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# Carrier-envelope-phase and helicity control of electron vortices in
photodetachment

M. M. Majczak  F. Cajiao Vélez<EMAIL_ADDRESS>J. Z. Kamiński
K. Krajewska<EMAIL_ADDRESS>Institute of Theoretical Physics,
Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland

(August 28, 2024)

###### Abstract

Formation of electron vortices and momentum spirals in photodetachment of the
H- anion driven by isolated ultrashort laser pulses of circular polarization
or by pairs of such pulses (of either corotating or counterrotating
polarizations) are analyzed under the scope of the strong-field approximation.
It is demonstrated that the carrier-envelope phase (CEP) and helicity of each
individual pulse can be used to actively manipulate and control the vortical
pattern in the probability amplitude of photodetachment. Specifically, the
two-dimensional mappings of probability amplitude can be rotated in the
polarization plane with changing the CEP of the driving pulse (or two
corotating pulses); thus, offering a new tool of field characterization.
Furthermore, it is shown that the formation of spirals or annihilation of
vortices relates directly to the time-reversal symmetry of the laser field,
which is realized by a pair of pulses with opposite helicities and CEPs.

##  I Introduction

During the last decade there has been an increasing interest towards the
electron momentum spirals (often referred as ‘vortices’) in laser-induced
photoionization Starace2015 ; Starace2016 ; Starace2017 ; Maxwell2020 ;
Bandrauk2016 ; Bandrauk2017 ; Faria2021 ; Wollenhaupt2017 ; Wollenhaupt2019a ;
Wollenhaupt2019b ; Wollenhaupt2020a ; Wollenhaupt2020b  or photodetachment
Circular . They manifest themselves, in the probability distribution of
photoelectrons, as zones of large probability which follow concentric Fermat
spirals with well-defined number of arms Starace2015 . However, other type of
structures, known as electron vortices, can also be found in the momentum
distribution of photoelectrons; they appear as continuous lines in the three-
dimensional momentum space where the probability amplitude of ionization (or
detachment) vanishes, and its phase changes from zero to integer multiples of
$2\pi$ around them Dirac1931 ; Circular ; VonKarmaan ; GroundState ; Trains .
Those two types of structures are fundamentally different as demonstrated by
their physical properties; for instance, electron vortices carry a
nonvanishing and quantized orbital-angular momentum (OAM) Bliokh2017 ;
Lloyd2017 , while spirals are characterized by a null OAM.

While electron vortices and spirals are formed due to the same physical
phenomena, i.e., by subtile interference effects in the probability amplitude
of ionization, whether one or the other are observed depends on the light
field configuration and the target atom (or ion) Maxwell2020 ; GroundState .
Several theoretical studies have predicted the formation of momentum spirals
in photoionization from a variety of atomic and molecular targets Starace2015
; Bandrauk2016 ; Bandrauk2017 ; Faria2021  and diverse laser field
configurations Starace2016 ; Starace2017 . For instance, it has been shown
that a sequence of two counterrotating circularly-polarized and ultrashort
laser pulses leads to momentum spirals, whereas single pulses or trains of
corotating pulses lead to the formation of electron vortices Circular .
However, corotating trains of bichromatic laser pulses may also lead to the
spiral formation Bandrauk2016 ; Bandrauk2017 .

According to the analysis presented in Ref. Starace2015 , which is performed
in the perturbation regime of laser-matter interactions, the energy spectra of
photoelectrons obtained in the photoionization of He atoms by two corotating
laser pulses is circularly-symmetric; i.e., annular zones of zero probability,
similar to Newton’s rings, are observed. Moreover, a change of the relative
carrier-envelope phase (CEP) between the driving pulses leaves the angular
symmetry unchanged. Hence, the effect of the CEP on the electron distribution
is, at most, detected as a variation of the rings radial locations or their
intensities. Note that in the nonperturbative regime, the probability
distribution of photoelectrons stimulated by corotating pulses is not
circularly-symmetric, and vortical structures are observed together with the
annular zones of zero probability Circular .

In this paper, we further advance the theoretical understanding of vortical
structures in photodetachment driven by circularly-polarized laser pulses.
Various pulse configurations are considered, including isolated pulses and
pairs of pulses in corotating and counterrotating schemes. The calculations
presented here are based on the strong-field approximation (SFA) Keldysh ;
Faisal ; Reiss  in a nonperturbative regime, which provides a remarkable
agreement with ab initio methods of solving numerically the Schrödinger
equation Circular ; VonKarmaan ; Trains . Our analysis focuses on the CEP
effects over the formation of vortex structures. We demonstrate that the
vortical pattern in the probability amplitude of photoelectrons rotates in the
polarization plane by the CEP of the driving pulse. The same is observed in
the corotating configuration of two identical pulses. For pulses with opposite
helicities, on the other hand, momentum spirals can be observed. More
specifically, we show that the formation of spirals is closely related to
annihilation of vortex-antivortex pairs, which occurs for laser fields with
the time-reversal symmetry. Since the manipulation of electron vortical
structures in photodetachment is sensitive to the handedness and the CEP of
the laser field, it might provide additional means of field characterization.

