# Signature of reentrant localization in collisional inhomogeneous spin-orbit
coupled condensates

Swarup K. Sarkar  Department of Physics, Indian Institute of Technology
Guwahati, Guwahati 781039, Assam, India  Rajamanickam Ravisankar  Department
of Physics, Zhejiang Normal University, Jinhua 321004, PR China  Zhejiang
Institute of Photoelectronics $\&$ Zhejiang Institute for Advanced Light
Source, Zhejiang Normal University, Jinhua, Zhejiang 321004, China  Tapan
Mishra  School of Physical Sciences, National Institute of Science Education
and Research, Jatni 752050, India  Homi Bhabha National Institute, Training
School Complex, Anushaktinagar, Mumbai 400094, India  Paulsamy Muruganandam
Department of Physics, Bharathidasan University, Tiruchirappalli 620024,
Tamilnadu, India  Pankaj Mishra  Department of Physics, Indian Institute of
Technology Guwahati, Guwahati 781039, Assam, India

###### Abstract

We study the localization transition in a spin-orbit (SO) coupled binary Bose-
Einstein condensates (BECs) with collisional inhomogeneous interaction trapped
in a one-dimensional quasiperiodic potential. Our numerical analysis shows
that the competition between the quasiperiodic disorder and inhomogeneous
interaction leads to a reentrant localization transition as the interaction
strength is tuned from attractive to repulsive in nature. Further, we analyse
the combined effect of the SO and Rabi coupling strengths on the localization
transition for different interaction strengths and obtain signatures of
reentrant localization transition as function of SO coupling in the regime of
weak interactions. We complement our numerical observation with the analytical
model using the variational approach. At the end we show how the reentrant
localization is manifested in the quench dynamics of the condensate. Our study
provides an indirect approach to achieve localization transition without
tuning the quasiperiodic potential strength, rather by tuning the
inhomogeneous interaction.

##  I Introduction

After the path breaking prediction of exponential localization of electronic
wavefunction in random media by Anderson, the study of localization transition
has attracted a great deal of attention in condensed matter [1, 2, 3, 4, 5].
Recent developments in the experimental front, such localization transitions
have been observed in various artificial systems such as electromagnetic waves
[6, 7, 8, 9, 10], microwaves [11, 12, 13, 14], acoustic waves [15], quantum
matter waves [16, 17], and in the non-interacting [18] and interacting Bose-
Einstein condensates(BECs) [19].

Among the various platforms, enormous progress has been made in studying
localization transition in systems of BECs in random and quasiperiodic
lattices due to the flexibility in controlling the geometry and interaction.
In this context, more complex systems have been considered recently, such as
the binary BECs [20, 21, 22], spin-orbit (SO) coupled spinor BECs, etc. [23,
24, 25, 26, 27, 28]. Theoretically, it has been demonstrated that the SO
coupling influences the appearance of localization, whereas the Rabi coupling
slightly promotes the delocalization [29]. Akin to SO coupling, interactions
among atoms play a vital role in dictating the localization-delocalization
phenomena in the condensate. For instance, several numerical studies show that
the increase of repulsive interaction induces the localization to
delocalization transitions for the condensate trapped in the disordered
potential [30, 31, 32, 33, 34, 35]. Subsequent studies have reported the
symmetry breaking of a localized binary condensate in the presence of inter-
species and intra-species atomic interactions [34, 36]. Santos and Cardoso
demonstrated that even with only one component subjected to the quasiperiodic
potential, the interspecies interaction induces the localization in the other
component [27].

On the other hand, localization transition has been achieved by spatial
variation of atomic scattering length leading to collisionally inhomogeneous
environment [37, 38]. A great deal of effort has been made in exploring areas
like adiabatic compression of matter waves [39, 40, 41, 42], dynamical
trapping of matter wave solutions [43], soliton oscillations [44, 45], BECs in
nonlinear optical lattices [46], and localization in the scalar [35, 47] or
binary BECs [48]. The investigations on localization in the presence of
collisionally inhomogeneous environment reveal the effect of spatially
dependent interaction towards localization as a result of the genesis of an
effective potential whereas, without the inhomogeneity, the atomic interaction
leads to delocalization of the condensate [48, 35, 47].

Apart from the standard delocalization-localization transition, a new class of
localization transition known as the reentrant localization transition in
quasiperiodic lattices has attracted a great deal of attention in recent
years. While in the conventional localization transition, the states after the
localization transition remain localized with an increase in the disorder in
the system, in the case of reentrant localization, the localized system
delocalizes again before getting localized for the second time as a function
of the quasiperiodic potential strength [49, 50, 51]. Such intriguing
reentrant localization has been studied extensively in recent years [52],
including in non-Hermitian lattices [53], leading to their experimental
observations [54, 55, 56, 57]. In the continuum systems, however, this kind of
reentrant localization has not been analyzed with the same rigour as those in
the strongly correlated system. Although a recent study has revealed the
reentrant of localization upon variation of SO coupling in a non-interacting
condensate trapped in a disordered quasiperiodic potential [25], a complete
understanding of the reentrant localization for weakly interacting BECs is
lacking.

In this work, we present an extensive numerical study that explores the impact
of inhomogeneous interactions in the localization of SO coupled BECs subjected
to a bichromatic quasiperiodic optical lattice potential. We consider the
spatial dependent interactions induced by the external optical fields to have
the same form as the bichromatic potential with the parameters depending on
the intensity of the lasers [58, 59]. This setting leads to the localization
of the condensate with the variation of interaction strength. However, by
implementing an additional phase shift $\pi/2$ between the inhomogeneous
interaction and trapping potential, we obtain the signature of reentrant
localization transition, where we find an intermediate delocalized phase
sandwiched between two localized phases with the variation of the strength of
the inhomogeneity of the nonlinearity. Furthermore, we complement this
reentrant behaviour in the localization through the Gaussian variational
approach. At last, we obtain the signatures of such localization properties
from the dynamical behaviour of the condensate.

The paper is structured as follows: Section II introduces a mean-field
theoretical model to examine SO coupled BECs in a quasiperiodic optical
lattice potential with spatially varying interactions. Section III.1 presents
the numerical results, starting from condensate densities depending on
inhomogeneous interactions and various coupling parameters. Here, we
characterize the localized and delocalized states using the form factor and
condensate width. Section III.2 explores the role of inhomogeneous
interactions in demonstrating the reentrant behaviour of condensate
localization. A variational approach is also employed to understand the
behaviour of the localization using the effective potential. Section III.4
investigates the dynamics of localized and delocalized condensates under
velocity perturbations and trap quenching. Finally, Section IV provides a
summary and conclusion. Appendices A and B provide detailed equations of
motion and discussions on variational approaches and Lagrangian details,
respectively, along with numerical and variational comparison and energy
analysis.

##  II Governing dynamical Mean-field equation and simulation details

We consider a pseudo spin-$1/2$ quasi-1D condensate with strong transverse
confinement, modelled using the coupled Gross-Pitaevskii equations (GPEs) as,

$\displaystyle\mathrm{i}\frac{\partial\psi_{1}}{\partial t}=$
$\displaystyle\Bigg{[}-\frac{1}{2}\frac{\partial^{2}}{\partial
x^{2}}+\left(g_{11}|\psi_{1}|^{2}+g_{12}|\psi_{2}|^{2}\right)+V(x)\Bigg{]}\psi_{1}$
$\displaystyle-\mathrm{i}k_{L}\frac{\partial\psi_{1}}{\partial
x}+\Omega\psi_{2},$ (1a)
$\displaystyle\mathrm{i}\frac{\partial\psi_{2}}{\partial t}=$
$\displaystyle\Bigg{[}-\frac{1}{2}\frac{\partial^{2}}{\partial
x^{2}}+\left(g_{22}|\psi_{2}|^{2}+g_{21}|\psi_{1}|^{2}\right)+V(x)\Bigg{]}\psi_{2}$
$\displaystyle+\mathrm{i}k_{L}\frac{\partial\psi_{2}}{\partial
x}+\Omega\psi_{1},$ (1b)

where $\psi_{1}$ and $\psi_{2}$ represent the wavefunctions for pseudo spin-up
and spin-down components, respectively. Here, $g_{11}$ and $g_{22}$ are the
intra-species nonlinearities for spin-up and spin-down components, and
$g_{12}$ represents the interspecies interaction strength. $k_{L}$ is the SO
coupling strength and $\Omega$ is the Rabi coupling strength. $V(x)$
represents the trapping potential. For our studies, we consider
$g_{11}=g_{22}=g$. The wave functions follow the normalization condition as

$\displaystyle\int_{-\infty}^{\infty}\left(|\psi_{1}|^{2}+|\psi_{2}|^{2}\right)dx=2.$
(2)

To obtain the non-dimensionalized Eq. 1, we consider the transverse harmonic
oscillator length $a_{\perp}=\sqrt{\hbar/(m\omega_{\perp})}$ as a
characteristic length scale with $\omega_{\perp}$ as the transverse harmonic
trapping frequency, $\omega^{-1}$ as the timescale and $\hbar\omega_{\perp}$
as the characteristic energy scale. The interaction parameters can be defined
in terms of $g=2Na_{11}/a_{\perp}$, and $g_{12}=2Na_{12}/a_{\perp}$, where
$a_{11}$ and $a_{12}$ represent the intra-component and inter-component
scattering lengths, respectively. The SO coupling is re-scaled as $k_{L}\equiv
k_{L}^{\prime}a_{\perp}$ and Rabi coupling as
$\Omega\equiv\Omega^{\prime}/(2\omega_{\perp})$. The wavefunction is rescaled
as $\psi_{1,2}(x,t)\equiv\psi^{\prime}_{1,2}(x,t)\sqrt{a_{\perp}}$. Here
variables with prime represent the dimensional quantities.

To analyze the characteristics and dynamics of the localization of the
condensate, we consider a quasiperiodic trapping potential $V(x)$ of the form
[25],

$\displaystyle V(x)=-V_{1}\cos(k_{1}x)-V_{2}\cos(k_{2}x),$ (3)

where $V_{1}$ and $V_{2}$ are the primary and secondary optical lattice
amplitudes, respectively. For all calculations, we consider the primary and
secondary lattice strengths as $V_{1}=1$ and $V_{2}=0.1$, respectively, with
the angular wavenumber ratio as the golden ratio, i.e.,
$k_{2}/k_{1}=(\sqrt{5}+1)/2$. With this angular wave number, the wavelength of
the laser beams will have the values as $\lambda_{1}\approx 3.1415a_{\perp}$
and $\lambda_{2}\approx 1.9416a_{\perp}$, respectively, where $a_{\perp}$
denotes the characteristic length scale approximately equal to $1.17\mu$m.
Note that in the experiment of Roati et al. [19], the ratio $k_{2}/k_{1}$ is
$1.1971$.

