#  All-optical generation of Abrikosov vortices by the Inverse Faraday Effect

V. D. Plastovets  Universite de Bordeaux, LOMA UMR-CNRS 5798, F-33405 Talence,
France  Institute for Physics of Microstructures, Russian Academy of Sciences,
607680 Nizhny Novgorod, GSP-105, Russia  I. D. Tokman  Institute for Physics
of Microstructures, Russian Academy of Sciences, 607680 Nizhny Novgorod,
GSP-105, Russia  B. Lounis  Universite de Bordeaux, LP2N, F-33405 Talence,
France  Institut d’Optique & CNRS, LP2N, F-33405 Talence, France  A. S.
Mel’nikov  Institute for Physics of Microstructures, Russian Academy of
Sciences, 607680 Nizhny Novgorod, GSP-105, Russia  Lobachevsky State
University of Nizhny Novgorod, 603950 Nizhny Novgorod, Russia  A. I. Buzdin
Universite de Bordeaux, LOMA UMR-CNRS 5798, F-33405 Talence, France

###### Abstract

Within the framework of the time-dependent Ginzburg-Landau theory we show that
circularly polarized THz or far-infrared radiation induces a dc supercurrent
that influences the dynamics of vortex-antivortex pair formation in a
mesoscopic superconductor undergoing rapid thermal quenching. The Lorentz
force arising from the supercurrents promotes vortex-antivortex separation and
allows survival of the vortex polarity defined by the helicity of light. Based
on this idea, we propose a two-stage irradiation scheme that provides a
powerful method for controlled all-optical generation of Abrikosov vortices.

##  I Introduction

Optical manipulation of Abrikosov vortices is an attractive topic due to the
simplicity of its experimental implementation and its perspectives for
superconducting optoelectronic devices. A simple mechanism of the light -
vortex interaction originates from the local heating of superconductor by a
tightly focused laser beam [1, 2, 3, 4]. The induced thermal gradient $\nabla
T$ [5, 6] offers a possibility of fast and precise manipulation of individual
Abrikosov vortices, demonstrated recently in Ref. [7]. Interestingly, a strong
laser pulse by itself is able to generate vortices in superconductors by the
Kibble-Zurek mechanism. First introduced in cosmology [8, 9] and later
generalized to the systems with broken U(1) symmetry [10, 11, 12], this
mechanism can describe the formation of topological defects during rapid
thermal quench after heating the sample with a thermal pulse. Such a scenario
has been observed in a superfluid helium [13, 14, 15], superconducting systems
[16, 17, 18, 19, 20] and cold atoms condensate [21, 22]. In superconductors,
Kibble-Zurek mechanism always results in the creation of vortex-antivortex
pairs which should annihilate during the post-quench dynamics. The presence of
pinning centers in the superconductor prevents the annihilation process making
it possible to experimentally visualize the generated vortex-antivortex pairs
[18, 19]. In order to generate the vortex with a desired polarity at a desired
position, one can use a focused laser pulse, which initiates the local rapid
quench of the superconductor in the presence of a weak magnetic field (see
Ref. [23]). The combined effect of the thermal force $f_{T}\sim-\nabla T$ and
the Lorentz force arising from Meissner currents $f_{L}\sim j_{M}$ is able to
separate vortex-antivortex pairs formed after quench with further transfer of
desired polarity at the position of the laser spot and expelling the opposite
fluxes to the edges of the superconductor.

The aim of this study is to consider an optical mechanism of inducing
supercurrent, which can be used for a separation of vortex-antivortex pairs in
a superconductor. The basic idea is to replace an external magnetic field with
a light radiation carrying a nonzero angular momentum. For instance, transfer
of the orbital angular momentum from Laguerre-Gaussian mode to the trapped
Bose-Einstein condensate can excite persistent currents, which have been
observed experimentally [24, 25]. On the other hand, it is expected that the
electromagnetic wave with a spin angular momentum or just characterized by the
circular polarization of a given helicity $\sigma_{\pm}$ can excite the
circulating dc currents in a superconductor. This problem is very similar to
the so-called Inverse Faraday Effect (IFE) consisting in generation of the
magnetic moment in the sample irradiated by a circularly polarized
electromagnetic wave [26, 27]. In the case of a superconducting system, the
light-induced dynamics of the order parameter comprises a nondissipative
oscillatory contribution, which arises from the imaginary part of the order
parameter relaxation time and creates the currents maintaining a nonzero
magnetic moment [28]. It was theoretically shown [29, 30] that the interplay
of the Kibble-Zurek mechanism and IFE in the case of a superconducting ring
leads to the generation of the circulating current states with the rotation
directions controlled by the external light polarization.

