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Low Temperature Physics

On the influence of the internal structure of the atom on Bose–Einstein condensationin an ideal gas of hydrogenlike atoms

Yu. V. Slyusarenko, Andrii Sotnikov

Published Jan 1, 2007• 10 citations
Publisher

Abstract

The features of Bose condensation in an equilibrium ideal gas consisting of two types of charged fermions and their bound states—hydrogenlike atoms—in the presence of equilibrium between the photons and matter are investigated. It is shown that under such conditions the main influence on the Bose–Einstein condensation comes from the existence of levels concerned with the hyperfine splitting of the ground state of the hydrogenlike atom. The critical temperature and condensate density as functions of magnetic field are determined by considering effects due to the additional splitting of the levels of the hyperfine structure of the ground state in an external uniform static magnetic field (the Zeeman and Paschen–Back effects). It is found that under conditions of total statistical equilibrium in the system, a condensate is formed only by atoms found in the lowest energy state. It is shown that in the absence of equilibrium between radiation and matter, in the region of ultralow temperatures and low densities, the system can be treated as a multicomponent ideal gas of hydrogenlike atoms. The existence of a hierarchy of individual transition temperatures of each of the samples to the state with Bose–Einstein condensation is established. Expressions are found for the critical temperatures and number densities of particles in the condensate for each of the system components.

