Detecting a logarithmic nonlinearity in the Schrödinger equation using Bose-Einstein condensates

We study the effect of a logarithmic nonlinearity in the Schrödinger equation (SE) on the dynamics of a freely expanding Bose-Einstein condensate (BEC). The logarithmic nonlinearity was one of the first proposed nonlinear extensions to the SE which emphasized the conservation of important physical properties of the linear theory, e.g., the separability of noninteracting states. Using this separability, we incorporate it into the description of a BEC obeying a logarithmic Gross-Pitaevskii equation. We investigate the dynamics of such BECs by using variational and numerical methods and find that, by using experimental techniques like $\delta$-kick collimation, experiments with extended free-fall times as available on microgravity platforms could be able to lower the bound on the strength of the logarithmic nonlinearity by at least one order of magnitude.

Figure 1

(left) Numerically simulated time evolution of a spherically symmetric BEC's width $\sigma \left(t\right)$ during free propagation for different values of $b$. The blue (dashed) curve represents the evolution under the current known upper bound on $b$. The two horizontal lines show the calculated maximum width from Eq. (12) for the two examples for which spatial confinement is visible. (right) Contour lines depicting the difference in width of linear and nonlinear expansion $\sigma (t;b=0)-\sigma (t;b=3.3\times {10}^{-15}\text{eV})$ for different $\chi$ [see Eq. (13)]. The parameter $\chi$ has been varied by changing the initial width $\sigma \left(0\right)$ while keeping $N=5\times {10}^4$, $a=90a_0$, and $\dot{\sigma }\left(0\right)=0$. For example, $\chi ={10}^3$ corresponds to roughly $1\mu \text{m}$ and $\chi ={10}^{-1}$ to $30\mu \text{m}$ initial width of the simulated BEC.

Figure 2

(left) Numerically simulated time evolution of $\sigma \left(t\right)$ with same parameter as in Fig. 1 but with application of a DKC pulse after 10 ms of free propagation. The inset plot shows the collimation process of the matter beam. (right) Contour lines depicting the difference in width $\sigma (t;b=0)-\sigma (t;b)$ for different $b$ of the same experiment using DKC as described for the left graph.