3.1. Excitation and Ionization
In
Figure 1 we demonstrate the calculated dependence on the laser frequency
$\omega $ of the population
${P}_{g}\left(\omega \right)$ of the ground state of the hydrogen atom after its interaction with a linearly polarized laser pulse of intensity
$I=$10
${}^{14}W/c{m}^{2}$. The results of calculations of the probabilities of excitation
${P}_{ex}\left(\omega \right)$ and ionization
${P}_{ion}\left(\omega \right)$ of the atom for the same laser frequencies, intensity and pulse duration are also presented in the same figure. Here, the total pulse duration is fixed at
${T}_{out}=NT=100\pi \phantom{\rule{3.33333pt}{0ex}}$a.u.
$\approx 7.6\phantom{\rule{3.33333pt}{0ex}}$fs which requires increased number of included optical cycles
N by increasing frequency
$\omega $. The populations of the ground state of the atom
${P}_{g}\left(\omega \right)$ were obtained with the standard procedure of projection at the end of the pulse (
$t={T}_{out}$) of the calculated electron wave-packet
$\psi (\mathbf{r},\omega ,t={T}_{out})$ onto the ground state
${\varphi}_{100}\left(\mathbf{r}\right)$ of the unperturbed atom
To evaluate the probability of excitation of an atom by a laser pulse
${P}_{ex}\left(\omega \right)={\sum}_{n>1}^{\infty}{P}_{n}\left(\omega \right)$ we applied the following procedure. The calculation of the populations
${P}_{n}\left(\omega \right)$ of
$2\le n\le 8$ states was carried out in exactly the same way as the population of the ground state (15). To take into account the populations
${P}_{n}\left(\omega \right)$ of states from
$n=9$ and above, we used the “interpolation” procedure proposed in our previous work [
26]. The probability of ionization of the atom
${P}_{ion}\left(\omega \right)$ is calculated using the formula
${P}_{ion}\left(\omega \right)=1-{P}_{g}\left(\omega \right)-{P}_{ex}\left(\omega \right)$ [
26].
The first thing that catches our eye in
Figure 1 is the resonant peaks in the probabilities
${P}_{ex}\left(\omega \right)$ near the frequencies
$\omega =0.37,\phantom{\rule{3.33333pt}{0ex}}0.44,\phantom{\rule{3.33333pt}{0ex}}0.47\phantom{\rule{3.33333pt}{0ex}}$a.u., defined by the resonant conditions
corresponding to the transitions
It should be noted that in the resonance condition (16) we do not take into account the perturbation of the excited state
${n}^{\prime}$ due to the dynamic Stark effect, which can give significant corrections with increasing field intensity, especially for highly excited states. The origin of the resonant peaks at
${P}_{ex}\left(\omega \right)$ due to one-photon transitions (17) is clearly confirmed by the calculated time-dynamics of the populations
${P}_{n}(\omega ,t)\stackrel{t\to {T}_{out}}{\to}{P}_{n}\left(\omega \right)$ of low-lying states (up to
$n=5$) of the hydrogen atom for some frequencies, including near resonant ones at
$\omega =$0.36a.u., 0.44a.u. and 0.48a.u. (see
Figure 2(c-e)). However, the position of the peak
$\omega =$0.24a.u. is not described by the resonance condition (16). Nevertheless, the calculated time-dynamics of populations
${P}_{n}\left(t\right)$ shown in
Figure 2(b) clearly demonstrates the dominance of transition
$n=1\to {n}^{\prime}=4$ (17) in the population
${P}_{ex}\left(\omega \right)$ at
$\omega =$0.24a.u.. It is clear that the resonant condition for this transition can be described by the formula
with
$2\hslash \omega \approx 0.47$a.u. for
$n=1$ and
${n}^{\prime}=4$. That is, the peak in
${P}_{ex}\left(\omega \right)$ at
$\omega =0.24$a.u. is formed due to a two-photon transition
$n=1\to {n}^{\prime}=4$. Some contribution of the state
$n=3$ in
${P}_{ex}\left(\omega \right)$ at this frequency is also considerable because
$2\hslash \omega =0.48$a.u. is also close to the resonant frequency of
$0.44$a.u. for the transition to the state
${n}^{\prime}=3$. As one can see in the
Figure 2(a), the excitation at the frequency
$\omega =0.22$a.u. has also a two-photon character due to the resonant transition to
${n}^{\prime}=3$.
