Non-stationary scattering theory and parametric resonance in ultracold Rydberg plasmas subjected to a perturbation by a terahertz acoustic wave

Abstract

Within the framework of a non-stationary scattering theory formulas for electron scattering cross sections on an arbitrary time-periodic potential are derived. These formulas to a weakly ionized ultracold Rydberg plasma subjected to a perturbation by a terahertz acoustic wave and a laser field are applied. Our result for laser-assisted electron-atom scattering generalizes the Kroll-Watson formula to the case of an elliptically polarized electromagnetic wave. Based on the kinetic approach we discuss a mechanism of the scattering of plasma charges on neutral atoms, vibrating in the field of the external acoustic wave, leading to a renormalization of the electron plasma frequency. Additionally, within the framework of the hydrodynamic approach, we demonstrate the possibility of a parametric resonance amplification of electron and ion density oscillations in weakly ionized two component ultracold Rydberg plasmas under the influence of a laser field and as high-frequency acoustic wave.

Graphic abstract

Appendix: Calculation the correction \(\delta f_e(\varvec{k}, \omega )\) to the Maxwellian electron distribution function

To find the correction $\delta f_e(\varvec{k}, \omega )$ we need to evaluate the integral in Eq. (56):

$$\begin{aligned} \varvec{J}= & {} \int \varvec{E}(\varvec{r},t)\mathrm{e}^{F(t)}dt = \int \varvec{E}(\varvec{r},t) \exp \left\{ (\nu ^{(0)} + i\varvec{k}\varvec{v})t \right. \nonumber \\&\left. + \sum _{n >-[n_k], n \ne 0}^{\infty }\frac{{{{\tilde{\nu }}}}_{ea}(v, n)}{n\omega _0} \sin (n\omega _0 t - \pi n/2)\right\} dt,\nonumber \\ \end{aligned}$$

(100)

which can be represented in the form

$$\begin{aligned} \varvec{J}= & {} \int \varvec{E}(\varvec{r},t)\mathrm{e}^{(\nu ^{(0)} + i\varvec{k}\varvec{v})t} \nonumber \\\times & {} \prod _{n >-[n_k], n \ne 0}^{\infty }\exp \left\{ \frac{{{{\tilde{\nu }}}}_{ea}(v, n)}{n\omega _0} \sin (n\omega _0 t - \pi n/2)\right\} dt.\nonumber \\ \end{aligned}$$

(101)

Using the well-known identity for Bessel functions

$$\begin{aligned} \mathrm{e}^{iz\sin (\varphi )} = \sum _{l = -\infty }^{\infty }J_l(z)\mathrm{e}^{il\varphi } \end{aligned}$$

(102)

and the Fourier transformation (58) for the electric field $\varvec{E}(\varvec{r},t)$, we obtain that

$$\begin{aligned} \varvec{J}= & {} \prod _{n >-[n_k], n \ne 0}^{\infty }\sum _{l = -\infty }^{\infty }(-1)^l J_l\left( i\frac{{{{\tilde{\nu }}}}_{ea}(v, n)}{n\omega _0}\right) \nonumber \\\times & {} \int _{-\infty }^{\infty } \frac{d\varvec{k}d\omega ^{'}}{(2\pi )^4}\varvec{E}(\varvec{k}, \omega ^{'}) \frac{\mathrm{e}^{(\nu ^{(0)} + i\varvec{k}\varvec{v} - i\omega ^{'} + il\omega _0 S)t + i\varvec{k}\varvec{r}}}{\nu ^{(0)} + i\varvec{k}\varvec{v} - i\omega ^{'} + il\omega _0 S} \,\nonumber \\ \end{aligned}$$

(103)

where $S = \sum _{n >-[n_k], n \ne 0}^{\infty }n$. It is obviously that $S \rightarrow +\infty$ and then, from Eq. (103), we have

$$\begin{aligned} \varvec{J}= & {} \prod _{n >-[n_k], n \ne 0}^{\infty }I_0\left( \frac{{{{\tilde{\nu }}}}_{ea}(v, n)}{n\omega _0}\right) \int _{-\infty }^{\infty }d\varvec{k}d\omega ^{'}\varvec{E}(\varvec{k}, \omega ^{'}) \nonumber \\&\times \frac{\mathrm{e}^{(\nu ^{(0)} + i\varvec{k}\varvec{v} - i\omega ^{'})t + i\varvec{k}\varvec{r}}}{\nu ^{(0)} + i\varvec{k}\varvec{v} - i\omega ^{'}} \, \end{aligned}$$

(104)

where we took into account that $J_0(iz) = I_0(z)$. Multiplying $\varvec{J}$ in Eq. (104) by $\exp \{-F(t)\}$ and using the inverse Fourier transformation (59) for the function $\delta f_e(\varvec{r}, t)$ in Eq. (56), we get the result (60).