DOI
10.1063/1.2409632

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The features of Bose condensation in an equilibrium ideal gas consisting of two types of charged fermions and their bound states—hydrogenlike atoms—in the presence of equilibrium between the photons and matter are investigated. It is shown that under such conditions the main influence on the Bose–Einstein condensation comes from the existence of levels concerned with the hyperfine splitting of the ground state of the hydrogenlike atom. The critical temperature and condensate density as functions of magnetic field are determined by considering effects due to the additional splitting of the levels of the hyperfine structure of the ground state in an external uniform static magnetic field (the Zeeman and Paschen–Back effects). It is found that under conditions of total statistical equilibrium in the system, a condensate is formed only by atoms found in the lowest energy state. It is shown that in the absence of equilibrium between radiation and matter, in the region of ultralow temperatures and low densities, the system can be treated as a multicomponent ideal gas of hydrogenlike atoms. The existence of a hierarchy of individual transition temperatures of each of the samples to the state with Bose–Einstein condensation is established. Expressions are found for the critical temperatures and number densities of particles in the condensate for each of the system components.
We study peculiarities of Bose-Einstein condensation of photons that are in thermodynamic equilibrium with atoms of noninteracting gases. General equations of the thermodynamic equilibrium of the system under study are obtained. We examine solutions of these equations in the case of high temperatures, when the atomic components of the system can be considered as nondegenerated ideal gases of atoms, and the photonic component can form a state with the Bose condensate. Transcendental equation for transition temperature and expression for the density of condensed photons in the considered system are derived. We also obtain analytical solutions of the equation for the critical temperature in a number of particular cases. The existence of two regimes of Bose condensation of photons, which differ significantly in nature of transition temperature dependence on the total density of photons pumped into the system, is revealed. In one case, this dependence is a traditional fractional-power law, and in another one it is the logarithmic law. Applying numerical methods, we determine boundaries of existence and implementation conditions for different regimes of condensation depending on the physical parameters of the system under study. We also show that for a large range of physical systems that are in equilibrium with photons (from ultracold gases of alkali metals to certain types of ideal plasma), the condensation of photons should occur according to the logarithmic regime.
The peculiar condensation in phase space of ideal Bose-Einstein gases is discussed when the internal energy states of the atoms or molecules is taken into account and when the gas of atoms without internal structure is placed in an external conservative field of force. The inclusion of the internal energy states decreases while the application of the external field increases the condensation temperature from its value in absence of either of these causes. The transition phenomenon, apart from the preceding change in the condensation temperature, proceeds in the same way as in the case of the free gas atoms without internal structure; in particular, the order of the transition remains invariable, as the continuity of the heat capacity at constant volume is maintained at the transition point.
A model of spin-polarized hydrogen (H\ensuremath{\uparrow}) is treated in which interactions between atoms are neglected while the single-atom Zeeman and hyperfine interactions are treated exactly. These magnetic terms in the Hamiltonian are found to affect substantially the Bose-Einstein condensation and the various thermodynamic variables. Computations are discussed of the condensation temperature, condensate density, and specific heat in order to indicate how changes in magnetic field strength might be expected to affect future measurements on this quantum system.
Since the experimental realization of Bose–Einstein condensation of ultra cold atoms, extensive studies have been conducted on boson systems. One of recent epochmaking technologies is the synthesis of a fictitious gauge field acting on neutral atoms, which opened another wide field of studies on bosons and/or fermions in a synthetic magnetic field. Charged bosons were studied more than half a century ago by Schafroth. Charged bosons in a magnetic field were then studied in some respects and a rigorous proof was given by Angelescu and Corviovei, who showed absence of Bose– Einstein condensation of free charged bosons in a uniform magnetic field. This is due to the divergence of the density of states at the lowest band energy, i.e., the lowest Landau level. However the situation changes in the presence of a periodic potential. Namely some external periodic potentials are able to restore the Bose–Einstein condensation of the free charge bosons in a magnetic field in threedimensions (3D), although there has been no actual calculation of the condensation temperature of bosons in a typical lattice structure under a magnetic field. According to these previous studies, it is expected that free cold bosonic atoms under a synthetic magnetic field, if realized experimentally, would not condensate in a continuum space but would have a finite Bose–Einstein condensation temperature in the presence of an optical lattice potential. The aim of this short note is to demonstrate the restoration of the Bose–Einstein condensation under a magnetic field by the introduction of a periodic potential, taking typical lattice structures as examples and calculating the condensation temperature analytically. In what follows, with focus on bosons in a 3D lattice under a uniform magnetic field, we calculate the condensation temperature for different lattices and different magnitude of the magnetic field. In particular we consider a layered structure of two dimensional lattices such as square, triangular and honeycomb ones, with a magnetic field applied in the direction normal to the two-dimensional (2D) lattices. The energy spectrum of a particle on a two dimensional lattice, which we assume to be a square lattice for a moment, under a uniform magnetic field shows a characteristic pattern, so-called Hofstadter’s diagram. The one-particle energy of this system is described by the following Hamiltonian in the tight-binding approximation.
We discuss the occurrence of Bose-Einstein condensation in systems of noninteracting charged particles in three and one dimensions and in presence of an external magnetic field. In the one dimensional, as well as in the magnetic field cases, although not a critical temperature, a characteristic temperature can be found, corresponding to the case in which the ground state density becomes a macroscopic fraction of the total density. The case of relativistic charged scalar and vector particles is studied. The results give support to the existence of superconductivity in extremely strong magnetic fields, and lead to the prediction of superconductive-ferromagnetic behavior in the vector field case, which might be of interest in condensed matter as well as in cosmology. Some features of the magnetization in the early universe are conjectured.
We have studied conditions of photon Bose-Einstein condensate formation that is in thermodynamic equilibrium with ideal gas of two-level Bose atoms below the degeneracy temperature.Equations describing thermodynamic equilibrium in the system were formulated; critical temperatures and densities of photonic and atomic gas subsystems were obtained analytically.Coexistence conditions of these photonic and atomic Bose-Einstein condensates were found.There was predicted the possibility of an abrupt type of photon condensation in the presence of Bose condensate of ground-state atoms: it was shown that the slightest decrease of the temperature could cause a significant gathering of photons in the condensate.This case could be treated as a simple model of the situation known as "stopped light" in cold atomic gas.We also showed how population inversion of atomic levels can be created by lowering the temperature.The latter situation looks promising for light accumulation in atomic vapor at very low temperatures.