# Laser-induced forces on atoms during ultrafast demagnetization

G. P. Zhang∗ Department of Physics, Indiana State University, Terre Haute,
Indiana 47809, USA  Y. H. Bai  Office of Information Technology, Indiana State
University, Terre Haute, Indiana 47809, USA

###### Abstract

Laser-induced femtosecond demagnetization has attracted a broad attention as a
possible candidate for information storage technology. However, whether or not
lattice vibration directly participates in demagnetization has been highly
controversial over a decade. A recent electron diffraction experiment
attributed the demagnetization to the polarized phonon effect, but a similar
x-ray diffraction experiment attributed it to the Einstein-de Haas effect.
Common to both experiments is that neither the angular momentum of the lattice
nor the rotation of the sample was directly probed. Here, we report our first
first-principles calculation of forces on atoms induced by an ultrafast laser
during ultrafast demagnetization. We employ two complementary methods: (i) the
frozen lattice with electronic excitation and (ii) frozen excitation but
moving the lattice. We find that the forces on atoms start at -50 fs and peak
around 30 fs. The magnitude of the force is far smaller than the empirical
estimates. Within the limit of our theory, our results suggest that the
polarized phonon effect and the Einstein-de Haas effect are unlikely to be the
main course of demagnetization. We expect that our finding has a profound
impact on the future direction of laser-induced dynamics in magnetic and
quantum materials.

##  I Introduction

In 1996, Beaurepaire and his coworkers eric  reported that a 60-fs laser pulse
can quench the magnetization of a nickel thin film more than 40% within 1 ps.
This pioneering discovery has forever changed the landscape of spin
manipulation, where a laser field, instead of a magnetic field, is employed.
The accelerated demagnetization, found in many magnetic materials hohlfeld1997
; aeschlimann1997 ; ju1998 ; roth2012 ; mann2012 ; sultan2012 ; chen2019 , has
attracted enormous attentions across several decades around the worldourreview
; rasingreview , for a historical account of the discovery of femtomagnetism,
see a recent book ourbook . This unprecedented ultrashort time scale involves
a large group of interactions among electrons, spins, orbitals and phonons,
either directly or indirectly, which significantly complicates femtomagnetism
ourreview . The first theory prl00  emphasized a crucial interplay between the
laser field and the spin-orbit coupling, so the spin symmetry can be broken
and the spin moment can be changed. Subsequent mechanisms invoke the phonon-
based Elliot-Yafet mechanism koopmans2010 ; born2021  with the spin flipping
through the spin-orbit coupling prl00 , spin transport with majority spin
moving out of the excited region battiato2012 , bands mirroring effect
eich2017  with the spin majority and minority switching their roles, spin-wave
excitation jap19  across many lattice sites to reduce the spin moment for the
entire sample, spin disordering chen2019w  within a big cell, ultrafast
electron correlations and memory effects prl20  through the dynamic exchange
kernel, optical inter-site spin transfer willems2020 , and electron-phonon
scattering baral2014 . The fact that so many mechanisms were proposed
demonstrates the complexity of femtomagnetism. To this end, no single
mechanism can explain all the experimental observations; and probably it may
never be possible, given the fact that diverse materials show different
dynamics. For instance, Fe and Gd have quite different demagnetization
processes sultan2012 .

Another line of thought is formulated around the spin angular momentum
transfer hennecke2019 ; dewhurst2021b ; zahn2021 , without considering the
fact that the total angular momentum operator is not a conserved quantity in
solids prb08 . Once this total angular momentum conservation is assumed, then
it is easy to see that if the spin angular momentum of the electron system
cannot transfer to the orbital angular momentum of the electron jpcc21 , it
must go to the lattice stamm2010 . Although this mechanism cannot exclude
other mechanisms such as spin quenching due to the spin misalignment, it
underlines two recent experimental investigations dornes2019 ; tauchert2022 .
However, to this end, no experiment ever directly measured the rotation of
samples or the lattice angular momentum as one should if this is due to the
Einstein-de Haas effect. Dornes et al. dornes2019  employed the time-resolved
X-ray diffraction as the probe and 40-fs laser pulse of 800 nm as the pump and
measured the transverse strain wave, from which they inferred through modeling
that 80% spin angular momentum in the Fe film went to the lattice within 200
fs. Tauchert et al. tauchert2022  carried out the electron diffraction
experiment and found the Bragg spot position from their Ni film did not change
with time within their experimental uncertainty, so there was no transient
strain or lattice expansion, contradicting the finding by Dornes et al.
dornes2019 . It is difficult to argue that the different conclusions reached
from these two experiments are due to the technique difference, as electron
diffraction should induce more strain than X-ray diffraction. Instead,
Tauchert et al. tauchert2022  found an anisotropic diffraction pattern that
follows the initial magnetization of their sample, which they attributed to
the polarized phonon involvement, but phonons are rarely magnetically active
baydin2022  and the long-lasting associated orbital moment of the electron
observed in He atoms watzel2022  cast doubts on this interpretation. The key
to resolve these difficult issues is to find the force on the atoms. If the
force is centripetal, then the postulate of the polarized phonon is valid.

Theory can help in this regard by tracing back to the initial stage of laser
excitation, where electrons are newly excited out of the Fermi sea and the
force starts to appear on atoms. In this paper, we develop a scheme that
combines the ground state density functional theory and the time-dependent
Liouville formalism in a supercell. This allows us to compute the force on the
atoms during the laser-induced ultrafast demagnetization. We employ one free-
standing Ni(001) monolayer as our first example. We find that the force on the
Ni atom starts around -50 fs, followed by a rapid oscillation due to the
strong charge fluctuation. The force ceases to change once the electronic
excitation configuration is formed. Quantitatively, we find that under
moderately strong laser pulse excitation, the amplitude of force is 0.01
mRy/a.u., which is far below the estimate based on the Tauchert’s parameter
tauchert2022 . Although the force does change directions during the laser
excitation, we do not detect evidence of the centripetal force as required by
the polarized phonon. Our results in fcc Ni further show that the directions
of forces on all four Ni atoms do not change. Therefore, our present
theoretical results, within the limit of our current theory, do not support
the picture of the polarized phonons. This finding is expected to motivate
further experimental and theoretical investigations in the future.

