· ¡ ¤ @ /† < Ɠ § : Ÿ x > Ó ü t o ƒ ½ ¨z ´, @ /½ ¨ 702-701



 ¹ Å 4 8 ý R  Ò Å] k ù ° Ë Ñ ¹ Åy ¢y ¢8 ý ± n É ¿ R <U ê s0 n É

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

 â

· ¡ ¤ @ /† < Ɠ § : Ÿ x > Ó ü t o ƒ  ½ ¨z  ´, @ /½ ¨ 702-701

~ ç

¡ ‘ 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¸   2 Z 4 27{ 9  ~ à Î6 £ §)

‘

: r ƒ  ½ ¨\ " f  H  ^ ‰> _  s  : r& h  ƒ  ½ ¨ ~ ½ ÓZ O _  
“   € ª œ  : Ÿ x > % i † < Æ& h  ~ ½ ÓZ O `  ¦ s 6   x # Œ ì ø ͕ ¸^ ‰ ? /

\

" f_  q ‚  + þ A F  g„  • ¸• ¸\  ¦ ½ ¨ % i  . q ‚  + þ A F  g„  • ¸• ¸  H Shen, Bloembergen 1 p x, Õ ªo “ ¦ Lee 1 p x \  _ K " f Ä »• ¸ ÷ &% 3 Ü ¼
 Shen_  „  • ¸• ¸ † < Êà º\ " f  H ¿ º — ¸× ¼_  „   l  \  @ /ô  Ç † ½ Ó, 7 £ ¤ (ω

1

+ ω

2

) s  Ÿ í

† <

ʝ ) a † ½ Ós 
 
t  · ú §€ Œ ¤Ü ¼ 9, Blembergen 1 p x_    õ \ " f  H „  • ¸• ¸ † < Êà º ? /\  ‚  ; Ÿ ¤ † < Êà º
 
 t

 · ú §€ Œ ¤ . Õ ªo “ ¦ Lee 1 p x \  _  # Œ ½ ¨K ”     õ \ " f  H ‚  ; Ÿ ¤† < Êà ºü < „   l  _   1 l x† < Êà º_  ½ + Ë$ í † ½ Ó s

 — ¸¿ º
 
t ë ß –   H   ~ ½ ÓZ O `  ¦ 2 [ % i  . ‘ : r  7 Hë  H \ " f  H   H   ~ ½ ÓZ O `  ¦ 2 [ t  · ú §“ ¦ þ j$   q ‚  + þ A

†

½ Ó  t  > í ß –ô  Ç   õ \  ¦ ™ è> hô  Ç .

PACS numbers: 70

Keywords: q ‚  + þ A F  g„  • ¸• ¸,  „   > , 6 £ x² ú š† < Êà º,  % ò ƒ  í ß – 

I. " e  ] Ø

“

¦^ ‰ F « Ñ_  „  • ¸• ¸(conductivity)  H F « Ñ? /_  „   [ þ t _  \  -t  ½ ¨› ¸ü < í ß –ê ø Íl ½ ¨\  _ ” > rô  Ç . „  • ¸• ¸  H ‚  + þ A Â

Òì  r ÷  rë ß –  m   q ‚  + þ A  Òì  r • ¸ Ÿ í† < Êô  Ç . œ íl _  ƒ  ½ ¨ [

þ t [1–6]“ É r ŠҖ Ð ‚  + þ A 6 £ x² ú š½ ¨› ¸\  ² D Gô  Ç÷ &% 3 “ ¦ q ‚  + þ A 6 £ x

²

ú š½ ¨› ¸\  @ /ô  Ç ƒ  ½ ¨ [7–11]  H B Ä º ] jô  Ç& h s % 3  . Õ ª Q
 s

] j ' ‘ é ß –F « Ñ_  µ 1 ϲ ú ˜– Ð q ‚  + þ A6 £ x² ú šs  ß ¼>  ‚ à Ð# Œ “ ¦ e ”

  H [ j> \  Ä ºo   H ¶ ú ˜“ ¦ e ”  . y © œ $ í F « Ñ? /_   l  o _  Ÿ í o, l ^ ‰? /_  ~ ½ ӄ  , p-n ] X ½ + Ë Ò_  „  l & h  : £ ¤$ í `  ¦ q  2

Ÿ

©ô  Ç ´ ú §“ É r q ‚  + þ A ‰ & ³ © œ1 p x s  Õ ª ˜ Ðl  s  .
  š ¸Z þ t

±

ú ˜\   H B | 9 ? /_  „   ½ ¨› ¸\  ¦ › ¸    H X < e ” # Q" f F  g„    z 

´+ « >s  “  l  e ”   H l Z O s  ÷ &% 3  . s  ì  r  \ " f  H q ‚  + þ A

„

 • ¸• ¸_  ƒ  ½ ¨ B Ä º ×  æ כ ¹  “ ¦ · ú ˜ 94 R e ”  . s  ì  r   _

 ƒ  ½ ¨_  ´ òr   H ] j 2 › ¸ o  Ò q t$ í \  @ /ô  Ç Franken 1 p x_  z 

´+ « >ƒ  ½ ¨ [12]ü < F  g  1 l x ™ D ¥½ + Ë\  @ /ô  Ç Bloembergen ƒ  ½ ¨

∗

E-mail: [email protected]

”

 _  s  : r& h  ƒ  ½ ¨ [9,10]s  .  ^ ‰> _  s  : r& h  ƒ  ½ ¨\ " f



 H € ª œ : Ÿ x > % i † < Æ& h  > í ß –~ ½ ÓZ O s  ¸ ú ˜ & h 6   x ) a  . Õ ª ×  æ \ " f

ƒ 

í ß –  @ /à º ~ ½ ÓZ O  (operator algebra technique )`  ¦ Ÿ í† < Ê ô 

Ç # Œ Q  ^ ‰ l Z O s  Ä »6   xô  Ç • ¸½ ¨– Ð “  & ñ ÷ &“ ¦ e ”   [4–6, 13,14,22]. ‘ : r ƒ  ½ ¨”  “ É r ‚  + þ Aõ  q ‚  + þ A 6 £ x² ú š\  @ / # Œ Y > 

