# 3D Mapping of Glacier Moulins: Challenges and lessons learned

William Dubois,1 Matěj Boxan,1 Johann Laconte,2 and François Pomerleau1
1Northern Robotics Laboratory, Université Laval, Québec City, Canada,
$\\{$william.dubois, matej.boxan, francois.pomerleau$\\}$
@norlab.ulaval.caUniversité Clermont Auvergne, INRAE, UR TSCF, 63000,
Clermont- Ferrand, France<EMAIL_ADDRESS>
###### Abstract

In this paper, we present a field report of the mapping of the Athabasca
Glacier, using a custom-made lidar-inertial mapping platform. With the
increasing autonomy of robotics, a wider spectrum of applications emerges.
Among these, the surveying of environmental areas presents arduous and
hazardous challenges for human operators. Leveraging automated platforms for
data collection holds the promise of unlocking new applications and a deeper
comprehension of the environment. Over the course of a week-long deployment,
we collected glacier data using a tailor-made measurement platform and
reflected on the inherent challenges associated with such experiments. We
focus on the insights gained and the forthcoming challenges that robotics must
surmount to effectively map these terrains.

##  I Introduction

Deploying robots in the cryosphere is still an open problem, particularly when
it comes to localization and mapping [1]. These deployments serve as crucial
sources of data collection and analysis, essential for understanding climate
change. However, accessing certain environments poses significant challenges
and risks to human safety. In this context, robotics emerges as a viable
solution, enabling the deployment of automated platforms to navigate hazardous
settings and gather environmental data. Notably, Dante was the first robot
successfully deployed in a remote Alaskan volcano [2], demonstrating the
feasibility of this approach. Similarly, glacier monitoring stands as a
critical endeavor due to the adverse impacts of melting ice.

Yet, conducting surveys within glaciers remains hazardous and logistically
arduous. Robotic systems have the growing potential to undertake such tasks.
For example, [3] describe a robotic deployment in degraded glacier settings,
focusing on the lidar-based mapping approach validated on the Athabasca
Glacier, Canada. In parallel, [4] further show the utility of robotics in
extreme environments, particularly in glacial moulin exploration in the Mer de
Glace glacier at Mont Blanc. A moulin is a near-vertical cave carved by
melting water streaming down a glacier. The study emphasizes the risks
associated with human-led data gathering and highlights the promising role of
robotic platforms in monitoring changes in glacial environments. Because of
the inherent challenges tied to these extreme environments, such as the lack
of distinct features, and the oscillation of the tethered robot, further
research is necessary before achieving true autonomy.

As such, we present in this short paper a deployment of a robotic platform in
the Athabasca Glacier, focusing on the forthcoming research opportunities and
lessons learned.

Figure 1: Deployment conducted on the Athabasca Glacier. The experimental
platform was lowered in a glacial moulin, mapping its surroundings.

##  II Experiments

The deployment took place on the Athabasca Glacier, located within the rugged
landscape of the Canadian Rockies. This glacier is characterized by its
complex features such as ice canyons, crevasses, and large glacial moulins,
making it a high-risk environment. Safety measures are crucial, requiring
careful anchoring systems to safely approach the edges of the glacier’s
moulins, which serve as entry points to its inner layers. The process of
descending into these icy chasms necessitates thorough preparation, including
the time-consuming task of establishing and securing lifelines and anchors,
ensuring the safety of personnel and equipment amidst the challenging
surroundings. In order to alleviate the risks, a data-gathering platform was
tethered inside the moulins, as shown in Figure 1.

The platform used to record data is depicted in Figure 2. It is equipped with
a lidar (Robosense RS-16), two Inertial Measurement Units (Xsens MTi-10,
Vectornav vn100), and a barometric pressure sensor (DPS310). The platform was
manually lowered inside the glacier’s moulin, with the operators staying at
least at arm’s length from the opening. A 3D map of the environment is
generated using the lidar data, with an example shown in Figure 3. Along with
the adverse conditions of the environment, the moulin displays little features
for Iterative Closest Point (ICP) to converge. In the next section, we discuss
the main challenges that remain to map such an environment, as well as the
practical issues linked to deployments in arctic regions.