Note also that the positions of peaks in ${P}_{ex}\left(\omega \right)$ exactly coincide with the positions of the minima in the population of the ground state ${P}_{g}\left(\omega \right)$ in frequency regions, especially in the regions near $\omega \sim $(0.22-0.24)a.u. and $\omega \sim $ (0.4-0.48)a.u. where ionization is suppressed. The marked areas of ionization suppression are near two-photon resonant excitation to the states $n=3\phantom{\rule{3.33333pt}{0ex}},4$ and one-photon excitations to the states $n=3\phantom{\rule{3.33333pt}{0ex}},4$ and 5. The suppression of ionization near the one-photon excitation to the state $n=$2 at $\omega =0.37$a.u. is only slightly noticeable. As the frequency increases and approaches the ionization threshold ${\omega}_{t}=$0.5a.u., ionization begins to increase sharply, up to a point ${\omega}_{i}=$0.52a.u. above the threshold, where ${P}_{ion}\left({\omega}_{i}\right)={P}_{g}\left({\omega}_{i}\right)$. Beyond this point, the probability of ionization decreases monotonically with increasing frequency. Note also that at the threshold point ${\omega}_{t}=$0.5a.u the probability of excitation is equal to the probability of ionization ${P}_{ex}\left({\omega}_{t}\right)={P}_{ion}\left({\omega}_{t}\right)$.
In
Figure 2 we present the calculated time dynamics of the population probabilities of low-lying states for various laser pulse frequencies. Frequencies
$\omega =$0.36a.u.,
$\omega =$0.44a.u. and
$\omega =$0.48a.u. in
Figure 2(c,d,e) correspond to one-photon excitation into atomic levels
$n=2,\phantom{\rule{3.33333pt}{0ex}}3,\phantom{\rule{3.33333pt}{0ex}}4$ and 5 (see also
Figure 1) while
Figure 2(a,b) related to
$\omega =$0.22a.u. and
$\omega =$0.24a.u. illustrate the time dynamics of population through a two-photon transition. Moreover, in the cases
$\omega =$0.22a.u., 0.24a.u., 0.44a.u. and 0.48a.u., the processes of excitation of an atom has the two-step character through an intermediate metastable state: the process begins with a transition to the first excited state of the atom
$n=2$, which begins to depopulate rapidly by transition to higher states due to the resonant interaction of the atom with the laser pulse. This two-step mechanism is most clearly visible in the cases
$\omega =$0.22a.u. and 0.24a.u.. The resonant population of the lowest excited state
$n=2$ which is observed at the frequency
$\omega =$0.36a.u. (
Figure 2(c)) is a one-step process and occurs directly without any transitions to an intermediate state.
Figure 2(f),
$\omega =$0.8a.u., illustrates a non-resonant case of atomic excitation and relates to the case of above ionization threshold excitation of the atom. It can be seen that in the latter case, all populated low-lying levels of the atom, with the exception of
$n=2$, are metastable. They are depopulated at the end of the laser pulse, after which only a small part of atoms remains in the excited
$n=2$ state.
3.2. Acceleration of Neutral Atoms
Figure 2 also shows the results of calculating the time-dynamics of the acceleration of a neutral hydrogen atom by a laser pulse of various frequencies. Here are the CM velocities of an atom in the direction of propagation of the laser pulse
${V}_{y}(\omega ,t)$ calculated as a function of time. It should be noted that in all cases considered, with the exception of
$\omega =$0.8a.u., the acceleration of the atom CM repeats the time dynamics of the population of the most populated level. In the case of the above threshold ionization with
$\omega =$ 0.8a.u., the velocity of the CM monotonically increases with time, reaching a maximum at the end of the pulse (
$t={T}_{out}=$314a.u.), while the populations of all low-lying atomic levels grow to the point of maximum intensity of the laser pulse (at the point
$t={T}_{out}/2$) and begin to depopulate after its passage. At the end of the laser pulse, only a small part of the atoms in the
$n=2$ state remains in the excited state.