To make the parameters considered in numerical simulations experimentally
feasible, we consider the two hyperfine states of 87Rb atom as pseudo spin-up
state $\ket{1}\equiv\ket{F=1,m_{F}=0}$, and pseudo spin-down state
$\ket{2}\equiv\ket{F=1,m_{F}=-1}$. Following the experiment of Lin et al.
[23], we consider $N\sim 1.8\times 10^{5}$ number of atoms in a harmonic trap
strongly confined along the perpendicular direction with frequency
$\omega_{\perp}\approx 2\pi\times 83.66$ Hz, and along the axial direction the
trapping frequency is chosen to be $\omega_{x}\approx 2\pi\times 7.07$ Hz. In
the experiment, the wavelength of the Raman lasers, $\lambda_{L}=804.1$nm and
its geometry provide a tool for tuning the SO-coupling strength ($k_{L}$). The
SO coupling strength $k_{L}$ can be tuned within the range
$k^{\prime}_{L}=[0.1-4]a_{\perp}$ using a pair of Raman lasers. On the other
hand, the intensity of the Raman lasers is attributed to the variation of Rabi
coupling frequency $\Omega$, where, in experiments, it can be adjusted within
the interval $\Omega^{\prime}=2\pi\times[105.13-669.28]$Hz, which is
associated to the dimensionless Rabi-coupling frequency $\Omega=[0.1-4]$.
Furthermore, by choosing the s-wave scattering length values between
$a_{11}=a_{22}=[-0.0866a_{0},0.0866a_{0}]$ the intra-species interaction
parameters can be adjusted between $g_{22}=g_{11}=[-1,1]$. Similarly, for
inter-species interaction parameters $g_{12}$, the scattering length $a_{12}$
can be varied from $-0.0866a_{0}$ to $0.0866a_{0}$, where $a_{0}$ is the Bohr
radius.

One of the main focuses of the present work is to show the role of
inhomogeneity in the interaction, describing the localization and
delocalization of the condensate. For that, here we provide a generalized
expression for the spatial dependence of the atomic scattering length
depending upon the intensity $I(x)$ and detuning parameter of the laser beam.
In general, the spatial dependence of the scattering length can be expressed
as [37, 38, 58, 47],

$\displaystyle a(x)=$ $\displaystyle\,a^{0}+\frac{\alpha
I(x)}{\gamma_{s}+\beta I(x)}$ (4a) $\displaystyle a_{12}(x)=$
$\displaystyle\,a_{12}^{0}+\frac{\alpha I(x)}{\gamma_{s}+\beta I(x)}$ (4b)

Here, $a^{0}$ and $a_{12}^{0}$ denote the intra- and interspecies s-wave
scattering lengths when the optical fields are absent. $\alpha$ and $\beta$
are the constants depending upon the laser detuning [38]. $\gamma_{s}$ is the
spontaneous decay rate between two atomic levels. In the experiment, the laser
detuning is considered to be larger than the spontaneous decay rate that
minimizes the decay of atomic state, which in turn helps to realize the
optical Feshbach resonance. Here, $I(x)$ is the intensity of the laser beam
generated due to the stimulated transition between two atomic states [60, 61,
62].

Theis et al. [38] experimentally demonstrated that the scattering length of
87Rb condensate can be measured upon variation of laser detuning utilizing the
optical Feshbach resonance technique. They estimated that the scattering
length $a$ is within the range $10a_{0}$ to $190a_{0}$, considering the
background scattering length value of 87Rb as $a^{0}=100a_{0}$ resulting in
the variation of the scattering length in the range $-90a_{0}$ to $+90a_{0}$.
One can tune the interaction from attractive to repulsive regions with the
above technique. The spatial dependence of the scattering length $a(x)$ can be
achieved using the space dependence intensity, $I(x)$ of the laser beam.
Without loss of generality, it is reasonable to consider the nonlinearity
proportional to the optical lattice potential $V(x)$, as it is created using
the laser field intensity [58, 48, 35]. In this work, we assume
$a_{11}(x)=a_{22}(x)=a(x)$. With this consideration, $g_{11}(x)$, $g_{22}(x)$
and $g_{12}(x)$ can be written as,

$\displaystyle g_{11}(x)=$ $\displaystyle\,\epsilon_{0}+\epsilon V(x),$ (5a)
$\displaystyle g_{22}(x)=$ $\displaystyle\,g_{11}(x),$ (5b) $\displaystyle
g_{12}(x)=$ $\displaystyle\,\eta_{0}+\eta V(x)$ (5c)

Here, $\epsilon_{0}$ and $\eta_{0}$ are the homogeneous interactions
associated with the background scattering length. Similarly, the $\epsilon$
and $\eta$ denote the inhomogeneity interaction strength. In the present work,
to investigate the sole impact of inhomogeneity, we choose
$\epsilon_{0}=\eta_{0}=0$ in all of our studies.

We also consider a scenario where the trapping potential (3) and spatially
dependent nonlinear interactions (5a)-(5c) have a phase difference $\theta$.
With this the nonlinear interactions take the form as:

$\displaystyle
g_{11}(x)=-\epsilon\sum_{l=1}^{2}V_{l}\cos\left(k_{l}x-\theta\right)$ (6a)
$\displaystyle
g_{12}(x)=-\eta\sum_{l=1}^{2}V_{l}\cos\left(k_{l}x+\theta\right)$ (6b)

As a second case, we have chosen a spatial inhomogeneity in the interspecies
interaction to be the same as that of the quasiperiodic potential but with a
phase difference $\theta=\pi/2$.

The dynamical equations [Eqs. 1(a) and 1(b)] are solved using the split-step
Crank-Nicholson method [63, 64, 65] with the box size as $[-102.4:102.4]$,
space step $\Delta x=0.025$ and time step $\Delta t=10^{-4}$. For all the
simulations, we consider the Gaussian wavefunction centered at zero as an
initial state for both the components and also assume the antisymmetric
condition, $\psi_{1}(x)=-\psi_{2}(-x)$ between the components. While solving
the dynamical equations, we employ an imaginary-time scheme to obtain the
ground state and a real-time propagation scheme for investigating the
condensate dynamics.

##  III  Results

Figure 1: Density profile of up (red line) and down (dashed black line)
components for different inhomogeneity parameter $\epsilon=\eta=-0.4$ (a1,b1),
$\epsilon=\eta=0$ (a2,b2), and $\epsilon=\eta=0.4$ (a3,b3) at $k_{L}=0$,
$\Omega=0.0$ (left column) and $k_{L}=0.6,\Omega=1$ (right panel). Insets in
[(a2, a3, b2, b3)] show the spatial variation of density profile in semilog
scale that explicitly displays the exponential tail of the localized
condensate in (a3) and (b3).

In this section, we present the effect of the inhomogeneous interaction and
the Rabi and SO coupling on the spatial profile of the ground state obtained
using the imaginary time propagation. First, we focus on analyzing the impact
of the inhomogeneity on the localization of the ground state. Following this,
we investigate the effect of phase shift between the interaction and potential
in bringing the interesting interplay between the localized and delocalized
state that finally leads to the manifestation of the reentrant features of the
localization with change in the interaction inhomogeneity. We complement our
numerical observation related to the reentrant behaviour by employing the
Gaussian variational approach. Finally, we present a detailed nature of the
dynamics of the localized and delocalized state induced by several means,
namely, by giving small velocity perturbations and quenching of the potential.

###  III.1 Effect of interaction inhomogeneity on the Localizatio-
delocalization transition of the condensate

In a recent work [25] Li et al. demonstrated that the condensate remains
localized for a higher ratio of secondary to the primary optical lattice
strength ($V_{2}/V_{1}$) for all the range of SO and Rabi coupling parameters.
However, for small $V_{2}/V_{1}(\sim 0.1)$, the condensate exhibits a
delocalized nature, which undergoes a transition to the localized state upon
either increasing the Rabi coupling for a fixed SO coupling or increasing the
SO coupling for a fixed Rabi coupling.

Figure 2: Pseudo color representation of the spatial variation of ground state
density of the spin-up component ($\lvert\psi_{1}\rvert^{2}$) with the
inhomogeneity parameter $\epsilon=\eta$ for (a) $k_{L}=0$, $\Omega=0$ and (b)
$k_{L}=0.6$, $\Omega=1.0$. For the zero coupling parameter, the condensate
exhibits a delocalized state for $\epsilon=\eta\lesssim 0.2$ and a localized
state for $\epsilon=\eta>0.2$. However, for finite coupling parameter
($\Omega=1$, $k_{L}=0.6$) the localization state is realized for
$\epsilon=\eta\gtrsim 0$.

While in the former case, the condensate remains delocalized with small value
of SO coupling strength, whereas, for moderate SO coupling strength the
condensate remain localized at high Rabi coupling. In the latter case, it
shows the reentrant nature of the localization with the change in the SO
coupling for small Rabi coupling. In this work, we aim to analyze the effect
of interaction inhomogeneity in tandem with the SO and Rabi coupling
parameters for the state, that is, delocalized, in the absence of
inhomogeneity and coupling parameters. In the following, we investigate the
effect of inhomogeneity and coupling parameters on the localization state of
the condensate by fixing $V_{2}/V_{1}=0.1$.

Figure 3: The form-factor $\chi$ (red circle marked solid line) and the width
$w$ (black square marked dashed line) of the condensate as a function of
inhomogeneity parameter $\epsilon(=\eta)$ for $k_{L}=0.6$ and $\Omega=1.0$.
The increase in $\chi$ is accompanied by a decrease in the width of the
condensate, which starts beyond the critical inhomogeneity parameter
$\epsilon(=\eta)\geq 0.1$, indicating a transition from the delocalized to
localized state. The localized range is shown with the grey-shaded area. Here,
the left and right vertical axis shows the variation of $\chi$ and $w$,
respectively.

