Planning Optical Networks for Unexpected Traffic Growth

Sai Kireet Patri(1,2), Achim Autenrieth(1),
Jörg-Peter Elbers(1), Carmen Mas Machuca(2)

(1) ADVA, Fraunhoferstr. 9A, 82152 Martinsried, Germany

(2) Chair of Communication Networks, TU Munich, Arcisstr. 21, 80331 Munich, Germany

A lightpath configuration algorithm considering a multi-period traffic model is presented and evaluated to a realistic German core network study. Strategically exploiting BVT excess capacity, we show that offered traffic can be carried up to an additional five years in all traffic growth scenarios.

§ INTRODUCTION

With 5G deployment becoming a reality across countries, core network operators are realising that deployed optical networks need to be upgraded to quickly capitalize gains. Due to the recent pandemic, many network operators (e.g., BT and Telefónica), have reported a 30-50% increase in broadband traffic in Q1 2020, compared to Q4 2019<cit.>. Given the worldwide impact on logistics, operators must invest in software configurable optical terminalsadva and Bandwidth Variable Transponders (BVTs), which offer opportunities to increase network throughput in order to carry the growth in offered traffic. We define offered traffic as the total throughput of the network if all demands are met by adding sufficient BVTs in the network.

Traditionally, lightpath configuration in network planning consists of algorithms exploiting Integer Linear Programming (ILP) as well as heuristics. A link state heuristic for optical networksWang2019 is of particular interest; however, no insights on its usability in a multi-period scenario are provided. The SNAP Algorithm Virgillito2019 offers progressive traffic loading, randomly allocating traffic between nodes without considering realistic growth. Both works also assume homogeneous spans across their physical topology, limitations of which have been discussed in our previous workPatri20.

In this work, we first present a traffic generator and discuss two traffic growth models viz., Expected and Unexpected. Using our developed solution (Scheme 1) on a German core topology, we show that the offered traffic can be met by strategically leveraging the configured BVTs' capacity, hence saving bandwidth for future growth. Scheme 1 is evaluated against a simple baseline (Scheme 2), which optimally adds new BVTs, instead of upgrading the deployed BVTs to a higher datarate when possible. Finally, results in terms of aggregate network throughput (carried traffic) and provisioned BVTs are discussed.

§ TRAFFIC GENERATOR AND GROWTH MODEL

As network operators do not share their offered traffic data openly due to confidentiality, network planners have relied on traditional empirical traffic models like the gravity modelDwivedi2000, which assumes traffic to be proportional to the human population of the source and destination. However, in present times, most of the traffic between two locations are exchanged between either Data Centers (DCs) or Internet Exchange Points (IXPs) and there exists a higher correlation of offered traffic to node degree as compared to human population. For e.g., Frankfurt and Duesseldorf are less populated cities as compared to Hamburg and Munich, but exchange traffic at higher ratesdecix. Based on these assumptions, we define initial traffic (time $t=0$), between source $i$ and destination $j$ as:

\begin{equation}\label{eq:initTrafModel}
\tau(i,j,0)[Gbps] =
\begin{cases}
2 \cdot \binom N2 \cdot \Delta_{i} \cdot  \Delta_{j} & \text{if~$N > 2 \cdot \bar{N}$}\\
N \cdot \Delta_{i} \cdot  \Delta_{j} & \text{$otherwise$}\\
\end{cases}\;
\end{equation}

where $\bar{N}$ is the average node degree of the topology $ N=\sum_{ \kappa  \in \{i,j\}} N_{\kappa}$ is the combined node degree, and $\Delta_{\kappa}$ is the absolute difference between the number of co-located DCs and IXPs at a given node $\kappa$dcmap.

For $t>0$, the offered traffic between nodes is given as $\tau(i,j,t)=\gamma_{t} \cdot \tau(i,j,0)$, where $\gamma_{t}$ is the expected growth in offered traffic<cit.>. For unexpected traffic growth, we modify $\gamma_{t}$ with an increase of 30-90% between 2023 and 2026, as compared to its original value. As seen in Fig. <ref>, this brings about an overall realistic increase of  40% in the aggregate offered traffic growth for the given network topology.

§ ALGORITHM AND MULTI-PERIOD PLANNING STUDY

Proposed solution flow in a multi-period scenario with two schemes and optional physical network upgrade

The multi-period planning algorithm is shown in Fig. <ref>. It takes as input the topology, along with information on the number of DCs and IXPs at each source destination pair ($i,j$). We then find the initial offered traffic using Eq. <ref> and also do a Routing Wavelength and Spectrum Assignment (RWSA), using a weighted probabilistic routing based on Yen's k-Shortest Path Algorithm and the number of continuous empty frequency slots in each of the paths. This heuristic for routing adds a randomness in choosing the candidate path list for each $(i,j)$. We use first-fit channel allocation strategy for spectrum assignment.

Aggregate Offered Traffic Growth models

Expected Traffic Throughput

Unexpected Traffic Throughput

Yearly aggregate throughput of Germany17 topology from 2020-2030

The objective of the algorithm is to minimize the number of lightpaths (LPs) added to the network for planning period $t$, while trying to meet the offered aggregate traffic. It is pertinent to note that one LP is associated to one BVT.  To begin, we take as input the candidate path list of all demands ($D_t$), all currently provisioned LPs in the network ($L_t$) and the traffic matrix ($T_t$) calculated using Eq. <ref> and Fig. <ref>. For each candidate path $d_{i,j,t}\in D_t$, we compute a list of LPs $L_{i,j,t}\subset L_t$ already present between nodes $(i,j)$ and additional offered traffic $\theta_{i,j,t}$, defined as $\tau(i,j,t)-\tau(i,j,0)$. Using a datarate granularity of 50 Gbps between 100-600 Gbps in steps of 50 Gbps with QAM values of QPSK, 8, 16, 32 and 64 QAM, we generate more than 60 different channel configurations (combination of $\{Datarate,QAM\}$) for each BVT. The list of valid channel configurations $C_{i,j,t}$ is then filtered using HeCSON<cit.>. For every provisioned LP ($\lambda \in L_{i,j,t}$), a set of channel configurations $\rho_{i,j,t}\subset C_{i,j,t}$ is found, having their channel bandwidth $BW_\rho$ less than or equal to the provisioned LP's channel bandwidth $BW_\lambda$. This ensures upgrade of provisioned LPs at the same central channel frequency, without additional bandwidth usage. We then iterate over $\rho_{i,j,t}$ sorted according to highest datarate (DR) to find the first $\rho$, which has a higher DR than $\lambda$. If found, $\lambda$ is updated with $\rho$ and the spare capacity generated can be used to satisfy some of the additional traffic, thereby reducing $\theta_{i,j,t}$.

