(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 69829, 2064]*) (*NotebookOutlinePosition[ 70483, 2087]*) (* CellTagsIndexPosition[ 70439, 2083]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["\<\ This tutorial demonstrates the behaviour of a two-level system upon action of \ periodic perturbation. Just evaluate the notebook. Then use the last input \ cell and change the parameters for your enjoyment! \ \>", "Title"], Cell["S. Rochester and D. Budker; October 6, 1999", "Subsubtitle"], Cell[CellGroupData[{ Cell["\<\ Here we define the Schroedinger Equation (SE) and the wave function in the \ 2-state basis:\ \>", "Subtitle"], Cell[BoxData[ \(SE[H_, \[CapitalPsi]_] := \[ImaginaryI]\ \[PartialD]\_t \[CapitalPsi] \ \[Equal] H . \ \[CapitalPsi]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[\[CapitalPsi][t_] = {a[t], b[t]}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(a[t]\)}, {\(b[t]\)} }], "\[NoBreak]", ")"}], MatrixForm[ { a[ t], b[ t]}]]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Explicit form of the Hamiltonian and the SE; note that the Hamiltonian is not quite Hermitian because of the \[ImaginaryI] \ \[CapitalGamma]/2 :\ \>", "Subtitle"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"H", "=", RowBox[{"(", GridBox[{ {"0", \(v0\ \[ExponentialE]\^\(\[ImaginaryI]\ \[Omega]\ t\)\)}, {\(v0\ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \[Omega]\ t\)\), \ \(\[Omega]0 - \[ImaginaryI]\ \[CapitalGamma]/2\)} }], ")"}]}], ";"}], "\[IndentingNewLine]", \(MatrixForm[ eqs = {\(LogicalExpand[ SE[H, \[CapitalPsi][t]]]\)[\([1]\)], \(LogicalExpand[ SE[H, \[CapitalPsi][t]]]\)[\([2]\)]}]\)}], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { RowBox[{ RowBox[{"\[ImaginaryI]", " ", RowBox[{ SuperscriptBox["a", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}], "==", \(\[ExponentialE]\^\(\[ImaginaryI]\ t\ \[Omega]\)\ v0\ \ b[t]\)}]}, { RowBox[{ RowBox[{"\[ImaginaryI]", " ", RowBox[{ SuperscriptBox["b", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}], "==", \(\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ t\ \[Omega]\)\ \ v0\ a[t] + \((\(-\(\(\[ImaginaryI]\ \[CapitalGamma]\)\/2\)\) + \[Omega]0)\)\ \ b[t]\)}]} }], "\[NoBreak]", ")"}], MatrixForm[ { Equal[ Times[ Complex[ 0, 1], Derivative[ 1][ a][ t]], Times[ Power[ E, Times[ Complex[ 0, 1], t, \[Omega]]], v0, b[ t]]], Equal[ Times[ Complex[ 0, 1], Derivative[ 1][ b][ t]], Plus[ Times[ Power[ E, Times[ Complex[ 0, -1], t, \[Omega]]], v0, a[ t]], Times[ Plus[ Times[ Complex[ 0, Rational[ -1, 2]], \[CapitalGamma]], \[Omega]0], b[ t]]]]}]]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Initial condition: all atoms are in the ground state at t=0 ", \ "Subtitle"], Cell[CellGroupData[{ Cell[BoxData[{ \(MatrixForm[\[CapitalPsi]0 = {1, 0}]\), "\[IndentingNewLine]", \(\(inits = {\(LogicalExpand[\[CapitalPsi][ 0] == \[CapitalPsi]0]\)[\([1]\)], \(LogicalExpand[\ \[CapitalPsi][0] == \[CapitalPsi]0]\)[\([2]\)]};\)\)}], "Input"], Cell[BoxData[ \(General::"spell1" \(\(:\)\(\ \)\) "Possible spelling error: new symbol name \"\!\(\[CapitalPsi]0\)\" is \ similar to existing symbol \"\!\(\[Omega]0\)\"."\)], "Message"], Cell[BoxData[ InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1"}, {"0"} }], "\[NoBreak]", ")"}], MatrixForm[ {1, 0}]]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Now we solve the SE explicitly. Here is the place where we put numerical \ values for various parameters. Start with \[CapitalGamma]=0, \ \[Omega]=\[Omega]0.\ \>", "Subtitle"], Cell[BoxData[ \(\(sols = \(NDSolve[ Union[eqs /. {\[CapitalGamma] \[Rule] 0, \[Omega]0 \[Rule] 20, \[Omega] \[Rule] 20, v0 \[Rule] 1}, inits], {a, b}, {t, 0, 100}, MaxSteps \[Rule] 10000]\)[\([1]\)];\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Now we plot the solution. Here we show the probability of finding the \ system in the excited state. We see the Rabi oscillations. Note that the \ oscillation (Rabi) frequency is ", Cell[BoxData[ \(TraditionalForm\`\(\(\[CapitalOmega]\_R\)\(=\)\)\)]], "2v0." }], "Subtitle"], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[{Abs[b[t]]\^2 /. sols}, {t, 0, 10}, PlotRange \[Rule] {0, 1}, AxesLabel \[Rule] \ {t, Probability}]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.0952381 0 0.618034 [ [.21429 -0.0125 -3 -9 ] [.21429 -0.0125 3 0 ] [.40476 -0.0125 -3 -9 ] [.40476 -0.0125 3 0 ] [.59524 -0.0125 -3 -9 ] [.59524 -0.0125 3 0 ] [.78571 -0.0125 -3 -9 ] [.78571 -0.0125 3 0 ] [.97619 -0.0125 -6 -9 ] [.97619 -0.0125 6 0 ] [1.025 0 0 -6.4375 ] [1.025 0 10 6.4375 ] [.01131 .12361 -18 -4.5 ] [.01131 .12361 0 4.5 ] [.01131 .24721 -18 -4.5 ] [.01131 .24721 0 4.5 ] [.01131 .37082 -18 -4.5 ] [.01131 .37082 0 4.5 ] [.01131 .49443 -18 -4.5 ] [.01131 .49443 0 4.5 ] [.01131 .61803 -6 -4.5 ] [.01131 .61803 0 4.5 ] [.02381 .64303 -35 0 ] [.02381 .64303 35 12.875 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .21429 0 m .21429 .00625 L s [(2)] .21429 -0.0125 0 1 Mshowa .40476 0 m .40476 .00625 L s [(4)] .40476 -0.0125 0 1 Mshowa .59524 0 m .59524 .00625 L s [(6)] .59524 -0.0125 0 1 Mshowa .78571 0 m .78571 .00625 L s [(8)] .78571 -0.0125 0 1 Mshowa .97619 0 m .97619 .00625 L s [(10)] .97619 -0.0125 0 1 Mshowa .125 Mabswid .07143 0 m .07143 .00375 L s .11905 0 m .11905 .00375 L s .16667 0 m .16667 .00375 L s .2619 0 m .2619 .00375 L s .30952 0 m .30952 .00375 L s .35714 0 m .35714 .00375 L s .45238 0 m .45238 .00375 L s .5 0 m .5 .00375 L s .54762 0 m .54762 .00375 L s .64286 0 m .64286 .00375 L s .69048 0 m .69048 .00375 L s .7381 0 m .7381 .00375 L s .83333 0 m .83333 .00375 L s .88095 0 m .88095 .00375 L s .92857 0 m .92857 .00375 L s .25 Mabswid 0 0 m 1 0 L s gsave 1.025 0 -61 -10.4375 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20.875 translate 1 -1 scale 63.000 13.