Nonadiabatic effects in the double ionization of atoms driven by a circularly polarized laser pulse

We study the double ionization of atoms subjected to circularly polarized (CP) laser pulses. We analyze two fundamental ionization processes: the sequential (SDI) and nonsequential (NSDI) double ionization in the light of the rotating frame (RF) which naturally embeds nonadiabatic effects in CP pulses. We use and compare two adiabatic approximations: The adiabatic approximation in the laboratory frame (LF) and the adiabatic approximation in the RF. The adiabatic approximation in the RF encapsulates the energy variations of the electrons on subcycle timescales happening in the LF and this, by fully taking into account the ion-electron interaction. This allows us to identify two nonadiabatic effects including the lowering of the threshold intensity at which over-the-barrier ionization happens and the lowering of the ionization time of the electrons. As a consequence, these nonadiabatic effects facilitate over-the-barrier ionization and recollision-induced ionizations. We analyze the outcomes of these nonadiabatic effects on the recollision mechanism. We show that the laser envelope plays an instrumental role in a recollision channel in CP pulses at the heart of NSDI.

Figure 1

Double ionization (DI) probability of Hamiltonian (1) in 2D as a function of the laser intensity $I$ (solid thick curves) for $\mathrm{Mg}$ (${\mathcal{E}}_g=-0.83$, $a=3$) and CP pulses. The laser envelope is trapezoidal with duration of ramp-up $\tau$, plateau $6T-\tau$, and ramp-down $2T$. The blue (dark gray), pink (gray), and purple (light gray) curves are for $\tau =2T$, $3T$, and $4T$, respectively. The thin dashed-dotted and dashed curves are the probability of NSDI and SDI, respectively. The lower panels are the correlated photoelectron radial momentum distributions for $\tau =2T$ with $p_{r,k}={\mathbf{r}}_k·{\mathbf{p}}_k/\left|{\mathbf{r}}_k\right|$ for $k=1,2$. Left and right panels are for $I={10}^{14}\mathrm{W}{\mathrm{cm}}^{-2}$ and $I=5\times {10}^{14}\mathrm{W}{\mathrm{cm}}^{-2}$ (indicated by the black vertical lines on the top panel), respectively. The dashed white lines are when $p_{r,1}=p_{r,2}$.

Figure 2

Double ionization (DI) probability of Hamiltonian (1) in 2D as a function of the laser intensity $I$ (solid thick curves) for $\mathrm{Ar}$ (${\mathcal{E}}_g=-1.60$, $a=1.5$) and CP pulses. The laser envelope is trapezoidal with duration of ramp-up $\tau$, plateau $6T-\tau$, and ramp-down $2T$. The blue (dark gray), pink (gray), and purple (light gray) curves are for $\tau =2T$, $3T$, and $4T$, respectively. The lower panels are the correlated photoelectron radial momentum distributions for $\tau =2T$ with $p_{r,k}={\mathbf{r}}_k·{\mathbf{p}}_k/\left|{\mathbf{r}}_k\right|$ for $k=1,2$. Left and right panels are for $I=4\times {10}^{15}\mathrm{W}{\mathrm{cm}}^{-2}$ and $I={10}^{16}\mathrm{W}{\mathrm{cm}}^{-2}$ (indicated by the black vertical lines on the top panel), respectively. The dashed white lines are when $p_{r,1}=p_{r,2}$.

Figure 3

Effective potentials in the LF given by Eq. (6) (left panel) and the RF given by Eq. (11) (right panel) for $I={10}^{14}\mathrm{W}{\mathrm{cm}}^{-2}$, $f=1$, $a=3$, $k=2$, and $\omega t=0$ (laser field along the $\widehat{\mathbf{x}}$ axis in the LF). The green (light gray) dots indicate the location of the saddle points in the LF $(-r_{\mathrm{s},k},h_{\mathrm{s},k})$ and in the RF $(-{\tilde{r}}_{\mathrm{s},k},{\tilde{h}}_{\mathrm{s},k})$. The magenta (gray) dots indicate the location of the stable points in the LF $(-r_{\mathrm{b},k},h_{\mathrm{b},k})$ and in the RF $(-{\tilde{r}}_{\mathrm{b},k},{\tilde{h}}_{\mathrm{b},k})$. The double arrows indicate the width of the barrier of the effective potentials in the LF, $W_{\mathrm{s},k}$, and in the RF, ${\tilde{W}}_{\mathrm{s},k}$.

