# A Novel Route to Reach a $p$-wave Superfluid Fermi Gas

T. Yamaguchi  D. Inotani  Y. Ohashi  Department of Physics, Keio University,
3-14-1 Hiyoshi, Yokohama 223-8522, Japan

###### Abstract

We theoretically propose an idea to realize a $p$-wave superfluid Fermi gas.
To overcome the experimental difficulty that a $p$-wave pairing interaction to
form $p$-wave Cooper pairs damages the system before the condensation growth,
we first prepare a $p$-wave pair amplitude ($\Phi_{p}$) in a spin-orbit
coupled $s$-wave superfluid Fermi gas, without any $p$-wave interaction. Then,
by suddenly changing the $s$-wave interaction with a $p$-wave one ($U_{p}$) by
using a Feshbach resonance, we reach the $p$-wave superfluid phase with the
$p$-wave order parameter being symbolically written as $\Delta_{p}\sim
U_{p}\Phi_{p}$. In this letter, we assess this scenario within the framework
of a time-dependent Bogoliubov-de Gennes theory. Our results would contribute
to the study toward the realization of unconventional pairing states in an
ultracold Fermi gas.

In cold Fermi gas physics, one of the most difficult and exciting challenges
is the achievement of a $p$-wave superfluid phase transition. Experimentally,
one can now tune the strength of a $p$-wave pairing interaction by adjusting
the threshold energy of a Feshbach resonancepwve1 ; pwve2 ; pwve3 ; pwve4 ;
pwve5 ; pwve6 ; pwve7 . The formation of $p$-wave molecules by using this
interaction has also been reportedpwve4 ; pwve12 ; pwve13 . In addition,
strong-coupling theories have quantitatively predicted the $p$-wave superfluid
phase transition temperature $T_{\rm c}$Ho ; Ohashi ; Iskin1 ; Iskin2 ;
Inotani . However, the realization of a $p$-wave superfluid Fermi gas has not
been reported yet.

This difficulty comes from the dilemma that, although a $p$-wave pairing
interaction is necessary to realize a $p$-wave Fermi gas superfluid, it also
damages the system. That is, a $p$-wave interaction causes dipolar
relaxationspwve4 , as well as three-body lossesGurarie1 ; Gurarie2 ; Castin ,
leading to very short lifetime of $p$-wave pairs ($=5\sim 20$ ms)Chevy . As a
result, $p$-wave pairs are destroyed before the $p$-wave condensate enough
grows ($=O(100~{}{\rm ms})$). At the same time, the particle loss from the
system destroys the system itself. Since all the current experiments toward
the realization of a $p$-wave superfluid Fermi gas are facing these problems,
a breakthrough seems necessary.

The purpose of this letter is to propose a possible route to reach a $p$-wave
superfluid Fermi gas. To explain our idea, it is convenient to recall that a
$p$-wave superfluid Fermi gas is well characterized by the $p$-wave superfluid
order parameter,

$\Delta_{\sigma,\sigma^{\prime}}^{p}({p})=\sum_{{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt}^{\prime}}U_{p}({p},{p}^{\prime})\Phi^{p}_{\sigma,\sigma^{\prime}}({p}^{\prime}),$
(1)

where $U_{p}({p},{p}^{\prime})$ is a $p$-wave pairing interaction.
$\Phi^{p}_{\sigma,\sigma^{\prime}}({p}^{\prime})=\langle c_{{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern 0.27995pt},\sigma}c_{-{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern 0.27995pt},\sigma^{\prime}}\rangle$ is
$p$-wave pair amplitude, where $c_{{\hbox{\scripti p}\kern-0.14003pt\kern
0.0pt\kern 0.27995pt},\sigma}$ is the annihilation operator of a Fermi atom
with pseudospin $\sigma=\uparrow,\downarrow$, describing two atomic hyperfine
states forming Cooper pairs. The current experiments always start from a Fermi
gas with a $p$-wave pairing interaction $U_{p}({p},{p}^{\prime})$, and then
cool down the gas to explore the $p$-wave superfluid phase where the pairing
interaction $U_{p}({p},{p}^{\prime})$ produces the pair amplitude
$\Phi^{p}_{\sigma,\sigma^{\prime}}({p}^{\prime})$, leading to non-vanishing
$p$-wave superfluid order parameter $\Delta_{\sigma,\sigma^{\prime}}^{p}({p})$
in Eq. (1). This very reasonable approach is, however, hampered by the above-
mentioned problems associated with the same $p$-wave interaction.

