††thanks: Currently at: Department of Physics, University of Haifa, 3498838
Haifa, Israel.

# Expansion of strongly interacting dipolar bosons in 1D optical lattices

Rhombik Roy<EMAIL_ADDRESS>Department of Physics, Presidency
University, 86/1 College Street, Kolkata 700073, India.  Andrea Trombettoni
<EMAIL_ADDRESS>Department of Physics, University of Trieste, Strada
Costiera 11, I-34151 Trieste, Italy.  CNR-IOM DEMOCRITOS Simulation Centre and
SISSA, Via Bonomea 265, I-34136 Trieste, Italy.  Barnali Chakrabarti
<EMAIL_ADDRESS>Department of Physics, Presidency University, 86/1 College
Street, Kolkata 700073, India.  The Abdus Salam International Center for
Theoretical Physics, 34100 Trieste, Italy.

###### Abstract

We numerically study the expansion dynamics of initially localized dipolar
bosons in a homogeneous 1D optical lattice for different initial states.
Comparison is made to interacting bosons with contact interaction. For shallow
lattices the expansion is unimodal and ballistic, while strong lattices
suppress tunneling. However for intermediate lattice depths a strong interplay
between dipolar interaction and lattice depth occurs. The expansion is found
to be bimodal, the central cloud expansion can be distinguished from the outer
halo structure. In the regime of strongly interactions dipolar bosons exhibit
two time scales, with an initial diffusion and then arrested transport in the
long time; while strongly interacting bosons in the fermionized limit exhibit
ballistic expansion. Our study highlights how different lattice depths and
initial states can be manipulated to control tunneling dynamics.

Expansion dynamics, Cold atom

††preprint: AIP/123-QED

##  I Introduction

Recent experimental developments allow a very precise control of properties of
ultracold atomic gases in optical lattices. This tunability makes the
nonequilibrium dynamics of strongly correlated many-body systems as one of the
most challenging problems for theoretical physics [1, 2]. External optical
lattices can be superimposed with typical experimental time-scales for the
dymaics faster than typical relaxation and decoherence rates. Thus, ultracold
atoms offer the possibility to study the transport properties and out-of-
equilibrium phenomena in well controlled way — number of atoms, inter-atomic
interaction, lattice depth all being controllable parameters. In the
understanding of non-equilibrium dynamics, special interest has been put for
expansion dynamics of strongly correlated bosons in 1D optical lattice [3, 4,
5, 6, 7, 8, 9]. The two prototypical transport mechanisms, ballistic and
diffusive transport, are now well confirmed in experiments.

The effect of interaction on expansion dynamics of confined bosonic and
fermionic gases on 1D and 2D latticed has been extensively studied
experimentally [3, 4, 10, 11, 12] as well theoretically [13, 14, 15, 16, 17,
18, 19]. Most of the studies in this direction considered the Mott insulator
state as the initial state and various aspects of the time dynamics after an
abrupt reduction of lattice height have been considered.

A very interesting experimental observation is the bimodal structure in the
expansion dynamics–diffusive dynamics in the center is surrounded by the
ballistic wings [4]. The interplay between the interaction and dimensionality
has been also the subject of considerable interest. Self trapping in the array
of ${}^{87}Rb$ atoms in 1D and quasi-1D tubes are also studied when the mean-
field energy gradient suppresses the tunneling and thus stops the expansion.
The self-trapping phenomenon, an effect induced by the nonlinear interaction
of the condensate, has been first experimentally observed in [20] for the
expansion in an optical lattice and in [21] for a double well potential. The
self-trapped phenomenon has been extensively studied in nondipolar BECs [22,
23, 24, 25, 7, 26, 27] and for dipolar bosons in the double well potential
[28, 29], see more references in [30].

In a typical experimental setup, atoms are first loaded in a 1D optical
lattice with an additional dipole trap, which forms an array of 2D
pancake.like BECs. Then the dipole trap is suddenly removed allowing the free
expansion of the bosons in the homogeneous lattice. In most of the studies of
the temporal expansion of clouds in optical lattices, the on-site interaction
is the most common assumption. However, it has been shown that the nearest
neighbour coupling rate is not negligible compared to atom tunneling rate near
a broad Feshbach resonance [31]. Rydberg dressing also gives rise to on-site
as well as long-range interactions [32], see the recent review [33]. Using
Ramsey spectroscopy quantum dynamics of disordered dipolar interacting
ultracold molecules in a partially filled optical lattice has also been
studied [34]. Using the Feshbach resonance technique, it is now possible to
considerably reduce the effects of short-range interactions and to study the
dominating effect introduced by the dipolar interaction [35, 36, 37]. This
facilitates additional control parameter to manipulate the tunneling dynamics
in optical lattices. Thus rapid experimental advances in controlling ultracold
atomic interactions make it possible to realize enticing states specially in
reduced dimension.

Dipolar ultracold atoms have also attracted much interest which is
corroborated by experimental realization of dipolar Bose Einstein condensate
consists of chromium [38], dysprosium [39] and erbium atoms [40]. The non-
local dipole-dipole interaction and its anisotropic nature leads to rich and
exotic many-body physics which are distinctly different from the BEC with
contact interaction [41, 42]. In the recent experiment [43], the dipolar
interaction is tuned which makes it possible to realize novel quantum phases
in the strongly correlated lattice with long-range interaction. Dipolar
interactions in reduced dimensions facilitate the exploration of fascinating
many-body phenomena. One such phenomenon is the crystallization in one- and
two-dimensional systems [44, 45, 46, 47]. Crystallization is the consequence
of strong repulsive long-ranged tail of the dipolar interaction when the
bosons exhibit maximal separation. The corresponding counterpart with contact
interaction is the fermionization when the strongly interacting bosons also
escape their spatial overlap. Crystallized bosons can be distinguished from
the fermionized bosons from the peak-to-peak separation in the density
distribution [47].

