# Elastic scattering of Airy electron packets on atoms

D. Grosman  School of Physics and Engineering, ITMO University, 197101 St.
Petersburg, Russia  N. Sheremet  School of Physics and Engineering, ITMO
University, 197101 St. Petersburg, Russia  I. Pavlov  School of Physics and
Engineering, ITMO University, 197101 St. Petersburg, Russia  D. Karlovets
School of Physics and Engineering, ITMO University, 197101 St. Petersburg,
Russia

(August 30, 2024)

###### Abstract

The problem of elastic scattering of the non-relativistic electron Airy beams
on potential fields is considered for a hydrogen atom in the ground 1s state.
It is demonstrated that the angular dependence of the scattering probability
density is in general azimuthally asymmetric. When the position of the atom
happens to coincide with one the minima of the probability density of the Airy
beam, the asymmetric pattern is represented by four separated peaks. We show
that this behaviour is very sensitive to the precision with which the relative
position of the atom and the minima is defined and study how uncertainty in
the position measured in terms of the wave-packet width affects observation of
the azimuthal asymmetry. Finally, we consider a spatially localized target and
discuss the difficulties of observing the asymmetry for targets with sizes
exceeding the critical value determined by the beam parameters and by the
position of the target center.

##  I Introduction

In this paper we aim to give a detailed analysis of elastic scattering of an
electron wave packet of a special shape by a potential field. Real particles
are localized both in the coordinate and momentum space, which results in
nonuniform densities and imposes the necessity of describing them as wave-
packets, however in a majority of scenarios they are represented as plane
waves, which is valid when the characteristic distances in the problem
considered are much smaller than the typical sizes of inhomogeneity, i.e, the
distances at which the nonuniform character of the density comes into play.
Nevertheless, there are examples of experimentally investigated scattering
processes where macroscopically large impact parameters greatly contributed to
the cross section [1]. In this case plane-wave description is no longer
sufficient and particles must be represented as wave-packets.

Non-Gaussian wave-packet solutions to the Shrödinger equation that are of
special interest due to the presence of phase have been known for over several
decades now. However, detailed analysis for massive particles (say, electrons)
is still rather scarce. In 2010-2011 it was reported on experimental
generation of vortex electron states carrying definite value $\hbar l$ of
intrinsic angular orbital momentum along the particle propagation axis where
the orbital quantum number $l$ can already be as high as one thousand [2, 3].
These states are characterized by spiraling current density proportional to
the orbital quantum number $l$ due to azimuthal dependence of the wave
function. Later, the electron Airy beams with energies up to 200 keV were
experimentally created by diffraction of Gaussian electrons on nano-scale
holograms [4]. The distinguishing feature of Airy beams is the cubic
dependence of the phase of the wave function on the particle momentum
$\varphi(\bm{k})\sim\xi_{x}^{3}k_{x}^{3}+\xi_{y}^{3}k_{y}^{3}$ with
$\bm{\xi}=\\{\xi_{x},\xi_{y}\\}$ being a 2D vector which parameterizes Airy
beams and transforms as coordinates under the Lorentz boosts [5]. In the limit
$\xi_{x,y}\to 0$ Airy packet coincides with a Gaussian one, while for
$\xi_{x,y}$ much larger than the size of the Gaussian packet it tends to a non
square integrable packet [6]. Among the features of the Airy packet can be
identified the presence of intrinsic quadrupole moment [7]. Currently,
electron Airy beams are not used as widely as their optical counterparts.
However, photon Airy beams have found various applications in optical
micromanipulation [8], optical trapping [9, 10], generation of plasma channels
[11], surface Airy plasmons [12, 13] and applications in lasers [14, 15].

Up to the present, most of the structured electron collision studies treated
twisted electrons. Scattering of free twisted particles has been researched in
[16] and [17]. Inelastic [18] and elastic [19, 20] scattering of vortex non-
relativistic electrons by atomic targets has also been considered in detail.
Interaction of twisted relativistic electrons with atomic targets has been
examined both in the Born approximation [21] and beyond it [22]. More complex
processes such as ionization of atomic targets using electron vortex beam
projectiles [23, 24, 25] and angular momentum transfer [26, 27] have been
thoroughly investigated. In the paper [28] the well-known Born approximation
was generalized for the case when the incident beam is a wave packet of a
general form. In the present paper we aim to apply this theory to the
scattering of Airy electron packets on an atomic potential.

The structure of the paper is as follows. In Sec.II A. we recall the standard
Born approximation including the well known formulas. Then we discuss
generalization of these formulas for scattering of wave-packets developed in
[28] and present the expression for the number of events. In Sec. II B. we
consider special case of the Airy beam and derive the scattering amplitude by
making use of the theory presented in previous sections. Then in Sec. II C.
and D. we consider the generalization of the number of events for scattering
on a macroscopic (infinitely wide) and mesoscopic (localized) target
respectively. In Sec. III we present density plots for different scattering
scenarios and introduce special points of the first and second type as well as
transitional points which are characterized by distinctive scattering
patterns. We analyze the feasibility of the scattering patterns and their
sensitivity to the parameters of the beam and scatterer. Then we discuss the
azimuthal dependence of the number of events and finally study how the
scattering patterns alter when switching to consideration of mesoscopic
targets.

##  II Theoretical background

###  II.1 Basic formulas

Consider the problem of scattering of a charged non-relativistic particle
(electron) off a spherically symmetric potential field $U(r)$ with a typical
radius of action $a$, which will be the Bohr radius in what follows. For the
initial and scattered electrons being plane waves, the S-matrix element for
the transition between states with momentum $\bm{p}_{i}$ and $\bm{p}_{f}$ is
expressed via the scattering amplitude $f(\varepsilon_{i},\theta,\varphi)$

$\displaystyle S_{fi}^{(pw)}=\bra{f}$ $\displaystyle
S\ket{i}=(2\pi)^{2}i\delta(\varepsilon_{i}-\varepsilon_{f})\frac{f(\varepsilon_{i},\theta,\varphi)}{m_{e}},$
(1)
$\displaystyle\varepsilon_{i}=\frac{\bm{p}_{i}^{2}}{2m_{e}},\;\varepsilon_{f}=\frac{\bm{p}_{f}^{2}}{2m_{e}},$

where $m_{e}$ is the electron mass, and the corresponding number of scattering
events and the differential cross section are

$\displaystyle
d\nu=N_{e}|\bra{f}S\ket{i}|^{2}\frac{d^{3}p_{f}}{(2\pi)^{3}},\;\frac{d\sigma}{d\Omega}=|f(\varepsilon_{i},\theta,\varphi)|^{2}.$
(2)

Here $N_{e}$ is the number of incident electrons.

For the initial state being a wave packet

$\ket{i}=\int\ket{\bm{k}}\Phi(\bm{k})\frac{d^{3}k}{(2\pi)^{3/2}},$ (3)

the matrix element for the transition into the plane wave state with momentum
$\bm{p}_{f}$ is given by integration of the plane wave amplitudes with the
packet’s wave function in momentum representation

$S_{fi}=\int\bra{\bm{p}_{f}}S\ket{\bm{k}}\Phi(\bm{k})\frac{d^{3}k}{(2\pi)^{3/2}}=\int
S_{fk}^{(pw)}\Phi(\bm{k})\frac{d^{3}k}{(2\pi)^{3/2}}.$ (4)

We will consider a wave packet propagating along the $z$ \- direction on
average

$\left\langle\bm{k}\right\rangle=(0,0,p_{i})$ (5)

with non-zero average of the absolute value of the transverse momentum

$\left\langle
k_{\perp}\right\rangle=\left\langle|\bm{k}_{\perp}|\right\rangle=\kappa_{0}=p_{i}\tan\theta_{k}.$
(6)

Here $\theta_{k}$ is the conical angle. We assume no coupling of the
longitudinal and transverse directions and hence factorization of the wave
function into transverse and longitudinal parts

$\Phi(\bm{k})=\Phi_{\perp}(\bm{k}_{\perp})\Phi_{long}(k_{z}).$ (7)

We also assume the dispersions to be small compared to the longitudinal
momentum [29, 30]:

$\Delta k_{x}=\Delta k_{y}\sim\frac{1}{\sigma_{\perp}}\ll p_{i},\;\Delta
k_{z}\sim\frac{1}{\sigma_{z}}\ll p_{i},$ (8)

where $\sigma_{\perp}$ and $\sigma_{z}$ are the transverse and longitudinal
averaged sizes of the electron packet. From the experimental point of view the
interesting case is when the packet’s size $\sigma_{z}$ is greater than the
field’s radius of action and yet still small enough to neglect the packet’s
spreading in the transverse plane during the collision:

$a\ll\sigma_{z}\ll\sigma_{\perp}\frac{p_{i}}{\kappa_{0}}.$ (9)

Provided (9) is fulfilled one can derive the following expression for the
scattering amplitude [28]:

$\displaystyle\frac{d\nu}{d\Omega}=\frac{N_{e}}{\cos\theta_{k}}|F(\bm{Q})|^{2},$
(10) $\displaystyle F(\bm{Q})=\int
f(\bm{Q}-\bm{k}_{\perp})\Phi_{\perp}(\bm{k}_{\perp})\frac{d^{2}k_{\perp}}{(2\pi)^{2}},$
$\displaystyle\bm{Q}=(p_{f}\sin\theta\cos\varphi,p_{f}\sin\theta\sin\varphi,p_{f}\cos\theta-
p_{i}),$

where $p_{f}=\sqrt{p_{i}^{2}+\kappa_{0}^{2}}$.

###  II.2 Scattering of the Airy packet

Figure 1: Spatial distribution of the Airy wave packet.
$\xi_{x}=\xi_{y}=2\sigma_{\perp}$, $b_{x}=b_{y}=0$.

In this section we consider the special case of a wave packet with a phase in
the momentum space — so-called Airy packet [4, 5], whose transverse wave
function in the momentum representation is

$\displaystyle\Phi_{\perp}(\bm{k}_{\perp})$
$\displaystyle=N\exp\left(-k_{x}^{2}\sigma_{\perp}^{2}-k_{y}^{2}\sigma_{\perp}^{2}\right)$
(11)
$\displaystyle\exp(\frac{i}{3}\xi_{x}^{3}k_{x}^{3}+\frac{i}{3}\xi_{y}^{3}k_{y}^{3}-ik_{x}b_{x}-ik_{y}b_{y}),$

where $N$ is the normalization constant defined by
$\int|\Phi_{\perp}(\bm{k}_{\perp})|^{2}\frac{d^{2}k}{2\pi}=1$ and
$\bm{b}=\\{b_{x},b_{y}\\}$ is the impact parameter introduced to account for
non head-on collision scenarios. The probability density in coordinate
representation is shown in Fig.1.