We use atomic units (a.u.) along this paper. For our theoretical derivations
we set $\hbar=1$ but show the electron charge, $e=-|e|$, and mass,
$m_{\mathrm{e}}$, explicitly. Our numerical illustrations are presented in
terms of the atomic units of momentum $p_{\rm at}=\alpha m_{\mathrm{e}}c$ and
energy $E_{\rm at}=(\alpha c)^{2}m_{\mathrm{e}}$, where $\alpha$ is the fine-
structure constant and $c$ is the speed of light. Furthermore, the atomic unit
of length corresponds to the Bohr radius, $a_{0}=\hbar/p_{\rm at}$, whereas
the atomic unit of electric field strength equals ${\cal E}_{\rm
at}=\alpha^{2}m_{\rm e}c^{2}/(|e|a_{0})$.

##  II Theoretical formulation

The theoretical derivations presented here are based on the SFA. As it was
shown in Refs. Circular ; VonKarmaan ; Trains , the SFA is an excellent
analytical tool for the treatment of photodetachment from negative ions. A
direct comparison between the results obtained within this framework and the
numerical solution of the time-dependent Schrödinger equation has shown
remarkable similarities. This is actually expected, as the SFA neglects the
Coulomb interaction between the freed electron and the parent ion. In laser-
induced photodetachment the residue is a neutral atom, hence the Coulomb
interaction during the electron evolution in the continuum is absent. In
contrast, in photoionization, the parent ion has a positive charge and the
Coulomb potential modifies the electron dynamics, particularly in the low-
energy regime. However, the SFA presents several advantages as compared to
other more sophisticated ab initio calculations, including a faster numerical
computation and simpler analytical expressions, which can be used to
understand, in a deeper way, the physical phenomena under consideration.

For the reasons stated above, we shall limit our analysis of photodetachment
from negative ions to the framework of the SFA. Even though the probability
amplitude of detachment was initially calculated by Gribakin and Kuchiev in
Ref. Gribakin1997 , and further explored elsewhere (see, e.g., GroundState ;
Trains ), here we shall present the most important results along its
derivation.

###  II.1 Probability amplitude of photodetachment

It is assumed that, in the remote past, the electron is found in the ground
state of the H- anion ($s$-electron) of energy $E_{0}$, which we denote as
$\Phi_{0}({\bm{r}},t)=\mathrm{e}^{-\mathrm{i}E_{0}t}\Phi_{0}({\bm{r}})$. By
the action of the laser field, which lasts for a time $T_{p}$, the electron is
promoted to the continuum. The probability amplitude of photodetachment, under
the scope of the SFA, is given by Gribakin1997 ; Circular ; VonKarmaan ;
GroundState ; Trains

${\mathcal{A}}({\bm{p}})=-\mathrm{i}\int_{0}^{T_{p}}\differential
t\int\differential^{3}r\,\psi^{*}_{\bm{p}}({\bm{r}},t)H_{\rm
I}({\bm{r}},t)\Phi_{0}({\bm{r}},t),$ (1)

where $\psi_{\bm{p}}({\bm{r}},t)$ is the Volkov state of the electron in the
laser field Wolkow1935  with an asymptotic momentum ${\bm{p}}$. In the
following, we shall consider only the length gauge for our calculations, as it
was suggested in Ref. Gribakin1997 . The Volkov solution, in this gauge, reads

$\psi_{\bm{p}}({\bm{r}},t)=\exp[\mathrm{i}[{\bm{p}}-e{\bm{A}}(t)]\cdot{\bm{r}}-\frac{\mathrm{i}}{2m_{\mathrm{e}}}\int_{0}^{t}\differential
t^{\prime}({\bm{p}}-e{\bm{A}}(t^{\prime}))^{2}\Big{]},$ (2)

with ${\bm{A}}(t)$ defining the vector potential of the laser field in the
dipole approximation. The interaction Hamiltonian, $H_{\rm I}({\bm{r}},t)$,
takes the form

${H}_{\rm I}({\bm{r}},t)=-e\bm{{\mathcal{E}}}(t)\cdot{\bm{r}},$ (3)

where $\bm{{\mathcal{E}}}(t)=-\partial_{t}{\bm{A}}(t)$ is the oscillating
electric field corresponding to the vector potential ${\bm{A}}(t)$.

The unperturbed ground-state wave function of the electron in the H- anion,
$\Phi_{0}({\bm{r}})$, is determined according to the zero-range potential
model Gribakin1997 ; Smirnov

$\Phi_{0}({\bm{r}})=\frac{A}{\sqrt{4\pi}a_{0}}\frac{\mathrm{e}^{-\kappa
r/a_{0}}}{r},$ (4)

where $A$ is a scaling factor chosen such that the properties of the anion
coincide with more advanced ab initio calculations or experimental data
Smirnov , and $\kappa$ relates to the ground state energy, $E_{0}=-(\alpha
c)^{2}m_{e}\kappa^{2}/2$. For our numerical illustrations we use the values
$A=0.75$ and $\kappa=0.2354$ Gribakin1997 , which correspond to an ionization
potential for H- equal to $|E_{0}|\approx 0.754$ eV.

