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arXiv:2503.01777 · Nature Communications

Probing non-ergodicity and symmetry via direct measurement of coherent scattering in a shaken rotor

Floriane Arrouas, Julien Hébraud, Nicolas Ombredane, Eloi Flament, Dominique Ronco, Nathan Dupont, Gabriel Lemarié, Bertrand Georgeot, Christian Miniatura, Juliette Billy, Bruno Peaudecerf, David Guéry-Odelin

Published Jun 17, 2026• 2 citations
arXivHTMLPDF

Abstract

The Coherent Backscattering (CBS) peak is a well-known interferential signature of weak localization in disordered or chaotic systems. More recently, a second interference feature—the Coherent Forward Scattering (CFS) peak—was predicted to emerge in the regime of strong localization. However, it has never been directly observed. Here we report the direct observation of the CFS peak and demonstrate its dual role as a signature of non-ergodicity and as a probe of symmetries in quantum chaotic systems. Using a shaken rotor model realized with a Bose-Einstein condensate (BEC) of ultracold atoms in a modulated optical lattice, we investigate dynamical localization in momentum space. The CFS peak emerges in the position distribution as a consequence of non-ergodic dynamics, while its growth timescale and width reveal the underlying localization length. By finely tuning the modulation, we control time-reversal and parity symmetries and measure their distinct effects on both CBS and CFS peaks in position space. Our results highlight the strong link of both temporal growth and contrast of the coherent scattering peaks with symmetry and localization. This work opens several perspectives for investigating coherent scattering signatures in a many-body context. Ergodicity (or lack thereof) plays a major role in the relaxation of quantum systems to equilibrium, yet the search of experimentally accessible quantities acting as its signatures is still ongoing. Here, the authors observe the sought-after features of the coherent forward scattering peak from a Bose-Einstein condensate within an optical lattice.

