# Attosecond-recollision-controlled selective fragmentation of polyatomic
molecules

Xinhua Xie1 Katharina Doblhoff-Dier2 Stefan Roither1 Markus S. Schöffler1
Daniil Kartashov1 Huailiang Xu1,3 Tim Rathje4,5 Gerhard G. Paulus4,5 Andrius
Baltuška1 Stefanie Gräfe2 Markus Kitzler1<EMAIL_ADDRESS>1Photonics Institute, Vienna University of Technology, A-1040 Vienna, Austria,
EU  2Institute for Theoretical Physics, Vienna University of Technology,
A-1040 Vienna, Austria, EU  3State Key Laboratory on Integrated
Optoelectronics, College of Electronic Science and Engineering, Jilin
University, Changchun 130012, China  4Institute of Optics and Quantum
Electronics, Friedrich-Schiller-University Jena, D-07743 Jena, Germany, EU
5Helmholtz Institute Jena, D-07743 Jena, Germany, EU

###### Abstract

Control over various fragmentation reactions of a series of polyatomic
molecules (acetylene, ethylene, 1,3-butadiene) by the optical waveform of
intense few-cycle laser pulses is demonstrated experimentally. We show both
experimentally and theoretically that the responsible mechanism is inelastic
ionization from inner-valence molecular orbitals by recolliding electron
wavepackets, whose recollision energy in few-cycle ionizing laser pulses
strongly depends on the optical waveform. Our work demonstrates an efficient
and selective way of pre-determining fragmentation and isomerization reactions
in polyatomic molecules on sub-femtosecond time-scales.

###### pacs:

33.80.Rv, 42.50.Hz, 82.50.Nd

Photodissociation of molecules by femtosecond light pulses is a fascinating
phenomenon both in terms of underlying physics and practical implications to
selective control of fragmentation pathway(s) through the shape of the optical
waveform. Triggered by optical field ionization, the outcome of the complex
electronic and subsequently nuclear dynamics leading to fragmentation is to a
large extent predetermined on a sub-cycle time scale because of a sharp
temporal localization of electron release to an attosecond interval at the
peak of an intense optical half-cycle. Consequently, as has been shown in the
case of diatomic molecules — D2 Kling  _et al._ (2006, 2008); Znakovskaya  _et
al._ (2012), H2 Kremer  _et al._ (2009); Fischer  _et al._ (2010), HD Kling
_et al._ (2008), DCl Znakovskaya  _et al._ (2011) and CO Znakovskaya  _et al._
(2009); Liu  _et al._ (2011) — dissociative ionization strongly depends on the
carrier-envelope phase (CEP) of a few-cycle ionizing laser pulse. The CEP-
dependence in the case of the lightest molecules, D2/H2, has been linked to a
field-driven population transfer between the binding ground state and the
dissociative excited state.

In this Letter, we present the first, to our knowledge, CEP control of
fragmentation pathways of polyatomic molecules (acetylene, ethylene,
1,3-butadiene). Given the increased number of participating nuclei and a
vastly more complex valence electron dynamics and the structure of energy
surfaces, it is counterintuitive to expect the CEP-dependence of fragmentation
pathways to survive in polyatomic molecules. Yet the experimental evidence
provided in this Letter reveals a prominent role of the CEP for all three
polyatomic species under examination and necessitates a search for a suitably
controllable attosecond physical mechanism that persists despite the increased
complexity of the system. We argue both experimentally and theoretically, that
such a universal attosecond mechanism of quasi-single-cycle fragmentation of
large molecules is the field control of the tunneled-out electronic wave
packet that is steered by the optical waveform before its recollision with the
parent ion. This scenario is indeed reminiscent to the role a light-field-
accelerated electron plays in the process of high-harmonic generation Corkum
(1993). The universality of the CEP-effect in polyatomic molecular
fragmentation is based on the fact that it is comparatively easy to “encode”
the detached electron wave packet with a necessary recollision momentum.
Conversely, it is difficult to encode many electronic degrees of freedom in a
complex molecule directly with the electric field of the femtosecond pulse.

In our experiments, we focus few-cycle laser pulses with sub-5 fs duration,
linearly polarized along the $z$-direction of our lab coordinate system, in an
ultra-high vacuum chamber onto a supersonic molecular gas jet, propagating
along $x$, of randomly aligned acetylene (C2H2), ethylene (C2H4), or
1,3-butadiene (C4H6) molecules. Details on the experiment can be found in the
Supplemental Material SOM . In short, the three-dimensional momentum vectors
of fragment ions created by laser-induced fragmentation of a single molecule
were measured by cold target recoil ion momentum spectroscopy Dörner  _et al._
(2000); Zhang  _et al._ (2012). The duration and the CEP of each few-cycle
laser pulse was measured on a single-shot basis by stereo-detection of
photoelectron spectra in a separate apparatus Wittmann  _et al._ (2009);
Sayler  _et al._ (2011a, b), and linked to the momentum of the ionic fragments
in the offline data analysis. The peak intensity of the laser pulses on target
was determined from separate calibration measurements using single ionization
of argon atoms in circularly polarized light Maharjan  _et al._ (2005).

