Direct Observation of Spin- and Charge-Density Waves in a Luttinger Liquid*

At low energy, interacting fermions in one dimension (e.g. electrons in quantum wires or fermionic atoms in 1D waveguides) should behave as Luttinger liquids.~ In stark contrast to Fermi liquids, the low-energy elementary excitations in Luttinger liquids are collective sound-like modes that propagate independently as spin-density and/or charge-density (i.e. particle-density) waves with generally unequal, and interaction-dependent, velocities.~ Here we aim to unambiguously confirm this hallmark feature of the Luttinger liquid -- the phenomenon of spin-charge separation -- by directly observing in real space the dynamics of spin-density and "charge"-density waves excited in an ultracold gas of spin-1/2 fermions confined in an array of 1D optical waveguides.~ Starting from a two-component mixture of $^{6}$Li atoms harmonically confined along each of the 1D waveguides, we excite low lying normal modes of the trapped system -- namely the spin dipole and density dipole and quadrupole modes -- and measure their frequency as a function of interaction strength.~ Luttinger liquid theory predicts that the spin dipole frequency is strongly dependent on interaction strength whereas the density dipole and quadrupole mode frequencies are relatively insensitive.~~ We will also discuss extending our approach to exciting localized spin density and particle density wavepackets which should propagate at different velocities.