Dispersive Limit, RWA, Bloch Equations

Some Functions

Commutator[a_,b_]:=a.b-b.a
(* general pure state in polar coordinates *)
State[θ_,ϕ_]:={Cos[θ/2],Sin[θ/2]Exp[I ϕ]}
(*normalized state with real first argument *)
StateNorm[x_List]:=Module[{arg1=Arg[x[[1]]],arg2=Arg[x[[2]]],normx=x/Norm[x],θ,ϕ},
θ=2Chop@ArcCos[normx[[1]]Exp[-I arg1]];
ϕ=Mod[arg2-arg1,2π];
{Cos[θ/2],Sin[θ/2]Exp[I ϕ]}]
(* outputs the polar coordinate from a state vector *)
blochcoord[state_]:=Module[{arg1=Arg[state[[1]]],arg2=Arg[state[[2]]],normx=state/Norm[state],θ,ϕ},
θ=Abs[2Chop@ArcCos[normx[[1]]Exp[-I arg1]]];
ϕ=Abs@Mod[arg2-arg1,2π];
{θ,ϕ}]

(* define a bloch sphere graph *)
With[{r=1},xy=ParametricPlot3D[{r Cos[θ],r Sin[θ],0},{θ,0,2 π},PlotStyle->{Dashed,Opacity[0.3]},Boxed->False,Axes->False];
yz=ParametricPlot3D[{0,r Cos[θ],r Sin[θ]},{θ,0,2 π},PlotStyle->{Dashed,Opacity[0.3]},Boxed->False,Axes->False];
xz=ParametricPlot3D[{r Cos[θ],0,r Sin[θ]},{θ,0,2 π},PlotStyle->{Dashed,Opacity[0.3]},Boxed->False,Axes->False];
bloch=Show[Graphics3D[{Blue,Opacity[.15],Sphere[{0,0,0},r],Black,Opacity[1],Thick,Line[{{0,r,0},{0,-r,0}}],Line[{{0,0,r},{0,0,-r}}],Line[{{r,0,0},{-r,0,0}}],Text[Style["z+",FontSize->16,FontWeight->Bold,FontColor->GrayLevel[.3]],{0,0,1.3 r}],Text[Style["z-",FontSize->16,FontWeight->Bold,FontColor->GrayLevel[.3]],{0,0,-1.3 r}],Text[Style["x+",FontSize->16,FontWeight->Bold,FontColor->GrayLevel[.3]],{1.3 r,0,0}],Text[Style["x-",FontSize->16,FontWeight->Bold,FontColor->GrayLevel[.3]],{-1.3 r,0,0}],Text[Style["y+",FontSize->16,FontWeight->Bold,FontColor->GrayLevel[.3]],{0,1.3 r,0}],Text[Style["y-",FontSize->16,FontWeight->Bold,FontColor->GrayLevel[.3]],{0,-1.3 r,0}]}],xy,xz,yz,Boxed->False]];
(* generate vector for bloch sphere representation *)
Options[blochvec]={color->Blue};
blochvec[{θ_,ϕ_},OptionsPattern[]]:=Graphics3D[{OptionValue[color],Thickness[0.02],Line[{{0,0,0},{Sin[θ]Cos[ϕ],Sin[θ]Sin[ϕ],Cos[θ]}}]}]
blocharrow[{θ_,ϕ_},opts:OptionsPattern[]]:=Arrow3D[{0,0,0},{Sin[θ]Cos[ϕ],Sin[θ]Sin[ϕ],Cos[θ]},opts]

σx=(({{0, 1}, {1, 0}}));σy=(({{0, -I}, {I, 0}}));σz=(({{1, 0}, {0, -1}}));
(* draw vector for bloch sphere taking an arbitrary pure input state *)
blochvecstate[state_]:=blochvec@blochcoord[state]
blocharrowstate[state_]:=blocharrow@blochcoord[state]
(*bloch vector for mixed states *)
blochvecmsphere[r_,θ_,ϕ_]:=Graphics3D[{Blue,Thickness[0.02],Line[{{0,0,0},r{Sin[θ]Cos[ϕ],Sin[θ]Sin[ϕ],Cos[θ]}}]}]
blochvecmsphere[{r_,θ_,ϕ_}]:=blochvec[r,θ,ϕ]
(* cartesian coordinates *)
blochvecmcart[x_,y_,z_]:=Graphics3D[{Blue,Thickness[0.02],Line[{{0,0,0},{x,y,z}}]}]
blochvecmcart[{x_,y_,z_}]:=blochvecmcart[x,y,z]
blochvecmarrowcart[{x_,y_,z_},opts:OptionsPattern[]]:=Arrow3D[{0,0,0},{x,y,z},opts]
(* show vector with sphere *)
vectoronsphere[vec_]:=Show[bloch,vec,SphericalRegion->True]
(* Bloch-sphere coordinates from arbitrary state *)
MatrixToBloch[x_]:={Tr[σx.x],Tr[σy.x],Tr[σz.x]}

