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.