\begin{equation*}
\begin{array}{ll@{}ll}
minimize~~ \sum_{c~\in~C_{i,j,t}} n_{c} &\\
subject~to:~~\theta_{i,j,t}\leq \sum n_{c}\cdot DR_{c}<\theta_{i,j,t}+\delta &\\
~~~~~~~~~~0< \sum n_{c}\cdot \eta_{NLI_c}\leq\sum_{\forall \lambda \in L_{i,j,t}} \eta_{NLI_\lambda}\\
\end{array}
\label{eq:ilp}\tag{2}
\end{equation*}

After the provisioned LPs are upgraded, we find the number of additional LPs $n_c$ of each valid channel configuration $c$, which are needed to carry the remaining $\theta_{i,j,t}$ using the ILP shown in Eq. <ref>. This ILP minimizes integer $n_c$, such that the configured total datarate can only be over-provisioned by $\delta$ Gbps. For our network study, we fix $\delta$ as 100 Gbps. We also use a power independent NLI co-efficient constraint, $\eta_{NLI}$, following definitions and calculations of ACF-EGN model Poggiolini19. This NLI
constraint restricts the total $\eta_{NLI}$ being added every $t$ to the sum of provisioned LPs' $\eta_{NLI}$. The LPs are then added to $L_{i,j,t+1}$, defined as a set of configured LPs between nodes $(i,j)$ for the next planning period.

After additions are completed, we also check on the number of empty frequency slots in the link. If more than 75% of the frequency slots have been filled up in any of the given links, it warns operators about upcoming frequency slot saturation in the configured fiber pair so that they plan a localized physical upgrade by exploring either additional bands or utilizing available dark fiber pairs<cit.>.

(dB) Virgillito2019
Germany 17
US Abilene

Reference topology informationphynwinfo

Using the discussed flow, we setup a multi-period network study for a 17 node German backbone network with single mode fibers having heterogeneous span lengthsphynwinfo. We assume single fiber pair C-Band operation with variable gain EDFAs having constant noise figure as shown in Tab. <ref>. The topological details, initial offered traffic and results for other topologies are also made publicly availablephynwinfo.

§ RESULTS AND DISCUSSION

We evaluate Scheme 1 and Scheme 2 in aforementioned traffic scenarios. It must be noted that an additional scheme, where all BVTs are placed according to their maximum possible datarate has been evaluated in previous work<cit.>, compared to which, both Scheme 1 and Scheme 2 provide a higher network throughput. For expected traffic growth shown in Fig. <ref>, we see that offered traffic can be satisfied by both schemes till 2025. After which, we observe that the overall expected throughput for Scheme 2 (dashed line with squares) cannot meet the offered traffic. The cause may be attributed to the lack of additional frequency slots on some links, which cause additional LPs ($n_{c}$) found in Eq. <ref> to remain unprovisioned. This translates to a loss of approximately 40 Tbps expected throughput at 2030. In the same scenario, we see that Scheme 1 allows us to meet offered traffic without the need of a network upgrade for the entire $t$.

BVTs vs Throughput for Germany 17 and US 12

Similarly, for unexpected traffic growth (shown in in Fig. <ref>), upgrading provisioned LPs using Scheme 1 enables the operators to utilize the C-Band for five additional years. However, Scheme 1 also shows a downward trend in provisioned LPs, beginning from 2029. We infer that in order to meet the offered traffic, network operators would have to upgrade their network, using new equipment and fibers or by exploring additional bands. The question then arises as to when must the operator upgrade their network. In the evaluated topology, we observe that three links cross the occupied slot threshold of 75% in the year 2027, which is when additional capacity must be planned for. Post upgrade, the offered traffic can be met by the algorithm.

In Fig. <ref> we see that Scheme 1 achieves a 40% increase in throughput for the same number of BVTs in Germany 17 topology. Conversely, similar throughput can be achieved by Scheme 1 utilizing 25% lesser BVTs. For the 12 node US Abilene topology (US12), a 30% increase in throughput by using 18% lesser BVTs is observed. The reduction may be because of long distance demands in the US12 topology, due to which offered traffic can only be met using more BVTs at lower modulation formats. However, Scheme 1 still provides savings on the number of BVTs.

§ CONCLUSIONS

With increased global travel restrictions, core network traffic is bound to grow at a rate higher than initial forecasts. Pragmatically, operators may use BVT over-provisioning to meet such surges in traffic. However, our proposed solution, evaluated with two network planning schemes, shows that a 40% overall increase in aggregate offered traffic for a realistic German core network can be carried up to five years, by efficiently planning the datarate and bandwidth usage of configured BVTs. For higher traffic growth, an estimate on when to plan a physical network upgrade is also provided.

§ ACKNOWLEDGEMENTS

This work is partially funded by Germany's Federal Ministry of Education and Research under project OptiCON (grant IDs #16KIS0989K and #16KIS0991). The authors would also like to thank Dr. Jose-Juan Pedreno-Manresa for fruitful discussions.

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

noteAccessed: Aug 2020
titleKeeping the Internet up and running in times of crisis
noteAccessed: Aug 2020
titleADVA Teraflex

journaltitleIEEE/OSA JOCN
titleLoad-aware Nonlinearity Estimation for Elastic Optical Network Resource Optimization and Management

MDPI AG

journaltitleMDPI Photonics
titleStatistical Assessment of Open Optical Networks

booktitleProc. OFC
titleHeCSON: Heuristic for Configuration Selection in Optical Network Planning

booktitleProc. OFC
titleTraffic model for USA long-distance optical network
journaltitleDE-CIX Management GmbH
noteAccessed: Aug 2020
titleGermany DE-CIX
noteAccessed: Aug 2020
titleData Center Map

booktitleAdvanced Photonics
titleRobust Optical Networks with Emerging Coherent Technologies and Traffic Trends

booktitleProc. OFC
titleAccurate Non-Linearity Fully-Closed-Form Formula based on the GN/EGN Model and Large-Data-Set Fitting
noteAccessed: Aug 2020
titlePhysical Network Information


Let us briefly review the critical large $N$ vector models in flat space (see
[85] for a more detailed review). We start with the following action as before

$\displaystyle S[\phi^{I}]=\int
d^{d}x\left[\frac{1}{2}\left(\partial_{\mu}\phi^{I}\right)^{2}+m_{0}^{2}(\phi^{I})^{2}+\frac{\lambda_{0}}{4N}\Lambda^{4-d}\left(\phi^{I}\phi^{I}\right)^{2}\right]\,,$
(4.103)

where $m_{0},\lambda_{0}$ are bare mass and coupling constants, which should
be fine-tuned to bring the theory to criticality. We introduce the Hubbard-
Stratonovich (HS) field $\sigma$ and integrate over the fields $\phi^{I}$ to
obtain the following effective action for $\sigma$,

$\displaystyle S[\sigma]=N\left[\frac{1}{2}\mathop{\rm
tr}\nolimits\log\left[-\Delta+\sigma\right]+\int
d^{d}x\left(\mu_{0}\sigma+\Lambda^{d-4}\frac{\sigma^{2}}{4\lambda_{0}}\right)\right]\,,\quad\mu_{0}=\frac{m_{0}}{\lambda_{0}}\,,$
(4.104)

which can be evaluated by saddle-point function in the large $N$ limit. The
saddle-point equation for $\sigma$, also known as the gap equation, reads

$\displaystyle\left[\frac{1}{-\Delta+\sigma}\right]_{\mathcal{R}}(x,x)=\mu_{0}+\Lambda^{d-4}\frac{\sigma}{4\lambda_{0}}\,,$
(4.105)

after choosing a regularization scheme ${\mathcal{R}}$. To reach critically,
we solve for $\mu_{0}=\mu_{\rm crit}$ by requiring that $\sigma=0$ in the flat
space time. One can check that if $4>d>2$ the second term on the RHS of
(4.105) is irrelevant in the IR and can be neglected. Thus to the leading
order in the large $N$ limit, it boils down to solving the following equation,

$\displaystyle\left[\frac{1}{-\Delta+\sigma}\right]_{\mathcal{R}}(x,x)=\mu_{\rm
crit}\,.$ (4.106)

Now we can apply this technique to our setup which involves the vector model
on a wedge geometry. We work with the physical dimension $d=3$. Note that the
metric and the HS field $\sigma$ on such a background take the following form

$\displaystyle
ds^{2}=dx_{\perp}^{2}+d\rho^{2}+\rho^{2}d\varphi^{2}\,,\quad\varphi\in[0,\theta]\,,\quad\sigma(r,\varphi)=\frac{1}{r^{2}}\sigma(\varphi)\,,$
(4.107)

where $\sigma(\varphi)$ is some unknown function and to be determined by
minimizing the effective action (free energy) for $\sigma$. Once
$\sigma(\varphi)$ is fixed, then in principle we can determine any correlator
in this geometry in the large $N$ limit. Below, we will calculate the wedge
free energy by taking the coincident point limit of the two-point function as
we have done in the previous subsections using $\epsilon$ expansion. We will
encounter some divergences in the process which we will regulate by working on
a lattice. We will then minimize this free energy to determine the profile of
$\sigma$ and hence get the free energy at the critical point.

The scalar propagator of $\phi^{I}$ satisfies the following equation,

$\displaystyle\left(-\partial_{z}^{2}-\partial_{\rho}^{2}-\frac{1}{\rho}\partial_{\rho}+\frac{-\partial_{\varphi}^{2}+\sigma(\varphi)}{\rho^{2}}\right)G(x_{\perp},\rho,\varphi,x^{\prime}_{\perp},\rho^{\prime},\varphi^{\prime})=\frac{1}{\rho}\delta(\rho-\rho^{\prime})\delta(\varphi-\varphi^{\prime})\delta(z-z^{\prime})\,.$

Note that the angular part is completely decoupled and we can find the
