Quantum Spins Mixing in Spinor Bose-Einstein Condensates

Abstract

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...

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

“…The experimental results are compared with a quantum calculation using the spin interaction Hamiltonian 22,[24][25][26]29 :…”

“…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 . The system is fully integrable in both the quantum 22 and classical 21,23 limits, and provides the opportunity to explore non-equilibrium quantum dynamics that are not captured by mean-field approaches, and can be solved exactly with Schrödinger's equation. The condensate is prepared in a state corresponding to a hyperbolic fixed point in the spin-nematic phase space.…”

“…The condensate is prepared in a state corresponding to a hyperbolic fixed point in the spin-nematic phase space. Although this state is non-evolving in the mean-field limit, the quantum solution yields intricate spin-mixing dynamics exhibiting non-linear quantum revivals for zero magnetic field 22 and a quantum carpet of highly non-Gaussian fluctuations 24 . At finite fields, the dynamics are similar 25,26 , although they occur on a timescale favourable for experimental observation.…”

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].…”

Extended two-site Bose–Hubbard model with pair tunneling: spontaneous symmetry breaking, effective ground state and fragmentation

“…This is a general relation that is also present in other fragmented systems. For example, in the antiferromagnetic ground state of spin-1 Bose gases [21][22][23], the quantum ground state is unique while its effective ground state has infinite degeneracy due to the intrinsic SU(2) symmetry. Thus the quantum ground state also corresponds to a fragmented condensate.…”