# Anisotropy parameters for two-color photoionization phases in randomly
oriented molecules: theory and experiment in methane and deuteromethane.

Dominik Ertel*  Physikalisches Institut, Albert-Ludwigs-Universität Freiburg,
Hermann-Herder-Straße 3, 79104 Freiburg, Germany.  David Busto  Physikalisches
Institut, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Straße 3, 79104
Freiburg, Germany.  Ioannis Makos  Physikalisches Institut, Albert-Ludwigs-
Universität Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany.
Marvin Schmoll  Physikalisches Institut, Albert-Ludwigs-Universität Freiburg,
Hermann-Herder-Straße 3, 79104 Freiburg, Germany.  Jakub Benda  Institute of
Theoretical Physics, Faculty of Mathematics and Physics, Charles University, V
Holešovičkách 2, 180 00, Prague 8, Czech Republic.  Francesca Bragheri
Istituto di Fotonica e Nanotecnologie, CNR, 20129 Milano, Italy.  Roberto
Osellame  Istituto di Fotonica e Nanotecnologie, CNR, 20129 Milano, Italy.
Eva Lindroth  Department of Physics, Stockholm University, AlbaNova University
Center, SE-106 91 Stockholm, Sweden.  Serguei Patchkovskii  Max Born
Institute, Max-Born-Str. 2A, D-12489 Berlin, Germany.  Zdeněk Mašín  Institute
of Theoretical Physics, Faculty of Mathematics and Physics, Charles
University, V Holešovičkách 2, 180 00, Prague 8, Czech Republic.
<EMAIL_ADDRESS>Giuseppe Sansone  Physikalisches Institut, Albert-
Ludwigs-Universität Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg,
Germany<EMAIL_ADDRESS>
###### Abstract

We present a combined theoretical and experimental work investigating the
angle-resolved phases of the photoionization process driven by a two-color
field consisting of an attosecond pulse train and an infrared pulse in an
ensemble of randomly oriented molecules. We derive a general form for the two-
color photoelectron (and time-delay) angular distribution valid also in the
case of chiral molecules and when relative polarizations of the photons
contributing to the attosecond photoelectron interferometer differ. We show a
comparison between the experimental data and the theoretical predictions in an
ensemble of methane and deuteromethane molecules, discussing the effect of
nuclear dynamics on the photoionization phases. Finally, we demonstrate that
the oscillating component and the phase of the two-color signal can be fitted
using complex asymmetry parameters, in perfect analogy with the atomic case.

###### keywords:

American Chemical Society, LaTeX

Department of Physics, Lund University, PO Box 118, SE-221 00 Lund, Sweden.
IR,NMR,UV 00footnotetext: * They contributed equally to this work

##  1 Introduction

The availability of isolated attosecond pulses and their trains 1 has enabled
the investigation of purely electronic processes occurring on a comparable
timescale 2, 3. Among several processes, the photoionization mechanism has
been widely investigated in the last years using attosecond technology to
derive information about the influence of the electronic structure on the
emission of photoelectron wave packets after the absorption of an extreme
ultraviolet (XUV) photon. In particular, the influence of electronic
correlation in atoms 4, the role played by shape resonances in molecules 5 and
the effect of an anisotropic molecular potential on the photoemission phases
6, 7 have been elucidated through attosecond photoelectron interferometry
based on the combination of XUV attosecond pulse trains and a synchronized
infrared (IR) field 8. When combined with photoelectron spectrometers with
angular resolution, this technique, usually indicated as reconstruction of
attosecond beating by interference of two-photon transitions (RABBIT), allows
one to retrieve three-dimensional momentum information about the emitted
photoelectrons, giving access, for example, to the angular dependence of the
photoemission phases or time delays. In atoms, this dependence has been
studied for the case of non-resonant photoionization, demonstrating the
presence of a minimum in the direction perpendicular to the common
polarization direction of the XUV and IR fields 9. Moreover, the angular-
resolved information gives the possibility of realizing a complete experiment,
with the determination of the amplitudes and relative phases of the partial
waves involved in the photoionization process 10. The flexibility offered by
the RABBIT approach, in which the polarization state and directions of the XUV
and IR fields can be independently controlled, has been exploited for the
investigation of attosecond circular dichroism in two-color photoionization in
atoms 11.

In molecular systems, the availability of information about the emission
direction of the photoelectron wave packet was used to elucidate the effect of
the molecular potential on the outgoing electron and to reconstruct attosecond
time delays in the recoil frame 12, 13, 6.

As demonstrated in ref. 14, in atoms, the angular dependence of the
photoionization phases and time delays can be parametrized using two complex
asymmetry parameters $\tilde{\beta}_{2}$ and $\tilde{\beta}_{4}$, in analogy
to the well-known formula describing the photoelectron angular distributions
$I(\theta)$ in a two-color field 15:

$I(\theta)=I_{0}\left[1+\beta_{2}P_{2}(\cos\theta)+\beta_{4}P_{4}(\cos\theta)\right]$
(1)

where $\theta$ indicates the photoemission angle with respect to the common
polarization of the XUV and IR fields, $\beta_{2}$ and $\beta_{4}$ are (real)
asymmetry parameters for a two-photon ionization process, and $P_{2}$ and
$P_{4}$ are the Legendre polynomials of the second and fourth order.

In this work, we demonstrate that the complex parametrization of the
photoionization phases can be extended to the case of an ensemble of randomly
oriented molecules in the laboratory frame (LF). This finding is consistent
with the observation that the photoelectron angular distributions (PADs)
generated in a molecular ensemble in similar experimental conditions are
described by an expression analogous to Eq. (1) 15. We present a general
derivation of the interfering terms involved in the RABBIT process,
considering also the case of chiral systems. Finally, we apply the formalism
to the angle-resolved photoionization phases measured in a randomly oriented
ensemble of CH4 and CD4 molecules, investigating the PADs and photoionization
phases associated with final non-dissociating states of the molecular cations
(CH4+ and CD4+).

##  2 Methods

###  2.1 Experimental setup

Measurements were conducted in an equal mixture of CH4 and CD4 molecules
utilizing a photoelectron-photoion coincidence spectrometer (Reaction
Microscope) 16, 17. XUV photons in the spectral range between 10 eV- and 50 eV
were delivered by a source of attosecond pulse trains operating at 50 kHz 18.
The sidebands of the main photoelectron lines were generated by a synchronized
IR pulse with an estimated intensity of IIR$\approx 5\times 10^{11}$ W/cm2.
The delay between the XUV and IR pulse was varied using a system of drilled
glass plates in a collinear configuration 19, 18. Figure 1a presents the
typical time-of-flight (TOF) mass spectrum generated by single XUV-
photoionization in the CH4-CD4 mixture. The TOF is characterized by
dissociating (CH3+ and CD3+) and non-dissociating channels (CH4+ and CD4+), in
agreement with spectroscopic information obtained with monochromatic XUV
radiation 20, 21, 22. The presence of a second dissociating channel (CH2+ and
CD2+) could not be identified due to the low branching ratio of this channel
in the photon energy range covered by our XUV source.

Figure 1: (a) XUV-only experimental TOF mass spectrum acquired in an equal
mixture of CH4 and CD4 molecules. XUV-only photoelectron angular distributions
measured in coincidence with the ionic fragments CH4+ (b) and CD4+ (c). The
laser polarization is along the horizontal axis. Adapted with permission from
ref. 23 . Copyright 2022 University of Freiburg.

