# Pattern Formation and Evidence of Quantum Turbulence in Binary Bose-Einstein
Condensates Interacting with a Pair of Laguerre-Gaussian Laser Beams

Madhura Ghosh Dastidar  Department of Physics, Indian Institute of Technology
Kharagpur, India  Subrata Das<EMAIL_ADDRESS>Department of Physics,
Indian Institute of Technology Kharagpur, India  Koushik Mukherjee  Department
of Physics, Indian Institute of Technology Kharagpur, India  Sonjoy Majumder
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology
Kharagpur, India

###### Abstract

We theoretically investigate the out-of-equilibrium dynamics in a binary Bose-
Einstein condensate confined within two-dimensional box potentials. One
species of the condensate interacts with a pair of oppositely wound, but
otherwise identical Laguerre-Gaussian laser pulses, while the other species is
influenced only via the interspecies interaction. Starting from the
Hamiltonian, we derive the equations of motion that accurately delineate the
behavior of the condensates during and after the light-matter interaction.
Depending on the number the helical windings (or the magnitude of topological
charge), the species directly participating in the interaction with lasers is
dynamically segmented into distinct parts which collide together as the pulses
gradually diminish. This collision event generates nonlinear structures in the
related species, coupled with the complementary structures produced in the
other species, due to the interspecies interaction. The long-time dynamics of
the optically perturbed species is found to develop the Kolmogorov-Saffman
scaling law in the incompressible kinetic energy spectrum, a characteristic
feature of the quantum turbulent state. However, the same scaling law is not
definitively exhibited in the other species. This study warrants the usage of
Laguerre-Gaussian beams for future experiments on quantum turbulence in Bose-
Einstein condensates.

Bose-Einstein condensate; Laguerre-Gaussian beam; Quantum Turbulence

##  I Introduction

Light-matter interaction utilizing laser beams is an essential tool to explore
the physics of ultracold atoms Chu (1998); Cohen-Tannoudji (1998); Phillips
(1998); Ritsch  _et al._ (2013). Most research activities in that direction
fall into two main categories. One category involves the quantum gases being
coupled to the electromagnetic mode of a high-quality optical resonator Slama
_et al._ (2007); Ritsch  _et al._ (2013). This picture of light-matter
interaction considers the quantum nature of both light and matter on equal
footing Walther  _et al._ (2006); Caballero-Benitez and Mekhov (2015); Dutra
(2005) thereby facilitating the most promising and controllable implementation
of various phenomena involving quantum many-particle physics Cooper  _et al._
(2019); Mivehvar  _et al._ (2021). In another category, spontaneous emission
of the involved laser light is drastically suppressed, leading to the creation
of an optical dipole potential resulting from the induced Stark shift Grimm
_et al._ (2000); Garraway and Minogin (2000); Frese  _et al._ (2000). This
latter regime, where the underlying electromagnetic field is treated
classically, and its spatial profile plays the most significant role in
determining the form of the dipole potential, has laid the foundation of the
trapping and the manipulation of cold bosonic and fermionic gases Cornell and
Wieman (2002); Ketterle (2002); Bloch  _et al._ (2008); Giorgini  _et al._
(2008) and mesoscopic particles Gordon and Ashkin (1980).

A special type of electromagnetic wave is the Laguerre-Gaussian (LG) laser
beam that possesses a transverse phase cross-section of $e^{il\phi}$, where
$\phi$ is the angular variable, and $l$ is the phase winding number (often
referred to as topological charge) of the beam Allen  _et al._ (1992); Yao and
Padgett (2011a); Padgett (2017a); Dennis  _et al._ (2009). Over the years,
remarkable success has been achieved in the field of generation Ren  _et al._
(2010); Ruffato  _et al._ (2014); Sueda  _et al._ (2004); Beijersbergen  _et
al._ (1993, 1994); Heckenberg  _et al._ (1992) and manipulation Mair  _et al._
(2001); Allen  _et al._ (1999); Yao and Padgett (2011b); Franke-Arnold  _et
al._ (2008); Shen  _et al._ (2019); Padgett (2017b) of LG beams, paving the
way to further studies of light-matter interaction Allen  _et al._ (1996);
Babiker  _et al._ (2002); Lloyd  _et al._ (2012); Power  _et al._ (1995);
Araoka  _et al._ (2005); Bougouffa and Babiker (2020); Mashhadi (2017);
Quinteiro  _et al._ (2017); Mukherjee  _et al._ (2017). In that direction, the
interaction of LG beams with atomic Bose-Einstein condensate(BEC) is well-
studied, both in and beyond the paraxial limit Andersen  _et al._ (2006);
Mondal  _et al._ (2014, 2015); Bhowmik  _et al._ (2016); Kanamoto  _et al._
(2007); Wright  _et al._ (2008); Bhowmik  _et al._ (2018); Bhowmik and
Majumder (2018); Das  _et al._ (2020). The optical trap created by the LG beam
has been used to confine BEC and investigate the quantized vortex states
Tempere  _et al._ (2001). Moreover, such LG modes transfer orbital angular
momentum from the optical field to the BEC, thus creating a single vortex
Marzlin  _et al._ (1997); Simula  _et al._ (2008); Nandi  _et al._ (2004) or a
superposition of vortices in the latter Bhowmik  _et al._ (2016). On-demand
particle transfer between two BECs has also been demonstrated, using a pair of
LG and Gaussian laser beams Mukherjee  _et al._ (2021).

The impacts of LG beam on the BEC have been discussed majorly in the context
of coherent particle transfer between different states and vortex imprinting
Mukherjee  _et al._ (2021). One largely unexplored arena is exploiting the
interference phenomenon using two LG beams Huang  _et al._ (2016); Tao  _et
al._ (2006); Yang  _et al._ (2011); Franke-Arnold  _et al._ (2007) incident
onto the BEC in 2D, and thereby, triggering out-of-equilibrium dynamics in the
latter. The out-of-equilibrium dynamics in BEC are interesting in their own
right. Many intrinsic condensate parameters Chin  _et al._ (2010); Köhler  _et
al._ (2006) and the dimensionality of the trapping potential Görlitz  _et al._
(2001) of the condensate can be remarkably controlled in the experiments Bloch
_et al._ (2008). Consequently, a plethora of density-wave-pattern Nicolin
(2011); Staliunas  _et al._ (2002); Engels  _et al._ (2007); Maity  _et al._
(2020); Zhang  _et al._ (2020) and non-linear structure forming phenomena
Anglin and Zurek (1999); Mukherjee  _et al._ (2020); Kevrekidis  _et al._
(2004); Kevrekidis and Frantzeskakis (2004); Fetter and Svidzinsky (2001)
stemming from dynamical means such as periodic modulation Maity  _et al._
(2020) or quenching a system parameter Law  _et al._ (2010) has been vastly
investigated. Such non-equilibrium dynamics often showcase the spectral
scaling law indicating the emergence of quantum turbulence Allen  _et al._
(2014); Tsatsos  _et al._ (2016); White  _et al._ (2012, 2014); Madeira  _et
al._ (2020) that is phenomenologically very much similar to classical
turbulence Kraichnan and Montgomery (1980). Infact, the convincing evidences
of this similarity have been demonstrated in the experiments Navon  _et al._
(2016, 2019).

Motivated by the studies mentioned above, we attempt to develop a light-matter
interaction mechanism that allows us to dynamically imprint the interference
profiles of a pair of LG beams onto the BEC and subsequently unravel the
emergent density patterns and energy transport phenomenon Horng  _et al._
(2009). Our starting point is a binary Bose-Einstein condensate, made of two
different atomic elements (alias species A and species B), confined in 2D box
potential of the same size. Species A interacts with the pair of LG pulses
applied simultaneously, while species B does not directly participate in this
interaction. Such selective interaction can be experimentally performed by
employing the so-called ”tune-in” and ”tune-out” approaches LeBlanc and
Thywissen (2007). Moreover, the involved LG pulses possess charges of the same
magnitude but an opposite sign, and as a result, their interference produces
azimuthally varied potentials Jesacher  _et al._ (2004)(see the discussion
below). We consider overlapping BECs Ao and Chui (1998) whose interspecies
interaction can be tuned through Feshbach resonance Chin  _et al._ (2010).
This framework allows exploring more non-trivial structures, such as vortex
Kasamatsu  _et al._ (2005) and vortex-bright Law  _et al._ (2010), and the
energy exchange between the species for varying interaction strengths. Note
that our objective in this paper is to identify the density patterns that
evolve dynamically due to the interference of two pulses onto species A. Non-
uniform density profiles stemming from harmonic confinement can make this
identification extremely difficult. Hence, we choose a box potential rendering
the initial density profiles of the BEC uniform. Most importantly, the 2D box
potentials have already been experimentally implemented Ville  _et al._
(2018); Chomaz  _et al._ (2015) and are accessible to the current state-of-
the-art experimental setting.

We consider two separate cases involving different magnitudes of charges of
the LG laser pulses. For single unit charge, species A is separated into two
parts after the pulses are applied, and the segments merge as the pulses
gradually diminish. This merging process creates soliton stripes which
eventually break into vortex-antivortex pairs Ma  _et al._ (2010). When a
finite interspecies interaction is considered (within the miscible regime),
the non-linear structures in species A are coupled by complementary structures
in species B. For example, in species B, a bright soliton is created precisely
at the same position where a vortex is located in species A Ma  _et al._
(2010). Moreover, we assert that a transfer of incompressible kinetic energy
Nore  _et al._ (1997a) from species A to species B is responsible for the
vortex creation in the latter. The corresponding incompressible spectrum
develops $k^{-5/3}$ and $k^{-4}$ power-law scalings at low and high momentum,
respectively Saffman (1971). However, such scaling is not apparent in species
B. The pattern formation and the spectral scaling law compose the main
products of this work. We observe that species A is segmented into four parts
for the double unit charge and gives rise to a similar to the above-described
overall dynamics, but the relevant phenomenology is less prominent than that
of the unit charged one.