Figure 2: Data gathering platform with which sensor measurements were recorded
to perform 3D localization and mapping, equipped with a lidar Robosense RS-16
(1), an Xsens MTi-10 IMU (2), a Vectornav vn100 IMU (3, behind the Xsens
MTi-10) and a barometric pressure sensor DPS310 (4, on the other side of the
platform)

##  III Challenges and Lessons Learned

While recording data in the extreme environment of the Athabasca Glacier, we
encountered numerous obstacles, necessitating robust solutions and meticulous
adherence to safety protocols. Weather conditions were very erratic. Within a
few hours, operators sustained sudden shifts ranging from hail, rain, and fog
to snow and freezing rain, highlighting the unpredictable nature of the
environment. To withstand such conditions, rugged laptops are indispensable;
however, a setback arose when our designated laptop malfunctioned prior to
deployment, compelling us to switch to an office laptop on the glacier. Even
with resilient, rugged equipment, it is thus necessary to always prepare a
spare set of equipment, which becomes more and more critical as the deployment
takes place in remote areas.

Furthermore, deploying rugged platforms emerged as imperative, with our
experience revealing a vulnerability in the form of a non-rugged computer
(Raspberry Pi 4B) on the platform. Although, as a whole, the platform
exhibited strong mechanical robustness, the connector for the SD card reader
weakened after multiple high-velocity shocks, leading to the system not
booting. A solution is to investigate embedded computers following the PC/104
standard. On one side, these computers were used successfully in micro-
satellites, but on the other, they are very specialized and hard to find with
a generic operating system such as Ubuntu. As the environment showcases
hazardous places, adherence to stringent safety measures remained non-
negotiable, with no exceptions tolerated. While such measures are necessary,
this clearly highlights the need to prepare the deployment in advance, with
every subsystem and procedure meticulously tested beforehand.

Additionally, the environmental impact of the deployment has to be taken into
account, ensuring no pollution will be left on-site, and no deterioration of
the environment will come from the field deployment. In our case, a metal cage
around the platform served as a precautionary measure in case of material loss
due to mechanical failure, wear, and tears.

The difficulties encountered also affected mapping solutions, as the lack of
distinct features in the environment led to under-constrained and degraded
solutions. Furthermore, as the platform was lowered in the moulin, sudden
movements and jerks of the platform, caused by collisions and inconsistent
lowering speeds, hindered the mapping capabilities, making it challenging to
map in real-time and thus impeding the on-site monitoring of the system’s
well-being. Ultimately, these conditions led to the necessity to compute the
map offline at a much lower rate, since the lack of features, and therefore
constraints, and the high-velocity movements prevented the ICP registration
from converging.

From this deployment, several key lessons emerged. Foremost among these was
the paramount importance of meticulous preparation, underscored by the
necessity of comprehensive testing and validation of equipment and procedures
prior to deployment. A meticulously documented experimental procedure is
essential, given the abundance of information to assimilate during deployment,
mitigating the risk of errors. Furthermore, a robust verification procedure is
paramount, as frequent on-site validation of the data will mitigate losses in
case of a system failure.

Figure 3: Map result (colored by elevation) from the moulin experiment. Left:
top view; Right: Side view. The lack of features in the moulin makes the
mapping of such environments challenging.

##  IV Conclusion

In this short paper, we describe the deployment of a mapping platform in the
Athabasca Glacier. The harsh conditions of arctic environments make
deployments arduous, with their specific set of challenges. Preparation,
resilience, and on-site monitoring of the system are key for the success of
the deployment. The specific environment of a glacier moulin led to
interesting challenges, which are the reflectiveness of the ice coupled with a
clear lack of features. Future work will focus on designing a new version of
the measurement platform and fusing information from several sensors, such as
atmospheric pressure, to improve the robustness of the mapping process.

## Acknowledgment

This research was supported by the Natural Sciences and Engineering Research
Council of Canada (NSERC) through the General Research Fund (GRF) from
Université Laval.