The PADs associated with the non-dissociating channels are shown in Fig. 1b
and c for CH4+ and CD4+, respectively. The plots are presented in the plane
$p_{r}-p_{z}$ corresponding to the components of the photoelectron momenta
$\mathbf{p}$ perpendicular ($p_{r}$) and parallel ($p_{z}$) to the
spectrometer axis. The polarization of the XUV radiation was oriented along
the spectrometer axis. The PADs are characterized by photoelectron peaks
corresponding to the absorption of a single XUV-photon. The width of a single
photoelectron peak is broader than the bandwidth of the corresponding XUV
harmonic ($\approx$ 200 meV (full-width at half maximum; FWHM)), due to the
finite resolution of the photoelectron spectrometer and the energy width of
the vibronic absorption band of methane and deuteromethane 21. In Fig. 1b and
1c, the sharp vertical artefacts indicate an information loss about the
photoelectrons momenta, known as magnetic nodes. This loss occurs when the
photoelectrons complete an exact integer multiple of cycles in their cyclotron
motion towards the detector. Therefore, to ensure the integrity of our
results, the analysis considers only experimental data that exhibits positive
momentum $p_{z}$ values, i.e., photoemission angles up to $90^{\circ}$ with
respect to the spectrometer axis.

For the two-color photoionization experiments, the IR and XUV pulses shared
the same polarization direction. The relative delay between the two fields was
changed in steps of $\Delta t\approx 200$ as. Sideband oscillations were
acquired in a range of 12 fs and the total acquisition time was about 96
hours.

###  2.2 Theory

In this section, we derive expressions for the complex $\tilde{\beta}$
parameters which fully characterize the angular dependency of the time delays.
The topic of PAD in both single- and multi-photon ionization has been
extensively studied both theoretically and experimentally, see e.g. the review
by Reid 15 and the references therein. The general expression for $n$-photon
PAD for randomly oriented molecules was presented earlier 24. While the result
presented there was for the direct $n$-photon contribution, the formula
remains identical for the interference term too if the product of the partial-
wave matrix elements is replaced with the product of dipoles coming from the
two interfering pathways. By itself, this expression is sufficient to show
that the atomic and molecular two-photon delays can be parametrized
equivalently for the simplest case of XUV and IR fields linearly polarized in
the same direction and for non-chiral molecules. Here, we outline the theory
for the special case of interference of two-photon transitions. In Appendices
A and B we perform the derivation in detail and analyze the resulting
expression further for the case of various polarizations of the two fields
including the case of chiral molecules (or atoms prepared in chiral states)
(see Appendix A). To reduce the complexity we specialize the discussion to co-
propagating XUV and IR pulses. In the following, we use Hartree atomic units
($m=e=\hbar=1$) unless noted otherwise.

The two-photon photoionization matrix elements employed here in applications
of the theory were calculated using the molecular R-matrix codes UKRMol+25.
The details of the calculations were described previously 26.

###  2.3 Photoelectron angular distributions in the RABBIT process

The momentum space two-photon matrix element has the form

$\displaystyle
M_{fi}^{(2)}=\langle\Psi_{f\bm{k}}^{(-)}|\hat{D}_{2}\hat{G}_{1}^{(+)}\hat{D}_{1}|\Psi_{i}\rangle,$
(2)

where $\ket{\Psi_{f\bm{k}}^{(-)}}$ is the final continuum wavefunction,
$\hat{D}_{1}$ and $\hat{D}_{2}$ are the dipole operators, $\ket{\Psi_{i}}$ is
the initial bound state wavefunction and $\hat{G}_{1}^{(+)}$ is the field-free
Green’s function of the system which generates the intermediate state at
energy $E_{i}+\omega$ (or $E_{i}-\omega$, depending on the arm of the
interferometer, see Fig. 2) after absorption of the XUV photon ($\omega$
indicates the central frequency of the IR field). A detailed description of
the calculation of this matrix element using the R-matrix method is given in
our previous work 24. The RABBIT signal is given by the interference term
originating from two two-photon pathways, see Fig. 2b, leading to the same
final continuum state:

$\displaystyle
I\sim|M_{fi}^{(2,A)}+M_{fi}^{(2,B)}|^{2}=|M_{fi}^{(2,A)}|^{2}+|M_{fi}^{(2,B)}|^{2}+\mathrm{Re}[2|M_{fi}^{(2,A)}||M_{fi}^{(2,B)}|e^{\mathrm{i}(\phi_{>}-\phi_{<})}e^{\mathrm{i}(\Phi_{>}-\Phi_{<})}],$
(3)

where $\Phi_{<}$ and $\Phi_{>}$, respectively the phases of the XUV spectral
components in the absorption and emission path, determine the relative timing
of the XUV and IR fields. The phases $\phi_{<}$ and $\phi_{>}$ are the
intrinsic phases of the involved two-photon matrix elements. The molecular-
frame RABBIT delay is then given by the phase of the exponential, i.e.

$\displaystyle\tau^{2\omega}(\theta_{MF},\phi_{MF})=\frac{(\phi_{>}-\phi_{<})}{2\omega}=\frac{\mathrm{arg}[M_{fi}^{(2,A)}M_{fi}^{(2,B)*}]}{2\omega}.$
(4)

Here the angles $\theta_{MF},\phi_{MF}$ specify the direction of the
photoelectron momentum in the molecular frame.

Figure 2: a) Direct two-photon above-threshold ionization. b) The RABBIT
interferometer comprises two arms A and B corresponding to the two-photon
amplitudes $M^{(2,A)}$ and $M^{(2,B)}$ which combine to produce the sideband
signal $I^{(p_{A1},p_{B1}),(p_{A2},p_{B2})}$. Each arm contains two photons
with polarizations $p_{X1}$, $p_{X2}$, where $X=A,B$. $I_{p}$ stands for the
ionization potential of the molecule.

Orientationally-averaged delays are computed from the phase of the
orientationally-averaged interference term written in the LF, where the
photoelectron momentum $\bm{k}$ and photon polarizations are defined:

$\displaystyle\tau^{2\omega}_{Av}(\theta,\phi)$ $\displaystyle=$
$\displaystyle\frac{\mathrm{arg}[I^{(p_{A1},p_{B1}),(p_{A2},p_{B2})}]}{2\omega},$
(5) $\displaystyle I^{(p_{A1},p_{B1}),(p_{A2},p_{B2})}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi^{2}}\int\mathrm{d}RM_{fi}^{(2,A)}(R)\left(M_{fi}^{(2,B)}(R)\right)^{*},$
(6)

where the angles $\theta,\phi$ indicate the direction of the photoelectron
momentum in the LF. The integration is over the three Euler angles which
rotate the molecule from the molecular frame to the laboratory frame and
$p_{A1}$, $p_{A2}$ and $p_{B1}$, $p_{B2}$ denote polarizations of the first
and the second photon in the spherical basis in the LF as shown in Fig. 2. We
note that in the case of orientational (or vibrational, see below) averaging,
the individual phases $\phi_{>}$ and $\phi_{<}$ are no longer well-defined,
unlike in the molecular-frame case. Their difference, given by the phase of
$I^{(p_{A1},p_{B1}),(p_{A2},p_{B2})}$, is however sufficient to define the
observables.

The final result expressed in the spherical basis for the four-photon
polarizations is

$\displaystyle
I^{(p_{A1},p_{B1}),(p_{A2},p_{B2})}=\sum_{L=0}^{4}b_{L,M}Y_{L,M}^{*}(\bm{k}),$
(7) $\displaystyle M=p_{B1}+p_{B2}-(p_{A1}+p_{A2}),$ (8)

where $b_{L}$ are coefficients whose explicit form is given in Eq. (13). In
most RABBIT experiments to date all four photon polarizations are linear and
equal implying $M=0$. Properties of the various other schemes employing
different photon polarizations are discussed in Appendix A and the complete
derivation leading up to Eq. (7) can be found in Appendix B.

