Collective mode damping and viscosity in a 1D unitary Fermi gas

We calculate the damping of the Bogoliubov-Anderson mode in a one-dimensional two-component attractive Fermi gas for arbitrary coupling strength within a quantum hydrodynamic approach. Using the Bethe-Ansatz solution of the 1D BCS-BEC crossover problem, we derive analytic results for the viscosity covering the full range from a Luther-Emery liquid of weakly bound pairs to a Lieb-Liniger gas of strongly bound bosonic dimers. At the unitarity point, the system is a Tonks-Girardeau gas with a universal constant $\alpha_{\zeta}=0.38$ in the viscosity $\zeta=\alpha_{\zeta}\hbar n$ for T=0. For the trapped case, we calculate the Q-factor of the breathing mode and show that the damping provides a sensitive measure of temperature in 1D Fermi gases.