corresponding eigenfunctions and eigenvalues

$\displaystyle\left(-\partial_{\varphi}^{2}+\sigma(\varphi)\right)\xi_{n}(\varphi)=E_{n}^{2}\xi_{n}(\varphi)\,,\quad\sum_{n}\xi^{*}_{n}(\varphi)\xi_{n}(\varphi^{\prime})=\delta(\varphi-\varphi^{\prime})\,.$
(4.108)

Then using these eigenfunctions, we can expand the propagator into the
following sum,

$\displaystyle
G(x_{\perp},\rho,\varphi,x^{\prime}_{\perp},\rho^{\prime},\varphi^{\prime})=\sum_{n}\xi^{*}_{n}(\varphi)\xi_{n}(\varphi^{\prime})\frac{1}{\sqrt{\rho\rho^{\prime}}}G_{n}(\rho,\rho^{\prime},x_{\perp},x_{\perp}^{\prime})\,,$
(4.109)

where $G_{n}$ is the propagator for a massive scalar field with mass
$\sqrt{E_{n}^{2}-1/4}$ on ${\mathbb{H}}^{2}$ which satisfies

$\displaystyle\left(-\rho^{2}(\partial_{x_{\perp}}^{2}+\partial_{\rho}^{2})+\left(E_{n}^{2}-1/4\right)\right)G_{n}(\rho,\rho^{\prime},x_{\perp},x_{\perp}^{\prime})=\rho^{2}\delta(\rho-\rho^{\prime})\delta(x_{\perp}-x_{\perp}^{\prime})\,.$
(4.110)

The explicit solutions are given by Legendre $Q$-functions [92]

$\displaystyle G_{n}(r,r^{\prime},x_{\perp},x_{\perp}^{\prime})\equiv
G_{n}(Z)=Q_{-\frac{1}{2}+E_{n}}(Z)\,,\quad Z\equiv
1+\frac{(\rho-\rho^{\prime})^{2}+(x_{\perp}-x_{\perp}^{\prime})^{2}}{2\rho\rho^{\prime}}\,.$
(4.111)

Expanded near $Z=1$, the above becomes [93]

$\displaystyle Q_{-\frac{1}{2}+E_{n}}(Z)=\frac{1}{2}\log
2-\gamma-\psi\left(E_{n}+\frac{1}{2}\right)-\frac{1}{2}\log(Z-1)+\mathcal{O}(Z-1)\,,$
(4.112)

where $\psi(x)$ is the digamma function. We will adopt the $\rm\overline{MS}$
scheme to regularize the propagator and obtain at coincident points,

$\displaystyle\rho
G(\rho,\varphi)=-\sum_{n}\xi_{n}^{*}(\varphi)\xi_{n}(\varphi)\psi\left(E_{n}+\frac{1}{2}\right)\,.$
(4.113)

This sum is still divergent, which we will handle with a lattice
regularization. We first consider the case when there is no wedge (i.e.
$\theta=2\pi$) and we approximate this circle by a lattice of $N$ sites
located at the angles $\varphi_{i}=\frac{2\pi i}{N}$. In this situation, we
can approximate the Schrödinger operator in (4.108) by the following finite
dimensional matrix $N\times N$,

$\displaystyle H=-\left[\partial_{\varphi}^{2}\right]_{N\times
N}+[\sigma(\varphi)]_{N\times N}\,,$ (4.114)

with

$\displaystyle\left[\partial_{\varphi}^{2}\right]_{N\times
N}=\frac{N^{2}}{(2\pi)^{2}}\begin{pmatrix}2&-1&0&\ldots&-1\\\
-1&2&-1&\ldots&0\\\ 0&-1&2&\ldots&0\\\ \ldots&\ldots&\ldots&\ldots&\ldots\\\
-1&0&0&\ldots&2\end{pmatrix}\,,\quad[\sigma(\varphi)]_{N\times
N}=\operatorname{diag}\left[\sigma\left(\frac{2\pi
i}{N}\right)\right]^{N}_{i=1}\,.$ (4.115)

Such a matrix can be diagonalized using Matlab to find energies $\hat{E}_{n}$
and eigenfunctions $\hat{\xi}_{n}(\varphi_{i})$, which are normalized as
$\sum\limits^{N}_{n=1}\hat{\xi}_{n}(\varphi_{i})\hat{\xi}^{*}_{n}(\varphi_{j})=N\delta_{ij}$.
The regularized propagator at coincident points then follows from

$\displaystyle\rho
G(\rho,\varphi_{i})=-\sum^{N}_{n=1}\hat{\xi}_{n}^{*}(\varphi_{i})\hat{\xi}_{n}(\varphi_{i})\psi\left(E_{n}+\frac{1}{2}\right)\,.$
(4.116)

In the large $N$ vector models, this regularized one-point function by the
virtue of the equations of motions is equal to the parameter $\mu$

$\displaystyle-\sum^{N}_{n=1}\hat{\xi}_{n}^{*}(\varphi_{i})\hat{\xi}_{n}(\varphi_{i})\psi\left(E_{n}+\frac{1}{2}\right)=\mu\,,$
(4.117)

and by setting $\mu$ to $\mu_{\rm crit}$ we tune the large $N$ vector model to
criticality. In the absence of wedge, this simply corresponds to setting
$\sigma(\varphi)\equiv 0$. In this situation we know the corresponding
eigenfunctions and energies,

$\displaystyle\hat{\xi}_{n}(\varphi_{i})=e^{\frac{2\pi in}{N}},\quad
E_{n}=\frac{N^{2}}{\pi^{2}}\sin^{2}\frac{\pi n}{N}\,,\quad\mu_{\rm
crit}=-\sum^{N}_{n=1}\psi\left(\frac{N^{2}}{\pi^{2}}\sin^{2}\frac{\pi
n}{N}+\frac{1}{2}\right)\,.$ (4.118)

Now we will consider that in this lattice realization of the $S^{1}$ we chop a
sub-lattice (a segment) of size $k$ and impose on the two ends the Dirichlet
(ordinary) boundary conditions.141414The Neumann (special) boundary conditions
could be implemented in the similar fashion, but the proposed numerical scheme
becomes very unstable. This would correspond to the study of the wedge CFT
with angle $\theta=\frac{2\pi k}{N}$. Then we can apply the same techniques as
before but instead of considering a Hamiltonian matrix of size $N\times N$, we
consider one of size $k\times k$. Again, we need to solve the following
spectral problem for new eigenvalues $\bar{E}_{n}$ and eigenfunctions
$\bar{\xi}_{n}(\phi_{i})$,

$\displaystyle H_{k\times
k}\bar{\xi}_{n}=\bar{E}_{n}\bar{\xi}_{n},\quad\sum^{k}_{n=1}\bar{\xi}^{*}_{n}(\phi_{i})\bar{\xi}_{n}(\phi_{j})=N\delta_{ij}\,,$
$\displaystyle-\sum^{k}_{n=1}\bar{\xi}_{n}^{*}(\varphi_{i})\bar{\xi}_{n}(\varphi_{i})\psi\left(\bar{E}_{n}+\frac{1}{2}\right)=\mu_{\rm
crit}=\sum^{N}_{n=1}\psi\left(\frac{N^{2}}{\pi^{2}}\sin^{2}\frac{\pi
n}{N}+\frac{1}{2}\right)\,.$ (4.119)

This equation could be solved using gradient descent by noticing that

$\displaystyle\frac{\partial
F}{\partial\sigma}=-\sum^{k}_{n=1}\bar{\xi}_{n}^{*}(\varphi_{i})\bar{\xi}_{n}(\varphi_{i})\psi\left(\bar{E}_{n}+\frac{1}{2}\right)-\mu_{\rm
crit}\,.$ (4.120)

In this way we can find numerically the $\sigma(\varphi)$ configuration for
any opening angle $\theta$. If we know the energy levels $E_{n}$, we can
extract the free energy as below,

$\displaystyle\frac{F}{A}=\sum^{k}_{n=1}\left[-\frac{\bar{E}_{n}}{4\pi}\log\Gamma\left(\frac{1}{2}+\bar{E}_{n}\right)+\frac{\psi^{(-2)}}{4\pi}\left(\frac{1}{2}+\bar{E}_{n}\right)\right]-\frac{\mu_{\rm
crit}}{8\pi}\int d\varphi\sigma(\varphi)-f_{0}\theta\,,$ (4.121)

where $f_{0}$ is a cosmological constant counterterm for the free energy. Let
us emphasize that in the above formula, there is always an contribution coming
from free energy of BCFT on the boundaries of the wedge (extensive in the
boundary directions). These contributions must be explicitly subtracted to get
the Casimir energy that captures the interactions between the boundaries.

Figure 7: On the left panel the dependence of the Casimir energy on the angle
of the wedge is displayed. On the right panel the typical profile of the HS
field $\sigma$ for $\theta=\frac{\pi}{2}$.

By implementing the above numerical procedure, we find that the typical
profile of the HS field $\sigma$ is of the form presented in the right panel
of the Figure 7. Near the boundary of the wedge, we should recover the usual
BCFT result. Indeed one can check that for small $\varphi$ the $\sigma$ field
behaves as $\sigma\approx-\frac{0.25}{\theta^{2}}$, that is in agreement with
the results of [51].151515For BCFT with ordinary boundary conditions, it was
shown in [51] that at large $N$, the saddle point value of $\sigma$ is
$(d-2)(d-4)/4$ which is $-1/4$ in $d=3$. In the left panel of the Figure 7 we
present the numerically computed Casimir energy as a function of the opening
angle of the wedge $\theta$. For small $\theta$, the dependence becomes power
like $F(\theta)=-\frac{(2.45\pm 0.05)\times 10^{-3}}{\theta^{2}}$. The
$1/\theta^{2}$ dependence is again as expected at small $\theta$ from general
considerations as discussed in Section 2.2.

##  5 Discussion

In this paper, we pursue a general investigation on the fusion of extended
defect operators in conformal field theory of dimension $d>2$, which
generalizes the more familiar operator product expansion for point operators.
We define a fusion product for a pair of parallel defects after subtracting
divergences at small transverse separation and explain general properties of
the corresponding fusion algebra for defects. The aforementioned divergences
are themselves physical observables that keep track of the interactions
between the pair of defect. In particular, the leading divergence captures the
Casimir energy due to the defect insertion. Furthermore we explain the
relation between parallel defect fusion and a wedge-like defect configuration
in CFT. The latter together with the utilization of hyperbolic spaces provides
an efficient tool to extract the Casimir energy between interfaces
(boundaries). To supplement the general discussion, we have determined these
defect fusion data (e.g. fusion algebra and Casimir energy) for a variety of
concrete examples including line defects and interfaces in the scalar $O(N)$
CFT and the fermionic Gross-Neveu(-Yukawa) CFT.

The ideas described here can be used to study the fusion of more general
defects in CFT. One obvious target is the monodromy defect for global
symmetries, such as the $O(N)$ monodromy defect studied in [48]. In the free
theory of a single complex scalar $\Phi$, the $U(1)$ monodromy defect
${\mathcal{D}}_{\vartheta}$ is defined by demanding that the scalar $\Phi$
picks up a phase as we go around the defect (using the same coordinates as in
(2.7) with the defect at $\rho=0$),

$\Phi(\vec{y},\rho,\varphi+2\pi)=e^{i\vartheta}\Phi(\vec{y},\rho,\varphi+2\pi)\,,\quad\vartheta\sim\vartheta+2\pi\,.$
(5.1)

Intuitively, the fusion product for such a monodromy defect is
${\mathcal{D}}_{\vartheta_{1}}\circ{\mathcal{D}}_{\vartheta_{2}}={\mathcal{D}}_{\vartheta_{1}+\vartheta_{2}}$.
However the Casimir energy is much less obvious. Another interesting example
in this class of CFTs is the magnetic line defect in the $O(N)^{3}$ tensor
