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Laser Spectroscopy

CORRELATIONS AND COLLECTIVE MODES IN FERMIONS ON LATTICES

P. D. Drummond, Joel Frederick Corney, Xia-Ji Liu, Haiping Hu

Published Dec 1, 2005• 0 citations
Publisher

Abstract

Experiments with ultracold atoms present an outstanding opportunity for implementing novel tests of theories of strongly correlated fermions. We present three new theoretical methods to treat these systems. Luttinger liquid theory and the exact one-dimensional solutions let us calculate collective mode frequencies at the metal-insulator transition in a lattice. A new Gaussian phase-space method for fermion systems can be used to simulate finite temperature atomic correlations. Finally, we introduce an approximate diagrammatic technique which correctly includes molecule-molecule interactions, and gives an accurate, quantitative theory of the BEC-BCS crossover regime.

DOI
10.1142/9789812701473_0017

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Experiments with ultracold atoms present an outstanding opportunity for implementing novel tests of theories of strongly correlated fermions. We present three new theoretical methods to treat these systems. Luttinger liquid theory and the exact one-dimensional solutions let us calculate collective mode frequencies at the metal-insulator transition in a lattice. A new Gaussian phase-space method for fermion systems can be used to simulate finite temperature atomic correlations. Finally, we introduce an approximate diagrammatic technique which correctly includes molecule-molecule interactions, and gives an accurate, quantitative theory of the BEC-BCS crossover regime.
Using the exact Bethe ansatz solution of the Hubbard model and Luttinger liquid theory, we investigate the density profiles and collective modes of one-dimensional ultracold fermions confined in an optical lattice with a harmonic trapping potential. We determine a generic phase diagram in terms of a characteristic filling factor and a dimensionless coupling constant. The collective oscillations of the atomic mass density, a technique that is commonly used in experiments, provide a signature of the quantum phase transition from the metallic phase to the Mott-insulator phase. A detailed experimental implementation is proposed.
A generic scenario of a crossover between weak-coupling and strong-coupling limits of a many-body quantum system is studied. Powerful numerical methods of investigating the strongly correlated regime are developed and applied to answering a number of important questions of the strongly correlated fermions physics. The internal structure of an exciton in an insulating medium is studied throughout the crossover between the (analytically solvable) strong- and weak-coupling limits. In the intermediate regime, accurate results for the exciton ground state are obtained and used to establish the limits of applicability of the corresponding limiting cases for the first time. A novel combination of the diagrammatic determinant Monte Carlo and the worm algorithm is developed and used for investigating the unitarity regime of the BEC-BCS crossover via simulating the dilute Fermi-Hubbard model. Superfluid phase transition and thermodynamic properties of the normal state are characterized for both lattice fermions and continuum unitary Fermi gas. The relation to current experiments with trapped ultracold quantum gases is discussed.
Ultracold atoms in optical lattices provide an extremely clean and controllable setting to explore quantum many-body phases of matter. Here we give a brief review of the strong-correlation physics that has been realized using such ultracold atoms in optical lattices ranging from the realization of Hubbard models to studies of quantum magnetism and the detection of single atoms with lattice site resolution. All this has opened up fundamentally new opportunities for the investigation of quantum many-body systems. Introduction Over the past years, ultracold atoms in optical lattices have emerged as versatile new systems to explore the physics of quantum many-body systems. On the one hand, they can be helpful in gaining a better understanding of known phases of matter and their dynamical behavior; on the other hand, they allow one to realize completely novel quantum systems that have not been studied before in nature [1, 2, 3]. Commonly, the approach of exploring quantum many-body systems in such a way is referred to as “quantum simulations.” Examples of some of the first strongly interacting many-body phases that have been realized both in lattices and in the continuum include the quantum phase transition from a superfluid to a Mott insulator [4, 5, 6], fermionic Mott insulators [7, 8], the achievement of a Tonks- Girardeau gas [9, 10], and the realization of the Bose-Einsten condensate (BEC)– Bardeen-Cooper-Schrieffer (BCS) crossover (see also Chapter 12) in Fermi gas mixture [11] using Feshbach resonances [12]. In almost all of these experiments, detection was limited to time-of-flight imaging or more refined derived techniques that mainly characterized the momentum distribution of the quantum gas [2]. For several years, researchers in the field have therefore aspired to employ in situ single-particle detection methods for the analysis of ultracold quantum gases. Only recently has it become possible to implement such imaging techniques, marking a milestone for the characterization and manipulation of ultracold quantum gases [13, 14, 15, 16, 17]. In our discussion, we will focus on one of the most successful of these techniques based on high-resolution fluorescence imaging. Despite being a rather new technique, such quantum gas microscopy has already proven to be an enabling technology for probing and manipulating quantum many-body systems. For the first time, controllable and strongly interacting many-body systems, as realized with ultracold atoms, could be observed on a local scale [17, 16].