The essence of our idea is, in a sense, to inverse this ordinary experimental
procedure. That is, as schematically shown in Fig. 1, when $t<0$, we first
prepare a Fermi gas with a non-vanishing $p$-wave pair amplitude
$\Phi^{p}_{\sigma,\sigma^{\prime}}({p})$ without a $p$-wave pairing
interaction ($U_{p}({p},{p}^{\prime})=0$). Recently, it has been shown that
such preparation is possible in a spin-orbit coupled $s$-wave superfluid Fermi
gasHu ; Yamaguchi . At this stage, since the $p$-wave interaction
$U_{p}({p},{p}^{\prime})$ vanishes, one does not suffer from the above-
mentioned serious difficulties. Once this first step is accomplished, the
$p$-wave superfluid order parameter is then immediately obtained, when one
suddenly changes the pairing interaction from the $s$-wave type to an
appropriate $p$-wave one ($t\geq 0$ in Fig. 1). This manipulation is also
possible in cold Fermi gases by adjusting an external magnetic field from an
$s$-wave Feshbach resonance to a $p$-wave one. Then, by definition, the system
is in the $p$-wave superfluid state with
$\Delta^{p}_{\sigma,\sigma^{\prime}}({p})$. Just after this manipulation, this
order parameter is constructed from the introduced $p$-wave interaction
$U_{p}({p},{p}^{\prime})$ and the $p$-wave pair amplitude
$\Phi^{p}_{\sigma,\sigma^{\prime}}({p})$ prepared in advance in the $s$-wave
state. Of course, the introduced $p$-wave interaction would start to damage
the system as in the ordinary approach. However, the advantage of our approach
is that the system is in the $p$-wave superfluid phase, at least just after
the $s$-wave interaction is replaced by a $p$-wave one.

We briefly note that the time evolution of a superfluid Fermi gas after the
sudden change of the interaction strength has recently been discussedt1 ; t2 ;
t3 ; t4 ; t5 ; t6 ; t7 ; t8 ; t9 . In a sense, our idea is an extension of
this so-called quench dynamics, to the case when the pairing symmetry is also
suddenly changed.

Figure 1: (Color online) Schematic picture of our idea to realize a $p$-wave
superfluid Fermi gas. When $t<0$, we prepare a $p$-wave pair amplitude
$\Phi^{p}_{\sigma,\sigma^{\prime}}({p})$ in a spin-orbit coupled $s$-wave
superfluid Fermi gas (where $U_{s}$ and $\lambda$ are an $s$-wave pairing
interaction and a spin-orbit coupling, respectively). At this stage, the
$p$-wave superfluid order parameter $\Delta_{\sigma,\sigma^{\prime}}^{p}({p})$
still vanishes, because of the vanishing $p$-wave interaction, $U_{p}=0$. At
$t=0$, we suddenly replace the $s$-wave interaction by a $p$-wave one
($U_{s}=0$ and $U_{p}\neq 0$). The product of the $p$-wave interaction and the
prepared $p$-wave pair amplitude $\Phi^{p}_{\sigma,\sigma^{\prime}}({p})$
immediately gives non-vanishing $p$-wave order parameter
$\Delta_{\sigma,\sigma^{\prime}}^{p}({p})$. The $s$-wave order parameter
vanishes when $t\geq 0$, because of $U_{s}=0$. As a result, the $p$-wave
superfluid state being characterized by $\Delta_{\sigma,\sigma^{\prime}}^{p}$
is realized.  Figure 2: (Color online) (a) Calculated Fermi chemical potential
$\mu$ in the BCS-BEC crossover regime of a spin-orbit coupled $s$-wave
superfluid Fermi gas at $T=0$. When $U_{s}=0$, $\mu+m\lambda^{2}/2$ means the
Fermi energy measured from the bottom of the energy band. The inset shows the
$s$-wave superfluid order parameter $\Delta_{s}$. We note that $\mu$ and
$\Delta_{s}$ are independent of the spin-orbit interaction $H_{\rm so}=\lambda
p_{z}\tau_{x}$Yamaguchi . (b) $N_{\rm c}\equiv\sum_{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern 0.27995pt}|\Phi^{\rm
t}_{\sigma,\sigma}({p})|^{2}$. A finite value of this quantity means the
induction of the triplet pair amplitude $\Phi^{\rm t}_{\sigma,\sigma}({p})$ in
the $s$-wave superfluid state. $v_{\rm F}$ is the Fermi velocity and
$\varepsilon_{\rm F}\equiv mv_{\rm F}^{2}/2$.

