# Generation of vortex electrons by atomic photoionization

I. I. Pavlov<EMAIL_ADDRESS>School of Physics and Engineering,
ITMO University, 197101 St. Petersburg, Russia  A. D. Chaikovskaia
<EMAIL_ADDRESS>School of Physics and Engineering, ITMO
University, 197101 St. Petersburg, Russia  D. V. Karlovets
<EMAIL_ADDRESS>School of Physics and Engineering, ITMO
University, 197101 St. Petersburg, Russia

###### Abstract

We explore the process of orbital angular momentum (OAM) transfer from a
twisted light beam to an electron in atomic ionization within the first Born
approximation. The characteristics of the ejected electron are studied
regardless of the detection scheme. We find that the outgoing electron
possesses a definite projection of OAM when a single atom is located on the
propagation axis of the photon, whereas the size of the electron wave packet
is solely determined by the energy of the photon rather than by its transverse
coherence length. Shifting the position of the atom yields a finite dispersion
of the electron OAM. We also study a more experimentally feasible scenario — a
localized finite-sized atomic target — and develop representative approaches
to describing coherent and incoherent regimes of photoionization.

Introduction. Atomic photoionization by twisted light was under investigation
in several recent papers [1, 2, 3, 4, 5, 6], with the focus staying on the
process cross-section and the angular distribution of photoelectrons. However,
there is a fundamental issue that was not directly addressed. In this Letter,
we theoretically demonstrate the principal possibility to generate twisted
photoelectrons by using the twisted laser beam, i.e. photons carrying quanta
of orbital angular momentum (OAM). We explore the resulting quantum state of
the electron in the photoionization process, the stages of which are depicted
in Fig. 1, and regardless of the detection protocol. Particular attention is
given to the OAM transfer from the incident photon to the photoelectron. The
incident vortex photon is taken in the form of a Bessel beam [7] and a
Laguerre-Gaussian wave packet [8], while for the target ”cathode” there are
also two toy-model options: a single (hydrogen) atom in $1s$ state and a
finite target consisting of symmetrically spread single atoms.

Photoemission of vortex electrons is a first step in one of the approaches
towards generating relativistic electron beams with OAM at particle
accelerators. Specifically, it is an ongoing experimental joint ITMO-JINR
project on relativistic vortex electron generation at Dzhelepov Laboratory of
Nuclear Problems of JINR [9] that can benefit from this study. Relativistic
beams of electrons, protons and other charged particles with OAM can become a
unique research tool not only in atomic and molecular physics, in the
diagnostics of nanomaterials and surfaces, where the orbital momentum allows
one to obtain new information about a sample, but also in nuclear physics,
spin and hadron physics, where such particles can be used to analyze the
proton spin, study nuclear forces at low energies, etc [10, 11, 12, 13, 14].

In the straightforward theoretical approach111Natural system of units is used
throughout the paper with $\hbar=c=1$ whereas the electron mass and charge are
denoted as $m$ and $e<0$, respectively. to photoionization one takes the
$S$-matrix element in the first order of perturbation theory [15], which
describes the transition between initial (bound) and final (continuum)
electron states and contains the energy conservation law:

$S_{fi}=2\pi i\delta(\varepsilon_{i}+\omega-\varepsilon_{f})M_{fi},$ (1)

where $\varepsilon_{i}$, $\varepsilon_{f}$ and $\omega$ are the energies of
the bound state, continuum state and the photon, respectively. The transition
amplitude $M_{fi}$ can be taken in the non-relativistic form as the energy of
the final electron cannot exceed that of the photon, which typically varies
from $10$ eV to $100$ eV (optical and ultraviolet range):

$M_{fi}=-i\frac{e}{m}\int\psi_{f}^{*}(\bm{r})\bm{A}(\bm{r})\cdot\bm{\nabla}\psi_{i}(\bm{r})\
\differential^{3}\bm{r}$ (2)

with $\psi_{i}(\bm{r})$ and $\psi_{f}(\bm{r})$ being the scalar wave
functions, and $\bm{A}$ – the vector potential of a photon. Employing the
first Born approximation is the next obvious step when energies $\omega$ are
well above the $1s$ ionization threshold (as is the case with
$\varepsilon_{i}=-1\operatorname{Ry}=-me^{4}/2\approx-13.6$ eV for the
hydrogen atom). Then, the outgoing electron is described with an asymptotic
state with a well-defined momentum, i.e. a plane wave, and such approach has
been successfully applied to describe the ionization by vortex light beams in
Refs. [1, 2, 3, 4]. A more complex method accounts for the modulation of a
free-electron state with the incident laser field by taking the emerging
electron wave function in the form of the Volkov-type solution (the so-called
Keldysh photoionization theory) [16, 17, 18, 19].

Figure 1: Photoionization of atomic target in three stages: left, twisted
light beam descends onto the cathode; central, the atom(s) in the cathode
become excited; right, a twisted photoelectron is emitted. A cathode, treated
as a target in the scattering process, can be a single atom or an atomic
ensemble.

A different approach to this problem is to consider the so-called evolved
state amplitude, without making projections onto the specific basis [20, 21].
One can also mention a nonperturbative approach based on numerical solution of
the exact time-dependent Schrödinger equation, which enables to track the
temporal dynamics of the electron without specifying the form of its final
state [22, 23, 24].

We aim to develop a measurement-strategy independent approach, which – on the
one hand – allows one to deduce the quantum state of the outgoing
photoelectron as it evolves from the process on its own, and – on the other
hand – involves well-known perturbative calculations rather than complex
numerical simulations.

We use the aforementioned evolved state formalism [20, 21] to derive the wave
function of a final photoelectron independently of the post-selection protocol
chosen for it, but taking into account full information on the interaction
itself and the initial state of the system. Let us call the bound state of the
electron $\ket{i}$, then, the evolved (pre-selected) state $\ket{\text{ev}}$
is connected to it via the $S$-matrix $\hat{S}$. Inserting a complete set of
the free electron states in the form of plane waves we obtain the expression

$\ket{\text{ev}}=\hat{S}\ket{i}=\int\frac{\differential^{3}p_{f}}{(2\pi)^{3}}\ket{\bm{p}_{f}}S_{fi}(\bm{p}_{f}).$
(3)

Projection onto a state with a definite momentum $\ket{\bm{p}}$ yields the
final state wave function in the momentum representation:

$\displaystyle\psi_{\text{ev}}(\bm{p})\equiv\innerproduct{\bm{p}}{\text{ev}}=\int\frac{\differential^{3}p_{f}}{(2\pi)^{3}}\innerproduct{\bm{p}}{\bm{p}_{f}}S_{fi}(\bm{p}_{f})$
$\displaystyle=\int\differential^{3}p_{f}\delta(\bm{p}-\bm{p}_{f})S_{fi}(\bm{p}_{f})=S_{fi}(\bm{p}).$
(4)

Thus, the wave function of the evolved state in momentum representation simply
coincides with the S-matrix element. The importance of the S-matrix element
phase, $\text{arg}\,S_{fi}$, is showcased in this approach, as the processes
in particle physics are commonly described with only the absolute value,
$|S_{fi}|$.

