Pretty good state transfer on double stars

Let A be the adjacency matrix of a graph X and suppose U(t)=exp(itA). We view A as acting on CxV(X) and take the standard basis of this space to be the vectors eu for u in V(X). hysicists say that we have perfect state transfer from vertex u to v at time τ if there is a scalar γ such that U(τ)eu=γev.