Quantum states in ultracold fermionic gases in optical lattices:Supersolid and dynamically generated antiferromagnetic states

近年来，凝聚态物理和超冷原子气体物理领域的相互作用取得了丰硕的成果。前者基于简化的模型(如哈伯德模型)描述物质的量子态已经有很长的历史，而后者现在设法模拟这些模型并产生激动人心的量子态。该研究项目的双重目标是更好地了解超固态量子态的稳定性和反铁磁(AF)量子态的动态产生。物质的超固态是一种同时具有晶序和超流体性质的特殊状态，已经被认为可以在限制在光学晶格中的吸引费米子的气体中实现。该研究项目将从理论上模拟这种状态，并分析其稳定性。这将通过使用数值重整化群作为杂质解算器的实空间动力学平均场计算来实现。第二部分讨论了一个关联的费米子系统，它通过相互作用猝灭被驱动到存在强AF不稳定性的情况。使用哈伯德模型，我们将开发和应用非平衡技术来理解由此产生的行为。对于具有公度填充的光学晶格中的超冷费米子气体来说，这种非平衡情况是可能实现的，该光学晶格突然调谐到费什巴赫共振的强排斥侧。在那里，AF不稳定性与分子形成过程竞争，分子形成过程也可能发生在Feshbach共振的这一边。通过考虑这两个过程，我们将从理论上研究系统的时间依赖响应，并预测其主导行为。目前的实验是在非常接近于这里描述的情况下进行的。预计在未来几年内实现的实验使它成为一个及时和高度相关的研究项目。

In recent years the fields of condensed matter physics and ultracold atomic gas physics have developed a fruitful interplay. The former has a long history of describing quantum states of matter based on simplified models, such as the Hubbard model, whilst the latter now manages to simulate these models and generate exciting quantum states. The twofold objective of the research project is to better understand the stability of the supersolid quantum state and the dynamic generation of antiferromagnetic (AF) quantum states. The supersolid state of matter, a peculiar state with simultaneous crystalline order and superfluid properties, has been proposed to be realizable in a gas of attractive fermions confined to an optical lattice. The research project will theoretically model such a state and analyze its stability. This will be achieved by real space dynamical mean field calculations utilizing the numerical renormalization group as an impurity solver. The second part deals with a correlated fermionic system, which by an interaction quench is driven into situation where a strong AF instability is present. Using the Hubbard model, we will develop and apply non-equilibrium techniques to understand the resulting behavior. A realization of this non-equilibrium situation is possible for an ultracold gas of fermions in an optical lattice with commensurate filling, which is tuned suddenly to the strongly repulsive side of a Feshbach resonance. There, the AF instability competes with processes of molecule formation, which can also occur on this side of the Feshbach resonance. By taking into account both processes we will investigate theoretically the time-dependent response of the system and predict its dominant behavior. Current experiments are performed in regimes very close to the situations described here. The expected experimental realizations within the next years make it a timely and highly relevant research project.