Appropriate conditions to realize a $p$-wave superfluid state starting from a spin-orbit-coupled $s$-wave superfluid Fermi gas

We theoretically investigate a spin-orbit-coupled $s$-wave superfluid Fermi gas, to examine the time evolution of the system, after an $s$-wave pairing interaction is replaced by a $p$-wave one at $t=0$. In our recent paper [T. Yamaguchi, D. Inotani, and Y. Ohashi, J. Phys. Soc. Jpn. 86, 013001 (2017)], we proposed that this manipulation may realize a $p$-wave superfluid Fermi gas because the $p$-wave pair amplitude that is induced in the $s$-wave superfluid state by a parity-broken antisymmetric spin-orbit interaction gives a nonvanishing $p$-wave superfluid order parameter, immediately after the $p$-wave interaction is turned on. In this paper, using a time-dependent Bogoliubov-de Gennes theory, we assess this idea under various conditions with respect to the $s$-wave and $p$-wave interaction strengths, as well as the spin-orbit coupling strength. From these, we clarify that the momentum distribution of Fermi atoms in the initial $s$-wave state ($t<0$) is a key to produce a large $p$-wave superfluid order parameter. Since the realization of a $p$-wave superfluid state is one of the most exciting and difficult challenges in cold Fermi gas physics, our results may provide a possible way to accomplish this.

Figure 1

(a) Proposed protocol to realize a $p$-wave superfluid Fermi gas. When $t<0$, a $p$-wave pair amplitude ${\mathrm{\Phi }}_p^{\sigma {\sigma }^{\prime }}\left(\mathit{p}\right)$ is induced in an equilibrium $s$-wave superfluid Fermi gas with a parity-broken spin-orbit interaction ($\lambda \ne 0$). At this stage, while the system is still in the $s$-wave superfluid state with the $s$-wave superfluid order parameter ${\mathrm{\Delta }}_s\ne 0$, the $p$-wave order parameter ${\mathrm{\Delta }}_p^{\sigma {\sigma }^{\prime }}\left(\mathit{p}\right)$ vanishes because of the vanishing $p$-wave interaction $g_p=0$. At $t=0$, we replace the $s$-wave interaction $g_s$ with a $p$-wave one by adjusting an external magnetic field from an $s$-wave Feshbach-resonance field to a $p$-wave one ($g_s=0,g_p\ne 0$). At the same time, we turn off the spin-orbit coupling $\lambda =0$. The product of the $p$-wave pair amplitude ${\mathrm{\Phi }}_p^{\sigma {\sigma }^{\prime }}\left(\mathit{p}\right)$ which has been prepared in the parity-broken $s$-wave superfluid state and the introduced $p$-wave interaction $g_p$ immediately gives a nonvanishing $p$-wave superfluid order parameter ${\mathrm{\Delta }}_p^{\sigma {\sigma }^{\prime }}\left(\mathit{p}\right)\ne 0$. Then, by definition, the system is in the $p$-wave superfluid state. The $s$-wave superfluid order parameter vanishes (${\mathrm{\Delta }}_s=0$) when $t\ge 0$ because the $s$-wave interaction is turned off (although the $s$-wave pair amplitude may remain). This $p$-wave superfluid state at $t\ge 0$ is generally not in the equilibrium state, so that ${\mathrm{\Delta }}_p^{\sigma {\sigma }^{\prime }}\left(\mathit{p}\right)$ may have time dependence, as schematically shown in (b).

Figure 2

(a1),(a2) BCS-Leggett solutions for a spin-orbit-coupled $s$-wave superfluid Fermi gas at $T=0$. (a1) Fermi chemical potential $\mu$. $\mu +m{\lambda }^2/2$ means the Fermi energy measured from the bottom of the energy band when $g_s=0$. (a2) Calculated $s$-wave superfluid order parameter ${\mathrm{\Delta }}_s$. We take $\lambda /v_{\mathrm{F}}=0.5$. The lower three panels show the intensity of the spin-triplet Cooper-pair amplitude ${\mathrm{\Phi }}_{\mathrm{t}}^{\uparrow \uparrow }(p_x,p_y=0,p_z)$ at A–C. (b1),(b2) BCS-Leggett solutions for an equilibrium $p_z$-wave superfluid Fermi gas when the basis function in Eq. (21) is given by ${\mathit{F}}_{\mathit{p}}=(0,0,F_{\mathit{p}}^z)$. (b1) Fermi chemical potential $\mu$. (b2) Calculated $p_z$-wave superfluid order parameter ${\mathrm{\Delta }}_{p_z:\mathrm{eq}}^{\uparrow \uparrow }$. The interaction strengths at A–C and D–F will be used as the initial $s$-wave interaction strengths ($t<0$) and the $p$-wave interaction strengths ($t\ge 0$), respectively.