Since we describe the electron with the Schrödinger equation and do not take
its spin into account, the OAM of the electron coincides with its total
angular momentum (TAM). The OAM projection operator in the momentum
representation acts as follows

$\bra{\bm{p}}\hat{L}_{z}\ket{\text{ev}}=-i\partialderivative{\varphi_{p}}\innerproduct{\bm{p}}{\text{ev}}.$
(5)

First, we reproduce the result for the amplitude (2) in the scenario with the
incident photon being a plane wave with momentum ${\bf
k}=(\bm{k}_{\perp},k_{z})=\omega(\sin{\theta_{k}}\cos{\varphi_{k}},\sin{\theta_{k}}\sin{\varphi_{k}},\cos{\theta_{k}})$
and helicity $\Lambda=\pm 1$, it is also convenient to take the Coulomb gauge

$A=(0,\bm{A}_{\bm{k}\Lambda}(\bm{r})),\
\bm{A}_{\bm{k}\Lambda}(\bm{r})=\bm{e}_{\bm{k}\Lambda}e^{i\bm{kr}}.$ (6)

From hereon we assume the initial electron to be described by the ground state
wave function of the “hydrogen” atom with charge $Ze$ [15]:
$\psi_{i}(\bm{r})=Z^{3/2}e^{-Zr/a}/\sqrt{\pi a^{3}},$ where $a=1/(me^{2})$ is
the Bohr radius. The asymptotics of the complete basis of the final electron
states can be chosen as plane waves:
$\psi_{f}(\bm{r})=e^{i\bm{p}\bm{r}},\;\bm{p}=(\bm{p}_{\perp},p_{z})=p(\sin{\theta_{p}}\cos{\varphi_{p}},\sin{\theta_{p}}\sin{\varphi_{p}},\cos{\theta_{p}}).$
This constitutes the first Born approximation, in which one neglects the
interaction of the final electron with the laser field and the electron–ion
attraction after the ionization process. In other words, the electron is
assumed to be emitted into the free space. The plane wave amplitude then
becomes [1]

$M_{fi}=\mathcal{N}p\frac{\bm{n}\bm{e}_{\bm{k}\Lambda}}{\left(\frac{Z^{2}}{a^{2}}+q^{2}\right)^{2}},$
(7)

where $\mathcal{N}=-\frac{8e}{m}\sqrt{\frac{Z^{5}\pi}{a^{5}}}$,
$\bm{n}=\bm{p}/p$ and $\bm{q}=\bm{p}-\bm{k}$ is the transferred momentum.

A Bessel beam. When the incoming photon is in a twisted state, a single plane
wave is replaced by a superposition of plane waves. Let us consider a Bessel
beam and also introduce the possible impact parameter $\bm{b}$:

$\displaystyle\bm{A}_{\kappa\ell k_{z}\Lambda}(\bm{r})$
$\displaystyle=\int\frac{\differential^{2}k_{\perp}}{(2\pi)^{2}}a_{\kappa\ell}(\bm{k}_{\perp})\bm{A}_{\bm{k}\Lambda}(\bm{r})e^{-i\bm{k}_{\perp}\bm{b}},$
(8) $\displaystyle a_{\kappa\ell}(\bm{k}_{\perp})$
$\displaystyle=\sqrt{\frac{2\pi}{\kappa}}(-i)^{\ell}e^{i\ell\varphi_{k}}\delta(\absolutevalue{\bm{k}_{\perp}}-\kappa)$
(9)

with $\kappa$ being the absolute value of the transverse momentum and
$\ell=0,\pm 1,\pm 2,\dots$ being the $z$-projection of the TAM. The amplitude,
and correspondingly the evolved wave function is obtained through the
integration of the plane wave amplitude (7) with the same weights as in Eq.
(8). Then, the amplitude for ionization by the Bessel beam representing the
process from Fig. 1 is

$\displaystyle
M_{fi}^{\text{\text{TW}}}=\int\frac{\differential^{2}k_{\perp}}{(2\pi)^{2}}a_{\kappa\ell}(\bm{k}_{\perp})e^{-i\bm{k}_{\perp}\bm{b}}(\bm{k}_{\perp})M_{fi}$
(10)
$\displaystyle=\mathcal{N}p\sqrt{\frac{\kappa}{2\pi}}(-i)^{\ell}\int\frac{\differential\varphi_{k}}{2\pi}e^{i\ell\varphi_{k}-i\kappa
b\cos(\varphi_{k}-\varphi_{b})}\frac{\bm{n}\bm{e}_{\bm{k}\Lambda}}{\left(\frac{Z^{2}}{a^{2}}+q^{2}\right)^{2}}.$

Introducing $\tilde{\varphi}\equiv\varphi_{k}-\varphi_{p}$ and rewriting
$\frac{Z^{2}}{a^{2}}+q^{2}=\alpha-\beta\cos{\tilde{\varphi}}$ in terms of the
variables $\alpha\equiv\frac{Z^{2}}{a^{2}}+p^{2}+k^{2}-2p_{z}k_{z}$ and
$\beta\equiv 2p_{\perp}k_{\perp}$ allows us to put the amplitude as follows:

$\displaystyle
M_{fi}^{\text{TW}}=\mathcal{N}\sqrt{\frac{\kappa}{2\pi}}(-i)^{\ell}e^{i\ell\varphi_{p}}\Bigg{\\{}\frac{p_{\perp}}{\sqrt{2}}\Big{[}d_{-1\Lambda}^{1}I_{\ell+1}(\alpha,\beta,\bm{b},\varphi_{p})$
$\displaystyle-
d_{1\Lambda}^{1}I_{\ell-1}(\alpha,\beta,\bm{b},\varphi_{p})\Big{]}+p_{z}d_{0\Lambda}^{1}I_{\ell}(\alpha,\beta,\bm{b},\varphi_{p})\Bigg{\\}},$
(11)

where $d^{J}_{MM^{\prime}}$ are the small Wigner matrices (see details in
Appendix A), and we define the function

$I_{\ell}(\alpha,\beta,\bm{b},\varphi_{p})\equiv\int\frac{\differential\tilde{\varphi}}{2\pi}e^{i\ell\tilde{\varphi}-i\kappa
b\cos(\tilde{\varphi}+\varphi_{p}-\varphi_{b})}\frac{1}{(\alpha-\beta\cos{\tilde{\varphi}})^{2}}.$
(12)

In fact, we can make an immediate conclusion about the evolved state of the
photoelectron when the ionizing beam falls onto the target atom with a
vanishing impact parameter. If $\bm{b}=0$, $I_{\ell}(\alpha,\beta,0)$ does not
depend on $\varphi_{p}$. Thus,

$-i\partialderivative{\varphi_{p}}M_{fi}^{\text{TW}}=\ell M_{fi}^{\text{TW}},$
(13)

leading to $-i\partialderivative{\varphi_{p}}S_{fi}^{\text{TW}}=\ell
S_{fi}^{\text{TW}}$ and, consequently,

$\hat{L}_{z}\ket{\text{ev}}=\ell\ket{\text{ev}}.$ (14)

This equation illustrates that the evolved state represents a vortex electron
with the OAM $z$-projection equal to $\ell$. A similar result can be expected
for the ionization of some excited state of the hydrogen-like atom with a
well-defined OAM projection $\ell_{e}$ onto the $z$ axis, as the conservation
of the OAM projection is a consequence of the system being axially symmetric.
In this case the electron OAM would clearly become $\ell+\ell_{e}$.

