$p$-wave stabilization of three-dimensional Bose-Fermi solitons

We explore bright soliton solutions of ultracold Bose-Fermi gases, showing that the presence of $p$-wave interactions can remove the usual collapse instability and support stable soliton solutions that are global energy minima. A variational model that incorporates the relevant $s$- and $p$-wave interactions in the system is established analytically and solved to probe the dependencies of the soliton stationary states on key experimental parameters. Under attractive $s$-wave interactions, bright solitons exist only as metastable states susceptible to collapse. Remarkably, the presence of repulsive $p$-wave interactions alleviates this collapse instability. This dramatically widens the range of experimentally achievable soliton solutions and indicates greatly enhanced robustness. While we focus specifically on the boson-fermion pairing of ${}^{87}$Rb and ${}^{40}$K, the stabilization inferred by repulsive $p$-wave interactions should apply to the wider remit of ultracold Bose-Fermi mixtures.

Figure 1

(a) The bands of soliton solutions in $a_{BFs}-N_F$ space for a ${}^{87}$Rb-${}^{40}$K mixture with zero $p$-wave interaction. We present a fixed number of bosons $N_B={10}^5$ and various trap frequencies ${\omega }_B/2\pi =10$ (green solid line), 100 (blue dotted line), and 1000 (black dashed line) Hz. The inset provides a close up of the soliton bands. (b) Energy landscapes (${\omega }_B/2\pi =10$ Hz, $N_F=500$, and $N_B={10}^5$) showing four distinct regimes: (i) well above the soliton band ($a_{BFs}=-6200a_0$), (ii) just above the band ($a_{BFs}=-6260a_0$), (iii) within the band ($a_{BFs}=-6296a_0$), and (iv) below the band ($a_{BFs}=-6350a_0$). These plots corresponds to the crosses in (a). In (iii) the soliton solution is highlighted by the white cross.

Figure 2

(a) The bands of soliton solutions in $a_{BFs}-N_F$ space for no $p$-wave contributions. We consider fixed trapping ${\omega }_B/2\pi =100$ Hz and various boson numbers of $N_B=500$ (green), 5000 (black), 50 000 (red), and 100 000 (blue). (b) Log-log plot of the soliton bands in (a), with the analytic predictions of Eqs. (11) (dashed line) and (12) (dotted line). The gray arrows indicate the direction of increasing $N_B.$

Figure 3

(a) Soliton bands in $a_F-N_F$ space for boson-fermion interaction $a_{BFs}=-215a_0$ in the presence of fermion-fermion $p$-wave interactions. The number of bosons is fixed to $N_B=100000$ and we present trap frequencies of ${\omega }_B/2\pi =10$ (horizontal hatch), 100 (vertical hatch), and 1000 Hz (gray region). (b) Energy landscapes (for ${\omega }_B/2\pi =100$ and $N_F=20000$) of (i) below the band ($a_F=50a_0$), (ii) within the band ($a_F=125a_0$), and (iii) above the band ($a_F=180a_0$). The locations of these cases are indicated in plot (a).

Figure 4

Soliton bands in $a_F-N_F$ space for fixed trap frequency ${\omega }_B/2\pi =100$ Hz and boson numbers of $N_B=10000$ (solid), $50000$ (dashed), and $100000$ (dotted).

Figure 5

Soliton bands (as shown by their boundary lines) in the parameter space of $a_F$ and $a_{BFs}$. (a) We fix $N_B=100000$ and $N_F=500$, and use ${\omega }_B/2\pi =10$ (red dashed line), 100 (green solid line), and 1000 (blue dot-dashed line) Hz. (b) Energy landscapes for the ${\omega }_B/2\pi =10$ case at the points (i) $a_F=-2000a_0$, $a_{BFs}=-6280a_0$; (ii) $a_F=-2000a_0$, $a_{BFs}=-6295a_0$; (iii) $a_F=-2000a_0$, $a_{BFs}=-6320a_0$; (iv) $a_F=2000a_0$, $a_{BFs}=-6280a_0$; (v) $a_F=3000a_0$, $a_{BFs}=-6320a_0$; and (iv) $a_F=4000a_0$, $a_{BFs}=-6280a_0$. These points are denoted in (a).

Figure 6

Soliton bands in the parameter space $a_F-a_{BFs}$ (shown by their boundary lines). We fix ${\omega }_B/2\pi =100$ Hz and $N_F=500$ and consider $N_B=1000$ (dashed line), $10000$ (solid line), and $100000$ (dotted line).