>

h_   ^ ‰> í ß –~ ½ ÓZ O `  ¦ ™ è> hÙ þ ¡  [4, 6, 14, 22]. s [ þ t s  : r

“ É

r ŠҖ Ð  % ò ~ ½ ÓZ O  (projection technique ) [22,23]`  ¦ l œ í

–

Ð “ ¦ e ”  . s  ~ ½ ÓZ O `  ¦ Y > Y >  ì ø ͕ ¸^ ‰? /_   l  F  g„  s  (magneto-optical transition ) \  & h 6   x # Œ Õ ª f  ¨ à º‚   — ¸

€

ª œ`  ¦ > í ß – # Œ Ÿ í 7 H_  ~ ½ ÓØ  ¦ õ  f  ¨ à º\  ¦ [ O " î   H X < $ í / B N

% i Ü ¼
 ‚  + þ A6 £ x² ú š\ ë ß – ² D Gô  ÇÙ þ ¡ . þ j  H \  Ÿ í 7 H õ  „    _

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

>

í ß –s % 3  . ‘ : r  7 Hë  H“ É r „    · Ÿ í 7 H > _  F  g„  • ¸ õ & ñ \ " f



 H  > í ß – t  · ú §“ ¦ þ j$   q ‚  + þ A† ½ Ó`  ¦ > í ß –   H ~ ½ ÓZ O `  ¦

™

è> hô  Ç . > í ß – õ & ñ `  ¦ q “ §& h   © œ[ j >  " fÕ ü t # Œ { 9 ì ø Í 1

l

q  [ þ t s  ~ 1 >  s K  ½ + É Ã º e ” >  % i  .

II. R  Ò Å] k ù  ¹ ÅM  ¹ Åy ¢y ¢

-203-

1. õ m Çy ¢ Ž ì ō ˜ m 

1) ¨ î + þ A © œI _  x 9 • ¸ ƒ  í ß – 

ì

ø ͕ ¸^ ‰? /\ " f # Œ Q t (Ÿ í 7 H ¢ ¸  H Ô  ¦í  HÓ ü t) © œ  ñ Œ •6   x

`

 ¦ “ ¦ 9 # Œ F  g„  • ¸• ¸(optical conductivity) J $ ™" f\  ¦ ½ ¨

l  0 AK " f  H €  $  q ¨ î + þ Aì  r Ÿ í\ " f o Ä ºq ~ ½ Ó& ñ d ” _  K 

\

 ¦ ½ ¨ # Œ x 9 • ¸ ƒ  í ß – (density operator) ρ int (t)\  ¦ % 3 “ ¦, Ä

ºo  " é ¶   H % i † < Æ& h    à º_  : Ÿ x > & h  ¨ î ç  H u (ensemble average)\  ¦ ½ ¨ô  Ç .

>

\  ü @ ҁ © œ\  _ ô  Ç  F G“ É r \ O  €   “ : r • ¸ T “   \ P & h 

¨ î

+ þ A  © œI _  x 9 • ¸ƒ  í ß – (grand canonical density opera- tor)  H € © œ © œ^  ¦ ¨ î ç  H(ensemble average) \  _ K " f  6 £ § õ 

° ú   .

ρ _eq = exp (βµN _e − βH eq ) /T _R {exp (βµN e − βH eq ) } . (1)

#

Œl \ " f β = 1/k B T s “ ¦, k B   H Boltzmann  © œÃ ºs  9, µ  H  o† < Æ( J $ ™[ > `  ¦
  · p . T ^R   H  ^ ‰> _  traces  9, H eq   H r ç ß –\  Á º › ' aô  Ç K x 9 ž Ðm î ß – s “ ¦, N ^e   H > \  ” > r F 

  H „  ^ ‰ „   à º\  @ /6 £ x   H ƒ  í ß – s  .

2) q ¨ î + þ A  © œI _  x 9 • ¸ ƒ  í ß – 

s

 > \  ü @ Ò F G s  Å Ò# Qt €   ¨ î + þ A © œI \ " f # Á # Q
> 

÷

& 9 s – Ð “   # Œ „  l „  • ¸• ¸ Ò q t|   . „  • ¸• ¸  H ‚  + þ A Â

Òì  r÷  rë ß –  m   q ‚  + þ A  Òì  r • ¸ Ÿ í† < Êô  Ç . s  כ `  ¦ ½ ¨  l

 0 A # Œ  6 £ § õ  ° ú  s  x 9 • ¸ƒ  í ß – \  ¦ ¨ î + þ A © œI  Òì  r õ  ü @ Â

Ò  F G \  _ K     o   H † ½ Ó ρ _int (t) – Ð S X ‰ © œr ~  ´ à º e ”  .

ρ(t) = ρ eq + ρ int (t). (2)

#

Œl \ " f s  > _  8 ú x K x 9 ž Ðm î ß – H(t)\  @ /6 £ x   H Li- ouville ƒ  í ß – \  ¦ L(t)   €   o Ä ºq  ~ ½ Ó& ñ d ”  (Liouville equation) d ” “ É r  6 £ § õ  ° ú   .

i~ ∂ρ(t)

∂t = [H(t), ρ(t)] = L(t)ρ(t). (3)

#

Œl \ " f

L(t) = L eq + L int (t) (4)

s

 9,

H(t) = H _eq + H _int (t) (5)

s

 . L eq = L _d + L _v   H H eq = H _d + V \ , H int (t)  H >  ü

< ü @ ҁ © œ_   © œ  ñ Œ •6   x† ½ Ó L int (t) \  @ /6 £ x   H o Ä ºq  ƒ   í

ß – s  . H ^d   H @ /y Œ •  Òì  r s “ ¦, V   H q  @ /y Œ •Â Òì  r( © œ  ñ



Œ

•6   x† ½ Ó)s  .   " f d ” (2)ü < d ” (4),(5)\  ¦ d ” (3)\  @ /{ 9  

€ 

  6 £ § õ  ° ú  “ É r   õ \  ¦ % 3 `  ¦ à º e ”  .