## References

  * [1] François Pomerleau  “Robotics in Snow and Ice”  In _Encyclopedia of Robotics_ Berlin, Heidelberg: Springer Berlin Heidelberg, 2023, pp. 1–6
  * [2] John E Bares and David S Wettergreen  “Dante II: Technical description, results, and lessons learned”  In _The International Journal of robotics research_ 18.7 SAGE Publications, 1999, pp. 621–649
  * [3] William Talbot et al.  “Principled ICP Covariance Modelling in Perceptually Degraded Environments for the EELS Mission Concept”  In _IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_ , 2023, pp. 10763–10770  IEEE
  * [4] Max Polzin and Josie Hughes  “Into the ice: Exploration and data capturing in glacial moulins by a tethered robot”  In _Journal of Field Robotics_ Wiley Online Library, 2024

  *[ICP]: Iterative Closest Point
  *[IMU]: Inertial Measurement Unit


# The Slow Growth of Entanglement in Long-range Interacting Spins Chains due
to Destructive Interference between quasi-Particles

Peyman Azodi<EMAIL_ADDRESS>Herschel A.Rabitz  Department of Chemistry,
Princeton University, Princeton, New Jersey 08544, USA

###### Abstract

The persistence of effective light cones in long-range interacting quantum
systems remains a significant and intriguing conundrum. In this paper, we
theoretically reveal a mechanism in Heisenberg spin chains at low
temperatures, wherein subsystems resist entanglement despite the onset of
entangling effects in quench scenarios. This mechanism is attributed to
destructive interference among quasi-particles outside an effective light
cone. Furthermore, we establish a necessary condition for the occurrence of
the destructive interference phenomenon. We demonstrate that for 1-D power-law
decay interactions, specifically those following $|d_{ij}|^{-p}$, destructive
interference occurs when $p>2$. As shown through theoretical and numerical
analysis, this destructive interference results in an unexpected acceleration
of entanglement propagation when the interaction range is truncated. This
prediction is proposed to be observable experimentally in chains of trapped
ions as a direct manifestation of the destructive interference effect.

††preprint: APS/123-QED

Introduction.- Understanding the propagation of entanglement in coherent out-
of-equilibrium many-body quantum systems is crucial for both theoretical and
practical purposes, particularly due to the recent advancements concerning the
connection between entanglement and thermodynamics in closed quantum systems.
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. Unlike in relativistic
quantum field theory, where correlation propagation is strictly limited within
a light cone, non-relativistic quantum theory allows for tunneling effects
that lead to immediate correlation build up [15]. Nevertheless, the Lieb-
Robinson (LR) bound establishes an effective light-cone structure in short-
range interacting quantum systems, beyond which correlations are required to
decay exponentially [16]. Over the past decades, this theorem has been
expanded [17, 18, 19, 20, 21], validated [22, 23, 24, 25, 26], and refined
[27, 28, 29, 30] in various ways. Considering the fundamental consequences and
the broad range of applications of this theorem, the LR bound has emerged as a
significant element in many-body quantum theory [31, 32, 33, 34].
Nevertheless, the counter-intuitive presence of an effective light cone in
long-range interacting quantum systems remains quite perplexing [35]. In these
systems, correlations could spread arbitrarily fast, potentially leading to
the absence of an effective light cone. Interestingly, various forms of linear
and non-linear LR bounds persist in the long-range limit [36, 28, 33, 37]. In
this study, we demonstrate that within a broad category of quantum systems,
specifically free particles, a mechanism of destructive interference operates,
causing particles, despite their rapid transport throughout the chain, to
interfere destructively and cancel out outside an effective light cone. A
similar phenomenon is observed in quantum electrodynamics, where photons
exceeding the speed of light cancel out, resulting in the formation of a light
cone [38].

Long-range interacting quantum lattices have been a focus of extensive studies
in the last two decades [33], not only due to their fundamental importance in
describing physical systems [39, 40, 33] but also for their interesting
properties, including dynamical phase transitions [41, 42, 43, 44], the
existence of long-lived meta-stable states [39, 45], and anomalous correlation
and entanglement dynamics [46, 47, 48, 49, 50, 51, 52, 53]. These systems are
mostly studied in the context of power law interactions, for which the
interactions between particles $i$ and $j$ decay as $\sim|d_{ij}|^{-p}$, where
$d_{ij}$ is inter-particle distance on a lattice, and $p\geq 0$ is a real
exponent, determining the length of interaction. In a $D$ dimensional system,
in contrast to the case of strong long-range interacting systems ($p<D$) [52,
50, 45], in the weak long-range interaction regime ($p>D$), the quasi-particle
formulation provides an accurate picture for entanglement propagation,
predicting various aspects of correlation propagation [54, 55, 56, 37, 36, 53,
57]. In particular, the existence of light cones for free particle systems has
been proved for $p>D+1$ [58]. Similarly, for $p>2D+1$, general linear light
cone structures have been proved to exist in generic long-range interacting
systems [28].

