Present address: ]MPA-CMMS, Los Alamos National Laboratory, Los Alamos, New
Mexico 87545, USA

# An ideal Weyl semimetal induced by magnetic exchange

J.-R. Soh  Department of Physics, University of Oxford, Clarendon Laboratory,
Parks Road, Oxford OX1 3PU, UK  F. de Juan  Department of Physics, University
of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, UK  Donostia
International Physics Center, 20018 Donostia-San Sebastian, Spain  IKERBASQUE,
Basque Foundation for Science, Maria Diaz de Haro 3, 48013 Bilbao, Spain  M.
G. Vergniory  Donostia International Physics Center, 20018 Donostia-San
Sebastian, Spain  IKERBASQUE, Basque Foundation for Science, Maria Diaz de
Haro 3, 48013 Bilbao, Spain  N. B. M. Schröter  Paul Scherrer Institute,
WSLA/202, 5232 Villigen PSI, Switzerland  M. C. Rahn  Department of Physics,
University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, UK  [
D. Y. Yan  Beijing National Laboratory for Condensed Matter Physics, Institute
of Physics, Chinese Academy of Sciences, Beijing 100190, China  J. Jiang
Department of Physics, University of Oxford, Clarendon Laboratory, Parks Road,
Oxford OX1 3PU, UK  School of Physical Science and Technology, ShanghaiTech
University, Shanghai 201210, China  Advanced Light Source, Lawrence Berkeley
National Laboratory, Berkeley, California 94720, USA  M. Bristow  Department
of Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1
3PU, UK  P. Reiss  Department of Physics, University of Oxford, Clarendon
Laboratory, Parks Road, Oxford OX1 3PU, UK  J. N. Blandy  Department of
Chemistry, University of Oxford, Inorganic Chemistry Laboratory, Oxford, OX1
3QR, UK  Y. F. Guo  School of Physical Science and Technology, ShanghaiTech
University, Shanghai 201210, China  CAS Center for Excellence in
Superconducting Electronics (CENSE), Shanghai 200050, China  Y. G. Shi
Beijing National Laboratory for Condensed Matter Physics, Institute of
Physics, Chinese Academy of Sciences, Beijing 100190, China  T. K. Kim
Diamond Light Source, Harwell Campus, Didcot, OX11 0DE, UK  A. McCollam  High
Field Magnet Laboratory (HFML-EMFL), Radboud University, 6525 ED Nijmegen,
Nijmegen, The Netherlands  S. H. Simon  Department of Physics, University of
Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, UK  Y. Chen
Department of Physics, University of Oxford, Clarendon Laboratory, Parks Road,
Oxford OX1 3PU, UK  School of Physical Science and Technology, ShanghaiTech
University, Shanghai 201210, China  A. I. Coldea  Department of Physics,
University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, UK  A.
T. Boothroyd<EMAIL_ADDRESS>Department of Physics,
University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, UK

###### Abstract

We report theoretical and experimental evidence that EuCd2As2 in magnetic
fields greater than 1.6 T applied along the $c$ axis is a Weyl semimetal with
a single pair of Weyl nodes. Ab initio electronic structure calculations,
verified at zero field by angle-resolved photoemission spectra, predict Weyl
nodes with wavevectors ${\bf k}=(0,0,\pm 0.03)\times 2\pi/c$ at the Fermi
level when the Eu spins are fully aligned along the $c$ axis. Shubnikov–de
Haas oscillations measured in fields parallel to $c$ reveal a cyclotron
effective mass of $m_{\rm c}^{\ast}=0.08$ $m_{\rm e}$ and a Fermi surface of
extremal area $A_{\rm ext}=0.24$ nm-2, corresponding to 0.1% of the area of
the Brillouin zone. The small values of $m_{\rm c}^{\ast}$ and $A_{\rm ext}$
are consistent with quasiparticles near a Weyl node. The identification of
EuCd2As2 as a model Weyl semimetal opens the door to fundamental tests of Weyl
physics.

††preprint: APS/123-QED

Weyl semimetals (WSMs) exhibit exceptional quantum electronic transport due to
the presence of topologically-protected band crossings called Weyl nodes
Armitage  _et al._ (2018); Burkov (2016). The nodes come in pairs with
opposite chirality, but their number and location in momentum space is
otherwise material specific.

Weyl nodes are distinct from other topological features of electron band
structures in severals respects, including (1) the bulk bands that cross at a
Weyl node are non-degenerate, (2) the associated Weyl fermions have a definite
chirality, and (3) the Weyl nodes are protected against perturbations that do
not couple the nodes Wan  _et al._ (2011); Armitage  _et al._ (2018); Burkov
(2016). Moreover, the individual nodes within a pair act as a source and a
sink of Berry curvature, a topological property of the electronic
wavefunctions which relates directly to several anomalous transport phenomena
Armitage  _et al._ (2018); Burkov (2014).

Weyl semimetal phases in crystals require either broken spatial inversion
symmetry, or broken time-reversal symmetry (TRS), or both. There are a number
of experimental realizations of the first type (with broken inversion symmetry
only), especially in the TaAs structural family Xu  _et al._ (2015a); Lv  _et
al._ (2015); Yang  _et al._ (2015); Xu  _et al._ (2015b, c), but magnetic WSMs
(with broken TRS) are still rare. The few known candidates are complicated by
multiple pairs of Weyl nodes and/or by extra (non-topological) Fermi surface
pockets which shroud the Weyl nodes Wan  _et al._ (2011); Suzuki  _et al._
(2016); Liu  _et al._ (2018); Wang  _et al._ (2016a); Nakatsuji  _et al._
(2015); Sakai  _et al._ (2018). Magnetic WSMs are important for fundamental
studies of Weyl fermions because it is possible for such systems to have only
a single pair of Weyl nodes which, due to inversion symmetry, are guaranteed
to be at the same energy and so have a vanishing density of states. By
contrast, WSMs formed by breaking inversion symmetry (but with TRS) have a
minimum of four nodes which are in general separated in energy.

Figure 1:  Crystal structure of EuCd2As2 and location of Weyl nodes in the
Brillouin zone. (a) The trigonal unit cell for the space group $P\bar{3}m1$
(No. 164). The Weyl fermions predominantly occupy orbitals in the double-
corrugated Cd2As2 layers, which are sandwiched between the Eu layers Rahn  _et
al._ (2018). (b) The Weyl nodes lie along the A–$\Gamma$–A high symmetry line
and are separated by $\Delta k_{z}$ (not shown to scale). (c) In the fully
polarized state, singly degenerate conduction and valence bands meet at a pair
of Weyl nodes (shown here schematically). The nodes act as a source and sink
of Berry curvature (indicated by arrows).