i~ ∂ρ eq

∂t = [H eq , ρ eq ] = 0 (6)

i~ ∂ρ int (t)

∂t = [H eq , ρ int (t)]+[H int (t), ρ eq ]+[H int (t), ρ int (t)]

(7) ρ int (t)\  ¦ ½ ¨ l  0 A # Œ Dirac³ ð‰ & ³\ " f x 9 • ¸ ƒ  í ß – \  ¦

ρ ^D _int (t) ≡ exp(iH eq t/~)ρ int (t)exp( −iH eq t/~) (8)

õ

 ° ú  s  & ñ _ ô  Ç . s  כ `  ¦ p ì  r “ ¦ d ” (3)\  ¦ “ ¦ 9 €  ,

i~ ∂ρ ^D _int (t)

∂t = exp(iH eq t/~)



−H eq ρ int (t) + ρ int (t)H eq + i~ ∂ρ int (t)

∂t



exp( −iH eq t/~)

= expp(iL _eq t/~)L _int (t)ρ _eq + exp(iL _eq t/~)L _int (t)ρ _int (t) (9)

s

  ) a  . r ç ß –`  ¦ t = −∞\ " f Ä ºo  · ú ˜“ ¦    H ì ø Í6 £ x s  { 9

# Qè ß – r ç ß – t = t t  & h ì  r “ ¦, −∞\ " f  H ü @ ҁ © œ\ 

% ò

† ¾ Ó`  ¦ ~ à Ît  · ú §  H  © œI  “ ¦ & ñ €  , ρ ^D _int ( −∞) ∼ = 0 – Ð Ñ

ü

t à º e ”     " f d ” (9)\  ¦ & h ì  r €  ,

i~ρ ^D _int (t) = Z t

−∞

duexp(iL eq u/~) {L int (u)ρ eq }

+ Z t

−∞

duexp(iL eq u/~) {L int (u)ρ int (u) } (10)

`

 ¦ % 3   H  . s  כ `  ¦ d ”  (8)\  @ /{ 9  €  ,

ρ int (t) = 1

i~ exp( −iL eq t/~) Z t

−∞

duexp(iL eq u/~) {L int (u)ρ eq }

+ 1

i~ exp( −iL eq t/~) Z t

−∞

duexp(iL eq u/~) {L int (u)ρ int (u) }

= 1 i~

Z t

−∞

duexp {−iL eq (t − u)/~}{L int (u)ρ eq }

+ 1 i~

Z t

−∞

duexp {−iL eq (t − u)/~}{L int (u)ρ int (u) } (11) s

  ) a  . # Œl \ " f t − u = t 1 – Ð ¿ º€  , d ” (11)`  ¦  6 £ § õ  ° ú  s   7 j þ t à º e ”  .

ρ int (t) = 1 i~

Z _∞

0

dt 1 exp( −iL eq t 1 /~) {L int (t − t 1 )ρ eq } + 1

i~

Z ∞ 0

dt 1 exp( −iL eq t 1 /~). {L int (t − t 1 )ρ int (t − t 1 ) } (12) s

 כ `  ¦ ì ø Í4 Ÿ ¤ ~ ½ ÓZ O Ü ¼– Ð æ ¼€  ,

ρ _int (t) = 1 i~

Z ∞ 0

dt ₁ exp( −iL eq t ₁ /~) {L int (t − t 1 )ρ _eq }

+  1 i~

 2 Z _∞

0

dt 1

Z _∞

0

dt 2 exp( −iL eq t 1 /~)L int (t − t 1 )

× exp(−iL eq t 2 /~)L int (t − t 1 − t 2 )ρ eq

+  1 i~

 ² Z ∞ 0

dt 1

Z ∞ 0

dt 2 exp( −iL eq t 1 /~)L int (t − t 1 )

× exp(−iL eq t ₂ /~)L _int (t − t 1 − t 2 )ρ _int (t − t 1 − t 2 )

+ · · · · (13)

s

 ÷ &“ ¦, s d ” `  ¦ & ñ o K ˜ Ѐ  , ρ _int (t) =

∞

X

n=1

1 (i~) ⁿ

Z ∞ 0

dt ₁ Z ∞

0

dt ₂ · · · Z ∞

0

dt _n

× exp(−iL eq t ₁ /~)L _int (t − t 1 )

× exp(−iL eq t ₂ /~)L _int (t − t 1 − t 2 ) · · ·

× exp(−iL eq t n /~)L int (t − t 1 − · · · − t n )ρ eq

≡ ρ ⁽¹⁾ (t) + ρ ⁽²⁾ (t) + · · · + ρ ⁽ⁿ⁾ (t) (14) s

  ) a  . # Œl \ " f ρ ⁽ⁿ⁾ (t)  H L int (t)\  ¦ n   Ÿ í† < Ê   H † ½ Ó

`

 ¦ _ p  “ ¦, ρ ⁽⁰⁾ (t) = ρ _eq s  .

2.  ¹ Œ ½ Ž ì ō ˜ m  1) „  À Ó_  l @ /u  r

ç ß – t\ " f x 9 • ¸ † < Êà º ρ int (t)\  ¦ · ú ˜€   ü @ ҁ © œ\  @ /ô  Ç ì ø Í 6

£

x› ' a8 £ ¤| ¾ ӓ   „  À Ó ƒ  í ß –  J_  l @ /u \  ¦ d ” (14)`  ¦  6   x 

#

Œ ½ ¨½ + É Ã º e ”  . „  À Ó_  l @ /u  ³ ð‰ & ³“ É r  6 £ § õ  ° ú   .

