‡ ˜ m; c " e ì Å À W ¥ M X ê s5 ; c" e8 ý V ê s„ ÆX c l “ Ó Þù p § U ° 6 ~ À W ¥ w Š ¹ Å ­ Žä à Šô p §8 ý

S

‡ ˜ m; c   " e
ì Å À W ¥  M X ê s5 ; c" e8 ý V ê s„ ÆX c l “ Ó Þù p § U  ° 6 ~ À W ¥ w Š ¹ Å ­ Žä à Šô p §8 ý

Ž Ò

ÞI í Ä] K ¡: Cosine ] K ¤• ¤  M X ê s vs. > H ¹ Å  M X ê s



¡) o £ Ó ^∗

Ø

 æ· ¡ ¤ @ /† < Ɠ § Ó ü t o † < Æõ  x 9  BK21 Ó ü t o  á Ԗ ÐÕ ªÏ þ ›, ' õ AÅ Ò 361-763 (2009¸   3 Z 4 17{ 9  ~ à Î6 £ §)

Cosine † < Êà º  l  © œ @ /’   r > ~ ½ ӆ ¾ Ó  r„    l  © œ`  ¦ G 6   x   H   H  Z O _  Ä »6   x$ í `  ¦  © œ  ñ Œ •6   x t  · ú §

 

H “ ¦„   Û ¼— 2 ;> _  — ¸4 S q`  ¦  6   x # Œ › ¸  % i  . Cosine † < Êà º  l  © œ_   â Ä º\   H Û ¼— 2 ; — ¸4 S q_  î  r1 l x

~

½ Ó& ñ d ” s  & ñ S X ‰ >  Û  ¦ o t  · ú §Ü ¼Ù ¼– Ð, s  כ `  ¦ à ºu & h ì  r`  ¦ : Ÿ x K " f Û  ¦% 3 Ü ¼ 9 Õ ª   õ \  ¦ r > ~ ½ ӆ ¾ Ó  r„  



l  © œ5 Å q \ " f_   l  o\  › ' a ô  Ç & ñ S X ‰ ô  Ç d ” s  Šҍ  H   õ ü < q “ § # Œ # QÖ ¼ & ñ • ¸ ¸ ú ˜ [ þ t # Q ´ ú   H t \  ¦ › ¸



 % i  . r ç ß –\    " f      H  l  © œ_  [ jl  €  •K   " f– Ð ¸ ú ˜ [ þ t # Q ´ ú `  ¦  כ s    H \ V © œ  A " f cosine † < Êà º  l  © œõ   r„    l  © œ_  ”  ; Ÿ ¤ s  B Ä º  Œ •“ É r  â Ä ºë ß –`  ¦ ƒ  ½ ¨ @ / © œÜ ¼– Ð  Œ ™€ Œ ¤ . Ä ºo _  \ V



© œ@ /– Ð cosine † < Êà º  l  © œõ   r„    l  © œ_  ”  ; Ÿ ¤ s  B Ä º  Œ •“ ¦ y Œ •”  1 l x à º• ¸ B Ä º  Œ •Ü ¼€   q ¨ î + þ A  l 



o_  x-$ í ì  r“ É r ° ú כs   _  { 9 u  “ ¦, y-$ í ì  r x 9  z-$ í ì  r • ¸ ° ú כs  " f– Ð  © œ{ © œy    H] X  % i  . Õ ª Q
 cosine

†

< Êà º  l  © œõ   r„    l  © œ_  ”  ; Ÿ ¤ s  B Ä º  Œ • 8 • ¸ / B N”   y Œ •”  1 l x à º   H % ƒ\ " f  H q ¨ î + þ A  l  o_  x- x 9

 y-$ í ì  r _   â Ä º\   ú ª“ É r r ç ß – à ºï  r \ " f  H ”  1 l x à º  H ° ú  `  ¦ t  • ¸ ² D G ™ è 4 Ÿ x Ä ºo (peak)\  ¦ ƒ    ô  Ç  > h

‚

 (envelope)_  ”  1 l x Å Òl   H cosine † < Êà º  l  © œ_   â Ä º\   r„   l  © œ_   â Ä º˜ Ð  2C  & ñ • ¸ ß ¼>  H † d`  ¦ µ

1 Ï|  % i  . q ¨ î + þ A  l  o_  z-$ í ì  r“ É r  > h‚  s  Ò q tl t  · ú §t ë ß – ”  1 l x Å Òl  cosine  l  © œ_   â Ä º\ 



r„   l  © œ_   â Ä º˜ Ð  2C  & ñ • ¸ ß ¼>  H † d`  ¦ µ 1 Ï|  % i  .  t } Œ •Ü ¼– Ð y Œ •”  1 l x à º B Ä º  H  â Ä º\  ¦ “ ¦ 9

% i   H X <, Z  t ³ 1 Ñ> • ¸ cosine † < Êà º  l  © œõ   r„    l  © œ_  ¿ º  â Ä º — ¸¿ º ° ú כs   _  { 9 u  % i  .