In Fig. 1, we show the spatial profile of the density for different sets of
spatial inhomogeneity parameters by considering the coupling parameters as
$\Omega=0$, $k_{L}=0$ (left column) and $\Omega=1$, $k_{L}=0.6$ (right
column). In the absence of the SO and Rabi coupling, the condensate density
profile exhibits a delocalized nature for zero ($\epsilon=\eta=0$) and
attractive inhomogeneity ($\epsilon=\eta=-0.4$) as depicted in the Figs. 1(a2)
and (a1), respectively. The condensate starts showing a tendency to
localization for the repulsive inhomogeneous interaction, quite evident with
the exponential tail as shown in the inset of Fig. 1(a3). As we consider the
SO and the Rabi couplings to a finite value, we find that the spatial spread
of the condensate becomes narrower compared to those with the coupling
parameters for the attractive and zero inhomogeneity parameters, which is
quite evident from the Fig. 1(b1) and (b2), respectively. Here, it is
interesting to note that even for repulsive inhomogeneity, the condensate
exhibits the localization [Fig. 1(b3)] contrary to the situation when
inhomogeneity is absent. Note that, in the absence of inhomogeneity, even a
small repulsive nonlinear interaction destroys the localization of the
condensate [66].

Figure 4: Pseudo color representation of the form-factor ($\chi$) in the
$\epsilon-\eta$ plane for $k_{L}=0.6$ and $\Omega=1.0$. For $\chi\gtrsim 0.5$,
the condensate is characterized in the localized state, while with $\chi<0.5$,
it is termed as delocalized. Three distinct regions, I, II, and III, represent
the delocalized, intermediate localized and localized states.

To investigate the effect of inhomogeneity in a more systematic way in Fig. 2,
we show the pseudocolour density profile of the condensate with a continuous
variation of the inhomogeneity parameter for zero [Fig. 2(a)] and finite
couplings, $\Omega=1$ and $k_{L}=0.6$ [Fig. 2(b)]. For this case, we find that
the condensate remains delocalized for the attractive interaction
inhomogeneity, while we witness localization for repulsive inhomogeneity
interaction. The overall features indicate that for zero coupling parameters
($\Omega=k_{L}=0$), the density gets localized for $\epsilon(=\eta)\gtrsim
0.2$. However, with $k_{L}=0.6$ and $\Omega=1$, the condensate display
localization even without interaction inhomogeneity ($\epsilon(=\eta)\gtrsim
0$). This particular feature suggests an interplay between interaction
inhomogeneity and the coupling parameters towards the localization of the
condensate. On the one hand, while the attractive inhomogeneity tends to
delocalize the condensate, the coupling parameters try to restore the
localization. To shed more light on this aspect, in the following, we will
present a systematic analysis of the role of these competing factors on
delocalization, which will be connected further to bring the reentrant feature
of the localization in the similar line observed in the many body system [49].

Figure 5: Pseudo color representation of the form factor $\chi$ in
$k_{L}-\Omega$ plane in three distinct regions of the inhomogeneity parameters
as indicated in the Fig. 4: (a) for $\epsilon=-0.4$, and $\eta=0.2$ in the
region I, (b) for $\epsilon=\eta=0.0$ in region II, and (c) for
$\epsilon=0.6,\eta=0.8$ in region III. Regions I and III remain in the
delocalized and localized states for all ranges of $\Omega$ and $k_{L}$.
Region II exhibits the reentrant localization upon tuning the SO and Rabi
coupling.

The extent of localization and delocalization of the condensate has been
characterized using two quantities, namely, the integral form factor ($\chi$)
and the width ($w$) of the condensate, which are defined as [25, 49],

$\displaystyle\chi_{j}$
$\displaystyle=\frac{1}{N_{j}^{2}}\int\lvert\psi_{j}\rvert^{4}dx,$ (7a)
$\displaystyle w_{j}^{2}$ $\displaystyle=2\int(x-x_{m})^{2}|\psi_{j}|^{2}dx,$
(7b)

where $N_{j}=\int_{-\infty}^{\infty}|\psi_{j}(x)|^{2}dx$ represents the
population of the $j^{\rm th}$ component of the condensate with
$j\in\\{1,2\\}$ and $x_{m}$ is the center of mass of the condensate. Since we
choose equal number of atoms in each component, the condensate density remains
same in both components, and hence, the form factor and width can be
considered as $\chi_{1}=\chi_{2}\equiv\chi$ and $w_{1}=w_{2}\equiv w$,
respectively. Consequently, we show the results pertaining solely to the spin-
up components only. We set the criteria that the condensate can be
characterized as in the localized state if $\chi\geq 0.5$. Otherwise, for the
lower form factor ($0.3\lesssim\chi\lesssim 0.5$), the condensate is in the
intermediate localized state and the delocalized state for ($\chi\lesssim
0.3$).

In Fig. 3, we illustrate the variation of the form factor (in the left
vertical axis) and width (in the right vertical axis) of the condensate with
respect to the change in the interaction inhomogeneity considering the equal
inter- and intra- interaction ($\epsilon=\eta$). In accordance with the above
discussion, we find that the $\chi$ remains close to zero for attractive
interaction inhomogeneity ($\epsilon=\eta<0$), indicating the delocalized
state of the condensate and increases to $\chi\sim 0.5$ as soon as
inhomogeneity gets tuned towards the repulsive region ($\epsilon=\eta\gtrsim
0$). This trend further continues upon increasing the interaction
inhomogeneity. Overall, we find that the slightly repulsive nature of the
interaction inhomogeneity is good enough to localize the condensate. This
feature of $\chi$ in characterizing the localization of the condensate is very
well complemented by the variation of the width ($w$) of the condensate as
shown using the black dashed squared line in Fig. 3. The width remains high
($w\gtrsim 5$) until ($\epsilon=\eta<0$), for which the condensate remains
delocalized. In the case of the localized condensate for repulsive
interactions, the condensate width becomes nearly equal to one.

To understand the effect of the interaction inhomogeneity on the localization
of the condensate, we perform an extensive simulation for different ranges of
interaction inhomogeneity ($\lvert\epsilon\rvert=\lvert\eta\rvert<1$) by
keeping the SO and Rabi coupling fixed at $k_{L}=0.6$ and $\Omega=1$. In Fig.
4, we present the pseudo color of form-factor $\chi$ in the $\epsilon-\eta$
plane. Here, we have used the criteria, $\chi\gtrsim 0.5$, to designate the
state of the condensate as a localized state. When both intraspecies
($\epsilon$) and interspecies ($\eta$) are positive, the condensate remains in
the localized state with $\chi\gtrsim 0.6$. However, if both interactions are
attractive, the form factor remains in the range $0.03\leq\chi\leq 0.1$,
indicating the delocalized state of the condensate.

Next, we present the effect of SO ($k_{L}$) and Rabi ($\Omega$) couplings on
the localization and delocalization of the condensate with spatial dependent
interactions. For this analysis, we select three sets of parameters
$\epsilon$, and $\eta$ of Fig. 4. These parameters are associated with three
different regions: region I with $\epsilon=-0.4$, $\eta=0.2$, region II with
$\epsilon=\eta=0.0$, and region III with $\epsilon=0.6$, $\eta=0.8$, as marked
in Fig. 4.

In Fig. 5(a) we show the phase diagram in $k_{L}-\Omega$ plane for
$\epsilon=-0.4$, and $\eta=0.2$. We notice that the form factor $\chi$ remains
within the range $0\lesssim\chi\lesssim 0.4$ for all ranges $k_{L}$ and
$\Omega$, revealing that the condensate still associates with the delocalized
state. Fig. 5(b) illustrates the $\chi$ in the $\Omega-k_{L}$ plane for the
region II ($\epsilon=\eta=0$) located at the phase boundary between the
delocalized and localized phases in the $\epsilon-\eta$ plane (see Fig. 4).
Furthermore, for this case, $\chi$ lies within the range
$0.1\lesssim\chi\lesssim 0.75$, implying a transition from delocalized to the
localized state of the condensate. We further find the reentrant of
localization as SO coupling is increased in the range $0\leq k_{L}\leq 4$
while keeping $\Omega$ fixed at a low value ($\Omega\sim 1$). Similar
reentrant features also have been reported earlier by Li et al. [25] by tuning
the $k_{L}$ for non-interacting condensate. After the comprehensive picture of
the condensate phase in $\Omega-k_{L}$ plane for the non-interacting case,
next, we consider repulsive inhomogeneous interactions with $\epsilon=0.6$,
$\eta=0.8$ marked as region III in Fig. 4. For this case, the $\chi$ value
lies within $0.60\lesssim\chi\lesssim 1.0$, suggesting a complete localization
of the condensate as shown in Fig. 5(c). Interestingly, we find that the
condensate becomes strongly localized ($\chi\sim 1$) for high SO and Rabi
coupling parameters, even when the interactions in the condensate are
repulsive in contrast to the homogeneous interaction where the repulsive
interaction weakens the localization [66]. In the second part of the paper, we
will connect this enhancement of the localization with an increase in the
repulsive interaction in the case of spatial inhomogeneity with the
enhancement of the depth of the effective potential of the condensate using
the variational approach.

###  III.2 Reentrant localization in presence of collisional inhomogeneities

In the previous section, we investigated the effect of spatially inhomogeneous
interaction for the case when the spatial inhomogeneity in the nonlinear
interaction is similar to those of the bichromatic lattice potential. Next, we
consider the situation when there happens to be $\pi/2$ phase difference
between the trapping potential and inhomogeneous interactions as given in Eqs.
(6a) and (6b).

Figure 6: Effect of inhomogeneity on the density profile of the condensate for
$\pi/2$ phase difference between the nonlinearity and potential with
$k_{L}=0.4$ and $\Omega=1.0$. (a) Pseudo color representation of density
spatial profile with respect to the inhomogeneous interaction parameter
$\epsilon(=-\eta)$. (b)-(d) spatial density profile for $\epsilon(=-\eta)=0.8$
(localized), $\epsilon(=-\eta)=0$ (delocalized), and
(d)$\epsilon(=-\eta)=-0.8$ (localized). Reentrant localization upon increasing
the strength of the inhomogeneity from the attractive to repulsive region.