model studied in [94]. In this case, there are a large zoo of conformal line
defects corresponding to different, inequivalent choices of the source term
$J_{abc}$ under the action of $O(N)^{3}$, and because of that we expect the
structure of defect fusion in the tensor models to be a lot more intricate.
For the fusion of codimension-one defects, it would be interesting to consider
cases with interactions localized on the boundary [95, 96, 97], which can
define a nontrivial boundary condition even when the bulk theory is free.
Another interesting phenomena to study will be the fusion of defects of
different co-dimensions, for instance, the fusion of line defects with
boundaries. Apart from exploring examples for defect fusion, it would also be
important to understand general constraints on the fusion algebra and the
Casimir energy from basic principles such as unitarity and locality.

We have used the wedge geometry to analyze the fusion of conformal boundaries
(also factorized interfaces) and in particular to extract the Casimir energy
in Section 4. However studying CFT on a wedge geometry is an interesting
problem in its own right, which has received some recent attention [77, 76].
In particular, as was pointed out in [76], one can write a crossing equation
by demanding that the one-point function of bulk operators, when expanded in
terms of operators on two different boundaries agree with each
other.161616There is one additional obvious channel where the bulk operator is
expanded into operators at the edge which leads to one more bootstrap equation
for the wedge setup. It will be interesting to exploit this crossing equation
in explicit interacting examples such as the large $N$ critical $O(N)$ model
or Gross-Neveu model. We expect the hyperbolic space method we have employed
here to calculate the wedge free energy will be useful in this context.

Finally, most of our calculations here in the explicit models utilized the
$\epsilon$ expansion. It will be useful to develop complimentary approaches
such as large $N$ expansion which may give us direct access to defect fusion
in the strongly coupled CFT at $d=3$. It is possible to study this numerically
directly in $d=3$ as we have done in Section 4.3 for the fusion of ordinary
boundary conditions in $d=3$ critical $O(N)$ model. But it will be desirable
to develop an analytic approach at large $N$ in arbitrary dimensions which can
serve as a cross-check of our results in $d=4-\epsilon$ which apply to
arbitrary $N$.

## Acknowledgements

The work of YW was supported in part by the NSF grant PHY-2210420 and by the
Simons Junior Faculty Fellows program. F.K.P. is currently a Simons Junior
Fellow at New York University and supported by a grant 855325FP from the
Simons Foundation.

## References

  * [1] J. Kondo, _Resistance Minimum in Dilute Magnetic Alloys_ , _Prog. Theor. Phys._ 32 (1964) 37.
  * [2] I. Affleck, _Conformal field theory approach to the Kondo effect_ , _Acta Phys. Polon. B_ 26 (1995) 1869 [cond-mat/9512099].
  * [3] I. Affleck, _A current algebra approach to the kondo effect_ , _Nuclear Physics B_ 336 (1990) 517.
  * [4] I. Affleck and A. W. Ludwig, _The kondo effect, conformal field theory and fusion rules_ , _Nuclear Physics B_ 352 (1991) 849.
  * [5] I. Affleck and A. W. Ludwig, _Critical theory of overscreened kondo fixed points_ , _Nuclear Physics B_ 360 (1991) 641.
  * [6] D. Poland, S. Rychkov and A. Vichi, _The Conformal Bootstrap: Theory, Numerical Techniques, and Applications_ , _Rev. Mod. Phys._ 91 (2019) 015002 [1805.04405].
  * [7] T. Hartman, D. Mazac, D. Simmons-Duffin and A. Zhiboedov, _Snowmass White Paper: The Analytic Conformal Bootstrap_ , in _2022 Snowmass Summer Study_ , 2, 2022, 2202.11012.
  * [8] D. Poland and D. Simmons-Duffin, _Snowmass White Paper: The Numerical Conformal Bootstrap_ , in _2022 Snowmass Summer Study_ , 3, 2022, 2203.08117.
  * [9] M. Billò, V. Gonçalves, E. Lauria and M. Meineri, _Defects in conformal field theory_ , _JHEP_ 04 (2016) 091 [1601.02883].
  * [10] A. Gadde, _Conformal constraints on defects_ , _JHEP_ 01 (2020) 038 [1602.06354].
  * [11] D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, _Generalized Global Symmetries_ , _JHEP_ 02 (2015) 172 [1412.5148].
  * [12] J. McGreevy, _Generalized Symmetries in Condensed Matter_ , 2204.03045.
  * [13] C. Cordova, T. T. Dumitrescu, K. Intriligator and S.-H. Shao, _Snowmass White Paper: Generalized Symmetries in Quantum Field Theory and Beyond_ , in _Snowmass 2021_ , 5, 2022, 2205.09545.
  * [14] S. Schafer-Nameki, _ICTP Lectures on (Non-)Invertible Generalized Symmetries_ , 2305.18296.
  * [15] T. D. Brennan and S. Hong, _Introduction to Generalized Global Symmetries in QFT and Particle Physics_ , 2306.00912.
  * [16] L. Bhardwaj, L. E. Bottini, L. Fraser-Taliente, L. Gladden, D. S. W. Gould, A. Platschorre et al., _Lectures on Generalized Symmetries_ , 2307.07547.
  * [17] S.-H. Shao, _What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetry_ , 2308.00747.
  * [18] B. Jones and C. Varma, _Critical point in the solution of the two magnetic impurity problem_ , _Physical Review B_ 40 (1989) 324.
  * [19] I. Affleck and A. W. W. Ludwig, _Exact critical theory of the two impurity Kondo model_ , _Phys. Rev. Lett._ 68 (1992) 1046.
  * [20] J. Gan, _Mapping the critical point of the two-impurity kondo model to a two-channel problem_ , _Physical review letters_ 74 (1995) 2583\.
  * [21] I. Affleck, A. W. Ludwig and B. A. Jones, _Conformal-field-theory approach to the two-impurity kondo problem: comparison with numerical renormalization-group results_ , _Physical Review B_ 52 (1995) 9528.
  * [22] J. Gan, _Solution of the two-impurity kondo model: Critical point, fermi-liquid phase, and crossover_ , _Physical Review B_ 51 (1995) 8287.
  * [23] A. Georges and A. M. Sengupta, _Solution of the two-impurity, two-channel kondo model_ , _Physical review letters_ 74 (1995) 2808.
  * [24] A. K. Mitchell and E. Sela, _Universal low-temperature crossover in two-channel kondo models_ , _Physical Review B_ 85 (2012) 235127\.
  * [25] A. O’Bannon, I. Papadimitriou and J. Probst, _A Holographic Two-Impurity Kondo Model_ , _JHEP_ 01 (2016) 103 [1510.08123].
  * [26] P. L. Lopes, I. Affleck and E. Sela, _Anyons in multichannel kondo systems_ , _Physical Review B_ 101 (2020) 085141.
  * [27] D. Gabay, C. Han, P. L. Lopes, I. Affleck and E. Sela, _Multi-impurity chiral kondo model: Correlation functions and anyon fusion rules_ , _Physical Review B_ 105 (2022) 035151.
  * [28] C. Bachas and M. Gaberdiel, _Loop operators and the Kondo problem_ , _JHEP_ 11 (2004) 065 [hep-th/0411067].
  * [29] C. Bachas and I. Brunner, _Fusion of conformal interfaces_ , _JHEP_ 02 (2008) 085 [0712.0076].
  * [30] C. Bachas, I. Brunner and D. Roggenkamp, _Fusion of Critical Defect Lines in the 2D Ising Model_ , _J. Stat. Mech._ 1308 (2013) P08008 [1303.3616].
  * [31] A. Konechny, _Fusion of conformal interfaces and bulk induced boundary RG flows_ , _JHEP_ 12 (2015) 114 [1509.07787].
  * [32] A. Söderberg, _Fusion of conformal defects in four dimensions_ , _JHEP_ 04 (2021) 087 [2102.00718].
  * [33] D. Rodriguez-Gomez, _A Scaling Limit for Line and Surface Defects_ , 2202.03471.
  * [34] A. Söderberg Rousu, _Fusion of conformal defects in interacting theories_ , _JHEP_ 10 (2023) 183 [2304.10239].
  * [35] M. Krech and S. Dietrich, _Free energy and specific heat of critical films and surfaces_ , _Phys. Rev. A_ 46 (1992) 1886.
  * [36] H. W. Diehl, D. Gruneberg and M. A. Shpot, _Fluctuation-induced forces in periodic slabs: Breakdown of epsilon expansion at the bulk critical point and revised field theory_ , _EPL_ 75 (2006) 241 [cond-mat/0605293].
  * [37] D. Gruneberg and H. W. Diehl, _Thermodynamic Casimir effects involving interacting field theories with zero modes_ , _Phys. Rev. B_ 77 (2008) 115409 [0710.4436].
  * [38] H. W. Diehl and F. M. Schmidt, _Critical Casimir effect in films for generic non-symmetry-breaking boundary conditions_ , _New J. Phys._ 13 (2011) 123025 [1110.1241].
  * [39] H. W. Diehl, D. Grüneberg, M. Hasenbusch, A. Hucht, S. B. Rutkevich and F. M. Schmidt, _Large- $n$ approach to thermodynamic Casimir effects in slabs with free surfaces_, _Phys. Rev. E_ 89 (2014) 062123 [1402.3510].
  * [40] O. Diatlyk, H. Khanchandani, F. Popov and Y. Wang, _to appear_ , .
  * [41] Y. Choi, C. Cordova, P.-S. Hsin, H. T. Lam and S.-H. Shao, _Non-invertible Condensation, Duality, and Triality Defects in 3+1 Dimensions_ , _Commun. Math. Phys._ 402 (2023) 489 [2204.09025].
  * [42] L. Bhardwaj, L. E. Bottini, S. Schafer-Nameki and A. Tiwari, _Non-invertible higher-categorical symmetries_ , _SciPost Phys._ 14 (2023) 007 [2204.06564].
  * [43] T. Bartsch, M. Bullimore, A. E. V. Ferrari and J. Pearson, _Non-invertible Symmetries and Higher Representation Theory II_ , 2212.07393.
  * [44] C. Copetti, M. Del Zotto, K. Ohmori and Y. Wang, _Higher Structure of Chiral Symmetry_ , 2305.18282.
  * [45] A. Kapustin and N. Saulina, _Surface operators in 3d Topological Field Theory and 2d Rational Conformal Field Theory_ , 1012.0911.
  * [46] K. Roumpedakis, S. Seifnashri and S.-H. Shao, _Higher Gauging and Non-invertible Condensation Defects_ , _Commun. Math. Phys._ 401 (2023) 3043 [2204.02407].
  * [47] D. Correa, J. Maldacena and A. Sever, _The quark anti-quark potential and the cusp anomalous dimension from a TBA equation_ , _JHEP_ 08 (2012) 134 [1203.1913].
