# Two-dimensional miscible-immiscible supersolid and droplet crystal state in
a homonuclear dipolar bosonic mixture

Soumyadeep Halder<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Technology Kharagpur, Kharagpur 721302, India  Subrata Das
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology
Kharagpur, Kharagpur 721302, India  Sonjoy Majumder<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur
721302, India

###### Abstract

The recent realization of binary dipolar BEC [Phys. Rev. Lett. 121, 213601
(2018)] opens new exciting aspects for studying quantum droplets and
supersolids in a binary mixture. Motivated by this experiment, we study
groundstate phases and dynamics of a Dy-Dy mixture. Dipolar bosonic mixture
exhibits qualitatively novel and rich physics. Relying on the three-
dimensional numerical simulations in the extended Gross-Pitaevskii framework,
we unravel the groundstate phase diagrams and characterize different
groundstate phases. The emergent phases include both miscible and immiscible
single droplet (SD), multiple droplets (MD), supersolid (SS), and superfluid
(SF) states. More intriguing mixed groundstates may occur for an imbalanced
binary mixture, including a combination of SS-SF, SS-MD, and SS-SS phases. We
observed the dynamical transition from a miscible MD state to an immiscible MD
state with multiple domains formed along the axial direction by tuning the
inter-species scattering length. Also by linear quenches of intra-species
scattering lengths across the aforementioned phases, we monitor the dynamical
formation of supersolid clusters and droplet lattices. Although we have
demonstrated the results for a Dy-Dy mixture and for a specific parameter
range of intra-species and inter-species scattering lengths, our results are
generally valid for other dipolar mixtures and may become an important
benchmark for future experimental scenarios.

##  I Introduction

Quantum droplets are dilute liquid-like clusters of atoms produced in a
quantum fluid where the dominant attractive mean-field-driven collapse is
arrested by the quantum fluctuations [1, 2]. The supersolid state is also an
intriguing state of matter in which the crystalline order of quantum droplets
and a global phase coherence [3, 4] coexist as a result of background
superfluid. Both of these states were initially predicted and searched for in
liquid helium [5, 6, 7, 8]. The ability to tune the interaction strength
between the particles of an ultracold atomic gas through the Feshbach
resonance [9] offers an excellent platform for studying a plethora of rich
physical phenomena. In recent years the quest for quantum droplets and
supersolid states in ultracold gases has attracted significant attention. Most
theoretical and experimental studies over the last few years reveal the
formation of droplets, mainly in two different types of ultracold bosonic
systems discussed below.

Quantum droplets in ultracold atomic gases have been first observed in single
component dipolar bosonic gases with sufficiently large magnetic dipole
moments like dysprosium (Dy) [10, 11, 12], and erbium (Er) [13, 14, 15]. In
this case, when the dipole-dipole interaction (DDI) dominates over the contact
interaction in that regime, the anisotropic and long-range characters of DDI
lead to the formation of self-bound quantum droplets [16, 17, 18, 19, 20, 21,
22] and supersolid states [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34].
These droplets with highly anisotropic properties have filament-like narrow
transverse widths and are elongated along the direction of the external
magnetic field.

Quantum droplets also have been realized in non-dipolar binary homonuclear
[35, 36, 37, 38, 39] and heteronuclear [40, 41] Bose mixtures. Binary mixtures
with an attractive inter-species interaction lead to the formation of miscible
droplets. Unlike the droplets formed in a single component dipolar BEC (dBEC)
due to the anisotropic and partial attractive nature of DDI, these droplets in
a binary system originate solely due to the contact interaction and,
therefore, spherical (isotropic) in nature.

These phases have been widely explored in various ultracold systems and
different experimental setups, ranging from rotating dipolar condensate [42,
43, 44, 45], dBEC under the influence of a rotating magnetic field [46, 47,
48, 49], optical lattice trapped dipolar condensate [50, 50], lattice trapped
atomic mixtures [51, 52, 53], Rydberg systems [54, 55], spin-orbit coupled
systems [56, 57], molecular BECs [58], and a binary mixture of dipolar-
nondipolar condensates [59, 60].

Recent experimental realization of binary dipolar condensates for the first
time [61], and the ability to control their intra-species and inter-species
interaction strength through the Feshbach resonance [62, 63] opens new
exciting aspects for the study of quantum droplets in a mixture of binary
dipolar condensates. Most of the recent theoretical works mainly focus on the
formation of a self-bound droplet state in a binary dipolar mixture without
any trapping confinement. In contrast to non-dipolar mixtures, due to the
anisotropic dipolar counterpart formation of a new class of self-bound
miscible, immiscible quantum droplets are predicted [64, 65, 66, 67, 68, 69].

In this article, we theoretically investigate the possibility of forming
different groundstate phases of a binary dBEC (Dy-Dy mixture) confined in a
quasi-two-dimensional harmonic trap. For a balanced system, we observe four
different groundstate phases: superfluid (SF), supersolid (SS), and single,
multiple droplets (SD, MD) that exist in both miscible and immiscible phases.
Both components form identical shapes in the miscible regime. Whereas, in the
immiscible domain, we observe axially immiscible SD and MD states, and
radially immiscible asymmetric SS and SF states. The energetically favored
groundstate depends on the number of atoms, intra- and inter-component
interactions, and the trap geometry. We depict the phase diagrams and demark
all these phases. For an imbalanced mixture, more intriguing states like a
mixture of SS-SF, SS-MD, and SS-SS states formed. We have also shown that in
an immiscible impurity regime, where one of the components consists of a very
small number of atoms (minor component), the major component with a larger
number of atoms can bind the impurity component along the axial direction and
form a self-bound droplet state for a small intra-species scattering length.
Whereas, for comparatively large intra- and inter-species scattering lengths
the major component cannot hold the minor component along the axial position.
Rather it is pushed along the radially outward direction in presence of the
harmonic trapping potential and forms an immiscible mixed state. Using the
time-dependent coupled eGPE, we also study the dynamics of a balanced binary
system across the above-mentioned phase boundaries.

This paper is structured as follows. Section II describes the theory and
formalism, including the coupled extended Gross-Pitaevskii equation (eGPE) and
the overlap integral to distinguish the miscible and immiscible phases. In
Sec. III, we extract the phase diagrams of the quasi-2D dipolar binary BEC.
Sec. IV characterizes different possible groundstates for an imbalanced binary
mixture. In Sec. V, we explore real-time dynamics and the formation of 2D
miscible-immiscible droplet and supersolid states by using the time-dependent
eGPE. A summary of our findings, together with future aspects, is provided in
Sec. VI. Appendix A describes the ingredients of our numerical simulations.
Appendix B is devoted to the variational solution within the same shape
approximation (SSA) framework. Appendix C delineates the contrast phase
diagrams to differentiate the superfluid, supersolid, and droplet phases. In
Appendix D, we describe the effective potential experienced by one condensate
due to the presence of the other condensate. Finally, in Appendix E, we have
shown the time evolution of density profiles and the overlap integral
following an interaction quench of a miscible SF state across the relevant
phase boundaries.