< J _i >=

∞

X

n=1

< J _i ⁽ⁿ⁾ >=

∞

X

n=1

T _R {ρ ⁽ⁿ⁾ (t)J _i } (15)

#

Œl \ " f

J i = X

γδ

(j i ) γδ a ^† _γ a δ (16)

s

 9, (j ⁱ ) γδ ≡< γ|j i |γ >, j i   H é ß –{ 9 „   _  „  À Ó ƒ  í ß –



(single electron current operator)s “ ¦ i ≡ x, y, zs 



. γ, δ  H „   _   © œI t à º\  ¦
  · p . „  À Ó l @ /u _  ' Í

  P : † ½ Ó`  ¦ ½ ¨ €   ‚  + þ A „  • ¸• ¸\  ¦ % 3 `  ¦ à º e ”   H X <, s

\  @ /K " f  H ´ ú §“ É r ƒ  ½ ¨ s À Ò# Q & ’ l  M :ë  H \  # Œl 

\

" f  H d ” (14)\  ¦ s 6   x # Œ q ‚  + þ A F  g„  • ¸• ¸ J $ ™" f\  ¦ ½ ¨ ô 

Ç . # Œl \ " f ‚  × þ ˜   H K x 9 ž Ðm î ß –“ É r H eq = H e +

H p + V = P

α < α |E α |α > a ^† α a α + P

q ~ ω q b ^† _q b q + P

q

P

α,β C α,β (q)a ^† _α a β (b q + b ^† _−q ), # Œl \ " f E ^α   H  © œI  α \ " f_  é ß –{ 9 „   _  \  -t  “ ¦Ä »u s “ ¦a ^† α (a _α )  H  © œI  α_  „   _  Ò q t$ í (™ èY > )ƒ  í ß – s  . ~ω ^q   H Ÿ í 7 H_  \  - t

 s “ ¦, b ^† q (b _q )   H  © œI  |q >\  e ”   H Ÿ í 7 H_  Ò q t$ í (™ èY > )

ƒ 

í ß –  s  . ω ^q   H  à º 7 ˜'  q “   Ÿ í 7 H_  ”  1 l x à º s  9, 

à º 7 ˜'  qü < ì  rF G t à º s\  @ /K  |q >≡ |q, s >s  . # Œ l

\ " f C α,β (q) = V q < α |exp(iq · r)|β >– Ð V q   H „   ü <

Ÿ

í 7 Hç ß –_   © œ  ñ  Œ •6   x ƒ  í ß – – Ð" f Ÿ í 7 H_  7 á x À Ó\     # Œ



Qt – Ð ² ú ˜ t  9, r“ É r „   _  0 Au  7 ˜' s  .

2) þ j$   q ‚  + þ A „  l  „  • ¸• ¸

q

‚  + þ A „  l y Œ ™Ã ºÖ  ¦`  ¦ % 3 l  0 A # Œ d ”  (14)_  ¿ º   P :

†

½ Ó

ρ ⁽²⁾ (t) =  1 i~

 ² Z ∞ 0

dt 1

Z ∞ 0

dt 2

× exp(−iL eq t ₁ /~) {L int (t − t 1 )

× exp(−iL eq t 2 /~) {L int (t − t 1 − t 2 )ρ eq } (17)

`

 ¦ d ”  (15)\  V , # Q" f > í ß – €    6 £ § õ  ° ú  s 
 è ­ q à º e ” 



.

< J _i ⁽²⁾ > = T _R {J i ρ ⁽²⁾ (t) }

=  1 i~

 ² Z ∞ 0

dt ₁ Z ∞

0

dt ₂ T _R { ρ eq [ exp(iL _eq t ₂ )/~

× [ exp(iL eq t 1 /~)J i , H int (t − t 1 ) ], H int (t − t 1 − t 2 ) ] }. (18)



© œ  ñ Œ •6   x K x 9 ž Ðm î ß –(interaction Hamilionian)“ É r Š © œF G     H   (dipole approximation)\  ¦ 2 [ €    6 £ § õ  ° ú  s  j þ t à º e ” 



.

H _int (t − t 1 ) = e X

j=x,y,z

X

α,β

(r _j ) _αβ a ⁺ _α a _β E _j exp {iω 1 (t − t 1 ) } + c.c. (19)

H int (t − t 1 − t 2 ) = e X

k=x,y,z

X

γ,δ

(r k ) γδ a ⁺ _γ a δ E k exp {iω 2 (t − t 1 − t 2 ) } + c.c. (20)

Õ

ªA " f d ”  (18)\  ¦  r  æ ¼€  

< J _i ⁽²⁾ > =  e i~

 2 X

j,k

X

α,β

X

γ,δ

(r j ) αβ (r k ) γδ E j (ω 1 )E k (ω 2 )

× T R {ρ eq

Z _∞

0

dt 2 exp {it 2 (L eq − ~¯ ω 2 )/~ }

×

Z _∞

0

dt 1 exp {it 1 (L eq − ~¯ ω 1 2)/~ }J i , a ^† _α a β ], a ^† _γ a δ



} (21)

s

  ) a  . # Œl \ " f ω 12 ≡ ω 1 + ω 2 s  9, ¯ ω 1 ≡ ω 1 − ib(b → 0 ⁺ ) õ  ¯ ω 2 ≡ ω 2 − ic(c → 0 ⁺ )\  ¦ “ ¦ 9 €  ,

< J _i ⁽²⁾ > =  e i~

 2 X

j,k

X

α,β

X

γ,δ

(r j ) αβ (r k ) γδ E j (ω 1 )E k (ω 2 )

× T R

 ρ _eq

 ~/i

~ ω ¯ ₂ − L eq

 ~/i

~¯ ω ₁₂ − L eq

J _i , a ^† _α a _β

 , a ^† _γ a _δ



(22)

¢

¸  H

< J _i ⁽²⁾ >= X

j,k

σ i,j,k E j (ω 1 )E k (ω 2 ) (23)

–

Ð j þ t à º e ” Ü ¼ 9, # Œl \ " f q ‚  + þ A F  g„  • ¸• ¸  H  6 £ § õ  ° ú   .

