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2 I d.e" -= K b " _(C;t4..1 g 'ti: tl.- -' :: r tot"i t:::j4 5t ---A = )-p.., -..S);.. \.J ;;, /".A -- A X -= K. (R «x -;J:I) -.--\.(... --(l F=- X' -4- -* t> - :-- fr I-U $. ::: : %.,...,i7; j"c :: a.. /lk Jy -4,."J % --::.::> p - ( VI') ( -"- ";,f\";- c- = --c X 0 - ';X- )(;::: cosx f,l "-S:-:w1XLJ..t /. I\T i,..0 '\. >' kj >:.In X c:t.lr4 "c.j S7-J( I.{.. 1- ('4 ';< t.,) LJ.:tI. 6,,E {3. C 0 _4--- oi-n) (My '1/3 f-4) - u\ = /1 - -J...l- (-0.-((&--0 t.j ( LJ d ef = -lj-o) AlJ _rt-'\ if ''-,,-ly --I.J - 'fq-\; ('1"' " r r(.j & )J t1"t.j -LL. J h.-a C A J,/."L.''-:-' (6 0 4 C o...n.: (j J1(.o) 0 1J')o..ak c(j4 4,- cj k,--»- --J).)t. -j e1(ut'a.1':tj;') tj..;c1 :=- 4: t;:\) At:zZf"--"t/., t-. 1 [,..(, j..j J wi f; f j10to1l 4 rl f{ oj

3 "'-" i..c'i 't c - L _c"d'.j ---" fi( \. - - ( ifi ; ii : (..JJ -l-_it : -c "')('-" (J -Y...,. LA L\ -- L ;;-*Wt: -;Q L O n -= A J" \,,'"; ;!;i i'\ I)( 0"""';: {.. 14 ';;;':c""''.-4, Cl -- 'l.. t At tj t.. UOt, t N r VI'll _C 1Ac.t..;""'J } ( ).4+::tia1S;Qt r --r:--"--'; " (I-CaY j; - -- S;t.1- \,- -r- 6,...('q5'j1:t;' '--S.,ot 6. :f Jid"--,-S1;.. 'JJ,. '/ ( >:.<\-t 1(1 tjl --- 1('Uxi 1<1' I( ( ) -K ---fc -((-CO S'i ",,1>-t:. -<!.O s; J2; :-,i-- (r--cos,qc --c co -o.l-,1)/ai-?,,"jt.

4 -.:... --,,",. :c.j I i '- [ > restartiwith(linalg): Warning, the protected names norm and trace have been redefined and unprotected > rot1:=matrix([[k*eo/sqrt(delta"2+k"2*eo"2),0,-delta/sqrt(delta"2+k "2*EO"2)], [0,1,0], [Delta/sqrt(Delta"2+k"2*EO"2),0,k*EO/sqrt(Delta" 2+k"2*EO"2)]])i keo "k2m2 rot I := t:!. 2.m2 0 0 t:!. -2-k2.m2 keo 2-m2 > rot2:=matrix([[1,0,0], [O,cos(Omega*t),-sin(Omega*t)], [0, sin (Omega* t),cos(omega*t)]])i 0 0 rot2 : cos(.q t) -sin(.q t) 0 sin(.q t) cos(.q t) > rot3:=matrix([[k*eo/sqrt(delta"2+k"2*eo"2),0,delta/sqrt(delta"2+k" 2*EO"2)], [0,1,0], [-Oelta/sqrt(Delta"2+k"2*EO"2),0,k*EO/sqrt(Delta" 2+k"2*EO"2)]])i keo 22.m2 0 t:!. 2-k2m2 rot3 := t:!. keo -0 2k2-m2 2m 2 > bloch:=rot3.rot2.rot1i k2 E02 t:!. 2 cos(.qt) t:!..sin(.q. t) k EO t:!. t:!. cos(.q. t)k EO + -+ t:!. 2 + k2 E02 t:!. 2 + k2 E02 2k2-m2 t:!. 2 + k2 E02 t:!. 2 + k2 E02 t:!. sin(.q t) k EO sin(.q. t) bloch m2 cos(.q. t).-2 - t:!. +k 2 EO 2 22.m2 L.l A 2+ k2 E0 2 k EO t:!. t:!. cos(.q t)k EO k EO sin(.q. t) t:!.2 k2 E02 cos(.q t) -t:!. 2 + k2 E02 + t:!. 2 + k2 E02 22-EO2 t:!. 2 + k2 E02 + t:!. 2 + k2 E02 > subs(delta=o,bloch)i \cj 'To? f'\."""'- ;i-., L- f 1 r- /\J-' :l(iaj. T' k 't- r at1.- "'" -Q"(.i, IA.. (.Jt pf'j- oj, '1J:lt 'Ic c- () I

