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Journal of Physics B Atomic Molecular and Optical Physics

Complete population return in a two-state system driven by a smooth asymmetric pulse

Nikolay V. Vitanov

Published Jan 28, 1995• 17 citations
Publisher

Abstract

A numerical study of the conditions for complete population return in a two-state system is presented. For the first time, a pulse with an infinitely smooth asymmetric envelope and a constant detuning is found which in principle can restore the system to its initial state. Several approximate population return values of the pulse parameters are given.

DOI
10.1088/0953-4075/28/2/002

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A numerical study of the conditions for complete population return in a two-state system is presented. For the first time, a pulse with an infinitely smooth asymmetric envelope and a constant detuning is found which in principle can restore the system to its initial state. Several approximate population return values of the pulse parameters are given.
In the concluding paragraph of a recent paper on two-state atoms driven by transients, Vitanov (1994) raised two questions. First, he queried whether or not, at non-zero detuning, there exist asymmetric pulses of arbitrary smoothness that can restore the system to its initial level. Second, he asked whether one could find symmetric pulses that fail to restore the system to its initial state, regardless of pulse area. In this letter, we respond to these speculations. We show that pulses with shapes that are hyperbolic secants modified by powers of the hyperbolic tangent can arbitrarily approach the restoring condition, while the existence of a symmetric pulse incapable of restoration implies unitarity violation. We also draw a distinction between the infinitely large set of restoring pulse areas that exist for symmetric pulses, as compared to the small number (typically no more than one) that can be found in the asymmetric case.
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.
A theoretical nonperturbative study of the coherent excitation of a two-state system by N consecutive equally spaced identical pulses is presented. General relations between the evolution matrix elements in the cases of one and N pulses are obtained in a closed form. For pulse envelopes allowing analytical solutions, these relations enable analytical treatment of pulse-train excitation; for pulses that can only be treated numerically, they shorten the computations by a factor of N. The relations show that the multiple-pulse excitation of a two-state system can be considered as a quantum analog of the diffraction grating. The interaction dynamics substantially depends on the way the train is produced because the phase shift that is accumulated by the probability amplitudes due to the free evolution of the system during and between the pulses differs. In the limit of weak excitation, the results recover earlier conclusions based on perturbative treatments. Simple formulas are derived for the conditions for complete population inversion (CPI) and complete population return (CPR), which differ from the single-pulse ones. These general results are applied to four particular cases that allow analytical solutions: resonant, rectangular, Rosen-Zener, and Allen-Eberly pulses. A common feature in all these cases is that the number and the amplitude of oscillations in the state populations increase with the number of pulses, while their width decreases. For rectangular pulses, CPI is possible below a given value of the ratio between the detuning and the pulse area; this value increases nearly linearly with the number of pulses.For the Rosen-Zener model, CPI is found to be possible for two and more pulses, while it is impossible for a single non-resonant pulse. It is shown that for a given number of pulses there is an upper limit on the detuning for which CPI can be observed; this limit increases logarithmically with the number of pulses. For the Allen-Eberly model, CPR is established to be possible for more than one pulse, while it is impossible for a single pulse. The cases of even and odd number of pulses are shown to lead to substantially different results. The limitations imposed on the detuning by the conditions for CPR and CPI are derived and discussed.
This paper presents an analytic study of the properties of a three-state system coupled to two external laser fields on two-photon resonance. The intermediate state is generally off single-photon resonance and can decay irreversibly out of the system. The two fields are assumed to have the same pulse-shaped time dependence but may have different strengths. Various general properties for arbitrary pulse shapes, detunings and decay rates are derived and illustrated with an exactly solvable analytic model. They include the conditions for complete population transfer and depletion, complete population loss or no loss, various limitations on the values of the populations, the effect of overdamping at large decay rates and others.