Dynamic Properties Of Condensate With Complex Interactions In Harmonic Potential

Recently, the studies of properties of Bose-Einstein condensates (BEC) in harmonic potential have gained great achievements, and find many new and interesting phenom-ena. However, these studies focus mainly on BEC with contact interaction trapped in harmonic potential. Actually BEC are trapped in three-dimension harmonic potential. Besides, it is known that the introduction of long-range interactions, i.e., dipole-dipole in-teractions (DDIs), can lead to a variety of unique properties that are absent in the system dominated by the short-range interaction, such as density fluctuations of the background gas resulting from introduction of impurity can cause polaron effects. However, up to now, the coupled effects of contact interaction and DDIs to impurity in 3D boson gas system are not clear or understood analytically. Hence, In this paper, based on simulta-neous consideration of DDIs with the very distinct properties-anisotropic and long-range, and contact interaction, we have studied the dynamics of the impurity trapped in a 3D harmonic potential interacting with a separately trapped Bose gas.Because of the Spin-Orbit Coupling(SOC) of electronic system, atom has abundant energy level structures. In addition, electronic system with SOC has other attractive phenomena such as the fractional quantum Hall effect and topological insulation phase. So the studying of the Spin-Orbit Coupled Bose-Einstein condensates(SOC-BEC) becomes a new hotspot in BEC field. More recently, the wave packet’s stabilities and oscillation dynamics of localized states of SOC-BEC with contact interaction are mainly studied in a quasiperiodic optical lattice potential. However, the studies of dynamic properties of the SOC-BEC with contact interaction in a harmonic potential are deficient. So that is the other topic included in the paper. The specific contents and conclusions of this paper are as follows:Firstly, the physical background and the related physical knowledge are briefly in-troduced, including Bose-Einstein condensates, harmonic potential, short-range and long-range interaction, dipolar BEC, impurity’s properties and significance studied in the Bose gas, the generating and properties of the SOC-BEC and so on.In the second part of our paper, we discuss the dynamics of the impurity interact-ing with Bose gas. With the synchronous consideration of short-range impurity-bosons coupling and DDI, we study breathing dynamics of a harmonically trapped impurity in-teracting with a separately trapped background of dipolar Bose gas feeling a parabolic dipolar potential in a cylindrically symmetric system. By using the variational approach, the breathing equations, the breathing frequencies and the effective potentials governing the breathing dynamics of the impurity in dipolar gas are obtained. The effects of DDI, impurity-bosons interaction, and two external trapping potentials on breathing dynamics of impurity are discussed. We find that, because of the anisotropic and long-range char-acters of DDI, the effects of DDI, impurity-bosons interaction, and two external trapping potentials on breathing dynamics of impurity are strongly coupled. DDI has significant modification on dynamics, which depends on two external trapping potentials’shape.In the third part of our paper, we discuss the dynamics of the SOC-BEC with con-tact interaction in a quasi-1D harmonic potential. With consideration of short-range bosons-bosons interaction, we study the dynamics of the BEC with spin-orbit coupling trapped in a quasi-one dimension harmonic potential using variational approach to solve a binary mean-field Gross-Pitaevskii equations with two pseudo-spin components. Firstly, the eight equations governing the evolutions of the atomic number of up preudo-spin, centre-of-mass coordinates and breathing widths of the condensate with two pseudo-spin components are deduced and solved by using precisely numerical methods. The evolu-tions of these variables are shown in the presence of the short-range interaction, spin-orbit and Rabi couplings. The results show complex characters:when SOC exists and Rabi couplings is absent the phase separated of two pseudo-spin components emerges and the amplitude of dipole oscillation increases with SOC, the phase mixed of them emerges and the amplitude of dipole oscillation is independent of SOC in the presence of medium Rabi couplings, and when the Rabi couplings become strong the phase separated of them emerges again and the amplitude of dipole oscillation increases only for strong SOC. Fur-ther, we find the spin-exchange process depends completely on synchronous existing of the SOC and the Rabi couplings, and the dynamics of polarization oscillation are closely relate to contact interaction and the initial phase difference of two pseudo-spin compo-nents. Finally, the evolutions of the breathing widths of two pseudo-spin components are shown, which the breathing widths increase with interaction, and it occurs synchronous quasi-periodic oscillation in the presence of SOC and interaction, but it don’t in the pres-ence of Rabi coupling and interaction, the coupling of SOC, Rabi coupling and contact interaction has significant modification on it. We summarize that these ample phenomena result from the coupling of SOC, Rabi coupling and contact interaction.Finally, we summarize the main results and give an outlook of the future in this field.