Coexistence of superfluid gap and pseudogap in the BCS-BEC crossover regime of a trapped Fermi gas below $T_{\mathrm{c}}$

We investigate strong pairing fluctuations and effects of a harmonic trap in the superfluid phase of an ultracold Fermi gas. Including amplitude and phase fluctuations of the inhomogeneous superfluid order parameter $\mathrm{\Delta }(r)$ in a trap within a combined $T$-matrix theory with the local density approximation, we examine local properties of single-particle excitations and a thermodynamic quantity in the BCS-BEC crossover region. Below the superfluid phase transition temperature $T_{\mathrm{c}}$, we show that inhomogeneous pairing fluctuations lead to a shell structure of the gas cloud in which the spatial region where the ordinary BCS-type superfluid density of states appears is surrounded by the region where the pseudogap associated with strong pairing fluctuations dominates single-particle excitations. The former spatial region enlarges to eventually cover the whole gas cloud far below $T_{\mathrm{c}}$. We also examine how this shell structure affects the photoemission spectrum, as well as the local pressure. Since a cold Fermi gas is always trapped in a harmonic potential, our results would be useful for the study of strong-coupling superfluid physics, including this realistic situation.