The rest of the paper is arranged as follows. Sec. II describes our setup, the
Hamiltonian and the equations governing the dynamics during and after the
incidence of light pulses. Here, we also discuss the observables which we
manipulate to monitor the dynamics of the system. In Sec III, we illustrate
the particle density profiles of the condensates during the dynamics, for both
singly and doubly charged LG beams. We calculate the time evolution of the
different parts of the kinetic energy and their densities in Sec. IV. We
characterize the scaling laws for the incompressible and compressible kinetic-
energy spectra in Sec. V. We conclude the paper in Sec. VI, discussing the
implications and scope of our results. We present the spatial and temporal
profiles of the electric field vectors associated with the LG laser pulses in
Appendix A, where we also highlight the strength of interaction between the
BEC and the optical beams. Finally, Appendix B provides the detailed
derivation of the equations of motion using the Hamiltonian describing our
system.

.

Figure 1: Schematic of the considered setup. (a) Binary BEC consisting of 87Rb
and 133Cs atoms is confined within a quasi-2D box potential. (b) Initial
densities of particles in the condensates for species A ($n_{A}$) and Species
B ($n_{B}$). (c) A pair of LG pulses, having the same temporal width and peak
position, (d) winding numbers with the same magnitude $\absolutevalue{l}$ but
opposite orientations, couple $\ket{1}$ and $\ket{2}$, the two internal states
of the 87Rb atom. The intraspecies interaction of each species is fixed, while
that of interspecies is tunable via Feshbach resonance.

##  II Setup

###  II.1 Hamiltonian and Equations of motion

We consider a binary mixture of BECs comprising an equal number of 87Rb and
133Cs atoms McCarron  _et al._ (2011) and both being confined in a 2D
homogeneous box potential of the same size; see the schematic 1(a). To trigger
the light-matter interaction, only the 87Rb atoms are exposed to a pair of LG
laser pulses possessing winding numbers $l_{1}$ and $l_{2}$, respectively,
whereas the 133Cs atoms are assumed to be uninfluenced by the external light
sources. The laser pulses are considered to be propagating parallel and co-
linear to the $z$-axis, and therefore, both the laser pulses and BEC share the
same transverse ($x$-$y$) plane. In particular, we consider that the applied
laser pulses couple two internal states, namely, $\ket{1}$ and $\ket{2}$, as
shown in Fig. 1(d). The frequencies of the pulses are considered to be far-
detuned from the resonant frequency between the two aforementioned internal
states. Since the detuning is large herein, photon scattering can be neglected
compared to the dipole force. Consequently, a negligible amount of population
resides at $\ket{2}$, which allows us to eliminate the excited state
adiabatically Grimm  _et al._ (2000).

Let $\hat{\Psi}^{\dagger}_{A,k}$[$\hat{\Psi}_{A,k}$] and
$\hat{\Psi}^{\dagger}_{B}$[$\hat{\Psi}_{B}$] be the bosonic
creation[annihilation] operators for the atoms of species A in
$\ket{k}(k=1,2)$ state and those of species B, respectively.

The Hamiltonian of the system reads Zeng  _et al._ (1995)

$\begin{split}H&=\int\mathrm{d}\mathbf{r}\Bigg{[}\hat{\Psi}^{\dagger}_{A,1}(\mathbf{r},t)\hat{h}_{A}\hat{\Psi}_{A,1}(\mathbf{r},t)+\hbar\Delta\hat{\Psi}_{A,2}^{\dagger}(\mathbf{r},t)\hat{\Psi}_{A,2}(\mathbf{r},t)+\hat{\Psi}^{\dagger}_{B}(\mathbf{r},t)\hat{h}_{B}\hat{\Psi}_{B}(\mathbf{r},t)+\Big{(}\hbar\Omega(\mathbf{r},t)e^{\mathrm{i}l_{1}\phi}\hat{\Psi}_{A,2}^{\dagger}(\mathbf{r},t)\hat{\Psi}_{A,1}(\mathbf{r},t)\\\
&+\hbar\Omega(\mathbf{r},t)e^{\mathrm{i}l_{2}\phi}\hat{\Psi}_{A,2}^{\dagger}(\mathbf{r},t)\hat{\Psi}_{A,1}(\mathbf{r},t)+\mathrm{h.c.}\Big{)}+\frac{U_{AA}}{2}\hat{\Psi}_{A,1}^{\dagger}(\mathbf{r},t)\hat{\Psi}_{A,1}^{\dagger}(\mathbf{r},t)\hat{\Psi}_{A,1}(\mathbf{r},t)\hat{\Psi}_{A,1}(\mathbf{r},t)+\frac{U_{BB}}{2}\hat{\Psi}_{B}^{\dagger}(\mathbf{r},t)\hat{\Psi}_{B}^{\dagger}(\mathbf{r},t)\\\
&\times\hat{\Psi}_{B}(\mathbf{r},t)\hat{\Psi}_{B}(\mathbf{r},t)+\frac{U_{AB}}{2}\hat{\Psi}_{B}^{\dagger}(\mathbf{r},t)\hat{\Psi}_{A,1}^{\dagger}(\mathbf{r},t)\hat{\Psi}_{B}(\mathbf{r},t)\hat{\Psi}_{A,1}(\mathbf{r},t)\Bigg{]}.\end{split}$
(1)

The first and second terms in the above Eq. (1) correspond to the kinetic
energy of the atoms in the states $\ket{1}$ and $\ket{2}$, respectively, of
species A, and the third term determines the kinetic energy of species B. The
notation $\hat{h}_{p}$ denotes the kinetic energy operator of the $p$-th
($p=A,B$) species. The forth and fifth terms designate the energies stemming
from the interaction of the species A with the LG laser modes having $+l_{1}$
and $+l_{2}$ units of charge, respectively. The interatomic interaction
energies within the species A and species B are characterized by the sixth and
seventh term, respectively, of Eq. (1). The last term denotes the interaction
energy between an atom of species A and that of species B. The coupling
between the states $\ket{1}$ and $\ket{2}$ through the LG laser mode of the
winding number $l_{\sigma}$ is characterized by the absolute value of the Rabi
frequency $\absolutevalue{\Omega_{l_{\sigma}}(\mathbf{r},t)}$($\sigma=1,2$),
which can be expressed via the relation
$|\Omega_{l_{\sigma}}(\mathbf{r},t)|^{2}=\Omega^{2}_{0}e^{-((t-\tau)^{2}/T^{2})}(x^{2}+y^{2})^{|l_{\sigma}|}e^{-2((x^{2}+y^{2})/w^{2})}$(see
the appendix A). In the previous expression, both laser modes have the same
profile, with the temporal peak position at $\tau$, pulse width $T$ and beam
waist $w$. Moreover, the coefficients of the intra- and interspecies
interaction strengths are given by $U_{pp}=4\pi\hbar^{2}a_{pp}/m_{p}$ and
$U_{pq}=2\pi\hbar^{2}a_{pq}/m_{pq}$, respectively Pethick and Smith (2008).
Here, $a_{pp}$ and $a_{pq}$ are the intra- and interspecies $s$-wave
scattering lengths of atoms, $m_{p}$ is the mass of the atom of the
$p$-th($p=A,B$) species, and $m_{pq}=m_{p}m_{q}/(m_{p}+m_{q})$ is the reduced
mass. Utilizing the commutation relations obeyed by the bosonic field
operators and Heisenberg equations of motion, one can derive the dynamical
equations for $\hat{\Psi}^{\dagger}_{A,1}$, $\hat{\Psi}^{\dagger}_{A,2}$ and
$\hat{\Psi}^{\dagger}_{B}$. Then, setting
$\partial\hat{\Psi}_{A,2}^{\dagger}/\partial t=0$, as negligible number of
atoms of species A populate the state $\ket{2}$ due to large detuning
$\Delta$, and replacing the field operator $\hat{\Psi}_{p}(\mathbf{r},t)$ by
$\Psi_{p}(\mathbf{r},t)$(to this end we will use $\Psi_{A}$ to refer to the
state $\Psi_{A,1}$ just for the notational convenience), we obtain (see the
Appendix B)

$\displaystyle
i\partialderivative{\Psi_{A}(\mathbf{r},t)}{t}=\big{[}-\frac{1}{2}\laplacian_{\perp}+\mathcal{G}_{AA}\absolutevalue{\hat{\Psi}_{A}(\mathbf{r},t)}^{2}+\mathcal{G}_{AB}\absolutevalue{\Psi_{B}(\mathbf{r},t)}^{2}$
$\displaystyle-\mathcal{V}(t)r^{(|l_{1}|+|l_{2}|)}e^{-2\frac{r^{2}}{w^{2}}}\cos^{2}{\frac{(l_{1}-l_{2})\phi}{2}}\big{]}\Psi_{A}(\mathbf{r},t),$
(2)

and

$\displaystyle i\partialderivative{\Psi_{B}(\mathbf{r},t)}{t}=$
$\displaystyle\big{[}-\frac{1}{2}\laplacian_{\perp}+\mathcal{G}_{BB}\absolutevalue{\hat{\Psi}_{B}(\mathbf{r},t)}^{2}+\mathcal{G}_{AB}\absolutevalue{\Psi_{A}(\mathbf{r},t)}^{2}\big{]}$
$\displaystyle.\times\Psi_{B}(\mathbf{r},t).$ (3)

Here, $r^{2}=x^{2}+y^{2}$, $\mathcal{V}(t)=\mathcal{V}_{\rm
max}\exp[-(t-\tau)^{2}/T^{2}]$, with $\mathcal{V}_{\rm
max}=\Omega^{2}_{0}m_{A}l^{2}_{s}/(\hbar\Delta)$ being related to maximum
Rabi-frequency. The intra- and interspecies coupling strengths are
$\mathcal{G}_{pp}=2N\sqrt{2\pi\lambda}a_{pp}/l_{s}$ and
$\mathcal{G}_{AB}=N\sqrt{2\pi\lambda}a_{AB}/l_{s}$, respectively Bandyopadhyay
_et al._ (2017); Mukherjee  _et al._ (2020). Here, $N$ is the particle number
in each species. The length scale $l_{s}$ determines the size of the box
potential, and $\lambda=l_{s}/l_{z}$ is an anisotropy parameter. The system is
tightly confined in the $z$-direction by applying a strong trapping frequency
$\omega_{z}/(2\pi)$ so that $l_{z}=\sqrt{\hbar/(m\omega_{z})}$ is much smaller
than $l_{s}$. Note that the Eqs. (II.1) and (II.1) are the dimensionless forms
of Eq. B and Eq. B(Appendix B) . To cast the equations into dimensionless
forms, we have scaled the spatial coordinates by $l_{s}$, time by
$m_{A}l^{2}_{s}/\hbar$, and the condensate wavefunctions by
$\sqrt{N/l^{3}_{s}}$.