In the present manuscript we aim at describing the expansion dynamics as to
the one in [4], but here with dipolar interaction, to study the expansion
dynamics of initially localized interacting dipolar bosons in 1D optical
lattices. We will compare our results with the ones for the expansion in 1D
optical lattices of bosons interacting via short-range interactions. We
consider diverse initial setups to unravel the intriguing effect of
interaction in the expansion and correlation dynamics. Our system is prepared
in the ground state of the many-body Hamiltonian. We then impose a harmonic
trap such that the bosons localize in the central wells. The choice of
interaction strength both for contact and dipolar interaction guarantees that
without harmonic trap localization the initial state is fermionized in the
first case and crystallized for the second case. The interaction strength is
kept fixed throughout the analysis. We have made scanning in the weak and
intermediate interaction strength, however the more interesting many-body
features in the dynamics are observed in the extreme limit of fermionized and
crystallized phases. We present numerically exact many-body expansion dynamics
by solving the time-dependent many-boson Schrödinger equation utilizing the
multiconfigurational time-dependent Hartree method for bosons (MCTDHB) [48,
49, 50, 51, 52, 53, 54], implemented in the MCTDH-X software [55, 56]. We
extract key measures such as one-body density dynamics and Glauber correlation
functions. We analyse results for $N=4$ interacting bosons in $S=21$ lattices.
We make a scan in lattice depth in the range of
$\\{V_{0}=0.1E_{r},10.0E_{r}\\}$, $V_{0}$ being the lattice strength and
$E_{r}$ the lattice recoil energy, and discuss four representative cases
having different initial set ups which lead to four distinct expansion
mechanisms. At time $t=0$, we instantaneously switch off the harmonic trap and
allow the bosons to expand. Sudden removal of the harmonic trap should
ballistically favor the tunneling into the outer wells, however the tunneling
dynamics is strongly influenced by the non-local interaction of dipolar
bosons. For very weak lattices, $(V_{0}=0.1E_{r})$, when one has effectively a
harmonic trap, we observe ballistic expansion in both cases but with distinct
many-body features. When the fermionized bosons loose their independent jet-
like structure quickly, the crystallized bosons carry the many-body features
for long time until they reach the lattice boundary. On the other hand, very
strong lattices, $(V_{0}=10.0E_{r})$ inhibits the tunneling. When the
fermionized bosons remain localized for quite long time and then start
tunneling in the adjacent wells; correspondigly, the fermionized bosons remain
localized for entire region of dynamics. Interplay between the lattice
potential and strong interparticle interaction controls the tunneling dynamics
for intermediate lattice depth potentials: $V_{0}=1.0E_{r}$ and
$V_{0}=5.0E_{r}$. We are able to observe dominating effect of non-local
interaction. When the bosons with strong contact interaction exhibits
ballistic expansion, dipolar bosons exhibit diffusive expansion in short time
and arrested transport at longer time. For intermediate lattice depths, the
expansion is bimodal; the expansion of high-density central cloud can be
distinguished from the incoherent outer cloud expansion with very intriguing
many-body features.

Self trapping is a well understood phenomenon in the context of non-dipolar
bosons and is caused due to competition between the lattice depth and mean-
field energy gradient. When the nonlinear effect suppresses the tunneling the
expansion is inhibited and causes localization of wave packets. As the root
cause of self trapping is the interatomic interaction, it is inherently a many
body phenomena, and mean-field Gross-Pitaevskii equation is commonly used to
study the self-trapping in three dimensions. However, in 1D gases, strong
quantum correlations are built up and strongly interacting bosons do not
macroscopically occupy single-particle wave function. The many-body wave
function is now a set of spatially distinct single particle wave function in
each lattice to avoid interaction. For dipolar bosons, there will be an
nterplay between lattice depth, strength of dipolar interaction as well as
long range correlation arising from long range part of dipolar interaction.
The scenario is now more complicated — the tunneling should be favored due to
long-range repulsion of the bosons as well as due to sudden removal of the
harmonic trap. However, due to very strong quantum correlation for dipolar
bosons, tunneling is quickly suppressed. There are strong mismatched
correlations between adjacent sites as the atoms are already well isolated due
to non-local interaction. To allow tunneling an atom needs to pay large
interaction energy to adjust this mismatched correlations. Thus dipolar bosons
favor localization instead of quantum tunneling which is described as arrested
transport in the manuscript.

The paper is organized as follows. In Sec. II, we introduce the theoretical
model, while in Sec III introduces the quantities of interest. In Sec IV, we
present the initial state and the setup. Sec V presents the quench dynamics
for four different initial set ups distributed over four subsections. Sec VI
is devoted to our conclusions and our final remarks.

