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Results in Physics

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

Jiajia Zha, Zhihao Qin, Na Cao, Qi Wei, Pingxiao Wang

Published Aug 10, 2022• 0 citations
Publisher

Abstract

Currently, the Magnus expansion method is widely used in studying Rydberg atoms in intense laser fields. In these studies, pulse lengths are usually set as an integer multiple of wavelength, at the same time, the initial phase of the laser pulse are 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. Herein, we investigated the effects of the non-integer-cycle pulse and laser initial phase in the calculation of the transition probability of Rydberg atoms in intense laser fields. Our calculation result shows 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 in the case of ultrashort laser pulses, these effects are obvious.

DOI
10.1016/j.rinp.2022.105882

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Currently, the Magnus expansion method is widely used in studying Rydberg atoms in intense laser fields. In these studies, pulse lengths are usually set as an integer multiple of wavelength, at the same time, the initial phase of the laser pulse are 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. Herein, we investigated the effects of the non-integer-cycle pulse and laser initial phase in the calculation of the transition probability of Rydberg atoms in intense laser fields. Our calculation result shows 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 in the case of ultrashort laser pulses, these effects are obvious.
Using a method based on the Magnus expansion, the survival of Rydberg atoms in intense laser fields is studied. Usually, for atoms in intense laser fields, effects beyond the electric dipole approximation need to be considered. In the past few years, researchers have primarily paid attention to the nondipole effects induced by the electric quadrupole term of the optical field. The quadrupole term plays a more important role than the electric dipole term in the calculation of the survival probability of Rydberg atoms in intense laser fields even though the magnitude of the former is smaller than that of the latter in terms of the multipole expansion of the optical vector potential. Meanwhile, the effects of higher-order corrections beyond the quadrupole term remain ambiguous. In this paper, we confirm that corrections containing higher-order terms, except for the quadrupole term, can be neglected in the calculation of the transitions of Rydberg atoms in external fields. We then use the second-order time evolution operator containing the quadrupole term to study the scaling law of the transition probability of Rydberg He in laser fields with different parameters.
We consider the interaction of Rydberg atoms with strong infrared laser pulses using an approach based on the Magnus expansion of the time evolution operator. First-order corrections beyond the electric dipole approximation are also included in the theory. We illustrate the dynamics of the interaction at the parameters of the experiment [Eichmann et al., Phys. Rev. Lett. 110, 203002 (2013)]. It emerges that the depletion of Rydberg atoms in this regime comes predominantly from the nondipole effects.
We treat the interaction of an atom with a short intense few-cycle laser pulse by the use of the Magnus expansion of the time-evolution operator. Terms of the Magnus expansion up to the third order in the pulse duration are evaluated explicitly, and expressions for the transition probability between atomic eigenstates are given. The transition operators up to this order are particularly simple, involving only momentum and position shifts arising from the laser field alone. The connection to other existing approaches is established. For a variety of generic short-pulse forms the results of the Magnus approximation are in excellent agreement with time-dependent transition probabilities obtained from accurate ab initio numerical calculations. However, the limitation of the Magnus expansion for pulses having both vanishing momentum and position shifts is demonstrated also.
In this paper we investigate how Rydberg atoms respond to perturbation by two contrapropagating ultrashort laser pulses. We consider the case where the durations of both pulses ${\ensuremath{\tau}}_{1}$ and ${\ensuremath{\tau}}_{2}$ are shorter than the inverse of the initial-state energy ${\ensuremath{\epsilon}}_{i}^{\ensuremath{-}1}$. When acting alone such a pulse passes through the atom without noticeable alteration in the atomic state. The situation is different if two such pulses interfere in the region of atom localization. In this case the atomic response is significantly enhanced. This is due to the nonzero momentum transferred to the electron by the interplay of the electric field of one pulse and the magnetic field of the other. The sudden perturbation approximation is used to evaluate the transition probabilities. They are shown to depend on the atom position with respect to the pulse interference region. This dependence is determined by the relationship between the atomic diameter di{d}_{i}di​ and the interference-region size $l=c({\ensuremath{\tau}}_{1}+{\ensuremath{\tau}}_{2})$ (ccc is the speed of light). If ${d}_{i}\ensuremath{\ll}l$ this dependence is sensitive to the function form of the pulses. For sufficiently strong fields the atoms can be ionized completely. In the opposite case of ${d}_{i}\ensuremath{\gg}l$ the transition probabilities are sensitive to the electron density distribution along the propagation direction. The probabilities of the initial-state destruction and atom ionization drop as l∕dil∕{d}_{i}l∕di​ irrespective of the characteristics of the pulses.
We study the ionization of Rydberg atoms by half-cycle electromagnetic pulses as a function of the duration of the pulses covering a broad range of time intervals connecting the adiabatic and sudden regimes. The threshold fields for ionization lie on a universal classical scaling invariant curve. We present quantum mechanical results which agree well with the universal curve. The n -2 threshold field scaling found experimentally is a linear approximation to the universal curve by the tangent in the region where the pulse duration is of the order of the orbital period of the atom. We predict the existence of an ultrashort pulse limit in which the threshold field scales as n -1 . Effects due to the non-hydrogenic core are negligible.
The absolute laser phase dependence of the time-dependent populations of the molecular states, including the steady-state (long time) populations of the states, associated with the interaction of a molecule with a pulsed laser is investigated using illustrative two-level examples. One-photon transitions, including the effects of permanent dipoles, are discussed as a function of the pulse duration, intensity, and (absolute) laser phase, for selected laser frequencies. The effects of laser phase can be large, depending on the values of the pulse duration for a given frequency and intensity. The effects of permanent dipoles, relative to no permanent dipoles, are significant for large laser field strengths ε0. When the laser-molecule coupling parameter b=μ12ε0/E21⩾0.2, where μ12 and E21 are the transition dipole and energy difference between the ground and excited states, respectively, the dynamics of the pulse-molecule interaction are (strongly) phase dependent, independent of pulse duration, whereas the corresponding steady-state populations of the molecular states may or may not be phase-dependent depending on the pulse duration. Analytical rotating wave approximations for pulsed laser-molecule interactions are useful for interpreting the dynamics and the steady-state results as a function of field strength and pulse duration, including the effects of permanent dipole moments. The results reported in this paper are based on molecular parameters associated with an S0→S1 electronic transition in a dipolar molecule. However, they are presented in reduced form and therefore can be scaled to other regions of the electromagnetic spectrum. Short, intense pulses at or beyond the limits of current laser technology will often be required for the types of absolute laser phase effects of this paper to be appreciable for electronic excitations. The discussion, in the UV-VIS, also suffers from the use of a two-level model and from the requirement of field intensities that can be beyond the Keldysh limit. For other spectral regions, these absolute laser phase effects will be much more readily applicable.