Inverting pump-probe spectroscopy for state tomography of excitonic systems

Stephan Hoyer and K. Birgitta Whaley

J. Chem. Phys. 138, 164102 (2013)

Derivation of analytical spectra of dimer [Section IV (A)]

Initial conditions and operators in Liouville space:

μ=(({{0, μgα, μgβ, 0}, {0, 0, 0, μαf}, {0, 0, 0, μβf}, {0, 0, 0, 0}}));
ρ0= (({{-(Subscript[r, 0]), 0, 0, 0}, {0, (1/2) ((Subscript[r, 0])+(Subscript[r, 3])), (1/2) ((Subscript[r, 1])-I (Subscript[r, 2])), 0}, {0, (1/2) ((Subscript[r, 1])+I (Subscript[r, 2])), (1/2) ((Subscript[r, 0])-(Subscript[r, 3])), 0}, {0, 0, 0, 0}}));
Gt=(({{0, 0, 0, 0}, {fα, 0, 0, 0}, {fβ, 0, 0, 0}, {0, fpβ, fpα, 0}}));
sitedipoles={μgα->Cos[θ]d1+Sin[θ]d2,μgβ->-Sin[θ]d1+Cos[θ]d2,μαf->Sin[θ]d1+Cos[θ]d2,μβf->Cos[θ]d1-Sin[θ]d2};

Derivation of Eq. (18):

S=(Tr[μ.(Gt(μ\[Transpose].ρ0-ρ0.μ\[Transpose]))]//FullSimplify)

Table[CoefficientList[2 S,{(Subscript[r, i])}][[2]]//FullSimplify,{i,0,3}]/.{I->-I}//MatrixForm

Note the switched sign assigned to i because of the complex conjugate when turning this from a ket- to a bra-vector.

Derivation of Eq. (19):

S2= Collect[2 S/.sitedipoles,{d1,d2}]/.{(d1^2)->1,(d2^2)->(δ^2),d1 d2->δ Cos[ϕ]}

Table[(CoefficientList[S2,{(Subscript[r, i])}][[2]]//FullSimplify)/.{Cos[θ] Sin[θ]->(1/2) Sin[2θ]},{i,0,3}]/.{I->-I}//MatrixForm