The main experimental results are summarized in Fig. 1. Panels (a)-(c) show
the ion yields of different two-body fragmentation channels from the doubly
charged ions of acetylene, ethylene and 1,3-butadiene as a function of the
CEP, $\varphi_{\text{CE}}$ (which exhibits a phase offset with respect to the
absolute CEP, $\varphi$; see Supplemental Material SOM ). The yields of all
identified fragmentation channels in Fig. 1 exhibit an extraordinarily strong
dependence on the CEP with the strongest modulation observed for acetylene
(close to 80%). The fragmentation yields in Figs. 1(a)-(c) are the integrals
over the three-dimensional momentum spectra of a certain fragmentation
channel, uniquely identified by momentum conservation (see Supplemental
Material SOM ). Figs. 1(d, e) show examples of momentum spectra for ethylene,
integrated over all CEP values. The mean momenta of the fragments show only a
slight dependence on the CEP. The yields of the singly and doubly charged
molecular ions, shown in Figs. 1(a)-(c), are also widely independent of the
CEP. The fragmentation yield, in contrast, is strongly modulated by the CEP,
as can be seen from the kinetic energy release spectra, exemplary shown for
the fragmentation channel C2H${}_{4}^{2+}\rightarrow$ CH${}_{2}^{+}+$
CH${}_{2}^{+}$ in Fig. 1(f).

Figure 1: (a)-(c) Measured fragmentation and ionization yields, normalized to
1 at their respective maxima, as a function of CEP for different fragmentation
channels of acetylene (a), ethylene (b), and 1,3-butadiene (c), measured at
the laser intensities indicated in the panels. The various fragmentation
reactions are C2H${}_{2}^{2+}\rightarrow$ CH${}^{+}+$ CH+ (red dots) and
C2H${}_{2}^{2+}\rightarrow$ H${}^{+}+$ C2H+ (blue squares) for acetylene (a);
C2H${}_{4}^{2+}\rightarrow$ CH${}_{2}^{+}+$ CH${}_{2}^{+}$ (red dots),
C2H${}_{4}^{2+}\rightarrow$ H${}^{+}+$ C2H${}_{3}^{+}$ (blue squares), and
C2H${}_{4}^{2+}\rightarrow$ C2H${}_{2}^{+}+$ H${}_{2}^{+}$ (green triangles)
for ethylene (b); C4H${}_{6}^{2+}\rightarrow$ C2H${}_{3}^{+}+$ C2H${}_{3}^{+}$
(red dots) and C4H${}_{6}^{2+}\rightarrow$ CH${}_{3}^{+}+$ C3H${}_{3}^{+}$
(blue squares) for 1,3-butadiene (c). The ionization yields of the singly and
doubly charged molecular ions are denoted by black dots and gray squares,
respectively. (d,e) Cuts through the measured three-dimensional momentum
spectra of CH${}_{2}^{+}$ (d) and H${}_{2}^{+}$ (e) associated with two of the
three fragmentation channels identified for ethylene shown in (b), in a plane
along ($p_{z}$) and perpendicular ($p_{y}$) the laser polarization direction,
for $|p_{x}|<10$ a.u. and integrated over all CEP values. (f) Kinetic energy
release (KER) spectrum of the fragmentation reaction
C2H${}_{4}^{2+}\rightarrow$ CH${}_{2}^{+}+$ CH${}_{2}^{+}$ as a function of
CEP. (g) Mean $p_{z}$ value (integrated over $p_{x}$ and $p_{y}$) of the
singly (black dots) and doubly (gray squares) charged molecular ions of
acetylene as a function of CEP, in comparison with the $p_{z}$-sum of the two
fragments from the two fragmentation channels shown in the same color and
point style as in (a).  Figure 2: (a) Electronic energy levels of the neutral,
cation and dication of acetylene as a function of C-C distance, calculated by
ab initio quantum-chemical methods as described in the Supplemental Material
SOM . TI, RI, and RE denote tunnel ionization, recollision ionization and
recollision excitation, respectively. (b) Electron recollision energy over the
time of ionization calculated for a 4.5 fs (FWHM) long laser pulse with an
intensity of $1.5\times 10^{14}$ W/cm2 for the three indicated values of the
CEP: 0 (red lines), $0.25\pi$ (green lines), and $0.5\pi$ (blue lines). The
contribution of each recollision event as determined by the ionization
probability is indicated by the color intensity. (c, d) Fragmentation yields
over CEP of the same channels as in Figs. 1(a) and (b) (same color and point
styles apply), but measured for a slightly higher intensity (as indicated).
(e) Measured intensity dependence of the yield of the fragmentation channel
C4H${}_{6}^{2+}\rightarrow$ CH${}_{3}^{+}+$ C3H${}_{3}^{+}$ over CEP. The
intensities are indicated in the figure.

Which light-field dependent mechanism can lead to such a strong CEP dependence
of the fragmentation yield? We will discuss this question for the example of
acetylene, for which the quantum chemically calculated energy level diagram
for the C-C stretch mode is shown in Fig. 2(a). Details about the quantum
chemical methods for calculating the energy levels are given in the
Supplemental Material SOM . The energy level diagram shows that the dicationic
ground state is bound. The observed fragmentation reactions, thus, have to
take place on one (or more) excited states. The first dissociative state lies
about 5 eV above the dicationic ground state and can lead to both
fragmentation channels observed for acetylene. Hence, to observe the strong
modulation of the fragmentation yield shown in Fig. 1, the population of the
involved excited state(s) has to be accomplished by a strongly CEP dependent
mechanism.

