Stationary solutions for the $2+1$ nonlinear Schrödinger equation modeling Bose-Einstein condensates in radial potentials

Stationary solutions for the $2+1$ cubic nonlinear Schrödinger equation modeling Bose-Einstein condensates (BEC) in a small potential are obtained via a form of perturbation. In particular, perturbations due to small potentials which either confine or repel the BECs are studied, and under arbitrary piecewise continuous potentials, we obtain the general representation for the perturbation theory of radial BEC solutions. Numerical results are also provided for regimes where perturbative results break down (i.e., the large-potential regime). Both repulsive and attractive BECs are considered under this framework. Solutions for many specific confining potentials of physical relevance to experiments on BECs are provided in order to demonstrate the approach. We make several observations regarding the influence of the particular small potentials on the behavior of the BECs.

Figure 1

Repulsive solutions under the constant potential $U\left(r\right)=\lambda$. When either $\lambda <0$ or $\epsilon \ll 1$, the solutions decay asymptotically as $r\to \infty$. The density associated to such solutions has a maximal value at the origin, with progressively smaller density maxima occurring as the radial distance from the origin increases. For the case of large $\epsilon$ and $\lambda >0$, solutions appear to become unstable. For small $\epsilon$, perturbation solutions are used. For large $\epsilon$, numerical solutions are used.

Figure 2

Repulsive solutions under the harmonic oscillator potential, which takes the form $U\left(r\right)=\lambda r^2$. Solutions for $\lambda >0$ are confined to values near the origin (with the increase in the graphs attributed to boundary effects). Solutions for $\lambda <0$ have a density maximum at the origin and smaller local density maxima decreasing in size radially from the origin. For small $\epsilon$, perturbation solutions are used. For large $\epsilon$, numerical solutions are used.

Figure 3

Repulsive solutions under the modified harmonic oscillator potential $U\left(r\right)=\lambda [r^2+\beta exp(-r^2)]$. Solutions for $\lambda >0$ are confined to the region near the origin (with the increase in the graphs attributed to boundary effects). When $\epsilon$ is small and $\lambda <0$, solutions have a density maximum at the origin and smaller local density maxima decreasing in size radially from the origin. When $\epsilon$ is large yet $\lambda <0$, the solutions attain a density maximum away from the origin, with successive smaller maxima at larger radius values. For small $\epsilon$, perturbation solutions are used. For large $\epsilon$, numerical solutions are used.

Figure 4

Repulsive solutions under the Morse potential $U\left(r\right)=\lambda (e^{-2Ar}-2e^{-Ar})$. Solutions for $\lambda >0$ are confined to values near the origin (with the increase in the graphs attributed to boundary effects). When $\epsilon$ is small and $\lambda <0$, solutions have a density maximum at the origin and smaller local density maxima decreasing in size radially from the origin. When $\epsilon$ is large, yet $\lambda <0$, the solutions attain a density maximum away from the origin, with successive smaller maxima at larger radius values. For small $\epsilon$, perturbation solutions are used. For large $\epsilon$, numerical solutions are used.

Figure 5

Repulsive solutions under the lattice potential $U\left(r\right)=\lambda [1-cos(r\left)\right]$. For small enough $\epsilon$, densities attain a maximal value at $r=0$ and gradually decay (with many smaller local density maxima occuring for larger $r)$. This behavior occurs no matter the sign of $\lambda$, since the lattice potential is weakly confining (it is bounded, even as $r\to \infty )$. Given larger values of $\epsilon$, the solutions increase without bound. However, physically this means that the solutions should be confined to the region near $r=0$, and the part of the function increasing without bound corresponds to a boundary or cutoff for the density since the solutions are confined to within the first potential well. For small $\epsilon$, perturbation solutions are used. For large $\epsilon$, numerical solutions are used.

Figure 6

Repulsive solutions under the double-well potential $U\left(r\right)=\lambda [{(r^2-1)}^2-\beta ]$. The interesting feature here is the parameter $\beta$. For small-enough $\beta$, densities attain a global maximum at the origin and then gradually decay (attaining many smaller local maxima for greater $r)$. For larger $\beta$, the first and largest density maximum occurs for some value of $r>0$ sufficiently far away from the origin. Small $\epsilon$ values are considered, hence perturbation solutions are used in the plots.

Figure 7

Repulsive solutions under the modified harmonic oscillator potential $U\left(r\right)=\lambda [r^2+\beta {cos}^2\left(r\right)]$. If $\beta <0$, the greatest density maxima occurs away from the origin, while when $\beta >0$ the largest density maxima occurs at the origin. Small $\epsilon$ values are considered, hence perturbation solutions are used in the plots.

