Theoretical Study On The Qubit-photon Interacting Models

The study of the interaction between atom and photon field is of central importance in quantum optics. The simplest and most basic interaction can be presented by the Rabi model, which describes a two-level atom interacting with a single-mode photon field. Since it couples a two-level system (qubit) to a bosonic mode, the Rabi model has wide applications ranging from quantum information and atomic physics to solid state and magnetic resonance. With the development of superconducting quantum circuit technique, now the ultrastrong qubit-photon coupling regime has been reached. In this regime, the Jaynes-Cummings model with the rotating-wave approximation breaks down, which is widely used in the weak coupling regime. However, the Rabi model works for the whole coupling regime. The work in this dissertation is based on the generalizations and extensions of the Rabi model, and includes following theoretical investigations and results.In Chapter two, we study the interaction between the two qubits and a single mode photon field in the whole qubit-photon coupling regime with the two-qubit Rabi model. We have analytically obtained the spectrum and eigenstates of the two-qubit Rabi model in two equivalent way:the Bargmann space and extended coherent states representation. The spectrum and eigenstates can be used to investigate the proper-ties of the system, and may give some possible theoretical insights to experimental re-searches. We also study the integrability of these models with the quantum integrability criterion proposed by Braak and the properties of the spectrum. The two-qubit system is fundamental to the construction of the universal quantum gate. Taking advantage of its ultrastrong coupling to the photon field, one can construct two-qubit quantum gate. On the other hand, the storage and transfer of the coherent quantum state can be real-ized via the coupling of two qubits mediated by a resonant cavity. In Fock space, the Hamiltonian for the model is generally infinite-dimensional with non-zero off diagonal elements and has no closed subspaces. However, using the Z2symmetry for the model, we have obtained its solution analytically. For identical couplings between two qubits and the photon field, we have found a series of special quasi-exact solutions whose photon part are formed by finite Fock states for some choices of parameters, and they correspond to closed subspaces in Fock space, which are resulted from the permutation symmetry of the qubit-photon coupling terms.Even more interestingly, for identical couplings between two qubits and the pho-ton field, there are very special quasi-exact solutions existing in the whole coupling regime with eigenenergies equal to single photon energy, independent of the coupling strength. The existing conditions for these solutions just depend on qubits and photon energies, which can be fine tuned in experiments in contrast to the coupling strengths, so these eigenstates are probably easy to prepare. At the same time, due to only con-nected with Fock states|0) and|1), they may find applications in single photon ex-periments. Their energies correspond to a horizontal line in the spectrum, similar to that for the famous "dark state". The so-called "dark state" is composed of a spin sin-glet formed by two qubits, and decouples from the photon field, so the eigenenergies are just the sum of the energies of the photon field and the qubits. However, the very special eigenstates we have found involve the coupling with the photon field, and the wavefunctions vary with coupling constants, so they are novel special eigenstates.In Chapter three, based on the two-qubit Rabi model, we consider some kinds of qubit-qubit interactions, including the dipole interaction, XXX and XYZ Heisen-berg interactions which are commonly used in generating entanglement, mandatory for application in quantum computation. Although entanglement between the qubits is produced naturally by their coupling to the radiation field, it is interesting to add a direct interaction term, which may provide more options to control the system pre-cisely. Considering of the same Z2symmetry as the two-qubit Rabi model, we obtain their spectra and eigenstates analytically within the Bargmann space. The integrability of these models are also studied with the quantum integrability criterion proposed by Braak and the properties of the spectrum. For identical couplings between two qubits and the photon field, on one hand, we find some special quasi-exact solutions for cer-tain parameters whose photon part are formed by finite Fock states. On the other hand, if the qubit-qubit interaction strengths, qubits and single photon energies satisfy cer-tain condition, there are two kinds of very special quasi-exact solutions which exist in the whole coupling regime with constant eigenenergy, corresponding to two horizontal lines in the spectrum.Next, in Chapter four, we generalized the two-photon Rabi model to study the qubit-photon interaction mediated by two-photon exchange process in superconducting quantum circuit, with the qubit static bias taken into account. Such non-linear optical process has some interesting properties being worthy to be studied, and has important application in two-photon laser, two-photon optical bistability and so on. The qubit static bias can be tuned in superconducting quantum circuit, so it can be used to give a fine control of the system. Using the Z2symmetry of this model, we have obtained its spectrum analytically in the Bargmann space. Some isolated exact solutions of the model are also obtained by a Bogoliubov transformation. These solutions are very important for getting the information about the structure and symmetries of the model. Compared with the former solutions in the Bargmann space, it is shown that both kinds of solutions coincide with each other very well. We also study the integrability of these models with the quantum integrability criterion proposed by Braak and the properties of the spectrum.Finally, we make a summary and give some outlooks in Chapter five.