From Eqs. (1) to (3), and after some algebraic manipulations (for details see,
e.g., Refs. GroundState ; Trains ), the probability amplitude of detachment
takes the form

${\mathcal{A}}({\bm{p}})=\mathrm{i}e\int_{0}^{T_{p}}\differential
t\,{\bm{\mathcal{E}}}(t)\cdot\tilde{\bm{\Phi}}_{0}({\bm{p}}-e{\bm{A}}(t))\mathrm{e}^{\mathrm{i}G_{\bm{p}}(t)}.$
(5)

Here, we have introduced the Fourier transform of the ground-state wave
function,

$\tilde{\bm{\Phi}}_{0}(\bm{p})=\mathrm{i}\nabla_{p}\tilde{\Phi}_{0}(\bm{p})=-\mathrm{i}\frac{4\sqrt{\pi}A}{(\kappa^{2}+\bm{p}^{2})^{2}}{\bm{p}},$
(6)

and the function $G_{\bm{p}}(t)$ defined as

$G_{\bm{p}}(t)=\frac{1}{2m_{\mathrm{e}}}\int_{0}^{t}\differential
t^{\prime}\Big{[}{\bm{p}}-e{\bm{A}}(t^{\prime})\Big{]}^{2}-E_{0}t.$ (7)

Hence, by integrating numerically Eq. (5) and taking into account Eqs. (6) and
(7), we calculate the probability amplitude of photodetachment. The driving
laser field used in our illustrations is described in Sec. II.3, after we
comment on the properties of ${\mathcal{A}}({\bm{p}})$ in the three-
dimensional (3D) momentum space.

###  II.2 Vortices vs Nodes

The differences between nodal surfaces and vortex lines have been thoroughly
analyzed in Ref. Circular . However, for the sake of clarity, we present here
their main properties. As presented in Ref. Bialynicki2000 , vortex lines
appear in the 3D momentum space as closed loops or continuous curves (starting
and ending at infinity). While the complex probability amplitude of detachment
${\cal A}({\bm{p}})$ vanishes along those lines, around them the amplitude’s
phase changes from $0$ to $2\pi m$, where $m=\pm 1,\pm 2,...$ is an integer
number called the topological charge (see, e.g., Refs. Lloyd2017 ; Bliokh2017
). In contrast, nodes appear as surfaces of vanishing probability, accross
which the phase of ${\cal A}({\bm{p}})$ jumps by $\pm\pi$. Note that in a two-
dimensional plane, known as Poincaré section, vortex lines are typically
visualized as single points, while nodal surfaces appear as curves of zero
probability.

As was elaborated in Ref. Circular , the appearance of vortices in
photodetachment can be quantified by the quantization condition,

$m=\frac{1}{2\pi}\oint_{\cal C}{\bm{v}}({\bm{p}})\cdot\differential{\bm{p}},$
(8)

where the velocity field ${\bm{v}}({\bm{p}})$ is related to the probability
amplitude of photodetachment ${\mathcal{A}}({\bm{p}})$,

${\bm{v}}({\bm{p}})=\frac{1}{|{\mathcal{A}}({\bm{p}})|^{2}}{\rm
Re}\,[{\mathcal{A}}^{*}({\bm{p}})(-\mathrm{i}{\bm{\nabla}}_{\bm{p}}){\mathcal{A}}({\bm{p}})].$
(9)

For simplicity, we assume that the Poincaré section, which contains the closed
path of integration ${\cal C}$ in Eq. (8), coincides with the
$p_{x}p_{y}$-plane (i.e., we set $p_{z}=0$). More precisely, we choose ${\cal
C}$ to be a circular contour of radius $p_{r}$ centered at momentum
${\bm{p}}_{0}=(p_{0x},p_{0y},0)$. Hence, it can be parametrized by an angle
$\varphi_{\bm{p}}$ such that

$p_{x}=p_{0x}+p_{r}\cos{\varphi_{\bm{p}}},\;p_{y}=p_{0y}+p_{r}\sin{\varphi_{\bm{p}}},\;p_{z}=0.$
(10)

It is straighforward to show in this case that the quantization condition (8)
becomes

$m(p_{r})=\frac{1}{2\pi}\int_{0}^{2\pi}\differential\varphi_{\bm{p}}\frac{{\rm
Im}\big{[}{\mathcal{A}}^{*}({\bm{p}})\partial_{\varphi_{\bm{p}}}{\mathcal{A}}({\bm{p}})\big{]}}{|{\mathcal{A}}({\bm{p}})|^{2}},$
(11)

where $m=m(p_{r})$. Actually, the contour ${\cal C}$ may contain multiple
vortices. In such case, $m(p_{r})$ represents a total topological charge
enclosed by the contour. If, however, ${\cal C}$ encircles an isolated vortex
point, which for instance can be realized if an individual vortex stays
enclosed by the contour in the limit $p_{r}\rightarrow 0$, Eq. (11) represents
its individual topological charge. This definition will be used in Secs. III
and IV to interpret our numerical results.

