Multi-sideband interference structures observed via high-order photon-induced continuum-continuum transitions in argon D Bharti1 H Srinivas1 F Shobeiry1 K R Hamilton2 R Moshammer1 T Pfeifer1 K Bartschat2 and A Harth13

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Multi-sideband interference structures observed via high-order photon-induced

continuum-continuum transitions in argon

D Bharti1, H Srinivas1, F Shobeiry1, K R Hamilton2, R Moshammer1, T Pfeifer1, K Bartschat2, and A Harth1,3∗

1Max-Planck-Institute for Nuclear Physics, D-69117 Heidelberg, Germany

2Department of Physics and Astronomy, Drake University, Des Moines, IA 50311, USA and

3Department of Optics and Mechatronics, Hochschule Aalen, D-73430 Aalen, Germany

(Dated: January 24, 2023)

We report a joint experimental and theoretical study of a three-sideband (3-SB) modiﬁcation of the

“reconstruction of attosecond beating by interference of two-photon transitions” (RABBIT) setup. The 3-SB

RABBIT scheme makes it possible to investigate phases resulting from interference between transitions of

different orders in the continuum. Furthermore, the strength of this method is its ability to focus on the

atomic phases only, independent of a chirp in the harmonics, by comparing the RABBIT phases extracted

from speciﬁc SB groups formed by two adjacent harmonics. We verify earlier predictions that the phases

and the corresponding time delays in the three SBs extracted from angle-integrated measurements become

similar with increasing photoelectron energy. A variation in the angle-dependence of the RABBIT phases in

the three SBs results from the distinct Wigner and continuum-continuum coupling phases associated with the

individual angular-momentum channels. A qualitative explanation of this dependence is attempted by invoking a

propensity rule. Comparison between the experimental data and predictions from an R-matrix (close-coupling)

with time dependence calculation shows qualitative agreement in most of the observed trends.

I. INTRODUCTION

The reconstruction of attosecond beating by interference of

two-photon transitions (RABBIT) is a widely employed tech-

nique to measure attosecond time delays in photoionization

processes [1–3]. The extraction of time information from

the RABBIT measurements usually involves retrieving atomic

phases encoded in the delay-dependent modulation of the

sideband (SB) yield. These SBs are traditionally formed

in the photoelectron spectrum by the interaction of two

photons (one pump, one probe) with the target. Spectral

harmonics from an attosecond pulse train (the pump photons)

form discrete photoelectron signal peaks. The presence of a

time-delayed infrared ﬁeld (the probe photon) then creates

a signal in between these main peaks that oscillates with

the time delay. The so retrieved atomic phase (∆φat)

from the RABBIT measurement can be separated into a

single-photon ionization contribution (∆η, Wigner phase [4])

and a continuum-continuum (cc) coupling phase (∆φcc) by

applying an “asymptotic approximation” [5–7].

Variations of the RABBIT scheme, such as 0-SB, 1-SB, and

2-SB, have been utilized to study dipole transition phases and

attosecond pulse shaping [8–10]. As the name suggests, in

a 3-SB RABBIT scheme, three SBs are formed between two

consecutive main photoelectron peaks [11, 12]. The delay-

dependent oscillation in the photoelectron signal of these three

SBs requires more than one transition in the continuum, i.e.,

the absorption or emission of several probe photons. For a

hydrogenic system, we recently [12] extended the asymptotic

approximation to a decomposition scheme, which expands

the phase of the Nth-order dipole matrix element M(N),

describing the absorption of an ionizing extreme ultraviolet

(XUV) photon followed by N−1infrared (IR) photon

∗bharti@mpi-hd.mpg.de; Anne.Harth@hs-aalen.de

exchange in the continuum, into a sum of the Wigner phase

and N−1cc phases.

