# Time delay in the recoiling valence-photoemission of Ar endohedrally
confined in C60

Gopal Dixit<EMAIL_ADDRESS>Center for Free-Electron Laser Science, DESY,
Notkestrasse 85, 22607 Hamburg, Germany  Max Born Institute, Max-Born-Strasse
2A, 12489 Berlin, Germany  Himadri S. Chakraborty<EMAIL_ADDRESS>Department of Natural Sciences, Center for Innovation and Entrepreneurship,
Northwest Missouri State University, Maryville, Missouri 64468, USA  Mohamed
El-Amine Madjet<EMAIL_ADDRESS>Center for Free-Electron Laser Science,
DESY, Notkestrasse 85, 22607 Hamburg, Germany  Qatar Energy and Environment
Research Institute (QEERI), Qatar Foundation, Doha, Qatar

###### Abstract

The effects of confinement and electron correlations on the relative time
delay between the 3s and 3p photoemissions of Ar confined endohedrally in C60
are investigated using the time dependent local density approximation - a
method that is also found to mostly agree with recent time delay measurements
between the 3s and 3p subshells in atomic Ar. At energies in the neighborhood
of 3p Cooper minimum, correlations with C60 electrons are found to induce
opposite temporal effects in the emission of Ar 3p hybridized symmetrically
versus that of Ar 3p hybridized antisymmetrically with C60. A recoil-type
interaction model mediated by the confinement is found to best describe the
phenomenon.

###### pacs:

32.80.Fb, 61.48.-c, 31.15.E-

With the tremendous advancement in technology for generating attosecond (as)
isolated pulses as well as attosecond pulse trains, it becomes possible to
study fundamental phenomena of light-matter interaction with unprecedented
precision on an as timescale hentschel ; goulielmakis1 ; krausz . In
particular, the relative time delay between the photoelectrons from different
subshells on as timescale, a subject of intense recent activities, is expected
to probe important aspects of electron correlations that predominantly
influence the photoelectron. Pump-probe experiments have been performed to
measure the relative delay in the photoemission processes, where extreme
ultra-violet (XUV) pulses are used to remove an electron from a particular
subshell and subsequently a weak infrared (IR) pulse accesses the temporal
information of the emission event pazourek .

Streaking measurements were carried out to probe photoemission from the
valence and the conduction band in single-crystalline magnesium
neppl2012attosecond  and tungsten cavalieri2007attosecond . A streaking
technique was also employed to measure the relative delay of approximately
21$\pm$5 as between the 2s and 2p subshells of atomic Ne at 106 eV photon
energy schultze2010delay . Despite several theoretical attempts
mauritsson2005accessing ; kheifets2010delay ; moore2011time ;
ivanov2011accurate ; nagele2012time ; dahlstrom2012diagrammatic ;
kheifets2013time  to explain this measured delay in Ne, only about a half of
the delay could be reproduced, keeping the time delay in Ne photoemissions
still an open problem. Recently, the relative delay between the 3s and 3p
subshells in Ar is measured at three photon energies by interferometric
technique using attosecond pulses klunder2011probing ; guenot2012photoemission
. Theoretical methods (e.g. time-dependent nonperturbative method
mauritsson2005accessing , diagrammatic many-body perturbation theory
dahlstrom2012diagrammatic , Random phase approximation with exchange (RPAE)
guenot2012photoemission ; kheifets2013time , and multi-configurational
Hartree-Fock (MCHF) carette2013multiconfigurational ) have been employed to
investigate this relative delay in Ar, although agreements between theory and
experiment is rather inconclusive. A ubiquitous understanding in all these
studies is the dominant influence of electron correlations to determine the
time behavior of outgoing electrons. Thus, it is fair to expect that the
process near a Cooper minimum or a resonance will be particularly nuanced.