###  II.3 Laser field

Figure 1: Modulus (upper panels) and phase (lower panels) of the probability
amplitude of photodetachment ${\mathcal{A}}({\bm{p}})$ [Eqs. (5) to (7)] in
the $p_{x}p_{y}$-plane ($p_{z}=0$). The modulus has been raised to the power
$\nu=0.5$ and the phase is presented in units of $\pi$, which is done only for
visual purposes. A circularly-polarized driving pulse ($N_{\rm rep}=1$)
comprising $N_{\rm osc}=3$ oscillations within a $\sin^{2}$ envelope [see,
Eqs. (12) to (14)] has been used. While we keep its peak intensity
($I=2.5\times 10^{11}$W/cm2) and wavelength ($\lambda=4000$ nm) fixed, each
column corresponds to a different CEP and helicity of the pulse. Specifically,
the left column (Case I) is for the laser field configuration
$(\chi,\sigma)=(0,+)$, meaning that the CEP is zero and the laser pulse
rotates counterclockwise. The middle column (Case II) corresponds to the
configuration $(\pi/3,+)$, whereas the right column (Case III) corresponds to
the configuration $(2\pi/3,-)$.

As we plan to demonstrate in this paper, the properties of the laser field are
imprinted onto vortical structures of photodetachment. For this reason, a
careful definition of the driving laser field is important. We define an
isolated $N_{\rm osc}$ cycle laser pulse, lasting for time $\tau_{p}$, by the
following time-dependent electric field,

$\bm{{\mathcal{E}}}_{\sigma\chi}(t)=\begin{cases}{\cal
E}_{0}\sin^{2}\bigl{(}\frac{\omega t}{2N_{\rm
osc}}\bigr{)}\bm{F}(t,\sigma,\chi),&0\leqslant t\leqslant\tau_{p},\cr
0,&\textrm{otherwise},\end{cases}$ (12)

with the circularly-polarized carrier wave,

$\bm{F}(t,\sigma,\chi)=\sin(\omega t+\chi){\bm{e}}_{x}-\sigma\cos(\omega
t+\chi){\bm{e}}_{y},$ (13)

and ${\cal E}_{0}={\cal E}_{\rm at}\sqrt{\frac{I}{2I_{\rm at}}}$. Here, $I$
denotes the peak intensity of the pulse expressed in units of $I_{\rm
at}=3.51\times 10^{16}$ W/cm2, $\omega=2\pi N_{\rm osc}/\tau_{p}$ is the
carrier frequency of the field, whereas $\chi$ is the carrier envelope phase
of the pulse. The polarization properties of the field are controlled by the
helicity, $\sigma$, and in our further analysis we shall choose circularly-
polarized pulses with either counterclockwise ($\sigma=+1$) or clockwise
($\sigma=-1$) orientations. We also define a train of $N_{\rm rep}$ such
pulses,

$\bm{{\mathcal{E}}}_{\sigma_{1}\chi_{1},\sigma_{2}\chi_{2},\ldots}(t)=\sum_{\ell=1}^{N_{\rm
rep}}\bm{{\mathcal{E}}}_{\sigma_{\ell}\chi_{\ell}}\Bigl{(}t-(\ell-1)\tau_{p}\Bigr{)},$
(14)

with, in principle, different $\sigma_{\ell}$ and $\chi_{\ell}$. (Note that,
under such conditions, the total duration of the train of pulses is
$T_{p}=N_{\rm rep}\tau_{p}$.) Hence, the electromagnetic vector potential
describing the laser field can be defined,

${\bm{A}}_{\sigma_{1}\chi_{1},\sigma_{2}\chi_{2},\ldots}(t)=-\int_{-\infty}^{t}\bm{{\mathcal{E}}}_{\sigma_{1}\chi_{1},\sigma_{2}\chi_{2},\ldots}(t^{\prime})\differential
t^{\prime},$ (15)

where it holds that

$\lim_{t\rightarrow\pm\infty}\bm{A}_{\sigma_{1}\chi_{1},\sigma_{2}\chi_{2},\ldots}(t)=\bm{0}.$
(16)

While those definitions are very general, in the following we shall consider
either an isolated pulse ($N_{\rm rep}=1$) or a sequence of two pulses
($N_{\rm rep}=2$) with corotating or counterrotating circular polarizations.
For our numerical illustrations we shall keep the peak intensity, wavelength,
and number of field oscillations within the $\sin^{2}$ envelope fixed
($I=2.5\times 10^{11}$W/cm2, $\lambda=4000$ nm, and $N_{\rm osc}=3$,
respectively), while allowing the CEP and helicity to vary independently for
each pulse in the train. Hence, we use the notation
$(\chi_{1},\sigma_{1};...;\chi_{\ell},\sigma_{\ell};...;\chi_{N_{\rm
rep}},\sigma_{N_{\rm rep}})$ to identify each field configuration. From
$\sigma_{\ell}$ we only retain its sign, i.e., we write $\sigma_{\ell}=\pm$.

