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Probabilistic representation and limit theorems for particle numbers of quasi-free states

This paper establishes a probabilistic representation of particle numbers in locally interacting bosonic quasi-free states as an infinite sum of independent geometrically distributed variables, thereby deriving exponential tail bounds, limit theorems, and a detailed description of quantum depletion statistics in Bose-Einstein condensates.

Original authors: Fanch Coudreuse, Simone Rademacher, Oliver Tse

Published 2026-09-11

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CC BY 4.0

Original authors: Fanch Coudreuse, Simone Rademacher, Oliver Tse

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the microscopic world of quantum physics, particles like atoms do not always behave as isolated individuals. Under the right conditions, vast numbers of them can synchronize into a single, collective state known as a Bose-Einstein condensate. In this state, the atoms lose their individual identities and act as a unified wave, a phenomenon that has fascinated scientists since it was first predicted in the 1920s and finally observed in laboratories decades later. However, even in this perfect unison, the system is not perfectly still. Quantum mechanics dictates that there are always tiny, unavoidable fluctuations, or jitters, in the number of particles occupying different energy levels. Understanding exactly how these numbers fluctuate is crucial for physicists trying to build precise quantum sensors or simulate complex materials, yet for a long time, the full statistical picture of these fluctuations remained elusive. While scientists could calculate the average number of particles or the size of typical variations, predicting the probability of rare, extreme events—where the number of particles deviates wildly from the average—was a formidable challenge.

A team of mathematicians and physicists has now cracked this problem for a broad class of these quantum systems. They have developed a rigorous method to describe the distribution of particle numbers in these interacting systems, revealing a surprising simplicity hidden beneath the complexity. The researchers proved that the total count of particles in these states can be broken down into a sum of independent, random components. Specifically, they showed that the particle number behaves exactly like a collection of independent dice rolls, where each "die" follows a specific statistical pattern known as a geometric distribution. This discovery confirms a long-held prediction from the physics community that had previously lacked a complete mathematical proof. By establishing this connection, the authors transformed a difficult quantum problem into a manageable statistical one, allowing them to calculate the likelihood of rare events with unprecedented precision.

The power of this new representation lies in its ability to predict how the system behaves under different conditions. The researchers explored three distinct scenarios, each corresponding to a different physical regime. In a regime where the system is sparse, meaning the particles are far apart, the fluctuations follow a "law of small numbers." In this case, the system rarely deviates from its ground state, and when it does, it typically involves the excitation of just a single pair of particles. The probability of seeing larger deviations drops off rapidly, following a predictable pattern that the authors were able to quantify. Conversely, in a dense regime where the particles are packed closely together, the fluctuations behave differently. Here, the random variations smooth out and begin to resemble a bell curve, a classic pattern known as the normal distribution. This means that in dense systems, the particle count fluctuates around an average in a way that is highly predictable and symmetric.

Perhaps most significantly, the team applied these findings to the study of quantum depletion in Bose-Einstein condensates. Quantum depletion refers to the number of particles that are kicked out of the main condensate due to interactions between them, even at absolute zero temperature. Using their new probabilistic framework, the researchers derived explicit formulas for the statistics of this depletion. They found that the number of these "stray" particles depends directly on a physical property called the scattering length, which measures how strongly the particles interact with one another. When the interaction is weak, the depletion is minimal and follows the sparse regime statistics. When the interaction is strong, the depletion grows, and the fluctuations transition into the dense regime behavior. The authors provided exact mathematical expressions for the average number of depleted particles and the size of their variations, linking these macroscopic observables directly to the microscopic details of the interaction potential.

This work does more than just confirm existing theories; it provides a complete toolkit for analyzing rare events in quantum systems. Before this study, calculating the probability of extreme fluctuations required approximations that were difficult to justify rigorously. Now, because the particle number is shown to be a sum of independent random variables, scientists can use standard statistical tools to determine the likelihood of any outcome, from the most common to the most extreme. The results offer a clear, quantitative description of how quantum fluctuations behave across different scales, bridging the gap between abstract mathematical theory and the physical reality of interacting quantum gases. By proving that these complex quantum systems can be understood through the lens of simple, independent random variables, the researchers have opened the door to a deeper understanding of quantum matter and its inherent unpredictability.

Technical Summary: Probabilistic Representation and Limit Theorems for Particle Numbers of Quasi-Free States

Problem Statement
The paper addresses the quantitative description of fluctuations in bosonic quasi-free states, a class of states central to quantum field theory, many-body quantum mechanics, and quantum information. While expectation values and low-order moments of observables are often accessible, the full probability distribution of physically relevant quantities, particularly the particle number, remains difficult to analyze. This difficulty arises because the particle-number observable in interacting systems involves contributions from infinitely many modes, resulting in infinite sums of dependent quantum observables. Specifically, the authors focus on locally interacting bosonic quasi-free states, which generalize free states to certain interacting systems where correlations are determined by two-point functions. The central challenge is to rigorously characterize the distribution of the particle number, including its tail behavior, rare-event statistics (large deviations), and asymptotic limits under various scaling regimes of the underlying lattice.