5 - :0.) fd1- I "Qj L co,r(, _O - [;,-t_l_- 0 t t -- L I 1.. j.-l;;4.,.;,' c ".,.,.:, ';"::':: '"../f; --;..; t l.t:-7:.-') (r1 D.;.v.'lfA - :\,.;:'J 2. ' = (-tr;;; 3) -t: H -.., -"1r (t"v,, '. -_.;- ci,?-;- L\A.1-- '-- (') : -I 0 r;r 1- v -;,..1-,,-:-, -0 :,:41:., -1;. tt-"\ i t. _0. "',,,c,.c,,!',., C :f, ll-ra-) :--'c"c" --_c--- "C- t -,-. A- -.-=:- Kr. t>.m c.,.cc 0 r D -{ ---, '\ () -J-4- --

6 MtJJ;&] :J'U -; 1" ia:tf' ( :::: i \ <:t5j of l") ---: \.J f.it ()" -- - _J r - t 4( 4J tl.t..a i..lj a (J.l(-tJ\1 - + < i\1:-- +' () - = i Q,,;A-il')(i -) ) -, ( A(ill) :d.s I --:..>u.- (. ---(6 ( R..,- /l - -(-,.tc _c_--- -'-- =) S C.=J- \..I;-:11.;., r- V, AffI(. ), :/1_\2.-,JIJ,. 0 If) --- r(t) =- N I""r (fy) -= N (ky- t C4-t-fl'bC.:c;;) '";.- t'-xnc'i+. - -=- 2- N k [t(ti.,..4)..; v(t,a)j,(,i (t,j'4.1 ([,) NJ J 4cdj (62 l(k "t; ;q4 J r t.k!tn {lada.--a --Ji\o..:$- 'J -f

7 _Cc A }'(4') a M G I Mf1.?(_4f) C-f6-6.')'"1;;' 2- (1r 1\ --t.) T: 4 ;Jofl.p.a-:L-1lJ.. ;Q.:;t (t;b:-""-1 Tz;)( '==" r"f'ri' (.4.;) - "- ;? = \ T;w'J..-dL1l.2. ". (24 -!-ts' of...pl).a/ I.-A2.,,<. i- --:.e... cl..c1 e.., "--c;c - I I -sf 0:;, l J- '" I.:t'1t- 1;1( =- ff/ ill- -p J '/,..,.It..J..t&,..-" L.) 2- IX- t- o(.:4j..1j:- j >l ) ') rf "'.. - l:.,. - )2. T -r; _c, C..'"...;.- N 't- "i:".c;'c,,;!. ;. ".cc, -."J;ci"","",,"!(T I'("-y 1- ji:( ) /[ -- ;=:: bt- L - 1r nit. d. 4' -n 1.A J- 1\_- " r.4- M t --- a.. cl- f d "-"'- 1/-0) 1) r r t; l4 /14J.-d "1r --

8 E(t) 11:/2 11: 0 t1 t )- 2 1/3 t 4 w T2*d) v Initial ground ---'-"" state population u w 11: pulse R u

9 -- -: > pi2pulse:=matrix([[1,-alpha,-alpha],[alpha,0,1], [-alpha,l,o]]); 1 ked ked > init :=Vector [column] ([0,0, -1] ); pi2pulse:= -0 ked 1 -keij" -1 0 init : : -1 ] > dephase:=matrix([[cos(delta*t21),-sin(delta*t21),0], [sin(delta*t21 ),cos(delta*t21),o], [0,0,1]]); > cos( t21) -sin( t21) 0 dephase:= sin(t21) cos( t21) > pipulse:=matrix([[1,0,-2*alpha], [0,-1,0], [-2*alpha,0,-1]]); ked pipulse := ked > rephase:=matrix([[cos(delta*t43),-sin(delta*t43),0], [sin(delta*t43 ), cas (Delta*t43), 0], [0,0,1]]) ; [COS( t43) -sin( 143) 0 ] rephase:= sin( t43) cos( t43) > phecho:=rephase.pipulse.dephase.pi2pulse.init; phecho := [(Cos( t43)cos( t21)+sin( t43) sin( t21)) ked + cos( t43) sin( t21) -sin( t43) co.( t21) ] [ (-{:os( t43) sin( t21)+sin( t43)cos( t21)), ked + sin( t43) sin( t21) + cos( t43) cos( t21) ] l [-2 2 cos( t21) -2 sin( t21) k2 E02 ked, [ > \ ]

10 b) 1-"-.11./ 'J Jt; 4 'f:c.t,o""-.1"1 H cd ;.L P--7 r?-.. /J vc1f,.. 1:..Jifp M.-z f?-.c... ad tb p;; '? 1r 7, CPtr J' f,e,.k I Jlio4 r.ltj; +-Q ( - fi, 7 JJ -, M bit. t..jw,l t:f! -.a- c1 f'l.p- O"""-c. G)-\J ;-;- / rh).-l-, ( COS4t -:s(.,t- 0' -') (/ ) '\ "'A t CuL A""+ 0. -t s'. \, t) 0 (/ sm..r\'\.jj. p...,1t"l..l t 2, pr-v --'- L-" <-I )1 j t... 0,..a M"'l-Q:., ;2.T/r,... I -t.j b-=.t tl.( # =1r(",e4J iif r-vk)/r: Lrr- J.. T -- - S6.<&0- AI Y-- iaj Z l.."l:/r l F=:, i I:' } 6'l -1 1 =1 );:) --- c

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