arXiv subjects

Cid Reyes-Bustos

Publications and source records attributed to Cid Reyes-Bustos.

13 recordsLinked to original sources

arXiv2026

Two-photon quantum Rabi models - Spectral degeneracy and symmetries

The quantum Rabi model occupies a distinguished role in the study of quantum light and matter, describing the most fundamental interactions. Elucidation of its spectral properties, along with those of its generalizations, is necessary for quantum optics and applications in areas such as quantum information technologies. In this paper, we explore a precise description of energy degeneracy of the two-photon asymmetric quantum Rabi model and the nature of the system's symmetries; as a result, we demonstrate that these have significant relationships with the structure of certain algebraic curves (hyperelliptic curves) and number theory. Concretely, we first prove the existence of degenerate eigenvalues using the monodromy data of the Fuchsian ODE, which draws the eigenvalue problem of the system, and the existence of associated symmetry operators, previously discovered and studied heuristically in physics. Subsequently, we give a complete characterization of spectral degeneracies and present a series of conjectures showing that the degeneracy may determine the entire energy spectral structure. More precisely, for sufficiently strong interaction, we obtain an excellent approximation of the energy curves by the zero locus of a two-variable polynomial associated to degenerate eigenstates. In addition, we postulate an explicit description of the relation between the missing symmetry at the Hamiltonian level (hidden symmetry) and the degeneracy of the two-photon quantum Rabi model.

math-ph↗

arXiv2024

Zeta limits for the spectrum of quantum Rabi models

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 parameters of the asymmetric quantum Rabi model (AQRM), a generalization obtained by adding a physically significant parameter to the QRM. In particular, we consider the limit corresponding to the growth of the coupling strength to infinity, recently studied using resolvent analysis. The limits obtained in this paper are given in terms of the Hurwitz zeta function and other $L$-functions, suggesting further relations between spectral zeta function of quantum interaction models and number theory.

math-ph↗

arXiv2024

Spacing distribution for quantum Rabi models

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 difference between consecutive eigenvalues of the AQRM in the limit of high energies, i.e. large quantum numbers. In the symmetric case, that is the quantum Rabi model (QRM), the spacing distribution for each parity (given by the $\mathbb{Z}_2$-symmetry) is fully clarified by an asymptotic expression derived by de Monvel and Zielinski, though some questions remain for the full spectrum spacing. However, in the general AQRM case, there is no parity decomposition for the eigenvalues. In connection with numerically exact studies for the first 40,000 eigenstates we describe the spacing distribution for the AQRM which is characterized by a new type of periodicity and symmetric behavior of the distribution with respect to the bias parameter. The results reflects the hidden symmetry of the AQRM known to appear for half-integer bias. In addition, we observe in the AQRM the excited state quantum phase transition for large values of the bias parameter, analogous to the QRM with large qubit energy, and an internal symmetry of the level spacing distribution for fixed bias. This novel symmetry is independent from the symmetry for half-integer bias and not explained by current theoretical knowledge.

math-ph↗

arXiv2023

Wolstenholme primes and group determinants of cyclic groups

A Wolstenholme prime is a prime number $p \geq 5$ that divides the numerator of the Bernoulli number $B_{p-3}$. A number of equivalent definitions for Wolstenholme primes are known, mostly related to congruences of harmonic sums or binomial coefficients. In this paper, we introduce an equivalent definition of Wolstelholme primes related the number of terms in the group determinant of cyclic groups, and equivalently, the cardinality of certain sets of restricted partitions.

math.NT↗

arXiv2023

The heat kernel of the asymmetric quantum Rabi model

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 based on path integrals or stochastic methods. In addition to the heat kernel formula, we present applications including the explicit formula for the partition function and the Weyl law for the distribution of the eigenvalues, obtained from the analytic continuation of the corresponding spectral zeta function.

math-ph↗

arXiv2023

Covering models of the asymmetric quantum Rabi model: $η$-shifted non-commutative harmonic oscillators