σ i,j,k (ω 1 , ω 2 ) = e ² X

α,β

X

γ,δ

X

lm

(r i ) α,β (r j ) γ,δ (j k ) l,m U α,β (¯ ω 1 , ¯ ω 2 ) (24)

U α,β (¯ ω 1 , ¯ ω 2 ) ≡ T R

 ρ eq

 1

~ ω ¯ 2 − L eq

 1

~ ω ¯ 12 − L eq

a ^† _l a m , a ^† _γ a δ

 , a ^† _α a β



(25)

III.  = k  W _ ËU ê s0 n É; c 8 ý” X ¢ 4  ˜ m

1.  W _ ËU ê s0 n É; c 8 ý” X ¢ U _αβ (¯ ω ₁ , ¯ ω ₂ )8 ý 4  ˜ m

s

 כ `  ¦ > í ß – l  0 A # Œ D h– Ðî  r  % ò ƒ  í ß –  P <sup>1</sup> õ  Q <sub>1</sub> `  ¦  6 £ § õ  ° ú  s  & ñ _ ô  Ç .

P ₁ X ≡ < X > ^γδ _αβ

< a ^† _l a _m > ^γδ _αβ a ^† _l a _m (26)

Q ₁ ≡ 1 − P 1 . (27)

#

Œl \ " f

< X > ^γδ _αβ ≡ T R {ρ eq [(~¯ ω 2 − L eq ) ⁻¹ [ X , a ^† _γ a δ ] , a ^† _α a β ] } (28) s

 . d ”  (25)_  o Ä ºq ƒ  í ß – \  d ”  (26),(27)\  ¦ & h 6   x 

“

¦, d ”  (25)_  î ß –A á ¤ L eq \  ¦ L eq (P 1 + Q 1 ) Ü ¼– Ð  Ë ¨“ ¦ † ½ Ó1 p x d ”



1 A − B = 1

A + 1 A B 1

A − B (29)

`

 ¦  6   x €  

1

~ ω ¯ 12 − L eq

a ^† _l a m = 1

~ ω ¯ 12 − L eq Q 1 − L eq P 1

a ^† _l a m

= 1

~¯ ω 12 − L eq Q 1

a ^† _l a m + 1

~¯ ω 12 − L eq Q 1

L eq P 1

1

~¯ ω 12 − L eq

a ^† _l a m (30)

s

 . Q ¹ a ^† _l a m = 0 s Ù ¼– Ð d ”  (25)  H

U _αβ (¯ ω ₁ , ¯ ω ₂ ) = 1

~ ω ¯ 12

< a ^† _l a _m > ^γδ _αβ + U αβ (¯ ω 1 , ¯ ω 2 )

< a ^† _l a m > ^γδ _αβ < 1

~¯ ω ₁₂ − L eq Q ₁ L eq a ^† _l a m > ^γδ _αβ (31) s

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

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 ) . (32)

#

Œl \ " f

W _αβ (¯ ω ₁ , ¯ ω ₂ ) =< 1

~ ω ¯ 12 − L eq Q 1

L _eq a ^† _l a _m > ^γδ _αβ (33) s

 .

2. W αβ (¯ ω 1 , ¯ ω 2 )8 ý 4  ˜ m

1) W αβ (¯ ω 1 , ¯ ω 2 )_  " é ¶ r & h  + þ AI 

s

\  ¦ 7 á §  8  © œ[ j >  > í ß – €  , [ Ò2 Ÿ ¤ A] \  _ K 

W _αβ (¯ ω ₁ , ¯ ω ₂ ) ≡ T R

 ρ _eq

 1

~ ω ¯ 2 − L eq

 1

~ ω ¯ 12 − L eq Q 1

L _eq a ^† _l a _m , a ^† _γ a _δ

 , a ^† _α a _β



= E lm

~¯ ω ₁₂ < a ^† _l a m > ^γδ _αβ + < 1

~¯ ω ₁₂ − L eq Q ₁ L v a ^† _l a m > ^γδ _αβ (34) s

 ÷ &“ ¦, d ”  (34)\  ¦ 7 á §  8  © œ[ jy  > í ß – l  0 AK 

W αβ (¯ ω 1 , ¯ ω 2 ) =< 1

~¯ ω ₁₂ − L eq Q ₁ L d a ^† _l a m > ^γδ _αβ + < 1

~¯ ω ₁₂ − L eq Q ₁ L v a ^† _l a m > ^γδ _αβ (35)

–

Ð „  >  “ ¦ [ Ò2 Ÿ ¤ A] ü < Q ₁ a ^† _l a _m `  ¦  6   x €  

W _αβ (¯ ω ₁ , ¯ ω ₂ ) = E lm

~ ω ¯ 12 < a ^† _l a m > ^γδ _αβ + < 1

~¯ ω ₁₂ − L eq Q ₁ L _v a ^† _l a _m > ^γδ _αβ (36) s

  ) a  .  r  d ”  (29)\  ¦ _{~¯ ω} ¹

2

−L

eq

\  & h 6   x €  , W αβ (¯ ω 1 , ¯ ω 2 ) = E lm

~ ω ¯ 12 < a ^† _l a m > ^γδ _αβ + 1

~¯ ω ₂ T R

 ρ eq

 1

~¯ ω ₁₂ − L eq Q ₁ L v a ^† _l a m , a ^† _γ a δ

 , a ^† _α a β



+ 1

~¯ ω ₂ T R

 ρ eq

 1

~ ω ¯ ₂ − L eq

 1

~¯ ω ₁₂ − L eq Q ₁ L v a ^† _l a m , a ^† _γ a δ

 , a ^† _α a β



(37)

s

“ ¦ [ Ò2 Ÿ ¤ B] \  _ K 

W αβ (¯ ω 1 , ¯ ω 2 ) = E lm

~¯ ω ₁₂ < a ^† _l a _m > ^γδ _αβ + M ₂ ^αβ (¯ ω ₁₂ )