The rest of the paper is arranged as follows. In Sec. II, we present our
theoretical algorithm, which combines the ground state calculation and the
time-dependent Liouville equation. Section III is devoted to (i) the results
in the Ni(001) monolayer, (ii) the effect of core states and the spin
polarization on the force, and (iii) the results in fcc Ni. We conclude this
paper in Sec. IV.

##  II Theoretical formalism

Laser-driven ultrafast demagnetization starts with the electronic excitation
which tip the balance of force on atoms. Figure 1 schematically shows a
typical laser excitation. A laser pulse excites the electron first and through
the electron-lattice interaction, atoms experience a net force. So the key to
resolve the above controversy is to directly compute the forces on atoms under
laser excitation. If the force is centripetal, then the atoms will undergo a
circular motion. However, such a calculation is difficult, if not impossible,
since a solid has highly coupled interactions among electron, spin and lattice
dynamics over several hundred fs in a supercell. This is part of the reason
why so far there has been no theory attempt to compute forces. We find a
method that integrates two complementary algorithms into a well-defined
formalism.

The top left of Fig. 1 shows that our density-functional theoretical
calculation starts with the ground state calculation, by solving the Kohn-Sham
equation self-consistently jpcm16 ,

$\left[-\frac{\hbar^{2}\nabla^{2}}{2m_{e}}+v_{eff}({\bf r})\right]\psi_{i{\bf
k}}({\bf r})=E_{i{\bf k}}\psi_{i{\bf k}}({\bf r}),$ (1)

where $\psi_{i{\bf k}}({\bf r})$ and $E_{i{\bf k}}$ are, respectively, the
eigenstate and eigenenergy of band $i$ and ${\bf k}$ point. $v_{eff}$ is
determined by

$v_{eff}({\bf r})=v({\bf r})+\int\frac{n({\bf r}^{\prime})}{|{\bf r}-{\bf
r}^{\prime}|}d{\bf r}^{\prime}+v_{xc}({\bf r}),$ (2)

where $v_{xc}({\bf r})$ is the exchange-correlation potential, $v_{xc}({\bf
r})=\delta E_{xc}[n]/\delta n({\bf r})$. We employ the Wien2k code, which uses
the full-potential augmented planewave basis. The spin-orbit coupling is
included through the second variational principle wien2k . We use the
generalized gradient approximation for the exchange-correlation energy
functional.

The forces on atom $\alpha$ contain several contributions. The Hellmann-
Feynman (HF) force in the atomic units is yu1991

${\bf
F}^{\alpha}=z_{\alpha}\frac{d}{d{\tau}_{\alpha}}\left(-\sum^{\prime}_{\beta}\sum_{\bf
R}\frac{z_{\beta}}{|\tau_{\alpha}-\tau_{\beta}+{\bf R}|}+\int\frac{n({\bf
r})}{|\tau_{\alpha}-{\bf r}|}d{\bf r}\right),$ (3)

where $z_{\alpha}$ is the atomic number, $\tau_{\alpha}$ is its position in a
unit cell, and ${\bf R}$ is the lattice vector. The prime over the summation
excludes the same atom. This force is common among all the methods. In LAPW,
the states are separated into core states, semicore state, and valence states.
The core correction to the HF force is

${\bf F}_{\rm core}^{\alpha}=-\int\rho_{c}^{\alpha}({\bf r})\nabla
v_{eff}({\bf r})d{\bf r},$ (4)

where $\rho_{c}^{\alpha}({\bf r})$ is the core charge density at atom
$\alpha$. The treatment of valence states is more complicated due to the
incomplete basis-function set (IBS) that is associated with the LAPW basis
function,

${\bf F}_{\rm
IBS}=-\sum_{i}\rho_{i}\left(\langle\psi_{i}^{\prime}|(H-\epsilon_{i})|\psi_{i}\rangle+\langle\psi_{i}|(H-\epsilon_{i})|\psi_{i}^{\prime}\rangle+{\bf
D}_{i}\right),$ (5)

where $\rho_{i}$ is the occupation number of valence state $i$, $\epsilon_{i}$
is the eigenvalue, $H$ is the Kohn-Sham Hamiltonian, $\psi_{i}$ is the
eigenfunction, and $\psi_{i}^{\prime}$ denotes its derivative with respect to
the atomic position $\tau_{\alpha}$. ${\bf D}_{i}$ is the kinetic energy
contribution due to the change in both the MT sphere and the interstitial
regions yu1991 ; madsen2001 . The total force is the sum of the HF force, the
core and IBS corrections. Wien2k has a limitation that its force has no
contribution from the spin-orbit coupling (SOC) even if we include SOC. Since
SOC in general is a weak contributor energetically, we do not expect that
missing force from SOC has a qualitative effect on our force.

Upon laser excitation, electrons are first excited out of the Fermi sea. We
employ the time-dependent Liouville equation np09 ; jpcm16

$i\hbar\frac{\partial\rho}{\partial t}=[H_{0}+H_{I},\rho],$ (6)

where $H_{0}$ is the unperturbed system Hamiltonian and $H_{I}$ is the
interaction between the laser field and the system np09 , $H_{I}=-{{\bf
p}\cdot{\bf A}(t)}/{m_{e}}$. The vector potential is ${\bf
A}(t)=A_{0}\exp(-t^{2}/\tau^{2})(\cos\omega t\hat{x}+\sin\omega t\hat{y})$,
where $A_{0}$ is the amplitude, $\tau$ is the laser pulse duration, $t$ is the
time, $\hat{x}$ and $\hat{y}$ are the unit vectors of the $x$ and $y$ axes,
respectively, and the laser polarization is within the $xy$ plane. We note in
passing that the fluence of laser field is computed from ourbook

${\cal
F}(\omega,\tau)=\sqrt{\frac{\pi}{2}}n(\omega)c\epsilon_{0}A_{0}^{2}\omega^{2}\tau,$
(7)

where $c$ is the speed of light, $n(\omega)$ is the index of refraction, and
$\tau$ is the pulse duration.