##  II Theory

We consider a mixture of two species of dipolar bosonic atoms with a large
magnetic dipole moment $\mu_{i}^{m}$ $(i=1,2)$ polarized along the $z$
direction by an external magnetic field and confined in a circular symmetric
harmonic trapping potential. In the ultracold regime, the atoms of species-$i$
are characterized by the macroscopic wave function $\psi_{i}$, whose temporal
evolution is described by the coupled eGPE:

$\displaystyle i\hbar\frac{\partial\psi_{i}(\textbf{r},t)}{\partial t}$
$\displaystyle=\Big{[}-\frac{\hbar^{2}}{2m_{i}}\nabla^{2}+V_{t}(\textbf{r})+\sum_{j=1}^{2}\Big{(}g_{ij}\absolutevalue{\psi_{j}(\bf{r},t)}^{2}+$
$\displaystyle\int
d\textbf{r}^{\prime}V_{ij}^{dd}(\textbf{r}-\textbf{r}^{\prime})\absolutevalue{\psi_{j}(\bf{r}^{\prime},t)}^{2}\Big{)}+\Delta\mu_{i}\Big{]}\psi_{i}(\textbf{r},t).$
(1)

Here,
$V_{t}(\textbf{r})=\frac{1}{2}m_{i}\omega^{2}(x^{2}+y^{2}+\lambda^{2}z^{2})$
is the harmonic trapping potential with angular frequencies
$\omega_{x}=\omega_{y}=\omega,\omega_{z}$; $m_{i}$ is the atomic mass of the
$i$’th species and $\lambda=\omega_{z}/\omega$ is the trap aspect ratio. The
short-range intra- and inter-component interaction strengths are given by
$g_{ii}=4\pi\hbar^{2}a_{ii}/m_{i}$ and $g_{ij}=2\pi\hbar^{2}a_{ij}/m_{ij}$,
respectively. Here, $a_{ii}$ and $a_{ij}$ are the intra- and inter-component
scattering length of atoms and $m_{ij}=m_{i}m_{j}/(m_{i}+m_{j})$ is the
reduced mass. Apart from the contact interaction, there exists a long-range
DDI between the atoms, and it takes the form

$V_{ij}^{dd}(\textbf{r})=\frac{3g_{ij}^{dd}}{4\pi}\left(\frac{1-3\cos^{2}\theta}{r^{3}}\right),$
(2)

where $g_{ij}^{dd}=2\pi\hbar^{2}a_{ij}^{dd}/m_{ij}$ is the DDI strength
between the atoms of $i$’th and $j$’th species, with the DDI length
$a_{ij}^{dd}=\mu_{0}\mu_{i}^{m}\mu_{j}^{m}m_{ij}/6\pi\hbar^{2}$, and $\theta$
is the angle between the axis linking the two particles and the dipole
polarization direction ($z$-axis). The last term appearing in Eq.(1)
represents the correction to the chemical potential resulting from the effect
of quantum fluctuation given by [64, 65, 68]

$\Delta\mu_{i}=\frac{m_{i}^{3/2}}{3\sqrt{2}\pi^{2}\hbar^{3}}\sum_{\pm}\int_{0}^{1}\differential
u~{}\Re{I_{i\pm}},$ (3)

where

$\displaystyle I_{1\pm}=\bigg{(}\tilde{U}_{11}\pm$
$\displaystyle\frac{\delta_{1}\tilde{U}_{11}+2\tilde{U}_{12}^{2}n_{2}}{\sqrt{\delta_{1}^{2}+4\tilde{U}_{12}^{2}n_{1}n_{2}}}\bigg{)}\bigg{(}n_{1}\tilde{U}_{11}+$
$\displaystyle
n_{2}\tilde{U}_{22}\pm\sqrt{\delta_{1}^{2}+4\tilde{U}_{12}^{2}n_{1}n_{2}}\bigg{)}^{3/2},$
(4)

with $\delta_{1}=n_{1}\tilde{U}_{11}-n_{2}\tilde{U}_{22}$, and
$\tilde{U}_{ij}(u)=g_{ij}[1+\epsilon_{ij}^{dd}(3u^{2}-1)]$, being the Fourier
transform of the total interaction potential and the dimensionless parameter
$\epsilon_{ij}^{dd}=a_{ij}^{dd}/a_{ij}$, quantifies the relative strength of
DDI to the contact interaction between the atoms in species-$i$ and $j$. A
similar expression for $\Delta\mu_{2}$ can be easily obtained with
$\delta_{2}=-\delta_{1}$. The order parameters of each of the condensates are
normalized to the total number of atoms in that species
$N_{i}=\int\differential\textbf{r}\absolutevalue{\psi_{i}(\textbf{r})}^{2}$.

###  II.1 Overlap integral

A binary dBEC can exhibit a miscible or immiscible phase. A well-known measure
to characterize these two phases is the overlap integral, defined as

$\Lambda=\frac{\left[\int
d\textbf{r}n_{1}(\textbf{r})n_{2}(\textbf{r})\right]^{2}}{\left[\int
d\textbf{r}n_{1}^{2}(\textbf{r})\right]\left[\int
d\textbf{r}n_{2}^{2}(\textbf{r})\right]},$ (5)

where $n_{i}(\textbf{r})=\absolutevalue{\psi_{i}(\textbf{r})}^{2}$ is the
densities of the species-$i$. $\Lambda=1$ implies maximal spatial overlap
between the condensates, i.e., the system is in a completely miscible state.
Whereas, a complete phase separation (immiscible phase) corresponds to
$\Lambda=0$.

##  III Groundstate phases of a balanced mixture

To illustrate the groundstate properties and explore different phases of a Dy-
Dy mixture111We consider both the species have equal mass m. This is a good
approximation for the mixture of 162Dy, 164Dy (suitable for experiments) with
a relative difference between mass extremes of less than 2%., we find that the
intriguing parameters are the intra- and inter-component scattering lengths
$(a_{ii},a_{ij})$, the trap aspect ratio ($\lambda$) and the number of atoms
in the condensate ($N_{i}$). Here, we consider a balanced mixture with equal
intra-species interactions ($a_{11}=a_{22},~{}a_{11}^{dd}=a_{22}^{dd}$) and an
equal number of particles in each species ($N_{1}=N_{2}=N$). We first evaluate
the groundstate of the binary mixture as a function of intra- and inter-
component scattering lengths and the number of atoms in the species $i=1,2$
keeping the trap aspect ratio fixed at $\lambda=2.95$. Subsequently, we also
investigate the effect of trap geometry on the groundstate phases by varying
the trap aspect ratio with the intra-species scattering length for a fixed
number of particles and inter-species scattering length.

A binary mixture can be either in a miscible or immiscible phase. We
differentiate the miscible and immiscible phases by numerically evaluating the
overlap integral $\Lambda$ (Eq. (5)). In the large $N$ limit, the effect of
quantum pressure is negligible compared to the non-linear interactions, and
the condensate can be well approximated by the Thomas-Fermi (TF)
approximation. Thus immiscibility is completely determined by the intra- and
inter-component scattering lengths for a balanced system (where we can apply
SSA, see the Appendix B) and the transition occurs when
$a_{12}=\sqrt{a_{11}a_{22}}$. However, when both condensates consist of a
small number of particles, quantum pressure makes a significant contribution
to the condition of immiscibility transition. Quantum pressure of individual
species is $P_{i}=\frac{\hbar^{2}}{2m_{i}\sqrt{n_{i}}}\nabla^{2}\sqrt{n_{i}}$,
where $n_{i}$ is the density of the species-$i$. This pressure describes the
attractive force due to spatial variation of density, which becomes maximum at
the interface when the two condensates are in an immiscible phase. As a
consequence, to minimize the quantum pressure energy for a small number of
particles, the miscible to immiscible transition boundary deviated from
$a_{12}=\sqrt{a_{11}a_{22}}$ and the binary system favors the miscible state
more, as can be seen in Fig. 1(a) and 1(b).

Due to the anisotropic DDI, the SF, SS, and droplet (SD and MD) phases emerge
in a dBEC. These phases are best characterized by the density contrast
$\mathcal{C}=(n_{\rm max}-n_{\rm min})/(n_{\rm max}+n_{\rm min})$ [60], where
$n_{\rm max}$ and $n_{\rm min}$ are the neighbouring maximum and minimum
densities as one moves on the $x-y$ plane (a plane perpendicular to the
polarization direction). This allows us to depict different phase domains in
the phase diagrams, where we take $\mathcal{C}=0$ to be a superfluid phase,
and consider $0<\mathcal{C}<0.96$ to be a supersolid and $\mathcal{C}>0.96$ as
a droplet state [60]. For a detailed discussion on the density contrast see
Appendix C.

