ _γδ L _eq P ˜ _γδ ^αβ 1

~¯ ω − L eq

a ^† _γ a _δ

= 1

~¯ ω a ^† _γ a _δ + 1

~¯ ω L _eq Q ˜ ^αβ _γδ 1

~ ω ¯ − L eq Q ˜ ^αβ _γδ a ^† _γ a _δ + 1

~ ω ¯ − L eq Q ˜ ^αβ _γδ L _eq P ˜ _γδ ^αβ 1

~ ω ¯ − L eq

a ^† _γ a _δ

= 1

~ ω ¯ a ^† _γ a δ + 1

~ ω ¯ − L eq Q ˜ ^αβ _γδ L eq

< (~¯ ω − L eq ) ⁻¹ a ^† _γ a _δ > _αβ

< a ^† γ a δ > α β a ^† _γ a δ

(25)

  ) a  . # Œl \ " f

(~¯ ω) ⁻¹ L eq Q ˜ ^αβ _γδ (~¯ ω − L eq Q ˜ ^αβ _γδ ) ⁻¹ a ^† _γ a δ = (~¯ ω) ⁻¹ (~¯ ω − L eq Q ˜ ^αβ _γδ ) ⁻¹ L eq Q ˜ ^αβ _γδ a ^† _γ a δ (26) ü

< ° ú  s  " f– Ð “ § ¨ 8 Š > í ß –s  0 p x  . s  כ `  ¦ > í ß – €  

Q ˜ ^αβ _γδ a ^† _γ a δ = (1 − ˜ P _γδ ^αβ )a ^† _γ a δ = a ^† _γ a δ − ˜ P _γδ ^αβ a ^† _γ a δ = a ^† _γ a δ − < a ^† _γ a _δ > _αβ

< a ^† γ a δ > αβ

a ^† _γ a δ = 0 (27)

s

Ù ¼– Ð,

(~¯ ω) ⁻¹ L _eq Q ˜ ^αβ _γδ (~¯ ω − L eq Q ˜ ^αβ _γδ ) ⁻¹ a ^† _γ a _δ = 0 (28) s

  ) a  .  À » d ” `  ¦ “ ¦ 9 # Œ d ”  (22)\  ¦  r  & h # Q˜ Ѐ  ,

A ˜ ^γδ _αβ (¯ ω) = < a ^† _γ a δ > αβ

~ ω ¯ − ~¯ ω ˜ B αβ (¯ ω)/ < a ^† γ a δ > αβ

(29) s

  ) a  . # Œl \ " f

B ˜ _αβ (¯ ω) =

* 1

~ ω ¯ − L eq Q ˜ ^αβ _γδ L _eq a ^† _γ a _δ +

αβ

(30)

s

 . d ”  (29)\  ¦ 7 á §  8  © œ[ jy  > í ß – €  ,

< a ^† _γ a δ > αβ = T R {ρ eq [a ^† _γ a δ , a ^† _α a β ] }

= T _R {ρ eq [ a ^† _γ a _β δ _δα − a ^† α a _δ δ _γβ ] }

= (f γ − f α )δ γβ δ δα (31)

s

 ÷ &“ ¦, ˜ B αβ (¯ ω)   H  6 £ § õ  ° ú  s  S X ‰ © œ 0 p x  .

B ˜ _αβ (¯ ω) =

* 1

~¯ ω − L eq Q ˜ ^αβ _γδ L _eq a ^† _γ a _δ +

αβ

= 1

~ ω ¯ < L _eq a ^† _γ a _δ > _αβ + 1

~¯ ω < L _eq Q ˜ ^αβ _γδ 1

~ ω ¯ − L eq Q ˜ ^αβ _γδ L _eq a ^† _γ a _δ > _αβ (32)