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 forms, elliptic curves and Eichler cohomology observed in the special values of the associated spectral zeta function. In addition, the Heun ODE picture of the eigenvalue problem of the NCHO reveals a connection with the quantum Rabi model (QRM), a fundamental interaction model from quantum optics. In this paper we introduce an $η$-shifted NCHO ($η$-NCHO) that has an analogous relation with the asymmetric quantum Rabi model (AQRM) and describe its basic properties. Even though the shift factor does not break the parity symmetry of the NCHO, a certain type of degeneracies appears for $η\in \frac12 \mathbb{Z}$, as if mirroring the situation of the AQRM. We give furthermore a detailed description of the confluence process, that we call iso-parallel confluence process due to the fact that it requires a parallel transformation of two parameters describing the spectrum of $η$-NCHO and representations of $\mathfrak{sl}_2(\mathbb{R})$. We relate the eigenvalues of the two models under the iso-parallel confluence process, including how the quasi-exact eigenfunctions of the $η$-NCHO correspond to Juddian solutions of the AQRM. From the point of view of this confluence process, a family of $η$-NCHO corresponds to a single AQRM, thus we may regard the $η$-NCHO as a covering of the AQRM. We expect the study of the $η$-NCHO and the AQRM from this point of view to be helpful for the clarification of several questions on the AQRM, including the hidden symmetry and the number of Juddian solutions.

math-ph↗

arXiv2021

Degeneracy and hidden symmetry -- an asymmetric quantum Rabi model with an integer bias

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, the crossings on the energy curves. In this paper we propose a conjectural relation between the symmetry and degeneracy for the ibQRM$_{\ell}$ given explicitly in terms of two polynomials appearing independently in the respective investigations. Concretely, one of the polynomials appears as the quotient of the constraint polynomials that assure the existence of degenerate solutions while the other determines a quadratic relation (in general, it defines a curve of hyperelliptic type) between the ibQRM$_{\ell}$ Hamiltonian and its basic commuting operator $J_\ell$. Following this conjecture, we derive several interesting structural insights of the whole spectrum. For instance, the energy curves are naturally shown to lie on a surface determined by the family of hyperelliptic curves by considering the coupling constant as a variable. This geometric picture contains the generalization of the parity decomposition of the symmetric quantum Rabi model. Moreover, it allows us to describe a remarkable approximation of the first $\ell$ energy curves by the zero-section of the corresponding hyperelliptic curve. These investigations naturally lead to a geometric picture of the (hyper-)elliptic surfaces given by the Kodaira-Néron type model for a family of curves over the projective line in connection with the energy curves, which may be expected to provide a complex analytic proof of the conjecture.

quant-ph↗

arXiv2021

Heat kernel for the quantum Rabi model II: propagators and spectral determinants

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 Schrödinger equation (integral kernel of the time evolution operator) for the Hamiltonian $H_{\text{Rabi}}$ and $H_{\pm}$ by the Wick rotation (meromorphic continuation) of the corresponding heat kernels. In addition, as in the case of the full Hamiltonian of the QRM, we show that for the Hamiltonians $H_{\pm}$, the spectral determinant is, up to a non-vanishing entire function, equal to the Braak $G$-function (for each parity) used to prove the integrability of the QRM. To do this, we show the meromorphic continuation of the spectral zeta function of the Hamiltonians $H_{\pm}$ and give some of its basic properties.

math-ph↗

arXiv2021

Remarks on the hidden symmetry of the asymmetric quantum Rabi model

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 under this operator, but shows nevertheless characteristic degeneracies of its spectrum for half-integer values of $ε$, the parameter governing the asymmetry. In an interesting recent work (arXiv:2010.02496), an operator has been identified which commutes with the Hamiltonian $H_ε$ of the asymmetric quantum Rabi model for $ε=\frac{\ell}{2} \, (\ell\in \mathbb{Z})$ and appears to be the analogue of the parity in the symmetric case. We prove several important properties of this operator, notably, that it is algebraically independent of the Hamiltonian $H_ε$ and that it essentially generates the commutant of $H_ε$. Then, the expected $\mathbb{Z}_2$-symmetry manifests the fact that the commuting operator can be captured in the two-fold cover of the algebra generated by $H_ε$, that is, the polynomial ring in $H_ε$.

