Quantum Phase Transition And Its Properties Of The Extended Hubbard Model

Bose-Hubbard model,a many-body system,is important to quantum simulation and quantum computation.With the development of the cold-atom technology,this Bose-Hubbard model can be realized in the experimental simulation by trapping the cold bosonic atoms in the optical lattice.The progress in the technique to manipulate the interaction between light and matter,such as the cavity quantum electrodynamics,superconducting quantum circuits,and so on,has opened another way to simulate the Bose-Hubbard model.When a real or an artificial atom is placed in an optical microcav-ity with a laser,the interaction between the atom and the field results in the Bose quasi-particles.Therefore,the coupled cavity array named the Jaynes-Cummings-Hubbard model,formed by coupling many such optical microcavities through a light field,also belongs to one type of the Bose-Hubbard model.The quantum phase transition of the Bose-Hubbard type formed by controlling the interaction between light and matter is the main interest of this paper.This extended Hubbard model not only expands the control-lable quantum system and provides new possibilities for quantum simulation,but also gives a new way for the storage and acquisition of quantum information.Firstly,we review a simple model of the interaction between light and matter—the Jaynes-Cummings model.Its nonlinear eigen-energy spectrum can promote the photon blockade effect,that is,a photon needs to overcome some energy to enter an optical cavity occupied by photons.For the Hubbard model,this repulsive effect between pho-tons can provide repulsive energy on a single lattice site and competes with the photon hopping term resulting in the phase transition from a Mott insulating phase to a super-fluid phase.By calculating the superfluid order parameters,the phase boundary from the Mott insulating phase to the superfluid phase can be given.Furthermore,for the Jaynes-Cummings model on the single lattice site,its excitation number exists a ladder structure.Therefore,Mott lobes are formed by the excitation number in the Jaynes-Cummings-Hubbard model.When the counter-rotating wave term is introduced,the（1）symmetry of the Jaynes-Cummings model is broken,leaving only(5₂symmetry.The ground state changes directly from zero excitation to non-zero excitation.In the Hubbard model,the photon hopping term breaks(5₂symmetry,resulting in a phase transition from a localized phase to a delocalized phase.In the localized phase,the su-perfluid order parameter is zero,and the number of photons in the cavity on a single lattice has no fluctuations;In the delocalized phase,the non-zero photon number fluc-tuation in the cavity on the single lattice site correspondings to the non-zero superfluid order parameter.When the interaction between light and matter is close to the deep strong coupling region,the diamagnetic term can not be ignored.The influence of the diamagnetic term on the quantum phase transition of the Jaynes-Cummings-Hubbard model is also studied.And we find that the diamagnetic term can compress the super-fluid phase.Secondly,we analytically and numerically study a nonlinear coupling model of the interaction between light and matter—the Buck-Sukumar model.This model is only well defined under a critical coupling.Beyond this critical coupling,the ground state exists an infinite excitation,which is no longer meaningful.On this critical coupling,the energy spectra of the model collapse.The counter-rotating wave term does not affect these properties,but reduces the critical coupling.This Buck-Sukumar model,which is well defined only under the critical coupling and whose energy-spectra-collapse on the critical coupling,is incomplete.By introducing a nonlinear photon term,the photon transition between energy levels is hindered and the energy spectra collapse is elimi-nated resulting in a complete Buck-Sukumar model.Furthermore,by calculating the excitation number and the second-order correlation function of the ground state,we study the photon blockade effect of the Buck-Sukumar model to make the construction of the extended-Hubbard model possible.Besides,we construct the Buck-Sukumar-Hubbard model to study its quantum phase transition from the Mott insulating to the superfluid phase by calculating two superfluid order parameters.The effects of the counter-rotating wave term,detuning,and the nonlinear photon term on the phase boundary are further studied.By ignoring the effect of the external driven field and the artificial chemical potential,the incomplete Buck-Sukumar-Hubbard model with the rotating wave approximation,is well defined only when the field frequency is positively detuned with the atom ones;The counter-rotating wave term breaks this limitation and makes it meaningful in the whole range of detuning.And the positive detuning promotes the single photon superfluid phase.For the complete Buck-Sukumar-Hubbard model,the nonlinear photon term promotes not only the Mott insulating phase but also the single photon superfluid phase with-out counter-rotating wave term while the counter-rotating wave term promotes the pair photon superfluid with counter-rotating wave term.Finally,considering another freedom degree of the atom,the atomic center-of-mass motion when atom interacts with the light field,the spin-orbit coupled effect is intro-duced into the Jaynes-Cummings model.We study the influence of spin-orbit coupling on the ground state properties.We find that the spin-orbit coupling effect makes the eigen-energy degenerate and enlarges the coupling regime of the ground state with zero excitation.When considering the influence of the counter-rotating wave term,the spin-orbit coupling makes the energy of the first excited state degenerate,which is helpful for the zero excitation.Simultaneously,by considering the influence of the spin-orbit coupling on the quantum phase transition of the Jaynes-Cummings-Hubbard model,we find that the spin-orbit coupling effect shrinks the superfluid phase.Our research not only provides different schemes for the experimental realization of the single photon source,but also paves a new path for the experimental simulation of the extended Bose-Hubbard model in the system of interaction between light and matter.