# Detangling the quantum tapestry of intra-channel interference in below-
threshold nonsequential double ionization with few-cycle laser pulses

S. Hashim1 R. Tenney1,2 C. Figueira de Morisson Faria1<EMAIL_ADDRESS>1Department of Physics & Astronomy, University College London
Gower Street London WC1E 6BT, United Kingdom
2Department of Mathematics, City, University of London,
Northampton Square, London EC1V 0HB, UK

###### Abstract

We perform a systematic analysis of single-channel quantum interference in
laser-induced nonsequential double ionization with few-cycle pulses, using the
strong-field approximation. We focus on a below-threshold intensity for which
the recollision-excitation with subsequent ionization (RESI) mechanism is
prevalent. We derive and classify several analytic interference conditions for
single-channel RESI in arbitrary driving fields, and address specific issues
for few-cycle pulses. Since the cycles in a short pulse are no longer
equivalent, there are several events whose dominance varies. We quantify this
dominance for single excitation channels by proposing a dominance parameter.
Moreover, there will be many more types of superimposed interference fringes
that must be taken into consideration. We find an intricate tapestry of
patterns arising from phase differences related to symmetrization, temporal
shifts and a combination of exchange and event interference.

###### pacs:

32.80.Rm

††preprint: APS/123-QED

##  I Introduction

The archetypal example of electron-electron correlation in intense laser
fields is laser-induced nonsequential double ionization (NSDI)[1, 2]. Thereby,
a returning electron, upon recollision, gives enough kinetic energy to its
parent ion to release another electron [3]. While a large amount of studies
interpret this correlation as classical, since the past few years quantum
effects in NSDI have gained increased attention. This interest started with
the theoretical findings that quantum interference is more robust than
initially anticipated [4, 5, 6], and, depending on the circumstances, may
survive integration over several degrees of freedom [4, 5] and even focal
averaging [6], followed by experiments [7]. Furthermore, recent work has shown
that entanglement may be unambiguously present in NSDI [8], by looking at the
correlation in the orbital angular momenta of the freed electrons. In [9],
this has been discussed as an example of entanglement in a Zerfall process.

The unexpected robustness of quantum interference was the first evidence that
NSDI is not classical. Nonetheless, before those findings, quantum effects in
NSDI have remained largely unexplored (for a few studies see, however, [10,
11, 12, 13]). This indifference may be justified by the huge success of
classical models in reproducing the key experimental findings in NSDI for
three decades (for reviews see, e.g., [1, 2]). Furthermore, early studies of
quantum-classical correspondence have shown that quantum interference gets
washed out upon integration over the electron momentum components
perpendicular to the laser-field polarization [14, 15, 16], which is the
typical scenario in experiments. Still, imprints of the type of electron-
electron interaction [14, 16], and of electronic bound states [17, 18, 19] are
present. Under particular circumstances, two-center interference in molecules
can also be embedded in classical NSDI models [20].

Classical approaches also comprise the overwhelming majority of NSDI studies
in tailored fields [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34,
35, 36], with good qualitative agreement with experiments. Furthermore,
whenever quantum methods such as the strong-field approximation (SFA) [18, 19,
37, 38, 39, 40] and the quantitative rescattering theory (QRS) [41, 42, 43,
44] have been used, the emphasis was on the shapes of the electron momentum
distributions. However, quantum interference and its connection to symmetry
[45, 46, 47, 48] have been extensively studied for high-order harmonic
generation [49, 50, 51] and strong-field ionization [45, 52, 53], and
photoelectron holography [54, 55].

The apparent contradiction regarding the quantum or classical nature of NSDI
lies in the different parameter ranges employed in each of the two sets of
studies. Classical behavior has been identified for driving-field intensities
in which the first electron, upon return, gives enough energy to the core so
that a second electron can be immediately released in the continuum by
overcoming the ionization potential of its parent ion. This physical mechanism
is known as electron-impact ionization (EI). On the other hand, if the kinetic
energy of the first electron is not sufficient to trigger EI, the second
electron is promoted to an excited state, and it is freed to the continuum
after a time delay. This mechanism is known as recollision-excitation with
subsequent ionization (RESI) and occurs for driving-field intensities in the
below-threshold regime. Although classical studies have recovered some
features specific to RESI, in this regime there is more room for quantum
interference.

In a quantum mechanical framework, transition amplitudes associated with
different excitation channels leading to the same final electron momenta will
interfere. Furthermore, even if a single channel is involved, quantum
interference may occur for events separated by specific time intervals, or
transition amplitudes stemming from the two electrons being indistinguishable.
The first paper in which quantum interference was studied in RESI focused on
inter-channel interference [4]. Using the strong-field approximation (SFA), it
established that a fourfold symmetry that exists for correlated electron
momentum distributions in the $p_{1\parallel}p_{2\parallel}$ plane spanned by
the electron momentum components parallel to the laser-field polarization,
could be broken by choosing appropriate coherent superpositions of excitation
pathways. These superpositions were employed to reproduce distributions
occupying the second and fourth quadrants of the parallel momentum plane.

These findings were substantially extended in our previous publications, in
which we have identified and classified intra- and interchannel two-electron
quantum interference in RESI with a linearly polarized monochromatic field
[5]. We have shown that, for a single excitation channel, quantum interference
manifests itself as hyperbolic fringes, and bright fringes at both diagonals
$p_{1\parallel}=\pm p_{2\parallel}$ of the $p_{1\parallel}p_{2\parallel}$
plane. For inter-channel interference, we have also identified hyperbolic
structures, although obtaining analytic expressions proved challenging.
Besides the excellent agreement of these expressions with our numerical
calculations, in [6], we specifically linked features observed in experiments
with the quantum interference in RESI predicted by our model. Our results have
also shown that additional phase shifts arise from the bound-state
wavefunction from which the excited electron tunnels. This builds upon
previous work, in which we have found that RESI has potential for imaging, as
the excited state wavefunction strongly influences the shape of the resulting
RESI electron momentum distributions [18, 19].

Nonetheless, several simplifications have been made in the abovementioned
computations. One of them is to assume that the driving field is monochromatic
[5, 6]. This is a good approximation for long enough pulses, and a reasonable
assumption for short pulses if the carrier-envelope phase (CEP) is averaged
over. However, in the latter case, one must compensate for the different
frequency widths. To that end, in [6] we have added extra amplitudes and
phases, whose existence has been justified using qualitative arguments and
averages in specific momentum ranges. These arguments backed the findings
that, for longer pulses, $d$ intermediate states are the most important, while
for few-cycle pulses $s$ states prevail.

Still, one must bear in mind that, for ultrashort pulses, the physics is
markedly different. First, the frequency and intensity widths introduced by a
pulse are expected to affect the dominant excitation channels. Second, the
field cycles will no longer be equivalent, which may hinder the fringe
contrast if different events within the pulse interfere or add to the
problem’s complexity. Third, the carrier-envelope phase (CEP) will play a
major role in determining the shapes of the distributions, which will be
linked to dominant events within a pulse. Fourth, there will be broken
symmetries for fixed CEPS, which will influence the electron-momentum
distributions and the quantum phase differences leading to interference
patterns. Evidence that the fourfold symmetry is broken is provided in
previous work [37, 38], where we have studied the influence of the CEP on
single-channel RESI distributions, and in experiments [23, 56]. Therein,
quantum interference was not included.

In the present work, we discuss quantum interference in RESI with few-cycle
pulses. We will focus on single-channel interference effects, as they depend
more critically on the driving field, while the target plays a more
significant role when more than one excitation channel is present. When
dealing with a pulse, a central question is what events dominate the
underlying RESI and which ones can be neglected. In [37, 38], we have assessed
the dominance of specific events in an intuitive, ad hoc way, but it is of
interest to seek a more systematic strategy. Furthermore, we generalize the
intra-channel interference conditions derived in [5] to a field of arbitrary
shape. We show that relaxing the assumptions stemming from the symmetries
specific to the monochromatic field adds complexity to the problem and gives
rise to a rich tapestry of phase shifts and quantum interference conditions.
Throughout, we employ the modified version of the strong-field approximation
(SFA) developed in [18, 57], in which excitation and electron-electron
correlation have been incorporated. Although the SFA makes significant
approximations, such as neglecting the residual binding potential in the
electron’s continuum propagation, it is useful for disentangling different
types of quantum interference. First, it can be constructed focusing on the
specific RESI process, without the presence of electron-impact ionization.
Second, if constructed in conjunction with saddle-point methods, the SFA is an
orbit-based approach that incorporates tunneling and quantum interference.
Thus, specific quantum pathways can be switched on and off at will. For a
recent review see [58].

The paper is organized as follows. In Sec. II, we revisit the theoretical
background necessary to understand the subsequent discussions. Thereafter, in
Sec. III, we bring the interference conditions generalized to an arbitrary
field. In Sec. IV, we investigate the pulse in detail, by systematically
finding dominant events (Sec. IV.1) and their influence on target-related
features (Sec. IV.2). Sec. V is devoted to calculating electron-momentum
distributions, bringing the studies of the previous sections together with
particular emphasis on quantum interference. Finally, in Sec. VI we state our
main conclusions. We will use atomic units throughout.

##  II Background

###  II.1 General expressions

In the SFA, the RESI transition amplitude related to the $\mathcal{C}$-the
excitation channel is given by

$\displaystyle
M^{(\mathcal{C})}(\mathbf{p}_{1},\mathbf{p}_{2})=\int_{-\infty}^{\infty}dt\int_{-\infty}^{t}dt^{{}^{\prime}}\int_{-\infty}^{t^{\prime}}dt^{{}^{\prime\prime}}\int
d^{3}k$ $\displaystyle\times
V^{(\mathcal{C})}_{\mathbf{p}_{2}e}V^{(\mathcal{C})}_{\mathbf{p}_{1}e,\mathbf{k}g}V^{(\mathcal{C})}_{\mathbf{k}g}\exp[iS^{(\mathcal{C})}(\mathbf{p}_{1},\mathbf{p}_{2},\mathbf{k},t,t^{\prime},t^{\prime\prime})],$
(1)

where the action

$\displaystyle
S^{(\mathcal{C})}(\mathbf{p}_{1},\mathbf{p}_{2},\mathbf{k},t,t^{\prime},t^{\prime\prime})=$
$\displaystyle\quad
E^{(\mathcal{C})}_{\mathrm{1g}}t^{\prime\prime}+E^{(\mathcal{C})}_{\mathrm{2g}}t^{\prime}+E^{(\mathcal{C})}_{\mathrm{2e}}(t-t^{\prime})-\int_{t^{\prime\prime}}^{t^{\prime}}\frac{[\mathbf{k}+\mathbf{A}(\tau)]^{2}}{2}d\tau$
$\displaystyle\quad-\int_{t^{\prime}}^{\infty}\frac{[\mathbf{p}_{1}+\mathbf{A}(\tau)]^{2}}{2}d\tau-\int_{t}^{\infty}\frac{[\mathbf{p}_{2}+\mathbf{A}(\tau)]^{2}}{2}d\tau$
(2)

describes the process in which an electron, initially at a bound state of
energy $-E^{(\mathcal{C})}_{1g}$, is released at $t^{\prime\prime}$, returns
to its parent ion at $t^{\prime}$ with intermediate momentum $\mathbf{k}$ and
excites a second electron from a state with energy $-E^{(\mathcal{C})}_{2g}$
to a state with energy $-E^{(\mathcal{C})}_{2e}$. Upon rescattering, the first
electron acquires the final momentum $\mathbf{p}_{1}$, while the second
electron is freed at a later time $t$ with final momentum $\mathbf{p}_{2}$.
The prefactors $V^{(\mathcal{C})}_{\mathbf{k}g}$,
$V^{(\mathcal{C})}_{\bm{p}_{1}e,\mathbf{k}g}$ and
$V^{(\mathcal{C})}_{\bm{p}_{2}e}$ are associated to the ionization of the
first electron, the recollision-excitation process and the tunnel ionization
of the second electron, respectively. Within the SFA, they contain all
information about the electronic bound states, and about all the interactions
[18, 57].

The ionization prefactor for the first electron is explicitly given by

$V^{(\mathcal{C})}_{\mathbf{k}g}=\bra{\mathbf{k}}V\ket{\psi_{1g}^{(\mathcal{C})}}=\frac{1}{(2\pi)^{3/2}}\int
d^{3}r_{1}e^{-i\mathbf{k}\cdot\mathbf{r}_{1}}V(\mathbf{r}_{1})\psi_{1g}^{(\mathcal{C})}(\mathbf{r}_{1}),$
(3)

where $V(\mathbf{r}_{1})$ is the neutral atom’s binding potential, typically
assumed to be of Coulomb-type, $V_{\mathrm{ion}}$ the potential for the singly
ionized target and $\psi_{1g}^{(\mathcal{C})}$ is the ground-state wave
function for the first electron. The excitation prefactor for the second
electron reads

$\displaystyle V^{(\mathcal{C})}_{\mathbf{p}_{1}e,\mathbf{k}g}$
$\displaystyle=$
$\displaystyle\bra{\mathbf{p}_{1},\psi_{2e}^{(\mathcal{C})}}V_{12}\ket{\mathbf{k},\psi_{2g}^{(\mathcal{C})}}$
$\displaystyle=$
$\displaystyle\frac{V_{12}(\mathbf{p}_{1}-\mathbf{k})}{(2\pi)^{3/2}}\int
d^{3}r_{2}e^{-i(\mathbf{p}_{1}-\mathbf{k})\cdot\mathbf{r}_{2}}\psi_{2e}^{*(\mathcal{C})}(\mathbf{r}_{2})\psi_{2g}^{(\mathcal{C})}(\mathbf{r}_{2}),$

where $\psi_{2e}^{(\mathcal{C})}(\mathbf{r}_{2})$ and
$\psi_{2g}^{(\mathcal{C})}$ are the excited and ground states wave functions
for the second electron. The electron-electron interaction reads

$V_{12}(\mathbf{p}_{1}-\mathbf{k})=\frac{1}{(2\pi)^{3/2}}\int
d^{3}rV_{12}(\mathbf{r})\exp[-i(\mathbf{p}_{1}-\mathbf{k})\cdot\mathbf{r}]$
(5)

where $\mathbf{r}=\mathbf{r}_{1}-\mathbf{r}_{2}$, and it is taken to be of
contact-type, as in [38, 6].

Finally, the ionization prefactor associated with the second electron is given
by

$\displaystyle V^{(\mathcal{C})}_{\mathbf{p}_{2}e}$ $\displaystyle=$
$\displaystyle\bra{\mathbf{p}}V_{\mathrm{ion}}\ket{\psi_{2e}^{(\mathcal{C})}}$
$\displaystyle=$ $\displaystyle\frac{1}{(2\pi)^{3/2}}\int
d^{3}r_{2}V_{\mathrm{ion}}(\mathbf{r}_{2})e^{-i\mathbf{p_{2}}\cdot\mathbf{r}_{2}}\psi_{2e}^{(\mathcal{C})}(\mathbf{r}_{2}).$

In the SFA, the target structure is incorporated via the prefactors
$V^{(\mathcal{C})}_{\mathbf{k}g}$,
$V^{(\mathcal{C})}_{\bm{p}_{1}e,\mathbf{k}g}$ and
$V^{(\mathcal{C})}_{\bm{p}_{2}e}$. In our calculations, we employ hydrogenic
wave functions $\psi_{nlm}(\mathbf{r})=R_{nl}(r)Y_{l}^{m}(\theta,\phi)$ for
the electronic bound states. The explicit expressions for these prefactors are
given in the appendix, and detailed derivations are given in our previous
publications [18, 5]. In all examples studied in this paper, we have taken the
magnetic quantum number $m=0$ to facilitate a comparison with our previous
work [5, 6] and with the existing results in the literature [4]. Furthermore,
we consider the ionization prefactors in the velocity gauge to avoid bound-
state singularities. Length-gauge prefactors contain additional momentum
shifts, which will not play an important role for the ionization prefactors
and will cancel out for the excitation prefactor. A detailed discussion has
been provided in [18].

