We consider the time-dependent two-level problem of quantum mechanics, where the levels are coupled by a radio-frequency pulse with an arbitrary time-dependent envelope
V(t). We derive an approximate solution for the system's transition amplitude $P(\ensuremath{\infty})$ which is correct to the third order of perturbation theory, and which applies to all pulses
V(t) with finite first and second moments which obey the following:
limt3V(t)=0, as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. Our form of solution for $P(\ensuremath{\infty})$ provides a criterion for judging the validity of a solution previously conjectured by Rosen and Zener, and it is generally useful for providing line-shape details in many cases of practical interest.
The problem of calculating transition probabilities in two-level systems is studied in the limit where the detuning is large compared to the inverse duration of the interaction. Coupling potentials whose Fourier transforms $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}(\ensuremath{\omega})$ are of the form $f(\ensuremath{\omega}){e}^{\ensuremath{-}(|b\ensuremath{\omega}|)}$ for large frequencies give rise to solutions which may be classified into families according to the form of $f(\ensuremath{\omega})$. Within each family transition probabilities may be calculated from formulas that differ only in the numerical value of a scaling parameter. In cases where the coupling function has a pole in the complex time plane, the families are identified with the order of this singularity. In particular, for poles of first order, a connection with the Rosen-Zener solution can be made. The analysis is performed via high-order perturbation expansions which are shown to always converge for two-level systems driven by coupling potentials of finite pulse area.
The paper presents a theory that solves analytically the time-dependent three-level problem of the scheme $\mathrm{discrete}\phantom{\rule{0.28em}{0ex}}|0\ensuremath{\rangle}\phantom{\rule{0.28em}{0ex}}\underset{\mathrm{as}\phantom{\rule{0.16em}{0ex}}\mathrm{pulse}\phantom{\rule{0.16em}{0ex}}{\ensuremath{\omega}}_{1}}{\overset{\mathrm{weak}\phantom{\rule{0.16em}{0ex}}}{\ensuremath{\rightarrow}}}\phantom{\rule{0.28em}{0ex}}\mathrm{resonance}\phantom{\rule{0.28em}{0ex}}|1\ensuremath{\rangle}\phantom{\rule{0.28em}{0ex}}\underset{\mathrm{fs}\phantom{\rule{0.16em}{0ex}}\mathrm{pulse}\phantom{\rule{0.16em}{0ex}}{\ensuremath{\omega}}_{2}}{\overset{\mathrm{moderately}\phantom{\rule{0.16em}{0ex}}\mathrm{strong}\phantom{\rule{0.16em}{0ex}}}{\ensuremath{\rightarrow}}}\phantom{\rule{0.28em}{0ex}}\mathrm{resonance}\phantom{\rule{0.28em}{0ex}}|2\ensuremath{\rangle}$, where $|0\ensuremath{\rangle}$ is a discrete state, and $|1\ensuremath{\rangle}$ and $|2\ensuremath{\rangle}$ are energy-normalized resonance states. The attosecond and femtosecond pulses with central frequencies ${\ensuremath{\omega}}_{1}$ and ${\ensuremath{\omega}}_{2}$ (${\ensuremath{\omega}}_{1}\ensuremath{\gg}{\ensuremath{\omega}}_{2}$) act with ultrashort time delay,
tD. The formalism leads to analytic expressions for photoelectron emission probabilities,
Pi(E,t),
i=1,2, in the energy region of the two resonance states, whose calculation in terms of N-electron matrix elements and external control parameters can be carried out economically for any such three-level system. The problem is first solved analytically, within the rotating wave approximation, for the case where the femtosecond pulse is rectangular. The weak attosecond pulse can have any temporal shape. The final solution for two pulses with arbitrary shape (Gaussian in our case) is then achieved by fitting numerically the shape of the femtosecond pulse to a sum of rectangular pulses. The formulas for
Pi(E,t) involve, among other things, the Fano discrete-resonance asymmetry parameter
q, and the resonance-resonance asymmetry parameter Q, defined by us in previous work. They also involve the complex energy poles of the two resonances, a result which accounts for the whole contribution of the open channels of energy-normalized scattering states into which they decay. The theory is applied to the calculation from first principles, using compact state-specific discrete and resonance wave functions, of observable aspects of the time-resolved dynamics associated with the processes
He 1s2 1S $\ensuremath{\rightarrow}$
(2s2p) 1Po $\ensuremath{\leftrightarrow}$
(2p2) 1D. Calculations show that the
(2p2) 1D resonance is 0.295 eV (4200 nm) below the energy of
(2s2p) 1Po, (60.15 eV), and its width is 0.067 eV. For the transition
1s2 1S $\ensuremath{\rightarrow}$
(2s2p) 1Po, the full width at half maximum (FWHM) of the Gaussian pulse is 160 as. The few-cycle midinfrared pulse coupling the resonance states is also Gaussian, whose duration is varied in the range of a few decades of femtoseconds. Its intensity is in the range $5\ifmmode\times\else\texttimes\fi{}{10}^{10}--5\ifmmode\times\else\texttimes\fi{}{10}^{11}$
W/cm2. The external control parameters are the intensity and the FWHM of the femtosecond pulse, as well as the
tD. For a given value of
tD, the FWHM of the second pulse is a crucial control parameter. The results include the time-resolved formation of either the
(2s2p) 1Po or the
(2p2) 1D resonance, while they are coupled with each other. For the
(2s2p) 1Po state, comparison is made with the case where the time-dependent buildup of its asymmetric profile, excited from
1s2 1S by the same attosecond pulse, is calculated in the absence of coupling to another resonance. The result is in harmony with that of our earlier work on this problem. For the
(2p2) 1D resonance excited from
(2s2p) 1Po, the time-resolved buildup of its profile does not develop into an asymmetric peak due to the fact that
Q has the very large value of $\ensuremath{\approx}\ensuremath{-}1300$. Instead, it finishes as an Autler-Townes doublet.