As the probability for field ionization depends exponentially on the field
strength and, therewith, is sensitive to the CEP of few-cycle pulses, simple
sequential double ionization could be a candidate for this mechanism. Indeed,
the fragmentation channels observed here have also been observed in other
experiments with similarly low laser pulse intensities, but longer pulse
durations ($\approx 25-50$ fs), e.g., for 1,3-butadiene Xu  _et al._ (2010).
In the present experiments, owing to the short duration and low intensity of
the laser pulses, and supported by previous measurements with longer pulses
Cornaggia and Hering (2000), we expect double ionization to occur
predominantly via electron recollision. Indeed, Fig. 1(g) shows that (i) the
measured momentum of C2H${}_{2}^{2+}$ along the laser polarization direction
is much larger than the one of C2H${}_{2}^{+}$, and that (ii) the two momenta
point into opposite directions for almost all CEP-values. Such CEP-dependence
has been shown to be caused by double ionization via electron recollision
Bergues  _et al._ (2012). Furthermore, as the sum momentum of the two emitted
fragments shows the same dependence on the CEP as C2H${}_{2}^{2+}$ [see Fig.
1(g)], the fragmentation reactions are also initiated by recollision
ionization (RI). We thus conclude that the observed CEP-dependence of the
fragmentation yields is most likely due to a strong CEP-dependence of RI into
the first dissociative state.

Figure 3: Comparison of simulated and measured fragmentation probability of
the channel C2H${}_{2}^{2+}\rightarrow$ CH${}^{+}+$ CH+. (a) Simulated
fragmentation probability encoded in color-scale as a function of CEP and
laser intensity, normalized to its maximum value and calculated as described
in the Supplemental Material SOM . The CEP and intensity dependent position of
maximum and minimum fragmentation yield is indicated by black upwards and gray
downwards pointing triangles, respectively. (b) Modulation depth of the
fragmentation probability [as defined in (c)], as a function of laser
intensity. (c) Simulated fragmentation probability over CEP (full green line)
obtained by averaging of the data in (a) over the intensity range
$1.2-1.4\times 10^{14}$ W/cm2 to account for the experimental intensity beam
profile [as indicated in (a) and (b)], in comparison with the experimental
data from Fig. 1(a) shifted by $60^{\circ}$.

In order to elucidate this point we plot in Fig. 2(b) the electron recollision
energy in the few-cycle pulses used in our experiments on acetylene, for
different CEP-values as a function of birth time, calculated using the simple
man’s model Paulus  _et al._ (1994); Lewenstein  _et al._ (1995). For this
intensity ($1.5\times 10^{14}$ W/cm2), the strongly CEP-dependent maximum
recollision energy of the dominant recollision event is only 22 eV for
$\varphi_{\text{CE}}=0$, while for $\varphi_{\text{CE}}=\pi/2$ roughly 26 eV
are reached. About 20 eV are necessary to reach the ground state of the
acetylene dication, see Fig. 2(a). Additional $\sim 5$ eV are necessary to
reach the first dissociative state that leads to the experimentally observed
correlated fragments CH+/CH+ and C2H+/H+. Thus, the probability for reaching
the dissociative state is highest for CEP-values around $\pi/2$, and small for
CE-phases around $0$. We therefore propose the following scenario [see Fig.
2(a)] for explaining the pronounced fragmentation yield modulations observed
in the experiment: An electronic wavepacket set free by field ionization
dominantly from the HOMO of the neutral molecule, is driven back by the laser
field and doubly ionizes the molecule by electron impact ionization. For CEP-
values that result in low recollision energy, the second electron will be
dominantly removed from the HOMO, leading to ionization to the non-
dissociative dication’s ground state. For CEP-values resulting in high
recollision energy, the electron can also be removed from a lower lying
orbital, which puts the molecular ion into a dissociative excited electronic
state. For example, the excited state associated with the fragmentation
products CH+/CH+ that we discussed above, is reached by removal of an electron
from the HOMO-1 orbital. Because the fragmentation reactions that we
investigated in the present experiments take place on only one electronically
excited state surface (see Supplemental Material SOM ), whose population is
controlled by the CEP, we observe the same CEP dependence for each
fragmentation channel of a certain molecule (Fig. 1).

We can test the proposed scenario experimentally simply by increasing the
laser peak intensity $I$, since the recollision energy scales linearly with
intensity as $E_{\textrm{rec}}\propto I/4\omega^{2}$ Paulus  _et al._ (1994);
Lewenstein  _et al._ (1995). If the recollision energy is high enough for all
CEP-values to reach the first dissociative state, the fragmentation yield is
expected to be independent of the CEP. Figs. 2(c) and (d) show the measured
yield as a function of CEP of the same fragmentation channels as in Figs. 1(a)
and (b), but with a twice higher laser intensity leading to twice the maximum
recollision energy. As expected, the fragmentation yields now show no
discernible modulation with the CEP. Thus, by precise adjustment of the laser
intensity, such that the energy threshold to the dissociative states is
overcome only for certain CEP-values, light-field control of the fragmentation
of polyatomic molecules becomes possible. This is demonstrated in Fig. 2(e),
for the example of the fragmentation channel C4H${}_{6}^{2+}\rightarrow$
CH${}_{3}^{+}+$ C3H${}_{3}^{+}$, where the laser intensity has been fine-tuned
in three steps over the dissociation energy threshold, resulting first in a
decreased modulation depth of the fragmentation yield, and for even higher
intensity in its complete disappearance.

