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3D Anderson Transition of Ultracold Atoms in state-dependent disorder : Direct observation and Numerical study of the energy-resolved transfer scheme

Hoa Quach

Published Sep 25, 2026• 0 citations
OpenAlex

Abstract

Anderson Localization - the complete halt of wave transport through multiple scattering and phase coherence - was discovered in 1958 and has been actively studied since then with several types of waves (light, µ-waves, ultrasound, acoustic waves, etc.). In three-dimensional systems, Anderson Localization manifests as a phase transition around a critical energy - the Mobility Edge - that separates the localized (insulator) states from the diffusive (metallic) states.Ultracold atoms are promising candidates for such experiments, and various groups have reported evidence of the 3D Anderson Transition, in particular, our group in 2012. Nevertheless, all these preliminary measurements were limited by the unwanted broadening of the energy distribution when the atoms were transferred into the disorder. Hence, no direct observation of the transition could be achieved.The quest for the direct observation of the 3D Anderson Transition has motivated us to develop a novel spectroscopic scheme using state-dependent disorder. The concept is to transfer, via RF coupling, a small fraction of a Bose-Einstein Condensate from a disorder-insensitive state to a disorder-sensitive state. By doing so, we selectively populate targeted energy levels in the disorder with a narrow energy distribution. This technique has paved the way for successfully achieving the direct observation of the 3D Anderson Transition in a 3D laser speckle. Moreover, it yields a precise measurement of the Mobility Edge, which is in excellent agreement with the numerical predictions.The heart of our achievement lies in the spectroscopic scheme. When transferring only a few atoms, we expect the coupling of a discrete state to a continuum, leading to a Fourier-limited energy distribution. This key assumption has so far been verified only indirectly, but never tested numerically. In the last part of my PhD, I have set up the framework for the numerical description of the RF transfer scheme. While the study remains preliminary, it already demonstrates that the dynamics can be more subtle than expected, and the model deserves further attention in the future.

Related papers

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Anderson localization of particles -- the complete halt of wave transport through multiple scattering and phase coherence -- is a paradigmatic manifestation of quantum interference in disordered media. In three dimensions, the scaling theory predicts a quantum phase transition at a critical energy, the mobility edge, separating localized from diffusive states and underpinning metal-insulator transitions in electronic systems. Despite decades of experimental efforts, a direct observation of this emblematic transition for matter waves has remained elusive. Previous attempts with ultracold atoms were hindered by strong and uncontrolled energy broadening, resulting in indirect, sometimes inaccurate, and model-dependent estimates of the mobility edge. Here we implement a novel energy-resolved scheme to prepare atomic matter waves with a narrow energy distribution and track their expansion dynamics over long timescales. This allows for a direct observation of the three-dimensional Anderson transition in a laser-speckle disordered potential, and for a precise measurement of the mobility edge that is independent of any underlying theoretical modeling. Our measurements show excellent agreement with state-of-the-art numerical predictions over a wide range of disorder strengths, resolving long-standing discrepancies between prior experiments and theory. Beyond the three-dimensional Anderson transition, our approach opens new avenues for quantitative investigations of quantum critical phenomena in spatially disordered systems, including the roles of dimensionality, symmetry class, and interactions.

arXiv:2602.07654•Feb 7, 2026•0 citations

3D Anderson Transition of Ultracold Atoms in state-dependent disorder : Direct observation and Numerical study of the energy-resolved transfer scheme

