We study the non-equilibrium condensation process in a holographic superconductor. When the temperature T is smaller than a critical temperature Tc, there are two black hole solutions, the Reissner-Nordström- AdS black hole and a black hole with a scalar hair. In the boundary theory, they can be regarded as the supercooled normal phase and the superconducting phase, respectively. We consider perturbations on supercooled Reissner-Nordström- AdS black holes and study their non-linear time evolution to know about physical phenomena associated with rapidly-cooled superconductors. We find that, for T < Tc, the initial perturbations grow exponentially and, eventually, spacetimes approach the hairy black holes. We also clarify how the relaxation process from a far-fromequilibrium state proceeds in the boundary theory by observing the time dependence of the superconducting order parameter. Finally, we study the time evolution of event and apparent horizons and discuss their correspondence with the entropy of the boundary theory. Our result gives a first step toward the holographic understanding of the non-equilibrium process in superconductors.
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