3 Bichromatic Control of Free Electron Wave Packets
57
atoms, the overlap of photoelectron wave packets from 3- and 4-photon ionization
is observed in a kinetic energy window centered around ε ≈ 1.3 eV. In this window,
we observe alternating patterns of destructive (negative asymmetry) and constructive (positive asymmetry) interference upon variation of the respective optical phase.
Scaling the ϕ 2 -axis in panel (f) by a factor of 1/3 and comparing the result to the
ϕ ce -map in panel (e) clearly shows the equivalence between the effect of the two
phases on the photoelectron asymmetry.
Equation (3.6) furthermore predicts that the photoelectron interference is sensitive to the time delay τ between the two colors. Experimentally, this time delay is
introduced by linear spectral phase modulation. While the asymmetry observed at
τ = 0 fs (see Fig. 3.6e, f) is energy-independent (flat) in the relevant energy window, for τ = −25 fs the interference pattern acquires an energy-dependent linear
tilt with negative slope in the ϕ ce -asymmetry map shown in Fig. 3.6g. The physical
mechanism behind this tilt was discussed in [23, 55, 57] in terms of the additional
chirp-dependent quantum phase. In the bichromatic MPI scenario considered here,
the slope of the tilt is inverted when the relative phase ϕ 2 is varied instead of the
CEP (cf. Fig. 3.6h). This result verifies the negative sign of the ϕ 2 -contribution to
ϕ in (3.6) relative to the CEP-dependent term. As a consequence, cycloidal bichromatic fields (cf. Sect. 3.2.1) rotate in opposite directions by variation of ϕ ce and ϕ 2 ,
respectively, as is described by (3.3). The interplay between the phase-sensitivity of
cycloidal bichromatic fields and the created photoelectron wave packets is discussed
in more detail in Sect. 3.3.4.
3.3.3 Single Color Electron Vortices
Ionization with two time-delayed CRCP pulses creates free electron wave packets,
that are shaped like an Archimedean spiral in the laser polarization plane [21, 59].
Inspired by the helical interference structures, the resulting PMD was termed an
‘electron vortex’ by Starace and coworkers [59]. This notion of an electron vortex
has to be distinguished from the traditional definition of vortex states in quantum
systems [60, 61], derived from the hydrodynamic formulation of quantum mechanics
[62]. Vortex states, which are defined by their non-vanishing azimuthal probability
current density of the wave function
j(r) ∝ ∝
ψ
∗
(r)∇ψ(r)
= |ψ(r)|
2
∇ arg ψ(r),
(3.7)
are subject of intense studies in collision physics [61, 63] and the generation of
electron vortex beams [64, 65]. For general N 1 - versus N 2 -photon ionization, using
CRCP laser pulses with a time delay τ the probability current density of the resulting
photoelectron wave packet reads
j(r) ∝
1
2i
|ψ|
2
iτ
e ε +
i(N 1 − N 2 )
ε sin(θ )
e φ
,
(3.8)
57
atoms, the overlap of photoelectron wave packets from 3- and 4-photon ionization
is observed in a kinetic energy window centered around ε ≈ 1.3 eV. In this window,
we observe alternating patterns of destructive (negative asymmetry) and constructive (positive asymmetry) interference upon variation of the respective optical phase.
Scaling the ϕ 2 -axis in panel (f) by a factor of 1/3 and comparing the result to the
ϕ ce -map in panel (e) clearly shows the equivalence between the effect of the two
phases on the photoelectron asymmetry.
Equation (3.6) furthermore predicts that the photoelectron interference is sensitive to the time delay τ between the two colors. Experimentally, this time delay is
introduced by linear spectral phase modulation. While the asymmetry observed at
τ = 0 fs (see Fig. 3.6e, f) is energy-independent (flat) in the relevant energy window, for τ = −25 fs the interference pattern acquires an energy-dependent linear
tilt with negative slope in the ϕ ce -asymmetry map shown in Fig. 3.6g. The physical
mechanism behind this tilt was discussed in [23, 55, 57] in terms of the additional
chirp-dependent quantum phase. In the bichromatic MPI scenario considered here,
the slope of the tilt is inverted when the relative phase ϕ 2 is varied instead of the
CEP (cf. Fig. 3.6h). This result verifies the negative sign of the ϕ 2 -contribution to
ϕ in (3.6) relative to the CEP-dependent term. As a consequence, cycloidal bichromatic fields (cf. Sect. 3.2.1) rotate in opposite directions by variation of ϕ ce and ϕ 2 ,
respectively, as is described by (3.3). The interplay between the phase-sensitivity of
cycloidal bichromatic fields and the created photoelectron wave packets is discussed
in more detail in Sect. 3.3.4.
3.3.3 Single Color Electron Vortices
Ionization with two time-delayed CRCP pulses creates free electron wave packets,
that are shaped like an Archimedean spiral in the laser polarization plane [21, 59].
Inspired by the helical interference structures, the resulting PMD was termed an
‘electron vortex’ by Starace and coworkers [59]. This notion of an electron vortex
has to be distinguished from the traditional definition of vortex states in quantum
systems [60, 61], derived from the hydrodynamic formulation of quantum mechanics
[62]. Vortex states, which are defined by their non-vanishing azimuthal probability
current density of the wave function
j(r) ∝ ∝
ψ
∗
(r)∇ψ(r)
= |ψ(r)|
2
∇ arg ψ(r),
(3.7)
are subject of intense studies in collision physics [61, 63] and the generation of
electron vortex beams [64, 65]. For general N 1 - versus N 2 -photon ionization, using
CRCP laser pulses with a time delay τ the probability current density of the resulting
photoelectron wave packet reads
j(r) ∝
1
2i
|ψ|
2
iτ
e ε +
i(N 1 − N 2 )
ε sin(θ )
e φ
,
(3.8)