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Apple Academic Press eBooks

Basics of the Bose–Einstein Condensation (BEC)

Mihai V. Putz

Published Jun 6, 2013• 0 citations
Publisher

Abstract

Since the nowadays growing interest in Bose-Einstein condensates (BEC) due to the expanded experimental evidence on various atomic systems within optical lattices in weak and strong coupling regimes, its many-body quantum bases and equations are elementarily presented emphasizing on the order parameter, pseudo-potential and mean field approaches, while emphasizing the common places with the Density Functional Theory, like the integral normalization of density or the chemical potential, as the main vehicles of both theories of matter (Putz, 2011d, 2011e).

DOI
10.1201/b14167-10

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Since the nowadays growing interest in Bose-Einstein condensates (BEC) due to the expanded experimental evidence on various atomic systems within optical lattices in weak and strong coupling regimes, its many-body quantum bases and equations are elementarily presented emphasizing on the order parameter, pseudo-potential and mean field approaches, while emphasizing the common places with the Density Functional Theory, like the integral normalization of density or the chemical potential, as the main vehicles of both theories of matter (Putz, 2011d, 2011e).
Potential ........................................................................................................... 143 6.7Exchange Density Functional of Homogeneous Systems-–Bosonic Case ..... 150 Keywords ................................................................................................................. 157 6.1 INTRODUCTION The basics of Density Functional Theory (DFT) and of Bose-Einstein Condensation (BEC) are merged such that the DFT enriches with the order parameter equation from the BEC modeling, while BEC acquires the new equation for the condensate density as provided by the DFT Hohenberg-KS analysis. Together, BEC-DFT may constitute as the next level of comprehension in modeling condensed matter in physics and chemical bond in chemistry, in particular, while providing the clue how the unified theory of physical-chemical phenomena should look like (Albus et al., 2003; Argaman and Band, 2011; Brand, 2004; Kim and Vetter, 1997; Nunes, 1999; Putz, 2011d; Zubarev, 2003).
This chapter introduces basic notions about Bose-Einstein condensates (BECs). One first studies ideal Bose gases, putting an emphasis on the role of the trapping geometry and of the dimensionality. One also introduces the correlation functions characterizing coherence. The second section deals with weakly-interacting BECs, which are described, in a mean-field approach, by the so-called Gross-Pitaevskii equation. This equation is derived and applied to a variety of experimentally relevant situations. The third section deals with beyond-mean-field effects that appear when the interactions are stronger, and introduces in particular the Bogolubov approximation. The last section is devoted to the study of BECs in double-well potentials, a situation inspired from condensed matter systems where beyond-mean-field effects can appear quite easily, and which remains simple enough to be studied theoretically in details.
Shorttitle: The first experimental achievement of Bose Einstein condensation (BEC) in dilute alkali gases in 1995 has developed a renewed interest both theoretical and experimental. Lot of work has already been done over the last decade to study the different aspects of this condensation, mainly based on mean-field theories, hyperspherical approaches and quantum Monte Carlo methods. This thesis work presents an approximate many-body approach to investigate mainly the static (ground state) properties of the condensate, incorporating realistic inter-atomic interactions. Potential harmonic expansion method (PHEM), which includes two-body correlations only and disregards three or higher body correlations, has been adopted to analyze this condensate. This is ideally suited for BEC, which is required to be extremely dilute for its experimental realization. This treatment and analysis should be useful for the physics community especially working with weakly interacting ultra cold atoms, bosons and fermions as well.
Since the nowadays growing interest in Bose-Einstein condensates due to the expanded experimental evidence on various atomic systems within optical lattices in weak and strong coupling regimes, the connection with Density Functional Theory is firstly advanced within the mean field framework at three levels of comprehension: the many-body normalization condition, Thomas-Fermi limit, and the chemical hardness closure with the inter-bosonic strength and universal Hohenberg-Kohn functional. As an application the traditional Heitler-London quantum mechanical description of the chemical bonding for homopolar atomic systems is reloaded within the non-linear Schrödinger (Gross-Pitaevsky) Hamiltonian; the results show that a two-fold energetic solution is registered either for bonding and antibonding states, with the bosonic contribution being driven by the square of the order parameter for the Bose-Einstein condensate density in free (gas) motion, while the associate wave functions remain as in classical molecular orbital model.
Introduction Bose–Einstein condensation (BEC) is one of the most intriguing phenomena in physics. The basic idea goes back to the 1920s, when Einstein made predictions for the behavior of atoms at sufficiently low temperatures and high densities. However, it took seven decades for experimental physicists to realize this phase under the low-density conditions envisioned by Einstein. In the intermediate period there were many experiments in which aspects of BEC played important roles. Therefore, phenomena such as superfluidity and superconductivity were connected to BEC long before its 1995 realization in ultra-cold, low-density atomic vapors. Since BEC can be discussed with very many different backgrounds and applications in mind, the subject can sometimes be overwhelming for interested students. There is a very good reason why a chapter on BEC does not belong in this book. In all the other chapters here, the processes described involve one or at most two atoms, and the interactions that play a role are between the particles that constitute the atom, namely between the nucleus (or nuclei) and their electrons. The interactions are described quantum mechanically by their Hamiltonian that can always be written exactly in non-relativistic terms, and in some cases can even be solved exactly. This makes atoms and molecules a branch of physics where comparisons between theory and experiments become very meaningful since theoretical predictions are often confirmed to very high accuracy. In BEC, it is the interactions among many atoms that are the key for understanding the phenomenon. Although in principle one could write a Hamiltonian for a Bose condensed gas, its large number of particles makes this not very practical. The starting point for the description of a BEC is in the area of many-body physics that belongs to the expertise of condensed-matter physicists. The interactions between ~10 23 electrons in a solid state with a lattice formed by ~10 23 ions resemble the kind of interactions in a BEC much more closely than the simple fewbody interactions in an atom or molecule. Thus all the theoretical models that have been described up to now are not applicable for the contents of this chapter.