###  II.2 Details of the Numerical Simulations

To investigate the LG modes assisted dynamics of the binary BECs, we
numerically solve Eqs. (II.1) and (II.1) by utilizing the well known split-
time Crank-Nicolson method Muruganandam and Adhikari (2009). Note that the
very first step of the computation is to obtain an initial wavefunction (in
the absence of the laser sources) for each species. To that purpose, we
consider $\mathcal{V}_{\rm max}=0$ in Eq. (II.1) and propagate the Eqs. (II.1)
and (II.1) in imaginary time, until the solution converges to an equilibrium
state. Moreover, the normalization of the species’ wavefunctions is preserved
by applying the following transformation
$\Psi_{p}\rightarrow\frac{\Psi_{p}}{\norm{\Psi_{p}}}$ to the $p$-species
wavefunction at each instant of the imaginary time propagation. Having
equipped with the initial state, we evolve the system in the presence of laser
pulses. For this, we solve Eqs. (II.1) and (II.1) in real time. The numerical
computations are carried out in a square grid of $400\times 400$ grid points
with grid spacing $\Delta x=\Delta y=0.05$, and the time step of integration
$\delta t=10^{-4}$. The intracomponent scattering lengths of the two
components are $a_{AA}=100.04a_{0}$ McCarron  _et al._ (2011) and
$a_{BB}=280a_{0}$, $a_{0}$ being the Bohr radius, and the total number of
atoms in each species is $N=10^{4}$. We choose $l_{s}=2.8\mu$m and
$\lambda=25$.

###  II.3 Observables

In the following, we introduce the observables used in the remainder of our
work to monitor various distinctive features of the system in the non-
equilibrium state.
To visualize the spatially resolved nature of the system during and after the
light-matter interaction, we resort to the particle densities $n_{p}$ of the
$p$-th species, defined as $n_{p}(x,y;t)=\absolutevalue{\Psi_{p}}^{2}(x,y;t)$.
The phase $\phi_{p}(x,y;t)$ of the $p$-th species’ wavefunction provides the
concrete signature of vortex formation during the dynamics. In particular,
$\phi_{p}(x,y;t)$ changes by an amount $\kappa\times 2\pi$ around each vortex
possessing a $\kappa$ unit charge. Moreover, the spatially resolved measure of
the vorticity can be elucidated by vorticity of the $p$-th species Mukherjee
_et al._ (2020),

$\displaystyle\bm{\Omega}_{p}=\bm{\nabla}\times\bm{J}_{p},$ (4)

where
$\bm{J}_{p}=\frac{i}{2}(\Psi^{*}_{p}\bm{\nabla}\Psi_{p}-\Psi_{p}\bm{\nabla}\Psi^{*}_{p})$
is the probability current. Defining the superfluid velocity Pethick and Smith
(2008) $\bm{u}_{p}=\bm{\nabla}\phi_{p}$, and on applying the Madelung
transformation Madelung (1927) $\Psi_{p}=n_{p}e^{i\phi_{p}}$, the Eq. (4) can
be cast into a form,

$\displaystyle\bm{\Omega}_{p}=\nabla\sqrt{n_{p}}\times\bm{u_{p}}+\sum_{j}2\pi
k_{j}\sqrt{n_{p}}\delta(\bf{r}-\bf{r}_{j})\hat{\bf{z}},$ (5)

where the delta function determines the location of the $j$-th vortex with
charge $k_{j}$. Interestingly, according to the Eq. (5), the vortical content
of the $p$ species arises from the two different origins. The first term in
Eq. (5) is related to the generation of vorticity due to a density gradient,
whereas the second term is the usual contribution in a superfluid made by the
quantized vortices (and therefore has no classical analog).
In order to characterize the dynamics from the perspectives of the generated
vortices and sound waves (acoustic waves) in the system, we calculate the
total kinetic energy term of the $p$-th species, given by Pethick and Smith
(2008),

$\displaystyle\frac{1}{2}\absolutevalue{\Psi_{p}}^{2}=\frac{1}{2}\Big{(}n_{p}\bm{u}^{2}_{p}+\absolutevalue{\bm{\nabla}\sqrt{n_{p}}}^{2}\Big{)}.$
(6)

The first and second terms in Eq. (6) represent the kinetic ($E_{kin,p}$) and
quantum pressure ($E_{q,p}$) energy, respectively, of the $p$-th species. The
velocity vector $\sqrt{n}_{p}\bm{u}_{p}$ is decomposed into a solenoidal part
(incompressible) $\bm{u}^{ic}_{p}$ and an irrotational (compressible) part
$\bm{u}^{c}_{p}$, i.e., $\bm{u}_{p}=\bm{u}^{ic}_{p}+\bm{u}^{c}_{p}$ such that
$\bm{\nabla}.\bm{u}^{ic}_{p}=0$ and $\bm{\nabla}\times\bm{u}^{c}_{p}=0$ Nore
_et al._ (1997b); Mithun  _et al._ (2021). This allows us to define the scalar
potential $\Phi_{p}$ and the vector potential $\bm{A}_{p}$ obeying the
relations $\sqrt{n}_{p}\bm{u}_{p}^{c}=\nabla\Phi_{p}$ and
$\sqrt{n}_{p}\bm{u}_{p}^{ic}=\nabla\times\bm{A}_{p}$, respectively. Taking the
divergence on both sides of the last expression, we arrive at the Poisson
equation for the scalar potential,
$\nabla^{2}\Phi_{p}=\nabla.(\sqrt{n_{p}}\bm{u}_{p})$, which is solvable to
determine $\Phi_{p}$ and subsequently, $\bm{A}_{p}$. Once $\Phi_{p}$ and
$\bm{A}_{p}$ are known the components $\bm{u}_{p}^{ic}$ and $\bm{u}_{p}^{c}$
can be easily determined. Therefore, the incompressible and compressible
kinetic energies are defined as

$\displaystyle E^{\rm ic}_{{\rm kin},p}=\frac{1}{2}\int
n_{p}\absolutevalue{(u^{ic}_{p})}^{2}dxdy,$ (7)

and

$\displaystyle E^{\rm c}_{{\rm kin},p}=\frac{1}{2}\int
n_{p}\absolutevalue{(u^{c}_{p})}^{2}dxdy,$ (8)

respectively. In the $k$-space (wave number), the angle-averaged
incompressible and compressible kinetic energy spectrum of the $p$-th species
is represented by Mithun  _et al._ (2021)

$\displaystyle E^{\rm ic[c]}_{{\rm
kin},p}(k)=\frac{k}{2}\sum_{j=x,y}\int_{0}^{2\pi}|F_{j}(\sqrt{n}\bm{u}_{j,p}^{ic[c]})|^{2}d\phi.$
(9)

where $F_{j}(\sqrt{n}\bm{u}_{p}^{ic[c]})$ denotes the Fourier transform of
$\sqrt{n}\bm{u}_{j,p}^{ic[c]}$ corresponding to the $j$-th component of
$\bm{u}_{p}=(u_{x,p},u_{y,p})$. Moreover, to perfrom the integral in Eq. (9),
we numerically add the grid points with $(k^{2}_{x}+k^{2}_{y})^{1/2}=k$, where
$k_{x}$ and $k_{y}$ are the Cartesian components of the wave vector $k$.

##  III Characteristics of the Light-matter interaction

In this section, we describe the dynamics ensued in the binary BEC when one
species is triggered by a pair of LG laser pulses with topological charges
$l_{1}=+l$ and $l_{2}=-l$, and having equal intensities characterized by the
$\mathcal{V}_{max}$, keeping the other species unaffected. In particular, the
system’s evolution is investigated for two specific values of charge, namely,
$\absolutevalue{l}=1$ and $\absolutevalue{l}=2$, and from zero to finite
repulsive interspecies scattering length $a_{AB}$. The dynamics is first
analyzed, in order to gain an overview of the dynamical evolution, by
employing the corresponding density $n_{p}(x,y,t)$, phase $\phi_{p}(x,y,t)$
and vorticity $\Omega_{p}(x,y,t)$ evolution of the participating components.
These observables serve as the foundations upon which we carry out a more
detailed analysis resorting to the different parts of their kinetic energy to
further our understanding of the system.

###  III.1 Singly charged LG laser pulses

We begin discussing our numerical results by analyzing the non-equilibrium
dynamics of the 2D binary bosonic system when both laser pulses possess one
unit charge but with opposite signs, namely, $l_{1}=1$ and $l_{2}=-1$. In
particular, two LG pulses are incident simultaneously along the $z$-direction,
on the species A lying in the $x$-$y$ plane. Moreover, the interaction
parameter $\mathcal{V}_{max}=500$ and the beam waist $w=30l_{s}$ have the same
values for both pulses. First, setting $a_{AB}=0$, we probe the dynamics of
the non-interacting species. Afterwards, we move to the case of finite
interspecies interaction and draw a comparison with the former to identify the
possible alterations.

