Atomes et lumière, interactions matière-rayonnement

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(1)Atomes et lumière, interactions matière-rayonnement Claude Fabre. To cite this version: Claude Fabre. Atomes et lumière, interactions matière-rayonnement. DEA. 2006. �cel-00092951�. HAL Id: cel-00092951 https://cel.archives-ouvertes.fr/cel-00092951 Submitted on 12 Sep 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

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(117) ! x b &, ')*\e * b n9 & ![*57![

(118) !*# "&$n'!w [ )!# ^ [ m*#*58 )**r,_&,!a n * mo)*%,)%5t),i; [.'!,k- o)!# xb m#)* wc b *#w$" ` !# * 9$"A! [.o \,x8 l b ,)'U'0 [ @[. _ \,imo)* !n* t)k [ ,*|w d * '* o! !#* &,!*,T+&')**#$ [.,!&, } o&,o [.*7 ω0 , [.,*, x * [ !aaU$ o)$nmaa&UE!* n,,! '!'m no & )' f i , Cb n,! w5,_)* '^ [[ } '' )*_ )',o)**1n! }> ` &,*7, >&,)**>>x b n^ [.2 +9*[ αe )' !*> } ,)* #&*^ [.& '&!,*, , *#) () !9ωi $ o) m# & d +9*. α e=. X. fi. q2 ωi − ω + iγdi 2 2mε0 ωi (ωi − ω)2 + γdi. ,* _\, mo)*Tn,)T5t), ,[.'!,. #. %$'&. )(+*,$. !& . i. $ . / * 2 v#2Ts#w# %*#)*U)* ,' **!nw #3[ k[ 9!# xb n^[.9 qN)w xt)\,R*tx! } ' )**, Cb &,*'() 8 +* a !\, *# )*#o!#t),* ,' *!#, *7 ,E>i*at)ix=! } a,b, ...!* x))**k[ ,*> ou\ k)* * 'o) eb !*&,))&,,* '&!,* ν = (E − E ) /h a b #.c b *#@ [ mEE,"&,*^ [\*,w !#1*Et ,*+!.*$U&,*#* .! t *wi5t), c b 5^t \,*@ [.*7U[$ uT&* ! =9!n*!i< )!R,*[^ [.# , #! 5)**@ [.*7 u c b &! m ,_ [. !U,.#.&, *(), C b &,*'() xt)+ , ^ [.,w/ * * *!a|!*T! ! [ A \, 9&,*^ [.&*))()!.

(119)

(120) 3

(121)

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(123). h /z5. . WAX^>3xURDOM ; i L

(124) ODOQ X r UrDTM A :=< ?wD?aXfQq>3xURDOMA< +9* .,'* ,1! x cb n^[ *eb !.! } *at)x!} #&()&,*&'& o)!9o. [ ), &,* *t)! ) *+[ ,* w /. } a& l &!,*,qr)|ai ro)!r$ *a)* ω0 =. |ai → |bi. w *)*. |ai!,* |bi *at)x! |bi >o)!i } ,)*. d. Eb − E a ~. w ! ')*" , l b &,n" c b n^[.E,'>xn&' '& b â` # N N a b #)mnm a&, eb ,!#n)*R#i*at)xn! } , 3+*R>* ' |ai |bi N a + Nb = 1 'J \k[ ,' _[.& l 1 f [. ,' !#o)& 4,*)#&,*! 4 - @b ()a 1'*#1^[.#, ')& &k! x))**#@ [.*7k,_ [. !x8w)*#!* !k [.*kx,&' &|E,*a& , 'on e b &*,()7 d . u (ν) =. dU dν.

(125) xb &,*'() ?t)) !Y[ !! f[. &,*))()!#|o)&, k/1*'* kb &,at)*7

(126) d l. U k&!n )*k#&*#^[. ³ ³ dNb dNa ω0 ´ ω0 ´ =− = −ANb + BNa u ν = − BNb u ν = dt dt 2π 2π. 258 l '@[. iH[ # "&,!n)* eb &,t) !# )*+&, ' xb &k[. ' * ! *#t)! !o&,!,wi*U)* )* n }#o&,z[ ,* n k[ ,*R!eb o,!# #& } [ @[. +9* o! ?n! &, 2 'o)*, *&,. u=0. A = γp. | n! } #?&'} n)* ro)! n)*,w!'w ^[[.

