Š= k ¹ Å 4 8 ý ° Ë Ñ ¹ Åy ¢y ¢5 8 ý P c p7 û ß Ã Å ; c 6 ” X ¢ ‘ ¤ ± n É ¿ R <U ê s0 n É8 ý ƒ º Z ÍX ì Ä R w ‹

w

Š= k ¹ Å 4 8 ý ° Ë Ñ ¹ Åy ¢y ¢5 8 ý P c p7 û ß Ã Å ; c 6 ” X ¢ ‘ ¤ ± n É ¿ R <U ê s0 n É8 ý ƒ º  Z ÍX ì Ä R w ‹



™

»€ Ða : @ · T g ` @+ ä  · L |„ ç ¡% ã <

 â

· ¡ ¤ @ /† < Ɠ § Ó ü t o † < Æõ , @ /½ ¨ 702-701

~ ç

¡ ‘ žó j u ^∗

x 9

€ ª œ@ /† < Ɠ § “ §€ ª œõ & ñ  Ò, x 9 € ª œ 627-702 (2004¸   3 Z 4 3{ 9  ~ à Î6 £ §)

l

” > r_  s  : r \ " f  6   xô  Ç  % ò ~ ½ ÓZ O õ    É r ~ ½ ÓZ O `  ¦ + ‹" f “ ¦^ ‰? /_  Ÿ í 7 H õ   © œ  ñ Œ •6   x   H „   [ þ t_  F  g

„

 • ¸• ¸\  @ /ô  Ç D h– Ðî  r s  : r`  ¦ ™ è> hô  Ç . s X O >  • ¸Ø  ¦ô  Ç „  • ¸• ¸  H s „  _  s  : r % ƒ! 3  ‚  + þ A† ½ Ó÷  rë ß – m 



 q ‚  + þ A† ½ ӕ ¸ Ÿ í† < Ê t ë ß –, # Œl \  Ÿ í† < Ê÷ &  H y Œ ™û Z“    „   ü < Ÿ í 7 H_  ì  r Ÿ í† < Êà º\  ¦ ˜ Ð  & h ] X y  Ÿ í

† <

Ê “ ¦ e ” # Q" f „   _  „  s  õ & ñ \ " f Ÿ í 7 H s  ~ ½ ÓØ  ¦ ÷ &“ ¦ f  ¨ à º÷ &  H € ª œ © œ`  ¦ [ O " î   H X < • ¸¹ ¡ §`  ¦ ï  r  . Õ ª A

" f s    õ   H s „  _    õ ˜ Ð  { 9 ì ø Í& h s    H  כ `  ¦ ˜ Ð# ŒÅ ғ ¦ e ”  .

PACS numbers: 71.38.+i, 72.10.-d, 72.20.-i Keywords: ƒ  í ß –  @ /à ºl Z O 

I. " e  ] Ø

ì

ø ͕ ¸^ ‰? /_  „   [ þ t_   1 l x \  @ /ô  Ç ƒ  ½ ¨  H € ª œ : Ÿ x > % i 

† <

Æ& h  ~ ½ ÓZ O `  ¦ æ ¼  H  כ s  ˜ Ð: Ÿ x s  . s  Qô  Ç + þ Ad ” \ " f  H „   l

„  • ¸• ¸ J $ ™" f Å Ò  ) a % i ½ + É`  ¦ ô  Ç . õ  _  ƒ  ½ ¨\ " f  H Å

Җ Ð ‚  + þ A6 £ x² ú š  Òì  rë ß –s  › ' a d ” _  @ / © œs  ÷ &# Q M ® o  [1-9].

‘

: r ƒ  ½ ¨”  “ É r ƒ  í ß –  @ /à º~ ½ ÓZ O _  ô  Ç > : Ÿ x“    % ò ~ ½ ÓZ O `  ¦ +

‹" f s  ƒ  ½ ¨\  ¦ ô  Ç  â + « >s  e ” % 3 Ü ¼
 Å Ò  ) a ƒ  ½ ¨  H ‚  + þ A6 £ x

²

ú š\  u Ä º5 g e ”  [4,6,7-9]. Õ ªA " f 2002¸   Ÿ í 7 H õ  „    _

  © œ  ñ Œ •6   x s  e ”   H q ‚  + þ A õ & ñ `  ¦ ™ è> hÙ þ ¡Ü ¼
   H  >  í

ß –s % 3  [14]. þ j  H \  D h– Ðî  r ~ ½ ÓZ O \  _  # Œ s  ë  H ] j\  ] X

  H K  ˜ Ѐ Œ ¤  H X <  ™ è   É r   õ \  ¦ % 3 % 3  [15]. ‘ : r ƒ  ½ ¨

\

" f  H s  ¿ º ~ ½ ÓZ O Ü ¼– Ð ½ ¨ô  Ç y Œ ™û Z“   \  ¦ Ÿ í F ‹ c& h Ü ¼– Ð q 

“

§ô  Ç .

‘

: r ƒ  ½ ¨\ " f ½ ¨ô  Ç   õ   H ‚  + þ A Òì  r÷  rë ß –  m   q ‚   + þ

A Òì  r • ¸ Ÿ í† < Ê “ ¦ „   -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† < Êô  Ç . y Œ •y Œ •_  í ß – ê

ø ͓   \  ¦ „   -„    ¢ ¸  H „   -Ô  ¦í  HÓ ü t  © œ  ñ Œ •6   x \  t 

&

h 6   x½ + É Ã º e ”   H t  › ¸ ô  Ç . ‚  + þ AF  g„  • ¸• ¸\  @ /K " f  H l

” > r_  ƒ  ½ ¨ [ þ t(Choi, Kawabata, Badjou 1 p x)_  + þ Ad ” [1- 9] õ  q “ § “ ¦ 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 ºü < \  -t  ç ß –  s  { 9 & ñ t  · ú §



 H ç ß –] X „  s + þ A\  › ' a K " f• ¸  7 H_ ô  Ç . q ‚  + þ A F  g„  • ¸• ¸

∗

E-mail: [email protected]

\

 @ /K " f  H Shen, Bloembergen, Suzuki 1 p x[10-13]_  + þ A d ”

õ  q “ § “ ¦ „   l  _   1 l x† < Êà º_  ½ + Ë$ í † ½ Ó\  Å Ò3 l qô  Ç



. ¢ ¸ ] j 2  q ‚  + þ A† ½ Ós  # Q‹ "  g 1 J s  | ¨ c  כ “  t \  ¦ \ V8 £ ¤ ô 

Ç . €  $  II] X \ " f  H ¿ º   õ \  ¦ % 3   H õ & ñ `  ¦ ™ è> h “ ¦ III] X õ  IV] X \ " f  H ¿ º ~ ½ ÓZ O _    õ \  ¦ ½ ¨^ ‰& h Ü ¼– Ð ½ ¨ 

“

¦ = å Q Ü ¼– Ð V] X \ " f  H ¿ º   õ \  ¦ q “ § €  " f ž Ð_  l 

– Ð ô  Ç .