DOI
10.1038/s41467-026-74302-7

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The Coherent Backscattering (CBS) peak is a well-known interferential signature of weak localization in disordered or chaotic systems. More recently, a second interference feature—the Coherent Forward Scattering (CFS) peak—was predicted to emerge in the regime of strong localization. However, it has never been directly observed. Here we report the direct observation of the CFS peak and demonstrate its dual role as a signature of non-ergodicity and as a probe of symmetries in quantum chaotic systems. Using a shaken rotor model realized with a Bose-Einstein condensate (BEC) of ultracold atoms in a modulated optical lattice, we investigate dynamical localization in momentum space. The CFS peak emerges in the position distribution as a consequence of non-ergodic dynamics, while its growth timescale and width reveal the underlying localization length. By finely tuning the modulation, we control time-reversal and parity symmetries and measure their distinct effects on both CBS and CFS peaks in position space. Our results highlight the strong link of both temporal growth and contrast of the coherent scattering peaks with symmetry and localization. This work opens several perspectives for investigating coherent scattering signatures in a many-body context. Ergodicity (or lack thereof) plays a major role in the relaxation of quantum systems to equilibrium, yet the search of experimentally accessible quantities acting as its signatures is still ongoing. Here, the authors observe the sought-after features of the coherent forward scattering peak from a Bose-Einstein condensate within an optical lattice.
We report on the direct observation of coherent backscattering (CBS) of ultracold atoms in a quasi-two-dimensional configuration. Launching atoms with a well-defined momentum in a laser speckle disordered potential, we follow the progressive build up of the momentum scattering pattern, consisting of a ring associated with multiple elastic scattering, and the CBS peak in the backward direction. Monitoring the depletion of the initial momentum component and the formation of the angular ring profile allows us to determine microscopic transport quantities. We also study the time evolution of the CBS peak and find it in fair agreement with predictions, at long times as well as at short times. The observation of CBS can be considered a direct signature of coherence in quantum transport of particles in disordered media. It is responsible for the so called weak localization phenomenon, which is the precursor of Anderson localization.
We study coherent forward scattering (CFS) in critical disordered systems, whose eigenstates are multifractals. We give general and simple arguments that make it possible to fully characterize the dynamics of the shape and height of the CFS peak. We show that the dynamics is governed by multifractal dimensions D_1 D 1 and D_2 D 2 , which suggests that CFS could be used as an experimental probe for quantum multifractality. Our predictions are universal and numerically verified in three paradigmatic models of quantum multifractality: Power-law Random Banded Matrices (PRBM), the Ruijsenaars-Schneider ensembles (RS), and the three-dimensional kicked-rotor (3DKR). In the strong multifractal regime, we show analytically that these universal predictions exactly coincide with results from standard perturbation theory applied to the PRBM and RS models.
The first successful experimental realization of a Bose-Einstein condensate (BEC) in 1995 sparked a still growing research activity that in the meantime has led to beautiful and surprising results, such as the interference of matter waves [4], the Mott-superfluid phase transition [49] or the direct observation of Anderson localization of matter waves [10]. The strong interest in BECs arises, among others, from the possibility to investigate known but hard to study quantum systems on a rather macroscopic scale: for example one can load ultra-cold bosons into an optical lattice created by a standing light wave, and thereby simulate a solid state device. When it comes to a measurement on the BEC, however, essentially all measurement techniques result in the destruction of the condensate: for example in the so called “time of flight method” the optical lattice is switched off to observe the expansion of the trapped atoms and thereby directly visualize their momentum distribution, which highlights the condensed component [11]. In this thesis we elaborate on a means to probe a Bose-Einstein condensate from outside, with a scattering particle. The interaction between the probe particle and the bosons allows energy exchange between them, i.e., this setup includes inelastic scattering. After the scattering process, the condensate is still confined in the optical lattice and can therefore be used for further investigations. Of course, this method of probing a BEC is only then a serious alternative if the properties of the BEC can be inferred from the scattering signal. Besides outlining the scattering formalism, the central aim of this work is to show how the properties of the BEC can be deduced from those quantities accessible in a scattering experiment, like for example the transmission or the inelastic cross section. The other, equally strong motivation of this work concerns the signature of the interaction between the bosons in the scattering quantities. The dynamics of an analog classical system depends strongly on the interaction strength, which can induce a complex structure of the underlying phase space like, e.g., the appearance of a separatrix or even the emergence of chaotic dynamics in a globally mixed regular-chaotic phase space. We are interested to which extent such phase-space metamorphosis is reflected in the properties of the scattering particle. Currently light scattering (Bragg spectroscopy) on an optical lattice [78, 77] or rather on an optical lattice embedded into a cavity [62,68,19] is under intense investigation. These efforts predominantly aim at a characterization of the Mott-superfluid transition by specific scattering properties. In the present thesis we focus on condensates loaded in two and three lattice sites, with a large number of bosons. These systems exhibit a diversity of important physical effects (as we will see below) and thus are of paradigmatic relevance for a detailed understanding of multiparticle quantum dynamics in periodic potentials.
We investigate the dynamics of a coherent matter wave packet launched in a random potential. As predicted by Cherroret et al. [Phys. Rev. A 85 011604(R) (2012)], the coherent backscattering signal in the momentum distribution freezes at the onset of Anderson localization. At the same time, a novel interference peak rises in the forward direction. An explanation in terms of chained maximally crossed diagrams predicts that the forward scattering peak emerges on a time scale τloc∗=2mξ2/ℏ\tau_\text{loc}^* = 2m\xi^2/\hbarτloc∗​=2mξ2/ℏ set by the localization length ξ\xiξ.
We numerically study time-dependent quantum transport of an interacting Bose-Einstein condensate based on the Gross-Pitaevskii equation in a two-dimensional correlated disorder potential. The scattering- and transport mean-free-path is numerically calculated and compared with a diagrammatic approach. Furthermore we introduce the phenomenon of coherent backscattering in the context of Bose-Einstein condensates. In the limit of a vanishing atom-atom interaction, a sharp cone in the angle-resolved density of the scattered matter wave is observed, arising from constructive interference between amplitudes propagating along reversed scattering paths. Weak interaction transforms this coherent backscattering peak into a pronounced dip, indicating destructive instead of constructive interference. This agrees with results obtained from a diagrammatic theory of weak localization in the presence of nonlinearity. For intermediate interaction strength we observe intrinsic time-dependent dynamic, which leads to a suppression of coherent backscattering.