The correlation between the time dynamics of the CM velocity of the atom
${V}_{y}(\omega ,t)$ and the population probability
${P}_{n}\left(t\right)$ of the most populated level demonstrated in
Figure 2 confirms the mechanism of acceleration of an atom by a laser pulse due to the acceleration of a spatially inhomogeneous electron cloud in excited states by ponderomotive forces for a frequency below the ionization threshold. In the region of laser frequencies exceeding the ionization threshold (see case
$\omega =$ 0.8a.u.), we observe acceleration of the atom CM even after passing the critical point
$t>{T}_{out}/2$, when the atom is depopulated. The growth in this region
${V}_{y}(\omega =0.8,t)$ occurs due to ionized electrons, the value of which is significant here (see
Figure 1).
Actually, presented in
Figure 3, the calculated dependencies on the laser frequency of the total probability of excitation and ionization of atom
${P}_{ex}\left(\omega \right)+{P}_{ion}\left(\omega \right)$ and the momentum
${P}_{y}\left(\omega \right)=M{V}_{y}\left(\omega \right)$ of its accelerated CM demonstrate their strong correlation with each other. This is a clear demonstration of what is the root cause of the acceleration of the CM of an atom by a laser field. This is the generation of a non-zero dipole moment between proton and electron cloud that, under the action of electromagnetic pulse, has transferred either to the excited state of the atom or to its continuum. Thus, we see that in the case of noticeable ionization of an atom, in addition to the acceleration of the neutral atom itself as a whole, the acceleration of the electron falling into the continuous spectrum also contributes to the acceleration of the atom CM. Naturally, for practical purposes of accelerating neutral atoms, one should use frequency regions of laser radiation in which ionization is suppressed compared to the excitation of the atom. In this regard, the frequencies near the two-photon resonances (
$n=3,4$)
$\omega \sim $(0.22-0.24)a.u and one-photon resonances (
$n=$3-5)
$\omega \sim $(0.42-0.48)a.u. (see
Figure 1) are promising.
In
Figure 4 we present the results of calculations for resonant frequencies
$\omega =$0.24a.u., 0.48a.u. and
$\omega =$0.8a.u. of the dependence of the velocity
${V}_{y}(\omega ,I)$ of the atom CM on the radiation intensity
I. Noteworthy is the linear dependence of the calculated curves for all given frequencies on this double-logarithmic scale. Moreover, the slopes of the presented curves, with the exception of the frequency
$\omega =$ 0.24a.u., are the same on a double logarithmic scale, which corresponds to the linear dependence of the the CM velocity on the intensity:
${V}_{y}(\omega ,I)\propto I$. However, the angle of inclination of the curve
$\omega =$ 0.24a.u. increases, which gives in this case exactly the quadratic dependence of the velocity of the atomic CM in the considered range of intensities:
${V}_{y}(\omega =0.24,I)\propto {I}^{2}$. The physical interpretation of the discovered effect is the following. All cases of acceleration of the CM of the atom considered here, except for
$\omega =$ 0.24a.u., correspond to the single-photon mechanism discussed above. In this case, the effect of accelerating the CM to velocity
${V}_{y}(\omega ,I)$ is proportional to the photon number density in the laser pulse (i.e. intensity
I), which is clearly demonstrated in
Figure 4. In the case of
$\omega =$ 0.24a.u., the process has two-photon origin, in which the acceleration of the CM to a velocity
${V}_{y}(\omega =0.24,I)$ is proportional to the square of the photon number density in the laser pulse, which leads to a quadratic dependence
${V}_{y}(\omega =0.24,I)\propto {I}^{2}$ which we observe in
Figure 4. It should be noted that the proportionality coefficients of
${V}_{y}(\omega =0.48,I)$ and
${V}_{y}(\omega =0.80,I)$ to the intensity
I are slightly decreasing by increasing intensity to
$I>{10}^{14}W/c{m}^{2}$. This feature can be interpreted as a complete depopulation of the ground state into excited states and a continuous spectrum of the atom at this region of intensities.