Pensieve header: The Benkart-Witherspoon representation; continues 2x2Rep.nb; continued pensieve://2017-07/.

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

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