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arXiv:2608.09924

A Quantum Coherence Microscope in the Hubbard Regime

Lin Su, Michal Szurek, Alec Douglas, Ceren B. Dag, Markus Greiner

quant-phcond-mat.quant-gascond-mat.str-elphysics.atom-phSubmitted Aug 10, 2026• Announced Aug 11, 2026• 0 citations
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Abstract

Quantum coherence underlies collective quantum phenomena and emerging quantum technologies. Quantum gas microscopes have transformed quantum simulation by providing projective snapshots of many-body states with single-atom resolution, but spatially resolved measurements of off-diagonal correlations have remained elusive. Here, using the Talbot effect, we introduce a quantum coherence microscope that maps off-diagonal correlations onto site-resolved density signals with near-single-site resolution. We use this technique to locally probe the superfluid-Mott transition in a layer of a three-dimensional optical lattice and to measure coherence beyond nearest neighbors in an engineered potential landscape. By mapping off-diagonal correlations onto density signals through controlled Talbot evolution, this work opens new possibilities for accessing observables beyond the density basis through tailored matter-wave evolution and recapture.

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Coherence properties are central to quantum systems and are at the heart of phenomena such as superconductivity. In this work, we studied coherence properties of an ultracold Bose gas in a two-dimensional optical lattice across the thermal phase transition. To infer the phase coherence and phase fluctuation profiles, we used direct matter-wave imaging of higher Talbot revivals and introduced a phase microscope based on a site-resolved mapping of phase fluctuations to density fluctuations during matter-wave imaging. We observed the algebraic decay of the phase correlations in the superfluid phase and a linear temperature increase of the exponent. These techniques may enable studying coherence properties in strongly correlated quantum systems with full spatial resolution.

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Ultracold quantum gases in optical lattices provide a rich experimental toolbox for simulating the physics of condensed matter systems. With atoms in the lattice playing the role of electrons or Cooper pairs in real materials, it is possible to experimentally realize condensed matter Hamiltonians in a controlled way. To realize the full potential of such quantum simulations, we leverage a quantum gas microscope which can spatially resolve the atoms in the optical lattice at the single site level and project arbitrary potential landscapes onto the atoms by combining the high resolution optics with static holographic masks or a spatial light modulator. The versatility of this system allows for a wide range of studies. In this thesis, we focus on three experiments that highlight the utility of a tunable platform and demonstrate the richness of physics that comes with interactions.\nIn the first, we observe on-site interparticle interactions introduce a non-linearity into the many-body energy spectrum of a harmonic oscillator. The excitation of a single atom into a higher band induces an energy shift in the excitation energy for the second atom, blockading the orbital excitation. We demonstrate a technique for deterministic number filtering based upon orbital excitation blockade. When applied to a high temperature Mott insulator, this number filtering realizes a form of algorithmic cooling, which can be used for initializing a large quantum register or creating low entropy initial states for many-body simulation.\nIn the second, we study the applicability of statistical physics in an isolated system governed by quantum mechanics. While the descriptive power of these two fields has been confirmed through repeated and varied empirical studies, their underlying axioms of entropy maximization and unitary evolution appear mutually exclusive. With our quantum gas microscope, we have the ability to probe local observables such as the on-site number statistics as well as measure the quantum mechanical purity through interference operations. With these capabilities, we are able to observe the local thermalization of a manifestly quantum mechanically pure many-body system.\nIn the last chapter, we report on the first cold atoms studies of an interacting system under an artificial gauge field. We consider the propagation of a pair of atoms in a $2\\times N$ ladder governed by the Harper-Hofstadter model. The eigenstates of this system include both scattering states, which can exhibit chiral propagation, and bound states. A superexchange energy shift in our initial state leads to imbalanced population in the two chiral scattering sectors, leading to chiral propagation dynamics. This effect is the result of both the presence of interactions and the artificial gauge fields. These two components are essential ingredients for fractional quantum Hall physics. Moreover, the dynamic tunable gauge field we develop and the control afforded by a quantum gas microscope make this platform well-suited for future studies in this regime.

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Quantum gas microscopes probe quantum many-body lattice states via projective measurements in the occupation basis, enabling access to various density and spin correlations. Phase information, however, cannot be directly obtained in these setups. Recent experiments went beyond this by measuring local current operators and local phase fluctuations. Here we propose how Fourier-space manipulation in a matter-wave microscope allows access to various long-range off-diagonal correlators in experimentally realistic settings, realizing a many-body phase microscope. We demonstrate in particular how the fermionic d-wave superconducting order parameter in arbitrary Hubbard-type models, the non-equal time Green's function yielding the spectral function, or the hidden order of composite bosons in a fractional Chern insulator can be directly measured. Our results show the great potential of matter-wave microscopy for accessing exotic correlators including phases and coherences and characterizing intriguing quantum many-body states.