It is therefore of spontaneous interest to extend the study to test the effect
of correlations on the temporal photoresponse of atoms in material
confinements. A brilliant natural laboratory for such is an atom endohedrally
captured in a fullerene shell; see Fig. 1 which envisions the process. There
are two compelling reasons for this choice: (i) such materials are highly
stable, have low-cost sustenance at room temperature and are enjoying a rapid
improvement in their synthesis techniques popov2013  and (ii) effects of
correlations of the central atom with the cage electrons have been predicted
to spectacularly influence the atomic valence photoionization madjet2007giant
. In this Letter, by considering Ar@C60, we show that a confinement-induced
correlation effect of C60 at energies surrounding the Ar 3p Cooper minimum
produces a faster and a slower emission of the Ar 3p electrons hybridized,
respectively, in a symmetric and an antisymmetric mode with a near-degenerate
C60 orbital.

Figure 1: (Color online). Schematic for probing the effects of correlations
from the confinement on the relative time delay in the emission of an atom
encaged endohedrally inside C60.

Time dependent local density approximation (TDLDA), with Leeuwen and Baerends
(LB) exchange-correlation functional to produce accurate asymptotic behavior
van1994exchange  of ground and continuum electrons, is employed to calculate
the dynamical response of the system to the external electromagnetic field. To
demonstrate the accuracy of the method for an isolated atom, the total
photoionization cross section and the partial 3s and 3p cross sections of Ar
are presented in Fig. 2a and compared with available experiments
mobus1993measurements ; samson2002precision . As seen, our TDLDA total and 3s
cross sections are in excellent agreement with experimental results and the
positions of the 3s and 3p Cooper minima at, respectively, 42 and 48 eV are
well reproduced. The dominance of 3p contribution over 3s in this energy range
(Fig. 2a) also automatically implies the accuracy of our TDLDA 3p result.

Figure 2: (Color online). Top: TDLDA 3p, 3s and total photoionization cross
sections for atomic Ar are compared with experiments for 3s
mobus1993measurements  and total samson2002precision . For 3s the computed
cross section is scaled to reproduce the measurement at the Cooper minimum.
Bottom: The relative TDLDA time delay between 3s and 3p of Ar and its
comparison with measurements (solid black circles, Ref.
guenot2012photoemission ; open red squares, Ref. klunder2011probing ). RPAE
results kheifets2013time  at three experimental energies are also cited.

The absolute time delay in Ar pump-probe photoemission contains two
contributions: one due to the absorption of XUV photon and the other due to
the probe pulse. Owing to the weak probe pulse, the probe-assisted delay
contributions can be estimated dahlstrom2012diagrammatic  as a function of the
kinetic energy of electrons from different Ar subshells. This allowed
evaluation of the relative delay in recent measurements klunder2011probing ;
guenot2012photoemission . This delay therefore connects to the energy
derivative of the quantum phase of complex photoionization amplitude
yakovlev2010attosecond  \- the Wigner-Smith time delay wigner1955lower ;
smith1960lifetime ; de2002time . Several methods ivanov2011accurate ;
nagele2012time ; dahlstrom2012theory ; ivanov2013extraction  have been
utilized to extract the Wigner-Smith time delay directly from the
measurements.

The photoionization amplitude from an initial bound state ($n_{i}l_{i}$) to a
final continuum state ($kl$) can be expressed as

$\displaystyle f(\hat{\bf{k}})$ $\displaystyle=$
$\displaystyle(8\pi)^{3/2}\sum_{\begin{subarray}{c}l=l_{i}\pm 1\\\
m=m_{i}\end{subarray}}(-i)^{l}e^{i\eta_{l}(\hat{\bf{k}})}Y_{lm}^{*}(\hat{\bf{k}})\langle\phi_{kl}||r+\delta
V||\phi_{n_{i}l_{i}}\rangle$ (5)
$\displaystyle\times\sqrt{(2l+1)(2l_{i}+1)}\left(\begin{array}[]{ccc}l&1&l_{i}\\\
0&0&0\end{array}\right)\left(\begin{array}[]{ccc}l&1&l_{i}\\\
-m&0&m_{i}\end{array}\right).$

Here, $\delta V$ is the complex induced potential which embodies TDLDA many-
body correlations. The phase $\eta_{l}$ includes contributions from both the
short range and Coulomb potentials, whereas the phase of the complex matrix
element in Eq. (5) is the correlation phase. For Ar, the correlation near
Cooper minima primarily arises from the coupling of 3p with 3s channels. The
total phase is the sum of these three contributions. The time delay profile is
computed by differentiating the TDLDA total phase in energy.

