Self-adjoint extensions of $k$-photon light-matter Hamiltonians

Abstract

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 $H=H_{\mathrm{mat}}\otimes I+I\otimes \ddot{I}‰a^{\ast }a+\hat{I}\pounds \otimes (a^{\ast }{)}^k+\hat{I}{\pounds }^{\ast }\otimes a^k$ on $\mathcal{H}\otimes L^2(\mathbb{R})$, coupling a single bosonic mode to an arbitrary matter system through a bounded operator $\hat{I}\pounds$. When $\hat{I}\pounds$ is normal and nonzero, we prove that $H$ is self-adjoint if and only if $k\le 2$; for $k\ge 3$ 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 $\hat{I}\pounds$. The normality of $\hat{I}\pounds$ 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.

17 pages