PACS numbers: 05.90.+m, 75.10.Hk, 75.74.Gb

Keywords: q ¨ î + þ A Û ¼— 2 ; 1 l x§ 4 † < Æ,  r„   ý a³ ð> , Bloch ~ ½ Ó& ñ d ” ,  l / B N" î

I. " e  ] Ø



$ í Ó ü t| 9 _  Ù þ ˜ l  / B N" î z  ´+ « >s  % ƒ6 £ § Ü ¼– Ð ' Ÿ K & ’ `  ¦ M :

\

  H &  8ê ø Í z-~ ½ ӆ ¾ Ó & ñ  l  © œõ   8Ô  ¦ # Q  r„    l  © œ ¢ ¸



 H cosine † < Êà º  l  © œ`  ¦ † < Êa  G 6   x % i t ë ß –, Õ ª Ê ê  © œ6   x



o  ) a z  ´+ « > © œu [ þ t“ É r  r„    l  © œ @ /’   cosine † < Êà º  l  © œ

`

 ¦ s 6   x “ ¦ e ”   [1–4]. s  Qô  Ç  $ í Ó ü t| 9 _  Ù þ ˜ l  / B N" î

‰

&

³ © œ`  ¦ s  : r& h Ü ¼– Ð K $ 3    H X < e ” # Q" f  © œ " é ¶ œ í& h “  

—

¸4 S q“ É r r ç ß –\    " f ”  1 l x   H cosine † < Êà º  l  © œ5 Å q _ 



© œ  ñ Œ •6   x t  · ú §  H “ ¦„   Û ¼— 2 ;[ þ t _  — ¸4 S qs  “ ¦ ½ + É Ã º e ” 



 H X <, Kubo [5]\  ¦ q 2 Ÿ © ô  Ç # Œ Q ƒ  ½ ¨ [ þ t s  s  Qô  Ç ‰ & ³ © œ

`

 ¦ q ¨ î + þ A : Ÿ x > % i † < Æ_  ë  H ] j– Ð ç ß –Å Ò “ ¦ ‚  + þ A6 £ x ² ú š s  : r _ 

„

 > hü < 6 £ x6   x \   6   x ô  Ç   e ”   [6–8]. Õ ª Q
 Ô  ¦' Ÿ  > • ¸ cosine † < Êà º  l  © œs     2 ;  â Ä º\   H î  r1 l x ~ ½ Ó& ñ d ” s  Û  ¦ o  t

 · ú §  H  . Õ ªo  # Œ # Œ Q  | à Ð[ þ t s  r > ~ ½ ӆ ¾ Ó(clockwise)



r„    l  © œõ  ì ø Ír > ~ ½ ӆ ¾ Ó(counterclockwise)  r„    l 

∗

E-mail: [email protected]



© œ`  ¦ ½ + Ëu €   cosine † < Êà º  l  © œs  ÷ &“ ¦, / B N”   ”  1 l x à º   H

%

ƒ\ " f  H ì ø Ír > ~ ½ ӆ ¾ Ó  l  © œ_  ´ òõ  Á ºr ½ + É ë ß –    H



r„      H  (rotating wave approximation)\  ¦ • ¸{ 9  # Œ r

> ~ ½ ӆ ¾ Ó  r„    l  © œë ß –Ü ¼– Ð cosine † < Êà º  l  © œ_  ´ òõ 

\

 ¦ [ O " î   H X <  6   x ô  Ç   e ”   [1,2,5].

Õ

ªX O t ë ß – s  Qô  Ç   H   # QÖ ¼ & ñ • ¸ b ” `  ¦ ë ß –ô  Çt   H  r

„

   l  © œ\ " f €  •ç ß – # Á # Qè ß –  â Ä º\ ë ß – · ú ˜ 94 R e ” “ ¦ { 9 ì ø Í

&

h “    â Ä º\   H Z > – Ð · ú ˜ 9t t  · ú §€ Œ ¤Ü ¼Ù ¼– Ð s  כ `  ¦ › ¸ 

  H  כ “ É r  © œ{ © œy  _ p e ”   H { 9 s  “ ¦  « Ñ  ) a   [9].   