II. ° Ë Ñ ¹ Åy ¢y ¢8 ý ‘ ¤ ] k ùÅ k Ä

r

ç ß –\  _ ” > r   H „  l  © œ E(t) K | 9  M : „   ü < Ÿ í



7 H s   © œ  ñ Œ •6   x   H  „   > _  Hamiltonian H(t)“ É r H(t) = H eq + H int (t), (1) s

 . # Œl " f H int (t)  H r ç ß –_ ” > r† ½ Ós “ ¦ H eq   H r ç ß –\  _ 

”

> r t  · ú §  H ¨ î + þ A © œI _  HamiltonianÜ ¼– Ð

H _eq = H _d + V = H _e + H _p + V (2) s

 . H e , H _p , V   H y Œ •y Œ • „   , Ÿ í 7 H, „   ü < Ÿ í 7 H_   © œ  ñ



Œ

•6   x \  @ /ô  Ç K x 9 ž Ðm î ß –s “ ¦ H e = X

α

E α a ⁺ _α a α (3)

H p = X

q

~ ω q b ⁺ _q b q (4)

-77-

V = X

q

X

α,µ

C _α,µ (q)a ⁺ _α a _µ (b _q + b ⁺ _−q ) (5)

–

Ð Å Ò# Q”   . # Œl " f a ⁺ α (a _α )  H \  -t  E α “   |α >  © œ I

\ " f_  „   _  Ò q t$ í (™ èY > )ƒ  í ß – s “ ¦ b q (b ⁺ _−q )  H \  - t

 ~ω ^q “   |q >  © œI _  Ÿ í 7 H_  Ò q t$ í (™ èY > )ƒ  í ß – s  9, ω q   H  à º 7 ˜'  q“   Ÿ í 7 H_  ”  1 l x à ºs  . Ÿ í 7 H_   à º 7 ˜'  q ü < ì  rF G t à º s\  @ /K " f q = (q, s)– Ð & ñ _ ÷ &“ ¦ C ^α,µ ≡<

α |C(q)|µ >  H „   ü < Ÿ í 7 H_   © œ  ñ Œ •6   xƒ  í ß –  C(q)_  ' Ÿ 

§ >

=כ ¹™ è\  ¦
  · p .

Liouville ~ ½ Ó& ñ d ”  i~ ∂ρ(t)

∂t = [H(t), ρ(t)] ≡ L(t)ρ(t). (6)

\

 ¦ + ‹" f x 9 • ¸ƒ  í ß – _  ρ(t)_  þ j$  † ½ Ó`  ¦ ½ ¨ €  [ref. 14

‚ à Г ¦]

ρ(t) = ρ eq + ρ ⁽¹⁾ (t) + ρ ⁽²⁾ (t)

= ρ eq + 1

~ Z _∞

0

dt 1 e ^−iL

^eq

^t

¹

^/~ L int (t − t 1 )ρ eq

+  1 i~

 ² Z ∞ 0

dt ₂ e ^−iL

^eq

^t

¹

^/~ L _int (t − t 1 )e ^−iL

^eq

^t

²

^/~

× L int (t − t 1 − t 2 )ρ eq (7) s

 . # Œl " f ρ _eq   H ¨ î + þ A © œI _  x 9 • ¸ƒ  í ß – s “ ¦ ρ ⁽ⁿ⁾ (t)  H L int \  ¦ n   Ÿ í† < Êô  Ç . L eq   H H eq \ , L int   H H int \  y Œ •y Œ • @ /6 £ x   H Liouville ƒ  í ß – s  . s  כ `  ¦ + ‹

"

f „  À Ӄ  í ß –  J ⁱ (i = x, y, z ¢ ¸  H +, −)_  l @ /u _  ‚   + þ

A Òì  r(< J _i ⁽¹⁾ >) õ  þ j$   q ‚  + þ A Òì  r(< J _i ⁽²⁾ >)

< J _i >=< J _i ⁽¹⁾ > + < J _i ⁽²⁾ > (8)

\

 ¦ ½ ¨ô  Ç .

ô 

Ǽ # , „  À Ӄ  í ß – _   ^ ‰³ ð‰ & ³“ É r J i = X

γδ

(j i ) γδ a ⁺ _γ a δ (9)

–

Ð j þ t à º e ”  . # Œl " f j i   H é ß –{ 9 „   _  „  À Ӄ    s “ ¦ γ, δ 1 p x“ É r é ß –{ 9 „   _   © œI t à ºs  . ¢ ¸ô  Ç, e ” _ _  ƒ  í ß –



 X\  @ /K 

(X) γδ ≡< γ|X|δ > (10)

–

Ð & ñ _ ô  Ç .

„ 

l  © œ 7 ˜' 

E _i (t) = E _i e ^iωt (11) _

 + þ AI “   „   l  \  @ /ô  Ç d ”  (1)_  r ç ß –_ ” > r K x 9 ž Ðm  î

ß – H(t) int   H H int (t) = e X

l=x,y,z

X

α,β

(x l ) αβ a ⁺ _α a β E l exp(iωt) + c.c. (12) s

 . e  H „   _  „   | ¾ Ó, x _l “ É r „   _  0 Au , ω  H „  l  © œ _  ”  1 l x à ºs “ ¦ c.c  H 4 Ÿ ¤ ™ è/ B NÓ  o`  ¦ _ p ô  Ç .

(a) ‚  + þ A Òì  r: ρ ⁽¹⁾ (t) \  @ /ô  Ç J _i _  € © œ © œ^  ¦ ¨ î ç  H`  ¦ 2 [ô  Ç



.

< J ⁽¹⁾

ⁱ

>= T R {ρ ⁽¹⁾ (t)J i } = X

k

σ ik (ω)E k (ω) (13)

#

Œl " f T R “ É r „   -Ÿ í 7 H > _  à ÔY Us Û ¼(trace)s  .

