Attosecond Dynamics of Photoexcitation of the Hydrogen Atom by Ultrashort Laser Pulses

Abstract

The dynamics of photoexcitation of the hydrogen atom in the discrete and continuous spectra under the action of laser pulses in the attosecond range of time and pulse durations has been analyzed using perturbation theory. It is shown that over time interval shorter than or on the order of pulse duration, the time dependence of the photoexcitation probability is generally oscillating by nature. It has been established that for certain values of parameters, the envelope of this dependence has a peak, the position of which is determined by the pulse duration and carrier frequency.

APPENDIX

We can write the following expression for the photoexcitation probability under the action of field E(t) (we assume that E(t → ±∞) = 0) in the dipole approximation in the first order of perturbation theory:

$$W(t) = \frac{1}{{{{\hbar }^{2}}}}\int\limits_{ - \infty }^t {dt{\kern 1pt} '\int\limits_{ - \infty }^t {dt{\kern 1pt} ''\langle \hat {d}(t{\kern 1pt} ')\hat {d}(t{\kern 1pt} '')\rangle E(t{\kern 1pt} ')E(t{\kern 1pt} '').} } $$

(A.1)

Angle brackets indicate averaging over the initial state of the target.

Expression (A.1) contains the dipole moment correlator (DMC) that can be written for a stationary system in form

$$\langle \hat {d}(t{\kern 1pt} ')\hat {d}(t{\kern 1pt} '')\rangle = K(t{\kern 1pt} ',t{\kern 1pt} '') = K(t{\kern 1pt} ''\, - t{\kern 1pt} ').$$

(A.2)

This gives

$$W(t) = \frac{1}{{{{\hbar }^{2}}}}\int\limits_{ - \infty }^t {dt{\kern 1pt} '\int\limits_{ - \infty }^t {dt{\kern 1pt} ''{\kern 1pt} K(t{\kern 1pt} ''\, - t{\kern 1pt} ')E(t{\kern 1pt} ')E(t{\kern 1pt} '').} } $$

(A.3)

Passing to the Fourier transform of the DMC, we obtain

$$K(t{\kern 1pt} ''\, - t{\kern 1pt} ') = \int\limits_{ - \infty }^\infty {\frac{{d\omega }}{{2\pi }}} \exp ( - i\omega (t{\kern 1pt} ''\, - t{\kern 1pt} '))K(\omega ).$$

(A.4)

Using the relation between the DMC and the photoexcitation cross section [15], we obtain

$$K(\omega ) = \frac{{\hbar c}}{{2\pi \omega }}\sigma (\omega ).$$

(A.5)

After simple transformations

$$\begin{gathered} W(t) = \frac{c}{{4{{\pi }^{2}}\hbar }}\int\limits_{ - \infty }^t {dt{\kern 1pt} '\int\limits_{ - \infty }^t {dt{\kern 1pt} ''\int {d\omega } } } \\ \times \exp ( - i\omega (t{\kern 1pt} ''\, - t{\kern 1pt} '))\frac{{\sigma (\omega )}}{\omega }E(t{\kern 1pt} ')E(t{\kern 1pt} '') \\ \end{gathered} $$

(A.6)

and

$$\begin{gathered} W(t) = \frac{c}{{4{{\pi }^{2}}}}\int\limits_0^\infty {d\omega \frac{{\sigma (\omega )}}{{\hbar \omega }}\int\limits_{ - \infty }^t {dt{\kern 1pt} '\int\limits_{ - \infty }^t {dt{\kern 1pt} ''} } } \\ \times \exp ( - i\omega (t{\kern 1pt} ''\, - t{\kern 1pt} '))E(t{\kern 1pt} ')E(t{\kern 1pt} '') \\ \end{gathered} $$

(A.7)

we pass to formula (1):

$$W(t) = \frac{c}{{4{{\pi }^{2}}}}\int\limits_0^\infty {d\omega \frac{{\sigma (\omega )}}{{\hbar \omega }}} {{\left| {\int\limits_{ - \infty }^t {dt{\kern 1pt} '{\text{exp}}(i\omega t{\kern 1pt} ')E(t{\kern 1pt} ')} } \right|}^{2}}.$$

(A.8)