Optical Lattice Effects on Shannon Information Entropy in Rotating Bose–Einstein Condensates

Abstract

We investigate the formation of Shannon information entropy in a rotating Bose–Einstein condensates confined in a harmonic potential combined with an optical lattice (OL) using the mean field Gross–Pitaevskii equation. With the increase of OL depth $V_0$, at the same rotational frequency $\Omega$, we show that the information entropy increases in momentum space $S_r$ and total entropy S and it decreases in position space $S_k$. We also calculate the Landsberg order parameter $\delta$ and its dependence on $\Omega$. We find that the critical points between the case of OL and non-OL move along the direction of decreasing $\Omega$ with the increase of $V_0$. In particular, the dynamics behaviors indicate that the periodicity of S and $S_{\text{ max }}$ loses due to the broken symmetry when OL is added.

Appendix A: Details for Calculating \(S_r\) and \(S_k\)

Gadre and Bendale[31] established a connection between $S_r$ and $S_k$ with the total kinetic energy T and mean square radius $\langle r^{2} \rangle$ of the system which has been derived using EUR [39].

$$\begin{aligned}&{S_r}_{\text{ min }} \leqslant S_r \leqslant {S_r}_{\text{ max }} , \end{aligned}$$

(A1)

$$\begin{aligned}&{S_k}_{\text{ min }} \leqslant S_k \leqslant {S_k}_{\text{ max }} ,\end{aligned}$$

(A2)

$$\begin{aligned}&{S}_{\text{ min }} \leqslant S \leqslant {S}_{\text{ max }}. \end{aligned}$$

(A3)

For density distribution normalized to unity, the above lower and upper limits took the form

$$\begin{aligned} {S_r}_{\text{ min }}= & {} \frac{3}{2}(1+ \ln \pi ) - \frac{3}{2} \ln \left( \frac{4}{3} T \right) , \end{aligned}$$

(A4a)

$$\begin{aligned} {S_r}_{\text{ max }}= & {} \frac{3}{2}(1+ \ln \pi ) + \frac{3}{2} \ln \left( \frac{2}{3} \langle r^{2} \rangle \right) , \end{aligned}$$

(A4b)

$$\begin{aligned} {S_k}_{\text{ min }}= & {} \frac{3}{2}(1+ \ln \pi ) - \frac{3}{2} \ln \left( \frac{2}{3} \langle r^{2} \rangle \right) , \end{aligned}$$

(A4c)

$$\begin{aligned} {S_k}_{\text{ max }}= & {} \frac{3}{2}(1+ \ln \pi ) + \frac{3}{2} \ln \left( \frac{4}{3} T \right) , \end{aligned}$$

(A4d)

$$\begin{aligned} {S}_{\text{ min }}= & {} 3(1+ \ln \pi ), \end{aligned}$$

(A4e)

$$\begin{aligned} {S}_{\text{ max }}= & {} 3(1+ \ln \pi ) + \frac{3}{2} \ln \left( \frac{8}{9} \langle r^{2} \rangle T \right) . \end{aligned}$$

(A4f)

Massen and Panos [38] presented the values of the lower and upper bound of S as in Eq. (A4). In the present work, we calculate numerically the values in Eq. (7) and the results are presented in Tables 1 and 2.