We have found a stable spin-structure of N [Formula: see text] 3 system in which three spin-3 atoms are trapped and coupled to total spin S [Formula: see text] 3. We have proved that a pair of this structure is nearly an exact solution for N [Formula: see text] 6 systems in a very broad district in the parameter-space. Comparing with the well-known singlet pairs, this pair is a more promising candidate to serve as a building block for large N systems with spin-3 atoms. This is because the spin-structure of the latter can be modified depending on the interactions to reduce the energy while the singlet pairs can not. In fact, we have proved that, for a specific set of strengths (a point in the parameter-space) the product state based on this pair is an exact solution of the N-body Hamiltonian. Thus, in the neighborhood of this point, the product state will appear as an approximate solution. However, how broad this neighborhood would be remains to be clarified.
We have found a stable spin-structure of N [Formula: see text] 3 system in which three spin-3 atoms are trapped and coupled to total spin S [Formula: see text] 3. We have proved that a pair of this structure is nearly an exact solution for N [Formula: see text] 6 systems in a very broad district in the parameter-space. Comparing with the well-known singlet pairs, this pair is a more promising candidate to serve as a building block for large N systems with spin-3 atoms. This is because the spin-structure of the latter can be modified depending on the interactions to reduce the energy while the singlet pairs can not. In fact, we have proved that, for a specific set of strengths (a point in the parameter-space) the product state based on this pair is an exact solution of the N-body Hamiltonian. Thus, in the neighborhood of this point, the product state will appear as an approximate solution. However, how broad this neighborhood would be remains to be clarified.
The spin-textures of bound medium-body systems with spin-[Formula: see text] atoms ([Formula: see text]) have been studied. The Hamiltonian is assumed to be dominated by the two-body interaction favoring parallel spins. The system with particle number [Formula: see text] and [Formula: see text] is first chosen, and the Hamiltonian is exactly diagonalized by using Fock-states as basis-states, thereby all the eigenenergies and eigenstates are obtained and a detailed analysis is made. Then the cases with [Formula: see text] and [Formula: see text] are further studied. Since the total spin S is conserved, the eigenstates having the same S form an S-group. Let the lowest (highest) energy state of an S-group be called a bottom-state (top-state). We found that all the bottom-states are bipartite product states with constituent states describing fully polarized subsystems containing [Formula: see text] and [Formula: see text] ([Formula: see text]) particles, respectively. For two bottom-states different in [Formula: see text], the one with a larger [Formula: see text] is higher. For two having the same [Formula: see text], the one with a smaller S is higher. Whereas all the top-states are found to be essentially a product state of the pairs, in each pair the two spins are coupled to [Formula: see text] if the strength of the [Formula: see text]-channel is more repulsive than the others. For the states belonging to an S-group, the higher one would contain more pieces. As the energy goes up, larger pieces (those containing more than two particles) will disappear.
We consider the case that four spin-3 atoms are confined in an optical trap. The temperature is so low that the spatial degrees of freedom have been frozen. Exact numerical and analytical solutions for the spin-states have been both obtained. Two kinds of phase-diagrams for the ground states (g.s.) have been plotted. In general, the eigen-states with the total-spin S (a good quantum number) can be expanded in terms of a few basis-states [Formula: see text]. Let [Formula: see text] be the probability of a pair of spins coupled to [Formula: see text], and 6 in the [Formula: see text] state. Obviously, when the strength [Formula: see text] of the [Formula: see text]-channel is more negative, the basis-state with the largest [Formula: see text] would be more preferred by the g.s.. When two strengths are more negative, the two basis-states with the two largest probabilities would be more important components. Thus, based on the probabilities, the spin-structures (described via the basis-states) can be understood. Furthermore, all the details in the phase-diagrams, say, the critical points of transition, can also be explained. Note that, for [Formula: see text], [Formula: see text] is completely determined by symmetry. Thus, symmetry plays a very important role in determining the spin-structure of the g.s..
Abstract The spin-textures of trapped medium-body cold systems of spin-4 atoms with the interaction dominated by the attractive λ = 4 channel (the two interacting spins are coupled to 4) have been studied via an exact diagonalization of the spin-dependent Hamiltonian under the single spatial mode approximation. Particle number N (3 ≤ N ≤ 13) is assumed. The eigenstates have been analyzed to find out the details of spin-structures and the inherent excitation modes, and the relation between all of them. The emphasis is to find out how the spectra depend on N. We found that the singlet-triplets (a 3-body subsystem where the three spins are coupled to zero) and various excited triplets are basic constituents and all the eigenstates are composed of them. For low-lying states, only a few particles are involved in the excitation modes, while the other particles would 3-by-3 keep themselves in singlet-triplets.
Abstract We have performed a quantum mechanic calculation (including solving the coupled Gross-Pitaevskii equations to obtain the spatial wave functions, and diagonalizing the spin-dependent Hamiltonian in the spin-space to obtain the total spin state) together with an analytical analysis based on a classical model. Then, according to the relative orientations of the spins SA, SB and SC of the three species, the spin-structures of the ground state can be classified into two types. In Type-I the three spins are either parallel or anti-parallel to each other, while in Type-II they point to different directions but remain to be coplanar. Moreover, according to the magnitudes of SA, SB and SC, the spin-structures can be further classified into four kinds, namely, p + p + p (all atoms of each species are in singlet-pairs), one species in f (fully polarized) and two species in q (a mixture of polarized atoms and singlet-pairs), two in f and one in q, and f + f + f. Other combinations are not allowed. The scopes of the parameters that supports a specific spin-structure have been specified. A number of spin-structure-transitions have been found. For Type-I, the critical values at which a transition takes place are given by simple analytical formulae, therefore these values can be predict.
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.