+ þ

Ad ”  I: d ”  (13)\  ¦ Õ ª@ /– Ð “ ¦ 9ô  Ç .

σ ik (ω) = −e X

αβ

(x k ) αβ A ^αβ _i (¯ ω) (14)

A ^αβ _i (¯ ω) =< (~¯ ω − L eq ) ⁻¹ J _i > _αβ (15) + þ

Ad ”  II: d ”  (15)\  d ”  (9)\  ¦ @ /{ 9  # Œ “ ¦ 9ô  Ç



.

σ ik (ω) = −e X

αβ

X

γδ

(x k ) αβ (j i ) γδ A ¯ ^γδ _αβ (¯ ω) (16)

A ¯ ^γδ _αβ (¯ ω) =< (~¯ ω − L eq ) ⁻¹ a ⁺ _γ a _δ > _αβ (17)

#

Œl " f  6 £ § õ  ° ú  s  & ñ _ ô  Ç .

< X > αβ ≡ T R {ρ eq [X, a ⁺ _α a α ] } (18) (b) q ‚  + þ A Òì  r: ρ ⁽²⁾ (t) \  @ /ô  Ç J i _  € © œ © œ^  ¦ ¨ î ç  H`  ¦ 2 [ ô 

Ç .

< J _i ⁽²⁾ >= T R {ρ ⁽²⁾ (t)J i } = X

jk

σ ijk (ω 1 , ω 2 )E j (ω 1 )E k (ω 2 ) (19) + þ

Ad ”  I: d ”  (19)\  ¦ Õ ª@ /– Ð “ ¦ 9ô  Ç .

σ _ijk (ω ₁ , ω ₂ ) = e ² X

ξ,

X

α,β

(r _j ) _ξ (r _k ) _αβ B _ξ ^αβ (¯ ω ₁ , ¯ ω ₂ ) (20)

B _ξ ^αβ (¯ ω 1 , ¯ ω 2 ) = T R {ρ eq [(~¯ ω 2 − L eq ) ⁻¹ T ξ (¯ ω), a ⁺ _α a β ] } (21) T ξ (¯ ω) = [(~¯ ω − L eq ) ⁻¹ )J i , a ⁺ _ξ a  ] (22) + þ

Ad ”  II: d ”  (22)\  d ”  (9)\  ¦ @ /{ 9  # Œ “ ¦ 9ô  Ç .

σ _ijk (ω ₁ , ω ₂ ) = e ² X

α,β

X

γ,δ

X

l,m

(r _j ) _αβ (r _k ) _γδ (j _i ) _lm U _αβ (¯ ω ₁ , ¯ ω ₂ )

(23)

U αβ (¯ ω 1 , ¯ ω 2 ) = T R {ρ eq [(~¯ ω 2 − L eq ) ⁻¹ [(~¯ ω 12 − L eq ) ⁻¹ a ⁺ _l a m , a ⁺ _γ a δ ], a ⁺ _α a β ] } (24)

#

Œl \ " f ω ≡ ω 1 + ω 2 , ¯ ω 1 ≡ ω 1 − ib(b → 0 ⁺ ), ¯ ω 2 ≡ ω 2 − ic(c → 0 ⁺ ) s  . + þ Ad ”  Iõ  + þ Ad ”  II_  s & h “ É r  6 £ § õ  ° ú  



; + þ Ad ”  I\ " f  H J i \  L d \  ¦ & h 6   x €   L _d J _i = c à º×J i  :

£

¤Z > ô  Ç  â Ä º\ ë ß – $ í w n  t ë ß – + þ Ad ”  II\ " f  H L d a ⁺ _l a m = (E _l − E m )a ⁺ _l a _m s  † ½ Ó © œ $ í w n ô  Ç .   " f + þ Ad ”  II + þ A d ”

 I\  q K  { 9 ì ø Í& h s   ½ + É Ã º e ”  ( [ jô  Ç  כ “ É r + '\ " f

 7 Hô  Ç ).

III. ] k ùÅ k Ä I8 ý 4  ˜ m

(a) ‚  + þ A Òì  r d ”

 (14)\  ¦ > í ß – l  0 AK  ] j 0   % ò ƒ  í ß –  P 0 , Q ₀ \  ¦ P 0 X ≡ < X > αβ

< J i > αβ

J i (25)

Q 0 ≡ 1 − P 0 (26)

–

Ð & ñ _  “ ¦ † ½ Ó1 p xd ” 

(A + B) ⁻¹ = A ⁻¹ + A ⁻¹ B(A + B) ⁻¹ (27)

`

 ¦  6   x # Œ d ”  (15)_  L eq _  š ¸ É rA á ¤ \  P ⁰ + Q 0 = 1\  ¦ @ / { 9

 # Œ „  > h €  , 7 £ ¤

(~ω − L eq )J i = (~ω − L eq Q 0 − L eq P 0 ) ⁻¹ J i

= (~ω − L eq Q 0 ) ⁻¹ J i + (~ω − L eq Q 0 ) ⁻¹

× L eq P ₀ (~ω − L eq ) ⁻¹ J _i (28)

`

 ¦  6   x # Œ A ^αβ i (¯ ω)\  ¦ > í ß – €  

A ^αβ _i (¯ ω) = < J i > αβ

~¯ ω − E β + E _α − iΓ αβ (¯ ω) (29)

  ) a  . # Œl " f P 0 J _i = J _i , Q ₀ J _i = 0`  ¦  6   xÙ þ ¡“ ¦ „    ü

< Ÿ í 7 H_   © œ  ñ Œ •6   x s  €  •  “ ¦ & ñ # Œ L v _  ] jY  L½ + É



t ë ß – “ ¦ 9 €  

iΓ αβ (¯ ω) ≈ T R {ρ eq [L v a ⁺ _α a β , (~¯ ω −L d ) ⁻¹ L v J i ] }/ < J i > αβ

(30) s

  ) a  . d ”  (30)“ É r  % ò ƒ  í ß – \  ¦ Ÿ í† < Ê t  · ú §“ ¦ ì  r — ¸

\

  H L v („   -Ÿ í 7 H  © œ  ñ Œ •6   x K x 9 ž Ðm î ß – V \  @ /6 £ x   H Liouville ƒ  í ß – ) † ½ Ós  \ O Ü ¼Ù ¼– Ð > í ß –s  0 p x  .   