Heisenberg spin-$\frac{1}{2}$ chains at low temperatures, the primary model
studied in this paper, can be described via free Holstein–Primakoff bosons
through the Linear Spin Wave (LSW) approximation [59]. Numerous prior studies
have reported the existence of entanglement light cones in the LSW
approximation [57, 60, 37].

In this study, we demonstrate that the persistence of the entanglement light
cone despite long-range interactions is a fundamental property of free
particles, which is rooted in an essential feature of this class of systems,
namely the destructive interference between the quasi-particles at long-
distances (or equivalently, outside a light cone structure). We identify a
necessary condition on generic long-range interaction functions, beyond just
the power-law decay functions, for the destructive interference phenomenon.
Although power-law decay interaction functions satisfy this necessary
condition for $p>1$, we numerically illustrate that for the case of one-
dimensional power-law decay interactions, destructive interference occurs when
$p>2$.

In addition to the main argument of the paper regarding the survival of an
effective light cone in long-range interacting chains, our results compliment
and align with multiple observations in this area. The $p>2$ condition for the
presence of the destructive interference justifies the observed dynamical
phase transition in multiple studies [37, 60, 28, 58, 57]. Notably, as will be
shown, the destructive interference phenomenon directly leads to the
independence of the entanglement edge velocity with respect to the exponent
$p$, when $p\gtrsim 3$. This counter intuitive phenomenon has been previously
reported in [61, 28, 18].

The destructive interference among particles leads to a unique and intriguing
phenomenon that should be experimentally observable. Our theoretical and
numerical analyzes demonstrate that when interactions are truncated, i.e.,
interactions beyond a certain cutoff length are omitted, the entanglement edge
propagates faster compared to scenarios with full long-range interactions.
This outcome results from the destructive interference and is counter-
intuitive, as one would expect correlations to develop more slowly when the
interaction range is reduced. Given the recent advancements in trapped ion
technologies, and the high level of controllability in these systems,
experimental demonstration of this unusual phenomenon in one-dimensional
chains of trapped ions is feasible [62, 25, 63, 64, 40, 65].

We will establish the aforementioned interference phenomenon in the context of
long-range interacting Heisenberg spin chains at low temperatures with power-
law coupling, by analytically obtaining the bipartite entanglement dynamics
between subsystems after a quench at $t=0$. Consider a continuous segment of
spins within the chain, referred to as subsystem $\mathcal{M}$, which
accompanies the rest of the system, labeled as $\mathcal{R}$ (see Figure 1). A
measure of bipartite entanglement of $\mathcal{M}$ is denoted by
$\mathcal{Q}_{\mathcal{M}}(t)$ (or $\tilde{\mathcal{Q}}_{\mathcal{M}}(s)$ in
the Laplace domain). This measure is the linear bipartite entropy of subsystem
$\mathcal{M}$. We will demonstrate the slow growth of
$\mathcal{Q}_{\mathcal{M}}(t)$ by finding the coefficients $\alpha_{r}$ in the
expansion $Q_{\mathcal{M}}(t)=\sum{\alpha_{r}}t^{2r}$. To put our analysis in
a clear context, consider the case of nearest neighbor interacting spins. In
this scenario, transient (i.e., outside and in the vicinity of the LR bound)
entanglement grows as $\mathcal{Q}_{\mathcal{M}}(t)\propto(t/q)^{2q}$, where
$q$ is the distance between the set of local spin quenches in $\mathcal{R}$
and the edge of $\mathcal{M}$ [66]. In this case $\alpha_{i}=0$ when $i<q$,
which is due to the bounded maximum group velocity of the dispersion relation.
This region, namely $i<q$ lies outside the effective light cone, or the LR
bound. By increasing the interaction range, a rapid increase of $|\alpha_{i}|$
is expected. On the contrary, we will demonstrate that a mechanism is
operative that highly suppresses $\alpha_{i}$ for $i<q$.