####  2.3.1 Direct two-photon ionization vs RABBIT

The scheme for the direct two-photon ionization process is compared in Fig. 2
to the RABBIT case. As shown in Appendix A the direct two-photon photoelectron
angular distribution will have the same shape as in RABBIT when equal
polarizations in the two arms of the interferometer are used, cf. Eq (1).
Terms with non-zero $M$ appear in the skewed-polarized case in both RABBIT 27
and direct two-photon ionization 28, 29. However, as shown in Appendix A (see
Eq. (14)), only in the case of RABBIT we have (at least in theory) the
possibility to control precisely the exact $M$ value of the contributing
spherical harmonic by fixing the polarizations of the four photons
independently. Furthermore, in RABBIT the quantum number $M$ can assume values
such that $|M|\leq 4$, whereas in the direct case it is limited to $|M|\leq
2$. In this sense, RABBIT is a more selective probe of photoionization
dynamics compared to direct two-photon ionization. Nevertheless, even the
usual RABBIT setup with equal polarizations is more informative than the
direct process since it provides access not only to the PAD but also to the
photoionization phase.

In this case, due to the interference between the two photoionization pathways
involving consecutive harmonics, the signal of the sideband (SB) of order $N$
is described by the relation:

$I^{SB}_{N}(\Delta t;E)=A_{0\omega}+A_{2\omega}\cos(2\omega\Delta
t-\Delta\varphi),$ (9)

where the parameters $A_{0\omega}$, $A_{2\omega}$ and $\Delta\varphi$ indicate
the constant and oscillating component of the RABBIT signal and the phase of
its oscillation. In our analysis, these three parameters depend on the
photoelectron energy $E$ and on the angle $\theta$, while $\Delta t$ indicates
the relative delay between the attosecond pulse train and the IR field. The
modulus of the interfering term corresponds to the amplitude of the
oscillating component of the sideband signal $A_{2\omega}$ and is proportional
to $I^{p_{1},p_{2}}$, a particular case of the general form introduced in Eq.
(7) whose expression is reported in Eq. (31). Moreover, the phase
$\Delta\varphi$ is equal to the numerator reported in Eq. (5).

For randomly oriented molecules, in analogy to atoms, the angular dependence
of the interference contribution to the sideband signal can be parametrized
using complex beta parameters $\tilde{\beta}_{2}$ and $\tilde{\beta}_{4}$ (see
Eq. (34) in Appendix B). As shown later in the case of the experiments in
methane and deuteromethane, these quantities can be extracted using the
relations:

$\displaystyle
A_{2\omega}(\theta)=\left|\frac{A_{int}}{4\pi}\left(1+\sum_{L=2,4}\tilde{\beta}_{L}P_{L}(\cos\theta)\right)\right|\,,$
(10)
$\displaystyle\Delta\varphi(\theta)=\mathrm{arg}\left[\frac{A_{int}}{4\pi}\left(1+\sum_{L=2,4}\tilde{\beta}_{L}P_{L}(\cos\theta)\right)\right]\,,$
(11)

where $A_{int}$ describes the total oscillating component $A_{2\omega}$
integrated over all angles.

###  2.4 Treatment of nuclear-motion effects and spectral simulations

The theory of nuclear-motion effects in RABBIT spectra and the numerical
details of the simulations in methane have been described elsewhere30, 26, and
need not be fully restated here. Briefly, the nuclear-motion effects encompass
three distinct, isotope-specific contributions: the change in the average
geometry upon ionization, vibronic energy redistribution in the cation, and
the coherent averaging over the spatial extent of the zero-point vibrational
wavefunction.

The average-geometry contribution is treated by considering electronic matrix
elements evaluated at the “characteristic” nuclear geometry, arising due to
the short-time dynamics on the cationic potential-energy surfaces upon
photoionization. For CH4 (CD4) these geometries are taken $1.17$ fs ($1.27$
fs) after the ionization event26. They exhibit $D_{2d}$ symmetry, lower than
the nominal $T_{d}$ symmetry of the equilibrium neutral species. The effects
of the vibronic-energy redistribution, leading to the finite energy width of
the photoelectron spectra, are treated within the autocorrelation formalism30.
These effects enter the electronic matrix elements via an effective ionization
potential, at each photoelectron energy and RABBIT sideband.

The zero-point effects are treated in the harmonic approximation for the
nuclear vibrational wavefunctions and second-order finite-difference expansion
of the electronic matrix elements around the high-symmetry initial geometry.
Due to the high cost of the matrix-element evaluation, the (small) zero-point
correction is only evaluated at one photoelectron energy within each sideband,
corresponding to the vertical IP in the valence photoionization band ($14.4$
eV).

For numerical simplicity, all nuclear-motion effects are evaluated in the
molecular frame, and then averaged over the molecular orientations
numerically, using the order-17 Lebedev grid for the $\alpha,\beta$ Euler
angles and a uniform grid of the matching order for the $\gamma$ Euler angle.
The complex $\tilde{\beta}$ coefficients were then determined by “observing”
the photoelectrons at three angles relative to the common polarization
direction of the laser fields, corresponding to the zeros of the $P_{2}$ and
$P_{4}$ Legendre polynomials and fitting the Eq. (10) to the results. We have
verified that the fit reproduces the calculated profile at other
photoelectron-detection angles exactly. In the absence of nuclear-motion
effects, the numerical procedure agrees exactly with the analytical result
given by the Eq. (7). The numerical results are illustrated below, in Fig. 3.

Figure 3: Calculated asymmetry parameters as a function of photoelectron
energy. On all panels, the lower horizontal axis gives photoelectron energy in
eV. The upper horizontal axis indicates the side-band number; the position of
the maximum of the photoelectron peaks correlating to non-fragmenting
CH4+/CD4+ photoions (IP$\approx 13.61$ eV) within each sideband are indicated
with asterisks. Magenta lines indicate values calculated for the fixed nuclei
at the equilibrium CH4 geometry. Gold lines with open squares (teal lines with
open circles) show the results including nuclear motion effects in CH4 (CD4).
The panels are: the real (a) and imaginary (b) part of $\tilde{\beta_{2}}$ and
the real (c) and imaginary (d) part of $\tilde{\beta_{4}}$ coefficient. See
text for the numerical details.

For the comparison with the experimental results presented in the following
sections, the theoretical predicted oscillating component, phase, and complex
asymmetry parameters are taken at the center of the associated ionic (CH4+ or
CD4+) band.

##  3 Results and discussion

###  3.1 XUV photoionization of CH4 and CD4

Figure 4 reports the comparison between the LF PADs measured in the XUV-only
case and the theoretical predictions for the energies corresponding to the
absorption of different harmonics orders from H13 to H19. The PADs measured in
coincidence with CH4+ and CD4+ are shown in the upper (a-d) and lower (e-h)
panels, respectively. The agreement between the experiment and theoretical
predictions (shown in red) is excellent. For high kinetic energy
photoelectrons ((H19, in Fig. 4d and h), small discrepancies above 60∘can be
observed. This might originate from a combination of low statistics and higher
sensitivity of high energy photoelectrons to magnetic nodes. The quality of
the agreement is further supported by the comparison between the $\beta_{2}$
parameter extracted from the fitting procedure of the experimental and
theoretical data reported in Fig. 5a and b for the CH4+ and CD4+ ions,
respectively.

Figure 4: Experimental LF PADs (blue points and error bars) measured in
coincidence with the ionic channels CH4+ (a-d, upper row) and CD4+ (e-h, lower
row) using a train of attosecond pulses consisting of the odd harmonics of the
fundamental radiation H13 (a,e), H15 (b,f), H17 (c,g), H19 (d,h). The error
bars are the standard deviations within the energy integration range. The
theoretical predictions, multiplied by $\sin(\theta)$ in order to take into
account the geometrical effect related to the solid angle of the detector, are
presented in red. Adapted with permission from ref. 23 . Copyright 2022
University of Freiburg. Figure 5: Comparison of the $\beta_{2}$ asymmetry
parameter for different harmonic orders derived from the fitting of the LF
PADs (blue open circles and solid lines) and from the R-matrix calculations 26
using UKRmol+ (red diamonds and dashed lines) for the PADs measured in
coincidence with the ionic channels CH4+ (a) and CD4+ (b). The error bars were
determined from the fitting procedure of the PADs. Adapted with permission
from ref. 23 . Copyright 2022 University of Freiburg.

