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Physical Review A

Landau-Zener-Stueckelberg theory for multiphoton intrashell transitions in Rydberg atoms: Bloch-Siegert shifts and widths

Morten Førre

Published Jul 13, 2004• 18 citations
Publisher

Abstract

We derive closed analytic expressions for intrashell transitions in Rydberg atoms exposed to linearly polarized or circularly polarized periodic electromagnetic fields. The resonance energies and transition probabilities are calculated using multichannel Landau-Zener-Stueckelberg theory. The theory provides formulas for the resonance widths and positions for arbitrary field strength and frequency. The formulas are in excellent agreement with numerical solution of the evolution equations.

DOI
10.1103/physreva.70.013406

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We derive closed analytic expressions for intrashell transitions in Rydberg atoms exposed to linearly polarized or circularly polarized periodic electromagnetic fields. The resonance energies and transition probabilities are calculated using multichannel Landau-Zener-Stueckelberg theory. The theory provides formulas for the resonance widths and positions for arbitrary field strength and frequency. The formulas are in excellent agreement with numerical solution of the evolution equations.
We measure for the first time, to our knowledge, NNN-photon intrashell resonances, widths, and Bloch-Siegert shifts up to N=23N=23N=23. Rydberg atoms of Li (n=25)(n=25)(n=25) in a rotating electric field and a static, in-plane electric or magnetic field are used. The measurements, in conjunction with theoretical estimates, precisely determine the characteristics of the resonances (shifts and widths) and their dependence on the strength of the rotating field.
Bloch-Siegert shifts and widths of resonances in two-level systems are evaluated within an adiabatic approximation applicable in the multiphoton regime. The level splitting is controlled by a static field. Most attention is paid to the case of a circular driving field with the static field in the plane of rotation. This nonconventional setup was used in a recent experiment on intrashell resonances in Rydberg atoms where shifts and widths for NNN-photon resonance with NNN as large as 23 were observed [D. Fregenal et al., Phys. Rev. A 69, 031401(R) (2004)]. For this in-plane configuration the resonance widths are drastically enhanced compared to their values in the conventional configuration (linearly polarized driving field perpendicular to static field), whereas the shifts are only weakly affected.
The Schrodinger equation for the interaction of a decaying two-level atom with an external oscillating field is separated via a series of matrix transformations into a time independent and a time dependent part; the former yields the radiofrequency Stark shift for decaying states, the latter is normally negligible for atomic beam resonance experiments. In the limit of stable states, or for the case of two levels of equal lifetime, the shift reduces to that of the Bloch-Siegert effect. Exact analytical expressions for the state amplitudes are obtained for the case when the time dependent matrix can be neglected; the resulting transition probabilities are investigated as functions of frequency, time, and interaction strength and are compared with the rotating field theory and the exact numerical solutions.
Multiphoton intrashell transitions in strongly driven Li (n=25n=25n=25) Rydberg atoms are studied experimentally. Orthogonal dc electric and magnetic fields lift the degeneracy of the nnn shell and define the eccentricity eee of the initial coherent elliptic states, which are formed by laser excitation and subsequent adiabatic transformation. The intrashell transitions are driven by a time-harmonic electric field linearly polarized parallel to the major axis of the ellipse. NNN-photon resonances with $N=1\ensuremath{-}9$ are studied as a function of eee. All resonances with $N\ensuremath{\geqslant}3$ are suppressed at certain eee values in between 000 and 111. A similar system was analyzed by Yabuzaki et al. [Phys. Rev. A 10, 1955 (1974)] who found a simple pattern of suppressions that applies also for the present experiments. The results of these experimentally confirm that each time NNN is increased by two, an additional suppression is observed.
The Bloch-Siegert-type shifts and multiple-quantum resonance shifts of a spin-one system subjected to an oscillating radio-frequency applied at right angles to a DC field have been calculated to sixth order in V, the amplitude of the oscillating field, for the single-quantum resonance, and to fourth order in V for the higher resonances. The level crossing criterion appropriate to two-level systems cannot be used here, and a resonance condition based on a previously derived analogue of Shirley's equation for spin-one systems is used instead. The calculation uses degenerate continued-fraction perturbation theory. Numerical calculations were performed in order to obtain the shape of the resonances, and partially check the analytic results. The atomic levels were assumed to be completely stable; the corrections due to finite atomic lifetimes are expected to be small.