Moreover, in this case the amplitude can be evaluated analytically as [25, 26]

$I_{\ell}(\alpha,\beta,0)=-\partialderivative{\alpha}\left\\{\left(\frac{\beta}{\alpha+\sqrt{\alpha^{2}-\beta^{2}}}\right)^{|\ell|}\frac{1}{\sqrt{\alpha^{2}-\beta^{2}}}\right\\}.$
(15)

For a non-vanishing impact parameter, we make an estimation supposing that
$\kappa b$ is a reasonable small parameter in our problem. For realistic
wavelengths of the laser field $\lambda\sim 10^{2}-10^{3}$ nm and the Bessel
beam opening angle $\theta_{k}\sim 1^{\circ}$ it means that we consider values
$b\ll 1-10$ $\mu$m, which are relevant to the beam axis positioning accuracy
in experiments with trapped atoms [27, 28, 29]. Expansion of the impact
parameter - dependent exponent in (12) up to the linear term:

$\displaystyle e^{-i\kappa
b\cos(\tilde{\varphi}+\varphi_{p}-\varphi_{b})}\approx$ $\displaystyle
1-\frac{i\kappa b}{2}e^{-i\varphi_{b}}e^{i(\tilde{\varphi}+\varphi_{p})}$ (16)
$\displaystyle-\frac{i\kappa
b}{2}e^{i\varphi_{b}}e^{-i(\tilde{\varphi}+\varphi_{p})}$

is equivalent to making the following replacement in Eq. (11):

$\displaystyle I_{\ell}(\alpha,\beta,\bm{b})\rightarrow$ $\displaystyle
I_{n}(\alpha,\beta,0)-\frac{i\kappa
b}{2}e^{-i\varphi_{b}}e^{i\varphi_{p}}I_{\ell+1}(\alpha,\beta,0)$ (17)
$\displaystyle-\frac{i\kappa
b}{2}e^{i\varphi_{b}}e^{-i\varphi_{p}}I_{\ell-1}(\alpha,\beta,0).$

We see that in this case the evolved state is a superposition of the twisted
states with OAM values $l$ and $l\pm 1$ rather then an eigenstate of
$\hat{L}_{z}$. Analogously, the expansion of exponent up to the arbitrary
$n$-th term gives rise to the new terms in the superposition with OAM $l\pm
n$. The coefficients at these terms are suppressed by the factor of $(\kappa
b)^{n}$. The fact that in this case the photoelectron no longer represents a
single twisted state is of no surprise since displacement of the target atom
with respect to center of the wavefront breaks the cylindrical symmetry of the
overall system.

A Laguerre-Gaussian beam. Let us now consider a more realistic model of a
twisted photon beam, represented by a Laguerre-Gaussian (LG) wave packet. The
vector potential is taken in the form

$\bm{A}(\bm{r})=\bm{e}_{\Lambda}u(\bm{r})e^{ikz},$ (18)

where the polarization vector $\bm{e}_{\Lambda}$ is supposed to be orthogonal
to the $z$ axis and the scalar amplitude $u(\bm{r})$ satisfies the paraxial
Helmholtz equation [30]. For the LG beam it is chosen as [8]

$\displaystyle
u\left(r_{\perp},\phi,z\right)=\frac{1}{w(z)}\left(\frac{\sqrt{2}r_{\perp}}{w(z)}\right)^{\ell}$
(19)
$\displaystyle\times\exp\left[-\frac{r_{\perp}^{2}}{w^{2}(z)}\right]L_{p}^{\ell}\left(\frac{2r_{\perp}^{2}}{w^{2}(z)}\right)$
$\displaystyle\times\exp\left[i\ell\varphi+\frac{ikr_{\perp}^{2}z}{2\left(z^{2}+z_{R}^{2}\right)}-i(2p+\ell+1)\arctan\left(\frac{z}{z_{R}}\right)\right].$

The beam is monochromatic and its TAM projection is $l+\Lambda$. The atom is
supposed to lie at the beam focus ($z=0$), so we can focus on the
modifications in the transverse plane. There, the LG profile has a
characteristic ringlike pattern with a finite number of rings determined by
the radial index $p$ in contrast to Bessel beams with an infinite number of
rings.

(a)

(b)

(c)

Figure 2: Normalized probability density
$|\psi_{\text{ev}}(r_{\perp},z,\varphi=0)|^{2}$ of the evolved state. Due to
the azimuthal symmetry of the problem there is no dependence on $\varphi$. The
incident photon represents Bessel beam with $\omega=1.2|\varepsilon_{i}|$,
opening angle $\theta_{k}=\arctan(\kappa/k_{z})=1^{\circ}$. Length is measured
in Bohr radii $a$. (a) $\ell=0$; (b) $\ell=1$; (c) $\ell=5$.

(a)

(b)

(c)

(c)

Figure 3: The same as in Fig. 2, but for the incident Laguerre-Gaussian beam
with $\omega=1.2|\varepsilon_{i}|$, $w_{0}=0.5$ mm. (a) $l=p=0$ (Gaussian
beam); (b) $\ell=3,p=1$; c) $\ell=5,p=3$; (d) $\ell=10,p=3$.

It is convenient to also represent the LG beam as a superposition of plane
waves:

$\bm{A}(\bm{r})=\int\frac{\differential^{2}k_{\perp}}{(2\pi)^{2}}U(\bm{k}_{\perp})\bm{e}_{\Lambda}e^{i\bm{k}\bm{r}}$
(20)

The modulus of the wave vector is kept fixed:
$k=\sqrt{k_{z}^{2}+k_{\perp}^{2}}=\operatorname{const}$. The function
$U(\bm{k}_{\perp})$ is connected to the amplitude $u(\bm{r})$ in the beam
focus by the two-dimensional Fourier transform:

$\displaystyle U(\bm{k}_{\perp})=$ $\displaystyle\displaystyle\int
u(r_{\perp},\varphi,z=0)e^{-i\bm{k}_{\perp}\bm{r}_{\perp}}\differential^{2}r_{\perp},$
(21) $\displaystyle U(\bm{k}_{\perp})=$
$\displaystyle\pi(-1)^{p}2^{-l/2}i^{-\ell}w_{0}^{\ell+1}k_{\perp}^{\ell}\exp\left[-\frac{k_{\perp}^{2}w_{0}^{2}}{4}\right]$
(22) $\displaystyle\times
L_{p}^{\ell}\left(\frac{k_{\perp}^{2}w_{0}^{2}}{2}\right)\exp\left[i\ell\varphi_{k}\right],$

where the beam waist $w_{0}\equiv w(z=0)$. However, such a vector potential
does not satisfy Coulomb gauge, because the superposition contains
nontransverse plane waves. To fix this, we make the following modification of
the vector potential (see details in [31]):

$\bm{A}_{\text{trans}}(\bm{r})=\int\frac{\differential^{2}k_{\perp}}{(2\pi)^{2}}U(\bm{k}_{\perp})e^{i\Lambda\varphi_{k}}\bm{e}_{\bm{k}\Lambda}e^{i\bm{k}\bm{r}}$
(23)

with the polarization vector defined in Eq. (37). Interestingly, a similar
issue on the ambiguity of the phase factor in the twisted wave function was
discussed in [21]).

