Quantum Simulation And Fundamental Quantum Mechanics Studying In Superconducting Circuit System

As the fundamental theory to describe physics in the microscopic world, quan-tum mechanics not only reveals the underlying principles of the world, but also enables modern science and technology. Unfortunately, many quantum prblems are difficult to solve. In order to study such hard problems of quantum simulation, Feynman formu-lated the idea to investigate one complex or inaccessible quantum system with another controllable one. Superconducting circuits play an important role in the study of quan-tum physics because its parameters are tunable in a wide range and it can be integrated monolithically and then scaled in complexity. At present, single qubit (two energy lev-el sytem) operation, two qubit coupling and strong qubit-field coupling in circuit QED with superconducting circuits have been realized in experiments.In this dissertation, we focus on using superconducting circuit systems for quan-tum simulation of fundamental problems in quantum mechanics. Specifically,1. We introduce basic concepts of superconducting circuit, its elements and the theoretical derivation of its operating principles, including the Josephson effect, super-conducting charge qubit, the coupling between qubits, quantization of superconducting transmission line resonator (TLR) and its Hamiltonian. Since the coupling between qubits plays an important role in quantum computation and quantum simulation, we introduce a tunable coupling between charge qubits. Transmission line resonator cou-pling with qubit can simulate the coupling of atom and photon field. Then, we introduce the basic concept of quantum simulation are introduced. Some proposals of supercon-ducting circuit as quantum simulator show in the first chapter. Finally, we explain how to use superconducing circuit system to study dynamical Casimir effect.2. We study the quantum phase transition in a spin chain with variable Ising inter-action and position-dependent coupling to a resonator field. Such a complicated model, usually not present in natural physical systems, can be simulated by an array of qubits based on man-made devices and exhibits interesting behavior. Our system is an array of charge qubit capacitively coupled to the transmission line resonator. Due to the macro-scopic size of the charge qubit array, they spread out along the entire TLR length which is also the wavelength of the EM modes that they couple to, and the position depen-dence of the coupling strength must be taken into account. By using large Josephson junctions inductively coupled to the charge qubits, we can induce strong and adjustable interactions between them, greatly enriching the physics of our system. We show that, when the coupling between the qubit and field is strong enough, a superradiant phase transition occurs, and it is possible to pick a particular field mode to undergo this phase transition by properly modulating the strength of the Ising interaction. We also study the impact of the resonator field on the magnetic properties of the spin chain, and find a rich set of phases characterized by distinctive qubit correlation functions.3. We investigate the possibility of observing in integrated solid-state systems the dynamical Casimir effect, in which photons are created out of vacuum. We use a trans-mission line resonator in a superconducting chip as the microwave cavity and modulate its properties by coupling it to carefully designed SQUID which is biased by an exteral time-dependent magnetic flux. Due to the time-dependent magnetic flux, the TLR has a time-dependent boundary condition. We introduce a set of instantaneous eignmodes, and use them to expand the field operator. Quantization can be accomplished by ex-panding the field operator and its conjugate momentum. In order to describe the system in Fock space, we introduce the instantaneous creation and annihilation operators, and use their Hamiltonian equation, get the effective Hamiltonian of the system. Because of the surface resistance of TLR and resistance of SQUID, we introduce a new kind of Bogolivbov transformation which contains the noise operators, it allows us to obtain the created photon number as well as squeezing. Using experimentally achievable pa-rameters, we calculated the evolution of photon number and squeezing and study how they depend on the dissipation.