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1 THE INTEACTION OF ADIATION AND MATTE: SEMICLASSICAL THEOY PAGE 26 III. EVIEW OF BASIC QUANTUM MECHANICS : TWO -LEVEL QUANTUM SYSTEMS : The lieaue of quanum opics and lase specoscop abounds wih discussions of he wo-level (wo-sae) ssem. This emphasis comes abou because he ineacion of such ssems wih he elecomagneic field ma be eaed in gea deail o obain valuable analic esuls and, hopefull, he analsis of wo-level ssems geneaes insighs ha ma be eended o moe ealisic siuaions. Founael, hee ae seveal impoan insances in which he applicaion of he wo-level model povides a ve good appoimaion o a moe complee heo. In he following, we label he uppe level of he ssem b he lee a and he lowe b he lee b. Fom Equaion [ I-12a ] we wie, specificall, he wave funcion of he wo level ssem as ψ(, ) = ψ( ) = u a ep i ω a + ( ) u b ep i ω b [ III-1 ] whee we know fom Equaion [ I-12b ] ha he ime vaing coefficiens saisf, in geneal, he following equaions: = i h C a E a H 1 ( ) E a + ( ) E a H 1 ( ) E b ep( i ω ab ) { } [ III-2a ] = i h C b E b H 1 ( ) E b + ( ) E b H 1 ( ) E a ep( i ω ba ) { } [ III-2b ] If we ake he ineacion o be he elecic dipole ineacion wih an applied elecic field we ma wie H 1 ( ) = p S E ( S,)= e E S, [ III-3a ]. Vico Jones, Mach 9, 2000

2 THE INTEACTION OF ADIATION AND MATTE: SEMICLASSICAL THEOY PAGE 27 whee S denoes he posiion of he cene of he wo-level ssem o aom. 14 Thus we wie E i H 1 ( ) E j = V ij = E i e E j E S, [ III-3b ] In all bu he mos biae cicumsances we ma use pesuasive smme agumens o eason ha Thus Equaions [ III-2 ] educe o E i e E i 0 C a ( ) = i h C bv ab ep( iω ab ) [ III-4a ] C b ( ) = i h C av ab ep ( iωab ) [ III-4b ] whee V ab = E a e E b = E E S, ( S,). ABI FLOPPING -- WITHOUT DAMPING : Fo an oscillao applied field V ab = E o cos ω = 1 E ep iω 2 o ( ) + c.c. [ III-5 ] we see, in he oaing-wave appoimaion, ha C a ( ) = i 2h E C o b ep i ( ω ab ω ) = i 2 o ( ) ep i ( ω ab ω ) [ III-6a ] 14 The use of his fom of ineacion needs consideable elaboaion, bu we defe ha discussion unil lae.. Vico Jones, Mach 9, 2000

3 THE INTEACTION OF ADIATION AND MATTE: SEMICLASSICAL THEOY PAGE 28 C b ( ) = i 2h E o Ca ( ) ep i( ω ab ω ) = i 2 o ( ) ep i( ω ab ω ) [ III-6b ] whee Ω o E o h defines he so called abi flopping fequenc which is, of couse, a measue of he sengh of he elecomagneic ineacion. Cleal, he coupling ems have maimum effec when he fequenc of he applied field is esonan wih he level spliing. In mos eamens he fequenc deuning of he field is epessed as δω = ω ab ω and he ssem's wave funcion - i.e. Equaion [ III-1 ] - is wien in ems of slighl modified ime vaing coefficiens b ansfoming o he oaing fame of efeence - vi. ψ(, ) = C a ( ) ep i ( ) = ( ) = 1 δω ω [ ] u 2 a a [ III-7a ] ( ) ep i 1 δω 2 [ III-7b ] + C b ( ) ep i 1 δω ω [ ( 2 b) ] u b ( ) [ III-7c ] ( ) ep i 1 δω 2 The coupling ems in his oaing fame of efeence become o in mai fom C a C b = 1 i δω 2 ( ) +Ω o C b ( ) = 1 2 i δω { } [ III-8a ] { C ( b ) + C o a ( ) } [ III-8b ] d d C = d d ( ) = i δω 2 Ω o Ω o δω ( ) = i 2 M C. [ III-8c ]. Vico Jones, Mach 9, 2000

4 THE INTEACTION OF ADIATION AND MATTE: SEMICLASSICAL THEOY PAGE 29 We look fo a soluion in he fom C ( ) = C ( 0) ep( 1 i 2 Ω ) whee is he genealiaion of he abi flopping fequenc. Theefoe, he condiion de[ M I] = 0 ields he genealied abi flopping fequenc = δω 2 +Ω o 2 [ III-9a ] and he geneal non-damped ime evolving wave funcion ( ) cos ( 1 2 Ω ) i δω = ( ) i Ω o sin 1 2 Ω sin ( 1 2 Ω ) ABI FLOPPING -- WITH DAMPING : i Ω o sin ( 1 2 Ω ) cos ( 1 2 Ω ) + i δω sin 1 2 Ω C a ( 0) C [ III-9b ] b ( 0) Negleced ineacions (e.g. sponaneous emission, collisions, hemal flucuaions) limi lifeime of a sae of a wo-level ssem. One class of lifeime limiing ineacions ma be descibed phenomenologicall b adding deca ems o he equaions of moion - i.e. o Equaions [ III-8 ] - as follows: C a ( ) = 1 γ 2( +iδω a ) ( ) + 1 i 2 o ( ) [ III-10a ] o C b ( ) = 1 γ 2( iδω b ) ( ) + 1 i 2 o ( ) [ III-10b ] d d C = d d ( ) = i δω iγ a 2 Ω o Ω o δω+ iγ b ( ) = i M C 2 [ III-10c ]. Vico Jones, Mach 9, 2000