< L v a ^† _γ a δ > α⠍  H b q ¢ ¸  H b ^† _−q ×  æ ô  Ç > h\  @ /ô  Ç € © œ © œ

^

 ¦ ¨ î ç  H s  ÷ &Ù ¼– Ð Õ ª ° ú כ“ É r % ò s  ÷ & 9, [ Ò2 Ÿ ¤ B] \  ¦ ‚ à Г ¦ 

#

Œ > í ß – €  ,

B ˜ _αβ (¯ ω) = 1

~ ω ¯ E _βα (f _β − f α ) + 1

~ ω ¯ < L _eq Q ˜ ^αβ _γδ 1

~ ω ¯ − L eq Q ˜ ^αβ _γδ L _d a ^† _β a _α > _αβ (33) s

 . s  d ” _  z ´ »† ½ Ó`  ¦ > í ß – l  0 A # Œ L ^v _  ] jY  L† ½ Ó  t ë ß –   H   “ ¦, L ^d J _k ^∗ = 0 s  ÷ & 9, T ^R {ρ eq [L eq Q ˜ ^αβ _γδ X, a ^† _α a β ] } = T _R {ρ eq [L _v a ^† _α a _β , X] }δ γβ δ _δα \  ¦ s 6   x €  ,

< L _eq Q ˜ ^αβ _γδ 1

~ ω ¯ − L eq Q ˜ ^αβ _γδ L _d a ^† _β a _α > _αβ = E _βα T _R {ρ eq [L _eq Q ˜ ^αβ _γδ (~¯ ω − L eq Q ˜ ^αβ _γδ ) ⁻¹ a ^† _β a _α , a ^† _α a _β ] }

∼ = T R {ρ eq [L v a ^† _α a β , (~¯ ω − L d ) ⁻¹ L v a ^† _β a α ] }δ γβ δ δα (34)

  ) a  .  À » d ” `  ¦ “ ¦ 9 # Œ Ä ºo  · ú ˜“ ¦    H ì  r K “   

\

 ¦  r  & h # Q ˜ Ѐ  

A ˜ ^γδ _αβ (¯ ω) = (f β − f α )

~ ω ¯ − E βα − Γ ^αβ γδ (¯ ω) (35)

  ) a  . # Œl \ " f y Œ ™û Z“   “   Γ ^αβ _γδ (¯ ω)   H

Γ ^αβ _γδ ∼ = T R {ρ eq [L _v a ^† _α a _β , (~¯ ω − L d ) ⁻¹ L _v a ^† _β a _α ] }δ γβ δ _δα (36) s

 . # Œl \ " f• ¸ L ^d ü < L v  & ñ K t €   ˜ A ^γδ _αβ (¯ ω)_  ½ ¨

^

‰& h “   + þ AI \  ¦ ½ ¨½ + É Ã º e ”  . ¿ º + þ Ad ” _  q “ §  H  6 £ § õ 

° ú   .

3. R w ‹



 " f ] j 1 + þ Ad ” _  „  • ¸• ¸ J $ ™" f  H σ _kl (ω) = −e X

α,β

(x _l ) _αβ (jk) _βα f β − f α

~ ω ¯ − E βα − i~Γ αβ

= −e X

α,β

(x l ) αβ (jk) βα

f β − f α

~ ω ¯ − E βα − < [L v a ^† α a β , (~¯ ω − L d ) ⁻¹ L v J k > 0 /< J k > αβ

(37)

 ÷ &“ ¦, ] j 2 + þ Ad ” _  „  • ¸• ¸ J $ ™" f  H σ _kl (ω) = −e X

α,β

(x _l ) _αβ (j _k ) _βα f _β − f α

~ ω ¯ − E βα − i~Γ αβ

= −e X

α,β

(x _l ) _αβ (j _k ) _βα f _β − f α

~¯ ω − E βα − < [L v a ^† α a β , (~¯ ω − L d ) ⁻¹ L v a ^† _β a α > 0 / < a ^† γ a β > αβ

(38)

s

 .