Hoa Quach

Transition d'Anderson 3D d'atomes ultrafroids dans un désordre dépendant de l'état : Observation directe et simulation numérique du principe du transfert résolu en énergie La localisation d'Anderson - l'arrêt complet du transport des ondes par diffusion multiple et cohérence de phase - a été découverte en 1958 et a depuis été activement étudiée pour plusieurs types d'ondes (lumière, micro-ondes, ultrasons, ondes acoustiques, etc.). Dans les systèmes tridimensionnels, la localisation d'Anderson se manifeste comme une transition de phase autour d'une énergie critique - le seuil de mobilité - qui sépare les états localisés (isolants) des états diffusifs (métalliques).Les atomes ultrafroids sont des candidats prometteurs pour de telles expériences, et plusieurs groupes ont rapporté des preuves de la transition d'Anderson 3D, en particulier notre groupe en 2012. Néanmoins, toutes ces mesures préliminaires étaient limitées par l'élargissement indésirable de la distribution en énergie lors du transfert des atomes dans le désordre. Ainsi, aucune observation directe de la transition n'a pu être réalisée.La quête d'une observation directe de la transition d'Anderson 3D nous a motivés à développer un nouveau schéma spectroscopique fondé sur un désordre dépendant de l'état. Le concept consiste à transférer, par couplage RF, une petite fraction d'un condensat de Bose-Einstein d'un état insensible au désordre vers un état sensible au désordre. Ce faisant, nous peuplons sélectivement des niveaux d'énergie ciblés dans le désordre avec une distribution en énergie étroite. Cette technique a ouvert la voie à l'observation directe, couronnée de succès, de la transition d'Anderson 3D dans un speckle laser tridimensionnel. De plus, elle fournit une mesure précise du seuil de mobilité, en excellent accord avec les prédictions numériques.Le cœur de notre résultat réside dans le schéma spectroscopique. Lors du transfert d'un petit nombre d'atomes, nous nous attendons au couplage d'un état discret à un continuum, conduisant à une distribution en énergie limitée par transformée de Fourier. Cette hypothèse essentielle n'a jusqu'à présent été vérifiée que de manière indirecte, sans jamais avoir été testée numériquement. Dans la dernière partie de ma thèse, j'ai mis en place le cadre pour la description numérique du schéma de transfert RF. Bien que cette étude demeure préliminaire, elle montre déjà que la dynamique peut être plus subtile que prévu, et que le modèle mérite une attention plus approfondie à l'avenir.

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Abstract In a recent experiment (Yu et al 2026 arXiv: 2602.07654 ), energy-resolved measurements of an atomic matter wave spreading in a speckle potential enabled the direct observation of the three-dimensional Anderson transition. In this work, we present a quantitative theoretical description of the matter-wave dynamics based on a tailored implementation of the self-consistent theory of localization, which incorporates both the spectral and spatial properties of the state prepared in the disorder. We benchmark this theoretical approach against ab initio numerical simulations, and use it to analyze the atom density profiles observed experimentally in the localized, diffusive, and critical regimes. Particular emphasis is placed on the key role of the atomic energy distribution, especially on the distinct contributions of Bose-condensed and thermal atoms to interpret the experimental profiles. Our framework provides a versatile and efficient theoretical toolbox for quantitatively describing wave-packet dynamics in three-dimensional disordered quantum systems, which remains challenging for state-of-the-art large-scale numerical simulations.

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Bichromatic state-dependent disordered potential for Anderson localization of ultracold atoms

Baptiste Lecoutre, Yukun Guo, Xudong Yu +5 more

Abstract: The ability to load ultracold atoms at a well-defined energy in a disordered potential is a crucial tool to study quantum transport, and in particular Anderson localization. In this paper, we present a new method for achieving that goal by rf transfer of atoms in an atomic Bose-Einstein condensate from a disorder-insensitive state to a disorder-sensitive state. It is based on a bichromatic laser speckle pattern, produced by two lasers whose frequencies are chosen so that their light-shifts cancel each other in the first state and add up in the second state. Moreover, the spontaneous scattering rate in the disorder-sensitive state is low enough to allow for long observation times of quantum transport in that state. We theoretically and experimentally study the characteristics of the resulting potential.

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Anderson Localization of Ultracold Atoms: Where is the Mobility Edge?

Michael Pasek, Giuliano Orso, Dominique Delande

Recent experiments in noninteracting ultracold atoms in three-dimensional speckle potentials have yielded conflicting results regarding the so-called mobility edge, i.e., the energy threshold separating Anderson localized from diffusive states. At the same time, there are theoretical indications that most experimental data overestimate this critical energy, sometimes by a large amount. Using extensive numerical simulations, we show that the effect of anisotropy in the spatial correlations of realistic disorder configurations alone is not sufficient to explain the experimental data. In particular, we find that the mobility edge obeys a universal scaling behavior, independently of the speckle geometry.

arXiv:1609.01065•Physical Review Letters•Apr 28, 2017•57 citations

Transport and disorder-induced localization of ultracold Fermi gases

William McGehee

We experimentally study localization and dynamics of ultracold fermions in speckle and optical lattice potentials to explore Anderson localization, many-body localization, and relaxation dynamics in strongly correlated systems. Anderson localization is probed by releasing non-interacting, spin-polarized gases into three dimensional, anisotropic disordered potentials produced from optical speckle. A fraction of the atoms are localized by the disorder, and a mobility edge is found separating localized from extended states. The length scale of the speckle is varied, and the localized state is found to scale linearly with the geometric mean of the speckle autocorrelation length. We realize the Fermi Hubbard model by loading atoms in a cubic optical lattice. Non-equilibrium momentum distributions are created via Raman transitions, and the excitation relaxation rate is measured in the lattice. Transport experiments were performed in a disordered optical lattice to explore the disordered Hubbard model. These experiments reveal localization in the presence of strong interactions and an interaction driven metal-to-insulator transition. The localized state is found to be insensitive to a doubling in the temperature of the gas and is consistent with predictions of many-body localization.