To obtain detailed insight into the light-determined fragmentation dynamics,
we now turn to simulations (detailed in the Supplemental Material SOM ),
performed exemplary for acetylene, C2H${}_{2}^{2+}$. Briefly, we use a one-
dimensional semi-classical model to simulate recollision-ionization into the
excited dissociative state. The resulting fragmentation probability as a
function of CEP and laser intensity is shown in Fig. 3(a). At very low laser
intensities below roughly $1.2\times 10^{14}$ W/cm2, the recollision energy of
the electron is too small to reach the dissociative state, independent of the
CEP. For slightly higher intensities, lower lying electrons can be ionized by
RI for certain values of the CEP. With increasing intensity the CEP values,
where maximum fragmentation probability is reached, shift. At the same time,
the modulation depth decreases [see Fig. 3(b)], since for higher laser
intensity and therewith higher electron recollision energy also recollisions
for other CEP values have enough energy to ionize a low lying electron. Taking
an average over $1.2-1.4\times 10^{14}$ W/cm2, thereby simulating the
different intensities within the experimental laser beam profile up to an
intensity very close to the experimentally determined value, we obtain the
modulation shown in Fig. 3(c), which agrees almost perfectly with the
experimental data shifted by $\sim 60^{\circ}$. The CEP-shift is attributed to
the influence of the long-range Coulomb field on the CEP-dependent energy of
electrons from single ionization Chelkowski  _et al._ (2004); Chelkowski and
Bandrauk (2005), which we have used in our experiment to arbitrarily calibrate
the CEP, $\varphi_{\text{CE}}$, and which is also not correctly reproduced in
the one-dimensional semi-classical model.

The numerical data shown in Fig. 3 have been calculated by neglecting field-
driven population transfer between the binding ground state and the
dissociative excited states, which in previous work on the dissociative
ionization of D2/H2 Kling  _et al._ (2006); Gräfe and Ivanov (2007); Kling
_et al._ (2008); Znakovskaya  _et al._ (2012) has been shown to be essential
in explaining the CEP-dependent charge-localization observed in the
experiments. Here, we can exclude a noticeable contribution of such a
mechanism to the experimentally observed CEP-dependence of the fragmentation
yield shown in Fig. 1, as it would be incompatible with the observation that
for each of the molecules all fragmentation channels show the same CEP-
dependence. Field-driven population dynamics, however, would happen on
considerably different timescales for breakage of the C-C and H-C bonds,
respectively, due to the large difference in the fragments’ reduced masses
($\text{m}_{\text{CC}}/\text{m}_{\text{HC}}\approx 6.8$). In order to confirm
this statement, we have performed reduced-dimensional quantum dynamical
calculations by solving the time-dependent nuclear Schrödinger equation for
the four strongest dipole-coupled electronic states of acetylene subjected to
the experimental laser parameters (see Supplemental Material SOM  for
details). The contribution of the field-driven population transfer to the CEP-
dependent modulation of the fragmentation yield due to recollision-induced
ionization was found to be very small ($<1\%$).

Electron recollision, however, can not only promote the second electron
directly into the continuum, but can also excite the molecular cation, which
may then be further ionized by the laser field Kopold  _et al._ (2000);
Feuerstein  _et al._ (2001); Rudenko  _et al._ (2004). We anticipate that this
mechanism, dubbed recollision-induced excitation (RE) and subsequent field
ionization (RESI), can, in principle, also be used for CEP-control of
polyatomic molecular fragmentation with few-cycle pulses even at peak
intensities that prohibit direct population of the first excited dicationic
state. In this case, however, as many of the closely spaced higher excited
states in the cation can be populated by RE, the CEP-selectivity becomes
dominantly dependent on the field-sensitive ionization rate during the field
ionization step. As a result, the yields of different fragmentation channels
as well as the yield of the dication are likely to show different CEP-
modulation depths and possibly different CEP-dependence. This is not observed
in our experiments. Furthermore, we have estimated that at the intensity of
our experiments and the channels observed, RESI is considerably less probable
than direct RI to the first dissociative excited state. We therefore conclude
that the contributions to the observed CEP-modulation of the fragmentation
yield due to RESI, while probably important for lower intensities, are
insignificant for the here demonstrated higher intensity case that permits
direct RI from lower lying orbitals.

In conclusion we have experimentally demonstrated for the first time control
over various fragmentation reactions of polyatomic molecules using intense
few-cycle laser pulses. Fragmentation reactions of polyatomic molecules are
essential building blocks of Chemistry. Equally important, however, are
preceding isomerization reactions. In our experiment we observe, for example,
two channels that involve molecular restructuring prior to the breakage of a
molecular bond: formation of H${}_{2}^{+}$ for ethylene and a proton migration
reaction for 1,3-butadiene [see Fig. 1]. These results demonstrate that RI
with intensity-tuned few-cycle laser pulses can be used as a very efficient
tool not only to initiate (or suppress) fragmentation but also isomerization
reactions in polyatomic molecules with high sensitivity. Our work, thus,
demonstrates an efficient and selective, yet straightforward way of pre-
determining fragmentation and isomerization reactions in polyatomic molecules
on sub-femtosecond time-scales.

