Excitation of Rydberg wave packets in the tunneling regime

In the tunneling regime for strong laser field ionization of atoms, experimental studies have shown that a substantial fraction of atoms survive the laser pulse in many Rydberg states. To explain the origin of such trapping of population into Rydberg states, two mechanisms have been proposed: the first involves ac-Stark-shifted multiphoton resonances, and the second, called frustrated tunneling ionization, leads to the recombination of tunneled electrons into Rydberg states. We use a very accurate spectral method based on complex Sturmian functions to solve the time-dependent Schrödinger equation for hydrogen in a linearly polarized infrared pulse and to calculate the tunneling probability in terms of the atomic ground-state width. We examine the probability of excitation into Rydberg states as a function of the peak intensity for various pulse durations and two wavelengths, 800 and 1800 nm, and we try to explain the results in light of the two aforementioned mechanisms. For long pulses of 800 nm wavelength, the extreme sensitivity of the trapping of population into high-lying Rydberg states to the peak intensity, the well-defined value, and parity of the angular momentum of the populated Rydberg states and the presence of Freeman resonances can be explained using a multiphotonic excitation mechanism. For strong pulses of 1800 nm wavelength, in the so-called adiabatic or quasistatic tunneling regime, the oscillations of the excitation probability as a function of intensity are in phase opposition to the ionization probability, and we observe a migration toward high values of the angular momentum with different distributions in the angular momentum at the maxima and minima of the oscillations. We also present a detailed study of how the excited-state wave packet builds up in time during the interaction of the atom with the pulse.

Figure 1

Ionization yield as a function of frequency for the interaction of atomic hydrogen with a two-optical-cycle cosine-squared pulse. Two peak intensities are considered: ${10}^{14}$ W/${\mathrm{cm}}^2$ (blue lines) and $4\times {10}^{14}$ W/${\mathrm{cm}}^2$ (red lines). The solid lines have been obtained by solving numerically the TDSE. The broken lines correspond to the adiabatic limit given by Eq. (1).

Figure 2

Probability of ionization (solid blue curve) and excitation (red curve) of atomic hydrogen exposed to a 40-cycle pulse of 800 nm wavelength as a function of the peak intensity, evaluated by solving the TDSE. The pulse has a two-cycle sine squared turn on and off with a total duration of 40 cycles. The dashed blue curve shows the probability of ionization in the adiabatic limit as given by Eq. (1). The vertical dashed lines indicate the position of the successive $N\mathrm{th}$-photon thresholds.

Figure 3

Bar chart of the excited-state population resulting from the interaction of atomic hydrogen with the same pulse as in Fig. 2. Populations are given as a function of $n$, the principal quantum number, and $\ell$, the angular quantum number for two values of the peak intensity just above the $14\mathrm{th}$ threshold: (a) $I_{\mathrm{peak}}=1.405\times {10}^{14}$ W/${\mathrm{cm}}^2$ and (b) $I_{\mathrm{peak}}=1.345\times {10}^{14}$ W/${\mathrm{cm}}^2$.

Figure 4

Bar chart of the excited-state population resulting from the interaction of atomic hydrogen with the same pulse as in Fig. 2. Populations are given as a function of $n$, the principal quantum number, and $\ell$, the angular quantum number for $I_{\mathrm{peak}}=1.335\times {10}^{14}$ W/${\mathrm{cm}}^2$, an intensity just above but almost at the 14th threshold.

Figure 5

Total probability of ionization (solid blue curve) and excitation (red curve) of atomic hydrogen exposed to a 10-cycle cosine-squared pulse of 800 nm wavelength as a function of the peak intensity, evaluated by solving the TDSE. The dashed blue curve shows the total probability of ionization in the adiabatic limit as given by Eq. (1). The vertical dashed lines indicate the position of the successive $N\mathrm{th}$-photon thresholds with $N$ given by the number at the bottom of each vertical line.

Figure 6

Total probability of ionization (blue lines), excitation (red line), and to stay in the ground state (black lines) for atomic hydrogen exposed to a two-cycle cosine-squared pulse of 800 nm wavelength as a function of the peak intensity. The solid lines are obtained by solving the TDSE and the dashed lines give the adiabatic limit [from Eq. (1)].

Figure 7

Total probability of ionization (solid blue curve) and excitation (red curve) of atomic hydrogen exposed to a six-cycle cosine-squared pulse of 1800 nm wavelength as a function of the peak intensity, evaluated by solving the TDSE. The blue dots show the total probability of ionization in the adiabatic limit as given by Eq. (1).

Figure 8

Excited-state population resulting from the interaction of atomic hydrogen with the same pulse as in Fig. 7. These populations are given as a function of the angular momentum quantum number for (a) $I_{\mathrm{peak}}=4\times {10}^{14}$ W/${\mathrm{cm}}^2$ and (c) $I_{\mathrm{peak}}=4.2\times {10}^{14}$ W/${\mathrm{cm}}^2$. In (b) and (d), these populations are shown a function of the principal quantum number for the same peak intensities, respectively.

Figure 9

Time evolution of the excited-state wave packet resulting from the interaction of atomic hydrogen with a four-cycle cosine-squared pulse of 800 nm wavelength and $4\times {10}^{14}$ W/${\mathrm{cm}}^2$ peak intensity. The left panels show a plot of the modulus squared of this wave packet in the $x-z$ plane at two times during the interaction with the pulse: (a) $t=-1.5$ optical cycles and (b) $t=0$ (middle of the pulse), for which the vector potential is zero. The arrow indicates the electric-field vector. The right panels (b) and (d) give the corresponding $n$ and $\ell$ distribution of the populations.

Figure 10

Time evolution of the excited-state wave packet resulting from the interaction of atomic hydrogen with a four-cycle cosine squared pulse of 800 nm wavelength and $4\times {10}^{14}$ W/${\mathrm{cm}}^2$ peak intensity. The left panels show a plot of the modulus squared of this wave packet in the $x-z$ plane at two times during the interaction with the pulse: (a) $t=1.5$ optical cycles and (b) $t=2$ optical cycles (end of the pulse) for which the vector potential is zero. The arrow indicates the electric-field vector. The right panels (b) and (d) give the corresponding $n$ and $\ell$ distribution of the populations.