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arXiv:quant-ph/0306022

Complete population transfer in degenerate n-state atoms

J. H. McGuire, Kh. Kh. Shakov, Kh. Yu. Rakhimov

Published Jun 3, 2003• 1 citation
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Abstract

We find a set of conditions 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 n-state atom, where the transitions are caused by an external interaction.
DOI
10.48550/arxiv.quant-ph/0306022

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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 nnn-state atom, where transitions are caused by an external interaction. In systems with degenerate states there is no time ordering. Analytic expressions have been found for transition probabilities in a degenerate nnn-state atom interacting with a strong external field that gives a common time dependence to all of the transition matrix elements. Except for solving a simple nthn^{th}nth order equation to determine eigenvalues of dressed states, the method is entirely analytic. These expressions may be used to control electron populations in degenerate nnn-state atoms. Examples are given for n=2n=2n=2 and n=3n=3n=3.
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.
We show that an arbitrary system described by two dipole moments exhibits coherent superpositions of internal states that can be completely decoupled from the dissipative interactions (responsible for decoherence) and an external driving laser field. These superpositions, known as dark or trapping states, can be completely stable or can coherently interact with the remaining states. We examine the master equation describing the dissipative evolution of the system and identify conditions for population trapping and also classify processes that can transfer the population to these undriven and nondecaying states. It is shown that coherent transfers are possible only if the two systems are nonidentical, that is the transitions have different frequencies and/or decay rates. In particular, we find that the trapping conditions can involve both coherent and dissipative interactions, and depending on the energy level structure of the system, the population can be trapped in a linear superposition of two or more bare states, a dressed state corresponding to an eigenstate of the system plus external fields or, in some cases, in one of the excited states of the system. A comprehensive analysis is presented of the different processes that are responsible for population trapping, and we illustrate these ideas with three examples of two coupled systems: single V- and $\ensuremath{\Lambda}$-type three-level atoms and two nonidentical two-level atoms, which are known to exhibit dark states. We show that the effect of population trapping does not necessarily require decoupling of the antisymmetric superposition from the dissipative interactions. We also find that the vacuum-induced coherent coupling between the systems could be easily observed in $\ensuremath{\Lambda}$-type atoms. Our analysis of the population trapping in two nonidentical atoms shows that the atoms can be driven into a maximally entangled state which is completely decoupled from the dissipative interaction.