This work was partly financed by the Austrian Science Fund (FWF), grants
P21463-N22, V193-N16, and I274-N16, by a SIRG grant from the ERC, and by a
grant from the National Natural Science Foundation of China (No. 11074098).

Supplemental Material

## I Experimental details

Sub-5 fs laser pulses were generated by spectral broadening of $\approx 25$ fs
(FWHM) laser pulses from a Titanium-Sapphire laser amplifier system, running
at 5 kHz repetition rate, in a 1 m long hollow-core glass capillary, filled
with neon, and subsequent recompression by several bounces from chirped
mirrors. The electric field of the laser pulses can be written as
$E(t)=\mathcal{E}(t)\cos(\omega t+\varphi)$, where $\mathcal{E}(t)$ is the
pulse envelope and $\omega$ and $\varphi$ are the light frequency and the
carrier-envelope phase (CEP), respectively. The few-cycle pulses, linearly
polarized along the $z$-direction of our lab coordinate system, with their
spectra centered around 750 nm, are directed into an ultra-high vacuum chamber
(background pressure $1.3\times 10^{-10}$ mbar), where they are focused onto a
supersonic molecular gas jet of randomly aligned acetylene (C2H2), ethylene
(C2H4), or 1,3-butadiene (C4H6) molecules by a spherical mirror with a focal
length of 150 mm. The gas jet, propagating along the $x$-direction, was
created by expanding the molecular gas samples, with a backing pressure of
around 0.5 bar (depending on the molecular species), through a nozzle of 10
$\mu$m in diameter into a vacuum chamber at $\sim 4\times 10^{-5}$ mbar
operating pressure and subsequent two-stage skimming and differential pumping.
To avoid smearing of the CEP due to the Gouy phase shift when the laser pulses
propagate along the $y$-direction through the gas jet Lindner  _et al._
(2004), which is $\approx 170$ $\mu$m in diameter, the jet is cut in width by
an adjustable slit before it reaches the interaction region. This reduces the
width of the jet to typically 30 $\mu$m. The laser focus is placed about half
the Rayleigh length before the center of the narrow jet. The Rayleigh length
is estimated as 150 $\mu$m.

Cold Target Recoil Ion Momentum Spectroscopy (COLTRIMS) Dörner  _et al._
(2000) is used to measure the three-dimensional momentum vector of fragment
ions emerging from the interaction of a single molecule with a few-cycle laser
pulse. The ions created by the laser pulses were guided over a distance of 5.7
cm by a weak homogeneous electric field of 16 V/cm, applied along the axis of
the spectrometer, $z$, onto a multi-hit capable RoentDek DLD 80 detector (80
mm in diameter), equipped with position sensitive delay line anodes. The ion
count rate was kept at 0.4 per laser shot in order to establish coincidence
conditions, which ensure that all observed processes take place within a
single molecule. From the measured time-of-flight and position of each
detected ion we calculate its three-dimensional momentum vector in the lab
frame Dörner  _et al._ (2000); Zhang  _et al._ (2012). The peak intensity of
the laser pulses on target was determined from separate calibration
measurements using single ionization of argon atoms in circularly polarized
light Maharjan  _et al._ (2005); Smeenk  _et al._ (2011).

As in our experiments we measure the ion momenta of fragments from only one
molecule created by ionization in a single laser pulse, it becomes possible to
perform single-shot CEP tagging Johnson  _et al._ (2011): We measure the
duration and the CEP, $\varphi$, of each few-cycle laser pulse on a single-
shot basis by exploiting the asymmetry of photoelectron spectra created by
above-threshold ionization (ATI), emitted to the left and right along the
laser polarization direction, using a stereo-ATI phase-meter Wittmann  _et
al._ (2009); Sayler  _et al._ (2011a, b). In the offline data analysis each
fragmentation event is then tagged with the corresponding CEP-value determined
from the photoelectron spectra Sayler  _et al._ (2011a, b). From the stereo-
ATI data we estimate the duration of our laser pulses as $4.5$ fs full width
at half maximum (FWHM).