Figure 2: (Color online) ($a_{1}$)-($a_{6}$) Density, ($b_{1}$)-($b_{6}$)
phase, and ($c_{1}$)-($c_{6}$) vorticity profiles of species A at various time
instants (see legends). The binary BEC consists of $N_{A}=N_{B}=10^{4}$ atoms
with zero interspecies interaction. Each species is confined within a 2D box
potential characterized by the length scale $l_{s}$. The intra- and
interspecies scattering lengths are $a_{AA}=100.04a_{0}$, $a_{BB}=280a_{0}$,
and $a_{AB}=0$, respectively. The dynamics is initiated by exposing species A
to a pair of LG laser pulses with wingdings $l_{1}=1$ and $l_{2}=-1$,
respectively.

We illustrate evolution of the density, phase, and vorticity profiles of
species A at different stages of the dynamics in Fig. 2. The initial state of
the system, at $t=0$ ms, showcases a uniform density distribution for both
species, like the ones shown in Fig. 1(b). Since we considered the case of
non-interacting species, the uniform density distribution of species B
remained unaltered throughout our observation( not shown for the brevity).

Focusing on species A, we can notice the changes in the density profiles
[Figs. 2($a_{1}$)-($a_{6}$)] as the optical potential contributed by the two
laser pulses gradually strengthens, modifying the initial uniform box geometry
of the species. For instance, at $t=129$ms when the pulses are just incident,
the density of species A shows a tendency of separation into two vertical
segments; see Fig. 2($a_{1}$). Particularly, owing to the Gaussian temporal
profiles of the pulses, the segregation of the density of species A occurs
gradually. Therefore, at $t=170$ms, the species A is noticed to be further
segregated into two distinguishable segments [Fig. 2($a_{2}$)]. Note that the
segregation is still not completed because LG pulses do not have the maximum
amplitudes yet. Moving further in time, at $t=326$ms, one can notice the broad
dark region at the center, and species A is largely accumulated on both sides
of the box [Fig. 2($a_{3}$)]. This phenomenon corresponds to the time instant
when the pulses have finally reached the maximum amplitude. Most importantly,
this dark region dividing species A is contributed by the optical potential,
which achieves the maximum values at $\phi=\pi/2$(or $3\pi/2$) when $l_{1}=1$
and $l_{2}=-1$. The density deformation, as per the first term of Eq. (5),
gives rise to vortices within the two segments; see the corresponding
vorticity profile in Fig. 2($c_{3}$). Note that these vortices are not
associated with the typical phase singularities of the wavefunction of species
A, which is evident from the continuous phase structure in the region (on the
sides of the box) where the non-negligible density of species A exists [Fig.
2($a_{3}$) and 2($b_{3}$)]. After that, as the pulses progressively reach
towards the end, the segments of species A gradually approach each other and
finally collide, for example, see Fig. 2($a_{4}$). This collision takes place
because the pulses’ intensity gradually diminishes, and the corresponding
optical potential is too weak to set up a barrier within species A.
Furthermore, as can be seen at $t=550$ ms [Fig. 2($a_{5}$)], species A
restores the original box geometry and forms what resembles a solitonic stripe
Mukherjee  _et al._ (2020) as a result of the collision event. Subsequently,
these stripes break into quantized vortices [see Fig. 2($b_{6}$)] that exhibit
very erratic behavior which persists even in the long time dynamics; see also
the Figs. 2($c_{5}$) and 2($c_{6}$). Finally, let us comment that since the
interspecies interaction is zero, species B is not affected by the light-
induced dynamics occurring in species A and retains its initial state
throughout the time evolution (densities not shown for brevity).

Figure 3: (Color online) Density profiles of ($a_{1}$)-($a_{6}$) species A and
($d_{1}$)-($d_{6}$) species B at various time instants (see legends). Shown
also corresponding phase [vorticity] profiles ($b_{1}$)-($b_{6}$)
[($c_{1}$)-($c_{6}$)] of species A and ($e_{1}$)-($e_{6}$)
[($f_{1}$)-($f_{6}$)] of species B. The binary BEC consists of
$N_{A}=N_{B}=10^{4}$ atoms. Each species is confined within a 2D box potential
characterized by the length scale $l_{s}$. The intra- and interspecies
scattering lengths are $a_{AA}=100.04a_{0}$, $a_{BB}=280a_{0}$, and
$a_{AB}=100a_{0}$, respectively. The dynamics is initiated by exposing species
A to a pair of LG laser pulses with wingdings $l_{1}=1$ and $l_{2}=-1$,
respectively.

Having explicated the behavior of the system in the absence of interspecies
interaction above, it is now imperative to assess the effect of a finite
interspecies interaction. Note that for the value, $a_{AB}=100a_{0}$, the
scattering length that we consider in the subsequent discussion ensures that
there is an overlap between both species in the initial state. The advantage
of such a miscible state is two folds. First, it captures within species B the
possible fingerprints of the light-matter interaction occurring in species A.
Second, the role played by species B in modifying the optical potential
induced dynamics of the other (species A) can also be elucidated in this
setting. Figure 3 presents the snapshots of the density, phase and vorticity
profiles of both species at different stages of the dynamics. In particular,
we have considered the same time instants as those in the case of no
interaction between the components to compare two scenarios. In fact, at
$t=129$ ms the density of species A displays the same tendency of separation
as that for the $a_{AB}=0$ case. Moreover, those segmented parts located on
the sides of the box act as the local potential humps (related to the term
$\mathcal{G}_{AB}\absolutevalue{\Psi_{A}}^{2}$) for the species B. Therefore,
species B mostly occupies the lower density region of species A, a result
which becomes increasingly evident as the pulses progress to reach the maxima
[compare Figs. 3($a_{1}$)-($a_{3}$) to Figs. 3($d_{1}$)-($d_{3}$)].
Furthermore, at $t=326$ms, as can be deduced from the Fig. 3($c_{3}$), vortex-
antivortex pairs emanating from the density distortions [Fig. 3($a_{3}$)]
develop within the segments of species A. Similarly, within species B, since
it is pushed away from both sides towards the central region of the geometry,
severe density distortion occurs, leading to the generation of vortex-
antivortex structures [Figs. 3($d_{3}$) and 3($f_{3}$)]. One can notice the
emergence of the quantized vortices within species B; see the circles in Figs.
3($d_{3}$) and 3($e_{3}$). Finally, when the optical potential is completely
removed, for instance, at $t=550$ ms, both the species are brought back to the
initial box potential, and the solitonic stripes form within the components.
Consequently, these stripes bend and break into quantized vortex-antivortex
structures; see the circles in Figs. 3($a_{6}$), 3($b_{6}$), 3($d_{6}$) and
3($e_{6}$). Furthermore, a closer inspection of the vorticity profiles [Figs.
3($d_{6}$), 3($f_{6}$)] reveals that vortices arising from the density
distortions have lesser magnitudes than those of the quantized vortices[the
circles in Figs. 3($d_{6}$), 3($f_{6}$)]; see also the phase profiles. Indeed,
in the long-time dynamics, both species are filled with vortex-antivortex
structures, which we shall expound on in the later sections while
characterizing the energetics of the system.

###  III.2 Doubly charged LG laser pulses

In the previous section, we have gained intuition regarding the fundamental
features of the light-induced dynamics in species A and its impact on species
B through the interspecies interaction. Next, we discuss the ensuing dynamics
when the applied LG laser pulses carry a charge of two units, more precisely,
$l_{1}=2$ and $l_{2}=-2$. We consider the interaction parameter
$\mathcal{V}_{max}=1500$, the beam waist $w=30l_{\rm s}$ to have the same
values for both the pulses. As in the previous discussion, we first examine
the behavior of the non-interacting species and then compare our findings to
that of the interacting ones.

To visualize the spatially resolved dynamics of species A, for $a_{AB}=0$, we
evoke the density, phase, and vorticity profiles illustrated in Fig. 4. We
find that the system phenomenologically exhibits the same dynamical behavior
compared to the case of singly charged LG beams. Note that the last term in
Eq. II.1, characterizing the optical potential, has maxima at
$\phi=(2n+1)\pi/4$ when $l_{1}=2$ and $l_{2}=-2$. Therefore, along the lines,
$x=y$ and $x=-y$, two potential barriers contributed by the laser pulses are
set up dynamically, and species A is segregated gradually into four distinct
lobes. From Figs. 4($a_{1}$)-($a_{6}$), this is indeed obvious that the
density of species A is sliced into four segments. As the pulses are gradually
removed, those segments move to regain their initial composition. This
ultimately leads to a collision among the segments that generates vortex-
antivortex pairs [Figs. 4($b_{5}$)-($b_{6}$) and Figs. 4($c_{5}$)-($c_{6}$)].
These pairs, being related to the density gradient, not the phase
singularities, are not quantized in nature [see Figs. 4($b_{5}$)-($b_{6}$)].

Figure 4: (Color online)($a_{1}$)-($a_{6}$) Density, ($b_{1}$)-($b_{6}$) phase
and ($c_{1}$)-($c_{6}$) vorticity profiles of species A at various time
instants (see legends). The binary BEC consists of $N_{A}=N_{B}=10^{4}$ atoms
with zero interspecies interaction. Each species is confined within a 2D box
potential characterized by the length scale $l_{s}$. The intra- and
interspecies scattering lengths are $a_{AA}=100.04a_{0}$, $a_{BB}=280a_{0}$,
and $a_{AB}=0$, respectively. The dynamics is initiated by exposing species A
to a pair of LG laser pulses with wingdings $l_{1}=2$ and $l_{2}=-2$,
respectively. Figure 5: (Color online) Density profiles of ($a_{1}$)-($a_{6}$)
species A and ($d_{1}$)-($d_{6}$) species B at various time instants (see
legends). The corresponding phase[vorticity] profiles ($b_{1}$)-($b_{6}$)
[($c_{1}$)-($c_{6}$)] of species A and ($e_{1}$)-($e_{6}$)
[($f_{1}$)-($f_{6}$)] of species B. The binary BEC consists of
$N_{A}=N_{B}=10^{4}$ atoms. Each species is confined within a 2D box potential
characterized by the length scale $l_{s}$. The intra-and interspecies
scattering lengths are $a_{AA}=100.04a_{0}$, $a_{BB}=280a_{0}$, and
$a_{AB}=100a_{0}$, respectively. The dynamics is initiated by exposing species
A to a pair of LG laser pulses with wingdings $l_{1}=2$ and $l_{2}=-2$,
respectively.