Destructive interference between quasi-particles — In this paper, we consider
the Heisenberg chain with a power-law long-range interaction and periodic
boundary conditions

$\mathbf{H}=\frac{1}{2}\sum_{j<l}^{N}\frac{-{J}}{|j-l|^{p}}\mathbf{S}_{j}\mathbf{S}_{l},$
(1)

where $\mathbf{S}_{j}$ is the spin operator at site $j$, and ${J}$ is the
exchange interaction amplitude. For future reference, we denote the
interaction function, in the lattice and momentum spaces, respectively, as:

$\displaystyle\mathbf{J}(n)=\begin{cases}\frac{J}{|n|^{p}},&n\in\mathbb{N}-\\{0\\}\\\
0,&n=0,\end{cases}$ (2)
$\displaystyle\tilde{\mathbf{J}}(k)=\sum_{n}e^{-ikn}\mathbf{J}_{n}.$ (3)

Here, $p$ determines the length of interaction and corresponds to nearest
neighbor interactions in the $p\rightarrow\infty$ limit. We are interested in
the characteristics of entanglement propagation throughout the chain in a
quench experiment for an arbitrarily large number (N) of spin-$\frac{1}{2}$
particles. For this purpose, we split the system into two contiguous parts
$\mathcal{R}$ and $\mathcal{M}$ (this is possible due to the periodic boundary
conditions), and quantify the evolution of bipartite entanglement upon
exciting quasi-particles through flipping a group of spins, initially in
ferromagnetic order. Figure 1.a schematically depicts this setting. Given the
fact that we will use the linear spin wave approximation of the ferromagnetic
Heisenberg model, our results are only valid in the limits of at low-
temperature and low (local and global) density of excitations. A high density
of excitations might lead to formation of bound states, which are neglected in
the LSW approximation [67, 25]. We will obtain the linear bipartite entropy of
subsystem $\mathcal{M}$, denoted by $Q_{\mathcal{M}}(t)$, which is
monotonically related to the second-order Rényi entropy
($\mathbf{H}^{(2)}(\rho_{\mathcal{M}})=-\ln\big({\Tr{\rho_{\mathcal{M}}^{2}}}\big{missing})$)
through
$Q_{\mathcal{M}}(t)=\frac{1}{2}\big{(}1-{\Tr{\rho_{\mathcal{M}}^{2}}}\big{)}=\frac{1}{2}\big{(}1-e^{-\mathbf{H}^{(2)}(\rho_{\mathcal{M}})}\big{)}$
[68]. Readers are referred to the supplementary materials for detailed
definition of $Q_{\mathcal{M}}(t)$.

We employ the Quantum Correlation Transfer Function (QCTF) [68] to develop a
formulation of bipartite entanglement, by analytically finding the evolution
of the entanglement within the LSW theory. We show how local quenches in
subsystem $\mathcal{R}$ lead to bipartite entanglement production and
characterize the associated collective effects. In particular, we show that
upon expansion of the entanglement measure as
$Q_{\mathcal{M}}(t)=\sum{\alpha_{r}}t^{2r}$, the coefficients $\alpha_{r}$
correspond to specific hoppings of bosons in the chain. The contribution of
these hoppings appears with different phases depending on their order, i.e.,
parity of the number of hoppings. This difference in parity leads to
destructive interference between the bosonic degrees of freedom.

