Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Scientific Reports

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

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

Published Feb 17, 2020• 6 citations
Publisher

Abstract

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.

DOI
10.1038/s41598-020-59540-z

Related papers

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

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

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 $S_A$, $S_B$ and $S_C$ 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 $S_A$, $S_B$ and $S_C$ the spin-textures 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-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.

arXiv:1910.04381•Oct 10, 2019•0 citations

Bose-Einstein condensate of a mixture of two species of spin-1 atoms

Ma Luo, Zhibing Li, Bao Cheng-guang

The interaction between two species of atoms in the condensate has been deduced and the trial wave functions that conserve the total spin have been constructed for the states of the ground band. A couple of generalized Gross-Pitaevskii equations have been derived for the spatial wave functions and the energies. The mixture of $^{23}\mathrm{Na}$ and $^{87}\mathrm{Rb}$ has been calculated as an example. The density of states is found to exhibit a ladder shape.

Physical Review A•Apr 12, 2007•37 citations

Polarized entangled Bose-Einstein condensation

Jinlong Wang, Yu Shi

We consider a mixture of two distinct species of atoms of pseudospin $\frac{1}{2}$ with both intraspecies and interspecies spin-exchange interactions and find all the ground states in a general case of the parameters in the effective Hamiltonian. In general, corresponding to the two-species and two-pseudospin states, there are four orbital wave functions into which the atoms condense. We find that in certain parameter regimes, the ground state is the so-called polarized entangled Bose-Einstein condensation; that is, in addition to condensation of interspecies singlet pairs, there are unpaired atoms with spins polarized in the same direction. The interspecies entanglement and polarization significantly affect the generalized Gross-Pitaevskii equations governing the four orbital wave functions into which the atoms condense, as an interesting interplay between spin and orbital degrees of freedom.

arXiv:1103.1021•Physical Review A•Dec 30, 2010•2 citations

Binary mixture of spinor atomic Bose-Einstein condensates

Zhi-Fang Xu, Yunbo Zhang, Li You

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 $(B)$ 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.

arXiv:0901.1505•Physical Review A•Feb 10, 2009•31 citations

Variation of the spin textures of 2-species spin-1 condensates studied beyond the single spatial mode approximation and the experimental identification of these textures

Y Z He, Y M Liu, Bao Cheng-guang

Abstract Based on the numerical solutions of the coupled Gross–Pitaevskii equations, the spin-textures of the ground state of a Bose–Einstein condensate with two kinds of spin-1 atoms have been studied. Besides, making use of the fractional parentage coefficients, the probabilities of an atom in spin component μ and of two correlated atoms one in μ and one in ν have been calculated. The previous results under the single spatial mode approximation (SMA) are found to be qualitatively consistent with the present results beyond the SMA. The variation of the probabilities against the input parameters is found to match the corresponding variation of the spin-textures excellently. In particular, once a spin-texture transition occurs, the associated probabilities will change suddenly. Since the probabilities are observable, the observation provides a way for the experimental identification of the spin-textures. Besides, sine the critical points (where a transition occurs) can be accurately identified, information on the input parameters (say, the very weak spin-dependent forces) can be extracted.

arXiv:1809.06617•Physica Scripta•Apr 4, 2019•4 citations

Vortex structure in spinor F=2 Bose-Einstein condensates

W. V. Pogosov, R. Kawate, Takeshi Mizushima +1 more

Extended Gross-Pitaevskii equations for the rotating $F=2$ condensate in a harmonic trap are solved both numerically and variationally using trial functions for each component of the wave function. Axially symmetric vortex solutions are analyzed and energies of polar and cyclic states are calculated. The equilibrium transitions between different phases with changing of the magnetization are studied. We show that at high magnetization the ground state of the system is determined by interaction in ``density'' channel, and at low magnetization spin interactions play a dominant role. Although there are five hyperfine states, all the particles are always condensed in one, two, or three states. Two interesting types of vortex structures are also discussed.

arXiv:cond-mat/0601539•Physical Review A•Dec 5, 2005•29 citations
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.
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.
The interaction between two species of atoms in the condensate has been deduced and the trial wave functions that conserve the total spin have been constructed for the states of the ground band. A couple of generalized Gross-Pitaevskii equations have been derived for the spatial wave functions and the energies. The mixture of 23Na^{23}\mathrm{Na}23Na and 87Rb^{87}\mathrm{Rb}87Rb has been calculated as an example. The density of states is found to exhibit a ladder shape.
We consider a mixture of two distinct species of atoms of pseudospin 12\frac{1}{2}21​ with both intraspecies and interspecies spin-exchange interactions and find all the ground states in a general case of the parameters in the effective Hamiltonian. In general, corresponding to the two-species and two-pseudospin states, there are four orbital wave functions into which the atoms condense. We find that in certain parameter regimes, the ground state is the so-called polarized entangled Bose-Einstein condensation; that is, in addition to condensation of interspecies singlet pairs, there are unpaired atoms with spins polarized in the same direction. The interspecies entanglement and polarization significantly affect the generalized Gross-Pitaevskii equations governing the four orbital wave functions into which the atoms condense, as an interesting interplay between spin and orbital degrees of freedom.
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.
Abstract Based on the numerical solutions of the coupled Gross–Pitaevskii equations, the spin-textures of the ground state of a Bose–Einstein condensate with two kinds of spin-1 atoms have been studied. Besides, making use of the fractional parentage coefficients, the probabilities of an atom in spin component μ and of two correlated atoms one in μ and one in ν have been calculated. The previous results under the single spatial mode approximation (SMA) are found to be qualitatively consistent with the present results beyond the SMA. The variation of the probabilities against the input parameters is found to match the corresponding variation of the spin-textures excellently. In particular, once a spin-texture transition occurs, the associated probabilities will change suddenly. Since the probabilities are observable, the observation provides a way for the experimental identification of the spin-textures. Besides, sine the critical points (where a transition occurs) can be accurately identified, information on the input parameters (say, the very weak spin-dependent forces) can be extracted.
Extended Gross-Pitaevskii equations for the rotating F=2F=2F=2 condensate in a harmonic trap are solved both numerically and variationally using trial functions for each component of the wave function. Axially symmetric vortex solutions are analyzed and energies of polar and cyclic states are calculated. The equilibrium transitions between different phases with changing of the magnetization are studied. We show that at high magnetization the ground state of the system is determined by interaction in ``density'' channel, and at low magnetization spin interactions play a dominant role. Although there are five hyperfine states, all the particles are always condensed in one, two, or three states. Two interesting types of vortex structures are also discussed.