Figure 4

RF-effective potential $V_{\mathrm{RF},k}(\tilde{\mathbf{r}},t)$ given by Eq. (10) for $I={10}^{14}\mathrm{W}{\mathrm{cm}}^{-2}$, $f=1$, $a=3$, and $k=2$. The green (light gray) ball indicates the location of the saddle point. The section corresponds to the surface ${\tilde{H}}_k(\tilde{\mathbf{r}},\tilde{\mathbf{p}},t)=-0.6$ in the plane $(\tilde{x},\tilde{y})$. The blue (light gray) colored areas correspond to the allowed positions of the electron.

Figure 5

(a–b) Poincaré sections of Hamiltonian (5) for $\tilde{y}=0$ and $\dot{\stackrel{̃}{y}}>0$ in the plane $(\tilde{x},{\tilde{p}}_x)$ for $\mathrm{Mg}$ ($a=3$), $k=2$ (inner electron), and $f=1$. Trajectories with quasiperiodic motions are in purple (dark gray). The color (gray) scale is the final distance of the electron from the core with initial condition given by $(\tilde{x},{\tilde{p}}_x)$ integrated over 10 laser cycles. The energy surface is ${\tilde{H}}_2(\tilde{\mathbf{r}},\tilde{\mathbf{p}})=-0.55$. The gray surfaces are the classical forbidden regions. The intensity is (a) $I={10}^{14}\mathrm{W}{\mathrm{cm}}^{-2}$ and (b) $I=5\times {10}^{14}\mathrm{W}{\mathrm{cm}}^{-2}$. (c) Position along $\widehat{\tilde{\mathbf{x}}}$ of the fixed points: the saddle point at $\tilde{x}=-{\tilde{r}}_{\mathrm{s},2}$ [green (light gray) curve] and the bottom of the well near the origin at $\tilde{x}=-{\tilde{r}}_{\mathrm{b},2}$ [magenta (gray) curve] as a function of the laser intensity. In panel (a), the fixed points are indicated by a green (light gray) cross (at $\tilde{x}=-{\tilde{r}}_{\mathrm{s},2}$) and a magenta (gray) dot (at $\tilde{x}=-{\tilde{r}}_{\mathrm{b},2}$).

Figure 6

Typical SDI of Hamiltonian (1) in the RF for Ar (${\mathcal{E}}_g=-1.60$, $a=1.5$), $I=5\times {10}^{14}\mathrm{W}{\mathrm{cm}}^{-2}$, and a trapezoidal envelope 2–4–2 (duration of ramp-up $\tau =2T$). The red (gray) and blue (dark gray) curves are the outer (first ionized) and inner (second ionized) electron trajectories for $t\in [t_{\mathrm{i},1},6T]$, respectively. (a) Projection in the plane $(\tilde{x},\tilde{y})$. A contour plot of the RF-effective potential $V_{\mathrm{RF},1}({\tilde{\mathbf{r}}}_1,{\tilde{\mathbf{p}}}_1,\tau )$ is displayed on the leftmost panel. (b–d) Snapshots of the electrons in the plane $(\tilde{x},{\tilde{H}}_k(\tilde{\mathbf{r}},\tilde{\mathbf{p}},t))$. The dark and light gray areas are the classical forbidden regions delimited by the RF-effective potential of the outer electron $V_{\mathrm{RF},1}(\tilde{x}\widehat{\tilde{\mathbf{x}}},t)$ and of the inner electron $V_{\mathrm{RF},2}(\tilde{x}\widehat{\tilde{\mathbf{x}}},t)$, respectively. The red (gray) and blue (dark gray) dots are the outer and inner electrons at times (b) $t_{\mathrm{i},1}\approx 0.8T$, (c) $t_{\mathrm{i},2}\approx 1.9T$, (d) $\tau =2T$. (d) The light red and blue dashed arrows indicate the subsequent ionization of the electrons.