We only propagate our time step from $t$ to $t+\Delta t$. The right figure of
Fig. 1 outlines the key steps of our implementation. The excited state density
$\rho$ is used to construct the excited state density and is fed back into Eq.
1 for an excited state self-consistent calculation jpcm16 . The shaded box
highlights this self-consistency. The converged density is looped back into
the Liouville equation (Eq. 6) to start a new time step.

##  III Results and discussions

Traditional force calculations in solids often adopt a single $\Gamma$ point
liu2021 , which is apparently unsuitable for metals. We propose a different
formalism to tackle this issue. We freeze the lattice in the same fashion as
the frozen phonon calculation, and then directly compute the forces on the
atoms during laser excitation. Regardless of whether angular momentum transfer
or electron-lattice coupling is central to demagnetization, the lattice must
experience a force in order to vibrate.

###  III.1 Nickel(001) monolayer

We take a free-standing nickel(001) monolayer as an example. Since its
primitive cell has no net force on atoms, we adopt a $2\times 2\times 1$
supercell, with two distinctive atoms Ni1 and Ni2 at (0,0,0) and (1/2,1/2,0),
respectively. We insert a vacuum layer of thickness 33 bohr along the $c$ axis
to eliminate the interaction between each layer. We choose a $k$ mesh of
$20\times 20\times 1$ in the crystal momentum space. The product of the
muffin-tin radius and the planewave cutoff $R_{\rm MT}K_{\rm max}$ is 7.

Forces in the ground state – We displace the Ni1 atom along the $x$ axis by a
small amount $\delta x_{1}$ from -5% to +5%, in the unit of the lattice
constant $a$. Figure 2(a) displays the total energy change $\Delta E$ as a
function of $\delta x_{1}$, where a typical parabola is observed. Here the
energy is referenced to the minimum energy. Forces on atoms are shown in Fig.
2(b). Since we move Ni1 along the $x$ axis, the main force is along the $x$
axis, while the forces along the $y$ are tiny (see the lines close to 0). The
empty (filled) circles denote the force $F_{1x}$ ($F_{2x}$) on Ni1 (Ni2).
$F_{1x}$ and $F_{2x}$ have an opposite slope as expected. The force slope
gives an estimate of the angular frequency of 10.76 THz for Ni, representing
an upper bound phonon frequency in the electronic ground state.

Forces in the excited state – We employ a circularly polarized pulse of
$\tau=60$ fs and the photon energy is $\hbar\omega=1.6$ eV. We choose a vector
field amplitude $A_{0}=0.03\rm~{}Vfs/\AA$, which corresponds to a fluence of
10.61 $\rm mJ/cm^{2}$, very typical experimentally eric ; jpcm10 ; dornes2019
; tauchert2022 . The full-width at half-maximum of the pulse is about 100 fs.
Figure 2(c) is the energy change as a function of time. We see the system
absorbs the energy quickly upon laser excitation; the entire process starts at
-100 fs and ends around 100 fs. Our interest is in the force. Figure 2(d)
shows the force on Ni1, and the force on Ni2 is similar and not shown.
$F_{1x}$ gains an appreciable value around -50 fs, and then it starts a rapid
oscillation. We verify that these rapid oscillations are from the charge
fluctuation during the time propagation of density. The spikes seen in the
force in the self-consistent step appear from one time step to next. Our time
step is 1/32 the laser period. We run four separate calculations under
different conditions, with or without the core states and with or without
spin-polarization rescaling jpcm16 , to verify that while details of these
rapid oscillations may differ from one to another, the general pattern is
highly reproducible. Naturally, atoms cannot respond to these rapid
oscillations, and the force after these rapid oscillations is important. If
the polarized phonon concept tauchert2022  is valid, we are supposed to see
the force direction change periodically. This direction change is barely
observed around 42 fs, where $F_{1y}$ transitions from a positive force to a
negative one, while $F_{1x}$ does the opposite. However, after 50 fs, both
force components settle down, and there is no evidence of centripetal force
from our data.

Tauchert et al. tauchert2022  carried out ultrafast electron diffraction to
detect the motion of Ni atoms in a nickel single crystal layer after optical
excitation. They found no change in the Bragg spot broadening and no change in
lattice displacement, but the intensity ratio between (200) and (020) planes
depends on the direction of the applied magnetic field. They attributed this
difference to the polarized phonon. They assumed that the 50% spin angular
momentum loss is converted to the circular motion of the Ni atom,

$\Delta L=M_{\rm Ni}\omega R^{2}=0.16\hbar,$ (8)

where $\omega$ is angular frequency, $M_{\rm Ni}$ is the atomic mass of Ni,
and $R$ is the radius of the circular motion of 0.019 $\rm\AA$ tauchert2022 .
Because the centripetal force on the atom is

$F=M_{\rm Ni}\omega^{2}R,$ (9)

we combine Eqs. 8 and 9 to rewrite the force as

$F=\omega\Delta L/R.$ (10)

This expression has a nice feature where the force is independent of mass,
thus eliminating the material difference between Dornes dornes2019  and
Tauchert tauchert2022 . The only remaining quantity is $\omega$.
Unfortunately, there is no information on $\omega$. Due to the translation
symmetry in solids, a single normal mode never undergoes a circular motion,
qualitatively different from molecular crystals prb21 . Tauchert et al.
tauchert2022  interpreted the highest phonon frequency of 8 THz found in the
nickel phonon spectrum as $\omega$, which in general is incorrect or
questionable. But using it, we can estimate the force from their data to be
10.84 mRy/bohr per atom, a value that we can compare with.

