Landau-Zener Tunneling In Spin-Orbit-Coupled Bose-Einstein Condensates In (?)(?) Potentials

Since the discovery of the supercooled atom Spin-Orbit coupling Bose-Einstein condensate in experiments,it has become an important experimental platform for quantum simulation and quantum manipulation due to its highly regulated parameters.Recently,it is found that the Hamiltonian of the system has a real number eigenspectrum in a certain range of parameter values when the Hamiltonian satisfies the Parity-Time-symmetry((?)(?) symmetry).Bose-Einstein condensates are a new field of (?)(?) symmetry.Therefore,This paper mainly studies Landau-Zener tunneling in Spin-Orbit-coupled Bose-Einstein condensates in (?)(?) periodic potentials.We use the mean field approximation to solve the coupled Gross-Pitaevskii equation(GP equation)in a two-color optical lattice with (?)(?) symmetry to obtain the analytical solution of the linear Bloch wave and draw the linear Bloch band.Then the effect of the modulation of the period of the potential function on the position of the singularity is studied when the spinorbit coupling strength,Raman coupling strength and detuning strength are all zero.Found when potential function don't change the cycle of real component,with the decrease of the imaginary part cycle (?)(?) symmetry broken sooner,and discover the potential function of the cycle of real and imaginary part of the same energy mismatch,SOC strength and Raman coupling strength of these parameters do not affect the location of the singular point,therefore mainly discussed the next cycle is not at the same time,The position changes of the singularities under these parameter modulation.In order to determine the relationship between the LZ tunneling probability and other parameters of the spin-orbit coupling Bose-Einstein condensate in the (?)(?) symmetry potential,we discuss the tunneling probabilities for the imaginary part,the SOC strength,the energy dissonance and the Raman coupling strength of the different (?)(?) symmetry potentials.The results show that the non-zero transition probability mainly occurs in the high Bloch band gaps.The LZ tunneling probabilities of the high band gaps increase with the imaginary part v2 of the (?)(?) symmetry potential increasing at different slopes.As the SOC strength ? increases,the LZ tunneling probability of the third band gap gradually increases,while the LZ tunneling probability of the fourth band gap decreases.As the Raman coupling strength ? increases from zero,the LZ tunneling rates of the odd multiple band gaps decrease rapidly from 1 to 0.Meanwhile,the LZ tunneling probabilities of the even number of band gaps increase with the increase of Raman coupling strength ?.We also find that the limited detuning can increase the probability of LZ tunneling.