The time-dependent theory of the absorption of three light pulses with an arbitrary duration in ladder-type four-level systems is proposed. The pulses are in resonance with the corresponding transitions. By using the time-dependent perturbation theory the spectral and temporal behaviour of the probability that the fourth level is excited at the moment t is found. We calculate and analyse the cases where the frequencies of the maxima of the two of three light pulses are fixed and the maximum of the remaining pules varies. Widths of possible absorption lines depend on different parameters. On the whole, three or four lines may exist, one of them corresponds to coherent contribution and the others do not.
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.
For the interaction of coherent radiation with a nondegenerate, homogeneously broadened multilevel system the nonlinear susceptibilities and excitations and their time dependence are evaluated using a perturbation approach to the density matrix up to fourth order. This is done for the "off-resonance" case, ${(\ensuremath{\Delta}{\ensuremath{\omega}}^{2}+{\ensuremath{\Gamma}}^{2})}^{\frac{1}{2}}\ensuremath{\gg}\frac{1}{\ensuremath{\tau}}$, as well as for the "on-resonance" case, ${(\ensuremath{\Delta}{\ensuremath{\omega}}^{2}+{\ensuremath{\Gamma}}^{2})}^{\frac{1}{2}}\ensuremath{\ll}\frac{1}{\ensuremath{\tau}}$, with $\ensuremath{\tau}$ being the pulse duration, by taking into account the temporal change of the electric field amplitude and the phase and energy relaxation times,
T2 and
T1. It is shown that the transient one-photon excitation known as adiabatic following, is not only important for a pulse duration $\ensuremath{\tau}<{T}_{2}$, but may also be important for ${T}_{2}\ensuremath{\lesssim}\ensuremath{\tau}<{T}_{1}$. In addition, expressions for the transient two-photon excitation in a multilevel system are derived. Finally, the two-level limit is evaluated to all orders of the perturbation including damping effects, which gives results valid also for large incident light intensities. For the "off-resonance" case an expression is obtained which describes not only the transient adiabatic following process, but also the usual one-photon absorption including saturation effects giving rise to power broadening.
Resonantly enhanced sum-frequency generation of pulsed laser radiation is treated theoretically for the case of a three-level system. The density-matrix equations of motion are solved using second-order, time-dependent perturbation theory with the inclusion of transient effects. The energy and spectrum of the generated radiation are calculated as functions of laser detunings, laser pulse lengths, and temporal overlap of the laser pulses. The tuning characteristics and output pulse characteristics are found to differ qualitatively depending on whether or not the excitation pulses satisfy the adiabatic following criteria generalized for the case of a three-level atom.
A theoretical analysis of the optical coherent transients that arise when a sample of two-level atoms is irradiated by a sequence of three broad-bandwidth pulses is presented. The first two pulses have a relative delay time of order of the correlation time of the pulse fluctuations and are sent into an atomic vapor from different directions. These pulses, whose temporal width is much greater than the delay time, can be correlated with one another and can be strong enough to saturate the two-level atomic transition. The third pulse is weak, noncorrelated with the first two, and is delayed in time so that it does not overlap them. We present a detailed examination of the transient signal that is produced when the third pulse is scattered by the spatial gratings in the population difference of atoms created by the first two pulses. Taking into account the effects of inhomogeneous and homogeneous broadening, we calculate the intensity of the transient signal, emitted in different directions, as a function of delay time. The signal is found to depend dramatically on the intensities of the excitation pulses. It is shown that, for strong excitation pulses, there is a direct dependence of the signal on the cross-correlation time of pulses ${\mathrm{\ensuremath{\tau}}}_{\mathit{c}}^{12}$ that does not exist when the pulses are weak. In particular, the strongest signals exhibit a peak of width of order ${\mathrm{\ensuremath{\tau}}}_{\mathit{c}}^{12}$. This peak can have a very narrow dip near its maximum whose width is much smaller than ${\mathrm{\ensuremath{\tau}}}_{\mathit{c}}^{12}$, if the pulses are fully correlated. We develop a representation of the time evolution of the Bloch vector of a two-level atom, driven by time-delayed pulses, that enables us to explain our results. In this representation, the two time-delayed pulses are replaced by two fully overlapping pulses having some effective amplitudes and atomic field detunings.