Stationary solutions for the nonlinear Schrödinger equation modeling three-dimensional spherical Bose-Einstein condensates in general potentials

Stationary solutions for the cubic nonlinear Schrödinger equation modeling Bose-Einstein condensates (BECs) confined in three spatial dimensions by general forms of a potential are studied through a perturbation method and also numerically. Note that we study both repulsive and attractive BECs under similar frameworks in order to deduce the effects of the potentials in each case. After outlining the general framework, solutions for a collection of 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 potentials on the behavior of the BECs in these cases, comparing and contrasting the qualitative behavior of the attractive and repulsive BECs for potentials of various strengths and forms. Finally, we consider the nonperturbative where the potential or the amplitude of the solutions is large, obtaining various qualitative results. When the kinetic energy term is small (relative to the nonlinearity and the confining potential), we recover the expected Thomas-Fermi approximation for the stationary solutions. Naturally, this also occurs in the large mass limit. Through all of these results, we are able to understand the qualitative behavior of spherical three-dimensional BECs in weak, intermediate, or strong confining potentials.

Figure 1

Density profiles for several repulsive solutions under the constant potential $U\left(r\right)=\lambda$.

Figure 2

Density profiles for several repulsive solutions under the harmonic oscillator potential, which takes the form $U\left(r\right)=\lambda r^2$.

Figure 3

Density profiles for several repulsive solutions under the modified harmonic oscillator potential $U\left(r\right)=\lambda [r^2+\beta exp\left(-r^2\right)]$.

Figure 4

Density profiles for several repulsive solutions under the Morse potential $U\left(r\right)=\lambda \left(e^{-2Ar}-2e^{-Ar}\right)$.

Figure 5

Repulsive solutions under the lattice potential $U\left(r\right)=\lambda [1-cos(r\left)\right]$.

Figure 6

Density profiles for several repulsive solutions under the double-well potential $U\left(r\right)=\lambda \left[{(r^2-1)}^2-\beta \right]$.

Figure 7

Density profiles for several repulsive solutions under the modified harmonic oscillator potential $U\left(r\right)=\lambda \left[r^2+\beta {cos}^2\left(r\right)\right]$.

Figure 8

Density profiles for several attractive solutions under the constant potential $U\left(r\right)=\lambda$.

Figure 9

Density profiles for several attractive solutions under the harmonic oscillator potential, which takes the quadratic form $U\left(r\right)=\lambda r^2$.

Figure 10

Density profiles for several attractive solutions under the modified harmonic oscillator potential $U\left(r\right)=\lambda \left[r^2+\beta exp\left(-r^2\right)\right]$.

Figure 11

Density profiles for several attractive solutions under the Morse potential $U\left(r\right)=\lambda \left(e^{-2Ar}-2e^{-Ar}\right)$.

Figure 12

Density profiles for several attractive solutions under the lattice potential $U\left(r\right)=\lambda [1-cos(r\left)\right]$.

Figure 13

Density profiles for several attractive solutions under the double-well potential $U\left(r\right)=\lambda \left[{(r^2-1)}^2-\beta \right]$.

Figure 14

Density profiles for several attractive solutions under the modified harmonic oscillator potential $U\left(r\right)=\lambda [r^2+\beta {cos}^2\left(r\right)]$.

Figure 15

Plot of the wave function solutions to (53) when $\epsilon =0$. This gives us the radial solutions to the $3+1$ cubic NLS equation. The results are in qualitative agreement with the density plots obtained in prior sections when both small amplitude wave functions and small potentials were considered. Initial conditions are taken to be $\mathrm{\Phi }\left(0\right)={\mathrm{\Phi }}_0=0.5$ and ${\mathrm{\Phi }}^{\prime }\left(0\right)=0$.