In the present paper we discuss the properties of IFE for the superconductor
beyond the perturbation theory considered in Ref. [28] and demonstrate the
possibility of using this effect for the vortex generation using the numerical
solution of the time-dependent Ginzburg-Landau (TDGL) equations. More
precisely we consider a mesoscopic superconductor homogenously quenched by a
strong laser pulse and exposed to the circularly polarized THz radiation which
provokes the IFE. We show that currents induced by the IFE lock up the
vortices with the polarity dependent on the circular polarization helicity,
thereby realizing the all-optical vortex generation.

Note that the question of the vortex generation due to the direct transfer of
the angular momentum to the superconducting condensate has already been
addressed in Ref. [31]. However, the analysis in Ref. [31] was based on the
linearized TDGL equation which can not properly describe an essentially
nonlinear problem of the vortex generation. The linearized model simply does
not allow selecting the stable solutions. Moreover, in Ref. [31] the purely
real relaxation constant is considered, but in this case, the IFE is merely
absent. In contrast, the present study proposes different mechanism of the
vortex generation, where the IFE plays the key role and the nonlinear TDGL
model was used for the correct description of all stages of nucleation and
evolution of vortices.

##  II Model

The temporal evolution of the complex-valued order parameter $\psi({\bf r},t)$
and the electric scalar potential $\varphi({\bf r},t)$ in a superconductor
square film is described by the modified TDGL equations:

$\displaystyle\tau_{\psi}\big{(}1+i\eta\big{)}\tilde{\partial}_{t}\psi=\big{\\{}a(t)-|\psi|^{2}-\xi^{2}{\bf
D}^{2}\big{\\}}\psi+f({\bf r},t),$ (1)
$\displaystyle\nabla^{2}\varphi+\frac{\hbar}{2e\tau_{\text{GL}}}\text{div}{\bf
j}_{s}=0,$ (2)

which are supplemented by the boundary conditions:

$\displaystyle{\bf D}\cdot{\bf n}\Big{|}_{S}\psi=0,\quad\nabla\varphi\cdot{\bf
n}\Big{|}_{S}=0.$ (3)

Here covariant operators ${\bf D}=(-i\bm{\nabla}-\frac{2\pi}{\Phi_{0}}{\bf
A})$ and $\tilde{\partial}_{t}=(\frac{\partial}{\partial
t}+\frac{2e}{\hbar}i\varphi)$ are introduced; $\psi$ is expressed in terms of
the equilibrium value of the order parameter in the absence of fields
$\psi_{0}$; ${\bf A}$ is a vector potential; ${\bf
j}_{s}=\text{Im}[\psi(\bm{\nabla}+i\frac{2\pi}{\Phi_{0}}{\bf A})\psi^{*}]$ is
a supercurrent density; $a(t)=(T_{c}-T(t))/(T_{c}-T_{0})$ is a temperature
profile created by the homogeneous laser heating. The parameter
$\tau_{\psi}=(\pi\hbar/8k_{B}T_{c})/(1-T_{0}/T_{c})$ is an order parameter
relaxation time at the temperature $T_{0}$. The crucial assumption which
allows one to describe the dynamics of $\psi$ in terms of the equations (1,2)
is valid for gapless superconducting systems. Here we also introduce an
imaginary part of the relaxation time of the order parameter proportional to a
certain parameter $\eta$. This is the key parameter responsible for the IFE
[28], which appears due to the broken electron-hole symmetry [32, 33, 34, 35].
As a length unit we use here the coherence length
$\xi=\xi_{0}/\sqrt{1-T_{0}/T_{c}}$ and the time unit is
$\tau_{\text{GL}}=\tau_{\psi}/u$, where $u$ is the dimensionless
characteristic time scale of the TDGL theory [32, 36]. Thermal fluctuations in
a superconductor can be simulated using a delta-correlated stochastic force
$f({\bf r},t)$ [33, 37, 38], which is normalized as
$\langle{f}(r,t){f}(r^{\prime},t^{\prime})\rangle\approx(4\pi
16\xi^{2}\lambda_{L}^{2}\tau_{\psi}T_{c}/\Phi^{2}_{0})\delta(r-r^{\prime})\delta(t-t^{\prime})$,
where $\lambda_{L}$ is the London penetration depth and $\Phi_{0}$ is a
magnetic flux quantum. The origin of the coordinate system is chosen in the
center of the sample, so that $\\{x,y\\}\in[-L/2,L/2]$.

