# Merging Dipolar Supersolids in a Double-Well Potential

Hui Li  JILA, NIST and Department of Physics, University of Colorado, Boulder,
Colorado 80309, USA<EMAIL_ADDRESS>Eli Halperin  JILA, NIST and
Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
Shai Ronen  Cleerly Inc., 110 16$th$ Street, Suite 1400, Denver, CO, 80218,
USA  John L. Bohn  JILA, NIST and Department of Physics, University of
Colorado, Boulder, Colorado 80309, USA

###### Abstract

We theoretically investigate the merging behaviour of two identical
supersolids through dipolar Bose-Einstein condensates confined within a
double-well potential. By adiabatically tuning the barrier height and the
spacing between the two wells for specific trap aspect ratios, the two
supersolids move toward each other and lead to the emergence of a variety of
ground state phases, including a supersolid state, a macrodroplet state, a
ring state, and a labyrinth state. We construct a phase diagram that
characterizes various states seen during the merging transition. Further, we
calculate the force required to pull the two portions of the gas apart,
finding that the merged supersolids act like a deformable plastic material.
Our work paves the way for future studies of layer structure in dipolar
supersolids and the interaction between them in experiments.

##  I Introduction

The formation of exotic quantum phases of matter unveils rich and fascinating
phenomena [1, 2, 3, 4, 5]. In particular, the emergence of supersolidity has
garnered significant attention [6, 7, 8, 9, 10, 11]. Supersolids exhibit both
spatial periodicity and frictionless flow, which are usually incompatible in
classical systems. Initially, supersolidity was proposed to arise in solid
4He, however, despite some controversial claims, no conclusive evidence of
supersolidity has been found in helium so far [12, 13].

Alternatively, ultracold atomic gases have become a platform for realizing and
studying supersolids [14, 15, 16, 17]. In particular, dipolar Bose-Einstein
condensates (BECs), composed of atoms with large magnetic or electric dipole
moments, offer a unique opportunity to engineer the interatomic interactions
which are crucial for supersolid formation. Dipolar interactions are long-
ranged and anisotropic, factors which can induce density modulations and roton
excitations in the BECs. Moreover, dipolar interactions can be tuned by
applying external fields or by changing the geometry of the system [18, 19,
20, 21, 3, 22].

Dipolar BECs have been achieved in highly magnetic atoms, such as chromium
[23], europium [24], dysprosium [25], and erbium [26]. Recent experiments have
observed the appearance of a supersolid phase in elongated one-dimensional
(1D) [27, 28, 29, 30] and 2D [31, 32] geometries with weakly magnetic dipole-
dipole interactions (DDIs) in these dipolar BECs.

Additionally, ultracold bosons confined in a bilayer have drawn much attention
because it is possible to control the inter- and intra-layer couplings [33].
For example, a BEC loaded into spin-dependent optical lattices, forming a
moiré structure, is the ideal scenario to investigate the physics behind
superconductivity in twisted-bilayer graphene [34]. Besides, dipolar gases in
a layer-structure have been achieved with the help of an optical lattice
potential [35, 36, 37, 38] and stack supersolid structures have been discussed
theoretically recently for an antidipolar single-component condensate [39],
mixture of antidipolar and nondipolar condenstaes [40] and a doubly dipolar
condensate with both electric and magnetic dipole moments [41] confined in a
standard harmonic trap.

Double-well potentials serve as a versatile platform in the study of
microscopic media, especially for atomic BECs, such as the quantum dynamics of
ultracold atoms [42, 43, 44, 45], self-trapping [46, 47, 48], stabilization of
purely dipolar BEC and formation of quasi 2D sheets [49], and spin-squeezing
[50]. A typical double well potential has been produced in experiments by
adiabatic radio frequency-induced splitting [51, 52, 53, 54] to study the
dynamics of a splitting or merging process of BECs.