In Fig. 6(a), we present the variation of the condensate density with the
inhomogeneous interaction strength $\epsilon$ and $\eta$ by assuming
$\epsilon=-\eta$. We find that the condensate exhibits localization for
$-0.4\gtrsim\epsilon(=-\eta)\gtrsim-1.0$, $0.4\lesssim\epsilon(=-\eta)\lesssim
1.0$ while, it displays delocalized nature
$-0.4\lesssim\epsilon(=-\eta)\lesssim 0.4$ as the coupling parameters are
fixed to $k_{L}=0.4$ and $\Omega=1$. In Figs. 6(b)-(d), we show the density
profile for three sets of parameters $\epsilon=-\eta=0.8$, $\epsilon=-\eta=0$,
and $\epsilon=-\eta=0.8$, respectively. In the case of Fig. 6(b), the
condensate is localized at $x_{0}\approx 0.3$, whereas in Fig.6(d), the
condensate is localized at $x_{0}\approx-0.3$. The shifting of the center of
mass of the condensate is explained further using the effective potential
using the variational approach in Sec. III.3. In contrast, Fig. 6(c)
illustrates a completely delocalized phase of the condensate in the absence of
inhomogeneity.

Figure 7: Pseudo color representation of the form factor $\chi$ in the
$\epsilon-\eta$ plane showing reentrant feature of the localization for
$\pi/2$ phase between the inhomogeneity and potential. The parameters are
$k_{L}=0.4$, $\theta=\pi/2$, and $\Omega=1.0$. Localized ($\chi\gtrsim 0.5$),
intermediate localized ($0.1<\chi\lesssim 0.5$) and delocalized ($\chi\lesssim
0.1$) have been observed. I, II, and III denote the three distinct regimes in
the inhomogeneous parameters space chosen to investigate the effect of
$\Omega$ and $k_{L}$ on the localization and delocalization. Figure 8:
Variation of (a) form-factor ($\chi$) and (b) width ($w$) of the condensate as
a function of the interaction inhomogeneity parameter $\epsilon=-\eta$ for
$k_{L}=0$ (red circles), $k_{L}=0.4$ (black triangle), and $k_{L}=0.6$ (green
pentagons) by fixing $\Omega=1$. Increasing $k_{L}$ reduces the range of
inhomogeneity parameters in which delocalization is observed. Figure 9: Pseudo
color representation of the form factor in the $\Omega-k_{L}$ plane in three
distinct regions (I, II, and III) of interaction inhomogeneity parameter space
as shown in the Fig. 7: (a) $\epsilon=-\eta=0.8$, region I,
(b)$\epsilon=-\eta=-0.1$, region II, and (c) $\epsilon=-\eta=-0.8$, region
III. Both regions I and III exhibit the localization for all ranges of
$\Omega$ and $k_{L}$, while region II displays reentrant localization by
varying $k_{L}$ for $\Omega$ close to $1$.

Next, to get a comprehensive picture of the localized and delocalized phases
due to the effect of the inhomogeneities, we show the pseudo color
representation of $\chi$ in the $\epsilon-\eta$ plane for $\Omega=1$ and
$k_{L}=0.4$ in Fig. 7. We noted that the delocalized region, where the $\chi$
remains near zero, is sandwiched between two localized regions with
$\chi\gtrsim 0.5$. This particular feature of localization-delocalization
transition with interaction inhomogeneities closely resembles the reentrant
feature of localization with the previous studies using the Aubrë-Andre models
[49].

Further, we analyze the effect of spin-orbit coupling parameters on the
reentrant localization. In Fig. 8(a), we show the variation of form factor
$\chi$ with the interaction inhomogeneity $\epsilon(=-\eta)$ at three
different values of $k_{L}=0.0,0.4$ and $0.6$. Conversely, Fig. 8(b)
illustrates the condensate width $w$ for the same values of $k_{L}$. In figure
(a), for the case of $k_{L}=0.4$, one can notice that the form factor
increases between $0.4\leq\chi\leq 0.6$ in between
$\epsilon(=-\eta)\lesssim-0.5$ and $\epsilon(=-\eta)\gtrsim 0.5$, resembling
the localized regions. On the other hand, the $\chi$ value remains almost
constant ($\chi\sim 0$) in between $-0.5\lesssim\epsilon(=-\eta)\lesssim 0.5$
depicting the delocalized region. Conversely, the condensate width $w$
complements the form factor $\chi$ by showing $w\lesssim 1$ in the localized
regions, and in the delocalized region, the width becomes $w\gg 1$. Another
notable fact is that with the increment of $k_{L}$ from zero, the minimum
value of $\chi$ in the delocalized region and the maximum value in the
localized region get increased. For instance, in the delocalized region, the
$\chi$ value remains in the order of $\sim\mathcal{O}(10^{-2})$, whereas the
localized region is featured with the larger $\chi$ value greater than $>0.5$.
Also, one may notice that the delocalized region (grey shaded region) shrinks
as the $k_{L}$ value increases with the variation of $\epsilon(=-\eta)$. On
the other hand, the condensate width $w$ (dashed line with open markers)
nicely complements the form-factor behaviour by showing a larger value of
$w\gtrsim 10.0$ in delocalized region and the smaller value $w\lesssim 1.0$ in
localized regions.

In order to get more comprehensive picture, we construct the phase diagram in
$k_{L}-\Omega$ plane by keeping $\epsilon$ and $\eta$ fixed at different
regions, i.e., (I), (II) and (III) as shown in Fig. 7. The phase diagram in
Fig. 9(a) corresponds to the localized region at (I) ($\epsilon=-\eta=0.8$)
where the $\chi$ value lies within the range $0.49\lesssim\chi\lesssim 0.88$,
clearly demonstrating the localized feature of the condensate. On the other
hand, when we decrease the interaction parameters towards delocalized region
such as region (II) ($\epsilon=-\eta=-0.1$), the associated form-factor in
Fig. 9(b) exhibits similar reentrant phenomenon as shown previously in Fig.
5(b). Next, Fig. 9(c) corresponds to the region (III) ($\epsilon=-\eta=-0.8$),
which shows the localized condensate for all the range of coupling parameters.

So far, our investigations have mainly focused on numerical analysis through
which we have characterized various localized and delocalized regions with
variations in interactions as well as the coupling strength of the system. To
understand reentrant localization, either by tuning the coupling parameters or
changing the interaction strengths, we adopt the Gaussian variational approach
in the following.

###  III.3 Role of inhomogeneity in the reentrant of localization

In this section, using the variation approach, we present a plausible
explanation for the observation of reentrant localization obtained using the
numerical simulation, as presented in Sec. III.2. In what follows, we discuss
the analytical model using the variational approach to disentangle the role of
different localizing and delocalizing factors in spatially inhomogeneous SO
coupled BECs.

####  III.3.1 Variational analysis

Using the time-dependent variational approach [24], the Lagrangian density of
Eqs. (1a) and (1b) can be written as,

$\displaystyle\mathcal{L}=$
$\displaystyle\sum_{j=1}^{2}\Bigg{\\{}\frac{i}{2}\left(\psi_{j}^{\ast}\dot{\psi_{j}}-\psi_{j}\dot{\psi_{j}^{\ast}}\right)-(-1)^{j}\frac{i}{2}k_{L}\left[\psi_{j}^{\ast}\psi_{j}^{\prime}-\psi_{j}(\psi_{j}^{\ast})^{\prime}\right]$
$\displaystyle\left.-\frac{1}{2}|\psi_{j}^{\prime}|^{2}-\frac{1}{2}g(x)|\psi_{j}|^{4}-V(x)|\psi_{j}|^{2}\Bigg{\\}}\right.$
$\displaystyle\left.-g_{12}(x)|\psi_{1}|^{2}|\psi_{2}|^{2}-\Omega\left(\psi_{1}^{\ast}\psi_{2}+\psi_{1}\psi_{2}^{\ast}\right)\right.$
(8)

where asterisk denotes the complex conjugate, prime represents the derivative
($d/dx$), and overdot denotes the time derivative ($d/dt$). Here, the notation
$j=1$ and $2$ refers to the spin-up and spin-down components, respectively.

To obtain the corresponding dynamical equations, we use the Gaussian ansatz
with time-dependent variational parameters $N_{j}$, $w_{j}$, $\beta_{j}$,
$x_{0_{j}}$ and $\phi_{j}$ of the form:

$\displaystyle\begin{split}\psi_{j}(x,t)=&\frac{1}{\pi^{1/4}}\sqrt{\frac{N_{j}}{w_{j}}}\exp[-\frac{(x-x_{0_{j}})^{2}}{2w_{j}^{2}}\Bigg{]}\\\
&\left.\times\exp[(-1)^{j}i\beta_{j}(x-x_{0_{j}})+i\phi_{j}\Bigg{]}\right.,\end{split}$
(9)

where $N_{j}$ is the number of atoms, $w_{j}$ represents the width of the
condensate centered at $x_{0_{j}}$, $\beta_{j}$ and $\phi_{j}$ are
respectively the chirp and phase of the condensate with Gaussian profile. The
effective Lagrangian with the Gaussian ansatz can be obtained by substituting
Eq. (9) into Eq. (8) and integrating over space variable with the potential
Eq. (3) and the nonlinearity Eq. (5). The detailed calculation for the
dynamical equation of motion using the Lagrangian is given in Appendix A.

Figure 10: Ground state density profile obtained using numerical (with lines)
and variational scheme (with markers) for (a) $k_{L}=0.91$,
$\epsilon(=\eta)=0$, and (b) $k_{L}=0.6$, $\epsilon(=\eta)=0.4$. The real and
imaginary parts of the spin-up condensate wavefunction obtained from the
numerical simulation and variational approach agree well.

