Quantum Simulation In Superconducting Quantum Circuits By The Parametric Conversion Method

Simulating a large quantum system using classical computer is a challeng-ing task because the required resource grows exponentially with the number of particles. Quantum simulation fights fire with fire, i. e. it employs a controlled quantum-mechanical device to mimic and investigate other quantum systems. Due to its potential applications and the recent advances in the coherent control of the artificial quantum systems, quantum simulation has been growing rapidly in the past few years. Taking the advantages of its flexibility, controllability, and tunability, superconducting quantum circuit is a promising candidate of quan-tum simulation. Its on-chip realization can support the fabrication of a large scale system with a widely tunable range of parameters and the site-resolved control and measurement. Moreover, the achieved strong coupling between su-perconducting transmissionline resonators (TLRs) and superconducting qubits can induce the effective strong photon-photon interaction, which is critical the study of the strongly correlated systems. The superconducting circuit based quantum simulator can not only simulate the existing solid-state systems, but al-so push the parameters to the extreme conditions which can hardly be achieved in conventional condensed matter materials.In this thesis, we study the quantum simulation in superconducting quan-tum circuits, including the following results:1. We propose a scheme of implementing artificial Abelian gauge fields via the parametric conversion method in a necklace of superconducting transmis-sion line resonators coupled by superconducting quantum interference devices (SQUIDs). The dynamic modulations of the SQUIDs can induce the photon hopping between neighboring TLRs by the parametric conversion, and in this process, the phase of the dynamic modulation is absorbed by the photon. When the photon travels along the ring, an non-trivial phase is accumulated which is similar to the Aharonov-Bohm phase of the electron moving in the presence of magnetic fields. Therefor, the non-trivial phase can be treated as the artificial gauge field. Since the phase of the dynamic modulations can be arbitrarily set, an extremely strong effective magnetic field for charge-neutral bosons can be synthesized in the necklace, which is difficult to achieve in conventional solid-state systems. To demonstrate the synthetic magnetic field, we study the realiza-tion and detection of the chiral photon flow dynamics in this architecture under the influence of decoherence. Taking the advantages of its simplicity and flex-ibility, this parametric scheme is feasible with state-of-the-art technology and may pave an alternative way for investigating the gauge theories with super-conducting quantum circuits. We further propose a quantitative measure for the chiral property of the photon flow. Beyond the level of qualitative description, the dependence of the chiral flow on external pumping parameters and cavity decay are characterized. We notice that, through a variety of generalizations, the artificial gauge fields with in situ tunability in a large-scale circuit quan-tum electrodynamic (QED) lattice can also be easily achieved by the parametric conversion method.2. Topology is an important degree of freedom in characterizing electronic systems. Recently, it also brings new theoretical frontiers and many potential applications in photonics. However, the verification of the topological nature is highly nontrivial in photonic systems as there is no direct analog of quan-tized Hall conductance for bosonic photons. We further propose a scheme of investigating topological photonics in superconducting quantum circuits by the the parametric conversion method, the flexibility of which can easily lead to the effective artificial gauge field for photons on a square lattice. We then study the detection of the topological phases of the photons. Our idea employs the exot-ic properties of the edge state modes which result in novel steady states of the lattice under the driving-dissipation competition. Through the pumping and the photon-number measurements of merely few sites on the lattice, not only the s-patial and the spectral characters, but also the momentums and even the integer topological quantum numbers with arbitrary values of the edge state modes can be directly probed, which reveal unambiguously the topological nature of pho-tons on the lattice. Due to the weak requirement of quantum coherence and the simple circuit needed for pumping and steady-state photon number detection, our scheme can then be examined with the state-of-art technology.3. The concept of flat band plays an important role in strongly-correlated many-body physics. However, the demonstration of the flat band physics is highly nontrivial due to intrinsic limitations in conventional condensed mat-ter materials. We propose a circuit quantum electrodynamics simulator of the 2D Lieb lattice exhibiting a flat middle band. Once again, by exploiting the parametric conversion method, we design a photonic Lieb lattice with in situ tunable hopping strengths in a 2D array of coupled TLRs. To unambiguously demonstrate the synthesized flat band, we further investigate the observation of the flat band localization of microwave photons through the pumping and the steady-state measurements of only few sites as we did previously. Requiring only current level of technique and being robust against imperfections in real-istic circuits, our scheme can be readily tested in experiments. Moreover, the flexibility of our proposal enables the immediate incorporation of both the ar-tificial gauge field and the strong photon-photon interaction through the strong coupling between qubits and TLRs. Therefore, this proposal offers a promising platform of realizing interacting photonic quantum Hall fluids.In summary, in this thesis we propose several superconducting circuit quantum simulators based on the parametric conversion method. Our work may pave an alternative promising way towards the future study of strong magnetic fields, topological insulators, and strong correlated physics.