Figure 1:  (a) Steady-state oscillations of the supercurrent
$j_{sy}(x=L/2,y=0,t)$ in the square superconductor with a side $L=7\xi$ with a
nonzero average $\langle j_{sy}\rangle_{T}$ (dashed lines) under the radiation
of an external field with a frequency $\omega\tau_{\text{GL}}=2$ (period
$T=\pi\tau_{\text{GL}}$) for parameters $u=1$, $\eta=0.3$. (b, c) Spatial
distribution of the averaged supercurrent $\langle j_{sy}(x,y=0)\rangle_{T}$
along a central section of the superconductor for different values of the
electric field penetration length $\ell_{E}$ and frequency $\omega$ for
$\eta=0.3$ ; the amplitude of the external field is different for the
different frequencies with the fixed relation
$E_{\text{ext}}/\omega=A_{\text{ext}}/c=0.75(E_{0}\tau_{\text{GL}})$.

For a rather small sample of the size $L$ much less than the wavelength of the
electromagnetic radiation one can assume the electric field ${\bf
E}_{\text{ext}}({\bf r},t)$ of the electromagnetic wave to be uniform over the
superconductor (see Fig. 3(a)), and the corresponding time-dependent
dimensionless vector potential can be written in the form:

${\bf A}_{\text{ext}}(t)=\text{Re}\big{[}-icE_{\text{ext}}/\omega({\bf
x}_{0}+\sigma_{\pm}i{\bf y}_{0})e^{-i\omega t}\big{]}.$

Here $c$ is the speed of light and the circular polarization is defined as
$\sigma_{\pm}=\pm 1$ for different helicity of the electromagnetic wave.
Dimensional unit for an electric field amplitude is
$E_{0}=\hbar/2e\tau_{\text{GL}}\xi$, for a supercurrent and magnetic moment
per unit area is $j_{0}=M_{0}c=\sigma_{n}E_{0}$, where $\sigma_{n}$ is a
conductivity of a superconductor in a normal state and $c$ is the speed of
light. We consider the case of small lateral sizes $L\ll\lambda_{L}^{2}/d$,
where $d$ is the sample thickness, therefore we can neglect the contribution
to the magnetic field induced by the supercurrents. This condition allows us
to treat the function ${\bf A}_{\text{ext}}(t)$ as an external source in the
Eqs. (1, 2) by using a direct substitution ${\bf A}\equiv{\bf
A}_{\text{ext}}$.

Numerical calculation is implemented as follows: for each moment of time, the
Poisson’s equation (2) for the potential $\varphi$ is solved using the Fourier
method; then using the value of $\varphi(x,y,t)$ we find the order parameter
from Eq. (1) in the next time step $\psi(x,y,t+\Delta t)$ using the semi-
implicit Crank-Nicolson scheme.

##  III Stationary regime of IFE

First, we address the stationary case - without heating and quench dynamics.
An alternating harmonic electric field of a circularly polarized THz radiation
of frequency $\omega$ induces a supercurrent with the density ${\bf
j}_{s}({\bf r},t)=\text{Re}\sum_{n}{\bf j}_{s,n}({\bf r})e^{in\omega t}$
including all harmonics $n\omega$ with an integer $n$. Note here that the
even-$n$ harmonics in the superconducting condensate response appear only for
a nonzero imaginary part of the order parameter relaxation time: $\eta\neq 0$.
An example of the multi-harmonic oscillations of the supercurrent ${\bf
j}_{s}$ is shown in Fig. 1(a). According to the IFE theory for the
superconducting condensate [28], the same parameter $\eta$ is responsible for
a nonzero averaged supercurrent induced by the electromagnetic wave:

