Proper phase imprinting method for a dark soliton excitation in a superfluid Fermi mixture

It is common knowledge that a dark soliton can be excited in an ultracold atomic gas by means of the phase imprinting method. We show that, for a superfluid fermionic mixture, the standard phase imprinting procedure applied to both components fails to create a state with symmetry properties identical to those of the dark soliton solution of the Bogoliubov–de Gennes equations. To produce a dark soliton in the BCS regime, a single component of the Fermi mixture should be phase imprinted only.

Figure 1

Panel (a) shows the squared overlap, $|\langle {\psi }_{\text{sol}}|\psi \left(t\right)\rangle {|}^2$, between the wave function corresponding to the stationary dark soliton solution, Eq. (11), and a similar wave function related to the pairing function $\Delta (\mathbf{r},t)$ that is obtained by means of the phase imprinting procedure applied to both components of the Fermi mixture. In panel (b) we present ${\left|\psi (0,0,z,t)\right|}^2$ at $t=10$, after the application of the phase imprinting procedure (black solid line) and for the stationary dark soliton state (red dashed line). Note that some fraction of superfluid fermions is lost during the excitation process, i.e., $\sqrt{\langle \Delta \left(0\right)\left|\Delta \right(0)\rangle }\approx 30$ while $\sqrt{\langle \Delta \left(t\right)\left|\Delta \right(t)\rangle }\approx 23$ at $t=10$. Panel (c) shows the phase of $\psi (0,0,z,t)$ at $t=10$. In the numerical simulation we have chosen interaction strength $g=2$, chemical potential $\mu =20$, length of the box along the transverse directions $L_{\perp }=3$, space grid in the $z$ direction $\delta z=0.08$, energy cutoff $E_c=70$, phase imprinting period $\tau ={10}^{-3}$, and $I_0$ fulfilling $I_0\tau =\pi /2$. Harmonic oscillator units (related to the harmonic trap in the $z$ direction) are used. The parameters correspond to the BCS regime, i.e., $1/\left(k_{F}a\right)=-1$ where $k_F=\sqrt{2\mu }$ and $a=-g/\left(4\pi \right)$.

Figure 2

Same as in Fig. 1 but for the case when the phase imprinting procedure is applied to one component of the Fermi mixture only. In the numerical simulation all parameters are identical to those in Fig. 1 except that $I_0$ is chosen so that $I_0\tau =\pi$. Note that the phase jump near $z=0,$ visible in panel (c), is well preserved during the temporal evolution, in contrast with the two-component phase imprinting procedure [see Fig. 1].