Theories of photoelectron correlation in laser-driven multiple atomic ionization

Experimental advances with laser intensities above $1\mathrm{TW}/{\mathrm{cm}}^2$, with pulse durations between roughly 50 and 5 fs, have led to the discovery of new atomic effects that include examples of startlingly high electron correlation. These phenomena have presented an unexpected theoretical challenge as they lie outside the domains of both of the nominally applicable theories, namely, straightforward perturbative radiation theory and quasistatic tunneling theory. The two liberated electrons present a new few-body collective effect. When they are not released independently, one by one, the term nonsequential double ionization has been adopted. Theoretical avenues of attack have emerged in two categories, which are strikingly different. They can be labeled as “all-at-once” and “step-by-step” approaches. Although different, even conceptually opposite in some ways, both approaches have been successful in confronting substantial parts of the experimental data. These approaches are examined and compared with their results in addressing key experimental data obtained over the past decade.

Figure 1

Dots and crosses show measured total ion yields of ${\mathrm{He}}^{+}$ and ${\mathrm{He}}^{2+}$ (the measured intensities are multiplied by 1.15). Solid and dashed lines with the data represent the results of single-active-electron and tunneling calculations. The solid line to the right shows the result expected on the basis of sequential ionization. The experimental results display the NSDI “knee” signature of the ion-count data. From 220.

Figure 2

An artist’s sketch of a recollision process leading from tunneling emission of one electron (a), through its laser guidance back onto the ion (b), to recollision and double ejection (c).

Figure 3

Two-dimensional momentum distributions $(P_{\parallel },P_{\perp })$ of ${\mathrm{Ne}}^{N+}$ ions ($N=1$, 2, 3) at peak intensities of $1.3\mathrm{PW}/{\mathrm{cm}}^2$ ($N=1$, 2) and $1.5\mathrm{PW}/{\mathrm{cm}}^2$ ($N=3$) and a wavelength of 795 nm [(a)–(c), respectively]. The distributions are integrated over the third Cartesian ion-momentum coordinate. Note how the distribution dramatically widens in the transition from single to multiple ionization. From 159.

Figure 4

Quantum TDSE (left) and classical TDNE (right) spatial distributions of positions of electrons undergoing NSDI midway through a laser pulse. The aligned-electron approximation (AEA; see Sec. 4a) and the same model potentials, laser wavelength, pulse shape, and intensity were employed for both calculations. Particularly notable is the common emergence of NSDI “jets” of electron pairs into the upper right quadrants. From 170.

Figure 5

Momentum-momentum correlation of the two electrons emitted in NSDI of argon below saturation by a Ti:sapphire laser with a peak intensity of $3.8\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$. From 228.

Figure 6

Longitudinal ion-momentum distribution for double ionization of (a)–(c) Ar and (d) Ne at 1300 nm (solid lines) and 800 nm (open circles). Intensities in $\mathrm{PW}/{\mathrm{cm}}^2$ are given in the panels. From 95.

Figure 7

Time evolution of the total energy of each NSDI electron during the first five laser cycles. The top panel of each column plots the time evolution of the laser electric field $E(t)$ with the peak amplitude $E_0$. The remaining panels of each column display the energy $E$ and the transverse displacement $x$ of the two NSDI electrons of a trajectory. The second and third rows depict typical NSDI events with comparatively long ($t_{\mathrm{rec}}-t_0>0.5T$) and short ($t_{\mathrm{rec}}-t_0<0.5T$) “travel” time intervals, respectively. The horizontal axes are labeled by multiples of the laser period $T$. The plots are obtained with the soft-core Coulombic $e\mathrm{\text{−}}e$ repulsion potential (16). The light and dark curves refer to the recolliding and the bound electron, respectively. From 97.