Analogously to Eq. (10), the amplitude of the photoionization process with an
incident LG beam is given by

$\displaystyle M_{fi}^{LG}=$
$\displaystyle\displaystyle\int\frac{\differential^{2}k_{\perp}}{(2\pi)^{2}}U(\bm{k}_{\perp})M_{fi}$
(24) $\displaystyle=$ $\displaystyle
p\mathcal{N}\int\frac{\differential^{2}k_{\perp}}{(2\pi)^{2}}U(\bm{k}_{\perp})e^{-i\bm{k}_{\perp}\bm{b}}\frac{\bm{n}\bm{e}_{\bm{k}\Lambda}}{\left(\frac{Z^{2}}{a^{2}}+q^{2}\right)^{2}}.$
(25)

Evaluation of the amplitude (24) is a bit more cumbersome than of that for the
Bessel beam as the integral over $k_{\perp}$ appears in the expression and
$k_{z}$ becomes a function of $k_{\perp}$. However, it is clear that the
result for the vanishing impact parameter holds and the evolved state of the
electron is twisted:

$-i\partialderivative{\varphi_{p}}M_{fi}^{LG}=(\ell+\Lambda)M_{fi}^{LG}$ (26)

Previous results on the OAM superposition in the photoelectron state for a
non-vanishing impact parameters also remain valid for the LG beam.

If the atom is placed not in the focal plane ($z\neq 0$) of the LG beam, every
plane wave in the superposition (23) should be multiplied by the corresponding
phase factor:

$\displaystyle
M_{fi}^{LG}=p\mathcal{N}\int\frac{\differential^{2}k_{\perp}}{(2\pi)^{2}}U(\bm{k}_{\perp})e^{-i\bm{k}_{\perp}\bm{b}-ib_{z}\sqrt{k^{2}-k_{\perp}^{2}}}\frac{\bm{n}\bm{e}_{\bm{k}\Lambda}}{\left(\frac{Z^{2}}{a^{2}}+q^{2}\right)^{2}}$
$\displaystyle\approx
p\mathcal{N}\int\frac{\differential^{2}k_{\perp}}{(2\pi)^{2}}U(\bm{k}_{\perp})e^{-i\bm{k}_{\perp}\bm{b}-ib_{z}k\left(1-k_{\perp}^{2}/(2k^{2})\right)}\frac{\bm{n}\bm{e}_{\bm{k}\Lambda}}{\left(\frac{Z^{2}}{a^{2}}+q^{2}\right)^{2}}.$

Such longitudinal shift does not alter the cylindrical symmetry and thus does
not affect the OAM of the evolved state.

(a)

(b)

Figure 4: Comparison of transverse coherence length of the photoelectron (blue
line) and characteristic size of the Bessel beam $\ell/\kappa$ for $\ell=3$
(orange line) depending on the photon energy $\omega$ for two different values
of the Bessel beam opening angle: a) $\theta_{k}=1^{\circ}$; b)
$\theta_{k}=30^{\circ}$. For larger energies of the photon the plots are
qualitatively the same with the exception that the probability density is more
localized in all directions.

Transverse coherence length of the photoelectron. Having derived the wave
function of the outgoing electron in momentum representation, we now want to
evaluate its Fourier transform and investigate the probability density in the
coordinate space. First, we rewrite the delta-function in (1) as

$\delta\left(\varepsilon_{i}+\omega-\frac{p^{2}}{2m}\right)=\frac{m}{p_{\perp}}\delta\left(p_{\perp}-\sqrt{2m(\varepsilon_{i}+\omega)-p_{z}^{2}}\right)$
(27)

Then for the Bessel photon and zero impact parameter, the Fourier integral
becomes

$\displaystyle\psi_{\text{ev}}(\bm{r})=e^{i\ell\varphi}\mathcal{N}m\sqrt{\frac{\kappa}{(2\pi)^{5}}}\int_{-\sqrt{2m(\varepsilon_{i}+\omega)}}^{\sqrt{2m(\varepsilon_{i}+\omega)}}J_{\ell}(p_{\perp}r_{\perp})$
$\displaystyle\times\Bigg{[}\frac{p_{\perp}}{\sqrt{2}}\big{[}d_{-1\Lambda}^{1}I_{\ell+1}(\alpha,\beta,0)-d_{1\Lambda}^{1}I_{\ell-1}(\alpha,\beta,0)\big{]}$
$\displaystyle+p_{z}d_{0\Lambda}^{1}I_{\ell}(\alpha,\beta,0)\Bigg{]}e^{ip_{z}z}dp_{z},$
(28)

where $p_{\perp}$, $\alpha$ and $\beta$ are the functions of the integration
variable $p_{z}$. The remaining integral over $p_{z}$ in Eq. (28) can be
computed numerically. For the incident LG photon the Fourier transform is done
analogously with an additional numerical integration over $k_{\perp}$.

Fig. 2 demonstrates the probability density $|\psi_{\text{ev}}(\bm{r})|^{2}$
of the photoelectron if the incident light is Bessel beam with different
values of OAM projection. Since we are not interested in the probability of
the process, the absolute value of the probability density is unimportant. For
this reason we normalize each distribution in a way that its maximal value
equals $1$. For zero impact parameter the probability density is azimuthally
symmetric, therefore we present only its dependence on radial coordinate
$r_{\perp}$ and $z$. As seen from the figure, the photoelectron represents a
wave packet localized around the position of the atom both in transverse and
longitudinal directions. In case of Bessel beam with zero OAM the maximum of
the probability density lies on the $z$ axis while for non-zero values of
$\ell$ the wave packet has a well-known “doughnut shape” inherent to vortex
states. It can also be seen that the radius of the first ring grows with the
increase of $\ell$.

In Fig. 3 we display the probability density of the photoelectron for the
incident LG beam with different values of the radial index $p$ and OAM $\ell$.
Its waist $w_{0}$ is chosen as $0.5$ $\mu$m, which approximately equals the
characteristic radius of the Bessel beam $1/\kappa$ for parameters listed
under the Fig. 2. The first graph, Fig. 3 (a), corresponds to a Gaussian mode
($l=p=0$) and does not qualitatively differ from the case of Bessel beam with
$\ell=0$. The Figs. 2 (b) and 3 (c) are also almost identical despite the key
difference between the Bessel beam and the LG one: the latter is localized in
the transverse plane whereas the former is not. It illustrates the fact that
the width of the photoelectron wave packet does not depend on the transverse
coherence of the incident light. This feature can be easily explained by the
Eq. (28) as the transverse coordinate $r_{\perp}$ is contained only in the
argument of the Bessel function $J_{\ell}(p_{\perp}r_{\perp})$ along with the
transverse momentum $p_{\perp}=\sqrt{2m(\varepsilon_{i}+\omega)-p_{z}^{2}}$,
which does not explicitly depend on $k_{\perp}$. Thus, we come to an important
conclusion that it is the photon energy $\omega$ and not its transverse
momentum $\kappa$ that determines the width of the photoelectron wave packet.

