Fock Space Research Articles

Uncertainty inequality for weighted Fock spaces

Uncertainty inequality for weighted Fock spaces

On the existence of an eigenvalue of the generalized Friedrichs model

We consider a family of bounded self-adjoint matrix operators (generalized Friedrichs models) acting on the direct sum of one-particle and two-particle subspaces of the Fock space. Conditions for the existence of eigenvalues of the matrix operators are obtained.

Generalized 𝑞-Fock spaces and structural identities

Using q q -calculus we study a family of reproducing kernel Hilbert spaces which interpolate between the Hardy space and the Fock space. We give characterizations of these spaces in terms of classical operators such as integration and backward-shift operators, and their q q -calculus counterparts. Furthermore, these new spaces allow us to study intertwining operators between classic backward-shift operators and the q-Jackson derivative.

Closed Range and Preserving Frames of Weighted Composition Operator on Quaternionic Fock Space

Closed Range and Preserving Frames of Weighted Composition Operator on Quaternionic Fock Space

Surjective and closed range differentiation operator

AbstractWe identify Fock-type spaces $$\mathcal {F}_{(m,p)}$$ F ( m , p ) on which the differentiation operator D has closed range. We prove that D has closed range only if it is surjective, and this happens if and only if $$m=1$$ m = 1 . Moreover, since the operator is unbounded on the classical Fock spaces, we consider the modified or the weighted composition–differentiation operator, $$D_{(u,\psi ,n)} f= u\cdot \big ( f^{(n)}\circ \psi \big )$$ D ( u , ψ , n ) f = u · ( f ( n ) ∘ ψ ) , on these spaces and describe conditions under which the operator admits closed range, surjective, and order bounded structures.

Spectral relations for a matrix model in fermionic Fock space

Spectral relations for a matrix model in fermionic Fock space

Complex symmetry in the Fock space of several variables

In this paper we study the complex symmetry in the several variable Fock space by using the techniques of weighted composition operators and semigroups. We characterize unbounded weighted composition operators that are (real) complex symmetric with respect to a concrete conjugation. Using this characterization, we study complex symmetric semigroups and their generators. We realize such generators as first-order differential operators.

Extended Cesàro companion operators on generalized Fock spaces

In this paper we give a characterizations of the bounded and compact extended Cesàro companion operators acting between the generalized Fock spaces. We also obtain asymptotic estimates for the norm of these operators in terms of certain integral transforms, which are related to some function-theoretic properties of the symbols that induce the operators. As an application, we prove that there exist no nontrivial compact extended Cesàro companion integral and multiplication operators on generalized Fock spaces.

Dual Toeplitz Operators on the Orthogonal Complement of the Generalized Fock Space

We characterize the boundedness and compactness of dual Toeplitz operators on the orthogonal complement of the generalized Fock space. We study the problem when the finite sum of the dual Toeplitz products is compact. Additionally, we also consider when the sum of the dual Toeplitz operators is equal to another dual Toeplitz operator.

Canonical integral operators on the Fock space

Canonical integral operators on the Fock space

Carleson measures and Berezin-type operators on Fock spaces

Carleson measures and Berezin-type operators on Fock spaces

Composition operators on weighted Fock spaces induced by $$A_{\infty }$$-type weights

Composition operators on weighted Fock spaces induced by $$A_{\infty }$$-type weights

Combinatorial Fock spaces and quantum symmetric pairs

A way to construct the natural representation of the quantized affine algebra U v ( s l ̂ l ) is via the deformed Fock space by Misra and Miwa. This relates the classes of Weyl modules for U q ( s l N ) were q is a root of unity to the action of U v ( s l ̂ l ) as N tends toward infinity. In this paper we investigate the situation outside of type A. In classical types, we construct embeddings of the Grothendieck group of finite dimensional U q ( g ) -modules into Fock spaces of different charges and define an action of an affine quantum symmetric pair that plays the role of the quantized affine algebra. We describe how the action is related to the linkage principal for quantum groups at a root of unity and tensor product multiplicities.

Taming the Rotating Wave Approximation

The interaction between light and matter is one of the oldest research areas of quantum mechanics, and a field that just keeps on delivering new insights and applications. With the arrival of cavity and circuit quantum electrodynamics we can now achieve strong light-matter couplings which form the basis of most implementations of quantum technology. But quantum information processing also has high demands requiring total error rates of fractions of percentage in order to be scalable (fault-tolerant) to useful applications. Since errors can also arise from modelling, this has brought into center stage one of the key approximations of quantum theory, the Rotating Wave Approximation (RWA) of the quantum Rabi model, leading to the Jaynes-Cummings Hamiltonian. While the RWA is often very good and incredibly useful to understand light-matter interactions, there is also growing experimental evidence of regimes where it is a bad approximation. Here, we ask and answer a harder question: for which experimental parameters is the RWA, although perhaps qualitatively adequate, already not good enough to match the demands of scalable quantum technology? For example, when is the error at least, and when at most, 1%? To answer this, we develop rigorous non-perturbative bounds taming the RWA. We find that these bounds not only depend, as expected, on the ratio of the coupling strength and the oscillator frequency, but also on the average number of photons in the initial state. This confirms recent experiments on photon-dressed Bloch-Siegert shifts. We argue that with experiments reporting controllable cavity states with hundreds of photons and with quantum error correcting codes exploring more and more of Fock space, this state-dependency of the RWA is increasingly relevant for the field of quantum computation, and our results pave the way towards a better understanding of those experiments.