Our TDLDA relative Wigner-Smith delay between Ar 3s and 3p,
$\tau_{\textrm{3s}}-\tau_{\textrm{3p}}$, is compared with the experimental
data of Guénot et al. guenot2012photoemission  and of Klünder et al.
klunder2011probing  in Fig. 2b. As seen, the relative delay is strongly energy
dependent. Note that the TDLDA results are in excellent agreement with both
sets of experimental results at 34.1 and 37.2 eV. The third measurement at
40.3 eV, which is in the vicinity of the 3s Cooper minimum, is negative in
Ref. klunder2011probing  in contrast to its positive value in Ref.
guenot2012photoemission . Note that our result captures the correct sign as in
Klünder et al. at 40.3 eV. In general, 3p$\rightarrow$kd photochannel is
dominant over 3p$\rightarrow$ks at most energies. Close to the 3p Cooper
minimum, however, 3p$\rightarrow$kd begins to rapidly decrease to its minimum
value, enabling 3p$\rightarrow$ks to significantly contribute to the net 3p
delay. The s- and d-wave emissions have different angular distributions but
their Wigner delays are independent of emission directions. Thus, assuming
that all 3p photoelectrons are detected (integration over solid angle), the
net 3p delay must be a statistical combination, that is, the sum of the delays
weighted by the channel’s individual cross section branching ratios. As
illustrated in Fig. 2b, upon including 3p$\rightarrow$ks along with
3p$\rightarrow$kd (purple curve) this way, the shape of the TDLDA delay
strikingly alters near 3p Cooper minimum. We stress that the delay near a
Cooper minimum needs to be addressed with great care which can reveal new
physics, as shown below for an endohedrally confined Ar atom.

We also include recent RPAE results kheifets2013time  for three experimental
energies in Fig. 2b. As seen, RPAE and experiments match only at 34.1 eV. The
superior performance of TDLDA in explaining the measurements is thus evident.
While both TDLDA and RPA are many-body linear response theories, they have
significant differences in the details, particularly, in treating electron
correlations onida2002electronic . Variants of the Kohn-Sham LDA+LB scheme
were successfully utilized to describe attosecond strong-field phenomena
petretti2010alignment ; heslar2011high ; farrell2011strong ; toffoil2012 ;
hellgren2013 , underscoring the reliability of many-body correlations that
TDLDA characteristically offers.

This success of TDLDA method for free Ar encouraged us to use the approach to
investigate the delay in an Ar atom endohedrally sequestered in C60. The
jellium model is employed for computing the relative delay madjet2010 . This
model enjoyed earlier successes in codiscovering with experimentalists a high-
energy plasmon resonance scully2005 , interpreting the energy-dependent
oscillations in C60 valence photo-intensities rudel2002 , and predicting giant
enhancements in the confined atom’s photoresponse from the coupling with C60
plasmons madjet2007giant . Significant ground state hybridization of Ar 3p is
found to occur with the C60 3p orbital, resulting in 3p[Ar+C60] and 3p[Ar-C60]
from, respectively, the symmetric and antisymmetric wavefunction mixing. These
are spherical analogs of bonding and antibonding states in molecules or
dimers. Such atom-fullerene hybridization was predicted before chakraborty2009
and detected in the photoemission experiment on multilayers of Ar@C60
morscher2010strong . In fact, the hybridization gap of 1.5 eV between
3p[Ar+C60] and 3p[Ar-C60] in our calculation is in good agreement with the
measured value of 1.6$\pm$0.2 eV morscher2010strong .

Figure 3: (Color online). TDLDA quantum phases for ionization via d-waves from
bonding 3p[Ar+C60] and antibonding 3p[Ar-C60] levels and via p-wave from Ar
3s@ are compared with their counterparts in free Ar.

