Bose-Einstein Condensation with Internal Degrees of Freedom in Alkali Atom Gases

Abstract

The Bogoliubov theory is extended to a Bose-Einstein condensation with internal degrees of freedom, realized recently in 23 Na gases where several hyperfine states are simultaneously cooled optically. Starting with a Hamiltonian constructed from general gauge and spin rotation symmetry principles, fundamental equations for condensate are derived. The ground state where time reversal symmetry is broken in some cases and low-lying collective modes, e.g. spin and density wave modes, are discussed. Novel vortex as a topological defect can be created experimentally. 4 He. One of the most notable differences in these two systems lies in their mutual interaction, either very weak in the former or very strong in the latter system. This enables us to construct a microscopic theory for BEC in the present system from first principles. In fact a mean-field theory or HartreeFock-Bogoliubov framework developed by Bogoliubov, 6)Gross, 7) Pitaevskii 8) and others has been quite successful in explaining fundamental physics of BEC realized in magnetically trapped atomic gases such as condensate fraction, transition temperature, or low-lying collective modes. Only the s-wave scattering length a is required to be fed into these microscopic theories as a material parameter, which itself is known rather accurately. 4, 5)Most experiments for BEC so far were performed by the magnetic traps which necessarily select one of several possible atomic ground states, or the so-called weak field seeking state, such as F z = −1 among F z = ±1, 0 of F = 1 in the 23 Na case, where the spin degrees of freedom are "frozen". Recently Stamper-Kurn et al.9) have succeeded in cooling 23 Na atoms purely optically in an optical dipole trap and achieved BEC, where the three substates F z = ±1, 0 are simultaneously "Bose-condensed". They demonstrated this by a "Stern-Gerlach" experiment: The F = 1 magnetic three sublevels are all visible as a split image by passing the condensate through a field gradient. This is remarkable. This finding opens an interesting possibility to explore BEC with internal degrees of freedom where not only the gauge symmetry U (1) but also "spin" symmetry SO(3) for the F = 1 case are involved, a situation similar to the superfluid 3 He problem where the orbital (l = 1) and spin (s = 1) degrees of freedom give rise to a 3 × 3 × 2 dimensional manifold of complex order parameter space. 10)The purposes of this paper are (1) to establish fundamental equations for describing such a situation, leading to an extended Gross-Pitaevskii equation, (2) to examine low-lying collective modes in order to extract basic physical properties of the ground state and finally (3) to point out several interesting possible topological excitations or textural spatial structures under special circumstances, e.g. when releasing BEC from an optically plugged quadrupole magnetic trap, a novel vortex may be created experimentally. Till date, no one has succeeded in observing a vortex even for one-component BEC although there have been several theoretical discuss...

Non-equilibrium dynamics of an unstable quantum pendulum explored in a spin-1 Bose–Einstein condensate

“…The experiment uses spin-1 atomic Bose condensates 6,18,19 with ferromagnetic interactions tightly confined in optical traps such that spin domain formation is energetically suppressed. In this case, the non-trivial dynamical evolution of the system occurs only in the internal spin variables, and the mean-field dynamics of the system can be described by a non-rigid pendulum similar to the two-site Bose-Hubbard model 20,21 .…”

Spinor Condensates and Light Scattering from Bose-Einstein Condensates

“…Such multi-component optically trapped condensates are represented by an order parameter which is a vector in hyperfine spin space, and are thus called spinor Bose-Einstein condensates. A variety of new phenomena are predicted for this new quantum fluid such as spin textures, spin waves, and coupling between atomic spin and superfluid flow [94][95][96].…”

“…The rotationally symmetric characterization of two-body collisions among atoms of hyperfine spin F 1 and F 2 can only depend on their total spin f = F 1 + F 2 and not on its orientation. Thus, in the s-wave limit, the interatomic interaction V int ( r 1 − r 2 ) is reduced to the form [94,95] …”

“…The spin-dependence of the interatomic interaction is thereby isolated in the rotationally-symmetric term g 2 F 1 · F 2 δ( r 1 − r 2 ). The Hamiltonian describing a weakly-interacting Bose gas of spin F = 1 now becomes [94,95] …”

“…Ho [94] and Ohmi and Machida [95] considered the ground state of a spinor condensate by an extension of mean-field theory (see also [101]). The N -particle condensate ground state is found by replacing the field operatorsΨ i in Eq.…”