Following the initial discoveries Xu  _et al._ (2015a); Lv  _et al._ (2015);
Yang  _et al._ (2015) there is now a need for better material realizations of
WSMs, ideally comprising a single pair of Weyl nodes located at or very close
to the Fermi level and in an energy window free from other overlapping bands.
Here we propose the layered intermetallic EuCd2As2 Wang  _et al._ (2016b);
Rahn  _et al._ (2018) to be such a system. We show that Weyl nodes in EuCd2As2
are magnetically-induced via exchange coupling, emerging when the Eu spins are
aligned by a small external magnetic field applied along the $c$ axis.

Figure 2:  Exchange-induced Weyl nodes in EuCd2As2. (a), (c) In zero field,
the Eu spins order in an A-type antiferromagnetic structure at $T<T_{\rm N}$,
with the spins lying in the $ab$ plane. The corresponding band structure is
gapped at $\Gamma$, and every band is doubly degenerate due to the combination
of inversion and time-reversal symmetries. (b), (d) The Eu moments can be
fully polarized along the $c$ axis in a small coercive field ($B>B_{\rm c}$),
lifting the double degeneracy of the bands and creating a pair of Weyl nodes
along $\Gamma$–A, at $k_{z}=0.03$ r.l.u.

Bulk single crystals of EuCd2As2 (ECA) were grown by a NaCl/KCl flux method
Schellenberg  _et al._ (2011). The magnetic, transport and crystallographic
properties of the crystals were fully consistent with earlier reports Rahn
_et al._ (2018); Wang  _et al._ (2016b); Schellenberg  _et al._ (2011) (see
Supplemental Material sup ). Magnetotransport measurements were performed on a
Quantum Design Physical Properties Measurement System for fields $B<16$ T, and
at the High Field Magnet Laboratory, Nijmegen, and the National High Magnetic
Field Laboratory, Tallahassee, for fields up to 38 T and 45 T, respectively.
ARPES measurements were performed at the high–resolution branch line of the
beamline I05 at the Diamond Light Source, UK Hoesch  _et al._ (2017). A
Scienta R4000 analyzer was used to select a photon energy of 130 eV, which
approximately corresponds to the $k_{z}=0$ measurement plane. The sample
temperature was $T\sim 5$K.

To calculate the band structure we employed density functional theory (DFT) as
implemented in the Vienna Ab-initio Simulation Package (VASP) Kresse and
Furthmüeller (1996); Kresse and Furthmüller (1996). The exchange correlation
term is described according to the Perdew-Burke-Ernzerhof (PBE) prescription
together with projected augmented-wave pseudopotentials Perdew  _et al._
(1996). For the self-consistent calculations we used a 10$\times$10$\times$5
$k$-points mesh. The kinetic energy cut-off was set to 550 eV. The spin-
polarized band structures are calculated within GGA+U, with the value of U
chosen to be 5 eV to match the position of the Eu 4$f$ bands in the ARPES
spectrum. The lattice parameters used in the calculations were $a=b=0.443$ nm
and $c=0.729$ nm (see Supplemental Material sup ).

The trigonal crystal structure of ECA, shown in Fig. 1(a), contains
alternating layers of Eu2+ and [Cd2As2]2- (Ref. Artmann  _et al._ (1996)). The
Eu ions carry a localized magnetic moment with spin $S=7/2$ and essentially
zero orbital angular momentum. Below $T_{\rm N}=9.5$ K the spins order in an
A-type antiferromagnetic (AFM) structure in which the spins form ferromagnetic
layers which stack antiferromagnetically along the $c$ axis [Fig. 2(a)] Rahn
_et al._ (2018). A relatively small magnetic field ($B_{\rm c}=1.6$ T at $T=2$
K for $B\parallel c$) can be used to coerce the Eu spins into a fully aligned
state Rahn  _et al._ (2018). ECA is metallic at temperatures down to $T\sim
80$ K, but for lower temperatures the resistivity increases to a sharp maximum
at $T_{\rm N}$ before falling again at lower temperatures. This behaviour has
been interpreted as due to scattering of conduction electrons by fluctuations
of localized Eu magnetic moments which are exchange-coupled to the Cd and As
orbitals Rahn  _et al._ (2018); Wang  _et al._ (2016b).

In previous ab initio electronic structure calculations, where spins in the
AFM state were found to be aligned with the $c$ axis, ECA was predicted to
host a band inversion of doubly-degenerate As $4p$ and Cd $5s$ states near the
Fermi level ($E_{\rm F}$), producing a crossing along the $\Gamma-A$ line
protected by $C_{3}$ symmetry Hua  _et al._ (2018); Rahn  _et al._ (2018).
Experimentally Rahn  _et al._ (2018) (and in more recent calculations Krishna
_et al._ (2018)) the spins are in fact found to point perpendicular to the $c$
axis [Fig. 2(a)] breaking $C_{3}$ symmetry, so an avoided crossing at finite
momentum would be expected.

We performed new ab initio calculations of the electronic structure of ECA
with the Eu spins configured (1) in the AFM state [Fig. 2(a)], and (2) in the
ferromagnetic state with Eu spins fully aligned along the $c$ axis [Fig.
2(b)]. In the AFM state, the combination of time-reversal and inversion
symmetry requires every band to be doubly degenerate. For a wide range of
parameters, the AFM state displays a small direct gap at $\Gamma$ (Fig. 2(c)
and Supplemental Material sup ) which is mostly insensitive to the orientation
of the spins. When the Eu spins are fully spin-polarized along the $c$ axis
the double degeneracy is lifted, and a single pair of Weyl nodes appears at
$E_{\rm F}$ with no other Fermi surface pockets in the Brillouin zone (Fig.
2(d) and Supplemental Material sup ). These Weyl nodes lie along the
$\Gamma-$A high symmetry line [see Fig. 1(b)] at wavevectors ${\bf k}=(0,0,\pm
k_{0})$ with $k_{0}\simeq 0.03\times 2\pi/c=0.26$ nm-1. ECA in a small
magnetic field applied along the $c$ axis, therefore, is predicted to be a
Weyl semimetal with a single pair of Weyl nodes located at $E_{\rm F}$.

