Many-body Phase Transition And Quantum Simulation In The Micro-cavities

Quantum phase transition is the topic of the condensed matter theory. The ground-breaking advances of experiment in the end of last century, has open up a new area of many-body physics. For example, the realization of Bose-Einstein condensate (BEC) in a dilute gas began a series of studying of BEC. People have investigating the interference of BEC clouds, observed rotating BEC and studied spinor condensates. With the development of laser cooling technique, the degenerate Fermi gas has been observed. Two Fermi atoms can form a cooper pair if they have opposite spins. At low temperature, cooper pairs may form a kind of condensate, which is Barden-Cooper-Schrieffer (BCS) super-liquid. The Feshbach resonance can modulate the interaction between two Fermi atoms. So it is possible to observe the transition from BCS to BEC. In these phase the short-range interaction is dominated. But in BEC-cavity system, there will be new phase which is dominated the long-range interaction. This long-range interaction was mediated by the cavity field. Therefore, the quantum phase transition from normal phase to super-radiant phase, have been observed in the BEC-cavity system. On the other hand, circuit QED on a chip has become a new platform or many-body physics. Many important quantum effects have been observed in this artificial atoms system.In this article, we mainly study the quantum phase transition in the cavity and quantum simulators. The main results are included as follows:(1) We have studied the ground state characters of the Rabi model with ultrastrong coupling. First we have introduced the generalized rotating-wave approximation method (GRWA). And we have shown that this method is invalid in the case of the positive detuning. Thus, we have presented a generalized variational method (GVM). Using GVM, we can get the explicit expression of the ground state energy and mean photon number in both negative and positive detuning case.(2) We have studied the finite-temperature properties of the Dicke model with a nonlinear atom-photon interaction. We have got the rich phase by the function path-integral approach. We have got the analytically expressions of the mean photon number and atomic population. We have shown that the specific heat has a large jump at the critical temperature where the system has a transition from the superradiant phase to normal phase.(3) We have proposed a Dicke-Ising model to describe the system with a superconducting qubit array and a transmission-line resonator. The completion between the nearest-neighbor spin-spin interaction and the spin-photon interaction leads a rich phase diagram. We have also found that the unconventional behavior of mean photon number. We believed that the phase diagram could be observed by detecting the phonon signature.