•Jul 14, 2015•0 citations
Anderson Localization - the complete halt of wave transport through multiple scattering and phase coherence - was discovered in 1958 and has been actively studied since then with several types of waves (light, µ-waves, ultrasound, acoustic waves, etc.). In three-dimensional systems, Anderson Localization manifests as a phase transition around a critical energy - the Mobility Edge - that separates the localized (insulator) states from the diffusive (metallic) states.Ultracold atoms are promising candidates for such experiments, and various groups have reported evidence of the 3D Anderson Transition, in particular, our group in 2012. Nevertheless, all these preliminary measurements were limited by the unwanted broadening of the energy distribution when the atoms were transferred into the disorder. Hence, no direct observation of the transition could be achieved.The quest for the direct observation of the 3D Anderson Transition has motivated us to develop a novel spectroscopic scheme using state-dependent disorder. The concept is to transfer, via RF coupling, a small fraction of a Bose-Einstein Condensate from a disorder-insensitive state to a disorder-sensitive state. By doing so, we selectively populate targeted energy levels in the disorder with a narrow energy distribution. This technique has paved the way for successfully achieving the direct observation of the 3D Anderson Transition in a 3D laser speckle. Moreover, it yields a precise measurement of the Mobility Edge, which is in excellent agreement with the numerical predictions.The heart of our achievement lies in the spectroscopic scheme. When transferring only a few atoms, we expect the coupling of a discrete state to a continuum, leading to a Fourier-limited energy distribution. This key assumption has so far been verified only indirectly, but never tested numerically. In the last part of my PhD, I have set up the framework for the numerical description of the RF transfer scheme. While the study remains preliminary, it already demonstrates that the dynamics can be more subtle than expected, and the model deserves further attention in the future.
Anderson localization of particles -- the complete halt of wave transport through multiple scattering and phase coherence -- is a paradigmatic manifestation of quantum interference in disordered media. In three dimensions, the scaling theory predicts a quantum phase transition at a critical energy, the mobility edge, separating localized from diffusive states and underpinning metal-insulator transitions in electronic systems. Despite decades of experimental efforts, a direct observation of this emblematic transition for matter waves has remained elusive. Previous attempts with ultracold atoms were hindered by strong and uncontrolled energy broadening, resulting in indirect, sometimes inaccurate, and model-dependent estimates of the mobility edge. Here we implement a novel energy-resolved scheme to prepare atomic matter waves with a narrow energy distribution and track their expansion dynamics over long timescales. This allows for a direct observation of the three-dimensional Anderson transition in a laser-speckle disordered potential, and for a precise measurement of the mobility edge that is independent of any underlying theoretical modeling. Our measurements show excellent agreement with state-of-the-art numerical predictions over a wide range of disorder strengths, resolving long-standing discrepancies between prior experiments and theory. Beyond the three-dimensional Anderson transition, our approach opens new avenues for quantitative investigations of quantum critical phenomena in spatially disordered systems, including the roles of dimensionality, symmetry class, and interactions.
Transition d'Anderson 3D d'atomes ultrafroids dans un désordre dépendant de l'état : Observation directe et simulation numérique du principe du transfert résolu en énergie La localisation d'Anderson - l'arrêt complet du transport des ondes par diffusion multiple et cohérence de phase - a été découverte en 1958 et a depuis été activement étudiée pour plusieurs types d'ondes (lumière, micro-ondes, ultrasons, ondes acoustiques, etc.). Dans les systèmes tridimensionnels, la localisation d'Anderson se manifeste comme une transition de phase autour d'une énergie critique - le seuil de mobilité - qui sépare les états localisés (isolants) des états diffusifs (métalliques).Les atomes ultrafroids sont des candidats prometteurs pour de telles expériences, et plusieurs groupes ont rapporté des preuves de la transition d'Anderson 3D, en particulier notre groupe en 2012. Néanmoins, toutes ces mesures préliminaires étaient limitées par l'élargissement indésirable de la distribution en énergie lors du transfert des atomes dans le désordre. Ainsi, aucune observation directe de la transition n'a pu être réalisée.La quête d'une observation directe de la transition d'Anderson 3D nous a motivés à développer un nouveau schéma spectroscopique fondé sur un désordre dépendant de l'état. Le concept consiste à transférer, par couplage RF, une petite fraction d'un condensat