(127) n! } *γp!aN*r$ '

(128) #&,&#*w$ f[ [.

(129) eb !*?' nat) γ&d uR$ !',*.%c b nf [ nw!_@& [.!* f[. *9)!# ,# γ,rad , )*,w e b !*.n'% ')**# w !_!a',!#>E&, }#a n * *#)*R n w7nN} @ [.

(130) ,! γ} col!#)*7|''&,iN , t m * !R'&! ,' n *%* kx C b !* ) # -

(131) 1!Cb ) )*7N,N!*& γp γd #!

(132) `&,* @8 l #! } z \@[.H [ nw')o) **,9 u w,*T,^ [ # x b nm)'#)* N a , # 5)**@ [.*7 c b nf [ nw!' !.*at)x! )*f [.*7 #*1|*t)! } a&)!kc b k `,k! x))**k [ ,* **7x A)8 l T#*# E_ [.w1#)o) )**, u w1&,a +&*#^ [\*Te b @& [. ' * , 'o [ !&wo)! *!anw x))**k [ ,*N b 9x b n^ [.nwo!#_'%! *t)! , 0. 0.

(133)

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(143) 8w _)o')**,Fu O~ > &o* u *#&x, '9ucb nf[ 9u! << fγ[.P /B*(N *7b )xstat 1 f [. H\ *,*

(144) 8w w #)*Nn!' w * , t), 1 + ~ > ` b ` # i$

(145) u '>> &o)*γ'P/B u

(146) H\(N )b ) stat r *7,*a&+* k(N [ an)stat ! !eb e* b .i2f [ n*N, [.A \, in* ,' n #! J\k[

(147) cb & n t), |ai x b &, h /h. |bi. wMI:=BmSVUr:xB=>4MA<cLJBlL@: A:-Qli >3pDONP:-> XEmG >+N

(148) ? >@?Y?CQoURWAX L. l

(149) f [ &f [ (*&, !

(150) n !&

(151) "!#*?&,*,()

(152) !oo[.*!nw7)!R!#( [.,*nw $%!n*a& ! )|!eb +.!#**a)*U t), , ~ω |bi #! ! } #,!#E@)8)! 0A78 l ,'!R2 8 w !* u !0w n!(^[.*7.cb &*|ai , ( !T f [.T5)**& *# )!# k ' * l .#! '*, * &'&"! , + [ +@ b &' #)*7 d P = −~ω0. µ. dNb dt. ¶. 2),3). = ~ω0 B (Nb − Na ) u. ³ω ´ 0. 2π.

(153)

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(157). +9* t) ! ' w"! ?

(158) !# !*7xw & *7

(159) *#t)! )*+P[ ,<* 0n l Nb am>')N#b )*[f[ *>*9%w , [U |ai &,*! * #! '! !&n1<{E,)*w l b &k[ )*J'o[ !& P > 0 N < N a b #^ [. *nw ,% [*+f[. )x * !J f[.J !& *#% x eb !#* *t),' )*+o)!#$n )* /*U)*, !# * w , [ A \,

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(162) !*7nat)

(163) ! )*)** [.,*i9 >!n'i#| ',w#' c b )* *7!#N ,i,)* *7 , u> , A B # '&!# k# [.'!n +9* ,)* n ,o*n*! [.A \, ",' *#^ [.,xw ^ # d 3 ` d ` cd #` d cd ! ^ â` # cb #` d cb c ^ d ` , b d wn! *7,'t ,***7 *#&,n k [.*

(164) * #)mz \@[.w! )!

