An outstanding problem in physics is how to understand strongly interacting quantum many-body systems such as the quark-gluon plasma, neutron stars, superfluid 4He, and the high-temperature superconducting cuprates. The physics approach to this problem is to reduce these complex systems to minimal models that are believed to retain relevant phenomenology. For example, the Hubbard model — the focus of this thesis — describes quantum particles tunneling between sites of a lattice with on-site interactions. The Hubbard model is conjectured to describe the low-energy charge and spin properties of high-temperature superconducting cuprates. Thus far, there are no analytic solutions to the Hubbard model, and numerical calculations are difficult and even impossible in some regimes (e.g., the Fermi-Hubbard model away from half-filling). Therefore, whether the Hubbard model is a minimal model for the cuprates remains unresolved. In the face of these difficulties, a new approach has emerged — quantum simulation. The premise of quan-tum simulation is to perform experiments on a quantum system that is well-described by the model we are trying to study, has tunable parameters, and is easily probed. Ultracold atoms trapped in optical lattices are an ideal candidate for quantum simulation of the Hubbard models. This thesis describes work on two such systems: a 87Rb (boson) optical lattice experiment in the group of Brian DeMarco at the University
An outstanding problem in physics is how to understand strongly interacting quantum many-body systems such as the quark-gluon plasma, neutron stars, superfluid 4He, and the high-temperature superconducting cuprates. The physics approach to this problem is to reduce these complex systems to minimal models that are believed to retain relevant phenomenology. For example, the Hubbard model — the focus of this thesis — describes quantum particles tunneling between sites of a lattice with on-site interactions. The Hubbard model is conjectured to describe the low-energy charge and spin properties of high-temperature superconducting cuprates. Thus far, there are no analytic solutions to the Hubbard model, and numerical calculations are difficult and even impossible in some regimes (e.g., the Fermi-Hubbard model away from half-filling). Therefore, whether the Hubbard model is a minimal model for the cuprates remains unresolved. In the face of these difficulties, a new approach has emerged — quantum simulation. The premise of quan-tum simulation is to perform experiments on a quantum system that is well-described by the model we are trying to study, has tunable parameters, and is easily probed. Ultracold atoms trapped in optical lattices are an ideal candidate for quantum simulation of the Hubbard models. This thesis describes work on two such systems: a 87Rb (boson) optical lattice experiment in the group of Brian DeMarco at the University
Over the past few decades, quantum simulation with ultracold atoms in optical lattices has proven to be a useful tool for understanding many-body Hamiltonians.One of its primary directions is the experimental study of the Fermi Hubbard Model (FHM), which aims to capture the fundamental properties of cuprate superconductors and has produced a wide range of intriguing phenomena, including the metal-to-Mott insulator transition, antiferromagnetism, strange metallicity, pseudogap physics, and is frequently used in the study of superconductivity. In recent years, new kinds of quantum simulators have emerged, and in particular we focus on two of them: 1) quantum simulators that study high symmetry fermions in optical lattices, and 2) optical tweezer arrays which provide a bottom-up approach to engineering the FHM. The first class of emerging quantum simulators utilizes alkaline-earth-like atoms (AEA’s) – which have large nuclear spin I (such as 173Yb and 87Sr) – to engineer the SU(N = 2I +1) symmetric FHM, where N = 2,..., 10. The second class corresponds to quantum simulators based on optical tweezer arrays which combine deterministic single tweezer ground state loading and tunneling coupled sites. Both classes of quantum simulators are interesting on their own. For example, the SU(N) FHM is generating a lot of interest due to its connections to multiorbital solid state systems and theoretical predictions of rich phase diagrams which exhibit exotic phases of matter, including unusual ones like spin liquids. On the other hand, optical tweezer arrays have gained a lot of attention since they can be used to study arbitrary geometries and access very low-entropy samples. This thesis reports on the numerical study of both quantum simulation architectures. Results of the SU(N) FHM at finite temperature, for different fillings and interaction strengths that span the non-interacting to the strongly interacting limit are obtained using Determinant Quantum Monte Carlo, Exact Diagonalization, and Numerical Linked Cluster Expansion. In particular, we focus on how thermodynamic and magnetic observables, such as the number of on-site pairs, energy, entropy, spin correlations, and structure factors depend on N, the interaction strength U/t, temperature T/t, and the anisotropy of tunneling rates. Our results demonstrate that in a homogeneous square lattice with one particle per site on average, thermodynamic observables as a function of temperature obey a universal scaling with N, and that short-range antiferromagnetic correlations are stronger for larger N and in lower dimensions. In addition, we present theory-experiment comparisons where possible, where we perform thermometry and provide a precise characterization of the equation of state of the SU(N) FHM. Our results emerge as a tool to perform thermometry in experiments from theory-experiment comparison and also provide guidance for future experiments with AEAs in optical lattices. Additionally, we present calculations using numerically precise discrete variable representation methods for two-dimensional stroboscopic tweezer arrays and compare the outcomes with experimental data. We quantify the effects of stroboscopic potentials on Hubbard parameters like the interaction strength U and the tunneling t in optical tweezer arrays and illustrate how heating from the stroboscopic potential relies on strobe frequency. Our calculations enable evaluation and optimization of two-dimensional tweezer array experiments. An important milestone of the results of this thesis corresponds to the record temperatures and entropies achieved with these quantum simulation architectures: 1) the coldest fermions ever created in nature in absolute temperature and in cold atoms, and 2) the lowest entropy per particle fermions ever created, albeit in a so-far small system.
