Quantum-Classical Correspondence in Response Theory

The correspondence principle between the quantum commutator [Â,B̂] and the classical Poisson brackets ιℏ{A,B} is examined in the context of response theory. The classical response function is obtained as the leading term of the ℏ expansion of the phase space representation of the response function in terms of Weyl-Wigner transformations and is shown to increase without bound at long times as a result of ignoring divergent higher-order contributions. Systematical inclusion of higher-order contributions improves the accuracy of the ℏ expansion at finite times. Resummation of all the higher-order terms establishes the classical-quantum correspondence ⟨v+n|α̂(t)|v⟩↔α_ne^ιnωt|_Jv+nℏ/2. The time interval of the validity of the simple classical limit [Â(t),B̂(0)]→ιℏ{A(t),B(0)} is estimated for quasiperiodic dynamics and is shown to be inversely proportional to anharmonicity.