The TDLDA Wigner-Smith phases for relevant ionization channels for confined
and free Ar are presented in Fig. 3. We use the symbol “@” to denote states
belonging to the confined Ar. The narrow resonance spikes below 40 eV are due
to single electron Rydberg-type excitations in C60. This energy zone also
includes the C60 plasmon resonances, although their effects are suppressed by
the Coulomb phase that dominates the extended region above ionization
thresholds. We note that the Ar 3s Cooper minimum shifts slightly lower in
energy to 36.5 eV from the confinement, but the confinement moves the two 3p
minima, each in the bonding and antibonding channels, somewhat higher in
energy. What is rather dramatic in Fig. 3 is that the quantum phase
corresponding to 3p[Ar+C60]$\rightarrow$kd@ (thick solid black) makes a
downward $\pi$ phase shift, whereas the phase associated with
3p[Ar-C60]$\rightarrow$kd@ (thick solid red) suffers a upward 2$\pi$ phase
shift at their respective Cooper minimum. Further note that both these
contributions together yield a net phase that shifts up by $\pi$ as in the
case of free-Ar 3p$\rightarrow$kd channel (dashed black curve in Fig. 3) at
its Cooper minimum.

This contrasting phase behavior between hybrid 3p emissions is likely the
effect of symmetric and antisymmetric wavefunction shapes on the matrix
elements through dynamical correlations. Using the well-known Fano scheme of
perturbative interchannel coupling fano1961  the lead contribution to the
matrix element $\langle\delta V\rangle$ (Eq. (5)) is javani2012

$\langle\delta V\rangle_{\alpha}(E)=\displaystyle\sum_{\beta}\int
dE^{\prime}\frac{\langle\Psi_{\beta}(E^{\prime})|\frac{1}{|{\bf
r}_{\alpha}-{\bf r}_{\beta}|}|\Psi_{\alpha}(E)\rangle}{E-E^{\prime}}\langle
z\rangle_{\beta}(E^{\prime}),$ (6)

where $\alpha$ denotes each of the 3p[Ar$\pm$ C60]$\rightarrow$kd@ channels.
$\Psi$ are channel-wavefunctions that involve both bound (hole) and continuum
(photoelectron) states, and $\langle z\rangle_{\beta}$ is the single channel
matrix element of each perturbing channel $\beta$. Thus, the summation over
channels incorporates bound states as the hole states. Two points can be
noted: First, $\langle\delta V\rangle$ dominates near the Cooper minimum of a
channel $\alpha$, since the “unperturbed” $\langle z\rangle_{\alpha}$ is
already small at these energies; second, $\langle\delta V\rangle$ depends on
the coupling matrix element in the numerator of Eq. (6) that involves overlaps
between the bound state of a $\alpha$ channel with that in a perturbing
$\beta$ channel. These overlaps are critical, since 3p[Ar+C60] wavefunction
has a structure completely opposite to that of 3p[Ar-C60] over the C60 shell
region where each of them strongly overlaps with a host of C60 wavefunctions
to build correlations. These opposing modes of overlap from one hybrid to
another flip the phase modification direction between two hybrid 3p emissions
around a respective Cooper minimum, as seen in Fig. 3.

Figure 4: (Color online). Top: Absolute time delay for ionizations in
3p[Ar$\pm$C60]$\rightarrow$kd@ and 3s@$\rightarrow$kp@ channels. For the two
hybrid channels, results modified by incorporating s-wave delays are also
shown. Bottom: Relative delays $\tau_{\textrm{3s@}}-\tau_{\textrm{3p
[Ar}\pm\textrm{C}_{60}]}$, including the s-wave contributions;
$\tau_{\textrm{3s}}-\tau_{\textrm{3p}}$ of free Ar is also shown for
comparison.