Let us now explicitly define the width (the transverse coherence length) of
the electron packet as the distance from the $z$ axis to the point where the
probability density is maximal. Further, we demonstrate the dependence of the
electron transverse coherence length on the energy of the incident photon and
compare it with the characteristic radius of the Bessel beam $\ell/\kappa$. As
seen from the Fig. 4, the width of the electron packet, which is many orders
of magnitude smaller than the Bessel beam radius, decreases with the increase
of the photon energy. For $\omega\sim 20|\varepsilon_{i}|$ the localization
becomes comparable to the Bohr radius $a$, which is the initial localization
distance in the atom. The unbounded growth of the width as
$\omega\to|\varepsilon_{i}|$ has little physical sense since the Born
approximation that we use becomes invalid in case of extremely small energies
of the outgoing electron.

Mesoscopic target. Up to this point, we have discussed the photoionization of
a single atom positioned at a specific impact parameter relative to the
propagation axis of the vortex beam. While recent advancements have made it
possible to localize the target in this way [28, 29], the majority of
photoionization experiments involve extended (mesoscopic) targets.

It is generally accepted in the related works that there is no coherence
between photoelectrons emitted from different atoms [32]. Thus, photoemission
from the localized target is described by averaging the cross section over a
target distribution function [3, 4, 2]. However, the coherent effects in
photoionization are also well-known: there are many theoretical and
experimental works studying photoemission from diatomic molecule, which is
regarded as analogue of Young’s double-slit experiment where the coherent
emission of the electrons originating from two spatially separated sources
gives rise to the interference patterns in the differential cross-section [33,
34, 35, 36, 2].

A peculiar feature of the evolved state arises in the hypothetical scenario
when the photoionization occurs coherently from all atoms in the target which
have the same phase of the wave function. It means that it is the amplitude of
the process that should be averaged over the target rather than its squared
absolute value (cross-section):

$\widetilde{M}_{fi}^{\text{TW}}=\int\differential^{2}b\>n(\bm{b})M_{fi}^{\text{TW}}=\int
b\>\differential b\>\differential\varphi_{b}n(\bm{b})M_{fi}^{\text{TW}}.$ (29)

Let us, in addition, assume the mesoscopic target to be axially symmetric. One
can notice that in this case the integration over $\varphi_{b}$ eliminates any
term containing the factor $e^{in\varphi_{b}}$ in the amplitude (11). Thus,
after the averaging over such a target the evolved state becomes a single-mode
twisted state, just as in the scenario with $\bm{b}=0$. From an experimental
viewpoint such a scenario could corresponds, for example, to the target
represented by a cold atomic gas, where the coherence effects can play an
important role [37, 38].

Incoherent photoionization: density matrix approach. Let us consider the
density matrix of the (pure) evolved state

$\hat{\rho}_{\text{ev}}=\ket{\text{ev}}\bra{\text{ev}}.$ (30)

Unlike the regular density matrices, $\Tr(\hat{\rho})\neq 1,$ as
$\Tr(\hat{\rho}_{\text{ev}})=\bra{\text{ev}}\ket{\text{ev}}=\int\frac{\differential^{3}p}{(2\pi)^{3}}|\psi_{\text{ev}}(\bm{p})|^{2}$,
which diverges. Thus, if we would like to evaluate the mean values of
observables using this density matrix, we should take the normalized
definition:
$\expectationvalue{\hat{A}}=\frac{\Tr\left(\hat{A}\hat{\rho}_{\text{ev}}\right)}{\Tr\left(\hat{\rho}_{\text{ev}}\right)}$.
We can check that thus defined average value of the OAM projection upon the
$z$ axis calculated with this density matrix equals $\ell$,
$\expectationvalue{\hat{L}_{z}}=\ell$, when the impact parameter equals zero,
see Appendix B.

Analogously, for a non-vanishing impact parameter we have

$\displaystyle\expectationvalue{\hat{L}_{z}}=$
$\displaystyle\frac{1}{\Tr(\hat{\rho}_{\text{ev}})}\sum_{\tilde{\ell}}\tilde{\ell}\int\frac{q_{\perp}\differential
q_{\perp}\differential
q_{z}\differential\varphi_{q}\differential\varphi_{q}^{\prime}}{(2\pi)^{6}}(2\pi)^{2}e^{i\tilde{\ell}(\varphi_{q}-\varphi_{q}^{\prime})}\psi_{\text{ev}}^{*}(q_{\perp},q_{z},\varphi_{q},\bm{b})\psi_{\text{ev}}(q_{\perp},q_{z},\varphi_{q}^{\prime},\bm{b})$
$\displaystyle=2\pi\mathcal{N}^{2}p^{2}\kappa\sum_{\tilde{\ell}}\tilde{\ell}\int\frac{q_{\perp}\differential
q_{\perp}\differential
q_{z}\differential\varphi_{q}\differential\varphi_{q}^{\prime}}{(2\pi)^{6}}(2\pi)^{2}e^{i(\tilde{\ell}-\ell)\varphi_{q}}e^{-i(\tilde{\ell}-\ell)\varphi_{q}^{\prime}}\delta\left(\varepsilon_{i}+\omega-\frac{q^{2}}{2m}\right)$
$\displaystyle\times\delta\left(\varepsilon_{i}+\omega-\frac{q^{\prime
2}}{2m}\right)\Bigg{[}\frac{q_{\perp}}{\sqrt{2}}\left[d_{-1\Lambda}^{1}I_{\ell+1}(\alpha,\beta,\bm{b},\varphi_{q})-d_{1\Lambda}^{1}I_{\ell-1}(\alpha,\beta,\bm{b},\varphi_{q})\right]+q_{z}d_{0\Lambda}^{1}I_{\ell}(\alpha,\beta,\bm{b},\varphi_{q})\Bigg{]}^{*}$
$\displaystyle\times\Bigg{[}\frac{q_{\perp}^{\prime}}{\sqrt{2}}\left[d_{-1\Lambda}^{1}I_{\ell+1}(\alpha^{\prime},\beta^{\prime},\bm{b},\varphi_{q}^{\prime})-d_{1\Lambda}^{1}I_{\ell-1}(\alpha^{\prime},\beta^{\prime},\bm{b},\varphi_{q}^{\prime})\right]+q_{z}^{\prime}d_{0\Lambda}^{1}I_{\ell}(\alpha^{\prime},\beta^{\prime},\bm{b},\varphi_{q}^{\prime})\Bigg{]}\bigg{/}{\Tr(\hat{\rho}_{\text{ev}})}.$
(31)

Let us again suppose that the target is small enough and expand expression
(31) in powers of $\kappa b$ up to the linear terms. Recall that

