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arXiv:1209.6568 · The European Physical Journal Plus

Beyond adiabatic elimination: A hierarchy of approximations for multi-photon processes

Vanessa Paulisch, Han Rui, Hui Khoon Ng, Berthold‐Georg Englert

Published Jan 1, 2014• 40 citations
arXivHTMLPDF
DOI
10.1140/epjp/i2014-14012-8

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In multi-level systems, the commonly used adiabatic elimination is a method for approximating the dynamics of the system by eliminating irrelevant, non-resonantly coupled levels. This procedure is, however, somewhat ambiguous and it is not clear how to improve on it systematically. We use an integro-differential equation for the probability amplitudes of the levels of interest, which is equivalent to the original Schrodinger equation for all probability amplitudes. In conjunction with a Markov approximation, the integro-differential equation is then used to generate a hierarchy of approximations, in which the zeroth order is the adiabatic-elimination approximation. It works well with a proper choice of interaction picture; the procedure suggests criteria for optimizing this choice. The first-order approximation in the hierarchy provides significant improvements over standard adiabatic elimination, without much increase in complexity, and is furthermore not so sensitive to the choice of interaction picture. We illustrate these points with several examples.
Driven Raman processes — nearly resonant two-photon transitions through an intermediate state that is non-resonantly coupled and does not acquire a sizeable population — are commonly treated with a simplified description in which the intermediate state is removed by adiabatic elimination. While the adiabatic-elimination approximation is reliable when the detuning of the intermediate state is quite large, it cannot be trusted in other situations, and it does not allow one to estimate the population in the eliminated state. We introduce an alternative method that keeps all states in the description, without increasing the complexity by much. An integro-differential equation of Lippmann-Schwinger type generates a hierarchy of approximations, but very accurate results are already obtained in the lowest order.
Driven Raman processes – nearly resonant two-photon transitions through an intermediate state that is non-resonantly coupled and does not acquire a sizeable population – are commonly treated with a simplified description in which the intermediate state is removed by adiabatic elimination. While the adiabatic-elimination approximation is reliable when the detuning of the intermediate state is quite large, it cannot be trusted in other situations, and it does not allow one to estimate the population in the eliminated state. We introduce an alternative method that keeps all states in the description, without increasing the complexity by much. An integro-differential equation of Lippmann–Schwinger type generates a hierarchy of approximations, but very accurate results are already obtained in the lowest order.