Finally, we analyze the light-induced dynamics for the finite interspecies
interaction with the scattering length $a_{AB}=100a_{0}$, particularly
focusing on the same time instants surveyed for the case of non-interacting
ones. As anticipated and observed in the previous case, the species A is
sliced into four distinct parts by the pair of LG pulses. However, one
noticeable difference is that segmented lobes of the species A possess higher
densities when compared to those of the non-interacting case; see Figs.
4($a_{1}$)-($a_{6}$) to Figs. 5($a_{1}$)-($a_{6}$). The species B, expectedly,
is affected so that a star-shaped density distribution complementing that of
species A form within it [Figs. 5($a_{1}$) and 5($d_{1}$)]. Moreover, such
density distribution of species B creates an effective potential(related to
the term $\mathcal{G}_{AB}\absolutevalue{\psi_{B}}^{2}$ in Eq. (II.1)) for
species A. This potential, along with the potential originating from the LG
pulses, pushes species A towards the edges, further enhancing its density.
Note that due to the enhancement of the density, the collision among the
segments is more severe for the $a_{AB}=100a_{0}$ case upon removing the light
pulses. Consequently, more vortex-antivortex pairs, including the quantized
ones [the circles in Figs. 5($d_{6}$), 5($e_{6}$) and 5($f_{6}$) indicating
the phase jump] are generated, enriching the vortical content of the system.

Before closing this section, let us briefly summarize our discussion. A pair
of LG pulses possessing charges of equal magnitude $\absolutevalue{l}$ but the
opposite sign can dynamically separate a homogeneous condensate into
2$\absolutevalue{l}$ segments and subsequently lead them to collide,
generating vortex-antivortex pairs. If a second species is present with finite
interspecies interaction, the densities of the segmented lobes of species A
are modified, which can significantly alter the outcome of the collision
event.

##  IV Kinetic energy contributions

In the last section, we have highlighted the features of the light-assisted
modulation dynamics manifesting within the density and vorticity profiles of
the condensates. The theme of this section is the time evolution of the
different parts of the kinetic energy, to shed further light on the previous
discussion. Recall that the incompressible kinetic energy reveals the presence
of the vortices associated with the phase singularities, whereas the
compressible kinetic energy conveys the presence of the acoustic waves Nore
_et al._ (1997b); Horng  _et al._ (2009); Mithun  _et al._ (2021); Mukherjee
_et al._ (2017).

Figure 6: (Color online) Snapshots of the density distribution of the
compressible, ($a_{1}$)-($a_{4}$) $\mathcal{E}^{c}_{{\rm kin},A}$ and
($b_{1}$)-($b_{4}$) $\mathcal{E}^{c}_{{\rm kin},B}$, and incompressible,
($c_{1}$)-($c_{4}$) $\mathcal{E}^{ic}_{{\rm kin},A}$ and ($d_{1}$)-($d_{4}$)
$\mathcal{E}^{ic}_{{\rm kin},B}$, parts of the kinetic energy of species A
[($a_{1}$)-($a_{4}$), ($b_{1}$)-($b_{4}$)] and species B [($c_{1}$)-($c_{4}$),
($d_{1}$)-($d_{4}$)] at few specific time instants. The dynamics is triggered
by the interaction between species A and a pair of LG laser pulses with
windings $l_{1}=1$ and $l_{2}=-1$, respectively. The intra-and interspecies
scattering lengths are given by $a_{AA}=100.04a_{0}$, $a_{BB}=280a_{0}$, and
$a_{AB}=100a_{0}$, respectively. Figure 7: (Color online) Time evolution of
the compressible, ($a$) $E^{c}_{{\rm kin},A}$ and ($c$) $E^{c}_{{\rm kin},B}$,
and incompressible, ($b$) $E^{\rm ic}_{{\rm kin},A}$ and ($d$) $E^{\rm
ic}_{{\rm kin},B}$, parts of the kinetic energy of the species A [($a$),
($b$)] and species B [($c$), ($d$)] for different interspecies scattering
lengths $a_{AB}$ (see legend) during the evolution. Each species is confined
within a uniform 2D box potential with fixed intraspecies scattering lengths
$a_{AA}=100.04a_{0}$, and $a_{BB}=280a_{0}$, respectively. A pair of LG laser
pulses windings $l_{1}=1$ and $l_{2}=-1$ is made to interact with species A.
Figure 8: (Color online) The time average of the ($a_{1}$)-($a_{2}$)
compressible, $\bar{E}^{\rm c}_{\rm kin,p}$ and ($b_{1}$)-($b_{2}$)
incompressible, $\bar{E}^{\rm ic}_{\rm kin,p}$, parts of the kinetic energy of
the $p-$th species[see legend] as a function of the interspecies scattering
length $a_{AB}$ for the two different cases involving LG laser pulses with
magnitude of the winding being $\absolutevalue{l}=1$ [($a$), ($c$)] and
$\absolutevalue{l}=2$ [($b$), ($d$)], respectively. The dynamics corresponding
to each case is triggered by employing a pair of LG laser pulses, with winding
of equal magnitude $\absolutevalue{l}$ but opposite orientation, to interact
with species A. The insets show $\bar{E}^{\rm ic}_{\rm kin,p}$ as a function
of $a_{AB}$ for ($c_{1}$) $\absolutevalue{l}=1$ and ($c_{2}$)
$\absolutevalue{l}=2$.

In Fig. 6 we display the compressible and incompressible kinetic energy
densities, defined as $\mathcal{E}^{ic[c]}_{kin,p}=n_{p}|u^{ic[c]}_{p}|^{2}$
$(p=A,B)$, of the two interacting condensates for the interspecies scattering
length $a_{AB}=100a_{0}$. The non-equilibrium dynamics stems from the
interaction of species A with a pair of laser pulses having winding $l_{1}=1$
and $l_{2}=-1$. At $t=433$ms [Fig. 6($a_{1}$)], we notice that the
compressible kinetic energy density of the species A, $\mathcal{E}^{c}_{{\rm
kin},A}$, is more pronounced only at the sides of the condensate. Notably,
these acoustic waves spring from the density deformation, which is indeed
evident from the Figs. 3($a_{3}$) and 3$(a_{4})$. However, shortly after the
parts of species A collide, when the pulses vanish, the $\mathcal{E}^{\rm
c}_{{\rm kin},A}$ is distributed all over the geometry indicating the further
generation of the acoustic waves resulting from the collision [Figs
6($a_{3}$)-($a_{4}$)]. Turning to the incompressible kinetic energy density of
the species A, $\mathcal{E}^{\rm ic}_{{\rm kin},A}$, we notice that, at
$t=433$ms [Fig. 6($b_{1}$)], it is majorly present in the location where the
particle density of species A is substantially low [Figs. 3($a_{3}$) and
3$(a_{4})$]. Note that smaller particle density leads to larger healing
length(that determines the size of the vortices) and, therefore, favoring the
initial development of the vortices at the middle of the geometry.
Subsequently, as most of the vortices annihilate each other, the
$\mathcal{E}^{\rm ic}_{{\rm kin},A}$ indicates the presence of only a few
vortices in the species A [Fig. 6($b_{2}$)]. These vortices, furthermore, in
the long-time dynamics annihilate themselves producing the acoustic waves; see
Figs. 6($a_{4}$) and 6($b_{4}$).

Surprisingly, as evident from the Figs. 6($c_{1}$)-($c_{4}$) and
6($d_{1}$)-($d_{4}$), the fabrics of the compressible and the incompressible
energy densities are more intricate within species B when compared to those of
species A, although the former is not directly acted upon by the light pulses.
As already discussed in the Sec. III.1, the species B is compressed towards
the $x=0$ plane, and subsequently, released producing accoustic waves with
larger energy densities; for instance, compare between Figs.
6($c_{1}$)-($c_{4}$) and Figs. 6($a_{1}$)-($a_{4}$). Furthermore, the
incompressible energy density, $\mathcal{E}^{\rm ic}_{{\rm kin},B}$, is
predominantly distributed at the center, as evident from the two bright spots
in Fig. 6($d_{1}$). Afterwards, these bright spots break down into several
parts, indicating the generation of several vortices. The final stage of the
dynamics having significantly reduced $\mathcal{E}^{\rm ic}_{{\rm kin},B}$,
contributed by only few surviving vortices, as shown in Fig. 6($c_{4}$).

It must be pointed out that the magnitude of the incompressible kinetic energy
is quite insignificant compared to that of compressible kinetic energy [Figs
7($a$), 7($b$), 7($c$), 7($d$) ]. Moreover, both kinetic energy contributions
are prevailing in species B. Therefore, one could be curious about how the
interspecies interaction is playing a role in the overall phenomenology.

In order to address the above question, we study the time evolution of the
different parts of the kinetic energy (by integrating over the respective
kinetic energy densities) in Fig. 7 for different interspecies scattering
lengths $a_{AB}$. The time evolution of $E^{\rm ic[c]}_{{\rm kin},A}(t)$
indicates that, for $t<400$ ms and all values of $a_{AB}$, different kinetic
components increase over time. This increase corresponds to the generation of
acoustic waves and short-lived vortices stemming from the separation of the
density of species A(see also the Fig. 3. For the later evolution time, $400$
ms $<t<600$ ms, both the $E^{\rm ic}_{{\rm kin},A}(t)$ and $E^{\rm c}_{{\rm
kin},A}(t)$ show decreasing tendency as the pulses continuously vanish [Figs.
7($a$) and 7($b$)]. Afterwards, for $t>700$ms, $E^{\rm c}_{{\rm kin},A}(t)$
and $E^{\rm ic}_{{\rm kin},A}(t)$ oscillate around a mean value. Note that, at
$t\approx 500$ms, when the pulses are present, $E^{\rm ic}_{{\rm kin},A}(t)$
acquires a maximum value, when $E^{\rm c}_{{\rm kin},A}(t)$ is minimized; see
Figs. 7($a$) and 7($b$). Therefore, the generation of vortices in the early
stage of the dynamics can be attributed to converting acoustic energy into
swirling energy. Similarly, when inspecting the longtime behavior of the
system, one can notice that acoustic waves are continuously generated due to
the annihilation of vortices by mutual collision or collision with the
boundaries of the potential box.