$\displaystyle\langle{\bf j}_{s}({\bf
r})\rangle_{T}=\frac{1}{T}\int_{0}^{T}{\bf j}_{s}({\bf r},t)dt,$ (4)

where $T=2\pi/\omega$ is a period of the electric field. The direction of the
current flow is determined by the helicity of the circular polarized wave
$\sigma_{\pm}$. The spatial distribution of the dc current is controlled by
two characteristic length scales: (i) the electric field penetration length
[32, 39] $\ell_{E}=\xi/\sqrt{u}$; (ii) the phenomenological frequency-
dependent length $\ell_{\omega}\sim\xi/\sqrt{\omega}$. While the first length
$\ell_{E}$ is the length of conversion of normal currents to the
superconducting ones, the $\ell_{\omega}$ value can be qualitatively
considered as a localization scale of the order parameter amplitude and phase
[28]. It is worth noting that the applicability of the TDGL model for
externally driven processes is provided by a condition
$\omega\tau_{\text{GL}}<T_{c}/(T_{c}-T_{0})$, where $T_{c}$ is a critical
temperature of a superconductor and $T_{0}$ is a substrate temperature. From
the general constraint on the time variation of the order parameter
$\omega_{\text{G}L}\equiv\tau_{\text{GL}}^{-1}\ll\tau_{\varepsilon}^{-1}$,
where $\tau_{\varepsilon}$ is the inelastic relaxation time of quasiparticles
[32, 39], we obtain an additional condition $\omega\ll
T_{c}/(T_{c}-T_{0})\tau_{\varepsilon}^{-1}$, which is always satisfied in the
vicinity of $T_{c}$.

The distribution of $y$-component of the dc current $\langle{\bf
j}_{s}\rangle_{T}$ induced by the radiation with $\sigma_{+}$ polarization for
different values of $\omega$ and $u$ is presented in Fig. 1. It is
straightforward from the fourfold symmetry of the problem that the
distribution $\langle j_{sx}(x=0,y)\rangle_{T}$ can be obtained by $\pi/2$
rotation. For convenience, the amplitude of the time-dependent vector
potential $A_{\text{ext}}=cE_{\text{ext}}/\omega$ is fixed for all plots:
$A_{\text{ext}}=0.75(cE_{0}\tau_{\text{GL}})$. We observe that the transition
from the adiabatic $\omega\tau_{\text{GL}}\lesssim 1$ to the nonadiabatic
$\omega\tau_{\text{GL}}\gg 1$ regime is accompanied by the strong decrease of
the localization length of the supercurrent $\langle{\bf j}_{s}({\bf
r})\rangle_{T}$ for $u\ll 1$ and rather weak decrease for $u\gg 1$. The
localization length of the supercurrent is determined by the frequency
$\omega$ when the largest length scale in the superconductor is $\ell_{E}\sim
L$ and mainly by the parameter $u$ when this length scale is
$\ell_{\omega}\lesssim L$ (see Fig. 1). Therefore the supercurrent is always
localized at the smallest length scale
$\sim\text{min}\\{\ell_{E},\ell_{\omega}\\}$.