###  3.2 Two-color photoionization of CH4 and CD4

####  3.2.1 Angle-resolved oscillating component and phase of the RABBIT
signal

It was not possible to extract the $\beta_{2}$ and $\beta_{4}$ parameters for
the two-color photoionization signal directly from the PADs of the sideband
signal, due to the large overlap between the single and two-color signal. To
isolate the latter term, we first extracted the oscillating component of the
photoelectron signal in the energy region centered around the sideband signal
by performing either a Fourier Transform and considering the term at frequency
$2\omega$ or a fit according to the Eq. (9). We have verified that both two
methods deliver consistent results.

In general, even though the single-photon (XUV) and the two-photon (XUV-IR)
paths can lead to the same final photoelectron energy, the absorption of the
initial XUV photon along the two pathways will lead to a different final ionic
state, thus preventing the observation of an interference effect. Moreover,
any possible interference term contributed by a single-photon and a two-photon
pathway is expected to wash out when averaging over several pulses, due to the
shot-to-shot variation of the carrier-envelope phase of the driving source
used in the experiment.

Figure 6 shows the angular dependence of the oscillating component
$A_{2\omega}(\theta)$ integrated over the energy range of the SB16 (blue
symbols and error bars), together with the corresponding quantity extracted
from the theoretical simulations for the ions CH4+ (a) and CD4+ (b) using the
Fourier transform of the photoelectron signal. The agreement in the two-color
case is also excellent.

Figure 6: Experimental LF angular dependence of the $A_{2\omega}$ components
of the photoelectron spectrum corresponding to the SB16 (blue points and error
bars) measured in coincidence with the ionic channels CH4+ (a) and CD4+ (b).
The error bars were determined as the standard deviation within the energy
integration interval. The red curves were obtained from the theoretical
predictions following the same procedure adopted for the experimental data.
Adapted with permission from ref. 23 . Copyright 2022 University of Freiburg.

A similar agreement was obtained with the direct fitting of the sideband
oscillations using Eq. (9), as presented in Fig. 7, which reports the angular
dependence of the $A_{2\omega}$ components for CH4+ (a) and CD4+ (b). The
curves corresponding to different sidebands have been vertically shifted for
visual clarity. The phase of the sideband oscillations, obtained additionally
through the fitting procedure, are depicted in Fig. 7c,d for CH4+ and CD4+,
respectively. Here, we subtracted from the sideband phase $\Delta\varphi$,
introduced in Eq. (9), the phase value obtained around $\theta=0^{\circ}$
(i.e., $\Delta\varphi(\theta=0^{\circ})$) for each sideband. Using this
approach, a comparison between the absolute phases of different sidebands is
not possible, but it makes the investigation of the angular dependence of the
phases and the comparison between consecutive sidebands easier. Moreover, this
also removes the influence of the attochirp, which adds a different angle-
independent phase offset to the different sidebands.

The comparison between experimental and theoretical data shows, in general,
excellent agreement in the interval from $0^{\circ}$ to $70^{\circ}$ (see Fig.
7). The low experimental sideband signal for larger angles makes a reliable
determination of the phases of the sideband oscillations around $90^{\circ}$
difficult. Nevertheless, the data indicate the presence of a minimum at this
angle for all sidebands. The presence of a minimum for the two-color
photoionization phase in the direction perpendicular to the laser polarization
in the randomly oriented ensemble of methane and deuteromethane is consistent
with the angular dependence of the photoemission phase in two-color fields
observed in atoms 9.

Figure 7: Experimental (open circles and error bars) and theoretical (dashed
lines) angular dependence of the $A_{2\omega}$ components of the photoelectron
spectrum and the sideband phases $\Delta\varphi$ measured in coincidence with
the ionic channels CH4+ (a, c) and CD4+ (b, d) for the sidebands SB14 (blue),
SB16 (purple), and SB18 (red). The theory values are taken at the center of
the CH4+/CD4+-associated band, with the nuclear motion treated as described in
ref.26. The grey solid lines correspond to the fitting of the distributions
obtained using Eqs. (10, 11). Adapted with permission from ref. 23 . Copyright
2022 University of Freiburg.

####  3.2.2 Parametrization of angle-resolved sideband phases

We retrieved the complex beta parameters $\tilde{\beta}_{2}$ and
$\tilde{\beta}_{4}$, by performing a simultaneous fit of the oscillating
component $A_{2\omega}$ and phase $\Delta\varphi$ according to Eqs. (10,11).
The fits of the experimental data are presented in Fig. 7, indicating that the
parametrization can reproduce qualitatively very well the angular evolution of
both the oscillating components and phases.

The numerical values of the complex beta parameters extracted from the
experimental data (Experiment) are reported in Table 1. We report in the same
table also those determined directly from the theoretical model including the
nuclear motion (Theory (nucl. motion)) and considering fixed nuclei (Theory
(fixed nuclei)). The comparison between the experimental and theoretical
results is shown in the complex plane representation in Fig. 8. In general,
the agreement is satisfactory. Some discrepancies are observed, in particular
for the imaginary parts of the $\tilde{\beta}_{4}$ parameters in CH4+ and, to
a less extent, in CD4+. Moreover, the values extracted from the theoretical
models with and without inclusion of nuclear effects present some differences,
which are more pronounced for the SB14, as also shown in Fig. 3. The isotopic
dependence of the real and imaginary parts of the asymmetry parameters derived
from the simulation is very limited, while the corresponding experimental
quantities are compatible within the error bars, except for the imaginary part
of $\tilde{\beta}_{4}$ for the SB16. The absence of a significant isotopic
effect in the photoionization phases is consistent with the results presented
in refs. 26, 31.

$\tilde{\beta_{L}}$ | SB | Data |  CH4+ |  CD4+
---|---|---|---|---
$\tilde{\beta}_{2}$ | 14 | Experiment | $(2.49\pm 0.21)+(0.45\pm 0.14)i$ | $(2.48\pm 0.19)+(0.40\pm 0.11)i$
Theory (nucl. motion) | $2.47+0.84i$ | $2.47+0.77i$
Theory (fixed nuclei) | $2.63+0.36i$ | $2.63+0.36i$
$\tilde{\beta}_{4}$ | 14 | Experiment | $(0.21\pm 0.44)+(0.05\pm 0.08)i$ | $(0.39\pm 0.35)+(0.16\pm 0.06)i$
Theory (nucl. motion) | $0.31+0.39i$ | $0.31+0.37i$
Theory (fixed nuclei) | $0.62+0.24i$ | $0.62+0.24i$
$\tilde{\beta}_{2}$ | 16 | Experiment | $(2.22\pm 0.36)+(0.20\pm 0.08)i$ | $(2.53\pm 0.10)+(0.30\pm 0.09)i$
Theory (nucl. motion) | $2.57+0.24i$ | $2.58+0.24i$
Theory (fixed nuclei) | $2.66+0.15i$ | $2.66+0.15i$
$\tilde{\beta}_{4}$ | 16 | Experiment | $(0.34\pm 0.53)+(0.02\pm 0.05)i$ | $(0.04\pm 0.10)+(0.14\pm 0.06)i$
Theory (nucl. motion) | $0.61+0.18i$ | $0.63+0.18i$
Theory (fixed nuclei) | $0.74+0.13i$ | $0.74+0.13i$
$\tilde{\beta}_{2}$ | 18 | Experiment | $(2.29\pm 0.24)+(0.26\pm 0.09)i$ | $(2.33\pm 0.09)+(0.35\pm 0.06)i$
Theory (nucl. motion) | $2.62+0.10i$ | $2.62+0.10i$
Theory (fixed nuclei) | $2.64+0.08i$ | $2.64+0.08i$
$\tilde{\beta}_{4}$ | 18 | Experiment | $(0.34\pm 0.37)+(-0.07\pm 0.05)i$ | $(0.57\pm 0.10)+(-0.01\pm 0.05)i$
Theory (nucl. motion) | $0.74+0.08i$ | $0.75+0.10i$
Theory (fixed nuclei) | $0.77+0.08i$ | $0.77+0.08i$