Urot[α_,t_,θ_,ϕ_]:=(({{Cos[α t/2] - I Cos[θ]Sin[α t /2], -I Exp[-I ϕ] Sin[θ] Sin[α t/2]}, {(-I Exp[I ϕ]) Sin[θ] Sin[α t/2], Cos[α t/2] + I Cos[θ]Sin[α t /2]}}))

RX[α_]:=Urot[α,1,π/2,0]
RY[α_]:=Urot[α,1,π/2,π/2]
RZ[α_]:=Urot[α,1,0,0]

SetOptions[ParametricPlot3D,PlotStyle->{Thick,Red},PlotRange->{{-1,1},{-1,1},{-1,1}},ImageSize->300,Axes->False,Boxed->False];
BlochSphere=Graphics3D[{Opacity[0.4],Sphere[{0,0,0},1]},Boxed->False];
BlochMesh=SphericalPlot3D[1,{θ,0,2 Pi},{ϕ,0,2Pi},MeshFunctions->{#4&},Mesh->{{Pi/2}},PlotStyle->Directive[Opacity[0]],Boxed->False,Axes->False];

Options[Arrow3D]={color->Red};
Arrow3D[a_,b_,OptionsPattern[]]:=Module[{l,n,rm,a3d,offset,col,arrowheadlength,thickness,arrowslope},
       col=OptionValue[color];
n=b-a;
l=Norm[n];
offset=l-1;
thickness=0.02;
arrowslope=0.3;
arrowheadlength=0.2;
rm=If[Abs[n]==l {0,0,1},Sign[n[[3]]]*IdentityMatrix[3],RotationMatrix[{{0,0,1},n}]];
a3d={ParametricPlot3D[rm.{(arrowslope v) Cos[u],(arrowslope v) Sin[u],1-v+offset},{u,0,2 π},{v,0,arrowheadlength},Mesh->None,PlotStyle->Directive[col]],ParametricPlot3D[rm.{v Cos[u],v Sin[u],1-arrowheadlength+offset},{u,0,2 π},{v,0,arrowslope*arrowheadlength},PlotStyle->Directive[col],Mesh->None],Graphics3D[{col,Cylinder[{a,b-arrowheadlength/l (b-a)},thickness]}]};
a3d];

Dispersive Limit of Jaynes-Cummings Model:

Example 1 : CPB Hamiltonian with perturbation in the polarization charge :

H0=((ℏ Ω)/2)σz; H=H0+ϵ σx;

Transform H to (e^S).H.(e^(-S)) such that [(Subscript[H, 0]),S]=(Subscript[H, 1]) and S being first order in ϵ:

S=a σy/.First@Solve[Commutator[a σy,H0]== -σx,a]

Commutator[S,σx]

Series[MatrixExp[ϵ S].H.MatrixExp[-ϵ S],{ϵ,0,2}]//FullSimplify;
%//MatrixForm

The perturbation provokes a level shift if small compared to the transition frequency!

Rotating Wave Approximation :

Larmor Precession of superposition state :

sp=(1/(Sqrt[2])){1,1};

Animate[Show[bloch,blocharrowstate[RZ[t].sp],SphericalRegion->True,ImageSize->300],{t,0,2π},AnimationRunning->False]

Time - dependent field :

Animate[Show[bloch,Arrow3D[{0,0,0},{Cos[t],0,0},color->Green],ImageSize->300],{t,0,2π},AnimationRunning->False]

Decompose state into field rotating clockwise and anti - clockwise:

Animate[Show[bloch,Arrow3D[{0,0,0},{Cos[t],0,0},color->Green],Arrow3D[{0,0,0},{1/2Cos[t],1/2Sin[t],0},color->Yellow],Arrow3D[{0,0,0},{1/2Cos[t],-1/2Sin[t],0},color->Blue],ImageSize->300],{t,0,2π-.1,0.1},AnimationRunning->False]

make a transformation into a frame rotating in the same direction as the Larmor precession (yellow arrow):

