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

This paper employs Wigner and Husimi phase-space distributions to compute a comprehensive set of information-theoretic measures for harmonically trapped Bose-Einstein condensates, revealing that stronger repulsive interactions drive increased phase-space delocalization and a systematic shift toward classical structures while clarifying that the observed mutual information reflects statistical dependence in the mean-field framework rather than genuine particle-particle entanglement.

The Gist

The Big Picture: Mapping a Quantum Cloud

Imagine you have a cloud of ultra-cold atoms (a Bose-Einstein Condensate) trapped inside a magnetic "bowl." These atoms are so cold and close together that they act like a single giant wave rather than individual particles.

The scientists in this paper wanted to understand how these atoms behave when they push against each other (repulsive interactions). To do this, they didn't just look at where the atoms are; they tried to map out a "weather map" of the entire system, showing both where the atoms are (position) and how fast they are moving (momentum) at the same time.

The Two Maps: The "Crystal Ball" vs. The "Fuzzy Photo"

To create this map, the researchers used two different mathematical tools, which they compare like two different ways of taking a photograph:

1. The Wigner Distribution (The Crystal Ball): This is a high-definition, "crystal ball" view of the quantum world. It shows everything, including the weird, invisible quantum tricks like interference patterns (where waves cancel each other out). However, because it shows these quantum tricks so clearly, the map sometimes has "negative" areas. In the real world, you can't have negative probability (you can't have -50% chance of an atom being there), so this map is mathematically tricky to use for standard statistics.
2. The Husimi Distribution (The Fuzzy Photo): This is the same map, but it has been run through a "blur" filter (Gaussian smoothing). It takes the sharp, weird quantum details and softens them out. The result is a perfectly smooth, positive map that looks more like a classical, everyday picture. It loses some of the "quantum magic" but is much easier to measure and understand.

The Experiment: Pushing the Atoms

The researchers simulated a cloud of Rubidium-85 atoms. They started with a calm cloud and then gradually increased the repulsive force between the atoms (making them push away from each other more strongly).

They used a toolbox of "Information Theory" measures—basically, ways to count how much "surprise," "disorder," or "connection" exists in the system. Here is what they found:

1. The Cloud Gets Fuzzier (Entropy Goes Up)

As the atoms pushed harder against each other, the cloud spread out more in space.

• The Analogy: Imagine a drop of ink in water. If you stir it gently, it stays in a tight spot. If you stir it violently (strong repulsion), the ink spreads out everywhere.
• The Result: The "Shannon Entropy" (a measure of disorder or spread) went up. The atoms became less predictable and more spread out in the trap. This happened in both the "Crystal Ball" (Wigner) and the "Fuzzy Photo" (Husimi) maps, but the Fuzzy Photo always showed slightly more disorder because the blur filter adds a little extra fuzziness.

2. The Sharpness Paradox (Fisher Information)

This was the most interesting finding. Usually, when things spread out, they get "blurry" and lose sharpness. But here, the researchers found a split personality:

• In Space: As the atoms pushed apart, the shape of the cloud in space actually developed sharper edges and more distinct features relative to its size. The "Fisher Information" (a measure of sharpness) increased.
• In Speed (Momentum): Because the atoms were moving in more complex ways to avoid each other, their speed distribution became smoother and less sharp. The Fisher Information here decreased.
• The Analogy: Imagine a crowd of people. If they all stand still in a tight group, they are hard to distinguish. If they start running away from each other (repulsion), the group spreads out (high disorder), but you can now clearly see the specific path each person is taking (high sharpness in position). However, because they are moving in so many different directions, it becomes harder to predict exactly how fast any single person is going (low sharpness in speed).

3. The "Connection" Between Position and Speed

The researchers measured "Mutual Information," which tells us how much knowing an atom's position helps you guess its speed.

• The Result: As the repulsion got stronger, this connection weakened. The atoms became so chaotic and spread out that knowing where they were didn't tell you much about how fast they were moving.
• The Convergence: Interestingly, as the repulsion got very strong, the "Crystal Ball" map and the "Fuzzy Photo" map started to look more similar. The quantum weirdness (interference) was smoothed out by the sheer chaos of the interaction, making the system look more "classical" (like a normal gas).

Important Clarification: What They Did Not Find

The paper is very careful to state what this study is not about.

• Not "Spooky Action at a Distance": In quantum physics, "entanglement" usually means two particles are linked across space. This study did not measure that.
• What they actually measured: They measured how the shape of the single giant wave (the whole cloud) changed. They looked at how the "position" part of the wave and the "speed" part of the wave were related to each other within that single cloud.
• The Limitation: Because they used a simplified model (the Gross-Pitaevskii equation), they treated the whole cloud as one big, smooth wave. They did not look at the complex, messy entanglement between individual atoms that happens in more advanced theories.

Summary

The paper shows that when you make a quantum gas push against itself:

1. It spreads out and becomes more disordered (higher entropy).
2. It becomes sharper in position but smoother in speed (a trade-off).
3. The link between where it is and how fast it moves gets weaker.
4. Eventually, the system looks less like a weird quantum object and more like a standard, classical gas, even though it's still made of atoms.

The authors used these "information maps" to prove that stronger interactions reshape the quantum world, turning a delicate, interference-heavy state into a broader, more classical-looking one.

