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arXiv:2610.06367

Quantum Fluctuations in Bose--Einstein Condensates:Microscopic Correlations and Moment Bounds

Mauris Chueng, Keheng Zhu, Hexiang Wang

Published Oct 5, 2026• 0 citations
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Abstract

We study higher moments of quantum fluctuations and microscopic correlation energy in the three-dimensional Gross--Pitaevskii regime. For a smooth, compactly supported, nonnegative radial interaction and an $N$-independent normalized initial condensate wave function in $H^6$, we propagate the third spectral moment of a positive correlated energy along the exact many-body evolution. A uniform initial bound controls the third moment of the depletion number and sixth moments of collective one-particle fluctuations on every compact time interval, independently of $N$. For a nonzero interaction and a smooth, compactly supported initial condensate, we also identify the excitation-energy measure at scale $N^2$, weighted by the first energy. This limit holds for norm-convergent pure-state families with a uniform bound on the initial third correlated-energy moment. Its positive-energy part is a spectral measure of the relative scattering operator $-2Δ+v$, multiplied by a time-dependent dressed contact. We compute the radial density and prove strict positivity of the contact for nonzero references with finite kinetic energy and finite mean excitation number. The same evolving states have bounded physical fluctuation moments and excitation-energy moments of order $p>1$ bounded below by a positive multiple of $N^{2p-2}$, in the extended spectral sense. Inverse powers of the positive initial energy construct admissible pure and mixed families. For the norm-convergent pure-state class, we determine the scaled interaction-energy moments for $1\le p<2$ and bound the second moment by $O(N^2)$. A finite cubic approximation has norm error $O(N^{-3/16})$ on the energy form domain and $O(N^{-5/4})$ on a stronger initial domain of its generator.

