Ionization from Rydberg atoms and wave packets by scaled terahertz single-cycle pulses

The strong-field ionization behavior when a Rydberg atom is exposed to a terahertz single-cycle pulse is studied. Fully three-dimensional time-dependent Schrödinger equations and classical trajectory Monte Carlo calculations are performed. Results from stationary eigenstates and Rydberg wave packets are presented, and it is found that the ionization properties can be different for the two cases. All of the pulse parameters and physical quantities are scaled versus the principal quantum number $n$. The ionized electron's scaled radial, energy, and angular distributions are investigated for different $n$, and the quantum results are interpreted using a semiclassical method. The scaling relations of quantum interference amplitudes are discussed.

Figure 1

Ionization probabilities from scaled classical and quantum calculations. The black dashed line is for classical calculations with an initial energy spread of $n_{\text{cl}}$ given in Eq. (11). The blue solid line is for classical calculations at a single value of $n$, and the ionization probability converges to $14.6\%$. The magenta dotted line is for quantum calculations up to $n=70$. The red thick line is a fitting for classical results with $n_{\text{cl}}$ spread. For $n_{\text{cl}}⩾60$, the fitting function is $P_{\text{ion}}=0.146+4573/n^{3.02}$. The inset figure gives the classical ionization probability when the pulse parameters are fixed at $n_0=15$ and the initial classical energy state is at a single value around $n_{\text{cl}}=15$, see Eq. (14). Scaling relations are given in Eq. (13). Pulse parameters can be found in the text. The initial angular momentum is fixed at $l=2$.

Figure 2

Full radial distributions at the final time of the pulse, from three different calculations. The initial angular momentum is $l=2$. The blue solid curve is from a classical calculation with the initial $n_{\text{cl}}$ fixed at 15. The black dashed curve is from a classical calculation with the initial $n_{\text{cl}}$ being a spread of 14.5–15.5. The magenta dotted curve is from a full quantum calculation starting in a $15d$ state. The ionization probabilities from the three calculations are 14.3%, 20.7%, and 21.0%, respectively. The inset is a magnification of the distribution with $r_f$ from 700 to 5500 a.u. Note that the probability density scale is different for the inset.

Figure 3

Radial distributions at the final time of the pulse from classical and quantum calculations. Only those from the ionized part of the distributions are plotted. The red solid curve with small extent around 1600 a.u. is from a classical calculation with $n_{\text{cl}}$ fixed at 14.5, and the ionization probability for this case is 0.77%. The green solid curve with the largest extent is from a classical calculation with $n_{\text{cl}}$ fixed at 15.5, and the ionization probability for this case is 37.2%. The other three curves are the same calculations as those introduced in the caption of Fig. 2.

Figure 4

Correlated distribution of the ionized electron's radial position and its emission angle at the final time of the pulse. Parameters of the field are given in the text of Sec. 3a. Figure (a) is from a full quantum calculation, figure (b) is from a classical calculation with $n_{\text{cl}}$ being a spread of 14.5–15.5, and figure (c) is from a classical calculation with $n_{\text{cl}}$ fixed at 15. The density distributions in all three figures are normalized to their respective ionization probabilities, which can be found in the caption of Fig. 2.

Figure 5

Angular distributions for ionized electrons from quantum and classical calculations. The initial angular momentum is $l=0$. Calculations are performed with different $n$ as indicated by the legends and different scaled pulse parameters as indicated in Eq. (13). At $n=15$, $F_m=500$ kV/cm and $t_w=500$ a.u. are used. Classical results at all three $n$ are the same.

Figure 6

(a) Classical action differences between two paths vs the final angle, at a scaled final energy. The final energy is scaled for different $n$, which are $E_f={(15/n)}^2\times 0.002$ a.u. The action differences are aligned as $\mathrm{\Delta }S=0$ at $cos{\theta }_f=1$. (b) Angular distributions at scaled final energies from quantum calculations.

Figure 7

The correlated energy and angular distributions with scaled pulse parameters at $n=15$, 30, and 45 for quantum calculations, and at $n=30$ for classical calculation. The maximum densities are normalized to 1.0 for all figures. The dashed lines are at $cos{\theta }_f=0.95$, while the circles are interference maxima calculated by the semiclassical method introduced in the text.

Figure 8

Radial wave functions of the hydrogen $15s$, “$15s+16s,$” and “$15s-16s$” states. Note that the $15s$ and $16s$ states do not have the equal weight ($1/\sqrt{2}$) in the superpositions, see Eq. (17). The “$15s+16s$” represents a wave packet of ${\phi }_0=0$, while “$15s-16s$” represents ${\phi }_0=\pi$.

Figure 9

Ionization probabilities for $15s+exp\left(i{\phi }_0\right)16s$ wave packets vs the superposition phase ${\phi }_0$ as given in Eq. (17). (a) A short-duration pulse process with $F_m=2000$ kV/cm and $t_w=606$ a.u. (b) A medium-duration pulse process with $F_m=60$ kV/cm and $t_w=3029$ a.u., where the pulse duration is approximately one Rydberg period for $15s$ state: $7.0t_w=T_{\text{Ryd}}\approx 21206$ a.u. The black dotted lines are weighted averages of ionization probabilities for stationary $15s$ and $16s$ states, separately. The averages are 11.7% and 13.1% for the two processes, respectively. The red points are from quantum calculations, while the red dashed lines are their fittings. The fitting functions are $0.116+0.112cos({\phi }_0-0.057)$ and $0.130+0.123cos({\phi }_0-0.245)$ for the two processes, respectively. Details for the fitting functions can be found in the Appendix.

Figure 10

Energy distributions for ionizations from $15s+exp\left(i{\phi }_0\right)16s$ wave packets as introduced in Eq. (17). A short-duration single-cycle pulse with parameters in the caption of Fig. 9 is used. The results are plotted for ${\phi }_0=0$, $\pi$, and averages of ${\phi }_0$ ranging from 0 to $2\pi$.

Figure 11

Ionization probability from classical calculations vs the radial position $r$ of electron at $t=0$. The electron is initiated at energy of $n=15$ and angular momentum of zero, where the distance of the classical outer turning point is approximately $2n^2=450$ a.u. See text for details of $r$. Short- and medium-duration pulses are defined in the caption of Fig. 9. The radial distribution from a wave packet of $15s-16s$ with ${\phi }_0=\pi$, as shown in Fig. 8 and Eq. (17), is plotted as a reference. The radial distribution is plotted in arbitrary units.