First, we focus on analyzing the stationary state upon solving the coupled
equations (Eq. 1a and Eq. 1b) considering equal atoms in both components
(i.e., $N_{1}=N_{2}$). In order to obtain the stationary conditions, we need
to make the time derivative of Eqs.(13b–13d) is equal to zero. As a result, we
obtain $\beta_{1}=\beta_{2}=\beta$, $w_{1}=w_{2}=w$, and
$x_{0_{1}}=-x_{0_{2}}=x_{0}$ as the initial condition. Using these initial
conditions, the equations of motion associated to $x_{0}$, $w$ and $\beta$
have the form,

$\displaystyle\begin{split}k_{L}-\beta-\sqrt{\frac{N_{3-j}}{N_{j}}}\hskip
5.69054pt\frac{\partial L_{\Omega}}{\partial\beta_{j}}=0\end{split}$ (10a)
$\displaystyle\frac{1}{2w^{3}}+\frac{1}{2\sqrt{2\pi}w^{2}}\Bigg{[}\epsilon_{0}-\frac{\epsilon}{4}\sum_{l=1}^{2}V_{l}\left(k_{l}^{2}w^{2}+4\right)\exp(-\frac{k_{l}^{2}w^{2}}{8})\times\cos\left(k_{l}x_{0}\right)\Bigg{]}-\frac{w}{2}\sum_{l=1}^{2}\left(V_{l}k_{l}^{2}\right)\cos\left(k_{l}x_{0}\right)\exp\left(-\frac{k_{l}^{2}w^{2}}{4}\right)$
$\displaystyle+\frac{\eta_{0}}{2\sqrt{2\pi}w^{4}}\left(w^{2}-4x_{0}^{2}\right)\exp\left(-\frac{2x_{0}^{2}}{w^{2}}\right)-\frac{\eta}{2\sqrt{2\pi}w^{4}}\sum_{l=1}^{2}V_{l}\left(-4x_{0}^{2}+w^{2}+\frac{k_{l}^{2}w^{4}}{4}\right)\exp(-\frac{k_{l}^{2}w^{2}}{8})$
$\displaystyle+2\Omega\exp\left(-\frac{x_{0}^{2}+\beta^{2}w^{4}}{w^{2}}\right)\left(\beta^{2}w+\frac{x_{0}^{2}}{w^{3}}\right)=0$
(10b)
$\displaystyle\frac{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(-1)^{j-1}}N_{j}\epsilon}{2\sqrt{2\pi}w}\sum_{l=1}^{2}\left(V_{l}k_{l}\right)\sin\left(k_{l}x_{0}\right)\exp\left(-\frac{k_{l}^{2}w^{2}}{8}\right)+\sum_{l=1}^{2}\left(V_{l}k_{l}\right)\sin\left(k_{l}x_{0}\right)\exp\left(-\frac{k_{l}^{2}w^{2}}{4}\right)-\frac{\sqrt{2}\eta_{0}N_{3-j}x_{0}}{\sqrt{\pi}w^{3}}$
$\displaystyle\times\exp\left(-\frac{2x_{0}^{2}}{w^{2}}\right)-\frac{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(-1)^{2j-1}}\eta
N_{3-j}}{2^{3/2}\sqrt{\pi}w^{3}}\exp\left(-\frac{2x_{0}^{2}}{w^{2}}\right)\sum_{l=1}^{2}(-1)^{j}4V_{l}x_{0}\exp\left(-\frac{k_{l}^{2}w^{2}}{8}\right)+\sqrt{\frac{N_{3-j}}{N_{j}}}(-1)^{j-1}\frac{\partial
L_{\Omega}}{\partial x_{0_{j}}}=0$ (10c)

Figure 11: Variation of the ground state energy as a function of the
inhomogeneity parameter ($\epsilon=\eta$). $E_{\rm num}$ (solid-blue line) and
$E_{\rm var}$ (dashed line with dots) represent the total energy obtained from
numerical simulation and analytical variational approach, respectively. Here
$k_{L}=0.6$ and $\Omega=1$. There is good agreement between the numerical and
variational energy in the localized region ($\epsilon=n>0$) where energy
increases towards more negative, indicating the formation of stronger
localization with an increase in the inhomogeneity. In the delocalized region,
$E_{\rm num}\sim-2.5$ for negative inhomogeneity parameter.

In Fig. 10 we show a comparison between the density profile of spin-up
component obtained using GPE (with solid line) and that using the variational
approach [Eq. (10a)–(10c] shown with open markers considering $k_{L}=0.91$,
$\epsilon=\eta=0$, and $\Omega=1.0$ [see Fig. 10(a)], and $\epsilon=\eta=0.4$,
$k_{L}=0.6$ and $\Omega=1.0$ [see Fig. 10(b)]. At stationary state,
$|\psi_{1}|^{2}=|\psi_{2}|^{2}$,
$\mathrm{Im}\,\psi_{1}=\mathrm{Im}\,\psi_{2}$, and
$\mathrm{Re}\,\psi_{1}=-\mathrm{Re}\,\psi_{2}$. Noting this, hereafter, we
consider the spin-up component for the discussion. The density profile
calculated using the numerical GPE and variational approach agree well in the
localized state. For the non-interacting case ($\epsilon=\eta=0$), the
variational approach (Eq. (10b)) reveals that the kinetic energy (first term)
is responsible for the delocalization. However, the Rabi-coupling and the
optical lattice potential contribute to localization of the condensate, which
is in line with the observations made earlier (refer to the discussion in Sec.
III.1). Further, to highlight the effect of inhomogeneities on the
localization in Fig. 10(b), we illustrate the density profile for
$\epsilon=\eta=0.4$, $k_{L}=0.6$ and $\Omega=1.0$. The density profile
obtained using the variational approach agrees well with those obtained using
GPE with slightly higher amplitude than those for non-interacting cases,
indicating a stronger localization with repulsive nonlinearity.

In Fig. 11 we show the variation of total energy ($E_{\rm total}$) with
interaction inhomogeneity as ($\epsilon=\eta$). For attractive interaction
($\epsilon=\eta\leq 0$), the total energy calculated using numerics remains
constant at $E_{\rm num}=-2.5$. On the other hand, for repulsive interactions
($\epsilon(=\eta)\geq 0$), the $E_{\rm num}$ decreases monotonically upon
increasing interaction strength, indicating that the localized condensate
minimizes the energy. However, the total energy $E_{\rm var}$ calculated using
the variational approach agrees well in the localized region with numerics.

The localization appears to arise from the interplay between interactions and
the bichromatic optical lattice in a collisionally inhomogeneous environment.
To account for this, we consider a phase difference of $\theta=\pi/2$ between
the interaction and the potential, as given in Eqs. (6a) and (6b),
respectively, which we further substitute into the Lagrangian (Eq. (8)). Using
the Euler-Lagrange formalism, we calculate the equations of motion with
respect to the variational parameters. The details of the Lagrangian are
provided in Appendix B.

####  III.3.2 Effective Potential for reentrant localization

To understand the role of inhomogeneity in the localization of the condensate,
we resort to derive the dynamical equation of the condensate moving under the
influence of effective potential mainly generated due to the spatial
inhomogeneity [34, 35].

The effective potential can be obtained using Eq. (13c) and Eq. (13e) (see
Appendix A and B for details), which yields:

$\displaystyle\frac{\partial^{2}x_{0_{j}}}{\partial
t^{2}}=-\dot{\beta}_{j}=-\frac{\partial V^{\mbox{eff}}_{j}}{\partial
x_{0_{j}}}$ (11)

Figure 12: Effective potential ($V^{\rm eff}_{(1,2)}$) profile for the
condensate obtained using variational approach [Eq. 14a]: (a) for different
inhomogeneous interaction parameter keeping $\Omega=1$ and $k_{L}=0.6$ and (b)
for different $\Omega$ keeping $k_{L}=0.6$ and inhomogeneity interaction as
$\epsilon(=\eta)=-1.0$. The depth of the effective potential at the central
point ($x_{0}\sim 0$) increases upon increasing the repulsive interaction
compared to those for $\epsilon=\eta=0$ indicating the localization for the
condensate for repulsive interaction inhomogeneity. However, the trend is
opposite for an increase in the attractive interaction, implying the
delocalization for them. Upon an increase of $\Omega$ in (b), the central
depth increases, suggesting the localization of condensate for this increment.

In Fig. 12(a), we show the profile of $V_{\rm eff}$ corresponding to the spin-
up component upon varying the inhomogeneous interactions while keeping the
intra and intercomponent interaction same for $k_{L}=0.6$ and $\Omega=1$. We
find that the effective potential attains a global minimum at $x_{0}=0$, and
the depth of the global minimum increases upon an increase in the spatial
repulsive inhomogeneous interaction. For instance, at $\epsilon=\eta=1$ (open
circled black solid line), a larger depth of effective potential ($V_{\rm
eff}\sim 2.5$) indicates that the condensate gets more localized as the
repulsive interaction increased. However, for attractive
interaction($\epsilon$ or $\eta<0$), the depth of minima at $x_{0}\sim 0$
becomes of the order of other minima of the effective potential, indicating
the delocalized state of the condensate. This observation aligns with the
earlier numerical results obtained with the one component BECs [35].
Additionally, this result closely matches the observed transition from
localized to delocalized states when interaction parameters are varied from
positive to negative (see Fig. 3).

Figure 13: (a) Variation of effective potential ($V^{eff}_{(1,2)}$) as
obtained in Eq. 15b of spin-components with $x_{0}$ for different
inhomogeneous interaction parameters $\epsilon(=-\eta)$ when the phase
difference between the spatial inhomogeneity and potential considered is
$\pi/2$. For $\epsilon=\eta=0$, the potential is peaked at $x_{0}=0$, which is
the delocalized state. However, for $\lvert\epsilon(-\eta)\rvert=0.5$ and $1$,
the central minima depth increases compared to those for zero inhomogeneity,
indicating the localization state. (b) Variation of effective potential with
$x_{0}$ for different $\Omega$ with $k_{L}=0.6$, $\epsilon=\eta=0$. As the
value of $\Omega$ increases, the depth of the central potential also
increases, implying that the Rabi coupling contributes towards the
localization of the condensate.  Figure 14: Pseudo color representation of the
velocity-induced dynamics of the spin-up component density for localized (a)
and delocalized (b) state of the case when the inhomogeneity has the same
phase as the potential. (c) shows the localized state for the case when the
inhomogeneous interaction and potential have $\pi/2$ phase difference. The
other parameters are $k_{L}=0.6$ and $\Omega=1$. Both the localized states (a)
and (c) do not show any change in shape and size upon perturbing with the
velocity, while the delocalized state (b) shows the spread in shape with time.

In the similar line as discussed above, to understand the role of Rabi
coupling in attaining the localization, we show the effective potential in
Fig. 12(b) as $\Omega$ is increased for the condensate with $\epsilon=\eta=-1$
and $k_{L}=0.6$. For $\Omega=0$, the minima of the effective potential at
$x_{0}=0$ is of the same width as of minima at other $x_{0}$, indicating the
delocalized state of the condensate. As the Rabi-coupling increased from zero,
the depth of effective potential at $x_{0}=0$ increases and becomes
significantly higher compared to the minima at other $x_{0}$ for
$\Omega\gtrsim 0.5$. This variation of the effective potential with increasing
$\Omega$ indicates that the role of Rabi coupling is to localize the
condensate in the presence of inhomogeneous interaction where it was initially
delocalized in line with the numerical observation made in the Sec. III.1.