  * [48] S. Giombi, E. Helfenberger, Z. Ji and H. Khanchandani, _Monodromy defects from hyperbolic space_ , _JHEP_ 02 (2022) 041 [2102.11815].
  * [49] A. Allais and S. Sachdev, _Spectral function of a localized fermion coupled to the Wilson-Fisher conformal field theory_ , _Phys. Rev. B_ 90 (2014) 035131 [1406.3022].
  * [50] G. Cuomo, Z. Komargodski and M. Mezei, _Localized magnetic field in the O(N) model_ , _JHEP_ 02 (2022) 134 [2112.10634].
  * [51] S. Giombi, E. Helfenberger and H. Khanchandani, _Line defects in fermionic CFTs_ , _JHEP_ 08 (2023) 224 [2211.11073].
  * [52] W. D. Goldberger and M. B. Wise, _Renormalization group flows for brane couplings_ , _Phys. Rev. D_ 65 (2002) 025011 [hep-th/0104170].
  * [53] S. Fredenhagen, M. R. Gaberdiel and C. A. Keller, _Bulk induced boundary perturbations_ , _J. Phys. A_ 40 (2007) F17 [hep-th/0609034].
  * [54] B. Michel, E. Mintun, J. Polchinski, A. Puhm and P. Saad, _Remarks on brane and antibrane dynamics_ , _JHEP_ 09 (2015) 021 [1412.5702].
  * [55] A. M. Polyakov, _Effect of strong interactions on vacuum polarization_ , _Zh. Eksp. Teor. Fiz._ 63 (1972) 24.
  * [56] M. A. Metlitski, _Boundary criticality of the O(N) model in d = 3 critically revisited_ , 2009.05119.
  * [57] A. Krishnan and M. A. Metlitski, _A plane defect in the 3d O(N) model_ , _SciPost Phys._ 15 (2023) 090 [2301.05728].
  * [58] S. Giombi and B. Liu, _Notes on a surface defect in the O(N) model_ , _JHEP_ 12 (2023) 004 [2305.11402].
  * [59] A. Raviv-Moshe and S. Zhong, _Phases of surface defects in Scalar Field Theories_ , _JHEP_ 08 (2023) 143 [2305.11370].
  * [60] M. Trépanier, _Surface defects in the O(N) model_ , _JHEP_ 09 (2023) 074 [2305.10486].
  * [61] H. W. Diehl, _Field-theoretical approach to critical behaviour at surfaces_ , in _Phase Transitions and Critical Phenomena_ , C. Domb and J. L. Lebowitz, eds., vol. 10, (London), pp. 75–267, Academic, (1986).
  * [62] H. W. Diehl, _The theory of boundary critical phenomena_ , _International Journal of Modern Physics B_ 11 (1997) 3503\.
  * [63] J. Cardy, _Scaling and Renormalization in Statistical Physics_ , Cambridge Lecture Notes in Physics. Cambridge University Press, 1996, 10.1017/CBO9781316036440.
  * [64] H. W. Diehl and S. Dietrich, _Field-theoretical approach to multicritical behavior near free surfaces_ , _Phys. Rev._ B24 (1981) 2878.
  * [65] D. M. McAvity and H. Osborn, _Energy momentum tensor in conformal field theories near a boundary_ , _Nucl. Phys._ B406 (1993) 655 [hep-th/9302068].
  * [66] D. M. McAvity and H. Osborn, _Conformal field theories near a boundary in general dimensions_ , _Nucl. Phys._ B455 (1995) 522 [cond-mat/9505127].
  * [67] A. J. Bray and M. A. Moore, _Surface critical exponents in terms of bulk exponents_ , _Phys. Rev. Lett._ 38 (1977) 1046.
  * [68] K. Ohno and Y. Okabe, _The 1/n Expansion for the n-Vector Model in the Semi-Infinite Space_ , _Progress of Theoretical Physics_ 70 (1983) 1226 [https://academic.oup.com/ptp/article-pdf/70/5/1226/5465952/70-5-1226.pdf].
  * [69] P. Liendo, L. Rastelli and B. C. van Rees, _The Bootstrap Program for Boundary CFT d_, _JHEP_ 07 (2013) 113 [1210.4258].
  * [70] F. Gliozzi, P. Liendo, M. Meineri and A. Rago, _Boundary and Interface CFTs from the Conformal Bootstrap_ , _JHEP_ 05 (2015) 036 [1502.07217].
  * [71] A. Bissi, T. Hansen and A. Söderberg, _Analytic Bootstrap for Boundary CFT_ , _JHEP_ 01 (2019) 010 [1808.08155].
  * [72] A. Kaviraj and M. F. Paulos, _The Functional Bootstrap for Boundary CFT_ , 1812.04034.
  * [73] D. Carmi, L. Di Pietro and S. Komatsu, _A Study of Quantum Field Theories in AdS at Finite Coupling_ , _JHEP_ 01 (2019) 200 [1810.04185].
  * [74] S. Giombi and H. Khanchandani, _CFT in AdS and boundary RG flows_ , _JHEP_ 11 (2020) 118 [2007.04955].
  * [75] J. L. Cardy, _Critical behaviour at an edge_ , _Journal of Physics A: Mathematical and General_ 16 (1983) 3617.
  * [76] A. Antunes, _Conformal bootstrap near the edge_ , _JHEP_ 10 (2021) 057 [2103.03132].
  * [77] A. Bissi, P. Dey, J. Sisti and A. Söderberg, _Interacting conformal scalar in a wedge_ , 2206.06326.
  * [78] M. A. Metlitski, C. A. Fuertes and S. Sachdev, _Entanglement Entropy in the O(N) model_ , _Phys. Rev. B_ 80 (2009) 115122 [0904.4477].
  * [79] R. Camporesi and A. Higuchi, _Spectral functions and zeta functions in hyperbolic spaces_ , _J. Math. Phys._ 35 (1994) 4217.
  * [80] A. A. Bytsenko, G. Cognola and S. Zerbini, _Quantum fields in hyperbolic space-times with finite spatial volume_ , _Class. Quant. Grav._ 14 (1997) 615 [hep-th/9605209].
  * [81] A. Erdélyi and F. G. Tricomi, _The asymptotic expansion of a ratio of gamma functions._ , _Pacific Journal of Mathematics_ 1 (1951) 133 .
  * [82] M. Abramowitz and I. A. Stegun, _Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing._ New York City, 1972\.
  * [83] V. I. Arnol’d, _Mathematical methods of classical mechanics_ , vol. 60. Springer Science & Business Media, 2013.
  * [84] D. J. Gross and A. Neveu, _Dynamical symmetry breaking in asymptotically free field theories_ , _Phys. Rev. D_ 10 (1974) 3235.
  * [85] M. Moshe and J. Zinn-Justin, _Quantum field theory in the large n limit: A review_ , _Physics Reports_ 385 (2003) 69.
  * [86] S. Giombi, E. Helfenberger and H. Khanchandani, _Fermions in AdS and Gross-Neveu BCFT_ , _JHEP_ 07 (2022) 018 [2110.04268].
  * [87] K. Johnson, _The M.I.T. Bag Model_ , _Acta Phys. Polon. B_ 6 (1975) 865.
  * [88] R. D. M. De Paola, R. B. Rodrigues and N. F. Svaiter, _Casimir energy of massless fermions in the slab bag_ , _Mod. Phys. Lett. A_ 14 (1999) 2353 [hep-th/9905039].
  * [89] P. W. Milonni, _The Quantum vacuum: An Introduction to quantum electrodynamics_. 1994.
  * [90] R. Camporesi, _The Spinor heat kernel in maximally symmetric spaces_ , _Commun. Math. Phys._ 148 (1992) 283.
  * [91] Y. Sato, _Free energy and defect $C$-theorem in free fermion_, _JHEP_ 05 (2021) 202 [2102.11468].
  * [92] A. Polyakov, _De sitter space and eternity_ , _Nuclear Physics B_ 797 (2008) 199.
  * [93] H. Bateman, _Higher transcendental functions [volumes i-iii]_ , vol. 1. McGRAW-HILL book company, 1953.
  * [94] F. K. Popov and Y. Wang, _Non-perturbative defects in tensor models from melonic trees_ , _JHEP_ 11 (2022) 057 [2206.14206].
  * [95] S. Giombi and H. Khanchandani, _$O(N)$ models with boundary interactions and their long range generalizations_, _JHEP_ 08 (2020) 010 [1912.08169].
  * [96] C. Behan, L. Di Pietro, E. Lauria and B. C. Van Rees, _Bootstrapping boundary-localized interactions_ , _JHEP_ 12 (2020) 182 [2009.03336].
  * [97] C. Behan, L. Di Pietro, E. Lauria and B. C. van Rees, _Bootstrapping boundary-localized interactions II. Minimal models at the boundary_ , _JHEP_ 03 (2022) 146 [2111.04747].


# Momentum scalar triple product as a measure of chirality in electron
ionization dynamics of strongly-driven atoms

G. P. Katsoulis  Department of Physics and Astronomy, University College
London, Gower Street, London WC1E 6BT, United Kingdom  Z. Dube  Joint
Attosecond Science Lab of the National Research Council and the University of
Ottawa, Ottawa, Ontario K1A 0R6, Canada  P. Corkum  Joint Attosecond Science
Lab of the National Research Council and the University of Ottawa, Ottawa,
Ontario K1A 0R6, Canada  A. Staudte  Joint Attosecond Science Lab of the
National Research Council and the University of Ottawa, Ottawa, Ontario K1A
0R6, Canada  A. Emmanouilidou  Department of Physics and Astronomy, University
College London, Gower Street, London WC1E 6BT, United Kingdom

###### Abstract

We formulate a transparent measure that quantifies chirality in single
electron ionization triggered in atoms, which are achiral systems. We do so in
the context of Ar driven by a new type of optical fields that consists of two
non-collinear laser beams giving rise to chirality that varies in space across
the focus of the beams. Our computations account for realistic experimental
conditions. To define this measure of chirality, we first find the sign of the
electron final momentum scalar triple product
$\mathrm{{\bf{p}}_{k}\cdot({\bf{p}}_{i}\times{\bf p}_{j})}$ and multiply it
with the probability for an electron to ionize with certain values for both
$\mathrm{p_{k}}$ and $\mathrm{p_{i}p_{j}}$. Then, we integrate this product
over all values of $\mathrm{p_{k}}$ and $\mathrm{p_{i}p_{j}}$. We show this to
be a robust measure of chirality in electron ionization triggered by globally
chiral electric fields.

Ultrafast phenomena in chiral molecules triggered by intense, infrared laser
pulses are at the forefront of laser-matter interactions Cireasa  _et al._
(2015); Beaulieu  _et al._ (2017, 2018); Comby  _et al._ (2018); Rozen  _et
al._ (2019). While ultrafast chiral processes can be studied using high
harmonic generation (HHG) Cireasa  _et al._ (2015); Ayuso  _et al._ (2019);
Baykusheva and Wörner (2018); Neufeld and Cohen (2018); Neufeld  _et al._
(2019); Heinrich  _et al._ (2021), the underlying recollision mechanism
entails that a stronger chiral response comes at the expense of a greatly
suppressed high harmonic signal Cireasa  _et al._ (2015). Hence, photoelectron
spectroscopy is a promising route to a robust signal from molecules driven by
intense chiral fields Lux  _et al._ (2012); Lehmann  _et al._ (2013); Boge
