Information Entropy for a Two-Dimensional Rotating Bose–Einstein Condensate

Abstract

We study the information entropy, order, disorder, and complexity for the two-dimensional (2D) rotating and nonrotating Bose–Einstein condensates. The choice of our system is a complete theoretical laboratory where the complexity is controlled by the two-body contact interaction strength and the rotation frequency ($\varOmega$) of the harmonic trap. The 2D nonrotating condensate shows the complexity of the category I where the disorder-order transition is triggered by the interaction strength. In the rotating condensates, $\varOmega$ is chosen as the disorder parameter when the interaction strength is fixed. With respect to $\varOmega$, the complexity shifts between maximum and minimum confirm the existence of category II complexity in the rotating condensate. Also, we consider the interaction strength as the disorder parameter when $\varOmega$ is unchanged and complexity as a function of interaction strength exhibits category III complexity. The present work also includes the calculation of upper bound and lower bound of entropy for 2D quantum systems.

Appendix A: Connection Between \(S_r\), \(S_k\) with the Total Kinetic Energy T and Mean Square Radius in Two Dimensions

Maximum value of entropy in momentum space for a 2D system is given by

$$\begin{aligned} S_{k_{\rho }} \le -\int n(\mathbf {k}_{\rho }) \ln n(\mathbf {k}_{\rho }) \hbox {d}\mathbf {k}_{\rho }. \end{aligned}$$

(10)

Dimensionless form of kinetic energy $T=\frac{1}{2}\int n(\mathbf {k}_{\rho }) {\mathbf {k}}_{\rho }^2 \hbox {d}\mathbf {k}_{\rho }$, where ${\mathbf {k}}_{\rho }^2\,=\, \mathbf {k}_x^2+\mathbf {k}_y^2$. We consider the density in momentum space $n(\mathbf {k}_{\rho })\,=\, A \exp [-\alpha {\mathbf {k}}_{\rho }^2]$, where A is the normalization constant and $\alpha$ is the appropriate Lagrange multiplier. The normalization of the density with respect to N particles is defined $\int _\infty ^ {-\infty } n(\mathbf {k}_{\rho }) \hbox {d}\mathbf {k}_{\rho }\, =\, N$. It calculates $A\,=\,\alpha N/\pi$ and $\alpha \,=\,N/2T$. Thus, maximum value of the momentum space is given by Eq. (10) and further simplification yields the maximum value of momentum space entropy is given by

$$\begin{aligned} S_{k_{\rho }} \le N (1+\ln \pi ) - N \ln N - N \ln \left( \frac{N}{2T}\right) . \end{aligned}$$

(11)

For the 2D model, we get the following relation from refs. [12, 13],

$$\begin{aligned} S_{\rho } + S_{k_{\rho }} \ge 2 N (1+\ln \pi ) - 2 N \ln N. \end{aligned}$$

(12)

From relations  (11) and (12), we obtain the lower bound to $S_{\rho }$

$$\begin{aligned} S_{\rho } \ge N (1+\ln \pi ) - N \ln N + N \ln \left( \frac{N}{2T}\right) . \end{aligned}$$

(13)

Addition of (11) and (13) provides the lower bound to the excess information entropy in the position space over that in the momentum space.

$$\begin{aligned} S_{\rho } - S_{k_{\rho }} \ge 2 N (1+\ln \pi ) - 2 N \ln (2T). \end{aligned}$$

(14)

Next, we calculate the upper as well as lower bounds for $S_{\rho }$ and $S_{k_{\rho }}$, respectively, in terms of $\langle {{{\rho }}}^2\rangle$

$$\begin{aligned} S_{\rho } \le N (1+\ln \pi ) - 2 N \ln N + N \ln \left( \langle {{\rho }}^2\rangle \right) , \end{aligned}$$

(15)

where ${{\rho }}^2=x^2+y^2$ and

$$\begin{aligned} S_{k_{\rho }} \ge N (1+\ln \pi ) +2 N \ln N - N \ln \left( \langle {{\rho }}^2\rangle \right) . \end{aligned}$$

(16)

For density distribution normalized to unity, the lower and upper limits of entropy in two dimensions took the form

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

(17a)

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

(17b)

$$\begin{aligned} {S_{k_{\rho }}}_{\text{ min }}= & {} (1+ \ln \pi ) - \ln \left( \langle {{\rho }}^{2} \rangle \right) , \end{aligned}$$

(17c)

$$\begin{aligned} {S_{k_{\rho }}}_{\text{ max }}= & {} (1+ \ln \pi ) + \ln \left( 2 T \right) , \end{aligned}$$

(17d)

$$\begin{aligned} {S}_{\text{ min }}= & {} 2(1+ \ln \pi ), \end{aligned}$$

(17e)

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

(17f)