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Physical Review A

Minimum area for complete inversion of a three-state atom

C. E. Carroll

Published Feb 1, 1989• 6 citations
Publisher

Abstract

Using a simple three-state model for an atom or molecule in which two transitions are driven by two laser beams, we demand complete transfer of the occupation probability from state 1 to state 2 to state 3. This complete transfer is produced by two laser pulses, which may be concurrent. The minimum area required for complete transfer is considered, although ``area'' has no obvious definition for the three-state model. If the areas of the two separate pulses are calculated following McCall and Hahn, it seems that the minimum area for each pulse is \ensuremath{\pi}.

DOI
10.1103/physreva.39.2243

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Using a simple three-state model for an atom or molecule in which two transitions are driven by two laser beams, we demand complete transfer of the occupation probability from state 1 to state 2 to state 3. This complete transfer is produced by two laser pulses, which may be concurrent. The minimum area required for complete transfer is considered, although ``area'' has no obvious definition for the three-state model. If the areas of the two separate pulses are calculated following McCall and Hahn, it seems that the minimum area for each pulse is \ensuremath{\pi}.
We use a three-state model for an atom or molecule in which two transitions are simultaneously driven by the oscillating electric fields of two laser beams. The amplitudes and detunings of the applied oscillating fields vary during the optical pulse, which can cause transitions from one state to another. For some special cases not previously known, the transition probabilities and probabilities of no transition are obtained analytically. We give conditions for complete transfer of the atomic population from one state to another, and for complete return to the initial state.
A simple model with only three quantum states is used for an atom or molecule in which two transitions are driven by two overlapping laser pulses. The amplitudes of the two pulses are not proportional to the same function of time. The authors impose the 'two-photon resonance' condition, but the two pulses separately are not applied at the resonant frequencies. For pulse pairs having certain shapes, the wavefunctions are written in terms of generalised hypergeometric functions. Simple formulae for transition probabilities are then obtained, are are used to discuss transfer of the three-state system from its ground state to its highest state.
For an atom or molecule in which two transitions are driven by laser beams, a three-state model is used. Transitions among the three states are caused only by the oscillating electric fields of the laser beams. The amplitudes and detunings of the two laser beams, which are various functions of the time, appear in the Schrödinger equation for the atom or molecule. In certain cases, the Schrödinger equation can be solved analytically, to find transition probabilities and the probability of no transition. This is done by using Clausen’s special function, or by assuming that the sum of the two detunings is zero at all times. Conditions for complete transfer of the population from the ground state to an excited state are obtained from the analytic solutions.
Transfer of molecules or atoms from an initial state to a certain final state is often wanted, and use of a single laser may be impractical. Transfer via an intermediate state of higher energy has been demonstrated, using laser pulses that overlap in time. We have proposed using the continuum in place of the intermediate state, and encouraging results have been computed for a simple model. Our latest calculations show the effects of operating this scheme near the bottom of the continuum or near a resonance having the Fano shape, and the effects of laser-driven continuum-continuum transitions.
We have found 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 three-state atom, where transitions are caused by an external interaction. Complete population transfer from an initially occupied state one to a designated state two 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. Some specific examples are discussed.