We show that in the presence of magnetic field, two superconducting phases with the center-of-mass momentum of Cooper pair parallel to the magnetic field are induced in spin-orbit-coupled superconductor Li$_2$Pd$_3$B. Specifically, at small magnetic field, the center-of-mass momentum is induced due to the energy-spectrum distortion and no unpairing region with vanishing singlet correlation appears. We refer to this superconducting state as the drift-BCS state. By further increasing the magnetic field, the superconducting state falls into the Fulde-Ferrell-Larkin-Ovchinnikov state with the emergence of the unpairing regions. The observed abrupt enhancement of the center-of-mass momenta and suppression on the order parameters during the crossover indicate the first-order phase transition. Enhanced Pauli limit and hence enlarged magnetic-field regime of the Fulde-Ferrell-Larkin-Ovchinnikov state, due to the spin-flip terms of the spin-orbit coupling, are revealed. We also address the triplet correlations induced by the spin-orbit coupling, and show that the Cooper-pair spin polarizations, generated by the magnetic field and center-of-mass momentum with the triplet correlations, exhibit totally different magnetic-field dependences between the drift-BCS and Fulde-Ferrell-Larkin-Ovchinnikov states.
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