  The Berezinskii-Kosterlitz-Thouless (BKT) phase transition is considered in
the condition of lowest temperatures, when thermal fluctuations give place to
quantum ones. For this goal, the critical dynamic of the Sine-Gordon model near
the quantum critical point is considered. The approach based on the
Keldysh-Schwinger technique of non-equilibrium dynamics description is used, as
well as the method of taking into account the crossover from the thermal
fluctuation regime to the quantum one in the renormalization group technique.
For the system with low vortex concentration, it is shown that the BKT
transition unavoidably occurs at a temperature above the crossover temperature
from the thermal fluctuation regime to quantum one. As a result at small
temperatures the critical exponent of the BKT transition, $\eta=1/4$, remains
unchanged, however, the universal relation of the superfluid density jump
adjusts a little from the well-known relation by Nelson and Kosterlitz.
