This book explores the physics of atoms frozen to ultralow temperatures and trapped in periodic light structures. It introduces the reader to the spectacular progress achieved on the field of ultracold gases and describes present and future challenges in condensed matter physics, high energy physics, and quantum computation.

Ultracold atoms in optical lattices are among the most developed platforms of interest for building quantum devices suitable for quantum simulation and quantum computation. Ultracold trapped atoms are advantageous because they are fundamentally indistinguishable qubits that can be prepared with high fidelity in well-defined states and read-out with similarly high fidelities. However, an outstanding challenge for ultracold atoms in optical lattices is to engineer interesting interactions and control the effects of heating that couple the system to states that lie outside the Hilbert space we wish to engineer. In this thesis, I describe a series of experiments and theoretical proposals that address several critical issues facing ultracold atoms in optical lattices. First, I describe experiments where the tunneling behavior of atoms in the lattice is modified to make our fundamentally neutral particles behave as though they are charged particles in a magnetic field. We show how engineering this interaction creates intrinsic degeneracy in the single particle spectrum of the many-body system and how to introduce strong interactions in the system with the goal of producing exotic many-body states such as a bosonic fractional quantum Hall states. Then, I discuss how this technique can be easily generalized to include spin and higher spatial dimensions in order to access a rich variety of new physics phenomena. Next, I report on the realization of a spin-1 Heisenberg Hamiltonian which emerges as the low energy effective theory describing spin ordering in the doubly-occupied Mott insulator of two spin components. This integer spin Heisenberg model is qualitatively different from the half-integer spin model because it contains a gapped, spin-insulating ground state for small inter-spin interaction energies which we call the spin Mott. Using a spin-dependent lattice to control the inter-spin interactions, we demonstrate high-fidelity, reversible loading of the spin-Mott phase and develop a probe of local spin correlations in order to demonstrate a spin entropy below 0.2 kB per spin. Progress on adiabatically driving the quantum phase transition from the spin Mott to the xy-ferromagnetic is discussed along with the progress towards the creation of a quantum gas microscope for single atom detection and manipulation..

This thesis explores ultracold quantum gases of bosonic and fermionic atoms in optical lattices. The highly controllable experimental setting discussed in this work, has opened the door to new insights into static and dynamical properties of ultracold quantum matter. One of the highlights reported here is the development and application of a novel time-resolved spectroscopy technique for quantum many-body systems. By following the dynamical evolution of a many-body system after a quantum quench, the author shows how the important energy scales of the underlying Hamiltonian can be measured with high precision. This achievement, its application, and many other exciting results make this thesis of interest to a broad audience ranging from quantum optics to condensed matter physics. A lucid style of writing accompanied by a series of excellent figures make the work accessible to readers outside the rapidly growing research field of ultracold atoms.

Ultracold atoms provide a platform that allows for pristine control of a physical system, and have found uses in both the fields of quantum measurement and quantum simulation. Optical lattices, created by the AC Stark shift of a coherent laser beam, are a versatile tool to control ultracold atoms and implement novel Hamiltonians. In this thesis, I report on three experiments using the bosonic species Rubidium-87 trapped in optical lattices. I first discuss our work in simulating the Harper-Hofstadter Hamiltonian, which describes charged particles in high magnetic fields, and has connections to topological physics. To simulate the charged particles, we use laser-assisted tunneling to add a complex phase to tunneling in the optical lattice. For the first time, we have condensed bosons into the ground state of the Harper-Hofstadter Hamiltonian. In addition, we have demonstrated that we can add strong on-site interactions to the effective Hamiltonian, opening the door to studies of interesting states near the Mott insulator transition. Next, I present a novel technique to preserve phase coherence between separated quantum systems, called superfluid shielding. Phase coherence is important for both quantum measurement and simulation, and is fundamentally limited by projection noise. When an interacting quantum system is split, frozen-in number fluctuations lead to fluctuations of the relative phase between separated subsystems. We cancel the effect of these fluctuations by immersing the separated subsystems in a common superfluid bath, and demonstrate that we can increase coherence lifetime beyond the projection noise limit. Finally, I discuss our efforts in simulating magnetic ordering in the spin-1 Heisen- berg Hamiltonian. It is hard to adiabatically ramp into magnetically ordered ground states, because they often have gapless excitations. Instead, we use a spin-dependent lattice to modify interspin interactions, allowing us to ramp into the spin Mott insulator, which has a gap and can therefore act as a cold starting point for exploration of the rest of the phase diagram. We have achieved a cold spin temperature in the spin Mott insulator, and I discuss plans to also achieve a cold charge temperature and then ramp to the the xy-ferromagnet, which has spin-charge separation.

The rapidly developing topic of ultracold atoms has many actual and potential applications for condensed-matter science, and the contributions to this book emphasize these connections. Ultracold Bose and Fermi quantum gases are introduced at a level appropriate for first-year graduate students and non-specialists such as more mature general physicists. The reader will find answers to questions like: how are experiments conducted and how are the results interpreted? What are the advantages and limitations of ultracold atoms in studying many-body physics? How do experiments on ultracold atoms facilitate novel scientific opportunities relevant to the condensed-matted community? This volume seeks to be comprehensible rather than comprehensive; it aims at the level of a colloquium, accessible to outside readers, containing only minimal equations and limited references. In large part, it relies on many beautiful experiments from the past fifteen years and their very fruitful interplay with basic theoretical ideas. In this particular context, phenomena most relevant to condensed-matter science have been emphasized. Introduces ultracold Bose and Fermi quantum gases at a level appropriate for non-specialists Discusses landmark experiments and their fruitful interplay with basic theoretical ideas Comprehensible rather than comprehensive, containing only minimal equations

Ultracold atoms in optical lattices are promising systems to realize and study novel quantum mechanical phases of matter with the control and precision offered by atomic physics. Towards this goal, as important as engineering new states of matter is the need to develop new techniques to probe these systems. I first describe our work on realizing Bragg scattering of infrared light from ultracold atoms in optical lattices. This is a detection technique which probes the spatial ordering of a crystalline system, and has led to our observation of Heisenberg limited wavefunction dynamics. Furthermore, we have observed the superfluid to Mott insulator transition through the matter wave Talbot effect. This technique will be particularly powerful for studying antiferromagnetic phases of matter due to its sensitivity to the crystalline composition. The second major component of this thesis describes a new scheme to realize the Harper Hamiltonian. The Harper Hamiltonian is a model system which effectively describes electrons in a solid immersed in a very high magnetic field. The effective magnetic field manifests itself as a position-dependent phase in the motion of the constituent particles, which can be related to gauge fields and has strong connections to topological properties of materials. We describe how we can engineer the Harper Hamiltonian in a two-dimensional optical lattice with neutral atoms by creating a linear potential tilt and inducing Raman transitions between localized states. In situ measurements provide evidence that we have successfully created the Harper Hamiltonian, but further evidence is needed to confirm the creation of the ground state of this Hamiltonian.