We calculate the multiple integrals in Eq. (1) using the steepest descent
method. This leads to the saddle-point equations

$[\mathbf{k}+\mathbf{A}(t^{\prime\prime})]^{2}=-2E^{(\mathcal{C})}_{1g},$ (7)
$\mathbf{k}=-\frac{1}{t^{\prime}-t^{\prime\prime}}\int_{t^{\prime\prime}}^{t^{\prime}}d\tau\mathbf{A}(\tau)$
(8)
$[p_{1\parallel}+A(t^{\prime})]^{2}+[\mathbf{p}_{1\perp}]^{2}=[\mathbf{k}+\mathbf{A}(t^{\prime})]^{2}-2(E^{(\mathcal{C})}_{2g}-E^{(\mathcal{C})}_{2e})$
(9)

and

$[p_{2\parallel}+A(t)]^{2}+[\mathbf{p}_{2\perp}]^{2}=-2E^{(\mathcal{C})}_{2e}.$
(10)

Eqs. (7) and (10) give the conservation of energy upon tunnel ionization for
the first and second electrons, respectively. Eq. (8) constrains the
intermediate momentum so that the first electron returns to the site of its
release, and Eq. (9) gives the rescattering event in which the first electron
gives part of the kinetic energy
$E_{k}(t^{\prime},t^{\prime\prime})=[\mathbf{k}+\mathbf{A}(t^{\prime})]^{2}/2$
to excite the second electron. We denote $p_{n\parallel}$ and
$\mathbf{p}_{n\perp}$, $(n=1,2)$, the momentum components parallel and
perpendicular to the laser-field polarization, respectively. The real parts of
the solutions of the saddle-point equations are directly related to the
classical recollision and ionization times. The integrals are then
approximated by sums over these stationary variables. For the second electron,
the saddles are well separated in all momentum regions, while, for the first
electron, pairs of saddles must be considered collectively using the uniform
approximation in [59].

Eq. (9) relates the kinetic energy $E_{k}(t^{\prime},t^{\prime\prime})$ of the
returning electron to the energy difference $\Delta
E^{(\mathcal{C})}=E^{(\mathcal{C})}_{2g}-E^{(\mathcal{C})}_{2e}$ between the
ground and excited state of the second electron. With regard to the space
spanned by the momenta $p_{1\parallel},\mathbf{p}_{1\perp}$ of the first
electron, this saddle-point equation represents a sphere, whose radius is
$\sqrt{2(E_{k}(t^{\prime},t^{\prime\prime})-\Delta E^{(\mathcal{C})})}$. If
$E_{k}(t^{\prime},t^{\prime\prime})>\Delta E^{(\mathcal{C})}$, this radius is
real and Eq. (9) may have a classical counterpart. If we are interested in
$p_{1\parallel}$ only, we can assume that $\mathbf{p}_{1\perp}^{2}/2$ is an
additional energy shift which will effectively alter the ionization threshold.
Therefore, assuming $\mathbf{p}_{1\perp}=\mathbf{0}$ helps to define an upper
bound for Eq. (9) in terms of the momentum range $p_{1\parallel}$ which may be
occupied, should rescattering have a classical counterpart. We call this
momentum range the classically allowed region (CAR). This concept has been
introduced in our previous publications [18, 57].

Furthermore, the saddle-point equations state that RESI may be viewed as two
time-ordered processes similar to above-threshold ionization (ATI) [18, 57].
The first electron behaves as if it were undergoing rescattered ATI, with the
difference that, for RESI, the process is inelastic. This means that the
maximal final kinetic energy of the first electron can approach the
rescattered ATI cutoff of $10U_{p}$, where $U_{p}$ is the ponderomotive
energy. Moreover, Eq. (10) resembles the saddle-point equation obtained by
direct ATI, for which the maximum classical kinetic energy, known as the
direct ATI cutoff, is 2$U_{p}$. This leads to the equation of a sphere in
terms of $p_{2\parallel}$, $\mathbf{p}_{2\perp}$, whose radius is
$2\sqrt{U_{p}}$ [60]. The ponderomotive energy is the time-averaged kinetic
energy acquired by an electron from the field. For a linearly polarized
monochromatic wave, it is related to the amplitude of the vector potential by
$A_{0}=2\sqrt{U_{p}}$.

###  II.2 Two-electron probability density

The quantity of interest is the correlated two-electron probability density as
a function of the momentum components $p_{n\parallel}$ $n=1,2$ parallel to the
driving-field polarization. This is given by

$\displaystyle\mathcal{P}(p_{1\parallel},p_{2\parallel})=\int\int
d^{2}p_{1\perp}d^{2}p_{2\perp}\mathcal{P}(\mathbf{p}_{1},\mathbf{p}_{2}),$
(11)

where $\mathcal{P}(\mathbf{p}_{1},\mathbf{p}_{2})$ is the fully resolved two-
electron momentum probability density, and the transverse momentum components
have been integrated over. When calculating this probability density, several
issues must be taken into consideration. First, both electrons are
indistinguishable, which means that Eq. (1) must be symmetrized upon
$\mathbf{p}_{1}\leftrightarrow\mathbf{p}_{2}$, that is, electron exchange.
Second, in a real target, there will be several excitation channels, each of
which will be associated with the transition amplitude Eq. (1). Third, there
will be several events within the pulse for the first and second electrons.
Thus, within the saddle-point approximation, the overall RESI amplitude must
contain sums over (i) transitions involving different excitation channels;
(ii) the symmetrization related to electron indistinguishability, which will
occur for each pair of excitation and ionization times; (iii) the events
within a pulse. Quantum mechanically, all these contributions add up
coherently.

The fully coherent sum over events, channels and symmetrization reads

$\mathcal{P}_{(\mathrm{ccc})}(\mathbf{p}_{1},\mathbf{p}_{2})=\left|\sum_{\varepsilon}\sum_{\mathcal{C}}M_{\varepsilon}^{(\mathcal{C})}(\mathbf{p}_{1},\mathbf{p}_{2})+M_{\varepsilon}^{(\mathcal{C})}(\mathbf{p}_{2},\mathbf{p}_{1})\right|^{2},$
(12)

where the symbols $\varepsilon$ and $\mathcal{C}$ denote event and channel,
respectively. In the present work, we will focus on single-channel
interference, so that sums over channels will not be taken into consideration.

For a single channel, a coherent sum over the events $\varepsilon$ in the
pulse and symmetrization leads to

$\mathcal{P}^{(\mathcal{C})}_{(\mathrm{cc})}(\mathbf{p}_{1},\mathbf{p}_{2})=\left|\sum_{\varepsilon}M_{\varepsilon}^{(\mathcal{C})}(\mathbf{p}_{1},\mathbf{p}_{2})+M_{\varepsilon}^{(\mathcal{C})}(\mathbf{p}_{2},\mathbf{p}_{1})\right|^{2},$
(13)

while the incoherent counterpart of Eq. (13) is given by

$\mathcal{P}^{(\mathcal{C})}_{(\mathrm{ii})}(\mathbf{p}_{1},\mathbf{p}_{2})=\sum_{\varepsilon}\left|M_{\varepsilon}^{(\mathcal{C})}(\mathbf{p}_{1},\mathbf{p}_{2})\right|^{2}+\left|M_{\varepsilon}^{(\mathcal{C})}(\mathbf{p}_{2},\mathbf{p}_{1})\right|^{2}.$
(14)

In the present work, we are interested in disentangling several types of
quantum interference. Therefore, we construct the two-electron probability
density in several ways, depending on the question we wish to address.

Throughout, we will use the notation
$\mathcal{P}^{(\mathcal{C})}_{(S\varepsilon)}$ where the indices $S$,
$\varepsilon,$ and $\mathcal{C}$ relate to the electron symmetrization, the
event, and the channel, respectively. The indices $c$ and $i$ on the left-hand
side stand for coherent and incoherent, respectively. Thus,
$\mathcal{P}^{(\mathcal{C})}_{(cc)}$ states that the sum considered in the
single-channel probability density is coherent over the pulse events and
symmetrization, and has been calculated for the $\mathcal{C}$-th channel. If
the symmetrization is done coherently, but the events are summed over
incoherently, this gives

$\mathcal{P}^{(\mathcal{C})}_{(\mathrm{ci})}(\mathbf{p}_{1},\mathbf{p}_{2})=\sum_{\varepsilon}\left|M_{\varepsilon}^{(\mathcal{C})}(\mathbf{p}_{1},\mathbf{p}_{2})+M_{\varepsilon}^{(\mathcal{C})}(\mathbf{p}_{2},\mathbf{p}_{1})\right|^{2}.$
(15)

Alternatively, one could sum the events within a pulse coherently and perform
the symmetrization incoherently. This yields

$\mathcal{P}^{(\mathcal{C})}_{(\mathrm{ic})}(\mathbf{p}_{1},\mathbf{p}_{2})=\left|\sum_{\varepsilon}M_{\varepsilon}^{(\mathcal{C})}(\mathbf{p}_{1},\mathbf{p}_{2})\right|^{2}+\left|\sum_{\varepsilon}M_{\varepsilon}^{(\mathcal{C})}(\mathbf{p}_{2},\mathbf{p}_{1})\right|^{2}.$
(16)

This notation differs from that employed in our previous work [5, 6], for
which the interference due to electron symmetrization was called “event
interference”. Apart from the combinations given above, given the wealth of
interference patterns, it may also be useful to sum transition amplitudes
pairwise. A discussion of the phase shifts that occur in these pairwise sums
is given in Sec. III.

Finally, under some circumstances, it is useful to compute partial momentum
distributions for each electron, given by

$M^{(n)}(p_{n\parallel})=\int d^{2}p_{n\perp}|M^{(n)}(\mathbf{p}_{n})|^{2},$
(17)

with $n=1,2$.

###  II.3 Target considerations

As a target, we will consider Argon, for which there are six main excitation
channels. The absolute values of ground-state energies associated with $3s$
and $3p$ are taken to be $E^{(1)}_{2g}=E^{(2)}_{2g}=1.016$ a.u. and the first
ionization potential is $E^{(\mathcal{C})}_{1g}=0.58$ a.u. For clarity, these
channels are provided in Table 1. The influence of the target occurs via the
bound-state energies and the prefactors. According to the saddle-point
equations (7)-(10), the bound-state energies will affect the ionization
probability of the first and second electron, and the classically allowed
region in momentum space for the rescattered electron. The more tightly bound
the electrons are, the lower their tunneling probability will be. Furthermore,
large (small) energy gaps $\Delta
E^{(\mathcal{C})}=(E^{(\mathcal{C})}_{2g}-E^{(\mathcal{C})}_{2e})$ stemming
from the second electron being excited will potentially favor small (large)
classically allowed regions for the first electron upon rescattering.

The bound-state geometry will introduce an additional momentum bias in the
problem via the prefactors. The shapes of the electron-momentum distributions
will mostly be influenced by the prefactor $V_{\mathbf{p}_{2}e}$ [18, 5].
Before integration over the transverse degrees of freedom, this prefactor will
exhibit radial and, for $l\neq 0$, angular nodes. The other prefactors will
have less influence in the shapes of the electron momentum distribution, due
to the time dependence of the intermediate momentum $\mathbf{k}$ (see saddle-
point equation (8)). This time dependence will cause a blurring in their
nodes. The radial and angular nodes of the prefactors may lead to additional
phase shifts, which will influence the overall interference patterns. A
detailed study of these features has been performed in [5] for a monochromatic
field, but some of these issues must be revisited for a pulse.

Channel | Excited-state configuration |  $E^{(\mathcal{C})}_{2e}$ (a.u.)
---|---|---
1 | $3s3p^{6}(3s\rightarrow 3p)$ | 0.52
2 | $3p^{5}3d(3p\rightarrow 3d)$ | 0.41
3 | $3p^{5}4d(3p\rightarrow 4d)$ | 0.18
4 | $3p^{5}4s(3p\rightarrow 4s)$ | 0.40
5 | $3p^{5}4p(3p\rightarrow 4p)$ | 0.31
6 | $3p^{5}5s(3p\rightarrow 5s)$ | 0.19
Table 1: Relevant excitation channels for $\mathrm{Ar}^{+}$, in order of
increasing principal and orbital quantum numbers. From left to right, the
first column gives the number associated with the channel, the second column
states the electronic configuration and the excitation pathway, and the third
column provides the excited-state energy in atomic units.

###  II.4 Momentum constraints for a pulse

The saddle-point Eqs. (9) and (10) can be used to determine constraints in
momentum space for each rescattering event within the pulse. The first
electron is released such that, for $\mathrm{Re}[t^{\prime\prime}]$, the
electric field is close to an extremum. For a specific event, the first
electron’s most prominent return, at $\mathrm{Re}[t^{\prime}]$, is near a
field zero crossing displaced by approximately three-quarters of a field
cycle. The second electron may then tunnel in any of the subsequent times $t$,
for which its real part is near a field extremum. However, due to bound-state
depletion, the prevailing ionization time occurs around the extremum
immediately after the zero crossing. Similarly, the first electron may return
following longer orbits, but such returns are expected to be suppressed due to
wave-packet spreading.

Using this information, we estimate that the final momentum of both electrons
will be located around
$(p_{1\parallel},p_{2\parallel})=(-A(t^{\prime}),-A(t))$. For times
$\mathrm{Re}[t^{\prime}]$ sufficiently close to the peaks of the field,
$-A(t^{\prime})\simeq\pm 2\sqrt{U_{p}}$ and $-A(t)\simeq 0$. The extension of
the momentum region occupied will depend on the electron’s kinetic energy upon
return at each rescattering event. This region may be large, small, or even
have no classical counterpart. Electron indistinguishability dictates that
there will also be events whose amplitudes are centered at
$(p_{1\parallel},p_{2\parallel})=(-A(t),-A(t^{\prime}))$. The final kinetic
energy for the first electron may extend to almost $10U_{p}$, while that of
the second electron may reach up to $2U_{p}$. Taking the perpendicular
momentum components to vanish provides an upper bound for the regions occupied
in the $p_{1\parallel}p_{2\parallel}$ plane [18, 57].

Similar to the procedure in [5] for a monochromatic wave, we consider a single
channel, neighboring events within the pulse, displaced by approximately half
a cycle, and those present due to the electron exchange symmetry of the
system. However, there are key differences for a few-cycle pulse: (i) the
half-cycle symmetry $\textbf{A}(t)=\pm\textbf{A}(t\pm T/2)$, where $T$ is a
field cycle, is broken; (ii) the cycles are not equivalent due to the presence
of the pulse envelope; (iii) the assumptions made upon $A(t^{\prime})$ and
$A(t)$ are approximate, and work better close to the center of the pulse.

This will lead to the transition amplitudes $M_{l}$, $M_{u}$, $M_{r}$, and
$M_{d}$, with
$M_{l}(\mathbf{p}_{1},\mathbf{p}_{2})=M_{d}(\mathbf{p}_{2},\mathbf{p}_{1})$
and
$M_{r}(\mathbf{p}_{1},\mathbf{p}_{2})=M_{u}(\mathbf{p}_{2},\mathbf{p}_{1})$,
where we have dropped the superscript $(\mathcal{C})$ for simplicity. In Fig.
1, we provide a schematic representation of the momentum constraints
associated with each of these transition amplitudes as the shaded regions.
Outside these constraints, the probability density is strongly suppressed, as
it has no classical counterpart. Additionally, for a monochromatic wave, the
half-cycle symmetry implies that
$M_{l}(\mathbf{p}_{1},\mathbf{p}_{2})=M_{r}(-\mathbf{p}_{1},-\mathbf{p}_{2})$
and
$M_{u}(\mathbf{p}_{1},\mathbf{p}_{2})=M_{d}(-\mathbf{p}_{1},-\mathbf{p}_{2})$.
This leads to a fourfold symmetry in the momentum-region constraints, as
outlined by the dashed rectangles in the figure, which, in the absence of
further momentum bias, causes the correlated two-electron distribution to be
cross-shaped [18, 57]. For a few-cycle pulse, the half-cycle symmetry is
broken, but the reflection symmetry about the diagonal
$p_{1\parallel}=p_{2\parallel}$ is retained.

In addition to that, Fig. 1 indicates a substantial overlap for (i) $M_{l}$
and $M_{d}$, (ii) $M_{r}$ and $M_{u}$, (iii) $M_{u}$ and $M_{l}$, and (iv)
$M_{r}$ and $M_{d}$. Therefore, their interference is expected to be
substantial and will be calculated in pairwise coherent sums. The sums
$M_{rl}=M_{l}+M_{r}$ and $M_{ud}=M_{u}+M_{d}$ of events separated by half a
cycle will also overlap and interfere, but this interference is expected to be
less relevant as the probability density is small outside the shaded regions
[18, 5].

Figure 1: Schematic representation of the momentum-space regions occupied by
the RESI transition amplitudes $M_{l}$, $M_{u}$, $M_{r}$ and $M_{d}$. The
dashed rectangles indicate the momentum constraints that would hold for a
monochromatic driving field, while the shaded areas show their counterparts
for few-cycle pulses. The overlapping areas indicate momentum regions for
which quantum interference will be significant. We have used the same color
for $M_{l}$ and $M_{d}$ (blue), and $M_{u}$ and $M_{r}$ (purple), to highlight
the property
$M_{l}(\mathbf{p}_{1},\mathbf{p}_{2})=M_{d}(\mathbf{p}_{2},\mathbf{p}_{1})$
and
$M_{r}(\mathbf{p}_{1},\mathbf{p}_{2})=M_{u}(\mathbf{p}_{2},\mathbf{p}_{1})$,
which gives a reflection symmetry about the diagonal
$p_{1\parallel}=p_{2\parallel}$. This diagonal is indicated by the dotted
black line. The negative (positive) signs in the center of the blue (purple)
shaded regions indicate the most probable momenta.

