Exotic Quantum Phase Transitions And Their Controllability Of Many-body Systems In The Micro-cavities

Quantum phase transitions and their controllability have attracted much attention in condensed-matter physics.In the earlier theoretical consideration, Dicke model for multi-atoms inside a cavity possesses a quantum phase transition from the normal to the superradiant phases.However,this predicted transition has never been observed in experiments.The main reason is that quantum fluctuation for each atom prevails,so all atoms can not interact identically with the photon.Recent experiment has reported that the collective strong coupling between the ultracold atoms and the photon can be achieved since all ultracold atoms has a peculiar property that the trapped Bose-Einstein condensate has the same quantum state in momentum space below a critical temperature.This experimental connection enables us to investigate conceptually light-matter interaction at ultimate quantum levels. In this composite system,the photon induced by a cavity mode can not only interact with ultracold atoms in near resonance but also act as a bus resulting in effective long-range interactions among ultracold atoms.Moreover,these generated long-range interactions have a high competition to the direct two-body interactions of ultracold atoms,which can lead to some exotic quantum effects.Based on current experimental techniques,in this dissertation,we mainly investigate exotic quantum phase transitions and their controllability for multi-atoms inside a cavity.Our results are included as follows:(1) We have obtained a general formula for geometric phase for Dicke model(proportional to the average photon number) and found that this ground-state geometric phase has an important relation with the superradiant quantum phase transition.Therefore,the geometric phase can be regarded as a good physical quantity to discover quantum phase transitions in experiments.(2) We have investigated quantum tunneling for finite-number Dicke model in the superradiant phase by instanton method.It shows that quantum-chaos-assisted tunneling has a high competition to pure quantum tunneling.As a result,the total tunneling is partly suppressed.(3) We have put forward a method to control the superradiant phase transition for Dicke model.In our proposal,the resonant frequency of atoms can be replaced by the frequency of an introduced driving external field,and therefore,the condition to realize the superradiant phase transition can be well realized.We have also suggested an experimentally-feasible scheme that the superconducting charge qubits couple with a cavity to simulate our proposal.(4) We have introduced an extended Dicke model with controllable long-range atom-atom interactions.By means of coherent-state path integral, this introduced model only for a few odd-numbers of atoms has an absolutely degenerate ground-state subspace and a large energy gap,which becomes larger with the increasing of the system-size.Therefore,the fault-tolerant quantum computing for suppressing decoherence can be achieved in this absolutely degenerate ground-state subspace. (5) We have realized a Dicke model in superconducting charge qubits interacting with the nanomechanical resonator,and then produced higher entangled cluster states under period evolution.(6) We have calculated the energy gap for the collective Lipkin-Meshkov-Glick model by means of spin-coherent-state path integral and found a new type of quantum phase transition induced by geometric phase.This predicted quantum phase transition of level-crossing may provide a possible technique to realize qubits for nonlinear collective spin model.(7) We have predicted a novel second-order phase transition from the superradiant to the Mott phases and a first-order superfluid phase transition in a Bose-Einstein condensate inside a cavity in the dispersive regime.By eliminating the freedom of photon,we can simulate the Lipkin-Meshkov-Glick model and obtain the topological quantum coherent effect.By using the spin-coherent-state path-integral,we can also find an interesting topological quantum transition for a special Lipkin-Meshkov-Glick model.(8) We have obtained a new type of controllable nonlinear coupling by the exchange of virtual photons in a Bose-Einstein condensate in cavity optomechanical system.It opens a new possibility for investigating the coherent nonlinear effect at the macroscopic scale and leads to a mechanical analogy of cavity quantum electrodynamics.Based on this nonlinear coupling we have also successfully simulated a Hamiltonian which has the collective spin four-body interaction.