Fock Bargmann Space Research Articles

Generalized Grassmann variables for quantum kit (k-level) systems and Barut–Girardello coherent states for su(r + 1) algebras

This paper concerns the construction of $su(r+1)$ Barut--Girardello coherent states in term of generalized Grassmann variables. We first introduce a generalized Weyl-Heisenberg algebra ${\cal A}(r)$ ($r \geq 1$) generated by $r$ pairs of creation and annihilation operators. This algebra provides a useful framework to describe qubit and qukit ($k$-level) systems. It includes the usual Weyl-Heisenberg and $su(2)$ algebras. We investigate the corresponding Fock representation space. The generalized Grassmann variables are introduced as variables spanning the Fock--Bargmann space associated with the algebra ${\cal A}(r)$. The Barut--Girardello coherent states for $su(r+1)$ algebras are explicitly derived and their over--completion properties are discussed.

Dynamics of two-cluster systems in phase space

We present a phase-space representation of quantum state vectors for two-cluster systems. Density distributions in the Fock–Bargmann space are constructed for bound and resonance states of 6,7Li and 7,8Be, provided that all these nuclei are treated within a microscopic two-cluster model. The density distribution in the phase space is compared with those in the coordinate and momentum representations. Bound states realize themselves in a compact area of the phase space, as also do narrow resonance states. We establish the quantitative boundaries of this region in the phase space for the nuclei under consideration. Quantum trajectories are demonstrated to approach their classical limit with increasing energy.

Concrete description of the (Г, χ)-theta Fock-Bargmann space for rank one in high dimension

We give concrete description of the -theta Fock-Bargmann space consisting of holomorphic automorphic functions associated to given discrete subgroup in of rank one and given character . It looks like a tensor product of a theta Fock-Bargmann space on the complex plane and the classical Fock-Bargmann space on . Moreover, we construct an orthonormal basis and we give explicit expression of its reproducing kernel function in terms of Riemann theta function.

A short note on the chaoticity of a weight shift on concrete orthonormal basis associated to some Fock-Bargmann space

Let $\mathfrak {F}_{\alpha }(\mathbb {C})$Fα(C) be the Hilbert space generated by the orthonormal basis ${e_{n}^{\alpha , \nu }(z) = (\frac{2\nu }{\pi })^{1/4}e^{\frac{\nu }{2}z^{2}}e^{-\frac{\pi ^{2}}{\nu }(n +\alpha )^{2} +2i\pi (n +\alpha )z}; n \in \mathbb {N}}$enα,ν(z)=(2νπ)1/4eν2z2e−π2ν(n+α)2+2iπ(n+α)z;n∈N where ν > 0 and α are real numbers, this space is a particular case of (Γ, χ)-theta Fock-Bargmann spaces recently constructed by Ghanmi-Intissar [J. Math. Phys. 54, 063514 (2013)]. In the present work, we consider on $\mathfrak {F}_{\alpha }(\mathbb {C})$Fα(C) the weight shift operator $\mathbb {M}$M defined by ${\mathbb {M}e_{n}^{\alpha ,\nu } = \omega _{n}e_{n+1}^{\alpha ,\nu }, n\in \mathbb {N}}$Menα,ν=ωnen+1α,ν,n∈N where $\omega _{n} = c_{\alpha ,\nu }e^{\frac{2\pi }{\nu }n}$ωn=cα,νe2πνn and $c_{\alpha , \nu }=e^{\frac{\pi }{\nu }+ 2\alpha }$cα,ν=eπν+2α and we show the chaoticity of the operator $\mathbb {H}_{p} = \mathbb {M}^{p}\mathbb {D}^{p+1}; p=0, 1, 2, ...$Hp=MpDp+1;p=0,1,2,... where $\mathbb {D}$D is adjoint of $\mathbb {M}$M.

Phase Portraits of Quantum Systems

We formulate a general approach to construct phase portraits of a quantum system in the Fock–Bargmann space. This approach is applied to simple model problems and two-cluster nuclei as well.