"

f d ”  (30)õ 

< J i > αβ = (f β − f α )(j i ) βα (31)

`

 ¦ d ”  (29)\  @ /{ 9  €   d ”  (14)  H

σ ik (ω) = −e X

αβ

(j i ) βα (r k ) αβ

f β − f α

~ ω ¯ − E βα − iΓ αβ (¯ ω) . (32) s

  ) a  . # Œl " f E βγ ≡ E β − E γ , f _α   H  © œI  α\  @ /ô  Ç „  



_  ` …Ø Ôp  ì  r Ÿ í† < Êà ºs “ ¦

iΓ αβ (¯ ω) = X

ξγ

X

q

 [(j i ) ξγ C βξ (q) − (j i ) βξ C ξγ (q)]C γα ( −q) (f β − f α )(j i ) βα



×  (1 + N _−q )f γ (1 − f β ) − N _−q f β (1 − f γ )

~¯ ω − E βγ − ~ω _−q − (1 + N q )f β (1 − f γ ) − N q f γ (1 − f β )

~¯ ω − E βγ + ~ω _q



− X

ξγ

X

q

 [(j i ) ξα C γξ (q) − (j i ) γξ C ξα (q)]C βγ ( −q) (f _β − f α )(j _i ) _βα



×  (1 + N _−q )f _α (1 − f γ ) − N _−q f _γ (1 − f α )

~ ω ¯ − E γα − ~ω −q − (1 + N _q )f _γ (1 − f α ) − N q f _α (1 − f γ )

~ ω ¯ − E γα + ~ω q



. (33)

s

 . N ^q   H Ÿ í 7 H \  @ /ô  Ç e  ¦| ½ Óß ¼ ì  r Ÿ í† < Êà ºs  . d ”  (33)_  Ó

ü

t o & h  _ p   H  6 £ § õ  ° ú   ; ' Í † ½ ӓ É r „    ×  æç ß – © œI 

γ \ " f þ j7 á x © œI  β– Ð „  s    H  כ `  ¦ _ p ô  Ç . 7 £ ¤, 1 +

N q   H Ÿ í 7 H_  ~ ½ ÓØ  ¦`  ¦ _ p  “ ¦ f γ (1 − f β )  H γ → β „  s 

\

 ¦ _ p ô  Ç . ì  r — ¸  H E γ + ~ω = E β + ~ω q s Ù ¼– Ð \  -t 

˜

Д > rZ O g Ë :s  $ í w n † < Ê`  ¦ _ p ô  Ç .
 Qt  † ½ Ó[ þ t • ¸ ° ú  “ É r ~ ½ Ó Z O

Ü ¼– Ð K $ 3 ½ + É Ã º e ” “ ¦ — ¸Ž  H † ½ Ó\ 
 
  H f β − f ⠍  H œ í l

 © œI  α\ " f þ j7 á x © œI  α– Ð „  s † < Ê`  ¦ _ p   .

(b) q ‚  + þ A Òì  r d ”

 (20)`  ¦ > í ß – l  0 AK  ] j 1   % ò ƒ  í ß –  P ¹ , Q 1 \  ¦ P 1 X ≡ < X > _αβ

< T ξ (¯ ω) > αβ

T ξ (¯ ω) (34)

Q 1 ≡ 1 − P 1 (35) ü

< ° ú  s  & ñ _ ô  Ç . < X > ^α⠍  H d ”  (18)õ  ° ú   . d ”  (28)`  ¦ Ä

»• ¸½ + É M :ü < ° ú  “ É r ~ ½ ÓZ O Ü ¼– Ð d ”  (22)_  L _eq _  š ¸ É rA á ¤ \  P 1 + Q 1 = 1`  ¦ @ /{ 9  “ ¦ d ”  (27)\  ¦  6   x # Œ „  > h €  

B _ξ ^αβ (¯ ω ₁ , ¯ ω ₂ ) = < T _ξ (¯ ω) >

~¯ ω ₂ − ~¯ ω ₂ D ^αβ _ξ (¯ ω ₁ , ¯ ω ₂ ) (36) s

  ) a  . # Œl " f P ¹ T ξ (¯ ω) = T ξ (¯ ω), Q 1 T ξ (¯ ω) = 0`  ¦   6

  xÙ þ ¡“ ¦

D _ξ ^αβ (¯ ω ₁ , ¯ ω ₂ ) < T _ξ (¯ ω) > _αβ

= − 1

~ ω ¯ 2

T R {ρ eq [[(~¯ ω − L eq Q 1 ) ⁻¹ J i , a ⁺ _ξ a  ], L eq a ⁺ _α a β ] }

− 1

~¯ ω ₂ T R {ρ eq [(~¯ ω − L eq Q 1 ) ⁻¹ L eq T ξ (¯ ω), L eq a ⁺ _α a β ] } + B ^αβ _ξ (¯ ω 1 , ¯ ω 2 )

~¯ ω ₂ < T _ξ (¯ ω) > _αβ T _R {ρ eq [T _ξ (¯ ω), L _eq a ⁺ _α a _β ] } (37) s

 . d ”  (36)õ  (37)_  < T ξ (¯ ω) >\  ¦ > í ß – l  0 AK  ] j 2   % ò ƒ  í ß –  P 2 , Q ₂ \  ¦  6 £ § õ  ° ú  s  & ñ _ ô  Ç .

P ₂ X ≡ < X > ^ξ

< J i > ^ξ J _i (38)

Q 2 ≡ 1 − P 2 (39)

#

Œl " f

< X > ^ξ ≡ T R {ρ eq [[X, a ⁺ _ξ a  ], a ⁺ _α a β ] } (40) s

 . d ”  (28)õ  d ”  (36)\  ¦ Ä »• ¸½ + É M :ü < ° ú  “ É r ~ ½ ÓZ O Ü ¼– Ð

< T ξ (¯ ω) >\  ¦ „  > h €  

< T _ξ (¯ ω) > _αβ = < J _i > ^ξ

~ ω ¯ − U _ξ ^αβ − W _ξ ^αβ (41) s

  ) a  . # Œl " f

U _ξ ^αβ = (f β − f α )(j _i ^βξ E βξ δ α − j i ^α E α δ ξβ )/ < J i > ^ξ

(42)