We will consider two potentials - Hydrogen-like atom and Yukawa potential

$\displaystyle
U_{\text{hyd}}(r)=-\frac{e^{2}}{r}\left(1+\frac{r}{a}\right)\exp(-\frac{2r}{a}),$
(12) $\displaystyle U_{\text{Yu}}(r)=\frac{V_{0}}{r}\exp(-\mu r).$

Here $\mu$ is the inverse effective radius of action of a Yukawa potential,
$V_{0}$ is the effective Yukawa potential amplitude. The corresponding Born
amplitudes for the potentials are

$\displaystyle
f_{\text{hyd}}(\bm{q})=\frac{a}{2}\left[\frac{1}{1+(qa/2)^{2}}+\frac{1}{\left(1+(qa/2)^{2}\right)^{2}}\right],$
(13) $\displaystyle f_{\text{Yu}}(\bm{q})=-\frac{2m_{e}V_{0}}{q^{2}+\mu^{2}},$

which can be written in a general form with the help of the function
$I(\eta,z)$ defined as follows

$\displaystyle I(\eta,z)=\int\limits_{0}^{\infty}(1+\eta
s)e^{-sz}ds=f_{0}\left(\frac{1}{z}+\frac{\eta}{z^{2}}\right),$ (14)
$\displaystyle
f_{\text{hyd}}(\bm{q})=\frac{a}{2}I\left(1,1+\frac{q^{2}a^{2}}{4}\right),$
$\displaystyle
f_{\text{Yu}}(\bm{q})=-\frac{2m_{e}V_{0}}{\mu^{2}}I\left(0,1+\frac{4q^{2}}{\mu^{2}}\right).$

Here $f_{0}$ is the amplitude of the potentials which is either
$f_{0}=\frac{a}{2}$ or $f_{0}=-\frac{2m_{e}V_{0}}{\mu^{2}}$. The amplitude for
scattering of an Airy packet on either one of the potentials of interest is
then also written in a general form

$\displaystyle F(\bm{Q},\bm{b},\eta,a,f_{0})=$ (15) $\displaystyle f_{0}\int
I\left(\eta,1+\frac{1}{4}(\bm{Q}-\bm{k}_{\perp})^{2}a^{2}\right)\Phi_{\perp}(\bm{k}_{\perp})\frac{d^{2}k_{\perp}}{(2\pi)^{2}}.$

Note that the function $F\left(\bm{Q},\bm{b},1,a,\frac{a}{2}\right)$ \- is the
amplitude for the scattering of an Airy packet on a ground state hydrogen atom
into a plane wave state and
$F\left(\bm{Q},\bm{b},0,\frac{2}{\mu},-\frac{2m_{e}V_{0}}{\mu^{2}}\right)$ \-
is the amplitude for the same process but involving Yukawa potential. The
final expression for the amplitude is given by a one dimensional integral

$\displaystyle
F(\bm{Q},\bm{b},\bm{\xi},\eta,a,f_{0})=f_{0}N\frac{(2\pi)^{2}}{\xi_{x}\xi_{y}}$
(16) $\displaystyle\times\int\limits_{0}^{\infty}ds(1+\eta
s)\exp\left\\{-s\left(1+\frac{1}{4}Q^{2}a^{2}\right)\right\\}$
$\displaystyle\times\exp\left\\{\frac{2}{3}\rho_{x}^{6}(s)-i\rho_{x}^{2}(s)\zeta_{x}(s)+\frac{2}{3}\rho_{y}^{6}(s)-i\rho_{y}^{2}(s)\zeta_{y}(s)\right\\}$
$\displaystyle\times\operatorname{Ai}\left[-i\zeta_{x}(s)+\rho_{x}^{4}(s)\right]\operatorname{Ai}\left[-i\zeta_{y}(s)+\rho_{y}^{4}(s)\right],$

where the expressions

$\displaystyle\rho_{x,y}^{2}(s)=\frac{1}{\xi^{2}_{x,y}}\left(a^{2}s/4+\sigma_{\perp}^{2}\right),$
(17)
$\displaystyle\zeta_{x,y}(s)=\frac{Q_{x,y}a^{2}s}{2\xi_{x,y}}-i\frac{b_{x,y}}{\xi_{x,y}},$

will be given physical interpretation shortly.

Expression (16) describes both the hydrogen atom and Yukawa potential. In the
formula for the scattering amplitude $\operatorname{Ai}(z)$ is the Airy
function properly defined for complex arguments:

$\operatorname{Ai}(z)=\frac{1}{2\pi
i}\int\limits_{C}\exp(zt-\frac{t^{3}}{3})dt,$ (18)

where the integral is over a path $C$ starting at a point at infinity with
$\frac{\pi}{2}<\text{arg(t)}<\frac{2\pi}{3}$ and ending at a point at infinity
with $\frac{7\pi}{6}<\text{arg(t)}<\frac{3\pi}{2}$.

For reason that will become clear shortly, it is convenient to rewrite
expression (16) in the following form

$\displaystyle F(\bm{Q},\bm{b},\bm{\xi},\eta,a,f_{0})=$ (19) $\displaystyle
f_{0}\int\limits_{0}^{\infty}(1+\eta
s)\exp{-s\left(1+\frac{1}{4}Q^{2}a^{2}\right)}\Psi_{\perp}(b_{x},b_{y},s)ds.$

Notice that the integrand in (19) taken at $s=0$ is simply the packet’s
transverse wave function in real space with the impact parameter being the
argument $\Psi_{\perp}(b_{x},b_{y})=\Psi_{\perp}(b_{x},b_{y},s=0)$ and
$|\Psi_{\perp}(x,y)|^{2}=|\Psi_{\perp}(x,y,s=0)|^{2}$ is the packet’s spatial
distribution presented in Fig.1

Had we considered a wave packet in momentum representation (11) with the
transverse size

$\sigma_{\perp}^{\prime
2}=\sigma_{\perp}^{2}+\frac{a^{2}s^{\prime}}{4}=\xi_{x,y}^{2}\rho^{2}_{x,y}(s^{\prime}),$
(20)

and a nonzero average projections of the momentum

$\left\langle
k_{x,y}\right\rangle=\frac{Q_{x,y}a^{2}s^{\prime}}{4\sigma^{2^{\prime}}_{\perp}}=\frac{1}{2\rho^{2}_{x,y}(s^{\prime})}Re(\zeta_{x,y}(s^{\prime}))$
(21)

the transverse wave function would have been exactly
$\Psi_{\perp}(b_{x},b_{y},s=s^{\prime})$. In the light of it, the amplitude
for scattering of a wave-packet on a hydrogen-like atom (or Yukawa potential)
is represented by superposition of the the wave-packets of the same shape, but
with different parameters. In this interpretation, $\rho^{2}_{x,y}(s)$ is a
dimensionless parameter determining the size of the packet and
$\zeta_{x,y}(s)$ is a dimensionless parameter the real part of which
determines the direction and the absolute value of the average transverse
momentum of the wave-packet and the imaginary part determines its average
position in space.

a) $\sigma_{\perp}=a,b_{x}=b_{y}=0$.

b) $\sigma_{\perp}=a,b_{x}=b_{y}\approx 4.8\sigma_{\perp}$.

c) $\sigma_{\perp}=5a,b_{x}=b_{y}\approx 4.8\sigma_{\perp}$.

d) $\sigma_{\perp}=10a,b_{x}=b_{y}\approx 4.8\sigma_{\perp}$.

e) $\sigma_{\perp}=a,b_{x}=b_{y}\approx 11.17\sigma_{\perp}$.

Figure 2: Scattering on a single atom: the dependence of the scattering
probability density $\frac{d\nu}{d\Omega}(\theta_{x},\theta_{y})$ on the flat
angles for the atom being located at different special points of the first
type defined by (31) - first minima (a,c,d), third minima (b) for different
transverse sizes of the packet - $\sigma_{\perp}=a,5a,10a$. Figure 3: Angular
distribution on flat angles $\theta_{x}$ and $\theta_{y}$ is almost equivalent
to the distribution with respect to cartesian coordinates in the plane of
detector.
$\cos\theta=\cos\theta_{x}\cos\theta_{y};\>\sin\varphi=\frac{\sin\theta_{y}}{\sqrt{1-\cos^{2}\theta_{x}\cos^{2}\theta_{y}}}$.

###  II.3 Scattering on a macroscopic target

After the discussion of scattering by a single potential, let us briefly
describe scattering on a macroscopic (infinitely wide) target, which consists
of randomly distributed potential centers. In this case we would need to
integrate over all potential centers’ positions and introduce the averaged
cross-section as the integration of the number of events over all the impact
parameters $\bm{b}$ and dividing the expression by the number of particles in
the incident packet. The result is

$\frac{d\bar{\sigma}}{d\Omega}=\frac{1}{N_{e}}\int\frac{d\nu}{d\Omega}d^{2}\bm{b}$
(22)

The wave function of the packet approaching a target at impact parameter
$\bm{b}$ can be written as

$\Phi_{\perp}({\bm{k_{\perp}}})=a(\bm{k_{\perp}})e^{-i\bm{k_{\perp}b}}$ (23)

Therefore, expression (22) is proportional to the integral

$I=\int F(\bm{Q})F^{*}(\bm{Q})d^{2}b,$ (24)

where

$F(\bm{Q})=\int
f(\bm{Q}-\bm{k}_{\perp})a(\bm{k_{\perp}})e^{-i\bm{k_{\perp}b}}\frac{d^{2}k_{\perp}}{(2\pi)^{2}}$
(25)

After the integration over $\bm{b}$ and $\bm{k_{\perp}}$ we obtain

$\frac{d\bar{\sigma}}{d\Omega}=\frac{1}{\cos\theta_{k}}\int|f(\bm{Q}-\bm{k}_{\perp})|^{2}|\Phi_{\perp}(\bm{k}_{\perp})|^{2}d^{2}k_{\perp}$
(26)

(see details in [28]). Importantly, there is no dependence on the phase in
expression (26) meaning that scattering of the wave-packet on a macroscopic
target is merely defined by its transverse probability density and for the
Airy packet it is the same as for the Gaussian one.

###  II.4 Scattering on a mesoscopic target

In a more realistic experimental scenario a focused electron beam collides
with a localized atomic target. In order to account for the geometrical
effects in such a scenario we describe the target as an incoherent
superposition of potential centers. The density of the scatterers in the
transverse plane is characterized by a distribution function $n(\bm{b})$,
which is normalized as follows:

$\int n(\bm{b})d^{2}\bm{b}=1.$ (27)

For the numerical analysis below we take $n(\bm{b})$ to be a Gaussian
function:

$n(\bm{b})=\frac{1}{2\pi\sigma_{b}^{2}}\exp(-\frac{(\bm{b}-\bm{b_{0}})^{2}}{2\sigma_{b}^{2}}).$
(28)

For such a scenario the number of events compared to (10) modifies in the
following way:

$\frac{d\nu}{d\Omega}=\frac{N_{e}}{\cos\theta_{k}}\int|F(\bm{Q},\bm{b})|^{2}n(\bm{b})d^{2}\bm{b}.$
(29)

a)$\sigma_{\perp}=5a,b_{x}\approx 4.8\sigma_{\perp},b_{y}=0$.

b)$\sigma_{\perp}=5a,b_{y}\approx 4.8\sigma_{\perp},b_{x}=0$.

c) $\sigma_{\perp}=a,b_{x}=b_{y}=4\sigma_{\perp}$.

Figure 4: Scattering on a single atom: the dependence of the scattering
probability density $\frac{d\nu}{d\Omega}(\theta_{x},\theta_{y})$ on the flat
angles for the atom being located at different special points of the second
type for the packet’s transverse size $\sigma_{\perp}=5a$ (a,b) and a
transitional point and the packet’s transverse size $\sigma_{\perp}=a$ (c).

##  III Results

All the figures in the following subsections are presented for
$p_{i}a=p_{f}a=10$. [28]. Such a momentum value corresponds to non-
relativistic, but still “fast” electrons. It can be seen from the expression
(16) that increasing the projectile momentum simply leads to faster
exponential decay of the scattering amplitude as a function of the polar angle
$\theta$, i.e. “narrowing” of the scattering pattern. For the scattering on a
hydrogen atom this corresponds to the kinetic energy $\varepsilon_{i}=1.36$
keV. We also assume that $\xi_{x,y}=2\sigma_{\perp}$ unless stated otherwise
[5, 4]. We normalize the packet’s wave function to 1, meaning that only one
particle collides the target. Thus, in the following sections it is more
correct to speak of scattering probability density rather than of the number
of events.

###  III.1 Scattering pattern for central collision

The final results are more illustrative when expressed via the so-called flat
angles (30) [31] presented in Fig.3

$\cos\theta=\cos\theta_{x}\cos\theta_{y},\sin\phi=\frac{\sin\theta_{y}}{\sqrt{1-\cos^{2}\theta_{x}\cos^{2}\theta_{y}}}.$
(30)

These angles are connected with the Cartesian coordinates on a surface of a
flat detector.