###  III.1 Phsae diagrams of binary dipolar condensate

####  III.1.1 Intra-species scattering length $(a_{ii})$ vs. population $(N)$

Here we construct a groundstate phase diagram with the intra-species
scattering length $a_{ii}~{}(i=1,2)$ and the number of particles
$N_{1}=N_{2}=N$ for a constant inter-species scattering length $a_{12}$ (see
Fig. 1(a)). To demonstrate the phase diagram, we fix the inter-species
scattering length at $a_{12}=90a_{B}$ and vary the intra-component scattering
length $a_{ii}$ from $60a_{B}$ to $120a_{B}$, and the number of atoms $N$ from
$10^{3}$ to $10^{5}$ of each species. The balanced binary mixture remains in a
miscible phase for a large value of $a_{ii}(>a_{12})$. Miscible to immiscible
transition for large number of particles, $N>4\times 10^{4}$, occurs at
$a_{ii}=90a_{B}$ $(a_{12}=\sqrt{a_{11}a_{22}})$. However as mentioned above,
for $N<4\times 10^{4}$, this transition occurs at smaller $a_{ii}(<90a_{B})$.
This transition is indicated by the solid white line corresponding to
$\Lambda=0.5$ in Fig. 1(a). For sufficiently large $a_{ii}$, due to the
dominated short-range contact type interactions over the DDI, the binary
dipolar mixture remains in a miscible SF state. It corresponds to a smooth
(non-modulated) quasi 2D TF density distribution with a low peak density (see
Fig. 2($\rm d$)). As we decrease $a_{ii}$ down to a critical value, each
component of the mixture undergoes an abrupt phase transition to a 2D SS state
(overlapping droplets) for a sufficiently large number of particles
$(N>2\times 10^{4}$). These droplets are coupled via a low-density superfluid.
In this regime, we get two coexisting miscible SS states (Fig. 2(c)) due to
$a_{ii}>a_{12}$. In contrast, for a small number of particles $N<2\times
10^{4}$, no droplet nucleation is observed in this regime, and both the
components of the binary mixture remain in a miscible superfluid state. When
further decreasing $a_{ii}$ below $90a_{B}$, two components become immiscible
due to comparatively large inter-component scattering length $(a_{12}>a_{ii})$
and the density overlap between the droplets vanishes rapidly. In this
sufficiently low $a_{ii}$ regime to minimize the DDI energy, atoms of each
species form multiple separate domains along the axial direction and the
binary mixture forms a multi-domain droplet state. Since we have taken a
balanced mixture with equal intra- and inter-species interaction strength, the
binary mixture forms a symmetric immiscible droplet state. For a small number
of particles, we observed an immiscible single droplet state (SD) (Fig. 1(c)).
In the case of a sufficiently large number of particles, we obtain an
immiscible multiple droplet state (MD) (Fig. 1(d)).

Figure 1: Groundstate phase diagrams in an oblate harmonic trap of a balanced
Dy-Dy mixture as a function of the number of particles and (a) intra- and (b)
inter-component s-wave scattering length. The colorbar corresponds to the
value of overlap integral $\Lambda$. The white solid contour has been drawn at
$\Lambda=0.5$ and it indicates the phase transition from a miscible to an
immiscible groundstate. The other two white dashed and dash-dotted lines are
drawn at $\Lambda=0.9$ and $\Lambda=0.1$, respectively. Whereas the black
dashed lines separate the phase diagram in superfluid (SF), supersolid (SS),
and single, multiple insulating droplets (SD, MD) states. Some examples of
immiscible 3D isosurfaces of (c) SD ( ), (d) MD ( ), (e) SS ( ), and (f) SF (
) states are highlighted by the symbols demarked in the groundstate phase
diagrams (a) and (b). Results are for the case of $N_{1}=N_{2}=N$, (a)
$a_{12}=90a_{B}$, (b) $a_{ii}=90a_{B}$ with
$a_{ii}^{dd}=131a_{B}~{}(\mu_{i}^{m}=9.93\mu_{B})$ where $a_{B}$ is the Bohr
radius, and $(\omega_{x},\omega_{y},\omega_{z})=2\pi\times(45,45,133)$Hz.
Figure 2: Groundstate density profiles of a miscible (a) SD ( ), (b) MD ( ),
and (c) SS ( ), and (d) SF ( ) states, are highlighted by the corresponding
marker demarked in the groundstate phase diagrams (see Fig.1(a), (b)). $n_{i}$
is the density of the species-$i$, where $i=1,2$ (both the components have
identical shapes). The first and second row represents the integrated density
profile in the $xy$ ($n_{i}(x,y)=\int\differential z~{}n_{i}(x,y,z)$) and $xz$
($n_{i}(x,z)=\int\differential y~{}n_{i}(x,y,z)$) planes, respectively.
Colorbar represents the density of each species in units of $10^{3}\mu
m^{-2}$. Results are for the case of a Dy-Dy mixture with
$N_{1}=N_{2}=N,~{}a_{11}=a_{22},~{}a_{ii}^{dd}=131a_{B}~{}(\mu_{i}^{m}=9.93\mu_{B}),$
and $\\{\omega_{x},\omega_{y},\omega_{z}\\}=2\pi\times\\{45,45,133\\}$Hz.
Other parameters are (a) $a_{ii}=90a_{B},~{}a_{12}=70a_{B},~{}N=3\times
10^{4}$, (b) $a_{ii}=90a_{B},~{}a_{12}=70a_{B},~{}N=6\times 10^{4}$, (c)
$a_{ii}=100a_{B},~{}a_{12}=90a_{B},~{}N=6\times 10^{4}$ and (d)
$a_{ii}=116a_{B},~{}a_{12}=90a_{B},~{}N=4\times 10^{4}$.

####  III.1.2 Inter-species scattering length $(a_{12})$ vs. population $(N)$

Now, in case of a fixed intra-component scattering length $(a_{ii}=90a_{B})$,
we construct a groundstate phase diagram (see Fig. 1(b)) by varying the inter-
species scattering length $a_{12}$ and the number of atoms $N$ in each
species. For a sufficiently large $a_{12}(>80a_{B})$ and a small number of
particles, the stationary state solution of the dipolar mixture is a miscible
SF state. The increase in the number of particles $N$ induces a transition to
an immiscible SF regime. In this case, since we have taken a balanced mixture,
there is no preference over which one particular component remains at the
center. So the groundstate of the balanced binary mixture has one domain of
each species and is separated along the $x-y$ plane (radial direction),
producing an asymmetric immiscible SF state (see Fig. 1(f)). As we further
increase the number of particles ($N>4\times 10^{4}$), the smooth non-
modulated density profile of each domain undergoes a phase transition and each
species develops a periodic density modulated pattern along the $x-y$ plane.
The density humps (droplets) are connected by lower-density regions
(superfluid). Both species unveil SS properties. However as we discussed
above, in this phase regime due to large $a_{12}~{}(>a_{ii})$, the phase of
the binary mixture is radially separated and we obtain an asymmetric
immiscible SS state (Fig. 1(e)). At a lower $a_{12}(<a_{ii})$, the density
overlap between the droplets in each species vanishes completely. Furthermore,
depending on the number of particles, the binary system displays a miscible SD
(small number of particles) and MD (large number of particles) state as
portrayed in Fig. 2(a) and 2(b).

Figure 3: Shows the groundstate phase diagram of a Dy-Dy mixture as a function
of intra-component s-wave scattering length ($a_{ii}$) and trap aspect ratio
($\lambda$). The shading represents the peak density of the combined binary
system. The black dashed lines indicate phase boundaries between the
superfluid state (SF) with a low-peak density, a supersolid (SS) and a droplet
state (SD and MD) with a high peak density. The large black dot encircled by a
red border demarks the critical point (CP) and the white solid line has been
drawn at $\Lambda=0.5$, representing the immiscibility phase boundary. Results
are for the case of $N_{1}=N_{2}=6\times 10^{4}$, 164Dy atoms with
$a_{ii}^{dd}=131a_{B}~{}(\mu_{i}^{m}=9.93\mu_{B})$ where $a_{B}$ is the Bohr
radius and $\mu_{B}$ is Bohr magneton.