Furthermore, the cycles within a pulse will not be identical and there may be
several comparable rescattering and ionization events, which will potentially
interfere. Thus, in principle, one must take their contributions to the
electron-momentum distributions into consideration. A schematic representation
is provided in Fig. 2, which shows the momentum constraints determined by four
events within a hypothetical pulse. Panel (a) illustrates the events whose
contributions must be added up, and whose momentum constraints are shown as
the red and yellow shaded areas. We assume that the event of each sub-panel
[(a)(i) and (a)(ii)] took place over a single cycle. The events in separate
sub-panels are displaced by at least a cycle. The shaded regions around the
same $p_{n\parallel}$, $n=1,2$ half axis (positive or negative) correspond to
events displaced by an integer number of cycles (different colors), or none
(same color), while those around opposite $p_{n\parallel}$, $n=1,2$ half-axis
and highlighted by the same color are displaced by half a cycle.

The momentum ranges occupied by the resulting events, shown in panel (b),
illustrate several overlapping regions, for which quantum interference will
occur. This interference may be pronounced, subtle, or negligible, depending
on whether the events are comparable or whether some of them prevail. Outside
the shaded regions, there will also be quantum interference, but due to the
transition amplitudes being suppressed, it is expected to be negligible. One
may control the dominant events by changing the pulse parameters.

Figure 2: Schematic representation of the momentum-space regions associated
with different events within the pulse [panel (a)], and those resulting from
their coherent superposition [panel (b)]. The dashed rectangles indicate the
momentum constraints that would hold for a monochromatic driving field, while
the shaded areas show their counterparts for few-cycle pulses. We assumed that
each subpanel in (a) gives the momentum regions occupied by events within a
single cycle, and the events in (a)(i) and (a)(ii) are summed coherently.
Overlapping shaded regions indicate that quantum interference is potentially
significant. We consider two events separated by a half cycle in each graph in
panel (a), which eventually interfere.

##  III Generalized intra-channel interference conditions

Below we generalize the interference conditions derived in [5] to an arbitrary
driving field, with vector potential $\mathbf{A}(\tau)$ with
$\tau=t,t^{\prime}$. We make no assumption about its component frequencies,
shape, or polarization, so the expressions below are entirely general. Due to
their level of complexity, we first provide the key assumptions behind the
derivations, together with a road map and a graphic representation to
facilitate their understanding.

###  III.1 Diagrammatic representation

Within a single cycle, quantum interference will stem from $|M_{ld}|^{2}$,
$|M_{ru}|^{2}$, $|M_{ul}|^{2}$, and $|M_{rd}|^{2}$, where
$M_{\mu\nu}=M_{\mu}+M_{\nu}$. If there is a shift of an integer number $n$ of
cycles $T=2\pi/\omega$ in a specific transition amplitude, we use the notation
$M_{\mu,nT}=M_{\mu}(t+nT)$. Each term in these coherent sums will be
approximated by $e^{iS_{\mu}}$, where $\mu=l,d,u,r$, and the phase differences
are given by $\alpha_{\mu\nu}=S_{\mu}-S_{\nu}$. This is justified by the
underlying assumption that the prefactors vary much more slowly than the
action, which is a key element of the approaches employed in this work.

Figure 3: Diagrammatic representation of different types of pairwise single-
channel interference that may occur for NSDI RESI. The momentum ranges
occupied by a specific process are indicated by the thick solid lines, while
the thin dashed lines are a simplified representation of the parallel momentum
axes $p_{n\parallel}$, $n=1,2$. Panel (a) shows the processes associated with
the electron momenta being swapped, but taking the same event. Panels (b)
consider the interference of events shifted by an odd [(b)(i)] and even
[(b)(ii)] number of half cycles, but without momentum exchange. Panels (c)
represent interfering processes for which there are temporal shifts and
momentum exchange, considering odd and even numbers of half cycles [panels
(c)(i) and (c)(ii), respectively]. In the diagrams, different colors indicate
that the events are shifted in time, and the same color indicates the same
event. The phase shifts $\alpha_{\mu\nu}$ are written in the figure. For
simplicity, for events displaced by more than a cycle, we have only stated
phase shifts associated with a single period, and, in (b)(ii), we omitted the
events giving $\alpha_{\mu,\nu T}$ for which the second argument is delayed by
a cycle. However, these events and those separated by longer times can be
inferred using this diagram.

In Fig. 3, we present a graphic representation of the three qualitatively
different types of intra-channel phase shifts that may occur, with the
interfering transition amplitudes being indicated by thick solid lines. One
should bear in mind that the phase differences $\alpha_{\mu\nu}$ illustrated
in the figure are a sample of what may occur and the maximal temporal
difference $\Delta\tau$ considered is a full cycle of the field. However, this
notation can be extrapolated to encompass larger time differences. The
diagrams are a simplification of the main momentum regions occupied by
specific transition amplitudes. They allow a quick assessment of what is going
on. Different colors indicate temporal shifts, while the same color indicates
simultaneous occurrence.

Fig. 3(a) shows the phase differences purely associated with the electrons
being exchanged. The phase shift $\alpha_{ld}$ ($\alpha_{ru}$) is associated
with the pairwise interference of $M_{l}$ and $M_{d}$ ($M_{r}$ and $M_{u}$).
These coherent sums are expected to populate mainly the first and third
quadrants of the $p_{1\parallel}p_{2\parallel}$ plane, and each pair
represents the same event, but in one of the contributions $\mathbf{p}_{1}$
and $\mathbf{p}_{2}$ are interchanged. This is indicated by using the same
colors for a pair.

Fig. 3(b) shows the phase differences uniquely given by temporal shifts,
without exchanging the electron momenta. In Fig. 3(b)(i), we plot a
diagrammatic representation of the pairwise interference of events displaced
by a half cycle. These events are located in opposite momentum half axes. The
phase difference $\alpha_{lr}$ is associated with the amplitudes $M_{l}$ and
$M_{r}$, located along the $p_{1\parallel}$ axis, and the phase shift
$\alpha_{du}$ stems from the difference of $M_{d}$ and $M_{u}$. This diagram
can be generalized to an odd number of half-cycles. The diagram in Fig.
3(b)(ii) represents the interference of events displayed by a full cycle,
which we indicate by including the subscript $T$ in $\alpha$. For example,
$\alpha_{lT,l}$ means that the interfering actions are
$S_{lT}=S_{l}(\mathbf{p}_{1},\mathbf{p}_{2},t+T,t^{\prime}+T,t^{\prime\prime}+T)$
and $S_{l}(\mathbf{p}_{1},\mathbf{p}_{2},t,t^{\prime},t^{\prime\prime})$, and
the same holds for the other scenarios illustrated in that figure. In this
case, the different events occupy the same parallel momentum half-axis, and
therefore their overlap is significant. A similar pattern is encountered if
the temporal shift is an arbitrary integer number of cycles.

Finally, the diagrams in Fig. 3(c) illustrate interfering processes for which
there are temporal shifts and electron exchange. Fig. 3(c)(i) shows the
interference of the amplitudes $M_{l}$ and $M_{u}$ ($M_{r}$ and $M_{d}$),
which correspond to processes displaced by half a cycle in which the two
electron momenta have been swapped. Finally, the schematic representations in
Fig. 3(c)(ii) depict interfering processes displaced by a full cycle, and
whose momenta were exchanged. In summary, there are a few common patterns in
Fig. 3: L-shaped structures indicate that the electron momenta were exchanged
in one of the interfering terms; straight lines indicate processes displaced
by either a full number of cycles (short lines), or an odd number of half
cycles (long lines); the same (different) colors indicate the absence
(presence) of temporal shifts. We will apply this notation to concrete
examples in subsequent sections, once we have outlined a way to select the
relevant events within a pulse. They are also useful to understand the
equations provided next.

###  III.2 Analytic expressions

In the following, we derive analytic expressions for a wide range of phase
differences that occur in the RESI amplitudes, and, as much as possible, we
discuss what patterns are expected from their interference.

####  III.2.1 Intra-cycle phase differences

Let us commence by exploring the phase differences that occur within a single
field cycle, and are illustrated by the diagram in Fig. 3. The phase
difference $\alpha_{ld}$ is purely associated with exchanging $\mathbf{p}_{1}$
and $\mathbf{p}_{2}$ and reads

$\alpha_{ld}=S_{l}-S_{d}=\alpha^{(\mathrm{exch})}_{\mathbf{p}_{1},\mathbf{p}_{2}}+\alpha_{\mathbf{p}_{1},\mathbf{p}_{2}}(t,t^{\prime}),$
(18)

where

$\alpha^{(\mathrm{exch})}_{\mathbf{p}_{1},\mathbf{p}_{2}}=\frac{1}{2}\left(\mathbf{p}_{2}^{2}-\mathbf{p}_{1}^{2}\right)(t-t^{\prime}),$
(19)
$\alpha_{\mathbf{p}_{1},\mathbf{p}_{2}}(t,t^{\prime})=(\mathbf{p}_{2}-\mathbf{p}_{1})\cdot\left[\mathbf{F}_{A}(t^{\prime})-\mathbf{F}_{A}(t)\right]$
(20)

and

$\mathbf{F}_{A}(t)=\int^{t}\mathbf{A}(\tau)d\tau.$ (21)

is the temporal integral of the vector potential.

Constructive interference between $M_{l}$ and $M_{d}$ will happen for
$\alpha_{ld}=2n\pi$, where $n$ is an integer, and, for simplicity, we will
investigate $n=0$. The phase shift
$\alpha^{(\mathrm{exch})}_{\mathbf{p}_{1},\mathbf{p}_{2}}$ is field-
independent, and requiring it to vanish gives equations of hyperbolae in the
$p_{1\parallel}p_{2\parallel}$ plane, whose asymptotes are $p_{1\parallel}=\pm
p_{2\parallel}$ and whose eccentricity is $\sqrt{2}$. The shift
$\alpha_{\mathbf{p}_{1},\mathbf{p}_{2}}(t,t^{\prime})$ is field dependent and
states that $\mathbf{p}_{2}-\mathbf{p}_{1}$ must be orthogonal to
$\mathbf{F}_{A}(t^{\prime})-\mathbf{F}_{A}(t)$. For linearly polarized light,
it reduces to $p_{1\parallel}=p_{2\parallel}$. The conditions upon the momenta
are identical to those obtained in [5] for a monochromatic field, and
determine that bright fringes must be present in the diagonal of the parallel
momentum plane. Their independence of the field profile makes these central
fringes robust against focal averaging, as found in our previous publication
[6]. Nonetheless, the asymptotes and axes of the hyperbolae may shift as they
depend on the transverse momentum integration.

The other phase difference solely related to electron momentum exchange is

$\alpha_{ru}=S_{r}-S_{u}=-\alpha^{(\mathrm{exch})}_{\mathbf{p}_{1},\mathbf{p}_{2}}+\alpha_{\mathbf{p}_{1},\mathbf{p}_{2}}(t+\frac{\pi}{\omega},t^{\prime}+\frac{\pi}{\omega}),$
(22)

which, if made to vanish, also leads to hyperbolae and a bright fringe at the
main diagonal for linearly polarized fields. One should note, however, that
the temporal shifts in the second term of Eq. (22) imply that
$|\alpha_{ru}|\neq|\alpha_{ld}|$ unless
$\mathbf{F}_{A}(t)=-\mathbf{F}_{A}(t+\pi/\omega)$. This condition, known as
the half-cycle symmetry, renders $\alpha_{ru}=-\alpha_{ld}$. If this condition
is broken, for instance for a driving field consisting of a wave and its
second harmonic, or for the few-cycle pulses discussed in Secs. IV and V, the
hyperbolae symmetric about the main diagonal will be asymmetric about
$(p_{1\parallel},p_{2\parallel})\leftrightarrow(-p_{1\parallel},-p_{2\parallel})$,
that is, in the first and third quadrants of the
$p_{1\parallel}p_{2\parallel}$ plane.

Next, let us calculate the phase difference $\alpha_{lr}$ ($\alpha_{du}$)
between the right and the left (upward and downward) peaks [see diagrams in
Fig. 3(b)(i)]. The action $S_{r}$ ($S_{u}$) gives events shifted from those
corresponding to $S_{l}$($S_{d}$) by half a cycle, but no momentum exchange
between the first and second electron takes place. This means that
$S_{l}(\mathbf{p}_{1},\mathbf{p}_{2},t,t^{\prime},t^{\prime\prime})=S_{r}(\mathbf{p}_{1},\mathbf{p}_{2},t+\pi/\omega,t^{\prime}+\pi/\omega,t^{\prime\prime}+\pi/\omega)$
and that
$S_{r}(\mathbf{p}_{1},\mathbf{p}_{2},t,t^{\prime},t^{\prime\prime})=S_{l}(\mathbf{p}_{1},\mathbf{p}_{2},t+\pi/\omega,t^{\prime}+\pi/\omega,t^{\prime\prime}+\pi/\omega)$.
For $\alpha_{lr}$, this gives

$\displaystyle\alpha_{lr}=S_{l}-S_{r}$
$\displaystyle=-\frac{1}{2(t^{\prime}-t^{\prime\prime})}\alpha^{(A^{2})}_{\pi/\omega}(t^{\prime},t^{\prime\prime})-\frac{1}{2}\alpha^{(\mathrm{pond})}_{\pi/\omega}(t,t^{\prime\prime})$
$\displaystyle-\alpha^{(\mathrm{ene})}_{\pi/\omega}-\alpha^{(\mathbf{p}_{1},\mathbf{p}_{2})}_{\pi/\omega}(t,t^{\prime}),$
(23)

where the purely temporal phase shifts are defined as

$\displaystyle\alpha^{(A^{2})}_{\Delta\tau}(t^{\prime},t^{\prime\prime})$
$\displaystyle=\left[\mathbf{F}_{A}(t^{\prime}+\Delta\tau)-\mathbf{F}_{A}(t^{\prime\prime}+\Delta\tau)\right]^{2}$
$\displaystyle-\left[\mathbf{F}_{A}(t^{\prime})-\mathbf{F}_{A}(t^{\prime\prime})\right]^{2},$
(24) $\displaystyle\alpha^{(\mathrm{pond})}_{\Delta\tau}(t,t^{\prime\prime})$
$\displaystyle=F_{A^{2}}(t+\Delta\tau)+F_{A^{2}}(t^{\prime\prime}+\Delta\tau)$
$\displaystyle-F_{A^{2}}(t)-F_{A^{2}}(t^{\prime\prime}),$ (25)

and

$\alpha^{(\mathrm{ene})}_{\Delta\tau}=\frac{\Delta\tau}{2}\left(2\text{E}_{2g}+2\text{E}_{1g}+\mathbf{p}_{1}^{2}+\mathbf{p}_{2}^{2}\right),$
(26)

with

$F_{A^{2}}(t)=\int^{t}\mathbf{A}^{2}(\tau)d\tau.$ (27)

One should note that Eqs. (24), (25) have temporal arguments, which indicate
that the values of $F_{A^{2}}$ and $F_{A}$ may vary within a pulse and this
must be taken into consideration, while (26) depends only on the time
difference $\Delta\tau$. The phase difference in Eq. (25) is expected to give
ponderomotive terms, which will add to that in Eq. (26). The latter is linear
in time and contains the bound-state and kinetic energies.

The phase difference
$\alpha^{(\mathbf{p}_{1},\mathbf{p}_{2})}_{\pi/\omega}(t,t^{\prime})$ is a
particular case of

$\displaystyle\alpha^{(\mathbf{p}_{1},\mathbf{p}_{2})}_{\Delta\tau}(t,t^{\prime})$
$\displaystyle=\mathbf{p}_{1}\cdot\left[\mathbf{F}_{A}(t^{\prime}+\Delta\tau)-\mathbf{F}_{A}(t^{\prime})\right]$
$\displaystyle+\mathbf{p}_{2}\cdot\left[\mathbf{F}_{A}(t+\Delta\tau)-\mathbf{F}_{A}(t)\right],$
(28)

which also incorporates the electron momenta. Similarly,

$\displaystyle\alpha_{du}=S_{d}-S_{u}$
$\displaystyle=-\frac{1}{2(t^{\prime}-t^{\prime\prime})}\alpha^{(A^{2})}_{\pi/\omega}(t^{\prime},t^{\prime\prime})-\frac{1}{2}\alpha^{(\mathrm{pond})}_{\pi/\omega}(t,t^{\prime\prime})$
$\displaystyle-\alpha^{(\mathrm{ene})}_{\pi/\omega}-\alpha^{(\mathbf{p}_{2},\mathbf{p}_{1})}_{\pi/\omega}(t,t^{\prime}),$
(29)

where $\alpha^{(\mathbf{p}_{2},\mathbf{p}_{1})}_{\pi/\omega}$ indicates that
the electron momenta have been swapped in Eq. (28), with
$\Delta\tau=\pi/\omega$. For a monochromatic wave, we have found that the
effect of these terms is minimal [18, 5].

