The multiple-linear-time-scale method is used to construct a perturbation theory for
N-level quantum systems subjected to time-dependent perturbations. The secular and small-denominator terms which plague conventional time-dependent perturbation theory are avoided; consequently, the theory is useful for treating long-term behavior and resonant interactions. The introduction of a self-energy operator allows the shifts in energy levels to be displayed explicitly. When the perturbation is the dipole interaction with an electromagnetic field, the successive approximations yield the well-known rotating-wave approximation, corrections due to counter rotating terms, and the Bloch-Siegert frequency shift. The formalism is applicable to arbitrary pulse shapes, provided that $\ensuremath{\parallel}{H}_{1}\ensuremath{\parallel}\ensuremath{\ll}\ensuremath{\hbar}\overline{\ensuremath{\omega}}$, where $\ensuremath{\parallel}{H}_{1}\ensuremath{\parallel}$ represents the inter-action strength and $\overline{\ensuremath{\omega}}$ is a characteristic frequency of the field. Rabi oscillations induced by multiphoton resonances are automatically included, and this effect is demonstrated in the case of a square pulse with frequency approximately one half the resonance frequency for a transition between two vibrational levels of a molecule. These calculations are compared to an exact numerical solution in order to find the limits of validity of the approximation. Even at high intensities ($I\ensuremath{\sim}{10}^{14}$ W/
cm2) the shifted resonance frequency is quite accurate; however, the approximate and exact solutions are slightly out of phase, so that the approximate solution can only be trusted for a few Rabi cycles. For much lower intensities ($I\ensuremath{\sim}{10}^{10}$ W/
cm2), the approximate solution is valid for many thousands of Rabi cycles.
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.
The response of a two-level atom to a smoothly varying near-resonant driving pulse can be described by the usual adiabatic-following approximation if the conditions $\frac{{T}_{1}^{\ensuremath{-}1},{T}_{2}^{\ensuremath{-}1}\ensuremath{\ll}{t}_{p}^{\ensuremath{-}1}\ensuremath{\ll}\ensuremath{\Omega}(t)}{2\ensuremath{\pi}}$ are satisfied throughout the pulse. Here, $\ensuremath{\Omega}(t)={[{\ensuremath{\Delta}}^{2}(t)+\frac{{p}^{2}{\mathcal{E}}^{2}(t)}{{\ensuremath{\hbar}}^{2}}]}^{\frac{1}{2}}$ is the atomic precession frequency, $\ensuremath{\Delta}(t)$ the detuning,
E(t) the field envelope,
tp the effective pulse width, and
T1 and
T2 the atomic level and phase relaxation times, respectively. From Bloch's equations, we have developed a generalized version of this approximation applicable to cases where
T1 and
T2 can be comparable to or less than
tp. It allows the condition ${T}_{1}^{\ensuremath{-}1}, {T}_{2}^{\ensuremath{-}1}\ensuremath{\ll}{t}_{p}^{\ensuremath{-}1}$ to be replaced by the weaker requirement ${T}_{1}^{\ensuremath{-}1}$,$\frac{{T}_{2}^{\ensuremath{-}1}{p}^{2}{\mathcal{E}}^{2}(t)}{{\ensuremath{\hbar}}^{2}{\ensuremath{\Omega}}^{2}(t)\ensuremath{\ll}\ensuremath{\Omega}(t)}$. If ${T}_{1}\ensuremath{\gg}{t}_{p}$, the atomic Bloch vector remains aligned nearly parallel to the effective driving field
E(t), 0, $\frac{\ensuremath{\hbar}\ensuremath{\Delta}(t)}{p}$) in the rotating reference frame (as in the pure adiabatic following case); however, it decays in length if
T2 is comparable to
tp. The approximation bridges the gap between pure adiabatic following behavior for ${t}_{p}^{\ensuremath{-}1}\ensuremath{\gg}{T}_{1}^{\ensuremath{-}1}, {T}_{2}^{\ensuremath{-}1}$ and rate-equation behavior for ${t}_{p}^{\ensuremath{-}1}\ensuremath{\ll}{T}_{2}^{\ensuremath{-}1}\ensuremath{\approx}\ensuremath{\Omega}(t)$, thereby allowing a more complete description of phenomena such as adiabatic inversion and resonantly enhanced self-focusing and defocusing.
A solution is presented for the damped optical Bloch equations under the excitation of a smooth pulse by first deriving three independent third-order equations of the Bloch vector components. Each equation is reduced to quadratures by assuming that the logarithmic time derivative of the field amplitude is small compared to the Rabi frequency. This results in an approximate summation of the infinite-order time-dependent perturbation in the field amplitude. The relaxation-dependent induced damping of the population inversion is calculated. Also calculated are additional relaxation-dependent contributions to the intensity-dependent refractive index. The time-integrated intensity contribution tends to cause line asymmetry, which becomes, at later times, linear in ${\ensuremath{\gamma}}_{2}$ when ${\ensuremath{\gamma}}_{2}\ensuremath{\gg}{\ensuremath{\gamma}}_{1}$ and zero when $2{\ensuremath{\gamma}}_{2}={\ensuremath{\gamma}}_{1}$, where ${\ensuremath{\gamma}}_{1}$ and ${\ensuremath{\gamma}}_{2}$ are the atomic energy and phase-changing relaxations, respectively. The dependence of the spectral broadening on pulse length, pressure, and length of the sample is discussed.
A class of molecular two-photon processes, occurring under transient or steady-state conditions, is analyzed exactly for certain cases using a density-matrix formalism. Our results are relevant to the recently observed phenomena of transient coherent Raman beats and steady-state two-photon absorption of oppositely directed laser beams. Lowest-order perturbation theories have shown that these processes are insensitive to the Doppler effect, and consequently the linewidth is unaffected by Doppler broadening and by elastic collisions that change molecular velocity. The solutions presented here are of the same form for the Raman and two-photon problems and reveal significant power-dependent frequency shifts and line broadening that correct the ideal Doppler-free solutions. For the transient case, we assume a molecular three-level quantum structure that can be switched in or out of optical resonance with cw laser radiation by means of a pulsed dc Stark field, but the solutions also apply when resonant optical pulses are introduced. During the resonant condition when the Stark pulse is on, the three levels are prepared in coherent superposition, and during the nonresonant condition following the Stark pulse, the laser field(s) stimulates the coherently prepared sample in a transient two-photon process. One of these processes, the Ramanbeat effect, has been observed in forward scattering, but the second one, transient two-photon emission, should be observable in backward scattering. Bloch-like equations are derived for this three-level problem that facilitate an exact pulse solution for state preparation. Following the pulse, transient decay is well-approximated by a perturbation calculation. For the steady-state case, an exact solution is also obtained and is of interest for continuous-wave spectroscopy or for transient experiments requiring an initial preparation of quantum states, prior to Stark switching. These solutions exhibit a power-dependent line broadening and a frequency shift of magnitude $\ensuremath{\sim}\frac{\ensuremath{\Delta}({\ensuremath{\alpha}}^{2}\ensuremath{-}{\ensuremath{\beta}}^{2})}{({\ensuremath{\Delta}}^{2}+\frac{1}{{T}_{2}^{2}})}$, in agreement with our earlier estimate, where $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ are the Rabi flopping frequencies for the two intermediate transitions and $\ensuremath{\Delta}$ is the off-resonance tuning behavior for one of them.