####  III.1.3 Intra-species scattering length vs. trap aspect ratio
($\lambda$)

So far, we have discussed the effect of intra- and inter-species contact
interactions on the groundstate of a binary dipolar mixture for different
numbers of atoms. However, the trap aspect ratio $\lambda=\omega_{z}/\omega$
(trap geometry) is also one of the key parameters to explore different
possible groundstate phases. Trap geometry influences the condensate shape as
well as the DDI energy. The average DDI energy changes from negative to
positive as the shape of the condensate changes from prolate to oblate. To
construct a phase diagram with $\lambda$ and intra-species scattering length
$a_{ii}(i=1,2)$, we fix the inter-species scattering length at
$a_{12}=90a_{B}$ and number of particles at $N_{1}=N_{2}=6\times 10^{4}$. In
Fig. 3, we plot the peak density corresponding to the groundstate of a binary
mixture as a function of $\lambda$ and $a_{ii}$. The peak density results
emphasize a significant change in the density among the SF and SS, droplet (SD
and MD) phases (The SS and droplet phases are approximately two orders of
magnitude denser than the SF phase). We demark all these phase boundaries by
black dashed lines. All these phase transition lines terminate at a critical
point (CP). Beyond this critical point, there is no abrupt phase transition.
Rather a smooth evolution among the above-mentioned phases is observed. A
similar kind of behavior was also observed for a single component dBEC [17].
Here, the immiscibility boundary is close to
$a_{12}=\sqrt{a_{11}a_{22}}=90a_{B}$ (as we discussed earlier for a large
number of atoms) marked by the white solid line drawn at $\Lambda=0.5$ in Fig.
3. The region below ($a_{ii}<a_{12}$) and above $(a_{ii}>a_{12})$ the white
solid line corresponds to the phase-separated (immiscible) and miscible phase
domains, respectively.

Figure 4: Miscible and immiscible SD state in an imbalanced mixture.
$\rm(a_{1}),~{}(a_{2})$ Shows the integrated axial density of species-1
(violet) and species-2 (green) of a miscible and immiscible SD state,
respectively. $\rm(b_{1}),~{}(b_{2})$ Show the corresponding chemical
potential densities along the axial direction (z-axis). Results are for the
case of $a_{11}=70a_{B},~{}a_{22}=80a_{B},~{}N_{1}=10^{4},~{}N_{2}=5\times
10^{3}$ and $(\rm a_{1}),(b_{1})$ $a_{12}=65a_{B}~{}(<\sqrt{a_{11}a_{22}})$,
$(\rm a_{2}),(b_{2})$ $a_{12}=75a_{B}~{}(>\sqrt{a_{11}a_{22}})$. In both cases
the binary mixture is confined in an oblate shape harmonic trap with
$(\omega_{x},\omega_{y},\omega_{z})=2\pi\times(45,45,133)$Hz.

##  IV Supersolid and droplets state in an imbalanced mixture

Now we consider an imbalanced binary mixture, where the intra-species
interactions and number of particles among the components are not equal
$(a_{11}\neq a_{22}$ and $N_{1}\neq N_{2}$). In addition to all the possible
groundstates discussed so far, some mixed states like a mixture of SS-SF, SS-
MD, and SS-SS states are formed in this case. Here, we consider a Dy-Dy
mixture with intra-species scattering lengths $a_{11}=70a_{B}$,
$a_{22}=80a_{B}$, and the condensates contain $N_{1}=10^{4}$ and
$N_{2}=5\times 10^{3}$ number of atoms. With these chosen values of
parameters, the binary mixture undergoes a miscible SD to immiscible SD phase
transition beyond $a_{12}\approx 75a_{B}$. To look into these miscible and
immiscible SD states of the imbalanced mixture, we depict the integrated
density profiles $n_{1z}$ and $n_{2z}$ of both species along the axial
direction in Figs. 4$(\rm a_{1})$ and 4$(\rm a_{2})$, respectively. In the
first scenario with $a_{12}=65a_{B}$ ($<\sqrt{a_{11}a_{22}}$), the density
profiles of both the species completely overlap with each other and form a
miscible SD state (see Fig. 4$(\rm a_{1}))$. However, as we increase $a_{12}$
beyond the miscible to immiscible transition value, species-1 (major
component) remains at the center, due to its larger population (atom number)
and smaller intra-species scattering length. The species-2 (minor component)
is pushed along the axial direction and resides at each extreme end of the
domain formed by the major component (see Fig. 4$(\rm a_{2}))$. See Appendix D
for the discussion on the effective potential experienced by each species due
to the presence of the other component.

Figure 5: Shows the 3D isosurface density profiles of (species-1 (red),
species-2 (violet)) immiscible groundstates in an imbalanced binary mixture,
drawn at $10\%$ of the respective peak densities. ($\rm a_{1}$) Immiscible SD
state along the $z$ direction, parameters remain same as of Fig. 4($\rm
a_{2}$). $\rm(a_{2}){-}(a_{4}$) show various mixed ground states like ($\rm
a_{2}$) SS-SF, ($\rm a_{3}$) SS-MD, and ($\rm a_{4}$) SS-SS states in $x-y$
plane. $\rm(b_{1}){-}(b_{4})$ show different pattern formation at the
interface of two species in an imbalanced mixture. Results are for the case of
a binary dBEC confined in a harmonic trap with
$\\{\omega_{x},\omega_{y},\omega_{z}\\}=2\pi\times\\{45,45,133\\}$ Hz,
$\rm(a_{2}){-}(a_{4})$ $a_{11}=90a_{B},~{}a_{22}=95a_{B},~{}a_{12}=95a_{B},$
and $N_{1}=6\times 10^{4}$, and for $\rm(b_{1}){-}(b_{4})$
$a_{11}=90a_{B},~{}a_{22}=100a_{B},~{}a_{12}=100a_{B}$. Other parameters are
$(\rm a_{2})$ $N_{2}=5\times 10^{3}$, $(\rm a_{3})$ $N_{2}=2\times 10^{4}$,
$(\rm a_{4})$ $N_{2}=5\times 10^{4}$ $(\rm b_{1})$ $N_{1}=6\times
10^{4},~{}N_{2}=10^{3}$, $(\rm b_{2})$ $N_{1}=6\times 10^{4},~{}N_{2}=5\times
10^{3}$, $(\rm b_{3})$ $N_{1}=8\times 10^{4},~{}N_{2}=1.5\times 10^{4},$ and
$(\rm b_{4})$ $N_{1}=10^{5},~{}N_{2}=3\times 10^{4}.$

The reason behind these kinds of density distributions can be clearly
understood from the chemical potential densities along the axial direction
($\mu_{z}$) of each species as shown in Figs. 4($\rm b_{1},~{}b_{2}$). In the
miscible SD state, the chemical potential densities of each component are
negative indicating that both components are self-bound. Despite having a
different number of particles and intra-species scattering lengths, the large
negative chemical potential of the major component sets the spatial width of
both species equal (see Fig. 4($\rm b_{1}$)). The chemical potential of each
species increases with $a_{12}$. In the immiscible SD and MD regimes, the
chemical potential density of the minor component becomes positive. However,
due to the negative chemical potential density of the major component along
the axial direction, the minor component is bound at each end of the domain
formed by the major component (Fig. 4($\rm b_{2}$)). We have shown the
corresponding 3D isosurface density profile of the immiscible SD state in Fig.
5($\rm a_{1}$). In absence of the major component, the minor component can not
bind itself in these axial positions. The total chemical potential of the
binary mixture in this state is still negative, which

Figure 6: (a)-(b) Shows the effective potential due to the presence of the
other component of a SS-SF mixed state in the x-y plane. Colorbar represents
the effective potential in units of $\hbar\omega_{x}$. (c) Shows the
corresponding chemical potential densities of species-1 (red), and species-2
(light-green) along the x-axis. Results are for the case of
$a_{11}=90a_{B},~{}a_{22}=100a_{B},~{}a_{12}=100a_{B},~{}N_{1}=10^{5},~{}N_{2}=3\times
10^{4}$, and the imbalanced binary mixture is confined in a harmonic trap
potential with $(\omega_{x},\omega_{y},\omega_{z})=2\pi\times(45,45,133)$Hz.

implies that together they form a self-bound immiscible droplet state.