$\displaystyle I_{\ell}(\alpha,\beta,\bm{b},\varphi_{p})\approx
I_{n}(\alpha,\beta,0)-\frac{i\kappa
b}{2}e^{-i\varphi_{b}}e^{i\varphi_{p}}I_{\ell+1}(\alpha,\beta,0)$
$\displaystyle-\frac{i\kappa
b}{2}e^{i\varphi_{b}}e^{-i\varphi_{p}}I_{\ell-1}(\alpha,\beta,0).$ (32)

Thus, any term in (31) that is proportional to $\kappa b$ has the form of
$\sim I_{n}(\alpha,\beta,0)I_{m}(\alpha^{\prime},\beta^{\prime},0)e^{\pm
i(\varphi_{b}-\varphi_{q})}$ or $\sim
I_{n}(\alpha,\beta,0)I_{m}(\alpha^{\prime},\beta^{\prime},0)e^{\pm
i(\varphi_{b}-\varphi_{q}^{\prime})}$. After the integration over
$\varphi_{q}$ (or $\varphi_{q}^{\prime}$) in Eq. (31) the Kronecker delta
$\delta_{\tilde{\ell},\ell\pm 1}$ appears. Then this term is eliminated by the
second azimuthal integral, as

$\int_{0}^{2\pi}e^{i(\tilde{\ell}-\ell)\varphi_{q}}=0$ (33)

when $\tilde{\ell}=\ell\pm 1$. Thus, we come to the conclusion that in the
first order expansion in the parameter $\kappa b$

$\expectationvalue{\hat{L}_{z}}=\ell$ (34)

despite the small, but non-vanishing impact parameter.

Importantly, the incoherent averaging over the target, which is generally
understood as the averaging of the differential cross-section, can be
interpreted in terms of the density matrix. In this case the outgoing electron
simply represents a mixed state of electron states scattered from different
atoms rather than a coherent superposition. We now introduce the averaged
density matrix

$\hat{\rho}_{\text{ev}}^{\prime}=\int\differential^{2}b\>n(\bm{b})\hat{\rho}_{\text{ev}}(\bm{b}).$
(35)

If the target is small enough in the sense that $\kappa b\ll 1$, the mean
value of the OAM projection of each state in the mixture approximately equals
$\ell$, and thus for the whole state we find that

$\expectationvalue{\hat{L}_{z}}^{\prime}=\frac{1}{\Tr(\hat{\rho}_{\text{ev}}^{\prime})}\Tr\left(\hat{L}_{z}\hat{\rho}_{\text{ev}}^{\prime}\right)=\ell.$
(36)

Although the described photoelectron state is not pure, it may be deemed
twisted in the sense of having a definite expectation value of the OAM
projection.

Conclusion. In summary, we have studied theoretically the final evolved state
of the electron that is emitted in the ionization of hydrogen–like atom by a
vortex light beam. The non-relativistic first-order perturbation theory was
applied to derive the state of the photoelectron as it evolves from the
process itself, without specifying any detection scheme. We have shown that
the emitted photoelectron generally represents a localized wave packet with a
well-defined projection of the orbital angular momentum (OAM) onto the
quantization axis of the photon TAM.

The transverse size (coherence length) of the electron packet is shown to be
solely governed by the energies of the initial bound electron and incident
photon. The ionization of an extended atomic target instead of a single atom
does not significantly violate the OAM of the resulting electron as long as
the size of the target does not exceed the transverse size of the light beam.
Although we have considered only nonrelativistic electron and have neglected
spin effects, it can be expected that in a rigorous relativistic framework the
angular momentum is also fully transferred from the photon to the electron.
However, in this case, the outgoing particle has a certain value of TAM
projection rather than OAM, and the splitting of the TAM into spin and orbital
parts is ambiguous for both particles [39].

Therefore, we propose that ionization by vortex light beams can be used to
generate vortex electrons in experimental setups where the usage of other
generation techniques (diffraction grating, phase plate, magnetic monopole,
etc. [11]) is hampered for some reason. For instance, such an approach could
be used to produce high-energy twisted electrons via irradiating photocathodes
at various accelerator facilities, such as LINAC-200 at JINR, Dubna. This
could open up opportunities for the experimental study of vortex state
collisions and various radiation processes [12, 20, 21, 10, 40, 41, 42, 13,
14].

More results related to the ITMO-JINR project on relativistic vortex electron
generation could be expected in near future. In particular, we plan to take
into account the possibility of OAM redistribution between the photoelectron
and the atomic center of mass (similarly to [29, 43, 44]), and the processes
of relaxation due to interaction with the adjacent atoms.

Acknowledgments. We are thankful to V. Ivanov for his initial contribution to
this project. We are also grateful to G. Sizykh and D. Grosman for many
valuable discussions and advice. The studies on ionization with the Bessel
beam are supported by the Ministry of Science and Higher Education of the
Russian Federation (agreement No. 075-15-2021-1349). The research on
photoionization with LG beams is supported by the Russian Science Foundation
(Project No. 23-62-10026; [9]). The developments on incoherent photoionization
by A.C. and D.K. are supported by the Government of the Russian Federation
through the ITMO Fellowship and Professorship Program. Analysis of the
transverse coherence length of the photoelectron by D.K. is also supported by
the Foundation for the Advancement of Theoretical Physics and Mathematics
“BASIS”.

##  Appendix A Polarization vector decomposition

Let us take a plane wave photon in representation (6) and write down its
polarization vector in terms of spherical angles $\theta_{k}$ and
$\varphi_{k}$ [7]

$\bm{e}_{\bm{k}\Lambda}=\sum\limits_{\sigma=0,\pm
1}e^{-i\sigma\varphi_{k}}d_{\sigma\Lambda}^{1}(\theta_{k})\bm{\chi}_{\sigma},$
(37)

where $d^{J}_{MM^{\prime}}(\theta)$ are the small Wigner functions [45] and
the basis vectors are defined as follows:

$\displaystyle\bm{\chi}_{0}=\begin{pmatrix}0\\\ 0\\\
1\end{pmatrix},\,\bm{\chi}_{\pm 1}=\mp\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\
\pm i\\\ 0\end{pmatrix};$ (38)
$\displaystyle\bm{\chi}_{1}\bm{\chi}_{-1}=\bm{\chi}_{-1}\bm{\chi}_{1}=-1,\
\bm{\chi}_{\pm 1}\bm{\chi}_{\pm 1}=0.$ (39)

Then, we can express the vector $\bm{n}=\bm{p}/p$ in the same basis as
polarization vector $\bm{e}_{\bm{k}\Lambda}$:

$\displaystyle\bm{n}=\bm{\chi}_{0}\cos{\theta_{p}}-\frac{1}{\sqrt{2}}(\bm{\chi}_{1}e^{-i\varphi_{p}}-\bm{\chi}_{-1}e^{i\varphi_{p}})\sin{\theta_{p}},$
(40)

and the scalar product in Eq. (10) can be decomposed as follows:

$\bm{e}_{\bm{k}\Lambda}\bm{n}=\sum\limits_{\sigma=0,\pm
1}e^{-i\sigma\varphi_{k}}d_{\sigma\Lambda}^{1}(\theta_{k})\bm{\chi}_{\sigma}\bm{n}\\\
=\frac{\sin\theta_{p}}{\sqrt{2}}e^{i(\varphi_{k}-\varphi_{p})}d_{-1\Lambda}^{1}(\theta_{k})+d_{0\Lambda}^{1}(\theta_{k})\cos\theta_{p}\\\
-\frac{\sin\theta_{p}}{\sqrt{2}}e^{-i(\varphi_{k}-\varphi_{p})}d_{1\Lambda}^{1}(\theta_{k}).$
(41)

##  Appendix B Calculating $\expectationvalue{\hat{L}_{z}}$ with density
matrix

Here we check that the average value of the OAM projection upon the $z$ axis
calculated with this density matrix equals $\ell$ when the impact parameter
equals zero.