Figure 8

Snapshots showing two-electron dynamics near the end of a double-ionization scenario, calculated in the aligned-electron approximation. In the first panel one electron (solid dot) is still deeply bound to the core, and the other has already been ejected and is returning from a far-left excursion under laser control, after it has been accelerated by the laser field during the preceding half cycle. It approaches, slowing down as it is now moving against the field and the laser potential rises, and collides in the fourth row with enough energy to liberate the other one, so that both exit together to the left in the last panel. The horizontal axis shows position and the vertical axis shows total energy, both in atomic units, while the arrow indicates the direction and strength of the laser field. The dashed and solid curves indicate the combined potential of nucleus, $e\mathrm{\text{−}}e$ repulsion, and laser field felt by the electrons, i.e., $V_{n,i}(x_i)+V_{ee}(x_{12})+x_{i}E(t)$ as a function of $x_i$, with the other electron being in the position indicated. The plots extend over slightly more than a quarter cycle. Time increases from left to right and from top to bottom, as illustrated by the arrow that specifies the electric field. From 171.

Figure 9

(a) The Feynman diagram proposed by 120 and by 12 that describes the quantum SFA amplitude (20), which implements the recollision–impact-ionization (RII) pathway, which is the Corkum recollision scenario. Time goes from left to right. The single lines represent the two electrons in their initial field-free bound states. Double lines represent electrons interacting with the field but not with the ion or each other (Volkov states). The vertical wavy line stands for the $e\mathrm{\text{−}}e$ interaction $V_{ee}$ (labeled $V_{12}$). (b) Diagram that describes the recollision-excitation and subsequent ionization (RESI) pathway: in the $e\mathrm{\text{−}}e$ interaction, the second electron is promoted into an excited bound state $|{\psi }_e^{(2)}⟩$, from which it tunnels out at the later time $t_1$. From 63.

Figure 10

The ability of a wide variety of NSDI theories demonstrated to recover the signature knee effect. From the top left they are taken from 35, 227, 13, 150, and 100 as discussed in the text.

Figure 11

Electron-momentum distribution for singly and doubly ionized helium at 390 nm and $0.8\mathrm{PW}/{\mathrm{cm}}^2$ (upper panel) and $1.1\mathrm{PW}/{\mathrm{cm}}^2$ (lower panel), experimental results (dashed curves) and TDSE (solid curves). For double ionization, the momentum of just one electron is plotted. From 175.

Figure 12

TDSE calculation of the distribution of the momenta $p_i=\sqrt{2E_i}$ ($i=1$, 2) of the two electrons emitted in double ionization at the end of a 7-period pulse with $1.0\mathrm{PW}/{\mathrm{cm}}^2$. The straight lines correspond to an apparent cutoff energy of $1.9U_P$ of the electrons coincident with single ionization. The circular arc is drawn at a total kinetic energy of $5.3U_P$ in order to serve as a cutoff for the total energy. From 175.

Figure 13

First quadrant of the electron momentum-momentum correlation for NSDI of helium at $4.5\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ and 800 nm. The lines specify momenta of $2\sqrt{U_P}$. The triangles and circles present the kinematical boundaries obtained from a simple classical $(e,2e)$ model. Namely, the recolliding first-liberated electron donates the energy corresponding to the second ionization potential to the second electron, which appears in the continuum with zero kinetic energy and is then accelerated by the field. In the process, the first electron either backscatters (dots) or forwardscatters (triangles). From 211.

Figure 14

Electron-electron correlations calculated from the TC model by 237, intended to model the experiments of 195 and 211. The laser intensity is $4.5\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ at 800 nm. See the text for further explanation and interpretation. From 237.

Figure 15

Distribution of the ion momentum $\mathbf{P}$ in NSDI of neon at 795 nm calculated from the FD approach with zero-range interaction for all potentials, to be compared with Fig. 3. For a better simulation of the experiment, for the intensity the lower value $8\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ was taken. (a) Linear-scale density plot with ten equidistant contour lines. The horizontal line at ${\mathbf{P}}_{\perp }=0$ and the four vertical lines at various values of $P_{\parallel }$ mark the positions of cuts along which the distribution is plotted in (b) and (c). From 115.