de Bose-Einstein d'un état insensible au désordre vers un état sensible au désordre. Ce faisant, nous peuplons sélectivement des niveaux d'énergie ciblés dans le désordre avec une distribution en énergie étroite. Cette technique a ouvert la voie à l'observation directe, couronnée de succès, de la transition d'Anderson 3D dans un speckle laser tridimensionnel. De plus, elle fournit une mesure précise du seuil de mobilité, en excellent accord avec les prédictions numériques.Le cœur de notre résultat réside dans le schéma spectroscopique. Lors du transfert d'un petit nombre d'atomes, nous nous attendons au couplage d'un état discret à un continuum, conduisant à une distribution en énergie limitée par transformée de Fourier. Cette hypothèse essentielle n'a jusqu'à présent été vérifiée que de manière indirecte, sans jamais avoir été testée numériquement. Dans la dernière partie de ma thèse, j'ai mis en place le cadre pour la description numérique du schéma de transfert RF. Bien que cette étude demeure préliminaire, elle montre déjà que la dynamique peut être plus subtile que prévu, et que le modèle mérite une attention plus approfondie à l'avenir.
Abstract In a recent experiment (Yu et al 2026 arXiv: 2602.07654 ), energy-resolved measurements of an atomic matter wave spreading in a speckle potential enabled the direct observation of the three-dimensional Anderson transition. In this work, we present a quantitative theoretical description of the matter-wave dynamics based on a tailored implementation of the self-consistent theory of localization, which incorporates both the spectral and spatial properties of the state prepared in the disorder. We benchmark this theoretical approach against ab initio numerical simulations, and use it to analyze the atom density profiles observed experimentally in the localized, diffusive, and critical regimes. Particular emphasis is placed on the key role of the atomic energy distribution, especially on the distinct contributions of Bose-condensed and thermal atoms to interpret the experimental profiles. Our framework provides a versatile and efficient theoretical toolbox for quantitatively describing wave-packet dynamics in three-dimensional disordered quantum systems, which remains challenging for state-of-the-art large-scale numerical simulations.
Abstract: The ability to load ultracold atoms at a well-defined energy in a disordered potential is a crucial tool to study quantum transport, and in particular Anderson localization. In this paper, we present a new method for achieving that goal by rf transfer of atoms in an atomic Bose-Einstein condensate from a disorder-insensitive state to a disorder-sensitive state. It is based on a bichromatic laser speckle pattern, produced by two lasers whose frequencies are chosen so that their light-shifts cancel each other in the first state and add up in the second state. Moreover, the spontaneous scattering rate in the disorder-sensitive state is low enough to allow for long observation times of quantum transport in that state. We theoretically and experimentally study the characteristics of the resulting potential.
Recent experiments in noninteracting ultracold atoms in three-dimensional speckle potentials have yielded conflicting results regarding the so-called mobility edge, i.e., the energy threshold separating Anderson localized from diffusive states. At the same time, there are theoretical indications that most experimental data overestimate this critical energy, sometimes by a large amount. Using extensive numerical simulations, we show that the effect of anisotropy in the spatial correlations of realistic disorder configurations alone is not sufficient to explain the experimental data. In particular, we find that the mobility edge obeys a universal scaling behavior, independently of the speckle geometry.
We experimentally study localization and dynamics of ultracold fermions in speckle and optical lattice potentials to explore Anderson localization, many-body localization, and relaxation dynamics in strongly correlated systems. Anderson localization is probed by releasing non-interacting, spin-polarized gases into three dimensional, anisotropic disordered potentials produced from optical speckle. A fraction of the atoms are localized by the disorder, and a mobility edge is found separating localized from extended states. The length scale of the speckle is varied, and the localized state is found to scale linearly with the geometric mean of the speckle autocorrelation length. We realize the Fermi Hubbard model by loading atoms in a cubic optical lattice. Non-equilibrium momentum distributions are created via Raman transitions, and the excitation relaxation rate is measured in the lattice. Transport experiments were performed in a disordered optical lattice to explore the disordered Hubbard model. These experiments reveal localization in the presence of strong interactions and an interaction driven metal-to-insulator transition. The localized state is found to be insensitive to a doubling in the temperature of the gas and is consistent with predictions of many-body localization.