x=(({{0, 1}, {0, 0}}));a=(({{r, 0}, {0, s}})); A=MatrixExp[ℏ α a]; y=(({{0, 0}, {1, 0}})); b=(({{s, 0}, {0, r}})); B=MatrixExp[ℏ β b];

eqns1=Thread[Flatten/@(x.y-y.x==((B-A)/((𝕖^(ℏ γ))-(𝕖^(-ℏ γ)))))]

eqns=eqns1

SolveAlways[eqns, {r,s,ℏ}]

Solve[eqns2⋃eqns3⋃eqns4⋃eqns5, {r,s,α,β,γ}]

{sol}=Solve[eqns, {q,r,s,t}]

NSolve[{True,-1==((-(E^(-0.2420748525683183` r ℏ))+(E^(-0.3740462780156224` t ℏ)))/ℏ),1==((-(E^(-0.2420748525683183` q ℏ))+(E^(-0.3740462780156224` s ℏ)))/ℏ),q-r==-0.3740462780156224`,-q+r==0.3740462780156224`,-s+t==-0.2420748525683183`},{q,r,s,t}]

sol /. C[1|2]->0

eqns /. Equal -> List

eqns /. Equal -> List/. (sol /. C[1|2]->0)