Figure 7

Ratio of the effective amplitude of the laser field in the nonadiabatic ${\tilde{E}}_k^{★}$ [see Eq. (23)] over the adiabatic $E_k^{★}$ [see Eq. (18)] for the first ionized electron ($k=1$) as a function of the parameter ${\mu }_k=2{\left|{\mathcal{E}}_k\right|}^3{\left(Z_k\omega \right)}^{-2}/27$. The upper axis indicates the ionization potential of the first electron $|{\mathcal{E}}_1|$ for $\omega =0.0584$ and $Z_1=1$. The shaded region indicates also the ratio ${\tilde{E}}_1^{★}/E_1^{★}$: The dark region is where the adiabatic approximation is similar to the nonadiabatic case, and the light region is where the adiabatic approximation fails to predict the ionization process of the electron.

Figure 8

Ionization times of the first and second electron in the double ionization of $\mathrm{Ar}$ for a Gaussian laser envelope of the form $f\left(t\right)=exp[-ln\left(2\right){(2t/t_{\mathrm{FWHM}})}^2]$. The experimental data (squares) for the ionization time of the first and second electrons are reproduced from Ref. [3] for a laser ellipticity $\xi =0.77$ (near-CP pulses). The laser amplitude is such that $E_0=\sqrt{I/({\xi }^2+1)}$. The solid and dotted lines are the theoretical classical predictions given by Eq. (25) and Eq. (18), respectively, for a laser wavelength $\lambda =788\mathrm{nm}$, ${\mathcal{E}}_1=-0.6$, $Z_1=1$, ${\mathcal{E}}_2=-1$, $Z_2=2$ [20]. The experimental measurements are given for $t_{\mathrm{FWHM}}=33\mathrm{fs}$ for a laser wavelength $\lambda =788\mathrm{nm}$ and for $t_{\mathrm{FWHM}}=7\mathrm{fs}$ for a laser wavelength $\lambda =740\mathrm{nm}$.

Figure 9

Typical NSDI of Hamiltonian (1) induced by a recollision in the RF for Mg (${\mathcal{E}}_g=-0.83$, $a=3$), $I={10}^{14}\mathrm{W}{\mathrm{cm}}^{-2}$, and a trapezoidal envelope 2–4–2 (duration of ramp-up $\tau =2T$). The red (gray) and blue (dark gray) curves are the recolliding and bound electron trajectories for $t\in [t_{\mathrm{i},1},5.7T]$, respectively. (a) Projection in the plane $(\tilde{x},\tilde{y})$. A contour plot of the RF-effective potential $V_{\mathrm{RF},1}(\tilde{\mathbf{r}},\tilde{\mathbf{p}},\tau )$ is displayed on the leftmost panel. (b–e) Snapshots of the electron in the plane $(\tilde{x},{\tilde{H}}_k(\tilde{\mathbf{r}},\tilde{\mathbf{p}},t))$. The dark and light gray areas are the classical forbidden regions delimited by the RF-effective potential of the recolliding electron $V_{\mathrm{RF},1}(\tilde{x}\widehat{\tilde{\mathbf{x}}},t)$ and of the bound electron $V_{\mathrm{RF},2}(\tilde{x}\widehat{\tilde{\mathbf{x}}},t)$, respectively. The red (gray) and blue (dark gray) dots are the recolliding and bound electrons at times (b) $t=t_{\mathrm{i},1}\approx 0.9T$, (c) $t=\tau =2T$, (d) $t=t_{\mathrm{r}}\approx 4.5T$, and (e) $t\approx 4.6T$. (d) The light red and blue dashed arrows indicate the subsequent ionization of the electrons.

