Complete population transfer in a degenerate three-state atom

Abstract

We find conditions required to achieve complete population transfer, via coherent population trapping, from an initial state to a designated final state at a designated time in a degenerate 3-level atom, where transitions are caused by an external interaction. Complete population transfer from an initially occupied state 1 to a designated state 2 occurs under two conditions. First, there is a constraint on the ratios of the transition matrix elements of the external interaction. Second, there is a constraint on the action integral over the interaction, or "area", corresponding to the phase shift induced by the external interaction. Both conditions may be expressed in terms of simple odd integers.

Analytic description of population transfer in a degeneraten-level atom

“…This yields P 2 = |a 2 | 2 = sin 2 (χ/ω sin(ωt)) and P 1 = 1 − P 2 . As noted previously [14,15], an interesting solution occurs when the action, A = χ/ω sin(ωt), is an odd integer multiple of π/2. Then the Probability versus time for transfer of an electron from the 2s to the 2p 0 level of atomic hydrogen in the limit that both energy levels are the same.…”

“…If ω 12 = E 2 − E 1 is finite, then a numerical calculation is required to solve equations (1), even for n = 2. The difference between degenerate (analytic) and non-degenerate (numerical) solutions can be estimated [14].…”

The results of an accurate analysis of EAO charts of the Moon marginal zone constructed on the basis of lunar occultations

“…Let Ü Ý be measured coordinates of photographic images of the points of the Moon limb, it follows that the standard coordinates are (Rakhimov, 1993)…”

“…of topocentric values of the optical libration with regard to the influence of physical libration from the tables Migus (1980) Charts EAO of the Moon marginal zone were constructed in graphical form by À È coordinates of more than 40000 points (Rakhimov, 1993). The lines of equal heights have been drawn on maps in 0 ¼ ¼ 2 intervals.…”

“…The Watts' maps (1963) corrected and edited by Morrison (1979), Morrison and Appleby (1981) are considered to be the most exact. The charts of the Moon marginal zone were constructed in a radical new way at EAO (Rakhimov, 1993). They are based on the large-scale star calibrated lunar photographs by Habibullin et al (1974).…”