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (t) show 1.000 setlinewidth grestore .02381 .12361 m .03006 .12361 L s [(0.2)] .01131 .12361 1 0 Mshowa .02381 .24721 m .03006 .24721 L s [(0.4)] .01131 .24721 1 0 Mshowa .02381 .37082 m .03006 .37082 L s [(0.6)] .01131 .37082 1 0 Mshowa .02381 .49443 m .03006 .49443 L s [(0.8)] .01131 .49443 1 0 Mshowa .02381 .61803 m .03006 .61803 L s [(1)] .01131 .61803 1 0 Mshowa .125 Mabswid .02381 .0309 m .02756 .0309 L s .02381 .0618 m .02756 .0618 L s .02381 .09271 m .02756 .09271 L s .02381 .15451 m .02756 .15451 L s .02381 .18541 m .02756 .18541 L s .02381 .21631 m .02756 .21631 L s .02381 .27812 m .02756 .27812 L s .02381 .30902 m .02756 .30902 L s .02381 .33992 m .02756 .33992 L s .02381 .40172 m .02756 .40172 L s .02381 .43262 m .02756 .43262 L s .02381 .46353 m .02756 .46353 L s .02381 .52533 m .02756 .52533 L s .02381 .55623 m .02756 .55623 L s .02381 .58713 m .02756 .58713 L s .25 Mabswid .02381 0 m .02381 .61803 L s gsave .02381 .64303 -96 -4 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20.875 translate 1 -1 scale 63.000 13.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (Probability) show 1.000 setlinewidth grestore 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .02381 0 m .02499 9e-005 L .02605 .00034 L .02729 .00082 L .02846 .00147 L .03053 .00307 L .03279 .00547 L .03527 .0089 L .0379 .01343 L .04262 .02379 L .05205 .05278 L .06244 .09625 L .0842 .21691 L .10458 .34763 L .12312 .46109 L .13291 .51281 L .14335 .55844 L .1491 .57858 L .15431 .59346 L .15683 .59943 L .15955 .60498 L .1621 .6093 L .16449 .61256 L .16691 .6151 L .1682 .61612 L .16957 .61697 L .17027 .6173 L .17102 .61758 L .17173 .61778 L .17239 .6179 L .17363 .61797 L .17495 .61781 L .17609 .61748 L .17731 .61694 L .17847 .61623 L .17953 .61543 L .18199 .61297 L .18458 .60951 L .18958 .60033 L .19412 .5892 L .20432 .55514 L .21377 .5135 L .22268 .46679 L .26115 .22605 L .28064 .11447 L .29171 .06438 L .29702 .04488 L .30207 .0294 L .30682 .01769 L .30933 .01267 L Mistroke .312 .00823 L .31447 .00496 L .3167 .00271 L .31778 .00186 L .31895 .00112 L .32005 .0006 L .32107 .00026 L .32225 4e-005 L .32352 2e-005 L .32471 .0002 L .3258 .00053 L .32708 .00113 L .32847 .00203 L .331 .00433 L .33364 .00766 L .33644 .0122 L .34147 .02293 L .35087 .05137 L .36098 .09312 L .37928 .1917 L .39896 .3164 L .42047 .45044 L .43042 .50433 L .44103 .5524 L .44619 .57153 L .45105 .58675 L .45575 .59872 L .46014 .60735 L .46238 .61078 L .46479 .61372 L .46616 .61504 L .46746 .61607 L .46862 .61679 L .46989 .61737 L .47112 .61772 L .47183 .61783 L .47248 .61787 L .47318 .61785 L .47381 .61778 L .47451 .61763 L .47524 .61741 L .47647 .61686 L .47761 .61618 L .4802 .61396 L .4826 .6111 L .48487 .60769 L .48912 .59949 L .4937 .58808 L .49866 .57279 L .50811 .53593 L Mistroke .51851 .48512 L .53955 .35972 L .5606 .22446 L .57986 .11437 L .58937 .07062 L .59439 .05124 L .59975 .03375 L .60489 .02022 L .60955 .01086 L .6118 .00736 L .61424 .00433 L .61535 .0032 L .61653 .0022 L .61763 .00143 L .61865 .00087 L .61987 .00037 L .62051 .0002 L .6212 7e-005 L .62235 0 L .62361 .00013 L .62474 .00043 L .62595 .00095 L .62709 .00162 L .62814 .00238 L .6305 .00467 L .63309 .00802 L .6383 .01746 L .6429 .02868 L .6478 .04353 L .65664 .07726 L .678 .18881 L .69816 .3163 L .71649 .43172 L .7269 .49029 L .73664 .53729 L .74624 .57428 L .75162 .59041 L .75653 .60201 L .75904 .60673 L .7617 .61084 L .76324 .61278 L .76463 .61427 L .76604 .61551 L .76733 .61641 L .76857 .61706 L .76987 .61752 L .77118 .61775 L .77192 .61778 L .77262 .61774 L .77387 .61749 L Mistroke .77502 .61708 L .77633 .61639 L .77758 .61552 L .77996 .61327 L .78219 .61047 L .78727 .60165 L .79222 .58985 L .79755 .57377 L .80717 .53647 L .81598 .49416 L .85568 .25045 L .8757 .13174 L .88442 .08862 L .89386 .05029 L .89913 .03321 L .90398 .02047 L .90862 .01108 L .91096 .00742 L .91353 .00423 L .91493 .00286 L .91626 .00181 L .91746 .00107 L .91876 .00048 L .91943 .00027 L .92017 .00011 L .92094 2e-005 L .92167 0 L .92297 .00016 L .9237 .00035 L .92437 .00059 L .92559 .00118 L .92672 .00191 L .9293 .00422 L .93181 .00733 L .93449 .01157 L .93982 .02277 L .94473 .03628 L .95583 .0772 L .97598 .1815 L .97619 .18275 L Mfstroke % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgOol00`00Oomoo`0ZOol01000Oomoogoo 0P000goo00<007ooOol08Goo00<007ooOol06goo0P000Woo0P002Goo00<007ooOol02goo1@001goo 000UOol01000Oomoo`00@7oo00<007ooOol00goo00<007ooOol0?Woo00<007ooOol00Woo00<007oo Ool0?Woo00<007ooOol017oo00<007ooOol05Goo00<007ooOol027oo000UOol01000Oomoo`00@7oo 00<007ooOol00goo00<007ooOol0?Woo00<007ooOol00goo00<007ooOol0?Goo00<007ooOol017oo 00<007ooOol087oo000UOol01@00Oomoogoo0000?Woo00<007ooOol01Goo00<007ooOol0?7oo00<0 07ooOol017oo00<007ooOol0?7oo00<007ooOol01Woo00<007ooOol07goo000UOol00`00Oomoo`02 Ool00`00Oomoo`0kOol00`00Oomoo`05Ool00`00Oomoo`0lOol00`00Oomoo`05Ool00`00Oomoo`0k Ool00`00Oomoo`06Ool00`00Oomoo`0OOol002Eoo`03001oogoo009oo`03001oogoo03Yoo`03001o ogoo00Ioo`03001oogoo03]oo`03001oogoo00Moo`03001oogoo03Uoo`03001oogoo00Moo`03001o ogoo01moo`009Goo0P0017oo00<007ooOol0>Goo00<007ooOol01goo00<007ooOol0>Woo00<007oo Ool01goo00<007ooOol0>Goo00<007ooOol027oo00<007ooOol07Woo000UOol00`00Oomoo`03Ool0 0`00Oomoo`0iOol00`00Oomoo`07Ool00`00Oomoo`0iOol00`00Oomoo`09Ool00`00Oomoo`0hOol0 0`00Oomoo`08Ool00`00Oomoo`0NOol002Eoo`03001oogoo00=oo`03001oogoo03Qoo`03001oogoo 00Qoo`03001oogoo03Uoo`03001oogoo00Uoo`03001oogoo03Qoo`03001oogoo00Qoo`03001oogoo 01ioo`009Goo00<007ooOol00goo00<007ooOol0>7oo00<007ooOol027oo00<007ooOol0>Goo00<0 