In this work, we combine concepts from dipolar BEC and double-well potentials.
We propose applying a double well potential in the polarization direction of
the atoms, then systematically bringing the two wells together while reducing
the barrier to merge two supersolids. Along the way, many intermediate
morphologies are seen.

The paper is organized as follows. In Sec. II, we outline the beyond mean-
field theory and the numerical strategy and introduce our system. Sec. III
investigates the merging process in two cases, distinguished by the
supersolids in the separated-well limit. Subsection III.1 considers
supersolids in a trap with aspect ratio $\omega_{z}=2\omega$ prior to merging,
while subsection III.3 considers a slightly different case, with initial
aspect ratio $\omega_{z}=3\omega$. In both cases, the various morphologies are
presented and summarized in phase diagrams in subsections III.2 and III.3,
respectively. A summary along with the future perspectives are given in Sec.
IV.

##  II Theory model

###  II.1 Formalism

Here we consider a dipolar BEC composed of $N$ atoms of mass $m$, confined in
an external potential to be defined below. The ground-state wave function
$\psi(\mathbf{r})$ satisfies the extended Gross-Pitaevskii equation (eGPE),
which includes the effects of beyond-mean-field quantum fluctuations [55, 56,
57, 58]. The energy functional (energy per atom) of eGPE can be expressed as,

$\displaystyle E(\psi)=$ $\displaystyle\int
d\mathbf{r}\Big{[}\frac{\hbar^{2}}{2m}|\nabla\psi(\mathbf{r})|^{2}+V(\mathbf{r})|\psi(\mathbf{r})|^{2}+\frac{gN}{2}\left|\psi(\mathbf{r})\right|^{4}$
$\displaystyle+\frac{N}{2}|\psi(\mathbf{r})|^{2}\int
d\mathbf{r}^{\prime}U_{dd}(\mathbf{r}-\mathbf{r}^{\prime})|\psi(\mathbf{r}^{\prime})|^{2}$
$\displaystyle+\frac{2N^{3/2}}{5}\gamma_{\mathrm{QF}}|\psi(\mathbf{r})|^{5}\Big{]},$
(1)

where $\hbar$ is the reduced Planck constant, the wave function is normalised
as $\int|\psi(\mathbf{r})|^{2}d\mathbf{r}=1$. The short-range two-body
interaction coupling constant $g=4\pi\hbar^{2}a_{s}/m$ is governed by the
$s$-wave scattering length $a_{s}$, and $U_{dd}(\mathbf{R})$ accounts for the
anistropic and long-range DDIs given by

$\displaystyle
U_{dd}(\mathbf{R})=\frac{3\hbar^{2}a_{\mathrm{dd}}}{m}\frac{1-3\mathrm{cos}^{2}\theta}{|\mathbf{r}-\mathbf{r}^{\prime}|^{3}},$
(2)

with the dipole length $a_{\mathrm{dd}}=\mu_{0}\mu_{m}^{2}m/12\pi\hbar^{2}$,
$\mu_{0}$ is the vacuum permeability and $\mu_{m}$ is magnetic moment,
$\mathbf{R}\equiv\mathbf{r}-\mathbf{r}^{\prime}$, we take $\hat{z}$ as the
polarization axis, then $\theta$ denotes the angle between $\mathbf{R}$ and
$\hat{z}$. The coefficient $\gamma_{\mathrm{QF}}$ represents the dipolar Lee-
Huang-Yang correction within the local density approximation [59, 60, 61, 62,
63, 3], induced by quantum fluctuation,

$\displaystyle\gamma_{\mathrm{QF}}=\frac{32}{3}g\sqrt{\frac{a_{s}^{3}}{\pi}}\left(1+\frac{3}{2}\varepsilon_{\mathrm{dd}}^{2}\right),$
(3)

the dimensionless parameter $\varepsilon_{\mathrm{dd}}=a_{\mathrm{dd}}/a_{s}$.