Let us start with the definition

$\displaystyle\expectationvalue{\hat{L}_{z}}$ $\displaystyle=$
$\displaystyle\frac{1}{\Tr(\hat{\rho}_{\text{ev}})}\Tr\left(\hat{L}_{z}\hat{\rho}_{\text{ev}}\right)$
(42) $\displaystyle=$
$\displaystyle\frac{1}{\Tr(\hat{\rho}_{\text{ev}})}\int\frac{\differential^{3}q}{(2\pi)^{3}}\bra{\bm{q}}\hat{L}_{z}\hat{\rho}\ket{\bm{q}}$
(43) $\displaystyle=$
$\displaystyle\frac{1}{\Tr(\hat{\rho}_{\text{ev}})}\int\frac{\differential^{3}q\differential^{3}q^{\prime}}{(2\pi)^{6}}\bra{\bm{q}}\hat{L}_{z}\ket{\bm{q}^{\prime}}\bra{\bm{q}^{\prime}}\hat{\rho}\ket{\bm{q}},$
(44)

where

$\displaystyle\bra{\bm{q}^{\prime}}\hat{\rho}\ket{\bm{q}}=$
$\displaystyle\psi_{\text{ev}}^{*}(\bm{q})\psi_{\text{ev}}(\bm{q}^{\prime}),$
(45) $\displaystyle\bra{\bm{q}}\hat{L}_{z}\ket{\bm{q}^{\prime}}=$
$\displaystyle\sum_{\tilde{\ell}}\int\frac{\tilde{q}_{\perp}\differential\tilde{q}_{\perp}\differential\tilde{q}_{z}}{(2\pi)^{2}}\bra{\bm{q}}\hat{L}_{z}\ket*{\tilde{q}_{\perp}\tilde{q}_{z}\tilde{\ell}}\bra*{\tilde{q}_{\perp}\tilde{q}_{z}\tilde{\ell}}\ket{\bm{q}^{\prime}}.$
(46)

With the projection of a Bessel vortex state on a plane wave being

$\bra*{\tilde{q}_{\perp}\tilde{q}_{z}\tilde{\ell}}\ket{\bm{q}}=e^{-i\tilde{\ell}\varphi_{q}}\frac{(2\pi)^{2}}{\sqrt{\tilde{q}_{\perp}}}\delta(\tilde{q}_{\perp}-q_{\perp})\delta(\tilde{q}_{z}-q_{z}),$
(47)

we can calculate

$\bra{\bm{q}}\hat{L}_{z}\ket{\bm{q}^{\prime}}=\sum_{\tilde{\ell}}\tilde{\ell}(2\pi)^{2}e^{i\tilde{\ell}(\varphi_{q}-\varphi_{q}^{\prime})}\delta(q_{\perp}-q_{\perp}^{\prime})\delta(q_{z}-q_{z}^{\prime})\frac{1}{q_{\perp}}.$
(48)

Then, with $\differential^{3}q=q_{\perp}\differential q_{\perp}\differential
q_{z}\differential\varphi_{q}$, we can show

$\expectationvalue{\hat{L}_{z}}=\frac{1}{\Tr(\hat{\rho}_{\text{ev}})}\int\frac{\differential^{3}q\differential^{3}q^{\prime}}{(2\pi)^{6}}\sum_{\tilde{\ell}}\tilde{\ell}(2\pi)^{2}e^{i\tilde{\ell}(\varphi_{q}-\varphi_{q}^{\prime})}\delta(q_{\perp}-q_{\perp}^{\prime})\delta(q_{z}-q_{z}^{\prime})\frac{1}{q_{\perp}}\psi_{\text{ev}}^{*}(\bm{q})\psi_{\text{ev}}(\bm{q}^{\prime})\\\
=\frac{1}{\Tr(\hat{\rho}_{\text{ev}})}\sum_{\tilde{\ell}}\tilde{\ell}\int\frac{q_{\perp}\differential
q_{\perp}\differential
q_{z}\differential\varphi_{q}\differential\varphi_{q}^{\prime}}{(2\pi)^{6}}(2\pi)^{2}e^{i\tilde{\ell}(\varphi_{q}-\varphi_{q}^{\prime})}\underbrace{\psi_{\text{ev}}^{*}(q_{\perp},q_{z},\varphi_{q})}_{\chi^{*}(q_{\perp},q_{z})e^{-i\ell\varphi_{q}}}\underbrace{\psi_{\text{ev}}(q_{\perp},q_{z},\varphi_{q}^{\prime})}_{\chi(q_{\perp},q_{z})e^{i\ell\varphi_{q}^{\prime}}}\\\
=\frac{1}{\Tr(\hat{\rho}_{\text{ev}})}\sum_{\tilde{\ell}}\tilde{\ell}\delta_{\ell\tilde{\ell}}\int\frac{q_{\perp}\differential
q_{\perp}\differential
q_{z}\differential\varphi_{q}}{(2\pi)^{6}}(2\pi)^{3}e^{i(\tilde{\ell}-\ell)\varphi_{q}}|\chi(q_{\perp},q_{z})|^{2}\\\
=\frac{1}{\Tr(\hat{\rho}_{\text{ev}})}\ell\int\frac{q_{\perp}\differential
q_{\perp}\differential
q_{z}\differential\varphi_{q}}{(2\pi)^{3}}|\psi_{\text{ev}}(\bm{q})|^{2}=\ell$
(49)