##  IV Magnetic moment

The averaged current $\langle{\bf j}_{s}\rangle_{T}$ produces a dc magnetic
moment per unit area ${\bf M}_{T}=L^{-2}\int[{\bf r}\times\langle{\bf
j}_{s}({\bf r})\rangle_{T}]d{\bf r}$, with a direction determined by the light
polarization. The dependencies of the modulus of the magnetic moment $|{\bf
M}_{T}|\equiv M_{T}$ on different parameters are shown in Fig. 2. Figure 2a
demonstrate that with a small amplitude of the vector potential, the moment
grows quadratically as $M_{T}\sim A_{\text{ext}}^{2}$ and after passing the
maximum value at $A_{\text{ext}}\approx 0.75(cE_{0}\tau_{\text{GL}})$ the
moment begins to decrease due to the suppression of the order parameter
$\psi$, shown in Fig. 2(b). The term $|{\bf A}|^{2}$ in the TDGL equation for
the order parameter can be treated as a negative contribution to the critical
temperature $T_{c}$, therefore superconductivity is destroyed and the moment
decreases to zero at $A_{\text{ext}}\approx 1.0(cE_{0}\tau_{\text{GL}})$,
which corresponds to
$E_{\text{ext}}\approx\omega\sqrt{0.5H_{c2}(T_{0})\Phi_{0}}/c^{2}$ in
dimensional units. At low frequencies and large amplitudes of the external
field, the distribution of the order parameter and, correspondingly, the
supercurrent ${\bf j}_{s}$ become strongly inhomogeneous, which leads to a
shift of the maximum of $M_{T}(E_{\text{ext}})$ at $\omega\tau_{\text{GL}}=1$
relative to the curves plotted for larger frequencies. The moment $M_{T}$ as a
function the frequency $\omega$ has a peak shown in Fig. 2(d). For the fixed
amplitude of the external filed $E_{\text{ext}}$ the moment grows linearly
$M_{T}\sim\omega$ at $\omega\tau_{\text{GL}}\ll 1$ and decreases as
$M_{T}\sim\omega^{-3}$ at $\omega\tau_{\text{GL}}\gtrsim 1$. This behavior is
in a good agreement with the perturbative analytical solution provided in
[28]. Using the optimal parameters one can achieve the most efficient
interaction of the dc current produced by IFE with Abrikosov vortices
generated by the thermal quench, which is discussed below.

Figure 2:  Dependence of the dc magnetic moment $M_{T}$ in the square
superconductor with a side $L=7\xi$ on different parameters (a, c, d). Subplot
(b) shows the dependence of the amplitude of the order parameter
$|\psi|(x=0,y=0)$ on the $E_{\text{ext}}$ and corresponds to (a). Sets of
parameters are chosen as follows: (a, b) $u=1$, $\eta=0.3$; (c) $u=1$,
$A_{\text{ext}}=0.6(cE_{0}\tau_{\text{GL}})$; (d) $\eta=0.3$,
$E_{\text{ext}}=0.06E_{0}$;

##  V Numerical simulation of the vortex generation

In order to implement the optical generation of the Abrikosov vortices with a
desired polarity, we consider a process consisting of two subsequent
illumination stages (see Fig.3(a)): (i) before the time instant $t=0$ the
superconductivity in the film is completely destroyed due to the sample
heating by a strong laser pulse with the beam radius well exceeding the size
$L$; (ii) a rapid thermal quench occurs at the second stage for $t>0$ in the
presence of a weak circularly polarized electromagnetic wave. We assume that
the temperature distribution over the film is uniform and its time evolution
can be described by the phenomenological expression [20]:
$T(t)=T_{0}+(T_{i}-T_{0})e^{-t/\tau_{q}}$, where $T_{i}$ is an initial
temperature of the superconductor and $\tau_{q}$ is a characteristic heat
drain time. Following this model and taking $T_{i}>T_{c}$ we assume the
homogeneous initial conditions $\psi({\bf r},t=0)=0$. After the start of the
quench at $t=0$, superconductivity begins to recover in the presence of the
thermal fluctuations $f({\bf r},t)$ and, according to the Kibble-Zurek
mechanism, the vortex-antivortex pairs appear throughout the sample. Further
dynamics of these vortex pairs at times $t\gg\tau_{q}$ is affected by the
circularly polarized radiation with the frequency $\omega$. Since the induced
current has both dc and ac components, the equation of the motion for the
vortex has a quite complicated form. The alternating electric field produces
local oscillations of the vortex position which are observable at
$\omega\lesssim\tau^{-1}_{\text{GL}}$ and are averaged at larger frequencies.
The averaged part of the current produces a Lorentz force $f_{L}\sim\langle
j\rangle_{T}$ acting on a single vortex. The direction of the force is defined
both by the sign of the polarization $\sigma_{\pm}$ and the vortex winding
number, or polarity: $n_{\text{v}}=\frac{1}{2\pi}\oint_{l}\nabla\arg(\psi)dl,$
where $l$ is the anticlockwise oriented contour around a single vortex. In the
following we use the term ’vortex’ for $n_{\text{v}}=1$ and ’antivortex’ for
$n_{\text{v}}=-1$. In the presence of the imaginary part of the relaxation
time $\eta\neq 0$ there is a Hall component of the vortex motion (see, e.g.,
[32]). Phenomenologically one can describe this effect as

