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arXiv:1307.0744 · New Journal of Physics

Spin fragmentation of Bose–Einstein condensates with antiferromagnetic interactions

L. De Sarlo, Lingxuan Shao, Vincent Le Corre, Tilman Zibold, David Jacob, Jean Dalibard, Fabrice Gerbier

Published Nov 19, 2013• 23 citations

Abstract

We study spin fragmentation of an antiferromagnetic spin 1 condensate in the presence of a quadratic Zeeman (QZ) effect breaking spin rotational symmetry. We describe how the QZ effect turns a fragmented spin state, with large fluctuations of the Zeeman populations, into a regular polar condensate, where the atoms all condense in the m = 0 state along the field direction. We calculate the average value and variance of the population of the Zeeman state m = 0 to illustrate clearly the crossover from a fragmented to an unfragmented state. The typical width of this crossover is q ∼ k B T / N , where q is the QZ energy, T the spin temperature and N the atom number. This shows that the spin fluctuations are a mesoscopic effect that will not survive in the thermodynamic limit N → ∞ , but are observable for a sufficiently small atom number.

DOI: 10.1088/1367-2630/15/11/113039

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We study spin fragmentation of an antiferromagnetic spin 1 condensate in the presence of a quadratic Zeeman (QZ) effect breaking spin rotational symmetry. We describe how the QZ effect turns a fragmented spin state, with large fluctuations of the Zeeman populations, into a regular polar condensate, where the atoms all condense in the m = 0 state along the field direction. We calculate the average value and variance of the population of the Zeeman state m = 0 to illustrate clearly the crossover from a fragmented to an unfragmented state. The typical width of this crossover is q ∼ k B T / N , where q is the QZ energy, T the spin temperature and N the atom number. This shows that the spin fluctuations are a mesoscopic effect that will not survive in the thermodynamic limit N → ∞ , but are observable for a sufficiently small atom number.

A Bose-Einstein condensate produced by a Hamiltonian which is rotationally or translationally symmetric may be fragmented as a direct result of these symmetries. A corresponding mean-field unfragmented state, with an identical energy to leading order in the number of particles, can generally be constructed. As a consequence, vanishingly weak symmetry-breaking perturbations destabilize the fragmented state, which would thus be extremely difficult to realize experimentally and lead to an unfragmented condensate.

We investigate dynamical behavior of fragmentation of a spinor Bose-Einstein condensate in both the semiclassical and quantum regimes. We show that an unfragmented condensate can become fragmented at later times due to the inherent interactions of elastic atomic collisions, when a certain phase relationship exists among the state amplitudes in different components initially. A close relationship between the fragmentation and population dynamics of atomic distributions within different spin components is found for certain initial states with zero magnetization. The type of fragmentation regime is characterized via atom-number fluctuations. We also address complications of our study when the condensate spatial mode function is different for each spin component.

We study experimentally the equilibrium phase diagram of a spin-1 Bose-Einstein condensate with antiferromagnetic interactions, in a regime where spin and spatial degrees of freedom are decoupled. For a given total magnetization

{m}_{z}

, we observe for low magnetic fields an ``antiferromagnetic'' phase where atoms condense in the $m=\ifmmode\pm\else\textpm\fi{}1$ Zeeman states, and occupation of the

m=0

state is suppressed. Conversely, for large enough magnetic fields, a phase transition to a ``broken-axisymmetry'' phase takes place: The

m=0

component becomes populated and rises sharply above a critical field

{B}_{c}({m}_{z})

. This behavior results from the competition between antiferromagnetic spin-dependent interactions (dominant at low fields) and the quadratic Zeeman energy (dominant at large fields). We compare the measured

{B}_{c}

as well as the global shape of the phase diagram with mean-field theory, and find good quantitative agreement.

We observe multistep condensation of sodium atoms with spin F=1, where the different Zeeman components m_{F}=0,±1 condense sequentially as the temperature decreases. The precise sequence changes drastically depending on the magnetization m_{z} and on the quadratic Zeeman energy q (QZE) in an applied magnetic field. For large QZE, the overall structure of the phase diagram is the same as for an ideal spin-1 gas, although the precise locations of the phase boundaries are significantly shifted by interactions. For small QZE, antiferromagnetic interactions qualitatively change the phase diagram with respect to the ideal case, leading, for instance, to condensation in m_{F}=±1, a phenomenon that cannot occur for an ideal gas with q>0.