  We study the phase-locking transition of two coupled low-dimensional
superfluids, either two-dimensional superfluids at finite temperature, or
one-dimensional superfluids at zero temperature. We find that the superfluids
have a strong tendency to phase-lock. The phase-locking is accompanied by a
sizeable increase of the transition temperature $T_{c}$ (in 2D systems) of the
resulting double-layer superfluid to thermal Bose gas transition, compared to
the Kosterlitz-Thouless temperature $T_{KT}$ of the uncoupled 2D systems, which
suggests a plausible way of observing the Kibble-Zurek mechanism in
two-dimensional cold atom systems by rapidly varying the tunneling rate between
the superfluids. If there is also interaction between atoms in different layers
present we find additional phases, while no sliding phase, characterized by
order or quasi long range order (QLRO) either in the symmetric or the
antisymmetric sector of the system.
