Identical Bosons, large occupation numbers and classical field description

This paper challenges the common assumption that large occupation numbers alone justify a classical field description for identical bosons, demonstrating instead that the validity of such a description depends critically on the quantum state's proximity to a coherent state rather than merely on the magnitude of the occupation number.

The Gist

Imagine you have a massive crowd of identical twins (these are the Bosons). In the world of quantum physics, when you have a huge number of these twins, scientists often assume the crowd starts acting like a single, smooth, predictable wave—like a calm ocean. This is called a classical field description.

For years, the rule of thumb has been: "If you have enough twins (a large occupation number), they will automatically behave like a smooth wave." This assumption is used to study things like Ultra-Light Dark Matter, a mysterious substance that might make up most of the universe.

However, this paper asks a simple but crucial question: Is having a huge crowd enough to guarantee they act like a smooth wave?

The Big Discovery: It's Not About the Crowd Size, It's About the Choreography

The author, Gaurav Goswami, ran a massive computer simulation to test this. He didn't just look at the number of particles; he looked at how they were arranged.

Here is the breakdown using a simple analogy:

1. The "Random Crowd" (Arbitrary States)
Imagine you throw a million people into a stadium and tell them to stand wherever they want. Even if the stadium is packed (a "large occupation number"), the crowd will look chaotic. Some people are jumping, some are sleeping, and there is no single rhythm.

• The Paper's Finding: If you pick a random quantum state with a huge number of particles, it is extremely unlikely to look like a smooth wave. The "noise" (quantum fluctuations) is too loud compared to the "signal" (the average wave). The crowd is too chaotic to be described by simple classical equations.

2. The "Perfectly Rehearsed Dance" (Coherent States)
Now, imagine that same million people, but they have been rehearsed for weeks. They all move in perfect unison, stepping left and right at the exact same time. This is a Coherent State.

• The Paper's Finding: When the particles are in this specific "rehearsed" state, they do behave like a smooth, classical wave. The noise is tiny compared to the movement.

3. The "Slightly Off-Beat" Test
The author then asked: How much can the dancers mess up before the performance stops looking like a smooth wave?

• He simulated crowds that were almost perfectly rehearsed but had small mistakes (deviations).
• The Result: Even tiny mistakes ruined the "smooth wave" effect. If the dancers were even slightly out of sync, the crowd looked chaotic again. The "smooth wave" behavior is incredibly fragile.

The Main Conclusion

The paper flips the common assumption on its head:

• Old Belief: "If the number of particles is huge, it acts like a classical wave."
• New Finding: "Having a huge number of particles is not enough. The particles must be in a very specific, special arrangement (a Coherent State) to act like a classical wave. If they are just randomly arranged, no matter how many there are, they remain quantum and chaotic."

Why This Matters for Dark Matter

The paper discusses how this affects our understanding of Ultra-Light Dark Matter.

Scientists often use simple classical equations to simulate how Dark Matter moves, assuming that because there are so many Dark Matter particles, they must act like a wave. This paper warns that this assumption is risky.

Just because the universe is full of these particles doesn't mean they are automatically "dancing in sync." For them to act like a smooth wave, there must be a specific physical mechanism (like a "rehearsal" or a connection to the environment) that forces them into that special state. Without knowing how they got into that state, we can't be sure our classical equations are actually correct.

In short: You can't just count the crowd and assume they are marching in step. You have to know if they are actually marching in step. If they aren't, the "classical" math you are using to describe them might be wrong.

Technical Summary

Technical Summary: Identical Bosons, Large Occupation Numbers, and Classical Field Description

Problem Statement
In the study of systems comprising a large number of identical Bosons, such as ultra-light dark matter (wave DM) or Bose-Einstein Condensates (BECs), it is a common assumption that a sufficiently large mean occupation number ($\langle N \rangle$) in a single-particle state automatically justifies a classical field description. This assumption underpins many simulations and experimental analyses of wave DM, where classical field equations are used to compute dynamics. However, the paper questions whether the largeness of $\langle N \rangle$ alone is sufficient to guarantee classical behavior, noting that in quantum optics, light can exist in various quantum states (e.g., Fock states) where high occupation numbers do not yield classical wave-like properties. The central question is: Does a large mean occupation number automatically imply the validity of classical field equations for arbitrary quantum states, or are additional constraints on the state vector required?

