Ultracold atomic gas systems provide a remarkably versatile platform for studying a wide range of physical phenomena, from analogue particle physics and gravity, to the emergence of subtle and profound order in many body and condensed matter systems. In addition, ultracold atomic gas systems can be used to perform a range of precision measurements, from time keeping to variations in the fine structure constant. In this dissertation, I describe our efforts to build a new apparatus capable testing a range of techniques for performing precision measurements in a magnetic storage ring for cold, possibly Bose-condensed, lithium and rubidium atoms. Next, I briefly touch upon our explorations of spin vortices in a ferromagnetic rubidium Bose-Einstein condensate before presenting an exhaustive account of our work using free-particle-like magnon excitations of the ferromagnetic gas to cool it and measure its temperature in a never-before-seen regime of low entropy. Using magnons as a thermometer, we measure temperatures as low as one nanokelvin in gases with an entropy per particle of about one thousandth of the Boltzmann constant, 0.001 k_B. I conclude by presenting the details of our procedure for calculating the entropy of our coldest, lowest entropy gases in the regime where the local density approximation does not apply.

Quantum simulation is the study of one quantum mechanical system via analog with another. In this thesis we explore a simple model of interacting bosons on a lattice, known as the Bose-Hubbard model, by experimental investigation of ultracold rubidium atoms in an optical lattice made from laser light.

We describe the construction and stabilization of an optical superlattice with threefold symmetry, and its use in studying the Bose-Hubbard model on triangular and trimerized kagome lattices. We study the short range phase coherence of a Mott insulator on the triangular lattice, and develop a scheme to mitigate out-of-equilibrium effects arising from the state preparation. We show the first experimental realization of an optical trimerized kagome lattice for cold atoms, and discuss experiments characterizing this lattice. Finally, we provide evidence for a Mott insulating state with fractional average particle number per site with measurements of the nearest-neighbor phase coherence of strongly interacting atoms in the trimerized kagome lattice.

At low temperatures, the thermodynamic properties of a many-body system are determined by the low energy excursions from the ground state. In quantum degenerate rubidium, the low energy excitations are phonons—phase and density modes—and magnons—spin modes. This thesis reports on the construction of an ultracold rubidium apparatus and techniques to create and image phonons and magnons. We propose and study matter wave interferometry of phonons in a ring trap as a potential rotation sensor in a compact sensor. Precision measurements of coherent magnons allow us to determine the magnon dispersion relation, including a gap induced by magnetic dipole-dipole interactions.

The development of quantum technology demands systems of many interacting quantum particles, each individually controlled. Neutral atom arrays are a leading platform in this pursuit, boasting scalable, configurable arrays of single atoms trapped in optical tweezer traps, with demonstrated mechanisms for generating local interactions. Neutral atoms coupled to an optical cavity provide complementary capabilities, enabled by cavity quantum electrodynamics: the control of single atoms with single photons, photon-mediated interactions between distant atoms, and cavity-assisted measurement.

This thesis presents a novel platform for cavity quantum electrodynamics with a neutral atom tweezer array. After motivating and describing the experimental apparatus, I present the details of a quantum sensing result, using single atoms as scanning probes to characterize optical fields of our cavity. I then discuss our study of single-atom cavity state detection, used to demonstrate a mid-circuit measurement within a coherent neutral atom array. The integration of the tweezer array and cavity QED platforms represents a small step toward quantum information processing with neutral atoms.

A cloud of ultracold atoms trapped within the confines of a high-finesse optical cavity shakes from the pressure of the light that probes it. This form of measurement backaction, a central component of quantum measurement theory, is the subject of this dissertation. Enlisting the collective motion of ultracold atoms as the mechanical degree of freedom in a cavity optomechanical system, we reach settings cold and quiet enough to allow for the effects of measurement backaction to manifest. We report predictions for and experimental observa- tions of the Standard Quantum Limit for force sensitivity, optical ponderomotive squeezing, and the possibility of complex squeezing through generalized optical correlations.

Motivated to develop quantum technologies and to study ever-more complex quantum systems,scientists are developing increasingly sophisticated experimental tools to control and measure quantum samples such as ensembles of ultracold atoms. A high-finesse optical cavity can be used to measure the state of atoms within its photonic mode with precision limited only by quantum uncertainty. Such a cavity can also be used to mediate interactions between different atoms within the cavity mode. High-resolution microscope objectives have been interfaced with ultracold atom experiments to allow researchers to image single atoms within optical lattices, to trap single atoms in microtweezer arrays, and to imprint arbitrary optical patterns onto atomic ensembles. Among other applications, these technologies have allowed researchers to investigate many-body quantum systems, engineer novel interactions, and realize high-fidelity quantum operations.

This dissertation presents the details of the design, construction, and operation of a new, versatileatomic physics apparatus that combines these two experimental tools. The apparatus includes a high-finesse optical cavity into which atoms are optically transported. In addition, there is a high-numerical-aperture objective aligned to image, with micron-scale resolution, atoms trapped within the center of the optical cavity. We present results demonstrating the capability of this apparatus to deliver and study ultracold atomic samples, ranging from single atoms to Bose-Einstein condensates. We demonstrate a dispersive shift of the cavity resonance due to the presence of atoms in the cavity mode and the trapping and imaging of single atoms in optical microtweezers. We also present an atomic scanning probe microscopy technique with which a single atom in a microtweezer is used to map out the spatial amplitude pattern of an optical cavity mode standing wave by monitoring the position-dependent scattering properties of the atom.