< J i > ^ξ = (f β − f α )(j ^βξ _i δ α − j i ^α δ ξβ ), (43) s

“ ¦ y Œ ™û Z“   (damping factor) W ξ ^α⠍  H

W _ξ ^αβ < J i > ^ξ ≡ < L eq Q 2 (~¯ ω − L eq Q 2 ) ⁻¹ L eq J i > ^ξ

= −T R {ρ eq [[(~¯ ω − L d ) ⁻¹ L _v J _i , L _v a ⁺ _ξ a _ ] } + T R {ρ eq [[(~¯ ω − L d ) ⁻¹ L v J i , a ⁺ _ξ a  ], L v a ⁺ _α a β ] }

(44) s

 . # Œl " f † ½ Ó1 p xd ” 

T R {ρ eq [[L eq X, A], B] } = 7T R {ρ eq [[L eq A, X], B] } + T _R {ρ eq [L _eq B, [X, A] } (45)

`

 ¦  6   xÙ þ ¡“ ¦ „   -Ÿ í 7 H  © œ  ñ Œ •6   x s  €  •  “ ¦ & ñ # Œ L v _  2 † ½ Ó t ë ß – “ ¦ 9Ù þ ¡ . d ”  (44)\   H  % ò ƒ    

Ÿ

í† < Ê÷ &# Q e ” t  · ú §“ ¦ ì  r — ¸\   H L v  \ O Ü ¼Ù ¼– Ð > í ß –s   0

p

xô  Ç + þ AI s  .   " f   õ   H

σ ijk (ω 1 , ω 2 ) = e ² X

j,k

X

ξ,

X

α,β

"

(r j ) ξα (r k ) αβ

[~¯ ω − E βξ − Γ 1 (¯ ω)]

(f β − f α )j _i ^βξ δ α

[~¯ ω 2 − E βα − Γ 2 (¯ ω 2 )]

− (r j ) βξ (r k ) αβ

[~¯ ω − E ξα − Γ 3 (¯ ω)]

(f β − f α )j _i ^α δ ξβ

[~¯ ω 2 − E βα − Γ 4 (¯ ω 2 )]



(46) s

 . # Œl " f

Γ 1 (¯ ω) = X

q

X

µ,γ

"

C µγ (q)C γξ ( −q)j i ^αµ − C βµ (q)C γξ ( −q)j ^µγ i

(f β − f α )j _i ^βξ

#

×  f β (1 − f γ ) + N q (f β − f γ )

~¯ ω + E _βγ − ~ω q

− f α (1 − f γ ) + N q (f α − f γ )

~ ω + E ¯ _βγ − ~ω q

+ f _γ (1 − f α ) + N _−q (f _γ − f α )

~ ω + E βγ + ~ω q

− f _γ (1 − f β ) + N _−q (f _γ − f β )

~ ω + E ¯ βγ + ~ω q



− X

q

X

µ,γ

"

C γµ (q)C βγ ( −q)j ^µξ i − C µξ (q)C βγ j _i ^µξ (f β − f α )j _i ^βξ

#

×  f _γ (1 − f α ) + N _q (f _γ − f α )

~ ω + E ¯ γξ − ~ω q − f _α (1 − f γ ) + N _−q (f _α − f γ )

~ ω + E ¯ γξ + ~ω q



(47)

Γ 2 (¯ ω 2 ) = X

q

X

γ

 C γβ (q)C βγ ( −q) (f β − f α )

  f γ (1 − f α ) + N q (f γ − f α )

~¯ ω 2 + E γα − ~ω q − f α (1 − f γ ) + N q (f α − f γ )

~¯ ω 2 + E γα + ~ω q



+ X

q

X

γ

 C αγ (q)C γα ( −q) (f _β − f α )

  f β (1 − f γ ) + N q (f β − f γ )

~¯ ω ₂ + E _βγ − ~ω q

− f γ (1 − f β ) + N _−q (f γ − f β )

~ ω ¯ ₂ + E _βγ + ~ω _q



. (48)

Γ 3 (¯ ω) = X

q

X

µ,γ

 C _µγ (q)C γ ( −q)j i ^µα − C µα (q)C γ ( −q)j ^γµ i

(f β − f α )j _i ^α



×  f γ (1 − f α ) + N q (f γ − f α )

~ ω + E ¯ _γα − ~ω q

− f γ (1 − f β ) + N q (f γ − f β )

~¯ ω + E _γα − ~ω q

+ f β (1 − f γ ) + N _−q (f β − f γ )

~ ω + E ¯ _γα + ~ω _q − f α (1 − f γ ) + N _−q (f α − f γ )

~¯ ω + E _γα + ~ω _q



− X

q

X

µ,γ

 C _µ (q)C _γα ( −q)j i ^µγ − C µγ (q)C _γα ( −q)j ^µ i

(f β − f α )j _i ^α



×  f β (1 − f γ ) + N q (f β − f γ )

~ ω + E ¯ γ − ~ω q − f γ (1 − f β ) + N _−q (f γ − f β )

~¯ ω + E γ + ~ω q



(49)

Γ 4 (¯ ω 2 ) = X

q

X

γ

 C _αγ (q)C _γα ( −q) (f β − f α )

  f _β (1 − f γ ) + N _q (f _β − f γ )

~ ω ¯ 2 + E βγ − ~ω q − f _γ (1 − f β ) + N _−q (f _γ − f β )

~ ω ¯ 2 + E βγ + ~ω q



+ X

q

X

γ

 C γβ (q)C βγ ( −q) (f _β − f α )

  f γ (1 − f α ) + N q (f γ − f α )

~¯ ω ₂ + E _γα − ~ω q

− f α (1 − f γ ) + N _−q (f α − f γ )

~ ω ¯ ₂ + E _γα + ~ω _−q



. (50)

d ”

 (46)-(50)`  ¦ Ä »• ¸½ + É M : ‚  + þ A Òì  r õ   ð ø Ít – Ð L v _  2 † ½ Ó  t ë ß – “ ¦ 9 % i “ ¦ / B N" î & h  Â Ò   H \ " f „  > hÙ þ ¡ . 7 £ ¤,

~ ω 2 ≈ E α − E β , ~ω ≈ E β − E ξ \  ¦ & ñ % i  . ¢ ¸ô  Ç, Q 1 L d J i = 0`  ¦ & ñ Ù þ ¡ . s  כ “ É r 1 p x \  -t  ç ß –  “    â Ä º

