Theoretical Study Of Magic And Tune-out Wavelengths Of Ba~+ Ions

Recently,the accuracies of optical atomic clocks cooled and trapped by lasers have reached the level of 10^-19.The atomic clocks have many important applications in the measurement standard,exploring the variation of physical constants and the navigation field,etc.Due to the relatively simple structure and the long-lived metastable states5d_3/2,5/2of Ba⁺ions,it is one of very good candidates for ionic clock experiments,quantum information techniques,and atomic parity-violation experiments.In theory,we urgently need to calculate high-precision atomic structure parameters and polarizability of Ba⁺ions to provide theoretical data support for relevant experiments.In this work,the energy levels and electric-dipole matrix elements are calculated by using a relativistic semiempirical method:the relativistic configuration interaction plus core polarization(RCICP)method.By using these structure parameters,static and dynamic dipole polarizabilities are further calculated.Finally,the tune-out wavelengths of the 6s_1/2state and magic wavelengths of the 6s_1/2?6p_1/2,3/2and 6s_1/2?5d_3/2,5/2transitions are also determined.By analyzing these tune-out wavelengths and magic wavelengths,We found that:1.For the tune-out wavelengths of the ground state 6s_1/2of Ba⁺ions,there is only one tune-out wavelength in the visible region,480.658(18)nm,which lies between the6s_1/2?6p_3/2and 6s_1/2?6p_3/2transitions.By analyzing the contributions of individual transitions to the dynamic polarizabilities at these tune-out wavelengths in detail,we propose that high-precision measurements on the tune-out wavelength 480.658(18)nm could be used to determine the f_6s1/2?6p3/2/f_6s1/2?6p1/2.If the measurement of tune-out wavelength is at a precision of 0.001 nm,the uncertainty of the predicted f_6s1/2?6p3/2/f_6s1/2?6p1/2should be reduced to 0.005%,which is ten times higher than the accuracy of experimental result.2.For the clock transition 6s_1/2?5d_5/2,there exists two magic wavelengths,652.8(1.0)nm and 480.78(2)nm,in the visible region.Especially,the 652.8(1.0)-nm magic wavelength is in good agreement with the latest experimental result 652.913(4)nm[Phys.Rev.A 101,042507(2020)].Moreover,the contributions to the dynamic polarizabilities at most of the magic wavelengths are dominated by only a few transitions.Therefore,it is proposed that the transition matrix elements or oscillator strengths among some excited states can be determined by using the high-accuracy measured magic wavelengths.For example,the oscillator strength of the 5d_5/2?4f_7/2transition is determined to be 0.466(15)by using the measured magic wavelength of652.913(4)nm[Phys.Rev.A 101,042507(2020)].Moreover,the measurements of the magic wavelengths for 6s_1/2?5d_3/2transition and its magnetic sublevel transitions could be used to determine the f_5d3/2?4f5/2,while the measurements of the magic wavelengths near 416 nm for the 6s_1/2?6p_3/2transition could be used to determine the f_6p3/2?6d5/2/f6p_3/2?6d_3/2.