A set of collective spin states is derived for a trapped Bose-Einstein condensate in which atoms have three internal hyperfine spins. These collective states minimize the interaction energy among condensate atoms, and they are characterized by strong spin correlations. We also examine the internal dynamics of an initially spin-polarized condensate. The time scale of spin mixing is predicted.[S0031-9007 (98)07921-6] PACS numbers: 03.75.Fi Bose-Einstein condensates (BEC) of atoms with internal degrees of freedom are new forms of macroscopically coherent matter which exhibit rich quantum structures. In the case of BEC with two internal spin states [1,2], theoretical studies have predicted interesting phenomena such as quantum entanglement of spins [3], suppression of quantum phase diffusion [4], and interference effects [5]. Recently, Stamper-Kurn et al. [6] have realized an optically trapped BEC in which all three hyperfine states in the lowest energy manifold of sodium atoms are involved. Such a three-component condensate raises new questions regarding the more complex ground state structure [7,8] and internal spin dynamics. One of the key features here is that there are spin exchange interactions which constantly mix different condensate spin components while the system as a whole remains in the ground state. For example, two atoms with respective hyperfine spins 11 and 21 interact and become two atoms with hyperfine spin 0. Therefore an important problem is to determine how atoms organize their spins in the ground state and how a spin-polarized BEC loses its polarization because of spin exchange interactions.In this paper we approach the questions using an algebraic method found in quantum optics. By excluding effects of noncondensate atoms, we identify the fact that the interaction between spin components in a BEC is analogous to 4-wave mixing in nonlinear optics. However, since the trap is like a matter wave cavity, a more appropriate optical analogy is the 4-wave mixing in a high finesse cavity (i.e., a cavity QED system). With the help of the methods developed in a related cavity QED problem [9,10], we are able to study the organization of spins in the condensate ground state. We find that there exists a class of quantum superposition states which minimize the interaction energy. These quantum states are recognized as collective spin states which are characterized by strong correlations among different spin components, and in some cases we find that the number of atoms in an individual spin component shows large fluctuations. In this paper we also examine the internal dynamics of the spin-mixing process arising from the nonlinear interactions between condensate atoms [11]. For an initially spin-polarized BEC, we predict the time scale at which spins become strongly mixed.To begin we consider a dilute gas of trapped bosonic atoms with hyperfine spin f 1. The second quantized Hamiltonian of the system is given by ͑h 1͒ H X a Z d 3 xĈ y a √ 2 = 2 2M 1 V T !Ĉ a 1 X a,b,m,n V abmn ZĈ y aĈ y bĈmĈn d 3 x , (1) wher...

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Quantum entanglement associated with transverse wave vectors of down conversion photons is investigated based on the Schmidt decomposition method. We show that transverse entanglement involves two variables: orbital angular momentum and transverse frequency. We show that in the monochromatic limit high values of entanglement are closely controlled by a single parameter resulting from the competition between (transverse) momentum conservation and longitudinal phase matching. We examine the features of the Schmidt eigenmodes, and indicate how entanglement can be enhanced by suitable mode selection methods.

We examine the quantum structure of continuum entanglement and in the context of short-pulse down-conversion we answer the open question of how many of the uncountably many frequency modes contribute effectively to the entanglement. We derive a set of two-photon mode functions that provide an exact, discrete, and effectively finite basis for characterizing pairwise entanglement. Our analysis provides a basis for entropy control in two-photon pulses generated from down-conversion.

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We consider the intrinsic stability of the vortex states of a pure Bose-Einstein condensate confined in a harmonic potential under the effects of coherent atom-atom interaction. We find that stable vortices can be supported, and that vortex stability can be controlled by changing the inter-particle interaction strength. At unstable regimes, a vortex will spontaneously disintegrate into states with different angular momenta even without external perturbations, with the lifetime determined by its imaginary excitation frequencies.pacs numbers: 03.75.Fi, 05.30.Jp