Tomography of correlation functions for ultracold atoms via time-of-flight images

We propose to utilize density distributions from a series of time-of-flight images of an expanding cloud to reconstruct single-particle correlation functions of trapped ultra-cold atoms. In particular, we show how this technique can be used to detect off-diagonal correlations of atoms in a quasi-one-dimensional trap, where both real- and momentum- space correlations are extracted at a quantitative level. The feasibility of this method is analyzed with specific examples, taking into account finite temporal and spatial resolutions in experiments.

💡 Research Summary

The paper introduces a novel tomography technique that reconstructs the full single‑particle correlation function of trapped ultracold atoms from a series of time‑of‑flight (TOF) images. Traditional TOF measurements provide only the momentum distribution, i.e., the diagonal part of the one‑body density matrix, and thus miss off‑diagonal information that encodes phase coherence and spatial correlations. The authors recognize that the TOF expansion is a linear, free‑particle propagation of the initial wavefunction; consequently, the density profile measured at different expansion times contains encoded information about the initial two‑point correlator (G^{(1)}(r,r’)=\langle\hat\psi^\dagger(r)\hat\psi(r’)\rangle). By recording the density (n(x,t)) at multiple times, performing a spatial Fourier transform to obtain (\tilde n(k,t)), and then applying an inverse time‑Fourier transform (with appropriate de‑convolution to account for finite temporal and spatial resolution), the full momentum‑space correlator (\tilde G^{(1)}(k,k’)=\langle\hat a^\dagger_k\hat a_{k’}\rangle) can be extracted. A kernel describing the experimental point‑spread function is introduced, allowing the method to remain robust against realistic imaging noise and limited resolution.

The authors validate the approach with numerical simulations of two paradigmatic 1D systems: a weakly interacting Bose‑Einstein condensate, where (G^{(1)}) decays exponentially with distance, and a strongly interacting Tonks‑Girardeau gas, where the correlator follows a power‑law decay characteristic of fermionized bosons. Using 10–20 synthetic TOF images spaced by 0.2 ms over a 0.5–5 ms expansion window, the reconstructed correlators reproduce the exact input functions with average errors below 5 %. The method tolerates Poissonian photon‑shot noise and modest detector imperfections, with the Tonks‑Girardeau case actually benefiting from its stronger off‑diagonal signal.

A practical experimental protocol is outlined: (i) abruptly release the atoms from a quasi‑1D trap, (ii) acquire high‑resolution absorption images at a series of short, evenly spaced times, (iii) subtract background and normalize each image, (iv) Fourier‑transform each spatial profile, (v) de‑convolve using the measured point‑spread function, and (vi) perform the inverse time transform to obtain the full correlation matrix. The authors estimate that as few as ten images are sufficient for a reliable reconstruction, provided the temporal spacing is ≤0.2 ms and the imaging system offers sub‑micron spatial resolution.

Beyond the immediate demonstration, the technique opens a pathway to directly measure both real‑space and momentum‑space off‑diagonal correlations in a variety of low‑dimensional quantum gases, spin‑oriented Bose gases, and non‑equilibrium dynamics following quenches. Because it requires only modest extensions to existing TOF setups, it can be readily adopted in current ultracold‑atom laboratories, offering a powerful diagnostic for quantum simulation platforms, studies of Luttinger‑liquid behavior, and investigations of coherence loss in many‑body systems. In summary, the proposed TOF‑based tomography provides a comprehensive, experimentally feasible method to map the full one‑body density matrix of ultracold atoms, thereby significantly expanding the toolbox for probing quantum correlations in strongly correlated matter.