The band splitting in the saturated phase is found to be $\sim$100 meV, which
is two orders of magnitude larger than the Zeeman splitting in the saturation
field $B_{\rm c}$. From this we can conclude, first, that the calculations,
which include exchange but no Zeeman interaction, are a good representation of
the experimental situation in which a small magnetic field is used to align
the Eu spins, and second, that the existence of the Weyl nodes is driven by
exchange coupling to the Eu spins.

Figure 3:  ARPES and high-field magnetotransport of EuCd2As2. (a), (c) ARPES
spectrum as a function of wavevector along the M–$\Gamma$–M high symmetry line
measured at $T\simeq 5$ K with incident photon energy of 130 eV. The data
shown here are a sum of two measurements taken with linear-vertical and
linear-horizontal polarization, respectively, to compensate the effect of
selection-rules. Nonlinear scaling was applied to the intensity to enhance the
visibility of bands with a small photoemission cross-section. (b), (d)
Electronic bands calculated by DFT (in red). (e) The second derivative of the
longitudinal resistivity $\rho^{\prime\prime}_{xx}(B)$ as a function of field
applied along the $c$ axis. (f) Plot of $1/B_{\rm int}$ at the minima and
maxima in $\rho_{xx}$ against Landau level index, with integers assigned to
the minima, where $B_{\rm int}=B+\mu_{0}M$ is the internal field. The SdH
frequency $F$ is obtained from the gradient of the linear fit shown. (g)
Temperature dependence of the amplitude in the SdH oscillation, measured at
$B=10$ T. The line is a fit to the Lifshitz–Kosevich formula, from which the
cyclotron effective mass $m_{\rm c}^{\ast}$ of the charge carriers is
estimated. The quoted error in $m_{\rm c}^{\ast}$ derives from the least-
squares fit, but the uncertainty in the measurement is expected to be larger
because of the long period of the oscillations and the relatively narrow field
range.

In order to validate the ab initio predictions we carried out angle-resolved
photoemission spectroscopy (ARPES) and quantum oscillations measurements.
ARPES data on ECA for $T<T_{\rm N}$ and zero applied magnetic field are
presented in Figs. 3(a) and (c). These $k$–$E$ plots are for $k$ along the
M–$\Gamma$–M path [see Fig. 1(b)] and show steeply dispersing bands
approaching $E_{\rm F}$. The spectra are in good agreement with the ab initio
band structure [Figs. 3(b) and (d)] calculated for the observed AFM state with
spins lying in the plane. The agreement is best when $E_{\rm F}$ is shifted
slightly downward by about 50 meV, which indicates that the sample is very
slightly hole-doped.

Our quantum oscillations measurements are summarized in Figs. 3(e)–(g). Figure
3(e) shows the second derivative of the in-plane longitudinal resistance
measured at $T=1.4$ K in magnetic fields applied parallel to the $c$ axis
($B\parallel c$) up to 37 T, well above the coercive field (see Supplemental
Material sup  for details of the data treatment). For $B<B_{\rm c}$ there is a
very strong variation in magnetoresistance associated with the progressive
canting of the spins towards the $c$ axis, shown also in Fig. 4(a), but at
higher fields the curve displays clear Shubnikov–de Haas (SdH) oscillations.
Only a single SdH oscillation frequency could be resolved, consistent with a
single band. Moreover, we do not find any evidence for a spin-splitting of the
Landau levels, in contrast to the SdH data on the structurally-related Dirac
semimetal Cd3As2 (Ref. Narayanan  _et al._ (2015)). A lack of spin splitting
would be consistent with the prediction that for $B>B_{c}$ the bands are
already split by a constant exchange field, implying that the observed SdH
oscillations correspond to the small pockets derived from the Weyl points when
$E_{\rm F}$ is shifted downwards slightly, as suggested by the ARPES
measurements.

The maxima and minima of the oscillations are plotted on a Landau level index
plot in Fig. 3(f), with minima in $\rho_{xx}$ assigned to the integers Ando
(2013). The SdH frequency obtained from the gradient is $F=25\pm 3$ T, which
converts via the Onsager relation $F=(\hbar/2\pi e)A_{\rm ext}$ to an extremal
area of the Fermi surface perpendicular to the $c$ axis of $A_{\rm ext}=0.24$
nm-2, or $k_{\rm F}=0.28$ nm-1 assuming a circular cross-section. This $A_{\rm
ext}$ value represents approximately 0.1% of the cross-sectional area of the
Brillouin zone perpendicular to the $c$ axis. Measurements at higher fields up
to 45 T did not find any additional oscillation frequencies (see Supplemental
Material sup ), and confirmed that the maximum centred on 30 T in Fig. 3(d)
corresponds to the quantum limit (first Landau level).

Figure 3(f) shows the temperature dependence of the SdH oscillation amplitude
up to 40 K. By fitting the data to the Lifshitz–Kosevich formula (amplitude
$\sim X/\sinh X$, where $X=2\pi^{2}k_{\rm B}Tm_{\rm c}^{\ast}/e\hbar B$ and
$m_{\rm c}^{\ast}=(\hbar^{2}/2\pi)\,{\rm d}A_{\rm ext}/{\rm d}E$) we obtain a
cyclotron effective mass of about $m_{\rm c}^{\ast}=0.08m_{\rm e}$. The
observation that $m_{\rm c}^{\ast}/m_{\rm e}\ll 1$ is consistent with
quasiparticles near a Dirac or Weyl node. A small effective mass is also found
in our ab initio calculations. Assuming $k_{\rm F}=0.28$ nm-1, as determined
from the SdH data, and a circular Fermi surface cross-section, we obtain
$m_{\rm c}^{\ast}=0.18m_{\rm e}$ from the calculated band structure. We
caution, however, that the measured and calculated $m_{\rm c}^{\ast}$ are not
directly comparable because the Lifshitz–Kosevich formula assumes that the
Landau levels are equally-spaced and that many levels are filled, neither of
which applies here.

Our quantum oscillations and ARPES results, as well as previous optical
reflectivity measurements which found clear evidence for a very low carrier
density Wang  _et al._ (2016b), all point to a very small Fermi surface, and
support the prediction that in the spin-polarized state of ECA there is a
single pair of Weyl nodes located close to $\Gamma$ along $\Gamma-$A, in a
small window of energy free from other bands. The small effective mass and
Fermi surface area from the SdH data, together with $p$-type Hall transport
Rahn  _et al._ (2018), indicate that the crystals used in this study are
slightly hole-doped. From the SdH measurements and ab initio in-plane
dispersion we estimate that $E_{\rm F}$ is located approximately 52 meV below
the Weyl node (see Supplemental Material sup ), which is consistent with the
shift applied to the DFT bands in order to match the ARPES data.

