Phase space approach to intense-laser-driven
electronic wavepacket propagation(Weizmann Institute) Norio Takemoto, Asaf Shimshovitz, David J. Tannor
We propose a new method to solve the time-dependent Schrödinger equation based ona phase space perspective [1]. The method employs the periodic von Neumann basis withbiorthogonal exchange (pvb basis) recently introduced for the calculation of the energy
eigenstates of time-independent quantum systems [2]. While the individual elements inthis basis set are time-independent, a small subset is chosen in a time-dependent mannerto adapt to the evolution of the wavepacket in phase space. The resulting method issimple, eﬃcient, accurate, ﬂexible, and stable. Our long-term goal is to apply the method
to simulate the correlated dynamics of multiple electrons in atoms and molecules in intenseand ultrashort laser pulses. As a ﬁrst demonstration, we present calculations of electron
dynamics in a 1D model atom interacting with the combined ﬁeld of intense near-infrared(NIR) and attosecond extreme-ultraviolet (XUV) laser pulses.
In Fig. 1, we show the simulated evolution of the 1D electronic wavepacket in phasespace. The NIR laser pulse (5 × 1013 W/cm2 , 800 nm, 4 fs) and the XUV laser pulse(1 × 1012 W/cm2 , 15 nm, 250 as) were applied to the electron in a 1D soft-core Coulomb
potential. In the pvb basis, the expansion coeﬃcients of a wavepacket are given by theprojection of the wavepacket to phase space Gaussians imposed with periodic boundary
conditions. Squared moduli of these coeﬃcients are shown by the red color scale on theellipses, which have the same center and aspect ratio as the corresponding phase space
Gaussians. A lattice of 64 × 64 Gaussians is required to span the rectangular boundary ofeach panel in Fig. 1, but only those Gaussians indicated by the ellipses have signiﬁcant
overlap with the wavepacket. The active Gaussians are predetermined by running classicaltrajectories for a simpliﬁed model. The representation with the full set of Gaussians isequivalent to the representation using the same number of Fourier grid points. However,the localized nature of the Gaussians allowed us to propagate the wavepacket using only
about 10 % of the pvb basis with negligible loss of accuracy.
Figure 2 shows the photoelectron momentum distribution that emerges from the simulation in Fig. 1 (blue solid line). Comparison with a simulation using the full pvb basis(red dashed line) indicates that the small active set not only reproduces the qualitativefeatures, such as the direct (N1 and N1’) and rescattered (N2 and N2’) NIR-photoelectron
cut-oﬀs as well as the NIR-streaked single-XUV-photon ionization peaks (X1 and X1’),but also has quantitative accuracy.

[1] N. Takemoto, A. Shimshovitz, and D.J. Tannor, J. Chem. Phys. 137, 011102 (2012).
[2] A. Shimshovitz and D.J. Tannor, Phys. Rev. Lett. (in press) [e-print arXiv:1201.2299v1].

8 (a)
64
20−2−4−6−88 (b)6420−2−4−6−88 (c)6420−2−4−6−8−600 −400 −200

Momentum p (a.u.)

Momentum p (a.u.)

Momentum p (a.u.)

10−12 10−9 10−6 10−30200

Position q (a.u.)

400600

Probability Density (a.u.)

FIG. 1. Snapshots of the wavepacket in phase space. The squared moduli of the overlaps of the wavepacket tothe phase space Gaussians are shown by the red color scale. The dark- and light-blue dots as well as dark-blue +marks represent the classical trajectories used to predetermine which Gaussians will have significant overlap.

10-4

FullReduced

10-6

X1'N2'

N1'

N1

X1

N2

10-8

10-10
10-12
10-14

−3−2−101

2

Photoelectron Momentum p (a.u.)

3FIG. 2. Photoelectron momentum distribution. The results with the reduced (blue solid line) and full (red dashedline) pvb bases are compared.