For atomic hydrogen, where numerical calculations with

high accuracy can be carried out by solving the time-

dependent Schr¨

odinger equation (TDSE) directly, we veriﬁed

that the decomposition approximation explains the RABBIT

phases in all three SBs qualitatively [12]. As expected,

its accuracy improves with increasing energy of the emitted

photoelectron. On the other hand, assuming ∆φcc to be

independent of the orbital angular momenta of the continuum

states leads to deviations from the analytical prediction,

particularly in the lower and the higher SB of the triplet at

low kinetic energies.

Even though starting with a 3pelectron still limits the

information that can be extracted due to the combined effect

of at least two Wigner and the cc phases, we decided to

perform the present proof-of-principle study on argon due to

its experimental advantages, including a signiﬁcantly lower

ionization potential than helium, which may be a viable

alternative to atomic hydrogen due to its quasi-one-electron

character, as long as one of the electrons remains in the 1s

orbital, i.e., away from doubly-excited resonance states. In

argon, the intermediate orbital angular momentum after the

XUV step is λ=0 or 2, while λ=1 in helium. For the latter

target, as for atomic hydrogen, the dependence on the Wigner

phase would drop out, and the 3-SB setup would provide

direct access to the phase associated with higher-order cc

transitions [11, 12]. Nevertheless, a signiﬁcant strength of our

current setup already lies in the fact that the results within each

group are independent of any chirp in the XUV pulse, because

the XUV harmonic pair is common to all three SBs.

This paper is organized as follows. We begin with a brief

review of the basic idea behind the 3-SB setup in Sec. II. This

is followed by a description of the experimental apparatus in

Sec. III and the accompanying theoretical R-matrix (close-

coupling) with time dependence (RMT) approach in Sec. IV.

In section V, we ﬁrst show angle-integrated data (Sec. V A)

arXiv:2210.09244v2 [physics.atom-ph] 20 Jan 2023

before focusing on the angle-dependence of the RABBIT

phases in the three SBs of each individual group in Sec. V B.

We ﬁnish with a summary and an outlook in Sec. VI.

II. THE 3-SB SCHEME

In this section, we brieﬂy review the 3-SB scheme

introduced in [11] and the analytical treatment presented

in [12] as applied to the 3-SB RABBIT experiment.

Mq-

ȁ ۧ

𝑖

FIG. 1. 3-SB RABBIT scheme. Mq−1and Mq+1 label the main

photoelectron peaks created directly by the odd harmonics (Hq−1

and Hq+1) of the frequency-doubled fundamental probe frequency

in the XUV pulse, while Sq,l,Sq,c , and Sq,h are the lower, central,

and higher SBs, respectively. These SBs are formed by emission

or absorption of probe photons by the quasi-free photoelectrons. |ii

denotes the initial state and Ipis the ionization potential.

Figure 1 illustrates only the two most dominant transition

paths for each SB contributing to the oscillation in their

respective yields. The lowest-order transition dominates the

yield, but its modulation requires interference between at least

two distinct paths leading to the same energy. This involves

two different XUV harmonics that are aided by absorption or

emission of near-infrared (NIR) photons. For the lower (l)

and higher (h) SBs, Sland Sh, the most important interfering

paths are of 2nd (one harmonic and one NIR) and 4th (one

harmonic and three NIR) order, which results in a weak

modulation of the yield. The lowest-order terms contributing

to the build-up of the central (c) SB, Sc, are both of 3rd order

(one harmonic and two NIR). Consequently, interference

between them exhibits the delay-dependent oscillation most

clearly.