Next, we turn our focus on analyzing the role of spatial inhomogeneity on the
reentrant localization that arises when the inhomogenous nonlinear interaction
carries the $\pi/2$ phase difference in spatial dependence as those for the
trapping quasiperiodic potential as described in Sec. III.2. The form of the
effective potential is provided in Appendix B.

Interestingly, the spatial variation of the effective potential in Fig. 13(a)
vividly illustrates the shifting of the minima with the increment of nonlinear
interactions $\epsilon(=-\eta)$ as a result of the captivating interplay
between the potential and spatial dependence of the interaction. For instance,
as we move from $\epsilon=\eta=0$, the local minima at $x_{0}=0$ shifts either
in the direction of positive or negative $x_{0}$, as depicted for
$\epsilon=-\eta=0.2$ (green dashed line) or $\epsilon=-\eta=-0.2$ (purple
dashed line), respectively. Notably, the displacement of the minima occurs
symmetrically on both sides of the center. For example, at
$\epsilon(=-\eta)=1.0$, the minima are displaced around $x_{0}=0.3$, whereas
at $\epsilon(=-\eta)=-1.0$, the minima are found to be at $x_{0}=-0.3$. Also,
the presence of disorder in the potential leads to a slightly larger depth of
the potential well located at $x_{0}=\pm 0.3$ compared to the other wells for
the aforementioned interactions, which is clearly distinguishable by the red
dotted line in the figure. These particular features highlight intriguing
competition between the potential and nonlinearity towards the reentrant
localization. Additionally, the effect of Rabi-coupling towards localization
is clearly evident when studying the magnitude of $V_{\rm eff}$ in Fig. 13(b).

###  III.4 Effect of spatial inhomogeneity on the dynamics of localized and
delocalized states

So far, we have analyzed the effect of the SO and Rabi coupling along with the
interaction inhomogeneity on the localization of the condensate and studied
their role using the variational approach. In this section, we present the
dynamics of those localized and delocalized states by perturbing the ground
state through velocity or performing a quench on the strength of the secondary
potential.

####  III.4.1 Velocity perturbed Dynamics

In this section, we present the dynamics that ensued in the condensate as a
result of the finite equal and opposite velocity imparted to the condensate in
the localized or delocalized state in the presence of the interaction
inhomogeneities. After imparting the velocity, we analyze the condensate
dynamics utilizing the real-time propagation scheme.

In Fig. 14, we show the temporal evolution of spin-up component condensate
density for various interaction inhomogeneities. In panel (a), we observe the
initial localized state obtained when the phase between spatial inhomogeneity
and potential is zero ($\theta=0$), with parameters $\epsilon=\eta=0.2$,
$k_{L}=0.6$, and $\Omega=1.0$. After applying a velocity perturbation at
$t=20$, we observe a slight change in the condensate. Only a small fraction
generates ripples in space, while the maximum density remains almost unchanged
compared to the ground state density. A similar behaviour is observed for the
localized condensate when the phase shift is
$\theta=\pi/2,\epsilon(=-\eta)=1.0$ (see panel (c)). In panel (b), the
delocalized condensate density demonstrates expansion in space over time after
the velocity perturbation at $t=20$ (after the red dashed line). In summary,
it can be said that localized condensates [see Fig. 14(a) and Fig. 14(c)]
remain stable against small perturbations, while delocalized condensates [see
Fig. 14(b)] exhibit instability under similar perturbations [35].

####  III.4.2 Quench induced dynamics

Figure 15: Time averaged mean-width ($\overline{w}$) of the condensate as a
function of the inhomogeneity $(\epsilon=-\eta)$ for different quenching of
the secondary potential ($V_{2}$) from initial value $0.1$ to the final value
0.0 (black rectangle) and 0.05 (red dots). Green triangle represents the case
when $V_{2}=0.1$. The other parameters are $k_{L}=0.6$, and $\Omega=1.0$. For
the case (a) when the inhomogeneity and potential have the same phase and (b)
when the phase between the potential and nonlinear inhomogeneity is $\pi/2$.

Apart from the analysis of the localized and delocalized phases depending upon
the ground state density with various coupling and interactions, now in this
section, we explore the dynamics of the different phases of the condensate
upon quenching of trap potentials [67, 68, 69]. Firstly, we analyze the
dynamics generated by applying the quench of the trap potential from initial
to different final strength. In Fig. 15, we show the temporal average of the
condensate width $\overline{w}=1/T\int_{0}^{T}w(t)dt$, for quenching of the
potential from initial ($V_{2}=0.1$) to lower trapping strength. In Fig. 15(a)
we show the variation of mean width with $\epsilon(=\eta)$ as the dynamics are
generated through two types of quenching: (i) by quenching the secondary
lattice strength $V_{2}=0.1\rightarrow 0.0$ (black solid line with square
markers) and (ii) by quenching $V_{2}=0.1\rightarrow 0.05$ (red dashed line
with circular markers). We also show the $\overline{w}$ (green dotted line
with triangular markers) corresponding to the ground state condensate. We find
that for repulsive interaction $\epsilon(=\eta)>0$, the $\overline{w}$ is of
the same order compared to the $\overline{w}$ obtained without quenching. This
particular feature implies that the condensate stability in the localized
region is primarily caused by the repulsive inhomogeneity present in the
system [35]. On the other hand, with attractive interaction
($\epsilon(=\eta)<0$), the $\overline{w}$ tend to increase towards a larger
value to the order of $\overline{w}\sim\mathcal{O}(10^{1})$. Moreover, in the
case of periodic quenching (when final $V_{2}=0$), the mean width is higher
compared to the case of quasiperiodic quenching (when $V_{2}=0.05$), which
indicates that even a low but finite disorder is good enough to resist the
expansion.

In Fig. 15(b), we present the mean condensate width with the variation of
$\epsilon(=-\eta)$ in the presence of the phase shift $\theta=\pi/2$. Similar
to $\theta=0$ here also, we observe that in the localized region,
$\overline{w}$ remains almost constant at the same order,
$\mathcal{O}(10^{-1})$, irrespective of the strength of quenching of the trap.
However in the delocalized region, the $\overline{w}$ follows similar behavior
as observed for $\theta=0$ (See Fig. 15(a) ). In general, we find that the
periodic quenching (final $V_{2}=0$) results in attaining the higher
$\overline{w}$ compared to those for quasiperiodic quenching (final
$V_{2}=0.05$). We also notice a slight shift of the delocalized regions
towards the left side of $\epsilon(=-\eta)<0$ for the delocalized regions
compare to the case of without quenching (dotted line with green triangles).
This feature can be attributed to the comparatively weak localization of the
condensate with phase shift $\pi/2$ compared to the localization without phase
shift. We also find the manifestation of the reentrant of the localization
with a change in inhomogeneity in the dynamics of the condensate captured
through the variation of the mean condensate width in Fig. 15(b) in comparison
to the results obtained using ground state in Fig. 8(b).

##  IV Conclusion

We have numerically investigated the effect of interaction inhomogeneity on
the localization-delocalization transition of the SO coupled binary quasi-one
dimensional BECs trapped in the quasiperiodic potentials. By assuming the form
of the spatial modulation of the interaction inhomogeneity same as that of the
quasiperiodic potential we have shown that the condensed undergoes a
localization to delocalization transition as the inhomogeneous interaction
strength varies from attractive to repulsive. However, when a phase difference
of $\pi/2$ is introduced between the inhomogeneity and the lattice potential,
we have shown a remarkable reentrant localization transition as a function of
the inhomogeneity strength. Furthermore, we have also analyzed the effect of
the SO and the Rabi coupling on the localization-delocalization transition for
different strength of the inhomogeneity. For weak inhomogeneity strength, the
increase in the SO coupling strength leads to a reentrant localization similar
to the one observed for non-interacting homogeneous SO coupled BECs [25].

Furthermore, we have utilized the variational approach to understand the
competing nature of the quasiperiodic disorder due to the inhomogeneity and
the potential responsible for the reentrant localization, which is revealed
clearly in the oscillatory nature of the depth of the effective potential as a
function of the inhomogeneity strength. We have also demonstrated the
manifestation of the reentrant localization in the quench dynamics of the
condensate. Our present analysis provides an alternate disorder parameter
other than the external potential to attain the reentrant localization, which
may motivate future experiments in BECs.

In the present work, we have considered the equal intraspecies interaction
between the components that restrict the same population among the components.
It would be intriguing to extend the formalism developed in the present work
to account for the effect of unequal intraspecies interactions on the
localization and delocalization of the individual component(s). Another
extension would include a completely random disordered potential, where the
condensate could exhibit complex phases in the localized and delocalized
state.

###### Acknowledgements.

R.R. acknowledges the postdoctoral fellowship supported by Zhejiang Normal
University, China, under Grants No. YS304023964. T.M. acknowledges support
from Science and Engineering Research Board (SERB), Govt. of India, through
project No. MTR/2022/000382 and STR/2022/000023. The work of P.M. is supported
by MoE RUSA 2.0 (Bharathidasan University - Physical Sciences). We gratefully
acknowledge our super-computing facility Param-Ishan and Param Kamrupa (IITG),
where all the simulation runs were performed.

