Formation of local and global currents in a toroidal Bose-Einstein condensate via an inhomogeneous artificial gauge field

We study the effects of a position-dependent artificial gauge field on an atomic Bose-Einstein condensate in quasi-one-dimensional and two-dimensional ring settings. The inhomogeneous artificial gauge field can induce global and local currents in the Bose-Einstein condensate via phase gradients along the ring and vortices, respectively. We observe two different regimes in the system depending on the radial size of the ring and the strength of the gauge field. For weak artificial gauge fields, the angular momentum increases, as expected, in a quantized manner; however, for stronger values of the fields, the angular momentum exhibits a linear (nonquantized) behavior. We also characterize the angular momentum for noncylindrically symmetric traps.

Figure 1

Vortex configurations. Schematic of vortices in a BEC trapped in a ring potential. We consider three possible scenarios: (a) an invisible vortex (gray dot) is located at the center of the ring, and therefore the BEC experiences a global current; in plot (b) a visible vortex (black dot) results in a local current at the vicinity of where the vortex is formed; and (c) a mixture of both visible and invisible vortices. The green arrows represent an intuitive representation of the phase winding number.

Figure 2

Schematic of a BEC cloud with wave function $\varphi (x,z)$ trapped in an external ring potential $V(x,z)$, with thickness $\delta r$ for the ring BEC. The cold gas is located in the vicinity of a dielectric prism having a reflective index of $n$. The orange line indicates the input laser field with wave vector $\mathbf{k}$ and the incident angle $\theta$.

Figure 3

(a) Artificial gauge field and (b) normalized magnetic field $B\left(z\right)/B_0$ is plotted as a function of the position $z$. The prism is located at $z=-20$. The green dashed ($-----$) and red dot-dashed ($-.-.-.-.$) lines correspond to an incident angle of $\theta -{\theta }_0=8\times {10}^{-4}$ rad and $s=10$ and $s=20$, respectively. The blue dotted ($..........$) curve corresponds to $\theta -{\theta }_0=4\times {10}^{-4}$ rad for $s=20$. The black line corresponds to our toy model $A_{\text{Model}}\left(z\right)=-0.1[arctan\left(\frac{z}{\beta }\right)-\frac{\pi }{2}]/\pi$, where $\beta =1.3$.

Figure 4

Quantized regime for a quasi-$1\mathrm{D}$ ring. The angular momentum $L_y$ of a narrow ring of the BEC is plotted with respect to the strength of the artificial gauge field $w_0$. The density plot of the BEC is shown with the location of invisible vortices for (a) $w_0=0$, (b) $w_0=1.00$, (c) $w_0=1.20$, and (d) $w_0=1.35$ with $\beta =\frac{1}{8}$. The trapping potential has $r_0=6.5$ and $v_0=5.66$. All parameters are given in harmonic oscillator units.

Figure 5

Visible and invisible vortices. The density (left figure) and phase (right figure) of the BEC trapped in a ring potential of parameters $v_0=1.34$, and radius of $r_0=6.5$. The strength of the gauge field is $w_0=1.5$ with $\beta =1/8$. The white dashes in the phase plot show the position of the trap, and the black circles indicate the locations of visible vortices.

Figure 6

Quantized and linear regimes of a $2\mathrm{D}$ ring. Density plot of the BEC ground state in a ring of radius $r_0=6.5$ that has a gauge field strength of (a) $w_0=0$, (b) $w_0=0.50$, (c) $w_0=0.80$, and (d) $w_0=1.50$ with $\beta =1/8$ and $v_0=1.34$. Panel (a) shows the density of the ground state of the cloud of the BEC when there is no artificial gauge field. As the strength of the gauge field increases, at first the invisible vortices enter the system, as can be observed in panel (b), and the angular momentum increases by quanta of one. In panel (c), there are five invisible vortices in the BEC which lead to a $10\pi$ phase. Finally, in panel (d), visible vortices enter the BEC and the angular momentum is $20\pi$. (e) presents the angular momentum of $L_y$ with respect to the strength of the artificial gauge field $w_0$. This plot shows both quantized and linear regimes of the angular momentum. For comparison, please refer to the angular momentum plot of the harmonic trap in Fig. 8, as it exhibits the same behavior.

Figure 7

Symmetry breaking of angular momentum. We plot the angular momentum of an elliptical ring versus ${\omega }_0$, as shown in Eq. (4) with $\eta =0.9, r_0=6.5, \gamma =\pi /2$, and $v_0=4$. The elliptical ring has semiminor and semimajor axes of (1,1.5), plotted for various angles: (a) $\alpha =0$, (b) $\alpha =\pi /4$, and (c) $\alpha =\pi /2$. All parameters are given in harmonic oscillator units.

Figure 8

Harmonic oscillator. Panels (a)–(d) show the density plot of a BEC trapped in a harmonic potential $V_{\text{harmonic}}=\frac{1}{2}{v}_0^2(x^2+z^2)$. The artificial gauge field has the same value as Fig. 6 with $v_0=0.22$. The strength of the gauge field is (a) $w_0=0$, (b) $w_0=0.40$, (c) $w_0=1.00$, and (d) $w_0=1.50$. In plot (e) the angular momentum $L_y$ is plotted with respect to the gauge field strength, $w_0$. All parameters are given in harmonic oscillator units.

Figure 9

Global current. Angular momentum $Ly$ as a function of the induced rotation, $\mathrm{\Omega }$, for different values of the angle of the ellipse in a potential [see Eq. (4)]. Parameters of the $2\mathrm{D}$ GPE are $g=120, V_0=15$, and $a/b=\sqrt{1.5}, a=25$, and length scale of $L=30$. For the circular ring we choose a radius of $r\approx 5.36$. All parameters are given in harmonic oscillator units.

Figure 10

Density for fixed ${\omega }_0$. Density cuts of the full density displaying the vortices along the radial direction. Insets show two corresponding full density images. Parameters of the system are $g=800$ and $w_0=0.65$, fixed for all curves.

Figure 11

Density for fixed $v_0$. Density cuts of the full density, displaying the transition from having no vortices to having vortices in the bulk. We compare the full density to the analytical Thomas-Fermi approximation. Parameters of the system are $g=800$, and fixed $v_0=0.25$ with a potential given by $v_0{(r-r_0)}^2$ and $r_0=6.5$ for all curves. All parameters are given in harmonic oscillator units.

Figure 12

Various trapping strengths. Angular momentum $L_y$ vs the strength of the gauge field, $w_0$, for different values of the potential $v_0$. Parameters of the system are $g=800$.