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arXiv:2606.02656 · Physica A Statistical Mechanics and its Applications

Probing information theoretic measures of nonlinear ultracold quantum gases using phase-space distributions

Mariyah Ughradar, Ramkumar Radhakrishnan, Siddharth Kumar Tiwari, Vikash Kumar Ojha

quant-phcond-mat.quant-gasSubmitted Jun 1, 2026• Announced Jun 2, 2026• 0 citations
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Abstract

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 $s$-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.

DOI
10.1016/j.physa.2026.131722

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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.
The local structure present in Wigner and Husimi phase-space distributions and their marginals are studied and quantified via information-theoretic quantities. Shannon, R\'enyi, and cumulative residual entropies of the Wigner and Husimi distributions are examined in the ground and excited states of a harmonic oscillator. The entropies of the Wigner function marginals are lower than the corresponding entropies of the Husimi function marginals, which illustrates how the nodal structure present in the Wigner function is lost upon consideration of the Husimi function. Shannon and cumulative residual entropies of the Wigner function yield entropies which are complex valued. Absolute values and real components of these quantities increase with quantum number, displaying a behavior which is consistent with their real-valued Husimi function counterparts. This comportment is also similar to the real-valued R\'enyi entropies of the Wigner function. The entropies of the Wigner function are observed to be lower than the corresponding Husimi function ones, in agreement with the results for the marginals. The real components of the Wigner function entropies are seen to be closer to the uncertainty relation bound compared to the corresponding Husimi function entropies. These real components are also closer to the bound when contrasted to the entropic sum of the marginal densities. The R\'enyi or collision entropy of the Wigner function sits exactly on the bound. Related statistical correlation measures show that the position-momentum correlation is larger in the Wigner function compared to the Husimi function and increases with quantum number.
We consider quasi-one-dimensional Bose–Einstein condensates and study the properties of matter-wave bright solitons within the framework information theoretic approach. In the limit of negligible trapping effect, we calculate Shannon entropy, Fisher information, and Rényi entropy for different mean-field interactions, and study the interplay between group velocity dispersion and nonlinear atomic interaction. We see that information theoretic approach can clearly reveal localization and delocalization properties of matter-wave density distribution. In all the cases, entropic uncertainties are found to be satisfied.
We investigate the quantum-information properties of a particle confined by the double Morse potential in position and momentum spaces. The quasi-exact solvability of the model gives analytical expressions for the first two bound states, allowing the corresponding probability densities to be analyzed directly. Shannon entropy, Onicescu energy, Fisher information, statistical complexity, and Fisher-Shannon products are evaluated as functions of the parameter AAA, which controls the transition from a well-separated double well to a merged single-well profile. The position distribution is more delocalized and structurally complex when the double-well character is pronounced, whereas the momentum distribution exhibits the complementary trend. As the wells merge, the ground-state Fisher--Shannon product approaches its Gaussian reference value, whereas the excited state retains stronger non-Gaussian structure.
Abstract We consider Shannon entropy, Fisher information, Rényi entropy, and Tsallis entropy to study the quantum droplet phase in Bose–Einstein condensates. In the beyond mean-field description, the Gross–Pitaevskii equation with Lee-Huang-Yang correction gives a family of quantum droplets with different chemical potentials. At a larger value of chemical potential, quantum droplet with sharp-top probability density distribution starts to form while it becomes flat top for a smaller value of chemical potential. We show that entropic measures can distinguish the shape change of the probability density distributions and thus can identify the onset of the droplet phase. During the onset of droplet phase, the Shannon entropy decreases gradually with the decrease of chemical potential and attains a minimum in the vicinity where a smooth transition from flat-top to sharp-top QDs occurs. At this stage, the Shannon entropy increases abruptly with the lowering of chemical potential. We observe an opposite trend in the case of Fisher information. These results are found to be consistent with the Rényi and Tsallis entropic measures.
Abstract We study the mean-field phase-space distribution of the ground state of an anisotropic, harmonically trapped Bose–Einstein condensate at zero temperature. First, we derive the Fourier transform of the ground-state wave function in the Thomas–Fermi limit in d dimensions. Second, we obtain analytical approximations for the Thomas–Fermi Wigner function in one and three dimensions with a factorization ansatz of the coherence function. These approximate expressions are compared with exact numerical simulations of the Gross–Pitaevskii equation.
Abstract The Shannon information entropy sum is examined as systems comprised of two-level harmonic oscillator states are passed through dephasing and depolarization quantum channels. We show how the entropy sum increases as the dephasing parameter is increased. The interpretation here is a loss of information in the system as the quantum coherences are removed. This result and interpretation also holds in the depolarization channel as the system transits to the maximally mixed state. The increase in the entropy sum is associated with a delocalization of both position and momentum densities. The behaviour of phase-space Wigner functions and their Shannon entropies is also addressed. Wigner negativity is seen to decrease and disappear with an increase of the dephasing parameter. Here, Wigner negativity is associated with the presence of quantum coherences. The imaginary component of the Wigner function entropy is shown to capture the changes in Wigner negativity. On the other hand, Wigner functions with diagonal input density matrices are non-negative in the depolarization channel. Their real-valued Shannon entropies increase with the depolarization parameter as the system approaches the maximally mixed state. The mutual information between the Wigner function and the product of position and momentum space densities is used to probe the statistical correlation between position and momentum variables. This measure decreases in both channels as the parameter increases. Thus, there is a loss of correlation as quantum coherences are eliminated in the dephasing case, and as the system transits to the maximally mixed case in the depolarizing channel.