A Note on State Decomposition Independent Local Invariants

We derive a set of invariants under local unitary transformations for arbitrary dimensional quantum systems. These invariants are given by hyperdeterminants and independent from the detailed pure state decompositions of a given quantum state. They also give rise to necessary conditions for the equivalence of quantum states under local unitary transformations

Validity of single-channel model for a spin-orbit coupled atomic Fermi gas near Feshbach resonances

We theoretically investigate a Rashba spin-orbit coupled Fermi gas near Feshbach resonances, by using mean-field theory and a two-channel model that takes into account explicitly Feshbach molecules in the close channel. In the absence of spin-orbit coupling, when the channel coupling $g$ between the closed and open channels is strong, it is widely accepted that the two-channel model is equivalent to a single-channel model that excludes Feshbach molecules. This is the so-called broad resonance limit, which is well-satisfied by ultracold atomic Fermi gases of $^{6}$Li atoms and $^{40}$K atoms in current experiments. Here, with Rashba spin-orbit coupling we find that the condition for equivalence becomes much more stringent. As a result, the single-channel model may already be insufficient to describe properly an atomic Fermi gas of $^{40}$K atoms at a moderate spin-orbit coupling. We determine a characteristic channel coupling strength $g_{c}$ as a function of the spin-orbit coupling strength, above which the single-channel and two-channel models are approximately equivalent. We also find that for narrow resonance with small channel coupling, the pairing gap and molecular fraction is strongly suppressed by SO coupling. Our results can be readily tested in $^{40}$K atoms by using optical molecular spectroscopy.Comment: 6 pages, 6 figure