## References

  * Matula  _et al._ [2013] O. Matula, A. G. Hayrapetyan, V. G. Serbo, A. Surzhykov, and S. Fritzsche, Journal of Physics B: Atomic, Molecular and Optical Physics 46, 205002 (2013).
  * Peshkov  _et al._ [2015] A. A. Peshkov, S. Fritzsche, and A. Surzhykov, Phys. Rev. A 92, 043415 (2015).
  * Müller  _et al._ [2016] R. A. Müller, D. Seipt, R. Beerwerth, M. Ornigotti, A. Szameit, S. Fritzsche, and A. Surzhykov, Phys. Rev. A 94, 041402 (2016).
  * Surzhykov  _et al._ [2016] A. Surzhykov, D. Seipt, and S. Fritzsche, Phys. Rev. A 94, 033420 (2016).
  * Kiselev  _et al._ [2023] M. D. Kiselev, E. V. Gryzlova, and A. N. Grum-Grzhimailo, Phys. Rev. A 108, 023117 (2023).
  * Kiselev  _et al._ [2024] M. D. Kiselev, E. V. Gryzlova, and A. N. Grum-Grzhimailo, Phys. Rev. A 109, 023108 (2024).
  * Knyazev and Serbo [2018] B. A. Knyazev and V. G. Serbo, Physics-Uspekhi 61, 449 (2018).
  * Allen  _et al._ [1992] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
  * [9] https://rscf.ru/en/project/23-62-10026/.
  * Ivanov [2022] I. P. Ivanov, Progress in Particle and Nuclear Physics 127, 103987 (2022).
  * Bliokh  _et al._ [2017] K. Y. Bliokh _et al._ , Phys. Rept. 690, 1 (2017), arXiv:1703.06879 [quant-ph] .
  * Ivanov  _et al._ [2020] I. P. Ivanov, N. Korchagin, A. Pimikov, and P. Zhang, Phys. Rev. Lett. 124, 192001 (2020).
  * Karlovets [2017] D. Karlovets, JHEP 03, 049 (2017), arXiv:1611.08302 [hep-ph] .
  * Karlovets and Serbo [2020] D. V. Karlovets and V. G. Serbo, Phys. Rev. D 101, 076009 (2020).
  * Berestetskii  _et al._ [1982] V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii,  _Quantum Electrodynamics: Volume 4_ , Vol. 4 (Butterworth-Heinemann, 1982).
  * Keldysh [2024] L. V. Keldysh, in  _Selected Papers of Leonid V Keldysh_ (World Scientific, 2024) pp. 56–63.
  * Zheltikov [2017] A. M. Zheltikov, Physics-Uspekhi 60, 1087 (2017).
  * Figueira de Morisson Faria  _et al._ [2002] C. Figueira de Morisson Faria, H. Schomerus, and W. Becker, Phys. Rev. A 66, 043413 (2002).
  * Maxwell  _et al._ [2021] A. S. Maxwell, G. S. J. Armstrong, M. F. Ciappina, E. Pisanty, Y. Kang, A. C. Brown, M. Lewenstein, and C. Figueira de Morisson Faria, Faraday Discuss. 228, 394 (2021).
  * Karlovets, D. V.  _et al._ [2022] Karlovets, D. V., Baturin, S. S., Geloni, G., Sizykh, G. K., and Serbo, V. G., Eur. Phys. J. C 82, 1008 (2022).
  * Karlovets  _et al._ [2023] D. V. Karlovets, S. S. Baturin, G. Geloni, G. K. Sizykh, and V. G. Serbo, The European Physical Journal C 83, 372 (2023).
  * Picón  _et al._ [2010a] A. Picón, J. Mompart, J. V. de Aldana, L. Plaja, G. Calvo, and L. Roso, Optics express 18, 3660 (2010a).
  * Picón  _et al._ [2010b] A. Picón, A. Benseny, J. Mompart, J. V. de Aldana, L. Plaja, G. F. Calvo, and L. Roso, New Journal of Physics 12, 083053 (2010b).
  * Chen  _et al._ [2023] Y. Chen, Y. Zhou, M. Li, K. Liu, M. F. Ciappina, and P. Lu, Phys. Rev. A 107, 033112 (2023).
  * Karlovets  _et al._ [2017] D. V. Karlovets, G. L. Kotkin, V. G. Serbo, and A. Surzhykov, Phys. Rev. A 95, 032703 (2017).
  * Serbo  _et al._ [2015] V. Serbo, I. P. Ivanov, S. Fritzsche, D. Seipt, and A. Surzhykov, Phys. Rev. A 92, 012705 (2015).
  * Goodwin  _et al._ [2016] J. F. Goodwin, G. Stutter, R. C. Thompson, and D. M. Segal, Phys. Rev. Lett. 116, 143002 (2016).
  * Schmiegelow  _et al._ [2016] C. T. Schmiegelow, J. Schulz, H. Kaufmann, T. Ruster, U. G. Poschinger, and F. Schmidt-Kaler, Nature communications 7, 12998 (2016).
  * Stopp  _et al._ [2022] F. Stopp, M. Verde, M. Katz, M. Drechsler, C. T. Schmiegelow, and F. Schmidt-Kaler, Phys. Rev. Lett. 129, 263603 (2022).
  * Andrews and Babiker [2012] D. L. Andrews and M. Babiker,  _The Angular Momentum of Light_ (Cambridge University Press, 2012).
  * Peshkov  _et al._ [2017] A. A. Peshkov, D. Seipt, A. Surzhykov, and S. Fritzsche, Phys. Rev. A 96, 023407 (2017).
  * Chambers [1992] S. A. Chambers, Surface Science Reports 16, 261 (1992).
  * Cohen and Fano [1966] H. D. Cohen and U. Fano, Phys. Rev. 150, 30 (1966).
  * Baltenkov  _et al._ [2012] A. Baltenkov, U. Becker, S. Manson, and A. Msezane, Journal of Physics B: Atomic, Molecular and Optical Physics 45, 035202 (2012).
  * Kunitski  _et al._ [2019] M. Kunitski _et al._ , Nature Commun. 10, 1 (2019).
  * Akoury  _et al._ [2007] D. Akoury _et al._ , Science 318, 949 (2007).
  * Claessens  _et al._ [2005] B. J. Claessens, S. B. van der Geer, G. Taban, E. J. D. Vredenbregt, and O. J. Luiten, Phys. Rev. Lett. 95, 164801 (2005).
  * McCulloch  _et al._ [2011] A. J. McCulloch, D. V. Sheludko, S. D. Saliba, S. C. Bell, M. Junker, K. A. Nugent, and R. E. Scholten, Nature Physics 7, 785 (2011).
  * Bauke  _et al._ [2014] H. Bauke, S. Ahrens, C. H. Keitel, and R. Grobe, Phys. Rev. A 89, 052101 (2014).
  * Ivanov  _et al._ [2023] V. K. Ivanov, A. D. Chaikovskaia, and D. V. Karlovets, Phys. Rev. A 108, 062803 (2023), arXiv:2305.12419 [physics.atom-ph] .
  * Chaikovskaia  _et al._ [2024] A. D. Chaikovskaia, D. V. Karlovets, and V. G. Serbo, Phys. Rev. A 109, 012222 (2024), arXiv:2310.09864 [quant-ph] .
  * Sheremet  _et al._ [2024] N. Sheremet, A. Chaikovskaia, D. Grosman, and D. Karlovets, (2024), arXiv:2404.11497 [physics.atom-ph] .
  * Peshkov  _et al._ [2023] A. A. Peshkov, Y. M. Bidasyuk, R. Lange, N. Huntemann, E. Peik, and A. Surzhykov, Phys. Rev. A 107, 023106 (2023).
  * Baturin and Volotka [2024] S. S. Baturin and A. V. Volotka, (2024), arXiv:2404.11558 [quant-ph] .
  * Varshalovich  _et al._ [1988] D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii,  _Quantum Theory of Angular Momentum_ (1988).