DOI
10.48550/arxiv.2610.06367

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We study higher moments of quantum fluctuations and microscopic correlation energy in the three-dimensional Gross--Pitaevskii regime. For a smooth, compactly supported, nonnegative radial interaction and an NNN-independent normalized initial condensate wave function in H6H^6H6, we propagate the third spectral moment of a positive correlated energy along the exact many-body evolution. A uniform initial bound controls the third moment of the depletion number and sixth moments of collective one-particle fluctuations on every compact time interval, independently of NNN. For a nonzero interaction and a smooth, compactly supported initial condensate, we also identify the excitation-energy measure at scale N2N^2N2, weighted by the first energy. This limit holds for norm-convergent pure-state families with a uniform bound on the initial third correlated-energy moment. Its positive-energy part is a spectral measure of the relative scattering operator −2Δ+v-2Δ+v−2Δ+v, multiplied by a time-dependent dressed contact. We compute the radial density and prove strict positivity of the contact for nonzero references with finite kinetic energy and finite mean excitation number. The same evolving states have bounded physical fluctuation moments and excitation-energy moments of order p>1p>1p>1 bounded below by a positive multiple of N2p−2N^{2p-2}N2p−2, in the extended spectral sense. Inverse powers of the positive initial energy construct admissible pure and mixed families. For the norm-convergent pure-state class, we determine the scaled interaction-energy moments for 1≤p<21\le p<21≤p<2 and bound the second moment by O(N2)O(N^2)O(N2). A finite cubic approximation has norm error O(N−3/16)O(N^{-3/16})O(N−3/16) on the energy form domain and O(N−5/4)O(N^{-5/4})O(N−5/4) on a stronger initial domain of its generator.
We consider the evolution of a gas of N bosons in the three-dimensional Gross–Pitaevskii regime (in which particles are initially trapped in a volume of order one and interact through a repulsive potential with scattering length of order 1/N ). We construct a quasi-free approximation of the many-body dynamics, whose distance to the solution of the Schrödinger equation converges to zero, as N \to \infty , in the L^{2} (\mathbb{R}^{3N}) -norm. To achieve this goal, we let the Bose–Einstein condensate evolve according to a time-dependent Gross–Pitaevskii equation. After factoring out the microscopic correlation structure, the evolution of the orthogonal excitations of the condensate is governed instead by a Bogoliubov dynamics, with a time-dependent generator quadratic in creation and annihilation operators. As an application, we show a central limit theorem for fluctuations of bounded observables around their expectation with respect to the Gross–Pitaevskii dynamics.
Bose-Einstein condensates have been experimentally realized in cold Bose gases for the first time in 1995.This discovery has renewed interest in the theory of interacting Bose gases; however, the mathematical understanding of Bose-Einstein condensation is still incomplete.In this thesis we derive effective theories for the dynamical and spectral properties of Bose gases.Our theoretical description is based on scaling regimes where the number of particles N becomes large and the two-body interaction potential scales as N 3β-1 V (N β •), for a fixed β ∈ (0, 1].The case β = 1 is known as Gross-Pitaevskii regime.Our first result concerns the time evolution of Bose-Einstein condensates after the trapping potential is removed.In previous works, it was shown that the many-body dynamics can be approximated by a one-body nonlinear Schrödinger equation at the level of reduced density matrices.We provide a more precise approximation of the time evolution: in a second-quantized setting, we construct a unitary effective dynamics, which, in the limit of large N , approximates the fully evolved state in Fock space norm.Our result comes with an explicit bound on the rate of convergence.It is valid for β < 1.Our second result addresses the properties of the ground state wave function.It has been shown before that the ground state wave function exhibits condensation in the limit N → ∞, i.e. the one-particle density matrix associated to the ground state approaches a projection.We prove the optimal bound for the convergence rate, for a translation invariant system.To do this, we estimate uniformly in N the number of excited particles.This result is valid for β = 1.Our last result concerns the ground state energy and the excitation spectrum of the translation invariant system.We prove the validity of Bogoliubov theory; this allows us to compute the ground state energy and the low-lying excitation spectrum up to an error that vanishes for large N .This result holds for β < 1.This thesis is based on the articles [11], [9] and [10].i
We study stationary quantum fluctuations around a mean field limit in trapped, dilute atomic gases of repulsively interacting bosons at zero temperature. Our goal is to describe quantum-mechanically the lowest macroscopic many-body bound state consistent with a microscopic Hamiltonian that accounts for inhomogeneous particle scattering processes. In the mean field limit, the wave function of the condensate (macroscopic quantum state) satisfies a defocusing cubic nonlinear Schrödinger-type equation, the Gross–Pitaevskii equation. We include consequences of pair excitation, i.e., the scattering of particles in pairs from the condensate to other states, proposed in [T. T. Wu, J. Math. Phys., 2 (1961), pp. 105–123]. Our derivations rely on an uncontrolled yet physically motivated assumption for the many-body wave function. By relaxing mathematical rigor, from a particle Hamiltonian with a spatially varying interaction strength we derive via heuristics an integro–partial differential equation for the pair collision kernel, K, under a stationary condensate wave function, Φ\PhiΦ. For a scattering length with periodic microstructure of subscale ϵ\epsilonϵ, we formally describe via classical homogenization the lowest many-body bound state in terms of Φ\PhiΦ and K up to second order in ϵ\epsilonϵ. If the external potential is slowly varying, we solve the homogenized equations via boundary layer theory. As an application, we describe the partial depletion of the condensate.
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Abstract We consider N bosons in a box with volume one, interacting through a two-body potential with scattering length of the order N−1+κN^{-1+\kappa }N−1+κ N - 1 + κ , for κ>0\kappa >0κ>0 κ > 0 . Assuming that κ∈(0;1/43)\kappa \in (0;1/43)κ∈(0;1/43) κ ∈ ( 0 ; 1 / 43 ) , we show that low-energy states exhibit Bose–Einstein condensation and we provide bounds on the expectation and on higher moments of the number of excitations.
By measuring the density fluctuations in a highly elongated weakly interacting Bose gas, we observe and quantify the transition from the ideal gas to a quasi-condensate regime throughout the dimensional crossover from a purely 1D to an almost 3D gas. We show that that the entire transition region and the dimensional crossover are described surprisingly well by the modified Yang-Yang model. Furthermore, we find that at low temperatures the linear density at the quasi-condensate transition scales according to an interaction-driven scenario of a longitudinally uniform 1D Bose gas, whereas at high temperatures it scales according to the degeneracy-driven critical scenario of transverse condensation of a 3D ideal gas. We also perform measurements of the third moment of atom number fluctuations. We find a positive skewness of the atom number distribution in the ideal gas regime and a reduced skewness compatible with zero in the quasi-condensate regime. For our parameters, the third moment is a thermodynamic quantity whose measurement constitutes a sensitive test of the equation of state and our results are in agreement with a modified Yang-Yang thermodynamic prediction. Moreover, we show that the measured skewness reveals the presence of true three body correlations in the system.