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Physical Review A

Discrete-resonance-resonance transitions induced by two laser pulses with an ultrashort time delay: A formal analytic solution and quantitative applications

Yannis Komninos, Theodoros Mercouris, Cleanthes A. Nicolaides

Published Aug 30, 2018• 2 citations
Publisher

Abstract

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, ${t}_{D}$. The formalism leads to analytic expressions for photoelectron emission probabilities, ${P}_{i}(E,t)$, $i=1,2$, 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 ${P}_{i}(E,t)$ 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 $\mathrm{He}$ $1{s}^{2}$ $^{1}S$ $\ensuremath{\rightarrow}$ $(2s2p)$ $^{1}P^{o}$ $\ensuremath{\leftrightarrow}$ $(2{p}^{2})$ $^{1}D$. Calculations show that the $(2{p}^{2})$ $^{1}D$ resonance is 0.295 eV (4200 nm) below the energy of $(2s2p)$ $^{1}P^{o}$, (60.15 eV), and its width is 0.067 eV. For the transition $1{s}^{2}$ $^{1}S$ $\ensuremath{\rightarrow}$ $(2s2p)$ $^{1}P^{o}$, 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 $5\ifmmode\times\else\texttimes\fi{}{10}^{10}--5\ifmmode\times\else\texttimes\fi{}{10}^{11}$ $\mathrm{W}/\mathrm{c}{\mathrm{m}}^{2}$. The external control parameters are the intensity and the FWHM of the femtosecond pulse, as well as the ${t}_{D}$. For a given value of ${t}_{D}$, the FWHM of the second pulse is a crucial control parameter. The results include the time-resolved formation of either the $(2s2p)$ $^{1}P^{o}$ or the $(2{p}^{2})$ $^{1}D$ resonance, while they are coupled with each other. For the $(2s2p)$ $^{1}P^{o}$ state, comparison is made with the case where the time-dependent buildup of its asymmetric profile, excited from $1{s}^{2}$ $^{1}S$ 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 $(2{p}^{2})$ $^{1}D$ resonance excited from $(2s2p)$ $^{1}P^{o}$, 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.