$\sigma n_{\text{v}}\langle{\bf j}_{s}\rangle_{T}\times{\bf
z}_{0}=\alpha_{1}{\bf v}_{L}+n_{\text{v}}\alpha_{2}(\eta){\bf v}_{L}\times{\bf
z}_{0},$

where ${\bf v}_{L}$ is the local vortex velocity, $\alpha_{1}{\bf v}_{L}$ is
the viscous drag force and $\alpha_{2}$ corresponds to the Hall effect. It is
useful to note that $\alpha_{2}(\eta=0)=0$ [34, 32, 35].

Figure 3:  (a) Sketch of the proposed experimental setup: superconductor
placed on a sapphire substrate is heated by an external laser pulse and then
quenched in the presence of a circularly polarized light. (Orange)blue arrows
show (anti-)vortices created after the rapid thermal quench. (c, d) Numerical
simulations of a vortex nucleation and dynamics in the presence of a
circularly polarized light with $\sigma_{+}$ and $\sigma_{-}$ polarization.
(b) Pinning of the vortex by a small square defect after vortex nucleation.
Panels (b-d) show the modulus $|\psi|$ and the phase $\phi$ of the order
parameter for different time instants. White circles with a dot(cross) denote
vortices with a polarity $n_{\text{v}}=1(-1)$. (e, f) Probability of the
creation of the vortices with a certain vorticity the for different
polarizations $\sigma_{+}$ and $\sigma_{-}$. The number of the implementations
(unique simulations) is $N_{\text{imp}}=20$ for each subplot. The set of
parameters used for the calculation (b-f): $L=80\xi$, $u=1$,
$\omega\tau_{\text{GL}}=10$, $\eta=0.3$ , $E_{\text{ext}}=7.5E_{0}$,
$\tau_{q}=1.0\tau_{\text{GL}}$.

The dynamics of the superconducting condensate during the quench process are
presented in Fig. 3(c,d) (see Supplementary movies [40]). At the initial stage
$t\approx 20\tau_{\text{GL}}$ we observe the nucleation of the vortex-
antivortex pairs which are distributed randomly over the superconductor area
since the quench is homogeneous. After that at $t\approx 50\tau_{\text{GL}}$
the part of the pairs annihilates and the remaining (anti)vortices begin to
move in the presence of the induced supercurrent. Since the (anti)vortex-
current interaction depends on the direction of the dc supercurrent, the
current acts selectively expelling the vortices with a certain polarity from
the sample. It is shown in Fig. 3(c,d) that for the $\sigma_{\pm}$
polarization only antivortices/vortices with $n_{\text{v}}=\mp 1$ survive in
the sample at the times $t\sim 700\tau_{\text{GL}}$. In the absence of the
pinning surviving (anti)vortices stay in the superconductor for quite a long
time: they escape from the superconductor only for $t\gg 700\tau_{\text{GL}}$.
The formation of vortex-antivortex pairs is controlled by a stochastic force
and in order to establish the correlation between a given polarization and the
polarity of vortices surviving at large times one needs to consider a
statistical dependence $N_{\text{v}}(\sigma)$, where $N_{\text{v}}=\sum
n_{\text{v}}$ is the sum over all vortices. This dependence is shown in Fig.
3(e,f) and it is clearly seen that $N_{v}(\sigma_{+})\leq 0$ and
$N_{v}(\sigma_{-})\geq 0$ for $N_{\text{imp}}=20$ implementations for each
polarization. Obviously, the distributions for $\sigma_{+}$ and $\sigma_{-}$
should be symmetric in the limit $N_{\text{imp}}\rightarrow\infty$. Note that
among the results of calculations we also observe the implementations with
$N_{v}=0$ when all the vortices and antivortices are either annihilated or
leaving the sample for the times $t\sim 700\tau_{\text{GL}}$. In the case of
linear polarization, which is the sum of two waves with opposite helicities,
we observe only $N_{v}=0$ for the times $t\gtrsim 700\tau_{\text{GL}}$, since
the IFE is absent.