The number of events for scattering on a single atom in reality turns out to
be very sensitive to the relative position of the atom and the wave-packet’s
probability density minima. As can be seen in Fig.(2,a), the scattering
pattern looks rather symmetric for the head-on collision, when the atom is
placed on the axis of the packet’s propagation with the impact parameters
being $b_{x}=b_{y}=0$. This can be interpreted in the following way: the first
maxima of the Airy packet is rather wide, about $5\sigma_{\perp}\approx 5a$,
which is why with the potential’s radius of action being equal to $a$ the atom
only feels the vicinity of the first maximum and does not feel the Airy nature
of the wave packet which manifests itself in oscillatory behaviour in one
quadrant of the $(x,y)$ \- plane. In a simple approximation we could say that
the atom feels the packet as a Gaussian one and the pattern then turns out to
be a symmetric circle as expected for a such case [28].

a)$b_{x,y}=4.4\sigma_{\perp}$.

b)$b_{x,y}=4.6\sigma_{\perp}$.

c)$b_{x,y}=4.8\sigma_{\perp}$.

d)$b_{x,y}=5\sigma_{\perp}$.

e)$b_{x,y}=5.2\sigma_{\perp}$.

f) $b_{x,y}=5.4\sigma_{\perp}$.

Figure 5: Scattering on a single atom: comparison of scattering patterns
within the vicinity of a special point of the first type for evaluating the
sensitivity with respect to inaccuracy of impact parameter value for packet
size $\sigma_{\perp}=2a$

###  III.2 Scattering pattern for special points of type 1

However, there are totally different scattering patterns that occur when the
atom is placed in the minima of the Airy packet’s probability density. These
points are defined conditions (31) that are both satisfied simultaneously:

$\displaystyle\operatorname{Ai}\left(-\frac{b_{x}}{\xi_{x}}+\frac{\sigma_{\perp}^{4}}{\xi_{x}^{4}}\right)=0,$
(31)
$\displaystyle\operatorname{Ai}\left(-\frac{b_{y}}{\xi_{y}}+\frac{\sigma_{\perp}^{4}}{\xi_{y}^{4}}\right)=0.$

In the following discussion we will refer to such points as special points of
the first type. For special points circular scattering pattern is replaced by
a 4-petal pattern.

As can be seen in Fig.2 the probability of the process vanishes for the
forward scattering, which corresponds to the flat angles being in the vicinity
of zero. In Fig.(2,b) we illustrate the scattering probability density for
scattering of a narrow packet when its transverse size is equal to the
potential’s radius of action ($\sigma_{\perp}=a$, e.g, Bohr radius in case of
hydrogen atom) on the atom that is placed in the first minima of the
transverse probability density ($b_{x}\approx 4.8\sigma_{\perp}$) and see that
the probability density is of order $10^{-3}-10^{-2}$. Increasing the wave-
packet’s transverse size leads to a relatively sharp decrease in the magnitude
of the probability density, e.g. increasing the packet’s size by a factor of 5
leads to a decrease in the probability density of the order of $10^{-7}$,
nevertheless, the scattering pattern qualitatively remains the same.

Switching the position of the atom from the first minima to a different one
keeps the scattering pattern visually the same, however, its the magnitude
decreases. As can be seen in Fig.(2,e) when the atom is placed in the third
minima with $b_{x}=b_{y}\approx 11.17\sigma_{\perp}$ the probability density
decreases by the order of $10^{-2}$. What else can be noticed is that figures
(2,b),(2,c),(2,d) are visually the same and the difference is only in the
values of the probability density, however, for a different minima Fig.(2,e)
the petals are spaced further apart from each other.

###  III.3 Scattering pattern for special point of type 2

a) $\sigma_{\perp}=a$

b) $\sigma_{\perp}=5a$

c) $b_{x,y}\approx 4.8\sigma_{\perp}$

d)
$\sigma_{\perp}=a,\>\operatorname{Ai}\left(-b_{x,y}/\xi_{x,y}+\sigma_{\perp}^{4}/\xi_{x_{y}}^{4}\right)=0$

Figure 6: Azimuthal dependence of the the ratio of probabilities
$d\nu(\theta,\varphi)/d\nu(\theta,\varphi=\pi/4)$ for different values of
impact parameter, packet’s size and parameters $\xi_{x,y}$. We assume that
$b_{x}=b_{y}$ and that outgoing electrons are detected at the polar angle
$\theta=0.1$ rad.

In reality the probability density is at its minima when either one of the
conditions (31) is fulfilled, thus one could ponder what would happen to the
scattering pattern if the atom is placed in a position that satisfies only one
of them. We will refer to such points as special points of the second type. We
illustrate such scattering patterns when the atom’s position satisfies the
condition of only the first or the second Airy function in (31) being equal to
zero in Fig.(4,a),(4,b) respectively. As can be seen in this case the circular
scattering pattern from Fig.(2,a) splits into a 2-petal pattern in two
orthogonal directions depending on which Airy function vanishes at that point.
Similar to the case of placing the atom in a special point of the first type
the probability density $\frac{d\nu}{d\Omega}(\theta_{x},\theta_{y})$ vanishes
for very small flat angles meaning vanishing probability of forward
scattering.

Combining observations of the scattering from special point of the first and
of the second type we can deduce that each of the two equations in (31)
behaves as a splitting mechanism that transforms circular-like parts of the
scattering patterns into two pieces.

Now, we could try to go beyond the cases considered thus far and study what
would happen to the scattering pattern in the vicinity of the special point.
In Fig.(4,c) we illustrate the case when impact parameter
$b_{x}=b_{y}=4\sigma_{\perp}$ is close to the special point of the first type
yet none of the conditions (31) is satisfied. For such a scenario we notice
transitional behaviour of the scattering pattern and will refer to such points
as transitional. Note that such a pattern also resembles the scattering of a
Gaussian wave-packet with nonzero impact parameter [28]. We could interpret
this result similarly to the central collision: the contribution of the main
maximum in this case significantly exceeds the contribution of other maxima.

###  III.4 Feasibility of scattering patterns and their sensitivity to
precision of atom’s placement

In Sec. III B. the density plots are given for impact parameters defined up to
$0.1\sigma_{\perp}$. For narrow packets with $\sigma_{\perp}=a$ this implies
the necessity to control the atoms position with the degree of precision of
$0.01-0.1$Å, which is hardly feasible nowadays. For larger packets with
$\sigma_{\perp}=5a,10a$ the required precision is insignificantly lower but
from experimental point of view it is still impossible to achieve such
accuracy and that is why we analyze the sensitivity of the scattering patterns
with respect to inaccuracy of atom’s position. In Fig.5 we present the series
of scattering patterns for values of impact parameter being gradually
increased by $0.2\sigma_{\perp}$. We see that for the values
$|\bm{b}|=4.6\sigma_{\perp},4.8\sigma_{\perp},5\sigma_{\perp},5.2\sigma_{\perp},6\sigma_{\perp}$
which are all in the vicinity of the special point of the first type the
scattering pattern deviates from the one corresponding to atom’s placement
precisely the minima of the probability density to a degree depending on the
value of impact parameter, and yet the 4-petal form is still more or less
recognizable, thus we can roughly estimate the required accuracy as $\sim
0.6\sigma_{\perp}=1.2a$. Moreover, for decreasing the required accuracy of
atom’s placement one could consider scattering on a Rydberg atom with the
radius of action $\tilde{a}=an^{2}$ which is much greater than $a$ for large
$n$. This would lead to rescaling of the whole problem keeping all the
expressions the same and the required accuracy could then be of the order
$1.2\tilde{a}\sim an^{2}\sim 10$ nm for $n=10$.

###  III.5 Azimuthal dependence for single atom scattering

After discussing the scattering patterns, let us dig deeper into the
scattering process for different scenarios. In reality the number of events is
a sharp function in terms of the polar angle and it grows sharper with the
increase of the momentum which we put equal $p_{i}a=p_{f}a=10$ as stated
before. Hence we fix the polar angle to be $\theta=0.1$ rad and study the
azimuthal dependence of the number of events for scattering on a single atom.

In Fig(6,a) we display the normalized number of events for scattering on an
atom placed at the special point of the first type ($b=4.8\sigma_{\perp}$ \-
orange line) and at two transitional points
($b=4.2\sigma_{\perp},b=5.4\sigma_{\perp}$). In the curve describing special
point we see 4 visibly equivalent maxima that correspond to the 4-petal
pattern. Shifting the impact parameter, two of the maxima flatten out and for
$b=4.2\sigma_{\perp}$ there are only 2 visible peaks like in a scattering
pattern Fig.(4,c) where the impact parameter is taken to be
$b=4\sigma_{\perp}$.

Increasing the size of the packet allows one to discover that the scattering
pattern for the special point is in general much more sensitive to the
azimuthal angle as can be seen in Fig(6,b) as for the size of the packet
$\sigma_{\perp}=5a$ the two curves for the transitional points (blue line for
$|b|=4.2\sigma_{\perp}$, black line for $|b|=5.4\sigma_{\perp}$) flatten out
compared to the curve describing the special point of the first type.

If we now compare the azimuthal dependence for special points with different
sizes of the packet Fig.(6,c), we see that the narrower the wave-packet is,
the smaller is the ratio of number of events as a function of the azimuthal
angle and the greater is its minimum value.

Finally, we consider scattering on an atom placed at a special point and study
how the dependence changes when we change the phase of the packet meaning
different $\xi_{x},\xi_{y}$ Fig.(6,d). The pattern of the dependence for
different $\xi_{x},\xi_{y}$ remains the same yet the ratio of probabilities
decreases. The idea behind it is that for smaller values of $\xi_{x},\xi_{y}$
the wave-packet is less and less distinguished as an Airy packet and starts to
resemble the Gaussian one more and more and this leads to the flattening of
the azimuthal dependence of the number of events.

###  III.6 Scattering pattern for a mesoscopic target

Figure 7: Scattering on a mesoscopic target: the dependence of the scattering
probability density $\frac{d\nu}{d\Omega}(\theta_{x},\theta_{y})$ on the flat
angles for scattering of the Airy packet with $\sigma_{\perp}=a$ on a
mesoscopic target centered at the special point of the first type
($b_{x,y}\approx 4.8\sigma_{\perp}$) with the width $\sigma_{b}=a$. This
corresponds to the case of a target consisting of very few atoms.

In this section we turn to a more realistic experimental scenario where we
investigate scattering on a mesocopic target of finite size. To study such a
process we describe the target with the distribution function (27). In the
case of incoherent scattering on a mesoscopic target, meaning that we average
the scattering probability density rather than the scattering amplitude, we
can think of the scattering pattern as an overlay of different scattering
patterns for different placings of a single atom. Thus, we will take into
account a continuum of points for atom’s placings that are neither special
points of the first nor of the second type the scattering pattern for which is
more or less circular and scattering pattern from special points with 4 or
2-petals and also the transitional cases. As a result, we expect the 4-petal
and 2-petal forms to smear out and end up with a pattern which is non-
vanishing for forward scattering.

In Fig.(7) we present the scattering pattern for the transverse size of the
beam and the width of the target being equal $\sigma_{\perp}=\sigma_{b}=a$ and
the center of the target located at the special point of the first type. As
can be seen in the figure, there are no separated peaks as expected, yet after
averaging over impact parameters the pattern looks ”quadratic” rather than
azimuthally symmetric even with a naked eye. The reason for it is the narrow
width of the target. As discussed above the resulting scattering pattern can
be thought of as an overlay of scattering patterns on the atom the position of
which is defined by the impact parameter $b_{x}=b_{y}=b$ with $b$ passing all
the values in the interval
$(4.8\sigma_{\perp}-\sigma_{b},4.8\sigma_{\perp}+\sigma_{b})=(3.8a,5,8a)$
which does not contain the point corresponding to the position of the main
maximum of the probability density ($|\bm{b}|\approx 2.16a$) of the Airy
packet the amplitude of the scattering pattern for which Fig(2,a) is of
several orders higher. Thus the effects of the internal structure of the
packet are still visible. However from the experimental point of view, the
size of target $\sigma_{b}=a$ is hardly achievable and the more realistic
scenario would imply $\sigma_{b}\sim 10-20a$. For such a target the interval
of the values of the impact parameters of overlaying patterns would inevitably
contain the main maxima and azimuthal asymmetry would be practically
invisible.