"

f ‘ : r  7 Hë  H \ " f  H 0 A\ " f ƒ  / å L ô  Ç   H  Z O s  \ O  
 & ñ S X ‰ ô

 Çt  · ú ˜ ˜ Ðl  0 AK " f cosine † < Êà º  l  © œs     2 ;  â Ä º

\

 à ºu & h ì  r`  ¦ à º' Ÿ  “ ¦ s  כ `  ¦ & ñ S X ‰ >  Û  ¦ o   H  r„  



l  © œs     2 ;  â Ä º_    õ ü < q “ § % i  .

II. { ¢¨ | Ê Ý Æ U ؎ Ò ÞU ê sX N ËÅ k Ä8 ý A 0

r

ç ß –\    " f      H  l  © œs     9 e ”   H N > h_  " f

-673-

–

Ð  © œ  ñ Œ •6   x`  ¦ t  · ú §  H “ ¦„   Û ¼— 2 ;>   H  6 £ § Hamilto- nian \  _ K " f l Õ ü t ) a  .

H = H ₀ + V (t). (1)

#

Œl " f H ⁰ = −h z P N

i=1 s ^z _i “  X <,    0 Au  i\  e ”   H z-

~

½ ӆ ¾ Ó Û ¼— 2 ; $ í ì  r s ^z _i ü <  l  © œ h z _   © œ  ñ Œ •6   x`  ¦
  · p .

Õ

ªo “ ¦ V (t)  H r ç ß –\    " f    o   H  l  © œ_  ´ òõ 

\

 ¦
 ? / ï  r  . 7 £ ¤, cosine † < Êà º  l  © œs     9 e ” Ü ¼€   V (t) = −h 0

N

X

i=1

s ^x _i cos ω 0 t (2)

 ÷ &“ ¦, r > ~ ½ ӆ ¾ Ó  r„    l  © œs     9 e ” Ü ¼€  

V (t) = −h ₀

N

X

i=1

(s ^x _i cos ω ₀ t − s ^y _i sin ω ₀ t) (3)

“

 X <, h ⁰   H y Œ •5 Å q • ¸ ω ⁰ – Ð  r„     H  l  © œ_  [ jl s  .

Õ

ªo “ ¦ + '\ " f  À ҍ  H î  r1 l x ~ ½ Ó& ñ d ” `  ¦ É Ò  H X < à ºu & h ì  r`  ¦



6   x ½ + É Ã º e ” • ¸2 Ÿ ¤ “ ¦„  & h  Û ¼— 2 ;> \  ¦ ‘ : r  7 Hë  H _  @ / © œÜ ¼– Ð



Œ

™l – Ð ô  Ç .    0 Au  i\  e ”   H “ ¦„  & h  Û ¼— 2 ; − → s i _  y Œ • $ í ì

 r“ É r

s ^x _i = sin θ i cos φ i , s ^y _i = sin θ i sin φ i ,

s ^z _i = cos θ _i . (4)

#

Œl " f θ ⁱ   H F G y Œ •(polar angle) Õ ªo “ ¦ φ ⁱ   H ~ ½ Ó0 A y

Œ

•(azimuthal angle)`  ¦ y Œ •y Œ •
  · p . 8 ú x Û ¼— 2 ;$ í ì  r S α = P N

i=1 s ^α _i (α = x, y, z)`  ¦ s 6   x # Œ Hamiltonian[ þ t`  ¦ ³ ð‰ & ³

€  , cosine † < Êà º  l  © œs     9 e ” `  ¦  â Ä º\   H

H = −h _z S _z − h 0 S _x cos ω ₀ t (5) s

“ ¦, r > ~ ½ ӆ ¾ Ó  r„    l  © œs     9 e ” Ü ¼€  

H = −h z S z − h 0 (S x cos ω 0 t − S y sin ω 0 t) (6)

  ) a  . s [ þ t Hamiltonian Ü ¼– РÒ'   6 £ § õ  ° ú  “ É r 8 ú x Û ¼

—

2 ; 7 ˜' $ í ì  r \  @ /ô  Ç î  r1 l x ~ ½ Ó& ñ d ” `  ¦ % 3   H  .

(1) Cosine † < Êà º  l  © œ_   â Ä º

dS _x

dt = h _z S _y , (7) dS y

dt = −h z S x + h 0 S z cos ω 0 t, (8) dS z

dt = −h 0 S y cos ω 0 t. (9)

· ú

¡\ " f ƒ  / å L ô  Ç  ü < ° ú  s , s  î  r1 l x ~ ½ Ó& ñ d ” “ É r & ñ S X ‰ y  Û  ¦ o

t  · ú §  H  . Õ ªA " f à ºu & h Ü ¼– Ð Û  ¦ # Q ë ß – ô  Ç .