As CEP-measurement and coincidence measurement of fragment ions are performed
in two separate apparatus, the laser beam, split into two arms, passes through
different amounts of glass and air. Thus, the CEP-value determined from the
stereo-ATI phase-meter, $\varphi$, is offset by a constant value,
$\varphi_{0}$, from the one of the pulses in the COLTRIMS device,
$\varphi_{\text{CE}}$. We arbitrarily calibrated the CEP in the COLTRIMS
device such that the mean momentum value of the singly charged molecular ion,
which due to momentum conservation is the inverse of the electron momentum,
peaks at $\varphi_{\text{CE}}=90^{\circ}/270^{\circ}$ and is zero at
$\varphi_{\text{CE}}=0^{\circ}/180^{\circ}$, see Fig. 1(g) in the Letter. This
CEP-dependence is predicted by the simple man’s model (SMM) of strong field
physics Lewenstein  _et al._ (1995), which neglects the influence of the long-
range Coulomb potential onto the departing electrons. Due to the Coulomb
influence, depending on the atomic or molecular species and on the laser
parameters, the CEP-values at which the maxima of the mean momentum are
observed, do not necessarily coincide with the predictions of the SMM
Chelkowski  _et al._ (2004); Chelkowski and Bandrauk (2005). As the Coulomb
influence and laser parameters are different for the separate measurements
shown in the Letter, $\varphi_{0}$ might also be different for each of them.
We would like to emphasize, though, that all explanations and conclusions
given in the Letter are unaffected by a constant CEP-offset,
$\varphi_{0}=\varphi-\varphi_{\text{CE}}$.

From the measured data a certain two-body fragmentation pathway is uniquely
identified using momentum conservation conditions in all three spatial
dimensions Hasegawa  _et al._ (2001); Xu  _et al._ (2010); Zhang  _et al._
(2012). From all detected ions per laser pulse we select only those sets of
two fragment ions, for which the absolute value of their momentum sum is
smaller than 3 a.u. along the $y$ and $z$ directions, and smaller than 5 a.u.
along the $x$ direction. By that we identified two different fragmentation
pathways for acetylene and 1,3-butadiene, and three different ones for
ethylene (see caption of Fig. 1 in the Letter).

## II Potential energy surfaces of acetylene from ab initio quantum chemical
methods

Figure 4-SM: Ab initio potential energy curves of C2H2, C2H${}_{2}^{+}$ and
C2H${}_{2}^{2+}$. The C-H distance, $R_{\text{CH}}$, is frozen at the
equilibrium distance of the dication, $R_{\text{CH}}=1.15\mathring{A}$. Figure
5-SM: Calculated potential energy surfaces of C2H${}_{2}^{2+}$. Electronic
ground state (a) and first excited state (${}^{3}\Pi$) (b) for the C-H stretch
modes with the C-C distance, $R_{\text{CC}}$, frozen at the equilibrium
distance of the dication, $R_{\text{CC}}=1.15\mathring{A}$. Figure 6-SM:
Selected cuts of the potential energy surfaces of C2H${}_{2}^{2+}$ for the C-C
stretching mode. Black curves: Electronic ground state and the first dipole
coupled state of the dication. Blue curves: the first excited electronic
states (weakly dipole coupled to the electronic ground state in the direction
perpendicular to the molecular axis). (a) For a symmetric C-H distance, the
${}^{3}\Pi$-states show a crossing. (b) At a slightly asymmetric C-H/C-H
distance, the two ${}^{3}\Pi$ states show an avoided crossing in 1 dimension
(red circle), and the first excited state becomes more pre-dissociative. (c)
Two-dimensional potential energy surfaces showing the conical intersection
between the ${}^{3}\Pi$ states as a function of one C-H and C-C distance. The
other C-H bond is kept frozen at $R_{\text{CH}}=1.15\mathring{A}$. Comparison
with Fig. 5-SM(b) shows, that this asymmetric C-H stretch will inevitably
happen, thus, opening up the fragmentation pathway.

The potential energy surfaces (PES) of acetylene (C2H2) and its ionic states
(C2H${}_{2}^{+}$, C2H${}_{2}^{2+}$) were calculated using high-level ab initio
quantum chemical methods. The calculations were carried out using a standard
correlation-consistent polarized double-zeta basis set with augmented
functions (aug-ccpVDZ) Dunning (1989). For the calculations of the potential
curves and surfaces, multi-reference methods (CASSCF Knowles and Werner (1985)
and MR-CIWerner and Knowles (1988); Knowles and Werner (1988)) were employed,
where all valence electrons of the molecule were taken into the active space,
the carbon 1$s$ orbitals were kept frozen. The valence electrons were
distributed in at least 8 orbitals, (8,8) CASSCF, but active spaces up to 11,
(8,11) CASSCF were also examined. In the CASSCF calculations, the state-
averaging procedure over the lowest five states was included. Subsequently,
MR-CISD calculations with a (8,8) reference space were carried out. All
calculations were performed in the frame of the C2v point group (with the
symmetry axis being along the C-C bond) in order to be able to treat all
strech vibrations examined here on the same footing. The ab initio calcuations
have been performed using the Columbus Lischka  _et al._ (2012, 1981, 2001)
software package.

Cuts of the PES of the neutral, singly, and doubly ionized acetylene
describing stretching along the C-C bond ($R_{\text{CC}}$) are shown in Fig.
2(a) in the Letter, and enlarged in Fig. 4-SM. The electronic ground state of
the neutral is ${}^{1}\Sigma_{g}^{+}$, ${}^{2}\Pi_{u}$ for the cation, and
${}^{3}\Sigma_{g}^{-}$ for the dication. Also shown in Fig. 4-SM are some of
the lowest lying doublet states of the cation, and some of the lowest lying
triplet states of the dication. The vertical ionization potential is found to
be 11.4 eV. In the cation, several higher lying doublet electronic states are
present with (vertical) excitation energies between 5 and 9 eV above the
doublet ground state (see also Ref. Perić  _et al._ (1998); Perić and Engels
(1998)), which may be populated optically or by electron recollisional
excitation (see Letter). The triplet groundstate of the dication
${}^{3}\Sigma_{g}^{-}$ is 31.5 eV above the neutral acetylene and is bound
along the C-C, as well as along the C-H stretch motion, see Fig. 5-SM(a). The
singlet groundstate (${}^{1}\Delta_{g}$) lies about 1 eV above the energy of
the triplet groundstate and is bound as well.