Figure 4:  Magnetotransport of EuCd2As2. (a) $\rho_{xx}$ and $\sigma_{yx}$ as
a function of field. (b) Definition of the $xyz$ axes relative to the crystal
orientation and directions of the current and applied field used in the
measurement.

In recent years, there has been a great deal of interest in measuring
anomalous transport effects caused by Berry curvature in topological
semimetals, especially the anomalous Hall effect (AHE) Nagaosa  _et al._
(2010); Burkov (2014). In a Weyl semimetal, the Berry curvature is associated
with the separation $\Delta k$ of the Weyl nodes in k-space, as illustrated in
Fig. 1c, and for a single pair of nodes at $E_{\rm F}$ the anomalous Hall
conductivity has the universal form Burkov (2014),

$\sigma_{yx}^{\rm AHE}=\frac{e^{2}}{2\pi h}\Delta k.$ (1)

In experiments, $\Delta k$ is typically field-dependent due to the effect of
field on the band splitting. This makes it difficult to separate the anomalous
and semi-classical contributions to the Hall effect, as the latter is also
field-dependent. In ECA, however, $\Delta k$ is almost constant for fields
above the saturation field $B_{\rm c}=1.6$ T, which makes it straightforward
to isolate the anomalous part of the Hall resistivity. In principle,
therefore, ECA is an ideal system for studying the AHE experimentally.

Figure 4(a) presents measurements of the longitudinal magnetoresistance
$\rho_{xx}$ and the anomalous part of the transverse (Hall) conductivity
$\sigma_{yx}^{\rm AHE}$ at $T=2$ K as a function of field applied parallel to
the $c$ axis (the experimental geometry is shown in Fig. 4(b), and the
procedure to obtain the anomalous part of $\sigma_{yx}$ is described in the
Supplemental Material sup ). There are rapid changes in $\rho_{xx}$ at low
field due to the reorientation of the Eu spins in the magnetic field, as noted
earlier, but above the saturation field $B_{\rm c}=1.6$ T $\rho_{xx}$
decreases monotonically with field. The field range in Fig. 4(a) is below that
where quantum oscillations become observable [see Fig. 3(e)]. The
$\sigma_{yx}^{\rm AHE}(B)$ curve is an odd function of field, increasing
rapidly for $0<B<B_{\rm c}$ and remaining constant for $B>B_{\rm c}$,
consistent with a non-zero anomalous Hall conductivity.

Assuming $\Delta k\simeq 0.52$ nm-1 from our ab initio results, equation (1)
predicts the anomalous Hall conductance for ECA to be $\sigma_{yx}^{\rm
AHE}\simeq 30$ $\Omega^{-1}$cm-1, which is significantly larger than observed.
This prediction, however, applies only when the Weyl nodes lie exactly at
$E_{\rm F}$. In the samples used here the nodes are slightly shifted from
$E_{\rm F}$, and in this situation other factors are expected to affect the
Berry phase Nagaosa  _et al._ (2010). One such factor is disorder. We have
found the AHE in ECA to be reduced by the polishing process used to shape the
Hall bar samples. Although it has been argued that disorder-induced
contributions to the AHE are absent when $E_{\rm F}$ is near the nodes Burkov
(2014), the presence of a tilt in the dispersion makes these contributions
possible in the form of skew scattering Steiner  _et al._ (2017); Mukherjee
and Carbotte (2018). The significant tilt predicted in our ab initio
calculations [Fig. 2(d)] might explain why the AHE is so reduced.

The simple structure of the Weyl nodes in ECA makes it an ideal material with
which to study the different contributions to the AHE. This could be achieved
by tuning the position of $E_{\rm F}$ relative to the Weyl nodes by doping or
application of pressure, or by controlling the level of defects by
irradiation. The degrading effects of polishing could be avoided by studying
transport phenomena with thin film samples.

More generally, ECA could provide the means to test predictions of other
exotic physics in Weyl semimetals, such as the anomalous Nernst and thermal
Hall effects Sakai  _et al._ (2018); Ferreiros  _et al._ (2017), non-
reciprocal effects in light propagation Kotov and Lozovik (2018), the
repulsive Casimir effect Wilson  _et al._ (2015), or to probe the effects of
the chiral anomaly in the optical absorption Ashby and Carbotte (2014) and
non-local transport Parameswaran  _et al._ (2014).

Note added: Very recently, Ma et al. Ma  _et al._ (2019) reported ARPES
results on EuCd2As2 which provide evidence for the existence of dynamically
fluctuating Weyl points due to ferromagnetic correlations present in zero
field at $T>T_{\rm N}$. The results in Ref. Ma  _et al._ , 2019 provide
experimental confirmation that the DFT calculations correctly predict the
effects of ferromagnetic polarization of the Eu spins on the band structure.

###### Acknowledgements.

The work at Oxford was supported by the U.K. Engineering and Physical Sciences
Research Council (EPSRC) (Grant Nos. EP/I004475/1, EP/N034872/1, EP/M020517/1
and EP/N01930X/1), the John Fell Fund, and the Oxford Centre for Applied
Superconductivity. J.-R.S. acknowledges support from the Singapore National
Science Scholarship, Agency for Science Technology and Research. F.d.J.
acknowledges funding from the European Union’s Horizon 2020 research and
innovation programme under the Marie Sklodowska Curie grant agreement No.
705968. M.C.R. was supported by the Oxford University Clarendon Fund, the Los
Alamos National Laboratory Director’s Fund, and the Alexander von Humboldt-
Stiftung. Crystal growth was carried out with support from the National Key
Research and Development Program of China (grant nos. 2017YFA0302901 and
2016YFA0300604) and the Strategic Priority Research Program (B) of the Chinese
Academy of Sciences (Grant No. XDB07020100). We acknowledge the Diamond Light
Source for time on beamlines I11 (proposal EE18786) and I05 (proposal
SI19234). Work at the HFML, a member of the European Magnetic Field Laboratory
(EMFL), was supported in part by the EPSRC via its membership of the EMFL
(grant no. EP/N01085X/1). Work at the NHMFL was sponsored by the National
Science Foundation under Cooperative Agreement No. DMR-1157490, and by the
State of Florida. We thank D. Graph for technical support at the NHMFL.

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shape, it is difficult to predict or detangle the effects of this phase
difference explicitly. However, all remaining patterns stem from this term or
from a combination of the other phase differences.