In this section, we present a simplified demonstration of the destructive
interference effect that fully captures the essence of this mechanism.
However, the full proof is provided in the supplementary materials. A key
result of the paper is the proof of the following relation:

$\alpha_{r}\propto\sum_{r_{1}+r_{2}=2r}\binom{2r}{r_{1}}(-1)^{r_{1}}\mathbf{J}^{*r_{1}}(q)\mathbf{J}^{*r_{2}}(q),$
(4)

where $\mathbf{J}^{*r_{1}}(q)$ denotes $r_{1}$ consecutive convolutions of
$\mathbf{J}(n)$ (given in (2)), evaluated at $q$. Here, $q$ is a measure of
distance along the chain between the quenched spins and subsystem
$\mathcal{M}$, which is precisely quantified in the proof. The convolution
operators incorporate all possible quasi-particle hoppings, and are convergent
for $p>1$. Moreover, we show that for the case of a power-law decaying
interaction (2), the convolution terms can be expanded as (see supplementary
materials for proof of the expansion and its convergence):

$\frac{{\mathbf{J}}^{*r}(q)}{J^{r}}=\frac{\binom{r}{1}}{(q-r+1)^{p}}+\frac{\binom{r}{2}}{\big{(}2(q-r)\big{)}^{p}}+\frac{\binom{r}{2}}{\big{(}3(q-r-1)\big{)}^{p}}+\cdots.$
(5)

We are interested in the $r<q$ regime, corresponding to the transient
entanglement evolution (outside the LR bound for short-range interacting
systems). In the regime of short-range interactions, characterized by large
$p$, the initial term in (5) predominates the series, and as the interaction
range extends, additional terms gain significance. To demonstrate how this
feature leads to the destructive interference phenomenon, here we will
consider the dominant (first) term in (5); The complete proof, which includes
higher order terms, is provided in the supplementary materials. In the
transient limit, the first term can be rewritten as:

$\frac{\binom{r}{1}}{(q-r+1)^{p}}=\frac{r}{q^{p}}\Big{(}1+p\frac{r-1}{q}+p(p+1)(\frac{r-1}{q})^{2}+\cdots\Big{)}.$
(6)

Utilizing the first two terms in this Taylor expression as approximations for
the high-order convolutions in (4), we see that $\alpha_{r}$ can be expressed
via terms of the following form:

$\begin{split}\frac{1}{q^{2p}}\sum_{r_{1}+r_{2}=2r}\binom{2r}{r_{1}}(-1)^{r_{1}}&\Big{(}r_{1}+\frac{pr_{1}(r_{1}-1)}{q}\Big{)}\\\
&\Big{(}r_{2}+\frac{pr_{2}(r_{2}-1)}{q}\Big{)}=0,\end{split}$ (7)

where we have used the binomial theorem. In this simplified version of the
proof, the primary nonzero term in the summation (4) for $\alpha_{r}$ comes
from the second-order term in the Taylor series expansion (6), which is
significantly suppressed in ($r/q$); Specifically, it is of the order
$\mathcal{O}\big{(}{q^{-2p}}(\frac{2r}{q})^{2}\big{)}$. In supplementary
materials, we demonstrate that this cancellation occurs for other terms in the
expansion (5) as well, and is not restricted to the first term. Consequently,
the primary contributions from quasi-particle hoppings cancel out due to
destructive interference.

(a)$\mathcal{M}$$\mathcal{R}$

0(b)${\cdots}$${\cdots}$$n$$\mathbf{J}(n)$$-\eta$0$\eta$(c)${\cdots}$${\cdots}$$n$$\mathbf{J}_{\eta}(n)$

Figure 1: (a) schematic illustration of a Heisenberg chain in a quench
experiment starting from ferromagnetic order. The arrows indicate quasi-
particle hoppings originating from a flipped spin and reaching the subsystem
$\mathcal{M}$. Various colors represent different phases that result in the
cancellation of terms in Equation (7). The green arrows depict a sequence of
short hops followed by a long hop, which corresponds to the initial term in
the expansion (5) of $\mathbf{J}^{*r}$. Figures (b) and (c) demonstrate the
full-range and truncated interaction functions, respectively.

Fast propagation of the entanglement edge for truncated interactions.—

The destructive interference between quasi-particles has a counter-intuitive
implication: The entanglement edge must propagate faster when the interaction
function is truncated, namely when the interaction terms that extend beyond
the range ($\eta$) are omitted from the interaction function. This phenomenon
is the direct consequence of quasi-particle cancellation in the transient
entanglement region due to the following reason. The destructive interference
effect occurs when all arrived quasi-particles (that have different phases)
are added together. Therefore, if part of the stream of quasi-particles is
obstructed due to truncation of the interaction function, then cancellation
can not take place; Hence, the entanglement edge propagates faster. This
phenomenon is mathematically proved in supplementary materials. Here, a
truncated interaction refers to:

$\mathbf{J}_{\eta}(n)=\begin{cases}\frac{J}{|n|^{p}},&n\in\\{-\eta,\cdots,\eta\\}\\\
0,&\text{otherwise},\end{cases}$ (8)

where $\eta=1$ gives nearest-neighbour interaction. Figures 1.b and c
schematically depict full-range and truncated interaction functions,
respectively.

