Kavli Affiliate: Felix Fischer
| Summary:
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 $H = H_rm matotimes I + Iotimesωa^ast a + Σotimes(a^ast)^k + Σ^astotimes a^k$ on $mathcalHotimes L^2(mathbbR)$, 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 $kleq2$; for $kgeq3$ 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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