~ ω ¯ 2 − E _αβ

~ ω ¯ 2

W αβ (¯ ω 1 , ¯ ω 2 ) − M ₁ ^αβ (¯ ω ₁ , ¯ ω ₂ )

~ ω ¯ 2

(38) s

  ) a  . # Œl \ " f

M ₂ ^αβ (¯ ω 12 ) = T R

 ρ eq

 1

~ ω ¯ 12 − L eq Q 1

L v a ^† _l a m , a ^† _γ a δ

 , a ^† _α a β



(39)

M ₁ ^αβ (¯ ω 1 , ¯ ω 2 ) = T R

 ρ eq

 1

~¯ ω 2 − L eq

 1

~¯ ω 12 − L eq Q 1

L v a ^† _l a m , a ^† _γ a δ



, L v a ^† _α a β



(40)

s

 . d ”  (38)\  d ”  (39), (40)\  ¦ V , # Q" f  r  & ñ o  €  ,

W αβ (¯ ω 1 , ¯ ω 2 ) = E _lm

~ ω ¯ 12

< a ^† _l a m > ^γδ _αβ + M ₂ ^αβ (¯ ω ₁₂ ) − M 1 ^αβ (¯ ω ₁ , ¯ ω ₂ )

~ ω ¯ 2

− E αβ

~ ω ¯ 2



W αβ (¯ ω 1 , ¯ ω 2 ) − 1

~ ω ¯ 12

E lm < a ^† _l a m > ^γδ _αβ



(41)

s

 .

2) M ₂ ^αβ _  > í ß –

M ₂ ^αβ (¯ ω 12 ) õ  M ₁ ^αβ (¯ ω 1 , ¯ ω 2 )\  ¦ 7 á §  8  [ jy  > í ß – €  , M ₂ ^αβ (¯ ω 12 ) = 1

~ ω ¯ 12

(E γδ + E αβ )M ₂ ^αβ (¯ ω 12 )

− 1

~¯ ω 12

T R

 ρ eq

 1

~ ω ¯ 12 − L eq Q 1

L v a ^† _l a m , a ^† _γ a δ

 , a ^† _α a β



− 1

~¯ ω ₁₂ T R

 ρ eq

 1

~ ω ¯ ₁₂ − L eq Q ₁ L v a ^† _l a m , a ^† _γ a δ



, L v a ^† _α a β



− E _lm

~ ω ¯ 12

W αβ (¯ ω 1 , ¯ ω 2 ) − _~¯ ^E _ω

^lm₁₂

< a ^† _l a m > ^γδ _αβ

< a ^† _l a _m > ^γδ _αβ < a ^† _l a _m > ₀ (42) s

  ) a  . s  כ `  ¦  r  & ñ o  €  

M ₂ ^αβ (¯ ω 12 ) = − 1

~ ω ¯ 12 + E γδ + E αβ

×

"

V _αβ (¯ ω ₁₂ ) + W αβ (¯ ω 1 , ¯ ω 2 ) − _~¯ ^E _ω

^lm₁₂

< a ^† _l a m > ^γδ _αβ

< a ^† _l a m > ^γδ _αβ E _lm < a ^† _l a _m > ₀

#

(43)

s

 ÷ &“ ¦, # Œl \ " f  HV αβ (¯ ω ₁₂ )  H  6 £ § õ  ° ú   .

V αβ (¯ ω 12 ) = T R

n ρ eq

hh

(~¯ ω 12 − L eq Q 1 ) ⁻¹ L v a ^† _l a m , L v a ^† _γ a δ

i , a ^† _α a β

io + T R

n ρ eq

hh

(~¯ ω 12 − L eq Q 1 ) ⁻¹ L v a ^† _l a m , a ^† _γ a δ

i

, L v a ^† _α a β

io

(44)

3) W _αβ (¯ ω ₁ , ¯ ω ₂ )_  & ñ o 

d ”

 (44)\  ¦ + ‹" f d ”  (41)`  ¦  r  + ‹˜ Ѐ  

W αβ (¯ ω 1 , ¯ ω 2 )

"

1 + E αβ

~¯ ω ₂ + E lm < a ^† _l a m > 0

~¯ ω ₁₂ + E _γδ + E _αβ 1

< a ^† _l a m > ^γδ _αβ

#

= 1

~ ω ¯ 12

E lm < a ^† _l a m > ^γδ _αβ − 1

~ ω ¯ 2

M ₁ ^αβ (¯ ω 1 , ¯ ω 2 ) + E αβ

~ ω ¯ 2

E lm

~ ω ¯ 12

< a ^† _l a m > ^γδ _αβ

− 1

~ ω ¯ 12 + E γδ + E αβ



V αβ (¯ ω 12 ) − E lm

~¯ ω 12

E lm < a ^† _l a m > 0



(45) s

  ) a  . # Œl " f / B N" î & h    H~ ½ Ó\ " f_  ì ø Í6 £ x`  ¦ › ' a8 £ ¤ô  Ç €   ~¯ ω ₂ ∼ = E β α s Ù ¼– Ð  6 £ § õ  ° ú  s  j þ t à º e ”  .

W _αβ (¯ ω ₁ , ¯ ω ₂ ) ~ ω ¯ ₁₂

< a ^† _l a _m > ^γδ _αβ = − ~ ω ¯ ₁₂ (~¯ ω ₁₂ + E _γδ + E _αβ )M ₁ ^αβ (¯ ω ₁ , ¯ ω ₂ )

< a ^† _l a _m > ₀ E _lm

− ~¯ ω ₁₂

< a ^† _l a _m > ₀ E _lm



V _αβ − E _lm

~ ω ¯ 12

E _lm < a ^† _l a _m > ₀



(46)

3. U αβ (¯ ω 1 , ¯ ω 2 )8 ý + s ÇÊ Ý Å k Ä

s

 © œ`  ¦ 7 á x½ + Ë “ ¦ / B N" î & h    H~ ½ Ó\ " f_  ì ø Í6 £ x`  ¦ › ' a8 £ ¤ €   ~¯ ω 12 ∼ = E lm s Ù ¼– Ð   õ & h Ü ¼– Ð d ” (25)  H  6 £ § õ  ° ú  s   ) a  .