07ooOol02Goo00<007ooOol0=goo00<007ooOol02Woo00<007ooOol07Goo000UOol00`00Oomoo`04 Ool00`00Oomoo`0gOol00`00Oomoo`09Ool00`00Oomoo`0gOol00`00Oomoo`0:Ool00`00Oomoo`0g Ool00`00Oomoo`0:Ool00`00Oomoo`0MOol002Eoo`03001oogoo00Aoo`03001oogoo03Ioo`03001o ogoo00Yoo`03001oogoo03Moo`03001oogoo00]oo`03001oogoo03Ioo`03001oogoo00Yoo`03001o ogoo01eoo`009Goo00<007ooOol017oo00<007ooOol0=Woo00<007ooOol02Woo00<007ooOol0=goo 00<007ooOol02goo00<007ooOol0=Woo00<007ooOol02goo00<007ooOol077oo000UOol20005Ool0 0`00Oomoo`0eOol00`00Oomoo`0Ool00`00Oomoo`0cOol00`00Oomoo`0>Ool00`00Oomo o`0cOol00`00Oomoo`0>Ool00`00Oomoo`0KOol002Eoo`8000Moo`03001oogoo039oo`03001oogoo 00moo`03001oogoo039oo`03001oogoo00moo`03001oogoo039oo`03001oogoo00moo`03001oogoo 01Yoo`009Goo00<007ooOol01Woo00<007ooOol0Ool00`00Oomoo`0QOol00`00Oomoo`0POol00`00Oomoo`0ROol00`00Oomoo`0OOol00`00 Oomoo`0QOol00`00Oomoo`0eOol002Eoo`03001oogoo00ioo`03001oogoo025oo`03001oogoo021o o`03001oogoo025oo`03001oogoo021oo`03001oogoo025oo`03001oogoo03Eoo`009Goo00<007oo Ool03Woo00<007ooOol08Goo00<007ooOol087oo00<007ooOol08Goo00<007ooOol08Goo00<007oo Ool087oo00<007ooOol0=Goo000UOol00`00Oomoo`0>Ool00`00Oomoo`0QOol00`00Oomoo`0QOol0 0`00Oomoo`0POol00`00Oomoo`0QOol00`00Oomoo`0POol00`00Oomoo`0eOol002Eoo`80011oo`03 001oogoo021oo`03001oogoo025oo`03001oogoo021oo`03001oogoo025oo`03001oogoo021oo`03 001oogoo03Eoo`009Goo00<007ooOol03goo00<007ooOol087oo00<007ooOol08Goo00<007ooOol0 87oo00<007ooOol08Goo00<007ooOol07goo00<007ooOol0=Woo000UOol00`00Oomoo`0?Ool00`00 Oomoo`0POol00`00Oomoo`0QOol00`00Oomoo`0POol00`00Oomoo`0QOol00`00Oomoo`0OOol00`00 Oomoo`0fOol002Eoo`03001oogoo00moo`03001oogoo01moo`03001oogoo029oo`03001oogoo01mo o`03001oogoo029oo`03001oogoo01moo`03001oogoo03Ioo`009Goo00<007ooOol03goo00<007oo Ool07goo00<007ooOol08Woo00<007ooOol07goo00<007ooOol08Woo00<007ooOol07goo00<007oo Ool0=Woo000UOol00`00Oomoo`0?Ool00`00Oomoo`0OOol00`00Oomoo`0SOol00`00Oomoo`0NOol0 0`00Oomoo`0SOol00`00Oomoo`0NOol00`00Oomoo`0fOol002Eoo`03001oogoo011oo`03001oogoo 01ioo`03001oogoo02=oo`03001oogoo01ioo`03001oogoo02=oo`03001oogoo01eoo`03001oogoo 03Moo`009Goo0P004Goo00<007ooOol07Woo00<007ooOol08goo00<007ooOol07Woo00<007ooOol0 8goo00<007ooOol07Goo00<007ooOol0=goo000UOol00`00Oomoo`0@Ool00`00Oomoo`0NOol00`00 Oomoo`0SOol00`00Oomoo`0NOol00`00Oomoo`0SOol00`00Oomoo`0MOol00`00Oomoo`0gOol002Eo o`03001oogoo011oo`03001oogoo01eoo`03001oogoo02Aoo`03001oogoo01eoo`03001oogoo02Ao o`03001oogoo01eoo`03001oogoo03Moo`009Goo00<007ooOol047oo00<007ooOol07Goo00<007oo Ool097oo00<007ooOol07Goo00<007ooOol097oo00<007ooOol07Goo00<007ooOol0=goo000UOol0 0`00Oomoo`0@Ool00`00Oomoo`0MOol00`00Oomoo`0UOol00`00Oomoo`0LOol00`00Oomoo`0TOol0 0`00Oomoo`0MOol00`00Oomoo`0gOol0019oo`8000Aoo`8000=oo`<000Eoo`03001oogoo015oo`03 001oogoo01aoo`03001oogoo02Eoo`03001oogoo01aoo`03001oogoo02Eoo`03001oogoo01]oo`03 001oogoo03Qoo`004Goo00@007ooOol000Qoo`04001oogoo0004Ool00`00Oomoo`0AOol00`00Oomo o`0LOol00`00Oomoo`0UOol00`00Oomoo`0LOol00`00Oomoo`0UOol00`00Oomoo`0KOol00`00Oomo o`0hOol0015oo`04001oogoo0008Ool01000Oomoo`0017oo0`004Goo00<007ooOol077oo00<007oo Ool09Goo00<007ooOol077oo00<007ooOol09Goo00<007ooOol06goo00<007ooOol0>7oo000AOol0 1000Oomoo`0027oo0`001Goo00<007ooOol04Goo00<007ooOol06goo00<007ooOol09Woo00<007oo Ool06goo00<007ooOol09Woo00<007ooOol06goo00<007ooOol0>7oo000AOol01000Oomoo`002Goo 00<007ooOol017oo00<007ooOol04Goo00<007ooOol06goo00<007ooOol09Woo00<007ooOol06goo 00<007ooOol09Woo00<007ooOol06goo00<007ooOol0>7oo000BOol2000:Ool30004Ool00`00Oomo o`0AOol00`00Oomoo`0KOol00`00Oomoo`0WOol00`00Oomoo`0JOol00`00Oomoo`0VOol00`00Oomo o`0KOol00`00Oomoo`0hOol002Eoo`03001oogoo019oo`03001oogoo01Yoo`03001oogoo02Moo`03 001oogoo01Yoo`03001oogoo02Ioo`03001oogoo01Yoo`03001oogoo03Uoo`009Goo00<007ooOol0 4Woo00<007ooOol06Woo00<007ooOol09goo00<007ooOol06Woo00<007ooOol09goo00<007ooOol0 6Goo00<007ooOol0>Goo000UOol00`00Oomoo`0BOol00`00Oomoo`0JOol00`00Oomoo`0WOol00`00 Oomoo`0JOol00`00Oomoo`0WOol00`00Oomoo`0IOol00`00Oomoo`0iOol002Eoo`03001oogoo019o o`03001oogoo01Yoo`03001oogoo02Moo`03001oogoo01Uoo`03001oogoo02Qoo`03001oogoo01Uo o`03001oogoo03Uoo`009Goo0P004goo00<007ooOol06Goo00<007ooOol0:7oo00<007ooOol06Goo 00<007ooOol0:7oo00<007ooOol06Goo00<007ooOol0>Goo000UOol00`00Oomoo`0COol00`00Oomo o`0HOol00`00Oomoo`0XOol00`00Oomoo`0IOol00`00Oomoo`0XOol00`00Oomoo`0IOol00`00Oomo o`0iOol002Eoo`03001oogoo01=oo`03001oogoo01Qoo`03001oogoo02Uoo`03001oogoo01Qoo`03 001oogoo02Qoo`03001oogoo01Qoo`03001oogoo03Yoo`009Goo00<007ooOol04goo00<007ooOol0 67oo00<007ooOol0:Goo00<007ooOol067oo00<007ooOol0:7oo00<007ooOol067oo00<007ooOol0 >Woo000UOol00`00Oomoo`0COol00`00Oomoo`0HOol00`00Oomoo`0YOol00`00Oomoo`0HOol00`00 Oomoo`0YOol00`00Oomoo`0GOol00`00Oomoo`0jOol002Eoo`03001oogoo01=oo`03001oogoo01Qo o`03001oogoo02Uoo`03001oogoo01Moo`03001oogoo02Yoo`03001oogoo01Moo`03001oogoo03Yo o`009Goo00<007ooOol057oo00<007ooOol05Woo00<007ooOol0:Woo00<007ooOol05goo00<007oo Ool0:Woo00<007ooOol05goo00<007ooOol0>Woo000UOol2000EOol00`00Oomoo`0FOol00`00Oomo o`0ZOol00`00Oomoo`0GOol00`00Oomoo`0ZOol00`00Oomoo`0FOol00`00Oomoo`0kOol002Eoo`03 001oogoo01Aoo`03001oogoo01Ioo`03001oogoo02]oo`03001oogoo01Ioo`03001oogoo02Yoo`03 001oogoo01Ioo`03001oogoo03]oo`009Goo00<007ooOol057oo00<007ooOol05Woo00<007ooOol0 :goo00<007ooOol05Woo00<007ooOol0:Woo00<007ooOol05Woo00<007ooOol0>goo000UOol00`00 Oomoo`0DOol00`00Oomoo`0FOol00`00Oomoo`0[Ool00`00Oomoo`0FOol00`00Oomoo`0ZOol00`00 Oomoo`0FOol00`00Oomoo`0kOol002Eoo`03001oogoo01Aoo`03001oogoo01Ioo`03001oogoo02]o o`03001oogoo01Eoo`03001oogoo02aoo`03001oogoo01Eoo`03001oogoo03]oo`009Goo00<007oo Ool05Goo00<007ooOol057oo00<007ooOol0;7oo00<007ooOol05Goo00<007ooOol0;7oo00<007oo Ool05Goo00<007ooOol0>goo000UOol00`00Oomoo`0EOol00`00Oomoo`0DOol00`00Oomoo`0/Ool0 