Depending on the upward (downward) shift in the quantum phase, the resulting
photoelectron exhibits positive (negative) time delay and hence emerges
slower(faster) from the ionization region. This is evident in Fig. 4a, which
features various absolute delays: Channels 3p[Ar+C60]$\rightarrow$ kd@ and
3p[Ar-C60]$\rightarrow$ kd@ exhibit, respectively, a fast and a slow emission
over relatively narrow ranges about their Cooper minima. Note that the peak
delay of the antibonding electron is approximately double to the peak
advancement (negative delay) of the bonding electron. The delay profile
becomes softer and broader in energy by including the contribution from
s-wave, but the general trend of a rapid and a slow ejection, respectively, in
the bonding and antibonding channels survives.

The conservation of the quantum phase, i.e., the net phase shift of $\pi$ in
the upward direction (as in the free Ar) for 3p in Ar@C60, can be understood
in the language of a collision type interaction between two hybrid 3p
electrons. The phase behaves like the linear momentum in a two-body collision
which is a conserved quantity. Its energy derivative, i.e., the time delay,
can be thought to be commensurate with the collision force, the time
derivative of the momentum, since time and energy are conjugate variables.
This implies, that if one hybrid electron goes through an advanced emission,
the other hybrid must delay or time-recoil appropriately to keep the net delay
roughly close to the delay of free Ar. Of course, here the process is
underpinned by the orbital mixing. Therefore, the phenomenon can be pictured
as the photo-liberation of two recoiling electrons in the temporal domain from
the atom-fullerene hybridization. Hence, it is also likely to exist in the
ionization of molecules, nanodimers, and fullerene onions that support hybrid
electrons.

The time delays in the photoionization of 3p hybrids (with s-wave contribution
included) relative to 3s@, $\tau_{\textrm{3s@}}-\tau_{\textrm{3p
[Ar}\pm\textrm{C}_{60}]}$, are presented in Fig. 4b. One notes in Fig. 4a that
3s@$\rightarrow$kp@ produces an absolute delay profile, which is negative for
most energies and, on an average, comparable to the absolute delay in
3p[Ar-C60]$\rightarrow$kd<EMAIL_ADDRESS>Consequently, their (fast) emergence at about
similar speeds keeps their relative delay close to $\tau=0$, but with a bias
toward negative values. On the other hand, for the
3p[Ar+C60]$\rightarrow$kd@+ks@ channel the relative delay remains mostly
strongly negative. However, the rich structures in the delay profiles
emphasize that the Cooper minimum regions are particularly attractive for time
delay studies.

The 3p bonding-antibonding gap of 1.5 eV requires the energy of the probe
pulse to be smaller than this gap. Otherwise, the sideband of one level will
begin to overlap with the harmonics of the other. Also, by varying the
polarization angle between XUV and IR pulses one can potentially probe both
independent contributions, i.e., the relative delay between 3s orbital and 3p
bonding/anti-bonding orbital, i.e., by extending the standard RABBIT method
veniard1995 , where the polarization of XUV pulse is the same as the IR pulse.
Therefore, techniques based on interferometry, such as RABBIT and PROOF
chini2010 , have potentials to probe the relative delay between 3p
bonding/antibonding and 3s electrons. One may also perform the streaking
experiments using IR as well as THz pulses for accessing the delay. We suggest
that future experiments be performed on the time delay in Ar and Ar@C60 over
broader photon energy ranges including the 3p Cooper minimum to unravel new
physics from confinement and correlations.

In conclusion, our TDLDA relative Wigner-Smith time delay between 3s and 3p
subshells in free Ar are in excellent agreement with the measured delay except
near the 3s Cooper minimum, where, however, the TDLDA is consistent with the
sign of one set of measurements. In the case of confined Ar, due to the
electron correlation, the delays of the 3p bonding and 3p antibonding
emissions are governed by a recoil-type emission in the time-domain mediated
by the host C60. It is found that the emission from the 3s@ level is slightly
faster than the emission from the 3p bonding level but is substantially
faster, by 100 as and above, than the emission from the 3p antibonding level.
We further demonstrate that the delay of Ar 3p electron, free or confined,
leads to significant modifications in the vicinity of the Cooper minimum by
including the s-wave photochannel.

###### Acknowledgements.

G. D. acknowledges Misha Ivanov, Tim Laarmann and Oliver Mücke for useful
discussions. The research is supported by the NSF, USA.

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