Spin-structures of the Bose-Einstein condensates with three kinds of spin-1 atoms

A mechanism with positive feedback governing the transitions in super-low temperature physics studied via a two-species condensate with spin-1 atoms

The ground states of two-species condensates with spin-1 atoms have been studied analytically and numerically. All the results from the analytical approach are checked by the latter. The [Formula: see text] channel has been neglected, where λ is the coupled spin of two different atoms. A complete phase diagram has been plotted against the whole parameter space which has been divided into regions associated with different phases. All the boundaries separating the regions have been analytically obtained. Let p- denote the phase of a species in which all the atoms are two by two in singlet-pairs, f- denote that all the spins are lying along a common direction (fully polarized), and q- denote a mixture of the above two phases, namely, some singlet-pairs together with a fully polarized subsystem. Then, we found that the p-phase cannot coexist with the q- or f-phase (the p-f or f-p phase could exist only if the species in the f-phase is a few-body system). The q-q phase also does not exist. Thus, once a few atoms of a species are aligning (i.e., in the q-phase), the singlet-pairs of the other species must all be broken and become fully polarized. This is an important feature of multi-species spin-1 systems arising from the basic character of the inter-species interaction via a mechanism with positive feedback. Note that, when both species are in the p-phase, there is no special direction. However, once a few of A-(or B-)atoms are aligning, a special direction is established, and the inter-species interactions from them will be unified and thereby promote the spin-alignment in the other pieces, and vice versa. This is the positive feedback, that is, the alignments in both species are mutually promoted. This explains why a q-phase causes the full polarization of the other species. In particular, the transition from p-p to q-f, f-q or f-f is governed by this mechanism. It leads to a sudden breaking of all the singlet-pairs and is expected to be popular in various multi-species spin-1 systems in the neighborhood of temperature zero.

Properties of a trapped multiple-species bosonic mixture at the infinite-particle-number limit: A solvable model

We investigate a trapped mixture of Bose-Einstein condensates consisting of a multiple number of P species. To be able to do so, an exactly solvable many-body model is called into play. This is the P-species harmonic-interaction model. After presenting the Hamiltonian, the ground-state energy and wavefunction are explicitly calculated. All properties of the mixture's ground state can, in principle, be obtained from the many-particle wavefunction. A scheme to integrate the all-particle density matrix is derived and implemented, leading to closed-form expressions for the reduced one-particle density matrices. Of particular interest is the infinite-particle-number limit, which is obtained when the numbers of bosons are taken to infinity while keeping the interaction parameters fixed. We first prove that at the infinite-particle-number limit all the species are 100% condensed. The mean-field solution of the P-species mixture is also obtained analytically and is used to show that the energy per particle and densities per particle computed at the many-body level of theory boil down to their mean-field counterparts. Despite these, correlations in the mixture exist at the infinite-particle-number limit. To this end, we obtain closed-form expressions for the correlation energy, namely, the difference between the mean-field and many-body energies, and the depletion of the species, i.e., the number of particles residing outside the condensed modes, at the infinite-particle-number limit. The depletion and the correlation energy per species are shown to critically depend on the number of species. Of separate interest is the entanglement between one species of bosons and the other P - 1 species. This quantity is governed by the coupling of the center-of-mass coordinates of the species and is obtained by the respective Schmidt decomposition of the P-species wavefunction. Interestingly, there is an optimal number of species, here P = 3, where the entanglement is maximal. Importantly, the manifestation of this interspecies entanglement in an observable is possible. It is the position-momentum uncertainty product of one species in the presence of the other P - 1 species, which is derived and demonstrated to correlate with the interspecies entanglement. All in all, we show and explain how correlations at the infinite-particle-number limit of a trapped multiple-species bosonic mixture depend on the interactions and how they evolve with the number of species. Generalizations and implications are briefly discussed.