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Physical Review A

Geometrical representation of coherent-excitation methods using delayed and detuned lasers

J. B. Kim, J. Lee, A. S. Choe, Y. J. Rhee

Published May 1, 1997• 11 citations
Publisher

Abstract

A precessing vector model has been developed for the geometrical interpretation of the adiabatic following and the optimal detuning methods, which play a decisive role in coherent population transfer in a three-level system. In our model, the Bloch vector defined by a population inversion and two-photon coherence components precesses about the torque vector consisting of an effective Rabi frequency and detuning components. The Bloch vector traces out different routes depending on combinations of the time delay of two laser pulses and detunings from the resonant frequencies. For the extreme cases where the optimal detuning with zero delay and the optimal delay with zero detunings are taken, our vector model with analytical solutions is compared with numerical solutions.

DOI
10.1103/physreva.55.3819

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A precessing vector model has been developed for the geometrical interpretation of the adiabatic following and the optimal detuning methods, which play a decisive role in coherent population transfer in a three-level system. In our model, the Bloch vector defined by a population inversion and two-photon coherence components precesses about the torque vector consisting of an effective Rabi frequency and detuning components. The Bloch vector traces out different routes depending on combinations of the time delay of two laser pulses and detunings from the resonant frequencies. For the extreme cases where the optimal detuning with zero delay and the optimal delay with zero detunings are taken, our vector model with analytical solutions is compared with numerical solutions.
We show that optimal detuning method describing population transfer passage in a three‐level system has an essential difference from the adiabatic following passage by counterintuititive sequence of two coherent laser pulses, by examining the dressed eigenstates and the Bloch vector precessing around a torque vector. In a four‐level Λ system, population dynamics is studied for the case where, on the optimal detuning condition, two‐photon coherences causing adiabtic transfer interfere each other. For a scheme of three degenerate levels, we show a possibility of complete population inversion by the optimal detuning method.
Highly selective excitations, which occur in a three-level system under the conditions of zero- and nonzero-delayed lasers, are examined. The detuning ranges which reach half inversion and their dependences on the laser conditions are obtained numerically using the Bloch equation. It is found that in the optimal detuning method which uses the zero-delayed lasers, the range is independent of the Rabi frequency and the detuning, but in the adiabatic passage which uses the nonzero-delayed lasers, the range is dependent on the Rabi frequency and the time delay. The inversion methods show strong dependences on the pulse length. It turns out that the optimal detuning method which has the narrow range is more appropriate to obtain highly selective excitation than the adiabatic passage. The mechanisms of population inversion and the decay effects between the methods are compared.
The problem of achieving population inversion adiabatically in an NNN-level system using one or more laser fields whose detunings and/or amplitudes are continuously varied is studied analytically and numerically. The SU(N)\mathrm{SU}(N)SU(N) coherence vector picture is shown to suggest unexpected inversion procedures and also to give a generalized interpretation of adiabatic following. It is shown that the (${N}^{2}\ensuremath{-}1$)-dimensional SU(N)\mathrm{SU}(N)SU(N) space contains an ($N\ensuremath{-}1$)--dimensional steady-state subspace $\mathit{\ensuremath{\Gamma}}(t)$ whose orthonormal basis vectors ${\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Gamma}}}_{1},\dots{}, {\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Gamma}}}_{N\ensuremath{-}1}$ are given explicitly in terms of the Hamiltonian matrix elements. The motion of the system can be interpreted as a "generalized precession" of $\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}$ about $\mathit{\ensuremath{\Gamma}}$. Multilevel adiabatic following occurs when the angle $\ensuremath{\chi}(t)$ between the coherence vector $\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}$ and its projection onto $\mathit{\ensuremath{\Gamma}}$ is very small. The multiple dimension of $\mathit{\ensuremath{\Gamma}}$ is shown to provide a variety of paths for adiabatic inversion. The adiabatic solution is obtained by solving $N\ensuremath{-}1$ simple equations for the directional cosines of $\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}$ on ${\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Gamma}}}_{i}$. The adiabatic solution and time scale and the state taken up by the atomic variable are discussed analytically and numerically for a three-level system.
For the purpose of an examination of coherence control, we study the dynamics of population inversion in the four-level λ system where initial populations are arbitrary coherently distributed in two lower levels. It is shown that the coherence control, which causes population transfer between the lower levels to be forbidden, is possible by using the ratio of Rabi frequencies. Under this condition, it is remarkable that the adiabatic following and the optimal detuning methods are useful to invert the four-level system completely, and it is shown that the two-photon coherences of the coupled three-level ladders are dispersive and absorptive in the former and the latter methods, respectively. From the geometrical interpretation, the Bloch vectors, describing the four-level inversions, could be defined as ones in a two-level system.
In a three-level atom interacting with nondelayed laser pulses, the inversion dynamics is studied to examine the detunings of the laser frequencies. When two Rabi frequencies are unequal, we show the interesting fact that complete inversion occurs if chirping is used, which causes the effective detuning of a two-photon resonance line to be zero. Because fixed optimal detunings allow an easier experimental approach, it is useful to find a method for obtaining the optimal detunings for realistic ratios of two Rabi frequencies. Using these optimal detunings, the process turns out to be a diadiabatic transition from investigation of the eigenvector and Bloch vector motions, whose behaviours are compared with their behaviour in the case of complete inversion.
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.