First excited band of a spinor Bose-Einstein condensate

Abstract

The analytical expression of the fractional parentage coefficients for the total spin-states of a spinor N-boson system has been derived. Thereby an S-conserved theory for the spinor BoseEinstein condensation has been proposed. A set of equations has been established to describe the first excited band of the condensates. Numerical solution for 23 Na has been given as an example.PACS numbers: 03.75. Fi, 03.65. Fd In recent years, accompanying the experimental realization of the spinor Bose-Einstein condensation (BEC) [1,2,3,4,5], corresponding theories based on the mean field theory have been developed [6,7,8,9,10,11,12,13]. The total spin S together with its Z-component are actually conserved in the spinor condensates. Furthermore, the interactions are in general spin-dependent. Therefore, the strength of the mean field g that acts on the macroscopically occupied quantum states might depend on S. In previous theories, the S-dependence of the strength g has not yet been perfectly clarified. A primary attempt along this line was proposed in Ref. [14] where the total spin-states have been introduced and the S-dependence of g has been derived analytically.The present paper is a continuation of the Ref. [14]. It is noted that the thermodynamical property of condensates at low temperature depends on their low-lying spectra, this is an interesting topic scarcely touched. The first aim of the paper is to generalize the study of Ref.[14] from the ground band (GB) to excited bands. Rigorous total spin-states with good quantum numbers, S and S Z , and with a specific permutation symmetry (see below) will be introduced, a set of equations is derived to describe, as a first step, the first excited band (FEB) of the spinor BEC.When the total spin states are adopted, how to achieve the matrix elements among them is crucial. The second aim of this paper is to develop a general tool, namely the fractional parentage coefficients, to facilitate the theoretical treatment of spinor N-boson systems. The case of 23 Na as an example will be studied numerically. Before going ahead, we need some knowledge from few-boson systems.Let the interaction between a pair of spin-one bosons beWhere P m and P q are projectors to the two-body spin-states with spin S ij = 0 and 2 (spin 1 is irrelevant since in this case both the spin wavefunction and the spatial wavefunction must be antisymmetric that prevents a point interaction), respectively, g m and g q are the associated strengths, F i is the spin operator of particle i, c 0 = (g m + 2g q )/3 , and c 2 = (g q − g m )/3. Let the N bosons (with mass m) be confined by a harmonic trap with frequency ω. When ω and mω are used as units of energy and length, respectively, the HamiltonianUsing the method as given in [15], the Hamiltonians for N = 3 and 4 two-dimensional systems have been diagonalized. Both the orbital angular momentum L and S are good quantum numbers, therefore the eigenenergies and eigenstates can be denoted as E LS and Ψ LS (there is actually a series of states having the same L an...

Evolution of the average populations of spin components of spin-1 Bose-Einstein condensates beyond mean-field theory

“…By using the analytical forms of the fractional parentage coefficients and Clebesh-Gordan coefficients [16,18], particle 1 can be extracted from the total spin-state as…”

Spin-Thermodynamics of Ultra-Cold Spin-1 Atoms

“…Note that the width of a band depends on the interactions and depends on the number of states contained in the band. Thus, both the g-band and the first excited band will be broadened with N [14,21]. Accordingly, the upper part of the g-band might overlap the lower part of the first excited band, this leads to E top > E ex 1 .…”

“…For the evaluation of T 3 which marks the freezing of the spatial DoF, the excitation energy of the lowest spatial mode, denoted as E ex 1 , has been calculated using the theory given in [14,21]. Obviously, the weight of the state associated with E ex 1 is P ex,1 = e −β E ex 1 /Z .…”

“…A crucial point is to define a temperature T 3 so that all the spatial DoF can be considered as being frozen when T < T 3 . This can be achieved using the theory given in [14] (see below). It is reminded that T 3 is different from the usually defined transition temperature T c (The latter is so defined that, when T > T c , the population of the ground state (g.s.)…”

Sensitivity of spin structures of small spin-1 condensates toward a magnetic field studied beyond the mean field theory

“…On the other hand, for small condensates, a theory goes beyond the MFT might be more appropriate. In this paper, the spin-structures of spin-1 small condensates under a magnetic field B are studied by using a many-body theory in which the parentage coefficients of spin-states are used as a tool [10,11]. The variation of B is considered as adiabatic and the discussion is limited to static behavior.…”