(165) !*\* t)!#

(166) &, !*T J \k[ n!'&'*1!eb !*E$n,)vr|& n!#7 ! [.A \, Ne b /*' *%,' {,^ [ m& Nc b oA \'9'!@& [.*7 *!#k*#r [.A \, !? f [ k& [ |@& [. ')*U'o [ !&9,' ,* '"xt) 9 f [.+ *, ,* +9*U')!#t)3 [ @[. *r, *N)!#tn()rx b H [ *.!? 4,n)* k& [ Nk& [ ' )*.'z [ ! & 9,*[n xt)> )*\**7" 4n -

(167) )!#

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(169) n -0A78, !>" o e b ' )* w! [ , )# &&,?)*!#$n) ' ,)'!,!$n "! 5)**@ [.*7xw C* b Un!!* ,*&, /3* )* w > *[5t),%

(170) [ Y \ "'&&,*7 C* b ,'|r&,m wi)* *?arr?!?'9*#r

(171) re b !#*U f [. *)* '&')**n*?xt)$>n*a *+nf [ !.8. #. * . ω 6= ω0. l b *!)*>i,_! } [.A\ &*#^[ &,*))()!#w+u?m ,* _&(7n _\, ^[. &k[ ,*,w*!

(172) .o,_[. |eb *!# kf[\,'ko[.o)'*7 o!kx b *,) * [ o \,9x))**#@ [.* d l | { nf [ ! → − % D , l k! n)* N , , +9* Tn!'1t! E_ {)aEN ,b,* 1"' }#n)* * +#*+ [ !# γ , γ d p #! #)mz \@[.wo!9)m#*?!* &,t))!#)* [.)*n)*"%x b ^ [. t),!* &, n )*#*w# *' ! o)!| ,k& n*() &*#(&, !,.

(173) .

(174) P ;

(175) ; P. l

(176) n**&,?@4E*|t ! c b 5^t \,*@ [.*7 , 9^ [.9$EuT&*!>=!* !nw,r,i'!k\|o!N} !N,*R ,! ,1$ '!,!9nf[ !< ! , #&, '9*#|?99,,'9& ?x b *, * 9x b ^ [.?xt)9!*U f[. &, ,,^ [ ()*&!#1o [.o)'&1 c b }&,!,w !#*N#@ [.o \, # )*' K u '!#,N*rx b f [ *,*

(177) c b n^ [.k , H [ ,r,)''o)**7 u"x b *, * , xt)!* + [ +&,^ [ (*&, !,$' !n +9* * n* ' #.!.$ [ a& ! xtnk- e b )* *^ [ w [ n!n ! J { e b # 4,k [ !*7! 4x8 w#! '!#1 _ f [. C* b ,'|N&

(178) # [ *#o \'

(179) !n*! ~ o@ b 5^t \,, ?)!# N!?,'9' # #)**!*,, *Rtn m ?#k#&*f [ \,*kn* v7z 3#k|,k,*())mow ,^ [[ 9*)! *i 9t) w# ,k,! }Tn ,

(180) #&,&#* "#. /. . & $. $. *. . +9*Ttn *?!#*['@ [. ,k@[.

(181) *&,() ()|)!#"*n,)* >xb ^[. 5t), xb }&, ,!k!#'!eb 5t)," f[.Tn !&. 50/z5 >4N UzQ pDTMAUR:JM a i UzM :=B=>VE"xURDOM oB=>k?A?a:=Qz<. ')!#t)T*"$+& &,*, - 258w ,>*"!* [ )*'[.,![-0@)8w !* } o)&

(182) & &|'! , +[ a*,*!,# m &, xb *,)*%*7 !#*R ^[.k!*7!

(183) N!* f [ . ' ! -

(184) )!N * ,#, , !, !, '&'!n|!# *!k')* !# ,|nk %'! /*J'&,*, C b !* f [ &,^ [ ()*#&,! ,$' !R#&' > >o *7 ?t),! → , 'x ' w !* + [ a*,* o)' m%o)! , − → − → − A ( r , t) U ( r , t) x b & ,,)*+k b &,' d +9*. ´ ´´2 ³ ³ ³ → − c − − − c c b= 1 → r ,t r ,t + qU → p − qA → H 2m. vN?f[. a)*a*.,'r,)'ro)!1!#N ) }._!()n ~ ,)**

(185) xb &,!n)*. [ 5)**n ~ ! i#*#9()

(186) !w *r,,# w l *7J ]?? )'-.

