This chapter discusses some of the “super” properties of lasers, superfluids, and superconductors in the context of quantum field theory, including their innate property of spontaneous coherence, which can be seen as the opposite of decoherence.

This chapter introduces the order parameter and the Gross--Pitaevskii equation. Although a Bose-Einstein C condensate (BEC) is a quantum many-body system which that can be described fully only using the many-body wavefunction, many of its simplest spatial dynamics can be understood using the order parameter and its dynamics. We first describe how the order parameter can be formally defined, and how this can be considered an effective spatial wavefunction of the BEC. We then derive the time-dependent evolution of the order parameter via the Gross--Pitaevskii equation, then and study various solutions of it. This includes plane-wave solutions, an infinite potential well solutions, and excited state solutions such as vortices and solitons. After introducing key concepts such as the Thomas--Fermi approximation and the healing length, we also discuss how the Gross--Pitaevskii equation can be cast in the form of hydrodynamic equations.

This chapter introduces the basic physics of Bose--Einstein condensation. We first start with why there is a difference between distinguishable and indistinguishable particles in terms of the number of states that are available in a composite system. Then Bose and Einstein's argument of why one expects a high probability of occupation of the ground state is discussed. This is derived then more rigorously for the grand canonical ensemble, showing that at some critical temperature there should be a macroscopic occupation of the ground state. Next, the low-lying energy spectrum of an interacting Bose--Einstein condensate is derived, leading to the Bogoliubov dispersion. The significance of the Bogoliubov dispersion as the origin of superfluidity is then discussed, in terms of superfluditysuperfluidity. Laudau'sLandau's criterion for superfluidity is derived, by general principles of Galilean transformations of Schrodinger's equation.