math-ph↗

arXiv2021

A simple continued fraction expansion for $e^n$

In this paper we present a family of continued fraction expansions for $e^n$, with $n\ge 1$, with a simple expression having partial denominators given by arithmetic progressions. We give an estimate for the convergence speed showing that the convergence is faster than the corresponding regular continued fractions. Moreover, we prove that the continued fractions for $e^n$ given in this paper are special cases of continued fraction expansions, different from the standard ones, of the confluent hypergeometric function, or equivalently, of the incomplete gamma function. In addition, using the same method we give a related family of continued fraction expansions of $e^{\ell/n}$ for positive integers $1\leq\ell< n $ that contains the case of integral exponent as a limit case.

math.NT↗

arXiv2021

Heat kernel for the quantum Rabi model

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 iterated integrals. The derivation of the formula is based on the direct evaluation of the Trotter-Kato product formula without the use of Feynman-Kac path integrals. More precisely, the infinite sum in the expression of the heat kernel arises from the reduction of the Trotter-Kato product formula into sums over the orbits of the action of the infinite symmetric group $\mathfrak{S}_\infty$ on the group $\mathbb{Z}_2^{\infty}$, and the iterated integrals are then considered as the orbital integral for each orbit. Here, the groups $ \mathbb{Z}_2^{\infty} $ and $\mathfrak{S}_\infty$ are the inductive limit of the families $\{\mathbb{Z}_2^n\}_{n\geq0}$ and $\{\mathfrak{S}_n\}_{n\geq0}$, respectively. In order to complete the reduction, an extensive study of harmonic (Fourier) analysis on the inductive family of abelian groups $\mathbb{Z}_2^n\, (n \geq0)$ together with a graph theoretical investigation is crucial. To the best knowledge of the authors, this is the first explicit computation for obtaining a closed formula of the heat kernel for a non-trivial realistic interacting quantum system. The heat kernel of this model is further given by a two-by-two matrix valued function and is expressed as a direct sum of two respective heat kernels representing the parity ($\mathbb{Z}_2$-symmetry) decomposition of the Hamiltonian by parity.

math-ph↗

arXiv2019

Determinant expressions of constraint polynomials and the spectrum of the asymmetric quantum Rabi model

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 $H^ε_{\text{Rabi}}$ of the AQRM is defined by adding the fluctuation term $εσ_x$, with $σ_x$ being the Pauli matrix, to the Hamiltonian of the quantum Rabi model, breaking its $\mathbb{Z}_{2}$-symmetry. The spectrum of $H^ε_{\text{Rabi}}$ contains a set of exceptional eigenvalues, considered to be remains of the eigenvalues of the uncoupled bosonic mode, which are further classified in two types: Juddian, associated with polynomial eigensolutions, and non-Juddian exceptional. We explicitly describe the constraint relations for allowing the model to have exceptional eigenvalues. By studying these relations we obtain the proof of the conjecture on constraint polynomials previously proposed by the third author. In fact, we prove that the spectrum of the AQRM possesses degeneracies if and only if the parameter $ε$ is a half-integer. Moreover, we show that non-Juddian exceptional eigenvalues do not contribute any degeneracy and we characterize exceptional eigenvalues by representations of $\mathfrak{sl}_2$. Upon these results, we draw the whole picture of the spectrum of the AQRM. Furthermore, generating functions of constraint polynomials from the viewpoint of confluent Heun equations are also discussed.

math-ph↗

arXiv2014

Cayley-type graphs for group-subgroup pairs

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 set some of the eigenvalues can be determined, including the largest eigenvalue of the graph. In particular, when this construction results in a bipartite regular graph we show a sufficient condition on the size of the generating sets that results on Ramanujan graphs for a fixed group-subgroup pair. Examples of Ramanujan pair-graphs that do not satisfy this condition are also provided, to show that the condition is not necessary.

math.CO↗