Structure and dynamics of a rotating superfluid Bose-Fermi mixture

We investigate the structure and dynamics of a rotating superfluid Bose-Fermi mixture (SBFM) made of superfluid bosons and two-component (spin-up and -down) superfluid fermions. A ground-state phase diagram for the nonrotating case of a SBFM with specific parameters is given, where the ground-state configuration of a nonrotating SBFM is mainly determined by the boson-fermion interaction. For the rotating case of a SBFM with a sufficiently large rotation frequency, we show that the system supports a mixed phase and three typical layer-separated phases. In particular, the visible vortex formation in the fermionic superfluid exhibits a remarkable hysteresis effect during the dynamical evolution of a rotating SBFM, which is evidently different from the case of rotating two-component Bose-Einstein condensates.

Figure 1

Ground-state phase diagram of a nonrotating superfluid ${}^7\mathrm{Li}\text{−}{}^6\mathrm{Li}$ (Bose-Fermi) mixture, where $R_1=a_{bf}/a_b$ and $R_2=a_f/a_b$. The parameters are $N_b=1000,N_p=100$, and $a_b=50$ nm. Regions I–IV represent the system collapse regime, mixed phase, layer-separated phase, and inlaid separated phase, respectively.

Figure 2

Density profiles (a)–(c) correspond to phases II–IV in Fig. 1, respectively. Here, 1 labels the bosonic superfluid and 2 denotes the fermionic one of the system. (a) $R_1=-0.4,R_2=-20,$ (b) $R_1=1,R_2=-20,$ and (c) $R_1=6,R_2=-20$. The other parameters are the same as those in Fig. 1. The darker color area indicates the lower density. $x$ and $y$ are in units of $d_0$.

Figure 3

Steady density profiles (the top two rows) and phase profiles (the bottom two rows) at $t=500$ after rotating the superfluid Bose-Fermi mixture with $\Omega =0.94$, where 1 and 3 denote the bosonic superfluid, while 2 and 4 represent the fermionic superfluid. The initial states corresponding to three cases of $a,b,$ and $c$ (from left to right) are illustrated in Figs. 2(a1,a2), Figs. 2(b1,b2), and Figs. 2(c1,c2), respectively. The other parameters are the same as those in Figs. 1 and 2. The value of phase varies continuously from 0 to $2\pi$. The darker color area indicates lower density or phase. Here, $x$ and $y$ are in units of $d_0$.

Figure 4

(a1),(a2) Ground-state density profiles of a static superfluid ${}^7\mathrm{Li}\text{−}{}^6\mathrm{Li}$ mixture with $N_b={10}^3$ and $N_p={10}^5$. (b1),(b2) The steady density profiles at $t=500$ for the superfluid Bose-Fermi system rotating with $\Omega =0.92$. (c1),(c2) The corresponding phase profiles. The relevant parameters are $a_b=5$ nm, $R_1=6$, and $R_2=-20$. Here, 1 denotes the bosonic superfluid, 2 represents the fermionic superfluid, and $x$ and $y$ are in units of $d_0$. The darker color area indicates lower density or phase.

Figure 5

(a1),(a2) Ground-state density profiles of a static superfluid ${}^7\mathrm{Li}\text{−}{}^6\mathrm{Li}$ mixture with $N_b={10}^4$ and $N_p={10}^5$. (b1),(b2) The steady density profiles at $t=500$ for the superfluid Bose-Fermi system rotating with $\Omega =0.96$. (c1),(c2) The corresponding phase profiles. The relevant parameters are $a_b=50$ nm, $R_1=6$, and $R_2=-0.1$. Here, 1 denotes the bosonic superfluid, 2 represents the fermionic superfluid, and $x$ and $y$ are in units of $d_0$. The darker color area indicates lower density or phase.

Figure 6

Time evolutions of the density profiles $|{\psi }_b{|}^2$ (row 1) and $|{\psi }_p{|}^2$ (row 2) and the corresponding phase profiles (rows 3 and 4) after the superfluid Bose-Fermi system rotates with $\Omega =0.96$, where the other parameters are the same as those in Fig. 5. The value of the phase varies continuously from 0 to $2\pi$. The time is (a1)–(a4) $t=11.5$, (b1)–(b4) $t=20$, and (c1)–(c4) $t=60$. The dark color area indicates the lower density or phase. Here $x$ and $y$ are in units of $d_0$, and $t$ is in units of $1/{\omega }_{bf}$.