This proposal focuses on theoretical study of the physical realizability of topological phases from two perspectives. The first is to perform numerical simulations of several proposed Hamiltonian models that have been put forward as candidates for supporting non-Abelian topological phases, with the general goals of i) identifying the conditions under which these emergent phases arise from physically realistic Hamiltonians, ii) characterizing these phases and analyzing their robustness, and iii) evaluating the effects of braiding operations. The second is to explore theoretically how to artificially engineer one such Hamiltonian model using cold trapped molecules. Since numerical simulations of emergent phenomena address new physics that is not well understood today, we need new computational tools to characterize and understand the subtleties of the resulting quantum many-body behavior. We shall employ a range of Quantum Monte Carlo (QMC) techniques to address these phenomena, with new variants introduced as required for specific problems.
Topologically ordered phases of matter are exotic phases of matter with unusual properties: they behave as quantum liquids with a non-local quantum order. Because of the unique properties due to non-local quantum entanglement present in these phases, they have been proposed as the basis of a naturally fault-tolerant quantum computer. However, topological phases are elusive in nature, motivating exploration of conditions for their existence as well as development of methods for emulating them with controllable physical systems. We explored the conditions for formation of such topological order in quantum dimer models, with dimer degrees of freedom located on links of a lattice and allowed to move around subject to well defined constraints, e.g., only one dimer at each vertex (Figure 1). Under certain conditions these systems possess topologically ordered dimer liquid ground states. We can map dimer models on lattices to loop models (Figure 2) and take advantage of the study of loop condensation, i.e., formation of a dense loop phase with loops fluctuating on all length scales, to gain insight into the formation of topological order. Thus a dimer crystal to dimer liquid transition is described as a transition from a loop crystal with short loops to a scale-invariant condensed loop liquid characterized by the distribution of the longest loop. Our analysis of these systems was made with a new quantum Monte Carlo code that we developed for studying ground states of locally constrained, frustrated systems. This allowed us to characterize geometric properties of loop condensates of quantum dimer and related Hamiltonian models for a wide range of parameters (see e.g., Figure 3). In our second research thrust on emulation (‘quantum simulation’) of topological phases, we proposed an experimental method for the robust generation of a topological phase in a set of trapped neutral atoms in an optical lattice (Figure 4). We developed a theoretical protocol to implement this for one of the simplest Hamiltonians known to generate a topological phase, using an addressable optical lattice and two-body interactions between qubits (pseudo-spins) defined on internal states of the trapped atoms (Figure 5). An integral component of this emulation scheme is the formulation of a cooling and thermalization protocol that allows topological phases to generated either in the ground state or a finite temperature. The procedure is general, applicable to n-body interactions and has been extended to more complex Hamiltonians, allowing emulation of both Abelian and non-Abelian topological phases, which differ in the way their excitations behave. In related work we developed Quantum Monte Monte Carlo simulations and an analytic functional path integral analysis of weakly coupled reservoirs of superfluid liquid helium with a view to understanding recent experiments showing macroscopic quantum effects that are induced by arrays of nanometer size junctions. These calculations allow us to explore whether unusual temperature and array size dependence of synchronicity in these phenomena are related to topological constraints induced by the junction array on the quantum properties of the helium. We have also applied the Quantum Monte Carlo simulation approach to the analysis of non-classical states of ultra-cold bosons in a three-dimensional double well potential (Figure 6), a system with significant potential for quantum metrology. Because of the multiple length and energy scales, realistic theoretical analysis is very challenging for these systems and reduced dimensionality models are commonly employed instead. Our calculations have revealed an intriguing non-monotonic dependence of the non-classical behavior on interaction strength that is not seen with reduced dimensionality models. This project enabled training of three graduate students and one postdoctoral fellow in quantum Monte Carlo methods. It contributed to education of three graduate students in topological quantum phases and resulted in one Ph.D. thesis and several publications. Broader impact beyond the education of graduate students and postdocs was achieved by the students participating in two outreach programs at Berkeley, the SMASH (Summer Math and Science Honors Academy) program and the Compass Project. In the SMASH program, our graduate students taught high school students from low income and high achieving Bay area families during summers. In the Compass Project, a multifaceted program aimed at improving retention and support for undergraduates in the physical sciences at Berkeley, with an emphasis on women and underrepresented minorities, our students acted as "principle investigators" for teams of Berkeley freshmen in courses on physics model-building, as mentors during the academic year and as teachers in a summer program. One student helped organize a workshop focused on reforming introductory physics classes, hosted at Berkeley by Compass and faculty from Florida International University and attended by faculty from MIT, Cornell, Stanford and other institutions. The PI taught and further developed the interdisciplinary (Physics, Chemistry, Engineering and Computer Science) undergraduate course in Quantum Information Science. In addition to these outreach, education and mentoring activities, we hosted a number of visitors from other institutions for seminars and research discussions.