\

 $ í w n ô  Ç (\ V\  ¦ [ þ t # Q  s 9 þ t – Ðà ԏ : r / B N" î ). þ j7 á x   õ 

“

  d ”  (46)-(50 )“ É r Suzuki 1 p x[12] õ  Shen 1 p x[11,13]_     õ

_  › ¸½ + ËÜ ¼– Ð K $ 3 ½ + É Ã º e ”  . 7 £ ¤, Suzuki 1 p x_    õ ˜ Ð



  H „   _  ì  r Ÿ í† < Êà º Ó ü t o & h Ü ¼– Ð K $ 3  l  ¼ # o  

•

¸2 Ÿ ¤ Ÿ í† < Ê÷ &# Q e ” “ ¦ Shen 1 p x_    õ ü <  H „  l „  • ¸• ¸_  + þ

AI  7 £ ¤, d ”  (46)“ É r q 5 p w t ë ß – Õ ª[ þ t_    õ \   H Å Ò# Qt  t

 · ú §“ É r Γ_  + þ AI \  ¦ ‘ : r  7 Hë  H \ " f  H ½ ¨^ ‰& h Ü ¼– Ð ½ ¨ % i 



[Shen 1 p x“ É r d ”  (47)-(50)`  ¦ ½ ¨ t  · ú §6 £ §].   " f ‘ : r ƒ  

½

¨_    õ \  _ K  Ÿ í 7 H õ  Ÿ í— : r_  f  ¨ à º , ~ ½ ÓØ  ¦ õ & ñ `  ¦ 7 á §  8 Ä

»l & h Ü ¼– Ð [ O " î | ¨ c à º e ” 6 £ §`  ¦ · ú ˜ à º e ”  . \ V\  ¦ [ þ t # Q d ”  (47)_  ' Í † ½ ӓ É r f β (1 − f γ ) + N q (f β − f γ )= (1 + N q )f β (1 −

f _γ ) −N q f _γ (1 −f β ) s Ù ¼– Ð Ÿ í 7 H`  ¦ ~ ½ ÓØ  ¦(1+N q ) €  " f β\ 

"

f γ– Ð „  s    H õ & ñ õ  Ÿ í 7 H`  ¦ f  ¨ à º €  " f(N ^q ) γ \ " f β – Ð „  s    H õ & ñ _  › ¸½ + ËÜ ¼– Ð K $ 3 ½ + É Ã º e ”  . Ñ ü t P : † ½ Ó f _α (1 −f α ) + N _q (f _α −f γ )= (1 + N _q )f _α (a −f γ ) −N q f _γ (a − f α )“ É r α  © œI ü < γ  © œI _  „  s \  ¦
  · p . é ß –, ì  r — ¸† ½ ӓ É r

~ ω 2 ≈ E α −E β s Ù ¼– Ð ~ω+E ^βγ −~ω q = ~ω 1 +E αγ −~ω q – Ð

¿

º# Q  ô  Ç .
 Qt  † ½ Ó[ þ t • ¸ q 5 p wô  Ç ~ ½ ÓZ O Ü ¼– Ð K $ 3 ½ + É Ã º e ”

 .

IV. ] k ùÅ k Ä II8 ý 4  ˜ m

(a) ‚  + þ A Òì  r

d ”

 (16)`  ¦ > í ß – l  0 AK  D h– Ðî  r  % ò ƒ  í ß –  P 0

⁰

, Q

⁰

₀ `  ¦



6 £ § õ  ° ú  s  & ñ _ ô  Ç .

P ₀

⁰

X ≡ < X > αβ

< a ⁺ γ a _δ > _αβ a ⁺ _γ a δ (51)

Q

⁰

₀ ≡ 1 − P 0

⁰

(52)

#

Œl " f < X > ^α⠍  H d ”  (18)õ  ° ú  “ ¦ + þ Ad ”  I_    õ “   d ”  (32) ü < (33)\  ¦ Ä »• ¸½ + É M :ü < ° ú  “ É r ~ ½ ÓZ O `  ¦ & h 6   x €    6 £ §

`

 ¦ % 3   H  .

σ ij (ω) = −e X

αβ

(r j ) αβ (j i ) βα

f β − f α

~ ω ¯ − E βα − Γ ⁽⁰⁾ _αβ (¯ ω) (53)

#

Œl " f

Γ ⁽⁰⁾ _αβ (¯ ω) = X

q

X

η

 |C ^ηα (q) | ² (f _β − f α )

  f β (1 − f η ) + N _−q (f β − f η )

~¯ ω − E βη − ~ω _−q − f η (1 − f β ) + N q (f η − f β )

~ ω ¯ − E βη − ~ω q



+ X

q

X

η

 |C ^βη (q) | ² (f β − f α )

  f α (1 − f η ) + N q (f α − f η )

~¯ ω − E ηα − ~ω q − f η (1 − f α ) + N _−q (f η − f α )

~¯ ω − E ηα + ~ω _−q



(54)

s

 . Ó ü t o & h “   K $ 3 “ É r + þ Ad ” õ  q 5 p w t ë ß – + þ Ad ”  II_     õ

 7 á §  8 ç ß –    .

(b) q ‚  + þ A Òì  r d ”

 (23)`  ¦ > í ß – l  0 AK   % ò ƒ  í ß –  P 1

⁰

, Q

⁰

₁ `  ¦  6 £ § õ 

°

ú  s  & ñ _ ô  Ç .

P ₁

⁰

X ≡ < X > ^γδ _αβ

< a ⁺ _l a _m > ^γδ _αβ a ⁺ _l a m (55)

Q

⁰

₁ ≡ 1 − P 1

⁰

(56)

#

Œl \ " f

< X > ^γδ _αβ ≡ T R {ρ eq [(~¯ ω 2 − L eq ) ⁻¹ [X, a ⁺ _γ a δ ], a ⁺ _α a β ] } (57) s

 . · ú ¡\ " fü < ° ú  “ É r ~ ½ ÓZ O Ü ¼– Ð „  > h €  

U _αβ (¯ ω ₁ , ¯ ω ₂ ) = < a ⁺ _l a m > ^γδ _αβ

~ ω ¯ 12 − _<a

+

^{~¯ ω}

¹²

l

a

_m

>

^γδ_αβ

W αβ (¯ ω 1 , ¯ ω 2 ) (58) W _αβ (¯ ω ₁ , ¯ ω ₂ ) = < (~¯ ω ₁₂ − L eq Q ₁ ) ⁻¹ L _eq a ⁺ _l a _m > ^γδ _αβ

