Quantum-Classical Correspondence in Response Theory

Abstract

The correspondence principle between the quantum commutator $[\widehat{A},\widehat{B}]$ and the classical Poisson brackets $\iota \hslash \{A,B\}$ is examined in the context of response theory. The classical response function is obtained as the leading term of the $\hslash$ 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 $\hslash$ expansion at finite times. Resummation of all the higher-order terms establishes the classical-quantum correspondence $⟨v+n|\widehat{\alpha }(t)|v⟩\leftrightarrow {\alpha }_n{e}^{\iota n\omega t}{|}_{J_v+n\hslash /2}$. The time interval of the validity of the simple classical limit $[\widehat{A}(t),\widehat{B}(0)]\to \iota \hslash \{A(t),B(0)\}$ is estimated for quasiperiodic dynamics and is shown to be inversely proportional to anharmonicity.