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Physical Review A

Nonlinear quantization of a degenerate charged Bose gas in an external Coulomb trap

Gilbert Reinisch

Published Sep 28, 2004• 14 citations
Publisher

Abstract

We consider a degenerate charged Bose-Coulomb gas populating several discrete stationary boson bound states that are located in a spherical-symmetrical central Coulombian potential. Each such state is defined, through appropriate boundary conditions and normalization, by a so-called ``nonlinear eigenstate'' that is actually a solution of the coupled (linear) stationary Schr\"odinger-like Gross-Pitaevskii differential equation and the (nonlinear) Poisson equation. The corresponding eigenvalues allow us to define the energies of these degenerate boson states, much like the Koopmans orbital energy in atomic physics. This theory applies surprisingly well (compared with the corresponding Hartree-Fock results) to spherical-symmetrical $s$ orbital states in atomic physics (i.e., bosonlike restricted orbital states where the additional spin degree of freedom is already integrated out). Finally the superposition of two such stationary nonlinear eigenstates is investigated and given a semiclassical physical significance similar to a Thomas-Fermi approach. The resulting concepts apply particularly well (namely within an average 1% error bar with respect to spectroscopic data) to the $1{s}^{2}\text{\penalty1000-\hskip0pt}2{s}^{2}$ orbital states of the $3\ensuremath{\leqslant}Z\ensuremath{\leqslant}9$ atomic subsystems.