Mixed groundstates can be formed when both the condensates of the binary
mixture have comparatively large intra-species scattering lengths and form
partially or completely phase-separated (immiscible) groundstates. Various
groundstates of mixed phases like SS-SF, SS-MD and SS-SS formed in a binary
dBEC depending upon the number of atoms, intra- and inter-species scattering
lengths. The 3D isosurface density profiles of these mixed states are shown in
Fig. 5($\rm a_{2}-a_{4}$). In this regime, beyond a critical value of $a_{11}$
and $a_{22}$ (here we consider $a_{11}<a_{22}$), both the components have a
slightly positive chemical potential. Further, the first species with smaller
intra-species interaction and larger number of atoms occupies the central
position of the trap, similar to the previous case. However, due to the
positive chemical potential, it (the major component) can not hold the second
species at each end along the axial direction. Rather in the presence of a
harmonic trap, the minor

Figure 7: Tuning the inter-species scattering lengths ($a_{12}$) for the fixed
values of $a_{ii}=90a_{B}$. Panel (a) shows the linear ramp with ramp time
$t_{on}=100$ms. From an initial value of $a_{12}=70a_{B}$, the inter-species
scattering lengths are tuned to the final values $a_{12}=95a_{B}$ and
$a_{12}=100a_{B}$, respectively, through this ramp. Panel (b) shows the
corresponding variation of the overlap integral with time. ($\rm c_{1}-f_{3}$)
Shows the snapshots of the density profile of species-1 $n_{1}(x,z,y=0)$ ($\rm
c_{1}-f_{1}$), species-2 $n_{2}(x,z,y=0)$ ($\rm c_{2}-f_{2}$), and of the
total system $n(x,z,y=0)$ ($\rm c_{3}-f_{3}$), following a quench from a
miscible MD state with $a_{12}=70a_{B}$ to an immiscible MD state with
$a_{12}=100a_{B}$. Both the condensate consist of $N=6\times 10^{4}$ atoms,
and the balanced binary mixture is confined in a harmonic trap with
$(\omega_{x},\omega_{y},\omega_{z})=2\pi\times(45,45,133)$Hz. The colorbar
denotes the density in units of $\mu\rm m^{-2}$.

component is pushed along the radially outward direction as illustrated in
Figs. 5$(\rm(a_{2}){-}(a_{4}))$ for the scattering lengths $a_{11}=90a_{B}$,
$a_{22}=95a_{B}$ and $a_{12}=95a_{B}$ and the species-1 contains
$N_{1}=6\times 10^{4}$ number of atoms. For different numbers of atoms in the
second species $N_{2}=10^{3},~{}2\times 10^{4}$ and $5\times 10^{4}$ we
observe different mixed phases like SS-SF (Fig. 5$(\rm a_{2})$), SS-MD (Fig.
5$(\rm a_{3})$) and SS-SS (Fig. 5$(\rm a_{4})$), respectively. The effective
potential experienced by each species due to the presence of the other species
plays a crucial role in determining the position of the condensates in the
trap. In Figs. 6(a), 6(b), we have shown the effective potential experienced
by each species in the $x-y$ plane for a SS-SF mixed state corresponding to
the density profile as shown in Fig. 5($\rm b_{4}$). The corresponding
chemical potential densities along the $x$-axis ($\mu_{x}$) are also shown in
Fig. 6(c). As we explained earlier, both the condensates have positive
chemical potential densities along the $x$-axis (see Fig. 6(c)). Moreover, the
first species experiences a minimum effective potential at the trap center
while the second species finds the same at the periphery of the first
condensate and forms a radially immiscible mixture.

Interestingly enough, in an SS-SF mixed state, various polygonal shape
patterns form at the interface of the two species depending upon the number of
droplets in the SS state, as shown in Figs. 5$\rm((b_{1}){-}(b_{4}$)). The
number of droplets can be varied by changing either the number of atoms or the
intra-species scattering lengths. For the visualization of these polygonal
patterns, we choose the intra-, and inter-species scattering lengths to be
$a_{11}=90a_{B},~{}a_{22}=100a_{B}$ and $a_{12}=100a_{B}$, and the number of
atoms $(N_{1},N_{2})$ to be $(6\times 10^{4},~{}10^{3}),~{}(6\times
10^{4},~{}5\times 10^{3}),~{}(8\times 10^{4},~{}1.5\times
10^{4}),~{}\rm{and}~{}(10^{5},3\times 10^{4})$ which are corresponding to the
triangular (Fig. 5($\rm b_{1}$)), rectangular (Fig. 5($\rm b_{2}$)),
pentagonal (Fig. 5($\rm b_{3}$)) and hexagonal (Fig. 5($\rm b_{4}$)) shapes
patterns, respectively, at the interface.

##  V Dynamics

So far we have discussed the phase diagrams of a balanced binary mixture and
different possible groundstates in an imbalanced binary mixture. Now we
explore the effect of tuning intra- and inter-species scattering lengths of a
balanced binary mixture in real-time dynamics. Consider the first case where
we initially prepare the dBEC in a miscible MD regime, with
$a_{ii}=90a_{B},~{}a_{12}=70a_{B}~{}\rm{and}~{}N_{1}=N_{2}=6\times 10^{4}$. We
then perform two different slow linear ramps for increasing the value of
$a_{12}$, one from $70a_{B}$ to $95a_{B}$, and the other from $70a_{B}$ to
$100a_{B}$ over a ramp time $t_{on}=100$ ms. After that, the inter-species
scattering length $a_{12}$ is kept constant to check the stability of the
evolved system (see Fig. 7(a)). We find both the evolutions produce
dynamically stable droplets, and these results are also consistent with the
formation of a self-bound droplet state in a trap-less system [68, 65]. In
Fig. 7(b) and 7(c), we have shown the time evolution of the overlap integral
($\Lambda$) and the density profile of each species in the $x-z$-plane
($n_{i}(x,z,y=0)$), respectively.

Figure 8: Shows the time evolution of the momentum space density ($\rm a_{1}$)
of species-1 $\tilde{n}_{1}(k_{x})$, ($\rm a_{2}$) species-2
$\tilde{n}_{2}(k_{x})$, and ($\rm a_{3}$) of the total system
$\tilde{n}(k_{x})$, following a linear quench with ramp time $t_{on}=100$ms
from a miscible SF state with $a_{ii}=120a_{B}$ to a immiscible MD state with
$a_{ii}=85a_{B}$. Both the condensate consists of $N=6\times 10^{4}$ atoms,
$a_{12}=90a_{B}$, and the balanced binary mixture is confined in a harmonic
trap with $(\omega_{x},\omega_{y},\omega_{z})=2\pi\times(45,45,133)$Hz. The
colorbar denotes the density in the $k_{x}$-space, in units of $10^{3}\mu\rm
m$.

Initially, while $a_{12}<a_{ii}$, the mixture forms a miscible MD state. As
soon as $a_{12}>a_{ii}$, the system undergoes a miscible to immiscible
transition. Near the transition time, the value of the overlap integral
rapidly changes from 1 to 0. Due to this sudden change, each component forms
multiple periodic segregated domains along the axial direction and forms a
completely phase-separated density profile. In this state, the density profile
of each component is complementary to the other, and together they form an
axially symmetric immiscible MD state.