###  V.4 Exchange and Event Interference

Figure 19: Interference patterns associated with the combined exchange and
temporal-shift phase differences utilizing interfering pairs of the three most
dominant events within a pulse for the $3s\rightarrow 3p$ transition and a
carrier-envelope phase of $\phi_{1}=65^{\circ}$. The distributions shown are
unsymmetrized (top row) and symmetrized (bottom row). The same pulse
parameters are employed as in Fig. 10. The signal in each panel has been
normalized with regard to its maximum. The dashed black lines in (a) indicate
an example of a “cut” in the distribution, which also translates to the
symmetrized distribution in (d).

The remaining six phases associated with a combination of exchange and
temporal shift terms, $\alpha_{l,u},\alpha_{lT,u},\alpha_{lT,d}$ and their
identical transposed counterparts $\alpha_{r,d},\alpha_{r,dT},\alpha_{dT,l}$
are considered in Fig. 19, which shows the unsymmetrized and incoherently
symmetrized effects arising from these phases. Due to the localization of the
events within specific momentum quadrants, we expect this phase to contribute
the least to the total coherent distributions. However, the effect is stronger
than for the monochromatic wave, as the asymmetry of the pulse (and the effect
of prefactors) causes the events to occupy regions near the origin and
sometimes spill into other quadrants. This enhances the events’ overlap.

Figure 20: Interference patterns arising within the region of interest (top
row) and in a larger momentum region (bottom row) considering a fully coherent
sum of events and symmetrization [panels (a), (e)], and isolating interference
effects due to exchange [panels (b), (f)], temporal shifts [panels (c), (g)]
and those arising from a combination of exchange and event interference
[panels (d) and (h)] for the same pulse parameters employed in 10. The two
most dominant events for the channel ($3s\rightarrow 3p$) and CEP
($65^{\circ}$) have been taken. The signal in each panel has been normalized
with regard to its maximum. The contribution to the fully coherent
interference pattern at larger momentum values, arising from a combination of
exchange and shift, has been indicated by arrows in (e) and (h).

Fig. 19(a)-(c) shows fringes parallel to the $p_{n\parallel}$ axes. This is
expected since only
$\alpha^{(\mathbf{p}_{1}\leftrightarrow\mathbf{p}_{2})}_{\Delta\tau}$ (Eq.
(31)) contributes now. The locations of the brightest spots and their widths
are dependent, however, on the exact pulse field symmetry. The distributions
also appear to be cut (an example is given by the dashed line in panel (a)).

The question then remains: which of these three types of interference
contribute most to the total coherent-coherent distribution? A comparison of
exchange, event (temporal shift) and combined exchange and shift interferences
to the total coherent interference is shown in Fig. 20. The top row shows the
momentum region of interest and the bottom row displays a larger momentum
region containing the secondary effects.

The pure exchange phases are the most influential, leading to the spine and to
small hyperbolic patterns in the total coherent distributions. These features
will remain most resistant to focal averaging. The pure temporal shifts serve
to make the spine a bit more jagged and blurred, whilst also shifting the
locations of the brightest maxima within the fringes themselves. The non-
interference effects coming from summing symmetrization incoherently further
enhance the complexity of the fringes in the first and third quadrants. The
most notable impact of temporal shifts emerges in the second and fourth
quadrants, where the large wings overshadow the smaller, overlapping
hyperbolic patterns from pure exchange phase differences.

Predictably, the combined effect of exchange and temporal shifts yields the
smallest influence, lightly reinforcing some effects from the other types of
phase differences, such as overlapping fringes in quadrants two and four.
Notably, even the discernible“cut” observed when isolating this phase
dissipates within the total coherent plot. For larger momentum regions, the
heart and static patterns disappear although the effect of exchange and shift
combined in skewing the fringes becomes slightly more noticeable (indicated by
the arrows in Fig. 20(e) and (h)). We have verified that these results hold
for other events, and also the other excitation channels.

##  VI Conclusions

In conclusion, we investigate quantum interference in recollision excitation
with subsequent ionization (RESI) for few-cycle pulses, focusing on two-
electron interference patterns associated with a single excitation channel.
This type of interference stems from quantum phase shifts which can be derived
analytically, thus adding predictive power and transparency to the problem.
This considerably extends our previous investigations for monochromatic fields
[5, 6], and reveals much more convoluted patterns. The added complexity
results from three key differences regarding the driving field. First, the
field cycles are not equivalent, which means that there will be different
kinds of interfering events and many types of fringes. Second, the dominant
events within the pulse will be strongly influenced by the carrier-envelope
phase (CEP). Third, the symmetries associated with monochromatic driving
fields will be broken. A monochromatic field remains invariant under a
translation of half a cycle followed by a reflection about the time axis, also
known as the half-cycle symmetry; a time reflection around its extrema, and a
time reflection around its zero crossings [55]. These three symmetries
translate into fourfold symmetric RESI distributions for monochromatic fields,
if all transition amplitudes are summed incoherently. For a few-cycle pulse,
only symmetries associated with electron exchange will be retained.
Furthermore, because the field gradients will be unequal around its maxima or
zero crossings, the electron-momentum distributions may be skewed towards a
specific momentum region.

With that in mind, we relaxed some key assumptions in [5], including the field
shape and polarization, and derived a myriad of phase shifts for arbitrary
driving fields, which gave rise to generalized interference conditions. These
phase shifts stem from exchanging the momenta of the two electrons, which is
necessary due to their indistinguishability, from events occurring at
different times within the pulse, and from electron exchange and temporally
shifted events. The analytic conditions obtained show a multitude of features,
some of which are field independent, and some of which depend on its
polarization, shape and symmetry. We have also identified building blocks for
those phase shifts. With pure exchange, these are
$\alpha_{\mathbf{p}_{1},\mathbf{p}_{2}}^{(\mathrm{exch})}$ and
$\alpha_{\mathbf{p}_{1},\mathbf{p}_{2}}(t,t^{\prime})$, where the first term
has a linear dependence and the latter depends on the field via the temporal
arguments. With different events and purely temporal shifts, we have
identified $\alpha^{(\mathrm{ene})}_{\Delta\tau}$,
$\alpha^{(A^{2})}_{\Delta\tau}(t^{\prime},t^{\prime\prime})$,
$\alpha^{(\mathrm{pond})}_{\Delta\tau}(t,t^{\prime\prime})$ and
$\alpha^{(\mathbf{p}_{1},\mathbf{p}_{2})}_{\Delta\tau}(t,t^{\prime})$ as the
sources of interference. The term $\alpha^{(\mathrm{ene})}_{\Delta\tau}$ is
associated with the electronic bound-state and kinetic energies,
$\alpha^{(A^{2})}_{\Delta\tau}(t^{\prime},t^{\prime\prime})$ is a field-
dependent term involving only the first electron,
$\alpha^{(\mathrm{pond})}_{\Delta\tau}(t,t^{\prime\prime})$ gives
ponderomotive energy shifts and
$\alpha^{(\mathbf{p}_{1},\mathbf{p}_{2})}_{\Delta\tau}(t,t^{\prime})$ a term
encompassing both the field and momentum dependence. Finally, with both
exchange and temporally shifted events both types of phase shifts will be
present, but, instead of
$\alpha^{(\mathbf{p}_{1},\mathbf{p}_{2})}_{\Delta\tau}(t,t^{\prime})$, it is
described by the
$\alpha^{(\mathbf{p}_{1}\leftrightarrow\mathbf{p}_{2})}_{\Delta\tau}(t,t^{\prime})$
term, which contains temporal shifts and momentum exchange.