##  Appendix A Details of the Variational approach Calculation

In this appendix, we provide the detailed steps of time dependent variational
approach. The Lagrangian for the zero phase difference between the
inhomogeneous interaction and potential is given by

$\displaystyle L=$
$\displaystyle{\sum_{j=1}^{2}}\Bigg{[}N_{j}(-1)^{j}\beta_{j}\dot{x}_{0_{j}}-N_{j}\dot{\phi_{j}}+N_{j}\beta_{j}k_{L}-\frac{N_{j}}{2}\left(\frac{1}{2w_{j}^{2}}+\beta_{j}^{2}\right)-\frac{N_{j}}{2^{3/2}\sqrt{\pi}w_{j}}\Bigg{[}\epsilon_{0}-\epsilon\sum_{l=1}^{2}V_{l}\cos\left(k_{l}x_{0_{j}}\right)\exp\left(-\frac{k_{l}^{2}w_{j}^{2}}{8}\right)\Bigg{]}$
$\displaystyle+N_{j}\sum_{l=1}^{2}V_{l}\cos\left(k_{l}x_{0_{j}}\right)\exp\left(-\frac{k_{l}^{2}w_{j}^{2}}{4}\right)\Bigg{]}-\frac{N_{1}N_{2}}{\sqrt{\pi}\sqrt{w_{1}^{2}+w_{2}^{2}}}\exp\left(-\frac{\left(x_{01}-x_{02}\right)^{2}}{w_{1}^{2}+w_{2}^{2}}\right)$
$\displaystyle\times\Bigg{[}\eta_{0}-\eta\sum_{l=1}^{2}V_{l}\cos\left(\frac{k_{l}\left(w_{1}^{2}x_{0_{2}}+w_{2}^{2}x_{0_{1}}\right)}{w_{1}^{2}+w_{2}^{2}}\right)\exp\left(-\frac{k_{l}^{2}w_{1}^{2}w_{2}^{2}}{4\left(w_{1}^{2}+w_{2}^{2}\right)}\right)\Bigg{]}-2\sqrt{N_{1}N_{2}}L_{\Omega},$
(12a) where $\displaystyle
L_{\Omega}=\Omega\sqrt{\frac{2w_{1}w_{2}}{\left(w_{1}^{2}+w_{2}^{2}\right)}}\cos\left[\frac{\left(\beta_{1}w_{1}^{2}-\beta_{2}w_{2}^{2}\right)\left(x_{0_{1}}-x_{0_{2}}\right)-\phi\left(w_{1}^{2}+w_{2}^{2}\right)}{w_{1}^{2}+w_{2}^{2}}\right]\exp\left(-\frac{(x_{0_{1}}-x_{0_{2}})^{2}+(\beta_{1}+\beta_{2})^{2}w_{1}^{2}w_{2}^{2}}{2(w_{1}^{2}+w_{2}^{2})}\right)$
(12b)

with $\phi=\phi_{2}-\phi_{1}$, and $\rho=x_{0_{1}}-x_{0_{2}}$ The Euler-
Lagrangian equation associated to variational parameters $\alpha_{j}$ is given
by

$\displaystyle\frac{\partial L}{\partial\alpha_{j}}-\frac{d}{dt}\frac{\partial
L}{\partial\dot{\alpha_{j}}}=0$ (13a) where $\alpha_{j}$ corresponds to the
variational parameters $N_{j}$, $\beta_{j}$, $\phi_{j}$, $x_{0_{j}}$, and
$w_{j}$. The Euler equation of motion corresponding to $\phi_{j}$ is gievn by
$\displaystyle\dot{\phi}_{j}=$
$\displaystyle-\frac{1}{4w_{j}^{2}}+\frac{\beta_{j}^{2}}{2}-\frac{N_{j}}{\sqrt{2\pi}w_{j}}\left[\epsilon_{0}-\epsilon\sum_{l=1}^{2}V_{l}\cos\left(k_{l}x_{0j}\right)\exp\left(-\frac{k_{l}^{2}w_{j}^{2}}{8}\right)\right]-\frac{N_{3-j}}{\sqrt{\pi\left(w_{1}^{2}+w_{2}^{2}\right)}}\exp\left(-\frac{\left(x_{0_{1}}-x_{0_{2}}\right)^{2}}{w_{1}^{2}+w_{2}^{2}}\right)$
$\displaystyle\left[\eta_{0}-\eta\sum_{l=1}^{2}V_{l}\cos\left(k_{l}\frac{w_{1}^{2}x_{0_{2}}+w_{2}^{2}x_{0_{1}}}{w_{1}^{2}+w_{2}^{2}}\right)\exp\left(-\frac{k_{l}^{2}w_{1}^{2}w_{2}^{2}}{4(w_{1}^{2}+w_{2}^{2})}\right)\right]-\sqrt{\frac{N_{3-j}}{N_{j}}}L_{\Omega}+\beta_{j}\sqrt{\frac{N_{3-j}}{N_{j}}}\frac{\partial
L_{\Omega}}{\partial\beta_{j}}$
$\displaystyle+\sum_{l=1}^{2}V_{l}\cos\left(k_{l}x_{0_{j}}\right)\exp\left(-\frac{k_{l}^{2}w_{j}^{2}}{4}\right)$
(13b) In the similar line the equation of motion associated to $x_{0_{j}}$ is
given by
$\displaystyle\begin{split}\dot{x}_{0_{j}}=(-1)^{1-j}\left[-k_{L}+\beta_{j}+\sqrt{\frac{N_{3-j}}{N_{j}}}\hskip
5.69054pt\frac{\partial L_{\Omega}}{\partial\beta_{j}}\right]\end{split}$
(13c) Further the equation of motion associated to $N_{j}$ can be written as:
$\displaystyle\dot{N}_{j}=(-1)^{j}2\Omega\sqrt{N_{1}N_{2}}\sqrt{\frac{2w_{1}w_{2}}{w_{1}^{2}+w_{2}^{2}}}\sin\left[\frac{\left(\beta_{1}w_{1}^{2}-\beta_{2}w_{2}^{2}\right)\left(x_{0_{1}}-x_{0_{2}}\right)}{\left(w_{1}^{2}+w_{2}^{2}\right)}-\phi\right]\exp\left[-\frac{\left(x_{0_{1}}-x_{0_{2}}\right)^{2}+\left(\beta_{1}+\beta_{2}\right)^{2}w_{1}^{2}w_{2}^{2}}{2\left(w_{1}^{2}+w_{2}^{2}\right)}\right]$
(13d) The Euler equation of motion corresponding to $\beta_{j}$ has the form
as $\displaystyle\dot{\beta}_{j}=$
$\displaystyle\frac{N_{j}\epsilon}{(-1)^{3-j}2\sqrt{2\pi}w_{j}}\sum_{l=1}^{2}\left(V_{l}k_{l}\right)\sin\left(k_{l}x_{0_{j}}\right)\exp\left(-\frac{k_{l}^{2}w_{j}^{2}}{8}\right)+\frac{1}{(-1)^{3-j}}\sum_{l=1}^{2}\left(V_{l}k_{l}\right)\sin\left(k_{l}x_{0j}\right)\exp\left(-\frac{k_{l}^{2}w_{j}^{2}}{4}\right)$
$\displaystyle-\frac{2\eta_{0}N_{3-j}\left(x_{0_{1}}-x_{0_{2}}\right)}{\sqrt{\pi}\left(w_{1}^{2}+w_{2}^{2}\right)^{3/2}}\exp\left(-\frac{\left(x_{0_{1}}-x_{0_{2}}\right)^{2}}{w_{1}^{2}+w_{2}^{2}}\right)-\frac{\eta
N_{3-j}(-1)^{j-3}}{\sqrt{\pi}\left(w_{1}^{2}+w_{2}^{2}\right)^{3/2}}\exp\left(-\frac{\left(x_{0_{1}}-x_{0}\right)^{2}}{w_{1}^{2}+w_{2}^{2}}\right)$
$\displaystyle\sum_{l=1}^{2}V_{l}\left[\left(-k_{l}w_{3-j}^{2}\right)\sin\left(k_{l}\frac{w_{1}^{2}x_{0_{2}}+w_{2}^{2}x_{0_{1}}}{w_{1}^{2}+w_{2}^{2}}\right)+(-1)^{j}2\left(x_{0_{1}}-x_{0_{2}}\right)\cos\left(k_{l}\frac{w_{1}^{2}x_{0_{2}}+w_{2}^{2}x_{0_{1}}}{w_{1}^{2}+w_{2}^{2}}\right)\right]$
$\displaystyle\times\exp\left(-\frac{k_{l}^{2}w_{1}^{2}w_{2}^{2}}{4\left(w_{1}^{2}+w_{2}^{2}\right)}\right)\sqrt{\frac{N_{3-j}}{N_{j}}}(-1)^{j-3}\frac{\partial
L_{\Omega}}{\partial x_{0_{j}}}-\frac{\dot{N}_{j}}{N_{j}}\beta_{j}$ (13e)
Finally, the Euler equation of motion with respect to $w_{j}$ is given by
$\displaystyle\begin{split}&\frac{N_{j}}{2w_{j}^{3}}+\frac{N_{j}^{2}}{2\sqrt{2\pi}w_{j}^{2}}\left[\epsilon_{0}-\frac{\epsilon}{4}\sum_{l=1}^{2}V_{l}\left(k_{l}^{2}w_{j}^{2}+4\right)\exp\left(-\frac{k_{l}^{2}w_{j}^{2}}{8}\right)\cos\left(k_{l}x_{0j}\right)\right]-\frac{N_{j}w_{j}}{2}\sum_{l=1}^{2}\left(V_{l}k_{l}^{2}\right)\cos\left(k_{l}x_{0_{j}}\right)\exp\left(-\frac{k^{2}_{l}w^{2}_{j}}{4}\right)\\\
&+\frac{N_{1}N_{2}\eta_{0}w_{j}}{\sqrt{\pi}\left(w_{1}^{2}+w_{2}^{2}\right)^{5/2}}\left(w_{1}^{2}+w_{2}^{2}-2\left(x_{0_{1}}-x_{0_{2}}\right)^{2}\right)\exp\left(-\frac{\left(x_{0_{1}}-x_{0_{2}}\right)^{2}}{w_{1}^{2}+w_{2}^{2}}\right)\\\
&-\frac{\eta
N_{1}N_{2}w_{j}}{2\sqrt{\pi}\left(w_{1}^{2}+w_{2}^{2}\right)^{5/2}}\exp\left(-\frac{\left(x_{0_{1}}-x_{0_{2}}\right)^{2}}{w_{1}^{2}+w_{2}^{2}}\right)\sum_{l=1}^{2}V_{l}\bigg{[}\left(-4\left(x_{0_{1}}-x_{0_{2}}\right)^{2}+2w_{1}^{2}+2w_{2}^{2}+k_{l}^{2}\omega_{3-j}^{4}\right)\cos\left(k_{l}\frac{w_{1}^{2}x_{0_{2}}+w_{2}^{2}x_{0_{1}}}{w_{1}^{2}+w_{2}^{2}}\right)\\\
&+(-1)^{j}4k_{l}\left(x_{0_{1}}-x_{0_{2}}\right)\omega_{3-j}^{2}\sin\left(k_{l}\frac{w_{1}^{2}x_{0_{2}}+w_{2}^{2}x_{0_{1}}}{w_{1}^{2}+w_{2}^{2}}\right)\bigg{]}\exp\left(-\frac{k_{l}^{2}w_{1}^{2}w_{2}^{2}}{4\left(w_{1}^{2}+w_{2}^{2}\right)}\right)-\sqrt{N_{j}N_{3-j}}\frac{\partial
L_{\Omega}}{\partial w_{j}}=0\end{split}$ (13f)

Note that the equation of motion corresponding to $w_{j}$ does not contain the
time derivative of the condensate width $w_{j}$ as like the other equations
(13b)-(13e). Due to this reason, the variational approach does not seem to fit
to analyze the dynamics of the width of the condensate.

