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

We extend our recent work [Y. Endo et al., Phys. Rev. A 92, 023610 (2015)] for a parity-mixing effect in a model of two-dimensional lattice fermions to a realistic three-dimensional ultracold Fermi gas. Including effects of broken local spatial inversion symmetry by a trap potential within the framework of the real-space Bogoliubov-de Gennes theory at $T=0$, we point out that an odd-parity $p\text{-wave}$ Cooper-pair amplitude is expected to have already been realized in previous experiments on an (even-parity) $s\text{-wave}$ superfluid Fermi gas with spin imbalance. This indicates that when one suddenly changes the $s\text{-wave}$ pairing interaction to an appropriate $p\text{-wave}$ one by using a Feshbach technique in this case, a nonvanishing $p\text{-wave}$ superfluid order parameter is immediately obtained, which is given by the product of the $p\text{-wave}$ interaction and the $p\text{-wave}$ pair amplitude that has already been induced in the spin-imbalanced $s\text{-wave}$ superfluid Fermi gas. Thus, by definition, the system is in the $p\text{-wave}$ superfluid state, at least just after this manipulation. Since the achievement of a $p\text{-wave}$ superfluid state is one of the most exciting challenges in cold Fermi gas physics, our results may provide an alternative approach to this unconventional pairing state. In addition, since the parity-mixing effect cannot be explained as far as one deals with a trap potential in the local density approximation (LDA), it is considered as a crucial example which requires us to go beyond the LDA.

Figure 1

Calculated (a) spin-triplet component pair amplitude as a function of the relative coordinate $r_{\mathrm{rel}}^x$ and $r_{\mathrm{rel}}^y$ in a trapped $s\text{-wave}$ superfluid Fermi gas with spin-imbalance, ${\mathrm{\Phi }}_{\mathrm{T}}\left(\mathit{R},{\mathit{r}}_{\mathrm{rel}}\right)R_{\mathrm{F}}^3$, where $R_{\mathrm{F}}=\sqrt{2{\varepsilon }_{\mathrm{F}}/\left(m\overline{\omega }\right)}$ is the Thomas-Fermi radius, and (b) spin-single component ${\mathrm{\Phi }}_{\mathrm{S}}\left(\mathit{R},{\mathit{r}}_{\mathrm{rel}}\right)R_{\mathrm{F}}^3$. We take ${\left(p_{\mathrm{F}}{a}_s\right)}^{-1}=-0.6$, $P\equiv (N_{\uparrow }-N_{\downarrow })/(N_{\uparrow }+N_{\downarrow })=0.2$, $r_{\mathrm{rel}}^z=0$, and $\mathit{R}=(0.8R_{\mathrm{F}},0,0)$. This parameter set is also used in Figs. 2 and 3. (c) Schematic spatial structure of the triplet Cooper-pair amplitude ${\mathrm{\Phi }}_{\mathrm{T}}\left(\mathit{R},{\mathit{r}}_{\mathrm{rel}}\right)$. At each center-of-mass position $\mathit{R}$ (open square), the ${\mathit{r}}_{\mathrm{rel}}$ dependence of ${\mathrm{\Phi }}_{\mathrm{T}}\left(\mathit{R},{\mathit{r}}_{\mathrm{rel}}\right)$ is shown.

Figure 2

Calculated (a) spin-triplet component ${\rho }_{\mathrm{T}}^{\mathrm{c}}\left(R\right)$ and (b) singlet component ${\rho }_{\mathrm{T}}^{\mathrm{c}}\left(R\right)$ of local condensate fraction in a trapped $s\text{-wave}$ superfluid Fermi gas with spin imbalance $P\equiv (N_{\uparrow }-N_{\downarrow })/(N_{\uparrow }+N_{\downarrow })=0.2$. (c) $s\text{-wave}$ superfluid order parameter $\mathrm{\Delta }\left(r\right)$. (d) Density profile $n_{\sigma }\left(r\right)$. We take ${\left(p_{\mathrm{F}}{a}_s\right)}^{-1}=-0.6$.

Figure 3

Calculated density of condensate fraction ${\rho }_L^{\mathrm{c}}\left(R\right)$ with the angular momentum $L$, as a function of the center-of-mass position $R$ of a Cooper pair. We set ${\left(p_{\mathrm{F}}{a}_s\right)}^{-1}=-0.6$, and $P=0.2$. In this case, the total $p\text{-wave}$ condensate fraction equals $N_{L=1}^{\mathrm{c}}/N_{\mathrm{total}}^{\mathrm{c}}\simeq 0.14$.

Figure 4

Same as Fig. 3, but when ${\left(p_{\mathrm{F}}{a}_s\right)}^{-1}=-0.6$ and $P=0.3$.

Figure 5

(a) Calculated condensate fraction $N_{\mathrm{T}}^{\mathrm{c}}$ of the spin-triplet component. (b) Spin-singlet component $N_{\mathrm{S}}^c$. In each panel, the intensity is renormalized by the total number of Fermi atoms $N=N_{\uparrow }+N_{\downarrow }$.

Figure 6

The upper panels show the case with mass imbalance. In (c) we take $m_{\uparrow }/m_{\downarrow }=0.5$ and ${\left(p_{\mathrm{F}}{a}_s\right)}^{-1}=-0.5$. The lower ones show the case with spin-dependent trap potential. In (f) we take ${\omega }_{\uparrow }/{\omega }_{\downarrow }=0.8$ and ${\left(a_s{p}_{\mathrm{F}}\right)}^{-1}=-0.9$. (a), (d) Spin-triplet component $N_{\mathrm{T}}^c$ of the total condensate fraction. (b), (e) Spin-singlet component $N_{\mathrm{S}}^c$ of the total condensate fraction. (c), (f) Triplet component ${\rho }_{\mathrm{T}}^{\mathrm{c}}\left(R\right)$ of the local condensate fraction.

Figure 7

Calculated condensate fraction in a trapped $s\text{-wave}$ superfluid Fermi gas with spin imbalance. (a) Spin-singlet component $N_{\mathrm{S}}^{\mathrm{c}}$. (b) Spin-triplet component $N_{\mathrm{T}}^{\mathrm{c}}$. We take ${\left(p_{\mathrm{F}}{a}_s\right)}^{-1}=0$. The solid squares are the results with $\gamma =0$ and the solid line shows the results when $\gamma =0.05{\varepsilon }_{\mathrm{F}}$. In obtaining the results with $\gamma =0$, we tuned the spin polarization $P=(N_{\uparrow }-N_{\downarrow })/(N_{\uparrow }+N_{\downarrow })$ by varying $N_{\downarrow }$ for $N_{\uparrow }=816$.