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 $H = H_{\rm mat}\otimes I + I\otimes\omega a^\ast a + \Sigma\otimes(a^\ast)^k + \Sigma^\ast\otimes a^k$ on $\mathcal{H}\otimes L^2(\mathbb{R})$, coupling a single bosonic mode to an arbitrary matter system through a bounded operator $\Sigma$. When $\Sigma$ is normal and nonzero, we prove that $H$ is self-adjoint if and only if $k\leq2$; for $k\geq3$ 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 $\Sigma$. The normality of $\Sigma$ 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.

Mathematical Physics (math-ph); Functional Analysis (math.FA); Quantum Physics (quant-ph)

arXiv:2607.22378 [math-ph]

(or arXiv:2607.22378v1 [math-ph] for this version)

https://doi.org/10.48550/arXiv.2607.22378 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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