Rotating quantum droplets confined in an anharmonic potential

We investigate the rotational properties of quantum droplets, which form in a mixture of two Bose-Einstein condensates, in the presence of an anharmonic trapping potential. We identify various phases as the atom number and the angular momentum or angular velocity of the trap vary. These phases include center-of-mass–like excitation (without or with vortices), vortices of single and multiple quantization, etc. Finally, we compare our results with those of the single-component problem.

Figure 1

(a)–(c) Density (left column) and phase (right column) of the droplet order parameter for $N=50$, $\omega =0.05$, $\lambda =0.05$, and (a) $\ell =0.0$, (b) $\ell =4.0$, and (c) $\ell =8.0$. Here the density is measured in units of ${\mathrm{\Psi }}_0^2$ and the length in units of $x_0$. (d) Corresponding dispersion relation as a function of $L$. Here the energy is measured in units of $E_0$ and the angular momentum in units of $\hslash$.

Figure 2

(a)–(d) Density (left column) and phase (right column) of the droplet order parameter for $N=100$, $\omega =0.05$, $\lambda =0.05$, and (a) $\ell =0.0$, (b) $\ell =0.5$, (c) $\ell =1.0$, and (d) $\ell =3.5$. Here the density is measured in units of ${\mathrm{\Psi }}_0^2$ and the length in units of $x_0$. (e) and (f) Same as (a)–(d) but for $\ell =4.0$, and 10.0, respectively. (g) Corresponding dispersion relation, in the rotating frame, i.e., $E_{\mathrm{rot}}\left(\ell \right)-E(\ell =0)$ as a function of $\ell$, with $\mathrm{\Omega }=0.054$. Here the energy is measured in units of $E_0$ and the angular momentum in units of $\hslash$.

Figure 3

(a)–(d) Density (left column) and phase (right column) of the droplet order parameter for $N=150$, $\omega =0.05$, $\lambda =0.05$, and (a) $\ell =2.0$, (b) $\ell =3.5$, (c) $\ell =5.0$, and (d) $\ell =8.0$. Here the density is measured in units of ${\mathrm{\Psi }}_0^2$ and the length in units of $x_0$. (e) Corresponding dispersion relation, in the rotating frame, i.e., $E_{\mathrm{rot}}\left(\ell \right)-E(\ell =0)$ as a function of $\ell$, with $\mathrm{\Omega }=0.054$ (solid curve) and $\mathrm{\Omega }=0.058$ (dashed curve). Here the energy is measured in units of $E_0$ and the angular momentum in units of $\hslash$.

Figure 4

(a)–(f) Density (left column) and phase (right column) of the droplet order parameter for $N=200$, $\omega =0.05$, $\lambda =0.05$, and (a) $\ell =0.0$, (b) $\ell =0.5$, (c) $\ell =1.0$, (d) $\ell =1.5$, (e) $\ell =2.0$, and (f) $\ell =2.5$. Here the density is measured in units of ${\mathrm{\Psi }}_0^2$ and the length in units of $x_0$. (g)–(l) Same as (a)–(f) but for $\ell =3.0$, 3.5, 4.0, 4.5, 5.0, and 5.5, respectively.

Figure 5

(a)–(c) Density (left column) and phase (right column) of the droplet order parameter for $N=200$, $\omega =0.05$, $\lambda =0.05$, and (a) $\ell =6.0$, (b) $\ell =6.5$, and (c) $\ell =8.0$. Here the density is measured in units of ${\mathrm{\Psi }}_0^2$ and the length in units of $x_0$. (d) Corresponding dispersion relation, in the rotating frame, i.e., $E_{\mathrm{rot}}\left(\ell \right)-E(\ell =0)$ as a function of $\ell$, with $\mathrm{\Omega }=0.054$. Here the energy is measured in units of $E_0$ and the angular momentum in units of $\hslash$.

Figure 6

Function $\ell =\ell \left(\mathrm{\Omega }\right)$ for the four values of (a) $N=50$, (b) $N=100$, (c) $N=150$, and (d) $N=200$ that we consider. Dashed vertical lines denote a transition from a mixed state to a localized state (see the text for details). Here the angular momentum is measured in units of $\hslash$.

Figure 7

Various phases that we have derived, on the $\mathrm{\Omega }/\omega -N$ plane, which correspond to the absolute minima of the energy in the rotating frame (see Sec. 3e), for (a) slow and (b) rapid rotation. On the horizontal axis is $\mathrm{\Omega }/\omega$ and on the vertical is $N$. Here different phases are denoted by the following symbols: $\circ$, the nonrotating ground state; $\times$, vortices of single quantization or vortex lattices; $✳$, vortices of multiple quantization; $•$, the mixed phase (see the text for details); and $•$, the center-of-mass–like localized state.