To study the merging process, we apply a double-well potential of the form

$\displaystyle
V(\mathbf{r})=\frac{1}{2}m\omega^{2}(x^{2}+y^{2})+a(z^{2}-z_{0}^{2})^{2},$ (4)

with $z_{0}>0$. $\omega$ is the harmonic trap frequency along the radial $x$
and $y$ directions. The parameters $a$ and $z_{0}$ control the central barrier
height as well as the relative distance between the two wells [64, 65, 66].
When the two wells are separated, the double-well trap potential near its
local minima $z=\pm z_{0}$ can be expanded with a harmonic frequency
$\omega_{z}$, one can obtain a harmonic form
$V_{\mathrm{ho}}(z)\doteq\frac{1}{2}m\omega_{z}^{2}(z\pm z_{\mathrm{0}})^{2}$,
with $m\omega_{z}^{2}=8az_{0}^{2}$, where $\omega_{z}$ corresponds to the trap
frequency along the axial $z$ direction. We can rewrite the double-well
potential with the help of $\omega_{z}$,

$\displaystyle
V(\mathbf{r})=\frac{1}{2}m\omega^{2}(x^{2}+y^{2})+\frac{m\omega_{z}^{2}}{8z_{0}^{2}}(z^{2}-z_{0}^{2})^{2},$
(5)

where the barrier height is $V_{h}=m\omega_{z}^{2}z_{0}^{2}/8$. Thus the two
wells each have the specified frequency $\omega_{z}$ when they are far from
each other, $i.e.$, $z_{0}\gg a_{ho}$ In this way one can specify the pre-
merged condensates by the separated-trap aspect ratio
$\lambda=\omega_{z}/\omega$.

###  II.2 Numerical strategy

Ground state wave functions can be obtained by numerically minimizing the
energy functional Eq. 1. We employ an optimization algorithm limited-memory
Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) method [67, 68], which has been
implemented in PyTorch [69]. Compared to other first-order gradient descent
algorithms, L-BFGS shows more stable and faster convergence, specifically, its
superior performance in terms of computational time can be enhanced when
running on a graphics processing unit (GPU).

The condensate wave function is discretized with an equally spaced 3D grid on
a cube box centered on the origin. Because the dipolar interaction integral
diverges in configuration space, the integration has been calculated in
momentum space using an analytical form of the interaction, with a spherical
cut-off [70]. We employed a cubic grid of $256\times 256\times 256$ grid
points with dimensions $56\times 56\times 56$ $a_{\mathrm{ho}}$, which ensures
a large enough grid to allow the various terms in the energy functional to be
computed with spectral accuracy.

In this work, we consider 162Dy in a trap with radial trap frequency
$\omega=2\pi\times 125$ Hz, $a_{s}=85a_{0}$ and $a_{\mathrm{dd}}=131a_{0}$,
here $a_{0}$ is the Bohr radius. The harmonic-oscillator length
$a_{\mathrm{ho}}=\sqrt{\hbar/m\omega}$ and an energy units of $\hbar\omega$
will be used to scale the length and energy.

Figure 1: 2D Slices of the density profiles for the dipolar BEC with atom
numbers of $N=10^{5}$ (a1), (a2), and $N=2\times 10^{5}$ (b1), (b2) 162Dy
confined in the harmonic trap with $\omega_{z}=2\omega$.

##  III Numerical Results

###  III.1 Characterization of merging process

Our goal in this work is to monitor the behavior of two supersolids as they
merge together. To this end, we begin with the situation at the beginning and
end of this process. In this section we consider, in the separated limit, two
“standard” supersolids such as the one shown in Figs. 1(a1) and (a2). This
supersolid consists of $N=10^{5}$ atoms in a single harmonic trap with
frequencies $\omega=2\pi\times 125$ Hz, and $\omega_{z}=2\omega$, defined by
the trap potential

$\displaystyle V(\mathbf{r})=\frac{1}{2}m\omega^{2}(x^{2}+y^{2}+4x^{2}).$ (6)

In these circumstances, dysprosium is well-known to display a six-fold
symmetric supersolid phase [71, 72, 31, 32, 73, 74].

