Creating superfluid vortex rings in artificial magnetic fields

Artificial gauge fields are versatile tools that allow the dynamics of ultracold atoms in Bose-Einstein condensates to be influenced. Here we discuss a method of artificial gauge field generation stemming from the evanescent fields of the curved surface of an optical nanofiber. The exponential decay of the evanescent fields leads to large gradients in the generalized Rabi frequency and therefore to the presence of geometric vector and scalar potentials. By solving the Gross-Pitaevskii equation in the presence of the artificial gauge fields originating from the fundamental Hybrid mode (${\mathrm{HE}}_{11}$) mode of the fiber, we show that vortex rings can be created in a controlled manner. We also calculate the magnetic fields resulting from the higher order ${\mathrm{HE}}_{21}$, Transverse electric mode (${\mathrm{TE}}_{01}$), and Transverse magnetic mode (${\mathrm{TM}}_{01}$) and compare them to the fundamental ${\mathrm{HE}}_{11}$ mode.

Figure 1

Magnetic fields strength $B(x,y)$ stemming from a linearly polarized ${\mathrm{HE}}_{11}$ mode in units of $B_0=\hslash k_0^2/2$. (a) $\widehat{r}$ and (b) $\widehat{\phi }$ components for polarization in the $x$ direction; (c) $\widehat{r}$ and (d) $\widehat{\phi }$ components for polarization along $y$. The wavelength and power of the blue-detuned light field are chosen as ${\lambda }_B=700$ nm and $P_B=30$ mW, and the components of the dipole moment are $d_r=1,d_{\phi }=0$, and $d_z=0$. Note that the axes in panels (c) and (d) are adjusted.

Figure 2

Magnetic field strength $B(x,y)$, in units of $B_0=\hslash k_0^2/2$, for atoms trapped outside a fiber of radius $a=200$ nm. The wavelength and power of blue- and red-detuned light fields are ${\lambda }_B=700$ nm, $P_B=30$ mW, and ${\lambda }_R=1060$ nm, $P_R=20$ mW. The components of the dipole moment are chosen to be $d_r=1,d_{\phi }=0$ and $d_z=0$. The polarizations of the input light fields are (a) both circular, (b) red circular and blue linear along $x$, (c) red linear along $y$ and blue linear along $x$, and (d) red linear along $y$ and blue linear along $y$. Note that the axes in panel (d) are adjusted.

Figure 3

Magnetic field $B\left(r\right)$ scaled in units of $B_0=\hslash k_0^2/2$ outside a fiber with radius $a=200$ nm for different values of the parameter $\tilde{s}$. Blue (with asterisk) curve: $\tilde{s}=0.3$; red (with dots) curve: $\tilde{s}=1$. For both values the components of the dipole moment are chosen to be $d_r=1,d_{\phi }=0$, and $d_z=0$. The inset (left axis) shows the change of the magnitude of the field maximum ($B_{\text{max}}/B_0$) as a function of $\tilde{s}$ (black dashed line). The right axis of the inset shows the location $r_{\text{max}}$ of maximum of the magnetic field (green solid line). The wavelengths and powers of the blue- and red-detuned light fields are the same as in Fig. 2.

Figure 4

(a) Magnetic field strength as a function of distance from the nanofiber surface. (b) Density profile for a BEC trapped in the harmonic potential on left and right sides outside an optical nanofiber. The vortices visible are the result of the presence of the artificial magnetic field created by the evanescent field outside the fiber. The artificial magnetic field used in the calculations corresponds to the laser parameters used in Fig. 2 with $\tilde{s}=0.7$ and $d_r=1,d_{\varphi }=0,d_z=0$.

Figure 5

Same as Fig. 4, but for $\tilde{s}=0.85$. One can see that the broader magnetic field distribution leads to the appearance of multiple vortices.

Figure 6

Magnetic fields $B\left(r\right)$, scaled in units of $B_0=\hslash k_0^2/2$, for the higher order modes ${\mathrm{HE}}_{21}$ (red line with dots), ${\mathrm{TE}}_{01}$ (green line with cross), and ${\mathrm{TM}}_{01}$ (black line with triangles), and compared to the fundamental ${\mathrm{HE}}_{11}$ mode (blue line with asterisk). The wavelength of blue-detuned light field is ${\lambda }_B=780$ nm and dipole moment components are $d_r=1,d_{\varphi }=1,d_z=1$ respectively, with parameter $\tilde{s}=5$. The fiber radius is chosen as $a=400$ nm.