In assessing our idea, a key is to clarify whether or not the $p$-wave pair
amplitude $\Phi_{\sigma,\sigma^{\prime}}^{p}({p})$ induced in the $s$-wave
state can really produce the $p$-wave superfluid order parameter
$\Delta^{p}_{\sigma,\sigma^{\prime}}({p})$ after the interaction change. To
see this, in this letter, we employ a time-dependent Bogoliubov-de Gennes
theory (TDBdG)timeT1 ; timeT2 , to examine the time evolution of the $p$-wave
order parameter after a $p$-wave interaction is switched on. Although TDBdG
cannot deal with the energy relaxation of the system, as well as the particle
loss, this approach is still convenient to examine the early behavior of the
$p$-wave superfluid after the $p$-wave interaction is turned on. We briefly
note that TDBdG has recently been applied to the quench dynamics of a
superfluid Fermi gast1 ; t2 ; t3 ; t4 ; t5 ; t6 ; t7 ; t8 ; t9 .

To prepare the $p$-wave pair amplitude, we employ a spin-orbit coupled
(equilibrium) $s$-wave superfluid Fermi gasHu ; Yamaguchi , described by the
model Hamiltonian,

$\displaystyle H_{s}$ $\displaystyle=$ $\displaystyle\sum_{{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt},\sigma,\sigma^{\prime}}\left[\xi_{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt}\delta_{\sigma,\sigma^{\prime}}+\lambda
p_{z}\tau_{x}^{\sigma,\sigma^{\prime}}\right]c_{{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt},\sigma}^{\dagger}c_{{\hbox{\scripti p}\kern-0.14003pt\kern
0.0pt\kern 0.27995pt},\sigma^{\prime}}$ (2) $\displaystyle-$ $\displaystyle
U_{s}\sum_{{\hbox{\scripti p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt},{\hbox{\scripti p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt}^{\prime},{\hbox{\scripti q}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt}}c_{{\hbox{\scripti p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt}+{\hbox{\scripti q}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt}/2,\uparrow}^{\dagger}c_{-{\hbox{\scripti p}\kern-0.14003pt\kern
0.0pt\kern 0.27995pt}+{\hbox{\scripti q}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt}/2,\downarrow}^{\dagger}c_{-{\hbox{\scripti p}\kern-0.14003pt\kern
0.0pt\kern 0.27995pt}^{\prime}+{\hbox{\scripti q}\kern-0.14003pt\kern
0.0pt\kern 0.27995pt}/2,\downarrow}c_{{\hbox{\scripti p}\kern-0.14003pt\kern
0.0pt\kern 0.27995pt}^{\prime}+{\hbox{\scripti q}\kern-0.14003pt\kern
0.0pt\kern 0.27995pt}/2,\uparrow}.$

In this letter, we take $\hbar=1$ and the the system volume $V$ is taken to be
unity, for simplicity. In Eq. (2), $\xi_{\hbox{\scripti p}\kern-0.14003pt\kern
0.0pt\kern 0.27995pt}=\varepsilon_{\hbox{\scripti p}\kern-0.14003pt\kern
0.0pt\kern 0.27995pt}-\mu={p}^{2}/(2m)-\mu$ is the kinetic energy of a Fermi
atom described by the creation operator $c_{{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern 0.27995pt},\sigma}^{\dagger}$, measured from
the Fermi chemical potential $\mu$. $-U_{s}~{}(<0)$ is a contact-type $s$-wave
pairing interaction, which is assumed to be tunable by a Feshbach-
resonanceChin . As usual in cold Fermi gas physics, we measure the $s$-wave
interaction strength in terms of the inverse $s$-wave scattering length
$(k_{\rm F}a_{s})^{-1}$, scaled by the Fermi momentum $k_{\rm F}$. In this
scale, $(k_{\rm F}a_{s})^{-1}\ \raise 1.29167pt\hbox{$<$}\kern-8.00003pt\lower
3.01385pt\hbox{$\sim$}\ -1$ and $(k_{\rm F}a_{s})^{-1}\ \raise
1.29167pt\hbox{$>$}\kern-8.00003pt\lower 3.01385pt\hbox{$\sim$}\ +1$
correspond to the weak-coupling BCS (Bardeen-Cooper-Schrieffer) regime and the
strong-coupling BEC (Bose-Einstein condensation) regime, respectively. The
region $-1\ \raise 1.29167pt\hbox{$<$}\kern-8.00003pt\lower
3.01385pt\hbox{$\sim$}\ (k_{\rm F}a_{s})^{-1}\ \raise
1.29167pt\hbox{$<$}\kern-8.00003pt\lower 3.01385pt\hbox{$\sim$}\ +1$ is called
the BCS-BEC crossover region. Treating the BCS-BEC crossover physics at $T=0$
within the framework of the BCS-Leggett theoryLeggett , we self-consistently
determine the chemical potential $\mu$, as well as the $s$-wave superfluid
order parameter $\Delta_{s}=U_{s}\sum_{\hbox{\scripti p}\kern-0.14003pt\kern
0.0pt\kern 0.27995pt}\langle c_{{\hbox{\scripti p}\kern-0.14003pt\kern
0.0pt\kern 0.27995pt},\uparrow}c_{-{\hbox{\scripti p}\kern-0.14003pt\kern
0.0pt\kern 0.27995pt},\downarrow}\rangle$, from the coupled BCS gap equation,

