Ramsey Fringes

David McIntyre
5 June 2012

Initialization

SetOptions[Plot,BaseStyle->{FontFamily->"Arial",FontSize->12},PlotStyle->{{Black,AbsoluteThickness[1.5]}},AxesStyle->{{Black,AbsoluteThickness[0.5]},{Black,AbsoluteThickness[0.5]}},TicksStyle->{{Black,AbsoluteThickness[0.5]},{Black,AbsoluteThickness[0.5]}}];

Off[General::spell]

Off[General::spell1]

ctrl-9 to start inline formula and ctrl-0 to end. ctrl-_ to begin subscript, ctrl-space to end. esc-alpha-esc to enter Greek letters.  Spell checker for symbol names turned off since many have similar names.

SetOptions[Plot,AxesStyle->Directive[FontFamily->"Arial",FontSize->12,Black,AbsoluteThickness[0.5],Arrowheads[0.04]]];

Ramsey Fringes

τ=1;
Ω=0.63/τ;
T=10*τ;

Plot[(((Ω*Sin[δ*τ/2]/δ))^2),{δ,-10,10},PlotRange->{0,0.11},AxesLabel->{"Detuning","𝒫"},PlotLabel->"One Pulse Probability",ImageSize->400]

Plot[(((2*Ω*Sin[δ*τ/2]/δ))^2)*((Cos[δ*T/2])^2),{δ,-10,10},PlotRange->{0,0.41},AxesLabel->{"Detuning","𝒫"},PlotLabel->"Two Pulse Probability",ImageSize->400]

Bloch Sphere

bv=-π/1.5;
va=π/6;
la=1.5;

p1=ParametricPlot3D[
  {{Cos[u], Sin[u], 0},{Cos[u] Cos[bv],Sin[u] Cos[bv],Sin[bv]},{0,Cos[u],Sin[u]}},
    {u, 0, 2Pi},PlotRange->{{-la,la},{-la,la},{-la,la}},ViewPoint->{10,0,5}]

p2=Show[Graphics3D[{AbsoluteThickness[4],Line[{{0,0,0},{-Cos[bv] Cos[π/4],-Cos[bv] Sin[π/4],Sin[bv]}}],Line[{{0,0,0},{0,0,-1}}]}]]

p2=Show[Graphics3D[{AbsoluteThickness[4],Line[{{0,0,0},{Cos[π/6] Cos[π/4],Cos[π/6] Sin[π/4],-Sin[π/6]}}],Line[{{0,0,0},{0,0,-1}}]}]];

p3=Show[Graphics3D[{AbsoluteThickness[2],Line[{{0,0,0},{la /Sqrt[2],-la /Sqrt[2],0}}],Line[{{0,0,0},{la /Sqrt[2],la /Sqrt[2],0}}],Line[{{0,0,0},{0,0,la}}]}]]

p7=Show[p1,p2,p3,Boxed->False,Axes->False,ViewPoint->{10,0,5}]

Show[GraphicsArray[{{p4,p4,p7},{p4,p6,p5}}]]

3.16 Bloch vector

vectorBloch[theta_,phi_]:= Graphics3D[
{Black,(*Thickness[0.01],*)Cylinder[{{0,0,0},Dynamic@{Sin[theta] Cos[phi],Sin[theta]Sin[phi],Cos[theta]}},.04]},Boxed->False,Axes->False,ImageSize->450] (*unit Bloch vector*)

thinvectorBloch[theta_,phi_]:= Graphics3D[
{Black,(*Thickness[0.01],*)Cylinder[{{0,0,0},Dynamic@{Sin[theta] Cos[phi],Sin[theta]Sin[phi],Cos[theta]}},.02]},Boxed->False,Axes->False,ImageSize->450] (*unit Bloch vector*)

xyz=ParametricPlot3D[{{Cos[t], Sin[t],0},{0, Cos[t], Sin[t]},{ Cos[t],0,Sin[t]}},{t,0,2 π},PlotStyle->{{Black,Thick,Dashed},{Black,Thick,Dashed},{Black,Thick,Dashed}},Boxed->False,Axes->False,ImageSize->450];

evolve[theta_]:=ParametricPlot3D[{Cos[phi]*Sin[theta], Sin[phi]*Sin[theta],Cos[theta]},{phi,0,2 π},PlotStyle->{{Black,AbsoluteThickness[4]}},Boxed->False,Axes->False,ImageSize->450]

poincareBlochdmc[theta_,phi_]:=
Graphics3D[{Opacity[0.4],Sphere[],Black,Thick,Opacity[1.],
PointSize[.02],Point[{0,0,0}],Point[{1,0,0}],Point[{-1,0,0}],Point[{0,1,0}],Point[{0,-1,0}],Point[{0,0,1}],Point[{0,0,-1}],
Line[{{0,1,0},{0,-1,0}}],
Line[{{0,0,1},{0,0,-1}}],
Line[{{1,0,0},{-1,0,0}}],
Text[Style["z", 14,Bold], {0,0,1.2 }],
Text[Style["y", 14,Bold], {0,1.2 ,0}],
Text[Style["x", 14,Bold], {1.2 ,0,0}],
Boxed->False,Axes->False,ImageSize->450}]

t1=π-π/3;
f1=1.0*π/2;
p11=Show[
poincareBlochdmc[t1,f1],
vectorBloch[t1,f1],
thinvectorBloch[π,f1]
,xyz,
evolve[t1],
Boxed->False,Axes->False,ImageSize->{400,400},SphericalRegion->True,ViewPoint->{1.3,2.5,1.5},ViewAngle->π/10]