Table 1: Comparison of the complex asymmetry parameters $\tilde{\beta}_{2}$
and $\tilde{\beta}_{4}$ for SB14-18 of CH4+ and CD4+. The experimental
parameters (first lines) were retrieved from the analytical fit of
$A_{2\omega}(\theta)$ and phase $\Delta\varphi(\theta)$ according to Eqs.
(10,11). The theory values (second line) are at the center of the
CH4+/CD4+-associated band (see Fig. 3), with the nuclear motion treated as
described in ref. 26. Fixed-nuclei results (third lines) are for the single
geometry of the ground state neutral CH4 molecule and for the effective
$I_{p}$ of 13.6 eV.

Figure 8: Comparison of the real and imaginary parts of the
$\tilde{\beta}_{2}$ (a,b) and $\tilde{\beta}_{4}$ (c,d) parameters derived
from the experimental data (full symbols with error bars) and theoretical
models with (crossed symbol) and without (open symbols) inclusion of the
effects of nuclear dynamics for the sidebands SB14 (black squares), SB16 (red
circles) and SB18 (green diamonds) for the ionic fragments CH4+ (a,c) and CD4+
(b,d).

##  4 Conclusions

We have shown that the oscillating components and the photoionization phases
of a RABBIT signal originating from an ensemble of randomly oriented molecules
can be described by two complex asymmetry parameters $\tilde{\beta_{2}}$ and
$\tilde{\beta_{4}}$, in full analogy with the atomic case. Using advanced
numerical simulations, we have demonstrated that the complex asymmetry
parameters extracted from the experimental and theoretical data are in
satisfactory agreement. We have derived a general formula, describing the
accessible magnetic and angular momentum quantum numbers for generic
polarization directions of the XUV and IR pulses.

Future developments of the theory will focus on inclusion of nuclear dynamics
in the electronic continuum which is presently neglected and only nuclear
dynamics in the ion is included. It is expected that nuclear dynamics in the
continuum becomes important in case of resonant photoionization unlike in this
case where strong resonant effects have not been observed. Development of such
theory and its numerical implementation are currently underway. Similarly,
applications of the present methods to larger polyatomic molecules presents a
further challenge for the theory. However, disentangling in the experiments
the different pathways leading to the same final ionic fragments becomes even
more challenging in larger systems. The implementation of coincidence
spectroscopy will be crucial in obtaining information on the molecule’s
photoionization and subsequent photodissociation paths.

This project has received funding from the European Union’s Horizon 2020
research and innovation program under the Marie Sklodowska-Curie grant
agreement no. 641789 MEDEA. D.B. and E.L. acknowledge support from the Swedish
Research Council (grant 2020-06384 and grant 2020-03315, respectively). D.E.
and G.S. acknowledge support and funding by the Deutsche
Forschungsgemeinschaft (DFG) project International Research Training Group
(IRTG) CoCo 2079 and INST 39/1079. I.M. and G.S. acknowledge financial support
from the BMBF project 05K19VF1, the DFG project Research Training Group DynCAM
(RTG 2717), and the Georg H. Endress Foundation. G.S. acknowledges financial
support from the DFG Priority Program 1840 (QUTIF) and grant 429805582
(Project SA 3470/4-1). Z.M. and J.B. acknowledge the support of the PRIMUS
(20/SCI/003) project and the Czech Science Foundation (20-15548Y).
Computational resources were supplied by the project “e-Infrastruktura CZ”
(e-INFRA LM2018140) provided within the program Projects of Large Research,
Development and Innovations Infrastructures. This work was also supported by
the Ministry of Education, Youth and Sports of the Czech Republic through the
e-INFRA CZ (ID:90254).

We thank C. D. Schröter, T. Pfeifer, and R. Moshammer from the Max Planck
Institute for Nuclear Physics in Heidelberg for their contribution to the
development of the Reaction Microscope.

##  Appendix A Appendix A: RABBIT with arbitrary photon polarizations

The particular form of the orientationally averaged PAD for RABBIT depends on
the choice of the four photon polarizations involved, see Fig. 2. The result
for the general case expressed in the spherical basis for the four photon
polarizations is

$\displaystyle
I^{(p_{A1},p_{B1}),(p_{A2},p_{B2})}=\sum_{L=0}^{4}b_{L,M}Y_{L,M}^{*}(\bm{k}),$
(12) $\displaystyle
b_{L,M}=(-1)^{p_{A1}+p_{A2}}\sum_{\begin{subarray}{c}l,m,q_{1},q_{2}\\\
l^{\prime},m^{\prime},q_{1}^{\prime},q_{2}^{\prime}\end{subarray}}\mathrm{i}^{-l+l^{\prime}}e^{\mathrm{i}(\sigma_{l}-\sigma_{l^{\prime}})}d^{q_{1},q_{2},(A)}_{if,l,m}\left[d^{q_{1}^{\prime},q_{2}^{\prime},(B)}_{if,l^{\prime},m^{\prime}}\right]^{*}\sum_{K_{1},K_{2}}(2K_{1}+1)(2K_{2}+1)$
$\displaystyle(-1)^{-q_{1}-q_{2}}\begin{pmatrix}1&1&K_{1}\\\
-q_{1}&q_{1}^{\prime}&M_{1}\end{pmatrix}\begin{pmatrix}1&1&K_{1}\\\
-p_{A1}&p_{B1}&M_{p_{1}}\end{pmatrix}\begin{pmatrix}1&1&K_{2}\\\
-q_{2}&q_{2}^{\prime}&M_{2}\end{pmatrix}\begin{pmatrix}1&1&K_{2}\\\
-p_{A2}&p_{B2}&M_{p_{2}}\end{pmatrix}$
$\displaystyle(-1)^{m^{\prime}}\sqrt{\frac{(2l+1)(2l^{\prime}+1)(2L+1)}{4\pi}}\begin{pmatrix}l&l^{\prime}&L\\\
m&-m^{\prime}&\mu\end{pmatrix}\begin{pmatrix}l&l^{\prime}&L\\\
0&0&0\end{pmatrix}$ $\displaystyle\begin{pmatrix}K_{1}&K_{2}&L\\\
M_{p_{1}}&M_{p_{2}}&M\end{pmatrix}\begin{pmatrix}K_{1}&K_{2}&L\\\
M_{1}&M_{2}&\mu\end{pmatrix},$ (13)

with

$\displaystyle M=p_{B1}+p_{B2}-(p_{A1}+p_{A2}).$ (14)

In the equations above $\bm{k}$ stands for the photoelectron momentum in the
LF, $\sigma_{l}$ is the Coulomb phase and $d^{q_{1},q_{2},(A/B)}_{if,l,m}$ are
the molecular-frame two-photon partial-wave dipole matrix elements. $K_{1}$
and $K_{2}$ denote the rank of the tensor operator formed from the combination
of two dipole interactions (each of rank one). $L$, which determines the
angular dependence, is the final rank obtained when $K_{1}$ and $K_{2}$ are
coupled. The tensor components, given by the integers $M_{1/2}$,
$M_{p_{1}/p_{2}}$, and $\mu$ are fixed by the standard selection rules for 3j
coefficients. The RABBIT delays are then obtained using Eq. (5). This is the
most general form of the orientationally averaged angle-resolved RABBIT time
delay. The last equation shows that terms in the angular distribution with
non-zero $M$ appear as a result of an imbalance of photon polarizations
between the two arms of the interferometer. The selection rule expressed in
Eq. (14) is also an indirect selection rule on $L$ too since $L\geq|M|$.
Therefore, if $M\neq 0$ then terms with $L\leq|M|$ don’t contribute and we see
that by controlling polarizations of the four photons it is possible to
partially disentangle the role of the individual $L,M$ contributions. E.g. in
the extreme case of $M=4$ only the spherical harmonic $Y_{4,4}^{*}(\bm{k})$
contributes. Furthermore, the angular distribution will contain spherical
harmonics with odd $L$ values only in chiral molecules and when at least one
of the photons is circularly polarized. This can be seen 32 by swapping the
signs of all magnetic quantum numbers and noticing that the overall sign of
the product of the 3j coefficients will change. Therefore, the total sum is
zero unless $d^{q_{1},q_{2}}_{if,l,m}\neq d^{-q_{1},-q_{2}}_{if,l,-m}$ which
happens only in chiral molecules 32.