Figure 9: Snapshots of the density profile of the composite binary system in
momentum space following a linear quench with ramp time $t_{on}=100$ms from a
miscible SF state with $a_{ii}=120a_{B}$ to an immiscible MD state with
$a_{ii}=85a_{B}$ keeping the inter-species scattering length $a_{12}$ fixed at
$90a_{B}$. (a) Initially at $t=0$ms, a single peak corresponds to the SF state
at $k_{x}=0~{}\mu\rm m^{-1}$, (b) at $t=55.7$ ms,appearance of two symmetric
roton peak at $k_{x}=\pm 3.05~{}\mu\rm m^{-1}$. The insets show the
corresponding density distribution $\tilde{n}(k_{x},k_{y},k_{z}{=}0)$ in
($k_{x},k_{y}$)-plane. Parameters are same as of Fig. 8.

Subsequently, we also explore the quench dynamics of a balanced binary
mixture, starting from a SF state in a miscible regime. In this case, dynamics
are triggered by reducing the intra-species scattering lengths into a miscible
and immiscible SS regime. For a fixed inter-species scattering length at
$a_{12}=90a_{B}$, we perform two interaction quenches by linearly reducing the
intra-species scattering length $a_{ii}$, one from $120a_{B}$ to $95a_{B}$,
and the other from $120a_{B}$ to $85a_{B}$ over a time period $t_{on}=100$ ms,
after which $a_{ii}$ is held constant (see Fig. 13(a) in Appendix E), and we
observe the time evolution of the binary system. In both quenching processes,
as $a_{ii}$ is reduced, the system undergoes a roton instability at
$a_{ii}\approx 100a_{B}$. In Fig. 8, we have shown the time evolution of
momentum space density $\tilde{n}(k_{x})$, following the quench
$a_{ii}=120a_{B}$ to $85a_{B}$. Initially, up to $t=55.7$ ms, the binary
mixture forms a miscible SF state which corresponds to a single density peak
at $(k_{x},k_{y})=(0,0)~{}\mu\rm m^{-1}$ (see Fig. 9(a)). Later following the
roton222The roton modes are characterized by the quantum number $m$ [70, 71,
72]. In the $(k_{x}-k_{y})$ plane, the roton population is spread over a ring
which corresponds to a radial roton mode with $m=0$. instability at the phase
boundary, a ring of radius $3.05~{}\mu\rm m^{-1}$ is readily visible in the
$k_{x}{-}k_{y}$ plane and for $k_{y}=0$, the density profile
$n(k_{x},k_{y}=0)$ corresponds to the appearance of two additional side peaks
in the momentum space (similar like a cigar-shaped trap geometry) (see Fig.
9(b)). The symmetric side peaks in the momentum space essentially indicate a
periodic density modulation in the real space. The binary dipolar mixture
forms a miscible SS state in the time interval $t=55.7$ ms
($a_{ii}=100.5a_{B}$) and $t=85.7$ ms ($a_{ii}=90a_{B}$). As the $a_{ii}$ is
reduced further the system enters into a MD phase domain and when
$a_{ii}<a_{12}$, the overlap integral ($\Lambda$) changes from 1 to 0 rapidly
(see Fig. 13(b)), and the system forms an immiscible MD state. The
characteristic density snapshots, while performing the quench of $a_{ii}$ from
$120a_{B}$ to $95a_{B}$, in the x-y plane are presented in Figs.
13($\rm(c_{1})-(e_{3})$ in Appendix E).

##  VI Conclusions and outlook

In this work, we have theoretically investigated the scope of formation of
two-dimensional supersolid and droplet lattice states in a binary dBEC. We
performed an in-depth investigation and demonstrated that a binary dipolar
mixture confined in a circular symmetric trap could exhibit a large variety of
groundstate phases with rich properties inaccessible for a non-dipolar binary
mixture and in a single component dBEC. The emergent phases include SF, SS,
SD, and MD states both in miscible and immiscible phase domains. The interplay
between intra-, inter-species contact interaction and the anisotropic dipole-
dipole interaction leads to the formation of all these phases. Numerically
solving the coupled eGPE, we obtain all these results. Besides the 3D
numerical simulation, we also employ a variational approach in the SSA
framework to validate our results. Although in this work we have demonstrated
the results for a Dy-Dy mixture and a specific range of atom-atom interaction
strengths, our analysis can be considered as one step forward in the direction
of the formation of more exciting new phases in binary dipolar BECs yet to be
revealed in ongoing and future research works.

We have examined different groundstate phases in a balanced binary dBEC, and
depicted the phase diagrams as a function of the number of particles, intra-,
and inter-species scattering lengths. We also monitor the effect of trap
geometry in terms of trap aspect ratio on the groundstate phases. More
intriguing mixed phases appear for an imbalanced mixture. In the miscible
phase domain, both condensates possess exactly identical shapes. Whereas in
the immiscible phase domain, we observe two types of immiscible phases: (i)
axially immiscible phase, and (ii) radially immiscible phase. The axially
immiscible phase for a self-bound droplet state without any trapping
confinement is predicted in some of the recent theoretical works. However, the
radially immiscible phases have not been reported so far to the best of our
knowledge.

In the self-bound immiscible droplet regime, due to the dominant anisotropic
dipole-dipole interaction, the component with a larger atom number and smaller
intra-species scattering length takes the central position of the droplet and
forms two potential minima at its two outer edges along the axial direction.
The second component with a slightly positive chemical potential energy is
docked at the above-mentioned position by the first component with a negative
chemical potential energy and forms an axially immiscible self-bound droplet
state. The chemical potential of each condensate increases with the increase
of intra- and inter-species scattering length. Hence in an immiscible regime
for a comparatively large value of intra-species scattering length, the
chemical potential of both the component becomes positive, and no longer the
major component can hold the minor component at the axial position. For an
imbalanced mixture in the presence of a circularly symmetric harmonic trap,
the minor component with a comparatively small number of atoms and large
intra-species scattering length is pushed along the radially outward direction
and forms a radially immiscible state. Depending on the value of intra-species
scattering lengths, each species can form a MD, SS, or SF state. Whereas for a
balanced system, none of the condensates have such biasness due to equal
interaction strength and hence forming a radially asymmetric phase-separated
state.

Utilizing our groundstate phase diagrams for a balanced binary mixture as a
reference, we explore the dynamics across the phase boundaries by tuning the
interaction strengths. The dynamical transition across the phase boundaries
initially governs some instability in the system, leading to the formation of
some metastable states in the intermediate time scale. In long-time dynamics,
we have shown the dynamical phase transition from a miscible droplet state to
an immiscible droplet state with multiple domains and a crossover from a SF
state to a MD droplet state via a SS state.

Our observations pave the way for several future research directions. In this
work, we have restricted our study to a particular Dy-Dy (homonuclear)
mixture. However, it would be intriguing to explore the formation of different
possible phases in a heteronuclear binary dipolar mixture like Er-Dy mixture
[61, 63, 62]. Furthermore, one straightforward option is to investigate the
lifetime of these phases by incorporating the effect of three-body interaction
loss [15, 34]. Another intriguing direction would be to consider the impact of
thermal fluctuation and unravel corresponding phases as well as dynamical
nucleation of the supersolid and droplet lattice in the finite temperature
limit [73, 74]. Moreover, the evaporation cooling mechanism is an alternative
approach to the interaction quench and provides the prospects of forming a
long-lived 2D supersolid state in a binary dipolar mixture [24]. Another vital
prospect would be to investigate quantum turbulence [75, 76, 77], pattern
formation [29, 78, 79, 80] and various topological excitations such as the
formation of vortex clusters and solitary waves in a binary dipolar
condensate. Finally, the observation discussed in this work would be equally
fascinating beyond the Lee-Huang-Yang description [81, 82].

