Nonsequential double ionization: a quasiclassical analysis of the Keldysh-type transition amplitude

Sergei P. Goreslavski; Sergei V. Popruzhenko

doi:10.1364/OE.8.000395

Optics Express
Vol. 8,
Issue 7,
pp. 395-400
(2001)
•https://doi.org/10.1364/OE.8.000395

Nonsequential double ionization: a quasiclassical analysis of the Keldysh-type transition amplitude

Sergei P. Goreslavski and Sergei V. Popruzhenko

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Sergei P. Goreslavski and Sergei V. Popruzhenko, "Nonsequential double ionization: a quasiclassical analysis of the Keldysh-type transition amplitude," Opt. Express 8, 395-400 (2001)

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Abstract

The S-matrix amplitude of nonsequential double ionization, written down in the Volkov state basis, has been calculated by a saddle-point method. The distribution in the total and relative momenta of the two emitted electrons is given in analytical form. The result obtained is discussed in the context of the recently measured differential momentum distributions in argon.

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Fig. 1. Momentum distribution of double ions along the polarization axis, obtained by integration of the distribution (6) over the relative momentum. Left panel is for He ²⁺ at intensity 6.6·10¹⁴ Wcm ^-2, full squares show the result of ref. [10]. Right panel is for Ar ²⁺ at intensities 2.0·10¹⁴ Wcm ^-2 (red), 3.8·10¹⁴ Wcm-2 (blue), 15·10¹⁴ Wcm ^-2 (black).

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Fig. 2. Momentum distribution of the two emitted electrons along the polarization axis in the case of P _⊥=q _⊥=0 for Ar atoms at laser intensity 2,0·10¹⁴ Wcm ^-2. Right panel presents the same distribution at a larger scale and here the quantum domain is shown by white color. Momenta are shown in atomic units and 2q_x =p _1x-p _2x in our notations.

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Fig. 3. The same as in Fig.2 but for intensity 3.8·10¹⁴ Wcm ^-2.

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Fig. 4. The same as in Fig.2 but for intensity 15·10¹⁴ Wcm ^-2.

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Equations (9)

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M^{(2 e)} (\vec{P}, \vec{q}) = i \sum_{k} \int_{- \infty}^{+ \infty} d t_{1} 〈 Ψ_{\vec{P} \vec{q}} (t_{1}) ∣ \frac{1}{r_{12}} ∣ Ψ_{\vec{k} 0 +} (t_{1}) 〉 M^{(e)} (\vec{k}, t_{1})

M^{(e)} (\vec{k}, t_{1}) = i \int_{- \infty}^{t_{1}} d t 〈 Ψ_{\vec{k} 0 +} (t) ∣ V_{F} (t) ∣ Ψ_{0} (t) 〉

Ψ_{\vec{k}} (\vec{r}, t) = \exp (i {\vec{Π}}_{\vec{k}} (t) \vec{r} - \frac{i}{2} \int_{- \infty}^{t} {\vec{Π}}_{\vec{k}}^{2} (τ) d τ)

\frac{I_{1}}{F \sin ω t_{0}} + \frac{F}{ω^{2}} [ω (t_{1} - t_{0}) \cos ω t_{0} - \sin ω t_{1} + \sin ω t_{0}] = 0

\frac{1}{2} {\vec{Π}}_{\vec{k} (t_{0})}^{2} (t_{1}) = I_{2} + {\vec{q}}^{2} + \frac{1}{4} {\vec{Σ}}_{\vec{P}}^{2} (t_{1})

d W^{(2 e)} (\vec{P}, \vec{q}) = (w_{-} + w_{+} - 2 {(w_{-} w_{+})}^{1 ⁄ 2} \sin S_{+ -}) d^{3} P d^{3} q

S = I_{1} t_{0} + I_{2} t_{1} + {\vec{q}}^{2} t_{1} + \frac{1}{4} \int_{- \infty}^{t_{1}} d t {\vec{Σ}}_{\vec{P}}^{2} (t) - \frac{1}{2} \int_{t_{0}}^{t_{1}} d t {\vec{Π}}_{\vec{k} (t_{0})}^{2} (t)

w (\vec{P}, \vec{q}) \propto {∣ A^{(e \to 2 e)} ({\vec{Σ}}_{\vec{P}} (t_{1}), \vec{q}, {\vec{Π}}_{\vec{k} (t_{0})} (t_{1})) ∣}^{2} \frac{\sin^{2} ω t_{0}}{Δ_{⊥}^{2} (t_{1}, t_{0}) ∣ D (t_{1}, t_{0}) ∣} \exp (- \frac{2 F_{a}}{3 F (t_{0})})

A^{(e \to 2 e)} ({\vec{P}}_{*}, {\vec{q}}_{*}, {\vec{k}}_{*}) \propto {(2 I_{2} + {({\vec{P}}_{*} - {\vec{k}}_{*})}^{2})}^{- 2} [{({\vec{k}}_{*} + {\vec{q}}_{*} - {\vec{P}}_{*} ⁄ 2)}^{- 2} + {({\vec{k}}_{*} - {\vec{q}}_{*} - {\vec{P}}_{*} ⁄ 2)}^{- 2}]

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Equations (9)

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