Figure 2(d) presents a different picture. Our force is only 0.01 mRy/bohr,
three orders of magnitude less than theirs. 10.84 mRy/bohr is also too large
for the ground state. If one used the same slope of the force as Fig. 2(b),
10.84 mRy/bohr would translate $\delta x_{1}$ to be 10%, or 0.3 $\rm\AA$,
which is really too large for nuclear vibrations. In $\rm C_{60}$, we
findprb21  the lattice displacement around 0.05 $\rm\AA$. Up to now, our
lattice is frozen. What if we release it in the electronic excited state? We
choose an electronic excited configuration around 566 fs, and then displace
Ni1 along the $x$ axis, just as we did for the ground state in Fig. 2(b).
Figure 2(e) shows the total energy change as a function of $\delta x_{1}$. We
find that in the electronic excited state, the parabola is steeper. For the
same $\delta x_{1}$, $\Delta E_{ex}=7\times 10^{-5}$ Ry, larger than $\Delta
E_{gs}=3.2\times 10^{-5}$ Ry. The force changes are shown in Fig. 2(f). It is
clear that the force change is more complex and no longer has a linear
relation with $\delta x_{1}$. If $\delta x_{1}$ is smaller than $\pm 1\%$, the
dependence is similar to the ground state (see Fig. 2(b)). But as soon as
$\delta x_{1}$ is larger than $\pm 2\%$, $F_{1x}$ decreases with $\delta
x_{1}$. $F_{2x}$ shows an opposite trend. In the ground state, $F_{1y}$ and
$F_{2y}$ are very small, but now in the excited state, they become
appreciable. If we compare the force magnitudes between Figs. 2(d) and 2(f),
we see that the force reaches 1 mRy/bohr. This shows that our excited state
force in Fig. 2(d) is a lower limit. But the force at the maximum is still 10
times smaller than Tauchert’s force.

###  III.2 Effect of core states and spin polarization scaling on the force

Since there has been no prior study on the laser-induced force on atoms during
demagnetization, we carry out four complementary tests to iron out some of
possible scenarios that Wien2k could provide.

We first examine the effect of core states on our force. The Ni-L edge has
eight core states ($3s^{2}3p^{6}$) for each Ni atom, which locate 65 eV below
the Fermi level. There are 10 valence electrons ($3d^{8}4s^{2}$ or
$3d^{9}4s^{1}$). A single excitation from 1.6-eV photons rarely affects the
core states, but multiple excitations do. Since core states have a larger
contribution to the force, their effect is of interest. For these reasons, a
direct test is necessary. Our time-dependent Liouville equation can
conveniently select a group of states with a particular window. Figure 3(a)
includes states from 5 ($3p^{6}$) to 100 (20 eV from the Fermi level), which
covers the core states, all the valence states and 66 conduction states. This
is the same data for our Fig. 2(d). By contrast, Fig. 3(b) includes all the
valence states but does not include core states. All the rest of the
parameters, including the laser parameters, are the same. We see that their
general trend remains similar, but there is a clear difference: Figure 3(a)
has a negative $F_{1x}$, but Figure 3(b) has a positive $F_{1x}$. The
direction of $F_{1y}$ is also exchanged. This shows that the core states do
contribute. We also understand why. Although the core states do not directly
participate in the optical excitation, the force from the core state contains
a correction term (Eq. 4) due to the effective potential $v_{eff}$. And this
$v_{eff}$ depends on the electron density $\rho({\bf r})$. So during the laser
excitation, once $\rho$ changes, the force due to the core state changes.

Next, we investigate whether the spin-polarization scaling factor jpcm16
affects the forces. This scaling is used in the spin-polarized potential. For
$\alpha$ electron, the potential is

$V_{\alpha}({\bf r},t)=\frac{1}{2}\left[(1+\xi(t))V_{\alpha}({\bf
r})+(1-\xi(t))V_{\beta}({\bf r})\right]$ (11)

where $\xi(t)=M(t)/M_{0}$ is computed from the spin moment $M(t)$. A similar
equation for $\beta$ electron. If we set $\xi(t)=1$, this means that we do not
have scaling. If $\xi(t)<1$, the potential is less spin-polarized. Figure 3(c)
has neither the core state nor the spin-polarization scaling. We see that
those rapid oscillations in the force are suppressed. The force directions are
now similar to Fig. 3(a). Finally, we remove the spin-polarization rescaling
but keep the core states. Figure 3(d) shows that these rapid oscillations
return, and in addition the force is smaller.

In summary, we should mention that our calculations are extremely time
consuming, 3-4 months to finish, since for each time step we self-consistently
converge the results. During the time when the laser peaks, the convergence
criteria are lowered because a large number of conduction states are occupied
and the self-consistency becomes difficult. In traditional TDDFT, there is no
self-consistency, but in our time-dependent Liouville density functional
theory, we have this extra step. There is another problem that is associated
with the Wien2k code. In the excited states, the electrons become very
delocalized, so we have to put some electrons as the background charge. This
background charge is not used for Figs. 3(a)-(c), but is used for Fig. 3(d).
For the moment, we do not find a viable method to overcome this difficulty. A
planewave code may help, but it often uses the pseudopotential, where the core
states are replaced by a pseudopotential and the effect of the core states on
the force on atoms during laser excitation is not possible to investigate.

###  III.3 fcc nickel

In order to demonstrate that our conclusion is not limited to the Ni(001)
monolayer, we also compute the force on bulk fcc nickel. For the same reason
as above, we adopt a simple cubic cell as a supercell for fcc Ni, where we
have four atoms per cell and remove all the symmetry operations, except the
identity matrix. Figure 4(a) shows the forces on atoms Ni1 and Ni2. Since
forces on Ni1 and Ni2 are similar, we only show that on Ni1. They do not gain
an appreciable value until -70 fs. $F_{1,x}$ points in the $x$ axis, while
$F_{1,y}$ points in the $-y$ axis. The magnitude of $F_{1,z}$ is much smaller.
At 0 fs when the laser peaks, the force drops to zero. The force peaks around
33 fs, after which it stabilizes around 0.5 $\mu$Ry/bohr. Figure 4(b) shows
the forces on Ni3 and Ni4. The general feature is similar. We can estimate how
far atoms move, provided the force remains constant. Take the force at 0.5
$\mu$Ry/bohr as an example, and we obtain the acceleration $a$ is
$0.2114\times 10^{12}$ m/s2. For 1 ps, the atom moves 0.1057 pm, far less than
1.9 pm postulated by Tauchert et al. tauchert2022 . If we used their time
scale of 125 fs (8 THz), the displacement would fall below the thermal
disturbance. Importantly, our theory predicts that $F_{1,x}$ and $F_{1,y}$ are
180∘ out of phase, with the resultant force along the $[1,\bar{1},0]$
direction. These forces are unlikely to lead to a circular motion of Ni atoms
around the lattice point, because they are not centripetal in any sense.
Instead, a regular vibration is expected. This raises a question whether their
polarized phonon really exists.