Trans=MatrixExp[I ((ωr t)/2)σz];
termplus=Trans.(Cos[ω t]σx + Sin[ω t] σy).Trans\[ConjugateTranspose]//Simplify[#,Assumptions->{t∈Reals,ωr ∈Reals,ωr==ω}]&
termminus=Trans.(Cos[ω t]σx - Sin[ω t] σy).Trans\[ConjugateTranspose]//Simplify[#,Assumptions->{t∈Reals,ωr ∈Reals,ωr==ω}]&;
1/2(Tr[termplus.σx]"σx"+Tr[termplus.σy]"σy")
1/2(Tr[termminus.σx]"σx"+Tr[termminus.σy]"σy")//ExpToTrig

Discarding the terms at 2ω amounts to making the Rotating Wave Approximation :

Animate[vectoronsphere@blocharrowstate@(RX[3t].{1,0}),{t,0,2π/3-0.03,0.03},AnimationRunning->False]

Without Rotating Wave Approximation :

No simple analytic solution, numerical simulation:

Hamiltonian :

Hnum[ϵ_,Ω_,ω_,ϕ0_,t_]=({ϵ Cos[ω t+ϕ0],0,Ω/2}.(({{σx}, {σy}, {σz}})))[[1]]

Solve the Schroedinger Equation numerically :

Block[{ϵ=3,Ω=50,ω=50,ϕ0=0},
solnrwa=u/.First@NDSolve[{Hnum[ϵ,Ω,ω,ϕ0,t].u[t]==I u'[t],u[0]=={1,0}},u,{t,0,2π/ϵ},StartingStepSize->0.01,MaxStepSize->(10^(-4)),MaxSteps->(10^5)]]

Block[{interv},
interv=1000;
xaxis=Table[t,{t,solnrwa[[1,1,1]],solnrwa[[1,1,2]],(solnrwa[[1,1,2]]-solnrwa[[1,1,1]])/interv}];
vecsolnrwa=Table[{Cos[#2]Sin[#1],Sin[#2]Sin[#1],Cos[#1],#1,#2}&[Sequence@@blochcoord@solnrwa[t]],{t,solnrwa[[1,1,1]],solnrwa[[1,1,2]],(solnrwa[[1,1,2]]-solnrwa[[1,1,1]])/interv}];
]
{xsolnrwa,ysolnrwa,zsolnrwa,θsolnrwa,ϕsolnrwa}=Table[Interpolation@MapThread[{#1,#2}&,{xaxis,vecsolnrwa[[All,jj]]}],{jj,1,5}]

Animate[
Show[ParametricPlot3D[{xsolnrwa[τ],ysolnrwa[τ],zsolnrwa[τ]},{τ,t-Min[.3,t],t}],
vectoronsphere@blocharrow[{θsolnrwa[t],ϕsolnrwa[t]}]],
{t,100$MachineEpsilon,2π/3-0.005},AnimationRate->0.1,AnimationRunning->False]

Bloch equations :

equs=MapThread[Equal,{{x'[τ],y'[τ],z'[τ]},Cross[{x[τ],y[τ],z[τ]},{ϵx[τ],ϵy[τ],ϵz[τ]}]-{((x[τ])/T2),((y[τ])/T2),(((z[τ]-zs))/T1)}}];
equs//MatrixForm

For (OverVector[ϵ])=(0,0,1), decay of excited state

Block[{ϵx,ϵy,ϵz,x,y,z,T1,T2,zs,ini,tini,tfinal,rx,ry,rz},

ϵx[τ_]=0;ϵy[τ_]=0;ϵz[τ_]=1;

zs=-1; (* steady state *)
ini={x[0]==0,y[0]==0,z[0]==1};

T1=1;
T2=∞;

tini=0; tfinal=2π;
vec1[τ_]={x[τ],y[τ],z[τ]}/.First@NDSolve[{Sequence@@equs,Sequence@@ini},{x[τ],y[τ],z[τ]},{τ,tini,tfinal}];
Animate[vectoronsphere@blochvecmarrowcart[vec1[t]],{t,tini,tfinal},AnimationRunning->False]
]