##  II The model

The equation of motion for $N$ interacting bosons in a time-dependent system
is governed by the time-dependent Schrödinger equation as
$\hat{H}\psi=i\frac{\partial\psi}{\partial t}$ (with $\hbar=1$). The total
Hamiltonian for the system takes the following form

$\hat{H}(x_{1},x_{2},\dots
x_{N})=\sum_{i=1}^{N}\hat{h}(x_{i})+\sum_{i<j=1}^{N}\hat{W}(x_{i}-x_{j}).$ (1)

The Hamiltonian $\hat{H}$ will be expressed in dimensionless units, see below
in Section IV. $\hat{h}(x)=\hat{T}(x)+\hat{V}_{trap}(x)$ is the one-body
Hamiltonian. $\hat{T}(x)$ is the kinetic energy operator and
$\hat{V}_{trap}(x)$ is the external trapping potential. $\hat{W}(x_{i}-x_{j})$
is the two-body interaction. $x_{i}$ is the coordinate of the $i$\- th boson.
The form of $\hat{V}_{trap}(x)$ and $\hat{W}(x_{i}-x_{j})$ will be specified
in Section IV.

To solve the time-dependent many-boson Schrödinger, we have deployed an in-
principle numerically exact, many-body, method, the MCTDHB, whose equation of
motions can be found in [48, 49, 50, 51]. We provide a brief description of
the implementation of MCTDHB in Appendix A.

##  III Quantities of interest

To explore the expansion dynamics in optical lattices of dipolar bosons, we
employ various measures to characterize the system’s behavior. These measures
provide insights into the spatial coherence and correlations within the
evolving bosonic cloud.

(i) The reduced one-body density matrix in coordinate space, denoted as
$\rho^{(1)}(x_{1}^{\prime}|x_{1};t)$, is a fundamental quantity of interest.
It captures the density distribution of the system by integrating over the
remaining coordinates, defined as:

$\begin{split}\rho^{(1)}(x_{1}^{\prime}|x_{1};t)=N\int
dx_{2},dx_{3}...dx_{N}\\\
\psi^{*}(x_{1}^{\prime},x_{2},\dots,x_{N};t)\psi(x_{1},x_{2},\dots,x_{N};t),\end{split}$
(2)

where $N$ represents the total number of particles. It is diagonal part gives
the one-body density $\rho(x,t)$.

(ii) The $p$-th order reduced density matrix in coordinate space, denoted as
$\rho^{(p)}(x_{1}^{\prime},\dots,x_{p}^{\prime}|x_{1},\dots,x_{p};t)$. This
quantity describes the joint probability distribution of finding $p$ particles
at specific positions and defined as

$\begin{split}\rho^{(p)}(x_{1}^{\prime},\dots,x_{p}^{\prime}|x_{1},\dots,x_{p};t)=\frac{N!}{(N-p)!}\int
dx_{p+1},...dx_{N}\\\
\psi^{*}(x_{1}^{\prime},\dots,x_{p}^{\prime},x_{p+1},\dots,x_{N};t)\psi(x_{1},\dots,x_{p},x_{p+1}\dots,x_{N};t).\end{split}$
(3)

(iii) The $p$-th order Glauber correlation function
$g^{(p)}(x_{1}^{\prime},\dots,x_{p}^{\prime},x_{1},\dots,x_{p};t)$ measures
degree of spatial coherence and correlations within the system. It is defined
as:

$\begin{split}g^{(p)}(x_{1}^{\prime},\dots,x_{p}^{\prime},x_{1},\dots,x_{p};t)=\\\
\frac{\rho^{(p)}(x_{1},\dots,x_{p}|x_{1}^{\prime},\dots,x_{p}^{\prime};t)}{\sqrt{\prod_{i=1}^{p}\rho^{(1)}(x_{i}|x_{i};t)\rho^{(1)}(x_{i}^{\prime}|x_{i}^{\prime};t)}}.\end{split}$
(4)

Similarly, the diagonal elements of
$g^{(p)}(x_{1}^{\prime},\dots,x_{p}^{\prime},x_{1},\dots,x_{p};t)$, denoted as
$g^{(p)}(x_{1},\dots,x_{p};t)$, provide a measure of $p$-th order coherence.
If $|g^{(p)}(x_{1},\dots,x_{p};t)|=1$, the system is fully coherent, while
deviations from unity indicate partial coherence. Specifically,
$g^{(p)}(x_{1},\dots,x_{p};t)>1$ implies correlated detection probabilities at
positions $x_{1},\dots,x_{p}$, while $g^{(p)}(x_{1},\dots,x_{p};t)<1$
indicates anti-correlations. These measures collectively provide valuable
insights into the evolving density profile, correlations, and coherence of the
bosonic system as it undergoes expansion in the shallow and deep optical
lattices.

Figure 1: Initial density profiles for contact and dipolar interactions in the
Hamiltonian of Eq. (5) for four bosons in $21$ lattice sites of various
lattice depths, $V=0.1E_{r},1.0E_{r},5.0E_{r},10.0E_{r}$ from top to bottom.
Left panel is for contact interaction ($\lambda=25$). Right panel is for
dipolar interaction ($g_{d}=25$). See the text for details. All quantities are
dimensionless.

##  IV Initial State

We prepare the initial state in a combined harmonic oscillator trap of
frequency $\omega$ and a 1D optical lattice. The many-body Hamiltonian reads
as

$\hat{H}=\sum_{i=1}^{N}\left(-\frac{1}{2}\frac{\partial^{2}}{\partial
x_{i}^{2}}+\frac{1}{2}\omega^{2}x_{i}^{2}+Vsin^{2}(kx_{i})\right)+\sum_{i<j}^{N}\hat{W}(x_{i}-x_{j})$
(5)

