Pensieve header: Simplifying the logos formula.

(Subscript[DP, x_->(Subscript[D, α_]),y_->(Subscript[D, β_])])[P_][f_]:=Total[CoefficientRules[P,{x,y}]/.({m_,n_}->c_):>c D[f,{α,m},{β,n}]]

(Subscript[OΛ, k_])[e_,l_,f_,α_,β_,δ_] := (Subscript[OΛ, k])[e,l,f,α,β,δ]=Module[
{ρh,ρe,ρl,ρf,eqn,a,b,c,sol,q,λ,ν},
ρh=(({{-1, 0}, {0, -1}}));  ρe=(({{0, 0}, {-ϵ, 0}}));  ρl=(({{-(1+1/ϵ)/2, 0}, {0, (1-1/ϵ)/2}})); ρf=(({{0, 1}, {0, 0}}));
eqn=MatrixExp[α ρf].MatrixExp[β ρe]==MatrixExp[a ρe].MatrixExp[c (ρh-2ϵ ρl)].MatrixExp[b ρf];
Echo[sol=Solve[Thread[Flatten/@eqn], {a,b,c}][[1]] /. C[1]->0];
λ=Simplify[(E^(-f α-e β+h α β))Normal@Series[(E^(c h + a e - 2ϵ c l + b f))/.sol, {ϵ,0,k}]];
q=(E^(ν(f α+e β-h α β+e f δ)));
Collect[(q^(-1))(Subscript[DP, α->(Subscript[D, f]),β->(Subscript[D, e])])[λ][q] /. ν->(((1+h δ))^(-1)),ϵ,Simplify]];

(Subscript[OΛ, 2])[e,l,f,α,β,δ]

FullSimplify[(E^(c h + a e - 2ϵ c l + b f))/.{a->-(β/(-1+α β ϵ)),b->-(α/(-1+α β ϵ)),c->((Log[1-α β ϵ])/ϵ)}]

(Subscript[Λ, k_])[e_,l_,f_,α_,β_,δ_] :=Module[{λ},
λ=Echo[Expand@Normal@Series[(E^((f α+e β)/(1-α β ϵ))) (((1-α β ϵ))^(-2 l+(h/ϵ))), {ϵ,0,k}]/.(E^(_))->1];
Collect[(Subscript[DP, α->(Subscript[D, f]),β->(Subscript[D, e])])[λ][(E^((f α+e β+e f δ)/(1+h δ)))]/.(E^(_))->1,ϵ,Simplify]];

Simplify[(Subscript[OΛ, 2])[e,l,f,α,β,δ]==(Subscript[Λ, 2])[e,l,f,α,β,δ]]

(Subscript[Λ, 1])[e,l,f,α,β,δ]

(Subscript[Λ, 2])[e,l,f,α,β,δ]

(Subscript[Λ, 3])[e,l,f,α,β,δ]

lΛ[k_] := Collect[Normal@Series[Log[(Subscript[Λ, k])[e,l,f,α,β,δ]], {ϵ,0,k}], ϵ, Simplify]

Exponent[lΛ[2] /. {x:(α|β|e|f):>ℏ x, l->(ℏ^2)l}, ℏ]

Exponent[lΛ[3] /. {x:(α|β|e|f):>ℏ x, l->(ℏ^2)l}, ℏ]

Exponent[lΛ[4] /. {x:(α|β|e|f):>ℏ x, l->(ℏ^2)l}, ℏ]