0`00Oomoo`0EOol00`00Oomoo`0/Ool00`00Oomoo`0DOol00`00Oomoo`0lOol002Eoo`8001Ioo`03 001oogoo01Aoo`03001oogoo02aoo`03001oogoo01Eoo`03001oogoo02aoo`03001oogoo01Aoo`03 001oogoo03aoo`009Goo00<007ooOol05Goo00<007ooOol057oo00<007ooOol0;Goo00<007ooOol0 57oo00<007ooOol0;7oo00<007ooOol057oo00<007ooOol0?7oo000UOol00`00Oomoo`0EOol00`00 Oomoo`0DOol00`00Oomoo`0]Ool00`00Oomoo`0DOol00`00Oomoo`0/Ool00`00Oomoo`0DOol00`00 Oomoo`0lOol002Eoo`03001oogoo01Ioo`03001oogoo01=oo`03001oogoo02eoo`03001oogoo01=o o`03001oogoo02ioo`03001oogoo01=oo`03001oogoo03aoo`009Goo00<007ooOol05Woo00<007oo Ool04Woo00<007ooOol0;Woo00<007ooOol04goo00<007ooOol0;Woo00<007ooOol04goo00<007oo Ool0?7oo000BOol20004Ool20004Ool20005Ool00`00Oomoo`0FOol00`00Oomoo`0BOol00`00Oomo o`0^Ool00`00Oomoo`0COol00`00Oomoo`0^Ool00`00Oomoo`0BOol00`00Oomoo`0mOol0015oo`04 001oogoo0008Ool01000Oomoo`0017oo00<007ooOol05Woo00<007ooOol04Woo00<007ooOol0;Woo 00<007ooOol04goo00<007ooOol0;Woo00<007ooOol04Woo00<007ooOol0?Goo000AOol01000Oomo o`0027oo00@007ooOol000Aoo`<001Ioo`03001oogoo019oo`03001oogoo02moo`03001oogoo019o o`03001oogoo02ioo`03001oogoo019oo`03001oogoo03eoo`004Goo00@007ooOol000Uoo`8000Eo o`03001oogoo01Ioo`03001oogoo019oo`03001oogoo02moo`03001oogoo019oo`03001oogoo02mo o`03001oogoo015oo`03001oogoo03eoo`004Goo00@007ooOol000Qoo`04001oogoo0004Ool00`00 Oomoo`0GOol00`00Oomoo`0AOol00`00Oomoo`0_Ool00`00Oomoo`0AOol00`00Oomoo`0`Ool00`00 Oomoo`0AOol00`00Oomoo`0mOol0019oo`8000Yoo`8000Eoo`03001oogoo01Moo`03001oogoo011o o`03001oogoo031oo`03001oogoo015oo`03001oogoo031oo`03001oogoo011oo`03001oogoo03io o`009Goo00<007ooOol05goo00<007ooOol047oo00<007ooOol0<7oo00<007ooOol04Goo00<007oo Ool0<7oo00<007ooOol047oo00<007ooOol0?Woo000UOol00`00Oomoo`0GOol00`00Oomoo`0@Ool0 0`00Oomoo`0aOol00`00Oomoo`0@Ool00`00Oomoo`0aOol00`00Oomoo`0?Ool00`00Oomoo`0nOol0 02Eoo`03001oogoo01Moo`03001oogoo011oo`03001oogoo035oo`03001oogoo011oo`03001oogoo 035oo`03001oogoo00moo`03001oogoo03ioo`009Goo00<007ooOol05goo00<007ooOol047oo00<0 07ooOol0Ool00`00Oomoo`0bOol00`00Oomoo`0?Ool00`00Oomoo`0bOol00`00 Oomoo`0>Ool00`00Oomoo`0oOol002Eoo`03001oogoo01Qoo`03001oogoo00ioo`03001oogoo03=o o`03001oogoo00ioo`03001oogoo03=oo`03001oogoo00eoo`03001oogoo03moo`009Goo00<007oo Ool067oo00<007ooOol03Woo00<007ooOol0Ool00`00Oomoo`0cOol0 0`00Oomoo`0>Ool00`00Oomoo`0cOol00`00Oomoo`0=Ool00`00Oomoo`0oOol002Eoo`03001oogoo 01Qoo`03001oogoo00ioo`03001oogoo03=oo`03001oogoo00eoo`03001oogoo03Aoo`03001oogoo 00eoo`03001oogoo03moo`009Goo00<007ooOol067oo00<007ooOol03Goo00<007ooOol0=Goo00<0 07ooOol037oo00<007ooOol0=Goo00<007ooOol02goo00<007ooOol0@7oo000UOol00`00Oomoo`0I Ool00`00Oomoo`07oo00<007ooOol02Goo00<007ooOol0@Goo000UOol00`00Oomoo`0KOol00`00Oomoo`08 Ool00`00Oomoo`0hOol00`00Oomoo`09Ool00`00Oomoo`0hOol00`00Oomoo`09Ool00`00Oomoo`11 Ool002Eoo`03001oogoo01]oo`03001oogoo00Qoo`03001oogoo03Uoo`03001oogoo00Qoo`03001o ogoo03Uoo`03001oogoo00Qoo`03001oogoo045oo`009Goo0P007Goo00<007ooOol01Woo00<007oo Ool0>Woo00<007ooOol01goo00<007ooOol0>Woo00<007ooOol01goo00<007ooOol0@Woo000UOol0 0`00Oomoo`0LOol00`00Oomoo`06Ool00`00Oomoo`0jOol00`00Oomoo`07Ool00`00Oomoo`0jOol0 0`00Oomoo`07Ool00`00Oomoo`12Ool002Eoo`03001oogoo01aoo`03001oogoo00Ioo`03001oogoo 03]oo`03001oogoo00Ioo`03001oogoo03Yoo`03001oogoo00Moo`03001oogoo049oo`009Goo00<0 07ooOol07Goo00<007ooOol017oo00<007ooOol0?7oo00<007ooOol01Goo00<007ooOol0?7oo00<0 07ooOol01Goo00<007ooOol0@goo000UOol00`00Oomoo`0MOol00`00Oomoo`04Ool00`00Oomoo`0m Ool00`00Oomoo`04Ool00`00Oomoo`0lOol00`00Oomoo`05Ool00`00Oomoo`13Ool002Eoo`03001o ogoo01eoo`03001oogoo00Aoo`03001oogoo03eoo`03001oogoo00=oo`03001oogoo03ioo`03001o ogoo00=oo`03001oogoo04Aoo`007Goo1@000goo00<007ooOol07Woo00<007ooOol00Woo00<007oo Ool0?goo00<007ooOol00Woo00<007ooOol0?goo00D007ooOomoo`0004Moo`007goo00<007ooOol0 0goo00<007ooOol07goo1@00@Goo0P0000=oo`000000@Woo0P0000=oo`000000Agoo000OOol00`00 Oomoo`03Ool3000POol20015Ool20015Ool20018Ool001moo`03001oogoo0?ioo`007goo00<007oo Ool0oWoo000NOol2003oOol1Ool00?moob5oo`00ogoo8Goo003oOolQOol00?moob5oo`00@Goo0P00 gGoo0012Ool00`00Oomoo`3KOol000Aoo`<000Aoo`@000=oo`80009oo`@000=oo`T000Aoo`<0009o o`<000Aoo`<000=oo`8000Eoo`03001oogoo0=Yoo`001Goo00<007ooOol017oo00<007ooOol00Woo 00@007ooOol0009oo`04001oogoo0002Ool40002Ool01000Oomoo`0017oo00<007ooOol00Woo00<0 07ooOol017oo00D007ooOomoo`00009oo`07001oogooOol007oo0000fgoo0005Ool30004Ool00`00 Oomoo`02Ool01000Oomoo`000Woo00@007ooOol000Eoo`04001oogoo0002Ool00`00Oomoo`02Ool0 0`00Oomoo`02Ool00`00Oomoo`04Ool01@00Oomoogoo00001Goo00@007ooOol00=]oo`001Goo00@0 07ooOol0009oo`@000=oo`8000=oo`<000Aoo`<0009oo`<000Aoo`8000Aoo`03001oogoo00=oo`80 009oo`D00005Ool00000001oo`02003JOol000Eoo`04001oogoo000>Ool00`00Oomoo`09Ool00`00 Oomoo`0:Ool00`00Oomoo`08Ool00`00Oomoo`3ROol000Aoo`@000ioo`8000Yoo`8000Moo`04001o ogooOol20006Ool00`00Oomoo`3VOol00?moob5oo`00ogoo8Goo003oOolQOol00001\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-1.6685, -0.0817938, \ 0.0443894, 0.00684034}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Now let us \"turn on\" relaxation. First let's try \[CapitalGamma]<", "Subtitle"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(sols = \(NDSolve[ Union[eqs /. {\[CapitalGamma] \[Rule] .3, \[Omega]0 \[Rule] 20, \[Omega] \[Rule] 20, v0 \[Rule] 1}, inits], {a, b}, {t, 0, 100}, MaxSteps \[Rule] 10000]\)[\([1]\)];\)\), "\[IndentingNewLine]", \(Plot[{Abs[b[t]]\^2 /. sols}, {t, 0, 10}, PlotRange \[Rule] {0, 1}, AxesLabel \[Rule] \ {t, Probability}]\)}], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.0952381 0 0.618034 [ [.21429 -0.0125 -3 -9 ] [.21429 -0.0125 3 0 ] [.40476 -0.0125 -3 -9 ] [.40476 -0.0125 3 0 ] [.59524 -0.0125 -3 -9 ] [.59524 -0.0125 3 0 ] [.78571 -0.0125 -3 -9 ] [.78571 -0.0125 3 0 ] [.97619 -0.0125 -6 -9 ] [.97619 -0.0125 6 0 ] [1.025 0 0 -6.4375 ] [1.025 0 10 6.4375 ] [.01131 .12361 -18 -4.5 ] [.01131 .12361 0 4.5 ] [.01131 .24721 -18 -4.5 ] [.01131 .24721 0 4.5 ] [.01131 .37082 -18 -4.5 ] [.01131 .37082 0 4.5 ] [.01131 .49443 -18 -4.5 ] [.01131 .49443 0 4.5 ] [.01131 .61803 -6 -4.5 ] [.01131 .61803 0 4.5 ] [.02381 .64303 -35 0 ] [.02381 .64303 35 12.875 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .21429 0 m .21429 .00625 L s [(2)] .21429 -0.0125 0 1 Mshowa .40476 0 m .40476 .00625 L s [(4)] .40476 -0.0125 0 1 Mshowa .59524 0 m .59524 .00625 L s [(6)] .59524 -0.0125 0 1 Mshowa .78571 0 m .78571 .00625 L s [(8)] .78571 -0.0125 0 1 Mshowa .97619 0 m .97619 .00625 L s [(10)] .97619 -0.0125 0 1 Mshowa .125 Mabswid .07143 0 m .07143 .00375 L s .11905 0 m .11905 .00375 L s .16667 0 m .16667 .00375 L s .2619 0 m .2619 .00375 L s .30952 0 m .30952 .00375 L s .35714 0 m .35714 .00375 L s .45238 0 m .45238 .00375 L s .5 0 m .5 .00375 L s .54762 0 m .54762 .00375 L s .64286 0 m .64286 .00375 L s .69048 0 m .69048 .00375 L s .7381 0 m .7381 .00375 L s .83333 0 m .83333 .00375 L s .88095 0 m .88095 .00375 L s .92857 0 m .92857 .00375 L s .25 Mabswid 0 0 m 1 0 L s gsave 1.025 0 -61 -10.4375 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20.875 translate 1 -1 scale 63.000 13.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (t) show 1.000 setlinewidth grestore .02381 .12361 m .03006 .12361 L s [(0.2)] .01131 .12361 1 0 Mshowa .02381 .24721 m .03006 .24721 L s [(0.4)] .01131 .24721 1 0 Mshowa .02381 .37082 m .03006 .37082 L s [(0.6)] .01131 .37082 1 0 Mshowa .02381 .49443 m .03006 .49443 L s [(0.8)] .01131 .49443 1 0 Mshowa .02381 .61803 m .03006 .61803 L s [(1)] .01131 .61803 1 0 Mshowa .125 Mabswid .02381 .0309 m .02756 .0309 L s .02381 .0618 m .02756 .0618 L s .02381 .09271 m .02756 .09271 L s .02381 .15451 m .02756 .15451 L s .02381 .18541 m .02756 .18541 L s .02381 .21631 m .02756 .21631 L s .02381 .27812 m .02756 .27812 L s .02381 .30902 m .02756 .30902 L s .02381 .33992 m .02756 .33992 L s .02381 .40172 m .02756 .40172 L s .02381 .43262 m .02756 .43262 L s .02381 .46353 m .02756 .46353 L s .02381 .52533 m .02756 .52533 L s .02381 .55623 m .02756 .55623 L s .02381 .58713 m .02756 .58713 L s .25 Mabswid .02381 0 m .02381 .61803 L s gsave .02381 .64303 -96 -4 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20.875 translate 1 -1 scale 63.000 13.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (Probability) show 1.000 setlinewidth grestore 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .02381 0 m .02499 9e-005 L .02605 .00034 L .02729 .00082 L .02846 .00146 L .03053 .00304 L .03279 .0054 L .03527 .00874 L .0379 .01313 L .04262 .0231 L .05205 .05049 L .06244 .09059 L .10458 .30654 L .11537 .36036 L .12562 .40557 L .1356 .442 L .14046 .45656 L .14487 .46779 L .14918 .47687 L .15384 .48452 L .1564 .48774 L .15877 .49007 L .16102 .49173 L .16216 .49236 L .1634 .49289 L .16405 .4931 L .16476 .49327 L .16605 .49345 L .1673 .49345 L .16846 .49329 L .1697 .49297 L .17041 .4927 L .17108 .49241 L .17244 .49166 L .17387 .49066 L .17632 .48843 L .17857 .48584 L .18364 .47809 L .18864 .46794 L .19318 .45668 L .20338 .42495 L .22175 .34978 L .26173 .15668 L .28005 .08138 L .29046 .04794 L .29555 .03456 L .3002 .02418 L .30478 .01572 L .30978 .00855 L .31252 .00553 L Mistroke .31506 .00332 L .31749 .00174 L .31884 .00108 L .32007 .00061 L .32139 .00026 L .32206 .00014 L .32279 5e-005 L .32408 0 L .32528 9e-005 L .32602 .0002 L .32671 .00034 L .328 .00072 L .3294 .00129 L .33091 .00209 L .33349 .00386 L .33622 .00633 L .34111 .01217 L .34602 .01978 L .35051 .02818 L .36063 .05155 L .37892 .10554 L .39948 .17415 L .41859 .23381 L .4281 .25879 L .43706 .27827 L .44202 .2871 L .44738 .29496 L .45226 .3005 L .45675 .30421 L .45921 .30566 L .46144 .30663 L .46268 .30703 L .46333 .30719 L .46403 .30733 L .4653 .30751 L .46647 .30758 L .46754 .30755 L .4687 .30744 L .46979 .30726 L .47079 .30701 L .4719 .30667 L .47311 .30619 L .47556 .30494 L .47826 .3031 L .48072 .30101 L .48625 .29494 L .49621 .27943 L .50618 .25869 L .51537 .23575 L .53616 .17516 L Mistroke .55649 .11299 L .57553 .06133 L .58488 .04053 L .59377 .02438 L .59867 .01714 L .60402 .01065 L .60888 .00607 L .61339 .00297 L .61603 .00166 L .61847 .00079 L .61974 .00046 L .62047 .00031 L .62114 .0002 L .62185 .00011 L .6226 5e-005 L .6233 1e-005 L .62396 0 L .62517 4e-005 L .62644 .00017 L .62753 .00035 L .62873 .00061 L .63002 .00098 L .63141 .00147 L .63392 .0026 L .63866 .00558 L .64363 .00984 L .65253 .02012 L .66209 .03444 L .67229 .05267 L .69066 .08998 L .71059 .1308 L .72108 .14996 L .73218 .1671 L .7421 .17898 L .74773 .18408 L .75043 .18608 L .753 .18771 L .75531 .18894 L .75783 .19002 L .76013 .19078 L .76115 .19104 L .76225 .19128 L .76355 .19149 L .76478 .19163 L .76597 .1917 L .76706 .19171 L .76825 .19166 L .76955 .19155 L .77021 .19146 L .77092 .19135 L Mistroke .77218 .19109 L .77476 .19037 L .77749 .1893 L .78239 .18663 L .78767 .18272 L .79348 .17721 L .80387 .16455 L .81362 .1499 L .85238 .07819 L .87208 .04363 L .88249 .02843 L .89359 .01543 L .89914 .01033 L .90438 .00643 L .90908 .00371 L .91151 .00259 L .91417 .0016 L .91667 .00088 L .91787 .00062 L .91901 .00041 L .9203 .00022 L .92148 .0001 L .92274 2e-005 L .92346 0 L .92412 0 L .92521 3e-005 L .92623 9e-005 L .92736 .00019 L .92856 .00035 L .92976 .00055 L .93103 .00081 L .93328 .00141 L .93828 .00328 L .94282 .00563 L .95302 .0129 L .96246 .0218 L .97138 .0317 L .97619 .03749 L Mfstroke % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHggoo00<007ooOol0 1goo00<007ooOol0>Goo00<007ooOol027oo0P0057oo00<007ooOol027oo000UOol01000Oomoo`00 ?goo00<007ooOol01Woo00<007ooOol0>Woo00<007ooOol027oo00<007ooOol0=goo00<007ooOol0 