_et al._ (2014); Beaulieu  _et al._ (2017, 2018); Rozen  _et al._ (2019).
However, the sensitivity of chiral photoelectron spectroscopy also struggles
with the fact that laser wavelengths are several orders of magnitude larger
than the molecular dimensions, i.e., the chiralities of the optical field and
the molecule are incommensurate.

Recently, Ayuso et al. proposed a new type of optical field which is chiral on
the atomic scale Ayuso  _et al._ (2019) and thereby holds the potential for
unprecedented chiral sensitivity. The chiral field is synthesized by combining
two orthogonally polarized two-color laser fields in a non-collinear geometry
as illustrated in Fig. 1. The non-collinear geometry creates an intensity and
ellipticity grating, and thereby causes the chirality of the laser field to
spatially vary across the focus. Thus, it is a fundamental challenge for
experiments to decipher the signatures of chirality in the photoelectron
spectra from these new laser fields.

Here, we provide a simple prescription on how to analyze experimental
photoelectron spectra produced from any type of chiral light. To this end, we
perform semi-classical simulations of strong-field ionization, taking into
account the focal volume distribution of the degree of light chirality. To
develop an understanding of chiral electron ionization we model atomic
photoionization, since ground-state atoms have spherical symmetry and are
intrinsically achiral systems. Thus, the chiral response of the escaping
electron that is imprinted on the ionization spectra in our model arises
solely from the dynamics triggered by the electric field of the laser.

Figure 1: Schematic plot of two non-collinear laser beams, each consisting of
$\omega$-$2\omega$ orthogonally polarized linear fields; $\omega$ is polarized
on the x-y plane and 2$\omega$ along the z-axis. Both beams propagate towards
the atom in the focus region (red-shaded ellipse). The resultant electric
field has chirality (square inset) that changes along the x-axis in the focus
region.

In order to analyze the resulting photoelectron spectra we identify a
transparent measure that quantifies chirality in electron ionization dynamics
ensuing from an atom strongly-driven by a chiral electric field. We construct
this measure by finding the probability P$(\mathrm{p_{k},p_{i}p_{j}})$ for an
electron to ionize with certain values for both $\mathrm{p_{k}}$ and
$\mathrm{p_{i}p_{j}}$, with pi,pj,pk being the components of the final
electron momentum. Next, we multiply this probability by the sign of the
momentum scalar triple product
$\mathrm{{\bf{p}}_{k}\cdot({\bf{p}}_{i}\times{\bf p}_{j})}$. Integrating over
the whole range of values of $\mathrm{p_{k}}$ and $\mathrm{p_{i}p_{j}}$, we
obtain a measure of chirality

$\displaystyle\begin{split}\mathcal{X}\mathrm{(p_{k},p_{i}p_{j})}&=\\\
&\mathrm{\iint sign(\mathrm{{\bf{p}}_{k}\cdot({\bf{p}}_{i}\times{\bf
p}_{j})})P(\mathrm{p_{k},p_{i}p_{j}})dp_{i}p_{j}dp_{k}}.\end{split}$ (1)

We show that $\mathcal{X}$ has an opposite sign for synthetic pulses with
opposite chirality, while it is zero for achiral synthetic pulses. This
measure of handedness of electron ionization dynamics is a general one and
accounts for chiral electron motion triggered by any chiral light.

We demonstrate that $\mathcal{X}$ is a measure of handedness of electron
ionization dynamics ensuing from atoms, in the context of Ar driven by two
non-collinear laser beams, see Fig. 1. Beams 1, 2 propagate on the x-y plane
with wavevectors $\mathbf{k}_{1}$, $\mathbf{k}_{2}$ forming an angle $\alpha$
with the y axis

$\displaystyle\begin{split}\mathbf{k}_{1}&=\mathrm{k}\sin(\alpha)\hat{\mathbf{x}}+\mathrm{k}\cos(\alpha)\hat{\mathbf{y}}\\\
\mathbf{k}_{2}&=-\mathrm{k}\sin(\alpha)\hat{\mathbf{x}}+\mathrm{k}\cos(\alpha)\hat{\mathbf{y}},\end{split}$
(2)

where $k=\mathrm{2\pi}/\lambda$. The electric field of each beam consists of
two orthogonally polarized $\omega$ and $2\omega$ laser fields. The $\omega$
field is polarized along the x-y plane and the $2\omega$ field is polarized
along the z-axis. Also, we take the 2$\omega$ field to have small intensity
compared to the $\omega$ field.

The resultant electric field is given by Ayuso  _et al._ (2019)

$\mathbf{E}(\mathbf{r},\mathrm{t})=2\mathrm{E}_{0}\exp\left[-\left(\frac{t}{\tau}\right)^{2}\right]\left(\mathrm{E_{x}}\hat{\mathbf{x}}+\mathrm{E_{y}}\hat{\mathbf{y}}+\mathrm{E_{z}}\hat{\mathbf{x}}\right),$
(3)

where

$\displaystyle\begin{split}\mathrm{E_{x/y}}&=\exp\left[-\left(\frac{\rho}{\mathrm{w_{0}}}\right)^{2}\right]\mathrm{f_{x/y}(x)}\cos\left[\mathrm{g(y,t)}\right]\\\
\mathrm{E_{z}}&=\exp\left[-\left(\frac{2\rho}{\mathrm{w_{0}}}\right)^{2}\right]\mathrm{f_{z}(x)}\cos\left[\mathrm{h(y,t)}\right],\end{split}$
(4)

and

$\displaystyle\begin{split}\mathrm{f_{x}(x)}&=\cos(\alpha)\cos\left[\mathrm{k}\sin(\alpha)x+\frac{\phi_{2}^{\omega}-\phi_{1}^{\omega}}{2}\right]\\\
\mathrm{f_{y}(x)}&=\sin(\alpha)\sin\left[\mathrm{k}\sin(\alpha)x+\frac{\phi_{2}^{\omega}-\phi_{1}^{\omega}}{2}\right]\\\
\mathrm{f_{z}(x)}&=\mathrm{r_{0}}\cos\left[2\mathrm{k}\sin(\alpha)x+(\phi_{2}^{2\omega}-\phi_{1}^{2\omega})\right]\\\
\mathrm{g(y,t)}&=\mathrm{k}\cos(\alpha)\mathrm{y}-\omega\mathrm{t}-\frac{\phi_{2}^{\omega}+\phi_{1}^{\omega}}{2}\\\
\mathrm{h(y,t)}&=2\mathrm{k}\cos(\alpha)\mathrm{y}-2\omega\mathrm{t}-(\phi_{2}^{2\omega}+\phi_{1}^{2\omega}).\end{split}$
(5)

We note that $\tau=25$ fs and $\tau\sqrt{2\ln(2)}$ is the full width at half
maximum of the pulse duration in intensity, while $\mathrm{E_{0}}$ is the
field strength corresponding to intensity $\mathrm{5\times
10^{13}\;W/cm^{2}}.$ Also, $\rho$ is the radial distance to the propagation
axis of each laser beam. Since $\alpha$ is small, $5^{\circ}$, it follows that
$\rho\approx\sqrt{\mathrm{x}^{2}+\mathrm{z}^{2}}$. Moreover,
$\mathrm{w_{0}=8.49\;\mu m}$ is the beam waist of the $\omega$ laser field,
and $\mathrm{r^{2}_{0}}$ is the intensity ratio of 1/100 of the 2$\omega$
versus the $\omega$ field. Finally, the wavelength $\lambda$ of the
$\mathrm{\omega}$ field is taken equal to 800 nm.

We treat single electron ionization of driven Ar by employing a three-
dimensional (3D) semi-classical model. The only approximation is the initial
state. One electron tunnel-ionizes through the field-lowered Coulomb-barrier
at time t0. To compute the tunnel-ionization rate, we employ the quantum
mechanical Ammosov-Delone-Krainov (ADK) formula Landau and Lifshitz (2013);
Delone and Krainov (1991). We use parabolic coordinates to obtain the exit
point of the tunneling electron along the laser-field direction Hu  _et al._
(1997). We set the electron momentum along the laser field equal to zero,
while we obtain the transverse momentum by a Gaussian distribution Landau and
Lifshitz (2013); Delone and Krainov (1991). The microcanonical distribution is
employed to describe the initial state of the initially bound electron Abrines
and Percival (1966).

We select the tunnel-ionization time, t0, randomly in the time interval
[-2$\tau$,2$\tau$]. Next, we specify at time $\mathrm{t_{0}}$ the initial
conditions for both electrons. Using the three-body Hamiltonian of the two
electrons with the nucleus kept fixed, we propagate classically in time the
position and momentum of each electron. All Coulomb forces and the interaction
of each electron with the electric field in Eq. (3) are fully accounted for
with no approximation. To account for the Coulomb singularity, we employ
regularized coordinates Kustaanheimo and Stiefel (1965). Here, we use atomic
units.

Previous successes of this model include identifying the mechanism underlying
the fingerlike structure in the correlated electron momenta for He driven by
800 nm laser fields Emmanouilidou (2008), see also Parker  _et al._ (2006);
Staudte  _et al._ (2007); Rudenko  _et al._ (2007). Moreover, we investigated
the direct versus the delayed pathway of non-sequential double ionization for
He driven by a 400 nm, long duration laser pulse and achieved excellent
agreement with fully ab-initio quantum mechanical calculations Emmanouilidou
_et al._ (2011). Also, for low intensities, we have identified a novel
mechanism of double ionization, namely, slingshot non-sequential double
ionization Katsoulis  _et al._ (2018). In addition, for several observables of
non-sequential double ionization, our results have good agreement with
experimental results for Ar when driven by near-single-cycle laser pulses at
800 nm Chen  _et al._ (2017).

It was previously shown Ayuso  _et al._ (2019) that the resultant electric
field is globally chiral if the relative phases of the $\omega$ and $2\omega$
laser fields in beams 1 and 2, i.e. $\phi_{1}^{2\omega}-\phi_{1}^{\omega}$ and
$\phi_{2}^{2\omega}-\phi_{2}^{\omega}$, satisfy the following condition

$\mathrm{\left(\phi^{2\omega}_{1}-\phi^{\omega}_{1}\right)-\left(\phi^{2\omega}_{2}-\phi^{\omega}_{2}\right)=\dfrac{\pi}{2}+n\pi,\text{with
n}\in}\mathbb{Z},$ (6)

while the resultant electric field is globally achiral when

$\mathrm{\left(\phi^{2\omega}_{1}-\phi^{\omega}_{1}\right)-\left(\phi^{2\omega}_{2}-\phi^{\omega}_{2}\right)=n\pi,\text{with
n}\in}\mathbb{Z}.$ (7)

To illustrate that $\mathcal{X}$ is a measure of chirality in electron
ionization of driven atoms, we perform six independent studies. Each study
corresponds to Ar being driven by one of six different resultant electric