For a wave-packet with $\xi_{x}=\xi_{y}=\xi$ and the position of the center of
the target $b_{0}$ being one of the special points of the first type the
following semi-empirical inequality

$-\frac{b}{\xi}+\frac{\sigma_{\perp}^{4}}{\xi^{4}}<-1.018$ (32)

has to be satisfied for all $b\in(b_{0}-\sigma_{b},b_{0}+\sigma_{b})$ for the
observation of the contribution of the internal structure of the wave-packet
to be possible. The inequality simply states that for all absolute values of
the impact parameter that contribute to the resulting scattering pattern the
argument of the Airy function in (31) should be less than the position of the
main maxima ($b\approx-1.018$), i.e., the main maxima does not contribute to
the pattern. With the use of (32) we could estimate the critical width of the
target $\sigma_{c}$ that could allow observation of the internal degrees of
freedom of the packet in the scattering pattern as the value that satisfies

$\displaystyle-\frac{b_{0}-\sigma_{c}}{\xi}+\frac{\sigma_{\perp}^{4}}{\xi^{4}}=-1.018,$
(33)
$\displaystyle\sigma_{c}=b_{0}-\frac{\sigma_{\perp}^{4}}{\xi^{3}}-1.018\xi.$

When $b_{0}=4.8\sigma_{\perp}$, which corresponds to the first special point,
and $\xi=2\sigma_{\perp}$ the critical size estimate is
$\sigma_{c}=2.64\sigma_{\perp}$ From expression (32) it is clear that to
achieve visible asymmetry in scattering patterns on reasonable-sized
mesoscopic targets one should place them at a father special points of the
first type. We remind that there are infinitely many special points of the
first type as a result of oscillatory behaviour of the Airy function with the
first three described by $b\approx 4.8\sigma_{\perp},b\approx
8.25\sigma_{\perp},b\approx 11.16\sigma_{\perp}$.

Nonetheless, for the fixed polar angle $\theta=0.1$ rad the variation of
theratio of probability densities
$d\nu(\theta,\varphi)/d\nu(\theta,\varphi=\pi/4)$ as of the function of the
azimuthal angle for $\sigma_{\perp}=\sigma_{b}=a$ is already of the order of
$0.05$. Increasing the size of the target inevitably leads to more and more
azimuthally symmetric (circular) scattering patterns as in the limit of a
macroscopic target the cross-section of the process no longer depends on the
phase of the wave-packet, as discussed in Sec.II D, and for
$\sigma_{b}\gtrsim\sigma_{c}=2.64\sigma_{\perp}=2.64a$ the asymmetry vanishes.
For the target width $\sigma_{b}=10a$ the asymuthal variation of the ratio of
probabilities is already of the order of $10^{-3}$ and for $\sigma_{b}=30$ it
decreases to $10^{-4}$.

##  IV Discussion and conclusion

Recalling the definition for the plane-wave scattering amplitude entering
expression (10) one can arrive at the following expression for the non-plane
wave amplitude

$F(\bm{Q})=-\frac{m_{e}}{2\pi}\int
U(r)\Phi(r_{\perp})e^{-i\bm{Q}\cdot\bm{r}}d^{3}r,$ (34)

which allows one to retrieve information about the transverse wave function of
the initial state by applying Fourier transform to derive

$\Phi(r_{\perp})=-\frac{U^{-1}(r)}{4m_{e}\pi^{2}}\int
F(\bm{q})e^{i\bm{q}\cdot\bm{r}}d^{3}q,$ (35)

which can be rewritten in the following form

$\Phi(r_{\perp})=-\frac{m_{e}}{2\pi}\frac{\int
F(\bm{q})e^{i\bm{q}\cdot\bm{r}}d^{3}q}{\int
f(\bm{q})e^{i\bm{q}\cdot\bm{r}}d^{3}q}.$ (36)

Technically, expression (36) solves the inverse scattering problem as it
allows one to derive the wave function of scattered particle from the
scattering amplitude. However, we stress that a fully satisfying result would
be the ability to retrieve at least a portion of information about the wave-
packet’s distribution solely from the absolute value of the scattering
amplitude as experimental measurements can in reality be insensitive to the
phase of the scattering amplitude in a majority of scenarios.

In summary, we have applied the generalized Born approximation to the problem
of elastic scattering of a non-relativistic charged Airy wave-packet on a
potential field. For single hydrogen atom the probability density sharply
decreases with the increase of the packet’s width. In particular, as it
changes from $\sigma_{\perp}=a=0.5\text{\AA}$ to $\sigma=5a=2.5\text{\AA}$ the
values drop from $0.01$ to $10^{-8}$. For the angular dependence we have found
that the probability density $d\nu/d\Omega(\theta_{x},\theta_{y})$ as a
function of flat angles (30) acquires the 4-petal shape when the atom is
placed at the special points of the first type and the 2-petal shape when
placed at the special points of the second type. These points correspond to
the packet’s density minimum, where 4-petal shapes correspond to satisfying
both conditions in (31) simultaneously and 2-petal shapes occur when only one
of them is satisfied. Characteristic scattering patterns allows one to get
insight into the internal structure of the wave-packet by placing the atom at
different points, i.e., performing tomography of the wave-packet using the
atom as a probing tool.

In reality, such specific placings of the atom require the precision of the
atom’s position of the order of $0.1\sigma_{\perp}\sim 0.1a-1a\sim 0.1-1$Å.
From experimental point of view it seems to be vary hard to control the
position of the atom with such accuracy. That is why we have introduced the
idea of using Rydberg atoms, which have much larger radius of action
$\tilde{a}=an^{2}\sim 10^{-8}-10^{-4}$ m, where $n$ is a principal quantum
number. This simply leads to rescaling of the whole problem keeping all the
expression and results intact. Thus, one can study scattering of wave-packets
with significantly larger sizes up to $\sigma_{\perp}\sim 10-100\;\mu m$ and
greatly decrease the required precision of atom’s placement to micrometer
scale since for Rydberg atoms $n$ can be as high as 1000 [32].

Analyzing the azimuthal dependence of the ratio of probabilities
$d\nu(\theta,\varphi)/d\nu(\theta,\varphi=\pi/4)$ we have shown that it is in
general much more sensitive to the azimuthal angle when the atom is placed at
the special point of the first type. For $b=4.8\sigma_{\perp}$, which
corresponds to the first special point, the ratio variation is of order of 1,
whereas for $b$ shifted by $0.6\sigma_{\perp}$ it is roughly ten times
smaller. This may turn out to be useful in different microscopy problems when
the information regarding some object is extracted from azimuthal dependence
of scattering patterns as setting the investigated object at the special point
would result in a more explicit behaviour.

Finally, we have discussed the most experimentally achievable scenario, i.e.,
scattering on a mesoscopic target. For a toy-model case of a narrow packet and
an equally narrow target the scattering pattern still contains information
about the internal structure of the wave-packet, which is manifested in
”quadratic” shape of the probability density. We have introduced the critical
size of target as the maximum size, which could enable observation of the
asymmetry in the scattering on a mesoscopic target, which for
$\xi_{x}=\xi_{y}=2\sigma_{\perp},b_{x}=b_{y}=4.8\sigma_{\perp},\sigma_{\perp}=a$
is

$\sigma_{c}=2.64a\;(\text{FWHM}=6.22a).$ (37)

From analysis of the general expression for $\sigma_{c}$ it follows that to
make the observation of azimuthal asymmetry more feasible apart from
considering Rydberg atoms rather than the hydrogen atom one could place the
target’s center at a special point remote from the packet’s main peak.

The nature of the studied effects and the problems that arise along the way
are not unique for Airy beams. They are the result of non-Gaussian profiles
and appear due to the presence of a phase $\varphi(\bm{p})$, where $\bm{p}$ is
the particle’s momentum. Similar scenarios could be studied for different
manifestations of phases such as vortex states, Pearcy beams and their various
generalizations [33, 34].

##  V Acknowledgements

We are grateful to S. Baturin, A. Volotka, V. Ivanov, A. Katanaeva, G. Sizykh,
and A. Surzhykov for fruitful discussions and criticism. The studies in Sec.II
are supported by the Russian Science Foundation (Project No. 21-42-04412). The
studies in Sec.III.A,B are supported by the Government of the Russian
Federation through the ITMO Fellowship and Professorship Program. The studies
in Sec.III.C,D are supported by the Ministry of Science and Higher Education
of the Russian Federation (agreement no. 075-15-2021-1349). The work in
Sec.III.E,F by D. Karlovets and D. Grosman was supported by the Foundation for
the Advancement of Theoretical Physics and Mathematics “BASIS”.

