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Scientific Reports

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

Y. Z. He, Yimin Liu, Bao Cheng-guang

Published Jan 22, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.1038/s41598-024-83865-8

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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.
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 SAS_ASA​, SBS_BSB​ and SCS_CSC​ of the three species, the spin-textures of the ground state can be classified into two types. In Type-I the three spins are either parallel or anti-parallel to each others, while in Type-II they point to different directions but remain to be coplanar. Moreover, according to the magnitudes of SAS_ASA​, SBS_BSB​ and SCS_CSC​ the spin-textures can be further classified into four kinds, namely, ppp+ppp+ppp (all atoms of each species are in singlet-pairs), one species in fff (fully polarized) and two species in qqq (a mixture of polarized atoms and singlet-pairs), two in fff and one in qqq, and fff+fff+fff. Other combinations are not allowed. The scopes of the parameters that supports a specific spin-texture have been specified. A number of spin-texture-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 predicted.
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.
Bose Einstein condensates of spin-1 atoms are known to exist in two different phases, both having spontaneously broken spin-rotation symmetry, a ferromagnetic and a polar condensate. Here we show that in two spatial dimensions it is possible to achieve a quantum phase transition from a polar condensate into a singlet phase symmetric under rotations in spin space. This can be done by using particle density as a tuning parameter. Starting from the polar phase at high density the system can be tuned into a strong-coupling intermediate-density point where the phase transition into a symmetric phase takes place. By further reducing the particle density the symmetric phase can be continuously deformed into a Bose-Einstein condensate of singlet atomic pairs. We calculate the region of the parameter space where such a molecular phase is stable against collapse.
We study the ground state and classify its phase diagram for a mixture of two spin-1 condensates in the absence of an external magnetic (B)(B)(B) field according to atomic parameters for intra- and interspecies spin-exchange coupling and singlet-pairing interaction. Ignoring the interspecies singlet-pairing interaction, the ground-state phases are found analytically. A numerical approach of simulated annealing is adopted when the singlet-pairing interaction is present. Our results on the phase diagram and the boundaries between phases allow for easy identifications of quantum phase transitions, which can be induced through the tuning of optical traps and atom numbers. They provide insight and guidance for several ongoing experiments on mixtures of spinor condensates.
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 study resonantly-paired s-wave superfluidity in a degenerate gas of two species (hyperfine states labeled by ↑, ↓) of fermionic atoms when the numbers N↑ and N↓ of the two species are unequal, i.e., the system is "polarized." We find that the continuous crossover from the Bose-Einstein condensate (BEC) limit of tightly-bound diatomic molecules to the Bardeen-Cooper-Schrieffer (BCS) limit of weakly correlated Cooper pairs, studied extensively at equal populations, is interrupted by a variety of distinct phenomena under an imposed population difference ΔN ≡ N↑ - N↓. Our findings are summarized by a "polarization" (ΔN) versus Feshbach-resonance detuning (δ) zero-temperature phase diagram, which exhibits regions of phase separation, a periodic FFLO superfluid, a polarized normal Fermi gas and a polarized molecular superfluid consisting of a molecular condensate and a fully polarized Fermi gas. We describe numerous experimental signatures of such phases and the transitions between them, in particular focusing on their spatial structure in the inhomogeneous environment of an atomic trap. © 2006 Elsevier Inc. All rights reserved.