Figure 16

Plot of the wave function solutions to (53) when $\epsilon =0.1, U\left(r\right)=-[1-cos(r\left)\right]$. This gives us perturbations to the radial solutions to the $3+1$ cubic NLS equation due to the small potential. The results are in qualitative agreement with Fig. 15 for the attractive and repulsive solutions. The primary difference is that the oscillations in the radial part of the wave function are amplified, due to the form of the potential. For the linear case $g=0$, the solution is drastically different, as we are no longer at a steady state. The linear case behaves qualitatively similarly to the repulsive case. Initial conditions are taken to be $\mathrm{\Phi }\left(0\right)={\mathrm{\Phi }}_0=0.5$ and ${\mathrm{\Phi }}^{\prime }\left(0\right)=0$.

Figure 17

Plot of the scaled density for wave function solutions to (60) in the case of both repulsive ($g<0$) and attractive ($g>0$) BECs confined by the large lattice potential (61). Initial conditions are taken to be ${\mathrm{\Phi }}_0\left(0\right)=0.5$ and ${\mathrm{\Phi }}_0^{\prime }\left(0\right)=0$. To recover the true density, one would multiply $|{\mathrm{\Phi }}_0{\left(R\right)|}^2$ by a factor of ${\epsilon }^{-1}$ [since $\mathrm{\Phi }\left(r\right)=O({\epsilon }^{-1/2}$ in this case], which would give ${\left|\mathrm{\Phi }\left(r\right)\right|}^2={\epsilon }^{-1}{\left|{\mathrm{\Phi }}_0(r/\sqrt{\epsilon })\right|}^2$. The large potential has a stronger influence on the density of the BECs than does the nonlinearity.

Figure 18

Plot of the scaled density for wave function solutions to (60) in the case of both repulsive ($g<0$) and attractive ($g>0$) BECs confined by the large Morse potential (62). Initial conditions are taken to be ${\mathrm{\Phi }}_0\left(0\right)=0.5$ and ${\mathrm{\Phi }}_0^{\prime }\left(0\right)=0$. To recover the true density, one would multiply $|{\mathrm{\Phi }}_0{\left(R\right)|}^2$ by a factor of ${\epsilon }^{-1}$ [since $\mathrm{\Phi }\left(r\right)=O\left({\epsilon }^{-1/2}\right)$ in this case], which would give ${\left|\mathrm{\Phi }\left(r\right)\right|}^2={\epsilon }^{-1}{\left|{\mathrm{\Phi }}_0(r/\sqrt{\epsilon })\right|}^2$. We see that the large potential dominates the dynamics of both solutions, with only minor differences seen between the attractive and repulsive BECs. The large potential confines most of the mass to near the origin.

Figure 19

Plot of the scaled radial eigenfunctions $q\left(r\right)$ for the repulsive BEC when the central density is $\mathrm{\Phi }\left(0\right)=\chi$, i.e., $\mathrm{\Phi }\left(r\right)=\chi q\left(r\right), q\left(0\right)=1$. We see that $\chi =\frac{\sqrt{2}}{2}$ is a critical value. For $0<\chi <\frac{\sqrt{2}}{2}$, the eigenfunctions will exhibit oscillatory decay toward zero density for large $r$. Meanwhile, if $\chi >\frac{\sqrt{2}}{2}$, the eigenfunctions will exhibit unbounded growth, making the solutions nonphysical. If $\chi =\frac{\sqrt{2}}{2}$, the eigenfunction is degenerate (in particular, a constant), which corresponds to a steady state in the space variable. Therefore, it makes sense to expand our perturbation solutions about the eigenfunctions which exhibit decay, and this is exactly what we have done in earlier sections.

Figure 20

Plot of the scaled radial eigenfunctions $q\left(r\right)$ for the attractive BEC when the central density is $\mathrm{\Phi }\left(0\right)=\chi$, i.e., $\mathrm{\Phi }\left(r\right)=\chi q\left(r\right), q\left(0\right)=1$. Solutions tend to either of the $\pm \frac{1}{\sqrt{2}\chi }$ spatial fixed points, and since the linear terms dominate (as we have shown in the above analysis) the solutions will exhibit damped oscillations as they tend toward these values.