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arXiv:quant-ph/0402150 · Physical Review A

General method for complete population transfer in degenerate systems

Jiangbin Gong, Stuart A. Rice

Published Jun 15, 2004• 23 citations

Abstract

A simple theoretical solution to the design of a control field that generates complete population transfer from an initial state, via $N$ 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.

DOI: 10.1103/physreva.69.063410

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A simple theoretical solution to the design of a control field that generates complete population transfer from an initial state, via

N

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.

We present an analytic solution to the "degenerate quantum control problem," which enables the transfer of any desired fraction of population between arbitrary initial and final pure wave packets, made up of nearly degenerate energy eigenstates. It consists of two two-photon adiabatic passage steps, in which the population of the initial wave packet is first transferred, via a number of nondegenerate intermediate states, to a single eigenstate and then returned to a different target superposition state. We demonstrate the approach by executing a stepwise isomerization of three Jahn-Teller states of the Al3O molecule, where a proper optical coupling can easily be established.

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

n

-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

n

-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

n^{th}

order equation to determine eigenvalues of dressed states, the method is entirely analytic. These expressions may be used to control electron populations in degenerate

n

-state atoms. Examples are given for

n=2

and

n=3

.

A five-level four-pulse phase-sensitive extended stimulated Raman adiabatic passage scheme is proposed to realize complete control of the population transfer branching ratio between two degenerate target states. The control is achieved via a three-node null eigenstate that can be correlated with an arbitrary superposition of the target states. Our results suggest that complete suppression of the yield of one of two degenerate product states, and therefore absolute selectivity in photochemistry, is achievable and predictable, even without studying the properties of the unwanted product state beforehand.

A five-level four-pulse phase-sensitive extended stimulated Raman adiabatic passage scheme is proposed to realize complete control of the population transfer branching ratio between two degenerate target states. The control is achieved via a three-node null eigenstate that can be correlated with an arbitrary superposition of the target states. Our results suggest that complete suppression of the yield of one of two degenerate product states, and therefore absolute selectivity in photochemistry, is achievable and predictable, even without studying the properties of the unwanted product state beforehand.

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.