We present the evolution in the number density and stellar mass functions of photometrically selected post-starburst galaxies in the UKIDSS Deep Survey (UDS), with redshifts of 0.5<z<2 and stellar masses logM>10. We find that this transitionary species of galaxy is rare at all redshifts, contributing ~5% of the total population at z~2, to <1% by z~0.5. By comparing the mass functions of quiescent galaxies to post-starburst galaxies at three cosmic epochs, we show that rapid quenching of star formation can account for 100% of quiescent galaxy formation, if the post-starburst spectral features are visible for ~250Myr. The flattening of the low mass end of the quiescent galaxy stellar mass function seen at z~1 can be entirely explained by the addition of rapidly quenched galaxies. Only if a significant fraction of post-starburst galaxies have features that are visible for longer than 250Myr, or they acquire new gas and return to the star-forming sequence, can there be significant growth of the red sequence from a slower quenching route. The shape of the mass function of these transitory post-starburst galaxies resembles that of quiescent galaxies at z~2, with a preferred stellar mass of logM~10.6, but evolves steadily to resemble that of star-forming galaxies at z<1. This leads us to propose a dual origin for post-starburst galaxies: (1) at z>2 they are exclusively massive galaxies that have formed the bulk of their stars during a rapid assembly period, followed by complete quenching of further star formation, (2) at z<1 they are caused by the rapid quenching of gas-rich star-forming galaxies, independent of stellar mass, possibly due to environment and/or gas-rich major mergers.
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