DOI
10.1103/physreva.70.033613

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We consider a degenerate charged Bose-Coulomb gas populating several discrete stationary boson bound states that are located in a spherical-symmetrical central Coulombian potential. Each such state is defined, through appropriate boundary conditions and normalization, by a so-called ``nonlinear eigenstate'' that is actually a solution of the coupled (linear) stationary Schr"odinger-like Gross-Pitaevskii differential equation and the (nonlinear) Poisson equation. The corresponding eigenvalues allow us to define the energies of these degenerate boson states, much like the Koopmans orbital energy in atomic physics. This theory applies surprisingly well (compared with the corresponding Hartree-Fock results) to spherical-symmetrical sss orbital states in atomic physics (i.e., bosonlike restricted orbital states where the additional spin degree of freedom is already integrated out). Finally the superposition of two such stationary nonlinear eigenstates is investigated and given a semiclassical physical significance similar to a Thomas-Fermi approach. The resulting concepts apply particularly well (namely within an average 1% error bar with respect to spectroscopic data) to the 1s2\penalty1000-2s21{s}^{2}\text{\penalty1000-\hskip0pt}2{s}^{2}1s2\penalty1000-2s2 orbital states of the 3\ensuremath⩽Z\ensuremath⩽93\ensuremath{\leqslant}Z\ensuremath{\leqslant}93\ensuremath⩽Z\ensuremath⩽9 atomic subsystems.
In recent work [Sakaguchi and Malomed, Phys. Rev. A 83, 013607 (2011)], a solution to the problem of the quantum collapse (fall onto the center) in the three-dimensional space with the attractive potential $\ensuremath{-}({U}_{0}/2){r}^{\ensuremath{-}2}$ was proposed, based on the replacement of the linear Schr\"odinger equation by the Gross-Pitaevskii (GP) equation with the repulsive cubic term. The model applies to a quantum gas of molecules carrying permanent electric dipole moments, with the attraction center representing a fixed electric charge. It was demonstrated that the repulsive nonlinearity suppresses the quantum collapse and creates the corresponding spherically symmetric ground state (GS), which was missing in the case of the linear Schr\"odinger equation. Here, we aim to extend the analysis to the cylindrical geometry and to eigenstates carrying angular momentum. The cylindrical anisotropy is imposed by a uniform dc field, which fixes the orientation of the dipole moments, thus altering the potential of the attraction to the center. First, we analyze the modification of the condition for the onset of the quantum collapse in the framework of the linear Schr\"odinger equation with the cylindrically symmetric potential for the states with azimuthal quantum numbers m=0m=0m=0 (the GS) and m=1m=1m=1, 2. The corresponding critical values of the strength of the attractive potential (U0)cr(m){({U}_{0})}_{\mathrm{cr}}(m)(U0​)cr​(m) are found. Next, a numerical solution of the nonlinear GP equation is developed, which demonstrates the replacement of the quantum collapse by the originally missing eigenstates with m=0,1,2m=0,1,2m=0,1,2. Their dynamical stability is verified by means of numerical simulations of the perturbed evolution. For m=0m=0m=0, the Thomas-Fermi approximation is presented too, in an analytical form. Crucially important for the solution is the proper choice of the boundary conditions at $r\ensuremath{\rightarrow}0$.
The stationary solutions of the Gross-Pitaevskii equation can be divided in two classes: those which reduce, in the limit of vanishing nonlinearity, to the eigenfunctions of the associated Schrödinger equation and those which do not have linear counterpart. Analytical and numerical results support an existence condition for the solutions of the first class in terms of the ratio between their proper frequency and the corresponding linear eigenvalue. For one-dimensional confined systems, we show that solutions without linear counterpart do exist in presence of a multi-well external potential. These solutions, which in the limit of strong nonlinearity have the form of chains of dark or bright solitons located near the extrema of the potential, represent macroscopically excited states of a Bose-Einstein condensate and are in principle experimentally observable.
We consider the photoassociation of a low-density gas of quantum-degenerate trapped fermionic atoms into bosonic molecules in a spherically symmetric harmonic potential. For a dilute system and the photoassociation coupling energy small compared to the level separation of the trap, only those fermions in the single shell with Fermi energy are coupled to the bosonic molecular field. Introducing a collective pseudospin operator formalism we show that this system can then be mapped onto the Tavis-Cummings Hamiltonian of quantum optics, with an additional pairing interaction. By exact diagonalization of the Hamiltonian, we examine the ground state and low excitations of the Bose-Fermi system, and study the dynamics of the coherent coupling between atoms and molecules. In a semiclassical description of the system, the pairing interaction between fermions is shown to result in a self-trapping transition in the photoassociation, with a sudden suppression of the coherent oscillations between atoms and molecules. We also show that the full quantum dynamics of the system is dominated by quantum fluctuations in the vicinity of the self-trapping solution.
Stationary solutions for the cubic nonlinear Schrödinger equation modeling Bose-Einstein condensates (BECs) confined in three spatial dimensions by general forms of a potential are studied through a perturbation method and also numerically. Note that we study both repulsive and attractive BECs under similar frameworks in order to deduce the effects of the potentials in each case. After outlining the general framework, solutions for a collection of specific confining potentials of physical relevance to experiments on BECs are provided in order to demonstrate the approach. We make several observations regarding the influence of the particular potentials on the behavior of the BECs in these cases, comparing and contrasting the qualitative behavior of the attractive and repulsive BECs for potentials of various strengths and forms. Finally, we consider the nonperturbative where the potential or the amplitude of the solutions is large, obtaining various qualitative results. When the kinetic energy term is small (relative to the nonlinearity and the confining potential), we recover the expected Thomas-Fermi approximation for the stationary solutions. Naturally, this also occurs in the large mass limit. Through all of these results, we are able to understand the qualitative behavior of spherical three-dimensional BECs in weak, intermediate, or strong confining potentials.
The ground-state energy of a finite inhomogeneous system of bosons located in a scalar external field was found within the framework of the self-consistent Hartree-Fock approximation, on the basis of the representation of the second quantization without using the formalism of anomalous averages. The ground-state wave function corresponds to the stationary Gross-Pitaevskii equation for the Bose-Einstein condensate wave function. It was shown that the ground-state energy can be found using the energy determined by the stationary Gross-Pitaevskii equation only for a system that satisfies the thermodynamic limit.
We introduce a nonlinear Schr\"odinger equation to describe the dynamics of a superfluid Bose gas in the crossover from the weak-coupling regime, where $a{n}^{1/3}\ensuremath{\ll}1$ with aaa the interatomic s\mathit{s}s-wave scattering length and nnn the bosonic density, to the unitarity limit, where $a\ensuremath{\rightarrow}+\ensuremath{\infty}$. We call this equation the unitarity Schr\"odinger equation (USE). The zero-temperature bulk equation of state of this USE is parametrized by the Lee-Yang-Huang low-density expansion and Jastrow calculations at unitarity. With the help of the USE we study the profiles of quantized vortices and vortex-core radius in a uniform Bose gas. We also consider quantized vortices in a Bose gas under cylindrically symmetric harmonic confinement and study their profile and chemical potential using the USE and compare the results with those obtained from the Gross-Pitaevskii-type equations valid in the weak-coupling limit. Finally, the USE is applied to calculate the breathing modes of the confined Bose gas as a function of the scattering length.