Computing the phase shifts $\alpha_{lu}$ and $\alpha_{rd}$ means that, in
addition to exchanging $\mathbf{p}_{1}$ and $\mathbf{p}_{2}$, we must consider
that one of the interfering events will be displaced by half a cycle [see Fig.
3(c)]. Specifically, we must note that
$S_{u}(\mathbf{p}_{1},\mathbf{p}_{2},t,t^{\prime},t^{\prime\prime})=S_{l}(\mathbf{p}_{2},\mathbf{p}_{1},t+\pi/\omega,t^{\prime}+\pi/\omega,t^{\prime\prime}+\pi/\omega)$
and that
$S_{r}(\mathbf{p}_{1},\mathbf{p}_{2},t,t^{\prime},t^{\prime\prime})=S_{d}(\mathbf{p}_{2},\mathbf{p}_{1},t+\pi/\omega,t^{\prime}+\pi/\omega,t^{\prime\prime}+\pi/\omega)$.
These phase differences will mainly lead to fringes in the second and fourth
quadrant of the $p_{1\parallel}p_{2\parallel}$ plane.

Explicitly, this gives

$\displaystyle\alpha_{lu}=S_{l}-S_{u}$
$\displaystyle=-\alpha^{(\mathrm{exch})}_{\mathbf{p}_{1},\mathbf{p}_{2}}-\frac{1}{2(t^{\prime}-t^{\prime\prime})}\alpha^{(A^{2})}_{\pi/\omega}(t^{\prime},t^{\prime\prime})$
$\displaystyle-\frac{1}{2}\alpha^{(\mathrm{pond})}_{\pi/\omega}(t,t^{\prime\prime})-\alpha^{(\mathrm{ene})}_{\pi/\omega}-\alpha^{(\mathbf{p}_{1}\leftrightarrow\mathbf{p}_{2})}_{\pi/\omega}(t,t^{\prime}),$
(30)

where the first four phase differences are given by Eqs. (19), (24), (25) and
(26). The term

$\displaystyle\alpha^{(\mathbf{p}_{1}\leftrightarrow\mathbf{p}_{2})}_{\Delta\tau}$
$\displaystyle(t,t^{\prime})=\mathbf{p}_{2}\cdot\mathbf{F}_{A}(t^{\prime}+\Delta\tau)+\mathbf{p}_{1}\cdot\mathbf{F}_{A}(t+\Delta\tau)$
$\displaystyle-\mathbf{p}_{2}\cdot\mathbf{F}_{A}(t)-\mathbf{p}_{1}\cdot\mathbf{F}_{A}(t^{\prime}),$
(31)

contains a momentum dependence and the double arrow in the superscript
indicates that the momentum in the second row of Eq. (31) are exchanged with
regard to the temporal arguments $t$ and $t^{\prime}$ of $\mathbf{F}_{A}$. We
have verified that $\alpha_{lu}=-\alpha_{rd}$ regardless of the field shape.
This guarantees symmetry of these contributions in the second and fourth
quadrant of the $p_{1\parallel}p_{2\parallel}$ plane.

Eq. (30) shows that, in addition to
$\alpha^{(\mathrm{exch})}_{\mathbf{p}_{1},\mathbf{p}_{2}}$, which is also
present for $\alpha_{ld}$ and leads to hyperbolic interference fringes, the
phase shift $\alpha_{lu}$ contains several terms which come from the temporal
shifts. The effects of these equations are more difficult to visualize. For
linearly polarized fields, the scalar products in Eq. (31) will reduce to the
products of $p_{2\parallel}$ and $p_{1\parallel}$ with field-dependent terms.
Imposing that
$\alpha^{(\mathbf{p}_{1}\leftrightarrow\mathbf{p}_{2})}_{\Delta\tau}(t,t^{\prime})=0$
will lead to conditions that depend on the field symmetry, although a first
inspection may give the impression that there should be fringes located at the
axes $p_{n\parallel}=0$.

For instance, one may verify that, for fields satisfying the half cycle
symmetry, the condition $\mathbf{F}_{A}(t)=-\mathbf{F}_{A}(t+\pi/\omega)$
leads to $\alpha^{(A^{2})}_{\pi/\omega}(t^{\prime},t^{\prime\prime})=0$ and

$\alpha^{(\mathbf{p}_{1}\leftrightarrow\mathbf{p}_{2})}_{\pi/\omega}(t,t^{\prime})=-(\mathbf{p}_{2}+\mathbf{p}_{1})\cdot\left[\mathbf{F}_{A}(t^{\prime})+\mathbf{F}_{A}(t)\right].$
(32)

For linearly polarized fields satisfying the half-cycle symmetry, the scalar
product stated above becomes
$(p_{2\parallel}+p_{1\parallel})\left[F_{A}(t^{\prime})+F_{A}(t)\right]$, and
the requirement that
$\alpha^{(\mathbf{p}_{1}\leftrightarrow\mathbf{p}_{2})}_{\pi/\omega}$ vanishes
for all times would imply that $p_{2\parallel}=-p_{1\parallel}$.

If, further to that, the field is taken to be monochromatic and linearly
polarized, that is $A(t)=2\sqrt{U_{p}}\cos\omega t$, one can show that

$\displaystyle\alpha^{(\mathrm{mono})}_{lu}$
$\displaystyle=-\frac{\pi}{2\omega}\left[4U_{p}+2E_{1g}+2E_{2g}+\mathbf{p}_{1}^{2}+\mathbf{p}_{2}^{2}\right]$
$\displaystyle-\frac{1}{2}\left(\mathbf{p}_{2}^{2}-\mathbf{p}_{1}^{2}\right)(t-t^{\prime})$
$\displaystyle+\frac{\sqrt{U_{p}}}{\omega}(p_{2\parallel}+p_{1\parallel})\left[\sin\omega
t^{\prime}+\sin\omega t\right],$ (33)

where the extra term in $U_{p}$ comes from
$\alpha^{(\mathrm{pond})}_{\pi/\omega}(t,t^{\prime\prime})=4U_{p}\pi/\omega$
in this case.

Due to several field- and energy-dependent terms in $\alpha_{lu}$, it is not
straightforward to derive simple analytic expressions for constructive
interference between $M_{u}$ and $M_{l}$. There is, however, a term in
$\mathbf{p}^{2}_{2}-\mathbf{p}^{2}_{1}$, which, if required to vanish, leads
to hyperbolic fringes. Interestingly, because, for a few-cycle pulse, the
half-cycle symmetry is broken, we cannot use the argument in [5], in which
imposing that the last term in (33) vanishes leads to a fringe along the anti-
diagonal $p_{1\parallel}=-p_{2\parallel}$.

####  III.2.2 Inter-cycle phase differences

Furthermore, in case we are considering more than one cycle of the field, we
must also assess what happens for contributions displaced by at least a full
cycle ($\Delta\tau\geq 2\pi/\omega$), without and with momentum exchange [see
Figs. 3(b)(ii) and (c)(ii), respectively]. The contributions from events
separated by a full number of cycles overlap considerably in momentum space,
which potentially makes their interference substantial. This is particularly
important in the present scenario, as the pulse envelope renders the cycles
not equivalent.

Shifting the action associated with the left peak in a full cycle gives
$S_{l,T}(\mathbf{p}_{1},\mathbf{p}_{2},t,t^{\prime},t^{\prime\prime})=S_{l}(\mathbf{p}_{1},\mathbf{p}_{2},t+2\pi/\omega,t^{\prime}+2\pi/\omega,t^{\prime\prime}+2\pi/\omega)$,
where we have defined $T=2\pi/\omega$. Similarly, for the downward peak one
obtains
$S_{d,T}(\mathbf{p}_{1},\mathbf{p}_{2},t,t^{\prime},t^{\prime\prime})=S_{d}(\mathbf{p}_{1},\mathbf{p}_{2},t+2\pi/\omega,t^{\prime}+2\pi/\omega,t^{\prime\prime}+2\pi/\omega)$.
The phase difference $\alpha_{lT,l}$ then reads

$\displaystyle\alpha_{lT,l}=S_{l,T}-S_{l}$
$\displaystyle=\frac{1}{2(t^{\prime}-t^{\prime\prime})}\alpha^{(A^{2})}_{2\pi/\omega}(t^{\prime},t^{\prime\prime})$
$\displaystyle+\frac{1}{2}\alpha^{(\mathrm{pond})}_{2\pi/\omega}(t,t^{\prime\prime})+\alpha^{(\mathrm{ene})}_{2\pi/\omega}+\alpha^{(\mathbf{p}_{1},\mathbf{p}_{2})}_{2\pi/\omega}(t,t^{\prime}),$
(34)

where the phase differences have been defined in Eqs. (24),(25), (26) and
(28). The phase difference $\alpha_{dT,d}=S_{d,T}-S_{d}$ resembles Eq. (34),
but with the momenta swapped in the last building block, i.e.,
$\alpha^{(\mathbf{p}_{2},\mathbf{p}_{1})}_{2\pi/\omega}(t,t^{\prime})$. The
phase differences $\alpha_{rT,r}=S_{r,T}-S_{r}$ and
$\alpha_{uT,u}=S_{u,T}-S_{u}$ are calculated similarly, but the arguments of
the functions $F_{A}$ and $F_{A^{2}}$ must be shifted in half a cycle with
regard to those used in Eq. (34). This implies that one must take
$\alpha^{(A^{2})}_{2\pi/\omega}(t^{\prime}+\pi/\omega,t^{\prime\prime}+\pi/\omega)$,
$\alpha^{(\mathrm{pond})}_{2\pi/\omega}(t+\pi/\omega,t^{\prime\prime}+\pi/\omega)$
and
$\alpha^{(\mathbf{p}_{i},\mathbf{p}_{j})}_{2\pi/\omega}(t+\pi/\omega,t^{\prime}+\pi/\omega)$
in the above equation.

For fields of period $T=2\pi/\omega$, $\alpha^{(A^{2})}_{2\pi/\omega}$ and
$\alpha^{(\mathbf{p}_{i},\mathbf{p}_{j})}_{2\pi/\omega}$, with $i\neq j$, will
vanish, while $\alpha^{(\mathrm{pond})}_{2\pi/\omega}$ will give ponderomotive
shifts which will add to $\alpha^{(\mathrm{ene})}_{2\pi/\omega}$. This will
happen regardless of whether the half-cycle symmetry is broken, but will not
apply to the pulse studied in this work. An example of this specific scenario
is a monochromatic field, for which

$\alpha^{\mathrm{mono}}_{lT,l}=\alpha^{\mathrm{mono}}_{dT,d}=\frac{\pi}{\omega}\left[4U_{p}+2E_{1g}+2E_{2g}+\mathbf{p}_{1}^{2}+\mathbf{p}_{2}^{2}\right]$
(35)

is the equation of a six-dimensional hypersphere in momentum space. Re-writing
the condition for interference maxima $\alpha^{\mathrm{mono}}_{lT,l}=2n\pi$ in
terms of the momentum components parallel to the driving-field polarization
yields

$p_{1\parallel}^{2}+p_{2\parallel}^{2}=2n\omega-(4U_{p}+2E_{1g}+2E_{2g}+p^{2}_{1\perp}+p^{2}_{2\perp}),$
(36)

which gives circular fringes if the integer $n$ is sufficiently large so that
the right-hand side is positive. Interference fringes taking into account
electron exchange and events displaced by a full cycle can be obtained by
considering, for instance,
$\alpha_{lT,d}=S_{l,T}-S_{d}=\alpha_{lT,l}+\alpha_{ld}$, or
$\alpha_{uT,l}=S_{u,T}-S_{l}=\alpha_{uT,u}+\alpha_{ul}$. The diagrams in Fig.
3 are useful as a guidance to construct phase shifts associated with events
displaced by even or odd numbers of half cycles. Alternatively, if all
ionization and rescattering times are computed directly, they may be used to
construct these shifts.

One should note that the time shifts employed in this section are an
approximation for the times calculated for different events in few-cycle laser
pulses. However, we verified that this works reasonably well for the dominant
events (not shown). Understanding these events and their dominance will be the
main topic of the next section.

##  IV Investigating the pulse

In the results that follow, we employ a linearly polarized few-cycle pulse
$\mathbf{E}(t)=-d\mathbf{A}(t)/dt$, whose vector potential is determined by

$\mathbf{A}(t)=2A_{0}\sin^{2}\left(\frac{\omega t}{2N}\right)\sin{(\omega
t+\phi)}\hat{e}_{z},$ (37)

where $A_{0}$ is the vector-potential amplitude, $N$ is the number of cycles,
$\omega$ is the field frequency, $\phi$ the carrier-envelope phase. Throughout
this work, we have made the approximation $A_{0}=2\sqrt{U_{p}}$. This is exact
for a monochromatic linearly polarized field, but not for a few cycle pulse.
By varying the parameters in (37), we can influence the main rescattering and
ionization events.

In Fig. 4, we plot the specific $\mathrm{sin}^{2}$ pulse employed in this
work. Its length ($N=4.3$) and carrier-envelope phases (CEPs) have been chosen
in order to facilitate a comparison with our previous publication [38],
although we have taken a lower peak intensity to ensure that RESI is the
dominant mechanism. This means that the maximal kinetic energy of the
returning first electron is lower than the second ionization potential.
Overall we observe that $E(t,\phi)=-E(t,\phi+\pi)$. This symmetry will
influence the electron momentum distributions so that
$\mathcal{P}(p_{1\parallel},p_{2\parallel},\phi)=\mathcal{P}(-p_{1\parallel},-p_{2\parallel},\phi+\pi)$.
Thus, without loss of generality, we may consider only two of the CEPs
employed in [38]. Here, we follow the same convention as in our previous
publication [38] and define the carrier-envelope phases as
$\phi=\phi_{1}-\phi_{0}$, where $\phi_{0}=60^{\circ}$ is an offset value. When
discussing the CEP in the following figures, we refer to $\phi_{1}$ without
the offset phase.

The classical counterparts $\mathrm{Re}[t^{\prime\prime}]$,
$\mathrm{Re}[t^{\prime}]$ and $\mathrm{Re}[t]$ of the ionization and
rescattering times computed with Eqs. (7)-(10) are indicated in the figure for
both electrons, where the orbits are classified in increasing numerical order
starting from the beginning of the pulse. This classification has been used in
[38]. The orbits associated with the first electron occur in pairs [37, 38]
and, for that reason, are indicated by $p_{i}$. They start after a maximum of
the field and end near a zero crossing roughly three quarters of a cycle later
(see arrows in the figure). Each pair is composed of a short and a long orbit,
whose classical ionization and times read
$(\mathrm{Re}[t^{\prime}_{s}],\mathrm{Re}[t^{\prime\prime}_{s}])$ and
$((\mathrm{Re}[t^{\prime}_{l}],\mathrm{Re}[t^{\prime\prime}_{l}])$,
respectively. The classical times shown in the figure have been estimated
using the tangent construction [61].

The most relevant orbits associated with the second electron start around the
field peak closest to the rescattering times, and are denoted as $o_{j}$. An
event encompassing the two electrons is classified as
$\varepsilon_{k}=p_{i}o_{j}$, where $i,j,k$ are integers. For example,
$\varepsilon_{1}=p_{3}o_{4}$ designates the first event in the pulse we
incorporate, in which the first electron returns following the pair or orbits
$p_{3}$ and the second electron is released along the orbit $o_{4}$ (see Fig.
4 for clarity). Note that, in this work, we only consider the ionization event
$o_{j}$ for the second electron immediately following the rescattering of the
first electron, as, due to bound-state depletion, the probability of later
events is smaller and therefore less important.