##  IV Conclusions

In contrast to all the prior studies, we have directly computed the laser-
induced forces on atoms during ultrafast demagnetization. We find that the
force arises around -50 fs before the laser pulse peaks at 0 fs. Due to the
strong charge fluctuation, the initial forces are highly oscillatory, and peak
around 50 fs, settling down after 100 fs. Although we have observed that the
force does change its direction, it does not appear to be centripetal, as
would be expected from the polarized phonon picture tauchert2022 .
Quantitatively, we find the force magnitude is at least one order of magnitude
weaker than the early estimate tauchert2022 . This conclusion is true for both
the Ni(001) monolayer and fcc nickel, within the limit of our current theory.
Interestingly, a recent study in He atoms reported the long-lasting orbital
moment of the electron watzel2022 , not phonon. Could the polarized phonon
turn out to be the electron contribution, not the lattice contribution? This
likelihood scenario is supported by our theory and also the fact that the
experiment employed the electron diffraction, but it can only be resolved with
further experimental and theoretical investigations jpcm22 ; mankovsky2022 .

###### Acknowledgements.

We would like to thank the helpful communication with Dr. Peter Baum
(Konstanz, Germany) on their paper tauchert2022 . GPZ and YHB were supported
by the U.S. Department of Energy under Contract No. DE-FG02-06ER46304. Part of
the work was done on Indiana State University’s quantum and obsidian clusters.
The research used resources of the National Energy Research Scientific
Computing Center, which is supported by the Office of Science of the U.S.
Department of Energy under Contract No. DE-AC02-05CH11231. ∗guo-
<EMAIL_ADDRESS>https://orcid.org/0000-0002-1792-2701

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Figure 1:  (Bottom left) Laser-induced force on atoms during the ultrafast
demagnetization has an enormous impact on subsequent lattice dynamics. We
employ a circularly polarized 60-fs laser pulse $\sigma_{xy}$. For a Ni(001)
monolayer, a supercell $(2\times 2\times 1)$ is used. (Top left) Schematic of
our theoretical formalism for our ground state calculation. (Right) Time-
dependent self-consistent Liouville density functional theory.  Figure 2: (a)
Total energy change $\Delta E_{gs}$ as a function of the Ni1 position $\delta
x_{1}$ in the electronic ground state. $\delta x_{1}$ is in the units of
lattice constant $a$. (b) Force as a function of $\delta x_{1}$. The open
circles denote $F_{1x}$ and the filled circles denote $F_{2x}$. The $y$
components are small, close to the zero. (c) Under laser excitation, the total
energy change $\Delta E(t)$ as a function of time $t$. The laser pulse has
duration of 60 fs, photon energy of 1.6 eV and the vector potential amplitude
of 0.03 $\rm Vfs/\AA$. (d) Force on Ni1. The solid line represents $F_{1x}$
and the dotted line $F_{1y}$. Force on Ni2 is similar, and not shown. (e) In
the electronic excited states, the total energy change $\Delta E_{ex}$ as a
function of $\delta x_{1}$. (f) In the electronic excited states, the forces
on atoms have a different dependence from the ground state counterpart. The
unfilled circles denote $F_{1x}$, and the filled ones $F_{2x}$. The unfilled
and filled boxes correspond to $F_{1y}$ and $F_{2y}$, respectively.  Figure 3:
Forces on atoms under four different conditions. (a) Energy window is between
[5,100], where the core states are included and the spin-polarization
rescaling is used. (b) Energy window is between [17,120] and does not include
the core states. The spin rescaling is used. (c) Energy window is between
[17,120], where the core states are not included and the spin-polarization
rescaling is not used. (d) Energy window is between [5,120], where the core
states are included and the spin-polarization rescaling is not used.  Figure
4: Forces in bulk fcc Ni calculated with a simple cubic cell and four atoms in
total. (a) Forces on Ni1. The force on Ni2 is similar and not shown. To
demonstrate the accuracy of our method, we remove all the symmetry operations,
except the identity matrix. The $k$ mesh is $14\times 14\times 14$. $R_{\rm
MT}K_{\rm max}$ is 7. $\tau=60$ fs. $A_{0}=0.015~{}\rm Vfs/\AA$.
$\hbar\omega=1.6$ eV. We use the circularly polarized light $\sigma_{xy}$. (b)
Forces on Ni3. The force on Ni4 is similar and not shown.


# Single-particle diffraction with a hydrodynamic pilot-wave model

Giuseppe Pucci1,2,3,∗, Antoine Bellaigue2, Alessia Cirimele4, Giuseppe Alì4,
Anand U. Oza5<EMAIL_ADDRESS><EMAIL_ADDRESS>1Consiglio Nazionale delle
Ricerche - Istituto di Nanotecnologia (CNR-Nanotec), Via P. Bucci 33C, 87036
Rende (CS), Italy,  2Univ Rennes, CNRS, IPR (Institut de Physique de
Rennes)—UMR 6251, F-35000 Rennes, France,  3INFN, Sezione di Lecce, Via per
Monteroni, Lecce 73100, Italy,  4Dipartimento di Fisica, Università della
Calabria, Via P. Bucci 31C, 87036 Rende (CS), Italy,  5Department of
Mathematical Sciences & Center for Applied Mathematics and Statistics, New
Jersey Institute of Technology, Newark, New Jersey 07102, USA

(August 28, 2024)

###### Abstract

A macroscopic hydrodynamic system that couples a particle and a wave has
recently renewed interest in the question as to what extent a classical system
may reproduce quantum phenomena. Here we investigate single-particle
diffraction with a pilot-wave model originally developed to describe the
hydrodynamic system. We study single-particle interactions with a barrier and
slits of increasing width by focusing on the near field. We find single-
particle diffraction arising as wavelike patterns in the particles’ position
statistics, which we compare to the predictions of quantum mechanics. We
provide a mechanism that rationalizes the diffractive behavior in our system.

