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arXiv:1303.7045 · Physical Review A

Correlated many-body calculation to study characteristics of Shannon information entropy for ultracold trapped interacting bosons

Sudip Kumar Haldar, Barnali Chakrabarti, Tapan Kumar Das, Anindya Biswas

Published Sep 3, 2013• 13 citations
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Abstract

A correlated many-body calculation is presented to characterize the Shannon information entropy of trapped interacting bosons. We reformulate the one-body Shannon information entropy in terms of the one-body probability density. The minimum limit of the entropy uncertainty relation is approached by making $N$ very small in our numerical work. We examine the effect of correlations in the calculation of information entropy. Comparison with the mean-field result shows that the correlated basis function is indeed required to characterize the important features of the information entropies. We also accurately calculate the point of critical instability of an attractive BEC, which is in close agreement with the experimental value. Next, we calculate two-body entropies in position and momentum spaces and study quantum correlations in the attractive BEC.

DOI
10.1103/physreva.88.033602

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A correlated many-body calculation is presented to characterize the Shannon information entropy of trapped interacting bosons. We reformulate the one-body Shannon information entropy in terms of the one-body probability density. The minimum limit of the entropy uncertainty relation is approached by making NNN very small in our numerical work. We examine the effect of correlations in the calculation of information entropy. Comparison with the mean-field result shows that the correlated basis function is indeed required to characterize the important features of the information entropies. We also accurately calculate the point of critical instability of an attractive BEC, which is in close agreement with the experimental value. Next, we calculate two-body entropies in position and momentum spaces and study quantum correlations in the attractive BEC.
We calculate Shannon information entropy of trapped interacting bosons in both the position and momentum spaces, SrS_rSr​ and SkS_kSk​ respectively. The total entropy maintains the fuctional form S=a+bln⁡NS=a + b \ln NS=a+blnN for repulsive bosons. At the noninteracting limit the lower bound of entropic uncertainty relation is also satisfied whereas the diverging behavior of SrS_rSr​ and SkS_kSk​ at the critical point of collapse for attractive condensate accurately calculates the stability factor. Next we study the dynamics of Shannon information entropy with varying interparticle potential. We numerically solve the time-dependent Gross-Pitaevskii equation and study the influence of increasing nonlinearity in the dynamics of entropy uncertainty relation (EUR). We observe that for small nonlinearity the dynamics is regular. With increase in nonlinearity although Shannon entropy shows large variation in amplitude of the oscillation, the EUR is maintained throughout time for all cases and it confirms its generality. We also study the dynamics in a very tight trap when the condensate becomes highly correlated and strongly inhomogeneous. Time evolution of total entropy exhibits aperiodic and fluctuating nature in very tight trap. We also calculate Landsberg's order parameter for various interaction strengths which supports earlier observation that entropy and order are decoupled.
We calculate Shannon information entropy of trapped interacting bosons in both the position and momentum spaces, Sr and Sk, respectively. The total entropy maintains the functional form S = a + b ln N for repulsive bosons. At the noninteracting limit the lower bound of entropic uncertainty relation is also satisfied whereas the diverging behavior of Sr and Sk at the critical point of collapse for attractive condensate accurately calculates the stability factor. Next we study the dynamics of Shannon information entropy with varying interparticle potential. We numerically solve the time-dependent Gross–Pitaevskii equation and study the influence of increasing nonlinearity in the dynamics of entropy uncertainty relation (EUR). We observe that for small nonlinearity the dynamics is regular. With increase in nonlinearity although Shannon entropy shows large variation in amplitude of the oscillation, the EUR is maintained throughout time for all cases and it confirms its generality. We also study the dynamics in a very tight trap when the condensate becomes highly correlated and strongly inhomogeneous. Time evolution of total entropy exhibits aperiodic and fluctuating nature in very tight trap. We also calculate Landsberg's order parameter for various interaction strengths which supports earlier observation that entropy and order are decoupled.
An overview of the Bose–Einstein condensation of correlated atoms in a trap is presented by examining the effect of interparticle correlations to one- and two-body properties of the above systems at zero temperature in the framework of the lowest order cluster expansion. Analytical expressions for the one- and two-body properties of the Bose gas are derived using Jastrow-type correlation function. In addition numerical calculations of the natural orbitals and natural occupation numbers are also carried out. Special effort is devoted for the calculation of various quantum information properties including Shannon entropy, Onicescu informational energy, Kullback–Leibler relative entropy and the recently proposed Jensen–Shannon divergence entropy. The above quantities are calculated for the trapped Bose gases by comparing the correlated and uncorrelated cases as a function of the strength of the short-range correlations. The Gross–Piatevskii equation is solved, giving the density distributions in position and momentum space, which are employed to calculate quantum information properties of the Bose gas.
We will study fermionic systems like atomic nuclei and bosonic systems like the correlated atoms in a trap from an information-theoretical point of view. The Shannon and Onicescu information measures are calculated for the above systems by comparing the correlated and uncorrelated cases as functions of the strength of the short range correlations. One-body and two-body density and momentum distributions are employed. Thus, the effect of short-range correlations on the information content is evaluated. The magnitude of distinguishability between the correlated and uncorrelated densities is also discussed employing suitable measures for the distance of states i.e. the well known Kullback–Leibler relative entropy and the recently proposed Jensen–Shannon divergence entropy. We will see that the same information-theoretical properties hold for quantum many-body systems obeying Bose–Einstein and Fermi–Dirac (statistics).
The universal property for the information entropy S = a + h In Ζ is verified for atoms using a systematic study with Roothaan-Hartree-Fock (RHF) wave functions with atomic number Ζ — 2 — 54. The above relation was proposed previously for atoms, nuclei, atomic clusters and correlated atoms in a trap. Kullback-Leibler relative entropy Κ and Jensen-Shannon divergence J are employed to compare RHF with Thomas-Fermi (TF) density of atoms as well as another phenomenological density obtained by Sagar et al. Two-body density distributions in position- and momentum-space are used to calculate and compare the corresponding information entropies for correlated and uncorrelated nuclei and bosonic systems (correlated atoms in a trap). It is seen that short-range correlations (SRC) increase the values of S. One-body information entropy entropy S\ is compared with two-body information entropy and a conjecture is made for TV-body information entropy SN- The entropy Κ and the divergence J are also used to evaluate the information distance between correlated and uncorrelated densities both at the one- and the two-body levels for nuclei and trapped Bose gases.
Calculation of the Shannon information entropy (S) and its connection with the order-disorder transition and with inter-particle interaction provide a challenging research area in the field of quantum information. Experimental progress with cold trapped atoms has corroborated this interest. In the present work, S is calculated for the Bose-Einstein condensate (BEC) with dominant dipolar interaction for different dipole strengths, trap aspect ratios, and number of particles (N). Trapped dipolar bosons in an anisotropic trap provide an example of a system where the effective interaction is strongly determined by the trap geometry. The main conclusion of the present calculation is that the anisotropic trap reduces the number of degrees of freedom, resulting in more ordered configurations. Landsberg's order parameter exhibits quick saturation with the increase in scattering length in both prolate and oblate traps. We also define the threshold scattering length which makes the system completely disordered. Unlike non-dipolar BEC in a spherical trap, we do not find a universal linear relation between S and lnN, and we, therefore, introduce a general quintic polynomial fit rather well working for a wide range of particle numbers.