##  III Laser-field control of vortical structures

In this Section, we reveal the relationship between the CEP of the driving
laser field and the formation of electron vortices in photodetachment. As
mentioned above, we fix the number of field oscillations within each
individual pulse, its peak intensity, and wavelength, while allowing to vary
the number of pulses in the train, the individual CEPs, and helicities. Our
main objective is to determine how such parameters influence the formation of
electron vortices in photodetachment and the overall probability distribution
of photoelectrons.

###  III.1 Isolated laser pulse

We start by analyzing the creation of electron vortices in photodetachment
driven by a single laser pulse [$N_{\rm rep}=1$ in Eq. (14)]. In Fig. 1, we
present the color mappings of detachment probability amplitude
${\mathcal{A}}({\bm{p}})$ [Eqs. (5) to (7)] calculated in the
$p_{x}p_{y}$-plane ($p_{z}=0)$. While in the upper panels we show the
amplitude’s modulus, i.e., $|{\mathcal{A}}({\bm{p}})|^{\nu}$, where $\nu=0.5$
has been chosen for visual purposes, in the lower panels we present its phase,
${\rm arg}[{\mathcal{A}}({\bm{p}})]/\pi$. We have considered pulses of
different CEPs, labelled as Case I (left column), Case II (middle column), and
Case III (right column), which correspond to the configurations $(0,+)$,
$(\pi/3,+)$, and $(2\pi/3,-)$, respectively.

Figure 2: The same as in Fig. 1 but for a pair of two identical laser pulses
($N_{\rm rep}=2$) [configuration $(\chi,\sigma;\chi,\sigma)$]. The left column
(Case ${\rm IV}$) is for the configuration $(0,+;0,+)$ (see, Ref. Circular ),
the middle column (Case V) is for $(\pi/3,+;\pi/3,+)$, whereas the right
column is for $(2\pi/3,-;2\pi/3,-)$.

In Case I (left column of Fig. 1), we observe the appearance of four vortices;
they manifest themselves as points of vanishing probability for which the
amplitude’s phase changes continuously from $-\pi$ to $\pi$ around them.
Hence, we assign to each of them the topological charge $m=1$. Two of such
vortices are located along the line $p_{y}=0$ while the other two appear at
$p_{x}=0.025\alpha m_{\mathrm{e}}c$ Comment2 . In the middle panels (Case II),
which correspond to a driving field of the type $(\pi/3,+)$, we observe the
exact same configuration of vortices, but rotated by an angle $\alpha=\pi/3$
with respect to the center of coordinates. Finally, in the right column of
Fig. 1 (Case III), we present the results corresponding to the driving pulse
configuration $(2\pi/3,-)$, such that the CEP takes now the value $2\pi/3$ and
the helicity is inverted, $\sigma=-1$ (clockwise field evolution). This time,
we deal with antivortices ($m=-1$) and the rotation angle
$\alpha=4\pi/3=-2\pi/3$. Hence, we conclude that the effect of the CEP and
helicity of the driving pulse ($N_{\rm rep}=1$) over the vortex pattern (or,
in general, over the probability distribution of photoelectrons in momentum
space) is an overall rotation by an angle $\alpha=\chi\sigma$ with respect to
the origin of coordinates.

###  III.2 Two corotating laser pulses

Up to now we have considered the vortex formation stimulated by isolated laser
pulses. However, it is interesting to analyze the case when a pair of pulses
[$N_{\rm rep}=2$ in Eq. (14)] interacts with the H- ion. We shall start by
exploring the photodetachment driven by such a train in the corotating
configuration ($\sigma_{1}=\sigma_{2}\equiv\sigma$) with identical CEPs
($\chi_{1}=\chi_{2}\equiv\chi$).