Figure 10

Typical recollision without NSDI of Hamiltonian (1) for $\mathrm{Mg}$ (${\mathcal{E}}_g=-0.83$, $a=3$), $I={10}^{13}\mathrm{W}{\mathrm{cm}}^{-2}$, and a trapezoidal envelope 2–4–2 ($\tau =2T$). (a) Projection onto the polarization plane in the RF $(\widehat{\tilde{\mathbf{x}}},\widehat{\tilde{\mathbf{y}}})$. (b) Projection onto the plane $(\tilde{x},{\tilde{H}}_k(\tilde{\mathbf{r}},\tilde{\mathbf{p}},t))$. The gray surfaces are the RF-effective potential of the red (dark gray surface) and blue (light gray surface) electrons. (c) Distance as a function of time per laser cycle, and interaction between electrons $V_{12}=\left(\right|{\tilde{\mathbf{r}}}_1-{\tilde{\mathbf{r}}}_2{{|}^2+1)}^{-1/2}$ as a function of time per laser cycle. The arrow on the lower panel indicates a recollision.

Figure 11

Probability density function of the ionization time $t_{\mathrm{i},1}$ (a–c) and return time $t_{\mathrm{r},1}$ (d–f) of the recolliding electrons triggering NSDI of Hamiltonian (1) as a function of the laser intensity $I$ for Mg (${\mathcal{E}}_g=-0.83$, $a=3$). The parameters are the same as in Fig. 1. The ramp-up duration is $\tau$, the plateau duration is $6T-\tau$ and the ramp-down is of duration $2T$ (indicated by horizontal black solid lines): in (a) and (d) $\tau =2T$, in (b) and (e) $\tau =3T$, in (e) and (f) $\tau =4T$. The blue (dark gray) dashed and solid curves are the theoretical prediction of the ionization time of the first electron for the hard-Coulomb potential given by Eq. (18) (adiabatic approximation in the LF) and Eq. (25) (adiabatic approximation in the RF), respectively. The light blue (light gray) dashed and solid curves are the theoretical prediction of the ionization time of the first electron for the soft-Coulomb potential with $a=3$. The vertical dashed black curves are $I={\tilde{I}}_{min}$ [see Eq. (29)]. The gray regions are intensities $I<{\tilde{I}}_{\mathrm{OTB},1}$. The vertical dashed black line indicates $I={\tilde{I}}_{max}$ given by Eq. (13). The gray solid curves are given by $t_{\mathrm{r}}=t_{\mathrm{i},1}+(n+1/2)T$, with $n=1,2,3,4,5$.

Figure 12

Distributions of the recolliding trajectories leading to NSDI of Hamiltonian (1) between their ionization time $t_{\mathrm{i}}$ and their return $t_r$ (where $|\mathbf{r}\left(t_{\mathrm{i}}\right)|=|\mathbf{r}\left(t_{\mathrm{r}}\right)|=5$) for Mg (${\mathcal{E}}_g=-0.83$, $a=3$), with a 2–4–2 trapezoidal envelope ($\tau =2T$) and $I={10}^{14}\mathrm{W}{\mathrm{cm}}^{-2}$. The parameters are the same as in Fig. 9. (a) Distribution of the trajectories in the plane $(\tilde{x},\tilde{y})$ for return times (from left to right) $t_{\mathrm{r}}\in [2.5T,3.75T]$, $t_{\mathrm{r}}\in [3.75T,5T]$, and $t_{\mathrm{r}}\in [5T,6T]$. The purple (gray) and black curves are RPOs of Hamiltonian (5). Distribution of (a) the distance $\left|\mathbf{r}\right|$ and (b) Jacobi value ${\tilde{H}}_1$ of the recolliding electron as a function of time. (a)–(c) The red (gray) curve is a typical recolliding trajectory leading to NSDI of Hamilonian (1). The dotted trajectory in (b) corresponds to the guiding-center trajectory of the recolliding electron whose dynamics is governed by Hamiltonian (31).