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

~ ω ¯ 12 − E lm + ^{(~ ¯ ω}

¹²

^+E

^γδ

^+E

^αβ

^)M

αβ

1

( ¯ ω

₁

, ¯ ω

₂

)+V

_αβ

( ¯ ω

₁₂

)

<a

^†_l

a

_m

>

₀

. (47)

#

Œl \ " f,

< a ^† _l a m > 0 = T R

n ρ eq

h

[a ^† _l a m , a ^† _γ a δ ], a ^† _α a β

io

= T R

n ρ eq

h

(a ^† _l a δ δ mγ − a ^† γ a m δ lδ ), a ^† _α a β

io

= (f l − f α )δ lβ δ δα δ mγ − (f γ − f α )δ γβ δ αm δ lδ

= (f _β − f α )δ _lβ δ _δα δ _mγ − (f β − f α )δ _γβ δ _αm δ _lδ (48) s

“ ¦,

< a ^† _l a _m > ^γδ _αβ = T _R

 ρ _eq

 1

~ ω ¯ 2 − L eq

[a ^† _l a _m , a ^† _γ a _δ ], a ^† _α a _β



= T R

 ρ eq

 1

~ ω ¯ 2 − L eq

a ^† _l a δ , a ^† _α a β



δ mγ − T R

 ρ eq

 1

~ ω ¯ 2 − L eq

a ^† _γ a m , a ^† _α a β



δ lδ

= f β − f α

~¯ ω ₂ − E βα − ⁰ Γ ^αβ _ph (¯ ω ₂ ) δ lβ δ δα δ mγ − f β − f α

~ ω ¯ 2 − E βα − ⁰ Γ ^α _ph β(¯ ω 2 ) δ γβ δ mα δ lδ (49) s

Ù ¼– Ð, d ”  (46)\  d ”  (48),(49)\  ¦ & h 6   x €   þ j7 á x& h Ü ¼– Ð  6 £ § õ  ° ú  s   7 j þ t à º e ”  .

U _αβ (¯ ω ₁ , ¯ ω ₂ ) = (f _β − f α )δ _lβ δ _δα δ _mγ

~ ω ¯ ₂ − E βα − ⁰ Γ ^αβ _ph (¯ ω ₂ )

1

~¯ ω ₁₂ − E βγ − ¹ Γ ^αβγ _ph (¯ ω ₁₂ )

− (f β − f α )δ γβ δ mα δ lδ

~ ω ¯ 2 − E βα − ⁰ Γ ^αβ _ph (¯ ω 2 )

1

~ ω ¯ 12 − E δα − ² Γ ^αβδ _ph (¯ ω 12 ) (50)

4.  ¹ ÅM P c p• ¤ ù o Ú8 ý + s ÇÊ Ý Å k Ä

d ”

(50)`  ¦ s 6   x # Œ q ‚  + þ A „  l y Œ ™Ã ºÖ  ¦  r  + ‹˜ Ѐ   σ i,j,k (ω 1 , ω 2 ) = e ² X

αβ

X

γδ

X

lm

(r i ) αβ (r j ) γδ (j k ) lm

× (f _β − f α )δ _lβ δ _δα δ _mγ

~ ω ¯ ₂ − E βα − ⁰ Γ ^αβ _ph (¯ ω ₂ )

1

~¯ ω ₁₂ − E βγ − ¹ Γ ^αβγ _ph (¯ ω ₁₂ )

− (f β − f α )δ γβ δ mα δ lδ

~ ω ¯ 2 − E βα − ⁰ Γ ^αβ _ph (¯ ω 2 )

1

~ ω ¯ 12 − E δα − ² Γ ^αβδ _ph (¯ ω 12 ) (51) s

  ) a  .   " f Ä ºo  · ú ˜“ ¦    H 6 £ x² ú š† < Êà º  H  6 £ § õ  ° ú  s  > í ß – 0 p xô  Ç + þ AI – Ð Å Ò# Q”   .

0 Γ ^αβ _ph (¯ ω 2 )(f β − f α ) ∼ = T R {ρ eq [L v a ^† _α a β , (~¯ ω − L d ) ⁻¹ L v a ^† _β a α ] } (52)

1 Γ ^αβγ _ph (¯ ω 12 )(f β − f α ) = T R

n ρ eq

hh

(~¯ ω 12 − L d ) ⁻¹ L v a ^† _β a γ , L v a ^† _γ a α

i , a ^† _α a β

io + T R

n ρ eq

hh

(~¯ ω 12 − L d ) ⁻¹ L v a ^† _β a γ , a ^† _γ a α

i

, L v a ^† _α a β

io

(53)

2 Γ ^αβδ _ph (¯ ω 12 )(f β − f α ) = T R

n ρ eq

hh

(~¯ ω 12 − L d ) ⁻¹ L v a ^† _δ a α , L v a ^† _β a δ

i , a ^† _α a β

io + T R

n ρ eq

hh

(~¯ ω 12 − L d ) ⁻¹ L v a ^† _γ a α , a ^† _β a γ

i

, L v a ^† _α a β

io

(54)

d ”

 (51)_  „  • ¸• ¸ + þ Ad ” “ É r Shen, Bloembergen 1 p x õ  Lee 1 p x _

   õ ü < Ä »   . Õ ª Q
 Shen_  „  • ¸• ¸\ " f  H ω 12 = ω ₁ +ω ₂ _  † ½ Ós 
 
t  · ú §€ Œ ¤“ ¦ Bloembergen 1 p x _  „  • ¸

•

¸\ " f  H damping term [ þ t s 
 
t  · ú §€ Œ ¤ . Lee 1 p x s

 s „  \  ½ ¨ô  Ç „  • ¸• ¸ü <  H f  ¨   t ë ß – damping term [

þ

t s   s `›   ç ß –  K  & ’  . s  כ “ É r  % ò ƒ  í ß – \  ¦  Ø Ô>  & ñ _  % i l  M :ë  H s  . z  ´] j „    -Ÿ í 7 H >    & ñ ÷ &€   & h  6

 

x s  0 p x  9 # Œ Q s  : r õ   © œ[ j >  q “ § | ¨ c à º e ”  .

