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Annalen der Physik

Control of Correlation Using Confinement in Case of Quantum System

Kirtee Kumar, Vinod Prasad

Published Sep 19, 2023• 12 citations

Abstract

Abstract This article investigates the behavior of a Moshinsky atom in a 1D harmonic trap. Focus is given on the theoretical foundations of confinement and its impact on the correlation between particles in the Moshinsky atom. The investigation begins by illustrating the (de)localization of the probability density function using Shannon entropy. The basics of correlation and interpretation of correlation using tools such as mutual information and statistical correlation coefficients and how these can be quantified are discussed. Then the concept of confinement is explored. The impact of interaction strength and confinement on Shannon entropy, statistical correlation coefficients, and mutual information is investigated. How interaction strength and confinement can be used to induce correlations between previously uncorrelated particles, as well as how they can be used to suppress correlations between previously correlated particles is discussed. Their implications for quantum information processing and quantum simulation are discussed. In conclusion, confinement is a powerful tool for controlling correlations in quantum systems, and its impact on correlation can be understood through theoretical models. The importance of experimental studies in this field, which provide insights into the behavior of quantum systems under confinement and pave the way for future applications in quantum technology is also emphasized.

DOI: 10.1002/andp.202300166

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Abstract This article investigates the behavior of a Moshinsky atom in a 1D harmonic trap. Focus is given on the theoretical foundations of confinement and its impact on the correlation between particles in the Moshinsky atom. The investigation begins by illustrating the (de)localization of the probability density function using Shannon entropy. The basics of correlation and interpretation of correlation using tools such as mutual information and statistical correlation coefficients and how these can be quantified are discussed. Then the concept of confinement is explored. The impact of interaction strength and confinement on Shannon entropy, statistical correlation coefficients, and mutual information is investigated. How interaction strength and confinement can be used to induce correlations between previously uncorrelated particles, as well as how they can be used to suppress correlations between previously correlated particles is discussed. Their implications for quantum information processing and quantum simulation are discussed. In conclusion, confinement is a powerful tool for controlling correlations in quantum systems, and its impact on correlation can be understood through theoretical models. The importance of experimental studies in this field, which provide insights into the behavior of quantum systems under confinement and pave the way for future applications in quantum technology is also emphasized.

We study the influence of the interparticle and confining potentials on statistical correlation via the correlation coefficient and mutual information in ground and some excited states of the Moshinsky atom in position and momentum space. The magnitude of the correlation between positions and between momenta is equal in the ground state. In excited states, the correlation between the momenta of the particles is greater than between their positions when they interact through an attractive potential whereas for repulsive interparticle potentials the opposite is true. Shannon entropies, and their sums (entropic formulations of the uncertainty principle), are also analyzed, showing that the one-particle entropy sum is dependent on the interparticle potential and thus able to detect the correlation between particles.

Position–momentum correlation in the Moshinsky atom is studied by examining the mutual information between Wigner functions and position/momentum space densities, at the one- and two-particle levels. We show that this correlation is independent of the strength of the interparticle potential at the two-particle level. At the one-particle level, the correlation decreases as the potential is increased. Shannon entropies of one- and two-particle Wigner functions are also discussed as measures of the localization in phase-space distributions.

A survey of recent ideas and goals, along with a brief history behind the quantification of uncertainties and statistical correlation in quantum systems is presented. The focus is on ideas and connections taken from information theory, in particular, the quantification of uncertainties via Shannon entropies, the entropic uncertainty relation, and statistical correlation by mutual information. A discussion of phase-space distributions and their use in information theory is also given. An incomplete list of applications, with emphasis on confined quantum systems, is provided. The article concludes by addressing future challenges in these directions.

We study the correlation of the ground state of an N-particle Moshinsky model by computing the Shannon entropy in both position and momentum spaces. We have derived the Shannon entropy and mutual information with analytical forms of such an N-particle Moshinsky model, and this helps us test the entropic uncertainty principle. The Shannon entropy in position space decreases as interaction strength increases. However, Shannon entropy in momentum space has the opposite trend. Shannon entropy of the whole system satisfies the equality of entropic uncertainty principle. Our results also indicate that, independent of the sizes of the two subsystems, the mutual information increases monotonically as the interaction strength increases.

Abstract This study investigates the impact of spatial confinement and the Hellmann potential on the Shannon entropy of a hydrogenic atom. A hydrogenic atom screened by the Hellmann potential and confined within an impenetrable spherical potential is analyzed. The Schrödinger equation is solved numerically using the finite difference method to determine energy eigenvalues and wavefunctions. These wavefunctions are examined in both position and momentum spaces to calculate the Shannon entropy in position space , the Shannon entropy in momentum space , and the total Shannon entropy . How the confinement radius and the screening parameter affect these entropies is investigated, and the results are compared to those of unconstrained systems. The findings are contrasted with previously reported results and discussed in relation to the Beckner–Bialynicki–Birula–Mycielski (BBM) inequality, offering insights into the behavior of quantum states under confinement. Significant variations in Shannon entropy and energy levels are observed with changes in confinement and screening parameters, providing a deeper understanding of quantum mechanics in confined systems. The study reveals a critical screening parameter at which the behavior of entropy transitions, shedding light on the interplay between spatial confinement, screening effects, and quantum uncertainties.