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Physical Review A

Nonadiabatic transitions in a two-level quantum system: Pulse-shape dependence of the transition probability for a two-level atom driven by a pulsed radiation field

P. R. Berman, Lixin Yan, Keng-Hwee Chiam, Ruwang Sung

Published Jan 1, 1998• 28 citations
Publisher

Abstract

The problem of a two-level atom interacting with a radiation pulse is studied in the limit that the atom-field detuning times the pulse duration is much greater than unity. Owing to the large atom-field detuning, transitions result from nonadiabatic coupling of the states by the field. The transition probability for the atom to be excited following the pulse is studied as a function of field strength for five different pulse shapes: hyperbolic secant, Lorentzian, hyperbolic secant squared, Lorentzian squared, and Gaussian. It is shown that the behavior of the transition probability differs qualitatively for these pulses. An explanation of this qualitative difference is given in terms of the Massey parameter. Numerical solutions are compared with asymptotic solutions and several anomalies are noted. In the limit of large field strength, a universal expression for the transition probability is found. An interesting feature of the solutions is that, in the limit of very large field strengths, the transition probability for a Gaussian pulse can approach unity despite the fact that the pulse has an exponentially small Fourier amplitude at the atom-field detuning. This apparent violation of the energy-time uncertainty principle is explained in terms of the nonlinear atom-field interactions.

DOI
10.1103/physreva.57.79

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The problem of a two-level atom interacting with a radiation pulse is studied in the limit that the atom-field detuning times the pulse duration is much greater than unity. Owing to the large atom-field detuning, transitions result from nonadiabatic coupling of the states by the field. The transition probability for the atom to be excited following the pulse is studied as a function of field strength for five different pulse shapes: hyperbolic secant, Lorentzian, hyperbolic secant squared, Lorentzian squared, and Gaussian. It is shown that the behavior of the transition probability differs qualitatively for these pulses. An explanation of this qualitative difference is given in terms of the Massey parameter. Numerical solutions are compared with asymptotic solutions and several anomalies are noted. In the limit of large field strength, a universal expression for the transition probability is found. An interesting feature of the solutions is that, in the limit of very large field strengths, the transition probability for a Gaussian pulse can approach unity despite the fact that the pulse has an exponentially small Fourier amplitude at the atom-field detuning. This apparent violation of the energy-time uncertainty principle is explained in terms of the nonlinear atom-field interactions.
We present an experimental study of the dynamics of a two-level system driven by strong nonresonant electromagnetic pulses as a function of pulse intensity and detuning. We have explored the qualitative and quantitative behavior of the transition probability as a function of pulse area for five different temporal profiles: Lorentzian, Lorentzian squared, hyperbolic secant, hyperbolic secant squared, and Gaussian. The two-level system consists of a fine-structure doublet in sodium Rydberg states coupled by Raman transitions driven through far-off-resonance intermediate states. The pulses are in the microwave regime and have high fidelity and uniform intensity. Experiments show that, despite the similarity in the pulse shapes, the behavior of the population transfer versus intensity depends dramatically on the temporal shape and that the spectral properties and area of the pulse do not adequately describe the response.
Excitation of an atomic system subjected to a slowly varying external electromagnetic field is studied using a two-level model. Time evolution of the system is found using an approach which is nonperturbative in the field strength. There is no constraint to small values of the applied field, that is, the field (in appropriate energy units) need not be small compared to the difference in energies of the two levels. Rather, we use the fact that the situation of interest to us is where the frequency of the exciting field is small compared to the frequency associated with the level difference. Transition probabilities and resonance conditions are found which circumscribe both the large- and small-field limits. In the weak-field limit the previous results of high-order perturbation theory are readily recovered. For a monochromatic field the characteristic features of resonance excitation at high harmonic number of the applied field are (a) extremely narrow resonance widths and (b) shifts in resonance positions which are strong functions of field intensity. Because of this sensitivity, we are able to demonstrate that when slow temporal evolution of the field amplitude is taken into account (e.g., due to finite pulse duration) the appropriate mean excitation rate is that due to the uncorrelated contribution of many resonances. The results of this analysis are used to estimate excitation rates in a specific atomic system, Cd12+{\mathrm{Cd}}^{12+}Cd12+, which are then compared to multiphoton-ionization rates. Our calculations suggest that the ionization rate exceeds the excitation rate by several orders of magnitude.
The problem of a two-level atom interacting with a nonmonochromatic radiation pulse is studied both with and without the resonant approximation. Using the time-dependent perturbation theory, the analytical expressions for transition probability of the atom are obtained. It is shown how, depending on the value of the field strength, the atom's excitation probability can be obtained with the needed accuracy. It is shown that the upper level population of the atom exhibits oscillations which are absent in the resonant approximation. It is shown that absorption probability and dipole moment spectrum are governed by interference between the spectral components of a non- monochromatic radiation involved in nonlinear interaction process. Spectral distribution symmetry with respect to the resonance frequency being destroyed, the atom can be excited even if there is no resonant component in the pulse spectrum.
We have studied a special class of time-symmetric pulses interacting with a two-level system. These pulses are built out of a class of asymmetric functions for which the analytic description of the interaction was possible. The time behavior of these pulses is given. We have evaluated the transition probability in a closed form. These pulses can be used to simulate the familiar Lorentzian and Gaussian pulse shapes remarkably well. Therefore, they can be used in computing the overall features of the interaction process. Comparisons of the transition amplitudes are made between the closed-form solution and numerically calculated Lorentzian and Gaussian cases.
The problem of a two-level atom interacting with a non-monochromatic radiation pulse is studied in the resonant approximation for arbitrary detunings, and without the resonant approximation. By using the time-dependent theory of perturbation the analytical expressions for the transition probability and induced dipole moment of the atom are obtained. It is shown that the upper level population of the atom exhibits oscillations that are absent in the resonant approximation. The temporal shape of the pulse is shown to affect significantly the atomic transition probability. The behaviour of the transition probability and dipole moment spectrum for arbitrary pulse shapes, phases and detunings is investigated. The interference between the spectral components arising in nonlinear processes is shown to play a considerable role in the formation of a dipole moment spectrum.
The problem of the evolution of a two-level quantum system in a radiation field of Gaussian shape with detuning is addressed by seeking an approximate expression for the transformation function governing the interaction. The transformation operator requires the evaluation of two independent matrix elements: the former is obtained by the Rosen-Zener conjecture; the latter is based on an expansion using the main parameters involved in the first one. The expressions obtained, which are supposed to hold in the case of small detunings, have been tested by comparison with a numerical integration. Then the solutions are modified to take into account decay phenomena to other levels. As a final example, the expressions dealing with the transition probability in the case of two separated oscillatory fields are given.