(187) 3

(188) ( *

(189)

(190) $ . t a,%%cb &,)*[* &,,*. → − *eb ' !# A ³ ´ → − → − p − q A /m * $,, * `b d"cd.] b ^ $ d. → − − v =→ p /m. wC[ n . → − v =. → − div A = 0 *i$n!#

(191)

(192) o,*ot),,,! → − ,'|&,!0u> ,4 ' n*'t),4.- !# #&, '&@[.*7n*'t),.*xb ,A,E.~)!' 8 −→ w % +[ Ex))** [.,*k' &,a , `^ d d # rnk*Ton*7 t)A!#⊥ −→ w| , )H[ ! , , A⊥ −−→ ∂A⊥ → → − − −→−→ E =− , B = rotA⊥ ∂t. )'T!*!#@[.*7 ,_[. eb &*,() ,)!^[ m** &, ,,')'n! &,& | ,!kcb n^[.

(193) b 2d. qU V (b r). ´ ³ ´ ³ q 2 −→ −→ 2 −→ − − − c c c b at − q → b= 1 → r =H p − q A⊥ + V → H A2 p . A⊥ + 2m m 2m ⊥ +9*T%* k`, → d −→ −→ c − − c p . A ⊥ = A ⊥ .→ p h i ~ −→ −→ −→ c − − c bi ⇒ → bi = ~ ∂i A p = div A⊥ = 0 p . A ⊥ − A ⊥ .→ pbi , A i i v-b ,' cb f[.a)*#* &,*^[[ & →

(194) ! e b − → −→ *Cb ,' !*E)o&,n!!n* − ” A . p ” !?!9n?!eb *+>@[. &

(195) n*i*U} 'A'⊥ )*U$tn'$m

(196) #9o)')* ,. xb o& !#. . ,' xb +[ *,* n^[. !nwx)*7 , tnn !# ,. → − → − 'r)*7

(197) |&*,()kc r cb nHbât[.9*&'&. *ieb !*E5))*#*@[.*7t ' mw4~pg w!7[. , )*#w )*%o!#)!#t),* 9 n '!*1z[ )na)*'!&@[.*7 0d5+b ,' xb ` 2 , ` #%c â` d b%cd w ?)*To!# $ ` #ocd ^ #bdb c #ocd /*[l `,xw *

(198) ,"x

(199) *[ → − h λ À h r i ´ #)*k@ [. , −→ ³→ −→ → − -→ − ,'N o)' )*%!.* 7n! kcb nf[ nw − r A⊥ c A⊥ ( 0 ) 0 '!o)'&?n* )!#

(200) ,)!'Nz [ [.ma58 l '* ,1,_ [.. )(!*7N*# b w! d H ,' ! ,)*T)',* −→ w,'?n )'|!#* 'o [. ?)* *7n ~ '' A⊥. ´ q −→ ³→ − c − c0 = H d p H A ⊥ 0 .→ at − m. < i *#()@[.*7| !(w )*Uo,!#k N9f[.*, u !#* n!#'9f[.a**$d ³ ´ ³ ´ → − → → − → − → − − c 00 → − c b b b H = Hat − q r . E 0 , t = Hat − D . E 0 , t.

(201) 3

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(203)

(204) $ . . → vjb Ncb f[.a)*#*%# o − − c Nxb o& !# o$'r& ,, !nw → c − c r'*7 , @[.*7_*{$ !an, D =* q,→ D &, ,,' !nwn)! → n w ! r * ) ! { # ! . a ' ) − → − ~ w [ )*7'E!# cb *7”E,.D ”* [ nz\%5))*#*@[.*7>' au '5} o[ *%_x b *,_ [.&#$n'r#!.# |{)r&,!#-

(205) )*,*3)*.!||f[. a)*a* , C* b ,' t $nm i! ')*.*&,() ()ik `,1 &,!.# *.# x b &, ,,)* V *#)*E !# ³ ´ ')!#,|!*+H [ e b &*,()3 [ n()*& !,* q b → − → − − m S. B 0 , t 9c b '5} o [ a)* +()*#+)*()!,!Ue b )* e* b TnUtn m nwr *#)!#t)! }%H [ ,@ b *!# ,*w)!$n*e b n!#'1()*,!1n,&,'! #"c b nf [ ! } &'t)&,?'!,t9

(206) f [ #

(207) &!