fields corresponding to six different synthetic pulses. For simplicity, we
refer to the resultant electric field of the synthetic pulse as electric
field. Each of the six synthetic pulses (cases 1-6) corresponds to a different
combination of $\phi_{1}^{2\omega}-\phi_{1}^{\omega}$ and
$\phi_{2}^{2\omega}-\phi_{2}^{\omega}$ for beams 1 and 2, respectively, see
Fig. 2(a). Using the conditions in Eq. (6) and Eq. (7), we select four
globally chiral electric fields, cases 1,2,4,5, and two globally achiral
fields, cases 3, 6, see Fig. 2. In Fig. 2(b), we show that the electric fields
which are globally chiral maintain the same handedness along the x-axis in the
focus region. That is, Ey(x)/Ex(x) and Ez(x) change sign at the same points in
space x. As a result, electric fields 1 and 4 have the same chirality (+) in
Fig. 2(b) and electric fields 2 and 5 have the same chirality (-) in Fig.
2(b). It follows that the pairs of electric fields (1,2) and (4,5) have
opposite chirality. Also, when Ey(x)/Ex(x) and Ez(x) change sign at different
points in space x as defined by Eq. (7), the chirality of the electric field
flips sign along the x-axis in the focus region. Hence, the electric field has
no overall chirality. This is the case for the globally achiral fields 3 and 6
shown in Fig. 2(b).

Figure 2: (a) Schematic plots of six combinations of two non-collinear beams;
(b) change of sign of Ey(x)/Ex(x) (black) and Ez(x) (blue) along the x-axis in
the focus region, at y=z=0, t=T/50. Cases 1,2 correspond to globally chiral
electric fields with opposite handedness, the same holds for cases 4 and 5.
Cases 3 and 6 correspond to globally achiral electric fields. For each case,
above beams 1 and 2, we denote $\phi_{1}^{2\omega}-\phi_{1}^{\omega}$ and
$\phi_{2}^{2\omega}-\phi_{2}^{\omega}$, respectively.

Next, we describe how we obtain the electron ionization spectra of Ar for each
of the six synthetic pulses (cases 1-6). For simplicity, for each case, we set
$\phi_{2}^{\omega}=0$. Since only the differences
$\phi_{1}^{2\omega}-\phi_{1}^{\omega}$ and
$\phi_{2}^{2\omega}-\phi_{2}^{\omega}$ are important, there is no loss of
generality. Moreover, for each of the six synthetic pulses, to simulate
realistic experimental conditions, we select 101 equally spaced values of the
phase $\mathrm{\phi_{1}^{\omega}}$ in the interval [0,2$\pi$). This allows us
to account for the nucleus being at different positions along the x-axis in
the focus region. Next, for each of the 101 values of
$\mathrm{\phi_{1}^{\omega}}$, we register the single ionisation events and
obtain the electron ionization spectra. Then, we average over all
$\mathrm{\phi_{1}^{\omega}}$ values and obtain the electron spectra
$\mathrm{P^{m}}(\mathrm{p_{x}},\mathrm{p_{y}p_{z}})$,
$\mathrm{P^{m}}(\mathrm{p_{y}},\mathrm{p_{z}p_{x}})$ and
$\mathrm{P^{m}}(\mathrm{p_{z}},\mathrm{p_{x}p_{y}})$. We normalize each one of
these spectra to one. The m index corresponds to the m electric field, i.e. to
case m and ranges from 1-6. For each synthetic pulse 1-6, the electron
ionization spectra are obtained using at least 107 singly ionizing
trajectories.

Figure 3: Probability distribution
$\mathrm{P^{m}}(\mathrm{p_{x}},\mathrm{p_{y}p_{z}})$ for the electron to
ionize with momenta $\mathrm{p_{x}}$ and $\mathrm{p_{y}p_{z}}$, with m=1,2,3
for electric fields 1,2,3, respectively. The sign in each quadrant corresponds
to the sign of $\mathrm{{\bf{p}}_{x}\cdot({\bf{p}}_{y}\times{\bf p}_{z})}$ in
this quadrant.

In Fig. 3, we plot the probability distribution
$\mathrm{P^{m}}(\mathrm{p_{x}},\mathrm{p_{y}p_{z}})$ for an electron to singly
ionize with momenta px and p$\mathrm{{}_{y}p_{z}}$ for the globally chiral 1,
2 and globally achiral 3 electric fields. In each quadrant we assign the sign
resulting from the scalar triple product
$\mathrm{{\bf{p}}_{x}\cdot({\bf{p}}_{y}\times{\bf p}_{z})}$. Next, using Eq.
(1), we compute the measure of chirality in electron ionization
$\mathcal{X}(\mathrm{p_{x}},\mathrm{p_{y}p_{z}})$. We find it to be equal to
$1.8\times 10^{-2},-1.9\times 10^{-2},-0.1\times 10^{-2}$ for electric fields
1,2,3, respectively. Indeed, a close inspection of Fig. 3 for case 1 reveals
that the probability distribution of the electron momenta px and
p$\mathrm{{}_{y}p_{z}}$ has larger values at the first and third quadrants,
where $\mathrm{{\bf{p}}_{x}\cdot({\bf{p}}_{y}\times{\bf p}_{z})}$ has a +
sign. It follows that $\mathcal{X}$ has a positive value when Ar is driven by
synthetic pulse 1. In contrast, in Fig. 3 for case 2 the probability
distribution of the electron momenta px and p$\mathrm{{}_{y}p_{z}}$ has larger
values at the second and fourth quadrants, where
$\mathrm{{\bf{p}}_{x}\cdot({\bf{p}}_{y}\times{\bf p}_{z})}$ has a - sign.
Hence, $\mathcal{X}$ has a negative value for case 2. The opposite signs of
$\mathcal{X}$ when Ar is driven by electric fields 1 and 2 are consistent with
the opposite chirality of these fields. Moreover, $|\mathcal{X}|$ is roughly
the same for cases 1 and 2. The small offset is due to the statistical error
introduced in our computations from the number of single ionization events
considered. This is also supported by $\mathcal{X}$ being equal to $-0.1\times
10^{-2}$, instead of zero, when Ar is driven by the achiral field 3.

Very interestingly, we find that all three measures of chirality
$\mathcal{X}\mathrm{(p_{x},p_{y}p_{z})}$,
$\mathcal{X}\mathrm{(p_{y},p_{z}p_{x})}$ and
$\mathcal{X}\mathrm{(p_{z},p_{x}p_{y})}$ have the same values for electric
field 1, 2 and 3. That is, all three measures are equal to $1.8\times 10^{-2}$
for electric field 1, equal to -$1.9\times 10^{-2}$ for field 2 and equal to
-$0.1\times 10^{-2}$ for field 3, see Table 1. The same is true for all three
measures of chirality $\mathcal{X}$ when Ar is driven by synthetic pulses
4,5,6. Indeed, all three $\mathcal{X}$ are equal to 1.$\times 10^{-2}$ for
electric field 4, equal to $-1.1\times 10^{-2}$ for electric field 5 and equal
to -$0.1\times 10^{-2}$ for electric field 6, see Table II. The above further
corroborate that $\mathcal{X}$ is a robust measure of chirality in electron
ionization triggered by a chiral field, yielding the same value for any of the
three combinations of the components of the final electron momentum.

Table 1: Measure of chirality $\mathcal{X}$ in electron ionization for globally chiral electric fields 1,2,4,5, and globally achiral fields 3,6. $\mathcal{X}$ is computed for each of the three $\mathrm{P^{m}}(\mathrm{p_{x}},\mathrm{p_{y}p_{z}})$, $\mathrm{P^{m}}(\mathrm{p_{y}},\mathrm{p_{z}p_{x}})$ and $\mathrm{P^{m}}(\mathrm{p_{z}},\mathrm{p_{x}p_{y}})$, and m ranges from 1-6. The values are expressed in $10^{-2}.$ Case | $\mathcal{X}\mathrm{\left(p_{x},p_{y}p_{z}\right)}$ | $\mathcal{X}\mathrm{\left(p_{y},p_{z}p_{x}\right)}$ | $\mathcal{X}\mathrm{\left(p_{z},p_{x}p_{y}\right)}$
---|---|---|---
1 | 1.8 |  1.8 | 1.8
2 | -1.9 |  -1.9 | -1.9
3 | -0.1 |  -0.1 | -0.1
4 | 1.0 |  1.0 | 1.0
5 | -1.1 |  -1.1 | -1.1
6 | -0.1 |  -0.1 | -0.1

Next, we outline a yet more transparent way to demonstrate chirality in
electron ionization. Namely, we plot
$\mathcal{P}^{\mathrm{m,n}}(\mathrm{p_{k},p_{i}p_{j}})$ defined as the
difference of the normalized probability distributions in the following way

$\mathcal{P}^{\mathrm{m,n}}(\mathrm{p_{k},p_{i}p_{j}})=\mathrm{P^{m}(p_{k},\mathrm{p_{i}p_{j}})}-\mathrm{P^{n}(\mathrm{p_{k},p_{i}p_{j}})},$
(8)

The corresponding measure of chirality is given by

$\displaystyle\begin{split}&\mathcal{X}d\mathrm{(p_{k},p_{i}p_{j})}=\\\
&\mathrm{\iint sign(\mathrm{{\bf{p}}_{k}\cdot({\bf{p}}_{i}\times{\bf
p}_{j})})\mathcal{P}^{\mathrm{m,n}}(\mathrm{p_{k},p_{i}p_{j}})dp_{i}p_{j}dp_{k}}.\end{split}$
(9)

In Figs. 4(a1)-4(a3), we plot the probability distribution
$\mathcal{P}^{\mathrm{m,n}}(\mathrm{p_{x}},\mathrm{p_{y}p_{z}})$ for the pair
of opposite chirality electric fields (1,2), i.e. Case1-Case 2 (Fig. 4(a1)),
and for the pairs of chiral-achiral electric fields (1,3) and (2,3), i.e. Case
1-Case 3 (Fig. 4(a2)) and Case 2-Case 3 ( Fig. 4(a3)). In each quadrant, we
assign the sign resulting from the scalar triple product
$\mathrm{{\bf{p}}_{x}\cdot({\bf{p}}_{y}\times{\bf p}_{z})}$. The yellow (blue)
color denotes positive (negative) values of
$\mathcal{P}^{\mathrm{m,n}}(\mathrm{p_{x}},\mathrm{p_{y}p_{z}})$,
corresponding to the electron being more (less) probable to ionize with
momenta px and p$\mathrm{{}_{y}p_{z}}$ due to pulse m rather than pulse n.
Next, in each quadrant, we multiply the $\pm$ sign (yellow/blue), resulting
from the distribution, with the $\pm$ sign of
$\mathrm{{\bf{p}}_{x}\cdot({\bf{p}}_{y}\times{\bf p}_{z})}$ and then sum up.
It easily follows that the measure of chirality
$\mathcal{X}d\mathrm{\left(p_{x},p_{y}p_{z}\right)}$ is larger and positive
($3.7\times 10^{-2}$) for the pair of opposite chirality fields (1,2), see
(Fig. 4(a1)). Also, $\mathcal{X}d$ is positive ($1.9\times 10^{-2}$) for the
pair of electric fields (1,3) and negative ($-1.8\times 10^{-2}$) for the pair
(2,3), with $1.9\times 10^{-2}-1.8\times 10^{-2}$ being roughly zero, since
pulses 1 and 2 have opposite chirality. As for chirality measures
$\mathcal{X}$, we find that all three measures of chirality
$\mathcal{X}d\mathrm{(p_{x},p_{y}p_{z})}$,
$\mathcal{X}d\mathrm{(p_{y},p_{z}p_{x})}$ and
$\mathcal{X}d\mathrm{(p_{z},p_{x}p_{y})}$ have the same value for each of the