Turning to the species B, for $a_{AB}=0$, both $E^{\rm ic}_{{\rm kin},B}(t)$
and $E^{\rm c}_{{\rm kin},B}(t)$ remain zero throughout the dynamics. However,
for nonzero $a_{AB}$, the density deformation occurring in species A also
affects the density of the species B [Figs. 7($c$) and 7($d$)]. Consequently,
$E^{\rm c}_{{\rm kin},B}(t)$ initially increases gradually, and then saturates
in the long time dynamics. On the other hand, $E^{ic}_{{\rm kin},B}(t)$, at
the early stage of the dynamics, displays a highly fluctuating and erratic
behavior with an overall increasing tendency followed by a gradual decrease
after $t\approx 750$ ms [Figs. 7($c$) and 7($d$)]. Moreover, with increasing
$a_{AB}$ both $E^{\rm ic}_{{\rm kin},B}(t)$ $E^{\rm c}_{{\rm kin},B}(t)$ have
increasing values at any instant of time, see Figs. 7($c$) and Figs. 7($d$).
Furthermore, one can notice that $E^{\rm ic}_{{\rm kin},A}(t)$ decreases for
increasing $a_{AB}$ [Figs. 7($b$)]. This suggests that swirling energy $E^{\rm
ic}_{{\rm kin},B}(t)$ develops at the expense of $E^{\rm ic}_{{\rm
kin},A}(t)$. Infact, for $a_{AB}=100a_{0}$, $E^{\rm ic}_{{\rm kin},B}(t)$
becomes greater than $E^{\rm ic}_{{\rm kin},A}(t)$ in the long time dynamics
[Fig. 7($b$) and Fig. 7($d$)]. Therefore, vortices located in species A not
only shares their energy to the acoustic waves, but also contribute to
creation of vortical structures in species B. On the other, when inspecting
the time evolution of the $E^{\rm c}_{{\rm kin},A}(t)$ as a function of
$a_{AB}$, we notice that the corresponding behavior is not monotonic [Fig.
7($a$)]. In particular, during the presence of the pulses, the increasing
values of $E^{\rm ic}_{{\rm kin},A}(t)$ are mostly independent of $a_{AB}$;
see during $t<350$ ms. However, after $t>700$ms, for $a_{AB}=100a_{0}$, the
values of $E^{\rm ic}_{{\rm kin},A}(t)$ are lower than those corresponding to
$a_{AB}=0$, but higher than those corresponding to $a_{AB}=50a_{0}$.

To unravel this peculiar behavior, we calculate the time average of different
parts of the kinetic energy of the $p-$th species, defined as $\bar{E}^{\rm
ic[c]}_{\rm kin,p}=(1/T_{f})\int_{0}^{T_{f}}E^{\rm ic[c]}_{\rm kin,p}dt$,
where $T_{f}$ is the total evolution time. In particular, such time-average
serves as a measure of the overall content of a particular kinetic energy
contribution developed during the dynamics. Inspecting the Fig. 8($a_{1}$) we
find that $\bar{E}^{\rm ic[c]}_{{\rm kin},A}$ decreases when $a_{AB}$ is
increased from zero. It reaches the minimum value, $\bar{E}^{\rm ic}_{{\rm
kin},A}=0.238$ at $a_{AB}=54a_{0}$, and then gradually increases with further
increase of $a_{AB}$; see the inset Fig. 8($c_{1}$) for a better
visualization. On the contrary, $\bar{E}^{\rm c}_{{\rm kin},B}$ exhibits a
monotonic behavior, namely, an increasing tendency with $a_{AB}$ [Fig.
8($a_{1}$)]. Infact, the value of $\bar{E}^{\rm c}_{{\rm kin},B}$ surpasses
that of $\bar{E}^{\rm c}_{{\rm kin},A}$ when $a_{AB}=26a_{0}$ [Fig.
8($a_{1}$)]. The time average of the incompressible kinetic energies,
$\bar{E}^{\rm ic}_{{\rm kin},A}$ and $\bar{E}^{\rm ic}_{{\rm kin},B}$,
showcase an extremely fluctuating behavior with respect to $a_{AB}$, since
they are the energies associated with the random chaotic motion of the
vortices [Fig. 8($b_{1}$)]. Moreover, $\bar{E}^{\rm ic}_{{\rm kin},A}$
possesses an increasing, while $\bar{E}^{\rm ic}_{{\rm kin},B}$ possesses a
decreasing tendency suggesting vortical energy transfer from species A to
species B. Remarkably, at $a_{AB}\approx 93a_{0}$, the overall vortical
content of species B exceeds that of species A [Fig. 8($b_{1}$)].

For the sake of completeness, we also present in Figs. 8($a_{2}$) and
8($b_{2}$) the time average of the kinetic energy contribution when the pair
of laser beams have winding $l_{1}=2$ and $l_{2}=-2$. Remarkably enough, the
overall phenomenology for $\absolutevalue{l}=2$ is same as that for
$\absolutevalue{l}=1$. The major difference between the two cases is that
$\bar{E}^{\rm ic[c]}_{\rm kin,p}$ is significantly suppressed for the former
when compared to the latter. Moreover, $\bar{E}^{\rm c}_{{\rm kin},B}$ exceeds
$\bar{E}^{\rm c}_{{\rm kin},A}$ at a smaller interspecies scattering length
$a_{AB}=13a_{0}$ [Fig. 8($a_{2}$)]. On the other hand, incompressible kinetic
energy transfer between the two species is much more dramatic and depends on
$a_{AB}$ in a non-trivial manner, since $\bar{E}^{\rm ic}_{{\rm kin},A}$ and
$\bar{E}^{\rm ic}_{{\rm kin},B}$ cross each other multiple times [Fig.
8($b_{2}$)]. However, a more detailed discussion of this behavior lie the
beyond the scope of this manuscript.

##  V Incompressible and compressible kinetic energy spectra

Figure 9: (Color online) Shows ($a$)-($b$) the incompressible $E^{\rm
ic}_{{\rm kin},p}(k)$ and ($c$)-($d$) compressible $E^{c}_{{\rm kin},p}(k)$
kinetic energy spectra(with $p=A,B$) for the species A[($a$), ($c$)] and
species B [($b$), ($d$)] at different instants of time (see the legends) for
the interspecies scattering length $a_{AB}=100a_{0}$. We have also shown the
time evolution of the exponents $\alpha_{i}$ with $i=1,2$, obtained from the
fit of the $E^{\rm ic}_{{\rm kin},p}(k)$ at low and high wave number for ($e$)
species A and ($f$) species B. The vertical lines in ($e$) and ($f$) indicate
the $-5/3$ and $-4$ scalings. The species A and species B comprises of 87Rb
and 133Cs atoms, respectively and are confined within a two dimensional box
geometry. The dynamics is triggered by applying a pair of LG laser pulses to
the 87Rb atoms.

To shed further light on the kinematics of the system, we perform the spectral
analysis of the compressible and incompressible kinetic energy of each
component according to the Eq (9) Nore  _et al._ (1997b); Horng  _et al._
(2009). Recall that the incompressible part being the divergence-free
component of the condensate velocity, is associated with the quantum vortices.
In contrast, the compressible part represents the energy associated with the
acoustic waves.

The log-log plot $E^{\rm ic}_{{\rm kin},p}(k)$ vs $k$ with $p=A,B$ for species
A and species B are shown in Fig. 9(a) and 9(b), respectively, corresponding
to $\absolutevalue{l}=1$ and $a_{AB}=100a_{0}$. In particular, we extract the
power-law exponents $\alpha_{1}$ and $\alpha_{2}$ ($E^{\rm ic}_{{\rm
kin},p}(k)\propto k^{-\alpha_{i}}$ with $i=1,2$) at low and high wave numbers,
respectively, of the incompressible kinetic energy. These exponents provide
information regarding the development of quantum turbulence in the system
Reeves  _et al._ (2012). As time progresses, $E^{\rm ic}_{{\rm kin},p}(k)$
evolves into a stationary form. For species A, we find that the spectrum at
$t=600$ms does not follow any power-law distribution [Fig. 9(a)]. However, in
the long-time dynamics, when $E^{\rm ic}_{{\rm kin},A}(k)$ becomes stationary,
we notice the appearance of two ranges of spectral scaling within the $E^{\rm
ic}_{{\rm kin},A}(k)$ [Fig. 9(a)]. The critical finding here is the
conformation of these scalings to the power-law form. For species A, the first
scaling appears over the range $0.3<k<3$, where the spectrum resembles the
well-known $k^{-5/3}$ Kolmogorov spectra; see Figs. 9(a). In the second one,
$E^{\rm ic}_{{\rm kin},A}(k)$ closely conform to the $k^{-4}$ power law over
the range $12<k<35$ [Fig 9(a)]. Additionally, a range of wavenumber exists,
namely $3\leq k\leq 12$, where $E^{\rm ic}_{{\rm kin},A}(k)$ does not exhibit
the afore-mentioned power-law scaling behavior [Fig. 9(a)]. Furthermore,
$E^{\rm ic}_{{\rm kin},A}(k)$ located at $k=1$ increases, while that located
at $k=10$ decreases with time [Fig. 9(a)], indicating that the incompressible
energy propagates towards larger $k$(small length scale) from small $k$(large
length scale), which is the signature of the so-called inverse energy cascade
Kraichnan (1967, 1975).