Unusual Manifestations of the Pauli Principle in Scattering of Atomic Nuclei

Within the microscopic model based on the algebraic version of the resonating group method, the role of the Pauli principle in the formation of the continuum wave function of the a +t + n nuclear system has been investigated. The norm kernel for the a + t + n three-cluster system has been constructed in the Fock–Bargmann space. The complete classification of the eigenfunctions and the eigenvalues of the 8Li norm kernel by the eigenvalues of the 7Li = a+t, 5He = a + n; and 4H = t + n binary subsystems has been given.

Phase-space path integral and Brownian motion

We formulate and examine a phase-space path integral representation of a quantum time evolution in terms of Fock–Bargmann spaces and stochastic line integrals. The Fock–Bargmann space \documentclass[12pt]{minimal}\begin{document}${{\cal H}_{\rm FB}}$\end{document}H FB is a closed subspace of \documentclass[12pt]{minimal}\begin{document}$L^2({\cal M})$\end{document}L2(M), where \documentclass[12pt]{minimal}\begin{document}${\cal M}$\end{document}M is the phase space. Let EFB be the orthogonal projection onto \documentclass[12pt]{minimal}\begin{document}${{\cal H}_{\rm FB}}$\end{document}H FB , and h the classical Hamiltonian, then H = EFBhEFB is understood as a quantization of h. The quantum time evolution e−itH is asymptotically represented by the solution of a variant of diffusion equation with the imaginary potential ih. Thus the quantum time evolution can be represented stochastically, in terms of Brownian motions and stochastic integrals.

Complex and real Hermite polynomials and related quantizations

It is known that the anti-Wick (or standard coherent state) quantization of the complex plane produces both canonical commutation rule and quantum spectrum of the harmonic oscillator (up to the addition of a constant). In this work, we show that these two issues are not necessarily coupled: there exists a family of separable Hilbert spaces, including the usual Fock–Bargmann space, and in each element in this family there exists an overcomplete set of unit-norm states resolving the unity. With the exception of the Fock–Bargmann case, they all produce non-canonical commutation relation whereas the quantum spectrum of the harmonic oscillator remains the same up to the addition of a constant. The statistical aspects of these non-equivalent coherent state quantizations are investigated. We also explore the localization aspects in the real line yielded by similar quantizations based on real Hermite polynomials.

Coherent-State Path Integrals and Brownian Motions

A stochastic approach to the rigorous foundation of the coherent-state (phase-space) path integral is given. Stochastic integrals and some generalizations of the Feynman–Kac theorem are used for this purpose. In this approach, quantum mechanics is described in terms of the Fock–Bargmann representation; a classical Hamiltonian is related to the corresponding quantum Hamiltonian on the Fock–Bargmann space, seen as a Hilbert subspace of \(L^{2}({\bf R}^{2})\). The coherent-state path integral is realized as a conditional expectation of a stochastic process defined by the exponential of the Fisk–Stratonovich integral of the fundamental 1-form along a path of Brownian motion on the phase space \({\bf R}^{2}\).

Coexistence of different cluster configurations in light neutron-rich nuclei

On the basis of the algebraic version of the resonating-group method (RGM) and within the framework of the discrete representation in the Fock-Bargmann space, a microscopic theory of nuclear reactions with due regard for a coexistence of different cluster configurations in a compound nucleus is realized. Fundamental tenets of the algebraic version of the RGM are stated both for a single binary cluster configuration and for a compound system, where several cluster configurations coexist. Several examples of norm kernels, their eigenvalues, phase shifts, and effective cross sections are given for a number of binary cluster systems.

Lowest Landau level functional and Bargmann spaces for Bose–Einstein condensates

A fast rotating Bose–Einstein condensate can be described by a complex valued wave function minimizing an energy restricted to the lowest Landau level or Fock–Bargmann space. Using some structures associated with this space, we study the distribution of zeroes of the minimizer and prove in particular that the number of zeroes is infinite. We relate their location to the combination of two problems: a confining problem producing an inverted parabola profile and the Abrikosov problem of minimizing an energy on a lattice, using Theta functions.