fields 1,2,3, see Table 2.

Figure 4: Probability distribution $\mathcal{P}^{\mathrm{m,n}}(\mathrm{p_{x},p_{y}p_{z}})$. The m,n indexes are 1,2 for Case 1-Case 2, 1,3 for Case 1-Case 3 and 2,3 for Case 2-Case 3. The sign in each quadrant corresponds to the sign of $\mathrm{{\bf{p}}_{x}\cdot({\bf{p}}_{y}\times{\bf p}_{z})}$ in this quadrant.  Table 2: Measure of chirality $\mathcal{X}d$ for globally chiral electric fields 1,2,4,5 and globally achiral fields 3,6. $\mathcal{X}d$ is computed for each of the three probability distributions $\mathcal{P}^{\mathrm{m,n}}(\mathrm{p_{x}},\mathrm{p_{y}p_{z}})$, $\mathcal{P}^{\mathrm{m,n}}(\mathrm{p_{y}},\mathrm{p_{z}p_{x}})$ and $\mathcal{P}^{\mathrm{m,n}}(\mathrm{p_{z}},\mathrm{p_{x}p_{y}})$. The values are expressed in $10^{-2}$.  m | n | $\mathcal{X}d\mathrm{\left(p_{x},p_{y}p_{z}\right)}$ | $\mathcal{X}d\mathrm{\left(p_{y},p_{z}p_{x}\right)}$ | $\mathcal{X}d\mathrm{\left(p_{z},p_{x}p_{y}\right)}$
---|---|---|---|---
1 | 2 |  3.7 | 3.7 |  3.7
1 | 3 |  1.9 | 1.9 |  1.9
2 | 3 |  -1.8 | -1.8 |  -1.8
4 | 5 |  2.1 | 2.1 |  2.1
4 | 6 |  1.1 | 1.1 |  1.1
5 | 6 |  -1.0 | -1.0 |  -1.0

A similar analysis holds for the measures of chirality in electron ionization
when Ar is driven by the globally chiral electric fields 4,5 and the globally
achiral field 6. Indeed, in Figs. 4(b1)-4(b3), we plot the probability
distribution $\mathcal{P}^{\mathrm{m,n}}(\mathrm{p_{x}},\mathrm{p_{y}p_{z}})$
corresponding to the pair of opposite chirality electric fields (4,5), Case
4-Case 5, and to the pairs of chiral-achiral pulses (4,6), Case 4-Case 6, and
(5,6), Case 5-Case 6. As for our results for electric fields 1,2,3, we find
that $\mathcal{X}d\mathrm{\left(p_{x},p_{y}p_{z}\right)}$ for the pair of
opposite chirality fields (4,5) has the largest value of $2.1\times 10^{-2}$.
Also, as expected, for the pairs (4,6) and (5,6), we find that $\mathcal{X}d$
has roughly opposite values, $1.1\times 10^{-2}$ and $-1.0\times 10^{-2}$. The
exact same results hold for $\mathcal{X}d$ obtained for the other two
combinations of momentum components for fields 4,5,6, see Table 2.

Summarizing, we identify a transparent and simple measure of chirality in
electron ionization triggered in atoms (Ar) by synthetic pulses. These pulses
can create electric fields which are globally chiral or achiral along the
focus region. Our computations account for realistic experimental conditions.
We define this measure by multiplying the sign of the final electron momentum
scalar triple product $\mathrm{{\bf{p}}_{k}\cdot({\bf{p}}_{i}\times{\bf
p}_{j})}$ with the probability for an electron to ionize with certain values
for both pk and pipj. Finally, we integrate over all values of pk and pipj.
Three such measures can be defined, corresponding to the three combinations of
pk and pipj. We find that all three measures of chirality have the same value
for a given electric field. This robust measure of chirality has opposite
values when the electron dynamics is triggered by fields with opposite
chirality and is zero for a globally achiral field. We expect that this
measure of chirality in electron ionization of atoms applies to any chiral
field.

###### Acknowledgements.

The authors A. E and G. P. Katsoulis acknowledge the use of the UCL Myriad
High Throughput Computing Facility (Myriad@UCL), and associated support
services, in the completion of this work.

## References

  * Cireasa  _et al._ (2015) R. Cireasa, A. E. Boguslavskiy, B. Pons, M. C. H. Wong, D. Descamps, S. Petit, H. Ruf, N. Thiré, A. Ferré, J. Suarez, J. Higuet, B. E. Schmidt, A. F. Alharbi, F. Légaré, V. Blanchet, B. Fabre, S. Patchkovskii, O. Smirnova, Y. Mairesse, and V. R. Bhardwaj, “Probing molecular chirality on a sub-femtosecond timescale,” Nat. Phys. 11, 654–658 (2015).
  * Beaulieu  _et al._ (2017) S. Beaulieu, A. Comby, A. Clergerie, J. Caillat, D. Descamps, N. Dudovich, B. Fabre, R. Géneaux, F. Légaré, S. Petit, B. Pons, G. Porat, T. Ruchon, R. Taïeb, V. Blanchet, and Y. Mairesse, “Attosecond-resolved photoionization of chiral molecules,” Science 358, 1288–1294 (2017).
  * Beaulieu  _et al._ (2018) S. Beaulieu, A. Comby, D. Descamps, B. Fabre, G. A. Garcia, R. Géneaux, A. G. Harvey, F. Légaré, Z. Mašín, L. Nahon, A. F. Ordonez, S. Petit, B. Pons, Y. Mairesse, O. Smirnova, and V. Blanchet, “Photoexcitation circular dichroism in chiral molecules,” Nat. Phys. 14, 484–489 (2018).
  * Comby  _et al._ (2018) A. Comby, E. Bloch, C. M. M. Bond, D. Descamps, J. Miles, S. Petit, S. Rozen, J. B. Greenwood, V. Blanchet, and Y. Mairesse, “Real-time determination of enantiomeric and isomeric content using photoelectron elliptical dichroism,” Nat. Commun. 9, 5212 (2018).
  * Rozen  _et al._ (2019) S. Rozen, A. Comby, E. Bloch, S. Beauvarlet, D. Descamps, B. Fabre, S. Petit, V. Blanchet, B. Pons, N. Dudovich, and Y. Mairesse, “Controlling subcycle optical chirality in the photoionization of chiral molecules,” Phys. Rev. X 9, 031004 (2019).
  * Ayuso  _et al._ (2019) D. Ayuso, O. Neufeld, A. F. Ordonez, P. Decleva, G. Lerner, O. Cohen, M. Ivanov, and O. Smirnova, “Synthetic chiral light for efficient control of chiral light–matter interaction,” Nat. Photonics 13, 866–871 (2019).
  * Baykusheva and Wörner (2018) D. Baykusheva and H. J. Wörner, “Chiral discrimination through bielliptical high-harmonic spectroscopy,” Phys. Rev. X 8, 031060 (2018).
  * Neufeld and Cohen (2018) O. Neufeld and O. Cohen, “Optical chirality in nonlinear optics: Application to high harmonic generation,” Phys. Rev. Lett. 120, 133206 (2018).
  * Neufeld  _et al._ (2019) O. Neufeld, D. Ayuso, P. Decleva, M. Y. Ivanov, O. Smirnova, and O. Cohen, “Ultrasensitive chiral spectroscopy by dynamical symmetry breaking in high harmonic generation,” Phys. Rev. X 9, 031002 (2019).
  * Heinrich  _et al._ (2021) T. Heinrich, M. Taucer, O. Kfir, P. B. Corkum, A. Staudte, C. Ropers, and M. Sivis, “Chiral high-harmonic generation and spectroscopy on solid surfaces using polarization-tailored strong fields,” Nat. Commun. 12, 3723 (2021).
  * Lux  _et al._ (2012) C. Lux, M. Wollenhaupt, T. Bolze, Q. Liang, J. Köhler, C. Sarpe, and T. Baumert, “Circular dichroism in the photoelectron angular distributions of camphor and fenchone from multiphoton ionization with femtosecond laser pulses,” Angew. Chem. Int. Ed. 51, 5001–5005 (2012).
  * Lehmann  _et al._ (2013) C. S. Lehmann, N. B. Ram, I. Powis, and M. H. M. Janssen, “Imaging photoelectron circular dichroism of chiral molecules by femtosecond multiphoton coincidence detection,” J. Chem. Phys. 139, 234307 (2013).
  * Boge  _et al._ (2014) R. Boge, S. Heuser, M. Sabbar, M. Lucchini, L. Gallmann, C. Cirelli, and U. Keller, “Revealing the time-dependent polarization of ultrashort pulses with sub-cycle resolution,” Opt. Express 22, 26967–26975 (2014).
  * Landau and Lifshitz (2013) L. D. Landau and E. M. Lifshitz,  _Quantum Mechanics: Non-Relativistic Theory_, Vol. 3 (Elsevier, 2013).
  * Delone and Krainov (1991) N. B. Delone and V. P. Krainov, “Energy and angular electron spectra for the tunnel ionization of atoms by strong low-frequency radiation,” J. Opt. Soc. Am. B 8, 1207 (1991).
  * Hu  _et al._ (1997) B. Hu, J. Liu, and S. G. Chen, “Plateau in above-threshold-ionization spectra and chaotic behavior in rescattering processes,” Phys. Lett. A 236, 533–542 (1997).
  * Abrines and Percival (1966) R. Abrines and I. C. Percival, “Classical theory of charge transfer and ionization of hydrogen atoms by protons,” Proc. Phys. Soc. 88, 861–872 (1966).
  * Kustaanheimo and Stiefel (1965) P. Kustaanheimo and E. Stiefel, “Perturbation theory of kepler motion based on spinor regularization,” J. Reine Angew. Math. 218, 204 (1965).
  * Emmanouilidou (2008) A. Emmanouilidou, “Recoil collisions as a portal to field-assisted ionization at near-uv frequencies in the strong-field double ionization of helium,” Phys. Rev. A 78, 2–5 (2008).
  * Parker  _et al._ (2006) J. S. Parker, B. J. S. Doherty, K. T. Taylor, K. D. Schultz, C. I. Blaga, and L. F. DiMauro, “High-energy cutoff in the spectrum of strong-field nonsequential double ionization,” Phys. Rev. Lett. 96, 133001 (2006).
  * Staudte  _et al._ (2007) A. Staudte, C. Ruiz, M. Schöffler, S. Schössler, D. Zeidler, Th. Weber, M. Meckel, D. M. Villeneuve, P. B. Corkum, A. Becker, and R. Dörner, “Binary and recoil collisions in strong field double ionization of helium,” Phys. Rev. Lett. 99, 263002 (2007).
  * Rudenko  _et al._ (2007) A. Rudenko, V. L. B. de Jesus, Th. Ergler, K. Zrost, B. Feuerstein, C. D. Schröter, R. Moshammer, and J. Ullrich, “Correlated two-electron momentum spectra for strong-field nonsequential double ionization of he at 800 nm,” Phys. Rev. Lett. 99, 263003 (2007).
  * Emmanouilidou  _et al._ (2011) A. Emmanouilidou, J. S. Parker, L. R. Moore, and K. T. Taylor, “Direct versus delayed pathways in strong-field non-sequential double ionization,” New J. Phys. 13, 043001 (2011).
  * Katsoulis  _et al._ (2018) G. P. Katsoulis, A. Hadjipittas, B. Bergues, M. F. Kling, and A. Emmanouilidou, “Slingshot nonsequential double ionization as a gate to anticorrelated two-electron escape,” Phys. Rev. Lett. 121, 263203 (2018).
  * Chen  _et al._ (2017) A. Chen, M. Kübel, B. Bergues, M. F. Kling, and A. Emmanouilidou, “Non-sequential double ionization with near-single cycle laser pulses,” Sci. Rep. 7, 7488 (2017).