Diffraction is a phenomenon that attracts great interest for its implications
in fundamental physics Feynman  _et al._ (1966) and wide range of applications
Bunaciu  _et al._ (2015); Filippetto  _et al._ (2022). In classical physics,
diffraction is fully captured as a wave phenomenon that occurs when a wave
meets an obstacle or an aperture in a screen Born and Wolf (2013). The
possibility of single-particle diffraction Taylor (1909) first arose with the
discovery of light quanta Planck (1901); Einstein (1905). The wave-like nature
of massive particles was postulated shortly thereafter De Broglie (1924), and
has been observed in landmark experiments on the diffraction of electrons
Merli  _et al._ (1976); Tonomura and Endo (1989); Bach  _et al._ (2013).

Over the last two decades, a classical macroscopic system that couples a
particle and a wave Couder  _et al._ (2005) has shown a number of analogies
with quantum particles Bush and Oza (2021), including quantized orbital radius
Fort  _et al._ (2010); Harris and Bush (2014); Blitstein  _et al._ (2024) and
angular momentum Perrard  _et al._ (2014); Durey and Milewski (2017), wavelike
statistics in cavities Harris  _et al._ (2013); Gilet (2016); Sáenz  _et al._
(2018), tunneling Eddi  _et al._ (2009); Hubert  _et al._ (2017); Nachbin  _et
al._ (2017); Tadrist  _et al._ (2020) and Friedel oscillations Sáenz  _et al._
(2020), among others Eddi  _et al._ (2012); Valani  _et al._ (2018); Frumkin
_et al._ (2022); Nachbin (2022); Papatryfonos  _et al._ (2022); Frumkin and
Bush (2023); Frumkin  _et al._ (2023). In this system, the particle is a
droplet coupled to the wave field that it generates by bouncing on the surface
of a vibrating liquid bath Couder  _et al._ (2005); Protière  _et al._ (2006);
Moláček and Bush (2013). The ensemble of drop and surface wave field has been
termed a walker. Seminal experiments and simulations with walkers suggested
that single-particle diffraction may be obtained when the walker crosses an
aperture between two submerged barriers Couder and Fort (2006, 2012). While
these studies prompted investigations of diffractive behavior in other
classical systems Schiebel  _et al._ (2019); Rieser  _et al._ (2019), they
were questioned on the basis of an insufficient number of data points Andersen
_et al._ (2015a); Bohr  _et al._ (2016), and later experiments with finer
control of experimental parameters could not reproduce the original results
Pucci  _et al._ (2018); Rode  _et al._ (2019); Ellegaard and Levinsen (2020,
2024). While these later experiments did not exclude the possibility of
obtaining statistical distributions similar to the first experiments and thus
incidentally to quantum particles in some corner of parameter space, the
similarities with quantum systems were limited. Indeed, the possibility of
observing quantum-like diffraction of walkers in experiments is likely limited
since the barriers are submerged under a relatively thin layer of liquid,
which causes specific particle-barrier interactions Faria (2017); Pucci  _et
al._ (2018); Ellegaard and Levinsen (2020, 2024). On the theoretical side, a
Greens-function model of the interaction of walkers with an aperture yielded
small scattering angles, so the associated statistical distributions were
different from both the walker system and quantum particles Dubertrand  _et
al._ (2016). Despite the large amount of work on this subject, it has remained
an open question as to what extent a hydrodynamic pilot-wave model inspired by
the walker system can exhibit quantum-like single-particle diffraction.

The walker system has motivated a number of hydrodynamic models Turton  _et
al._ (2018); Bush and Oza (2021) with different degrees of success in
describing quantum analogs. The so-called stroboscopic model Oza  _et al._
(2013) successfully reproduces hydrodynamic analogs of two canonical quantum
phenomena, namely, the quantized states exhibited by particles in a rotating
frame Oza  _et al._ (2014a, b) and in a simple harmonic potential Labousse
_et al._ (2016); Kurianski  _et al._ (2017), as well many other related
systems Valani (2022); Perrard and Labousse (2018); Budanur and Fleury (2019);
Montes  _et al._ (2021); Tambasco and Bush (2018); Valani and Slim (2018);
Couchman and Bush (2020); Thomson  _et al._ (2020a, b, 2021); Barnes  _et al._
(2020); Couchman  _et al._ (2022); Sáenz  _et al._ (2021). While the model
exhibits excellent agreement with experiments in unbounded geometries Oza  _et
al._ (2014a, b); Labousse  _et al._ (2016); Thomson  _et al._ (2020a);
Couchman  _et al._ (2019); Couchman and Bush (2020), it has not yet been used
to model the interactions of particles with barriers.

Figure 1: Simulation of walkers interacting with a linear barrier made of a
series of secondary sources. (a) Snapshot of a trajectory with the full wave
field $\mathcal{H}(\boldsymbol{x},t)$; see also Supplementary Movie 1. (b)
Selection of trajectories color-coded by their position $x_{s}$ on a screen
located at $y=\lambda_{F}$, which highlights the self-focusing mechanism.
(c–f) Wavefields $\mathcal{H}_{s}(\boldsymbol{x},t^{*})$ due to the secondary
sources alone for four trajectories, with insets showing the effective source
potential $\mathcal{V}(x)$ (not to scale); see also Supplementary Movie 2.
Colorbars are in arbitrary units.

We here model the interaction of single walkers with linear barriers described
by series of secondary sources, similarly to the pioneering simulations on
this subject Couder and Fort (2006) but using the stroboscopic model Oza  _et
al._ (2013). Previous studies have searched for diffraction-like behavior in
the far field, in the so-called Fraunhofer limit Couder and Fort (2006, 2012);
Andersen  _et al._ (2015b); Bohr  _et al._ (2016); Dubertrand  _et al._
(2016); Pucci  _et al._ (2018); Ellegaard and Levinsen (2020, 2024). We here
investigate the near field, that is, a few wavelengths away from the barriers,
and find that the distribution of particle positions is wavelike. In the
single-slit geometry, the number of peaks in the distribution increases with
the slit’s width. We rationalize the emergence of the distributions in terms
of the wavefield generated by the secondary sources, which behaves as an
effective transient potential for the particle. We provide a detailed
comparison with diffraction in quantum mechanics. The distributions are
qualitatively similar, with the periodicity of the walker’s distribution being
half that of quantum particles.

