Abstract
The variance of the position operator is associated with how wide or narrow a wave-packet is, the momentum variance is similarly correlated with the size of a wave-packet in momentum space, and the angular-momentum variance quantifies to what extent a wave-packet is non-spherically symmetric. We examine an interacting three-dimensional trapped Bose–Einstein condensate at the limit of an infinite number of particles, and investigate its position, momentum, and angular-momentum anisotropies. Computing the variances of the three Cartesian components of the position, momentum, and angular-momentum operators we present simple scenarios where the anisotropy of a Bose–Einstein condensate is different at the many-body and mean-field levels of theory, despite having the same many-body and mean-field densities per particle. This suggests a way to classify correlations via the morphology of 100% condensed bosons in a three-dimensional trap at the limit of an infinite number of particles. Implications are briefly discussed.
Highlights
There has been an increasing interest in the theory and properties of trapped Bose–Einstein condensates at the limit of an infinite number of particles [1,2,3,4,5,6,7,8,9,10,11,12]
The first group of research questions deals with rigorous results, mainly proving when many-body and mean-field, Gross–Pitaevskii theories coincide at this limit, whereas the second group of questions deals with characterizing correlations in a trapped Bose–Einstein condensate based on the difference between many-body and mean-field properties at the infiniteparticle-number limit
Summarizing, we have demonstrated in a simple scenario, of an out-of-equilibrium breathing dynamics, the emergence of anisotropy classes other than {1}, namely, {1, 2} and {1, 2, 3}, for the many-particle position as well as manyparticle momentum operators of a trapped Bose–Einstein condensate at the limit of an infinite number of particles
Summary
There has been an increasing interest in the theory and properties of trapped Bose–. Einstein condensates at the limit of an infinite number of particles [1,2,3,4,5,6,7,8,9,10,11,12]. The difference between many-body and mean-field theories at the limit of an infinite number of particles, which as stated above coincide at the level of the energy, densities, and reduced density matrices per particle, starts to show up in variances of many-particle observables [6,7]. The position variance per particle in a thin annulus can exhibit a different dimensionality [17] and both the position and momentum variances can exhibit opposite anisotropies [18] when computed at the many-body and mean-field levels of theory in an out-of-equilibrium quench dynamics. The available permutations between the three Cartesian components of a many-particle operator, such as the position and momentum operators, allow one for various different anisotropies of the respective mean-field and many-body variances than in two spatial dimensions [18].
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