Methodology
The authors employ a combination of operator algebra, probabilistic representation, and asymptotic analysis:

Probabilistic Representation: The core methodological innovation is the rigorous derivation of a probabilistic representation for the particle number operator. By utilizing the structure of the locally interacting quasi-free vector $\Psi_\nu = \exp(A_\nu)\Omega$ , where $A_\nu$ is a sum of local Bogoliubov transformations, the authors demonstrate that the global particle number operator $N_\Lambda$ can be decomposed into a sum of local operators $N_p$ .
Disentanglement and Local Analysis: Using a disentanglement formula for the exponential of $su(1,1)$ generators (specifically the operator $A_{loc} = a^*_+ a^*_- - a_+ a_-$ ), the authors explicitly construct the local state $\Psi_{\nu_p}$ . They show that the local number operator $N_p$ in this state follows a specific distribution related to geometric random variables.
Independence Argument: Leveraging the commutativity of local operators at distinct sites, the authors prove that the local particle numbers $N_p$ are mutually independent random variables. This reduces the analysis of the global observable to the study of an infinite sum of independent geometric random variables.
Asymptotic Analysis: Based on this probabilistic representation, the authors analyze the system in three distinct regimes defined by the lattice spacing parameter $\kappa$ $κ$ :
- Intermediate Regime ( $\kappa$ fixed): Analysis of tail bounds using Chernoff-type estimates.
- Sparse Regime ( $\kappa \to +\infty$ ): Investigation of rare events using log-Laplace transforms and large deviation principles at logarithmic speed.
- Dense Regime ( $\kappa \to 0$ ): Application of a generalized Lindeberg Central Limit Theorem to establish Gaussian fluctuations.

Key Contributions and Results

Probabilistic Representation (Theorem 2.4):
The primary result confirms a prediction from the physics literature: the particle number $N_\Lambda$ in a locally interacting quasi-free state is distributed as $2G_\Lambda$ , where $G_\Lambda = \sum_{p \in \Lambda} G_p$ is a sum of independent geometrically distributed random variables. Specifically, $G_p \sim \text{Geo}(\text{sech}^2(\nu_p))$ . This representation provides explicit formulas for the moment generating function, mean, and variance, transforming a complex quantum many-body problem into a classical probabilistic one.
Tail Bounds and Large Deviations:
- Intermediate Regime: The paper establishes exponential tail bounds for the particle number distribution (Theorem 2.5). It shows that the probability of the particle number exceeding $\lambda$ times its mean decays exponentially, with a rate determined by the geometric variable with the largest mean.
- Sparse Regime ( $\kappa \to \infty$ ): The authors prove a "law of small numbers" (Theorem 2.6), showing that the scaled particle number converges to a Poisson distribution with vanishing intensity. Furthermore, they establish a Large Deviation Principle (LDP) at logarithmic speed (Theorem 2.7), characterizing the probability of rare events where the particle number deviates from zero.
Central Limit Theorem (Dense Regime):
In the limit where the lattice spacing vanishes ( $\kappa \to 0$ ), the paper proves a Central Limit Theorem (Theorem 2.8). The rescaled and centered particle number converges weakly to a standard normal distribution, along with convergence of moments. This result is derived using a generalization of Lindeberg's CLT rather than just characteristic function arguments.
Application to Bose–Einstein Condensates (BEC):
The theory is applied to the quantum depletion in Bose–Einstein condensates (Theorem 3.3). The authors show that in the large-particle limit, the statistics of particles outside the condensate (quantum depletion) are governed by the same probabilistic structure derived for the abstract lattice model.
- Weak Coupling ( $a \to 0$ ): Corresponds to the sparse regime. The depletion follows a law of small numbers, where the system is almost entirely condensed, with rare excitations of particle pairs (Theorem 3.4).
- Strong Coupling ( $a \to \infty$ ): Corresponds to the dense regime. The fluctuations of the quantum depletion satisfy a Central Limit Theorem, with explicit asymptotic formulas for the mean and variance in terms of the scattering length $a$ (Theorem 3.7).
- Intermediate Coupling: Explicit tail bounds are provided for the quantum depletion (Theorem 3.9).

Significance and Claims
The paper claims to provide the first rigorous probabilistic description of the full particle-number distribution for locally interacting quasi-free states. While heuristic arguments and structural properties in the physics literature suggested that particle numbers might decompose into independent geometric variables, this work offers a mathematically complete formulation.

The significance lies in:

Rigorous Confirmation: It validates the "mode-wise" independence structure often assumed in physics heuristics.
Quantitative Fluctuation Theory: It fills a gap in the quantitative description of fluctuations, moving beyond moments to full distributional limits (LDPs, CLTs) and tail bounds.
Explicit Constants: In the application to Bose–Einstein condensates, the authors provide explicit asymptotic formulas for the mean and variance of quantum depletion in terms of the scattering length, refining previous implicit characterizations.
Universality: The results suggest that the probabilistic structure of particle numbers in these states is robust across different scaling regimes, transitioning from Poissonian (rare events) to Gaussian (dense fluctuations) behavior depending on the system's parameters.

The authors maintain a modest tone, noting that while the results extend to general families of discrete sets, they restrict their attention to the punctured cubic lattice to provide the simplest non-trivial example that captures the essential difficulties of singular behavior near excluded sets and behavior at infinity. The work does not propose new experimental setups but rather provides a theoretical framework to interpret high-precision measurements of fluctuations and rare events in existing quantum systems.

Technical Summary: Probabilistic Representation and Limit Theorems for Particle Numbers of Quasi-Free States

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