$V$ is the depth of the optical lattice, $k$ is the wave vector and
$l=\frac{\pi}{k}$ is the periodicity of the optical lattice. The many-body
Hamiltonian is rescaled in the unit of recoil energy
$E_{r}=\frac{\hbar^{2}k^{2}}{2m}$. Thus the unit of time, length, and
interaction strength are $\hbar E_{r}^{-1}$, $k^{-1}$ and $2E_{r}k^{-1}$
respectively. The spatial extension of the optical lattice is $x_{+}=10\pi$ to
$x_{-}=-10\pi$ to restrict the number of lattice sites to $21$. We impose the
hard wall boundary condition. The Hamiltonian can be written in dimensionless
form by dividing the dimensionful Hamiltonian by $\frac{\hbar^{2}}{mL^{2}}$,
$L$ is the arbitrary length scale which leads to $\hbar$= $m$= $k$ =1. The
initial state is the ground state of the many-body Hamiltonian with harmonic
trap frequency $\omega=0.22$ and lattice depth is chosen as $V=0.1E_{r}$,
$V=1.0E_{r}$, $V=5.0E_{r}$ and $V=10.0E_{r}$ to study the dynamics in the
entire range of lattice height which can be controlled experimentally.
The harmonic trap leads to localization towards the central wells of the
lattice. To ensure that the system is in quasi-1D regime, we assume strong
transverse confinement. The two-body interaction for contact interaction reads
as

$\hat{W}(x_{i}-x_{j})=\lambda\delta(x_{i}-x_{j})$ (6)

where $\lambda$ is the interaction strength determined by the scattering
length $a_{s}$ and the transverse confinement frequency. For dipolar
interaction,

$\hat{W}(x_{i}-x_{j})=\frac{g_{d}}{|(x_{i}-x_{j})|^{3}+\alpha}$ (7)

$g_{d}$ is the pure dipolar interaction strength;
$g_{d}=\frac{d_{m}^{2}}{4\pi\epsilon_{0}}$for electric dipoles and
$g_{d}=\frac{d_{m}^{2}\mu_{0}}{4\pi}$ for magnetic dipoles, $d_{m}$ being the
dipole moment, $\epsilon_{0}$ is the vacuum permittivity, and $\mu_{0}$ is the
vacuum permeability. $\alpha$ is the short-range cutoff to avoid singularity
at $x_{i}=x_{j}$. We choose $\alpha$ = 0.05, which corresponds to
$a_{\perp}=0.37$ and an aspect ratio = 42.5 [57, 58]. In general, the dipole-
dipole interaction potential in 1D also includes a contact term owing to the
transverse confinement, however that can be safely neglected for strong
interaction strengths [59].

We consider expansion dynamics of $N=4$ strongly interacting bosons
interacting via contact interaction of strength ($\lambda=25.0$) and dipolar
interaction strength ($g_{d}=25.0$). However, the physics remains same for
$N=3$ and $N=5$. We restrict to the smaller number of particles to ensure
convergence in the observed many-body dynamics. We solve the set of coupled
equations using MCTDH-X software (R-MCTDHB package) [55, 56]. For the relaxed
state, propagation is done in imaginary time with an initial guess which
converges to the ground state of the system.

In Fig. 1 , we plot the initial state both for contact and dipolar interaction
for four different choices of lattice depth. For $V=0.1E_{r}$, when the
effective trap is almost a HO trap, the one-body density for $\lambda=25$, is
the fermionic density distribution. The emergence of four maxima corresponds
to four strongly interacting bosons in the fermionized limit. The two
innermost humps are pronounced at the center of the trap where the effective
potential is zero, the two outermost humps are less pronounced due to the
larger distance from the center of the trap. The corresponding one-body
density for $g_{d}=25$, exhibits a crystal phase. The four humps again signify
the four strongly interacting dipolar bosons, this is the similarity between a
crystal phase and a fermionized phase, in both cases the strongly interacting
bosons escape their spatial overlap. However, unlike the contact interaction
where the humps were ”non isolated”, in the crystal phase they are ”isolated”
due to the long range tail of the dipolar interaction. For crystallized
dipolar bosons, the value of the density at the minima between the humps tends
to zero and the spreading of the density profile is broadened. Thus the
density modulation in the crystallized phase is significantly different from
the fermionized phase. Now increasing the lattice depth, the HO trap and
lattice height interplay and the effective lattice structure becomes
inhomogeneous. For $V=1.0E_{r}$, with contact interaction, three distinct
humps are distributed over three lattice sites around the central lattice and
two weak humps are observed in the outer lattice. Whereas the initial state is
less significantly affected by the dipolar interaction for this weak lattice.
Four distinct humps with some modulation in the density are observed as
before. For much higher lattice depth, $V=5.0E_{r}$, four tightly trapped and
completely isolated peaks around the central lattice with a very weak outer
peak are observed in the density profile for contact interaction. For dipolar
interaction, we observe five tightly bound peaks distributed over five
consecutive lattices around the central lattice. It is to be noted that the
weak outer peak in the density profile for contact interaction and the strong
outer peaks for dipolar interaction will play significant role in the
tunneling dynamics. For lattice depth $V=10.0E_{r}$, there is no further
effect on density modulation for contact interaction; only the peaks become
more strongly bound due to larger height of lattice. Whereas density pattern
for dipolar bosons are strongly affected. We observe four distinct peaks in
the density distribution but long range interaction strongly interplay with
lattice depth and one peak settles to a far right lattice site escaping a near
one.

##  V Expansion dynamics in the lattice

Figure 2:  Density dynamics on sudden removal of trap in the Hamiltonian
(Eq.(5)), for $N=4$ bosons, orbital $M=15$, lattice sites $S=21$, lattice
depth $V=0.1E_{r}$. The upper panel depicts post quench density dynamics for
contact interaction ($\lambda=25$), while the lower panel shows the same for
dipolar interaction ($g_{d}=25$).

