In[]:=
ClearAll
Out[]=
ClearAll
In[]:=
ClearAll
Out[]=
ClearAll
In[]:=
<<AtomicDensityMatrix`
In[]:=
system={AtomicState[1,NaturalWidthΓ1,BranchingRatio[4]0,BranchingRatio[5]0,ParityEven,Energy->0],AtomicState[2,NaturalWidthΓ2,BranchingRatio[1]1,BranchingRatio[5]0,ParityOdd],AtomicState[3,NaturalWidthΓ3,BranchingRatio[2]1,ParityEven],AtomicState[4,NaturalWidthΓ4,BranchingRatio[3]1,BranchingRatio[1]0,ParityOdd],AtomicState[5,NaturalWidthΓ5,BranchingRatio[4]1,BranchingRatio[2]0,ParityEven]};
In[]:=
SetOptions[DensityMatrix,TimeDependenceFalse];
In[]:=
DensityMatrix[system]//MatrixForm
Out[]//MatrixForm=
ρ 1,1 | ρ 1,2 | ρ 1,3 | ρ 1,4 | ρ 1,5 |
ρ 2,1 | ρ 2,2 | ρ 2,3 | ρ 2,4 | ρ 2,5 |
ρ 3,1 | ρ 3,2 | ρ 3,3 | ρ 3,4 | ρ 3,5 |
ρ 4,1 | ρ 4,2 | ρ 4,3 | ρ 4,4 | ρ 4,5 |
ρ 5,1 | ρ 5,2 | ρ 5,3 | ρ 5,4 | ρ 5,5 |
In[]:=
fieldp=OpticalField[ωp,Ωp/ReducedME[1,{Dipole,1},2]]
Out[]=
Ωp,0,0
-tωp
ReducedME[1,{Dipole,1},2]
In[]:=
fieldd=OpticalField[ωd,Ωd/ReducedME[2,{Dipole,1},3]]
Out[]=
Ωd,0,0
-tωd
ReducedME[2,{Dipole,1},3]
In[]:=
fieldr=OpticalField[ωr,Ωr/ReducedME[3,{Dipole,1},4]]
Out[]=
Ωr,0,0
-tωr
ReducedME[3,{Dipole,1},4]
In[]:=
fieldthz=OpticalField[ωthz,Ωthz/ReducedME[4,{Dipole,1},5]]
Out[]=
Ωthz,0,0
-tωthz
ReducedME[4,{Dipole,1},5]
In[]:=
MatrixForm[H=Hamiltonian[system,{{ElectricFieldfieldp,AllowedCouplings{12}},{ElectricFieldfieldd,AllowedCouplings{23}},{ElectricFieldfieldr,AllowedCouplings{34}},{ElectricFieldfieldthz,AllowedCouplings{45}}}]]
Out[]//MatrixForm=
0 | -ΩpCos[tωp] | 0 | 0 | 0 |
-ΩpCos[tωp] | Energy[2] | -ΩdCos[tωd] | 0 | 0 |
0 | -ΩdCos[tωd] | Energy[3] | -ΩrCos[tωr] | 0 |
0 | 0 | -ΩrCos[tωr] | Energy[4] | -ΩthzCos[tωthz] |
0 | 0 | 0 | -ΩthzCos[tωthz] | Energy[5] |
In[]:=
LevelDiagram[system,H/.{Energy[2]1,Energy[3]2,Energy[4]3,Energy[5]4}]
Out[]=
In[]:=
(Hrwa=RotatingWaveApproximation[system,H,{{ωp,12,Δp},{ωd,23,Δd},{ωr,34,Δr},{ωthz,45,Δthz}}])//MatrixForm
Out[]//MatrixForm=
0 | - Ωp 2 | 0 | 0 | 0 |
- Ωp 2 | -Δp | - Ωd 2 | 0 | 0 |
0 | - Ωd 2 | -Δd-Δp | - Ωr 2 | 0 |
0 | 0 | - Ωr 2 | -Δd-Δp-Δr | - Ωthz 2 |
0 | 0 | 0 | - Ωthz 2 | -Δd-Δp-Δr-Δthz |
In[]:=
MatrixForm[relax=IntrinsicRelaxation[system]+TransitRelaxation[system,γ]]
Out[]//MatrixForm=
γ+Γ1 | 0 | 0 | 0 | 0 |
0 | γ+Γ2 | 0 | 0 | 0 |
0 | 0 | γ+Γ3 | 0 | 0 |
0 | 0 | 0 | γ+Γ4 | 0 |
0 | 0 | 0 | 0 | γ+Γ5 |
In[]:=
MatrixForm[repop=OpticalRepopulation[system]+TransitRepopulation[system,γ]]
Out[]//MatrixForm=
Γ2 ρ 2,2 | 0 | 0 | 0 | 0 |
0 | Γ3 ρ 3,3 | 0 | 0 | 0 |
0 | 0 | Γ4 ρ 4,4 | 0 | 0 |
0 | 0 | 0 | Γ5 ρ 5,5 | 0 |
0 | 0 | 0 | 0 | 0 |
In[]:=
TableForm[eqs=LiouvilleEquation[system,Hrwa,relax,repop]]
Out[]//TableForm=
In[]:=
obs=Observables[system,Energy[1],Ωp/ReducedME[1,{Dipole,1},2],TransformMatrixMatrixExp[-ItDiagonalMatrix[{Energy[1],0,0,0,0}]]]/.Energy[1]2π/λ//FullSimplify
Out[]=
,,{0,0}
2--2Im[]ReducedME[1,{Dipole,1},4]
2
π
3Γ2Im
ρ
1,2
3
Energy[2]
3
Γ2
3
Energy[2]
ρ
1,4
λΩp
2+2Re[]ReducedME[1,{Dipole,1},4]
2
π
3Γ2Re
ρ
1,2
3
Energy[2]
3
Γ2
3
Energy[2]
ρ
1,4
λΩp
In[]:=
SetOptions[RowReduce,Method->"CofactorExpansion"];sol=Solve[eqs,DMVariables[system]][[1]];SetOptions[RowReduce,Method->Automatic];
With[{coherence=Im[Subscript[ρ,2,1]]/.sol/.{Γ11,γ0,λ1}},Manipulate[Plot[coherence,{Δc,-8,8},PlotRange{-.2,.2},FrameTrue,ImageSizeMedium,FrameLabel{"Probe detuning \!\(\*SubscriptBox[\(Δ\), \(a\)]\)/Γ",Im[Subscript[ρ,1,2]]}],{{Ωp,.1},0.0001,5},{{Ωd,.5},0.0001,5},{{Ωr,.1},0.0001,5},{{Ωt,.1},0.0001,5},{{Δp,.5},-5,5},{{Δd,.5},-5,5},{{Δt,.5},-5,5}]]