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arXiv:quant-ph/0306157 · Physical Review A

Complete population transfer in a degenerate three-state atom

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

Published Feb 19, 2004• 3 citations
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Abstract

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.

DOI
10.1103/physreva.69.023405

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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 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.
A simple theoretical solution to the design of a control field that generates complete population transfer from an initial state, via NNN nondegenerate intermediate states, to one arbitrary member of $M\phantom{\rule{0.3em}{0ex}}(M\ensuremath{\leqslant}N)$ degenerate states is constructed. The full control field exploits an $(M+N\ensuremath{-}1)$-node null adiabatic state, created by designing the relative phases and amplitudes of the component fields that together make up the full field. The solution found is universal in the sense that it does not depend on the exact number of the unwanted degenerate states or their properties. The results obtained suggest that a class of multilevel quantum systems with degenerate states can be completely controllable, even under extremely strong constraints, e.g., never populating a Hilbert subspace that is only a few dimensions smaller than the whole Hilbert space.
Ongoing work aimed at developing highly efficient methods of populating a chosen sublevel of an energy level highlights the need to understand off-resonant effects in coherent excitation. This motivated us to re-examine some aspects of the theory of coherent excitation in a three-state system with a view to obtaining algebraic expressions for off-resonant eigenvalues and eigenvectors. Earlier work gives simple closed-form expressions for the eigenvalues this system, expressions applying even when the system is not on two-photon resonance. We present here expressions of similar simplicity for the components of the normalised eigenvectors. The analytic properties of these components explain the observed sensitivity of the stimulated-Raman-adiabatic-passage process (STIRAP) to the condition of two-photon resonance. None of the eigenstates is ‘trapped’ or ‘dark’ unless the system is on two-photon resonance; off resonance, all states have nonzero projections on the unperturbed intermediate state. A simple argument shows that no dressed state can be adiabatically connected to both the unperturbed initial and final states when the system is off two-photon resonance. That is, adiabatic transfer from initial to final state requires that these be degenerate before and after the STIRAP pulse sequence, and this implies zero two-photon detuning. However, substantial transfer probabilities are observed experimentally for very small two-photon detunings. We show that such systems are characterised by very sharp avoided crossings of two eigenvalues, and that the observed population transfer can be understood as an effect of non adiabatic transitions occurring at the avoided crossings.
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.