2goo00<007ooOol077oo000UOol01@00Oomoogoo0000?Goo00<007ooOol01goo00<007ooOol0>Goo 00<007ooOol02Woo00<007ooOol0=Goo00<007ooOol03Goo00<007ooOol06goo000UOol00`00Oomo o`02Ool00`00Oomoo`0iOol00`00Oomoo`09Ool00`00Oomoo`0gOol00`00Oomoo`0Ool00`00Oomoo`0` Ool00`00Oomoo`0DOol00`00Oomoo`0GOol002Eoo`03001oogoo00=oo`03001oogoo03Moo`03001o ogoo00aoo`03001oogoo03=oo`03001oogoo011oo`03001oogoo02moo`03001oogoo02ioo`009Goo 00<007ooOol017oo00<007ooOol0=Goo00<007ooOol03Goo00<007ooOol0Ool00`00Oomoo`0aOol00`00Oomoo`0BOol00`00Oomoo`0/Ool00`00Oomoo`0`Ool002Eoo`03 001oogoo00Aoo`03001oogoo03Eoo`03001oogoo00ioo`03001oogoo035oo`03001oogoo019oo`03 001oogoo02aoo`03001oogoo031oo`009Goo00<007ooOol017oo00<007ooOol0=Goo00<007ooOol0 3Woo00<007ooOol0<7oo00<007ooOol057oo00<007ooOol0:Woo00<007ooOol07oo000UOol00`00Oomoo`07Ool00`00Oomoo`0^Ool00`00 Oomoo`0GOol00`00Oomoo`0VOol00`00Oomoo`0POol00`00Oomoo`0LOol00`00Oomoo`0hOol0019o o`8000Aoo`8000=oo`@000Aoo`03001oogoo00Moo`03001oogoo02ioo`03001oogoo01Moo`03001o ogoo02Ioo`03001oogoo021oo`03001oogoo01]oo`03001oogoo03Uoo`004Goo00@007ooOol000Qo o`03001oogoo00Eoo`03001oogoo00Moo`03001oogoo02ioo`03001oogoo01Moo`03001oogoo02Eo o`03001oogoo029oo`03001oogoo01Yoo`03001oogoo03Uoo`004Goo00@007ooOol000Uoo`03001o ogoo00Aoo`<000Qoo`03001oogoo02aoo`03001oogoo01Qoo`03001oogoo02Eoo`03001oogoo029o o`03001oogoo01Uoo`03001oogoo03Yoo`004Goo00@007ooOol000Yoo`03001oogoo00=oo`03001o ogoo00Qoo`03001oogoo02aoo`03001oogoo01Uoo`03001oogoo02Aoo`03001oogoo02=oo`03001o ogoo01Qoo`03001oogoo03Yoo`004Goo00@007ooOol000Qoo`04001oogoo0004Ool00`00Oomoo`08 Ool00`00Oomoo`0/Ool00`00Oomoo`0IOol00`00Oomoo`0SOol00`00Oomoo`0TOol00`00Oomoo`0G Ool00`00Oomoo`0kOol0019oo`8000Yoo`8000Eoo`03001oogoo00Qoo`03001oogoo02]oo`03001o ogoo01Yoo`03001oogoo02=oo`03001oogoo02Eoo`03001oogoo01Ioo`03001oogoo03]oo`009Goo 00<007ooOol027oo00<007ooOol0:goo00<007ooOol06goo00<007ooOol08Woo00<007ooOol09Goo 00<007ooOol05Goo00<007ooOol0?7oo000UOol00`00Oomoo`09Ool00`00Oomoo`0ZOol00`00Oomo o`0KOol00`00Oomoo`0QOol00`00Oomoo`0WOol00`00Oomoo`0DOol00`00Oomoo`0lOol002Eoo`03 001oogoo00Uoo`03001oogoo02Yoo`03001oogoo01]oo`03001oogoo025oo`03001oogoo02Moo`03 001oogoo01=oo`03001oogoo03eoo`009Goo0P002Woo00<007ooOol0:Goo00<007ooOol07Goo00<0 07ooOol087oo00<007ooOol0:7oo00<007ooOol04Goo00<007ooOol0?Woo000UOol00`00Oomoo`09 Ool00`00Oomoo`0YOol00`00Oomoo`0MOol00`00Oomoo`0OOol00`00Oomoo`0ZOol00`00Oomoo`0@ Ool00`00Oomoo`0nOol002Eoo`03001oogoo00Uoo`03001oogoo02Uoo`03001oogoo01eoo`03001o ogoo01moo`03001oogoo02Yoo`03001oogoo00moo`03001oogoo03moo`009Goo00<007ooOol02Woo 00<007ooOol0:7oo00<007ooOol07Goo00<007ooOol07goo00<007ooOol0:goo00<007ooOol03Goo 00<007ooOol0@7oo000UOol00`00Oomoo`0:Ool00`00Oomoo`0WOol00`00Oomoo`0OOol00`00Oomo o`0MOol00`00Oomoo`0]Ool00`00Oomoo`0;Ool00`00Oomoo`11Ool002Eoo`03001oogoo00Yoo`03 001oogoo02Moo`03001oogoo01moo`03001oogoo01eoo`03001oogoo02eoo`03001oogoo00Yoo`03 001oogoo049oo`009Goo00<007ooOol02Woo00<007ooOol09goo00<007ooOol07goo00<007ooOol0 7Goo00<007ooOol0;Woo00<007ooOol02Goo00<007ooOol0@Woo000UOol2000;Ool00`00Oomoo`0W Ool00`00Oomoo`0POol00`00Oomoo`0KOol00`00Oomoo`0`Ool00`00Oomoo`07Ool00`00Oomoo`13 Ool002Eoo`03001oogoo00]oo`03001oogoo02Ioo`03001oogoo021oo`03001oogoo01]oo`03001o ogoo035oo`<000=oo`<004Ioo`009Goo00<007ooOol02goo00<007ooOol09Goo00<007ooOol08Goo 00<007ooOol06goo00<007ooOol0Ool00`00Oomoo`0OOol00`00Oomoo`0Z Ool00`00Oomoo`0?Ool00`00Oomoo`26Ool0019oo`8000]oo`8000Aoo`03001oogoo00ioo`03001o ogoo01moo`03001oogoo02Yoo`03001oogoo00moo`03001oogoo08Ioo`009Goo00<007ooOol03Woo 00<007ooOol07goo00<007ooOol0:goo00<007ooOol03Woo00<007ooOol0QWoo000UOol00`00Oomo o`0>Ool00`00Oomoo`0OOol00`00Oomoo`0[Ool00`00Oomoo`0=Ool00`00Oomoo`27Ool002Eoo`03 001oogoo00ioo`03001oogoo01moo`03001oogoo02aoo`03001oogoo00aoo`03001oogoo08Moo`00 9Goo0P0047oo00<007ooOol07Goo00<007ooOol0;Goo00<007ooOol02goo00<007ooOol0R7oo000U Ool00`00Oomoo`0?Ool00`00Oomoo`0MOol00`00Oomoo`0^Ool00`00Oomoo`0:Ool00`00Oomoo`28 Ool002Eoo`03001oogoo00moo`03001oogoo01eoo`03001oogoo02ioo`03001oogoo00Uoo`03001o ogoo08Uoo`009Goo00<007ooOol03goo00<007ooOol07Goo00<007ooOol0;goo00<007ooOol01goo 00<007ooOol0RWoo000UOol00`00Oomoo`0?Ool00`00Oomoo`0MOol00`00Oomoo`0_Ool00`00Oomo o`06Ool00`00Oomoo`2;Ool002Eoo`03001oogoo011oo`03001oogoo01]oo`03001oogoo035oo`03 001oogoo00Eoo`03001oogoo08]oo`009Goo00<007ooOol047oo00<007ooOol06goo00<007ooOol0 Ool002Eoo`03001oogoo01Qoo`03001oogoo00aoo`03001oogoo09oo`0017oo 10003Woo0P002Woo0P001goo00@007ooOomoo`8000Ioo`03001oogoo0>Ioo`00ogoo8Goo003oOolQ Ool00?moob5oo`00\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-1.6685, -0.0817938, \ 0.0443894, 0.00684034}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ We see damped oscillation. Now try \[CapitalGamma]>>v0:\ \>", "Subtitle"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(sols = \(NDSolve[ Union[eqs /. {\[CapitalGamma] \[Rule] 3, \[Omega]0 \[Rule] 20, \[Omega] \[Rule] 20, v0 \[Rule] .3}, inits], {a, b}, {t, 0, 100}, MaxSteps \[Rule] 10000]\)[\([1]\)];\)\), "\[IndentingNewLine]", \(Plot[{Abs[b[t]]\^2 /. sols}, {t, 0, 10}, PlotRange \[Rule] {0, .1}, AxesLabel \[Rule] \ {t, Probability}]\)}], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.0952381 0 6.18034 [ [.21429 -0.0125 -3 -9 ] [.21429 -0.0125 3 0 ] [.40476 -0.0125 -3 -9 ] [.40476 -0.0125 3 0 ] [.59524 -0.0125 -3 -9 ] [.59524 -0.0125 3 0 ] [.78571 -0.0125 -3 -9 ] [.78571 -0.0125 3 0 ] [.97619 -0.0125 -6 -9 ] [.97619 -0.0125 6 0 ] [1.025 0 0 -6.4375 ] [1.025 0 10 6.4375 ] [.01131 .12361 -24 -4.5 ] [.01131 .12361 0 4.5 ] [.01131 .24721 -24 -4.5 ] [.01131 .24721 0 4.5 ] [.01131 .37082 -24 -4.5 ] [.01131 .37082 0 4.5 ] [.01131 .49443 -24 -4.5 ] [.01131 .49443 0 4.5 ] [.01131 .61803 -18 -4.5 ] [.01131 .61803 