###  V.1 Expansion dynamics in weak lattice: $V=0.1E_{r}$

Here we prepare the initial state in the combination of harmonic trap
($\omega=0.22$) and lattice with depth $V=0.1E_{r}$. To induce dynamics, we
suddenly switch off the harmonic trap and allow the bosons to tunnel in the
outer wells. The many-body expansion dynamics is studied for $M=15$ orbitals.
On increasing the number of orbitals to $M=16$, the occupation in the highest
natural orbital becomes insignificant, it assures the convergence in the
dynamical evolution. The one-body density evolution is presented in the Fig.
2. The potential is effectively a harmonic oscillator potential, the lattice
is sub relevant and the initial state is a fermionized phase for contact
interaction and a crystal phase for dipolar interaction Fig. 1. Sudden quench
to harmonic trap frequency to zero makes the bosons trapless and allows
ballistic expansion as shown in Fig 2. However, both for the contact and
dipolar interaction, density dynamics exhibit distinguished many-body features
in the expansion dynamics. At $t=0$, for contact interaction, four bright red
spots signify four strongly interacting bosons. On switching off the trap they
propagate as four bright jets till time $t=5.0$, the two inner jets are
brighter than the two outer jets which agree well with the initial state
density presented in Fig. 1. After time $t=5.0$, the structure melts and the
many-body features disappear, the cloud expands as a whole. In contrast, for
the dipolar interaction, the jets are very bright and well separated as shown
in lower panel of Fig. 2. The distinct many-body feature is maintained in the
expansion of crystallized bosons until they hit the wall, only with time of
propagation the intensity of the jets is reduced. The diverging nature in the
expansion dynamics of dipolar bosons signify the unbounded energy for dipolar
interaction, in contrast, the expansion dynamics for contact interaction is
bounded. The corresponding root mean square radius calculated from
$\sqrt{\int{x^{2}\rho(x)dx}}$ determines the average size of the expanding
cloud and is plotted in Fig. 3(a). Both exhibit ballistic expansion from time
$t=5.0$, the initial expansion (just after release of the trap) has some
nonlinear behaviour which looks the initial expansion is diffusive like. The
cloud radius for dipolar bosons is more as expected.

Figure 3: Time evolution of the root mean square (rms) radius of the bosonic
cloud for different lattice depths. Simulation is done with $N=4$ bosons in
$S=21$ lattice sites and $M=15$. Contact interaction strength is fixed to
$\lambda=25$ and dipolar interaction strength is kept fixed to $g_{d}=25$. (a)
Expansion in a very shallow lattice ($V=0.1E_{r}$); both for contact as well
as dipolar interaction the cloud expands, the dipolar bosons expand fast. (b)
Expansion in an intermediate lattice ($V=1.0E_{r}$); for contact interaction,
the expansion is ballistic. For dipolar interaction, there is an initial
expansion followed by the saturation of the cloud’s radius which results to
suppressed tunneling. (c) Expansion in a stronger lattice ($V=5.0E_{r}$); in
the case of contact interaction, the bosonic cloud expands over time. However,
for dipolar interaction the localization happens, the corresponding rms radius
remains constant. (d) Expansion in a deep lattice ($V=10.0$); for contact
interaction the cloud remains arrested for some time and then expands. Whereas
for dipolar interaction the transport remains arrested throughout the
dynamics. All quantities are dimensionless.

###  V.2 Expansion dynamics in intermediate lattice: $V=1.0E_{r}$

Figure 4:  Density dynamics on sudden removal of trap in the Hamiltonian
(Eq.(5)), for $N=4$ bosons, orbital $M=15$, lattice sites $S=21$, lattice
depth $V=1.0E_{r}$. The upper panel depicts post quench density dynamics for
contact interaction ($\lambda=25$), while the lower panel shows the same for
dipolar interaction ($g_{d}=25$). All quantities are dimensionless. See the
text for discussion.

Expansion dynamics presented in Fig. 4 exhibit bimodal structure both for the
contact and dipolar interaction. The central cloud expansion which is coherent
is surrounded by incoherent halo cloud in both cases. For contact interaction,
the high density central cloud consists of three inner peaks (as shown in Fig.
1), expands slowly. Whereas the outer low density halo cloud consists of two
weak outer peaks (as shown in Fig. 1) tunnelled out to the neighbouring
lattice sites ballistically. The corresponding cloud radius as shown in Fig.
3(b) also exhibit ballistic expansion, the cloud radius almost linearly grows
up with time. The expansion dynamics of dipolar bosons is distinctly different
in time scale as shown in the lower panel of Fig. 4. As the initial state as
shown in Fig. 1 consists of four peaks almost of same intensity, the expansion
dynamics is basically controlled by the expansion of central cloud surrounded
by extremely fade halo structure. We observe expansion up to time $t=25$ as
shown in Fig. 3(b). At the very initial releasing point the expansion exhibits
diffusive like for very short time and then becomes ballistic and remains till
time $t=25$. However at longer time, the expansion speed is reduced and
finally the transport is suppressed. This happens due to large mismatch in the
energy in neighbouring sites. The interaction energy costs tunneling and when
there is a significant mismatch in energy in the adjacent lattice sites,
tunneling of the bosons from central cloud is inhibited. Due to long range
correlation in dipolar interaction, the bosons experience more mismatched
correlation and favor to stay in the central lattice rather than tunneling.
The localization also happens for contact interaction but at a much longer
scale not shown in Fig. 4.