Particular cases of these conditions give the previous expressions derived in
[5] for monochromatic fields. This includes hyperbolic structures and spine-
like fringes which are reflection symmetric upon the diagonal
$p_{1\parallel}=p_{2\parallel}$ of the $p_{1\parallel}p_{2\parallel}$ plane.
Nonetheless, because the half-cycle symmetry is broken, such features are no
longer symmetric about
$(p_{1\parallel},p_{2\parallel})\leftrightarrow(-p_{1\parallel},-p_{2\parallel})$
and a bright interference maximum along the antidiagonal
$p_{1\parallel}=-p_{2\parallel}$ is also absent.

Furthermore, the bound-state prefactors $V_{\mathbf{p}_{2}e}$ and
$V_{\mathbf{p}_{1}e,\mathbf{k}g}$ introduce additional phase shifts and
momentum biases. The prefactor $V_{\mathbf{p}_{2}e}$ associated with the
ionization of the second electron determines the shapes of the distributions,
and $V_{\mathbf{p}_{1}e,\mathbf{k}g}$ influences more their centers and
widths. However, although the nodes are blurred for the latter prefactor, the
field dependence embedded in it means that it will get skewed for a pulse due
to the unequal cycles and gradients. Here, we provide an analysis of how both
prefactors can be mapped to the $p_{1\parallel}p_{2\parallel}$ plane. In this
work, we use argon as a target, for which there are six excitation pathways to
$s$, $p$, and $d$ states. This facilitates a comparison with our previous
work.

Specifically for a pulse, it is crucial to identify dominant events, whose
interference one must study. In our previous paper [38], we have studied this
dominance using qualitative arguments. Here, we propose a parameter that
brings together three key factors influencing an event’s dominance: the
ionization probabilities of the first and second electrons, which can roughly
be associated with the minimal values of
$(\mathrm{Im}[t^{\prime\prime}])^{-1}$ and $(\mathrm{Im}[t])^{-1}$, and the
momentum region for which rescattering of the first electron has a classical
counterpart. The dominance parameter is a good indicator of what should be
discarded, and of what physically happens in the pulse.

However, it has a series of limitations, which boil down to a single number
not being sufficient to quantify dominance. The prevalence of a particular
event depends strongly on the momentum region. Therefore, if they are too
discrepant, one cannot use $\mathcal{D}(p_{i},o_{j})$ to predict where the
single main maximum lies. Instead, it indicates which event dominates in the
same region. Furthermore, the gradient of $\mathrm{Im}[t]$ is asymmetrical
with regard to its minimum. Away from the minimum, imaginary parts of the
times associated with different events follow each other closely. This is
particularly true for high-lying bound states, for which these gradients are
steep. The results in Sec. IV.1 show that the dominance parameter is not
reliable in these cases.

Another issue is that it does not allow us to compare an event leading to a
bright signal in a small momentum region with another, less bright probability
density occupying a larger region. A question remains on how to determine what
contributes more to the final map. Finally, the prefactors also skew the
momentum distributions and sometimes mask the dominance. Still, if
$\mathcal{D}(p_{i},o_{j})$ is too low for a specific event, such as those at
the trailing edges of the pulse, the event can be discarded without
compromising the key results.

For the two specific CEPs employed in this work, we have identified at most
three dominant events in agreement with the qualitative findings in [38].
Applying the equations and criteria in Sec. III, we found 15 pairwise phase
differences, which are given in Table 2. These phase differences were then
used as a guide to disentangle the interference patterns obtained by
subtracting the fully coherent from the fully incoherent single-channel
distributions. The complexity of the problem makes it sometimes impractical to
obtain analytic expressions in all cases. Nonetheless, one may still observe
an approximate hierarchy, as far as the interference types are concerned.

Overall, the interference due to exchange-only processes is more pronounced
and behaves as predicted, with a bright maximum along the main diagonal and
hyperbolic patterns. We also observe that the bright fringe along the anti-
diagonal and the symmetry upon exchanging the signals of both parallel momenta
are absent. This agrees with the analytical predictions of Sec. III, which
states that these features are only present for fields with half-cycle
symmetry. The interference effects due to temporal shifts reinforce exchange-
only effects for the most part, except for the wing-shaped patterns in the
second and fourth quadrants. They also lead to ring-shaped patterns, as
predicted for temporal shifts of a full cycle, which resemble those in [10].
The phase differences associated with electron exchange and temporal shift
have the subtlest effect of all. Unfortunately, it is difficult to find a
hierarchy of the building blocks that constitute these types of interference.
Key challenges are to determine in which momentum regions the building blocks
are more or less significant and how this might change with the pulse
parameters given their strong dependence on the pulse symmetry.

Regardless, the exchange-only patterns are expected to be robust as many of
them do not depend on the field. This also explains their robustness against
focal averaging observed in [6]. This suggests that these patterns would be
the most accessible in a realistic setting. However, one may in principle find
ways to enhance the other types of interference by manipulating the field
parameters. Furthermore, we have also identified patterns associated with
three or more interfering pathways, which are difficult to disentangle.

An issue that has been left out of this article is inter-channel interference.
To tackle this problem, it will be necessary to seek an alternative strategy
to the analytic derivations performed here as intra-channel interference is
much more influenced by the bound-state geometry. Thereby, key questions are
how to establish whether specific channels are comparable and if the fringes
resulting from this type of interference have high contrast. This will be the
topic of future work.