IV. + s Ç Â ] Ø

#

ŒI  t  Ÿ í 7 H õ   © œ  ñ  Œ •6   x   H „   [ þ t s  F  g„  s \  ¦ 



 H õ & ñ \ " f
 
  H F  g„  • ¸• ¸_  q ‚  + þ A ´ òõ _  þ j$ 

\  ¦ > í ß –   H ~ ½ ÓZ O `  ¦ ™ è> h Ù þ ¡ .  ^ ‰> í ß –\   H  % ò ~ ½ Ó Z O

\  l œ í\  ¦ é  H ƒ  í ß –  @ /à º ~ ½ ÓZ O `  ¦  6   x % i  . þ j$ 

 q ‚  + þ A F  g„  • ¸ y Œ ™Ã ºÖ  ¦ \   H ] j 2 _  6 £ x² ú š† < Êà º Ÿ í† < Ê

÷

&# Q e ”   H X < s  כ “ É r ‚  + þ A6 £ x² ú š\ " f_  ‚  ; Ÿ ¤† < Êà º\  @ /6 £ xô  Ç



. > _  ½ ¨^ ‰& h “   ½ ¨› ¸ & ñ K t €   s  כ `  ¦  © œ[ jy  > í ß –

  H  כ s  0 p x  “ ¦ Ò q ty Œ • ) a  . s  ë  H ] j  H  6 £ §_  ƒ  ½ ¨

\

" f à º' Ÿ | ¨ c כ s  . ‘ : r  7 Hë  H s  ì ø ͕ ¸^ ‰\  € ª œ : Ÿ x > % i † < Æ

`

 ¦ & h 6   x # Œ z  ´] j ‰ & ³ © œ`  ¦ K $ 3   9  H ƒ  ½ ¨\  • ¸¹ ¡ § s  ÷ &

U

 ´  ê ø Í .

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

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

[2] A. Suzuki and D. Dunn, Phys. Rev. B25, 7754 (1982)

[3] H. Kobori, T. Ohyama and E. Otsuka, J. Phys. Soc.

Japan 59, 2141 (1990).

[4] Y. J. Cho and S. D. Choi, Phys. Rev. B47, 9273 (1993).

[5] P. N. Argyres, Phys. Rev. B39, 2982 (1989).

[6] N. L. Kang and S. D. Choi, J. Kor. Phys. Soc. 36, 219 (2000).

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[8] K. Tani, Progr. Theor. Phys. 32, 167 (1964).

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

[10] N. Bloembergen and Y. R. Shen, Phys. Rev. A37, (1964).

[11] A. Suzuki and M. Ashikawa, Phys. Rev. E58 , 4307 (1998).

[12] P. A. Franken, A. E. Hill, C. W. Peters and G. Wein- reich, Phys. Rev. Lett. 7, 118 (1961).

[13] Y. J. Choi and S. D. Choi, Phys. Rev. B53, 6896 (1996).

[14] N. L. Kang, J. Y. Sug and S. D. Choi, Nuovo Ci- mento 20D, 55 (1998).

[15] M. H. Lee, J. Math. Phys. 24, 2512 (1983).

[16] M. H. Lee, J. Hong, and J. Florencio, Jr., Phys. Scr.

T19, 498 (1987).

[17] R. Karplus and J. Schwinger, Phys. Rev. 73, 1020 (1948).

[18] J. Hong, J. Kor. Phys. Soc. 22 , 145 (1989).

[19] H. Mori, Prog. Theor. Phys. 34, 399 (1965).

[20] J. Y. Ryu and S. D. Choi, Phys. Rev. B44, 11328 (1991).

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B16, 5394 (1977).

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

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[24] Y. R. Shen, ”The Principles of Nonlinear of Optics”

(John Wiley and Sons, Inc. New York, 1984), Chap- ter 2.

 Ò2 Ÿ ¤ A

L eq (a ^† _α a β ) = (L d + L v )a ^† _α a β

L _d a ^† _α a _β (E _l − E m )a ^† _l a _m = E _lm a ^† _l a _m

L v a ^† _α a β = 0 (55)

 Ò2 Ÿ ¤ B

T R {ρ eq [L eq A, B], } = T R {ρ eq [L eq B, A] }

T R {ρ eq [[L eq A, B], C] } = T R {ρ eq [[L eq B, A], C] } + T R {ρ eq [L eq C , [ A , B ] ] }

= −T R {ρ eq [[A, L eq B], C] } − T R {ρ eq [[ A , B ] , L eq C ] } (56)

A Quantum Statistical Approach to Nonlinear Optical Conductivity for a Many Electron System

Hyun Jung Lee, Yun Ju Lee, Jai Hoon Lee and So Yeun Kim Statistical Physics 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 27 February 2003)

A quantum-statistical method is introduced for derivation of nonlinear optical conductivity for electrons interacting with phonons in semiconductors. The nonlinear terms appear in the average of the many electron current in the system, in which the lowest order part is calculated by using many-body projectors. The result is similar in form to those of Bloembergen et al, Shen, and H, J. Lee et al. with some difference in the damping terms, the reason lying in the difference in the way of perturbative expansion. The damping terms can be further calculated if the proper form of electron-phonon interaction hamiltonian is chosen.

PACS numbers: 70

Keywords: Nonlinear optical conductivity, Many electron system, Response function, Projection operator

∗

E-mail: [email protected]