Figure 2 provides a numerical demonstration of this phenomenon in a chain
consisting of $2.5\times 10^{3}$ spins. For simplicity, we examine the single
spin quench scenario, where the first spin in the chain is quenched to
$\ket{\downarrow}$ from the fully ferromagnetic state
$\ket{\uparrow\cdots\uparrow}$. In the linear spin-wave approximation, magnon
interactions are neglected, making this simple scenario sufficient to capture
the phenomenon of interest. Figure 2 shows the Entanglement Edge Times
(E.E.Ts) at a distance of $10^{3}$ from the quenched spin as a function of the
interaction range ($\eta$) and for different values of $p$. E.E.T represents
the time required for entangling effects, beyond quantum tunneling, to reach
spins at the given distance. Prolonged E.E.Ts for the full-range interaction
(denoted by $\eta=\infty$), compared to truncated interaction ranges
demonstrate the destructive interference effect, which is apparent for $p\geq
2$ among the simulated cases. In Figure 2, the red dashed lines show extended
E.E.Ts when $\eta=\infty$ is contrasted with $\eta=14$. Destructive
interference also manifests itself in the upward slopes of the curves,
highlighted in red, which indicate an increase in the entanglement edge time
as the interaction range grows.

Another notable aspect in Figure 2 is the lack of dependence of the
$\eta=\infty$-E.E.T on the variable $p$ for $p\gtrsim 3$. This can be seen in
the flattening of the curves as $p$ increases, which is caused by the
destructive interference that creates the previously mentioned upward slopes.
Consequently, for $p\gtrsim 3$, as $\eta$ increases and the destructive
interference appears, the E.E.Ts quickly revert to the nearest neighbor
interaction time ($\eta=1$), indicated by the horizontal dashed black line.
Thus, in this region, the E.E.Ts are not only independent of $p$, but also
become increasingly independent of $\eta$ when $\eta$ exceeds a certain value.
To illustrate this phenomenon more clearly, the inset in figure 2 displays the
E.E.Ts for the full-range interaction length as a function of $p$.

135791113$\infty$$0$$100$$200$$300$$400$$500$$p=1.7$$p=1.9$$p=2$$p=2.1$$p=2.3$$p=2.5$$p=2.9$$p=3.5$Interaction
range-$\eta$Entanglement Edge Time (E.E.T)
$(\frac{\hbar}{J})$$\backslash$$\backslash$$\backslash$$\backslash$2345$\infty$$300$$400$$500$$p$E.E.T
($\eta=\infty$)$\backslash$$\backslash$$\backslash$$\backslash$ Figure 2:
Entanglement Edge Times (E.E.Ts) for various interaction ranges ($\eta$) and
different values of $p$. The red-highlighted upward slopes of the curves
illustrate the unexpected increase in E.E.Ts as the interaction range ($\eta$)
grows for $p\gtrsim 2$. Furthermore, the prolonged E.E.Ts for full-range
interaction (represented by $\eta=\infty$) compared to truncated interaction
values are shown by red dashed lines. Both of these unusual phenomena result
from the destructive interference between particles, which leads to the
formation of an effective light cone in long-range interacting free particles.
The inset shows the progress of the $\eta=\infty$-E.E.Ts versus $p$. This
figures shows the quick convergence of E.E.Ts to the nearest-neighbor value
for $p\gtrsim 3$.

