Large spin systems can exhibit unconventional types of magnetic ordering different from the ferromagnetic or Neel-like antiferromagnetic order commonly found in spin 1/2 systems. Spin-nematic phases, for instance, do not break time-reversal invarianc
e and their magnetic order parameter is characterized by a second rank tensor with the symmetry of an ellipsoid. Here we show direct experimental evidence for spin-nematic ordering in a spin-1 Bose-Einstein condensate of sodium atoms with antiferromagnetic interactions. In a mean field description this order is enforced by locking the relative phase between spin components. We reveal this mechanism by studying the spin noise after a spin rotation, which is shown to contain information hidden when looking only at averages. The method should be applicable to high spin systems in order to reveal complex magnetic phases.
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 Zee
mans populations, into a regular polar condensate, where atoms all condense in the $m=0$ state along the field direction. We calculate the average value and variance 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 $q sim k_B T/N$, where $q$ is the QZ energy, $T$ the spin temperature and $N$ the atom number. This shows that spin fluctuations are a mesoscopic effect that will not survive in the thermodynamic limit $Nrightarrow infty$, but are observable for sufficiently small atom number.