The first excited triplet state in the dication (${}^{3}\Pi_{u}$) lies
energetically about 5 eV above the ${}^{3}\Sigma_{g}^{-}$ groundstate. This
state is dissociative along the C-H stretch (and also along the $D_{\infty h}$
distorting modes Zyubina  _et al._ (2005)), leading to the fragmentation
products H+/C2H+, see Fig. 5-SM(b). It is not immediately obvious, however,
that the ${}^{3}\Pi_{u}$ state is also dissociative along the C-C stretching
mode, since the first excited triplet state has a symmetry-allowed level
crossing with the higher excited ${}^{3}\Pi_{g}$ state, see Fig. 6-SM(a). This
effectively leads to the ${}^{3}\Pi_{u}$ state being bound along the mirror-
symmetric C-C stretching mode. For non-symmetric configurations, however, the
degeneracy is lifted [see Figs. 6-SM(b) and 6-SM(c)] and the state crossing
between the two ${}^{3}\Pi$ states becomes avoided (in one dimension). As a
result, fragmentation into the products CH+/CH+ becomes possible. Breaking of
the symmetry will inevitably happen since the state is dissociative along the
antisymmetric C-H stretch motion, as well as for distortions from the linear
geometry (where the ${}^{3}\Pi$ state splits up into two components with
${}^{3}A^{\prime}$ and ${}^{3}A^{\prime\prime}$ symmetry, respectively).
Please note, that breakage of the C-C bond may also proceed via different
electronic states. In particular the ${}^{1}\Sigma_{g}^{+}$ state with a small
dissociation barrier of about 3 eV, that can be overcome for higher
recollision energies, leads to the fragmentation products CH+/ CH+ Duflot  _et
al._ (1995).

From this analysis we can deduce that an excitation to the first few excited
states, about 25-26 eV above the cationic groundstate, will suffice to lead to
a fragmentation into both fragmentation channels observed in the experiment.

## III Quantum dynamical simulations of field-driven fragmentation dynamics

With the ab initio potential energy surfaces $V_{i}(R_{1},R_{2},\dots)$ and
transition dipole moments $\vec{\mu}_{ij}(R_{1},R_{2},\dots)$ calculated as
described in Section II, we have performed quantum dynamical calculations,
integrating numerically the time-dependent nuclear Schrödinger equation in the
presence of an intense, external laser field. The calculations included the
lowest four electronic states, two ${}^{3}\Sigma$ states, as well as the two
${}^{3}\Pi$ states. The ${}^{3}\Sigma$ (and ${}^{3}\Pi$) states have a large
transition dipole moment along the internuclear axis, while the transition
dipole moment between the $\Sigma$ and $\Pi$ states of different symmetry are
very small. Thus, only the coupling from the electronic ground state to the
lowest lying excited electronic state,
${}^{3}\Sigma_{g}^{-}\rightarrow^{3}\Pi_{u}$, is included. The nuclear
Schrödinger equation reads as (atomic units are employed here and in what
follows)

$i\frac{\partial}{\partial t}\left(\begin{array}[]{c}\psi_{1}(\vec{R},t)\\\
\psi_{\Pi 1}(\vec{R},t)\\\ \psi_{\Pi 2}(\vec{R},t)\\\ \psi_{2}(\vec{R},t)\\\
\end{array}\right)=H\left(\begin{array}[]{c}\psi_{1}(\vec{R},t)\\\ \psi_{\Pi
1}(\vec{R},t)\\\ \psi_{\Pi 2}(\vec{R},t)\\\ \psi_{2}(\vec{R},t)\\\
\end{array}\right),$ (1)

where the Hamiltonian is given by

$\displaystyle H$ $\displaystyle=T+$
$\displaystyle\quad\left(\begin{array}[]{cccc}V_{1}(\vec{R})&-\mu_{1,\Pi
1}(\vec{R})\vec{E}(t)&0&-\mu_{1,2}(\vec{R})\vec{E}(t)\\\ -\mu_{1,\Pi
1}(\vec{R})\vec{E}(t)&V_{\Pi 1}(\vec{R})&-\mu_{\Pi 1,\Pi
2}(\vec{R})\vec{E}(t)&0\\\ 0&-\mu_{\Pi 2,\Pi 1}(\vec{R})\vec{E}(t)&V_{\Pi
2}(\vec{R})&0\\\ -\mu_{2,1}(\vec{R})\vec{E}(t)&0&0&V_{2}(\vec{R})\\\
\end{array}\right).$ (6)