Many outstanding problems in quantum physics, such as high-Tc superconductivity or quark confinement, are still - after decades of research - awaiting commonly accepted explanations. One reason is that such systems are often difficult to control, show an intermingling of several effects, or are not easily accessible to measurement. To arrive at a deeper understanding of the physics at work, researchers typically derive simplified models designed to capture the most striking phenomena of the system under consideration. However, due to the exponential complexity of Hilbert space, even some of the simplest of such models pose formidable challenges to analytical and numerical calculations. In 1982, Feynman proposed to solve such quantum models with experimental simulation on a physically distinct, specifically engineered quantum system [Int. J. Theor.Phys. 21, 467]. Designed to be governed by the same underlying equations as the original model, it is hoped that direct measurements on these so called quantum simulators (QSs) will allow to gather deep insights into outstanding problems of physics and beyond. In this thesis, we identify four requirements that a useful QS has to fulfill, relevance, control, reliability, and efficiency. Focusing on these, we review the state of the art of two popular approaches, digital QSs (i.e., special purpose quantum computers) and analog QSs (devices with always-on interactions). Further, focusing on possibilities to increase control over QSs, we discuss a scheme to engineer quantum correlations between mesoscopic numbers of spinful particles in optical lattices. This technique, based on quantum polarization spectroscopy, may be useful for state preparation and quantum information protocols. Additionally, employing several analytical and numerical methods for the calculation of many-body ground states, we demonstrate the variety of condensed-matter problems that can be attacked with QSs consisting of ultracold ions or neutral atoms in optical lattices. The chosen examples, some of which have already been realized in experiment, include such diverse settings as frustrated antiferromagnetism, quantum phase transitions in exotic lattice geometries, topological insulators, non-Abelian gauge-fields, orbital order of ultracold Fermions, and systems with long-range interactions. The experimental realization of all of these models requires techniques which go beyond standard optical lattices, e.g., time-periodic driving of lattices with exotic geometry, loading ultracold atoms into higher bands, or immersing trapped ions into an optical lattice. The chosen models, motivated by important open questions of quantum physics, pose difficult problems for classical computers, but they may be amenable in the near future to quantum simulation with ultracold atoms or ions. While the experimental control over relevant models has increased dramatically in the last years, the reliability and efficiency of QSs has received considerably less attention. As a second important part of this thesis, we emphasize the need to consider these aspects under realistic experimental conditions. We discuss specific situations where terms that have typically been neglected in the description of the QS introduce systematic errors and even lead to novel physics. Further, we characterize in a generic example the influence of quenched disorder on an analog QS. Its performance for simulating universal behavior near a quantum phase transition seems satisfactory for low disorder. Moreover, our results suggest a connection between the reliability and efficiency of a QS: it works less reliable exactly in those interesting regimes where classical calculations are less efficient. If QSs fulfill all of our four requirements, they may revolutionize our approach to quantum-mechanical problems, allowing to solve the behavior of complex Hamiltonians, and to design nano-scale materials and chemical compounds from the ground up.
The present thesis is devoted to the study of physical phenomena emerging from strong correlations in strongly interacting quantum many-body systems with several components. Hubbard models are widely used as minimal models which take into account the interactions between particles and they have been studied in relation to phenomena such as Mott localization, unconventional superconductivity, quantum magnetism and many others. All of these striking phenomena share their origin from the strong correlations among fermions induced by their mutual interactions. Furthermore, condensed matter models are usually realized only in an approximate fashion in actual solid-state systems, making the situation all the more puzzling and hard to be treated analytically or numerically. Therefore, a great effort has been performed to simulate Hubbard models in a system of atoms cooled down to ultra low temperatures and trapped in optical lattices. The most peculiar feature of cold atoms experiments consists in the possibility of tuning relevant physical parameters of the systems, as the density or the interactions among atoms, using laser and/or magnetic fields. This paved the way to the observation of fundamental quantum states of matter as the weakly interacting Bose-Einstein condensate, the super fluid to Mott insulator transition, the super fluid BEC-BCS crossover, the Mott transition in systems of composite fermions and so on. Hence, it is considered of great interest establishing connections between the quantum simulations cold atomic toolbox and systems realized in solid-state physics...
