Cid Reyes Bustos

NTT Communication Science Laboratories · NTT Institute for Fundamental Mathematics

The non-commutative harmonic oscillator (NCHO) is a matrix valued differential operator originally introduced as a generalization of the quantum harmonic oscillator having a weaker \(\mathfrak {sl}_2({\mathbb {R}})\)-symmetry. The spectrum of the NCHO has remarkable properties, including the presence of number theoretical structures such as modular...

The purpose of the present paper is to study the exceptional eigenvalues of the asymmetric quantum Rabi models (AQRM), specifically, to determine the degeneracy of the exceptional eigenstates. Exceptional eigenvalues are labelled by certain integers and are considered to be remains of the eigenvalues of the uncoupled bosonic mode (i.e. the quantum...

In this paper we discuss a recently discovered continued fraction expansion \[ e = 3 - \cfrac{1}{4 - \cfrac{2}{5 - \cfrac{3}{6 - \cfrac{4}{7 - \cdots}}} }, \] and its convergence properties. We show that this expansion is a particular case of a continued fraction expansion of $e^n$, for positive integer power $n$, and more generally, it is a specia...

The quantum Rabi model (QRM), one of the fundamental models used to describe light and matter interaction, has a deep mathematical structure revealed by the study of its spectrum. In this paper, from the explicit formulas for the partition function we directly derive various limits of the spectral zeta function with respect to the systems parameter...

Originating from the unified treatment of probability distributions, the α-determinant is a one-parameter interpolation of the determinant and permanent of a matrix. While it generally does not have the invariance properties of the determinant, the irreducible representations of the associated general linear (Lie) groups define interesting invarian...

Knowledge of the partition and spectral zeta function of a quantum system is fundamental for both physics and mathematics, and the positions these functions occupy in their respective fields share a common philosophy. In this article, we describe the number theoretic structures hidden behind light and matter interaction models, focusing on the part...

The asymmetric quantum Rabi model (AQRM) is a fundamental model in quantum optics describing the interaction of light and matter. Besides its immediate physical interest, the AQRM possesses an intriguing mathematical structure which is far from being completely understood. In this paper, we focus on the distribution of the level spacing, the differ...

In this paper we derive an explicit formula for the heat kernel of the asymmetric quantum Rabi model (AQRM), a symmetry breaking generalization of the quantum Rabi model (QRM). The method described here is the extension of a recently developed method for the heat kernel of the QRM that uses the Trotter-Kato product formula instead of path integrals...

The quantum Rabi model (QRM), one of the fundamental models used to describe light and matter interaction, has a deep mathematical structure revealed by the study of its spectrum. In this paper, directly from the explicit formulas for the partition function we derive various limits of the spectral zeta function with respect to the systems parameter...

The non-commutative harmonic oscillator (NCHO) is a matrix valued differential operator introduced as a generalization of the quantum harmonic oscillator. The spectrum of the NCHO has remarkable properties, including the presence of a number theoretical structure such as modular forms, elliptic curves, Eichler cohomology observed in the special val...

The hidden symmetry of the asymmetric quantum Rabi model (AQRM) with a half-integral bias (ibQRM$_{\ell}$) was uncovered in recent studies by the explicit construction of operators $J_\ell$ commuting with the Hamiltonian. The existence of such symmetry has been widely believed to cause the degeneration of the spectrum, that is, crossings on the ene...

The symmetric quantum Rabi model (QRM) is integrable due to a discrete Z2 -symmetry of the Hamiltonian. This symmetry is generated by a known involution operator, measuring the parity of the eigenfunctions. An experimentally relevant modification of the QRM, the asymmetric (or biased) quantum Rabi model (AQRM) is no longer invariant under this oper...

The quantum Rabi model (QRM) is widely recognized as an important model in quantum systems, particularly in quantum optics. The Hamiltonian $H_{\text{Rabi}}$ is known to have a parity decomposition $H_{\text{Rabi}} = H_{+} \oplus H_{-}$. In this paper, we give the explicit formulas for the propagator of the Schr\"odinger equation (integral kernel o...

The symmetric quantum Rabi model (QRM) is integrable due to a discrete $\mathbb{Z}_2$-symmetry of the Hamiltonian. This symmetry is generated by a known involution operator, measuring the parity of the eigenfunctions. An experimentally relevant modification of the QRM, the asymmetric (or biased) quantum Rabi model (AQRM) is no longer invariant unde...

In this paper we derive an explicit formula for the heat kernel of the asymmetric quantum Rabi model (AQRM), a symmetry breaking generalization of the quantum Rabi model (QRM). The method described here is an extension of the recently developed one for the heat kernel of the QRM based on the Trotter-Kato formula. In particular, the method is not ba...

The quantum Rabi model (QRM) is widely regarded as one of the fundamental models of quantum optics. One of its generalizations is the asymmetric quantum Rabi model (AQRM), obtained by introducing a symmetry-breaking term depending on a parameter 𝜀∈ℝ to the Hamiltonian of the QRM. The AQRM was shown to possess degeneracies in the spectrum for values...

The quantum Rabi model (QRM) is widely recognized as an important model in quantum systems, particularly in quantum optics. The Hamiltonian $H_{\text{Rabi}}$ is known to have a parity decomposition $H_{\text{Rabi}} = H_{+} \oplus H_{-}$. In this paper, we give the explicit formulas for the propagator of the Schr\"odinger equation (integral kernel o...

The quantum Rabi model (QRM) is widely recognized as a particularly important model in quantum optics. It is considered to be the simplest and most fundamental system describing quantum light-matter interaction. The objective of the paper is to give an analytical formula of the heat kernel of the Hamiltonian explicitly by infinite series of iterate...

The aim of this article is to investigate certain family of (so-called constraint) polynomials which determine the quasi-exact spectrum of the asymmetric quantum Rabi model. The quantum Rabi model appears ubiquitously in various quantum systems and its potential applications include quantum computing and quantum cryptography. In (Wakayama, Symmetry...

The purpose of this paper is to study the exceptional eigenvalues of the asymmetric quantum Rabi models (AQRM), specifically, to determine the degeneracy of their eigenstates. Here, the Hamiltonian $H^{\epsilon}_{\text{Rabi}}$ of the AQRM is defined by adding the fluctuation term $\epsilon \sigma_x$, with $\sigma_x$ being the Pauli matrix, to the H...

In this paper we introduce a Cayley-type graph for group-subgroup pairs and present some elementary properties of such graphs, including connectedness, their degree and partition structure, and vertex-transitivity. We relate these properties to those of the underlying group-subgroup pair. From the properties of the group, subgroup and generating se...