= T _R {ρ eq [(~¯ ω ₂ − L eq ) ⁻¹

× [(~¯ ω 12 − L eq Q

⁰

₁ ) ⁻¹ L eq a ⁺ _l a m , a ⁺ _γ a δ ], a ⁺ _α a β ] } (59)

s

  ) a  . # Œl " f P 1

⁰

a ⁺ _l a m = a ⁺ _l a m , Q

⁰

₁ a ^m _l = 0`  ¦  6   xÙ þ ¡



. d ”  (59)`  ¦ 7 á §  8 > í ß – l  0 AK  d ”  (27)`  ¦  6   x # Œ d ”  (59)_  ~ω 2 − L eq \  ¦ ô  Ç    8 „  > hô  Ç (+ þ Ad ”  Iõ   H ² ú ˜o 

¿

º   P :  % ò ƒ  í ß – \  ¦  6   x t  · ú §  H  ). Õ ª Q€  

W _αβ ^γδ (¯ ω 1 , ¯ ω 2 ) = E γδ

~¯ ω 2

< a ⁺ _l a m > ^γδ _αβ + M ₂ ^αβ (¯ ω) − M 1 ^αβ (¯ ω 1 , ¯ ω 2 )

~¯ ω

− E αβ

~¯ ω ₂



W _αβ ^γδ (¯ ω 1 , ¯ ω 2 ) − E γδ

~ ω ¯ ₂ < a ⁺ _l a m > ^γδ _αβ

 (60)

#

Œl " f

M ₂ ^αβ (¯ ω) ≡ T R {ρ eq [[(~¯ ω −L eq Q

⁰

₁ ) ⁻¹ L v a ⁺ _l a m , a ⁺ _γ a δ ], a ⁺ _α a β ] } (61)

M ₁ ^αβ (¯ ω ₁ , ¯ ω ₂ ) ≡ T R {ρ eq [(~¯ ω ₂ − L eq ) ⁻¹ [(~¯ ω − L eq Q

⁰

₁ ) ⁻¹ L _v a ⁺ _l a _m , a ⁺ _γ a _δ ] } (62)

s

 . # Œl " f Q ¹ L d a ⁺ _l a m = 0`  ¦  6   xÙ þ ¡“ ¦ L ^v _  2 † ½ Ó  t

ë ß – “ ¦ 9Ù þ ¡ . d ”  (58)õ  (60)\  e ”   H < a ⁺ _l a m > ^γδ _α⠍  H   6

£

§ õ  ° ú  s  ~ 1 >  > í ß – ) a  .

< a ⁺ _l a _m > ^γδ _αβ ≡ T R {ρ eq [(~¯ ω ₂ − L eq ) ⁻¹ [a ⁺ _l a _m , a ⁺ _γ a _δ ], a ⁺ _α a _β ] }

= (f β − f α )δ lβ δ δα δ mγ

~ ω ¯ 2 − E βα − Γ ⁽⁰⁾ _αβ (¯ ω 2 )

− (f β − f α )δ γβ δ mα δ lδ

~ ω ¯ 2 − E βα − Γ ⁽⁰⁾ αβ (¯ ω 2 )

(63)



 " f d ”  (58)-(63)`  ¦ d ”  (23)\  @ /{ 9  €  

σ ijk (ω 1 , ω 2 ) = e ² X

αβ

X

γδ

X

lm

(r i ) αβ (r j ) γδ (r k ) lm

×

"

(f β − f α )δ lβ δ δα δ mγ

~ ω ¯ ₂ − E βα − Γ ⁽⁰⁾ _αβ (¯ ω ₂ )

1

~ ω ¯ − E βγ − Γ ⁽¹⁾ _αβγ (¯ ω) − (f β − f α )δ γβ δ mα δ lδ

~¯ ω ₂ − E βα − Γ ⁽⁰⁾ _αβ (¯ ω ₂ )

1

~¯ ω − E βα Γ ⁽²⁾ _αβδ (¯ ω)

# (64)

s

  ) a  . # Œl " f

Γ ⁽¹⁾ _αβγ (¯ ω) = X

q

X

µ

 C _γµ (q)C µγ ( −q) (f β − f α )



×  f β (1 − f µ ) + N q (f β − f µ )

~¯ ω ₁₂ + E _βµ + ~ω _q − f α (1 − f µ ) + N _−q (f α − f µ )

~¯ ω ₁₂ + E _βµ − ~ω q

+ f _µ (1 − f α ) + N _−q (f _µ − f α )

~ ω ¯ 12 + E βµ − ~ω −q − f _µ (1 − f β ) + N _−q (f _µ − f β )

~ ω ¯ 12 + E βµ − ~ω −q



− X

q

X

µ

 C _µβ (q)C _βµ ( −q) (f β − f α )



×  f α (1 − f µ ) + N q (f α − f µ )

~ ω ¯ 12 + E µγ + ~ω q − f µ (1 − f α ) + N _−q (f µ − f α )

~ ω ¯ 12 + E µγ + ~ω q



(65)

Γ ⁽²⁾ _αβδ (¯ ω) = X

q

X

µ

 C µδ (q)C δµ ( −q) (f _β − f α )



×  f _µ (1 − f α ) + N _q (f _µ − f α )

~ ω ¯ 12 + E µα + ~ω q − f _µ (1 − f β ) + N _−q (f _µ − f β )

~ ω ¯ 12 + E µα − ~ω q

+ f β (1 − f µ ) + N _−q (f β − f µ )

~ ω ¯ 12 + E µα − ~ω _−q − f α (1 − f µ ) + N _−q (f α − f µ )

~ ω ¯ 12 + E µα − ~ω _−q



− X

q

X

µ

 C αµ (q)C µα ( −q) (f β − f α )



×  f β (1 − f µ ) + N q (f β − f µ )

~¯ ω ₁₂ + E _δµ + ~ω _q − f µ (1 − f β ) + N _−q (f µ − f β )

~¯ ω ₁₂ + E _δµ + ~ω _q



(66)

s

 . d ”  (64)-(66)`  ¦ Ä »• ¸½ + É M : + þ Ad ”  Iõ   ð ø Ít – Ð / B N" î

&

h  Â Ò   H, 7 £ ¤ ~ω ≈ E ξ − E  õ  ~ω 2 ≈ E β − E α \  ¦ & ñ Ù þ ¡“ ¦ L v _  2 † ½ Ó t ë ß – “ ¦ 9Ù þ ¡ . + þ Ad ”  Iõ _   H s & h “ É r   6

£

§ õ  ° ú   . + þ Ad ”  II\ " f  H + þ Ad ”  I\ " f  6   xô  Ç ¿ º   P :  

% ò

ƒ  í ß – “   d ”  (38)-(39)\  @ /6 £ x   H  % ò ƒ  í ß –  € 9 כ ¹

\ O

l  M :ë  H \  ç ß –   “ ¦ ¢ ¸ô  Ç, + þ Ad ”  I\ " f  H Q 1 L _d J _i = 0 s  :

£

¤Z > ô  Ç  â Ä º\ ë ß – $ í w n  t ë ß – + þ Ad ”  II\ " f  H & ñ S X ‰ô  Ç d ” “   Q 1 L d a ⁺ _l a m = 0`  ¦  6   xÙ þ ¡l  M :ë  H \  { 9 ì ø Í& h s    H  כ s 



.

