Effects of the noninteger-cycle pulse and the initial phase on the Magnus approximation to study Rydberg atoms in intense laser fields

The Magnus expansion method is popular for the study of Rydberg atoms in intense laser fields. In such studies, the pulse length is usually set as an integer multiple of wavelength, at the same time, the initial phase of the laser pulse is usually set as zero, i.e., without considering the effect of the initial phase in the calculation. However, this will result in some items multiplied by zero. Then, contributions from these items will be lost. Here, we investigated the effects of the noninteger-cycle pulse and initial phase of the laser on the calculated transition probability of Rydberg atoms in intense laser fields. Our calculation results show that these effects are trivial when the nondipole approximation is adopted. In dipole approximation i.e., when the wavelength of the laser field exceeds the characteristic size of atomic or molecular orbitals, these effects are trivial only in the case of a long pulse duration and a large principal quantum number, however for ultrashort laser pulses (pulse duration that lasts less than five light cycles), these effects are evident.