###  V.3 Expansion dynamics in stronger lattice: $V=5.0E_{r}$

Figure 5:  Density dynamics on sudden removal of trap in the Hamiltonian
(Eq.(5)), for $N=4$ bosons, orbital $M=15$, lattice sites $S=21$, lattice
depth $V=5.0E_{r}$. The upper panel depicts post quench density dynamics for
contact interaction ($\lambda=25$), while the lower panel shows the same for
dipolar interaction ($g_{d}=25$). All quantities are dimensionless. See the
text for discussion

The expansion dynamics of strongly interacting bosons in a lattice with depth
$V=5.0E_{r}$ is presented in Fig. 5. For this choice of lattice depth, the
effective lattice becomes inhomogeneous and has consequence in configuring the
initial states for contact interaction and dipolar interaction as shown in
Fig. 1. We also observe distinct difference in the expansion dynamics due to
dipolar interaction. For contact interaction, the central cloud consists of
four bright jets propagate uniformly till time $t=30$. Up to this time
tunneling does not happen in the outer lattice, the corresponding rms radius
remains constant as shown in Fig. 3(c). Then the central cloud disappears and
quick tunneling happens in the outer lattice sites which is exhibited by the
halo structure which further expands uniformly as a bright wave front. The
corresponding rms radius also uniformly increases. Thus we find two
independent modes in the expansion dynamics. Initially it is contributed by
the high density central cloud and later by low density outer halo. In
contrast, for dipolar bosons, the four high intensity peaks in the initial
state remains like this all throughout the dynamics as shown in the lower
panel of Fig. 5. The long range correlation causes mismatch in the energy
between the adjacent sites as argued before, however the stronger lattice
depth also inhibits the propagation which causes localization even in the long
time dynamics. The corresponding rms radius is also constant as shown in Fig.
3(c).

###  V.4 Expansion dynamics in very strong lattice: $V=10.0E_{r}$

Figure 6:  Density dynamics on sudden removal of trap in the Hamiltonian
(Eq.(5)), for $N=4$ bosons, orbital $M=15$, lattice sites $S=21$, lattice
depth $V=10.0E_{r}$. The upper panel depicts post quench density dynamics for
contact interaction ($\lambda=25$), while the lower panel shows the same for
dipolar interaction ($g_{d}=25$). All quantities are dimensionless. See the
text for discussion.

The dynamics is studied in a deep lattice of lattice depth $V=10.0E_{r}$. The
corresponding density dynamics is presented in Fig 6. Arrested transport is
expected for both types of interaction as strong lattice plays the dominating
role and will inhibit the tunneling. However, for contact interaction bosons
remain localized up to certain time and then the cloud starts to tunnel in the
adjacent wells only. Where as, for dipolar interaction we observe complete
localization. The highly correlated dipolar bosons need to pay much larger
interaction energy for tunneling. The cost of larger interaction energy is
justified as dipolar bosons feel more mismatch in the adjacent site
correlation due to asymmetry in the density compared to the non-dipolar
boosns. The same dynamical features are concluded from Fig 3(d) where cloud
radius is initially constant and then slowly increases for contact interaction
whereas for dipolar interaction the rms radius remains constant throughout the
entire dynamics.

##  VI Conclusion

We have investigated the expansion dynamics of strongly correlated dipolar
bosons in crystallized limit in 1D homogeneous lattices. Our protocol consists
of preparing initial states in a different lattice depth superimposed with a
harmonic trap and then suddenly switching off the trap. The dynamics is
studied by key measures of one-body density dynamics, rms radius of the
expanding cloud, one-body and two-body correlation dynamics. All results are
compared with the expansion dynamics for the strongly interacting bosons in
fermionized limit. We maintain the same quench protocol for fermionized and
crystallized boson expansion dynamics. We observe very interesting and
distinguished many-body features in the expansion dynamics for all initial
setups. For very weak lattice depths, when the harmonic potential trap plays
the dominating role to configure the initial set up, both the crystallized and
fermionized bosons exhibit ballistic expansion. However the ballistic
expansion in two cases can be distinguished utilizing the many-body features.
The long range tail in dipolar interaction allows the crystallized bosons to
carry on the independent jet like structure until it hits the boundary of the
lattice. Whereas the fermionized bosons quickly loose the many body features,
and the cloud expands ballistically. On the other habd, very strong lattice
inhibits the transport as expected. The most intriguing many-body features are
observed for intermediate lattice; we observe strong interplay between lattice
depth, strong interaction and long range correlation. The cloud expansion is
separated into central cloud expansion with distinct many body features and a
halo low density cloud expansion without many body features. The expansion is
bimodal, however the many body features are different for the crystallized
bosons from those of fermionized bosons. With increase in lattice depth, the
dipolar bosons exhibit localization as they are completely isolated due to
long range interaction and the correlation in the adjacent wells are strongly
mismatched. Thus tunneling is not favored at the cost of large interaction
energy. Instead of paying large cost, dipolar bosons like to stay in the well
which results to arrested transport. Thus our present study highlights how to
manipulate the initial state configuration and interaction to control the
tunneling dynamics.

The kind of initial states prescribed in this manuscript should be established
in the current experimental setups and to study the many-body correlation in
the strongly interacting limit. In the future, the similar approach can be
made for localized fermions both for the attractive and repulsive interaction.
Our analysis can be extended to study the other crystallization phenomena in
higher dimensions or for other kind of long-range interactions, e.g. mediated
by cavities [60, 61]. We expect that several of the qualitative results
discussed in this paper hold also for larger values of $N$, and the
exploration of regimes with large $N$ is an interesting subject of future
investigation. Finally, we observe that the expansion dynamics may be strongly
influenced in the presence of disordered potential, which is an interesting
topic for future work.

