Periodic excitations of bilinear quantum systems

A well-known method of transferring the population of a quantum system from an eigenspace of the free Hamiltonian to another is to use a periodic control law with an angular frequency equal to the difference of the eigenvalues. For finite dimensional quantum systems, the classical theory of averaging provides a rigorous explanation of this experimentally validated result. This paper extends this finite dimensional result, known as the Rotating Wave Approximation, to infinite dimensional systems and provides explicit convergence estimates.

💡 Research Summary

The paper addresses a fundamental problem in quantum control: how to drive a quantum system from one eigenspace of its free Hamiltonian to another using a periodic external field. In finite‑dimensional settings, the Rotating Wave Approximation (RWA) – which states that a control oscillating at the Bohr frequency (the difference of two eigenvalues) induces a resonant transition – is well understood through classical averaging theory. The novelty of this work lies in extending the RWA rigorously to infinite‑dimensional bilinear quantum systems, where the free Hamiltonian typically has an unbounded spectrum and the interaction operator may be unbounded as well.

The authors first recast the Schrödinger equation with a control term as a bilinear evolution equation on a separable Hilbert space (H): \