(2) r > ~ ½ ӆ ¾ Ó  r„    l  © œ_   â Ä º

dS _x

dt = h _z S _y + h ₀ S _z sin ω ₀ t, (10) dS y

dt = −h _z S _x + h ₀ S _z cos ω ₀ t, (11) dS z

dt = −h 0 S y cos ω 0 t − h 0 S x sin ω 0 t. (12)

‚ Ã

Г ¦ë  H‰  ³ [11]\   H s  î  r1 l x ~ ½ Ó& ñ d ” `  ¦ É Ò  H ~ ½ ÓZ O õ   6 £ § õ

 ° ú  “ É r & ñ S X ‰ ô  Ç K  Å Ò# Q4 R e ”  .

S x (t) = S x (0)[{cos ² γ + sin ² γ cos(h ef f t)} cos(ω 0 t) − sin γ sin(h ef f t) sin(ω 0 t)]

+S y (0)[sin γ sin(h ef f t) cos(ω 0 t) + cos(h ef f t) sin(ω 0 t)]

+S _z (0)[sin γ cos γ{1 − cos(h _{ef f} t)} cos(ω ₀ t) + cos γ sin(h _{ef f} t) sin(ω ₀ t)], (13) S y (t) = S x (0)[−{cos ² γ + sin ² γ cos(h ef f t)} sin(ω 0 t) − sin γ sin(h ef f t) cos(ω 0 t)]

+S y (0)[− sin γ sin(h ef f t) sin(ω 0 t) + cos(h ef f t) cos(ω 0 t)]

+S _z (0)[− sin γ cos γ{1 − cos(h _{ef f} t)} sin(ω ₀ t) + cos γ sin(h _{ef f} t) cos(ω ₀ t)], (14) S z (t) = S x (0) sin γ cos γ{1 − cos(h ef f t)}

+S _y (0) cos γ sin(h _{ef f} t)

+S z (0){sin ² γ + cos ² γ cos(h ef f t).} (15)

#

Œl " f h ef f = ph ² ₀ + (h z − ω) ² , sin γ = ^h _h

^−ω

ef f

, Õ ªo “ ¦ cos γ = _h ^h

ef f

s  .

III. ç g Ë] k ù õ m Í R ç g Ë] k ù V ê s? 0; c" e8 ý  M × D

Û

¼— 2 ;> _  œ íl › ¸| s  ¨ î + þ A © œI – Ð Å Ò# Qt €  , ‚ à Г ¦ë  H‰  ³ [11] \  Å Ò# Q”    ü < ° ú  s   l  o $ í ì  r m ^eq _α ≡< S α > eq /N (α = x, y, z)  H  6 £ § õ  ° ú   .

m ^eq _x = < S _x > _eq N

= b

√ a ² + b ² coth p

a ² + b ² − b

a ² + b ² , (16) m ^eq _y = < S _y > _eq

N = 0, (17)

m ^eq _z = < S z > eq

N

= a

√

a ² + b ² coth p

a ² + b ² − a

a ² + b ² (18)

#

Œl " f a = βh z , b = βh 0 , β = 1/k B T , Õ ªo “ ¦ k B   H Boltzmann  © œÃ ºs  .

0

A_  œ íl › ¸| `  ¦ s 6   x # Œ r ç ß –\    " f      H  l 



© œs     9e ” `  ¦ M :_  q ¨ î + þ A  l  o\  ¦ ½ ¨K ˜ Ðl – Ð ô  Ç .

(1) Cosine † < Êà º  l  © œ_   â Ä º d ”

 (7), (8), (9) ‚  + þ A ƒ  w n  p ì  r ~ ½ Ó& ñ d ” s Ù ¼– Ð, s [ þ t

~

½ Ó& ñ d ” \  q ¨ î + þ A € © œ © œ^  ¦ ¨ î ç  H`  ¦ 2 [ “ ¦ NÜ ¼– Ð
¾ º€   q

¨ î + þ A  l  o\  › ' a ô  Ç  6 £ § _  ƒ  w n  p ì  r ~ ½ Ó& ñ d ” `  ¦ % 3   H



.

dm _x

dt = h _z m _y , (19) dm y

dt = −h _z m _x + h ₀ m _z cos ω ₀ t, (20) dm z

dt = −h 0 m y cos ω 0 t. (21) s

[ þ t î  r1 l x ~ ½ Ó& ñ d ” • ¸ & ñ S X ‰ y  Û  ¦ o t  · ú §  H  . Õ ªA " f 4  Runge-Kutta ~ ½ ÓZ O  [10]`  ¦ + ‹" f à ºu & h Ü ¼– Ð Û  ¦ l – Ð   H X

< ¨ î + þ A  l  o $ í ì  r[ þ t`  ¦ œ íl › ¸| Ü ¼– Ð  6   x l – Ð ô  Ç



.