Figure A.1:  Comparison of the condensate density profile obtained numerically
(solid line) and using the variational approach (open markers) for different
inhomogeneity parameters when the phase difference between the nonlinear
inhomogeneity and potential is $\pi/2$. (a) For $\epsilon=-\eta=-0.8$, and (b)
For $\epsilon=-\eta=0.8$. The other parameters are
$k_{L}=0.6,\theta_{l}=\pi/2k_{l}$, and $\Omega=1.0$. Incorporating the phase
shift $\pi/2$ between the potential and inhomogeneity results in the
condensate localizing at an off-center position at (a) $x=-0.3$ and (b)
$x=0.3$. The real (Re) and imaginary (Im) parts of the condensate wavefunction
show good agreement between the wave function profile obtained from numerical
simulation and that from using the variational approach.

The detailed equation for the effective potential $V^{\rm eff}_{j}$ can be
derived from the Eq. (13c) and Eq. (13e) and it will assume the form as

$\displaystyle V^{\mbox{eff}}_{j}=$
$\displaystyle-\frac{\epsilon}{2^{3/2}\sqrt{\pi}w_{j}}\sum_{l=1}^{2}V_{l}\cos\left(k_{l}x_{0_{j}}\right)\exp\left(-\frac{k_{l}^{2}w_{j}^{2}}{8}\right)-\sum_{l=1}^{2}V_{l}\cos\left(k_{l}x_{0_{j}}\right)\exp\left(-\frac{k_{l}^{2}w_{j}^{2}}{4}\right)+\frac{(-1)^{3-j}\eta_{0}}{\sqrt{\pi}\left(w_{1}^{2}+w_{2}^{2}\right)^{1/2}}$
$\displaystyle\times\exp\left(-\frac{\left(x_{0_{1}}-x_{0_{2}}\right)^{2}}{w_{1}^{2}+w_{2}^{2}}\right)-\frac{\eta}{\sqrt{\pi}\sqrt{w_{1}^{2}+w_{2}^{2}}}\exp\left(-\frac{\left(x_{01}-x_{02}\right)^{2}}{w_{1}^{2}+w_{2}^{2}}\right)\sum_{l=1}^{2}V_{l}\cos\left(\frac{k_{l}\left(w_{1}^{2}x_{0_{2}}+w_{2}^{2}x_{0_{1}}\right)}{w_{1}^{2}+w_{2}^{2}}\right)$
$\displaystyle\times\exp\left(-\frac{k_{l}^{2}w_{1}^{2}w_{2}^{2}}{4\left(w_{1}^{2}+w_{2}^{2}\right)}\right)+L_{\Omega}$
(14a)

The effective potential obtained in Eq. (14a) is mainly composed of five
different terms. The first term involving interactions $\epsilon$ contributes
towards localization in case of repulsive interactions. The second term,
containing the bichromatic optical lattice potential, is responsible for the
localization of the condensate. The third term contains $\eta_{0}$, which
determines the state of localization or delocalization depending upon the
component and the sign of $\eta_{0}$. The last term containing the Rabi-
coupling [See Eq.(14a)] contributes towards either localization or
delocalization for in-phase or out-of-phase state of the condensate,
respectively. For our calculation, we choose $\phi=\pi$ between two components
in all cases which implies that the role of Rabi coupling towards
localization. Further, the condensate width $w$, and chirp $\beta$ appearing
in the effective potential can be estimated by solving Eqs. (10a–10c). After
substituting the $w_{j}$, and $\beta_{j}$, we compute the $V^{\rm eff}(x_{0})$
for various spatial inhomogeneity and coupling parameters, which has been
depicted in Fig. 12.

##  Appendix B Details of the Lagrangian and effective potential for the phase
shift $\pi/2$ between optical lattice potential and nonlinearity

In this appendix section, we provide the Lagrangian and the corresponding
equation of the effective potential for the case of $\theta=\pi/2$ between
optical lattice and interaction inhomogeneity. Here, $j=1,2$ represents the
spin-up and spin-down components, respectively.

Figure B.1: Variation of numerical ($E_{\rm num}$) and variational ($E_{\rm
var}$) total energy with inhomogeneity $\epsilon(=-\eta)$ with $k_{L}=0.6$,
and $\Omega=1.0$. Tuning the inhomogeneity leads to a monotonic decrease in
the total energy in the localized regions, that is
$-1\lesssim\epsilon(=-\eta)\lesssim-0.3$ or
$0.3\lesssim\epsilon(=-\eta)\lesssim 1.0$ whereas, in the delocalized region
$-0.3\lesssim|\epsilon(=-\eta)|\lesssim 0.3$, the total energy remains almost
constant at $E_{\rm num}\sim-2.56$. In both cases, energy follow a symmetric
nature in either side of delocalization regions.

$\displaystyle\begin{split}L=&{\sum_{j=1}^{2}}\Bigg{[}N_{j}(-1)^{j}\beta_{j}\dot{x}_{0_{j}}-N_{j}\dot{\phi_{j}}+N_{j}\beta_{j}k_{L}-\frac{N_{j}}{2}\left(\frac{1}{2w_{j}^{2}}+\beta_{j}^{2}\right)-\frac{N_{j}}{2^{3/2}\sqrt{\pi}w_{j}}\Bigg{[}\epsilon_{0}-\epsilon\sum_{l=1}^{2}V_{l}\sin\left(k_{l}x_{0_{j}}\right)\exp\left(-\frac{k_{l}^{2}w_{j}^{2}}{8}\right)\Bigg{]}\\\
&\left.+N_{j}\sum_{l=1}^{2}V_{l}\cos\left(k_{l}x_{0_{j}}\right)\exp\left(-\frac{k_{l}^{2}w_{j}^{2}}{4}\right)\Bigg{]}-\frac{N_{1}N_{2}}{\sqrt{\pi}\sqrt{w_{1}^{2}+w_{2}^{2}}}\exp\left(-\frac{\left(x_{01}-x_{02}\right)^{2}}{w_{1}^{2}+w_{2}^{2}}\right)\right.\\\
&\left.\times\Bigg{[}\eta_{0}+\eta\sum_{l=1}^{2}V_{l}\sin\left(\frac{k_{l}\left(w_{1}^{2}x_{0_{2}}+w_{2}^{2}x_{0_{1}}\right)}{w_{1}^{2}+w_{2}^{2}}\right)\exp\left(-\frac{k_{l}^{2}w_{1}^{2}w_{2}^{2}}{4\left(w_{1}^{2}+w_{2}^{2}\right)}\right)\Bigg{]}-2\sqrt{N_{1}N_{2}}L_{\Omega}\right.\end{split}$
(15a) In the above equation $L_{\Omega}$ has the same form as in Eq. (12b).
The effective potential $V_{\rm eff}$ assumes the form as, $\displaystyle
V^{\mbox{eff}}_{j}=$
$\displaystyle-\frac{\epsilon}{2^{3/2}\sqrt{\pi}w_{j}}\sum_{l=1}^{2}V_{l}\sin\left(k_{l}x_{0_{j}}\right)\exp\left(-\frac{k_{l}^{2}w_{j}^{2}}{8}\right)-\sum_{l=1}^{2}V_{l}\cos\left(k_{l}x_{0_{j}}\right)\exp\left(-\frac{k_{l}^{2}w_{j}^{2}}{4}\right)-\frac{\eta_{0}}{\sqrt{\pi}\left(w_{1}^{2}+w_{2}^{2}\right)^{1/2}}$
$\displaystyle\times\exp\left(-\frac{\left(x_{0_{1}}-x_{0_{2}}\right)^{2}}{w_{1}^{2}+w_{2}^{2}}\right)+\frac{\eta}{\sqrt{\pi}\sqrt{w_{1}^{2}+w_{2}^{2}}}\exp\left(-\frac{\left(x_{01}-x_{02}\right)^{2}}{w_{1}^{2}+w_{2}^{2}}\right)\sum_{l=1}^{2}V_{l}\sin\left(\frac{k_{l}\left(w_{1}^{2}x_{0_{2}}+w_{2}^{2}x_{0_{1}}\right)}{w_{1}^{2}+w_{2}^{2}}\right)$
$\displaystyle\times\exp\left(-\frac{k_{l}^{2}w_{1}^{2}w_{2}^{2}}{4\left(w_{1}^{2}+w_{2}^{2}\right)}\right)+L_{\Omega}$
(15b)

In Fig. A.1(a) and Fig. A.1(b), we show a comparison between the density
profiles obtained from the numerical simulation (solid lines) and that from
the variational approach (marked with open markers) for $\epsilon=-0.8$,
$\eta=0.8$ and $\epsilon=0.8$, $\eta=-0.8$, respectively. For both the
combination of the strength of the inhomogeneity, the condensate gets
localized at off-center. For instance, in Fig. A.1(a), the condensate gets
localized at $x\approx-0.3$, while, in Fig. A.1(b), the localization happens
at $x\approx 0.3$. To unravel the reason for this particular nature of
localization of the condensate at the off-center position in space, we analyze
the nature of effective potential as depicted in Fig. 13(a). We find that
increasing the strength of interaction results in shifting the minima of the
effective potential $V^{\rm eff}$ either left or right, which depends solely
on the nature of the interaction inhomogeneity. The left shift happens for the
attractive ($\epsilon=-\eta=-1.0$) while the right shift happens for the
repulsive intraspecies ($\epsilon=-\eta=1.0$) interactions.

To get more insight into the probable cause for the localization and
delocalization transition, in Fig. B.1, we show a variation of the total
energy as a function of the interaction inhomogeneity obtained from the
numerical simulation and using the variational approach. The energy shows a
monotonically decreasing trend with the interaction inhomogeneity strength for
$0.3\lesssim\lvert\epsilon(=-\eta)\rvert\lesssim 1$. However, for
$0\lesssim\lvert\epsilon(=-\eta)\rvert\lesssim 0.3$ the energy assumes the
constant value around $E_{\rm num}\approx-2.56$. Variational energy $E_{\rm
var}$ qualitatively follows a similar trend as that of $E_{\rm num}$ in both
localized and delocalized regions.

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