Figure 3

Calculated time evolution of the magnitude of the $p_z$-wave superfluid order parameter $|{\mathrm{\Delta }}_{p_z}^{\uparrow \uparrow }(t\ge 0)|$. The $s$-wave interaction strength ($t<0$) and the $p$-wave interaction strength ($t\ge 0$) are shown as, for example, $\mathrm{A}\to$ D. Here, A–C and D–F, respectively, represent the $s$-wave and $p$-wave interaction strengths shown in Fig. 2. The solid circles show $|{\mathrm{\Delta }}_{p_z}^{\uparrow \uparrow }(t=0)|$. The insets in (b1) and (c1) show the long-time and short-time behaviors of $|{\mathrm{\Delta }}_{p_z}^{\uparrow \uparrow }\left(t\right)|$, respectively.

Figure 4

(a1)–(a3) Time-averaged $p_z$-wave superfluid order parameter $\langle {\mathrm{\Delta }}_{p_z}^{\uparrow \uparrow }\rangle$ in Eq. (35). The initial $s$-wave interaction strength is (a1) ${\left(k_{\mathrm{F}}{a}_s\right)}^{-1}=-1$, (a2) ${\left(k_{\mathrm{F}}{a}_s\right)}^{-1}=0$, and (a3) ${\left(k_{\mathrm{F}}{a}_s\right)}^{-1}=1$. At the vertical dotted line $[{\left(k_{\mathrm{F}}^3{v}_p\right)}^{-1}=0.447]$, the Fermi chemical potential $\mu$ changes its sign in the equilibrium $p_z$-wave superfluid state. (b1)–(b3) The same plots as (a1)–(a3), where $\langle {\mathrm{\Delta }}_{p_z}^{\uparrow \uparrow }\rangle$ is normalized by the equilibrium value ${\mathrm{\Delta }}_{p_z:\mathrm{eq}}^{\uparrow \uparrow }$.

Figure 5

Calculated momentum distribution function $n_{\mathit{p}}^{\uparrow }\left(t\right)$ in the $p_z$-wave superfluid phase, given in Eq. (36). ${\tilde{n}}_{\mathit{p}}^{\uparrow }$ is the equilibrium result, given in Eq. (37). We set ${\left(k_{\mathrm{F}}{a}_s\right)}^{-1}=-1,\lambda /v_{\mathrm{F}}=0.1$, and $\mathit{p}=(0,0,p_z)$. (a) ${\left(k_{\mathrm{F}}^3{v}_p\right)}^{-1}=-6$. (b) ${\left(k_{\mathrm{F}}^3{v}_p\right)}^{-1}=-3$. (c) ${\left(k_{\mathrm{F}}^3{v}_p\right)}^{-1}=0$. (d) The magnitude of the averaged $p_z$-wave superfluid order parameter $\langle {\mathrm{\Delta }}_{p_z}^{\uparrow \uparrow }\rangle$ in each case.

Figure 6

Calculated atomic momentum distribution function $n_{\mathit{p}}^{\uparrow }\left(t\right)$ with $\mathit{p}=(0,0,p_z)$. We take $\lambda /v_{\mathrm{F}}=0.5$. ${\tilde{n}}_{\mathit{p}}^{\uparrow }$ is the momentum distribution in the equilibrium case. The steplike structures seen around $p_z/k_{\mathrm{F}}=1$ in (a1)–(c1), reflect the momentum distribution in a spin-orbit-coupled Fermi gas.

Figure 7

Calculated atomic momentum distribution function $n_{\mathit{p}}^{\uparrow }\left(t\right)$ with $\mathit{p}=(0,0,p_z)$. ${\tilde{n}}_{\mathit{p}}^{\uparrow }$ shows the equilibrium result when ${\left(k_{\mathrm{F}}^3{v}_p\right)}^{-1}=6$ (case F). Since the time dependence of $n_{\mathit{p}}^{\uparrow }\left(t\right)$ is weak, we only show the results at $t=0$.

Figure 8

(a) Quench dynamics of the magnitude of the $p_z$-wave superfluid order parameter $|{\mathrm{\Delta }}_{p_z}^{\uparrow \uparrow }\left(t\right)|$. In this calculation, the system is initially in the equilibrium strong-coupling $p_z$-wave superfluid state with ${\left(k_{\mathrm{F}}^3{v}_p\right)}^{-1}=8$. At $t=0$, the interaction strength is suddenly tuned. (b) Time evolution of $|{\mathrm{\Delta }}_{p_z}^{\uparrow \uparrow }\left(t\right)|$, when the $s$-wave interaction is replaced by the $p$-wave one at $t=0$, which is followed by the interaction quench at $t{\varepsilon }_{\mathrm{F}}=20(=t_0{\varepsilon }_{\mathrm{F}})$. We take $\lambda /v_{\mathrm{F}}=0.5$ in the equilibrium $s$-wave state at $t<0$. The inset shows the momentum distribution function $n_{\mathit{p}}^{\uparrow }\left(t\right)$, with the dashed line the case of $|{\mathrm{\Delta }}_{p_z}^{\uparrow \uparrow }\left(t\right)|$ in (b), where the equilibrium result ${\tilde{n}}_{\mathit{p}}^{\uparrow }$ is in the case ${\left(k_{\mathrm{F}}^3{v}_p\right)}^{-1}=8$.