(2) r > ~ ½ ӆ ¾ Ó  r„    l  © œ_   â Ä º

¨ î

+ þ A © œI \ " f m y (0) = 0e ” `  ¦ s 6   x €    l  o $ í ì  r _

 r ç ß –\    É r ”   o  H d ”  (10), (11), (12)– РÒ' 

m x (t) = m ^eq _x [{cos ² γ + sin ² γ cos(h ef f t)} cos(ω 0 t) − sin γ sin(h ef f t) sin(ω 0 t)]

+m ^eq _z [sin γ cos γ{1 − cos(h _{ef f} t)} cos(ω ₀ t) + cos γ sin(h _{ef f} t) sin(ω ₀ t)], (22) m y (t) = m ^eq _x [−{cos ² γ + sin ² γ cos(h ef f t)} sin(ω 0 t) − sin γ sin(h ef f t) cos(ω 0 t)]

+m ^eq _z [− sin γ cos γ{1 − cos(h _{ef f} t)} sin(ω ₀ t) + cos γ sin(h _{ef f} t) cos(ω ₀ t)], (23) m z (t) = m ^eq _x sin γ cos γ(1 − cos(h ef f t)) + m ^eq _z (sin ² γ + cos ² γ cos(h ef f t)) (24) ü < ° ú   .

IV. À X Ø 8 ý

Cosine † < Êà º  l  © œs     9e ”   H  â Ä ºü <  r„    l  © œs 

 

 9e ”   H  â Ä º_  Hamiltonian`  ¦ q “ §K  ˜ Ѐ    r„    l 



© œ_  [ jl  B Ä º €  • “ ¦ y Œ •”  1 l x à º ω 0  B Ä º  Œ •Ü ¼€   ¿ º Hamiltonian >  " f– Ð q 5 p w K t t ë ß – y Œ •”  1 l x à º & t 

€

  & h & h   8 s 
>  H † d`  ¦ ~ 1 >  · ú ˜ à º e ”  . Õ ªX O t ë ß –



r„    l  © œ_  [ jl  B Ä º €  • €   Z  t ³ 1 Ñ> • ¸ / B N" î y Œ •”   1

l

x à º\ " f  H r > ~ ½ ӆ ¾ Ó  r„   l  © œ_   â Ä ºü < cosine † < Êà º



l  © œ_   â Ä º\  " f– Ð ' Ÿ 1 l x s  q 5 p w K  ”     H  r„      H



(rotating wave approximatio) t è ß – Y > z  ¸   1 l x î ß – V ,  o

 æ ¼“     e ”  .

s

] j cosine † < Êà º  l  © œs     9e ” `  ¦ M :_  q ¨ î + þ A   l

 oü <  r„    l  © œs     9e ” `  ¦ M :_  q ¨ î + þ A  l  o

#

Q‹ "   â Ä º\  " f– Ð q 5 p w ô  Çt \  ¦ €  x 9  >  › ¸ K  ˜ Ðl – Ð ô

 Ç . · ú ¡\ " f ƒ  / å L ô  Ç  ü < ° ú  s  cosine † < Êà º  l  © œs    

9e ”   H  â Ä º\   H î  r1 l x ~ ½ Ó& ñ d ” s  & ñ S X ‰ y  Û  ¦ o t  · ú §Ü ¼Ù ¼– Ð 4  Runge-Kutta ~ ½ ÓZ O `  ¦ G 6   x # Œ p ì  r ~ ½ Ó& ñ d ” `  ¦ à ºu 

&

h Ü ¼– Ð Û  ¦ l – Ð ô  Ç . — ¸Ž  H r ç ß – ½ ¨ç ß –\ " f  l  o ™ èà º

&

h  s   þ j™ èô  Ç # Œ$ Á  o  t  & ñ S X ‰$ í s  Ä »t ÷ &• ¸2 Ÿ ¤ r  ç

ß – ç ß –  “ É r ∆t = 0.01(h z _  é ß –0 A\  ¦  6   x) – Ð ¸ ú š€ Œ ¤ .