$1=-{4\pi a_{s}\over m}\sum_{\hbox{\scripti p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt}\left[{1\over 2}\sum_{\alpha=\pm}{1\over 2E_{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern 0.27995pt}^{\alpha}}-{1\over
2\varepsilon_{\hbox{\scripti p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt}}\right],$ (3)

with the equation for the number $N$ of Fermi atoms,

$N={1\over 2}\sum_{{\hbox{\scripti p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt},\alpha=\pm}\left[1-{\xi_{\hbox{\scripti p}\kern-0.14003pt\kern
0.0pt\kern 0.27995pt}^{\alpha}\over E_{\hbox{\scripti p}\kern-0.14003pt\kern
0.0pt\kern 0.27995pt}^{\alpha}}\right].$ (4)

Here, $E_{\hbox{\scripti p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt}^{\pm}=\sqrt{(\xi_{\hbox{\scripti p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt}^{\alpha})^{2}+\Delta_{s}^{2}}$ (where $\xi_{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern 0.27995pt}^{\alpha}=\xi_{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern 0.27995pt}+\alpha\lambda|p_{z}|$) describes
Bogoliubov single-particle excitations in the presence of the one-component
spin-orbit interaction, $H_{\rm so}=\lambda
p_{z}\tau_{x}^{\sigma,\sigma^{\prime}}$ (where $\tau_{x}$ is the Pauli
matrix). We briefly note that $H_{\rm so}$ has recently been realized in
6LiCheuk  and 40KWang  Fermi gases, by using a synthetic gauge field
technique.

As pointed out in Refs.Hu ; Yamaguchi , the spin-orbit interaction $H_{\rm
so}$ induces the spin-triplet pair amplitudes in the $s$-wave superfluid
state, given by

$\Phi^{\rm t}_{\uparrow,\uparrow}({p})=-\Phi^{\rm
t}_{\downarrow,\downarrow}({p})=-{p_{z}\over|p_{z}|}\sum_{\alpha=\pm}{\alpha\Delta_{s}\over
2E_{\hbox{\scripti p}\kern-0.14003pt\kern 0.0pt\kern 0.27995pt}^{\alpha}}.$
(5)

In Fig. 2, we show the quantity $N_{\rm c}\equiv\sum_{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern 0.27995pt}|\Phi^{\rm
t}_{\sigma,\sigma}({p})|^{2}$ in the BCS-BEC crossover region, together with
$\Delta_{s}$ and $\mu$ (that are self-consistently determined from the coupled
equations (3) and (4)). The induced spin-triplet pair amplitude $\Phi^{\rm
t}_{\sigma,\sigma}({p})$ has the $p_{z}$-wave component
$\Phi^{p_{z}}_{\sigma,\sigma}({p})\propto p_{z}$Yamaguchi , which may also be
expected from the prefactor of Eq. (5).

To connect the above BCS-Leggett scheme to the TDBdG formalism, it is useful
to note that the former result is also obtained as a steady-state solution of
the TDBdG equation,

${\rm i}\frac{\partial}{\partial
t}{\tilde{u}}(t)={\hat{H}}_{s}{\tilde{u}}(t),$ (6)

where

$\displaystyle{\hat{H}}_{s}=\left(\begin{array}[]{cccc}\varepsilon_{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern 0.27995pt}&\lambda
p_{z}&0&{\tilde{\Delta}}_{s}(t)\\\ \lambda p_{z}&\varepsilon_{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern 0.27995pt}&-{\tilde{\Delta}}_{s}(t)&0\\\
0&-{\tilde{\Delta}}_{s}^{*}(t)&-\varepsilon_{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern 0.27995pt}&\lambda p_{z}\\\
{\tilde{\Delta}}_{s}^{*}(t)&0&\lambda p_{z}&-\varepsilon_{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern 0.27995pt}\end{array}\right).$ (11)

