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A. Pasquale

Publications and source records attributed to A. Pasquale.

16 recordsLinked to original sources
arXiv2024

Transmon qubit modeling and characterization for Dark Matter search

This study presents the design, simulation, and experimental characterization of a superconducting transmon qubit circuit prototype for potential applications in dark matter detection experiments. We describe a planar circuit design featuring two non-interacting transmon qubits, one with fixed frequency and the other flux tunable. Finite-element simulations were employed to extract key Hamiltonian parameters and optimize component geometries. The qubit was fabricated and then characterized at $20$ mK, allowing for a comparison between simulated and measured qubit parameters. Good agreement was found for transition frequencies and anharmonicities (within 1\% and 10\% respectively) while coupling strengths exhibited larger discrepancies (30\%). We discuss potential causes for measured coherence times falling below expectations ($T_1\sim\,$1-2 \textmu s) and propose strategies for future design improvements. Notably, we demonstrate the application of a hybrid 3D-2D simulation approach for energy participation ratio evaluation, yielding a more accurate estimation of dielectric losses. This work represents an important first step in developing planar Quantum Non-Demolition (QND) single-photon counters for dark matter searches, particularly for axion and dark photon detection schemes.

quant-ph↗
arXiv2023

Symmetry breaking operator for the reductive dual pair $(\mathrm{U}_l,\mathrm{U}_{l'})$

We consider the dual pair $(G,G')=(\mathrm{U}_l,\mathrm{U}_{l'})$ in the symplectic group $\mathrm{Sp}_{2ll'}(\mathbb{R})$. Fix a Weil representation of the metaplectic group $\tilde{\mathrm{Sp}}_{2ll'}(\mathbb{R})$. Let $\tilde{G\,}$ and $\tilde{G'}$ be the preimages of $G$ and $G'$ under the metaplectic cover $\tilde{\mathrm{Sp}}_{2ll'}(\mathbb{R})\to \mathrm{Sp}_{2ll'}(\mathbb{R})$, and let $Π\otimesΠ'$ be a genuine irreducible representation of $\tilde{G\,}\times\tilde{G'}$. We study the Weyl symbol $f_{Π\otimesΠ'}$ of the (unique up to a possibly zero constant) symmetry breaking operator (SBO) intertwining the Weil representation with $Π\otimesΠ'$. This SBO coincides with the orthogonal projection of the space of the Weil representation onto its $Π$-isotypic component and also with the orthogonal projection onto its $Π'$-isotypic component. Hence $f_{Π\otimesΠ'}$ can be computed in two different ways, one using $Π$ and the other using $Π'$. By matching the results, we recover Weyl's theorem stating that $Π\otimesΠ'$ occurs in the Weil representation with multiplicity at most one and we also recover the complete list of the representations $Π\otimesΠ'$ occurring in Howe's correspondence.

math.RT↗
arXiv2023

The wave front set correspondence for dual pairs with one member compact

Let W be a real symplectic space and (G,G') an irreducible dual pair in Sp(W), in the sense of Howe, with G compact. Let $\widetilde{\mathrm{G}}$ be the preimage of G in the metaplectic group $\widetilde{\mathrm{Sp}}(\mathrm{W})$. Given an irreducible unitary representation $Π$ of $\widetilde{\mathrm{G}}$ that occurs in the restriction of the Weil representation to $\widetilde{\mathrm{G}}$, let $Θ_Π$ denote its character. We prove that, for the embedding $T$ of $\widetilde{\mathrm{Sp}}(\mathrm{W})$ in the space of tempered distributions on W given by the Weil representation, the distribution $T(\checkΘ_Π)$ has an asymptotic limit. This limit is an orbital integral over a nilpotent orbit $\mathcal O_m\subseteq \mathrm{W}$. The closure of the image of $\mathcal O_m$ in $\mathfrak{g}'$ under the moment map is the wave front set of $Π'$, the representation of $\widetilde{\mathrm{G}'}$ dual to $Π$.