## References

  * Kotkin et al. [1992] G. Kotkin, V. Serbo, and A. Schiller, International Journal of Modern Physics A 07, 4707 (1992), eprint https://doi.org/10.1142/S0217751X92002131, URL https://doi.org/10.1142/S0217751X92002131.
  * Uchida and Tonomura [2010] M. Uchida and A. Tonomura, Nature 464, 737 (2010), ISSN 1476-4687, URL https://doi.org/10.1038/nature08904.
  * McMorran et al. [2011] B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, Science (New York, N.Y.) 331, 192 (2011).
  * Voloch-Bloch et al. [2013] N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, Nature 494, 331 (2013).
  * Karlovets [2017] D. V. Karlovets, Journal of High Energy Physics 2017, 49 (2017), ISSN 1029-8479, URL https://doi.org/10.1007/JHEP03(2017)049.
  * Berry and Balazs [1979] M. V. Berry and N. L. Balazs, American Journal of Physics 47, 264 (1979), eprint https://doi.org/10.1119/1.11855, URL https://doi.org/10.1119/1.11855.
  * Karlovets and Zhevlakov [2019] D. Karlovets and A. Zhevlakov, Phys. Rev. A 99, 022103 (2019), URL https://link.aps.org/doi/10.1103/PhysRevA.99.022103.
  * Baumgartl et al. [2008] J. Baumgartl, M. Mazilu, and K. Dholakia, Nature Photonics 2, 675 (2008), ISSN 1749-4893, URL https://doi.org/10.1038/nphoton.2008.201.
  * Zheng et al. [2011] Z. Zheng, B.-F. Zhang, H. Chen, J. Ding, and H.-T. Wang, Appl. Opt. 50, 43 (2011), URL https://opg.optica.org/ao/abstract.cfm?URI=ao-50-1-43.
  * Zhang et al. [2011] P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, Opt. Lett. 36, 2883 (2011), URL https://opg.optica.org/ol/abstract.cfm?URI=ol-36-15-2883.
  * Polynkin et al. [2009] P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009), eprint https://www.science.org/doi/pdf/10.1126/science.1169544, URL https://www.science.org/doi/abs/10.1126/science.1169544.
  * Salandrino and Christodoulides [2010] A. Salandrino and D. N. Christodoulides, Opt. Lett. 35, 2082 (2010), URL https://opg.optica.org/ol/abstract.cfm?URI=ol-35-12-2082.
  * Minovich et al. [2011] A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, Phys. Rev. Lett. 107, 116802 (2011), URL https://link.aps.org/doi/10.1103/PhysRevLett.107.116802.
  * Porat et al. [2011] G. Porat, I. Dolev, O. Barlev, and A. Arie, Opt. Lett. 36, 4119 (2011), URL https://opg.optica.org/ol/abstract.cfm?URI=ol-36-20-4119.
  * Longhi [2011] S. Longhi, Opt. Lett. 36, 716 (2011), URL https://opg.optica.org/ol/abstract.cfm?URI=ol-36-5-716.
  * Ivanov and Serbo [2011] I. P. Ivanov and V. G. Serbo, Phys. Rev. A 84, 033804 (2011), URL https://link.aps.org/doi/10.1103/PhysRevA.84.033804.
  * Ivanov et al. [2016] I. P. Ivanov, D. Seipt, A. Surzhykov, and S. Fritzsche, Phys. Rev. D 94, 076001 (2016), URL https://link.aps.org/doi/10.1103/PhysRevD.94.076001.
  * Van Boxem et al. [2015] R. Van Boxem, B. Partoens, and J. Verbeeck, Phys. Rev. A 91, 032703 (2015), URL https://link.aps.org/doi/10.1103/PhysRevA.91.032703.
  * Van Boxem et al. [2014] R. Van Boxem, B. Partoens, and J. Verbeeck, Phys. Rev. A 89, 032715 (2014), URL https://link.aps.org/doi/10.1103/PhysRevA.89.032715.
  * Karlovets et al. [2017] D. V. Karlovets, G. L. Kotkin, V. G. Serbo, and A. Surzhykov, Phys. Rev. A 95, 032703 (2017), URL https://link.aps.org/doi/10.1103/PhysRevA.95.032703.
  * Serbo et al. [2015] V. Serbo, I. P. Ivanov, S. Fritzsche, D. Seipt, and A. Surzhykov, Phys. Rev. A 92, 012705 (2015), URL https://link.aps.org/doi/10.1103/PhysRevA.92.012705.
  * Kosheleva et al. [2018] V. P. Kosheleva, V. A. Zaytsev, A. Surzhykov, V. M. Shabaev, and T. Stöhlker, Phys. Rev. A 98, 022706 (2018), URL https://link.aps.org/doi/10.1103/PhysRevA.98.022706.
  * Harris et al. [2019] A. L. Harris, A. Plumadore, and Z. Smozhanyk, Journal of Physics B: Atomic, Molecular and Optical Physics 52, 094001 (2019), URL https://dx.doi.org/10.1088/1361-6455/ab12f3.
  * Plumadore and Harris [2020] A. Plumadore and A. L. Harris, Journal of Physics B: Atomic, Molecular and Optical Physics 53, 205205 (2020), URL https://dx.doi.org/10.1088/1361-6455/abb3ac.
  * Dhankhar et al. [2020] N. Dhankhar, A. Mandal, and R. Choubisa, Journal of Physics B: Atomic, Molecular and Optical Physics 53, 155203 (2020), URL https://dx.doi.org/10.1088/1361-6455/ab8718.
  * Lloyd et al. [2012a] S. Lloyd, M. Babiker, and J. Yuan, Phys. Rev. Lett. 108, 074802 (2012a), URL https://link.aps.org/doi/10.1103/PhysRevLett.108.074802.
  * Lloyd et al. [2012b] S. M. Lloyd, M. Babiker, and J. Yuan, Phys. Rev. A 86, 023816 (2012b), URL https://link.aps.org/doi/10.1103/PhysRevA.86.023816.
  * Karlovets et al. [2015] D. V. Karlovets, G. L. Kotkin, and V. G. Serbo, Phys. Rev. A 92, 052703 (2015), eprint 1508.00026.
  * Hwang et al. [2014] J.-G. Hwang, E.-S. Kim, H.-J. Kim, and D.-O. Jeon, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 767, 153 (2014), ISSN 0168-9002, URL https://www.sciencedirect.com/science/article/pii/S0168900214009383.
  * Sun et al. [2004] Y.-E. Sun, P. Piot, K.-J. Kim, N. Barov, S. Lidia, J. Santucci, R. Tikhoplav, and J. Wennerberg, Phys. Rev. ST Accel. Beams 7, 123501 (2004), URL https://link.aps.org/doi/10.1103/PhysRevSTAB.7.123501.
  * Ivanov and Karlovets [2013] I. P. Ivanov and D. V. Karlovets, Phys. Rev. A 88, 043840 (2013), URL https://link.aps.org/doi/10.1103/PhysRevA.88.043840.
  * Aasen et al. [2022] T. H. Aasen, D. H. Zeiner-Gundersen, S. Zeiner-Gundersen, P. Ohlckers, and K. Wang, Journal of Cluster Science 33, 839 (2022), ISSN 1572-8862, URL https://doi.org/10.1007/s10876-021-02031-6.
  * He et al. [2022] X. He, H. Huang, C. Xu, X. Deng, Y. Wu, H. Tang, D. Xu, S. Luo, J. Liu, R. Wu, et al., Annalen der Physik 534, 2200238 (2022), ISSN 0003-3804, URL https://doi.org/10.1002/andp.202200238.
  * Huang et al. [2022] H. Huang, H. Tang, X. He, Y. Wu, X. Cai, X. Deng, and D. Deng, Annalen der Physik 534, 2200202 (2022), ISSN 0003-3804, URL https://doi.org/10.1002/andp.202200202.


# Multitasking Framework for Unsupervised Simple Definition Generation

Cunliang Kong134, Yun Chen2, Hengyuan Zhang1, Liner Yang134, Erhong Yang134
1School of Information Science, Beijing Language and Culture University
2School of Information Management & Engineering,
Shanghai University of Finance and Economics
3National Language Resources Monitoring and Research Center Print Media
Branch,
Beijing Language and Culture University
4Beijing Advanced Innovation Center for Language Resources,
Beijing Language and Culture University  Correspondence to: Liner Yang.

###### Abstract

The definition generation task can help language learners by providing
explanations for unfamiliar words. This task has attracted much attention in
recent years. We propose a novel task of Simple Definition Generation (SDG) to
help language learners and low literacy readers. A significant challenge of
this task is the lack of learner’s dictionaries in many languages, and
therefore the lack of data for supervised training. We explore this task and
propose a multitasking framework SimpDefiner that only requires a standard
dictionary with complex definitions and a corpus containing arbitrary simple
texts. We disentangle the complexity factors from the text by carefully
designing a parameter sharing scheme between two decoders. By jointly training
these components, the framework can generate both complex and simple
definitions simultaneously. We demonstrate that the framework can generate
relevant, simple definitions for the target words through automatic and manual
evaluations on English and Chinese datasets. Our method outperforms the
baseline model by a 1.77 SARI score on the English dataset, and raises the
proportion of the low level (HSK level 1-3) words in Chinese definitions by
3.87% 111Code can be found at https://github.com/blcuicall/
SimpDefiner..

##  1 Introduction

Helping language learners understand words in doubt is an important topic in
the field of Intelligent Computer-Assisted Language Learning (ICALL) (Segler
et al., 2002; Enayati and Gilakjani, 2020; Lolita et al., 2020). In recent
years, researchers attempted to automatically generate definitions for words
rather than formulating predefined word-definition inventories (Ishiwatari et
al., 2019; Yang et al., 2020; Huang et al., 2021). There are two reasons for
this. Firstly, it can be difficult for users to distinguish which sense is
appropriate in the current context because of the cognitively inaccurate
nature of discrete sense boundaries (Rosch and Mervis, 1975; Kilgarriff, 1997;
Tyler and Evans, 2001). Secondly, the predefined inventories need to be
updated manually by lexicographers, which is time-consuming and causes
dictionaries to lag behind the ever-changing language usage.

Figure 1: Different definitions for advertisement in the Oxford Dictionary
(OD) and Oxford Advanced Learner’s Dictionary (OALD).

Different from previous work (Noraset et al., 2017; Gadetsky et al., 2018;
Mickus et al., 2019; Kong et al., 2020) that focused only on how to generate
definitions, we further propose a novel task of Simple Definition Generation
(SDG). Making the definitions easier to read and understand could benefit the
language learners, low literacy readers, as well as helping people with
aphasia or dyslexia. For example, compared with the Oxford Dictionary (OD),
the Oxford Advanced Learner’s Dictionary (OALD) has simpler definitions, which
are specifically designed for language learners. As shown in Figure 1, the
definition of the word advertisement in OALD does not contain difficult words
or phrases such as announcement and public medium.

The goal of SDG task is to generate simple definitions for languages that lack
learner’s dictionary. For example, Chinese as Second Language (CSL) learners
do not have suitable dictionaries. As Zhang (2011) pointed out, since the
difficulty of definitions is not considered, the existing dictionary cannot
meet CSL learner’s needs.

The SDG task is challenging because it requires a model to learn from a
standard dictionary containing complex definitions and then generate simple
ones, and hence fully unsupervised. A seemingly feasible solution is to
generate definitions first and then simplify them, i.e., the generation-
simplification pipeline. However, the simplification task requires dataset
with complex-simple sentence pairs, and such data is also difficult to find in
languages other than English Martin et al. (2020). Besides, the pipeline
methods do not perform well due to accumulated errors (Section 6.1).

To solve this dilemma and bridge the gap between practical needs for simple
definitions and current trivial definition generation systems, we present a
novel method for the SDG task. As illustrated in Figure 2, our method
leverages a multitasking framework SimpDefiner to generate simple definitions
by performing three sub-tasks at the same time, which are definition
generation, text reconstruction, and language modeling tasks. The framework
consists of a fully shared encoder and two partially shared decoders. We
disentangle the complexity factors from the text by designing a parameter
sharing scheme. Particularly, we share parameters in Complexity-Dependent
Layer Normalization and Complexity-Dependent Query Projection of the
transformer architecture (Vaswani et al., 2017) to control the complexity
(Section 3.3). Through joint learning and sharing parameters between the
decoders, the SimpDefiner is able to generate complex and simple definitions
simultaneously.

Main contributions of our paper are listed below:

  * •

For the first time, we propose the task of SDG to generate simple definitions
without supervised training data.

  * •

We propose a multitasking framework SimpDefiner to tackle this task. Through
joint training three sub-tasks, the framework can generate complex and simple
definitions simultaneously.

  * •

Both automatic and manual evaluations demonstrate the effectiveness of
SimpDefiner. The framework outperforms the baseline model by 1.77 SARI score
on the English test set. And the proportion of low level words (HSK level 1-3)
in generated definitions raised by 3.87% on the Chinese test set.

Figure 2:  The SimpDefiner consists of three sub-tasks. Gen. means generation
and Rec. means reconstruction. The solid black lines indicate the data-flow
during training, and the dashed black lines indicate the data-flow during
inference.

##  2 Related Work

###  2.1 Definition Generation

The definition generation task is first introduced by Noraset et al. (2017).
Although this task is proposed as a potentially useful tool for explainable
AI, many subsequent works believe that it can assist language learning by
giving definitions for words in the text Ishiwatari et al. (2019); Mickus et
al. (2019); Yang et al. (2020).

Various studies attempted to generate multiple different definitions for
polysemous words. Gadetsky et al. (2018) tackled this problem by computing the
AdaGram vectors Bartunov et al. (2016) of input words, which are capable of
learning different representations at desired semantic resolutions. However,
generating different definitions based on contexts, i.e., example sentences,
became the mainstream method Chang et al. (2018); Reid et al. (2020); Li et
al. (2020); Bevilacqua et al. (2020). Among them, some studies used pre-
trained language models to obtain contextualized embeddings. Reid et al.
(2020) initialized encoders with BERT Devlin et al. (2019) and employed
variational inference for estimation and leveraged contextualized word
embeddings for improved performance. Bevilacqua et al. (2020) employed a novel
span-based encoding scheme to fine-tune a pre-trained English encoder-decoder
system to generate definitions. Huang et al. (2021) leveraged the T5 Raffel et
al. (2019) model for this task and introduced a re-ranking mechanism to model
specificity in definitions.