Figure 4: Few-cycle pulse given by (37), with peak intensity $I=1.5\times
10^{14}\mathrm{W/cm}^{2}$, wavelength $\lambda=800$ nm ($\omega=0.057$ a.u.),
$N=4.3$ and carrier-envelope phases
$\phi_{1}=65^{\circ},155^{\circ},245^{\circ},335^{\circ}$ [panels (a), (b),
(c) and (d), respectively]. The ionization and rescattering times at the peak
and zero crossings of the field are indicated by the red and green dots. The
three main events $p_{i}o_{j}$ towards the center of the pulse are labeled
with their corresponding pair and orbit numbers. The ionization and return
times associated with the pairs of orbits $p_{3}$, $p_{4}$, $p_{5}$ of the
first electron are indicated by arrows, and the most relevant ionization times
for the orbits $o_{j}$ of the second electron, with $(j=4,5,6)$, are
signposted by rectangles. The initial numbers chosen for the indices $i,j$
refer to the extremum of the field for which the counting starts. Matching
styles and colors have been used for different events
$\varepsilon_{k}=p_{i}o_{j}$.

###  IV.1 Finding dominant orbits

A central question when dealing with pulses is how to find the events whose
contributions to the single-channel probability density
$\mathcal{P}^{(\mathcal{C})}(p_{1\parallel},p_{2\parallel})$ are dominant.
This provides a roadmap of what can be neglected or incorporated without
compromising the physics. In previous publications [37, 38], we have performed
qualitative studies of this issue. Here, we aim to quantify the dominance of
an event within a single excitation channel by proposing a dominance
parameter. An orbit’s contribution will prevail in
$\mathcal{P}^{(\mathcal{C})}(p_{1\parallel},p_{2\parallel})$ if three
conditions are satisfied.

First, the ionization probability for the first electron must be high [15,
38]. For a specific orbit, the ionization probability is proportional to
$\exp[-2\mathrm{Im}S^{(\mathcal{C})}(\mathbf{p}_{1},\mathbf{p}_{2},\mathbf{k},t,t^{\prime},t^{\prime\prime})]$.
In order to consider the first electron individually, one must note that the
action will be proportional to the ionization times. Thus,
$\exp[-2\mathrm{Im}[t^{\prime\prime}]]$ is a good indicator of whether
ionization is significant for the first electron along a specific orbit [see
also the saddle-point equation (7]. A small imaginary part for
$t^{\prime\prime}$ means a high ionization probability. For simplicity, one
may consider $|1/\mathrm{Im}[t^{\prime\prime}]|$.

Second, the classically allowed region (CAR) for the (returning) first
electron must be large111The classically allowed region can be identified by
nearly vanishing $\mathrm{Im}[t^{\prime}]$. At the boundary of this region,
there is a sudden increase in $|\mathrm{Im}[t^{\prime}]|$ and
$\mathrm{Re}[t^{\prime}]$ nearly coalesce. For details see [38]. . This means
that the first electron has returned with enough kinetic energy for the
process to have a classical counterpart over a large momentum range.
Therefore, according to Eq. (9), there is a substantial energy transfer to the
core and the excitation of the second electron is very likely. We take a rough
estimate of the width of the region in momentum space, where the difference
between imaginary parts of the rescattering times $t^{\prime}$ for the long
and short orbits is minimal. The width is found by taking the difference
between the lowest and highest classically allowed momenta
$p^{(\mathrm{min})}_{1\parallel}$ and $p^{(\mathrm{max})}_{1\parallel}$ for
$p_{1\perp}=0$. A more precise value of these momenta can be found by solving
the saddle-point equation (9) for $p_{1\perp}=0$. However, we found that this
added precision did not affect the dominance parameter value significantly so
it is sufficient to take an estimate. On the other hand, it may happen that
the complex times obtained from solving the saddle-point equations have no
classical counterpart and this region collapses. In that case, we take the
most probable momentum. This momentum is chosen so that
$|\mathrm{Im}[t^{\prime}]|$ has a minimum. Physically, this implies that
rescattering is most probable at that time, although it has no classical
equivalent [15, 38]. Because we are taking the momentum for which rescattering
is most probable, this is likely an overestimation of the actual probability
for this process.

Third, there must be a high ionization probability for the second electron.
This is inferred similarly to the first condition, with the difference that,
instead, we take $|1/\mathrm{Im}[t]|$ and the saddle-point equation (10). One
should also prioritize the first ionization event after recollision, as those
in subsequent half-cycles will be suppressed by bound-state depletion.

Putting all these criteria together, we can define the dominance parameter as
a product of the partial dominance parameters for the rescattered and direct
electrons

$\mathcal{D}(p_{i},o_{j})=\mathcal{D}^{(1)}(p_{i})\mathcal{D}^{(2)}(o_{j}),$
(38)

where the indices $p_{i},o_{j}$ refer to specific events (classified as pairs
and orbits, respectively - see Fig 4) and

$\mathcal{D}^{(1)}(p_{i})=\frac{|\mathrm{Im}[\bar{t}_{p_{i}}^{\prime\prime}]|^{-1}||p^{(\mathrm{max})}_{1\parallel}-p^{(\mathrm{min})}_{1\parallel}|}{\mathcal{D}^{(\mathrm{e_{1}})}_{\mathrm{max}}}$
(39)
$\mathcal{D}^{(2)}(o_{j})=\frac{|\mathrm{Im}[t_{o_{j}}]^{(min)}|^{-1}}{\mathcal{D}^{(\mathrm{e_{2}})}_{\mathrm{max}}}$
(40)

The denominator $\mathcal{D}^{(e_{n})}_{(\mathrm{max})}$, $n=1,2$, is the
maximum dominance parameter for the corresponding electron, for the channel in
question, which we have used as a normalization. In the partial dominance
parameter $\mathcal{D}^{(1)}(p_{i})$ for the first electron, as the orbits
occur in pairs, we have considered the average of
$\mathrm{Im}[t_{\varepsilon_{1}}^{\prime\prime}]$ within a pair $p_{i}$ of
orbits at vanishing parallel and perpendicular momenta. Vanishing
perpendicular momenta are used to obtain the largest possible classically
allowed region as non-zero values simply shift the ionization potential [18].
For the second electron, we have taken the minimum of the imaginary part of
the tunneling time as this will lead to the greatest probability.

Figure 5: Estimates of the partial dominance parameter
$\mathcal{D}^{(1)}(p_{j})$ (top row), the real part of the rescattering time
calculated for $p_{1\perp}=0$(middle row) and the partial momentum
distribution [Eq. (17)] calculated for the first electron as a function of the
parallel momentum without prefactors (bottom row) for three events in the
center of a pulse with CEP $\phi=65$, for the $3p\rightarrow 4p$ (panels
(a)-(c)), $3p\rightarrow 4d$ (panels (d)-(f)), $3p\rightarrow 4s$ (panels
(g)-(i)) and $3s\rightarrow 3p$ (panels (j)-(l)) transitions. The values of
the dominance parameter and the momentum map have been normalized with respect
to the most dominant event for each channel.

We will now discuss the dominance parameter for each electron separately. Fig.
5 shows the dominance parameter for the first electron and four of the six
excitation channels, together with the real parts of the ionization times and
the partial momentum distributions for the first electron (first, second, and
third row, respectively). The $3p\rightarrow 5s$ and $3p\rightarrow 3d$
transitions have been omitted in Fig. 5 because they strongly resemble (are
almost identical to) the $3p\rightarrow 4d$ and $3p\rightarrow 4s$
transitions, given their very similar energy gaps. This renders the analysis
redundant unless prefactors are incorporated. Indeed, we have verified that
the partial distributions and dominance of pairs are almost identical.

For all channels, the least dominant contribution comes from $p_{5}$ arising
from the rescattering of the first electron towards the edge of the pulse.
This can be seen from the real parts of the rescattering times, displayed in
the middle row of Fig. 5. The behavior of these times provides a good
illustration of whether there is or not a classically allowed region (CAR). If
a classically allowed region is present, $\mathrm{Re}[t^{\prime}]$ will
surround the momentum value for which rescattering is most probable, and
nearly coalesce at two values of $p_{1\parallel}$ which mark the minimal and
maximal momentum for which rescattering has a classical counterpart. If there
is no classical counterpart, $\mathrm{Re}[t^{\prime}]$ associated with a
particular pair of orbits will be practically the same [62]. For
$3s\rightarrow 3p$, we see a classically allowed region for $p_{5}$ owing to
the relatively small energy gap $\Delta E^{(\mathcal{C})}$ [Fig. 5(b)].

For the $3p\rightarrow 4d$, $3p\rightarrow 4s$ and $3p\rightarrow 4p$
transitions the energy gap between the ground and excited state of the second
electron is larger, and the classically allowed region collapses. This is
reflected in the value of the dominance parameter [Figs. 5(d), (g), (j)],
which is lower than when a classically allowed region exists. For Fig. 5(g),
pair 5 has the lowest dominance parameter. This happens because the
classically allowed region is substantial for the other pairs [see Fig. 5(h)],
which brings it down due to the normalization employed. Still, one should note
that Fig. 5(i) exhibits a faint blue peak associated with $p_{5}$, while the
probability density associated with this event is vanishingly small in Fig.
5(f). As the energy gap between the ground and excited states in Table 1
increases, the residual kinetic energy for the first electron, and,
consequently the CAR, will decrease. This will bring the dominance parameter
associated with pair 3 slightly up. We do not take pair 5 into account in the
final momentum maps if the CAR collapses.

The contributions from $p_{3}$ and $p_{4}$ appear to be strongly competing for
all channels studied - this is also indicated by comparable widths of the
classically allowed regions for all channels. This can be intuitively
understood by looking at Fig. 4(a), as these are the two events closest to the
peak of the pulse. The ratio of the dominance parameter values for $p_{3}$ and
$p_{4}$ is directly proportional to the difference in width of the classically
allowed regions for these events.

However, the partial momentum distributions $M^{(1)}(p_{1\parallel})$ for the
three events, displayed in the bottom row of Fig. 5, show that, in reality,
the contributions from $p_{4}$ prevail in all channels. This agrees with Fig.
4(a), which shows that the local extremum associated with $p_{4}$ leads to the
largest absolute value of the electric field $E(t)$ and intuitively translates
into a large ionization probability for the first electron. However,
$\mathrm{Im}[t^{\prime\prime}]$ is very similar for both $p_{3}$ and $p_{4}$,
and this leads to competing dominance parameters. On the other hand, the
partial distributions illustrate the collapse of the classically allowed
region for $p_{5}$ accurately. This region is present in Fig. 5(b), which
leads to a small, but visible peak in Fig. 5(c) associated with $p_{5}$. In
Fig. 5(i) the peak is suppressed as the CAR collapses [see Fig. 5(h)].
Furthermore, the peaks of the distributions are roughly centered at the most
probable momenta, in agreement with [38].

The results discussed above indicate that the dominance parameter is an
oversimplification of the actual contributions. However, it provides a good
indication of which events to exclude in the final momentum distributions.
When the dominance parameter is less than $0.4$, we can neglect the
contribution of that event. However, when the value is greater than $0.9$, it
is difficult to ascertain which pair is most dominant.

Figure 6: Estimates of the dominance parameter $\mathcal{D}^{(2)}(o_{j})$ for
the second electron using the minimum imaginary ionization time $t$ (top row),
the dominance parameter as a function of the parallel momentum of the second
electron (second row), the imaginary part of the ionization time calculated
for $p_{2\perp}=0$ (third row) and the partial momentum distribution as a
function of the parallel momentum without prefactors (bottom row) for three
events in the center of a pulse with CEP $\phi=65$, for the $3p\rightarrow 3p$
(panels (a)-(d)) and $3p\rightarrow 4p$ (panels (e)-(h)) transitions. The
values of the dominance parameters and the momentum map have been normalized
to the most dominant event for each channel.

Next, we will consider the dominance parameter for the second electron.
Estimates of this parameter are shown in Fig. 6 for the $3s\rightarrow 3p$
[panel (a)] and $3p\rightarrow 4d$ [panel (e)] transitions. In both cases, we
see that $o_{6}$ is the least dominant orbit, followed by $o_{5}$ then
$o_{4}$. Physically, this agrees with Fig. 4(a) which shows that $o_{6}$ lies
towards the edge of the pulse whilst $o_{4}$ is at the center. It is
noteworthy that the ratio between $o_{4}$ and $o_{5}$ values is larger than
for the corresponding events for the first electron, with values for $o_{5}$
around 0.8. This implies that these events are not competing, and we can state
that $o_{4}$ has a higher tunneling probability than $o_{5}$. This was not the
case for $p_{3}$ and $p_{4}$. Because we are taking the minimum imaginary time
$t$ to calculate the dominance parameter, we find that the plots look almost
identical for both excitation pathways, despite the difference in their
binding energies.

Nonetheless, upon closer inspection of how $1/\mathrm{Im}[t]$ normalized
concerning the maximum value varies with $p_{2\parallel}$ for each of these
orbits [Fig. 6(b)], we can see that, for $o_{4}$, it is centered and almost
completely symmetric about the origin $p_{2\parallel}=0$. This is expected as
$o_{4}$ is located near the center of the pulse. However, for $o_{5}$ and
$o_{6}$, the maxima deviate from the origin, explained by the asymmetry of the
pulse. More importantly, we note that the gradient is not the same on either
side of the maxima. In the negative (positive) momentum region, the normalized
values of $1/\mathrm{Im}[t]$ approach each other for $o_{5}$ and $o_{6}$
($o_{4}$ and $o_{5}$), with a large deviation in the positive (negative)
region. This implies that the dominance of orbits is momentum dependent and a
single value, as in Figs. 6(a) and (e), is not enough to capture the entire
physics. Therefore the dominance parameter must be used as an indicator of
which events to exclude from the momentum map rather than a way to find the
location of the brightest primary maxima.

The asymmetry effect is much stronger for the $3p\rightarrow 4d$ pathway [Fig.
6(f)] than for $3s\rightarrow 3p$ [Fig. 6(b)], because the second electron
tunnels from a much more loosely bound state, leading to a steeper gradient.
Thus, the imaginary parts of the tunneling times will follow each other more
closely once they move away from the origin [see Fig. 6(g) in comparison to
(c)]. This broadens the dominant partial momentum distribution
$M^{(2)}(p_{2\parallel})$ from $o_{4}$, and enhances the remaining
contributions, as expected from a narrower potential barrier. Interestingly,
practically identical dominance parameters lead to different distributions.
Regardless, the estimates arising from the dominance parameter give a clear
hierarchy of which events contribute the most.

Figure 7: Estimates of the total dominance parameter
$\mathcal{D}(\varepsilon_{1},\varepsilon_{2})$ for three events in the center
of a pulse with CEP $\phi_{1}=65^{\circ}$ [panel (a)] and
$\phi_{1}=155^{\circ}$ [panel (b)] for the $3p\rightarrow 4p$, $3p\rightarrow
4d$, $3p\rightarrow 4s$ and $3s\rightarrow 3p$ transitions. For each pair
$p_{3}$, $p_{4}$ and $p_{5}$, only the ionization event immediately after has
been taken. The incoherent-incoherent momentum maps without prefactors have
been calculated for each channel and are displayed in panels (c)-(f) for
$\phi_{1}=65^{\circ}$ and panels (g)-(j) for $\phi_{1}=155^{\circ}$. The axes
are indicated with white dashed lines. The values of the dominance parameters
and the momentum map have been normalized with respect to the most dominant
event for each channel.

In Fig. 7, we inspect the combined estimates for both electrons (note that we
consider degenerate energies $E_{2g}$ for $3s$, $3p$ here). Figs. 7(a) and (b)
show the total dominance parameter for the channels discussed previously, for
CEP $\phi=65^{\circ}$ and $\phi=155^{\circ}$, respectively. For
$\phi=65^{\circ}$, according to Fig. 7(a) the $p_{3}o_{4}$ event is prevalent,
followed by $p_{4}o_{5}$ and $p_{5}o_{6}$ for all channels. This happens
because, for $p_{3}o_{4}$, the second electron is freed near a very prominent
field maximum [see red rectangle in Fig. 4(a)] bringing up the dominance of
this event significantly. For $p_{5}o_{6}$, the ionization time for the second
electron is too close to the end of the pulse for it to be as dominant as the
other two events, decreasing this value. The dominance of this event is larger
for $3s\rightarrow 3p$, as the collapse of the CAR suppresses this event for
the other excitation pathways. The second electron brings down the value for
the $p_{4}o_{5}$ event for all channels too. However, there are also
differences due to the excited states’ binding energies. The dominance of
$p_{4}o_{5}$ for the $3p\rightarrow 4d$ pathway is smaller than that of the
other channels, due to a smaller CAR for this event.