Obviously, the escape of vortices from the superconductor can be additionally
prevented by introduction of pinning centers. In order to strengthen the
influence of pinning we should place these centers near the edges of the
superconductor, where the dc supercurrent $\langle{\bf j}_{s}\rangle_{T}$ is
maximal and plays, thus, a stronger role in separation of vortex-antivortex
pairs (see Fig. 2(b,c)). An example of such a process is shown in Fig. 3(b)
for the case of the square defects with locally suppressed superconducting
critical temperature $T_{c}$ [40]. Numerical simulation shows that the
polarity of the pinned vortices is consistent with the helicity of the light
polarization, according to the statistical dependence $N_{\text{v}}(\sigma)$.
These observations prove the possibility of creation of vortices with a
certain polarity in the absence of the applied magnetic field only by the
circularly polarized electromagnetic wave. Generated vortices contribute to
the dc magnetic moment providing, thus, a possibility to observe the enhanced
IFE.

##  VI Discussion

Reduction in the parameter $\eta$, used in the simulation of the vortex
dynamics above, leads to a decrease in the amplitude of the averaged current
(see Fig. 2(c)), which makes locking of a vortex with a desired polarity less
likely. Therefore, an experimental observation of the light-induced vortex
generation is possible in materials with relatively large imaginary part of
the superconducting relaxation time $\eta\sim T_{c}/E_{F}\lesssim 1$. Since
the parameter $\eta$ is also responsible for the Hall effect and the Hall-
anomaly in the vortex state of type-II superconductors [35, 34, 32], promising
candidates for an experiment can be high-$T_{c}$ compounds, where studies
indicate relatively large Hall effect [41, 42, 43, 44, 45]. Among other
possible candidates with quite a large relation $T_{c}/E_{F}\sim 0.3$ one can
mention the class of actively studied iron selenides [46, 47].

Consider a specific example of a thin YBCO sample with the size $L\sim 0.1-4$
$\mu$m and $d\sim 10$nm. For the substrate temperature $T_{0}\approx
0.98T_{c}$ (with $T_{c}\approx 90$ K) the typical frequency of the circularly
polarized radiation used in the calculation corresponds to the far infrared
range $\omega\sim 10/\tau_{\text{GL}}\sim 50$ THz. Corresponding intensity of
the polarized radiation at which the effect is the most pronounced is
$\mathcal{I}\approx 5\cdot 10^{-2}$ $\mu$W/$\mu$m2. Note, that low temperature
materials Nb or FeSe with $T_{c}\sim 9$ K require a terahertz frequency range
$\omega\sim 1-10$ THz. The control of the quench time $\tau_{q}$ in an
experiment is possible due to good heat removal from the superconductor
ensured by, for example, sapphire substrate film (see Fig. 1) with a typical
thickness $\sim 1$ $\mu$m [23]. It provides large thermal conductivity $\sim
10^{3}$ W/mK [48], which ensures the heat transfer of a surface power density
of the order of $\sim 10$ $\mu$W/$\mu$m2 at the temperature change of the
superconductor $\Delta T\sim 10^{-2}$ K.

Vortex polarity can be detected with the local vortex imaging provided by the
SQUID measurements with sub-micron spatial resolution [49, 50, 51], the
scanning magnetometry with nitrogen-vacancy centers in diamonds [52, 53], the
magneto-optical imaging based on Faraday rotation of light polarization [54,
23, 7] or magnetic force microscopy technique [55, 56]. It is also possible to
use an array of superconducting disks [57] simultaneously irradiated with
polarized radiation, while the average magnetic moment can be measured using a
standard SQUID magnetometer technique [58].

In summary, we theoretically showed that vortex-antivortex pairs created by a
thermal laser pulse in a superconductor can be separated by the dc
supercurrent induced by an external circularly polarized radiation due to IFE.
This leads to effective locking of vortices with a certain polarity inside the
superconductor, determined by the light polarization. The findings of this
research can be applied in experiments on a fast vortex manipulation in
mesoscopic superconductors.

##  VII Acknowledgements

The work has been supported by the ANR OPTOFLUXONICS, the LIGHT S$\&$T
Graduate Program, the Russian Science Foundation (Grant No. 21-72-10161) and
the IdEx of the University of Bordeaux/Grand Research Program ”GPR LIGHT”.

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