Mathematically, the angle-integrated yield in the three

SBs, considering only two prominent transition paths, can be

written as

Sq,l ∝X

`,m



˜

Eq+1 ˜

E∗3

ωM(4,e)

`,m (kl,q)+ ˜

Eq−1˜

EωM(2,a)

`,m (kl,q)



=Il

0+X

`,m

`,m cos(4 ωτ −∆φq

Ω−∆φl,at

`,m)

=Il

0+Il

1cos(4 ωτ −φl

R+π); (1a)

Sq,c ∝X

`,m



˜

Eq+1 ˜

E∗2

ωM(3,e)

`,m (kc,q)+ ˜

Eq−1˜

ωM(3,a)

`,m (kc,q)



=Ic

0+X

`,m

`,m cos(4 ωτ −∆φq

Ω−∆φc,at

`,m )

=Ic

0+Ic

1cos(4 ωτ −φc

R); (1b)

Sq,h ∝X

`,m



˜

Eq+1 ˜

E∗

ωM(2,e)

`,m (kh,q)+ ˜

Eq−1˜

ωM(4,a)

`,m (kh,q)



=Ih

0+X

`,m

`,m cos(4 ωτ −∆φq

Ω−∆φh,at

`,m )

=Ih

0+Ih

1cos(4 ωτ −φh

R+π)(1c)

Here qlabels the SB group, while kl,q ,kc,q , and kh,q

denote the ﬁnal linear momenta of the ejected electron in

the lower, central, and higher sidebands in each group. The

subscript `denotes one of generally several allowed orbital

angular momenta of the ejected electron in the ﬁnal state

and mlabels the magnetic quantum number, which can be 0

or ±1for the electron starting in the 3psubshell. Note that m

is a conserved quantity for all orders nof the transition matrix

element M(n)

`,m due to our use of linearly polarized light.

Furthermore, ˜

EΩ=EΩei φΩand ˜

Eω=Eωei ωτ (for

absorption) are the complex electric-ﬁeld amplitudes of the

co-linearly polarized XUV-pump (Ω) and NIR-probe (ω)

pulses, respectively. ∆φat

`,m = arg[M(a)

`,mM∗(e)

`,m ]is the phase

difference between the two matrix elements and a(e)denotes

the pathway involving absorption (emission) of the probe

photons. Finally, ∆φq

Ωis the spectral phase difference (XUV

chirp) of two neighbouring harmonics.

As seen from Eqs. (1), the yield of each SB is separated into

an average part I0and another term I1that oscillates at 4ω

with the delay. As discussed in [12], every dipole transition

also adds a phase of π/2. Since the two dominant interfering

terms in Sland Share of different orders (2nd and 4th), this

leads to an additional πphase in Sland Shrelative to Sc,

where both interfering terms are of the same (3rd) order.

The RABBIT phase (φR) includes the spectral phase differ-

ence of the two harmonics and the channel-resolved atomic

phases weighted according to their transition amplitudes. It

is a complex inverse trigonometric function involving many

parameters and hence is best determined by ﬁtting the signal

to the known analytic form given above. Since the three SBs

involve the same pair of harmonics, the contribution of the

XUV group delay (i.e., the chirp) to the oscillation phase is

the same in all three SBs. This is a key advantage of the

3-SB method, since it removes the inﬂuence of the XUV chirp

when we compare the phases of the three SBs only within a

particular group.

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Multi-sidebandinterferencestructuresobservedviahigh-orderphoton-inducedcontinuum-continuumtransitionsinargonDBharti1,HSrinivas1,FShobeiry1,KRHamilton2,RMoshammer1,TPfeifer1,KBartschat2,andAHarth1;31Max-Planck-InstituteforNuclearPhysics,D-69117Heidelberg,Germany2DepartmentofPhysicsandAstronomy,Drake...

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Multi-sideband interference structures observed via high-order photon-induced continuum-continuum transitions in argon D Bharti1 H Srinivas1 F Shobeiry1 K R Hamilton2 R Moshammer1 T Pfeifer1 K Bartschat2 and A Harth13.pdf

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Multi-sideband interference structures observed via high-order photon-induced continuum-continuum transitions in argon D Bharti1 H Srinivas1 F Shobeiry1 K R Hamilton2 R Moshammer1 T Pfeifer1 K Bartschat2 and A Harth13

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