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Quantum simulations with ultracold atoms in optical lattices open up an exciting path toward understanding strongly interacting quantum systems. Atom gas microscopes are crucial for this as they offer single-site density resolution, unparalleled in other quantum many-body systems. However, currently a direct measurement of local coherent currents is out of reach. In this Letter, we show how to achieve that by measuring densities that are altered in response to quenches to noninteracting dynamics, e.g., after tilting the optical lattice. For this, we establish a data analysis method solving the closed set of equations relating tunneling currents and atom number dynamics, allowing us to reliably recover the full covariance matrix, including off-diagonal terms representing coherent currents. The signal processing builds upon semidefinite optimization, providing bona fide covariance matrices optimally matching the observed data. We demonstrate how the obtained information about noncommuting observables allows one to quantify entanglement at finite temperature, which opens up the possibility to study quantum correlations in quantum simulations going beyond classical capabilities.

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Probing the Hubbard Model With Single-Site Resolution

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Site-resolved measurement of the spin-correlation function in the Fermi-Hubbard model

Maxwell F. Parsons, Anton Mazurenko, Christie Chiu +3 more

Exotic phases of matter can emerge from strong correlations in quantum many-body systems. Quantum gas microscopy affords the opportunity to study these correlations with unprecedented detail. Here, we report site-resolved observations of antiferromagnetic correlations in a two-dimensional, Hubbard-regime optical lattice and demonstrate the ability to measure the spin-correlation function over any distance. We measure the in situ distributions of the particle density and magnetic correlations, extract thermodynamic quantities from comparisons to theory, and observe statistically significant correlations over three lattice sites. The temperatures that we reach approach the limits of available numerical simulations. The direct access to many-body physics at the single-particle level demonstrated by our results will further our understanding of how the interplay of motion and magnetism gives rise to new states of matter.