(208) j [ n()*& !, d + b '

(209) 4,&t) o@ [.*7 [ !#o)a 4%- t)) i - 2 88 . 50/ h. 1. ?Y?CBlD YUzN>xURDOM. . X< Y<7JN : 7h MAU

(210) O:l> X. |o)! &,n ')' &,n ,, w5Cb &,*'() , w b at H ,)* *!!A[ l b &,

(211) nf[ !!k|ai ()&,,*|bi &,, ..., kb &,a )*#d Ea , Eb , ... |Ei Z X −iEi t/~ |Ψi = ci (t)e |ii + c (E) (t)e−iEt/~ |Ei dE i=a,b,.... o! l ,{, .*7{N #&f[ o)' )* ')* &' !{ _[. −iE t/~ c (t)e i ! 9 . ,* ')|*'* ,* cb m'*, Cb *,)*+5t), f[. n !&.- 4,''&,c*i (t) n)* eb *, )* 48 l b ,n"t

(212) mo k, &,n n [.!#r |9z [.*' )* *3)*w|# . -u [ n* !# , l ?)m#o\@[.

(213) 9'o[ , ! '&***7 , # )ko* '!o)'? f [. # !& → − → − E = E ε cos (ωt − ϕ) '! $k'! n*a * - &,!*,irqr 8 l )*, *"e b )*# 'R&,^ [.o)Un )'"!*|ai !#→@ [.|bi *7%'! >,! } *tx! } ω0 , w ,>)!> , |ai |bi n!#'k*t)! }+o,!#t)*7 ()*)'&! d i. |Ψi ≈ ca (t) e−iEa t/~ |ai + cb (t) e−iEb t/~ |bi. +9* '%+[ \,*%)juR!* '% o[.*# * @ w xb f[ *,* ^[. ! kb &,' ')! _[.eb !*[ n d 2×2 4 ¸ · ¸ · 4 Ea 0 4 b Hat = = ~ω0 0 Eb # [ @[.! x b )o&n,!| o) '9&, ' ! → − c D=. " −→ daa − → d∗ab. − → # dab − → dbb.

(214) 3

(215) ( *

(216)

(217) $ . ?*r!*U ^[. → − → − wxnr *eb "N?# ,, )* 't &(&"-

(218) c hi| D |ii = _\@[.1 {*t * ' n 0)*8!+9*"'!o)' ! −→ &,o)!# 'o[. ),, dab +9*TE) d 4 ¸ ·4 ¸ · 4 b= 4 H − 4 E cos (ωt − ϕ) ~ω0 xt) − →− d = dab .→ ε. ?*1 9!anw))*E!#o), #k!!. 50/ .

(219) ODOQ X3xUrDTM |X7<Y<=NP:. |ai. cb &,n )*f[.*#|cb nf[ n. l b &,!n)* # *#()|@b &,

(220) n )'!d i~. d b |Ψ(t)i |Ψ(t)i = H dt. +9*%*%&!# ,_&!n )*#,)!&,eb &t)) ! *%# . , ,*. ca (t). . d. cb (t). dca dEe−iφ dEeiφ = − ei(ω−ω0 )t ca − e−i(ω+ω0 )t cb dt 2 2 dEeiφ dEe−iφ dcb −i(ω−ω0 )t = − e ca − ei(ω+ω0 )t cb i~ dt 2 2. i~. vN J\k[ '+o[. ) )*# #&nm k[ ,*E'|cb )* ^ ` 2 ` # ! b. ^ ` d w tn m7u+xn!..$U!n &,)**.,*'.$ '&!*# #! f[. , , ?xb ^[. / ?*'3u *#, &() (),rxb l` |,|_[.i!U o ±i(ω+ω )t w , *7 @[.*7+xt) '&!,*,T) !,wi u *# (),eU! , H [ ,|)' *7r ,*k [ ,*- )!U'n! u '&')**, }n,x8wo, &r,_[. '&,!$n ,wn! o ±i(ω−ω )t /*El ` ,,!*i !{ r n*()@ [.,*rtn'$m# - 4,n(*k 'e& &,*7 !*n*" 4j 8 d 0. 0. δt. ca (t) = cã (t)ei 2. δt. cb (t) = c˜b (t)e−i 2. w))*E)m ,*1 _!*E''_\@[. eb &!n *1! &, u, . *7 δ = ω − ω 0 ,)* *7d δ ΩR e−iφ d˜ ca = cã + c˜b dt 2 2 ΩR eiφ δ d˜ cb = cã − c˜b i dt 2 2. i.

