Relativistic ionization characteristics of laser-driven hydrogenlike ions

In this contribution, we investigate the relativistic ionization characteristics of highly charged hydrogenlike ions in short intense laser pulses as a function of the laser pulse parameters by means of the numerical solution of the time-dependent Dirac equation and the time-dependent Klein-Gordon equation as well as by the classical phase-space averaging method. For this purpose, we generalize the phase-space averaging method such that it is applicable to relativistically driven particles in arbitrary central potentials. If the ionization probability is not too small, quantum mechanical and classical methods give similar results for laser wavelengths in the range from the near-infrared to soft x-ray radiation. We find that ionization in few-cycle intense laser pulses depends sensitively on the pulses’ peak intensity but little on the pulse tails and on the pulse energy. The ionization probability is shown to be strongly linked to the peak intensity allowing for an estimation of the laser intensity via ionization yields.

Figure 1

Illustration of the employed envelope functions (3), from top to bottom $w_{\mathrm{rect}}(\eta ,10)$, $w_{\sin {}^2}(\eta ,10)$, and $w_{\mathrm{ramp}}(\eta ,10,2)$.

Figure 2

Ionization probability $p_{\mathrm{ion}}$ as a function of the laser intensity $I$ for an electron in a two-dimensional soft-core potential (5) calculated by two different computational approaches: the solution of the time-dependent Dirac equation and the relativistic phase-space averaging method (RPSAM). Calculations are performed with different atomic numbers $Z$. Soft-core parameters $\zeta$ were chosen such that the ground-state energy equals the ground-state energy of the Coulomb potential with the same $Z$. In all cases, the laser field is made up of a pulse with envelope $w_{\mathrm{rect}}(\eta ,2)$ and has a wavelength of $\lambda =6.5\hspace{0.5em}\mathrm{nm}$.

Figure 3

Ionization probability $p_{\mathrm{ion}}$ as a function of the laser intensity $I$ for an electron in a one-dimensional soft-core potential (left column) and in a two-dimensional soft-core potential (right column) calculated by two different computational approaches: the solution of the time-dependent Klein-Gordon equation and the relativistic phase-space averaging method (RPSAM). Calculations are performed with the atomic number $Z=20$ and the soft-core parameters $\zeta =0.06$ (one-dimensional soft-core potential) and $\zeta =0.032$ (two-dimensional soft-core potential). In all cases, the laser field is made up of a pulse with envelope $w_{\mathrm{rect}}(\eta ,1)$. Wavelengths lie in the near-infrared ($\lambda =911\hspace{0.5em}\mathrm{nm}$), in the ultraviolet ($\lambda =91\hspace{0.5em}\mathrm{nm}$), and in the soft x-ray range ($\lambda =5\hspace{0.5em}\mathrm{nm}$).

Figure 4

Snapshots of the Dirac electron probability density for an electron in a soft-core potential (5) with $Z=10$ and $\zeta =0.079a_0$ on a logarithmic scale for a field strength of $E_{L,\max }=5.14\times {10}^{13}\hspace{0.5em}\mathrm{V}/\mathrm{m}$ corresponding to an average intensity of $3.52\times {10}^{20}\hspace{0.5em}\mathrm{W}/{\mathrm{cm}}^2$ at different points in time: (a) $t=1/4\hspace{0.5em}T$, (b) $t=2/4\hspace{0.5em}T$, (c) $t=3/4\hspace{0.5em}T$, and (d) $t=4/4\hspace{0.5em}T$, with $T=\lambda /c$ and $\lambda =17.6\hspace{0.5em}\mathrm{nm}$. Length scales of the propagation direction ($x$ axis) and the polarization direction ($y$ axis) are given in units of the Bohr radius $a_0$; the pulse shape is formed by the envelope function $w_{\mathrm{ramp}}(\eta ,4,1)$.

Figure 5

Snapshots of the Dirac electron probability density for an electron in a soft-core potential (5) with $Z=10$ and $\zeta =0.079a_0$ on a logarithmic scale for various field strengths and intensities, respectively: (a) $E_{L,\max }=3.07\times {10}^{13}\hspace{0.5em}\mathrm{V}/\mathrm{m}$, $I=1.25\times {10}^{20}\hspace{0.5em}\mathrm{W}/{\mathrm{cm}}^2$; (b) $E_{L,\max }=5.14\times {10}^{13}\hspace{0.5em}\mathrm{V}/\mathrm{m}$, $I=3.51\times {10}^{20}\hspace{0.5em}\mathrm{W}/{\mathrm{cm}}^2$; (c) $E_{L,\max }=1.03\times {10}^{14}\hspace{0.5em}\mathrm{V}/\mathrm{m}$, $I=1.41\times {10}^{21}\hspace{0.5em}\mathrm{W}/{\mathrm{cm}}^2$; (d) $E_{L,\max }=1.54\times {10}^{14}\hspace{0.5em}\mathrm{V}/\mathrm{m}$, $I=3.15\times {10}^{21}\hspace{0.5em}\mathrm{W}/{\mathrm{cm}}^2$. The snapshots were taken at $T/4$ with $T=\lambda /c$ and $\lambda =17.6\hspace{0.5em}\mathrm{nm}$, length scales are given in units of the Bohr radius $a_0$, and the pulse shape is formed by the envelope function $w_{\mathrm{rect}}(\eta ,2)$.