5 THE INTEACTION OF ADIATION AND MATTE: SEMICLASSICAL THEOY PAGE 30 Again we look fo a soluion in he fom C ( ) = C ( 0) ep( 1 i Λ 2 ) so ha de [ M Λ I] = 0 o Λ 2 + i ( γ b + γ a )Λ ( δω iγ a )( δω+ iγ b )+Ω 2 o = 0. Theefoe Λ= i 1 γ 2( + γ a b)± δω 1 i γ γ 2 a b 2 + Ω 2 o [ III-11a ] -- whee he we efe o γ ab = 1 γ 2( + γ a b) as he aveage deca ae consan and = [ δω 1 i γ γ 2 ( a b) ] 2 +Ω 2 o as he genealied comple abi flopping fequenc -- and he geneal ime evolving wave funcion -- in he oaion fame -- ma be wien ( ) cos 1 2 = ( ) 1 [ 2( γ γ a b) + i δω] i Ω o sin 1 2 sin ( 1 2 ) i Ω o sin 1 2 cos ( 1 2 ) + 1 γ γ 2 a b + i δω sin 1 2 ep( 1 γ 2 ab ) C a 0 C [ III-11b ] b 0 DENSITY MATIX TEATMENT OF A TWO -L EVEL S YSTEMS : ecall fom Equaion [ II-30 ] ρ = i [ h ρ,h ] = i { h ρ H H ρ } Using a a + b b = 1 we ma wie { } [ III-12a ] ρ = i h ρ ( a a + b b )H H ( a a + b b )ρ. Vico Jones, Mach 9, 2000

6 THE INTEACTION OF ADIATION AND MATTE: SEMICLASSICAL THEOY PAGE 31 If H = H 0 + H 1, hen { [ E a a a + E b b b + H 1 a a + H 1 b b ] ρ} ρ = i h ρ E a a + E b b + a a H + b b H a b 1 1 [ III-12b ] Wiing mai elemens -- i.e. epesenaives -- of he densi opeao a ρ a = i { h a ρ b b H a a H b b ρ a 1 1 } [ III-13a ] b ρ b = i { h b ρ a a H b b H a a ρ b 1 1 } [ III-13b ] { a ρ b = i h E E a b a H 1 b a ρ b + a ρ a b ρ b + a ρ b [ b H 1 b a H 1 a ]} [ III-13c ] We ma again assume, b smme agumens, ha a H 1 a = b H 1 equaions of moion fo he elemens of he densi educe o b = 0 so ha he ρ bb ρ aa = [ i h 1 2V ab ρ ba + c.c. ] [ III-14a ] ρ ab = i ω ab ρ ab +ih 1 V ab [ ρ aa ρ bb ] [ III-14b ] To inoduce an elemen of eali, we add o hese equaions a pai of he mos inuiivel saisfing damping ems (a useful aibue of densi mai fomulaions) -- vi. ρ bb ρ ρaa = [ i h 1 2V ab ρ ba + c.c. ] Γ ρ bb ρ aa [ III-15a ] ( ρ bb ρ aa ) 0. Vico Jones, Mach 9, 2000

7 THE INTEACTION OF ADIATION AND MATTE: SEMICLASSICAL THEOY PAGE 32 ρ ab = i ω ab ρ ab +ih 1 V ab [ ρ aa ρ bb ] γ ρ ab [ III-15b ] whee Tansfom hese equaions o a oaing fame b aking ρ ab = ρ ab ep i ω ρ ab is assumed o be a slowl vaing funcion of ime which saisfies he equaions of moion ρ ab = i δω ρ ab + i h 1 V ab ep ( i ω ) [ ρ aa ρ bb ] γ ρ ab [ III-16a ] Γ ( ρ bb ρ aa ) ( ρ ρ bb aa ) 0 ρ bb ρ aa = i h 1 2V ab ep( iω ) ρ ba + c.c. [ III-16b ] Wih V ab = [ 1 2h o ep ( iω ) + c.c. ] and ignoing ems popoional o ep ( ± i 2ω ) -- i.e. he oaing wave appoimaion - we find ha ρ ab = i ( δω iγ) ρ ab i 1 ( 2 Ω o ) [ ρ aa ρ bb ] [ III-17a ] ρ bb ρ aa = [ i Ω o ρ ba + c.c. ] Γ ρ bb ρ aa [ III-17b ] ( ρ bb ρ aa ) 0 Sead sae behavio: If we ake all ime deivaives in hese equaions equal o eo, we obain ρ ab = i o ρ bb ρ aa 2 i δω+γ = o 2( δω 2 +γ 2 ) ( δω+ i γ) ρ bb ρ aa [ III-18a ] o ( ρ bb ρ aa ) = ρ bb ρ δω ( aa ) 2 +γ 2 0 δω 2 +γ 2 1+Ω 2 o Γγ [ III-18b ]. Vico Jones, Mach 9, 2000