In Eq. (6), ${\tilde{u}}(t)=({\tilde{u}}_{{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt},\uparrow}^{\alpha}(t),{\tilde{u}}_{{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt},\downarrow}^{\alpha}(t),{\tilde{v}}_{{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt},\uparrow}^{\alpha}(t),{\tilde{v}}_{{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern 0.27995pt},\downarrow}^{\alpha}(t))^{T}$
consists of the coefficients of the Bogoliubov transformation,

$c_{{\hbox{\scripti p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt},\sigma}(t)=\sum_{\alpha}\left[{\tilde{u}}_{{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt},\sigma}^{\alpha}(t)\gamma_{{\hbox{\scripti p}\kern-0.14003pt\kern
0.0pt\kern 0.27995pt},\alpha}+{\tilde{v}}_{-{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt},\sigma}^{\alpha*}(t)\gamma_{-{\hbox{\scripti p}\kern-0.14003pt\kern
0.0pt\kern 0.27995pt},\alpha}^{\dagger}\right].$ (12)

Here, the normalization condition $|{\tilde{u}}_{{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt},\sigma}^{\alpha}(t)|^{2}+|{\tilde{v}}_{{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern 0.27995pt},\sigma}^{\alpha}(t)|^{2}=1$ is
imposed. The Fermi operator $\gamma_{{\hbox{\scripti p}\kern-0.14003pt\kern
0.0pt\kern 0.27995pt},\alpha}$ describes Bogoliubov excitations. Substituting
${\tilde{u}}_{{\hbox{\scripti p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt},\sigma}^{\alpha}(t)=u_{{\hbox{\scripti p}\kern-0.14003pt\kern
0.0pt\kern 0.27995pt},\sigma}^{\alpha}e^{-{\rm i}(E+\mu)t}$,
${\tilde{v}}_{{\hbox{\scripti p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt},\sigma}^{\alpha}(t)=v_{{\hbox{\scripti p}\kern-0.14003pt\kern
0.0pt\kern 0.27995pt},\sigma}^{\alpha}e^{-{\rm i}(E-\mu)t}$, and
${\tilde{\Delta}}_{s}(t)=\Delta_{s}e^{-2{\rm i}\mu t}$ (where $\Delta_{s}$ is
assumed to be real) into Eq. (6), we obtain the ordinary (time-independent)
BdG equation as,

$\displaystyle\left(\begin{array}[]{cccc}\xi_{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern 0.27995pt}&\lambda p_{z}&0&\Delta_{s}\\\
\lambda p_{z}&\xi_{\hbox{\scripti p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt}&-\Delta_{s}&0\\\ 0&-\Delta_{s}&-\xi_{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern 0.27995pt}&\lambda p_{z}\\\
\Delta_{s}&0&\lambda p_{z}&-\xi_{\hbox{\scripti p}\kern-0.14003pt\kern
0.0pt\kern 0.27995pt}\end{array}\right){u}_{0}=E{u}_{0},$ (17)

where ${u}_{0}=(u_{{\hbox{\scripti p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt},\uparrow}^{\alpha},u_{{\hbox{\scripti p}\kern-0.14003pt\kern
0.0pt\kern 0.27995pt},\downarrow}^{\alpha},v_{{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt},\uparrow}^{\alpha},v_{{\hbox{\scripti p}\kern-0.14003pt\kern
0.0pt\kern 0.27995pt},\downarrow}^{\alpha})^{T}$. The $s$-wave order parameter
$\Delta_{s}=-(U_{s}/2)\sum_{{\hbox{\scripti p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt},\alpha}u_{{\hbox{\scripti p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt},\uparrow}^{\alpha}v_{{\hbox{\scripti p}\kern-0.14003pt\kern
0.0pt\kern 0.27995pt},\downarrow}^{\alpha*}$ and the number equation
$N=(1/2)\sum_{{\hbox{\scripti p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt},\sigma,\alpha}|v_{{\hbox{\scripti p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt},\sigma}^{\alpha}|^{2}$ in the BdG formalism reproduce Eqs. (3) and
(4), respectively. The eigen-energy equals $E_{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern 0.27995pt}^{\pm}$ given below Eq. (4).