A generic necessary condition for the destructive interference phenomenon

We now develop a necessary condition on the interaction functions, extending
beyond power law interactions. We will pinpoint the initial cause of the
destructive interference phenomenon in the context of power-law functions and
subsequently demonstrate that this criterion can be generalized to other types
of interaction functions. Observe that equation (4) for $\alpha_{r}$ is
derived for general interaction functions $\mathbf{J}(n)$. Thus, the task is
to determine the types of $\mathbf{J}(n)$ in (4) that can cause destructive
interference, as indicated by the cancellation of terms in (7). By tracing
back from equation (7), cancellation between terms arises due to the
predominance of the first term in the expansion (5) for $\mathbf{J}^{*r}(q)$.
(Note that the high order convolution $\mathbf{J}^{*r}(q)$ incorporates all
possible series of $r$ quasi-particles hoppings, with varying jumps, that
reach distance $q$). The first term in (5) represents a sequence of shorter
particle jumps, along with one extended jump (depicted by green arrows in
Figure 1.a). In the supplementary materials, we demonstrate that such
sequences of jumps can dominate the series only when
$\big{(}(\mathbf{J}^{\prime})^{2}-\mathbf{J}\mathbf{J}^{\prime\prime}\big{)}>0,\forall
n$, where ′ indicates the derivative with respect to $n$, and $n$ is treated
as continuous. For power-law decay interaction functions, the necessary
condition is also sufficient when $p\gtrsim 2$, as shown and detailed in our
numerical simulations.

We should highlight the connection between the experiment proposal provided in
the previous Section and the aforementioned necessary condition. In the
experimental proposal, by truncating the interaction function, the
differentiability of the interaction function is violated, therefore the
necessary condition
$\big{(}(\mathbf{J}^{\prime})^{2}-\mathbf{J}\mathbf{J}^{\prime\prime}\big{)}>0$
does not hold.

Concluding remarks- In this study, we revealed a mechanism that explains the
unusual persistence of the effective light cone in long-range interacting spin
chains. We demonstrated that the primary portion of quasi-particles, despite
moving rapidly due to the long-range nature of the interaction function,
cancel out through a destructive interference process. This results in an
inductive character of entanglement generation in long-range interacting
chains. By inductive, we draw a parallel to the initial resistance to current
flow in an inductor, upon applying a voltage.

Our results show that the $p=2$ dynamical phase transition in spin systems,
observed in previous studies [54, 56, 37, 58], is the result of the break down
of the destructive interference. Likewise, the presence of a strictly linear
light cone for $p>3$ [28] is consistent with our demonstration that in this
regime, the $\eta=\infty$ E.E.T converges to the nearest neighbour E.E.T as a
result of the destructive interference phenomenon. The latter concept also
justifies the $p-$ independence of the E.E.T for $p\gtrsim 3$ reported in
previous studies [61, 28, 18]. The extensive consistency of our findings with
existing research highlights the importance of the identified destructive
interference.

Although our main analysis focused on Heisenberg chains at low temperatures,
the proof of the destructive interference phenomenon holds for the wide class
of free particle quantum systems. This class compromises a collection of
significant integrable Hamiltonians including long-range Ising, and a variety
of Heisenberg chains at low temperatures [69, 37, 70, 71, 60]. We emphasize
that the LSW approximation of the Heisenberg chain is accurate in the low-
temperature limit, where the density of excitations is low [71]. This happens
because in the LSW approximation, the magnon-magnon interaction terms are
neglected. Such interactions can lead to formation of bound states [67, 25],
that can alter the characteristics of entanglement propagation.

Experimental observation of the destructive interference among particles can
be achieved through the use of truncated interactions. This prediction entails
an anomalous acceleration of entanglement growth in the presence of truncated
interactions functions. Due to their high coherence times, as well as high
levels of interaction controllability and readout precision, chains of trapped
ions provide an excellent framework to experimentally test the prediction of
the theory in this paper [25, 72, 40, 65].

Future research on the destructive interference effect should explore its
behavior in higher (two and three)-dimensional lattices, examine the
robustness of this phenomenon when magnon-magnon interactions are present, and
conduct a more detailed analysis of the $p=2$ phase transition mechanism.

###### Acknowledgements.

P.A would like to thank Hans Halvorson, Nicolo Defenu and Alessio Lerose for
fruitful discussions. P.A acknowledges partial support from the Princeton
Program in Plasma Science and Technology (PPST). H.R and P.A acknowledge
partial support from the U.S Department Of Energy (DOE) grant (DE-
FG02-02ER15344).

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