Temporal dynamics of resonant scattering of an ultrashort laser pulse by an atom

Abstract

Within the framework of the second order of perturbation theory and the dipole approximation, we derived simple formulas describing the temporal dynamics of ultrashort laser pulse scattering in terms of the scattering tensor and the Fourier transform of the strength of the electric field in a pulse. This expression was used for description of resonant scattering of an ultrashort pulse by an atom. We study the dependence of the spectral scattering probability as a function of time and scattering frequency. In particular, it is shown that the time dependence of the spectral scattering probability for certain values of parameters is of oscillatory character, while the integral scattering probability is always a monotonically increasing function of time.

Appendix

Using the standard approach in the second order of the perturbation theory and the dipole approximation, we have the following expression for the amplitude of a two-photon process (in atomic units) [24]:

$$A_{fi}^{\left( 2 \right)} \left( t \right) = \left( { - i} \right)^{2} \,\sum\limits_{n} {\int\limits_{ - \infty }^{t} {dt^{\prime}\,\int\limits_{ - \infty }^{{t^{\prime}}} {dt^{\prime\prime}\,\exp \left( {i\,\tilde{\omega }_{fn} \,t^{\prime} + i\,\tilde{\omega }_{ni} \,t^{\prime\prime}} \right)\,V_{fn} \left( {t^{\prime}} \right)\,V_{ni} \left( {t^{\prime\prime}} \right)} } } .$$

(15)

In case of the spontaneous USP scattering by an atom, the operator of sum of the electromagnetic perturbation is the two terms corresponding to interaction of the atom with the vacuum electromagnetic field and the USP field:

$$\hat{V} = \hat{V}^{vac} + \hat{V}^{E} ;\hat{V} = {\hat{\mathbf{d}}}\,\,{\mathbf{E}}^{vac} \left( t \right) + {\hat{\mathbf{d}}}\,\,{\mathbf{E}}\left( t \right)$$

(16)

$${\mathbf{E}}^{vac} \left( t \right) = {\mathbf{e^{\prime}}}^{ * } E_{0}^{vac} \left( {\omega^{\prime}} \right)\,\exp \left( {i\,\omega^{\prime}\,t} \right)$$

(17)

here ${\hat{\mathbf{d}}}$ is the operator of the electric dipole moment, ${\mathbf{E}}\left( t \right)$ and ${\mathbf{E}}^{vac} \left( t \right)$ are the strengths of the electric field in an USP and the scattered radiation field, ${\mathbf{e^{\prime}}}$ is the unit polarization vector of the scattered radiation. The vacuum field corresponds to the spontaneous photon emission during the USP scattering by an atom. In the dipole approximation, its amplitude looks like

$$E_{0}^{vac} \left( {\omega^{\prime}} \right) = \sqrt {\frac{{2\,\pi \,\omega^{\prime}}}{\Lambda }} ,$$

(18)

here $\Lambda$ is the normalization volume. We present the strength of the electric field in an USP in terms of its Fourier transform:

$${\mathbf{E}}\left( t \right) = {\mathbf{e}}\,E\left( t \right) = {\mathbf{e}}\int\limits_{ - \infty }^{\infty } {E\left( {\tilde{\omega }} \right)\,\exp \left( { - i\,\tilde{\omega }\,t} \right)\,{{d\tilde{\omega }} \mathord{\left/ {\vphantom {{d\tilde{\omega }} {2\,\pi }}} \right. \kern-\nulldelimiterspace} {2\,\pi }}}$$

(19)

For the spontaneous scattering amplitude with no change of a target state, it follows from (15):

$$A_{11}^{sc} \left( t \right) = \left( { - i} \right)^{2} \,\sum\limits_{n\,M} {\int\limits_{ - \infty }^{t} {dt^{\prime}\,\int\limits_{ - \infty }^{{t^{\prime}}} {dt^{\prime\prime}\,\exp \left( {i\,\tilde{\omega }_{n1} \,\left( {t^{\prime\prime} - t^{\prime}} \right)} \right)\,\left[ {V_{1n}^{vac} \left( {t^{\prime}} \right)\,V_{n1}^{E} \left( {t^{\prime\prime}} \right) + V_{1n}^{E} \left( {t^{\prime}} \right)\,V_{n1}^{vac} \left( {t^{\prime\prime}} \right)} \right]} } }$$

(20)

Here we single out the summation over the quantum numbers of the projection of the angular momentum M and introduce the notation

$$\tilde{\omega }_{n1} = \omega_{n1} - i\,\gamma_{n1} .$$

(21)

Summing over the angular quantum numbers M, we have:

$$A_{11}^{sc} \left( t \right) = - \frac{{\left( {{\mathbf{e^{\prime}}}^{ * } \,{\mathbf{e}}} \right)}}{2\,\pi }E_{0}^{vac} \left( {A_{1} + A_{2} } \right)$$

(22)

where

$$A_{1} \left( t \right) = \sum\limits_{n} {\int\limits_{ - \infty }^{t} {dt^{\prime}} \int\limits_{ - \infty }^{{t^{\prime}}} {dt^{\prime\prime}} \,\left| {d_{n1} } \right|^{2} } \exp \left( {i\,\omega^{\prime}\,t^{\prime}} \right)\,\int\limits_{ - \infty }^{\infty } {d\tilde{\omega }\,\exp \left( { - i\,\tilde{\omega }\,t^{\prime\prime}} \right)\,E\left( {\tilde{\omega }} \right)\,\exp \left[ {i\,\tilde{\omega }_{n1} \,\left( {t^{\prime\prime} - t^{\prime}} \right)} \right]}$$