A single such supersolid in this trap is depicted in Figure 1. Fig. 1(a1)
shows a cut through the density at $z=0$ of the 3D density
$|\psi(x,y,0)|^{2}$, and from the side at $y=0$ in Fig. 1(a2). The energy per
atom is $E=22\ \hbar\omega$ and the chemical potential is $\mu_{0}=31\
\hbar\omega$. In one limit of the simulation, we envision two such
supersolids, centered at the vertical coordinates $\pm z_{0}$, far enough away
that they only weakly perturb one another.

Figure 2: 2D Slices of the density profiles of the dipolar BEC of $N=2\times
10^{5}$ 162Dy atoms confined in the double well trap with
$\omega_{z}=2\omega$. The upper panels show the cut of the $x-y$ plane at the
local minima of $z$ with (a1) $z_{0}=10\ a_{\mathrm{ho}}$, (b1) $z_{0}=11\
a_{\mathrm{ho}}$, and (c1) $z_{0}=12\ a_{\mathrm{ho}}$. The lower panels
illustrate the corresponding cut along the $x-z$ plane at $y=0$.

In the merged limit where $z_{0}=0$, a different BEC configuration would be
realized, as shown in Figs. 1(b1) and (b2). In this case the confining
potential is a single harmonic trap as described by Eq. 6, with familiar
characteristics $\omega=2\pi\times 125$ Hz, $\omega_{z}=2\omega$, whereas the
merged BEC now contains $N=2\times 10^{5}$ atoms, and is therefore in a
different part of the supersolid phase space. In this case it is more like a
labyrinth, as demonstrated in Ref. [75], higher atom numbers at the same
scattering length can break up the supersolid droplets into labyrinth phase.
Figures 1(b1) and (b2) therefore represent the anticipated result of fully
merging the two supersolids into a single layer.

Figure 3: 2D Slices of the density profiles of the dipolar BEC of $N=2\times
10^{5}$ 162Dy atoms confined in the double well trap with
$\omega_{z}=2\omega$. The upper panels show the cut of the $x-y$ plane at the
local minima of $z$ with (a1) $z_{0}=5\ a_{\mathrm{ho}}$, (b1) $z_{0}=7\
a_{\mathrm{ho}}$, and (c1) $z_{0}=9\ a_{\mathrm{ho}}$. The lower panels
illustrate the corresponding cut along the $x-z$ plane at $y=0$.

The initial stages of the merger, for relatively large $z_{0}$, are shown in
Fig. 2. In these panels the values of $z_{0}$ are, from left to right,
$z_{0}=10,11,12\ a_{\mathrm{ho}}$. Shown are side view cuts through $y=0$
(lower panels) and top view cuts through $z=z_{0}$ (upper panels). The double
well potential is given by Eqs. 4 and 5.

Figure 4: 2D Slices of the density profiles of the dipolar BEC of $N=2\times
10^{5}$ 162Dy atoms confined in the double well trap with
$\omega_{z}=2\omega$. The upper panels show the cut of the $x-y$ plane at the
local minima of $z$ with (a1) $z_{0}=0.5\ a_{\mathrm{ho}}$, (b1) $z_{0}=2.2\
a_{\mathrm{ho}}$, and (c1) $z_{0}=3\ a_{\mathrm{ho}}$. The lower panels
illustrate the corresponding cut along the $x-z$ plane at $y=0$.

At the largest separation $z_{0}=12\ a_{\mathrm{ho}}$ [e.g., Figs. 2(c1) and
(c2)], the gas is still fundamentally separated into two supersolids. Their
long-range attraction to each other is evident in the way the central density
maximum in each is drawn toward the center at $z=0$. Between $z_{0}=11\
a_{\mathrm{ho}}$ [e.g., Figs. 2(b1) and (b2)] and $z_{0}=10\ a_{\mathrm{ho}}$
[e.g., Figs. 2(a1) and (a2)] this attraction becomes overwhelming, and the BEC
easily tunnels through the barrier; this is the occasion of merging. Once this
pathway is open between the upper and lower components, atomic density floods
into this central tendril, reducing the density in the six local density
maxima on the periphery.