(221) 3

(222) ( *

(223)

(224) $ . ,'? b^ ` w5t)7d. #. ΩR = −dE/~. cdhi`$ vr ''J\k[ >.o)! #! n)*k'),

(225) , !*&. ±Ω. q Ω = Ω2R + δ 2. +9*+')!#t) kn &k[.*k!E !# )*U()&,*&@b &,ad ¶ µ ΩR Ωt δ Ωt Ωt ca (t = 0) − i e−iφ sin cb (t = 0) cã (t) = cos − i sin 2 Ω 2 Ω 2 ¶ µ ΩR iφ Ωt Ωt δ Ωt cb (t = 0) c˜b (t) = −i e sin ca (t = 0) + cos + i sin Ω 2 2 Ω 2. cb )* ,*\^[[.)*#)*k* a. w. w )*. m#*7 ,t ,!'!t *o)! n)* ca (t = 0) = 12 cb (t = 0) = 02 w , Nb (t) = |cb | Na = |ca | E #!T# |{)n^[.!# D→ d − D (t) = Re (2dca c∗b ) Nb = 1 − N a = e = d ReD. e = d ImD. Ω2R 2 Ωt sin Ω2 2. δΩR (1 − cos Ωt) + δ2. Ω2R. ΩR sin Ωt Ω. #&'*7& !$<)()!T- @2 8?cb &,t) !# *[@[.o)' U N b )'!" J\k[ > 9*#?xb & n *f[.*7 u.cb * *7 +9*\Im t)De =!0 ! > 'J \k[ &,,*%!#* f [ o) k[ ,*?)' |ai $n*

(226) do ')*9 t, ^ â` # cdih|` $ f ~ e* b En # '&()z [ i n )*#* 'i)! ),& n*x^gk@[ !)*#1! w ! ,'?o)' )**# w?"' ()* o)&nIw u.$E!n* n* &,&,> Im x b nDe^ [..n! f [. w *() ()*E)! " @ [. ,"o&, #") g m d *# k&*,()&!# n|'*nat)@ [.*7i ^ [.rt)' 9 f [.+,k! + [ Ut) knf [ , :J&')*n* }#,w )*" o)!# w5, π c (t ) = 0 t = (2m + 1) a m m o! - 5t) 0 ,*' 8 d u>,i *#' *7Ω 3 $` #cb (t " d m ),= `0^ 0 π t m m = 2m #!T 'mJ \k[ t), Ω c b & n w! |*)! t)x!Tn* x b &, +9*. 0. R. 0. R. |bi. |ai.

(227) 3

(228) ( *

(229)

(230) $ . ).

(231) ( " . #. $ ) $*f& $. . Nb. $+& $ . . .

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(236) kb *&,?!ebyu"!* ''J\k[. u>@*at)x! }w ρ!#* ρ [ n , d 2×2. ·. ρaa ρ∗ba ρba ρbb. ¸. x t) +9*U&, w d#,9)*7| , bâ` # , ρ + ρ = 1 ρ = N ρ = N aa bb aa a bb b #! }+*at)x!} w!# *#)! xt))*k,**&, *k![.A\9eb /1*'* ρba d # d w ! ' '@[.,* ' &! |{) d ¿ À ³ → − ´ → − → − c c D = T rρ D = Re 2 d ρba.

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(245) , T * k . [ o \ ' . n x} o[ n* 7 u ]&'w, )* ±2iωt. +9* e. ¢¡ ¢ d dE ¡ iωt ρbb = i e + e−iωt ρe∗ba eiωt − ρeba e−iωt dt 2~ ¢ dE ¡ iωt d ρeba = −i (ω0 − ω) ρeba + e + e−iωt (ρbb − ρaa ) eiωt dt 2i~.

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