Our proposed SimpDefiner also takes the given word and context as input.
Differently, our main focus is to generate definitions with appropriate
complexity to better help language learners. Besides, our model is based on
MASS Song et al. (2019), which is a pre-trained encoder-decoder model and is
suitable for generation tasks.

###  2.2 Sentence Simplification

Researchers usually regard the sentence simplification task as a monolingual
variant of machine translation (MT) Wubben et al. (2012). Benefiting from the
advancement of neural machine translation, this task has also made great
progress in recent years.

Lately, many works built upon the Seq2Seq MT model Sutskever et al. (2014)
performed well. First attempted by Nisioi et al. (2017), the Seq2Seq models
for this task are able to perform lexical simplification and content reduction
simultaneously by training on complex-simple sentence pairs. This method was
inherited and improved by many subsequent works, such as combining with the
reinforcement learning method by setting a simplification reward Zhang and
Lapata (2017), augmenting memory capacities Vu et al. (2018) or training with
multitasking on entailment and paraphrase generation Guo et al. (2018). Martin
et al. (2019) proposed to prepend additional prompt tokens to source sentences
at train time, which enables the end-users to condition the simplifications
returned by the model on attributes like length, lexical complexity, and
syntactic complexity. This controllable simplification system (called ACCESS)
and its improved version MUSS Martin et al. (2020) achieved SOTA results on
the Turk corpus in terms of the SARI metric Xu et al. (2016).

The generation-simplification pipeline methods are used as baselines of the
SDG task, and we use both ACCESS and MUSS models for the simplification.
Unlike the baseline, the SimpDefiner can generate simple definitions directly,
alleviating the accumulated errors.

###  2.3 Unsupervised Style Transfer

Style transfer aims to change the style attributes while preserving the
content. Our work is related to unsupervised style transfer by regarding the
text complexity as one of the style attributes Kawashima and Takagi (2019).

Dumoulin et al. (2017) demonstrated that the neural networks can capture the
artistic style of a diversity of paintings. The authors discovered that
adjusting parameters in the layer normalization mechanism leads to different
artistic styles. This method permits users to transform images to arbitrary
styles learned from individual paintings. Jin et al. (2020) successfully
applied this method to the task of headline generation, allowing the model to
generate headlines of a specific style, such as humorous, romantic or click-
baity, in an unsupervised manner.

By treating the task of simplification as a variant of style transfer, we
borrow the insight of learning complexity-dependent parameters in the Layer
Normalization mechanism. Additionally, we introduce the language modeling task
into SimpDefiner, which is to enhance the decoder and make it more sensitive
to text complexity.

##  3 Method

We integrate three sub-tasks of definition generation, text reconstruction,
and language modeling into the SimpDefiner. This section first gives a formal
definition of the SDG task, then introduces each sub-task, and finally the
parameter sharing scheme.

###  3.1 Task Formulation

The SDG task is to generate a simple definition $\bm{d}^{sim}$ for a given
word and context $(w^{*},\bm{c})$, where
$\bm{c}=[w_{1},\dots,w^{*},\dots,w_{n}]$ is a sentence containing $w^{*}$.
This task is challenging because there is no corpus like
$\\{(w_{i}^{*},\bm{c}_{i},\bm{d}_{i}^{sim})\\}_{i=1}^{N}$ and hence it is
fully unsupervised.

The only data available in this work include a standard dictionary dataset
$G=\\{(w_{i}^{*},\bm{c}_{i},\bm{d}_{i}^{com})\\}_{i=1}^{N}$ and a simple text
corpus $Y=\\{\bm{y}_{i}\\}_{i=1}^{M}$. Note that we use $\bm{d}^{com}$ for
complex definitions and $\bm{d}^{sim}$ for simple ones.

###  3.2 Multitasking Framework

We design the three sub-tasks in the SimpDefiner to learn different abilities.
Cooperating with each other, the entire framework obtains the ability to
compute the conditional probability $P(\bm{d}^{sim}|w^{*},\bm{c})$ of simple
definitions in a zero-shot manner.

Specifically, the definition generation task aims to model the probability of
a complex definition given the word and context $P(\bm{d}^{com}|w^{*},\bm{c})$
(Section 3.2.1). And the text reconstruction task aims to model the
probability of a simple sentence given the corrupted version
$P(\bm{y}|\tilde{\bm{y}})$ (Section 3.2.2). As we can see, neither task can
directly get the $P(\bm{d}^{sim}|w^{*},\bm{c})$. To solve the problem, we
assume that complexity and semantic information are controlled by different
parameters in the decoders, and we attempt to disentangle the complexity
factors from the text through a carefully designed parameter sharing scheme.
In the inference stage, we obtain a simple definition by feeding the encoded
hidden state into the reconstruction decoder as in Figure 2. The detailed
parameter sharing scheme is in Section 3.3.

Nevertheless, the complexity information may still be kept in some shared
parameters, resulting in the reconstruction decoder fail to generate simple
definitions occasionally. Eliminating the complexity information in all shared
parameters is obviously technically impossible. Instead, we introduce the
language modeling task (Section 3.2.3) to enhance the reconstruction decoder
and make it more focused on simple text generation. The experiment results in
Section 6 confirm our assumption.

####  3.2.1 Definition Generation Task

We follow the mainstream method (Yang et al., 2020; Kong et al., 2020; Reid et
al., 2020) to concatenate the word and context together with a special token
[SEP] as $\mathbf{x}=(w^{*};$ [SEP]$;\bm{c})$. The entire sequence is then fed
into SimpDefiner, and the definition is obtained by the following language
model:

$\displaystyle
P(\bm{d}^{com}|\mathbf{x};\bm{\theta}_{g})=\prod_{t}P\left(\bm{d}^{com}_{t}|\bm{d}^{com}_{<t},\mathbf{x};\bm{\theta}_{g}\right),$
(1)

where $\bm{d}^{com}_{t}$ is the $t$-th token of the definition, and
$\bm{\theta}_{g}$ is the set of parameters. The model is optimized using the
following loss function.

$\mathcal{L}_{gen}(\bm{\theta}_{g})=-\sum_{(\mathbf{x},\bm{d}^{com})\in G}\log
P(\bm{d}^{com}|\mathbf{x};\bm{\theta}_{g})$ (2)

####  3.2.2 Text Reconstruction Task

We corrupt each sentence in the corpus $Y$ by randomly deleting or blanking
some words and shuffling the word orders. And then we obtain a new corpus
$\tilde{Y}=\\{(\tilde{\bm{y}_{i}},\bm{y}_{i})\\}_{i=1}^{M}$, and
$\tilde{\bm{y}}$ is a corrupted version of $\bm{y}$. We input $\tilde{\bm{y}}$
into SimpDefiner and obtain $\bm{y}$ by solving a self-supervised task of

$P(\bm{y}|\tilde{\bm{y}};\bm{\theta}_{r})=\prod_{t}P\left(\bm{y}_{t}|\bm{y}_{<t},\tilde{\bm{y}};\bm{\theta}_{r}\right),$
(3)

where $\bm{y}_{t}$ is the $t$-th token of the sentence, and $\bm{\theta}_{r}$
is a set of parameters. The loss function of this task is as follows:

$\mathcal{L}_{rec}(\bm{\theta}_{r})=-\sum_{(\bm{y},\tilde{\bm{y}})\in\tilde{Y}}\log
P(\bm{y}|\tilde{\bm{y}};\bm{\theta}_{r}).$ (4)

####  3.2.3 Language Modeling Task

This task facilitates zero-shot generation of $P(\bm{d}^{sim}|\mathbf{x})$ by
jointly training the reconstruction decoder as a language model. Once the
model captures correct complexity that guides the model to generate the
desired simple texts, it’s more likely for the model to ignore the wrongly
shared complexity information. Similar to Eq. 3, we have:

$P(\bm{y}|\bm{\theta}_{l})=\prod_{t}P\left(\bm{y}_{t}|\bm{y}_{<t};\bm{\theta}_{l}\right).$
(5)

It is equivalent to masking the encoder out and ignoring the attention modules
between the encoder and reconstruction decoder. The model is optimized by the
following loss function:

$\mathcal{L}_{lm}(\bm{\theta}_{l})=-\sum_{\bm{y}\in Y}\log
P(\bm{y}|\bm{\theta}_{l}).$ (6)

Finally, we train the entire SimpDefiner by jointly minimizing the weighted
sum of all above mentioned loss functions. And the overall loss function is
calculated as:

$\mathcal{L}=\lambda_{\alpha}\mathcal{L}_{gen}+\lambda_{\beta}\mathcal{L}_{rec}+\lambda_{\gamma}\mathcal{L}_{lm},$
(7)

where $\lambda_{\alpha}$, $\lambda_{\beta}$, $\lambda_{\gamma}$ are hyper-
parameters.

Figure 3:  The parameter-sharing scheme between decoders. Parameters in blue
layers are shared, and those in green are not.

###  3.3 Parameter-Sharing Scheme

For parameters in the decoders, we divided them into two parts, which are
complexity-independent and complexity-dependent parameters. The former ones
are shared between decoders, and the latter ones are not, as illustrated in
Figure 3.

We now introduce the complexity-dependent layers, namely Complexity-Dependent
Layer Normalization and Complexity-Dependent Query Projection.

##### Complexity-Dependent Layer Normalization

Previous works (Dumoulin et al., 2017; Jin et al., 2020) demonstrated that the
layer normalization is related to the style of the target texts. We further
argue that as an attribute of style, the complexity can be retained by
independent layer normalization. Thus, we make the scaling and shifting
parameters for layer normalization not shared in both decoders. This approach
is to transform a layer activation $\bm{x}$ into a complexity-specific
normalized activation $\bm{z}$ as:

$\bm{z}=\gamma_{c}(\frac{\bm{x}-\mu}{\sigma})-\beta_{c},$ (8)

where $\mu$, $\sigma$ are the mean and standard deviation of the batch of
$\bm{x}$, and $\gamma_{c}$, $\beta_{c}$ are learnable parameters specific to
complexity $c$. Note that $c$ is a binary variable indicating different
decoders. This mechanism is used in all decoder layers.

##### Complexity-Dependent Query Projection

The decoder layers extract information from encoded hidden states through
cross-attention mechanism. We believe that the required information may vary
for different complexity. Therefore, the parameters of the linear mapping used
for the query transformation in the cross-attention are not shared among
decoders. This calculation is as follows:

$\bm{Q}=\bm{\hat{Q}}\cdot\bm{W}_{c}^{q},$ (9)

where $\bm{W}_{c}^{q}$ is the query transformation matrix specific to
complexity $c$. The obtained query vector $\bm{Q}$ is then fed into the cross-
attention mechanism. By using this approach, the model can obtain different
information from the encoded hidden states for different complexities.

| OD |  |  OALD |  | CWN
---|---|---|---|---|---
|  Train |  Valid |  Test |  |  Test |  |  Train |  Valid |  Test
Words |  33,128 |  8,867 |  3,881 |  |  3,881 |  |  6,574 |  823 |  823
Entries |  97,855 |  12,232 |  5,111 |  |  5,111 |  |  67,861 |  8,082 |  8,599
Context Length |  17.74 |  17.80 |  16.24 |  |  16.24 |  |  34.49 |  34.73 |  34.06
Def. Length |  11.02 |  10.99 |  10.03 |  |  12.74 |  |  14.76 |  14.60 |  14.72
Table 1: Statistics of the OD (English) dataset, OALD (English) test set, and CWN (Chinese) dataset. The rows are number of words and entries, and the average length of contexts and definitions. |  Sents |  Tokens |  Avg. Len
---|---|---|---
English |  32,395 |  392,625 |  12.12
Chinese |  58,867 |  860,761 |  14.62
Table 2: Statistics of simple text corpora. The columns are number of
sentences and tokens, and the average length of sentences.