math.RT↗
arXiv2021

Symmetry breaking operators for dual pairs with one member compact

We consider a dual pair $(G, G')$, in the sense of Howe, with G compact acting on $L^2(\mathbb{R}^n)$, for an appropriate $n$, via the Weil representation $\omega$. Let $\tilde{\mathrm{G}}$ be the preimage of G in the metaplectic group. Given a genuine irreducible unitary representation $\Pi$ of $\tilde{\mathrm{G}}$, let $\Pi'$ be the corresponding irreducible unitary representation of $\tilde{\mathrm{G}'}$ in the Howe duality. The orthogonal projection onto $L^2(\mathbb{R}^n)_\Pi$, the $\Pi$-isotypic component, is the essentially unique symmetry breaking operator in $\mathrm{Hom}_{\tilde{\mathrm{G}}\tilde{\mathrm{G}'}}(\mathcal{H}_\omega^{\infty}, \mathcal{H}_\Pi^{\infty}\otimes \mathcal{H}_{\Pi'}^{\infty})$. We study this operator by computing its Weyl symbol. Our results allow us to recover the known list of highest weights of irreducible representations of $\tilde{\mathrm{G}}$ occurring in Howe's correspondence when the rank of $\tilde{\mathrm{G}}$ is strictly bigger than the rank of $\tilde{\mathrm{G'}}$. They also allow us to compute the wavefront set of $\Pi'$ by elementary means.

math.RT↗
arXiv2021

Hypergeometric functions of type $BC$ and standard multiplicities

We study the Heckman-Opdam hypergeometric functions associated to a root system of type $BC$ and a multiplicity function which is allowed to assume some non-positive values (a standard multiplicity function). For such functions, we obtain positivity properties and sharp estimates which imply a characterization of the bounded hypergeometric functions. As an application, our results extend known properties of Harish-Chandra's spherical functions on Riemannian symmetric spaces of the non-compact type $G/K$ to spherical functions over homogeneous vector bundles on $G/K$ which are associated to certain small $K-$types.

math.RT↗
arXiv2017

Differential operators, radial parts and a one-parameter family of hypergeometric functions of type BC

We introduce the symmetric (respectively, non-symmetric) $τ_{-\ell}-$hypergeometric functions associated with a root system of type $BC$ as joint eigenfunctions of a commutative algebra of differential (respectively, differential-reflection) operators. Under certain conditions on the real parameter $\ell$, we derive their properties (positivity, estimates, asymptotics and boundedness) by establishing the analogous properties for the Heckman-Opdam (symmetric and non-symmetric) hypergeometric functions corresponding to (not necessarily positive) multiplicity functions which are standard.

math.RT↗
arXiv2016

Resonances for the Laplacian on Riemannian symmetric spaces: the case of SL(3,$\mathbb{R}$)/SO(3)

We show that the resolvent of the Laplacian on SL(3,$\mathbb{R}$)/SO(3) can be lifted to a meromorphic function on a Riemann surface which is a branched covering of $\mathbb{C}$. The poles of this function are called the resonances of the Laplacian. We determine all resonances and show that the corresponding residue operators are given by convolution with spherical functions parameterized by the resonances. The ranges of these operators are infinite dimensional irreducible SL(3,$\mathbb{R}$)-representations. We determine their Langlands parameters and wave front sets. Also, we show that precisely one of these representations is unitarizable. Alternatively, they are given by the differential equations which determine the image of the Poisson transform associated with the resonance.

math.RT↗
arXiv2015

Resonances for the Laplacian on products of two rank one Riemannian symmetric spaces

Let $X=X_1 \times X_2$ be a direct product of two rank-one Riemannian symmetric spaces of the noncompact type. We show that when at least one of the two spaces is isomorphic to a real hyperbolic space of odd dimension, the resolvent of the Laplacian of $X$ can be lifted to a holomorphic function on a Riemann surface which is a branched covering of $\mathbb C$. In all other cases, the resolvent of the Laplacian of $X$ admits a singular meromorphic lift. The poles of this function are called the resonances of the Laplacian. We determine all resonances and show that the corresponding residue operators are given by convolution with spherical functions parameterized by the resonances. The ranges of these operators are finite dimensional and explicitly realized as direct sums of finite-dimensional irreducible spherical representations of the group of the isometries of $X$.

math.RT↗
arXiv2015

Weyl calculus and dual pairs

We consider a dual pair $(G,G')$, in the sense of Howe, with $G$ compact acting on $L^2(\mathbb R^n)$ for an appropriate $n$ via the Weil Representation. Let $\widetilde{G}$ be the preimage of $G$ in the metaplectic group. Given a genuine irreducible unitary representation $Π$ of $\widetilde{G}$ we compute the Weyl symbol of orthogonal projection onto $L^2(\mathbb R^n)_Π$, the $Π$-isotypic component. We apply the result to obtain an explicit formula for the character of the corresponding irreducible unitary representation $Π'$ of $\widetilde{G'}$ and to compute of the wave front set of $Π'$ by elementary means.

