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

• URL: http://arxiv.org/abs/2607.22378v1
• Date: Fri, 24 Jul 2026 15:04:35 GMT
• Title: Self-adjoint extensions of $k$-photon light-matter Hamiltonians
• Abstract summary: Multiphoton light-matter interactions are a source of genuine nonlinearity in quantum optics.<n>We study the class of operators $H = H_rm matotimes I + Iotimesaast a + otimes(aast)k + astotimes ak$ on $mathcalHotimes L2(mathbbR)$.
• Score: 0.7077641025321454
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $H = H_{\rm mat}\otimes I + I\otimesωa^\ast a + Σ\otimes(a^\ast)^k + Σ^\ast\otimes a^k$ on $\mathcal{H}\otimes L^2(\mathbb{R})$, coupling a single bosonic mode to an arbitrary matter system through a bounded operator $Σ$. When $Σ$ 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 $Σ$. 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.

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