We now examine the time-evolution of the system, after the sudden change of
the pairing interaction from the $s$-wave type to a $p$-wave one ($t\geq 0$).
As a possible situation in an ultracold Fermi gas, we consider the case when
an external magnetic field is rapidly tuned from an $s$-wave Feshbach
resonance field (which gives $U_{s}$ in Eq. (2)) to a $p$-wave Feshbach
resonance field of the $\uparrow$-spin component, giving the $p$-wave pairing
interaction,

$V_{p}={1\over 2}\sum_{{\hbox{\scripti p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt},{\hbox{\scripti p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt}^{\prime},{\hbox{\scripti q}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt}}U_{p}({p},{p}^{\prime})c_{{\hbox{\scripti p}\kern-0.14003pt\kern
0.0pt\kern 0.27995pt}+{\hbox{\scripti q}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt}/2,\uparrow}^{\dagger}c_{-{\hbox{\scripti p}\kern-0.14003pt\kern
0.0pt\kern 0.27995pt}+{\hbox{\scripti q}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt}/2,\uparrow}^{\dagger}c_{-{\hbox{\scripti p}\kern-0.14003pt\kern
0.0pt\kern 0.27995pt}^{\prime}+{\hbox{\scripti q}\kern-0.14003pt\kern
0.0pt\kern 0.27995pt}/2,\uparrow}c_{-{\hbox{\scripti p}\kern-0.14003pt\kern
0.0pt\kern 0.27995pt}^{\prime}+{\hbox{\scripti q}\kern-0.14003pt\kern
0.0pt\kern 0.27995pt}/2,\uparrow},$ (18)

where $U_{p}({p},{p}^{\prime})=-4\pi g_{p}\sum_{L_{z}=0,\pm
1}F_{\hbox{\scripti p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt}^{*}F_{{\hbox{\scripti p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt}^{\prime}}Y_{1,L_{z}}^{*}({\hat{p}})Y_{1,L_{z}}({\hat{p}}^{\prime})$Ho
; Iskin2 , and $Y_{1,L_{z}}({\hat{p}})$ are spherical harmonics, with
${\hat{p}}={p}/|{p}|$. $F_{\hbox{\scripti p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt}=|{p}|p_{0}/[|{p}|^{2}+p_{0}^{2}]$, where $p_{0}$ is a cutoff
momentum, which we take $p_{0}=10k_{\rm F}$ in this letterLiao ; note . The
coupling constant $g_{p}$ is related to the $p$-wave scattering volume $v_{p}$
as,

${4\pi v_{p}p_{0}^{2}\over m}=-{g_{p}\over 1-g_{p}\sum_{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern 0.27995pt}F_{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern 0.27995pt}^{2}/(2\varepsilon_{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern 0.27995pt})}.$ (19)

Among the three $p$-wave Cooper channels ($L_{z}=0,\pm 1$), the
$(L_{z}=0)$-component only produces a non-vanishing $p$-wave superfluid order
parameter $\Delta_{\uparrow,\uparrow}^{p}({p})$ in the presence of triplet
pair amplitude $\Phi_{\uparrow,\uparrow}^{\rm t}({p})$ in Eq. (5). (Note that
$Y_{1,0}\propto p_{z}$.) Thus, we only retain this channelIskin1 . In
addition, we turn off the spin-orbit coupling ($\lambda=0$) when $t\geq 0$,
and also assume the absence of any other residual interactions, for
simplicity. The TDBdG equation for $t\geq 0$ is then given by Eq. (6), where
${\hat{H}}_{s}$ is replaced by

$\displaystyle{\hat{H}}_{p}=\left(\begin{array}[]{cccc}\varepsilon_{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt}&0&\Delta_{\uparrow,\uparrow}^{p}({p},t)&0\\\
0&\varepsilon_{\hbox{\scripti p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt}&0&0\\\
\Delta_{\uparrow,\uparrow}^{p*}({p},t)&0&-\varepsilon_{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern 0.27995pt}&0\\\
0&0&0&-\varepsilon_{\hbox{\scripti p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt}\end{array}\right).$ (24)

The initial condition is given by ${\tilde{u}}(0)={u}_{0}$, where ${u}_{0}$ is
obtained from the BdG equation (17) for the $s$-wave superfluid state. The
$p$-wave superfluid order parameter
$\Delta_{\uparrow,\uparrow}^{p}({p},t)=F_{p}^{*}Y_{1,0}^{*}({\hat{p}})\Delta_{p}(t)$
in Eq. (24) is calculated from

$\Delta_{p}(t)=-2\pi g_{p}\sum_{\hbox{\scripti p}\kern-0.14003pt\kern
0.0pt\kern 0.27995pt}F_{\hbox{\scripti p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt}Y_{1,0}({\hat{p}})\sum_{\alpha=\pm}{\tilde{u}}_{{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt},\uparrow}^{\alpha}(t){\tilde{v}}_{{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern 0.27995pt},\uparrow}^{\alpha*}(t).$ (25)

At $t=0$, the $p$-wave superfluid order parameter $\Delta_{p}(t=0)$ is
evaluated by using ${u}_{0}$ prepared in the $s$-wave superfluid state. Since
the $\uparrow$-spin component is completely disconnected from the
$\downarrow$-spin component when $t\geq 0$, one may only deal with the
$\uparrow$-spin component in the TDBdG equation (6), when $t\geq 0$.