Figure 8

Attractive solutions under the constant potential $U\left(r\right)=\lambda$. The densities tend toward an asymptotic value of ${\left|\varphi \right|}^2\to ({\epsilon }^{-1}+\lambda )/2$ as $r\to \infty$, while the densities oscillate around this value for finite $r$. When the boundary condition $\varphi \left(0\right)$ is such that ${\left|\varphi \left(0\right)\right|}^2=({\epsilon }^{-1}+\lambda )/2$ then the density remains constant.

Figure 9

Attractive solutions under the harmonic oscillator potential, which takes the quadratic form $U\left(r\right)=\lambda r^2$. Since the potential is unbounded, the solutions do not satisfy the asymptotic behavior outlined in Sec. 6. Rather, the behavior of solutions mirrors that of the repulsive case. Therefore, in the presence of the harminic oscillator potential, solution behavior is qualitatively the same in both repulsive and attractive cases.

Figure 10

Attractive solutions under the modified harmonic oscillator potential $U\left(r\right)=\lambda [r^2+\beta exp(-r^2)]$. Since this potential is a modification of the harmonic oscillator potential, we again find that there is little qualitative difference between the repulsive and the attractive solutions.

Figure 11

Attractive solutions under the Morse potential $U\left(r\right)=\lambda (e^{-2Ar}-2e^{-Ar})$. In contrast to the repulsive solutions in the presence of a Morse potential, the attractive solutions under the Morse potential tend toward a positive density value in the asymptotic large-$r$ limit assuming $A>0$ (as is standard). Indeed, for $A>0$, $U\left(r\right)\to 0$ as $r\to \infty$ and hence ${\left|\varphi \right|}^2\to {\left(2\epsilon \right)}^{-1}$ in this limit. On the other hand, for $A<0$, the Morse potential is radially unbounded, so solutions behave as would be expected in the repulsive case.

Figure 12

Attractive solutions under the lattice potential $U\left(r\right)=\lambda [1-cos(r\left)\right]$. The density exhibits nonuniform oscillations throughout the radial domain. The potential neither grows in an unbounded manner nor does it decay to some finite limit as $r\to \infty$. For this reason, the density will continue to undergo such oscillations even as $r\to \infty$.

Figure 13

Attractive solutions under the double-well potential $U\left(r\right)=\lambda [{(r^2-1)}^2-\beta ]$. Since this potential is radially unbounded, solutions will either tend to zero density as $r\to \infty$ or they will be confined to those values near $r=0$. In the plots shown (corresponding to $\lambda <0)$, the densities all eventually decay to zero. Whether the absolute maximal density occurs at $r=0$ or for some $r>0$ will depend strongly on the choice of parameters taken.

Figure 14

Attractive solutions under the modified harmonic oscillator potential $U\left(r\right)=\lambda [r^2+\beta {cos}^2\left(r\right)]$. Since this is another example of a radially unbounded potential, we see that the solutions for this potential behave similarly to those in the repulsive case.

Figure 15

Plots of the density corresponding to leading-order solutions ${\widehat{\varphi }}_0\left(R\right)$ to (61) and (62) for various large potentials. For these plots, we considered the case where the spatial scaling was large $\left({\epsilon }^{-1/2}\right)$. For cases where the spatial scaling is much smaller than the amplitude (which also scales as ${\epsilon }^{-1/2})$, one would have the Thomas-Fermi approximation and the solutions would behave as shown in (63). In all plots, $\mathrm{sgn}\left(g\right)=-1$.

Figure 16

Comparison of the density profiles obtained using perturbation and numerical methods for the repulsive BECs under the Morse potential $U\left(r\right)=(e^{-2r}-2e^{-r})$. For the analytical solution, only a first-order perturbation expansion is considered. However, for small $\epsilon$ even this first-order expansion is rather accurate. As $\epsilon$ increases, the analytical and numerical solutions begin to differ in exact value, although many of the qualitative features are preserved in the analytical solutions. As discussed before, the perturbative results are useful in the small-$\epsilon$ regime, which corresponds effectively to weak confining potentials. In the case of strong confining potentials, numerical simulations are necessary.

Figure 17

The same as in Fig. 16, only under the lattice potential $U\left(r\right)=-[1-cos(r\left)\right]$. Again, the first-order perturbation expansion agrees nicely with the numerical solution when $\epsilon$ is sufficiently small.