Figure 13

Stable ${\mathcal{W}}^s$ (white solid lines) and unstable ${\mathcal{W}}^u$ (gray solid lines) manifolds of fixed point [purple (gray) cross in panel (a)] under the Poincaré section $\dot{\stackrel{̃}{x}}=0$ and $\ddot{\stackrel{̃}{x}}>0$ of Hamiltonian (5) in the plane $(\tilde{x},\tilde{y})$ for $I={10}^{14}\mathrm{W}{\mathrm{cm}}^{-2}$, $f=1$, $a=3$, ${\tilde{H}}_k(\tilde{\mathbf{r}},\tilde{\mathbf{p}})=0$, $k=1$. $(\tilde{x},\tilde{y})$. (a) Final distance of the electron as a function of the initial conditions $(\tilde{x},\tilde{y})$ on the Poincaré section. The purple (gray) and black crosses are fixed points under this Poincaré section corresponding to the purple (gray) and black periodic orbits depicted in Fig. 12. The gray region is the forbidden region on this Poincaré section. The trajectories are integrated for 10 laser periods with a constant laser envelope $f=1$. (b) Time when the electron crosses $\left|\mathbf{r}\right|=5$ as a function of the initial conditions $(\tilde{x},\tilde{y})$ in the dashed square of panel (a). (c) The gray surface is the RF-effective potential at the end of the ramp-up $V_{\mathrm{RF},1}(\tilde{\mathbf{r}},\tau )$. The red (gray) solid line is a a typical recolliding trajectory leading to NSDI of Hamiltonian (1) in the RF [same trajectory as the red (gray) one depicted in Fig. 12] for Mg ($a=3$ and ${\mathcal{E}}_g=-0.83$), $I={10}^{14}\mathrm{W}{\mathrm{cm}}^{-2}$ and $\tau =2T$. The solid red (gray) curve is the trajectory before the end of the ramp-up. The location of the electron at the end of the ramp-up is indicated by a red (gray) ball [also indicated in panel (a) by a red (gray) dot]. The transparent red (gray) curve is the trajectory after the end of the ramp-up. The transparent red (gray) ball is the location of the electron at the return time.

Figure 14

Conditions on the laser wavelength $\lambda$ and the ionization potential $|{\mathcal{E}}_1|$ for which the atom can undergo NSDI in CP [green (light gray) region] or not [red (dark gray) region]. The black line is ${\text{IP}}_c\approx \sigma {\omega }^{2/3}$ [Eq. (34)]. The square ($\square$) is $\mathrm{He}$ and up triangle ($△$) is $\mathrm{Ne}$ for $\lambda =614\mathrm{nm}$ reproduced from Ref. [36]. The diamond ($\diamond$) is $\mathrm{Ar}$ and star ($★$) is $\mathrm{Xe}$ for $\lambda =800\mathrm{nm}$ reproduced from Ref. [37]. The yellow circle ($\circ$) is $\mathrm{Mg}$ for $\lambda =780\mathrm{nm}$ reproduced from Ref. [24]. The down triangle ($▽$) is $\mathrm{Mg}$ for $\lambda =1030\mathrm{nm}$, the right triangle ($▹$) is $\mathrm{Zn}$, and the blue hexagon ($*$) is $\mathrm{Mg}$ for $\lambda =800\mathrm{nm}$ reproduced from Ref. [19].

Figure 15

Maximum of the probability of nonsequential double ionization (PNSDI) (upper panel) and intensity $I^{\left(c\right)}$ at which the PNSDI is maximum (lower panel) as a function of the laser wavelength $\lambda$, computed from Hamiltonian (1). The laser envelope is trapezoidal 2–4–2. The parameters are ($\mathrm{Mg}$) ${\mathcal{E}}_g=-0.83$ and $a=3$, ($\mathrm{Xe}$) ${\mathcal{E}}_g=-1.20$ and $a=2$, ($\mathrm{Ar}$) ${\mathcal{E}}_g=-1.60$ and $a=1.5$, ($\mathrm{Ne}$) ${\mathcal{E}}_g=-2.30$ and $a=1$, ($\mathrm{He}$) ${\mathcal{E}}_g=-2.90$ and $a=0.8$. The maximum of PNSDI becomes smaller than ${10}^{-5}$ for laser wavelength ($\mathrm{Mg}$) ${\lambda }_c\approx 1050\mathrm{nm}$, ($\mathrm{Xe}$, $a=2$) ${\lambda }_c\approx 770\mathrm{nm}$, ($\mathrm{Ar}$) ${\lambda }_c\approx 510\mathrm{nm}$, ($\mathrm{Ne}$) ${\lambda }_c\approx 320\mathrm{nm}$, ($\mathrm{He}$) ${\lambda }_c\approx 180\mathrm{nm}$.