The high flexibility of ultracold atoms allows for the design, control and thorough investigation of quantum matter. In this work we consider two one-dimensional Ramsey tunnel-coupled Bose gases and elaborate on its relation to the Prokrovsky-Talapov model. Further, we study the ground state with variational wave-functions and explore the experimental relevant finite system size effects. Finally, we discuss possible quench protocols in order to study the non-equilibrium evolution of the Pokrovsky-Talapov model and predict experimental relevant quantities such as the phase and the density.
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