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Elementary excitations in a spin-orbit-coupled spin-1 Bose-Einstein condensate

While a spin-orbit-coupled spin-1 Bose-Einstein condensate has been experimentally observed, its elementary excitations remain unclear in the stripe phase. Here, we systematically study the elementary excitations in three distinct phases of a spin-orbit-coupled spin-1 Bose-Einstein condensate. We find that the excitation spectrum as well as the corresponding static response function and structure factor depend strongly on spin-orbit coupling parameters such as the quadratic Zeeman field and the Rabi frequency. In the stripe phase, besides two gapless Goldstone modes, we show the existence of roton excitations. Finally, we demonstrate that quantum phase transitions between these different phases including the zero-momentum, plane wave and stripe phases are characterized by the sound velocities and the quantum depletion.