Methodology
The author employs a numerical analysis based on the formalism of quantized fields for non-relativistic identical Bosons. The study focuses on a single mode of the field, treating the system as a collection of Bosonic oscillators.

1. Criterion for Classicality: The paper adopts the criterion $2\sigma_\phi < |\langle \phi \rangle|$ (or equivalently $\sigma_\phi / |\langle \phi \rangle| < 1/2$) to define a classical field state. Here, $\langle \phi \rangle$ is the expectation value of the field operator, and $\sigma_\phi$ is the standard deviation of the field value in a given quantum state. A state is considered classical if the quantum fluctuations are small relative to the mean field value.
2. State Space Construction: The system is modeled using a truncated Fock basis $|n\rangle$ up to a dimension $N_{dim}$. Arbitrary normalized states are constructed as superpositions $|\psi\rangle = \sum c_n |n\rangle$.
3. Numerical Sampling: The author generates large samples of state vectors ($N_{sample}$) with varying coefficient distributions to test the robustness of the classicality assumption:
   • Case I: Coefficients $c_n$ are uniformly distributed pseudo-random numbers in $(-1, 1)$.
   • Case II: Variation of $N_{dim}$ and $N_{sample}$ to test sensitivity.
   • Case III: Coefficients $c_n$ are uniformly distributed in $(0, 2)$ (positive coefficients only).
   • Case IV: Coefficients are generated as normal distributions centered around the coefficients of a coherent state ($c_n^{coh} = \alpha^n / \sqrt{n!} e^{-|\alpha|^2/2}$), with a spread controlled by a factor $f$.

Key Results
The numerical simulations yield the following findings regarding the relationship between occupation number and classical behavior:

• Arbitrary States (Case I): For states with randomly chosen coefficients (even with large $\langle N \rangle \approx 25$), the ratio $\sigma_\phi / |\langle \phi \rangle|$ is typically much greater than 1. In these cases, $\langle \phi \rangle$ is often close to zero, and the condition for classicality is not met. This demonstrates that a large occupation number alone does not ensure classical behavior.
• Positive Coefficients (Case III): Restricting coefficients to be positive introduces correlations between $\langle N \rangle$, $\langle \phi \rangle$, and $\sigma_\phi$. While $\sigma_\phi / |\langle \phi \rangle|$ drops below 1 for a fraction of states, it rarely falls below $1/2$. The probability of randomly finding a state with classical behavior comparable to a coherent state is extremely low ($< 2.5 \times 10^{-4}$).
• Proximity to Coherent States (Case IV): The analysis reveals that classical behavior is highly sensitive to the proximity of the state to a coherent state.
  • When the deviation from a coherent state is small ($f=0.1$), the majority of states satisfy the classicality criterion ($\sigma_\phi / |\langle \phi \rangle| < 1/2$).
  • As the deviation increases ($f=0.5$), the fraction of classical states drops significantly (to $\approx 22\%$).
  • For larger deviations ($f=1$ or $f=2$), the classicality criterion is almost never satisfied, regardless of the mean occupation number.

Key Contributions

• Refutation of the "Large Occupation" Heuristic: The paper provides quantitative evidence that the assumption "large $\langle N \rangle \implies$ classical field" is invalid for arbitrary quantum states.
• Identification of the True Criterion: The study establishes that the validity of the classical field description depends primarily on the state's proximity to a coherent state with a large amplitude, rather than the magnitude of the occupation number itself.
• Quantification of State Space Rarity: The results suggest that states exhibiting quasi-classical behavior form a "small island" within the vast Hilbert space of a Bosonic system. Classical states are low-dimensional approximations that are statistically rare without specific preparation mechanisms.

Significance and Implications
The paper argues that the use of classical field equations for ultra-light dark matter (wave DM) requires justification beyond the mere existence of a large occupation number. Since classical behavior is not an automatic consequence of high density, phenomenological constraints on wave DM models (particularly those relying on small particle masses) may need to be revisited.

The author posits that for a dark matter system to behave classically, there must be a physical mechanism—such as quantum decoherence due to environmental coupling or generation via linear coupling to classical sources—that prepares and maintains the system in a state close to a coherent state. Without such mechanisms, the assumption of classicality in wave DM simulations remains unjustified. The paper concludes that future models should account for the specific state preparation dynamics rather than assuming classicality based solely on particle density.