The process of merging continues in Fig. 3, which shows similar density plots
for (from left to right) $z_{0}=5,7,9\ a_{\mathrm{ho}}$. By $z_{0}=9\
a_{\mathrm{ho}}$, the original six density minima around the periphery of the
supersolid have all but vanished, leaving an elongated cloud that thins in the
middle as if it were a stretched piece of taffy. Upon further merging to
smaller $z_{0}$, tunneling becomes easier, the gas conforms to a more
cylindrical shape, and ultimately forms a cylindrical shell, a shape familiar
from previous work [72]. At even smaller $z_{0}$ the labyrinth states begin to
appear, two examples of which are illustrated in Figs. 4(b1), (b2), and (c1),
(c2). Finally, the emergence of a pancake shaped BEC state [e.g., Figs. 4(a1)
and (a2)] has been observed when $z_{0}$ close to 0.

Note that in Fig. 2, the upper and lower supersolids are displayed radially so
that the features above are aligned with the features below. This is done to
simplify the figure; in fact, the DDI is sufficiently weak in this limit that
no such alignment necessarily occurs. We verified this by artificially
rotating the upper supersolid relative to the lower one, finding the energy
variation to be only the order of $10^{-5}\ \hbar\omega$.

###  III.2 Phase diagram

Thus various morphologies appear during the merging process, as summarized by
the colored regions in Fig. 5(a). To assist in the description of what happens
as the gas evolves between these morphologies, we also plot the energy per
atom $E$ (black), the chemical potential $\mu$ (blue), and the height of the
barrier separating the upper and lower wells (red).

Perhaps the most important part of this diagram is around $z_{0}=11\
a_{\mathrm{ho}}$, where the transition occurs between separated supersolids
(pink region) and those that have merged (tan region), as shown in Figs.
2(a1), (a2) and (b1), (b2). For the separated supersolids, both the energy and
chemical potential diminish very slightly as the two clouds are brought
together, presumably due to the long-range dipole attraction between them.

Figure 5: (a) Phase diagram of the dipolar BEC of $N=2\times 10^{5}$ 162Dy
atoms confined in the double well trap with $\omega_{z}=2\omega$, the energy
per atom (thick black curve) and chemical potential (thick blue curve) and the
barrier height as a function of the local minima parameter $z_{0}$ are shown.
The black and blue dashed lines are twice of energy per atom and chemical
potential for $N=10^{5}$ atoms confined in the single harmonic trap with the
same trap aspect ratio. (b) The stretch force, explained in the context, as a
function of $z_{0}$ around $z_{0}=11\ a_{\mathrm{ho}}$.  Figure 6: 2D Slices
of the density profiles of the dipolar BEC of $N=2\times 10^{5}$ 162Dy atoms
confined in the double well trap while holding fixed $\omega_{z}=3\omega$. The
upper panels show the cut of the $x-y$ plane at the local minima of $z$ with
(a1) $z_{0}=8\ a_{\mathrm{ho}}$, (b1) $z_{0}=8.2\ a_{\mathrm{ho}}$, (c1)
$z_{0}=9\ a_{\mathrm{ho}}$, and (d1) $z_{0}=10\ a_{\mathrm{ho}}$. The lower
panels illustrate the corresponding cut along the $x-z$ plane at $y=0$.