Figure 16

Correlation function of the electron-momentum components parallel to the laser field for NSDI, plotted on the scale of $\sqrt{U_P}$, with the transverse components not measured (in the data) or integrated over (in the theory). (a) Measurements for neon at $1.0\mathrm{PW}/{\mathrm{cm}}^2$ and 800 nm [rescaled from 161]. Middle panels: three-dimensional (b) and two-dimensional (c) TDNE results for a two-electron “neon” atom exposed to a 780-nm 8-cycle trapezoidal laser pulse with an intensity of $0.3\mathrm{PW}/{\mathrm{cm}}^2$. The TDNE calculation is based on the analysis of more than $10\times {10}^6$ two-electron trajectories with the $e\mathrm{\text{−}}e$ smoothing parameter $b=0.1$ in the Hamiltonian (4) [from 97]. Lower panels: FD results computed from the $S$-matrix theory of Sec. 4c for $\omega =1.55\mathrm{eV}$, $I=5.5\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ ($U_P=32.6\mathrm{eV}$), and the ionization potentials $I_{P1}=21.5\mathrm{eV}$ and $I_{P2}=41\mathrm{eV}$ corresponding to neon. The electron-electron interaction is modeled by the contact interaction (23) (d) or by the Coulomb interaction (22) (e). From 77.

Figure 17

Momentum distributions of the doubly charged neon ion from three-dimensional (left column) and two-dimensional (middle column) TDNE calculations and from the measurements of 242 (right column). The data were collected for a peak intensity of $1.5\mathrm{PW}/{\mathrm{cm}}^2$ at 795 nm, the intensity used for the simulations was $0.3\mathrm{PW}/{\mathrm{cm}}^2$. The uppermost row exhibits the two-dimensional ion-momentum distributions with one component not observed (in the data) or integrated over (in the 3D simulations). The middle row displays the distribution of the longitudinal momentum, and the lowermost row the distribution of the pertinent transverse component. From 97.

Figure 18

Time evolution of the total energy, including dipole-interaction energy, of each NSDI electron during the first eight laser cycles. There are four clear stages of an NSDI process, as described in the text. From 103.

Figure 19

COLTRIMS data from neon and argon are “matched” to TDNE results by hand, while full-blown TDNE calculations at two different laser intensities give similar results to the two helium panels, with an exaggeration of the Z contribution. From 103.

Figure 20

Parallel-momentum correlation of the two electrons emitted in NSDI of argon (a) ($3\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$) and neon (c) ($1.5\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$). (b) Transverse-momentum distributions in coincidence with ${\mathrm{Ar}}^{1+}$ (dotted line) and ${\mathrm{Ar}}^{2+}$ at $3\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$ (solid line) and $7\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$ (dashed solid line) as well as (d) for ${\mathrm{Ne}}^{1+}$ and ${\mathrm{Ne}}^{2+}$ at $1.5\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$. From 146.

Figure 21

Recollision events under elliptical polarization. Top: A nearly perfectly elliptical trajectory by one electron makes a recollision in the first postionization cycle, and both electrons are freed. Bottom: A sequence of near-elliptical circulations is pulled by nuclear attraction into a double-ionizing recollision after several near misses. From 225.

Figure 22

Top panels: Predicted longitudinal and transverse ion-momentum distributions from 223 for $\epsilon =0.5$. Bottom panels: Experimental results from 148 for Ne and Ar atoms and $\epsilon \sim 0.8$. The experimental distributions have only a single broad peak along the $P_x$ axis, consistent with the theoretical curve in the top left panel. The arrows indicate four transverse $P_y$ peak positions of the type predicted in the top right panel.

Figure 23

Experimental (circles) ${\mathrm{Ne}}^{+}$ to ${\mathrm{Ne}}^{8+}$ (left to right) yields and the corresponding ADK yields (dashed lines) as well as a calculation including the relativistic drift (solid lines). Knee signatures are clearly visible for ${\mathrm{Ne}}^{2+}$ to ${\mathrm{Ne}}^{5+}$. From 168.

Figure 24

Two-electron and three-electron longitudinal ion-momentum distributions for double (DI) and triple (TI) ionization of argon by 780 nm laser pulses at $0.8\mathrm{PW}/{\mathrm{cm}}^2$. The left-hand panel shows the results of TDNE calculations, the right-hand panel the data of 242. From 101.

Figure 25

Distribution of the longitudinal ion momentum for NSQI of argon at $1.2\mathrm{PW}/{\mathrm{cm}}^2$ (upper panel) and $1.5\mathrm{PW}/{\mathrm{cm}}^2$ (lower panel). The curves (squares) are the experimental data from Fig. 1 of 242. The remaining curves are calculated from Eq. (29) for $\Delta t=0$ and $\Delta t=0.265$. From 142.