We will now discuss the properties of the angle-resolved time delays for
different choices of the photon polarizations. The complete formulas
specialized to selected combinations of photon polarizations are given in
Appendix B. Here we limit ourselves to qualitative aspects of the different
polarization choices.

####  A.0.1 Parallel linear polarizations of the XUV and IR photons

This is the most frequently chosen configuration corresponding to $p_{i}=0$
and the interference term specialized to this case is given by Eq. (31) in
Appendix B. Concretely, the 3j coefficient $\begin{pmatrix}1&1&K_{i}\\\
-p_{i}&p_{i}&0\end{pmatrix}=\begin{pmatrix}1&1&K_{i}\\\ 0&0&0\end{pmatrix}$
forces $K_{i}$ to be even and therefore $\begin{pmatrix}K_{1}&K_{2}&L\\\
0&0&0\end{pmatrix}$ implies that $L$ is even too (since both $K_{1}$ and
$K_{2}$ are), i.e. $L=0,2,4$ and the angular distribution has the form of Eq.
(1), where only even-order $\tilde{\beta}$ parameters appear which is
equivalent to the atomic case 14 and this result does not depend on the
chirality of the system.

####  A.0.2 Circular polarization and chirality

If at least one of the photons is circularly polarized then $K_{i}$ may be odd
and the selection rule $K_{1}+K_{2}+L=2n$ allows $L$ to be odd too. Therefore
$\tilde{\beta}_{1}$ and $\tilde{\beta}_{3}$ components may appear. Let’s
assume that $L$ is odd and for simplicity set $p_{A1}=p_{B1}\neq 0$ and
$p_{A2}=p_{B2}=0$. This implies the following:

  1. 1.

The 3j coefficient $\begin{pmatrix}K_{1}&K_{2}&L\\\ 0&0&0\end{pmatrix}$
implies that $K_{1}+K_{2}$ is odd.

  2. 2.

Changing $m\rightarrow-m$ and $m^{\prime}\rightarrow-m^{\prime}$ does not
change the sign of the 3j coefficient $\begin{pmatrix}l&l^{\prime}&L\\\
m&-m^{\prime}&\mu\end{pmatrix}$ but requires changing the sign of $\mu$.

  3. 3.

Changing the sign of $\mu$ implies $M_{1}\rightarrow-M_{1}$ and
$M_{2}\rightarrow-M_{2}$, i.e. changing signs of all $q$s.

  4. 4.

The combined sign change upon the replacements above is
$(-1)^{K_{1}+K_{2}}=-1$ due to the first point and the combined sign change of
the two $q$-related 3j coefficients.

Therefore, the whole summation reduces to zero (i.e. $\beta_{1}=\beta_{3}=0$)
if the two-photon partial wave dipoles are symmetric with respect to a change
of all magnetic quantum numbers

$\displaystyle d_{if,l,m}^{q_{1},q_{2}}=d_{if,l,-m}^{-q_{1},-q_{2}}.$ (15)

As mentioned above, this is precisely the case in non-chiral molecules. In
chiral molecules the summation is generally non-zero since the symmetry of the
dipole matrix elements is broken as in the one-photon case 32, 33, 34.
Therefore we obtain a straightforward generalization of the one-photon
photoemission circular dichroism (PECD) to multi-photon above-threshold PECD.
A similar case of ($n+1$) multi-photon ionization of chiral molecules has been
studied before 35. A mixed case of circular and linear polarization in various
configurations has been studied for the case of atoms recently 36.

####  A.0.3 Polarization-skewed measurement

In this configuration, polarizations of the XUV and NIR photons are linear but
skewed at an angle $\Theta$. This setup has been realized recently 27. The PAD
29 and the RABBIT delays 27 then include spherical harmonic components with
$|M|\leq 2$. The exact form of the PAD is obtained as the appropriate linear
combination of the general interference term according to Eq. (7) formally
including contributions of circularly polarized photons. The appearance of odd
L terms in this case follows the rules for the circular polarization above.

##  Appendix B Appendix B: orientation averaging

In this Appendix we perform an explicit derivation of the orientationally
averaged PAD, Eq. (13), for the case of two-photon interfering pathways. For
the general case of $n$-photon above-threshold ionization we refer to the
result presented in our earlier work 24. Throughout we employ the spherical
harmonics and rotational matrices as defined in Varshalovich 37.

The final continuum wavefunction in the molecular frame is given by the
expression

$\displaystyle\ket{\Psi_{\bm{k^{\prime}}}^{(-)}}=\sum_{\begin{subarray}{c}l,m\end{subarray}}\mathrm{i}^{l}e^{-\mathrm{i}\sigma_{l}}Y_{l,m}^{*}(\mathbf{k^{\prime}})\ket{\psi_{l,m}^{(-)}}_{M},$
(16)

where $\sigma_{l}$ is the Coulomb phase. The Wigner rotation matrix transforms
angular momentum eigenstates $\Psi_{JM}$ between the laboratory and molecular
frames

$\displaystyle\Psi_{JM}(\mathbf{r^{\prime}})$ $\displaystyle=$
$\displaystyle\sum_{M^{\prime}}D^{J}_{M^{\prime},M}(R)\Psi_{JM^{\prime}}(\mathbf{r}),$
(17) $\displaystyle\Psi_{JM}(\mathbf{r})$ $\displaystyle=$
$\displaystyle\sum_{M^{\prime}}[D^{-1}(R)]^{J}_{M^{\prime},M}\Psi_{JM^{\prime}}(\mathbf{r}_{M})=\sum_{M^{\prime}}[D^{J}_{M,M^{\prime}}(R)]^{*}\Psi_{JM^{\prime}}(\mathbf{r^{\prime}})$
(18) $\displaystyle=$
$\displaystyle\sum_{M^{\prime}}(-1)^{M-M^{\prime}}D^{J}_{-M,-M^{\prime}}(R)\Psi_{JM^{\prime}}(\mathbf{r^{\prime}}),$

where $\mathbf{r^{\prime}}$ and $\mathbf{r}$ stand for the molecular and
laboratory frame coordinates, respectively. In the following we will suppress
the argument $R$ of the rotation matrices which specifies the particular
rotation of the molecular frame. Photon polarization and the detector
photoelectron momentum are both fixed vectors defined in the lab-frame while
the photoelectron wavefunction (and the dipole matrix elements) are calculated
in the molecular frame, cf. Eq. (2) and Eq. (16),

$\displaystyle M_{\bm{k^{\prime}},q_{1},q_{2}}^{(2)}$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}l,m\end{subarray}}\mathrm{i}^{-l}e^{\mathrm{i}\sigma_{l}}Y_{l,m}(\bm{k^{\prime}})d^{q_{1},q_{2}}_{if,l,m},$
(19) $\displaystyle d^{q_{1},q_{2}}_{if,l,m}$ $\displaystyle=$
$\displaystyle\braket{\psi_{l,m}^{(-)}}{\hat{D}_{q_{2}}\hat{G}_{1}^{(+)}\hat{D}_{q_{1}}}{\Psi_{i}}.$
(20)