This dissertation summarizes my recent work regarding systems of strongly interacting fermionic atoms in optical lattices. This work addresses the combination of two experimental techniques that have been the subject of much recent research in ultracold atom physics. One is the use of optical lattices, which provide a means to realize diverse interaction configurations within a clean, controllable system. The other is the use of magnetically tunable Feshbach resonances to control the strength of the interatomic interaction. Together, these techniques offer the possibility of an experimental realization of many important model Hamiltonians of condensed matter physics, and may also lead to the discovery of new physics. Recent study of this system has shown that strong interactions near Feshbach resonance will lead to the population of multiple lattice bands, and that collisions between atoms on neighboring sites cannot be neglected. These effects lead to a complicated Hamiltonian, but one which can be simplified to an effective single-band model equivalent to the generalized Hubbard model (GHM), which is an extension of the Hubbard model that includes correlated hopping terms. My main results concern the study of this model. The strong correlations between the particles make it difficult to definitively determine the many body physics of the GHM. As a first approach to understanding the GHM in optical lattices, I focus mainly on cases where the problem is greatly simplified by allowing interactions among only small groups of lattice sites. This restriction can be implemented in experiments using an optical superlattice potential. Our results include a proposed scheme (based on double-well superlattices) to empirically verify that the GHM describes this system and to directly measure the various parameters of this model. Other results include exact solutions on four-site square plaquettes, which demonstrate that d-wave excitations can occur in the low-energy states. By using a superlattice to give an array of weakly coupled plaquettes, one can thus produce a d-wave superfluid state. This is of relevance to the study of high-Tc superconductors, although I note certain key differences between the sort of d-wave superfluid described here and that of the superconductors.
This thesis describes experiments performed on a gas of lithium atoms in an optical lattice which are relevant to the low-temperature physics of the Hubbard model. The Hubbard model is perhaps the most intensively studied model of fermionic particles on a lattice, both because it is the simplest such model which seriously challenges modern theoretical tools, and because it is a candidate model for describing high-temperature superconductivity in cuprate materials. The challenges associated with numerical simulations of the Hubbard model have motivated “quantum simulations” of the type described in this thesis, in which a well-controlled physical system is engineered to mimic the model, allowing its properties to be determined through experiments. Historically, however, such experiments could not access the low-temperature regime of the Hubbard model most relevant to open questions on materials like cuprates, where modern computational techniques cannot yet definitively characterize the physics. We developed a novel set of cooling techniques which allowed our experiment to access this regime. These techniques are based on adiabatic conversion between a band insulator, which can be prepared with very low entropies using single-atom-resolved control and detection techniques now common in ultracold atomic physics, and various quantum fluids, including Hubbard systems. We then performed the first experiments in this new regime of the Hubbard model, focusing on the anomalous “pseudogapped” metallic state. This state is predicted to exist by computational methods, and is believed to mimic the eponymous state in cuprates which gives rise to high-temperature superconductivity upon cooling. Nevertheless, its properties and mechanisms remain poorly understood. We developed several new measurements that enabled an observation of the pseudogap and which yielded novel data on its thermodynamics, linking the pseudogap to an anomaly in the equation of state. These experiments demonstrate a new mode of operating fermionic quantum gas experiments, in which single-atom-resolved controls, rather than bulk techniques like evaporative cooling, are leveraged to access low temperatures in a sample of several hundred atoms. They also signal the arrival of quantum simulators in a regime of poorly-understood emergent low-temperature phenomena that are relevant to the electronic properties of solids, concretely demonstrating the utility of quantum simulation in addressing frontier problems in quantum many-body physics.
Ultracold atoms in optical lattices can provide faithful analog realizations of the Hubbard model and many of its variants. Recently, these simulators have been augmented with improved programmable controls that enable new routes to state preparation, new kinds of dynamics, and, by extension, access to new observables. I will discuss two examples where such tools have enabled simulations that challenge classical numerical techniques: first, a sampling problem involving non-interacting bosons on a lattice that is closely related to boson sampling; and second, the preparation of low temperature equilibrium states of interacting spinful fermions on a lattice. These simulations have the potential to enhance our understanding of quantum many body physics, including practical problems of interest in material design relating to strongly correlated quantum materials. However, fully harnessing their capabilities in the regime where exact classical numerical simulations are absent requires new, precision probes of the relevant microscopic or thermodynamic quantities.