The Research Of Quantum Phase Transition In Extended Jaynes-Cummings-Hubbard Model

Coupled-cavity array system is the most typical many-body system for studying the light-matter interaction in quantum optics.Compared with the cold atom system in optical lattices,it is easy to study the long-range interaction between atoms and does not depend on the distance between atoms.Through photonatom coupling and photons exchange between cavities,the long-range interaction between ultracold atoms can be realized indirectly.It can not only explore the collective dynamics of photons,but also study more complex quantum many-body phenomena and novel states.The coupled-cavity array system can be realized not only by the microcavity coupling with ultracold atoms,but also by the superconducting circuit system.Due to the rapid development of technology and the diversity of experimental schemes,more and more researchers explore the quantum many-body effect of the system in different cross fields.One of the most commonly models in coupled-cavity array systems is the extended JCH model.Although this model has been studied widely,there are still some unclear problems to be solved.This thesis explores the Superfluid-Mottinsulator quantum phase transition behaviors in several extended JCH models,whether there are novel quantum phases,and whether the phase transition from ergodic phase to non-ergodic phase will occur in the standard JCH model.The specific research contents are listed as follows:The first study of this thesis is to discuss the quantum phase transition of Superfluid-Mott-insulator in two-dimensional hybrid optomechanical coupledcavity array systems by using the mean-field method.By adjusting the frequency detuning of the cavity field and the movable oscillator,we obtain an intersection between different critical chemical potentials,and certain Mott lobes physically disappear.The pronounced optomechanical coupling strength and a large number of phonons favor the Mott-insulator phase due to the enhanced effective on-site interactions.Additionally,the Kerr-nonlinearity reduces the average excitation of the high Mott-insulator state and diminishes the superfluid regime.The results obtained here provide a novel and complete image for characterizing the quantum phase transitions in optomechanical coupled-cavity array systems.We study the steady-state quantum phase transition of the JCH model with quantized center-of-mass motions as the second research content.By comparing the results of approximate and non-approximate treatment of the center-of-mass motions,the applicable conditions of Lamb-Dicke regime are obtained.Four different quantum phases emerge due to the interplay between the atom-photon interaction modulated by the center-of-mass motion and the tunnelling of photons.Importantly,as the amplitude of the atomic oscillations increases,the phase boundaries will become more and more blurred.We also observed that the phase diagram changes periodically with the change of the relative position between the trap center and the cavity field.Through further analysis,we get the influence of the center-of-mass motions on quantum phase transition and know which conditions can ignore its impact.The third study is the ground state quantum phase of the one-dimensional JCH model including atomic dipole-dipole interactions between nearest neighbors by using the quantum Monte Carlo method of stochastic series expansion.Through the characterization of order parameters,it can be seen that the unstable density wave phase exists under the small size and weak dipole-dipole interaction,and the numerical results fluctuate greatly and have strong size dependence,thus it is difficult to obtain stable results.It is found that there are only stable Mottinsulator phase and Superfluid phase in the large size,which overturns the idea of looking for density wave phase and Haldane insulation in one-dimensional extended JCH model.Secondly,the same method is used to study the quantum phase transition of the two-dimensional square lattice extend JCH model.We find that under the weak dipole-dipole interaction,the system only has Chessboard and Superfluid phases,but under the strong dipole-dipole interaction,there is also a Supersolid phase,which does not change with the system size,and a wealth of quantum phase transitions are obtained.The results verify the Supersolid phase obtained by the cluster mean field method,and overturn the result that there is no Supersolid phase in the soft-core JCH model predicted by worm quantum Monte Carlo method for a small-size hard-core case.The fourth research content is to study ergodic and nonergodic behaviors of a clean JCH chain for different parameters based on the average level spacing ratios and the generalized fractal dimensions of eigenstates by using exact diagonalization.It can be found that a transition from ergodic to nonergodic regimes happens when increasing the atom-photon detuning,and the nonergodic phases should be exist in the thermodynamic limit.We also find that the nonergodic phase violates the eigenstate thermalization hypothesis and displays MBL-like behavior.Finally,we study the many-body multifractality of the ground state and find that the derivative of the generalized fractal dimensions can determine the critical point of the Superfluid-Mott-insulator phase transition in a small range of parameters under different boundary conditions and there is no ergodicity for the ground state.Our results provide a new factor leading to ergodic-nonergodic phase transition,and the derivative of generalized fractal dimensions can reduce the dependence on large-scale numerical methods to study the critical point of the ground state phase transition.Based on the above research on the influence of the optomechanical cavity,atomic center-of-mass motions,dipole-dipole interaction on the quantum phase transition in the extended JCH model and the energy spectra phase transition in the JCH model,we give the corresponding novel phase transition behaviors and phase diagrams under different theoretical methods,the deepen understanding of the quantum phase transition in the extended JCH model,provide valuable insights for the quantum simulation in the coupled-cavity array system and should promote the development of quantum phase transition.