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Physical Review A

Coherent dynamics and complete population depletion of a special three-level quantum system

Ning Lü, Edward J. Robinson, P. R. Berman

Published Jun 1, 1987• 13 citations
Publisher

Abstract

By constructing a vector model we study the coherent dynamics of a special three-level quantum system--one with equal detunings and equal Rabi frequencies. This three-level problem can be fully related to a two-level problem. We find that complete population depletion can be achieved even in the case of constant nonzero detunings when the three-level system is prepared initially in the common level of the two transitions. Complete population depletion with constant detunings can also occur in other odd-number multilevel systems. An explanation in terms of an equivalent resonant excitation is presented.

DOI
10.1103/physreva.35.5088

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By constructing a vector model we study the coherent dynamics of a special three-level quantum system--one with equal detunings and equal Rabi frequencies. This three-level problem can be fully related to a two-level problem. We find that complete population depletion can be achieved even in the case of constant nonzero detunings when the three-level system is prepared initially in the common level of the two transitions. Complete population depletion with constant detunings can also occur in other odd-number multilevel systems. An explanation in terms of an equivalent resonant excitation is presented.
It is shown that any pulse with a time-asymmetric envelope and a constant non-zero detuning excites a two-level quantum system with a probability of nearly one-half provided the pulse is very asymmetric, adiabatic in the slowly rising part of its envelope, sudden in the steep part and the peak Rabi frequency is much greater than the detuning. Three particular cases are considered analytically: the asymmetric hyperbolic-secant, the Bambini-Berman and the exponential pulse, which illustrate the general conclusions. The results are applied to a special three-level 'vee' system whose two transitions are driven by two asymmetric laser pulses of the same Rabi frequencies and the same detunings. Analysis shows that if such a system is prepared initially in its ground state, effective population depletion of this state can be achieved, the results being quite insensitive to variations in the Rabi frequency and the detuning.
We study the effects of statistical broadening and population loss on the dynamics of strongly excited three-level quantum systems. The results of both analytic and numerical treatments of the three-level Schr\"odinger equation are displayed. In order to allow for statistical broadening, solutions are required for arbitrary detuning of the two very intense monochromatic lasers. The most efficient statistically averaged population depletion occurs when the Rabi frequency of the second transition is greater than that of the first.
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.
Previously published papers have noted the existence of two-level periodicities in idealized theoretical models of monochromatic excitation of multilevel systems. The present paper provides a theoretical derivation of such two-level dynamics from the more general NNN-level rotating-wave-approximation Schr\"odinger equation and applies the resulting formulas to different extremes of excitation in a nondegenerate system of sequential (ladderlike) excitation: first, when intermediate levels have much larger cumulative detunings than Rabi frequencies, and second, when intermediate Rabi frequencies are much larger than detunings. In each case one has simple analytic expressions for the effective two-level Rabi frequencies and detunings.
The population of a three-level system in the lambda configuration interacting with three classical fields is studied. The system is essentially a Raman coupled system with pump, Stokes and anti-Stokes fields. One channel between the two ground states involves the pump and Stokes field while the other involves the pump and anti-Stokes field. Since exact solutions are not generally available the author studies the dynamics numerically. But when the excited state is highly detuned, it may be adiabatically removed so that the system becomes effectively a two-level system but with an effective Rabi frequency that depends on the relative phases of the driving fields. For some choices of the phases this frequency can be made to vanish, an effect of the interference between the two channels. For the special case of equal and opposite detunings, an exact solution of the full three-level equations is presented. The effects of ionization of the excited state are also briefly considered.
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.