In Fig. 2 we show the same as in Fig. 1 but this time the driving field
consists of two corotating pulses ($N_{\rm rep}=2$) and we vary the CEP, which
is common for both pulses. According to our convention, the driving fields
considered here are labelled as $(\chi,\sigma;\chi,\sigma)$. In the left
column of Fig. 2 (Case ${\rm IV}$), which corresponds to a configuration
$(0,+;0,+)$, we observe the formation of the same four vortices as in Case I
(see, the left column of Fig. 1) with additional nodal surfaces. The latter
appear as annular zones of zero probability located at different radii (see,
Ref. Circular ). Note that the regions of high probability resemble the
Newton’s rings reported in the perturbative photoionization of He atoms
Starace2015 . In the middle column of Fig. 2 (Case ${\rm V}$), which is for
the configuration $(\pi/3,+;\pi/3,+)$ we see that the very same system of
vortices and nodal surfaces is formed, but it is rotated by an angle
$\alpha=\pi/3$ with respect to the origin of coordinates. Moreover, in the
right column of Fig. 2 (Case VI) we present our results for the train
configuration $(2\pi/3,-;2\pi/3,-)$, namely, both pulses are oriented
clockwise, and both CEPs are equal to $2\pi/3$. Here we observe an identical
pattern as in Case ${\rm IV}$ but rotated by an angle $\alpha=4\pi/3=-2\pi/3$
mod $2\pi$, while vortices are transformed into antivortices. This indicates
that the addition of a nonzero CEP, common to both pulses in the corotating
train, leads to a rotation of the probability distribution of photoelectrons
by an angle $\alpha=\sigma\chi$ in the polarization plane. The same has been
seen in Fig. 1 for a single pulse. Note also that a similar dependence of the
distribution of photoelectrons with the CEP has been observed in
photoionization of molecules by trains of bichromatic laser pulses
Bandrauk2016 .

A more interesting situation is met when one of the CEPs is kept constant (say
$\chi_{1}$) and we allow the other one ($\chi$) to vary. In the Supplemental
Material we show the magnitude of the probability amplitude (Animation 1) and
its phase (Animation 2) for photodetachment driven by a train of two pulses
($N_{\rm rep}=2$) in the configurations $(\pi/3,+;\chi,+)$, where $\chi$
changes from $0$ to $2\pi$. From both magnitude and phase we can see that, at
$\chi=0$, four well defined vortices with the same topological charge ($m=1$)
and one antivortex ($m=-1$) are present. Also, an open ring of low probability
is visible at larger photoelectron momenta. By changing the CEP there is a
continuous expansion of the open ring, together with antivortex migration
towards larger $|{\bm{p}}|$. When both antivortex and open ring merge, a full
nodal surface (annular region of zero probability) is formed. This can be seen
in Animation 1 for $\chi\approx 0.9$ rad. Another interesting effect is the
creation of vortex-antivortex pairs while increasing $\chi$ (see Animation 2),
which leads to the formation of new open rings at smaller momentum. Hence,
vortex-antivortex pair creation, their migration, and further recombination
lead to a cyclic formation of nodal lines propagating outwards in the
$p_{x}p_{y}$-plane while changing $\chi$. As expected, when $\chi=2\pi$, we
end up with the same momentum pattern as for $\chi=0$. The formation of nodal
surfaces (visualized as closed rings) suggests that the open annular zones
propagate together with a single vortex. As a result, vortex-antivortex
annihilation leads to the formation of the closed rings in the 2D space. Note
that, in contrast to the results presented in Ref. Starace2015  (perturbative
regime), the probability amplitude of photodetachment in Animation 1
(nonperturbative regime) is not circularly symmetric. Hence, the formation of
vortex patterns and the rotation of the high-probability structures can be
observed with increasing the relative CEP.

###  III.3 Two counterrotating laser pulses

We have seen that single driving pulses and pairs of two corotationg pulses
with the same CEP lead to vortex structures and nodal surfaces in the
probability amplitude of detachment. A change in the CEP (or helicity) has the
effect of a rotation of the probability amplitude by an angle
$\alpha=\sigma\chi$ with respect to the origin of coordinates. In this
Section, we shall analyze the electron photodetachment driven by a pair of
counterrotating pulses ($\sigma_{1}=1$ and $\sigma_{2}=-1$). First we assume a
common CEP, i.e., we set $\chi_{1}=\chi_{2}\equiv\chi$, meaning that the laser
field configurations are of the type $(\chi,+;\chi,-)$. In the Supplemental
Material we show the modulus of the probability amplitude of detachment
(Animation 3) and its phase (Animation 4) with $\chi$ changing from $0$ to
$\pi$. This time, the effects of the CEP are much more complex than a simple
rotation of momentum distribution in the $p_{x}p_{y}$-plane. For $\chi=0$, we
observe the presence of momentum spirals (see, Ref. Circular ) with no
vortices. However, a continuous formation and annihilation of vortex-
antivortex pairs with topological charges $m=1$ and $m=-1$, respectively,
occurs for increasing $\chi$. Finally, at $\chi=\pi$ the original momentum
spiral is recovered, but rotated by an angle $\pi$ in the polarization plane.
Hence, vortical structures are observed for all CEPs, except for the
particular cases $\chi=0$ or $\chi=\pi$.

Figure 3: The same as in Figs. 1 and 2 but for the following laser field
configurations: the left column (Case VII) corresponds to photodetachment
driven by a single laser pulse of the type $(\pi/3,+)$; the middle column
(Case VIII) is obtained from a single pulse in the configuration $(-\pi/3,-)$;
and the right column (Case IX) corresponds to photodetachment driven by a pair
of such pulses in the configuration $(\pi/3,+;-\pi/3,-)$.

