Softening of roton and phonon modes in a Bose-Einstein condensate with spin-orbit coupling.

Roton-type excitations usually emerge from strong correlations or long-range interactions, as in superfluid helium or dipolar ultracold atoms. However, in a weakly short-range interacting quantum gas, the recently synthesized spin-orbit (SO) coupling can lead to various unconventional phases of superfluidity and give rise to an excitation spectrum of roton-maxon character. Using Bragg spectroscopy, we study a SO-coupled Bose-Einstein condensate of ^{87}Rb atoms and show that the excitation spectrum in a "magnetized" phase clearly possesses a two-branch and roton-maxon structure. As Raman coupling strength Ω is decreased, a roton-mode softening is observed, as a precursor of the phase transition to a stripe phase that spontaneously breaks spatially translational symmetry. The measured roton gaps agree well with theoretical calculations. Furthermore, we determine sound velocities both in the magnetized and in the nonmagnetized phases, and a phonon-mode softening is observed around the phase transition in between. The validity of the f-sum rule is examined.