In classical 2D turbulence, the inverse energy cascade occurs due to the
conservation of enstrophy(defined as mean-squared-vorticity), which is a
fundamental feature Kraichnan and Montgomery (1980); Sreenivasan (1999).
Besides, in classical 2D turbulence, there exists a direct cascade of
enstrophy from large to small length scale. According to Kraichnan-Batchelor
theory Kraichnan (1967); Batchelor (1969), this dual cascade of energy and
entropy leads to the logarithmically bilinear spectrum in the incompressible
kinetic energy, with the $k^{-5/3}$ dependence for large $k$ and $k^{-3}$
dependence for small $k$, for the 2D turbulence. Since vortex-antivortex pairs
in the quantum fluid can annihilate each other, the enstrophy is not a
conserved quantity here. This non-conservation of enstrophy makes the inverse
energy cascade a debatable issue in 2D quantum turbulence Parker and Adams
(2005); Numasato  _et al._ (2010); White  _et al._ (2012).

Notably, the $k^{-3}$ power-law fails to develop in the incompressible sector
of our system. The value of the scaling, instead, is close to $-4$ [Fig.
9(a)]. Note that observation of the $k^{-4}$ spectrum, also known as the
Saffman $-4$ power law Saffman (1971), has been reported in the previous works
Horng  _et al._ (2009); Reeves  _et al._ (2017). This $k^{-4}$ scaling
behavior is associated with randomly distributed vortex discontinuities, as
shown in Fig. 6$(b_{3})$-$(b_{6})$. Moreover, it has been shown that,
$\alpha_{2}$, the exponent of the incompressible energy spectrum at high
wavenumber can undergo a transition from $\alpha_{2}=-4$ early to
$\alpha_{2}=-3$ at late time dynamics Reeves  _et al._ (2017); Brachet  _et
al._ (1988). We remark that our system does not display such a transition.
Furthermore, in Saffman’s theory Saffman (1971), the $k^{-4}$ spectrum is
known as the ’dissipation spectrum.’ However, we do not include any
dissipative term in GP simulation. Therefore, the question regarding whether
this $k^{-4}$ scaling appears in our system due to the so-called dissipation
has to be addressed cautiously. Recall that our system does not include any
vortical structure at the initial state. These vortices have been created by
the ’chunk’ of acoustic energy generated by the collision of segments of
species A. Moreover, acoustic waves are emitted when a pair of vortex and
antivortex annihilate each other, and the energy of the vortices is dissipated
to the sound waves. Thus, the large amount of compressible energy can be
regarded as both source and sink of the incompressible energy. In this sense,
our work closely follows what reported in Ref. Horng  _et al._ (2009), where
the compressible field, in a similar manner, is held responsible for the
appearance of $k^{-4}$ scaling.

Surprisingly, we notice that $k^{-5/3}$ scaling in $E^{\rm ic}_{{\rm
kin},B}(k)$ is not well developed for species B, and it comes into sight over
the range $0.8<k<3$ [Fig. 9(b)]. It is also observed that, unlike species A,
the $k^{-4}$ scaling for species B is not pronounced at the high wavenumber
region; see the range $0.3<k<10$, for instance in Fig. 9(b). Furthermore, this
indicates that both scaling regimes are shifted towards higher $k$ for species
B when compared to those of species A.

Figures 9(c)-(d) presents the compressible energy spectra, $E^{c}_{{\rm
kin},p}(k)$ (with $p=A,B$) for the same parameter set like that for Figs.
9(a)-(b). As time progresses, a power-law region with $E^{c}_{{\rm
kin},p}(k)\propto k$ develops on the low-$k$ side of the peak located at
$k\approx 21$ [Figs. 9(c)-(d)]. The power-law relation $E^{c}_{{\rm
kin},p}(k)\propto k$ shows excellent agreement with the relation that
expresses the frequencies of Bogoliubov’s elementary excitations at low-wave
number. This resemblance suggests that the $k<20$ region in the stationary
state corresponds to the equilibration of the phonons. The $E^{c}_{{\rm
kin},p}(k)$ spectrum falls steeply after the peak position [Figs. 9(c)-(d)].

Another important aspect of the above-mentioned scaling exponents involves
their development in time for varying interspecies interaction Figs. 9(e)-(f).
For species A, $\alpha_{1}(t)$ and $\alpha_{2}(t)$ reside close to the values
$-5/3$ and $-4$, respectively, from $t=0.8$s onward, irrespective of the
interspecies interaction considered. The corresponding results for species B
are shown in Fig. 9(f). As previously noted, the exponents significantly
deviate from the values $-5/3$ and $-4$ for most of the simulation. This
deviation is particularly dependent on $a_{AB}$. For example, for larger
$a_{AB}$, $\alpha_{1}(t)$ seems to approach the value $-5/3$. However, a
similar conclusion can not be drawn for the $\alpha_{2}(t)$ since it fails to
exhibit any systematic behavior for varying $a_{AB}$.

##  VI Conclusions

We have investigated LG beams-induced dynamics in a homogeneous binary BEC
considering a mass difference between the two species. A pair of LG laser
pulses possessing equal magnitude but oppositely oriented winding are applied
to species A, coupling its two internal states, while the pulses do not
directly impact species B. Notably, the interaction of the pulses with the
laser creates in the latter an azimuthally dependent inhomogeneous potential
that leads an event involving dynamical segregation and collision of the
segments of species A. We have demonstrated the outcome of this event, which
results in the generation of vortices in species A, and their effects on
species B. A particular focus of this work has been to describe the role
played by different windings of the laser light and that of interspecies
interaction. To gain insight into the dynamical response of the condensates,
we have utilized several diagnostics such as density, vorticity, and the time
evolution of compressible and incompressible parts of the kinetic energy in
real and momentum space.

In the first part of our work, we have focused on the density and vortices to
exemplify various events during and after the light-matter interaction. The
density of species A is segregated into two segments for unit-charged LG laser
pulses. After merging two segments, we observe the generation of the soliton
stripes and their ultimate relaxation into vortex-antivortex pairs only within
species A for zero interspecies interaction. Focusing on the non-zero
interspecies interaction, the density disturbances in species A leave their
foot-prints on species B, producing complementary density patterns and,
ultimately, creating vortex-antivortex structures in the latter. In contrast,
the doubly charged pulses initially separate species A into four parts but
produce phenomenologically similar dynamical events that are less dramatic
than their unit-charged counterparts.

As the next step, we examined the dynamics of the system predominantly by
invoking the compressible and incompressible parts of the kinetic energy. This
description provides further supports to the phenomena already depicted
through the density and vorticity profiles. For example, we characterise the
transfer of the compressible and incompressible kinetic energies from species
A to species B, and essentially, by calculating the time average of these
energy components, we have found out the value of interspecies scattering
lengths beyond which swirling and acoustic energies are more dominant in
species B than species A. However, the most novel and highlighting part of
this work addresses different power-law scaling of the kinetic energy spectra.
We have observed $-5/3$ and $-4$ power-law scaling in the low and high
wavenumber regime of the incompressible energy spectra of species A for any
interspecies scattering length in the miscible regime. However, similar
scaling is not properly manifested in species B. Since these scalings
substantiate the emergence of quantum turbulence, we can claim that species A
has been projected onto a turbulent regime due to its interaction with LG
beams. Additionally, both species exhibit equilibration of sound waves in the
long-time dynamics.

There are many promising research directions ushered by the present work to be
pursued in future endeavors. A straightforward one is to consider the
asymmetrical LG beam Kovalev  _et al._ (2016); Das  _et al._ (2020) and
investigate the emergent density pattern and kinematics during light-matter
interaction. More systematic studies of the dynamics involving the finite
temperature effect would be interesting in its own right Proukakis and Jackson
(2008). Another critical prospect would be to employ dipolar BECs Aikawa  _et
al._ (2012), where one could further unravel the role of the long-range
interaction Lahaye  _et al._ (2009). In that direction, the additional long-
range interaction would induce additional terms in the interaction Hamiltonian
that might lead to novel density patterns and scaling law in the different
kinetic energy spectra, not captured by the contact interaction. Furthermore,
these calculations can also be performed beyond the paraxial limit Bhowmik and
Majumder (2018); Bhowmik  _et al._ (2016). Finally, the studies mentioned
above would also be equally fascinating at the beyond-mean field level, where
the correlations among particles become significant Cao  _et al._ (2017).

M.G.D, S.D. and K.M. have contributed equally to this work.

###### Acknowledgements.

K.M. gratefully acknowledges S. I. Mistakidis for insightful discussions. We
also acknowledge useful discussions with K. Seshasayanan.

##  Appendix A Electric field of vortex beams

The expression of electric field vectors involved here can be written as (for
$\sigma=1$ and $2$)

$\begin{split}E_{\sigma}(x,y,t)=&\frac{\hat{\epsilon_{\sigma}}\mathcal{E}_{\sigma}(t)}{\sqrt{|l_{\sigma}|!}}\bigg{(}\frac{2}{w}\bigg{)}^{\frac{|l_{\sigma}|}{2}}(x^{2}+y^{2})^{\frac{|l_{\sigma}|}{2}}\\\
&\times
e^{-\frac{(x^{2}+y^{2})}{w_{\sigma}^{2}}}e^{-i(k_{\sigma}z-\omega_{\sigma}t)},\end{split}$
(10)

where, $r=\sqrt{(x^{2}+y^{2})}$. Moreover,
$\mathcal{E}_{\sigma}(t),\hat{\epsilon}_{\sigma},k_{\sigma},\omega_{\sigma}$
and $w_{\sigma}$ are the corresponding amplitude profile, polarization vector,
wave number, frequency, and beam waist of the $\sigma$-th laser pulse. The
spatial amplitude profile arises due to LG beams propagating along $z$
direction, and the $\sigma$-th pulse is having topological charge of
$l_{\sigma}$. If the temporal amplitude profiles of the laser pulses having
the same form are considered, then,

$\displaystyle\mathcal{E}_{\sigma}(t)=\mathcal{E}_{\rm
max}e^{-(\frac{t-\tau_{\sigma}}{T})^{2}}.$ (11)

We consider both LG laser pulses to have the same maximum intensity
$\varepsilon_{max}$, width $w_{1}=w_{2}=w$, and centered temporally at
$\tau_{1}=\tau_{2}=\tau$ and having pulse duration $T$.