DOI
10.1103/physreva.98.023428

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The paper presents a theory that solves analytically the time-dependent three-level problem of the scheme discrete∣0\ensuremath⟩\ensuremath→weakaspulse\ensuremathω1resonance∣1\ensuremath⟩\ensuremath→moderatelystrongfspulse\ensuremathω2resonance∣2\ensuremath⟩\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}discrete∣0\ensuremath⟩aspulse\ensuremathω1​\ensuremath→weak​​resonance∣1\ensuremath⟩fspulse\ensuremathω2​\ensuremath→moderatelystrong​​resonance∣2\ensuremath⟩, where ∣0\ensuremath⟩|0\ensuremath{\rangle}∣0\ensuremath⟩ is a discrete state, and ∣1\ensuremath⟩|1\ensuremath{\rangle}∣1\ensuremath⟩ and ∣2\ensuremath⟩|2\ensuremath{\rangle}∣2\ensuremath⟩ are energy-normalized resonance states. The attosecond and femtosecond pulses with central frequencies \ensuremathω1{\ensuremath{\omega}}_{1}\ensuremathω1​ and \ensuremathω2{\ensuremath{\omega}}_{2}\ensuremathω2​ (\ensuremathω1\ensuremath≫\ensuremathω2{\ensuremath{\omega}}_{1}\ensuremath{\gg}{\ensuremath{\omega}}_{2}\ensuremathω1​\ensuremath≫\ensuremathω2​) act with ultrashort time delay, tD{t}_{D}tD​. The formalism leads to analytic expressions for photoelectron emission probabilities, Pi(E,t){P}_{i}(E,t)Pi​(E,t), i=1,2i=1,2i=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){P}_{i}(E,t)Pi​(E,t) involve, among other things, the Fano discrete-resonance asymmetry parameter qqq, 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\mathrm{He}He 1s21{s}^{2}1s2 1S^{1}S1S \ensuremath→\ensuremath{\rightarrow}\ensuremath→ (2s2p)(2s2p)(2s2p) 1Po^{1}P^{o}1Po \ensuremath↔\ensuremath{\leftrightarrow}\ensuremath↔ (2p2)(2{p}^{2})(2p2) 1D^{1}D1D. Calculations show that the (2p2)(2{p}^{2})(2p2) 1D^{1}D1D resonance is 0.295 eV (4200 nm) below the energy of (2s2p)(2s2p)(2s2p) 1Po^{1}P^{o}1Po, (60.15 eV), and its width is 0.067 eV. For the transition 1s21{s}^{2}1s2 1S^{1}S1S \ensuremath→\ensuremath{\rightarrow}\ensuremath→ (2s2p)(2s2p)(2s2p) 1Po^{1}P^{o}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×\else\texttimes\fi1010−−5\ifmmode×\else\texttimes\fi10115\ifmmode\times\else\texttimes\fi{}{10}^{10}--5\ifmmode\times\else\texttimes\fi{}{10}^{11}5\ifmmode×\else\texttimes\fi1010−−5\ifmmode×\else\texttimes\fi1011 W/cm2\mathrm{W}/\mathrm{c}{\mathrm{m}}^{2}W/cm2. The external control parameters are the intensity and the FWHM of the femtosecond pulse, as well as the tD{t}_{D}tD​. For a given value of tD{t}_{D}tD​, the FWHM of the second pulse is a crucial control parameter. The results include the time-resolved formation of either the (2s2p)(2s2p)(2s2p) 1Po^{1}P^{o}1Po or the (2p2)(2{p}^{2})(2p2) 1D^{1}D1D resonance, while they are coupled with each other. For the (2s2p)(2s2p)(2s2p) 1Po^{1}P^{o}1Po state, comparison is made with the case where the time-dependent buildup of its asymmetric profile, excited from 1s21{s}^{2}1s2 1S^{1}S1S 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)(2{p}^{2})(2p2) 1D^{1}D1D resonance excited from (2s2p)(2s2p)(2s2p) 1Po^{1}P^{o}1Po, the time-resolved buildup of its profile does not develop into an asymmetric peak due to the fact that QQQ has the very large value of \ensuremath≈\ensuremath−1300\ensuremath{\approx}\ensuremath{-}1300\ensuremath≈\ensuremath−1300. Instead, it finishes as an Autler-Townes doublet.
The multiple-linear-time-scale method is used to construct a perturbation theory for NNN-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{\mathrm{cm}}^{2}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{\mathrm{cm}}^{2}cm2), the approximate solution is valid for many thousands of Rabi cycles.
The coherent interaction of two smoothly varying, near-resonant, two-photon pulses with a three-level system can be described by "two-photon damped Bloch equations" which are analogous to those for a one-photon transition in a two-level system except for the presence of a two-photon coupling and a frequency shift. These equations are solved for the cases ${\ensuremath{\gamma}}_{1}$, ${\ensuremath{\gamma}}_{2}\ensuremath{\ll}\ensuremath{\Omega}$, ${\ensuremath{\gamma}}_{1}={\ensuremath{\gamma}}_{2}$, and $\frac{{\ensuremath{\gamma}}_{2}{k}^{2}{\ensuremath{\epsilon}}^{4}}{{\ensuremath{\Omega}}^{2}}$, ${\ensuremath{\gamma}}_{1}\ensuremath{\ll}\ensuremath{\Omega}$, where ${\ensuremath{\gamma}}_{1}$ and ${\ensuremath{\gamma}}_{2}$ are the atomic energy and phase relaxation widths, respectively, and $\ensuremath{\Omega}$ is the Rabi frequency. The leading contribution to the refractive index is intensity dependent, caused by the level shifts inherent in multiphoton processes; it includes a relaxation dependent part which is important at times shorter than ${\ensuremath{\gamma}}_{1}^{\ensuremath{-}1}$. The second-order contributions depend on the square of the intensity and the time-integrated square of the intensity. The latter contribution, which is relaxation dependent, causes line asymmetry at the long-wavelength wing; it consists of a term proportional to ${\ensuremath{\gamma}}_{2}\ensuremath{-}{\ensuremath{\gamma}}_{1}$ and only important at early times and a term proportional to $2{\ensuremath{\gamma}}_{2}\ensuremath{-}{\ensuremath{\gamma}}_{1}$.
Abstract We predict quantitatively that it is possible to use few-femtosecond positive or negative time delays , Δ t , between two XUV Gaussian few-femtosecond pulses of moderate intensities and central frequencies ω 1 and ω 2 , in order to control the probability of ionization at clearly defined exit energies corresponding to the sums of photon frequencies ( ω 1 + ω 1 ), ( ω 2 + ω 2 ), and ( ω 1 + ω 2 , or ω 2 + ω 1 ). The phenomenon is demonstrated quantitatively by obtaining and using nonperturbative solutions of the time-dependent Schrödinger equation for a time-dependent scheme involving the process of two-photon resonant ionization of Helium via the 1 s 2 p P o 1 (58.4 nm) and 1 s 4 p P o 1 (52.22 nm) excited states. The calculations used wavefunctions which are state-specific for the discrete spectrum, (up to the 1 s 7 g
We study the Rabi flopping of the population between the ground and excited 2p states of the hydrogen atom, induced by intense short laser pulses of different shapes and of carrier frequency ω=0.375 a.u. which resonantly couples the two states, and manifestations of this dynamics in the energy spectra of photoelectrons produced in the subsequent ionization of the atom from the excited state. It is found that, for Gaussian, half-Gaussian and rectangular pulses, characterized by the same pulse area, the final populations take the same values and the spectra consist of similar patterns having the same number of peaks and approximately the same separation between the prominent edge (Autler–Townes) peaks. The additional analysis in terms of dressed states showed that the mechanism of formation of multiple-peak structures during the photoionization process is the same regardless of the pulse shape. These facts disprove the hypothesis proposed in earlier studies with Gaussian pulse, that the multiple-peak pattern appears due to dynamic interference of the photoelectrons emitted with a time delay at the rising and falling sides of the pulse, since the hypothesis is not applicable to either a half-Gaussian pulse that has no rising part or a rectangular pulse whose intensity is constant.