##  4 Datasets

We evaluate the proposed multitasking framework on both English and Chinese
datasets. Each language has a definition generation dataset and a simple text
corpus.

###  4.1 English Dataset

The English datasets are constructed from the Oxford Dictionary (OD) and
Oxford Advanced Learner’s Dictionary (OALD). Since the OALD is for language
learners, it has much simpler definitions than OD. Therefore, we use the OD
for the definition generation training, and use the OALD for validation of
simple definition generation. Note that the words used for testing are
excluded from the training and validation sets.

For the definition generation dataset, we directly use the OD dataset
published by Gadetsky et al. (2018). The training set has 33,128 words and
97,855 entries. Each entry consists of a triplet of
$(w^{*},\bm{c},\bm{d}^{com})$. For testing, we align the words and context in
OD with the definitions in OALD through manual annotation. The annotated test
set includes 3,881 words and 5,111 entries, which is used for automatic
evaluation in experiments. Each entry in the test set has both golden complex
and simple definitions from OD and OALD, respectively. Detailed statistics are
listed in Table 1.

We extract the OALD definitions that are not in the test set for constructing
the simple text corpus. This corpus has 32,395 sentences with an average
length of 12.12. We list more statistics in Table 2.

During training, the definition generation dataset and the simple text corpus
are randomly sampled as mini-batches respectively. And there is no alignment
between the two mini-batches at each step.

###  4.2 Chinese Dataset

For the definition generation dataset, we use the Chinese WordNet (CWN) (Huang
et al., 2010), which is a semantic lexicon aiming to provide a knowledge base
of sense distinction.222Chinese WordNet:
http://lope.linguistics.ntu.edu.tw/cwn2 We use the corresponding words,
contexts, and definitions in CWN for the definition generation task. We split
the entire dataset into training, validation, and test sets roughly according
to the ratio of 8:1:1. The training set contains 6,574 words and 67,861
entries. Statistics are listed in Table 1.

For the simple text corpus, we extract 58,867 sentences from a number of
primary level Chinese as Second Language textbooks, with an average sentence
length of 14.62.

Since no suitable dictionary can be used for evaluation, there are no golden
simple definitions in Chinese Dataset. In the experiments, we count the
difficulty level of words in definitions to estimate if they are simple. We
also organize a manual evaluation to score the accuracy and simplicity of
definitions.

##  5 Experiments

This section presents the experimental settings and evaluation methods.

###  5.1 Settings

##### Baselines

We compare the SimpDefiner with generation-simplification pipelines. We first
employ LOG-CaD (Ishiwatari et al., 2019) and MASS (Song et al., 2019) models
to generate definitions, and then employ ACCESS (Martin et al., 2019) and MUSS
(Martin et al., 2020) models to simplify them. Thus, we have four different
pipeline baselines. Since these models are not available in Chinese, we only
apply these pipelines to English datasets. For the Chinese SDG task, we
specially pretrained a MASS-ZH model from scratch using the Chinese Gigaword
Fifth Edition333https://catalog.ldc.upenn.edu/LDC2011T13 corpus. Note that we
set the learning rate to 3e-4, warmup steps to 500 when fine-tuning both MASS
and MASS-ZH.

##### SimpDefiner

We use the parameters in the MASS model to initialize the encoder and two
decoders in SimpDefiner. For the sentence corruption in the text
reconstruction task, we randomly delete or blank words with a uniform
probability of 0.2, and randomly shuffle the order of words within 5 tokens.
For the language modeling task, we set the input representations to $\bm{0}$
and use the simplified text as the target output. We tune the $\lambda$
parameters in Eq. 7 on the validation set and adopt the same hyper-parameters
as the baseline for comparison. We set 5 different random seeds as and report
the average result of multiple runs. Each run takes 7.68 GPU hours on 4
GeFource RTX 2080 Ti GPUs.

###  5.2 Evaluation

Evaluation of the generated definitions mainly focuses on two aspects, i.e.,
accuracy and simplicity. We perform both automatic and manual evaluations for
each aspect.

We first introduce these automatic metrics, and then the manual evaluation
method.

##### BLEU

Previous definition generation studies Noraset et al. (2017); Yang et al.
(2020); Kong et al. (2020) used the BLEU Papineni et al. (2002) score to
measure the closeness of generated results to the standard answers, and to
evaluate the accuracy of results. Since the English test set is manually
annotated, we calculate the BLEU score of both complex and simple definitions,
respectively.

##### Semantic Similarity

In addition to the BLEU score, we use the sentence-transformers toolkit
Reimers and Gurevych (2020) to convert the generated definitions and
references into sentence vectors, and calculate cosine similarity between
them.

##### SARI

SARI Xu et al. (2016) is a lexical simplicity metric that measures how good
are the words added, deleted and kept by a simplification model. This metric
compares the model output to simplification references and the original
sentence. We use the SARI implementation in the EASSE
toolkit444https://github.com/feralvam/easse.

##### HSK Level

HSK, namely Chinese Proficiency Test, is set up to test the proficiency of
non-native speakers555http://www.chinesetest.cn. It has nine levels, from easy
to hard, and each level corresponds to a vocabulary. We count the proportion
of words at levels 1-3 and 7+ in the generated definitions. The higher the
proportion of words in levels 1-3 (7+), the easier (more challenging) the
definitions are understood.

##### Manual Evaluation

We randomly select 200 words and contexts from the Chinese test set and let
the MASS and SimpDefiner generate definitions for them one by one. We mix the
two generated definitions and the golden complex definition and then ask three
native-speaker annotators to score them. Specifically, each annotator
evaluates the definitions on two criteria of accuracy and simplicity. Both
criteria have a range of 1-3. For accuracy, the annotators are asked to
evaluate how semantically relevant the definitions are to the word. For
simplicity, the annotators are asked to evaluate how simple the definitions
are. After collecting evaluation results, we average the scores as final
score.

##  6 Results and Analysis

| Complex | Simple |  SARI
---|---|---|---
|  BLEU |  SSim |  BLEU |  SSim |
LOG-CaD |  19.04 |  40.32 |  – |  – |  –
\+ ACCESS |  – |  – |  12.32 |  32.63 |  38.02
\+ MUSS |  – |  – |  11.74 |  27.66 |  36.53
MASS |  24.00 |  52.78 |  – |  – |  –
\+ ACCESS |  – |  – |  12.95 |  38.53 |  38.59
\+ MUSS |  – |  – |  12.58 |  37.49 |  38.48
SimpDefiner |  24.17 |  53.87 |  15.05 |  46.99 |  40.36
Table 3: Main results on the English test set. LOG-CaD (Ishiwatari et al., 2019) is a definition generation model. |  L1-3 (%) |  L7+ (%)
---|---|---
MASS |  44.16 |  37.05
SimpDefiner |  48.03 |  36.59
Table 4: Main results on the Chinese test set. |  |  #1 |  #2 |  #3 |  Avg.
---|---|---|---|---|---
Acc. | Golden |  3.00 |  2.93 |  2.98 |  2.97
MASS |  1.26 |  1.30 |  1.38 |  1.31
SimpDefiner |  1.48 |  1.47 |  1.59 |  1.51
Sim. | Golden |  2.04 |  2.06 |  2.11 |  2.07
MASS |  1.92 |  2.03 |  1.89 |  1.95
SimpDefiner |  2.14 |  2.04 |  2.21 |  2.13
Table 5: Manual evaluation results on the Chinese test set. Accuracy and
simplicity scores are listed in the table. The last column are averaged scores
among all three annotators.

###  6.1 Main Results

Table 3 and Table 4 present the experiment results on the English and Chinese
test sets respectively. Results show that our proposed SimpDefiner
significantly outperforms baseline methods of generation-simplification
pipelines on both English and Chinese datasets.

For English results, the performance of simple definition generation improves
2.1 and 8.46 on the BLEU and SemSim metrics respectively, and improves 1.77 on
the SARI metric. This indicates that both accuracy and simplicity are
effectively improved comparing with the baseline. We also observe that complex
definition generation also improves by 0.17 on BLEU and 1.09 on SemSim. This
shows that SimpDefiner improves the ability to generate both complex and
simple definitions.

For Chinese results, we compute the HSK Level metric on generated simple
definitions. We can see that the proportion of low-level (HSK level 1-3) words
increases by 3.87%, and that of high-level (HSK level 7+) words decreases by
0.46%. The lexical complexity of the SimpDefiner generated definitions are
significantly reduced.

Besides, we also conduct a manual evaluation on the Chinese test set, and the
results are listed in Table 5. From the averaged scores, we observe that
SimpDefiner outperforms MASS by 0.2 in terms of accuracy (more accurate) and
0.18 in terms of simplicity (more straightforward). On the accuracy score, all
three annotators agree that SimpDefiner has higher accuracy than MASS, which
shows the superiority of our framework. As expected, the golden definitions
have the highest accuracy in the table, far exceeding the definitions
generated by the two models. We believe this is caused by insufficient
knowledge in the model, and this can be solved by using larger pretrained
models, such as BART Lewis et al. (2019). On the simplicity score, three
annotators agree that SimpDefiner generates simpler definitions than MASS, and
two of three annotators think SimpDefiner generates simpler definitions than
the golden ones.

ID |  Model | Complex | Simple |  SARI
---|---|---|---|---
|  |  BLEU |  SSim |  BLEU |  SSim |
① |  SimpDefiner |  24.31 |  53.60 |  15.24 |  47.05 |  40.19
② |  ① – LM |  23.83 |  53.04 |  14.82 |  45.74 |  39.63
③ |  ② – TR |  25.02 |  53.80 |  13.66 |  44.01 |  38.58
④ |  ① – LN |  24.45 |  53.76 |  13.87 |  44.66 |  38.61
⑤ |  ① – QP |  23.40 |  52.95 |  14.61 |  45.57 |  39.87
⑥ |  ④ – QP |  24.80 |  53.95 |  13.90 |  44.77 |  38.52
Table 6: Ablation study on the English test set. LM: the language modeling
task. TR: the text reconstruction task. LN: complexity-dependent layer
normalization. QP: complexity-dependent query projection.

###  6.2 Ablation Study

We conduct ablation experiment to demonstrate the effectiveness of SimpDefiner
components and the parameter sharing scheme. For the language modeling (LM)
and text reconstruction (TR) tasks, we ablate them by setting their weights to
0. For the layer normalization (LN) and query projection (QP) as parameter-
shared layers, we ablate them by sharing their parameters between models. We
illustrate the experiment results in Table 6.

In general, ablating any of the components or parameter-shared layers reduces
the performance in terms of simple definitions, which indicates that the
SimpDefiner benefits from both components and parameter sharing scheme. We
also observe that the performance of ablation experiments have slight
disturbance on complex definitions. But since we pay more attention to the
performance on simple definitions, we argue that the benefits of SimpDefiner
far outweigh the losses.

($\lambda_{\alpha}$, $\lambda_{\beta}$, $\lambda_{\gamma}$) | Complex | Simple |  SARI
---|---|---|---
|  BLEU |  SSim |  BLEU |  SSim |
(0.8,0.1,0.1) |  24.31 |  53.60 |  15.24 |  47.05 |  40.19
(0.6,0.2,0.2) |  23.27 |  53.19 |  15.01 |  46.85 |  40.49
(0.4,0.3,0.3) |  21.92 |  51.82 |  15.11 |  46.54 |  40.74
Table 7: Different hyper-parameter settings on the English test set.

###  6.3 Analysis on Hyper-Parameters

Furthermore, we conduct additional experiments on the English dataset to study
how hyper-parameters affect the performance. By setting different $\lambda$ to
each model, we observe the relationship between the performance and these
weights.