###### Acknowledgements.

We would like to thank N. Defenu, T. Macri’, and A. Smerzi for useful
discussions. BC acknowledges the financial support and hospitality of ICTP.

##  Appendix A  The Multiconfigurational Time dependent Hartree method for
bosons

The ansatz for the many-body wave function is the linear combination of time
dependent permanents

$|\Psi(t)\rangle=\sum_{\bar{n}}C_{\bar{n}}(t)|\bar{n};t\rangle,$ (8)

The vector $\vec{n}=(n_{1},n_{2},\dots,n_{M})$ represents the occupation of
the orbitals, with the constraint that $n_{1}+n_{2}+\dots+n_{M}=N$, which
ensures the preservation of the total number of particles. Distributing $N$
bosons over $M$ time dependent orbitals, the number of permanents become
$\left(\begin{array}[]{c}N+M-1\\\ N\end{array}\right)$. The permanents are
constructed over $M$ time-dependent single-particle wave functions, called
orbitals, as

$|\bar{n};t\rangle=\prod^{M}_{k=1}\left[\frac{(\hat{b}_{k}^{\dagger}(t))^{n_{k}}}{\sqrt{n_{k}!}}\right]|0\rangle$
(9)

Where $|0\rangle$ is the vacuum state and $\hat{b}_{k}^{\dagger}(t)$ denotes
the time-dependent operator that creates one boson in the $k$-th working
orbital $\psi_{k}(x)$, i.e.:

$\displaystyle\hat{b}_{k}^{\dagger}(t)$ $\displaystyle=$
$\displaystyle\int\mathrm{d}x\>\psi^{*}_{k}(x;t)\hat{\Psi}^{\dagger}(x;t)\>$
(10) $\displaystyle\hat{\Psi}^{\dagger}(x;t)$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{M}\hat{b}^{\dagger}_{k}(t)\psi_{k}(x;t).$ (11)

The accuracy of the algorithm depends on the number of orbitals $M$. $M=1$
corresponds to the mean field Gross-Pitaevskii equation. If
$M\rightarrow\infty$, the wave function becomes exact, with the set
$|n_{1},n_{2},\dots,n_{M}\rangle$ spanning the complete Hilbert space for $N$
particles. However, due to computational limitations, the number of orbitals
is restricted to a desired value to ensure proper convergence in the measured
quantities. It is important to note that both the expansion coefficients
$\left\\{C_{\bar{n}}(t)\right\\}$ and the working orbitals $\psi_{i}(x,t)$
that are used to construct the permanents $|\bar{n};t\rangle$ are fully
variationally optimised and time-dependent quantities [62, 63, 64, 65]. The
time-dependent optimization of the orbitals and expansion coefficients ensure
that the MCTDHB method can accurately describe the dynamics of strongly
interacting bosons. As the permanents are time-dependent, a given degree of
accuracy is achieved with shorter time compared to a time-independent basis.
The time-adaptive many-body basis set employed in MCTDHB allows for the
dynamic tracking of correlations that arise from inter-particle interactions.

##  Appendix B Measures of correlation

In this appendix, we present the emergent correlation properties by the
measures of one-body and two-body Glauber correlation functions as defined in
Eqs.(2) to (4). The parameters for quench dynamics remain the same as
described in the expansion dynamics. The results for very weak lattice
$(V=0.1E_{r})$ are presented in Figs.7 and 8 for some selected time points
$t=0.0$, $10.0$ and $15.0$. At $t=0$ (pre-quench state), the diagonal of the
first order correlation for contact interaction (Fig. 7(a)) shows four non
isolated bright regions where $|g^{(1)}|^{2}$ $\simeq$ $1$. This describes
correlation between four fermionized bosons trapped in the HO potential. The
corresponding for dipolar bosons (Fig. 7(b)), shows four completely separated
bright lobes. This describes correlation between four crystallized bosons, it
is specifically to be noted that bright lobes are completely isolated due to
long range interaction. The corresponding $|g^{(2)}|$ in Figs. 8(a) (contact
interaction) and (b) (dipolar interaction) shows four extinguished lobes,
which are termed as ’correlation holes’. It signifies that the probability of
finding two bosons in the same place $(x_{1}=x_{2})$ is zero. On sudden
removal of the trap, the correlations, both one- and two-body exhibit
expansion for contact as well as dipolar interaction. The structure of
diagonal correlation in $g^{(1)}$ is maintained during expansion, whereas the
width of the correlation hole in $g^{(2)}$ also expands maintaining the
distinguished features.

Figure 7: Dynamics of the one-body Glauber correlation function
$g^{(1)}(x_{1},x_{2})$ for lattice depth $V=0.1E_{r}$. Figs. (a), (c), and (e)
are for contact interaction with $\lambda=25$, while Figs. (b), (d), and (f)
are for dipolar interaction with $g_{d}=25$. The pre-quench and post quench
parameters remain same as in the expansion dynamics. The plots show the
correlation functions at three different times: $t=0.0$, $t=10.0$, and
$t=15.0$ (top to bottom). Figure 8: Dynamics of the two-body Glauber
correlation function $g^{(2)}(x_{1},x_{2})$ for lattice depth $V=0.1E_{r}$.
Figs. (a), (c), and (e) are for contact interaction with $\lambda=25$, while
Figs. (b), (d), and (f) are for dipolar interaction with $g_{d}=25$. The pre-
quench and post quench parameters remain same as in the expansion dynamics.
The plots show the correlation function at three different times: $t=0.0$,
$t=10.0$, and $t=15.0$ (top to bottom). Figure 9: Dynamics of the one-body
Glauber correlation function $g^{(1)}(x_{1},x_{2})$ for lattice depth
$V=1.0E_{r}$. Figs. (a), (c), and (e) are for contact interaction with
$\lambda=25$, while Figs. (b), (d), and (f) are for dipolar interaction with
$g_{d}=25$. The pre-quench and post quench parameters remain same as in the
expansion dynamics. The plots show the correlation functions at three
different times: $t=0.0$, $t=5.0$, and $t=10.0$ (top to bottom).

