2026/07/24 by Felix Fischer, Felix Knapp, Daniel Burgarth +1
Mathematics · #math-ph #math.FA #math.MP #quant-ph
Multiphoton light-matter interactions, in which a bosonic mode exchanges k excitations at a time with a quantum system, are a source of genuine nonlinearity in quantum optics and are increasingly accessible experimentally. Here we study the class of operators H = H\rm mat⊗ I + I⊗ωa^∗ a + Σ⊗(a^∗)k + Σ^∗⊗ ak on H⊗ L2(ℝ), coupling a single bosonic mode to an arbitrary matter system through a bounded operator Σ. When Σ is normal and nonzero, we prove that H is self-adjoint if and only if k≤2; for k≥3 we compute the deficiency indices, parametrise all self-adjoint extensions, and show that every extension has purely discrete spectrum whenever the matter system is finite-dimensional. Our analysis rests on a block Jacobi decomposition paired with a suitable unitary transformation depending on the polar decomposition of Σ. The normality of Σ is optimal: a k-photon Jaynes-Cummings model, with non-normal coupling, remains self-adjoint for every k. We illustrate our results on the k-photon Rabi and Dicke models.