The Stability And Sound Wave Excitation Of The Spin-orbit Coupled Bose-Einstein Condensates In The Optical Latice

Recently,the investigations for the spin-orbit(SO)coupled Bose-Einstein condensate(BEC)trapped in an optical lattice have attracted great interest in theories and experiments.Meanwhile many interesting phenomena have been largely explored,including the study on the superfluidity and the transition between the different phases in the system.For the SO coupled BEC,new phases occur,for example the stripe phase and the plane-wave phase.Except that the researchers concentrate on the properties of low-energy excitations of the SO coupled BEC,the phonon and roton are two kinds of collective excitations.The measurement on the collective excitations can help us well understand the physics of the manybody systems.In our work we fix on the sound wave and the stability of the SO coupled condensates trapped in weak and deep optical lattice,respectively.Firstly,we introduce the physical background and the related physical knowledge about our work,such as the BEC,sound speed,superfluidity,optical lattice,SO coupling,low-energy excitations and the rapid development and the values of the exploration about the SO coupled BEC.In the second part of our paper,we discuss the sound wave of the SO coupled BECs trapped in 1D weak optical lattice by means of calculating the sound speed of the system.Furthermore,the superfluidity of the SO coupled system is also studied.The sound speed vanishes at the critical value of the Raman coupling,which is called the soften of the phonon.The place for the sound speed of the system vanishing exactly indicates that the superfluidity of the system is destroyed and the system becomes dynamically unstable.The existence of the optical lattice enhances this kind of softening.The larger the atomic interaction is,the more stable the system will be.Further investigation indicates that the system shows the same superfluidity for sufficiently larger atomic interaction.In brief,the effects on the condensates from the optical lattice,SO coupling,Raman coupling and the atomic interaction couple together.Obviously,the analysis and the investigation play very important roles in further exploring the BEC system.In the third part of our paper,considering the atomic interaction and using the Bogoliubov method,we analyze the stability of the SO coupled BEC trapped in a deep optical lattice,which includes the energetic stability and the dynamical stability.Specially,we show the dispersion relation of the SO coupled system in the case of the atomic interaction absenting.It shows that the dispersion relation consists of two branches.For the lower branch,the two spin states merge into a single minimum with the increasing of the Zeeman splitting.In the presence of the atomic interaction,due to the SO coupling,a new dynamical instability region occurs when the momentum is smaller than the critical value of the momentum.However,a new stable region occurs when the momentum is larger than the critical momentum,which is really different from the case of the weak regular optical lattice.In all,the existence of the SO coupling modifies the stability of the system.Finally,we briefly summarize the main conclusion and give an outlook of the future work in this field.