Spinor Bose Einstein condensates have widely been sought after as perfect emulators of condensed matter phenomena, providing widely tunable and highly controlled systems. Utilizing a novel spin sensitive phase contrast imaging technique, the vector magnetization is measured in-situ with high spatial and temporal resolution and applied to a number of experiments. Using optically trapped F = 1 ⁸⁷Rb spinor condensates, the equilibrium phase diagram of a spin-1 Bose gas is quantitatively explored by observing the evolution of unmagnetized spin textures and their thermal equilibrium properties. Spin domain coarsening and a strong dependence of the spin configuration on the quadratic Zeeman shift is observed, supporting the predicted mean-field equilibrium phase diagram for small values of the quadratic shift. Additionally, spinor Bose gases are demonstrated to be an effective tool in calibrating and characterizing experimental imaging systems. Sinusoidal test patterns of varying pitch are created and used to extract the modulation transfer function and quantify optical aberrations which are of immense importance in systems which claim to have high spatial resolution. Lastly we realize an optical kagome geometry in a two-dimensional optical superlattice with a scalar Bose gases. The optical superlattice can be tuned between various geometries, including kagome, one-dimensional stripe, and decorated triangular lattice. Using atom optics we characterize the various geometries and demonstrate the versatility of this optical superlattice. The kagome geometry presents a new experimental arena for studies of geometrically frustrated systems.

Geometric frustration is a phenomenon where local energy constraints cannot be simultaneously satisfied because of geometric reasons. This leads to a manifold of degenerate ground states in which ordering can be highly nontrivial and exotic phases of matter could emerge. In a kagome lattice, both interaction energies of spins and kinetic energies of particles can be frustrated. In the latter case, a single-particle band structure with a flat (dispersionless) energy band results.

In this dissertation, we report our experimental studies of interacting bosonic systems in kagome lattices. Such studies are performed with a quantum simulator based on ultracold atoms in optical lattices. We describe upgrades and development of the apparatus that constructs optical kagome lattices in our experiment. We briefly discuss results of an experiment studying the Bose-Hubbard model in a trimerized kagome lattice. We then report the details of an experiment investigating the interplay of kinetic frustration and interactions by probing the band structure of an optical kagome lattice with interacting ultracold bosonic atoms. Finally, we propose experimental ideas for exploring the properties of the singular band touching point in the kagome flat band.

Ultracold atomic gases trapped at the interference of coherent beams of light constitute an artificial material. This optical lattice material may be used for controlled quantum simulations of condensed matter theories. The bulk of this dissertation concerns the construction and calibration of an optical superlattice that can form the triangular, honeycomb and kagome crystal structures. The properties and geometry of this artificial material may be dynamically changed, allowing for the experiments discussed in this thesis that would be impossible with real materials.

The use of ultracold atoms in optical lattices for quantitative tests is challenging because of the novelty of many of the techniques in the field, and because of the myriad experimental differences between these artificial materials and true materials. This thesis reports the development of a method to characterize optical lattice potentials using matter-wave diffraction. We observe an enhancement of inversion asymmetry in matter-wave diffraction from a honeycomb lattice, which we explain using a time-independent perturbative treatment of the single-particle band structure of the lattice. Our experiment also provides new insight into a commonly used detection technique.

This thesis culminates in the development and experimental realization of a quantitative test of a condensed-matter theory. The test is insensitive to the experimental differences between artificial materials and real materials. We focus on a prediction from a mean-field treatment of the Bose-Hubbard model that concerns the difference in behavior of itinerant particles on lattices that are identical but for their geometry. Using the tunable geometry of our quantum simulator, we measure the properties of ultracold atomic gases trapped in the triangular and kagome lattices under otherwise identical conditions and find that they are consistent with the mean-field scaling prediction.

The quantum nature of light makes it a basic component for models of quantum measurement and information exchange between disparate quantum modes, pioneered in the field of cavity quantum electrodynamics. The interaction of atomic ensembles with the mode of an optical cavity provides a flexible platform for exploring the coherent interaction of light with diverse macroscopic dynamics, such as collective motion and spin. This dissertation presents experimental results and theoretical models for continuous measurement and control of the center of mass motion and collective spin precession of an atomic ensemble, mediated by coupling to a high-finesse optical cavity. First, the theory of dispersive coupling between the cavity mode and the collective motion and spin of an atomic ensemble is derived, and then a general time-domain formalism is developed for theoretical analysis of multi-mode optodynamical systems. Single-mode optodynamical effects are introduced through experimental demonstrations of measurement and control of the collective atomic spin, providing a close analogy to cavity optomechanics.

Next, multiple collective atomic modes are considered within a single cavity, in order to assemble optically mediated interactions within multi-mode optodynamical systems. A demonstration of optodynamical interactions between the center of mass motion of two atomic ensembles is presented, coupled through an optical spring mediated by the cavity mode. Then simultaneous coupling of the center of mass motion and total spin precession of a single ensemble of atoms is described, yielding an experimental realization of a negative-mass instability, facilitated by the novel resource of the spin ensembles inverted state. A theoretical analysis of the negative-mass instability is presented, which indicates the possibility of generating two-mode squeezed states in the absence of excess incoherent noise. Finally, linear state retrodiction from the optodynamical signals is discussed, providing background and supplemental material for a forthcoming manuscript.