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Faraday patterns in spin-orbit coupled Bose-Einstein condensates

We study the Faraday patterns generated by spin-orbit-coupling induced parametric resonance in a spinor Bose-Einstein condensate with repulsive interaction. The collective elementary excitations of the Bose-Einstein condensate, including density waves and spin waves, are coupled as the result of the Raman-induced spin-orbit coupling and a quench of the relative phase of two Raman lasers without the modulation of any of the system's parameters. We observed several higher parametric resonance tongues at integer multiples of the driving frequency and investigated the interplay between Faraday instabilities and modulation instabilities when we quench the spin-orbit-coupled Bose-Einstein condensate from zero-momentum phase to plane-wave phase. If the detuning is equal to zero, the wave number of combination resonance barely changes as the strength of spin-orbit coupling increases. If the detuning is not equal to zero after a quench, a single combination resonance tongue will split into two parts.