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arXiv:cond-mat/0409197 · Physical Review A

Ground band and a generalized Gross-Pitaevskii equation for spinor Bose-Einstein condensates

Bao Cheng-guang, Zhibing Li

Published Oct 20, 2004• 22 citations
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Abstract

For spinor Bose-Einstein condensates both the total spin $S$ and its $Z$ component ${S}_{Z}$ should be conserved. However, in existing theories, only the conservation of ${S}_{Z}$ has been taken into account. To remedy this, here we attempt to take the conservation of both $S$ and ${S}_{Z}$ into account. For this purpose, a total spin state with the good quantum numbers $S$ and ${S}_{Z}$ is introduced in the trial wave function; thereby a generalized Gross-Pitaevskii equation has been derived. With this equation, the ground bands of the $^{23}\mathrm{Na}$ and $^{87}\mathrm{Rb}$ condensates have been studied, where the levels that are distinct in $S$ split. It was found that the level density is extremely dense in the bottom of the ground band of $^{23}\mathrm{Na}$, i.e., in the vicinity of the ground state. On the contrary, for $^{87}\mathrm{Rb}$, the levels are extremely dense in the top of the ground band.

DOI
10.1103/physreva.70.043620

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For spinor Bose-Einstein condensates both the total spin SSS and its ZZZ component SZ{S}_{Z}SZ​ should be conserved. However, in existing theories, only the conservation of SZ{S}_{Z}SZ​ has been taken into account. To remedy this, here we attempt to take the conservation of both SSS and SZ{S}_{Z}SZ​ into account. For this purpose, a total spin state with the good quantum numbers SSS and SZ{S}_{Z}SZ​ is introduced in the trial wave function; thereby a generalized Gross-Pitaevskii equation has been derived. With this equation, the ground bands of the 23Na^{23}\mathrm{Na}23Na and 87Rb^{87}\mathrm{Rb}87Rb condensates have been studied, where the levels that are distinct in SSS split. It was found that the level density is extremely dense in the bottom of the ground band of 23Na^{23}\mathrm{Na}23Na, i.e., in the vicinity of the ground state. On the contrary, for 87Rb^{87}\mathrm{Rb}87Rb, the levels are extremely dense in the top of the ground band.
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.
In this paper, we systematically review mathematical models, theories and numerical methods for ground states and dynamics of spinor Bose-Einstein condensates (BECs) based on the coupled Gross-Pitaevskii equations (GPEs). We start with a pseudo spin-1/2 BEC system with/without an internal atomic Josephson junction and spin-orbit coupling including (i) existence and uniqueness as well as non-existence of ground states under different parameter regimes, (ii) ground state structures under different limiting parameter regimes, (iii) dynamical properties, and (iv) efficient and accurate numerical methods for computing ground states and dynamics. Then we extend these results to spin-1 BEC and spin-2 BEC. Finally, extensions to dipolar spinor systems and/or general spin-F (F>=3) BEC are discussed.
We demonstrate that the ground state of a trapped spin-1 and spin-2 spinor ferromagnetic Bose-Einstein condensate (BEC) can be well approximated by a single decoupled Gross-Pitaevskii (GP) equation. Useful analytic models for the ground-state densities of ferromagnetic BECs are obtained from the Thomas-Fermi approximation (TFA) to this decoupled equation. Similarly, for the ground states of spin-1 antiferromagnetic and spin-2 antiferromagnetic and cyclic BECs, some of the spin-component densities are zero, which reduces the coupled GP equation to a simple reduced form. Analytic models for ground-state densities are also obtained for antiferromagnetic and cyclic BECs from the TFA to the respective reduced GP equations. The analytic densities are illustrated and compared with the full numerical solution of the GP equation with realistic experimental parameters.
This PhD thesis contains new results on the mathematical study of Bose-Einstein condensation and the main part of it is devoted to mixtures of condensates, i.e., systems composed of multiple bosonic species in interaction. We prove the validity of effective ground state theories for mixtures in the Gross-Pitaevskii and mean-field regime. We show that the ground state energy asymptotics, in the large-N limit, is captured by the minimum of a suitable one-body functional. Moreover, we prove that in the ground state all species exhibit Bose-Einstein condensation onto the minimizer of that functional. For mixtures in the mean-field regime, we provide a rigorous justification of Bogoliubov’s theory. This is done by computing the contribution to the ground state energy which is due to excited particles. We also prove a norm approximation for the ground state vector, in the Fock space norm. From the time-dependent viewpoint, we prove for the first time the validity of the effective equations that were previously known due to heuristic physical arguments, and that are confirmed by robust experimental evidence. Our results show that, for mixtures in the Gross-Pitaevskii and mean-field regime, the effective dynamics is governed by a system of non-linear Schrödinger equations, one for each species of the mixture. In the final part of the thesis we present additional results on problems and models related to the study on mixtures. We were able to derive the effective dynamics for spinor- and pseudo-spinor condensates. The equations that we obtain are precisely those of modern experiments with ultra-cold spin bosons. We also show that the mean-field model provide a time-dependent control of condensation that is very accurate for the typical duration times of experiments. A further result is the global well-posedness in the energy space of the singular Hartree equation. Last, we present new remarks on the adaptation of known techniques that one needs in order to prove the derivation of the magnetic Gross-Pitaevskii equation.
The spatial structure of the spinor Bose-Einstein condensates with the spin degrees of freedom is analyzed based on the generalized Gross-Pitaevskii (GP) equation in light of the present spin-domain experiment on ${m}_{F}=\ifmmode\pm\else\textpm\fi{}1,$ and 000 of the hyperfine state F=1F=1F=1 of 23Na{}^{23}\mathrm{Na}23Na atom gases. The GP solutions in three- and one-spatial dimensional cases reproduce the observed spin-domain structures, revealing the length scale associated with the existence of the weak interaction of the spin-spin channel, other than the ordinary coherence length related to the density-density channel. The obtained domain structure in GP is compared with the result in the Thomas-Fermi approximation. The former solution is found to better describe the observed features than the latter.