In Animations 5 and 6 of the Supplemental Materials we show the modulus and
phase of the probability amplitude of detachment ${\mathcal{A}}({\bm{p}})$,
respectively, for laser fields in the configuration $(\chi_{1},+;\chi,-)$,
with $\chi_{1}=\pi/3$. While the CEP of the first pulse is fixed, the CEP of
the second pulse, $\chi$, varies from $0$ to $2\pi$. At $\chi=0$, both spirals
and vortex-antivortex pairs are observed within the region of momentum
considered here. With increasing $\chi$, the spirals start to rotate and
vortex-antivortex pairs are continuously created or annihilated. It is worth
mentioning that a CEP $\chi=-\pi/3=5\pi/3\approx 5.2$ rad leads to the total
annihilation of votex-antivortex pairs and only the spiral remains. The
disappearance of vortical structures occurs when $\chi=-\chi_{1}$, which is
directly related to the time-reversal properties of the laser field. We note
that a train of pulses in the configuration $(0,+;0,-)$ also leads exclusively
to momentum spirals (no vortices) which are rotated by an angle
$-\chi_{1}=-\pi/3$ with respect to the configuration $(\pi/3,+;-\pi/3,-)$
(see, the right panels of Fig. 4 in Ref. Circular ). The time-reversal
symmetry leading to mutual annihilation of vortex-antivortex pairs, and the
observation of spirals, will be discussed next.

##  IV Vortex-antivortex annihilation and time reversibility

In this Section, we focus on the case when detachment is driven by a pair of
counterrotating circularly-polarized laser pulses with opposite CEPs [i.e.,
the laser fields are in the configurations $(\chi,\sigma)$ and
$(-\chi,-\sigma)$ for single pulses or $(\chi,\sigma;-\chi,-\sigma)$ for
trains of two pulses]. As an example, in Fig. 3 we plot the color mappings of
probability amplitude of ionization ${\cal A}({\bm{p}})$ for: an isolated
pulse with $\sigma=1$ and $\chi=\pi/3$ (left column, Case VII), the same with
$\sigma=-1$ and $\chi=-\pi/3$ (middle column, Case VIII), and finally the
sequence of those pulses (right column, Case IX). For the latter, we obtain
the exactly same pattern irrespectively of the order of pulses in a sequence.
As before, we keep $p_{z}=0$, and we present separately the modulus and the
phase of ${\cal A}({\bm{p}})$ in each case.

When comparing the results for individual pulses (left and middle columns of
Fig. 3), the absolute values of probability amplitudes turn out to be
identical. Specifically, the zeroes of ${\cal A}(\bm{p})$ in both cases are
located at the exactly same momenta. As it follows from the mappings of the
phase of probability amplitudes, those points have a vortex character. More
specifically, in both cases we observe two pairs of vortex points, with $m=1$
in Case VII and $m=-1$ in Case VIII. Those properties can be explained if one
realizes that the respective probability amplitudes, in Cases VII and VIII,
are related by the time-reversal transformation: $t\longmapsto-t$. Namely, it
follows from Eq. (13) that $\bm{F}(t,-\sigma,-\chi)=-\bm{F}(-t,\sigma,\chi)$
and, hence,
$\bm{{\mathcal{E}}}_{-\sigma-\chi}(t)\longmapsto-\bm{{\mathcal{E}}}_{\sigma\chi}(t-\tau_{p})$.
Taking into account that under the time-reversal symmetry the wave functions
are transformed into their complex conjugates [i.e.,
$\psi_{\bm{p}}({\bm{r}},t)\longmapsto\psi_{\bm{p}}^{*}({\bm{r}},t)$ and
$\Phi_{0}({\bm{r}},t)\longmapsto\Phi_{0}^{*}({\bm{r}},t)$ in Eq. (1)], one
concludes that the probability amplitudes of ionization driven by pulses of
opposite helicities and CEPs are related such that

${\cal A}({\bm{p}};-\sigma,-\chi)=[{\cal
A}({\bm{p}};\sigma,\chi)]^{*}\,\mathrm{e}^{-\mathrm{i}\tau_{p}({\bm{p}}^{2}/2m_{\rm
e}-E_{0})}.$ (17)

Here, we have explicitly distinguished between the probability amplitudes
calculated for those pulses. It follows from Eq. (17) that $|{\cal
A}({\bm{p}};-\sigma,-\chi)|=|{\cal A}({\bm{p}};\sigma,\chi)|$, whereas ${\rm
arg}[{\cal A}({\bm{p}};-\sigma,-\chi)]=-{\rm arg}[{\cal
A}({\bm{p}};\sigma,\chi)-\tau_{p}({\bm{p}}^{2}/2m_{\rm e}-E_{0})$. Note that
an extra phase term here does not contribute to the contour integral in Eq.
(11). This explains why the vortex-like points, while located at the same
momenta, will carry opposite topological charges in both cases.