V. + s Ç Â ] Ø

t

F K  t  „   ü < Ÿ í 7 H_   © œ  ñ Œ •6   x s  e ”   H  „   > _ 

„ 

l „  • ¸• ¸\  ¦ þ j$   q ‚  + þ A† ½ Ó t  > í ß –   H ¿ º t  ~ ½ Ó Z O

`  ¦ ¶ ú ˜( R˜ Ѐ Œ ¤ .   õ   H ¿ º ~ ½ ÓZ O  — ¸¿ º \  -t  ˜ Д > rZ O g Ë : s

 $ í w n  “ ¦ „   ü < Ÿ í 7 H_  ì  r Ÿ í† < Êà º\  ¦ & h ] X y  Ÿ í† < Ê 

“

¦ e ” # Q" f   É r s  : r[ þ t[10-13] ˜ Ð  Ó ü t o & h Ü ¼– Ð K $ 3  l 

 ¼ # o     H  © œ& h s  e ”  . + þ Ad ”  Iõ  + þ Ad ”  II\  ¦ q “ § 

€ 

 ‚  + þ A Òì  r“    â Ä º  H q 5 p w # Œ  © œé ß –& h `  ¦  7 H½ + É Ã º \ O t 

ë

ß – + þ Ad ”  II_    õ  ç ß –   # Œ à ºu & h ì  r`  ¦ l \   H ¼ #  o

    H  © œ& h s  e ”  . ¢ ¸ô  Ç q ‚  + þ A Òì  r“    â Ä º + þ Ad ”  II

   H  \  ¦ W =  % i l  M :ë  H \  7 á §  8 { 9 ì ø Í& h s   ½ + É Ã º e ” “ ¦

%

i r  ç ß –      H  © œ& h s  e ”  . t ë ß – ‘ : r ƒ  ½ ¨\ " f  H Ÿ í



7 H õ _   © œ  ñ Œ •6   x s  €  •  “ ¦ & ñ % i “ ¦ / B N" î & h  Â Ò   H \ 

"

f „  > h % i l  M :ë  H \  z  ´] j– Ð z  ´+ « >u ü < q “ § % i `  ¦ M :



 H # QÖ ¼ s  : r s  ¸ ú ˜ ´ ú `  ¦ t   H  © œ{ Œ ™½ + É Ã º \ O  . · ú ¡Ü ¼– Ð_ 

ƒ 

½ ¨  H ‘ : r ƒ  ½ ¨\ " f ½ ¨ô  Ç   õ \  ¦ 4 Ÿ ¤¸ ú š t ë ß – z  ´] j> \ 

&

h 6   x # Œ # QÖ ¼ s  : r s  7 á §  8
“ É r t  q “ § “ ¦ ¢ ¸ô  Ç, ‘ : r ƒ  

½

¨\ " f  H “ ¦ 9 t  · ú §“ É r „   ü < „   , „   ü < Ô  ¦í  HÓ ü t  © œ  

ñ Œ •6   x s  e ”   H  â Ä º\   t  s  : r`  ¦ S X ‰ © œ   H  כ s  | ¨ c כ s

 .

Y c

p w Š à U Ø ”  ô

[1] R. Kubo, J. Phys. Soc. Jpn. 12, 570(1957).

[2] A. Kawabata, J. Phys. Soc. Jpn. 23, 999 (1967).

[3] A. Lodder and S. Fujita, J. Phys. Soc. Jpn. 25, 774(1968).

[4] S. D. Choi and O. H. Chung, Solid State Commun.

46, 717(1983).

[5] S. Badjou and P. N. Argyres, Phys. Rev. B 5 5964 (1987).

[6] S. N. Yi, O. H. Chung and S. D. Choi, Progr. Theor.

Phys. 77, 429(1987).

[7] J. Y. Ryu and S. D. Choi, Phys. Rev. B 44, 11 328 (1991).

[8] Y. J. Cho and S. D. Choi, Phys. Rev. B 47, 9273(1993-I).

[9] N. L. Kang, Y. J. Cho, and S. D. Choi, Prog. Theor.

Phys. 96, 307 (1996); J. Korean Phys. Soc. 29, 628 (1996).

[10] J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).

[11] N. Bloembergen and Y. R. Shen, Phys. Rev. 133, A 37 (1964).

[12] A. Suzuki and M. Ashikawa, Phys. Rev. E 58, 4307 (1998).

[13] Y. R. Shen, T he P rinciples of N onlinear Optics (John Wily and Sons, New York, 1984), Chap 2.

[14] H. J. Lee, N. L. Kang, J. Y. Sug, and S. D. Choi, Phys. Rev. B 65, 195113 (2002).

[15] N. L. Kang, H. J. Lee and S. D. Choi, unpublished note(2003).

Overall Comparison of Two Approaches to Damping Factors in Optical Conductivity for Electron Systems in Solids

So Youn Kim, Hyun Jung Lee and Sang Don Choi

Department of Physics, Kyungpook National University, Daegu 702-701 Nam Lyong Kang

Faculty of Liberal Arts, Miryang National University, Miryang 627-702 (Received 3 March 2004)

A new theory, which uses projectors different from the previous ones is introduced for the optical conductivity of a system of electrons interacting with phonons in solids. The conductivity derived includes nonlinear terms, as well as linear terms, and the damping factors in both parts contain the electron and phonon distribution functions properly, which helps with the interpretations of the phonon emissions and absorptions in all the electron transition processes. The result is shown to be more general than the previous one based on a different projection technique.

PACS numbers: 71.38.+i, 72.10.-d, 72.20.-i

Keywords: Operator algebra technique