  We study a transition between a homogeneous and an inhomogeneous phase in a
system of one-dimensional, Raman tunnel-coupled Bose gases. The homogeneous
phase shows a flat density and phase profile, whereas the inhomogeneous ground
state is characterized by periodic density ripples, and a soliton staircase in
the phase difference. We show that under experimentally viable conditions the
transition can be tuned by the wavevector difference $Q$ of the Raman beams and
can be described by the Pokrovsky-Talapov model for the relative phase between
the two condensates. Local imaging available in atom chip experiments allows to
observe the soliton lattice directly, while modulation spectroscopy can be used
to explore collective modes, such as the phonon mode arising from breaking of
translation symmetry by the soliton lattice. In addition, we investigate
regimes where the cold atom experiment deviates from the Pokrovsky-Talapov
field theory. We predict unusual mesoscopic effects arising from the finite
size of the system, such as quantized injection of solitons upon increasing
$Q$, or the system size. For moderate values of $Q$ above criticality, we find
that the density modulations in the two gases interplay with the relative phase
profile and introduce novel features in the spatial structure of the mode
wave-functions. Using an inhomogeneous Bogoliubov theory, we show that spatial
quantum fluctuations are intertwined with the emerging soliton staircase.
Finally, we comment on the prospects of the ultra-cold atom setup as a tunable
platform studying quantum aspects of the Pokrovsky-Talapov theory in and
out-of-equilibrium.