Combinatorial aspects of weighted free Poisson random variables

This paper will be devoted to the study of weighted (deformed) free Poisson random variables from the viewpoint of orthogonal polynomials and statistics of non-crossing partitions. A family of weighted (deformed) free Poisson random variables will be defined in a sense by the sum of weighted (deformed) free creation, annihilation, scalar, and intermediate operators with certain parameters on a weighted (deformed) free Fock space together with the vacuum expectation. We shall provide a combinatorial moment formula of non-commutative Poisson random variables. This formula gives us a very nice combinatorial interpretation to two parameters of weights. One can see that the deformation treated in this paper interpolates free and boolean Poisson random variables, their distributions and moments, and yields some conditionally free Poisson distribution by taking limit of the parameter.

Gaussian RBF kernels via Fock spaces: quaternionic and several complex variables settings

In this paper, we study two extensions of the complex-valued Gaussian radial basis function (RBF) kernel and discuss their connections with Fock spaces in two different settings. First, we introduce the quaternionic Gaussian RBF kernel constructed using the theory of slice hyperholomorphic functions. Then, we consider the case of Gaussian RBF kernels in several complex variables.

Toeplitz Operators on Harmonic Fock Spaces with Radial Symbols

The main aim of this paper is to study new features and specific properties of the Toeplitz operator with radial symbols in harmonic Fock spaces. A new spectral decomposition of a Toeplitz operator with Wick symbols is also established.

Quantum de Sitter Geometry

Quantum de Sitter geometry is discussed using elementary field operator algebras in Krein space quantization from an observer-independent point of view, i.e., ambient space formalism. In quantum geometry, the conformal sector of the metric becomes a dynamical degree of freedom, which can be written in terms of a massless minimally coupled scalar field. The elementary fields necessary for the construction of quantum geometry are introduced and classified. A complete Krein–Fock space structure for elementary fields is presented using field operator algebras. We conclude that since quantum de Sitter geometry can be constructed by elementary fields operators, the geometry quantum state is immersed in the Krein–Fock space and evolves in it. The total number of accessible quantum states in the universe is chosen as a parameter of quantum state evolution, which has a relationship with the universe’s entropy. Inspired by the Wheeler–DeWitt constraint equation in cosmology, the evolution equation of the geometry quantum state is formulated in terms of the Lagrangian density of interaction fields in ambient space formalism.

Generalization of Gisin’s theorem to quantum fields

We generalize Gisin’s theorem on the relation between the entanglement of pure states and Bell non-classicality to the case of mode entanglement of separated groups of modes of quantum fields extending the theorem to cover also states with undefined particle number. We show that any pure state of the field which contains entanglement between two groups of separated modes violates some Clauser–Horne (CH) inequality. In order to construct the observables leading to a violation in the first step, we show an isomorphism between the Fock space built from a single-particle space involving two separated groups of modes and a tensor product of two abstract separable Hilbert spaces spanned by formal monomials of creation operators. In the second step, we perform a Schmidt decomposition of a given entangled state mapped to this tensor product space and then we map back the obtained Schmidt decomposition to the original Fock space of the system under consideration. Such obtained Schmidt decomposition in Fock space allows for construction of observables leading to a violation of the CH inequality. We also show that our generalization of Gisin’s theorem holds for the case of states on non-separable Hilbert spaces, which physically represent states with actually infinite number of particles. Such states emerge, for example, in the discussion of quantum phase transitions. Finally, we discuss the experimental feasibility of constructed Bell test and provide a necessary condition for realizability of this test within the realm of passive linear optics.

A matrix acting between Fock spaces

If Hν=(νn,k)n,k≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{H}_{\ u}=(\ u _{n,k})_{n,k\\geq 0}$\\end{document} is the matrix with entries νn,k=∫[0,∞)tn+kn!dν(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\ u _{n,k}=\\int _{[0,\\infty )}\\frac{ t^{n+k}}{n!}\\,d\ u (t)$\\end{document}, where ν is a nonnegative Borel measure on the interval [0,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$[0,\\infty )$\\end{document}, the matrix Hν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{H}_{\ u}$\\end{document} acts on the space of all entire functions f(z)=∑n=0∞anzn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$f(z) =\\sum_{n=0}^{\\infty} a_{n} z^{n}$\\end{document} and induces formally the operator in the following way: Hν(f)(z)=∑n=0∞(∑k=0∞νn,kak)zn.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{H}_{\ u}(f) (z)=\\sum_{n=0}^{\\infty} \\Biggl(\\sum_{k=0}^{\\infty}\ u _{n,k}a_{k}\\Biggr)z^{n}. $$\\end{document} In this paper, for 0<p≤∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$0< p\\leq \\infty $\\end{document}, we classify for which measures the operator Hν(f)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{H}_{\ u}(f)$\\end{document} is well defined on Fp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$F^{p}$\\end{document} and also gets an integral representation, and among them we characterize those for which Hν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{H}_{\ u}$\\end{document} is a bounded (resp., compact) operator between Fp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$F^{p} $\\end{document} and F∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$F^{\\infty }$\\end{document}.