< x > ₀   H X _  ¨ î + þ A € © œ © œ^  ¦ ¨ î ç  H s  . ë ß –€  • L _v \  @ /6 £ x   H í ß –ê ø Í ( J $ ™[ >  V \  @ /K " f

< [L _v a ^† _α a _β , (~¯ ω − L d ) ⁻¹ L _v J _k ] > ₀ < a ^† _γ a _β > _αβ = < [L _v a ^† _α a _β , (~¯ ω − L d ) ⁻¹ L _v a ^† _α a _β ] > ₀ < J _k > _αβ (39)

–

Ð ì  r K  | ¨ c à º e ” Ü ¼€   ¿ º + þ Ad ” “ É r ° ú   ”   . \ V\  ¦ [ þ t # Q „    -Ÿ í 7 H > \ " f  H { 9 ì ø Í& h Ü ¼– Ð V = X

q

X

α,µ

C α,µ (q)a ^† _α a µ (b q + b ^† _−q ) (40)

1 l

x| ¾ Ós  ~q, −~q “   Ÿ í 7 H_  ™ èY > , Ò q t$ í ƒ  í ß – s  .) J k   H d ”

 (4)– Ð Å Ò# Qt Ù ¼– Ð ~ 1 >  d ”  (39)– Ð “  à ºì  r K  ÷ &t  · ú §  H



. [ d ”  (39) _  L _v \  Å Ò3 l q €   ~ 1 >  · ú ˜ à º e ”  .]   " f

¿

º + þ Ad ” “ É r  Ø Ô .

ì

ø ͕ ¸^ ‰\  & ñ  l  © œ`  ¦ “  r v €   „   [ þ t“ É r { ? / „  s  ü

< { ç ß –„  s \  ¦ { 9 Ü ¼†   . & ñ l  © œ`  ¦ Ù þ ¡`  ¦  â Ä º { 9 # Q




 H \  -t  ç ß –  s  { 9 & ñ ô  Ç  â Ä º\  ô  Ç # Œ  H ] j 1+ þ Ad ” s   8



À Òl  ~ 1  “ ¦ · ú ˜ 94 R e ”   [21–23]. Õ ª Q
 \  -t  ç ß –

 

s  { 9 & ñ ô  Ç f ” ] X „  s + þ A  s 9 þ t – Ðà ԏ : r / B N" î õ  ° ú  s  { 9 

&

ñ t  · ú §“ É r  â Ä º\ " f  H ] j 2+ þ Ad ” s   8  | à Ðf ”   . ] j 2+ þ Ad ” “ É r L d J _k ^∗ = 0 s  ÷ &Ù ¼– Ð { 9 ì ø Í& h Ü ¼– Ð   & h 6   x 0 p x 



.

III. + s Ç Â ] Ø

#

Œl \ " f Ä ºo   H ¿ ºt   % ò ƒ  í ß –  ~ ½ ÓZ O `  ¦ s 6   x 

#

Œ y Œ •y Œ • ‚  + þ A „  l  „  • ¸• ¸\  ¦ ½ ¨ % i  . y Œ •y Œ •_  „  • ¸• ¸  H L _v _  ] jY  L† ½ Ó  t    H   # Œ ½ ¨ % i  . s  כ “ É r e ” _ _  x 9

• ¸ ƒ  í ß – _  „   -Ÿ í 7 H  © œ  ñ Œ •6   x K x 9 ž Ðm î ß –“ É r  © œ  ñ



Œ

•6   x s  €  •ô  Ç > _  F  g„  • ¸• ¸_  : £ ¤f ç \   H ß ¼>  % ò † ¾ Ó`  ¦ p  u

t  3 l w l  M :ë  H s  . y Œ ™û Z† ½ Ó\ 
 
 e ”   H F  g„  • ¸• ¸ J $

™" f\   H „   ü < Ÿ í 7 H  s _  Ø  æ[  t õ & ñ ÷  rë ß –  m   ü @ Â

Ò\ " f Å Ò# Q”   y n C_  Å Ò à º_  / B N‰  ³ ´ òõ  ì ø Í% ò ÷ &# Q e ” 



. ¿ º + þ Ad ” \  e ” # Q" f L v  & ñ K t €   y Œ ™û Z“   \  ¦ ½ ¨ 

“

¦   " f F  g„  • ¸• ¸ J $ ™" f\  ¦ ½ ¨½ + É Ã º e ”  . Õ ª Q
 # QÖ ¼ A

á

¤ s   8 Ä »6   x >  æ ¼# Œt   H t \  @ /ô  Ç  7 H_   H L v _  + þ AI 

\

     \  ¦  כ Ü ¼– Ð l @ /ô  Ç . s \  @ /ô  Ç  © œ[ jô  Ç › ¸ 



 H · ú ¡Ü ¼– Ð Ã º' Ÿ ½ + É ƒ  ½ ¨ õ ] js  .

