Generalized Bose–Einstein Condensation

Abstract

Generalized Bose-Einstein condensation (GBEC) involves condensates appearing simultaneously in multiple states. We review examples of the three types in an ideal Bose gas with different geometries. In Type I there is a discrete number of quantum states each having macroscopic occupation; Type II has condensation into a continuous band of states, with each state having macroscopic occupation; in Type III each state is microscopically occupied while the entire condensate band is macroscopically occupied. We begin by discussing Type I or "normal" BEC into a single state for an isotropic harmonic oscillator potential. Other geometries and external potentials are then considered: the "channel" potential (harmonic in one dimension and hard-wall in the other), which displays Type II, the "cigar trap" (anisotropic harmonic potential), and the "Casimir prism" (an elongated box), the latter two having Type III condensations. General box geometries are considered in an appendix. We particularly focus on the cigar trap, which Van Druten and Ketterle first showed had a two-step condensation: a GBEC into a band of states at a temperature Tc and another "one-dimensional" transition at a lower temperature T1 into the ground state. In a thermodynamic limit in which the ratio of the dimensions of the anisotropic harmonic trap is kept fixed, T1 merges with the upper transition, which then becomes a normal BEC. However, in the thermodynamic limit of Beau and Zagrebnov, in which the ratio of the boundary lengths increases exponentially, T1 becomes fixed at the temperature of a true Type I phase transition. The effects of interactions on GBEC are discussed and we show that there is evidence that Type III condensation may have been observed in the cigar trap.

Precision measurements of temperature and chemical potential of quantum gases

“…This system is also analytically convenient because the average number of particles and the Fisher information relative to the chemical potential of the two-dimensional homogeneous gas have simple expressions (47,48). Note also the formal similarity with the ideal gas confined in a cigar-like harmonic potential [78,87,88,90,91], because both the two-dimensional ideal gas in a box potential and the one-dimensional ideal gas in a harmonic potential have a constant density of states.…”

“…Thus, the contributions of the condensate must be singled out from the integral (41) in the thermodynamic averages. For instance, the average number of particles is [90,91] …”

“…We emphasize that the one-dimensional ideal gas trapped in a harmonic potential has mathematically similar properties as the two dimensional ideal gas in an infinite square potential. In particular, a three-dimensional harmonically trapped gas with frequencies ω x,y,z in the three directions can be confined to the x direction if ω x ω y,z , as studied in [78,87,88,90,91]. The equations for the average number of particles and the Fisher matrix in the condensed phase are the sum of three-and one-dimensional contributions, similarly to the above discussed case of homogeneous gas.…”

Quantum-enhanced measurements without entanglement

“…A hierarchy of condensations to subsequent lower-dimensional gases is possible. These Bose-Einstein condensates have been studied both at finite size and in the thermodynamic limit focusing on mathematical structures and general properties (Beau and Zagrebnov, 2010;van den Berg, 1983;van den Berg and Lewis, 1982;van den Berg et al, 1986a,b;Casimir, 1968;van Druten and Ketterle, 1982;Girardeau, 1960Girardeau, , 1965Ketterle and van Druten, 1982;Krueger, 1968;Mullin, 1997;Mullin and Sakhel, 2012;Rehr, 1970;Zobay and Garraway, 2004), in connection with liquid helium in thin films (Douglass et al, 1964;Goble and Trainor, 1965, 1966, 1967Khorana and Douglass, 1965;Mills, 1964;Osborne, 1949), magnetic flux of superconducting rings (Sonin, 1969), and gravito-optical traps (Wallis, 1996). Experimental realizations of effective low dimensional Bose-Einstein condensates with trapped atoms have been reported in (van Amerongen, 2008;van Amerongen et al, 2008;Armijo et al, 2011;Bouchoule et al, 2011;Esteve et al, 2006;Görlitz et al, 2001;Greiner et al, 2001).…”

Bose–Einstein condensation in the three-sphere and in the infinite slab: Analytical results

“…Actually, it is well known that the one-dimensional box does not possess BEC, but once we have all the mathematical apparatus set up, the study of this system requires no extra effort. The infinite slab is an example of generalized BEC, in which the particles condense into a low lying set of states rather than the ground state, as described early on in [3] (and which the work in [4] already hinted at) and later systematically studied by van den Berg and collaborators [5,6,7] (see [8,9] for recent articles on this topic). From the systems we study here, the only one having the usual form of BEC is the three-sphere.…”