Quantum Simulation Of Many-body Frustrated Models

Richard Feynman first proposed the concept of quantum simulation. Quantum simulation is that one quantum system with more controllable simulates another. Quan-tum simulation is devoted to solving some of the most challenging computational prob-lems. These problems are usually about strongly correlated quantum many-body sys-tems which play a fundamental role in condensed matter physics, material science, high-energy physics or quantum chemistry. Now, some quantum simulations have been realized using ultracold neutral atoms, ions or photons. Some experiments have already addressed problems which are not solvable by supercomputers. Several important prob-lems have been identified where quantum simulations could have an important impact and help. These problems contain the underlying mechanisms of high-temperature superconductivity, quantum magnetism, dynamical evolutions of quantum many-body systems and some models relevant in high-energy physics. The field of quantum simu-lation is expanding rapidly and has interdisciplinary connections to other fields.One of the major challenges in the field of quantum simulation is the realization and study of quantum magnetism, especially the frustrated spin systems. Frustrated spin systems attracted great attention due to their fascinating and rich low-temperature behaviors in condensed matter physics. Frustration has two major causes, competing in-teraction and lattice geometry. Quantum fluctuation can greatly influence the properties of the frustrated spin systems and make the ground states different from the classical Neel state. For instance, it was pointed out that two neighbor spins can arrange them-selves to form a singlet state, which consists of the basic unit of many exotic quantum phases, such as valence-bond crystals and resonating valence-bond spin-liquid states, the latter especially is usually argued to play a very important role in understanding the properties of high-T oxides superconductivity. On the other hand, due to the high ground-state degeneracy, theoretical treatment numerical simulation based on quantum Monte Carlo method in frustrated systems usually suffers from the sign problem, which make the understanding of such systems a big challenge for theory and experiment.Currently, it is possible to study quantum magnetism even frustrated spin models by using quantum simulators. In the optical lattice, an antiferromagnetic Ising chain has been realized. Compared to the approach using two internal states of the atoms, they use the motional degree of the atoms to represent the effective spin. However, it is difficult to be extended to two dimensional case and realize the complete Heisenberg model. Moreover, simulators based on photons can only probe the problems of few lattices. Although trapped ions system can simulate several kinds of spin models, the interactions are network structure and can not be designed to an arbitrary geometry.In this thesis, we focus on how to simulate frustrated spin systems and frustrated tunneling of ultracold atoms. We propose to simulate one-dimensional Heisenberg spin chains with next nearest neighbor interactions in coupled cavities. Moreover, we design a special two-dimensional optical lattice and use it to simulate the J1-J2Heisenberg model and the checkerboard antiferromagnet model. Finally, we realize frustrated tun-neling of ultracold atoms in a state-dependent optical lattice. The detail goes as follows:1. We propose a scheme to simulate one-dimensional Heisenberg spin models with competing interactions between nearest neighbors (NNs) and next-NNs in coupled cavities. By taking advantage of the many controls available and using a smart idea of interaction cancellation and enhancement, we can adjust at will the ratio between the NN and next-NN interaction strengths. Such a powerful capability allows us to simulate frustration phenomena and disorder behaviors in one-dimensional systems arising from next-NN interactions. Our scheme is robust because spontaneous emission of internal states and cavity are strongly suppressed.2. We design a special optical lattices that allow quantum simulation of spin frus-tration in two-dimensional systems. By carefully overlaying optical lattices with differ-ent periods and orientations we are able to adjust the ratio between the NN and next-NN interaction strengths in a square spin lattice and realize frustration effects. We show that only laser beams of a single frequency are required and the parameter space reachable in our design is broad enough to study the important phases in the J1-J2frustrated Heisenberg model and checkerboard antiferromagnet model. By using the polariza-tion spectroscopy for detection, distinct quantum phases and quantum phase transition points can be characterized straightforwardly. Our design thus offers a suitable setup for simulation of frustrated spin systems.3. We propose a general method to realize frustrated tunneling of ultracold atoms in a state-dependent optical lattice. Two typical lattice configurations are considered, the square lattice with competing interaction and the kagome lattice with geometrical frustration. The ideal can be extended to implement frustrated tunneling of ultracold atoms in various geometries, which enable us to investigate physics of frustration in both bosonic and spin systems. We study the mean-field phase diagrams of the consid-ered models and the experimental situations are also discussed.