0 4.5 ] [.02381 .64303 -35 0 ] [.02381 .64303 35 12.875 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .21429 0 m .21429 .00625 L s [(2)] .21429 -0.0125 0 1 Mshowa .40476 0 m .40476 .00625 L s [(4)] .40476 -0.0125 0 1 Mshowa .59524 0 m .59524 .00625 L s [(6)] .59524 -0.0125 0 1 Mshowa .78571 0 m .78571 .00625 L s [(8)] .78571 -0.0125 0 1 Mshowa .97619 0 m .97619 .00625 L s [(10)] .97619 -0.0125 0 1 Mshowa .125 Mabswid .07143 0 m .07143 .00375 L s .11905 0 m .11905 .00375 L s .16667 0 m .16667 .00375 L s .2619 0 m .2619 .00375 L s .30952 0 m .30952 .00375 L s .35714 0 m .35714 .00375 L s .45238 0 m .45238 .00375 L s .5 0 m .5 .00375 L s .54762 0 m .54762 .00375 L s .64286 0 m .64286 .00375 L s .69048 0 m .69048 .00375 L s .7381 0 m .7381 .00375 L s .83333 0 m .83333 .00375 L s .88095 0 m .88095 .00375 L s .92857 0 m .92857 .00375 L s .25 Mabswid 0 0 m 1 0 L s gsave 1.025 0 -61 -10.4375 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20.875 translate 1 -1 scale 63.000 13.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (t) show 1.000 setlinewidth grestore .02381 .12361 m .03006 .12361 L s [(0.02)] .01131 .12361 1 0 Mshowa .02381 .24721 m .03006 .24721 L s [(0.04)] .01131 .24721 1 0 Mshowa .02381 .37082 m .03006 .37082 L s [(0.06)] .01131 .37082 1 0 Mshowa .02381 .49443 m .03006 .49443 L s [(0.08)] .01131 .49443 1 0 Mshowa .02381 .61803 m .03006 .61803 L s [(0.1)] .01131 .61803 1 0 Mshowa .125 Mabswid .02381 .0309 m .02756 .0309 L s .02381 .0618 m .02756 .0618 L s .02381 .09271 m .02756 .09271 L s .02381 .15451 m .02756 .15451 L s .02381 .18541 m .02756 .18541 L s .02381 .21631 m .02756 .21631 L s .02381 .27812 m .02756 .27812 L s .02381 .30902 m .02756 .30902 L s .02381 .33992 m .02756 .33992 L s .02381 .40172 m .02756 .40172 L s .02381 .43262 m .02756 .43262 L s .02381 .46353 m .02756 .46353 L s .02381 .52533 m .02756 .52533 L s .02381 .55623 m .02756 .55623 L s .02381 .58713 m .02756 .58713 L s .25 Mabswid .02381 0 m .02381 .61803 L s gsave .02381 .64303 -96 -4 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20.875 translate 1 -1 scale 63.000 13.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (Probability) show 1.000 setlinewidth grestore 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .02381 0 m .06244 .05111 L .08255 .08904 L .10458 .1254 L .12507 .152 L .14415 .17056 L .15465 .17848 L .16457 .18465 L .18314 .19321 L .19284 .19635 L .20312 .19885 L .21189 .20039 L .21688 .20105 L .22156 .20153 L .22652 .20191 L .22933 .20208 L .23194 .20219 L .23323 .20224 L .23444 .20227 L .23555 .2023 L .23671 .20233 L .23799 .20234 L .23916 .20235 L .24042 .20236 L .24114 .20236 L .24181 .20236 L .24296 .20235 L .24421 .20234 L .24539 .20232 L .24648 .2023 L .24855 .20225 L .25075 .20218 L .2553 .20197 L .26033 .20167 L .27042 .20086 L .27968 .19988 L .30053 .19706 L .34016 .19013 L .37827 .18245 L .41883 .1739 L .45787 .16567 L .49937 .15715 L .53935 .14924 L .57782 .14195 L .61874 .13455 L .65814 .12778 L .69603 .12158 L .73637 .1153 L .77519 .10957 L .81647 .10378 L Mistroke .85623 .0985 L .89447 .09367 L .93517 .08879 L .97435 .08433 L .97619 .08413 L Mfstroke % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgeoo`03001oogoo00Qoo`009Goo00<007oo0000n7oo000UOol0 1000Oomoo`00mgoo000UOol01@00Oomoogoo0000mWoo000UOol00`00Oomoo`02Ool00`00Oomoo`3c Ool002Eoo`8000=oo`03001oogoo0?=oo`009Goo00<007ooOol00goo00<007ooOol0lWoo000UOol0 0`00Oomoo`04Ool00`00Oomoo`3aOol002Eoo`03001oogoo00Eoo`03001oogoo0?1oo`009Goo00<0 07ooOol01Goo00<007ooOol0l7oo000UOol00`00Oomoo`06Ool00`00Oomoo`3_Ool002Eoo`03001o ogoo00Moo`03001oogoo0>ioo`009Goo0P0027oo00<007ooOol0kWoo000UOol00`00Oomoo`08Ool0 0`00Oomoo`3]Ool002Eoo`03001oogoo00Qoo`03001oogoo0>eoo`009Goo00<007ooOol02Goo00<0 07ooOol0k7oo000UOol00`00Oomoo`09Ool00`00Oomoo`3/Ool002Eoo`03001oogoo00Yoo`03001o ogoo0goo1@00VWoo 000UOol00`00Oomoo`0LOol00`00Oomoo`0eOol5002OOol002Eoo`8001ioo`<002eoo`L00:Aoo`00 9Goo00<007ooOol087oo0P0097oo1`00Zgoo000UOol00`00Oomoo`0ROol2000NOol4002bOol002Eo o`03001oogoo02Aoo`<001Ioo`D00;Ioo`009Goo00<007ooOol09goo5P00^goo000UOol00`00Oomo o`3hOol002Eoo`03001oogoo0?Qoo`009Goo00<007ooOol0n7oo000UOol2003iOol002Eoo`03001o ogoo0?Qoo`009Goo00<007ooOol0n7oo000UOol00`00Oomoo`3hOol002Eoo`03001oogoo0?Qoo`00 37oo0P0017oo0P0017oo0P001Goo0`000goo00<007ooOol0n7oo000;Ool01000Oomoo`0027oo00@0 07ooOol000Eoo`03001oogoo009oo`03001oogoo0?Qoo`002goo00@007ooOol000Qoo`04001oogoo 0002Ool50003Ool3003hOol000]oo`04001oogoo0008Ool01000Oomoo`000Woo00@007ooOol000Ao o`03001oogoo0?Qoo`002goo00@007ooOol000Qoo`04001oogoo0003Ool00`00Ool00004Ool00`00 Oomoo`3hOol000aoo`8000Yoo`8000Eoo`8000Aoo`03001oogoo0?Qoo`009Goo00<007ooOol0n7oo 000UOol00`00Oomoo`3hOol002Eoo`03001oogoo0?Qoo`009Goo0P00nGoo000UOol00`00Oomoo`3h Ool002Eoo`03001oogoo0?Qoo`009Goo00<007ooOol0n7oo000UOol00`00Oomoo`3hOol002Eoo`03 001oogoo0?Qoo`009Goo00<007ooOol0n7oo000UOol00`00Oomoo`3hOol002Eoo`800?Uoo`009Goo 00<007ooOol0n7oo000UOol00`00Oomoo`3hOol002Eoo`03001oogoo0?Qoo`009Goo00<007ooOol0 n7oo000UOol00`00Oomoo`3hOol002Eoo`03001oogoo0?Qoo`009Goo0P00nGoo000UOol00`00Oomo o`3hOol002Eoo`03001oogoo0?Qoo`009Goo00<007ooOol0n7oo000UOol00`00Oomoo`3hOol000ao o`8000Aoo`8000Aoo`8000=oo`<000Eoo`03001oogoo0?Qoo`002goo00@007ooOol000Qoo`04001o ogoo0002Ool01000Oomoo`0017oo00<007ooOol0n7oo000;Ool01000Oomoo`0027oo00@007ooOol0 009oo`04001oogoo0004Ool3003hOol000]oo`04001oogoo0008Ool01000Oomoo`000Woo0`001Goo 00<007ooOol0n7oo000;Ool01000Oomoo`0027oo00@007ooOol000=oo`03001oogoo00Aoo`03001o ogoo0?Qoo`0037oo0P002Woo0P0017oo0`0017oo00<007ooOol0n7oo000UOol00`00Oomoo`3hOol0 02Eoo`03001oogoo0?Qoo`009Goo00<007ooOol0n7oo000UOol00`00Oomoo`3hOol002Eoo`800?Uo o`009Goo00<007ooOol0n7oo000UOol00`00Oomoo`3hOol002Eoo`03001oogoo0?Qoo`009Goo00<0 