P c

p 8 ý ò k >

‘

: r ƒ  ½ ¨  H † < Ʋ D G ½ + ÉÕ ü t”  < É ª F é ß – ƒ  ½ ¨t " é ¶ (KRF-2002- 015-DP0122) _  t " é ¶ Ü ¼– Ð s À Ò# Q & ’ Ü ¼ 9 s \  y Œ ™ × ¼w n  m

 .

Y c

p w Š à U Ø ”  ô

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APPENDIX A

T R ρ eq [L eq A, a ^† _α a β ],

= T R {ρ eq [H eq A − AH eq , a ^† _α a β ] }

= T R {ρ eq ((H eq Aa ^† _α a β − AH eq a ^† _α a β ) }

− T R {ρ eq (a ^† _α a β H eq A − a ^† α a β AH eq ) } (A1) [H eq , ρ eq ] = 0 s “ ¦ T ^R {ABC} = T R {BCA} `  ¦ & h 6   x

€  ,

T R ρ eq [L eq A, a ^† _α a β ],

= T R {ρ eq ((Aa ^† _α a β H eq − AH eq a ^† _α a β ) }

− T R {ρ eq (a ^† _α a β H eq A − H eq a ^† _α a β A) } (A2)

 ÷ & 9, þ j7 á x& h Ü ¼– Ð  6 £ § õ  ° ú  “ É r   õ \  ¦ % 3   H  .

T R {ρ eq [L eq A, a ^† _α a β ] } = T R {ρ eq [L eq a ^† _α a β , A] } (A3)

APPENDIX B

< L eq a ^† _γ a δ > αβ = < L d a ^† _γ a δ > αβ + < L v a ^† _γ a δ > αβ

= < L e a ^† _γ a δ > αβ + < L p a ^† _γ a δ > αβ

+ < L v a ^† _γ a δ > αβ (B1)

< L _d a ^† _γ a _δ > _αβ = T _R ρ eq [ (L _e + L _p )a ^† _γ a _δ , a ^† _α a _β ]

= T R ρ eq [ L e a ^† _γ a δ , a ^† _α a β ]

= E γδ T R ρ eq [ a ^† _γ a δ , a ^† _α a β ]

= E γδ (f γ − f α )δ γβ δ αδ (B2)

< L p a ^† _γ a δ > αβ = 0 (B3)

< L _v a ^† _γ a _δ > _αβ = 0 (B4)

Comparison of Damping Factors in the Optical Conductivity of Many-electron System in Solids by Using Two Projection Techniques

Hyun Jung Lee, Yun Ju Lee, Jai Hoon Lee and So Youn Kim Optical Transition Laboratory, Kyungpook National University, Daegu 702-701

Nam Lyong Kang

Faculty of Liberal Arts, Miryang National University, Miryang 627-702

Jung Young Sug

Faculty of Electronic and Electrical Engineering, Kyungpook National University, Daegu 702-701

Sang Gyu Jo and Sang Don Choi ^∗

Department of Physics, Kyungpook National University, Daegu 702-701 (Received 17 April 2003)

Two different projection techniques are applied to derive the damping factors contained in the optical conductivity tensors of a many-electron system in solids. The damping terms contain the effect of the external radiation, as well as the electron phonon interaction. The two results turn out to be different form each other in general. The first form is well known to be applicable to cyclotron transitions in systems with constant energy separations. The second one seems to be more suitable for transitions with irregular energy separations.

PACS numbers: 72

Keywords: Optical conductivity, Many-electron system, Projection technique, Damping term

∗

E-mail: [email protected]