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

We calculate the damping of the Bogoliubov - Anderson mode in a one-dimensional (1D) 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)(h) over barn 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.