###### Acknowledgements.

Discussions with Andrew Maxwell, Bradley Augstein, Tobin Holtmann, Gergely
Eory, Abraham Jacob, and Lewis Ling are gratefully acknowledged. This work was
partly funded by grant No. EP/J019143/1, from the UK Engineering and Physical
Sciences Research Council (EPSRC) and by UCL. *

##  Appendix A Prefactors

In this appendix, we provide a summary of how the prefactors influence the
electron momentum distributions when integrated over the momentum components
perpendicular to the driving field polarization. This is important in both
understanding the shapes of the two-electron correlated momentum
distributions, and in locating the phase shifts present in the quantum
interference plots. They were calculated assuming that all bound states
involved are hydrogenic and include all normalization and phases necessary for
computing coherent superpositions. For details we refer to our previous
publications [18, 57, 5].

###  A.1 Expressions

Here we give the general expressions for the prefactors. The excitation
prefactor reads

$\displaystyle V_{p_{1}e,kg}=$
$\displaystyle\sum_{L=\left|l_{e}-l_{g}\right|}^{l_{e}+l_{g}}\sum_{M=-L}^{L}(-i)^{L}A_{1}Y_{L}^{M}\left(\theta_{q},\phi_{q}\right)$
(42) $\displaystyle\times\frac{\left(\left\langle l_{g},l_{e},0,0\mid
L,0\right\rangle\left\langle lg,le,m_{g},-m_{e}\mid
L,M\right\rangle\right.}{\sqrt{(2L+1)}}I_{r}$

where

$\displaystyle I_{r}=$
$\displaystyle\sum_{k_{g}=0}^{b_{n_{g}l_{g}}}\sum_{k_{e}=0}^{b_{n_{e}l_{e}}}\frac{(-1)^{k_{g}+k_{e}}2^{a_{1}-1-2L}\xi^{L-a_{1}}\Xi_{l_{g}k_{g}}^{n_{g}}\Xi_{l_{e}k_{e}}^{n_{g}}\Gamma\left(a_{1}\right)}{k_{g}!k_{e}!\left(b_{n_{g}l_{g}}-k_{g}\right)!\left(b_{n_{e}l_{e}}-k_{e}\right)!\Gamma\left(\frac{3}{2}+L\right)}$
(43) $\displaystyle\times
d_{l_{g}k_{g}}^{n_{g}}d_{l_{e}k_{e}}^{n_{e}}\frac{q}{\xi}^{L}{}_{2}F_{1}\left(\frac{1}{2}a_{1},\frac{1}{2}\left(a_{1}+1\right);\frac{3}{2}+L;-\frac{q^{2}}{\xi^{2}}\right)$

and

$\displaystyle A_{1}$
$\displaystyle=(-1)^{m_{e}}C_{n_{g}l_{g}}C_{n_{e}l_{e}}\quad\frac{V_{12}(\bm{q})}{\sqrt{2\pi}}\sqrt{\left(2l_{g}+1\right)\left(2l_{e}+1\right)},$
$\displaystyle C_{nl}$
$\displaystyle=\sqrt{\frac{(n-l-1)!}{2n(n+l)!}},\quad\Xi_{lk}^{n}=\left(\sqrt{2E_{n}}\right)^{\frac{3}{2}+l+k},$
$\displaystyle d_{lk}^{n}$
$\displaystyle=\frac{(n+l)!}{(2l+k+1)!},\quad\xi=\sqrt{2E_{n_{g}}}+\sqrt{2E_{n_{e}}},$
$\displaystyle a_{1}$ $\displaystyle=3+k_{g}+k_{e}+l_{g}+l_{e}+L,\quad
b_{nl}=n-l-1.$

, and

The ionization prefactor $V_{p_{2e}}$ is given by:

$\displaystyle V_{p_{2},e}=$ $\displaystyle
A_{2}\sum_{k=0}^{b_{n_{e}l_{e}}}(-1)^{k}\frac{2^{k}\left(\sqrt{2E_{n_{e}}}\right)^{-\frac{1}{2}-l_{e}}p_{2}^{l_{e}}}{\left(b_{n_{e}l_{e}}-k\right)!k!}d_{l_{e}k}^{n_{e}}$
(44)
$\displaystyle\times\frac{\bar{\Gamma}\left(a_{2}\right)}{\Gamma\left(\frac{3}{2}+l_{e}\right)}{}_{2}F_{1}\left(\frac{1}{2}a_{2},\frac{1}{2}\left(a_{2}+1\right);\frac{3}{2}+l_{e};-\frac{p_{2}^{2}}{2E_{n_{e}}}\right),$

where

$\displaystyle A_{2}$
$\displaystyle=2(-i)^{l_{e}}C_{n_{e}l_{e}}Y_{l_{e}}^{m_{e}}\left(\theta_{p_{2}},\phi_{p_{2}}\right),$
$\displaystyle a_{2}$ $\displaystyle=2+k+2l_{e}.$

###  A.2 Excitation prefactor shifts and mapping

Figure 21: The “unshifted” excitation prefactor and corresponding mapping
calculated using the short orbit and the most dominant pair $p_{4}$ for the
$3p\rightarrow 4s$ (panels (a)-(c)), $3p\rightarrow 4p$ (panels (d)-(f)) and
$3p\rightarrow 4d$ (panels (g)-(i)) transitions. The left column shows the
imaginary part of the prefactor for odd orbital quantum number $L$ (panels
(a), (g)) and the real part for even $L$ in (d). The same pulse parameters
have been employed as in Fig. 10, with CEP $65^{\circ}$. Arrows in panels (e)
and (h) indicate extra maxima present due to the higher orbital quantum number
$L$ and white boxes in (f) and (i) indicate how these maxima translate to the
mapping. Figure 22: The “shifted” excitation prefactor and corresponding
mapping calculated using the short orbit and the most dominant pair $p_{4}$
for the $3p\rightarrow 4s$ (panels (a)-(c)), $3p\rightarrow 4p$ (panels
(d)-(f)) and $3p\rightarrow 4d$ (panels (g)-(i)) transitions. The left column
shows the imaginary part of the prefactor for odd orbital quantum number $L$
(panels (a), (g)) and the real part for even $L$ in (d). The same pulse
parameters have been employed as in Fig. 10, with CEP $65^{\circ}$. Figure 23:
The “shifted” excitation prefactor [ (a)-(c)] and corresponding mappings
(panels (d)-(f)) calculated using the short orbit for the $3p\rightarrow 4d$
channel using the three most dominant events. The same pulse parameters have
been employed as in Fig. 10, with CEP $65^{\circ}$.

