Connected correlations in cold atom experiments

TL;DR

The paper surveys how connected correlations, quantified by cumulants $\kappa_n$, and full counting statistics enable direct access to high-order correlations in ultracold atom systems. It argues that nonzero $\kappa_n$ for $n>2$ signal non-Gaussian states and reveal genuine $n$-particle clusters, complementing traditional two-point observables. Across bosons, fermions, and spin models, the work shows how single-atom-resolved probes extract Bogoliubov pairing, antiferromagnetic and charge-density-wave order, and higher-order spin-charge correlations, highlighting regimes where Gaussian theories fail. The discussion underscores the methodological impact of measuring higher-order cumulants for diagnosing strong correlations, identifying microscopic clustering, and informing metrological gains, with implications for entanglement characterization in many-body quantum systems.

Abstract

The recent development of single-atom-resolved probes has made full counting statistics measurements accessible in quantum gas experiments. This capability provides access to high-order moments of physical observables, from which cumulants, or equivalently connected correlations, can be precisely determined. Through a selection of recent cold atom experiments, this article illustrates the significance of connected correlations in characterizing ensembles of interacting quantum particles. First, non-zero connected correlations of order n > 2 unambiguously identify non-Gaussian quantum states. Second, connected correlations of order n identify clusters made of n elements whose statistical properties are irreducible to combinations of smaller clusters. The ability to identify such multi-particle clusters offers a an interesting perspective on strongly correlated quantum states of matter at the microscopic scale.

Figures (8)

• Figure 1: Illustration of the decomposition of cumulants of order $n$ as the difference between the total correlations and all the contributions from cumulants of order $n'<n$rispoli:2019.
• Figure 2: (a) Contact interactions and four-wave mixing in momentum space. Here $g=4 \pi \hbar^2 a_s/m$. (b) Mean-field description of interactions: two-body interactions within the BEC (c) Linearised quantum fluctuations induced by interactions: creation of momentum-correlated pairs exiting the BEC (d) Beyond linearised quantum fluctuations: generation of momentum correlation between 3 or 4 different momentum modes.
• Figure 3: Connected correlations $G_{\rm c}^{(2)}({\bm k},{\bm k}')$ in the depletion of a weakly interacting condensate as a function of ${\bm k}+ {\bm k}'$. Inset: illustration of two-mode squeezing induced by interactions. A mode in the momentum space occupies a volume $\Delta k^3$ where $\Delta k = 2 \pi/L$ with $L$ the size of the considered system. The Bose-Einstein condensate (BEC) occupies the mode at ${\bm k}={\bm 0}$ and interactions promote Bogoliubov pairs in modes with opposite momenta $\pm \bm k$.
• Figure 4: a. Single-atom spin-resolved measurements of a trapped Fermi gas with $N=12$ particles in momentum space. The dashed circle indicates the Fermi surface. Momenta are expressed in natural units of the harmonic trap $p_\mathrm{HO} = \sqrt{\hbar m\omega}$. b. Connected correlations $C^{(2)}(\bm p_\uparrow, \bm p_\downarrow)$, where the momentum of the $\downarrow$-fermion is fixed (cross mark). c. Single-atom resolved measurements of a dilute, strongly interacting Fermi gas in position space. d. Normalised correlation function $g_{\uparrow\downarrow}^{(2)}(\bm r)$ as a function of the distance between opposite-spin fermions, showing a strong peak at short distances and anticorrelations at intermediate distances that is not captured by the mean-field BCS theory (dashed line). The panels a and b were adapted from holten:2022, while c and d were adapted from daix:2025.
• Figure 5: a. Quantum gas microscopy of a repulsive Fermi-Hubbard system close to half-filling, with one spin component removed. b. Resulting spin correlation map, computed using Eq. \ref{['eq:eqSpinCorrelations']}, displaying significant correlations across the entire system. Panels a and b were extracted and adapted from mazurenko:2017a. c. Quantum gas microscopy of an attractive Fermi-Hubbard system. d. Resulting density correlations, exhibiting a checkerboard pattern characteristic of CDW ordering, analogous to the AFM ordering of the repulsive case. Note that this connected correlator takes negative values precisely because the disconnected part is subtracted. Panels c and d were extracted and adapted from hartke:2023