Figure 1“ É r " é ¶ s  \ O   H “ : r • ¸ βh ^z = 1.00 \ " f " é ¶ s

 \ O   H  l  © œ_  [ jl  h ⁰ /h z = 0.01 s “ ¦ ω ⁰ = 0.01

rad/s“   y Œ •”  1 l x à º\ " f_   l  o $ í ì  r`  ¦ ˜ Ð# Œï  r  .  l 

Fig. 1. Magnetization components (a) m x , (b) m y , and (c) m z vs. time at a dimensionless inverse temperature of βh z = 1.00 for a dimensionless magnetic field intensity h 0 /h z = 0.01 at an angular frequency of ω 0 = 0.01 rad/s.



o_  x-$ í ì  r(a)“ É r ¿ º  â Ä º — ¸¿ º Rabi y Œ •”  1 l x à º ω ^R = h 0 = 0.01 rad/s – Ð ”  1 l x “ ¦ " f– Ð ¸ ú ˜ { 9 u † < Ê`  ¦ ˜ Ð# Œï  r  .  l 



o_  y-$ í ì  r(b) _   â Ä º\   H cosine  l  © œs    o €    _ 



  t  · ú §  H X <,  r„   l  © œs    o €   B Ä º  Œ •“ É r ”  ; Ÿ ¤`  ¦

t “ ¦ Rabi y Œ •”  1 l x à º 0.01 rad/s– Ð ”  1 l x`  ¦ ô  Ç . Õ ªo “ ¦



l  © œ_  z-$ í ì  r(c) _   â Ä º\   H Õ ªü < ì ø Í@ /– Ð cosine † < Êà º



l  © œs    o €   B Ä º  Œ •“ É r ”  ; Ÿ ¤`  ¦ t “ ¦ Rabi y Œ •”  1 l x Ã

º_  2C    É r ”  1 l x à º– Ð ”  1 l x`  ¦ “ ¦,  r„    l  © œs     o

€    _     t  · ú §  H  .   " f Fig. 1_    õ   H cosine

†

< Êà º  l  © œõ   r„    l  © œ_  [ jl  B Ä º €  • “ ¦ y Œ •”  

Fig. 2. Magnetization components (a) m x , (b) m y , and (c) m z vs. time at a dimensionless inverse temperature of βh z = 1.00 for a dimensionless magnetic field intensity h 0 /h z = 0.01 at the resonant angular frequency ω 0 = h z = 1.00 rad/s.

1 l

x à º• ¸ B Ä º  Œ •Ü ¼€   ¿ º  â Ä º  _  { 9 u   ) a ' Ÿ 1 l x`  ¦ ½ + É  כ s

   H Ä ºo _  \ V © œõ  ¸ ú ˜[ þ t # Q ´ ú   H  .

Figure 2  H " é ¶ s  \ O   H “ : r • ¸ βh ^z = 1.00 \ " f " é ¶ s

 \ O   H  l  © œ_  [ jl  h ⁰ /h z = 0.01 s “ ¦ ω ⁰ = h z = 1.00 rad/s“   / B N" î y Œ •”  1 l x à º\ " f_   l  o $ í ì  r`  ¦ ˜ Ð# Œ ï

 r  .  l  o_  x-$ í ì  r(a)“ É r ¿ º  â Ä º  ú ª“ É r r ç ß – à ºï  r \ 

"

f  H — ¸¿ º ° ú  s  1.00 rad/s_  y Œ •”  1 l x à º\  ¦ t “ ¦ ”  1 l x 



 H X < q K " f, |   r ç ß – à ºï  r \ " f  H  r„    l  © œ_   > h

‚

 (envelope)s  Rabi y Œ •”  1 l x à º 0.01 rad/s– Ð cos  l  © œ _

  > h‚  ˜ Ð  2C   Ø Ô>  ”  1 l x ô  Ç .  l  o_  y-$ í ì  r • ¸

Fig. 3. Magnetization components (a) m x , (b) m y , and (c) m z vs. time at a dimensionless inverse temperature of βh z = 1.00 for a dimensionless magnetic field h 0 /h z = 0.01 at an angular frequency ω 0 = 100.00 rad/s.