Technical Summary

Technical Summary: Probing Information Theoretic Measures of Nonlinear Ultracold Quantum Gases Using Phase-Space Distributions

Problem Statement
The paper addresses the challenge of characterizing the dynamics and correlation structures of strongly interacting quantum systems, specifically harmonically trapped Bose–Einstein condensates (BECs). While ultracold gases provide a controllable platform for studying collective dynamics and coherence, there is a need for systematic diagnostics that quantify how interaction strength reshapes phase-space structure. The authors focus on the transition from quantum to semiclassical behavior as a function of the s-wave scattering length, utilizing information-theoretic measures to probe localization, delocalization, and coherence within the mean-field Gross–Pitaevskii (GP) framework.

Methodology
The study employs a numerical and analytical investigation of a one-dimensional effective BEC of Rubidium-85 (Rb-85) atoms confined in a harmonic trap. The system is described by the time-dependent Gross–Pitaevskii equation, with the ground state wavefunction derived analytically as a sech-profile in the axial direction and a Gaussian in the radial direction.

The core methodology involves:

1. Phase-Space Representation: Constructing both the Wigner quasi-probability distribution ($W$) and the Husimi distribution ($H$), the latter being a Gaussian-smoothed version of the former.
2. Marginal Analysis: Deriving position and momentum space marginals from both distributions to compute survival functions.
3. Information-Theoretic Metrics: Calculating a comprehensive suite of measures for both distributions and their marginals, including:
   • Entropies: Shannon, Wehrl (for Husimi), Rényi, and Cumulative Residual Entropies.
   • Fisher Information: In both position and momentum spaces.
   • Divergences: Kullback–Leibler (KL), Jeffreys-type (based on survival functions), Cauchy–Schwarz, and Rényi divergences.
   • Correlation Measures: Mutual information and Cross Cumulative Residual Entropy.
4. Parameter Variation: Systematically varying the s-wave scattering length ($a$) within the repulsive regime ($a > 0$) to observe interaction-induced effects, while keeping the particle number ($N=10$) and trap parameters fixed.

Key Contributions and Results
The paper provides a comparative analysis of Wigner and Husimi representations under varying interaction strengths, yielding the following specific results:

• Entropy and Delocalization: As the repulsive interaction strength (scattering length $a$) increases, both Shannon and Wehrl entropies exhibit monotonic growth. This indicates increased phase-space delocalization and a broader effective volume occupied by the condensate. The Husimi entropy is consistently higher than the Wigner entropy due to the coarse-graining inherent in the Gaussian smoothing.
• Fisher Information Trends: The Fisher information displays a complementary trend to the entropies. It increases in position space while decreasing in momentum space as $a$ increases. This behavior satisfies the global Fisher uncertainty bound ($F_x \cdot F_k \ge 4$). The authors interpret the increase in position-space Fisher information as a result of interaction-induced sharpening of relative spatial modulations within the broadening wavefunction envelope, rather than a contradiction to the broadening effect.
• Divergence Measures:
  • KL and Jeffreys Divergences: These measures, which quantify the difference between Wigner and Husimi distributions, increase with scattering length. This suggests that stronger interactions generate increasingly non-Gaussian phase-space structures that are more distinct from the smoothed Husimi representation.
  • Cauchy–Schwarz Divergence: Conversely, this measure decreases with increasing $a$, reflecting an increase in the global overlap between the distributions as they broaden.
  • Rényi Divergence: Increases with interaction strength, consistent with the KL trend.
• Mutual Information and Correlations: The mutual information between conjugate variables (position and momentum) decreases as interaction strength increases. This indicates that stronger repulsive interactions lead to a factorization of the joint phase-space distribution, reducing the statistical dependence between $r$ and $k$. The Wigner and Husimi mutual information curves converge at larger interaction strengths, signaling a transition toward a more semiclassical phase-space structure where quantum interference effects are suppressed.
• Survival Functions: The survival functions derived from the Husimi distribution decay more rapidly than those from the Wigner distribution, reflecting the loss of fine structure and information due to Gaussian smoothing.

Significance and Claims
The authors explicitly frame the significance of their work within the constraints of the mean-field approximation. They emphasize that because the Gross–Pitaevskii equation treats the many-body state as a pure product state, the computed mutual information and correlations quantify statistical dependence between conjugate phase-space variables of the effective one-body distribution, rather than genuine particle-particle entanglement.

The paper claims that these information-theoretic measures serve as physically transparent diagnostics for:

1. Interaction-Induced Restructuring: Quantifying how repulsive interactions drive delocalization and suppress non-classical interference features in phase space.
2. Semiclassical Transition: Demonstrating the convergence of Wigner and Husimi measures at high interaction strengths, indicating a shift toward classical phase-space structures.
3. Resource Theory Context: The authors note that while the specific ground state studied here possesses a positive-definite Wigner function (lacking "Wigner negativity" or "MAGIC"), the framework establishes a baseline for future studies in regimes where negativity emerges (e.g., attractive condensates or driven systems). In such regimes, these entropic and divergence measures could complement direct quantifiers of non-classicality.

The work concludes that these measures are relevant to quantum thermodynamics (characterizing coherence loss and irreversibility) and provide a systematic tool for analyzing the complexity of interacting many-body systems without requiring beyond-mean-field calculations.