When the CEP is increased to $\phi_{1}=155^{\circ}$ [Fig. 7(b)], the dominance
parameter indicates that, for most excitation channels, $p_{4}o_{5}$ prevails,
followed by $p_{3}o_{4}$. Furthermore, the values obtained for the least
dominant event $p_{5}o_{6}$ are higher than for $\phi_{1}=65^{\circ}$.
Physically, this may be understood by inspecting Fig. 4(b), which shows that
$p_{4}o_{5}$ has moved towards the center of the pulse, with an increase in
the field amplitude for the second electron’s ionization time (see black
rectangle therein). Although $p_{3}$ has moved further to the left edge of the
pulse, $o_{4}$ is still towards the center, bringing up
$\mathcal{D}(p_{3},o_{4})$. Meanwhile, $p_{5}o_{6}$ has a larger field
amplitude for both electrons’ ionization times. For this phase, we also have a
classically allowed region for the $p_{5}o_{6}$ event for all channels except
$3p\rightarrow 4d$ (which is the most loosely bound), explaining the increase
in the relative size of the $p_{5}o_{6}$ dominance parameter.

To assess the validity of the dominance parameter estimates, we look at the
correlated two-electron momentum distributions without any prefactors [Figs.
7(c)-(f) and (g)-(j)]. This will give a good idea of the key variables used to
construct this parameter without introducing further biases, leading to
distributions whose shapes are roughly described by the constraints in Fig. 1.
For $\phi_{1}=65^{\circ}$, an overall feature is that the agreement is better
for more deeply bound excited states. Concretely, the distribution for the
$3s\rightarrow 3p$ pathway, shown in Fig. 7(c), agrees with the prediction of
the dominance parameter. The distribution is brightest in the lower left
quadrant, associated with contributions from $p_{3}o_{4}$ and $p_{5}o_{6}$.
The strong tail in the positive momentum region comes from the $p_{4}o_{5}$
event. Similarly, the $3p\rightarrow 4s$ transition populates both negative
and positive momentum regions [see the probability density displayed in Fig.
7(e)]. Since the CAR collapses for this region, the loss of relevance of
$p_{5}$ and $o_{5}$ becoming more prominent leads to distributions almost
equally occupying the four quadrants of $p_{1\parallel},p_{2\parallel}$ plane,
although they are still slightly brighter in the lower left quadrant.

For the $3p\rightarrow 4p$ transition [Fig. 7(f)], we see a similar
distribution. However, now the brightest intensity has shifted to the positive
momentum region despite $p_{3}o_{4}$ predicted to be the most dominant. This
shift is even more pronounced for $3p\rightarrow 3d$ distribution, depicted in
Fig. 7(d), for which the intensity is more strongly located in the top right
quadrant, despite the strong tail in the negative momentum region. These
shifts can be accounted for by the gradient arising from the $o_{5}$ event not
being taken into account by the dominance parameter. As discussed previously
this effect is most strongly observed for smaller binding energies. Further
evidence of a strongly asymmetric gradient is the fact that the maxima of the
move further and further away from the axes $p_{n\parallel}=0$ as the binding
energy decreases. From these distributions, a main limitation of the dominance
parameter becomes evident: a single value is not sufficient to explain the
dominance of events in a few-cycle pulse, given that the variables used to
construct it are strongly dependent on the momentum region. For instance, the
dominance parameter $\mathcal{D}^{(2)}(o_{n})$ for the second electron
considers only the minimum $\mathrm{Im}[t]$, which, for the SFA, is around
$p_{2\parallel}=0$, but this quantity varies drastically away from this value.
Furthermore, the threshold for when orbits become ‘competing’ or large enough
to contribute is fairly arbitrary for the combined dominance parameter. When
the value is less than 0.2, we can safely omit the pair from the distribution.
If the value is greater than 0.6, the events are competing but due to how
sensitive the parameter is with respect to the estimates of the width of the
CAR, the parameter is prone to over or under-estimation.

For $\phi_{1}=155^{\circ}$ [bottom panels in Fig. 7], the top-right quadrant
of the momentum plane is occupied for all channels arising from the dominance
of the $p_{4}o_{5}$ event. In all cases, we see a tail in the negative
momentum region arising from the $p_{3}o_{4}$ event. As the binding energy
increases, this tail becomes longer and more localized along the axes, so that
the entire distribution becomes more cross-shaped. However, because, for a
few-cycle pulse, the gradient of $\mathrm{Im}[t]$ is not symmetric around its
minima, as it would be for a monochromatic field, there will never be a
perfect symmetry around the axes. With decreasing binding energy, the
gradients of $\mathrm{Im}[t]$ as functions of $p_{2\parallel}$ become steeper
and we see the maxima of the distribution deviate away from the axes. It
should be noted, however, that the parameter does not consider, at least
explicitly, the brightness of the distributions.

###  IV.2 Prefactor effects and mapping

The shape of the momentum distributions in Fig. 7(c)-(j) is altered by the
ionization ($V_{\mathbf{p}_{2}e}$) and excitation ($V_{\mathbf{p}_{1}e,kg}$)
prefactors, whose expressions are given in the appendix. For full derivations
see [18, 5]. The prefactors introduce additional momentum biases which will
modify the length, width and maxima of the correlated distributions. They also
bring additional phase shifts to those discussed in Sec. III.

The six channels investigated in this study (Table 1) involve the excitation
of the second electron to states of very different spatial geometry. This
geometry is reflected in the prefactor shapes via the radial and angular
nodes. However, these nodes do not directly map onto the two-electron momentum
distributions. In this section we perform a more detailed analysis on the
mapping of the $V_{\mathbf{p}_{2}e}$ and $V_{\mathbf{p}_{1}e,kg}$ prefactors
into the $p_{1\parallel}p_{2\parallel}$ plane to understand how the maxima and
nodes translate to the momentum distributions explicitly.

First, we will consider the ionization prefactor $V_{\mathbf{p}_{2}e}$,
displayed in Fig. 8(a)-(f) for the six channels. This prefactor is the most
important in determining the shapes of the electron momentum distributions,
and has specific nodes. There are $l_{e}$ angular nodes associated with the
spherical harmonics in the wave function
$\psi_{n_{e}l_{e}m_{e}}(\mathbf{r}_{2})=R_{n_{e}l_{e}}(r_{2})Y_{l_{e}}^{m_{e}}(\theta_{2},\phi_{2})$
of the second electron, where the subscripts $e$ indicate that it belongs to
an excited state. The figure shows that, for $s$ excited states [Figs. 8(d)
and (f)], there are no angular nodes for the prefactor, as expected. For $p$
states [Figs. 8(a) and (e)], there is a single angular node and a strong
suppression along the $p_{2\perp}$ axis, which leads to a phase shift with
regard to $p_{2\parallel}\rightarrow-p_{2\parallel}$. Finally, for $d$ states
[Figs. 8(b) and (c)], there are two nodes that cross each other at the origin
approximately at angles $\pm 55^{\circ}$. Clear positive and negative regions
can be seen from the imaginary part of the prefactor, corresponding to phase
shifts of $\pi$.

Further to angular nodes, there is an $n_{e}-l_{e}-1$ number of radial nodes
which manifest as circles in the $p_{2\parallel}p_{2\perp}$ plane and a phase
jump of $\pi$ if a radial node is crossed. The relevant radial nodes for our
problem must be located at energies lower than that of the direct ATI cutoff,
indicated by the white circle at
$p^{2}_{2\parallel}+\mathbf{p}^{2}_{2\perp}=4U_{p}$. Changes in the shapes of
the momentum distributions will be caused by nodes within the direct ATI
cutoff. The greater the number of nodes, the more phase shifts are present.

The mapping to the $p_{1\parallel}p_{2\parallel}$ plane is performed by
integrating $|V_{p_{2e}}|^{2}$ over the plane spanned by the momenta
perpendicular to the laser-field polarization. We account for particle
exchange leading to four-fold symmetry in the plots. Explicitly, we consider

$|V_{p_{2\parallel e}}|^{2}=\iint
dp_{1\perp}dp_{2\perp}\left|p_{1\perp}\right|\left|p_{2\perp}\right|\left(\left|V_{\mathbf{p}_{2e}}\left(\mathbf{p}_{1}\right)\right|^{2}+\left|V_{\mathbf{p}_{2e}}\left(\mathbf{p}_{2}\right)\right|^{2}\right).$
(41)

A similar mapping can be done for the excitation prefactor, resulting in
$|V_{p_{1\parallel e,kg}}|^{2}$ with the difference that the integrand in (41)
is replaced by $V_{\mathbf{p}_{1}e,\mathbf{k}g}(\mathbf{p}_{1})$ and its
symmetrized version taking $\mathbf{p}_{1}\leftrightarrow\mathbf{p}_{2}$.

The outcomes of these integrals are displayed in the bottom row of Fig. 8.
From the mappings of Figs. 8(g)-(i), we see that, in agreement with [5], $s$
states lead to cross-shaped distributions in momentum space, with maxima at
the origin and along the axes [Figs. 8(j) and (l)]. However, their widths will
depend on the number and energy positions of the radial nodes. Figs. 8(g) and
(k) show that, for $p$ states, the strong suppression along $p_{2\parallel}=0$
survives the transverse momentum integration. For these states there are
pronounced maxima along all diagonals $p_{1\parallel}=\pm p_{2\parallel}$,
between $\pm 2\sqrt{U_{p}}$ and $4\sqrt{U_{p}}$. In the region of interest
(i.e. within the ATI cutoff), there are maxima at around $\sqrt{U_{p}}$ and
$\sqrt{U_{p}}/2$ for these states respectively. For $d$ states, the x-shaped
nodes intersecting at the origin are mapped to maxima along the axes with the
brightest point at the origin similar to the $s$ states. However, unlike the
$s$ states, there are additional secondary k maxima parallel to the axes as
well. For every orbital angular momentum, any additional radial node within
the ATI cut-off leads to narrower distributions. This effect is particularly
prominent for highly excited states with small binding energies. For example,
Fig. 8(k) corresponding to a $5s$ transition, shows a narrower cross compared
to Fig. 8(j) with a $4s$ transition.

Figure 8: The imaginary part of the ionization prefactor $V_{\mathbf{p}_{2}e}$
for the second electron, as a function of the momentum components
$p_{2\parallel},p_{2\perp}$ (upper panels), and corresponding mapping into the
$p_{1\parallel},p_{2\parallel}$ plane. From left to right, the panels refer to
channel 1 (panels (a) and (g)), channel 2 (panels (b) and (h)), channel 3
(panels (c) and (i)), channel 4 (panels (d) and (j)), channel 5 (panels (e)
and (k)), channel 6 (panels (f) and (l)). The white circle shows the direct
ATI cut-off. Figure 9: Absolute values of the excitation contribution
($V_{ec}$) accounting for the excitation prefactors $V_{\mathbf{p}_{1}ekg}$
for the first electron (panels (a)-(c)) and its corresponding mapping into the
$p_{1\parallel}p_{2\parallel}$ plane (panels (d)-(h)) for three different
pairs $p_{3},p_{4},p_{5}$ respectively. The ionization and rescattering times
associated with the short orbit were used to compute these plots. Panels
(a)-(f) are associated with channel 3, whilst (g) and (h) show mappings for
channels 4 and 5 with $p_{4}$ as this is most dominant for
$\phi_{1}=65^{\circ}$. The mapping axes have been transposed relative to the
other figures to make a clearer connection with the location of the events in
the pulse (upper or lower half). The orbital angular momenta L indicated in
panels (e), (g), and (h) are obtained from the sums over $l_{g}$ and $l_{e}$
given in Eq. (42) in the first Appendix.

Next, we consider the excitation prefactor. From the explicit expression of
$V_{\mathbf{p}_{1}ekg}$ given in the appendix, we can see that the shape of
this prefactor is not trivial given that
$\mathbf{q}=\mathbf{p}_{1}-\mathbf{k}$ where $\mathbf{k}$ has an implicit
dependence on the times $t^{\prime}$ and $t^{\prime\prime}$ [see Eq. (8)].
This makes this prefactor orbit dependent, giving very different shapes for
the long and short orbits. These pairs of orbits are incorporated collectively
in a uniform saddle-point approximation, as they nearly coalesce at the
boundaries of the classically allowed region [59]. The nodes of this prefactor
are mostly washed out, even before integration over transverse momentum is
performed [18, 5]. However, $V_{\mathbf{p}_{1}ekg}$ moves the centers of the
electron momentum distributions, and care must be taken if the driving field
is a pulse. The contribution of this prefactor to the overall momentum
distribution can be better understood by looking at how it changes the
transition amplitude, when incorporated in the uniform approximation. This
contribution (referred to as ‘excitation contribution’ ($V_{ec}$) henceforth)
is obtained by subtracting the amplitude calculated using the coefficient with
the prefactor from the amplitude calculated using the coefficient without the
prefactor. The mapping is done as in Eq. (41).

In $V_{ce}$, any radial or angular nodes from the excitation prefactor are
washed out with insignificant differences between different excitation
pathways. Instead, the prefactor serves to cause a slight shift of the
transition amplitude up or down, depending on the location of the event in the
pulse. Fig. 9(a)-(c) shows $V_{ec}$ for three different pairs, for the
$3p\rightarrow 4d$ pathway. The locations of these rings are pulse-dependent,
with $p_{3}$ and $p_{5}$ located in the region with negative $p_{2\parallel}$
and $p_{4}$ in the positive $p_{2\parallel}$ region. This is implied by Fig.
4(a) for this phase. The size of the rings is also dependent on the width of
the CAR for each pair and agrees with Fig. 5(e). The mappings for each pair in
Figs. 9(d)-(f) are therefore also pulse-dependent, with sizes proportional to
the width of the CAR. This can be explicitly seen in Figs. 9(g) and (h), which
show the mapping of the excitation contribution with $p_{4}$, the most
dominant pair, for two more excitation channels, whose energy gaps are smaller
than that considered in Fig. 9(e). The shapes are almost identical, but occupy
broader regions in momentum space. This indicates the excitation prefactor has
little impact on the shapes of the final momentum distributions, other than
through narrowing them - the amount it narrows is proportional to the binding
energy.

##  V Photoelectron momentum distributions

In this section, we will focus on the correlated electron momentum
distributions for each of the channels in Table 1 and the few-cycle pulse
given by Eq. (37), bringing together the dominant events, the different types
of intra-channel interference and the phase shifts and momentum biases
associated with prefactors. We will start by discussing the main features
[Sec. V.1], and then delve deeper into quantum interference [Secs. V.2, V.3
and V.4]. Guided by the results from Sec. IV.1, we take the three most
dominant (combined) events for both CEP $65^{\circ}$ and CEPs $155^{\circ}$.
Although the event $p_{5}o_{6}$ does not have a CAR for most channels computed
with CEP $65^{\circ}$, it has been included in this investigation for
completeness and contributes very little to the total momentum distribution.

###  V.1 Prefactors, dominance and quantum interference

Figs. 10 and 11 exhibit the fully incoherent [Eq. (14)] and coherent [Eq.
(13)] single-channel sums for CEPs $65^{\circ}$ and $155^{\circ}$,
respectively. The incoherent sums, plotted in panels (a)-(f) of both figures,
exhibit the symmetry about the diagonal $p_{1\parallel}=p_{2\parallel}$,
although the four-fold symmetry associated with monochromatic fields is
absent. Furthermore, they display the expected shapes, depending on the second
electron’s excited states. Distributions computed for $s$ excited states are
localized at the axes $p_{n\parallel}=0$ [see Figs. 10 and 11, panels (d) and
(f)], distributions involving $p$ excited states exhibit a strong suppression
at the axes and maxima around the diagonals $p_{1\parallel}=\pm
p_{2\parallel}$ [see Figs. 10 and 11, panels (a) and (e)], and distributions
for which the second electron leaves from a $d$ state, shown in Figs. 10 and
11, panels (b) and (c), exhibit maxima around the axes and the diagonals.

Moreover, the distributions including the excitation and ionization prefactors
are much narrower than those in Sec. IV.1, where only the momentum space
constraints were taken. This is due to the $n_{e}-l_{e}-1$ radial nodes and
$l_{e}$ angular nodes in $V_{\mathbf{p}_{2},e}$, which will be mapped to
vertical or horizontal nodes in $|V_{p_{2\parallel},e}|^{2}$ (see Fig. 8).
Many nodes in the momentum regions of interest will split the momentum
distributions, or translate into narrower electron momentum distributions. For
instance, according to Fig. 8, narrow distributions are expected for channels
3, 4, 5 and 6. This is very visible in Figs. 10(c) and (e), for which the
distributions split into several maxima, and in Figs. 10(d) and (f), for which
the distributions are very narrow. Due to a higher number of nodes, the
distribution in Fig. 10(f) has more substructure and is narrower than that in
Fig. 10(d).

