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We extend our recent work [Y. Endo et. al., Phys. Rev. 92, 023610 (2015)] fora parity-mixing effect in a model two-dimensional lattice fermions to arealistic three-dimensional ultracold Fermi gas. Including effects of brokenlocal spatial inversion symmetry by a trap potential within the framework ofthe real-space Bogoliubov-de Gennes theory at $T=0$, we point out that anodd-parity $p$-wave Cooper-pair amplitude is expected to have already beenrealized in previous experiments on an (even-parity) $s$-wave superfluid Fermigas with spin imbalance. This indicates that, when one suddenly changes the$s$-wave pairing interaction to an appropriate $p$-wave one by using a Feshbachtechnique in this case, a non-vanishing $p$-wave superfluid order parameter isimmediately obtained, which is given by the product of the $p$-wave interactionand the $p$-wave pair amplitude that has already been induced in thespin-imbalanced $s$-wave superfluid Fermi gas. Thus, by definition, the systemis in the $p$-wave superfluid state, at least just after this manipulation.Since the achievement of a $p$-wave superfluid state is one of the mostexciting challenges in cold Fermi gas physics, our results may provide analternative approach to this unconventional pairing state. In addition, sincethe parity-mixing effect cannot be explained as far as one deals with a trappotential in the local density approximation (LDA), it is considered as acrucial example which requires us to go beyond LDA.

Triplet pair amplitude in a trapped $s$-wave superfluid Fermi gas with broken spin rotation symmetry. II. Three dimensional continuum case