8 THE INTEACTION OF ADIATION AND MATTE: SEMICLASSICAL THEOY PAGE 33 ρ ab = 1 2 Ω o 0 [ δω+ i γ] ρ bb ρ aa δω 2 +γ 2 1+ Ω 2 o Γγ [ III-18c ] We have gaphed Equaion [ III-18b ] a esonance - i.e. δω= 0 - and, fom he following epession, we see how he oscillao polaiaion is sauaed a high elecomagneic powes. In geneal, he macoscopic polaiaion is hen given b Tha is o sa P = ( P ρ) aa + ( P ρ) bb = P aa ρ aa + P ab ρ ba + P ba ρ ab + P bb ρ bb = P ab ρ ba + P ba ρ ab since P aa = P bb = 0 b smme.. Vico Jones, Mach 9, 2000

9 THE INTEACTION OF ADIATION AND MATTE: SEMICLASSICAL THEOY PAGE 34 If we gaph Equaion [ III-18b ] a esonance - i.e. δω= 0 - we see how he oscillao polaiaion is sauaed a high elecomagneic powes - vi P = N dipole momen of ssem = N ρ ba + c.c. [ III-19a ] P = N o ( δω i γ) ρ bb ρ aa 2 δω 2 +γ 2 ep( i ω ) + c.c. [ III-19b ] 0 P = N 2 [ δω i γ] ρ bb ρ aa 2h δω 2 +γ 2 1+Ω 2 o Γγ E o ep ( i ω ) + c.c. [ III-19c ] THE VECTO MODEL OF THE DENSITY MATIX : Thee is a se of agumens b analog which is eceedingl valuable in eaing ansien eciaion poblems in opics. The basis fo he analog lies in he fac ha Equaions [ III-16] ae idenical in fom o he famous Bloch equaions of magneic esonance. 16 If we make a ansfomaion o a oaing fame -- i.e., M ± = M ± ep mi ω -- he Bloch equaions ake on he fom 16 ecall fom he heo of magneic esonance d M d = γ mag M H whee γ mag is he gomagneic aio. This equaion of moion is geal simplified if i is wien in ems of he cicula polaiaions M ± = M ± i M and H ± = H ± i H -- vi. M ± = mi γ mag M ± H ± i γ mag M H ± and M = i γ mag 2 [ M + H M H + ] To hese Bloch added he phenomenological longiudinal (hemal) elaaion ime T 1 and ansvese. Vico Jones, Mach 9, 2000

10 THE INTEACTION OF ADIATION AND MATTE: SEMICLASSICAL THEOY PAGE 35 M ± = mi δω M ± ± i γ mag M ep ( ±i ω ) H ± M ± T 2 [ III-20a ] M = i γ mag 2 H ep i ω M + + c.c. M M 0 T 1 [ III-20b ] whee δω =ω γ mag H. B compaing hese Bloch equaions wih Equaions [ III-16 ] and [ III-17 ] we see ha we have idenical poblems if we make he idenificaions M + ρ ab [ III-21a ] M ρ bb ρ aa [ III-21b ] γ mag H ω ab [ III-21c ] γ mag H + h 1 V ab = 1 2Ω o [ III-21d ] 1 T 1 = Γ [ III-21e ] 1 T 2 = γ [ III-21f ] In ohe wods we have he equivalen equaion of moion M = M Ω eff γ ( M ˆ M ) Γ ˆ M [ III-22 ] whee he effecive abi pecession field Ω eff is given b Ω eff = Ω o ˆ δω ˆ [ III-23 ] dephasing ime T 2 so ha M ± = mi γ mag M ± H ± i γ mag M H ± M ± T 2 M = i γ mag 2 [ M + H M H + ] ( M M 0 ) T 1. Vico Jones, Mach 9, 2000

11 THE INTEACTION OF ADIATION AND MATTE: SEMICLASSICAL THEOY PAGE 36 "abi field" "abi field" Iniial sae of ssem Effec of ansien eciaion - pecession of sae veco Sae afe 90 pulse "abi field" off -sa of "fee deca" peiod Effec o inhomogeneiies in level spliing Sae of ssem afe 180 pulse "Echo" sae. Vico Jones, Mach 9, 2000

12 THE INTEACTION OF ADIATION AND MATTE: SEMICLASSICAL THEOY PAGE 26. Vico Jones, Mach 9, 2000

13 THE INTEACTION OF ADIATION AND MATTE: SEMICLASSICAL THEOY PAGE 26. Vico Jones, Mach 9, 2000

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