Phase Diagram of a 1-D Spin-Imbalanced Fermi Gas*

Spin-imbalance in a quantum gas typically destroys superfluidity, but theory suggests that there exists an exotic superfliud state (FFLO) where pairs form with finite center-of-mass momentum. In 3D, the FFLO state is predicted to occupy only a small region of the phase diagram, while in 1D the FFLO state is pervasive\footnote{K. Yang, \textit{Phys. Rev. B.} \textbf{63}, 140511(R) (2001); G. Orso, \textit{Phys. Rev. Lett.} \textbf{98}, 070402 (2007).}. We create a spin-imbalanced Fermi gas of $^{6}$Li in 1D tubes by using a 2-D optical lattice. The central region of each tube is a partially spin-polarized gas surrounded by wings of either a fully polarized or fully paired gas depending on the spin imbalance\footnote{Y. Liao \textit{et al.}, arXiv:0912.0092 (2009).}. The phase diagram is a function of polarization and chemical potential extracted from \textit{in situ} images of the spatial distribution. This diagram is well described by finite temperature Bethe Ansatz theory, suggesting that these partially polarized regions are the 1D equivalent of FFLO states.