x-$ í ì  r õ  ° ú  “ É r ' Ÿ I \  ¦ ˜ Г   . Õ ª Q
  l  © œ_  z-$ í ì  r _

  â Ä º\   H  > h‚  “ É r \ O t ë ß –  r„    l  © œ_   l  o

Rabi y Œ •”  1 l x à º 0.01 rad/s– Ð cosine † < Êà º  l  © œ_   l 



o ˜ Ð  2C   Ø Ô>  ”  1 l x ô  Ç .   " f & ñ $ í & h Ü ¼– Ѝ  H B  Ä

º  ú ª“ É r r ç ß – à ºï  r \ " f  H cosine  l  © œ @ /’   r > ~ ½ ӆ ¾ Ó



r„    l  © œ_    õ \  ¦  6   x K • ¸ ÷ &t ë ß –, |   r ç ß – à ºï  r \ 

"

f  H  6   x ½ + É Ã º \ O Ü ¼ 9  r„      H    H & h ] X  t  · ú §6 £ §`  ¦

· ú

˜ à º e ”  .

Figure 3“ É r " é ¶ s  \ O   H “ : r • ¸ βh z = 1.00 \ " f " é ¶ s 

\ O

  H  l  © œ_  [ jl  h 0 /h z = 0.01 s “ ¦ ω 0 = 100.00

rad/s“   y Œ •”  1 l x à º\ " f_   l  o $ í ì  r`  ¦ ˜ Ð# Œï  r  .  l 



o_  x-$ í ì  r(a)“ É r  r„    l  © œõ  cosine † < Êà º  l  © œ ¿ º

 â

Ä º — ¸¿ º y Œ •”  1 l x à º 1.00 rad/s– Ð ”  1 l x “ ¦ " f– Ð ¸ ú ˜ { 9 u 

†

< Ê`  ¦ ˜ Ð# Œï  r  .  l  o_  y-$ í ì  r(b) _   â Ä º\ • ¸ q 5 p w ô  Ç '

Ÿ I \  ¦ ˜ Г   . Õ ªo “ ¦  l  © œ_  z-$ í ì  r(c) _   â Ä º\   H



r„    l  © œs     2 ;  â Ä º    t  · ú §  H X < q  # Œ cosine

†

< Êà º  l  © œs     2 ;  â Ä º  H p €  • t ë ß – €  •ç ß –_     o\  ¦

˜

Ð{ 9 ÷  r  _  ° ú  “ É r ° ú כ`  ¦ Ä »t ô  Ç .   " f cosine † < Êà º   l

 © œ@ /’    r„    l  © œ`  ¦ + ‹• ¸ ë  H ] j \ O  .

V. CONCLUSION

Ä

ºo   H ‘ : r  7 Hë  H \ " f cosine † < Êà º  l  © œ @ /’   r > ~ ½ Ó

†

¾ Ó  r„    l  © œ`  ¦ # Q‹ "   â Ä º\  j þ t à º e ”   H t \  ¦  © œ  ñ Œ •6   x

t  · ú §  H “ ¦„   Û ¼— 2 ;> _  — ¸4 S q`  ¦  6   x # Œ › ¸  % i  .

r

ç ß –\    " f      H  l  © œ_  [ jl  €  •K   " f– Ð ¸ ú ˜ [

þ

t # Q ´ ú `  ¦  כ s    H \ V © œ  A " f cosine † < Êà º  l  © œõ 



r„    l  © œ_  ”  ; Ÿ ¤ s  B Ä º  Œ •“ É r  â Ä ºë ß –`  ¦ ƒ  ½ ¨ @ / © œÜ ¼

–

Ð  Œ ™€ Œ ¤ .

Cosine † < Êà º  l  © œõ   r„    l  © œ_  ”  ; Ÿ ¤ s  B Ä º  Œ •

“

¦ y Œ •”  1 l x à º• ¸ B Ä º  Œ •Ü ¼€   q ¨ î + þ A  l  o_  x-$ í ì  r“ É r

° ú

כs   _  { 9 u  “ ¦, y-$ í ì  r x 9  z-$ í ì  r • ¸ ° ú כs  " f– Ð  © œ{ © œ y

   H] X  # Œ cosine † < Êà º  l  © œ @ /’    r„    l  © œ`  ¦ + ‹

•

¸ H † d`  ¦ · ú ˜ à º e ” % 3  . Õ ª Q
 / B N”   y Œ •”  1 l x à º   H % ƒ\ " f  H q

