  Formation of topological defects during symmetry breaking phase transitions
via the {\it Kibble mechanism} is extensively used in systems ranging from
condensed matter physics to the early stages of the universe. Kibble mechanism
uses topological arguments and predicts equal probabilities for the formation
of defects and anti-defects. Certain situations, however, require a net bias in
the production of defects (or antidefects) during the transition, for example,
superfluid transition in a rotating vessel, or flux tubes formation in a
superconducting transition in the presence of external magnetic field. In this
paper we present a modified Kibble mechanism for a specific system, $^4$He
superfluid transition in a rotating vessel, which can produce the required bias
of vortices over antivortices. Our results make distinctive predictions which
can be tested in superfluid $^4$He experiments. These results also have
important implications for superfluid phase transitions in rotating neutron
stars and also for any superfluid phases of QCD arising in the non-central low
energy heavy-ion collision experiment due to an overall rotation.