Once they merge, however, the energy and chemical potential drop much more
rapidly as $z_{0}$ decreases, and the barrier height (red line) drops below
twice the chemical potential of a single, isolated supersolid (blue dashed
line). Here, a long, thin, continuous strip of dipoles aligned along the
strip’s axis defines a tendril that connects the lower and upper branches of
the condensate. Viewing the gradient of energy with respect to $z_{0}$ as a
force, Fig. 5(b) plots the force required to pull the two condensates apart.
This force decreases as the distance increases and the tendril becomes
thinner, behavior reminiscent of stretching a plasticine material such as
taffy. At the phase transition, the tendril breaks and the force has a
different dependence on density. This dependence is due to the usual long-term
dipolar attraction.

The next milestone occurs as a transition from the merged supersolids (tan
region) to the single elongated object in Figs. 3(b1) and (b2) (skyblue
region). This is seen to occur when the barrier height (red line) drops below
twice of the energy per atom of a single, isolated supersolid (black dashed
line). At this point it is no longer energetically favorable for the gas to
form the six density maxima at the periphery of the supersolid. The hexagonal
pattern associated with the supersolid is gone.

Next, the BEC passes into the cylindrical morphology of Figs 3 (a1) and (a2)
(orange region). At this point the BEC, being shortened in the vertical
direction, experiences increasing dipolar repulsion, thus the total energy and
chemical potential now rise with decreasing $z_{0}$. This shift in the energy
balance occasions the transformation to the hollow cylinder, in which the
dipoles repel one another, in the same way that the electrons in an
electrically charged conductor flow to the conductor’s surface.

When $z_{0}$ is further reduced, the hollow cylinder breaks up their symmetry
and enter the labyrinthine phase (red region), where various near degenerate
states coexist in this regime as examples shown in Figs. 4 (b1), (b2) and
(c1), (c2) . Finally, in the $z_{0}\rightarrow 0$ limit, the gas assumes a
simple pancake-like shape as Figs. 4 (a1) and (a2) (green region). Note that
in this case, the confinement in the vertical direction is given by a nearly
quartic, rather than quadratic, potential. Because of this difference, the
result of merging is actually distinct from the labyrinth predicted in Figs.
1(b1) and (b2).

###  III.3 Increasing the axial trap frequency

Here, we turn to investigate the merging behavior by increasing the vertical
confinement frequency to $\omega_{z}=3\omega$ with the same frequency of
$\omega=2\pi\times 125$ Hz for the radial directions. In this situation, the
trap aspect ratio is more oblate than in the previous section, which allows a
more elaborate pattern to develop even in a single layer. This is shown in
Figs. A.1(a1) and (a2) which show the density profile of $N=1\times 10^{5}$,
in this case a 12-droplet hexagonal supersolid state, which has been predicted
in Ref. [72]. By contrast, in the the merged limit where $N=2\times 10^{5}$
atoms are present, a labyrinthine state is seen, in Figs. A.1(b1) and (b2).

Upon merging the supersolids by reducing $z_{0}$, an even greater variety of
morphologies is seen. Similar to the $\omega_{z}=2\omega$ case, the gas is
separated into two supersolids at large $z_{0}$ at first, as shown in Fig. 6.
As $z_{0}$ decreases, the number of distinct droplets in the pattern drops
from 12 at $z_{0}=10a_{\mathrm{ho}}$, [Figs. 6(d1) and (d2)] transitions to 10
droplets at $z_{0}=9a_{\mathrm{ho}}$ [Figs. 6(c1) and (c2)], then to 7
droplets (with lesser features on the periphery) at $z_{0}=8.2a_{\mathrm{ho}}$
[Figs. 6(b1) and (b2)]. Finally, by $z_{0}=8a_{\mathrm{ho}}$ the upper and
lower supersolids have merged [Figs. 6(a1) and (a2)], surrounded by seven
additional features, as compared to the six features surrounding the central
tendril in the $\omega_{z}=2\omega$ case.

Figure 7: Phase diagram of the dipolar BEC of $N=2\times 10^{5}$ 162Dy atoms
confined in the double well trap with $\omega_{z}=3\omega$. The chemical
potential (thick blue curve) and the energy per atom (thick black curve) and
the barrier height (red) as a function of the local minima parameter $z_{0}$.
The blue and black dashed lines are twice of the chemical potential and energy
per atom for $N=10^{5}$ atoms confined in the harmonic traps. The two white
dashed lines divide the 7-, 10-, and 12-droplets regimes from left to right.

