Barut-Girardello States Research Articles

Coherent state excitons in anisotropic quantum dots: classical and quantum correlations

We investigate the entanglement dynamics of excitons confined in two adjacent quantum dots, which we describe through their algebraic properties using in the z-direction. We use two explicit forms of coherent states: the Perelomov and Barut–Girardello states to represent the electronic component of the excitonic state in the z-direction. Our results show that in a coherent state basis, the concurrence of an excitonic-qubit pair shows subtle variations which are dependent on the algebraic parameters of the group. These variations also appear in the classical and quantum correlations, present in the qubit–qubit density matrix that is associated with the purely Förster-coupled excitonic-qubit pair. A brief discussion of a plausible experimental technique of detecting excitonic coherent states is provided.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.

Coherent states and their time dependence in fractional dimensions

We construct representations of the Lie algebra using representations of the momentum and position operators satisfying the R-deformed Heisenberg relations, in which the fractional dimension d and angular momentum ℓ appear as parameters. The Bargmann index κ, which characterizes representations of the positive discrete series of , can take any positive value. We construct coherent states in fractional dimensions, in particular we extend the two well-known analytic representations of coherent states for , Perelomov and Barut–Girardello states, from dimension one to any dimension d. We generalize this construction to time-dependent coherent states by means of the symmetries of the quantum time-dependent harmonic oscillator in fractional dimensions. We investigate the uncertainty relations of the momentum and position operators with respect to these coherent states, and their dependence on the dimension.

Analytic representations in quantum mechanics

Various Euclidean, hyperbolic and elliptic analytic representations are introduced and relations among them are discussed. The Bargmann analytic representation in the complex plane is considered and its relation to other phase-space methods for the harmonic oscillator is reviewed. The general theory that relates the growth of analytic functions with the density of their zeros is applied to Bargmann functions and it leads to theorems on the completeness of sequences of Glauber coherent states. Two hyperbolic analytic representations in the unit disc, based on SU(1, 1) coherent states and also on phase states are introduced. A third analytic representation in the complex plane based on Barut–Girardello states is also considered and transformations which relate it to the other ones are studied. In the case of systems with finite-dimensional Hilbert space, an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced. The Berezin formalism in the Euclidean, hyperbolic and elliptic cases is discussed. Contour analytic representations in these three cases are also presented.

Coherent state polarons in quantum wells

We investigate the polaronic effects of an electron confined in a quantum well, which we describe through its algebraic properties using su(1,1), taking into account the electron-bulk longitudinal-optical phonon interaction. We construct the variational wave function as the direct product of an electronic part and a part describing coherent phonons generated by the Low–Lee–Pines transformation from the vacuum state. We use two explicit forms of coherent states, Perelomov and Barut-Girardello states, to represent the electronic part in the quantum well spectrum. Our results show that in a coherent state basis for electrons the basic polaron parameters such as the energy gap shift and effective mass are further enhanced compared to those obtained with the conventional sinusoidal form of the basis. The difference between the two types of quantum well coherent states appears in polaronic interactions in quantum wells. We extend the calculations in order to estimate polaron lifetimes for a variety of different material systems.

Photon states associated with the Holstein-Primakoff realization of the SU(1,1) Lie algebra

Statistical and phase properties and number-phase uncertainty relations are systematically investigated for photon states associated with the Holstein-Primakoff realization of the SU(1,1) Lie algebra. Perelomov's SU(1,1) coherent states and the eigenstates of the SU(1,1) lowering generator (the Barut-Girardello states) are discussed. A recently developed formalism, based on the antinormal ordering of exponential phase operators, is used for studying phase properties and number-phase uncertainty relations. This study shows essential differences between properties of the Barut-Girardello states and the SU(1,1) coherent states. The philophase states, defined as states with simple phase-state representations, relate the quantum description of the optical phase to the properties of the SU(1,1) Lie group. A modified Holstein-Primakoff realization is derived, and eigenstates of the corresponding lowering generator are discussed. These stares are shown to contract, in a proper limit, to the familiar Glauber coherent states.

Subcoherent p-representation for non-classical photon states

The Barut-Girardello states (the eigenstates of the SU(1, 1) lowering generator K- are considered in the harmonic oscillator Hilbert space. They are found to have sub-Poissonian photon statistics. By using these states, the diagonal P-representation of the density operator is constructed, and it is shown to be well behaved for non-classical photon states.