  A quantum phase transition from the miscible to the immiscible phase of a
quasi-one-dimensional binary Bose-Einstein condensate is driven by ramping down
the coupling amplitude of its two hyperfine states. It results in a random
pattern of spatial domains where the symmetry is broken separated by defects.
In distinction to previous studies [J. Sabbatini et al., Phys. Rev. Lett. 107,
230402 (2011), New J. Phys. 14 095030 (2012)], we include nonzero detuning
between the light field and the energy difference of the states, which provides
a bias towards one of the states. Using the truncated Wigner method, we test
the biased version of the quantum Kibble-Zurek mechanism [M. Rams et al., Phys.
Rev. Lett. 123, 130603 (2019)] and observe a crossover to the adiabatic regime
when the quench is sufficiently fast to dominate the effect of the bias. We
verify a universal power law for the population imbalance in the nonadiabatic
regime both at the critical point and by the end of the ramp. Shrinking and
annihilation of domains of the unfavourable phase after the ramp, that is,
already in the broken symmetry phase, enlarges the defect-free sections by the
end of the ramp. The consequences of this phase-ordering effect can be captured
by a phenomenological power law.