A very distinct spiral pattern is observed when ionization occurs in a
sequence of both pulses (right column in Fig. 3, Case IX). In this case, there
is no vortices. Instead, we observe the nodal lines of the probability
amplitude, with its phase jumping by $\pm\pi$ accross those lines. These
features can be explained by the formula,

$\displaystyle{\cal A}({\bm{p}})$ $\displaystyle=$ $\displaystyle{\cal
A}({\bm{p}};\sigma,\chi)+{\cal A}({\bm{p}};-\sigma,-\chi)$ (18)
$\displaystyle=$ $\displaystyle 2|{\cal
A}({\bm{p}};\sigma,\chi)|\mathrm{e}^{-\frac{\mathrm{i}\tau_{p}}{2}({\bm{p}}^{2}/2m_{\rm
e}-E_{0})}$ $\displaystyle\times$ $\displaystyle\cos[{\rm arg}[{\cal
A}({\bm{p}};\sigma,\chi)]+\frac{\tau_{p}}{2}\Bigl{(}\frac{{\bm{p}}^{2}}{2m_{\rm
e}}-E_{0}\Bigr{)}\Bigr{]},$

which defines the probability amplitude of ionization by a sequence of time-
reversed pulses. It shows that the previously observed vortices and
antivortices, located at the same momenta but with opposite topological
charges, annihilate each other into simple nodes. At the same time, new nodes
appear as indicated by the cosine function in Eq. (18). Since ${\rm arg}[{\cal
A}({\bm{p}};\sigma,\chi)]$ changes by $2\pi$ (or by $-2\pi$) around an old
vortex (antivortex), the cosine function has only two zeroes in their close
vicinity at the fixed ${\bm{p}}^{2}$. With increasing ${\bm{p}}^{2}$, the
nodal curve is formed. The cosine function also contributes to the phase of
the total probability amplitude in Eq. (18). Namely, it changes the sign while
passing across the node, which results in a disconuity of ${\rm arg}[{\cal
A}({\bm{p}};\sigma,\chi)]$ by $\pm\pi$ across the nodal line. This explains
how the momentum spirals in photodetachment are formed.

##  V Conclusions

We have presented an in-depth analysis of the formation of electron vortices
and momentum spirals in photodetachment driven by either an isolated laser
pulse or a pair of pulses with circular polarization. We have seen that, by
changing the CEP and helicity of the individual pulses, it is possible to
control the vortical structures and the appearance of spirals in the momentum
distributions of photoelectrons.

We have shown that, in the presence of a circularly-polarized laser pulse or
two corotating pulses, the probability distribution of photoelectrons exhibits
only vortices or circular nodal lines (no spirals are observed). In addition,
a rotation of such vortex structures by an angle $\alpha$ is consistent with
the introduction of a nonvanishing CEP [configurations $(\chi,\sigma)$ and
$(\chi,\sigma;\chi,\sigma)$ with $\chi\neq 0$] such that $\alpha=\sigma\chi$.
This, in principle, can find applications in ultrashort laser pulse
diagnostics, as the CEP can be directly inferred from the vortex pattern
orientation in the two-dimensional momentum distribution of photoelectrons.
Hence, it is expected that the CEP can be experimentally determined with great
accuracy by a careful measurement of the rotation angle $\alpha$.

When the driving field consists of two corotating pulses with different CEPs
[configuration $(\chi_{1},\sigma;\chi,\sigma)$ where $\chi_{1}$ is kept
unchanged], the formation of vortices and antivortices is observed. Moreover,
an increasing $\chi$ leads to a cyclic outwards propagation of nodal rings in
the probability distribution of photoelectrons.

We have observed that spirals do only appear in the case of counterrotating
laser pulses, which is in agreement with Ref. Circular . More generally, the
driving field configurations of the type $(\chi,+;\chi,-)$ lead to generation
of spirals and vortex-antivortex pairs. The latter are created and annihilated
with increasing $\chi$. However, at $\chi=0$ or $\chi=\pi$ the distributions
of photoelectrons exhibit only spirals, and they are rotated by an angle $\pi$
with respect to one another.

Finally, the laser fields labelled as $(\chi_{1},+;\chi,-)$, with $\chi_{1}$
constant, lead to rotating spirals while increasing $\chi,$ together with
vortex-antivortex generation. Again, only spirals are observed when
$\chi=-\chi_{1}$, which is a direct consequence of the time-reversal symmetry
of the pulse train.

We have shown analytically that spectra of photoelectrons containing only
spirals are formed when the driving field comprises two pulses with time-
reversal symmetry. In such case, the vortex-antivortex patterns superimpose
each other leading to spirals. We have also demonstrated that the phase of
${\cal A}({\bm{p}})$ changes by $\pm\pi$ while crossing the spiral arms.

## Acknowledgements

This work is supported by the National Science Centre (Poland) under Grant
Nos. 2018/31/B/ST2/01251 (F.C.V.) and 2018/30/Q/ST2/00236 (J.Z.K. and K.K.).

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