Paolo Tommasini

The hydrogen atom as an entangled electron–proton system

We illustrate the description of correlated subsystems by studying the simple two-body hydrogen atom. We study the entanglement of the electron and proton coordinates in the exact analytical solution. This entanglement, which we quantify in the framework of the density matrix formalism, describes correlations in the electron–proton motion.

Gaussian Time-Dependent Variational Principle for Bosons

Abstract We investigate the Dirac time-dependent variational method for a system of non-ideal Bosons interacting through an arbitrary two body potential. The method produces a set of non-linear time dependent equations for the variational parameters. In particular we have considered small oscillations about equilibrium. We obtain generalized RPA equations that can be understood as interacting quasi-bosons, usually mentioned in the literature as having a gap. The result of this interaction provides us with scattering properties of these quasi-bosons including possible bound-states, which can include zero modes. In fact the zero mode bound state can be interpreted as a new quasi-boson with a gapless dispersion relation. Utilizing these results we discuss a straightforward scheme for introducing temperature.

Methods for a Nonuniform Bose Gas

We review mathematical methods for the treatment of a system of Bose particles with nonuniform density. The use of the pseudopotential is explained, especially with respect to negative scattering lengths. It is emphasized that the delta-function potential produces no scattering in three dimensions, and should not be used in the Bogoliubov self-consistent field method, which is variational in nature. A common misuse of the Bogoliubov method at finite temperatures is pointed out. A Gaussian variational method is proposed.

Gaussian time-dependent variational principle for bosons: Contact interaction in one dimension

We investigate the Dirac time-dependent variational method using a Gaussian trial functional for an infinite one-dimensional system of bosons interacting through a repulsive contact interaction. The method produces a set of nonlinear time-dependent equations for the variational parameters. By solving the static equations we have calculated the ground state energy per particle. We have also considered small oscillations about the equilibrium and obtain mode equations which lead us to a gapless dispersion relation. The existence of an exact numerical solution for the ground-state energy and excitations obtained by Lieb allow us to compare with the Gaussian results. We can also, as the system becomes less dilute, see the improvement of the results as compared with the Bogoliubov scheme. {copyright} {ital 1997} {ital The American Physical Society}

Non-ideal Boson System in the Gaussian Approximation

Abstract We investigate ground-state and thermal properties of a system of non-relativistic bosons interacting through repulsive, two-body interactions in a self-consistent Gaussian mean-field approximation which consists in writing the variationally determined density operator as the most general Gaussian functional of the quantized field operators. Finite temperature results are obtained in a grand canonical framework. Contact is made with the results of Lee, Yang, and Huang in terms of particular truncations of the Gaussian approximation. The full Gaussian approximation supports a free phase or a thermodynamically unstable phase when contact forces and a standard renormalization scheme are used. When applied to a Hamiltonian with zero range forces interpreted as an effective theory with a high momentum cutoff, the full Gaussian approximation generates a quasi-particle spectrum having an energy gap, in conflict with perturbation theory results.

Superfluidity and Feshbach Resonances in BEC

The experimental realisation of Bose-Einstein condensation (BEC) in trapped dilute alkali gases in 1995 has generated an exciting new field in physics. Since the initial experiments on 87Rb at JILA, 1 7Li at Rice University,2 and 23Na at MIT,3 many new groups have joined the BEC club, mostly in 87Rb and 23Na systems, although, condensation was recently observed in a spin-polarized hydrogen gas.4

Extended Hartree-Fock-Bogoliubov theory for degenerate Bose systems

An extension of the Hartree–Fock–Bogoliubov (HFB) theory of degenerate Bose systems in which the coupling between one and two quasi-particles is taken into account is developed. The excitation operators are written as linear combinations of one and two HFB quasi-particles. Excitation energies and quasi-particle amplitudes are given by generalized Bogoliubov equations. The excitation spectrum has two branches. The first one is a discrete branch which is gapless and has a phonon character at large wavelength and, contrarily to HFB, is always stable. This branch is detached from a second, continuum branch whose threshold, at fixed total momentum, coincides with the two quasi-particle threshold of the HFB theory. The gap between the two branches at P = 0 is twice the HFB gap, which thus provides for the relevant energy scale. Numerical results for a specific case are given.

Molecular signatures in hybrid atomic-molecular Bose-Einstein condensates

As it was proposed and recently verified experimentally, the mechanism of Feshbach resonance in a condensate can create a second condensate component of molecules that coexists with the atomic condensate. In this work we investigate signatures of the presence of the molecular condensate through the equilibrium properties and collective excitations of the hybrid system of atoms and molecules, subjected to a trap, employing a time-dependent variational ansatz. We show that the shape of the condensate changes significantly by the presence of the molecules and that modes unique to this hybrid system appear at observable frequencies.