In the equation for $M_{\bm{k^{\prime}},q_{1},q_{2}}^{(2)}$ we have explicitly
included both photon polarization indices $q_{1}$ and $q_{2}$. The dipole
operator for lab-frame polarization $p$ is defined as

$\displaystyle\hat{D}_{p}=\sqrt{\frac{4\pi}{3}}\sum_{i}r_{i}Y_{1,p}(\bm{r}_{i}),$
(21)

where the sum runs over all electrons in the system. Expressing the lab-frame
spherical harmonic (photon polarization) in the molecular frame using Eq. (18)
and inserting it into Eq. (19) we obtain the dipole matrix element for the
lab-frame photon polarizations $p_{1}$ and $p_{2}$

$\displaystyle M_{\bm{k}_{fM}i,p_{1},p_{2}}^{(2)}$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}l,m,q_{1},q_{2}\end{subarray}}\mathrm{i}^{-l}e^{\mathrm{i}\sigma_{l}}Y_{l,m}(\bm{k^{\prime}})d^{q_{1},q_{2}}_{if,l,m}(-1)^{p_{2}-q_{2}}D^{1}_{-p_{2},-q_{2}}(-1)^{p_{1}-q_{1}}D^{1}_{-p_{1},-q_{1}}.$
(22)

Using this result to express the RABBIT interference term we get

$\displaystyle
M_{\bm{k}_{fM}i,p_{A1},p_{A2}}^{(2,A)}M_{\bm{k}_{fL}i,p_{B1},p_{B2}}^{(2,B)*}=\sum_{\begin{subarray}{c}l,m,q_{1},q_{2}\\\
l^{\prime},m^{\prime},q_{1}^{\prime},q_{2}^{\prime}\end{subarray}}\mathrm{i}^{-l+l^{\prime}}e^{\mathrm{i}(\sigma_{l}-\sigma_{l^{\prime}})}Y_{l,m}(\bm{k^{\prime}})Y_{l^{\prime},m^{\prime}}^{*}(\bm{k^{\prime}})$
$\displaystyle
d^{q_{1},q_{2},(A)}_{if,l,m}\left[d^{q_{1}^{\prime},q_{2}^{\prime},(B)}_{if,l^{\prime},m^{\prime}}\right]^{*}(-1)^{-q_{1}-q_{1}^{\prime}}(-1)^{-q_{2}-q_{2}^{\prime}}D^{1}_{-p_{A2},-q_{2}}D^{1*}_{-p_{B2},-q_{2}^{\prime}}D^{1}_{-p_{A1},-q_{1}}D^{1*}_{-p_{B1},-q_{1}^{\prime}}$
$\displaystyle(-1)^{p_{A1}+p_{A2}},$ (23)

where we’ve included the sub- and superscripts $A$ and $B$ to differentiate
between the two interferometric pathways and the corresponding photon
polarizations. Next we simplify the pairs of products of the rotation matrices
using the identity

$\displaystyle
D^{1}_{-p,-q}[D^{1}_{-p^{\prime},-q^{\prime}}]^{*}=(-1)^{-p^{\prime}+q^{\prime}}\sum_{K_{1}}(2K_{1}+1)\begin{pmatrix}1&1&K_{1}\\\
-q&q^{\prime}&M\end{pmatrix}\begin{pmatrix}1&1&K_{1}\\\
-p&p^{\prime}&M_{p}\end{pmatrix}[D^{K_{1}}_{M_{p},M}]^{*},$ (24)

contract the product of the two spherical harmonics and use Eq. (17) to
transform them into the lab-frame

$\displaystyle
Y_{l,m}(\bm{k^{\prime}})Y_{l^{\prime},m^{\prime}}^{*}(\bm{k^{\prime}})$
$\displaystyle=$
$\displaystyle(-1)^{m^{\prime}}\sum_{L}\sqrt{\frac{(2l+1)(2l^{\prime}+1)(2L+1)}{4\pi}}\begin{pmatrix}l&l^{\prime}&L\\\
m&-m^{\prime}&\mu\end{pmatrix}\begin{pmatrix}l&l^{\prime}&L\\\
0&0&0\end{pmatrix}$ (25) $\displaystyle\times$ $\displaystyle
Y_{L,\mu}^{*}(\bm{k^{\prime}}),$ $\displaystyle
Y_{L,\mu}^{*}(\bm{k^{\prime}})$ $\displaystyle=$
$\displaystyle\sum_{M}[D^{L}_{M,\mu}(R)]^{*}Y_{L,M}^{*}(\bm{k}).$ (26)

Inserting those expressions into Eq. (23) we arrive at the expression for the
lab-frame interference term for a fixed orientation of the molecule given by
the rotation $R$

$\displaystyle
M_{\bm{k}_{fM}i,p_{A1},p_{A2}}^{(2,A)}M_{\bm{k}_{fL}i,p_{B1},p_{B2}}^{(2,B)*}=\sum_{\begin{subarray}{c}l,m,q_{1},q_{2}\\\
l^{\prime},m^{\prime},q_{1}^{\prime},q_{2}^{\prime}\end{subarray}}\mathrm{i}^{-l+l^{\prime}}e^{\mathrm{i}(\sigma_{l}-\sigma_{l^{\prime}})}d^{q_{1},q_{2},(A)}_{if,l,m}\left[d^{q_{1}^{\prime},q_{2}^{\prime},(B)}_{if,l^{\prime},m^{\prime}}\right]^{*}\sum_{K_{1},K_{2}}(2K_{1}+1)(2K_{2}+1)$
$\displaystyle(-1)^{-q_{1}-q_{2}}(-1)^{p_{A1}+p_{A2}}\begin{pmatrix}1&1&K_{1}\\\
-q_{1}&q_{1}^{\prime}&M_{1}\end{pmatrix}\begin{pmatrix}1&1&K_{1}\\\
-p_{A1}&p_{B1}&M_{p_{1}}\end{pmatrix}\begin{pmatrix}1&1&K_{2}\\\
-q_{2}&q_{2}^{\prime}&M_{2}\end{pmatrix}\begin{pmatrix}1&1&K_{2}\\\
-p_{A2}&p_{B2}&M_{p_{2}}\end{pmatrix}$
$\displaystyle(-1)^{m^{\prime}}\sum_{L}\sqrt{\frac{(2l+1)(2l^{\prime}+1)(2L+1)}{4\pi}}\begin{pmatrix}l&l^{\prime}&L\\\
m&-m^{\prime}&\mu\end{pmatrix}\begin{pmatrix}l&l^{\prime}&L\\\
0&0&0\end{pmatrix}$
$\displaystyle\sum_{M}[D_{M_{p_{1}},M_{1}}^{K_{1}}]^{*}[D_{M_{p_{2}},M_{2}}^{K_{2}}]^{*}[D^{L}_{M,\mu}]^{*}Y_{L,M}^{*}(\bm{k}).$
(27)

Orientation averaging is now performed by integrating over all molecular
orientations

$\displaystyle\frac{1}{8\pi^{2}}\int\mathrm{d}R[D_{M_{p_{1}},M_{1}}^{K_{1}}(R)]^{*}[D_{M_{p_{2}},M_{2}}^{K_{2}}(R)]^{*}[D^{L}_{M,\mu}(R)]^{*}=\begin{pmatrix}K_{1}&K_{2}&L\\\
M_{p_{1}}&M_{p_{2}}&M\end{pmatrix}\begin{pmatrix}K_{1}&K_{2}&L\\\
M_{1}&M_{2}&\mu\end{pmatrix}.$ (28)