arXiv:1605.02704•Science•Sep 15, 2016•335 citations
Quantum coherence underlies collective quantum phenomena and emerging quantum technologies. Quantum gas microscopes have transformed quantum simulation by providing projective snapshots of many-body states with single-atom resolution, but spatially resolved measurements of off-diagonal correlations have remained elusive. Here, using the Talbot effect, we introduce a quantum coherence microscope that maps off-diagonal correlations onto site-resolved density signals with near-single-site resolution. We use this technique to locally probe the superfluid-Mott transition in a layer of a three-dimensional optical lattice and to measure coherence beyond nearest neighbors in an engineered potential landscape. By mapping off-diagonal correlations onto density signals through controlled Talbot evolution, this work opens new possibilities for accessing observables beyond the density basis through tailored matter-wave evolution and recapture.
Coherence properties are central to quantum systems and are at the heart of phenomena such as superconductivity. In this work, we studied coherence properties of an ultracold Bose gas in a two-dimensional optical lattice across the thermal phase transition. To infer the phase coherence and phase fluctuation profiles, we used direct matter-wave imaging of higher Talbot revivals and introduced a phase microscope based on a site-resolved mapping of phase fluctuations to density fluctuations during matter-wave imaging. We observed the algebraic decay of the phase correlations in the superfluid phase and a linear temperature increase of the exponent. These techniques may enable studying coherence properties in strongly correlated quantum systems with full spatial resolution.
Ultracold quantum gases in optical lattices provide a rich experimental toolbox for simulating the physics of condensed matter systems. With atoms in the lattice playing the role of electrons or Cooper pairs in real materials, it is possible to experimentally realize condensed matter Hamiltonians in a controlled way. To realize the full potential of such quantum simulations, we leverage a quantum gas microscope which can spatially resolve the atoms in the optical lattice at the single site level and project arbitrary potential landscapes onto the atoms by combining the high resolution optics with static holographic masks or a spatial light modulator. The versatility of this system allows for a wide range of studies. In this thesis, we focus on three experiments that highlight the utility of a tunable platform and demonstrate the richness of physics that comes with interactions.\nIn the first, we observe on-site interparticle interactions introduce a non-linearity into the many-body energy spectrum of a harmonic oscillator. The excitation of a single atom into a higher band induces an energy shift in the excitation energy for the second atom, blockading the orbital excitation. We demonstrate a technique for deterministic number filtering based upon orbital excitation blockade. When applied to a high temperature Mott insulator, this number filtering realizes a form of algorithmic cooling, which can be used for initializing a large quantum register or creating low entropy initial states for many-body simulation.\nIn the second, we study the applicability of statistical physics in an isolated system governed by quantum mechanics. While the descriptive power of these two fields has been confirmed through repeated and varied empirical studies, their underlying axioms of entropy maximization and unitary evolution appear mutually exclusive. With our quantum gas microscope, we have the ability to probe local observables such as the on-site number statistics as well as measure the quantum mechanical purity through interference operations. With these capabilities, we are able to observe the local thermalization of a manifestly quantum mechanically pure many-body system.\nIn the last chapter, we report on the first cold atoms studies of an interacting system under an artificial gauge field. We consider the propagation of a pair of atoms in a 2timesN2\\times N2timesN ladder governed by the Harper-Hofstadter model. The eigenstates of this system include both scattering states, which can exhibit chiral propagation, and bound states. A superexchange energy shift in our initial state leads to imbalanced population in the two chiral scattering sectors, leading to chiral propagation dynamics. This effect is the result of both the presence of interactions and the artificial gauge fields. These two components are essential ingredients for fractional quantum Hall physics. Moreover, the dynamic tunable gauge field we develop and the control afforded by a quantum gas microscope make this platform well-suited for future studies in this regime.
Quantum gas microscopes probe quantum many-body lattice states via projective measurements in the occupation basis, enabling access to various density and spin correlations. Phase information, however, cannot be directly obtained in these setups. Recent experiments went beyond this by measuring local current operators and local phase fluctuations. Here we propose how Fourier-space manipulation in a matter-wave microscope allows access to various long-range off-diagonal correlators in experimentally realistic settings, realizing a many-body phase microscope. We demonstrate in particular how the fermionic d-wave superconducting order parameter in arbitrary Hubbard-type models, the non-equal time Green's function yielding the spectral function, or the hidden order of composite bosons in a fractional Chern insulator can be directly measured. Our results show the great potential of matter-wave microscopy for accessing exotic correlators including phases and coherences and characterizing intriguing quantum many-body states.
Quantum simulations with ultracold atoms in optical lattices open up an exciting path toward understanding strongly interacting quantum systems. Atom gas microscopes are crucial for this as they offer single-site density resolution, unparalleled in other quantum many-body systems. However, currently a direct measurement of local coherent currents is out of reach. In this Letter, we show how to achieve that by measuring densities that are altered in response to quenches to noninteracting dynamics, e.g., after tilting the optical lattice. For this, we establish a data analysis method solving the closed set of equations relating tunneling currents and atom number dynamics, allowing us to reliably recover the full covariance matrix, including off-diagonal terms representing coherent currents. The signal processing builds upon semidefinite optimization, providing bona fide covariance matrices optimally matching the observed data. We demonstrate how the obtained information about noncommuting observables allows one to quantify entanglement at finite temperature, which opens up the possibility to study quantum correlations in quantum simulations going beyond classical capabilities.
Strongly-correlated electron systems generate some of the richest phenomena and most challenging theoretical problems studied in physics. One approach to understanding these systems is with ultracold fermionic atoms in optical lattices, which can provide a level of control and ways of observing strongly-correlated fermionic systems that are not accessible with conventional materials. This thesis describes the development of an experimental technique where a quantum gas of fermionic 6Li atoms is prepared in a two-dimensional optical lattice and each atom can be frozen in place and imaged with single-site resolution. Combining a vacuum-compatible large numerical aperture microscope with Raman sideband cooling enables site-resolved fluorescence imaging with high fidelity. We observe several phases of the Hubbard model, including band and Mott insulators. The observed in-situ occupation distributions of atoms in the lattice are compared to theory with unprecedented detail and are used to determine the thermodynamic properties of the system. By combining site-resolved imaging with a spin-removal technique, we observe antiferromagnetic correlations in the Hubbard model with single-site resolution. We observe, for the first time in cold atom systems, beyond-nearest-neighbor magnetic correlations, which provide a direct measurement of the correlation length. We also present detailed measurements of the formation of correlations during lattice loading.
Exotic phases of matter can emerge from strong correlations in quantum many-body systems. Quantum gas microscopy affords the opportunity to study these correlations with unprecedented detail. Here, we report site-resolved observations of antiferromagnetic correlations in a two-dimensional, Hubbard-regime optical lattice and demonstrate the ability to measure the spin-correlation function over any distance. We measure the in situ distributions of the particle density and magnetic correlations, extract thermodynamic quantities from comparisons to theory, and observe statistically significant correlations over three lattice sites. The temperatures that we reach approach the limits of available numerical simulations. The direct access to many-body physics at the single-particle level demonstrated by our results will further our understanding of how the interplay of motion and magnetism gives rise to new states of matter.