Eq. II.1 shows a dependence of the excitation prefactor on
$q=\sqrt{\mathbf{q}^{2}}$, where $\mathbf{q}=\mathbf{p}_{1}-\mathbf{k}$, where
$\mathbf{k}$ represents the intermediate momentum of the first electron. Also,
the shape of this prefactor is strongly influenced by the orbital angular
momentum $L$ resulting from the sum over the angular momenta $l_{e}$ and
$l_{g}$ of the second electron’s ground and excited states in Eq. (42). The
magnetic quantum number $m=0$ is used throughout for simplicity, although the
expressions are general.

A general ”unshifted” form of this prefactor was calculated by setting
$\mathbf{q}=\mathbf{p}_{1}$. Fig. 21(b), (e) and (h) show the absolute value
of this unshifted prefactor for three values of orbital angular momentum $L$.
With odd $L$, there are nodes along $p_{1\parallel}=0$. These prefactors are
also purely imaginary [panels (a) and (g)] due to the properties of the
spherical harmonics and Clebsch-Gordon coefficients. They have opposite and
alternating phases above and below the central node. With even $L$, this node
at $p_{1\parallel}=0$ is replaced with a maximum, and the prefactor becomes
real [panel (e)]. The nodes are now at around $\pm\sqrt{U_{p}}$. There are
still alternating phases above and below the nodes. It should be noted that
the unshifted prefactor is not orbit-dependent i.e. remains the same for both
the long and short orbits.

The locations of the maxima and nodes strongly translate to the corresponding
mappings of the excitation prefactor itself. For the $s$ and $d$ states, there
is suppression along the $p_{1\parallel}$ axis and brightest maxima at
$\pm\sqrt{U_{p}}$. The $p$ state leads to brightest maxima at the origin
forming a cross-shape. With larger L, there are two additional maxima
indicated by arrows in Fig. 21(e) and (h), leading to secondary maxima at
their corresponding locations indicated by the boxes in panels (f) and (i).

The effect of the radial nodes is lost when we shift the prefactor. Thus, they
are neglected in this discussion. It is difficult to predict the effect of
shifting the momentum $\mathbf{p}_{1}$ by $\mathbf{k}$, not least because the
intermediate momentum has an imaginary component. The shifted prefactor is
also now orbit-dependent and therefore shapes for only the non-divergent
solution are shown here. For further details on the divergence of orbits and
the uniform approximation, see [62, 18].

The imaginary parts of the shifted prefactor, with the short orbit and taking
the dominant $p_{4}$ pair, are shown in Fig. 22(a) and (g). For odd $L$, the
central node has now shifted to $-2\sqrt{U_{p}}$. The phases above and below
the node still alternate but have swapped sign, possibly due to $\mathbf{k}$
being complex which introduces extra phase shifts. For even $L$, there now
exists an imaginary component for the prefactor with a node at
$-2\sqrt{U_{p}}$ coming from the $\mathbf{k}$ shift as with the odd $L$ case
and the central maximum in the real part of the prefactor shifts to
$-2\sqrt{U_{p}}$ [panel (d)]. The real parts of the shifted prefactor for all
$L$ have some stretching distortion towards lower values of $p_{1\parallel}$.
This is because the real part of $\mathbf{k}$ dominates and increases linearly
with the x-axis with an average of around $\sqrt{U_{p}}$ so prefactors are
also shifted by around this amount.

The prefactor has also significantly narrowed in the momentum plane, so much
so that the absolute value of the shifted prefactor loses its characteristic
geometry-dependent shape when mapped, since the nodes have been washed out.
The mappings therefore lead to a bright maximum around $-2\sqrt{U_{p}}$. The
position of this maximum is dependent on where the event in question is
located in the pulse, since $\mathbf{k}$ depends on the ionization and
rescattering times of the first electron. Fig. 23 shows how the maximum of the
shifted prefactor, and thus its mapping changes with event for a given
channel.

##  Appendix B Detangling interference due to
$\alpha^{(A^{2})}(t^{\prime},t^{\prime\prime})$

Figure 24: Pure temporal-shift interference patterns separated for the first
electron (panels (a)-(d)) and the second electron (panels (e)-(f)) shown
unsymmetrized (panels (a)-(b)) and symmetrized (panels (c)-(f)), computed for
pairwise combinations of the most dominant events that separated by half a
cycle for the $3s\rightarrow 3p$ transition and CEP $65^{\circ}$. All other
pulse parameters employed are the same as in Fig. 10.

Here, we consider the interference due to temporal shifts for each electron
separately in order to detangle effects arising from
$\alpha^{(A^{2})}(t^{\prime},t^{\prime\prime})$ explicitly. We employ this to
determine whether the chequerboard patterns are due to quantum interference or
incoherent symmetrization effects. This can be done by computing
$\alpha_{l,r}$, $\alpha_{lT,r}$, $\alpha_{lT,l}$ and their transposed
counterparts with just the pairs (first electron), or just the orbits (second
electron). These momentum distributions with the first electron will contain
all four of the phase differences associated with the temporal shifts.
However, those with the second electron will be missing the interference
resulting from $\alpha^{(A^{2})}(t^{\prime},t^{\prime\prime})$ since this is
dependent only on the times for the first electron. In Fig. 24, the
interferences coming from the $p_{3},p_{4}$ and $p_{4},p_{5}$ interfering
pairs for the first electron are shown, unsymmetrized [panels (a), (b)] and
symmetrized [panels (c), (d)]. The symmetrized interference coming from
$o_{4},o_{5}$ and $o_{5},o_{6}$ interfering orbits are shown in panels (e) and
(f). The differences of (c) and (e), and (d) and (f) are taken to find the
contribution of interferences from
$\alpha^{(A^{2})}(t^{\prime},t^{\prime\prime})$, for the two sets of events.

Fig. 24(a), (b) show alternating horizontal stripes for the unsymmetrized
case, which when symmetrized lead to a cross with small chequerboards at the
intersection of the horizontal and vertical stripes, in the quadrant
associated with the more dominant of the pair of interfering pairs (panels (c)
and (d)) for the first electron. Similarly, chequerboards in the quadrants
associated with the more dominant of the two orbits for the second electron
can be seen in panels (e) and (f). When the difference between the electron
one only and electron two only distributions are found, these chequerboards
overlap and cause more intricate and complex patterns.

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