###### Acknowledgements.

We thank Koushik Mukherjee and S.I.Mistakidis for fruitful discussions. We
also acknowledge the National Supercomputing Mission (NSM) for providing
computing resources of ‘PARAM Shakti’ at IIT Kharagpur, which is implemented
by C-DAC and supported by the Ministry of Electronics and Information
Technology (MeitY) and Department of Science and Technology (DST), Government
of India.

##  Appendix A Numerical Methods

Results in this work are based on three-dimensional numerical simulations in
the coupled eGPE (Eq. (1)) framework. For the sake of the convenience of
numerical simulations and better computational precision, we cast the coupled
eGPE into a dimensionless form. This is achieved by rescaling the length scale
and time scale in terms of oscillator length
$l_{osc}=\sqrt{\hbar/m\omega_{x}}$ and $\omega_{x}$ trapping frequency along
the x direction. Under this transformation, the wave function of species-$i$
obeys
$\psi_{i}(\mathbf{r}^{\prime})=\sqrt{l_{osc}^{3}/N_{i}}\psi_{i}(\mathbf{r})$,
where $N_{i}$ is the number of particles in species-$i$. After the
transformation of variables into dimensionless quantities the coupled eGPE is
solved by split-step-Crank-Nicolson scheme [83]. Since the dipolar potential
has a singularity at $r=0$ (see Eq. (2)), it is numerically evaluated in
Fourier space and we obtain the real space contribution through the
application of the convolution theorem. The groundstates of binary dipolar
condensate are obtained by propagating the relevant equations in imaginary
time until the relative deviations of the wave functions (calculated at every
grid point) and energy of each condensate between successive time steps are
less than $10^{-6}$ and $10^{-7}$, respectively. Furthermore, we fix the
normalization of each species at every time instant of the imaginary time
propagation. Using this groundstate solution as an initial state, at $t=0$,
and by changing the interaction strengths we monitor their evolution in real-
time. Our simulations are performed within a 3D box grid containing
$(256\times 256\times 256)$ grid points, with the spatial grid spacing
$\Delta_{x}=\Delta_{y}=\Delta_{z}=0.1l_{osc}$ while the time step
$\Delta_{t}=10^{-4}/\omega_{x}$.

Figure 10: Shows the variation of energy with the inter-species scattering
length $a_{12}$ of a balanced binary mixture. The mixture is confined in a
harmonic trap with
$(\omega_{x},\omega_{y},\omega_{z})=2\pi\times(45,45,133)$Hz and the other
parameters fixed at $a_{11}=a_{22}=90a_{B}$ and $N_{1}=N_{2}=10^{4}$ number of
atoms. The results are obtained through a variational approach within the SSA
framework (dashed line), and the coupled eGPE solutions (solid line) are in
agreement.

##  Appendix B Variational solution within same shape approximation (SSA)
framework

In addition to numerical 3D simulations of Eq. (1), we employ a simple
variational approach in the regime where both components are miscible and take
the exact same shape (i.e., $\psi_{1}=\psi_{2}$), and this is only possible
when both the condensate have an equal number of atoms and equal intra-species
interaction. In this regime, the Hamiltonian of the $i$’th condensate is
reduced to an effective single-species Hamiltonian given by

$\displaystyle\hat{\mathcal{H}}_{SSA}^{i}=-\frac{\hbar^{2}}{2m_{i}}\nabla^{2}+V_{t}(\textbf{r})+\mathcal{G}_{SSA}\absolutevalue{\psi_{i}(\textbf{r})}^{2}+$
$\displaystyle\mathcal{G}_{SSA}^{dd}\int
d\textbf{r}^{\prime}U^{dd}(\textbf{r}-\textbf{r}^{\prime})\absolutevalue{\psi_{i}(\textbf{r}^{\prime})}^{2}+\gamma_{SSA}^{QF}(\epsilon_{ii}^{dd})\absolutevalue{\psi_{i}(\textbf{r})}^{3}.$

Here $\mathcal{G}_{SSA}=g_{ii}+g_{12}$,
$\mathcal{G}_{SSA}^{dd}=g_{ii}^{dd}+g_{12}^{dd}$ are the effective strengths
of contact interaction and DDI, respectively. The last term of Eq. B denotes
the contribution of quantum fluctuation. We remark that within this SSA
framework, quantum fluctuations depend on the density $n_{i}^{3/2}$, where
$n_{i}=\absolutevalue{\psi_{i}}^{2}$. The coefficient of quantum fluctuations
$\gamma_{SSA}^{QF}$ is well approximated by the known form of a single-species
dBEC [84]:

$\displaystyle\gamma_{SSA}^{QF}(\epsilon_{SSA}^{dd})=$
$\displaystyle\frac{32}{3}\mathcal{G}_{SSA}\sqrt{\frac{a_{SSA}^{3}}{\pi}}$
$\displaystyle\Re{\int_{0}^{1}\differential
u[1+\epsilon_{SSA}^{dd}(3u^{2}-1)]^{5/2}},$

where $a_{SSA}=a_{ii}+a_{12}$ and the dimensionless parameter
$\epsilon_{SSA}^{dd}=\mathcal{G}_{SSA}^{dd}/\mathcal{G}_{SSA}$ quantifies the
effective relative strength of the DDI to the contact interaction. Within this
SSA framework, the total energy of the $i$’th species

$\displaystyle
E_{i}=\int\bigg{[}\frac{\hbar^{2}}{2m_{i}}\absolutevalue{\nabla\psi_{i}}^{2}+v_{t}(\textbf{r})\absolutevalue{\psi_{i}(\textbf{r})}^{2}+\frac{\mathcal{G}_{SSA}}{2}\absolutevalue{\psi_{i}(\textbf{r})}^{4}$
$\displaystyle+\frac{\mathcal{G}_{SSA}^{dd}}{2}\int
d\textbf{r}^{\prime}U^{dd}(\textbf{r}-\textbf{r}^{\prime})\absolutevalue{\psi_{i}(\textbf{r}^{\prime})}^{4}+\frac{2}{5}\gamma_{SSA}^{QF}\absolutevalue{\psi_{i}(\textbf{r})}^{5}\bigg{]}d\textbf{r}.$

Figure 11: Groundstate phase diagrams in an oblate harmonic trap of a Dy-Dy
mixture as a function of number of particles in each condensate and (a)
intra-, (b) inter-species scattering length. The color domains on the phase
plots correspond to the different contrast regimes as demarked above in the
phase diagrams. Other parameters remain same as of Fig. 1.

A qualitative and to some extent quantitative insight into the droplet and
supersolid physics in the miscible SSA regime may be gained from a simplified
Gaussian ansatz

$\psi_{i}(\textbf{r})=\sqrt{\frac{8N_{i}}{\pi^{3/2}\sigma_{\rho}^{2}\sigma_{z}}}\prod_{\eta=\rho,z}\exp(-\frac{2\eta^{2}}{\sigma_{\eta}^{2}}),$
(9)

where the variational parameters are the condensate widths $\sigma_{\eta}$ in
the $\eta=\rho,z$ direction. We insert the ansatz (9) into Eq. (B) and obtain