For (OverVector[ϵ])=(0,0,1), decay of ground state

Block[{ϵx,ϵy,ϵz,x,y,z,T1,T2,zs,ini,tini,tfinal},

ϵx[τ_]=0;ϵy[τ_]=0;ϵz[τ_]=1;

zs=-1; (* steady state *)
ini={x[0]==0,y[0]==0,z[0]==-1};

T1=1;
T2=∞;

tini=0; tfinal=2π;
vec2[τ_]={x[τ],y[τ],z[τ]}/.First@NDSolve[{Sequence@@equs,Sequence@@ini},{x[τ],y[τ],z[τ]},{τ,tini,tfinal}];
Animate[vectoronsphere@blochvecmarrowcart[vec2[t]],{t,tini,tfinal},AnimationRunning->False]
]

For (OverVector[ϵ])=(0,0,1), dephasing of superposition state

Block[{ϵx,ϵy,ϵz,x,y,z,T1,T2,zs,ini,tini,tfinal},

ϵx[τ_]=0;ϵy[τ_]=0;ϵz[τ_]=3;

zs=-1; (* steady state *)
ini={x[0]==0,y[0]==1,z[0]==0};

T1=∞;
T2=2;

tini=0.; tfinal=2π;
vec3[τ_]={x[τ],y[τ],z[τ]}/.First@NDSolve[{Sequence@@equs,Sequence@@ini},{x[τ],y[τ],z[τ]},{τ,tini,tfinal}];
Animate[Show[ParametricPlot3D[vec3[θ],{θ,0,a}],vectoronsphere@blochvecmarrowcart[vec3[a]]],{{a,$MachineEpsilon},tini,tfinal},AnimationRunning->False]
]

For (OverVector[ϵ])=(0,0,1), dephasing and relaxation of superposition state

Block[{ϵx,ϵy,ϵz,x,y,z,T1,T2,zs,ini,tini,tfinal},

ϵx[τ_]=0;ϵy[τ_]=0;ϵz[τ_]=3;

zs=-1; (* steady state *)
ini={x[0]==0,y[0]==1,z[0]==0};

T1=3;
T2=1.5;

tini=0.; tfinal=4π;
vec4[τ_]={x[τ],y[τ],z[τ]}/.First@NDSolve[{Sequence@@equs,Sequence@@ini},{x[τ],y[τ],z[τ]},{τ,tini,tfinal}];
Animate[Show[ParametricPlot3D[vec4[θ],{θ,0,a}],vectoronsphere@blochvecmarrowcart[vec4[a]]],{{a,$MachineEpsilon},tini,tfinal},AnimationRunning->False]
]

Steady state solutions at resonance and in rotating frame at the driving frequency

Block[{ϵx,ϵy,ϵz,x,y,z,T1,T2,zs,ini,tini,tfinal},

ϵx[τ_]=1;ϵy[τ_]=0;ϵz[τ_]=.0;

zs=-1; (* steady state *)
ini={x[0]==0,y[0]==0,z[0]==-1};

T1=1;
T2=1;

tini=0.; tfinal=π;
(* For small ϵx: *)
vec5s[τ_]={x[τ],y[τ],z[τ]}/.First@NDSolve[{Sequence@@equs,Sequence@@ini},{x[τ],y[τ],z[τ]},{τ,tini,tfinal}];

(* For medium ϵx: *)
ϵx[τ_]=2;
vec5m[τ_]={x[τ],y[τ],z[τ]}/.First@NDSolve[{Sequence@@equs,Sequence@@ini},{x[τ],y[τ],z[τ]},{τ,tini,tfinal}];

(* For large ϵx: *)
ϵx[τ_]=20;
vec5l[τ_]={x[τ],y[τ],z[τ]}/.First@NDSolve[{Sequence@@equs,Sequence@@ini},{x[τ],y[τ],z[τ]},{τ,tini,tfinal}];

Animate[Show[ParametricPlot3D[vec5s[θ],{θ,0,a+$MachineEpsilon}],vectoronsphere@blochvecmarrowcart[vec5s[a]],
ParametricPlot3D[vec5m[θ],{θ,0,a+$MachineEpsilon},PlotStyle->Directive[Yellow]],
blochvecmarrowcart[vec5m[a],color->Yellow],
ParametricPlot3D[vec5l[θ],{θ,0,a+$MachineEpsilon},PlotStyle->Directive[Green]],
blochvecmarrowcart[vec5l[a],color->Green]],{{a,100$MachineEpsilon},tini+100$MachineEpsilon,tfinal},AnimationRunning->False]
]

© Stefan Filipp, 11/2008