Therefore the final orientationally averaged expression for the interference
term is

$\displaystyle
I^{(p_{A1},p_{B1}),(p_{A2},p_{B2})}=(-1)^{p_{A1}+p_{A2}}\sum_{\begin{subarray}{c}l,m,q_{1},q_{2}\\\
l^{\prime},m^{\prime},q_{1}^{\prime},q_{2}^{\prime}\end{subarray}}\mathrm{i}^{-l+l^{\prime}}e^{\mathrm{i}(\sigma_{l}-\sigma_{l^{\prime}})}d^{q_{1},q_{2},(A)}_{if,l,m}\left[d^{q_{1}^{\prime},q_{2}^{\prime},(B)}_{if,l^{\prime},m^{\prime}}\right]^{*}\sum_{K_{1},K_{2}}(2K_{1}+1)(2K_{2}+1)$
$\displaystyle(-1)^{-q_{1}-q_{2}}\begin{pmatrix}1&1&K_{1}\\\
-q_{1}&q_{1}^{\prime}&M_{1}\end{pmatrix}\begin{pmatrix}1&1&K_{1}\\\
-p_{A1}&p_{B1}&M_{p_{1}}\end{pmatrix}\begin{pmatrix}1&1&K_{2}\\\
-q_{2}&q_{2}^{\prime}&M_{2}\end{pmatrix}\begin{pmatrix}1&1&K_{2}\\\
-p_{A2}&p_{B2}&M_{p_{2}}\end{pmatrix}$
$\displaystyle(-1)^{m^{\prime}}\sum_{L}\sqrt{\frac{(2l+1)(2l^{\prime}+1)(2L+1)}{4\pi}}\begin{pmatrix}l&l^{\prime}&L\\\
m&-m^{\prime}&\mu\end{pmatrix}\begin{pmatrix}l&l^{\prime}&L\\\
0&0&0\end{pmatrix}$ $\displaystyle\begin{pmatrix}K_{1}&K_{2}&L\\\
M_{p_{1}}&M_{p_{2}}&M\end{pmatrix}\begin{pmatrix}K_{1}&K_{2}&L\\\
M_{1}&M_{2}&\mu\end{pmatrix}Y_{L,M}^{*}(\bm{k}).$ (29)

The first 3j coefficient on the last line implies

$\displaystyle M=-M_{p_{1}}-M_{p_{2}}=p_{B1}+p_{B2}-(p_{A1}+p_{A2}).$ (30)

In case of identical polarizations of the first and the second photon in both
arms we have $p_{A1}=p_{B1}=p_{1}$ and $p_{A2}=p_{B2}=p_{2}$ giving $M=0$ and
the interference term is

$\displaystyle I^{p_{1},p_{2}}$ $\displaystyle=$
$\displaystyle(-1)^{p_{1}+p_{2}}\sum_{\begin{subarray}{c}l,m,q_{1},q_{2}\\\
l^{\prime},m^{\prime},q_{1}^{\prime},q_{2}^{\prime}\end{subarray}}\mathrm{i}^{-l+l^{\prime}}e^{\mathrm{i}(\sigma_{l}-\sigma_{l^{\prime}})}d^{q_{1},q_{2},(A)}_{if,l,m}\left[d^{q_{1}^{\prime},q_{2}^{\prime},(B)}_{if,l^{\prime},m^{\prime}}\right]^{*}\sum_{K_{1},K_{2}}(2K_{1}+1)(2K_{2}+1)$
(31) $\displaystyle\times$
$\displaystyle(-1)^{-q_{1}-q_{2}}\begin{pmatrix}1&1&K_{1}\\\
-q_{1}&q_{1}^{\prime}&M_{1}\end{pmatrix}\begin{pmatrix}1&1&K_{1}\\\
-p_{1}&p_{1}&0\end{pmatrix}\begin{pmatrix}1&1&K_{2}\\\
-q_{2}&q_{2}^{\prime}&M_{2}\end{pmatrix}\begin{pmatrix}1&1&K_{2}\\\
-p_{2}&p_{2}&0\end{pmatrix}$ $\displaystyle\times$
$\displaystyle(-1)^{m^{\prime}}\sum_{L}\sqrt{\frac{(2l+1)(2l^{\prime}+1)(2L+1)}{4\pi}}\begin{pmatrix}l&l^{\prime}&L\\\
m&-m^{\prime}&\mu\end{pmatrix}\begin{pmatrix}l&l^{\prime}&L\\\
0&0&0\end{pmatrix}$ $\displaystyle\times$
$\displaystyle\begin{pmatrix}K_{1}&K_{2}&L\\\
0&0&0\end{pmatrix}\begin{pmatrix}K_{1}&K_{2}&L\\\
M_{1}&M_{2}&\mu\end{pmatrix}Y_{L,0}^{*}(\bm{k}),$

which shows that the angle-resolved RABBIT delays contain contributions of
spherical harmonics with only $M=0$ and obviously $L=0,\dots,4$. This is
equivalent to the result in atoms 14. Expressing the spherical harmonic using
Legendre polynomials the equation above reduces to

$\displaystyle
I^{p_{1},p_{2}}=\frac{1}{4\pi}\sum_{L=0}^{4}b_{L}^{p_{1},p_{2}}P_{L}(\cos\theta),$
$\displaystyle
b_{L}^{p_{1},p_{2}}=(2L+1)\sum_{\begin{subarray}{c}l,m,q_{1},q_{2}\\\
l^{\prime},m^{\prime},q_{1}^{\prime},q_{2}^{\prime}\end{subarray}}\mathrm{i}^{-l+l^{\prime}}e^{\mathrm{i}(\sigma_{l}-\sigma_{l^{\prime}})}d^{q_{1},q_{2},(A)}_{if,l,m}\left[d^{q_{1}^{\prime},q_{2}^{\prime},(B)}_{if,l,m}\right]^{*}\sum_{K_{1},K_{2}}(2K_{1}+1)(2K_{2}+1)$
$\displaystyle\times$
$\displaystyle(-1)^{-q_{1}-q_{2}}(-1)^{p_{1}+p_{2}}\begin{pmatrix}1&1&K_{1}\\\
-q_{1}&q_{1}^{\prime}&M_{1}\end{pmatrix}\begin{pmatrix}1&1&K_{1}\\\
-p_{1}&p_{1}&0\end{pmatrix}\begin{pmatrix}1&1&K_{2}\\\
-q_{2}&q_{2}^{\prime}&M_{2}\end{pmatrix}\begin{pmatrix}1&1&K_{2}\\\
-p_{2}&p_{2}&0\end{pmatrix}$ $\displaystyle\times$
$\displaystyle(-1)^{m^{\prime}}\sum_{L}\sqrt{(2l+1)(2l^{\prime}+1)}\begin{pmatrix}l&l^{\prime}&L\\\
m&-m^{\prime}&\mu\end{pmatrix}\begin{pmatrix}l&l^{\prime}&L\\\
0&0&0\end{pmatrix}\begin{pmatrix}K_{1}&K_{2}&L\\\
0&0&0\end{pmatrix}\begin{pmatrix}K_{1}&K_{2}&L\\\
M_{1}&M_{2}&\mu\end{pmatrix},$

where $b_{L}$ is generally a complex number since the dipole matrix elements
are. This result is equivalent to Eq (66) from our previous work 24 for the
case of direct multi-photon ionization and to the result derived in 38 for the
case of RABBIT with equal polarization of both photons ($p_{1}=p_{2}=p$). The
complex $\tilde{\beta_{L}}$ parameters are calculated from Eq. (B) as

$\displaystyle\tilde{\beta_{L}}=\frac{b_{L}}{b_{0}}.$ (34)

Finally, in all cases above contributions with odd $L$ appear only in chiral
molecules when at least one circularly polarized photon participates 32.

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