Bose-Einstein Condensation

abstract

Bose-Einstein condensation (BEC) refers to a prediction of quantum statistical mechanics (Bose [1], Einstein [2]) where an ideal gas of identical bosons undergoes a phase transition when the thermal de Broglie wavelength exceeds the mean spacing between the particles. Under these conditions, bosons are stimulated by the presence of other bosons in the lowest energy state to occupy that state as well, resulting in a macroscopic occupation of a single quantum state. The condensate that forms constitutes a macroscopic quantum-mechanical object. BEC was first observed in 1995, seventy years after the initial predictions, and resulted in the award of 2001 Nobel Prize in Physics to Cornell, Ketterle and Weiman. The experimental observation of BEC was achieved in a dilute gas of alkali atoms in a magnetic trap. The first experiments used ⁸⁷Rb atoms [3], ²³Na [4], ⁷Li [5], and H [6] more recently metastable He has been condensed [7]. The list of BEC atoms now includes molecular systems such as Rb₂ [8], Li2 [9] and Cs2 [10]. In order to cool the atoms to the required temperature (~200 nK) and densities (10¹³–10¹⁴ cm^–3) for the observation of BEC a combination of optical cooling and evaporative cooling were employed. Early experiments used magnetic traps but now optical dipole traps are also common. Condensates containing up to 5x10⁹ atoms have been achieved for atoms with a positive scattering length (repulsive interaction), but small condensates have also been achieved with only a few hundred atoms. In recent years Fermi degenerate gases have been produced [11], but we will not discuss these in this chapter.

BECs are now routinely produced in dozens of laboratories around the world. They have provided a wonderful test bed for condensed matter physicswith stunning experimental demonstrations of, among other things, interference between condensates, superfluidity and vortices. More recently they have been used to create optically nonlinear media to demonstrate electromagnetically induced transparency and neutral atom arrays in an optical lattice via a Mott insulator transition.

Many experiments on BECs are well described by a semiclassical theory discussed below. Typically these involve condensates with a large number of atoms, and in some ways are analogous to describing a laser in terms of a semiclassical mean field. More recent experiments however have begun to probe quantum 397 398 19 Bose-Einstein Condensation properties of the condensate, and are related to the fundamental discreteness of the field and nonlinear quantum dynamics. In this chapter, we discuss some of these quantum properties of the condensate. We shall make use of “few mode” approximations which treat only essential condensate modes and ignore all noncondensate modes. This enables us to use techniques developed for treating quantum optical systems described in earlier chapters of this book.