Figure 6

Ionization probability as a function of the mean laser intensity $I$ and the maximal electric field strength $E_{L,\max }$ for hydrogenlike ions of different atomic numbers $Z$ and a laser pulse with wavelength of $\lambda =6.5\hspace{0.5em}\mathrm{nm}$. The atomic number $Z$ increases from left to right. The pulse shape is formed by the envelope function $w_{\mathrm{rect}}(\eta ,2)$. Ionization profiles have been obtained by averaging over 10 000 trajectories or more per data point. Kinks in the ionization profiles correspond to changes in the decay rate of the occupation probability $1-p_{\mathrm{ion}}$, see Sec. 4b2.

Figure 7

Occupation probability $1-p_{\mathrm{ion}}$ as a function of the scaled intensity $I/Z^6$ for hydrogenlike ions of different nuclear numbers $Z$ and a laser pulse with scaled wavelength of $\lambda =(1\hspace{0.5em}\mu m)/Z^2$. Ionization profiles for small $Z$ (nonrelativistic limit) collapse onto a universal curve; deviations from this curve are caused by relativistic effects. The pulse shape is formed by the envelope function $w_{\mathrm{rect}}(\eta ,2)$. The noise in the tails of the distributions is due to the statistical uncertainties by the relativistic phase-space averaging method. Ionization profiles have been obtained by averaging over 50 000 trajectories per data point.

Figure 8

Occupation probability $1-p_{\mathrm{ion}}$ as a function of the intensity $I$ for hydrogenlike ions with $Z=30$. The applied laser pulse has a shape function $w_{\mathrm{rect}}(\eta ,2)$ and wavelengths of various multiples of ${\lambda }_0=6.5$ nm. The intensity range is expected to be accessible to high-intensity laser facilities of the near future.

Figure 9

Probability histogram $\rho (t_{\mathrm{ion}})$ of the moment of ionization for ionization of hydrogenlike ions with $Z=25$ in laser pulses of wavelength $\lambda =1054$ nm, peak field strength $E_{L,\max }=7\times {10}^{14}\hspace{0.5em}\mathrm{V}/\mathrm{m}$, and intensity profile proportional to $[w_{\mathrm{ramp}}(\eta ,4,1)\sin (\eta )]{}^2$ (solid line, left axis) and the intensity near the ionic core (dashed line, right axis). $T$ denotes the laser period $T=\lambda /c$. Ionization happens most likely at the steepest intensity ascent.

Figure 10

(a) Pair of intensity profiles with the same pulse energy. (b), (c) Two pairs of laser pulses with similar intensity profile in the pulse’s bulk but with different tails. For comparison of the laser pulses, the horizontal positions of the pulses in (a) and (c) have been shifted such that the pulse maxima are at the same position.

Figure 11

Occupation probability $1-p_{\mathrm{ion}}$ as a function of the intensity $I=c{\varepsilon }_0{E}_{L,\max }^2/2$ for hydrogenlike ions with $Z=30$ and laser pulses of wavelength $\lambda =6.5$ nm and pulse shapes shown in Fig. 10a. Laser pulses with same pulse energy but different intensity profiles result in distinct ionization profiles.

Figure 12

Occupation probability $1-p_{\mathrm{ion}}$ as a function of the intensity $I=c{\varepsilon }_0{E}_{L,\max }^2/2$ for hydrogenlike ions with $Z=30$ and laser pulses of wavelength $\lambda =6.5$ nm and pulse shapes shown in Figs. 10b and 10c. Laser pulses with similar intensity profiles in the pulse bulk result in similar ionization profiles.

Figure 13

Laser pulses with the intensity profiles proportional to $[w_{\sin {}^2}(\eta ,3)\sin (\eta +{\alpha }_L)]{}^2$ but with different phases ${\alpha }_L$. Despite the different phases, all three pulses carry the same energy.

Figure 14

Occupation probability $1-p_{\mathrm{ion}}$ as a function of the intensity $I=c{\varepsilon }_0{E}_{L,\max }^2/2$ for hydrogenlike ions with $Z=30$ and laser pulses of wavelength $\lambda =6.5$ nm and pulse shapes shown in Fig. 13. Ionization profiles are affected marginally by the carrier phase.

Figure 15

Occupation probability $1-p_{\mathrm{ion}}$ as a function of the intensity $I$ for $Z=30$. The solid line shows results predicted by the WKB theory, while the dashed line show $1-p_{\mathrm{ion}}$ as obtained by the application of the relativistic phase-space averaging method (RPSAM).