States of matter, such as the familiar gas, liquid and solid phases, are characterized by certain specific correlations between particles. For instance, the solid phase is characterized by the existence of a periodicity in the atomic density n(x) = (Ψ∧†( x)Ψ∧(x)), such that the Fourier transform of n(x) signals the periodic lattice structure of the solid. This kind of order is called diagonal long-range order, because the periodic structure that extends over the whole size of the solid shows itself in the diagonal elements of the one-particle density matrix n(x, x′) = (Ψ∧† (x) Ψ∧(x′)). As we soon see, in the state of matter that is known as a Bose-Einstein condensate the long-range order is actually off-diagonal in the one-particle density matrix, which makes the Bose-Einstein condensed gas behave very differently from the other phases of matter that we have encountered so far. In particular, the intrinsic quantum-mechanical nature of this many-body state results in intriguing properties such as the possibility for the gas to flow without friction, i.e. superfluidity.
The subjects of this chapter are the Bose-Einstein condensation (BEC) and the superfluidity of ultracold atomic Bose gases, as first observed in gases of rubidium , lithium , sodium  and hydrogen . The experimental realization of BEC for rubidium is shown in Fig. 4.4. We can describe Bose-Einstein condensed gases elegantly using statistical field theory, allowing for a treatment of interaction effects, which leads to a quantitative comparison with experiments. We start with a discussion of the order parameter for Bose-Einstein condensation, after which we give a criterion due to Landau, telling us when a quasiparticle dispersion gives rise to superfluid flow. Using the functional form of the Bogoliubov theory for Bose-Einstein condensation, we then derive the quasiparticle dispersion for excitations above the ground state, showing that it satisfies the Landau criterion. We also obtain the celebrated Gross-Pitaevskii equation for the condensate wavefunc-tion, the Bogliubov-de Gennes equation that describes the inhomogeneous case, the Popov theory that takes fluctuations into account, and the collective modes using a hydrodynamic-like approach. Finally, we briefly discuss what happens when we try to bring the Bose-Einstein condensed gas into rotation, and what happens when the condensate has effectively attractive interactions, such that it is metastable.
KeywordsCondensate Density Landau Free Energy Bogoliubov Theory Bogoliubov Approximation Landau Criterion
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