The dominance of specific events within a pulse potentially dictates the
length of the distributions, which is associated with the classically allowed
region (CAR), and the brightest peaks. For example, for the $3s\rightarrow 3p$
transition and CEP $65^{\circ}$ [Fig. 10(b)], the combination of a classically
allowed region for $p_{5}$ [Fig. 5(b)] and a high ionization probability for
the $o_{4}$ event of the second electron [Fig. 6(a)] causes the primary maxima
of the correlated two-electron distribution to be stronger in negative
momentum regions. In contrast, the distribution resulting from the
$3p\rightarrow 4p$ transition [Figs. 10(d)] shows much stronger maxima in the
top right quadrant (the positive momentum region) because the CAR for the pair
$p_{5}$ has collapsed and the $p_{4}o_{5}$ event dominates. The $p_{3}o_{4}$
transition, by itself, is not strong enough to cause the spread of the maxima
in the lower half quadrant, as understood by the PMD in Fig. 5(l). Competing
events, together with the prefactor, may give the impression that a specific
distribution is fourfold symmetric. This is the case, for instance, in Fig.
10(d), for the $3p\rightarrow 4s$ transition.

Varying the CEP changes the spread of the signal quite drastically for all
channels, and can change the quadrant in which the maxima is localized. This
is directly linked to the change in dominance of events as discussed
previously. The overall locations of the maxima and nodes in the distributions
remain the same as with $\phi_{1}=65^{\circ}$. However, the shapes for almost
all the distributions in Fig. 11, most notably panels (d) and (f) (the $s$
states) become much shorter; we no longer see the long extended distributions
along the axes. Instead, they are concentrated mostly at the origin with some
secondary maxima along the positive axes. Similarly, in Figs. 11(c) and (e),
we have less of a tail in the lower-left quadrant, and the majority of the
distribution, though narrower, is in the positive momentum region. This is
reflective of the $p_{4}o_{5}$ event being much stronger for this phase,
compared to their $\phi_{1}=65^{\circ}$ counterparts where the $p_{3}o_{4}$
and $p_{4}o_{5}$ events are competing.

Figure 10: Correlated two-electron momentum distributions as functions of the
momentum components $p_{1\parallel}$,$p_{2\parallel}$ parallel to the driving
field polarization, calculated for a single excitation channel using a
$\sin^{2}$ pulse of 4.3 cycles, intensity $I=1.5\times
10^{14}\mathrm{W/cm}^{2}$, angular frequency $\omega=0.057$ a.u. and CEP
$\phi_{1}=65^{\circ}$. The first and second rows correspond to fully
incoherent and coherent sums of amplitudes as given in Eqs (14), and (13),
respectively. The third row shows the difference between the maps in the first
two rows, showing the interference arising from summing events and particle
exchange terms coherently. All six RESI channels described in Table 1 are
shown from panels (a)-(f), (g)-(l) and (m)-(r). The signal in each panel has
been normalized with regard to its maximum. The diagonals
$p_{1\parallel}=p_{2\parallel}$ and the $p_{1\parallel}$ axes are indicated
with red dashed lines in each panel of the upper and middle rows. Figure 11:
Correlated two-electron momentum distributions as functions of the momentum
components $p_{1\parallel}$,$p_{2\parallel}$ parallel to the driving field
polarization, calculated for a single excitation channel for the same pulse
parameters employed in 10 and $\phi_{1}=155^{\circ}$. The first and second
rows correspond to fully incoherent and coherent sums of amplitudes as given
in Eqs (14), and (13), respectively. The third row shows the difference
between the first two rows, showing the interference arising from summing
events and particle exchange terms coherently. All six RESI channels described
in Table 1 are shown from panels (a)-(f) and (g)-(l). The signal in each panel
has been normalized with regard to its maximum. The diagonals
$p_{1\parallel}=p_{2\parallel}$ and the $p_{1\parallel}$ axes are indicated
with red dashed lines in each panel of the upper and middle rows.

Panels (g)-(l) of Figs. 10 and 11 show coherent sums over symmetrization and
events for $\phi_{1}=65,155^{\circ}$, given by Eq. (13). This means that, for
the dominant events of the pulse considered here, the interference types
outlined in Fig. 3 and the phase types specified in Sec. III are present.
Throughout, one can see superimposed interference fringes near the diagonals
and the origin. To visualize the fringes more clearly, the difference between
the fully coherent and fully incoherent sums is plotted in the third row of
these figures.

Overall, there exists an intricate tapestry of patterns, which, however, share
some common features. In all panels, we can see a bright fringe along the
diagonal, surrounded by hyperbolas, and are reflection-symmetric about the
diagonal $p_{1\parallel}=p_{2\parallel}$. These features are particularly
clear in the third quadrant of Figs. 10 (m), (n), and the first quadrants of
Figs. 11(m), (n), (o), and (q). Furthermore, because the half-cycle symmetry
is broken, these features are not symmetric about
$(p_{1\parallel},p_{2\parallel})\rightarrow(-p_{1\parallel},-p_{2\parallel})$
and there is no bright interference fringe along the anti-diagonal
$p_{1\parallel}=-p_{2\parallel}$. These are key differences from the single-
channel interference observed for monochromatic fields [5, 6]. Other features
are phase shifts associated with the prefactor, which can be seen very clearly
for excitations to $p$ states [see panels (m) and (q) in Figs. 10 and 11], and
convoluted patterns in the second and fourth quadrant.

Next, we will have a closer look at the different types of interference
present in Figs. 10 and 11. According to Sec. IV.1, there are at most three
relevant events within the pulse, $p_{3}o_{4}$, $p_{4}o_{5}$ and $p_{5}o_{6}$,
each of which is roughly shifted by half a cycle from the previous one.
Pairwise, they may interfere in fifteen main ways, which are listed in Table.
2. This table can be used as a roadmap to understand the subsequent
discussions. The left column of Table. 2 gives the phase differences discussed
in Sec. III. The second column illustrates schematically how these phases can
be applied to the specific pulse studied in this work, following the
convention used in Fig. 3. The colors and labelling of events are as in Sec.
IV.1. The third column details the specific sources of pairwise interference,
including electron exchange, and the temporal shift, $\Delta\tau$ which
quantifies the relative (approximate) position of interfering events in a
pulse. Another analytical device employed to temporally order the events
within the pulse is the temporal locations $\tau_{\mu}$. They are not directly
employed in the numerical computations, but serve as a measure of where the
times $t,t^{\prime},t^{\prime\prime}$ are located for a specific event. For
instance, for $p_{3}o_{4}$ we take $\tau_{\mu}$ where $\mu=l,d$ to be zero
given that this is the first dominant event in the pulse to be considered. The
later event $p_{4}o_{5}$ is separated by half a cycle from $p_{3}o_{4}$, so
that $t+\pi/\omega,t^{\prime}+\pi/\omega,t^{\prime\prime}+\pi/\omega$, where
$\tau_{\mu}=\pi/\omega$ and $\mu=r,u$. Similarly for $p_{5}o_{6}$,
$\tau_{\mu}=2\pi/\omega$ where $\mu=lT,dT$.

There exist three main types of interference. Phase differences numbered from
1 to 3 are associated with individual events within the pulse, but for which
the electron momenta have been exchanged. Because we are only considering one
event, the temporal shift $\Delta\tau=0$. Next, there are phase differences
associated solely with temporal shifts, numbered 4 to 9. For pairs of events
that are separated by half a cycle (i.e. $p_{3}o_{4}$+$p_{4}o_{5}$ and
$p_{4}o_{5}$+$p_{5}o_{6}$), the relative temporal shift associated is,
therefore, $\Delta\tau=\pi/\omega$. For the inter-cycle events, the relative
temporal shift is $\Delta\tau=2\pi/\omega$. Finally, the last five phase
differences (10 to 15) account for a combination of electron exchange and
temporal shifts.

Phases | Schematic Representation | Type | S-E
---|---|---|---
1. $\alpha_{l,d}$ 2. $\alpha_{r,u}$ 3. $\alpha_{lT,dT}$ |  |
---
| Intracycle Exchange
---
No temporal shift $\Delta\tau=0$
1. $\tau_{l}=0$, $\tau_{d}=0$ 2. $\tau_{r}=\pi/\omega$, $\tau_{u}=\pi/\omega$ 3. $\tau_{lT}=2\pi/\omega$, $\tau_{dT}=2\pi/\omega$
| ci, cc
---
4. $\alpha_{l,r}$ 5. $\alpha_{d,u}$
6. $\alpha_{lT,r}$ 7. $\alpha_{dT,u}$
8. $\alpha_{lT,l}$ 9. $\alpha_{dT,d}$ |  |
---
| No Exchange
---
a) Intracycle shift $\Delta\tau=\pi/\omega$
4. $\tau_{l}=0$, $\tau_{r}=\pi/\omega$ 5. $\tau_{d}=0$, $\tau_{u}=\pi/\omega$ 6. $\tau_{lT}=2\pi/\omega,\tau_{r}=\pi/\omega$ 7. $\tau_{dT}=2\pi/\omega,\tau_{u}=\pi/\omega$

b) Intercycle shift $\Delta\tau=2\pi/\omega$
8. $\tau_{lT}=2\pi/\omega$, $\tau_{l}=0$ 9. $\tau_{dT}=2\pi/\omega$, $\tau_{d}=0$

| ic, cc
---
10. $\alpha_{l,u}$ 11. $\alpha_{r,d}$
12. $\alpha_{lT,u}$ 13. $\alpha_{dT,r}$
14. $\alpha_{lT,d}$ 15. $\alpha_{dT,l}$ |  |
---
| a) Intracycle shift $\Delta\tau=\pi/\omega$
---
10. $\tau_{l}=0$, $\tau_{u}=\pi/\omega$ 11. $\tau_{d}=0$, $\tau_{r}=\pi/\omega$
12. $\tau_{lT}=2\pi/\omega,\tau_{u}=\pi/\omega$ 13. $\tau_{dT}=2\pi/\omega,\tau_{r}=\pi/\omega$

b) Intercycle shift $\Delta\tau=2\pi/\omega$
14. $\tau_{lT}=2\pi/\omega$, $\tau_{d}=0$ 15. $\tau_{dT}=2\pi/\omega$, $\tau_{l}=0$

| cc
---
Table 2: All the 15 phase differences present for the few-cycle pulse,
associated with just exchange (row 1), temporal shifts (row 2) and a
combination of both exchange and shifts (row 3). From left to right, the first
column provides the phases for the interfering processes involving the three
most relevant events in the pulse given by Eq. (37), $p_{3}o_{4}$,
$p_{4}o_{5}$ and $p_{5}o_{6}$. The second column shows a schematic
representation of these phases. The colors used in the diagrams and the
labeling of the events are the same as in the section discussing the dominant
orbits, i.e., $p_{3}o_{4}$ is represented in red, $p_{4}o_{5}$ in black and
$p_{5}o_{6}$ in blue. When more than one event is interfering, the two events
are placed between the corresponding diagrams. The third column indicates the
type of interference i.e. whether it transpires due to particle exchange,
temporal shifts of either $\pi/\omega$ or $2\pi/\omega$ or a combination of
both. The specific shifts, $\tau_{\mu}$ of $t,t^{\prime},t^{\prime\prime}$ for
each phase $\mu$ are stated - the shifts associated $p_{3}o_{4}$
($\tau_{l},\tau_{d}$) are taken to be 0. The final column indicates when the
corresponding type of interference can be observed i.e. with a fully coherent
sum of symmetrization or events, or combination of incoherent and coherent
symmetrization and events.

###  V.2 Exchange Interference

Figure 12: Correlated two-electron momentum distributions as functions of the
momentum components $p_{1\parallel}$,$p_{2\parallel}$ parallel to the driving
field polarization, calculated for a single excitation channel for the same
pulse parameters employed in 10. The first row corresponds to amplitudes where
events are summed incoherently, and symmetrization is done coherently as given
in Eq (15), calculated for $\phi_{1}=65^{\circ}$, and illustrates that the
interference fringes are different from those in 10. The second and third rows
shows the difference between the first row and the corresponding fully
incoherent sums [see panels (a)-(f) in Figs.10, 11], with CEP $65^{\circ}$ and
$155^{\circ}$. These rows show the interference arising from summing particle
exchange terms coherently and events incoherently. All six RESI channels
described in Table 1 are shown from panels (a)-(f) and (g)-(l). The signal in
each panel has been normalized with regard to its maximum.

Fig. 12 shows similar coherent sums and difference plots as in Figs. 10 and
11, but summing only the exchange processes coherently. This isolates the
interference effects arising from the
$\alpha_{\mathbf{p}_{1},\mathbf{p}_{2}}^{(\mathrm{exch})}$ and
$\alpha_{\mathbf{p}_{1},\mathbf{p}_{2}}(t,t^{\prime})$ phase differences in
Eqs. (19), (20). In all plots, we observe superimposed hyperbolas and a bright
fringe along the diagonal. Upon inspection of Eq. (19), field-independent
hyperbolae arise when
$\alpha_{\mathbf{p}_{1},\mathbf{p}_{2}}^{(\mathrm{exch})}=2n\pi$ whilst the
bright fringe (the “spine”), which increases in width along the diagonal can
be attributed to the field-dependent
$\alpha_{\mathbf{p}_{1},\mathbf{p}_{2}}(t,t^{\prime})$ phase. As expected for
a pulse with no half cycle symmetry, the structures are asymmetric with regard
to
$(p_{1\parallel},p_{2\parallel})\leftrightarrow(-p_{1\parallel},-p_{2\parallel})$
and there is no bright fringe along the anti-diagonal. Interestingly, the
asymptotes and focal points of the hyberbolae do not lie at the diagonals, but
at the axes. This feature was also present for monochromatic driving fields
[5], and it is likely caused by the transverse momentum integration. It is
also noteworthy that many of the intricate patterns in the second and fourth
quadrants of the $p_{1\parallel}p_{2\parallel}$ plane are absent, which is
evidence that they stem from other types of quantum interference. We note that
for CEP $65^{\circ}$ [Figs. 12(e)-(h)] the fringes are well distributed in the
four quadrants and the patterns are more intricate suggesting competing events
within the pulse, whilst the distributions and the interference structures for
CEP $155^{\circ}$ [Fig. 12(i)-(l)] are primarily located in the third quadrant
of the parallel momentum plane, and their cleanness suggests a dominant event.

To understand the effects in Fig. 12(d)-(i) in greater detail, the
interferences associated with $\alpha_{l,d}$, $\alpha_{r,u}$ and
$\alpha_{lT,dT}$ are separated. Therein, we choose the $3s\rightarrow 3p$
channel as Fig. 12(e) shows obvious hyperbolic structures and central nodes at
the axes, and the $3p\rightarrow 4p$ excitation channel for which there are
also visible hyperbolas and extra nodes stemming from the higher principal
quantum number $n_{e}=4$ [Fig. 12(g)].

Figure 13: Interference patterns associated with the purely exchange phase
differences for the three most dominant events within a pulse for the
$3s\rightarrow 3p$ transition incorporating both the excitation and ionization
prefactors (top row) and the $3p\rightarrow 4p$ transition, with both
prefactors (middle row) and with neither prefactor (bottom row). The same
pulse parameters are employed as in Fig. 10, and the carrier-envelope phase is
$\phi_{1}=65^{\circ}$. The signal in each panel has been normalized with
regard to its maximum. The diagonals $p_{1\parallel}=p_{2\parallel}$ are
indicated with black dashed lines in the bottom row. The locations of the
nodes and phase shifts arising from the prefactors are marked by red dotted
lines in (a), (b), (d), (e). The box in panel (f) shows the location of finely
spaced hyperbole.

Figs. 13(a)-(c) and Figs. 13(d)-(f) show $\alpha_{l,d},\alpha_{r,u}$ and
$\alpha_{lT,dT}$ for the two chosen excitation pathways incorporating both the
$V_{\mathbf{p}_{1},ekg}$ and $V_{\mathbf{p}_{2},e}$ prefactors. Both
prefactors have been omitted in Figs. 13(g)-(i) for the excitation channel
$3p\rightarrow 4p$ to highlight the phase jumps and nodes associated with the
ionization prefactor, arising from locations of the radial nodes and the
mapping of the angular nodes of $V_{\mathbf{p}_{2},e}$ [Fig. 8(g)]
respectively. These are indicated by the red dotted lines in Figs. 13(a), (b),
(d) and (e). The effects of the prefactor are present and the same for all
three interference types discussed in this work.