Integral equations in the Fock-Bargmann space and the generator-coordinate method

In the generator-coordinate method, the norm kernel built as a Slater determinant of the Brink-Bloch orbitals plays a role of unity for the complete set of orthonormalized states defined in the Fock-Bargmann space and satisfying the Pauli exclusion principle. The closure condition is then written as the Hilbert-Schmidt expansion of the norm kernel and can be derived by a simple algebraic procedure. A basis of eigenstates of the norm kernel is shown to be a powerful tool in resonating-group calculations.

Norm Kernels and the Closeness Relation for Pauli-Allowed Basis Functions

The norm kernel of the generator-coordinate method is shown to be a symmetric kernel of an integral equation with eigenfunctions defined in the Fock--Bargmann space and forming a complete set of orthonormalized states (classified with the use of SU(3) symmetry indices) satisfying the Pauli exclusion principle. This interpretation allows to develop a method which, even in the presence of the SU(3) degeneracy, provides for a consistent way to introduce additional quantum numbers for the classification of the basis states. In order to set the asymptotic boundary conditions for the expansion coefficients of a wave function in the SU(3) basis, a complementary basis of functions with partial angular momenta as good quantum numbers is needed. Norm kernels of the binary systems 6He+p, 6He+n, 6He+4He, and 8He+4He are considered in detail.

Fock–Bargmann space and SU(3) models

The aim of this review is to present an approach to unitary models that has been developed during the last ten years. The approach uses wave functions and operators defined in Fock–Bargmann space. A solution for the SU(3) shell model is introduced, as well as some extensions that are related to a classical treatment of SU(3) theory.

Microscopic SUπ(3)⊗SUν(3) interpretation of a two triaxial rotor model

A description of the relative motion of neutrons and protons in strongly deformed even-even nuclei is reduced to that of two coupled triaxial rotors, one representing the neutrons and the other the protons, by using generating function methods. The two rotor theory is interpreted as a generalization of the Elliott SU(3) model with a neutron–proton direct product SUν(3)×SUπ(3) substructure. The second order Casimir operator for this system is constructed and some of its eigenfunctions are found in explicit form in the Fock–Bargmann space representation. Analytical results for isovector M1 transitions probabilities are obtained and the results compared with experimental data and pseudo-SU(3) calculations. Graphs picturing the relative orientation of neutron-proton rotors in Ne22 are presented.

On the complete basis of Pauli-allowed states of three-cluster systems in the Fock–Bargmann space

In the Fock–Bargmann space, the complete harmonic-oscillator basis for the three-cluster system is constructed. The indices of the reduction U(6)⊇U(2)×U(3) are chosen as the quantum numbers. The Pauli-forbidden states are eliminated by the orthogonal transformation of basis functions. The basis states are obtained in terms of hypergeometric functions and the spherical Wigner functions. Their simple form allows one to solve the problem of calculating the matrix elements of the microscopic Hamiltonian needed for the study of three-cluster systems within the algebraic version of RGM.

Amplitude squared squeezed states in the Fock–Bargmann space

AbstractSqueezed states are normally studied in configuration space or using the Glauber formalism for the coherent states. Here we show that the use of the Fock–Bargmann space can lead to more general solutions, especially in the case of squeezed states associated with the square of field amplitude. The relations that we obtain are completely general and several well‐known, approximate results can be obtained as the limit cases of our approach.© 1993 John Wiley & Sons, Inc.

Squeezed states and the Franck-Condon overlap

A simple and direct approach to studying squeezed states is proposed. It is based on treating the corresponding differential equations and their solutions directly on the Fock-Bargmann space. The method is exemplified by giving quite general solutions to the case of amplitude-squared squeezed states. We show the importance of this approach by deriving the overlap between these solutions and connecting them with the overlap integral associated with the Franck-Condon principle.

Poincare contraction of SU(1,1) Fock-Bargmann structure

Some aspects of the contraction process SO0(1,2) to Poincare are studied. The starting point is the choice of a suitable parametrization for the de Sitterian phase space SO0(1,2)/SO(2) approximately=SU(1,1)/U(1). The authors show that the contraction to Poincare must be realized by restricting the Fock-Bargmann space to a specific subspace. This constraint is necessary to make the divergent terms disappear. In particular, the classical result according to which the discrete series representation of SU(1,1) contracts onto the Wigner representation P(m) is described at a global level.