Pensieve header: The Benkart-Witherspoon representation; continues pensieve://2017-06/.

ρ=0; σ=β; {ℏ=((σ-ρ)/(γ β)), q=(E^(ℏ γ β)), -ℏ γ,-ℏ β}

y=(({{0, 0}, {-(E^ρ), 0}})); a=(γ/(ρ-σ))(({{ρ, 0}, {0, σ}}));x=(((E^(-σ))-(E^(-ρ)))/ℏ)(({{0, 1}, {0, 0}})); b=(β/(σ-ρ))(({{σ, 0}, {0, ρ}}));
ME=MatrixExp; MF=MatrixForm;
MF /@ {y,a,x,b,t=Simplify[β a-γ b],A=ME[-ℏ β a], B=ME[-ℏ γ b], T=ME[ℏ t]}

{(({{0, 0}, {-1, 0}})),(({{0, 0}, {0, -γ}})),(({{0, (-1+(E^(-β))) γ}, {0, 0}})),(({{β, 0}, {0, 0}})),(({{-β γ, 0}, {0, -β γ}})),(({{1, 0}, {0, (E^β)}})),(({{(E^(-β)), 0}, {0, 1}})),(({{(E^(-β)), 0}, {0, (E^(-β))}}))}

{a.x-x.a==γ x, x.A==q A.x, a.y-y.a==-γ y, b.y-y.b==-β y, x.y-q y.x==(1/ℏ)((({{1, 0}, {0, 1}}))-T.A.A)} // Simplify

ME[ξ x].ME[η y] // Simplify // MF

ME[η y].ME[ξ x] // Simplify // MF

ME[ξ x].ME[η y].ME[-ξ x].ME[-η y] // Simplify // MF

x.y-y.x // MF

eqn0=ME[ξ0 x].ME[η0 y]==ME[η y].ME[α a].ME[ξ x].ME[τ t]

MF /@ eqn0

{sol0}=Solve[1+(E^(-β)) (-1+(E^β)) γ η0 ξ0==(E^(-β γ τ)),τ] /. C[1]->0

MF/@(eqn1=Simplify/@ (eqn0 /. sol0))

{sol1}=Solve[Thread[Flatten/@eqn1], {η,α,ξ}]

Column@Series[{η,α,ξ,τ} /.sol1/.sol0, {β,0,2}]