(23)

$$A_{2} \left( t \right) = \sum\limits_{n} {\int\limits_{ - \infty }^{t} {dt^{\prime}} \int\limits_{ - \infty }^{{t^{\prime}}} {dt^{\prime\prime}} \,\left| {d_{n1} } \right|^{2} } \exp \left( {i\,\omega^{\prime}\,t^{\prime\prime}} \right)\,\int\limits_{ - \infty }^{\infty } {d\tilde{\omega }\,\exp \left( { - i\,\tilde{\omega }\,t^{\prime}} \right)\,E\left( {\tilde{\omega }} \right)\,\exp \left[ {i\,\tilde{\omega }_{n1} \,\left( {t^{\prime\prime} - t^{\prime}} \right)} \right]}$$

(24)

After integration with respect to the time $t^{\prime\prime}$, we arrive at the expressions

$$A_{1} \left( t \right) = - i\,\sum\limits_{n} {\int\limits_{ - \infty }^{t} {dt^{\prime}} \,} \exp \left( {i\,\omega^{\prime}\,t^{\prime}} \right)\,\int\limits_{ - \infty }^{\infty } {d\tilde{\omega }\,E\left( {\tilde{\omega }} \right)\exp \left( { - i\,\tilde{\omega }\,t^{\prime}} \right)\,\frac{{\left| {d_{n1} } \right|^{2} }}{{\tilde{\omega }_{n1} - \tilde{\omega }}}}$$

(25)

$$A_{2} \left( t \right) = - i\sum\limits_{n} {\int\limits_{ - \infty }^{t} {dt^{\prime}} \,} \exp \left( {i\,\omega^{\prime}\,t^{\prime}} \right)\,\int\limits_{ - \infty }^{\infty } {d\tilde{\omega }\,E\left( {\tilde{\omega }} \right)\exp \left( { - i\,\tilde{\omega }\,t^{\prime}} \right)\,\frac{{\left| {d_{n1} } \right|^{2} }}{{\tilde{\omega }_{n1} + \omega^{\prime}}}}$$

(26)

Substituting (25) and (26) in (22), we obtain:

$$A_{11}^{sc} \left( t \right) = - i\left( {{\mathbf{e^{\prime}}}^{ * } \,{\mathbf{e}}} \right)\,E_{0}^{vac} \left( {\omega^{\prime}} \right)\,D\left( {t,\omega^{\prime}} \right)$$

(27)

$$D\left( {t,\omega^{\prime}} \right) = \int\limits_{ - \infty }^{t} {dt^{\prime}\,\exp \left( {i\,\omega^{\prime}\,t^{\prime}} \right)\,\int\limits_{ - \infty }^{\infty } {c_{11} \left( {\omega^{\prime},\tilde{\omega }} \right)\,E\left( {\tilde{\omega },\omega ,\tau } \right)\,\exp \left( { - i\,\tilde{\omega }\,t^{\prime}} \right)\frac{{d\tilde{\omega }}}{2\,\pi }\,\,} }$$

(28)

It should be noted that the function (14) is an incomplete Fourier transform of the dipole moment induced by an USP.

The scattering tensor c₁₁ in the formula (28) is equal to

$$c_{11} \left( {\omega^{\prime},\tilde{\omega }} \right) = \sum\limits_{n} {\left| {d_{n1} } \right|^{2} \,\left\{ {\frac{1}{{\omega_{n1} - \tilde{\omega } - i\,\gamma_{n1} }} + \frac{1}{{\omega_{n1} + \omega^{\prime} - i\,\gamma_{n1} }}} \right\}}$$

(29)

For the differential scattering probability, after averaging over the polarization of the scattered radiation we have

$$dW\left( t \right) = \left\langle {\left| {A_{11}^{sc} \left( t \right)} \right|^{2} } \right\rangle_{{{\mathbf{e^{\prime}}}}} \,d\Gamma^{\prime}$$

(30)

$$\left\langle {\left| {A_{11}^{sc} \left( t \right)} \right|^{2} } \right\rangle_{{{\mathbf{e^{\prime}}}}} \, = \left( {1 - {\mathbf{n^{\prime}e}}} \right)\frac{{\omega^{\prime}}}{\Lambda }\,\left| {D\left( {t,\omega^{\prime}} \right)} \right|^{2}$$

(31)

where ${\mathbf{n^{\prime}}}$ is the unit vector in the direction of scattered radiation,

$$d\Gamma^{\prime} = \Lambda \,\frac{{\omega^{{\prime}{2}} }}{{8\,\pi^{3} \,c^{3} }}\,d\omega^{\prime}\,d\Omega_{{{\mathbf{n^{\prime}}}}} ,$$

(32)

is the differential statistical weight corresponding to scattered radiation emitted in a given spectral range and a unit solid angle around the ${\mathbf{n^{\prime}}}$ direction. After the integration over the solid angle, we finally arrive at the expression

$$\frac{dW\left( t \right)}{{d\omega^{\prime}}} = \frac{{2\,\omega^{{\prime}{3}} }}{{3\,\pi \,c^{3} }}\,\left| {D\left( {t,\omega^{\prime}} \right)} \right|^{2}$$

(33)

This equation, in view of (28), coincides with the formula (1).