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Dynamics of information entropy in a interacting trapped Bose gas

Sudip Kumar Haldar, Barnali Chakrabarti

Published Nov 29, 2011• 0 citations
OpenAlex

Abstract

We calculate Shannon information entropy of trapped interacting bosons in both the position and momentum spaces, $S_r$ and $S_k$ respectively. The total entropy maintains the fuctional 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 $S_r$ and $S_k$ 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.

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In this paper, we study the properties of information entropy in spin–orbit coupled spin-1 Bose–Einstein condensates. Our discussion is divided into two parts, stationary state and dynamics. For the former, we find that the coordinate component Sr and total entropy S increase with the strength of the spin–orbit coupling [Formula: see text]. The momentum component Sk shows a discontinuity behavior, it increases with [Formula: see text] when [Formula: see text] is lesser than a critical value [Formula: see text], while it decreases with [Formula: see text] when [Formula: see text] is greater than [Formula: see text]. In order to characterize the disorder-to-order transition, we introduce an order parameter [Formula: see text]. Numerical results display that [Formula: see text] gradually increases with increasing [Formula: see text] and reaches its saturation, which implies that the system becomes more and more ordered. For the latter, the Sr increases and Sk decreases with increasing [Formula: see text] at the same time. The entropic uncertainty relation is well obeyed at fixed [Formula: see text]. Moreover, [Formula: see text] decreases as a function of t. The strength effect of spin–orbit coupling on order-to-disorder transition becomes weak.

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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.
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.
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 use phase space distributions, specifically the Wigner and Husimi quasi probability distributions, to study harmonically trapped Bose--Einstein condensate described by the Gross Pitaevskii equation. From the mean field ground state wavefunction we construct both distributions and their position and momentum space marginals and we use these to compute a comprehensive set of information theoretic measures: Shannon, Wehrl, and Rényi entropies; Fisher information; cumulative and cross cumulative residual entropies; mutual information; and Kullback--Leibler, Jeffreys, Cauchy Schwarz, and Rényi divergences. Studying these quantities as a function of the sss-wave scattering length for a representative Rb-85 condensate, we find that stronger repulsive interactions drive increased phase space delocalization, seen by a monotonic growth of Shannon and Wehrl entropies, while the Fisher information shows the complementary trend -- increasing in position space and decreasing in momentum space in a manner consistent with the global Fisher uncertainty bound. Rényi entropies and divergence measures further reveal a systematic suppression of non classical interference and a shift toward more classical phase space structure in moving from the Wigner to the Husimi representation, with Wigner and Husimi based mutual informations converging at larger interaction strength. We note that, because the Gross Pitaevskii framework treats the many body state as a mean field product, the mutual information computed here quantifies statistical dependence between the conjugate phase space variables of the effective one body distribution rather than genuine particle particle entanglement.
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.
In this paper, we study the properties of information entropy in spin–orbit coupled spin-1 Bose–Einstein condensates. Our discussion is divided into two parts, stationary state and dynamics. For the former, we find that the coordinate component Sr and total entropy S increase with the strength of the spin–orbit coupling [Formula: see text]. The momentum component Sk shows a discontinuity behavior, it increases with [Formula: see text] when [Formula: see text] is lesser than a critical value [Formula: see text], while it decreases with [Formula: see text] when [Formula: see text] is greater than [Formula: see text]. In order to characterize the disorder-to-order transition, we introduce an order parameter [Formula: see text]. Numerical results display that [Formula: see text] gradually increases with increasing [Formula: see text] and reaches its saturation, which implies that the system becomes more and more ordered. For the latter, the Sr increases and Sk decreases with increasing [Formula: see text] at the same time. The entropic uncertainty relation is well obeyed at fixed [Formula: see text]. Moreover, [Formula: see text] decreases as a function of t. The strength effect of spin–orbit coupling on order-to-disorder transition becomes weak.