Figure 3: (Color online) Time evolution of the $p$-wave superfluid order
parameter $\Delta_{p}(t\geq 0)$, after the $s$-wave pairing interaction is
suddenly replaced by the $p$-wave one $V_{p}$ in Eq. (18). (a) A$\to$D. (b)
B$\to$E. (c) C$\to$F. Here, (A, B, C) and (D, E, F) represent the $s$-wave
interaction strengths and the $p$-wave interaction strengths shown in Figs. 2
and 4, respectively. In each case, the filled circle at $t=0$ shows the
initial value $|\Delta_{p}(t=0)|$. The values of $\lambda$ are the spin-orbit
coupling strengths when $t<0$. (Note that $\lambda=0$ when $t\geq 0$.) In
panel (b), although $|\Delta_{p}(t)|$ looks increasing with the time evolution
when $\lambda/v_{\rm F}=0.1$, it actually oscillates as shown in the inset.
Figure 4: (Color online) Calculated Fermi chemical potential $\mu$ in the
$p$-wave BCS ground state with the pairing interaction $V_{p}$ in Eq. (18).
The inset shows the $p$-wave superfluid order parameter $\Delta_{p}$. These
are obtained within the framework of the BCS-Leggett theory. In the inset, the
solid squares are $|\Delta_{p}(t\varepsilon_{\rm F}=100)|$ in Fig. 3 when
$\lambda/v_{\rm F}=0.5$.

Figure 3 shows the time evolution of the $p$-wave order parameter
$\Delta_{p}(t\geq 0)$. In this figure, the expected non-vanishing $p$-wave
superfluid order parameter is found to really be obtained at $t=0$ from the
triplet pair amplitude $\Phi^{\rm t}_{\uparrow,\uparrow}({p})$ which has
already been induced in the $s$-wave superfluid state (solid circles in Fig.
3). When the interaction is tuned from the $s$-wave weak-coupling BCS regime
(“A” in Fig. 2, where $\mu+m\lambda^{2}/2\sim\varepsilon_{\rm F}$) to the
$p$-wave weak-coupling regime (“D” in Fig. 4, where $\mu\sim\varepsilon_{\rm
F}$), Fig. 3(a) shows that, although $\Delta_{p}(t)$ initially increases with
the time evolution, it decreases to vanish when $\lambda/v_{\rm F}=0.1$. Thus,
our idea does not work in this case. On the other hand, $\Delta_{p}(t)$
survives even at $t\varepsilon_{\rm F}=160$ for $\lambda/v_{\rm F}=0.5,~{}1$.
When we use the typical value $\varepsilon_{\rm F}=1$ $\mu$K in an ultracold
Fermi gas, one finds that $t\varepsilon_{\rm F}\sim 10^{-2}$ ms. Thus, these
two cases can be used to produce a $p$-wave superfluid Fermi gas within the
short lifetime of $p$-wave pairs ($=5\sim 20$ msChevy ).

Figures 3(b) and (c), respectively, show the cases when the interaction is
tuned from the $s$-wave intermediate-coupling regime (“B” in Fig. 2, where
$(k_{\rm F}a_{s})^{-1}=0$) to the $p$-wave intermediate-coupling regime (“E”
in Fig. 4, where $\mu\sim 0$), and the $s$-wave strong-coupling BEC regime
(“C” in Fig. 2, where $\mu+m\lambda^{2}/2<0$) to the $p$-wave strong-coupling
regime (“F” in Fig. 4, where $\mu<0$). These results confirm the validity of
our idea, that is, apart from the oscillating behaviors seen in panel (b),
$\Delta_{p}(t)$ approaches a nearly constant value within shorter time scale
than the lifetime of $p$-wave pairs.

Figure 5: (Color online) Calculated momentum distribution $n_{{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt},\uparrow}(t)=(1/2)\sum_{\alpha}|\tilde{v}_{{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern 0.27995pt},\uparrow}^{\alpha}(t)|^{2}$ in
the case of Fig. 3(b) when $\lambda/v_{\rm F}=0.5$.
${\tilde{n}}_{{\hbox{\scripti p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt},\uparrow}$ is the momentum distribution in the $p$-wave BCS ground
state.