07ooOol0n7oo000UOol00`00Oomoo`3hOol002Eoo`03001oogoo0?Qoo`009Goo0P00nGoo000UOol0 0`00Oomoo`3hOol002Eoo`03001oogoo0?Qoo`009Goo00<007ooOol0n7oo000UOol00`00Oomoo`3h Ool002Eoo`03001oogoo0?Qoo`009Goo00<007ooOol0n7oo000UOol2003iOol002Eoo`03001oogoo 0?Qoo`009Goo00<007ooOol0n7oo000UOol00`00Oomoo`3hOol002Eoo`03001oogoo0?Qoo`0037oo 0P0017oo0P0017oo0P0017oo0P001Goo00<007ooOol0n7oo000;Ool01000Oomoo`0027oo00@007oo Ool0009oo`04001oogoo0004Ool00`00Oomoo`3hOol000]oo`04001oogoo0008Ool01000Oomoo`00 0Woo00@007ooOol000Aoo`<00?Qoo`002goo00@007ooOol000Qoo`04001oogoo0003Ool20005Ool0 0`00Oomoo`3hOol000]oo`04001oogoo0008Ool01000Oomoo`000Woo00@007ooOol000Aoo`03001o ogoo0?Qoo`0037oo0P002Woo0P0017oo0P001Goo00<007ooOol0n7oo000UOol00`00Oomoo`3hOol0 02Eoo`03001oogoo0?Qoo`009Goo00<007ooOol0n7oo000UOol00`00Oomoo`3hOol002Eoo`800?Uo o`009Goo00<007ooOol0n7oo000UOol00`00Oomoo`3hOol002Eoo`03001oogoo0?Qoo`009Goo00<0 07ooOol0n7oo000UOol00`00Oomoo`3hOol002Eoo`03001oogoo0?Qoo`009Goo0P00nGoo000UOol0 0`00Oomoo`3hOol002Eoo`03001oogoo0?Qoo`009Goo00<007ooOol0n7oo000UOol00`00Oomoo`3h Ool002Eoo`03001oogoo0?Qoo`009Goo00<007ooOol0n7oo000UOol2003iOol002Eoo`03001oogoo 0?Qoo`009Goo00<007ooOol0n7oo000UOol00`00Oomoo`3hOol002Eoo`03001oogoo0?Qoo`009Goo 00<007ooOol0n7oo000BOol20004Ool20003Ool50003Ool00`00Oomoo`3hOol0015oo`04001oogoo 000:Ool00`00Oomoo`03Ool00`00Oomoo`3hOol0015oo`04001oogoo000:Ool00`00Oomoo`03Ool3 003hOol0015oo`04001oogoo000:Ool00`00Oomoo`3nOol0015oo`04001oogoo000:Ool00`00Oomo o`3nOol0019oo`8000Yoo`800?moo`5oo`00ogoo8Goo003oOolQOol00?moob5oo`00ogoo8Goo0011 Ool2003MOol0049oo`03001oogoo0=]oo`0017oo0`0017oo10000goo0P000Woo10000goo2@0017oo 0`000Woo0`0017oo0`000goo0P001Goo00<007ooOol0fWoo0005Ool00`00Oomoo`04Ool00`00Oomo o`02Ool01000Oomoo`000Woo00@007ooOol0009oo`@0009oo`04001oogoo0004Ool00`00Oomoo`02 Ool00`00Oomoo`04Ool01@00Oomoogoo00000Woo00L007ooOomoo`00Ool0003KOol000Eoo`<000Ao o`03001oogoo009oo`04001oogoo0002Ool01000Oomoo`001Goo00@007ooOol0009oo`03001oogoo 009oo`03001oogoo009oo`03001oogoo00Aoo`05001oogooOol00005Ool01000Oomoo`00fgoo0005 Ool01000Oomoo`000Woo10000goo0P000goo0`0017oo0`000Woo0`0017oo0P0017oo00<007ooOol0 0goo0P000Woo1@0000Eoo`00000007oo00800=Yoo`001Goo00@007ooOol000ioo`03001oogoo00Uo o`03001oogoo00Yoo`03001oogoo00Qoo`03001oogoo0>9oo`0017oo10003Woo0P002Woo0P001goo 00@007ooOomoo`8000Ioo`03001oogoo0>Ioo`00ogoo8Goo003oOolQOol00?moob5oo`00\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-1.6685, -0.00817938, \ 0.0443894, 0.000684034}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Now we see that there are no more Rabi oscillations. This the \"overdamped\ \" regime. Note that at first the upper state population grows as if there \ was no relaxation, but then it \"saturates\" at a small level ", Cell[BoxData[ \(TraditionalForm\`P\_max\)]], "~", Cell[BoxData[ \(TraditionalForm\`\((\(2 v0\)\/\[CapitalGamma])\)\^2\)]], ", and then eventually decays away. The maximum upper state population \ occurs at a time ", Cell[BoxData[ \(TraditionalForm\`t\_max\)]], "~2\[Pi]/\[CapitalGamma]." }], "Subtitle"], Cell["\<\ More things to try and think about: 1. What happens if the frequency \[Omega] is different from the \ resonance frequency \[Omega]0 ? 2. For \[Omega]=\[Omega]0, what is the critical value of v0/\ \[CapitalGamma] at which the \"overdamped\" regime turns into oscillatory \ regime? 3. In the above, we assumed that the relaxation occurs due to the decay \ into unobserved states. What will change if the excited state decays back to \ the ground state?\ \>", "Subtitle", TextAlignment->AlignmentMarker] }, Open ]] }, Open ]] }, FrontEndVersion->"4.0 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 695}}, WindowSize->{1016, 668}, WindowMargins->{{0, Automatic}, {Automatic, 0}} ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. ***********************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1739, 51, 233, 4, 240, "Title"], Cell[1975, 57, 66, 0, 53, "Subsubtitle"], Cell[CellGroupData[{ Cell[2066, 61, 119, 3, 93, "Subtitle"], Cell[2188, 66, 133, 2, 30, "Input"], Cell[CellGroupData[{ Cell[2346, 72, 77, 1, 30, "Input"], Cell[2426, 75, 228, 8, 41, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[2703, 89, 172, 4, 93, "Subtitle"], Cell[CellGroupData[{ Cell[2900, 97, 525, 10, 71, "Input"], Cell[3428, 109, 1564, 46, 55, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[5041, 161, 82, 1, 64, "Subtitle"], Cell[CellGroupData[{ Cell[5148, 166, 266, 4, 50, "Input"], Cell[5417, 172, 196, 3, 24, "Message"], Cell[5616, 177, 192, 6, 41, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[5857, 189, 185, 4, 93, "Subtitle"], Cell[6045, 195, 279, 5, 30, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[6361, 205, 303, 7, 122, "Subtitle"], Cell[CellGroupData[{ Cell[6689, 216, 144, 2, 31, "Input"], Cell[6836, 220, 27639, 714, 186, 7077, 456, "GraphicsData", "PostScript", \ "Graphics"], Cell[34478, 936, 130, 3, 29, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[34657, 945, 100, 2, 64, "Subtitle"], Cell[CellGroupData[{ Cell[34782, 951, 428, 7, 51, "Input"], Cell[35213, 960, 20424, 624, 186, 7042, 454, "GraphicsData", "PostScript", \ "Graphics"], Cell[55640, 1586, 130, 3, 29, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[55819, 1595, 83, 2, 64, "Subtitle"], Cell[CellGroupData[{ Cell[55927, 1601, 430, 7, 51, "Input"], Cell[56360, 1610, 12158, 412, 186, 4955, 319, "GraphicsData", "PostScript", \ "Graphics"], Cell[68521, 2024, 130, 3, 29, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[68700, 2033, 572, 14, 157, "Subtitle"], Cell[69275, 2049, 526, 11, 209, "Subtitle"] }, Open ]] }, Open ]] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)