¨ î + þ A  l  o_  x- x 9  y-$ í ì  r _   â Ä º\   ú ª“ É r r ç ß – à ºï  r

\

" f  H ”  1 l x à º  H ° ú  `  ¦ t  • ¸ ² D G ™ è 4 Ÿ x Ä ºo (peak)\  ¦ ƒ     ô

 Ç  > h‚  (envelope)_  ”  1 l x Å Òl  cosine † < Êà º  l  © œ_ 

 â

Ä º\   r„   l  © œ_   â Ä º˜ Ð  2C  & ñ • ¸ ß ¼>  H † d`  ¦ µ 1 Ï| 

% i  .  l  o_  z-$ í ì  r“ É r  > h‚  s  Ò q tl t  · ú §t ë ß – ”   1

l

x Å Òl  cosine  l  © œ_   â Ä º\   r„   l  © œ_   â Ä º˜ Ð



 2C  & ñ • ¸ ß ¼>  H † d`  ¦ µ 1 Ï|  % i  . 7 £ ¤, s   â Ä º\   H s 

„

 _  Šҁ © œ [9]õ   H  Ø Ô>   r„      H  \  ¦  6   x ½ + É Ã º \ O 



.  t } Œ •Ü ¼– Ð y Œ •”  1 l x à º B Ä º  H  â Ä º\  ¦ “ ¦ 9 % i   H X

<, Z  t ³ 1 Ñ> • ¸ cosine † < Êà º  l  © œõ   r„    l  © œ_  ¿ º  â Ä

º — ¸¿ º ° ú כs   _  { 9 u  # Œ cosine † < Êà º  l  © œ@ /’    r

„

   l  © œ`  ¦ + ‹• ¸  ) a  .

P

c p 8 ý ò k >

s

  7 Hë  H“ É r 2008 † < Ƹ  • ¸ Ø  æ· ¡ ¤ @ /† < Ɠ § † < ÆÕ ü t t " é ¶ƒ  ½ ¨ \ O  _

 ƒ  ½ ¨q  t " é ¶ \  _  # Œ ƒ  ½ ¨÷ &% 3 _ þ v m  .

Y

c p w Š à U Ø ”  ô

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[2] F. Bloch, Phys. Rev. 70, 460 (1946); R. K.

Wangsness and F. Bloch, Phys. Rev. 89, 728 (1953);

F. Bloch and I. I. Rabi, Rev. Mod. Phys. 17, 237 (1945).

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Hioe, Phys. Rev. A 30, 2100 (1984); A. V. Alekseev and N. V. Sushilov, Phys. Rev. 46, 351 (1992); H.

K. Kim and S. P. Kim, J. Korean Phys. Soc. 48, 119 (2006); C. Brouder, J. Phys. A.: Math. Theor. 40, 9455 (2007).

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Mod. Phys. 26, 167 (1954).

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[6] G. F. Mazenko, Nonequilibrium Statistical Mechan- ics (Wiley-VCH, Weinheim, 2006).

[7] S. W. Lovesey, Condensed Matter Physics: Dynamic Correlations, 2nd ed. (Benjamin/Cummings, Menro Park, 1986).

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S. H. Autler and C. H. Townes, Phys. Rev. 100, 703 (1955).

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K. Oh, SAEMULLI (New Phys.) 57, 16 (2008).

Dynamics of Noninteracting Classical Spins in Time-dependent Magnetic Fields: Cosine Field vs. Rotating Field

Suhk Kun Oh ^∗

BK21 Physics Program and Department of Physics, Chungbuk National University, Cheongju 361-763

(Received 17 March 2009)

The validity of adopting a cosine field on behalf of a rotating magnetic field is examined via a model of noninteracting classical spins. Since the case of a cosine field is not exactly solvable, we solve the equations of motion for nonequilibrium magnetization components numerically and compare the results with those of the exactly solvable rotating field case. We also confine our attention to the case of a time-dependent magnetic field with a small amplitude because in this case, we expect the minimal difference between the cosine field and rotating field. When the angular frequency is very small, the x-component of the nonequilibrium magnetizations in cosine and the rotating fields coincide with each other whereas the y- and the z-components in these fields are quite close to each other. However, at the resonant angular frequency, the x- and the y-components of the magnetization oscillates with the same frequency, but the peaks of these components form envelopes, and they oscillate with different frequencies; i.e., the envelope for the rotating field oscillates twice as fast as than that for the cosine field. Meanwhile, the z-components in these fields show no envelope, but the envelope for the rotating field oscillates with a frequency three that for the cosine field. Finally, when the angular frequency is extremely large, the magnetization components in the cosine and rotating fields almost coincide with each other, to our surprise.

PACS numbers: 05.90.+m, 75.10.Hk, 75.74.Gb

Keywords: Nonequilibrium spin dynamics, Rotating coordinate system, Bloch equation, Magnetic resonance

∗

E-mail: [email protected]