Figure 2: Comparison between the probability density functions (PDFs) of
walkers (a,b) and quantum particles (c,d) interacting with an edge. (a) PDF
$p(x,y)$ of walkers. (b) PDF of walker impacts on a screen placed at a
distance $\lambda_{F}$ from the edge (white dashed line in (a)). (c)
Unnormalized probability density $|\psi|^{2}$ of quantum particles. (d)
Variation in $|\psi|^{2}$ at a distance $\lambda$ from the edge, where
$\lambda$ is the de Broglie wavelength.

Model description.—We consider two geometries: (1) a semi-infinite linear
barrier, or an edge, as shown in Fig. 1, and (2) two semi-infinite linear
barriers separated by a distance $L$, or a single-slit, as shown in Fig. 3.
Both geometries are represented by linear arrays of pointlike sources
separated by a distance $\Delta x=\lambda_{F}/2$, specifically, at
$\boldsymbol{x}_{j}=(j\Delta x,0)$ for $j\in\mathbb{N}$ for the edge [Fig.
1(a)] and $\boldsymbol{x}_{j}=\pm(L/2+j\Delta x,0)$ for the single slit [Fig.
3], where $\lambda_{F}$ is the wavelength of the subthreshold Faraday waves
generated by the walker Faraday (1831). The statistical behavior we report
herein is qualitatively independent of $\Delta x$, the effect of which is
analyzed in (Sup, , §III). In a typical walker experiment, a drop of mass $m$
with horizontal position $\boldsymbol{X}_{p}(t)=(X_{p}(t),Y_{p}(t))$ bounces
periodically with period $T_{F}=2/f$ in the presence of a gravitational
acceleration $g$ on the surface of a fluid bath vibrating with frequency $f$
and peak acceleration $\gamma<\gamma_{F}$, $\gamma_{F}$ being the so-called
Faraday instability threshold Faraday (1831). The stroboscopic model reads Oza
_et al._ (2013)

$\displaystyle m\ddot{\boldsymbol{X}}_{p}+D\dot{\boldsymbol{X}}_{p}$
$\displaystyle=-mg\boldsymbol{\nabla}\mathcal{H}(\boldsymbol{X}_{p},t),$
$\displaystyle\text{where }\mathcal{H}(\boldsymbol{x},t)$
$\displaystyle=\frac{1}{T_{F}}\int_{-\infty}^{t}h(\boldsymbol{x},s)\mathrm{e}^{-(t-s)/T_{M}}\,{\rm
d}s$ (1)

and $T_{M}$ is the “memory" timescale over which the waves generated by the
walker decay Eddi  _et al._ (2011); Moláček and Bush (2013). That is, the drop
moves in response to two horizontal forces: a wave force
$-mg\boldsymbol{\nabla}\mathcal{H}(\boldsymbol{X}_{p},t)$ proportional to the
local slope of the wave height $\mathcal{H}$, and a drag
$-D\dot{\boldsymbol{X}}_{p}$ experienced during impact and flight Moláček and
Bush (2013). To model the wave field, we first assume that, in the absence of
barriers (free space), the impact of a walker at the origin generates a
standing wave $h_{0}(r)$ that is monochromatic of wavelength $\lambda_{F}$ in
the near field and decays exponentially over a lengthscale $d$ (Sup, , Eq.
(1)). The wave profile due to a droplet impact at $\boldsymbol{X}_{p}(t)$ at
time $t$ in the presence of barriers is thus

$\displaystyle h(\boldsymbol{x},t)$
$\displaystyle=h_{0}(|\boldsymbol{x}-\boldsymbol{X}_{p}(t)|)+h_{s}(\boldsymbol{x},t),$
(2)

where the wave field $h_{s}(\boldsymbol{x},t)$ due to the barriers is
determined by imposing the Dirichlet boundary condition

$\displaystyle h(\boldsymbol{x}_{j},t)=0$ (3)

on each barrier; that is, the wave field is zero for all time on the secondary
sources. The constants $D$, $\lambda_{F}$, $d$ and $T_{M}$ are known in terms
of fluid parameters (Sup, , §I.A). We note that, while a linear array of
secondary sources separated by $\lambda_{F}/2$ was also used in the original
simulations of Couder & Fort Couder and Fort (2006), they did not enforce the
boundary conditions (3) exactly. We simulate $10^{4}$ trajectories per
geometry, discarding those that cross the barrier (Sup, , Table I), and report
here the results obtained for $\gamma/\gamma_{F}=0.95$, for which $d\approx
3\lambda_{F}$. Results for lower and higher forcings are reported in the
Supplementary Material (Sup, , §V).

Wave-like diffraction from an edge.—In our first simulations, particles are
launched one-by-one with normal incidence at an edge, as shown in Fig. 1(a,b).
We observe that the trajectories focus towards certain regions of space, while
other regions of space are relatively devoid of trajectories. This self-
focusing mechanism is evident in Fig. 1(b). In the vicinity of the edge, some
of these trajectories are deflected to the left away from the edge, while
others are deflected behind the edge.

To obtain physical insight into the self-focusing mechanism, we consider the
wave field
$\mathcal{H}_{s}(\boldsymbol{x},t)\equiv\int_{-\infty}^{t}h_{s}(\boldsymbol{x},s)\mathrm{exp}[-(t-s)/T_{M}]\,{\rm
d}s$ due to the secondary sources for four different trajectories. We note
from Eq. (1) that $\mathcal{H}_{s}$ plays the role of a potential energy due
to the secondary sources. $\mathcal{H}_{s}(\boldsymbol{x},t)$ reaches the
highest amplitude and momentarily “freezes” when the drop crosses the plane of
the edge, $\boldsymbol{X}_{p}(t^{*})\equiv(x^{*},0)$ (Supplementary Movie 2).
This critical time $t=t^{*}$ is when the waves emitted by the sources most
constructively interfere and are thus responsible for most of the walker’s
deflection (Sup, , §II). From Figs. 1(c–f) we observe that the walkers are
deflected by the local gradient in $\mathcal{H}_{s}(\boldsymbol{x},t^{*})$,
with two trajectories being deflected leftward [Fig. 1(d,f)] and the other two
rightward [Fig. 1(c,e)]. The insets, which show the effective source potential
$\mathcal{V}(x)\equiv\mathcal{H}_{s}((x,0),t^{*})$, serve to further highlight
this mechanism, as each walker is pushed laterally toward the nearest trough
in $\mathcal{V}(x)$.