$\displaystyle\frac{E_{i}}{N_{i}\hbar\omega_{z}}$ $\displaystyle=$
$\displaystyle\left(\frac{2l_{z}^{2}}{\sigma_{\rho}^{2}}+\frac{l_{z}^{2}}{\sigma_{z}^{2}}\right)+\left(\frac{1}{8\lambda^{2}}\frac{\sigma_{\rho}^{2}}{l_{z}^{2}}+\frac{\sigma_{z}^{2}}{16l_{z}^{2}}\right)$
(10)
$\displaystyle+\frac{4N_{i}\left(\mathcal{G}_{SSA}-\mathcal{G}_{SSA}^{dd}f(\frac{\sigma_{\rho}}{\sigma_{z}})\right)}{(2\pi)^{3/2}\sigma_{\rho}^{2}\sigma_{z}\hbar\omega_{z}}$
$\displaystyle+\frac{128N_{i}^{3/2}\gamma_{SSA}^{QF}}{25\sqrt{5}\pi^{9/4}\sigma_{\rho}^{3}\sigma_{z}^{3/2}\hbar\omega_{z}},$

where,

$f(k)=\frac{1+2k^{2}}{1-k^{2}}-\frac{3k^{2}\tanh^{-1}{\sqrt{1-k^{2}}}}{(1-k^{2})^{3/2}},$
(11)

Figure 12: Shows the effective potential due to the presence of the other
component of an immiscible SD state in the x-z plane. Colorbar represents the
effective potential in units of $\hbar\omega_{x}$. Other parameters remain
same as of Fig. 4($\rm a_{2}$).

and $l_{z}=\sqrt{\hbar/m_{i}\omega_{z}}$. We find the stationary solutions by
numerically locating the values of $\sigma_{\rho}$ and $\sigma_{z}$ that
minimize the variational energy (10). However, the ansatz (9) is inappropriate
for immiscible and imbalanced droplets and supersolid states (see ([64]) for
alternative ansatz).

##  Appendix C Density contrast

The groundstate phase diagrams for a balanced binary mixture are depicted in
Fig. 1. The binary mixture can be in one of the three phases: a SF state, a SS
state with periodic density modulation, and a 2D array of isolated droplets.
These distinct phases are best characterized by the density contrast, defined
as [60]

$\mathcal{C}=\frac{(n_{\rm max}-n_{\rm min})}{(n_{\rm max}+n_{\rm min})}.$
(12)

Here $n_{max}$ and $n_{min}$ are the neighboring maxima and minima as one
moves on the x-y plane. A SF state corresponds to a smooth density
distribution with $n_{\rm max}=n_{\rm min}$ which implies $\mathcal{C}=0$. In
an insulating droplet state when there is no overlap between the droplets
($n_{\rm min}\approx 0$), the Eq. 12 gives $\mathcal{C}\approx 1$. Whereas in
a SS state, the droplets are connected by a low-density superfluid ($n_{\rm
min}\neq 0$) and the density contrast $\mathcal{C}$ attains an intermediate
value between 0 and 1. In this work, we consider [24] $\mathcal{C}=0$ to be a
superfluid phase, $0<\mathcal{C}<0.96$ to be a supersolid, and
$\mathcal{C}>0.96$ to be a droplet state. In Fig. 11, we plot the different
contrast ($\mathcal{C}$) regimes as a function of intra-, inter-species
scattering length and the number of particles.

##  Appendix D Effective potential

Each condensate experiences an effective potential due to the presence of the
other component. The effective potential experienced by species-$i$ due to the
presence of species-$j$ is given by,

$\displaystyle V_{eff}^{ij}(\textbf{r})=$ $\displaystyle
g_{ij}\absolutevalue{\psi_{j}(\textbf{r})}^{2}+\int
d\textbf{r}^{\prime}V_{ij}^{dd}(\textbf{r}-\textbf{r}^{\prime})\absolutevalue{\psi_{j}(\bf{r}^{\prime})}^{2}.$
(13)

Figure 13: Quenching intra-species scattering lengths for a fixed values of
$a_{12}=90a_{B}$. Panel (a) shows the linear ramp with ramp time
$t_{on}=100$ms. From an initial value of $a_{ii}=120a_{B}$, the intra-species
scattering lengths are quenched to a final value $a_{ii}=85a_{B}$ and
$a_{ii}=95a_{B}$, respectively through this ramp. Panel (b) shows the
corresponding variation of the overlap integral with time. The snapshots of
the density profile of species-1 $n_{1}(x,y,z=0)$ ($c_{1}-e_{1}$), species-2
$n_{2}(x,y,z=0)$ ($c_{2}-e_{2}$), and of the total system $n(x,y,z=0)$
($c_{3}-e_{3}$), following a quench from a miscible SF state
($a_{ii}=120a_{B}$) to a miscible SS state ($a_{ii}=95a_{B}$). Both the
condensate consists of $N=6\times 10^{4}$ number atoms, and the balanced
binary mixture is confined in a harmonic trap with
$(\omega_{x},\omega_{y},\omega_{z})=2\pi\times(45,45,133)$Hz. The colorbar
denotes the density in units of $\mu\rm m^{-2}$.

In the main text, we have shown the density profiles of an axially immiscible
SD state in an imbalanced binary mixture (see Fig. 4($\rm a_{2}$)). Where we
observe that the major333In an imbalanced binary mixture, the species with a
larger number of atoms is referred to as the major component and the other
species as the minor component. component with smaller $a_{ii}$ and a larger
population acquires the central position and the minor component with
comparatively larger $a_{ii}$ and a smaller number of atoms is bound at each
end along the axial direction. Here in Fig. 12, we have shown the
corresponding effective potentials experienced by each species due to the
presence of other species. The species-1 encountered a minimum potential at
the trap center (which is elongated along the axial direction (z-axis)),
whereas the second species experienced a maximum effective potential there but
a minimum effective potential at each end of the minimum effective potential
domain formed by the condensate-1.

##  Appendix E Quench dynamics

To track the emergent features of the intra-species interaction quench as we
discussed in the main text, here we have shown the time evolution of the
density profiles and the phase of the binary mixture. In Figs. 13(a), we have
shown two different interaction quenches of $a_{ii}$, one from $120a_{B}$ to
$95a_{B}$, and the other from $120a_{B}$ to $85a_{B}$. Following these
interaction quenches, the corresponding time evolution of the overlap integral
$\Lambda$ is shown in Fig. 13(b). Initially, at $t=0$ ms the mixture forms a
miscible SF state with a smooth 2D TF distribution (see Fig. 13($\rm
c_{1}-c_{3}$)) which corresponds to a global phase coherence as can be seen
from Fig. 14(a). However, since the quench is performed across the phase
boundary, it excites the roton instability in the binary system leading to
ring-shaped density structures as can be seen in Fig. 13($\rm d_{1}-d_{3}$).
The appearance of the roton mode is readily visible in the momentum space. Due
to the circular symmetry of the trap geometry
($\omega_{x}=\omega_{y}=\omega$), the roton population is spread over a ring
in the $k_{x}-k_{y}$ plane and for $k_{y}=0$ it corresponds to the appearance
of two prominent side peaks as discussed in the main text (see Fig. 9(b)).

Figure 14: Characteristic phase profiles (a)-(c) of the composite binary
mixture in the x-y plane at a specific time instant following a linear quench
with ramp time $100$ms from a miscible SF state with $a_{ii}=120a_{B}$ to an
immiscible MD state with $a_{ii}=85a_{B}$ keeping the inter-species scattering
length ($a_{12}$) fixed at $90a_{B}$. The black dashed circles delineate the
edges of the dipolar binary mixture. The phase profiles correspond to (a)
miscible SF state, (b) miscible SS state, and (c) immiscible MD state. The
parameters are same as in the Fig. 8.

As we decrease the intra-species scattering lengths more, the ring-shaped
density structure breaks into several overlapping density humps (droplets) and
the binary mixture forms a miscible SS state (Fig. 13($\rm e_{1}-e_{3}$)).
This SS state corresponds to almost a perfect global phase coherence with a
very small fluctuation in the phase observed due to the interaction quench
performed across the phase boundary (see Fig. 14(b)). Instead of interaction
quench by the evaporative cooling mechanism directly into the SS state, one
could produce a SS state with robust global phase coherence as demonstrated in
[24]. Further decreasing the $a_{ii}$, the phase coherence between these
droplets is completely lost (see Fig. 14(c)) and the binary mixture forms a 2D
array of immiscible MD crystal (not shown here).

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