For $3s\rightarrow 3p$, one can see that the fringes stemming from the
$p_{3}o_{4}$ event, are the brightest and occupy the largest momentum region,
located primarily in the third quadrant of the $p_{1\parallel}p_{2\parallel}$
plane, but also spilling in the other quadrants [see the left column in Fig.
13]. This is consistent with the findings in Fig. 7, which indicate that this
event is dominant. Figs. 13(b) and (c) shed light on the other events. The
fringes arising from the second dominant event, i.e., $p_{4}o_{5}$ lead to
well-defined hyperbolas in the first quadrant of the parallel momentum plane.
These occupy a slightly smaller momentum region but are also quite bright, in
agreement with the findings that both events are comparable. Lastly,
$p_{5}o_{6}$ is much less dominant than the other two events, which is
reflected in the much smaller momentum region in which the fringes caused by
$\alpha_{lT,dT}$ are significant. For the excitation pathway $3p\rightarrow
4p$ [Figs. 13(d)-(f)], the RESI distributions are more symmetric. This is due
to the events $p_{3}o_{4}$ and $p_{4}o_{5}$ competing and the collapse of the
CAR for $p_{5}o_{6}$ due to a higher energy gap between the ground and excited
states [see Fig. 7(a) and (f)], together with the influence of the prefactors,
which have additional radial nodes.

We retain the diagonal symmetry of the interference patterns for all three
phases, indicated by the black dashed lines in Fig. 13(g)-(i). Additionally,
within the direct ATI cut-off, the hyperbolae associated with alpha exchange
can only be seen for $n=0$ \- these are most clear in quadrants opposite to
where the events are dominant - an example is indicated by the box in Fig.
13(f). This is because $\alpha_{\mathbf{p}_{1},\mathbf{p}_{2}}(t,t^{\prime})$
plays the smallest role in these regions, and thus the bright spine no longer
masks or skews the hyperbolae caused by
$\alpha_{\mathbf{p}_{1},\mathbf{p}_{2}}^{(\mathrm{exch})}$.

Figure 14: Interference patterns associated with the purely exchange phase
differences for the three most dominant events within a pulse for the
$3s\rightarrow 3p$ transition (top row) and the $3p\rightarrow 4p$ transition
(bottom row), for a larger momentum region than in Fig. 16. The same pulse
parameters are employed as in Fig. 10, and the carrier-envelope phase is
$\phi_{1}=65^{\circ}$. The signal in each panel has been normalized with
regard to its maximum. The locations of phase shifts arising from the
prefactors are marked by red dotted lines in (a). Purple dashed boxes in (b)
show the different hyperbole arising from
$\alpha_{\mathbf{p}_{1},\mathbf{p}_{2}}^{(\mathrm{exch})}=2n\pi$ with $n>0$.
The arrows indicate where these hyperbolae meet leading to convex shapes in
the interference pattern. The asymptotes of one such hyperbola are marked in
(d) with purple dashed lines along the diagonal and parallel to the anti-
diagonal. Panel (e) marks a cross-shape with purple lines, between which we
see the finest fringes.

Further information about the hyperbolae can be gleaned from the interference
patterns at larger momentum values. In Fig. 14, we see additional hyperbolae
with shifted centers, presumably arising from greater values of $n$ indicated
by the purple boxes in panel (b). These hyperbolae have asymptotes at or
parallel to the diagonal and antidiagonal [see Fig. 14(d)] with decreasing
fringe spacing. At the locations where two hyperbolae meet, we see a convex
shape [see arrows in Fig. 14(b)]. The narrowest fringe spacings form a rough
cross [marked in Fig. 14(e)], the shape of which is not unlike the mapping of
the excitation prefactor in Fig. 9 with the center of the cross located in the
momentum quadrant where the associated event is dominant. We also note that at
locations of phase shifts, the fringes become slightly jagged [see red dotted
line in Fig. 14(a)].

Finally, this analysis reveals several additional key observations regarding
the combination of the three events (and thus the three exchange phases) shown
in Figs. 12(d)-(k).

First, faint hyperbolae observed in the second and fourth quadrants result
from the incoherent combination of events and are associated with
$\alpha_{\mathbf{p}_{1},\mathbf{p}_{2}}^{(\mathrm{exch})}$. These shapes are
therefore not actually interference effects but a consequence of combining the
interference effects from $\alpha_{l,d},\alpha_{r,u}$ and $\alpha_{lT,dT}$
incoherently, since their fringe direction in the second and fourth quadrants
varies depending on the events’ locations in the pulse. The spine, along with
alternating intense positive and negative fringes in the diagonal (both of
which arise from $\alpha_{\mathbf{p}_{1},\mathbf{p}_{2}}(t,t^{\prime})$)
contribute to the first and third quadrants, with the most dominant events
leading to the brightest and biggest contributions.

Second, in the quadrants where both
$\alpha_{\mathbf{p}_{1},\mathbf{p}_{2}}^{(\mathrm{exch})}$ and
$\alpha_{\mathbf{p}_{1},\mathbf{p}_{2}}(t,t^{\prime})$ are present, the effect
of $\alpha_{\mathbf{p}_{1},\mathbf{p}_{2}}(t,t^{\prime})$ tends to be stronger
causing the bright fringes to obfuscate the hyperbolae. However,
$\alpha_{\mathbf{p}_{1},\mathbf{p}_{2}}^{(\mathrm{exch})}$ appears to bend
these parallel fringes creating large bright hyperbole-like structures on
either side of the spine. Where
$\alpha_{\mathbf{p}_{1},\mathbf{p}_{2}}(t,t^{\prime})$ is less strong (i.e. in
the second and fourth quadrants) there are more clearly observable hyperbolae.

###  V.3 Event Interference

Figure 15: Correlated two-electron momentum distributions as functions of the
momentum components $p_{1\parallel}$,$p_{2\parallel}$ parallel to the driving
field polarization, calculated for a single excitation channel for the same
pulse parameters employed in 10. The first row corresponds to amplitudes where
events are summed coherently, and symmetrization is performed incoherently as
given in Eq. (16), and was calculated for $\phi_{1}=65^{\circ}$. The second
and third rows show the difference between the first row and the corresponding
fully incoherent sums [see panels (a)-(f) in Figs.10, 11], with CEP
$65^{\circ}$ and $155^{\circ}$. These rows show the interference arising from
summing particle-exchange terms incoherently and events coherently. All six
RESI channels described in Table 1 are shown from panels (a)-(f) and (g)-(l).
The signal in each panel has been normalized with regard to its maximum.

Next, we isolate the effects of the phase differences associated with the
temporal shifts: $\alpha_{l,r},\alpha_{lT,r},\alpha_{lT,l}$ and their
identical but transposed (symmetrized) counterparts:
$\alpha_{d,u},\alpha_{dT,u},\alpha_{dT,d}$. Fig. 15 shows the momentum
distribution with incoherent symmetrization and coherent events, as well as
the differences between these distributions and the fully incoherent sum for
both CEPs. This enables us to investigate interference effects emerging from
four different phase differences: $\alpha^{(\mathrm{ene})}_{\Delta\tau}$,
$\alpha^{(A^{2})}_{\Delta\tau}(t^{\prime},t^{\prime\prime})$,
$\alpha^{(\mathrm{pond})}_{\Delta\tau}(t,t^{\prime\prime})$ and
$\alpha^{(\mathbf{p}_{1},\mathbf{p}_{2})}_{\Delta\tau}(t,t^{\prime})$. Figs.
15(e)-(l) show distributions with alternating positive and negative fringes,
that are brightest in the region(s) associated with the dominant event(s) for
each excitation pathway, for both CEPs. There appears to be a darker “box” in
the center of the distributions - most prominent for channels with a higher
number of angular nodes. All other prefactor effects, such as nodes and
relative distribution size remain. A faint circular substructure is also
somewhat visible, particularly in Fig. 15(k). This is associated with
$\alpha^{(\mathrm{ene})}_{\Delta\tau}$ [Eq. (26)] which gives rise to circular
fringes with a target-dependent radius. This is expected because Eq. (26) is
that of a hypersphere. The radius of the largest circular fringe is obtained
at vanishing perpendicular momenta. These distributions occupy similar
momentum regions to the fully coherent distributions in Fig. 10(m)-(r) and
Fig. 11(m)-(r).

Figure 16: Interference patterns associated with the temporal-shift phase
differences utilizing interfering pairs of the three most dominant events
within a pulse for the $3s\rightarrow 3p$ transition (top, middle rows) and
the $3p\rightarrow 4p$ transition (bottom row). The top row shows the
unsymmetrized distributions, whilst the middle and bottom rows account for the
transposed counterpart of the phase differences. The same pulse parameters are
employed as in Fig. 10, using a CEP of $65^{\circ}$ for the $3s\rightarrow 3p$
transition [top and middle row] and a CEP of $155^{\circ}$ for the
$3p\rightarrow 4p$ transition [bottom row]. The signal in each panel has been
normalized with regard to its maximum. The gradients of the fringes are marked
in (b) and (e) with solid black lines. The red dotted box in (b) indicates the
location of fine fringes coming from
$\alpha^{(\mathbf{p}_{1},\mathbf{p}_{2})}_{\Delta\tau}(t,t^{\prime})$, which
are obfuscated by the larger ’wings’ in the symmetrized distributions in (e)
and (h). The solid blue boxes in (d) indicate the non-interference
chequerboard patterns coming from
$\alpha^{(A^{2})}_{\Delta\tau}(t^{\prime},t^{\prime\prime})$. Purple dashed
circles in (f) indicate the largest circular fringes resulting from
$\alpha^{(\mathrm{ene})}_{\Delta\tau}=2\pi n$ where $n=3,4$. The symmetry
along the diagonal of the symmetrized distributions can also be seen by the
dash-dot lines in (g)-(i).

As with the exchange-only case, we now isolate the effects of
$\alpha_{l,r},\alpha_{lT,r},\alpha_{lT,l}$ with the $3s\rightarrow 3p$ with
CEP $65^{\circ}$ [Fig. 15(e)]. However, this time $3p\rightarrow 4p$ is taken
with CEP $155^{\circ}$ [Fig. 15(k)]. Figs. 16(a)-(c) show the separated
phases, $\alpha_{u,d},\alpha_{dT,u},\alpha_{dT,d}$ with the $3s\rightarrow 3p$
transition. The gradient of some of the fringes is indicated by the lines in
Fig. 16(b). As expected for up-down interference, the majority of the lines in
Figs. 16(a),(b) are roughly located in the up-down direction. In quadrants two
and four (depending on the events in question), we see thinner fringes, coming
from different directions and combining [Fig. 16(b)]. For their identical but
transposed phases, we would expect these distributions to also be transposed.
An incoherent sum of the temporal-shift phases with their transposed
counterparts,
$\alpha_{u,d}+\alpha_{l,r},\alpha_{dT,u}+\alpha_{lT,r},\alpha_{dT,d}+\alpha_{lT,l}$
is provided in the second row for $3s\rightarrow 3p$, and in the third row for
$3p\rightarrow 4p$. In the first quadrants of Figs. 16(e), (h), we see
widening v-shaped fringes which overlap to form complex fine criss-crossed
patterns. These are actually not due to interference, but arise as a
consequence of incoherent symmetrization - in panel (e), the gradients of the
fringes associated with $\alpha_{dT,u}$ and $\alpha_{lT,r}$ individually are
indicated with lines. The fine fringes in the red box in panel (b) are almost
completely obfuscated by the thicker fringes in that quadrant coming from the
symmetrized phase difference, forming a “wing” shape along the anti-diagonal
in panels (d), (e), (g) and (h). These patterns occur mostly for intra-cycle
shifts as in the first two columns of this figure.

With inter-cycle shifts i.e. $\alpha_{dT,d}$ and $\alpha_{lT,l}$, almost
circular fringes are observed arising from
$\alpha^{(\mathrm{ene})}_{\Delta\tau}$, the largest radius of which is
positive only for $n>2$ for the target in question. The largest fringes are
plotted with $n=3,4$ in Fig. 16(f) and appear to be a fairly good match. The
other rings may arise from varying values of transverse momenta. We note that
the fringes are not centered at the exact origin, since the temporal shift of
$2\pi/\omega$ from $p_{3}o_{4}$ to $p_{5}o_{6}$ is an approximation for the
few-cycle pulse.

Figure 17: Interference patterns associated with a pure temporal-shift phase
difference, $\alpha^{(A^{2})}_{\Delta\tau}$ 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.

We note that $\alpha^{(A^{2})}_{\Delta\tau}(t^{\prime},t^{\prime\prime})$
depends only on the ionization and rescattering times of the first electron.
Therefore, we can detangle the interference stemming from
$\alpha^{(A^{2})}_{\Delta\tau}(t^{\prime},t^{\prime\prime})$ by calculating
$\alpha_{l,r}(+\alpha_{u,d})$ and $\alpha_{lT,r}(+\alpha_{dT,u})$ for the
first and second electron separately. The distribution with the first electron
only contains effects from all four phase differences associated with temporal
shifts including $\alpha^{(A^{2})}_{\Delta\tau}(t^{\prime},t^{\prime\prime})$
whilst the distribution from the second electron excludes the
$\alpha^{(A^{2})}_{\Delta\tau}(t^{\prime},t^{\prime\prime})$ contribution. For
more details, see Fig. 24 in the second AppendixB.

The one-electron unsymmetrized distributions (i.e. associated with
interference along either the $p_{1\parallel}$ or $p_{2\parallel}$ axes)
consist of straight fringes parallel to the associated axis, varying in
intensity. When incoherently symmetrized, these fringes form a cross shape. At
the point where the fringes intersect, we see a chequerboard-type pattern.
When $\alpha^{(A^{2})}_{\Delta\tau}(t^{\prime},t^{\prime\prime})$ is isolated,
as in Fig. 17, this pattern becomes even more detailed and complex due to the
subtraction of the two individual electrons’ fringes. We can therefore deduce
that this chequerboard, indicated by the blue boxes in Fig. 16d), for
instance, is not actually an interference effect but a symptom of incoherent
symmetrization.

The wings of the distribution are likely to be brought about by
$\alpha^{(\mathbf{p}_{1},\mathbf{p}_{2})}_{\Delta\tau}(t,t^{\prime})$
(Eq.(28)), the only purely temporal-shift phase which is both momentum and
field dependent. Thus, intuitively, when the momenta are large and have the
same sign (as in quadrants one and three), one expects that this phase
difference will be large and the fringes will be finer. In contrast, if both
momenta are small (close to the origin), or if they are large but have
opposing signs (second and fourth quadrant), a small
$\alpha^{(\mathbf{p}_{1},\mathbf{p}_{2})}_{\Delta\tau}(t,t^{\prime})$ will
lead to thicker fringes. This effect contributes to the appearance of the dark
”box” visible in some distributions of Fig. 15 \- there are simply fewer
interference fringes at play around the origin.

Figure 18: Interference patterns associated with the temporal-shift phase
differences utilizing interfering pairs of the three most dominant events
within a pulse for the $3s\rightarrow 3p$ transition (top row) and the
$3p\rightarrow 4p$ transition (bottom row). A larger momentum region is taken
than in Fig. 16. The same pulse parameters are employed as in Fig. 10, using a
CEP of $65^{\circ}$ for the $3s\rightarrow 3p$ transition [top row] and a CEP
of $155^{\circ}$ for the $3p\rightarrow 4p$ transition [bottom row]. The
signal in each panel has been normalized with regard to its maximum. The blue
lines in (c) along the axes are drawn to emphasize the presence of the
slightly off-axes circular fringes presumably coming from
$\alpha^{(\mathrm{ene})}_{\Delta\tau}$. Panel (e) annotates the heart shape of
the interference which is also present in (a) and (b).

Fig. 18 shows the temporal-shift phases for both channels, in a larger
momentum space. As before, the wings can be seen. In addition to the wings,
the intracycle shifts also exhibit heart shapes in the first or third
quadrants (depending on the dominance of events involved), with ’static’ like
patterns in the opposing quadrant. In panel (d) the heart disappears,
presumably because for CEP $155^{\circ}$, $p_{3}o_{4}$ loses dominance. The
origin of this shape is unclear. However, this behavior indicates that they
may arise from a combination of the field-dependent temporal-shift phase
differences. Chequerboards appear only for small momentum values indicating
$\alpha^{(A^{2})}_{\Delta\tau}(t^{\prime},t^{\prime\prime})$ is only
influential in the regions where the events are strongest, which makes sense
as this phase difference is field-dependent only.

The inter-cycle shifts (Fig. 18(c), (f)) show multiple sets of circular
fringes due to $\alpha^{(\mathrm{ene})}_{\Delta\tau}$, located along the axes,
and faintly even in the quadrants. They are centered slightly off-axis due to
the asymmetry of the pulse. Since $\alpha^{(\mathrm{ene})}_{\Delta\tau}$
[Eq.(26)] gives the equation of a hypersphere, it is not trivial to obtain the
centers and sizes of these extra circular fringes. Additionally, due to the
strong dependence of
$\alpha^{(\mathrm{pond})}_{\Delta\tau}(t,t^{\prime\prime})$ on the field