At intermediate values of $z_{0}$, the cylindrical form of the connecting
tendril appears, similar to what was observed in the case with
$\omega_{z}=2\omega$ above [Fig. B.1]. Further reducing $z_{0}$ results in a
small window of labyrinthine patterns before entering the cylindrical shell
regime, and then the other labyrinthine. Finally, in the merged limit,
$z_{0}\leq 2.5$, concentric rings appear, the outermost distributed in a large
number of droplets [eg, Figs. B.2(b1) and (b2)]. A phase diagram articulating
these morphologies is shown in Fig. 7. The different colors indicate the
similar phase state region as described in Sec. III.2.

##  IV Conclusions and outlooks

In addition to the vast array of patterns known to occur for individual
dipolar BECs in pancake-shaped traps, we have now demonstrated that even a
greater variety exists when two such BECs are allowed to influence one
another. This influence already occurs at a distance when the clouds are
relatively far apart, but results in a greater variety after they have merged.

We have further shown that, close to the merging transition, the BECs are
connected by a thin tendril of dipolar material with an internal cohesion.
This tendril behaves much like a deformable, plasticine material, whose
cohesive force diminishes as the tendril lengthens and thins. The merging
transition is characterized by the abrupt change by a discontinuity in the
derivative of this minute force, as the tendril breaks.

The ground-state phase diagram have been identified, and typical density
profile cutoffs are showcased. Building upon these results, the quench
dynamics and the collective Bogoliubov excitation spectrum across the phase
transition will be fascinating for better understanding the merge process.
With the help of a double well potential, it is also interesting to exploit
the collision process of two supersolids. Besides, The similar behaviour of
two-components or antidipolar single-component supersolids can be a promising
research directions in both theory and experiment.

## Acknowledgements

This material is based upon work supported by the National Science Foundation
under Grants No. PHY-2317149 and PHY-2110327.

##  Appendix A Plots for Harmonic Trap with $\omega_{z}=3\omega$

Figure A.1: 2D Slices of the density profiles of the dipolar BEC of $N=10^{5}$
(a1), (a2), and $N=2\times 10^{5}$ (b1) and (b2) 162Dy atoms confined in the
harmonic traps with $\omega_{z}=3\omega$.

##  Appendix B Plots for Double Well Trap with $\omega_{z}=3\omega$

Figure B.1: 2D Slices of the density profiles of the dipolar BEC of $N=2\times
10^{5}$ 162Dy atoms confined in the double well trap with
$\omega_{z}=3\omega$. he upper panels show the cut of the $x-y$ plane at the
local minima of $z$ with (a1) $z_{0}=5.0\ a_{\mathrm{ho}}$, (b1) $z_{0}=5.9\
a_{\mathrm{ho}}$, and (c1) $z_{0}=6.6\ a_{\mathrm{ho}}$. The lower panels
illustrate the corresponding cut along the $x-z$ plane at $y=0$. Figure B.2:
2D Slices of the density profiles of the dipolar BEC of $N=2\times 10^{5}$
162Dy atoms confined in the double well trap with $\omega_{z}=3\omega$. The
upper panels show the cut of the $x-y$ plane at the local minima of $z$ with
(a1) $z_{0}=1.0\ a_{\mathrm{ho}}$, (b1) $z_{0}=1.2\ a_{\mathrm{ho}}$, (c1)
$z_{0}=2.4\ a_{\mathrm{ho}}$, (d1) $z_{0}=2.8\ a_{\mathrm{ho}}$, and (e1)
$z_{0}=3.0\ a_{\mathrm{ho}}$. The lower panels illustrate the corresponding
cut along the $x-z$ plane at $y=0$.

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