In the last equation, $T$ is the kinetic energy operator, $\vec{\mu}_{ij}$ the
transition dipole moment between states $i,j$ with components parallel and
perpendicular to the molecular axis, and $\vec{E}(t)$ denotes the vectorial
laser electric field. Denoting the angle between the laser polarization
direction and the molecular axis with $\vartheta$,
$-\mu_{ij}^{z}\cos\vartheta\,E(t)$ and $-\mu_{ij}^{\perp}\sin\vartheta\,E(t)$
then represent the projections of the interaction with the electric field onto
the directions along and perpendicular to the molecular axis, respectively. In
Eq. III, $\vec{R}$ represents one or more selected vibrational coordinates,
either the C-C or the C-H vibrations. The reduced nuclear mass
$M_{red}=(M_{a}M_{b})/(M_{a}+M_{b})$ contains the nuclear masses $M_{a,b}$ of
the atoms $a,b$ of the selected degree-of-freedom. The time-dependent nuclear
Schrödinger equation is solved numerically on a grid using the split-operator
method Feit  _et al._ (1982).

For the laser field, we applied a few-cycle pulse with a wavelength of 800 nm
and an intensity of $1.4\times 10^{14}$ W/cm2 with a duration of 5 fs (FWHM).
Different angles $\vartheta$ of the field relative to the molecular axis have
been simulated. The initial conditions were also varied, mimicking either (i)
initial nuclear wavepackets in field-dressed states or, (ii) nuclear
wavepackets with an initial population ratio of the ground and excited state
as given by the recollisional ionization estimate described in Section IV. The
population dynamics $P_{i}(t)=\int|\psi_{i}(R,t)|^{2}dR$ features pronounced
oscillations following closely the oscillations of the laser field, with
maximum values of the population of the excited state, $P_{2}$, up to 0.025.
The oscillations in the population dynamics of the $\Pi$ states, $P_{\Pi
1},P_{\Pi 2}$, although following the oscillations of the laser field as well,
are much less pronounced and reach maximum values in the population of about
10 times weaker than the values found for the state labeled with 2.

The final population of the ${}^{3}\Pi$ state(s) was almost independent of the
CEP for all angles $\vartheta$ and all examined pulses with different pulse
lengths and intensities. Population transfer in the dication induced by the
laser field seems to be of minor importance to describe the large differences
in the fragmentation yield as a function of CEP observed in the experiment.
Hence, only the different initial population of the $\Pi$ state being
populated by electron recollisional ionization of the cation can explain the
large variation in the fragmentation yield – additional field excitation in
the dication only gives a minor contribution.

## IV Modeling of CEP-dependent fragmentation via recollision-ionization into
the excited dissociative state

To simulate the mechanism for CEP-control of fragmentation based on
recollision-ionization (as outlined in the Letter), we use a one-dimensional
semi-classical model. We consider fragmentations into C2H+/H+ (H-C stretching
mode) and CH+/CH+ (C-C stretching mode) that were observed in the experiment.
As detailed in Section II, the first states leading to a strong dissociation
into C2H+/H+ and CH+/CH+ lie $\sim 25$ eV above the ionic groundstate, and
population of these states will lead to a fragmentation into both channels. In
the model we assume that the first electron is set free by field ionization,
with an ionization rate according to tunneling theory Ammosov  _et al._
(1986),

$\Gamma=\text{const}\cdot\left(\frac{2(2I_{p})^{3/2}}{|E(t_{b})|}\right)^{2n-|m|-1}\,\exp\left(-\frac{2(2I_{p})^{3/2}}{|E(t_{b})|}\right),$
(7)

for an ionization potential of $I_{p}=11.4$ eV and quantum numbers $l=m=1$, as
suggested in Ref. Tong  _et al._ (2002) for ionization from $\pi$-bonds. The
electron is placed at a distance $z_{0}=I_{p}/E(t_{b})$, with $E(t_{b})$ the
electric field at birth time $t_{b}$, from the nucleus with a Gaussian
momentum distribution parallel to the laser polarization direction, $p_{z}$,
according to tunneling theory Krainov (1997). Neglecting the initial
longitudinal velocity distribution, however, did not change the results
markedly. Wavepacket spreading is estimated via a quadratic increase of its
size $A$ perpendicular to the laser polarization direction with time Delone
and Krainov (1991):

$A=\pi\frac{|E|}{2\sqrt{2I_{p}}}(t-t_{b})^{2}.$ (8)

The electron, after its birth at time $t_{b}$, is propagated classically in
the laser field and a soft core Coulomb potential $V(z)=-1/\sqrt{(1+z^{2})}$
by numerically solving Newton’s equations. The electron is defined as
recolliding with the molecule, if it returns to the ion’s position. Only the
first recollision event is taken into account. The corresponding recollision
energies for different values of the CEP are displayed in Fig. 2(b) of the
Letter. Clearly, the recollision energy features a strong dependence on the
CEP.

The probability for recollision-ionization (RI) to the excited ionic states is
derived from the energy dependent impact ionization cross-section $\sigma$
Medvedev  _et al._ (2010), scaled by $\sigma/A$. The cross-section is zero
below 25 eV, the minimum excitation energy to the first dissociative state,
followed by an approximately linear rise as a function of energy Medvedev  _et
al._ (2010). Arbitrary scaling factors of $\sigma$ are unimportant since we
are only interested in relative changes over CEP. The resulting fragmentation
probability as a function of CEP and laser intensity is shown in Fig. 3(a) of
the Letter.

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