The experiment results are listed in Table 7. From the table, we observe the
inconsistency between metrics. As the definition generation task weight
declines, the BLEU and SemSim metrics are generally declining, but the SARI
metric is increasing. Since the BLEU and SemSim measure the accuracy and the
SARI measures simplicity, we consider this phenomenon as a seesaw between the
two attributes of accuracy and simplicity. The balance between them can be
achieved by conditioning the hyper-parameters.

Word |  commander
---|---
Context |  Military commanders have warned coalition troops in the south.
Golden |  A person who is in charge of sth, especially an officer in charge of a particular group of soldiers or a military operation.
Baseline |  An officer of the highest rank is a country in a country.
SimpDefiner |  The head of a military force.
Word |  督促 (supervise and urge)
Context |  我很感谢他的支持、鼓励与督促。 I appreciate his support, encouragement and supervision.
Golden |  监督他人并促使后述事件发生。 Supervise others and promote the occurrence of the following events.
Baseline |  以后述对象为凭借进行特定事件。 Sb. is used as a reference for specific events.
SimpDefiner |  要求后述对象赶快行动。 Ask sb. to act quickly.
Table 8: Cases of generated simple definitions.

###  6.4 Case Study

Table 8 shows two generation cases from English and Chinese test set
respectively. In both cases, the golden definition is a long sentence with
quite complicated syntax. The baseline generated definitions contains
difficult words and often wrongly defines the given word. In the English case,
the word commander is defined by the baseline as an officer of the highest
rank in a country, which is incorrect in most cases. In the Chinese case, the
baseline generated definition contains difficult words like 凭借 (reference) and
特定事件 (specific events). On the other hand, the SimpDefiner generates simple
and accurate definitions in both cases.

##  7 Conclusion

In this work, we propose the SDG task, a novel task of generating simplified
definitions in a zero-shot manner. To this end, we leverage a multitasking
framework SimpDefiner to tackle this task. We introduce a text reconstruction
task to the framework to control the text complexity, and a language modeling
task to enhance the decoder. For evaluation, we construct a novel test set in
English by manually aligning the two dictionaries of OD and OALD. The
automatic and manual evaluations indicate that the our proposed framework can
generate more accurate and more straightforward definitions than other models
and the generation-simplification pipelines. In the future, we will try to
combine the current method with prompt learning methods, aiming to let users
condition the complexity of generated definitions.

## Acknowledgements

This work was supported by the funds of Beijing Advanced Innovation Center for
Language Resources (No. TYZ19005), Research Project of the National Language
Commission (No. ZDI135-131) and National Natural Science Foundation of China
(No. 62106138, No. 61872402). We would like to thank Xiaowan Wang, Chenhui
Xie, and Junhui Zhu for their manual evaluation and all anonymous reviewers
for their valuable comments and suggestions on this work.

## References

  * Bartunov et al. (2016) Sergey Bartunov, Dmitry Kondrashkin, Anton Osokin, and Dmitry Vetrov. 2016.  Breaking sticks and ambiguities with adaptive skip-gram.  In _Proceedings of the 19th International Conference on Artificial Intelligence and Statistics_ , volume 51.
  * Bevilacqua et al. (2020) Michele Bevilacqua, Marco Maru, and Roberto Navigli. 2020.  Generationary or “how we went beyond word sense inventories and learned to gloss”.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing_.
  * Chang et al. (2018) Ting-Yun Chang, Ta-Chung Chi, Shang-Chi Tsai, and Yun-Nung Chen. 2018.  xsense: Learning sense-separated sparse representations and textual definitions for explainable word sense networks.  _CoRR_ , abs/1809.03348.
  * Devlin et al. (2019) Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2019.  BERT: Pre-training of deep bidirectional transformers for language understanding.  In _Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_.
  * Dumoulin et al. (2017) Vincent Dumoulin, Jonathon Shlens, and Manjunath Kudlur. 2017.  A learned representation for artistic style.  In _5th International Conference on Learning Representations, ICLR 2017_.
  * Enayati and Gilakjani (2020) Fatemeh Enayati and Abbas Pourhosein Gilakjani. 2020.  The impact of computer assisted language learning (CALL) on improving intermediate EFL learners’ vocabulary learning.  _International Journal of Language Education_ , 4(2).
  * Gadetsky et al. (2018) Artyom Gadetsky, Ilya Yakubovskiy, and Dmitry Vetrov. 2018.  Conditional generators of words definitions.  In _Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics_.
  * Guo et al. (2018) Han Guo, Ramakanth Pasunuru, and Mohit Bansal. 2018.  Dynamic multi-level multi-task learning for sentence simplification.  _CoRR_ , abs/1806.07304.
  * Huang et al. (2010) Chu-Ren Huang, S. Hsieh, Jia-Fei Hong, Yun-Zhu Chen, I. Su, Yong-Xiang Chen, and Sheng-Wei Huang. 2010.  Chinese wordnet : design, implementation, and application of an infrastructure for cross-lingual knowledge processing.  _Journal of Chinese information processing_ , 24.
  * Huang et al. (2021) Han Huang, Tomoyuki Kajiwara, and Yuki Arase. 2021.  Definition modelling for appropriate specificity.  In _Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing_.
  * Ishiwatari et al. (2019) Shonosuke Ishiwatari, Hiroaki Hayashi, Naoki Yoshinaga, Graham Neubig, Shoetsu Sato, Masashi Toyoda, and Masaru Kitsuregawa. 2019.  Learning to describe unknown phrases with local and global contexts.  In _Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_. Association for Computational Linguistics.
  * Jin et al. (2020) Di Jin, Zhijing Jin, Joey Tianyi Zhou, Lisa Orii, and Peter Szolovits. 2020.  Hooks in the headline: Learning to generate headlines with controlled styles.  In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_.
  * Kawashima and Takagi (2019) Takashi Kawashima and Tomohiro Takagi. 2019.  Sentence simplification from non-parallel corpus with adversarial learning.  In _2019 IEEE/WIC/ACM International Conference on Web Intelligence_. IEEE.
  * Kilgarriff (1997) Adam Kilgarriff. 1997.  I don’t believe in word senses.  _Computers and the Humanities_ , 31(2).
  * Kong et al. (2020) Cunliang Kong, Liner Yang, Tianzuo Zhang, Qinan Fan, Zhenghao Liu, Yun Chen, and Erhong Yang. 2020.  Toward cross-lingual definition generation for language learners.  _CoRR_ , abs/2010.05533.
  * Lewis et al. (2019) Mike Lewis, Yinhan Liu, Naman Goyal, Marjan Ghazvininejad, Abdelrahman Mohamed, Omer Levy, Veselin Stoyanov, and Luke Zettlemoyer. 2019.  BART: denoising sequence-to-sequence pre-training for natural language generation, translation, and comprehension.  _CoRR_ , abs/1910.13461.
  * Li et al. (2020) Jiahuan Li, Yu Bao, Shujian Huang, Xinyu Dai, and Jiajun Chen. 2020.  Explicit semantic decomposition for definition generation.  In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_.
  * Lolita et al. (2020) Yuri Lolita, Endry Boeriswati, and Ninuk Lustyantie. 2020.  The impact of computer assisted language learning (CALL) use of english vocabulary enhancement.  _Linguistic, English Education and Art Journal_ , 4(1).
  * Martin et al. (2020) Louis Martin, Angela Fan, Éric de la Clergerie, Antoine Bordes, and Benoît Sagot. 2020.  Multilingual unsupervised sentence simplification.  _CoRR_ , abs/2005.00352.
  * Martin et al. (2019) Louis Martin, Benoît Sagot, Éric de la Clergerie, and Antoine Bordes. 2019.  Controllable sentence simplification.  _CoRR_ , abs/1910.02677.
  * Mickus et al. (2019) Timothee Mickus, Denis Paperno, and Matthieu Constant. 2019.  Mark my word: A sequence-to-sequence approach to definition modeling.  In _Proceedings of the First NLPL Workshop on Deep Learning for Natural Language Processing_ , Turku, Finland. Linköping University Electronic Press.
  * Nisioi et al. (2017) Sergiu Nisioi, Sanja Štajner, Simone Paolo Ponzetto, and Liviu P. Dinu. 2017\.  Exploring neural text simplification models.  In _Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics_.
  * Noraset et al. (2017) Thanapon Noraset, Chen Liang, Larry Birnbaum, and Doug Downey. 2017.  Definition modeling: Learning to define word embeddings in natural language.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 31.
  * Papineni et al. (2002) Kishore Papineni, Salim Roukos, Todd Ward, and Wei-Jing Zhu. 2002.  Bleu: a method for automatic evaluation of machine translation.  In _Proceedings of the 40th Annual Meeting of the Association for Computational Linguistics_.
  * Raffel et al. (2019) Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J. Liu. 2019.  Exploring the limits of transfer learning with a unified text-to-text transformer.  _CoRR_ , abs/1910.10683.
  * Reid et al. (2020) Machel Reid, Edison Marrese-Taylor, and Yutaka Matsuo. 2020.  VCDM: Leveraging Variational Bi-encoding and Deep Contextualized Word Representations for Improved Definition Modeling.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing_.
  * Reimers and Gurevych (2020) Nils Reimers and Iryna Gurevych. 2020.  Making monolingual sentence embeddings multilingual using knowledge distillation.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing_. Association for Computational Linguistics.
  * Rosch and Mervis (1975) Eleanor Rosch and Carolyn B Mervis. 1975.  Family resemblances: Studies in the internal structure of categories.  _Cognitive Psychology_ , 7(4).
  * Segler et al. (2002) Thomas M Segler, Helen Pain, and Antonella Sorace. 2002.  Second language vocabulary acquisition and learning strategies in ICALL environments.  _Computer Assisted Language Learning_ , 15(4).
  * Song et al. (2019) Kaitao Song, Xu Tan, Tao Qin, Jianfeng Lu, and Tie-Yan Liu. 2019.  Mass: Masked sequence to sequence pre-training for language generation.  In _International Conference on Machine Learning_. PMLR.
  * Sutskever et al. (2014) Ilya Sutskever, Oriol Vinyals, and Quoc V Le. 2014.  Sequence to sequence learning with neural networks.  In _Advances in neural information processing systems_.
  * Tyler and Evans (2001) Andrea Tyler and Vyvyan Evans. 2001.  Reconsidering prepositional polysemy networks: The case of over.  _Language_.
  * Vaswani et al. (2017) Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. 2017.  Attention is all you need.  In _Proceedings of the 31st International Conference on Neural Information Processing Systems_.
  * Vu et al. (2018) Tu Vu, Baotian Hu, Tsendsuren Munkhdalai, and Hong Yu. 2018.  Sentence simplification with memory-augmented neural networks.  In _Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_.
  * Wubben et al. (2012) Sander Wubben, Antal van den Bosch, and Emiel Krahmer. 2012.  Sentence simplification by monolingual machine translation.  In _Proceedings of the 50th Annual Meeting of the Association for Computational Linguistics_.
  * Xu et al. (2016) Wei Xu, Courtney Napoles, Ellie Pavlick, Quanze Chen, and Chris Callison-Burch. 2016\.  Optimizing statistical machine translation for text simplification.  _Transactions of the Association for Computational Linguistics_ , 4\.
  * Yang et al. (2020) Liner Yang, Cunliang Kong, Yun Chen, Yang Liu, Qinan Fan, and Erhong Yang. 2020\.  Incorporating sememes into chinese definition modeling.  _IEEE/ACM Transactions on Audio, Speech, and Language Processing_ , 28.
  * Zhang and Lapata (2017) Xingxing Zhang and Mirella Lapata. 2017.  Sentence simplification with deep reinforcement learning.  In _Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing_.
  * Zhang (2011) Yihua Zhang. 2011.  Discussion on the Definitions in Chinese Learner’s Dictionaries: Comparative Study of Domestic and Foreign Learner Dictionaries (Translated from Chinese).  _Chinese Teaching in the World_.