Entropic uncertainty relations for the ground state of a coupled system

There is a renewed interest in the uncertainty principle, reformulated from the information theoretic point of view, called the entropic uncertainty relations. They have been studied for various integrable systems as a function of their quantum numbers. In this work, focussing on the ground state of a nonlinear, coupled Hamiltonian system, we show that approximate eigenstates can be constructed within the framework of adiabatic theory. Using the adiabatic eigenstates, we estimate the information entropies and their sum as a function of the nonlinearity parameter. We also briefly look at the information entropies for the highly excited states in the system.

Figure 1

The contours of the potential as a function of $x$ and $y$ (in arb. units) for $\alpha =0.2$ and $k_1,k_2=1$. Each contour represents a particular value of energy. For large values of energy, the potential develops “channels”, as seen in the figure.

Figure 2

Entropy for the ground state of the Hamiltonian in Eq. (6) with $k_1=k_2=1$ as a function of $\alpha$. (a) Momentum entropy and (b) spatial entropy. Circles are numerical values of information entropies obtained without any approximation. Solid lines are theoretical curves of Eqs. (25) and (28).

Figure 3

Entropy sum for the ground state of the Hamiltonian in Eq. (6) with $k_1=k_2=1$ as a function of $\alpha$. Circles are numerical values and solid line is the quadratic fit to the numerical data.

Figure 4

(a) Entropy in momentum representation and in (b) spatial representation for the Hamiltonian in Eq. (6) with $k_1=k_2=1$ as a function of $\alpha$ ranging from 30 to 90. Circles are the numerical entropies and solid line is the best fit to the numerical data. Crosses represent the theoretical result based on the adiabatic approach.

Figure 5

(a) Entropy sum for the ground state of the Hamiltonian in Eq. (6) with $k_1=k_2=1$ as a function of $\alpha$. Circles are numerical values and solid line is the best fit to the numerical data as given by Eq. (48).

Figure 6

Spatial (circle) and momentum (square) entropy for a highly excited state characterized by approximate quantum number pair (110,0) with $k_1=k_2=1$ as a function of $\alpha$.