Now, we can derive the Rabi frequency for the transition involving the above
electric field as Mukherjee  _et al._ (2021),

$|\Omega(\mathbf{r},t)|^{2}=\bigg{(}\frac{\mathcal{E}_{\rm
max}d}{\hbar}\bigg{)}^{2}e^{-\frac{(t-\tau)^{2}}{T^{2}}}r^{2|l_{\sigma}|}e^{-2\frac{(x^{2}+y^{2})}{w^{2}}},$
(12)

where $d$ is the transition dipole moment.

##  Appendix B Derivation of equations of motion

We consider the condensate prepared with two different atomic elements. One
species, named species A, has two internal states $\ket{1}$ and $\ket{2}$.
Initially, all the atoms of species A are in the state $\ket{1}$. The
intermediate energy level is denoted by $\ket{2}$ to which atoms of species A
are transferred when the LG pulses are simultaneously applied to the
condensate. The internal states of the atoms of species B is irrelevant here
since they does not take part directly in light-matter interaction. We use
bosonic field operators to represent the atoms in a particular state. Now, the
Hamiltonian (refer to Eq. (1)) for such a system, in the rotating wave
approximation, is shown in the main text.

The creation and annihilation field operators for species A and B commute with
each other. Commutation relations obeyed by the bosonic field operators for
species A are:

$\displaystyle\quantity[\hat{\Psi}_{A,j}(\mathbf{r}),\hat{\Psi}_{A,k}^{\dagger}(\mathbf{r^{\prime}})]=\delta\quantity(\mathbf{r}-\mathbf{r^{\prime}})\delta_{jk},$
(13)
$\displaystyle\quantity[\hat{\Psi}_{A,j}(\mathbf{r}),\hat{\Psi}_{A,k}(\mathbf{r^{\prime}})]=0,$
(14)
$\displaystyle\quantity[\hat{\Psi}_{A,j}^{\dagger}(\mathbf{r}),\hat{\Psi}_{A,k}^{\dagger}(\mathbf{r^{\prime}})]=0.$
(15)

For the above-defined field operators [$\hat{\Psi}_{A,k}($k=1,2$)$ and
$\hat{\Psi}_{B})$], we may define the Heisenberg equations of motion as

$\mathrm{i}\hbar\partialderivative{\hat{\Psi}_{A,k}(\mathbf{r},t)}{t}=[\hat{\Psi}_{A,k}(\mathbf{r},t),\hat{H}],$
(16)
$\mathrm{i}\hbar\partialderivative{\hat{\Psi}_{B}(\mathbf{r},t)}{t}=[\hat{\Psi}_{B}(\mathbf{r},t),\hat{H}].$
(17)

Now, using the commutation relations shown above, we find the explicit forms
of the Heisenberg equations of motion for $\hat{\Psi}_{A,1}$,
$\hat{\Psi}_{A,2}$ and $\hat{\Psi}_{B}$, respectively, under the action of the
Hamiltonian given by Eq. 1 as,

$\displaystyle\mathrm{i}\hbar\partialderivative{\hat{\Psi}_{A,1}(\mathbf{r},t)}{t}$
$\displaystyle=\hat{h}_{A}\hat{\Psi}_{A,1}(\mathbf{r},t)+U_{AA}\hat{\Psi}^{\dagger}_{A,1}(\mathbf{r},t)\hat{\Psi}_{A,1}(\mathbf{r},t)$
$\displaystyle\times\hat{\Psi}_{A,1}(\mathbf{r},t)$
$\displaystyle+U_{AB}\hat{\Psi}^{\dagger}_{B}(\mathbf{r},t)\hat{\Psi}_{B}(\mathbf{r},t)\hat{\Psi}_{A,1}(\mathbf{r},t)$
$\displaystyle+\hbar\Omega(\mathbf{r},t)\hat{\Psi}_{A,2}(\mathbf{r},t)(e^{-il_{1}\phi}+e^{-il_{2}\phi}),$
(18)

$\displaystyle\mathrm{i}\hbar\partialderivative{\hat{\Psi}_{A,2}(\mathbf{r},t)}{t}=$
$\displaystyle\hbar\Delta\hat{\Psi}_{A,2}(\mathbf{r},t)+\hbar\Omega(\mathbf{r},t)\hat{\Psi}_{A,1}(\mathbf{r},t)$
$\displaystyle\times(e^{il_{1}\phi}+e^{il_{2}\phi}),$ (19)

and

$\displaystyle\mathrm{i}\hbar\partialderivative{\hat{\Psi}_{B}(\mathbf{r},t)}{t}=\hat{h}_{B}\hat{\Psi}_{B}(\mathbf{r},t)+U_{BB}\hat{\Psi}_{B}(\mathbf{r},t)\hat{\Psi}^{\dagger}_{B}(\mathbf{r},t)$
$\displaystyle\times\hat{\Psi}_{B}(\mathbf{r},t)+U_{AB}\hat{\Psi}^{\dagger}_{A,1}(\mathbf{r},t)\hat{\Psi}_{A,1}(\mathbf{r},t)\hat{\Psi}_{B}(\mathbf{r},t).$
(20)

Equation (B) gives us the governing equation for species B. We adiabatically
eliminate the field operator $\hat{\Psi}_{A,2}(\mathbf{r},t)$ (as it is the
intermediate state in our case), by setting,

$\displaystyle\mathrm{i}\hbar\partialderivative{\hat{\Psi}_{A,2}(\mathbf{r},t)}{t}=0,$
(21)

and get,

$\displaystyle\hat{\Psi}_{A,2}(\mathbf{r},t)=$
$\displaystyle-\frac{\Omega(\mathbf{r},t)}{\Delta}(e^{il_{1}\phi}+e^{il_{2}\phi})\hat{\Psi}_{A,1}(\mathbf{r},t).$
(22)

On substituting the above in (B) and using trigonometric identities, we arrive
at the result:

$\displaystyle\mathrm{i}\hbar\partialderivative{\hat{\Psi}_{A,1}(\mathbf{r},t)}{t}=$
$\displaystyle\hat{h}_{A}\hat{\Psi}_{A,1}(\mathbf{r},t)+U_{AA}\hat{\Psi}^{\dagger}_{A,1}(\mathbf{r},t)\hat{\Psi}_{A,1}(\mathbf{r},t)$
$\displaystyle\times\hat{\Psi}_{A,1}(\mathbf{r},t)$
$\displaystyle+U_{AB}\hat{\Psi}^{\dagger}_{B}(\mathbf{r},t)\hat{\Psi}_{B}(\mathbf{r},t)\hat{\Psi}_{A,1}(\mathbf{r},t)$
$\displaystyle-\frac{4\hbar}{\Delta}\absolutevalue{\Omega(\mathbf{r},t)}^{2}\cos^{2}{\frac{(l_{1}-l_{2})\phi}{2}}\hat{\Psi}_{A,1}(\mathbf{r},t).$
(23)

Therefore, the two Eqs. (B) and (B) govern the time evolution of the bosonic
field operators for species A and B, respectively. At $t=0$, if we assume the
low-energy $s$-wave scattering, neglect quantum fluctuations and suppress
thermal fluctuations, we can replace the field operators by their
corresponding wave functions. Thus, the Eqs. (B) and (B) become:

$\displaystyle\mathrm{i}\hbar\partialderivative{\Psi_{A,1}(\mathbf{r},t)}{t}=$
$\displaystyle\big{[}-\frac{\hbar^{2}}{2m_{A}}\laplacian_{\perp}+V(\mathbf{r})\big{]}\Psi_{A,1}(\mathbf{r},t)$
$\displaystyle+U_{AA}\absolutevalue{\Psi_{A,1}(\mathbf{r},t)}^{2}\Psi_{A,1}(\mathbf{r},t)$
$\displaystyle+U_{AB}\absolutevalue{\Psi_{B}(\mathbf{r},t)}^{2}\Psi_{A,1}(\mathbf{r},t)$
$\displaystyle-\frac{4\hbar}{\Delta}\absolutevalue{\Omega(\mathbf{r},t)}^{2}\cos^{2}{\frac{(l_{1}-l_{2})\phi}{2}}\Psi_{A,1}(\mathbf{r},t),$
(24)

and,

$\displaystyle\mathrm{i}\hbar\partialderivative{\Psi_{B}(\mathbf{r},t)}{t}=$
$\displaystyle\big{[}-\frac{\hbar^{2}}{2m_{B}}\laplacian_{\perp}+V(\mathbf{r})\big{]}\Psi_{B}(\mathbf{r},t)$
$\displaystyle+U_{BB}\absolutevalue{\Psi_{B}(\mathbf{r},t)}^{2}\Psi_{B}(\mathbf{r},t)$
$\displaystyle+U_{AB}\absolutevalue{\Psi_{A,1}(\mathbf{r},t)}^{2}\Psi_{B}(\mathbf{r},t),$
(25)

where $m_{A}$ and $m_{B}$ are the atomic masses of species A and B,
respectively, and $V(\mathbf{r})$ is the trapping potential of the BEC. For
our problem, assuming the trapping frequency to be very weak, we take
$V(\mathbf{r})\approx 0$, i.e., the condensate is present in a homogeneous
potential box in the x-y plane. In particular, we assume that z-direction is
tightly confined compared to the x and y directions. As a result the dynamics
along the $z$-direction is frozen and the effective dynamics of the
condensates can be described in the 2D x-y plane by integrating the
wavefunction along the z-directions; see Ref. Mukherjee  _et al._ (2020);
Bandyopadhyay  _et al._ (2017) for the relevant methodology. Thereafter,
substituting the expression for the Rabi frequency as given by Eq. (12) and
non-dimensionalizing the above Eqs. (B) and (B), we can arrive at the coupled
Gross-Pitaevskii Eqs. (II.1) and (II.1), respectively.

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