Analytic Solution And Dynamics Of The Multi-qubit Rabi Model

The interaction of quantum matter and quantum harmonic oscillators plays a central role in the dynamics of quantum optics and condensed matter physics, where the Rabi model is the one of the simplest and most typical model. In the rotating wave approximation, the Rabi model can be transformed to the Jaynes-Cummings (J-C) model. In recent years, the realizations of the ultra-strong coupling regime have made remarkable achievements on the application aspects of the fundament quantum physics and quantum information. The study of the interaction between the qubit and the single mode radiation field is the essential resource in the process of quantum information. In this paper, we mainly research the N-qubit Rabi model, where we focus on the eigensolutions, dynamics and Berry phase of the system.First, we study the analytic solution of the two-qubit quantum Rabi model with inhomogeneous coupling and transition frequencies. The significant advantage of the displaced oscillator basis enables us to apply the same truncation techniques adopted in the Rabi model to the two qubits system. The derived analytical spectra match perfectly with the numerical solutions in the parameter regime where the qubits'transition frequencies are far off-resonance with the field frequency and the interaction strengths reach the ultra-strong coupling regime. By means of the fidelity of quantum states we identify several nontrivial level crossing points in the same parity subspace, which implies that the Rabi model with two identical qubits is integrable. We further explore the dynamical behavior of the two qubits as well as the evolution of entanglement. The time evolutions of the probability for the qubits show that the collapse-revival phenomena emerge, survive and finally disappear when one coupling strength increases from weak to strong coupling regimes and the other coupling strength is well into the ultras-trong coupling regime. The inhomogeneous coupling system exhibits new dynamics, which are different from the homogeneous coupling case.Next, we study the geometric curvature and the geometric phase for the Rabi model. Under the rotating-wave approximation (RWA), we apply the gauge independent Berry curvature over a surface integral to calculate the Berry phase of the eigenstate for both single and two-qubit systems. We extend the idea to define a vacuum-induced geometric curvature when the system starts from an initial state with pure vacuum bosonic field. The induced geometric phase is related to the average photon number in a period which is possible to measure in the qubit-cavity system. Moreover, we also calculate the geometric phase beyond the RWA and find an anomalous sudden change, which implies the breakdown of the adiabatic theorem and the Berry phases in an adiabatic cyclic evolution are ill-defined near the anti-crossing point in the spectrum.Finally, we give an analytical description of the dynamics of the three-qubit Dicke model using the adiabatic approximation in the ultra-strong coupling regimes. Qualitative differences arise when comparing to the single-and two-qubit systems. Simple analytic formulas show that three revival sequences produce a three-frequency beat note in the time evolution of the population. We find an explicit way to estimate the dynamics for the qubit-fleld and qubit-qubit entanglement inside the three-qubit system in the ultra-strong coupling regime, and the resistance to the sudden death proves that the entanglement in GHZ state is more robust than that in W state.