We note that, although non-vanishing $\Delta_{p}(t)$ is obtained, it is
smaller than that expected in the $p$-wave BCS ground state (see the inset in
Fig. 4). This is because TDBdG ignores the relaxation of the system into the
ground state. Indeed, as shown in Fig. 5, the momentum distribution
$n_{{\hbox{\scripti p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt},\uparrow}(t)=(1/2)\sum_{\alpha}|\tilde{v}_{{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern 0.27995pt},\uparrow}^{\alpha}(t)|^{2}$ is
different from that in the $p$-wave BCS ground state, even at
$t\varepsilon_{\rm F}=100$. Instead, $n_{{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern 0.27995pt},\uparrow}(t\varepsilon_{\rm
F}=100)$ is rather close to the initial distribution $n_{{\hbox{\scripti
p}\kern-0.14003pt\kern 0.0pt\kern 0.27995pt},\uparrow}(t=0)$, which is just
the momentum distribution in the $s$-wave superfluid state when $t<0$.

To summarize, we have discussed a possible idea to realize a $p$-wave
superfluid Fermi gas. To overcome the difficulty that a $p$-wave pairing
interaction damages the system before the enough growth of $p$-wave
condensate, our idea first prepares a $p$-wave pair amplitude in a spin-orbit
coupled $s$-wave superfluid Fermi gas, without relying on any $p$-wave
interactions. Then, by suddenly change the pairing interaction from the
$s$-wave type to an appropriate $p$-wave one, we can immediately reach the
$p$-wave superfluid phase, at least just after this manipulation. In this
letter, we numerically assessed this scenario within the analyses by a time-
dependent Bogoliubov-de Gennes equation (TDBdG) at $T=0$.

Since TDBdG cannot treat the energy relaxation of the system into the ground
state, the $p$-wave superfluid phase obtained in this letter is different from
the equilibrium BCS ground state. Although it is an interesting future problem
to include this relaxation effect, since the magnitude of the $p$-wave
superfluid order parameter obtained in this letter is smaller than the
equilibrium mean-field BCS value (see the inset in Fig. 4), this effect is
expected to enhance $\Delta_{p}(t)$ compared to the present result.

Including the lifetime of $p$-wave pairs is also another crucial future
challenge, in order to assess our proposal quantitatively. However, at least
qualitatively, TDBdG analyses confirmed the possibility that one may produce
$p$-wave superfluid order parameter without suffering from the serious
lifetime effect that all the current experiments are facing, our results would
be still important for the research toward the realization of unconventional
pairing states in an ultracold Fermi gas.

In this letter, we have focused on the assessment of our idea, so that we did
not discuss detailed physical properties of the $p$-wave state at $t\geq 0$,
such as the oscillation of $\Delta_{p}(t)$ seen in Fig. 3, as well as the dip
structure of $n_{{\hbox{\scripti p}\kern-0.14003pt\kern 0.0pt\kern
0.27995pt},\uparrow}(t\varepsilon_{\rm F}=100)$ seen in Fig. 5. Since our idea
does not always work as shown in Fig. 3(a), we should also clarify the
condition for this idea. We will discuss these issues in our future papers.

###### Acknowledgements.

This work was supported by KiPAS project in Keio University, as well as Grant-
in-aid for Scientific Research from MEXT and JSPS in Japan (No.JP16K17773,
No.JP15H00840, No.JP15K00178, No.JP16K05503).

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  * (37) Choosing this value of $p_{0}$ is simply due to computational problems. The cutoff-momentum $p_{0}$ is related to the effective range $p_{\rm eff}$ asHo ; Iskin2 , $p_{\rm eff}=-(\pi p_{0}^{2}/m^{2})\sum_{\hbox{\scripti p}\kern-0.14003pt\kern 0.0pt\kern 0.27995pt}F_{\hbox{\scripti p}\kern-0.14003pt\kern 0.0pt\kern 0.27995pt}^{2}/\varepsilon_{\hbox{\scripti p}\kern-0.14003pt\kern 0.0pt\kern 0.27995pt}^{2}-2/(p_{0}^{2}v_{p})$. In a 40K Fermi gaspwve2 , as well as in a 6Li Fermi gaspwve7 , $p_{\rm eff}$ has been estimated as $p_{\rm eff}\cong-0.05a_{\rm B}^{-1}\sim-0.02a_{\rm B}^{-1}$ near a $p$-wave Feshbach resonance (where $a_{\rm B}$ is the Bohr radius). This gives $p_{0}\sim 10^{2}k_{\rm F}\gg 10k_{\rm F}$ for the typical value of the Fermi momentum, $k_{\rm F}^{-1}\sim 100$ nm. Although our smaller $p_{0}=10k_{\rm F}$ is expected to qualitatively not affect our results, $|\Delta_{p}(t)|$ is considered to be underestimatedIskin1 .