Limit of spin squeezing in condensates

Interactions between atoms with two internal states within a condensate allow to create dynamically quantum correlations that might be useful for metrology (spin squeezing). For the first time, with a multimode description, we have determined the maximum spin squeezing that can be obtained in a non zero temperature condensate by this method. The results differ strongly from those of the two mode theory currently used.
  Spin squeezing is a very active field that knew recently significant experimental achievements. To quantify the metrology gain, one introduces a spin squeezing parameter : ξ2. The smaller ξ2, the larger the metrology gain due to quantum correlations. A state is squeezed if ξ2<1, while ξ2=1 corresponds to an uncorrelated state.

A crucial question is the behavior of ξ2 with the atom number N for large N. While the two mode theory predicts ξ2 ≈ 1/N2/3 for N->, we find that ξ2 is non zero in the thermodynamic limit  : it is the product of (ρa3)1/2 and a universal function of kBT/(ρg) that we calculate explicitly for an homogeneous system. Here ρg is the T=0 mean field chemical potential, ρ is the spatial density, g=4πℏ2a/m is the coupling constant and a is the scattering length.

Figure : spin squeezing as a function of the temperature in the thermodynamic limit
Spin squeezing parameter and non condensed fraction, divided by (ρa3)1/2, as a function of kBT/(ρg). Here ρg is the T=0 mean field chemical potential.
For kBT/(ρg) >> 1, the spin squeezing parameter ξ2 approaches the non condensed fraction in the system.

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