Figure 3: Diffraction of walkers past a single slit for three different slit
widths: $L=3\lambda_{F}$ (a,d,g); $L=5\lambda_{F}$ (b,e,h) and
$L=10\lambda_{F}$ (c,f,i); see also Supplementary Movie 3. (a–c) Trajectories
are color-coded by their impact position $x_{s}$ on a screen located at
$y=\lambda_{F}$ (dashed line). (d–f) PDFs of walkers. (g–i) PDFs of walker
impacts on the screen. (j–l) Variation in $|\psi|^{2}$ for quantum particles
at a distance $\lambda$ from the slit.

The trajectories can be used to construct a probability density function (PDF)
$p(x,y)$ of walker positions (Fig. 2(a,b)). As a comparison, we show in Fig.
2(c,d) the solution for the diffraction of a quantum mechanical plane wave
$\psi$ by an edge in 2D, as obtained by solving Schrödinger’s equation with
the appropriate boundary conditions (Sup, , §IV). There are qualitative
similarities between the two systems: both exhibit spatial oscillations in
amplitude, which in the walker system reflect the self-focusing of
trajectories observed in Fig. 1(b). There is a pronounced bright beam in both
systems that emanates from the edge and curves leftward [Fig. 2(a,c)], which
in the walker system indicates a relatively large density of walkers which
cross that particular path. Both systems also have a small (but nonzero)
amplitude shadow region behind the edge. The 1D position PDFs [Fig. 2(b,d)] of
both the walker and quantum systems exhibit relatively large amplitudes near
the edge, with oscillations that decay farther away from the edge. The
oscillations decay algebraically in the quantum system but exponentially in
the walker system because the latter is dissipative and thus has spatially
damped waves. We observe that the quantum probability density oscillates on
the de Broglie wavelength $\lambda$ [Fig. 2(d)], while the walker screen
position PDF oscillates on half the Faraday wavelength, $\lambda_{F}/2$ [Fig.
2(b)]. The half-wavelength periodicity of the walker statistics may be
intuited by observing that the secondary source wavefield
$\mathcal{H}_{s}(\boldsymbol{x},t^{*})$ flips sign as $x^{*}$ is increased by
$\lambda_{F}/2$ [Fig. 1(c,e)], suggesting that the source potential
$\mathcal{V}(x^{*})$ is $\lambda_{F}/2$-periodic in $x^{*}$ (see (Sup, , §III)
for a quantitative explanation). We note that in the quantum system the
regions of relatively large intensity [yellow regions in Fig. 2(c)] curve to
the left, while they are mostly straight upward in the walker system.

Wave-like diffraction by a single slit.—In the next set of simulations,
walkers were launched one-by-one with normal incidence towards two linear
barriers separated by a distance $L$, as depicted in Fig. 3(a–c). The self-
focusing mechanism present for the edge [Fig. 1] is also evident for the
single-slit [Fig. 3]. Trajectories near the center are funneled into preferred
regions of space, as evidenced by the yellow streaks in the 2D PDFs [Fig.
3(d–f)], while trajectories with impact parameters near the barriers are
strongly deflected. The PDFs of screen impact positions $x_{s}$ [Fig. 3(g–i)]
may be compared with their counterparts from quantum mechanics [Fig. 3(j–l)],
the most salient feature being that the number of oscillations in the PDF
increases with the slit width $L$. A mechanism from this phenomenon may be
inferred from the trajectories in Fig. 3(a–c), as their $x_{s}$–values
oscillate within a larger range as $L$ is increased progressively. Moreover,
as the screen is placed farther from the barriers, the oscillations in the
central region of the PDF are suppressed in both amplitude and lateral extent,
in agreement with the predictions of quantum mechanics (Sup, , Fig. 6). There
are again some differences between the walker and quantum systems; most
notably, the walker PDFs exhibit oscillations on $\lambda_{F}/2$ while the
quantum intensity oscillates on $\lambda$, a behavior that may be rationalized
using the argument presented for the edge geometry in the preceding section.

Discussion.—We have used a hydrodynamic pilot-wave model to investigate
single-particle diffraction by linear barriers. Because the system is
dissipative and thus the walker wavefield decays exponentially in space, we
focused on the near field, that is, a few wavelengths away from the barriers,
a regime that was not explored in prior studies Couder and Fort (2006, 2012);
Andersen  _et al._ (2015b); Bohr  _et al._ (2016); Dubertrand  _et al._
(2016); Pucci  _et al._ (2018); Ellegaard and Levinsen (2020, 2024). We found
wavelike particle statistics that exhibit oscillations on half the wavelength
of the walker wave field, that is, $\lambda_{F}/2$. We have rationalized this
behavior in terms of the wave field generated by the barriers, which creates
an effective transient potential responsible for walker deflection and self-
focusing. We have compared our results to the diffraction of a plane wave in
the same geometries as described by quantum theory. The distributions are
qualitatively similar but differ in their periodicity, which for quantum
particles is generally on the order of one de Broglie wavelength. We note that
the diffractive behavior in our system arises from a non-chaotic dynamics, in
which the position at which the walker hits a screen generally has a
predictable dependence on the initial conditions (Sup, , Figs. 7 and 8). Our
results establish the possibility of obtaining quantum-like diffraction with a
classical wave-driven system. The results presented here can inspire future
works on single-particle diffraction with generalized Bush and Oza (2021);
Durey  _et al._ (2020) or walker-inspired pilot-wave theories Durey and Bush
(2020); Durey (2020); Dagan and Bush (2020); Darrow and Bush (2024), which
could yield results even closer to quantum mechanics.

###### Acknowledgements.

A. O. acknowledges support from NSF DMS-2108839. G. P. acknowledges the CNRS
Momentum program, and thanks Pedro Sáenz for fruitful discussions.

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