math.RT↗
arXiv2015

Resonances for the Laplacian: the cases $BC_2$ and $C_2$ (except $SO_0(p,2)$ with $p>2$ odd)

Let $X=G/K$ be a Riemannian symmetric space of the noncompact type and restricted root system $BC_2$ or $C_2$ (except $G=SO_0(p,2)$ with $p>2$ odd). The analysis of the meromorphic continuation of the resolvent of the Laplacian of $X$ is reduced from the analysis of the same problem for a direct product of two isomorphic rank-one Riemannian symmetric spaces of the noncompact type which are not isomorphic to real hyperbolic spaces. We prove that the resolvent of the Laplacian of $X$ can be lifted to a meromorphic function on a Riemann surface which is a branched covering of the complex plane. Its poles, that is the resonances of the Laplacian, are explicitly located on this Riemann surface. The residue operators at the resonances have finite rank. Their images are finite direct sums of finite-dimensional irreducible spherical representations of $G$.

math.RT↗
arXiv2014

Semisimple orbital integrals on the symplectic space for a real reductive dual pair

We prove a Weyl Harish-Chandra integration formula for the action of a reductive dual pair on the corresponding symplectic space $W$. As an intermediate step, we introduce a notion of a Cartan subspace and a notion of an almost semisimple element in the symplectic space $W$. We prove that the almost semisimple elements are dense in $W$. Finally, we provide estimates for the orbital integrals associated with the different Cartan subspaces in $W$.

math.RT↗
arXiv2013

Asymptotics of Harish-Chandra expansions, bounded hypergeometric functions associated with root systems, and applications

A series expansion for Heckman-Opdam hypergeometric functions $φ_λ$ is obtained for all $λ\in \mathfrak a^*_{\mathbb C}.$ As a consequence, estimates for $φ_λ$ away from the walls of a Weyl chamber are established. We also characterize the bounded hypergeometric functions and thus prove an analogue of the celebrated theorem of Helgason and Johnson on the bounded spherical functions on a Riemannian symmetric space of the noncompact type. The $L^p$-theory for the hypergeometric Fourier transform is developed for $0<p<2$. In particular, an inversion formula is proved when $1\leq p <2$.

math.RT↗
arXiv2012

Analytic and Group-Theoretic Aspects of the Cosine Transform

This is a brief survey of recent results by the authors devoted to one of the most important operators of integral geometry. Basic facts about the analytic family of cosine transforms on the unit sphere and the corresponding Funk transform are extended to the "higher-rank" case for functions on Stiefel and Grassmann manifolds. The main topics are the analytic continuation and the structure of polar sets, the connection with the Fourier transform on the space of rectangular matrices, inversion formulas and spectral analysis, and the group-theoretic realization as an intertwining operator between generalized principal series representations of SL(n, R).

math.FA↗
arXiv2004

Support properties and Holmgren's uniqueness theorem for differential operators with hyperplane singularities

Let W be a finite Coxeter group acting linearly on $R^n$. In this article we study support properties of W-invariant partial differential operator D on $\R^n$ with real analytic coefficients. Our assumption is that the principal symbol of D has a special form, related to the root system corresponding to W. In particular the zeros of the principal symbol are supposed to be located on hyperplanes fixed by reflections in W. We show that conv(supp Df) = conv(supp f) holds for all compactly supported smooth functions f so that conv(supp f) is W-invariant. The main tools in the proof are Holmgren's uniqueness theorem and some elementary convex geometry. Several examples and applications linked to the theory of special functions associated with root systems are presented.

math.AP↗
arXiv2003

Paley-Wiener theorems for the $Θ$-spherical transform: an overview

The aim of this article is to give an overview of several types of Paley-Wiener theorems occuring in harmonic analysis related to symmetric spaces. This will serve as a motivation for the introduction of the $Θ$-spherical functions, the correspondint $Θ$-spherical Fourier transform, and the Paley-Wiener theorem for this transform. Up to now such a theorem has only been proven in very special cases, and still, its formulation and proof are very technical. In this paper we do not go into details of the proofs, but present an overview which explains the different examples which have inspired and motivated the theory of $Θ$-spherical functions.

math.RT↗