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arXiv:2003.11669 · Physical Review Letters

Doublon-Hole Correlations and Fluctuation Thermometry in a Fermi-Hubbard Gas

Thomas R. Hartke, Botond Oreg, Ningyuan Jia, Martin W. Zwierlein

Published Sep 9, 2020• 89 citations
arXivHTMLPDF

Abstract

We report on the single atom and single site-resolved detection of the total density in a cold atom realization of the 2D Fermi-Hubbard model. Fluorescence imaging of doublons is achieved by splitting each lattice site into a double well, thereby separating atom pairs. Full density readout yields a direct measurement of the equation of state, including direct thermometry via the fluctuation-dissipation theorem. Site-resolved density correlations reveal the Pauli hole at low filling, and strong doublon-hole correlations near half filling. These are shown to account for the difference between local and nonlocal density fluctuations in the Mott insulator. Our technique enables the study of atom-resolved charge transport in the Fermi-Hubbard model, the site-resolved observation of molecules, and the creation of bilayer Fermi-Hubbard systems.

DOI
10.1103/physrevlett.125.113601

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We report on the single atom and single site-resolved detection of the total density in a cold atom realization of the 2D Fermi-Hubbard model. Fluorescence imaging of doublons is achieved by splitting each lattice site into a double well, thereby separating atom pairs. Full density readout yields a direct measurement of the equation of state, including direct thermometry via the fluctuation-dissipation theorem. Site-resolved density correlations reveal the Pauli hole at low filling, and strong doublon-hole correlations near half filling. These are shown to account for the difference between local and nonlocal density fluctuations in the Mott insulator. Our technique enables the study of atom-resolved charge transport in the Fermi-Hubbard model, the site-resolved observation of molecules, and the creation of bilayer Fermi-Hubbard systems.
Strong electron correlations lie at the origin of high-temperature superconductivity. Its essence is believed to be captured by the Fermi-Hubbard model of repulsively interacting fermions on a lattice. Here we report on the site-resolved observation of charge and spin correlations in the two-dimensional (2D) Fermi-Hubbard model realized with ultracold atoms. Antiferromagnetic spin correlations are maximal at half-filling and weaken monotonically upon doping. At large doping, nearest-neighbor correlations between singly charged sites are negative, revealing the formation of a correlation hole, the suppressed probability of finding two fermions near each other. As the doping is reduced, the correlations become positive, signaling strong bunching of doublons and holes, in agreement with numerical calculations. The dynamics of the doublon-hole correlations should play an important role for transport in the Fermi-Hubbard model.
We demonstrate atom-resolved detection of itinerant bosonic ^{23}Na and fermionic ^{6}Li quantum gases, enabling the direct in situ measurement of interparticle correlations. In contrast to prior work on lattice-trapped gases, here we realize microscopy of quantum gases in the continuum. We reveal Bose-Einstein condensation with single-atom resolution, measure the enhancement of two-particle g^{(2)} correlations of thermal bosons, and observe the suppression of g^{(2)} for fermions; the Fermi or exchange hole. For strongly interacting Fermi gases confined to two dimensions, we directly observe nonlocal fermion pairs in the BEC-BCS crossover. We obtain the pair size and the short-range contact directly from the pair correlations. In situ thermometry is enabled via the fluctuation-dissipation theorem. Our technique opens the door to the atom-resolved study of strongly correlated quantum gases of bosons, fermions, and their mixtures.
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.
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Cold atom systems provide a rich platform to realize strongly interacting condensed matter systems, and recent progress in fluorescence imaging technique has enabled identification of nontrivial doublon, singlon, and holon correlation functions. We show that these correlators can be used to identify the conditions under which local moments form in an interacting electronic system. Toward this end, we report a Determinantal Quantum Monte Carlo (DQMC) study of such correlation functions in the two-dimensional repulsive Fermi Hubbard model on a square lattice as a function of doping, interaction strength and temperature. We find definite signatures of the crossover from small U(band regime) to large U(correlated insulator regime). Our key findings are: (1) An opening of a charge gap in the thermodynamic density of states is accompanied by the appearance of temperature insensitive points in the equation of state at finite doping, which can be used to distinguish the band regime in cold atom experiments. (2) Nearest neighbor doublon holon correlations track the opening of charge gap; these compete with density correlations to generate moment moment correlations that show different behavior in the metallic and correlated insulator regime. (3) Non local correlation functions can be used to distinguish between the two regimes, both at and away from half filling. Our results allow comparisons of different correlation functions with recent experimental findings and guide further experimental investigations.
Recent advances in fluorescence imaging have enabled direct measurements of doublon, singlon, and holon correlation functions in cold-atom experiments. We demonstrate that these correlators provide a powerful probe of the extended crossover from metallic to insulating behavior at intermediate and high temperatures. To this end, we carry out determinantal quantum Monte Carlo simulations of the two-dimensional repulsive Fermi-Hubbard model on a square lattice, varying doping, interaction strength UUU, and temperature TTT. Our results reveal clear signatures of a crossover from the metallic regime at small UUU to the Mott insulating regime at large UUU. Specifically, we note the following: (1) At half-filling, we distinguish three regimes (metallic, non-Fermi-liquid, and Mott insulating) by analyzing the temperature dependence of the thermodynamic density of states $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\kappa}}=\frac{\ensuremath{\partial}n}{\ensuremath{\partial}\ensuremath{\mu}}$ in comparison with the low-energy single-particle density of states $N(\ensuremath{\omega})$. (2) At finite doping, although $N(\ensuremath{\omega})$ remains gapless, a sign change in $\frac{\ensuremath{\partial}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\kappa}}}{\ensuremath{\partial}T}$ marks a transition from insulating to metallic behavior at a critical density ncr(U,T){n}_{\text{cr}}(U,T)ncr​(U,T). (3) These crossovers can also be tracked using experimentally accessible correlators, such as density-density, moment-moment, and doublon-holon correlations, providing valuable diagnostics in settings where direct spectral information is difficult to obtain.