Fig. 9 displays the spatial first-order correlation function for a lattice
depth $V=1.0E_{r}$. Figs. 9(a), (c), and (e) depict the correlation function
for contact interaction with $\lambda=25$, while Figs. 9(b), (d), and (f) show
the correlation function for dipolar interaction with $g_{d}=25$. The plots
are presented at three different times: $t=0.0$, $t=5.0$, and $t=10.0$. At
$t=0$, the diagonal of the first-order correlation function exhibits coherent
regions both for contact Fig. 9(a) and dipolar interaction Fig. 9(b). In the
case of contact interaction, two small red patches at top left corner and down
right corner appear. It signifies the two small peaks as observed in the
initial state (Fig. 1). Fig. 9(b) signifies that the dipolar bosons are
initially more isolated than the case of contact interaction. As time
progresses, the correlations expand and occupy the allotted space and in
consequence the off diagonal coherence fades up gradually for the case of
contact interaction; Fig. 9(c) and Fig. 9(e). In comparison, Fig. 9(d) and (f)
show that the correlation for dipolar bosons expands very fast and without
loss of off diagonal correlation. Thus the correlation quickly reaches the
boundary and the further expansion in correlation dynamics stops. Thus at much
longer time (not shown here), while the correlation expands for contact
interaction, that for dipolar interaction reaches its saturation value. To
determine the degree of spatial second-order coherence, we study the
normalized two-body correlation function $g^{(2)}(x_{1},x_{2})$ and present
for selected evolution times as before in the Fig.10. Figs.10(a), (c), and (e)
correspond to contact interactions, while Figs. 10(b), (d), and (f) correspond
to dipolar interaction. At the initial time $t=0.0$, the emergence of a
correlation hole along the diagonal of the normalized two-body correlation
function $g^{(2)}(x_{1},x_{2}=x_{1})$ with
$g^{(2)}(x_{1},x_{2}=x_{1})\rightarrow 0$ is a characteristic feature of the
many-body system. This indicates that the bosons tend to maximize their
spatial separation in the presence of strong repulsive interactions. The width
of the correlation hole is broader for the dipolar interaction case. As the
trap is switched off, the system expands and the two-body correlations spread.
During expansion, the dynamics of contact and dipolar interactions can be
distinguished by a single feature: the dipolar interaction leads to a larger
width of the correlation hole along the diagonal. Unlike the case of one-body
correlation, here we do not find any signature of the loss of off diagonal
correlation.

Figure 10: Dynamics of the two-body Glauber correlation function
$g^{(2)}(x_{1},x_{2})$ for lattice depth $V=1.0E_{r}$. Figs. (a), (c), and (e)
are for contact interaction with $\lambda=25$, while Figs. (b), (d), and (f)
are for dipolar interaction with $g_{d}=25$. The pre-quench and post quench
parameters remain same as in the expansion dynamics. The plots show the
correlation function at three different times: $t=0.0$, $t=5.0$, and $t=10.0$
(top to bottom). Figure 11: Dynamics of the one-body Glauber correlation
function $g^{(1)}(x_{1},x_{2})$ for lattice depth $V=5.0E_{r}$. Figs. (a),
(c), and (e) are for contact interaction with $\lambda=25$, while Figs. (b),
(d), and (f) are for dipolar interaction with $g_{d}=25$. The pre-quench and
post quench parameters remain same as in the expansion dynamics. The plots
show the correlation functions at three different times: $t=0.0$, $t=50.0$,
and $t=100.0$ (top to bottom).

In Fig. 11, we present the one-body correlation for contact and dipolar
interactions at different times. Figs.11(a), (c), and (e) display the
correlation function for contact interactions with $\lambda=25$, while
Figs.11(b), (d), and (f) depict the correlation function for dipolar
interactions with $g_{d}=25$. The plots are shown at three different times:
$t=0.0$, $t=50.0$, and $t=100.0$. At $t=0$, the normalized first-order
correlation function for contact interaction (Fig.11(a)) exhibits a structure
along the diagonal which signify the distinct peaks in the initial state
(Figs.1) and $[g^{(1)}(x_{1},x_{2})]^{2}=0$ on the off-diagonal. On expansion,
diagonal correlation structure is completely maintained. On the other hand,
for dipolar interaction, at $t=0$, the diagonal structure is very prominent
(Fig.11(b)). On switching of the trap, the correlation does not expand even
for long time dynamics. During this ’arrested transport’ regime, the one-body
correlation along the diagonal is also maintained, without showing any
expansion (Figs.11((d) and (f)).

Figure 12: Dynamics of the two-body Glauber correlation function
$g^{(2)}(x_{1},x_{2})$ for lattice depth $V=5.0E_{r}$. Figs. (a), (c), and (e)
are for contact interaction with $\lambda=25$, while Figs. (b), (d), and (f)
are for dipolar interaction with $g_{d}=25$. The prequench and post quench
parameters remain same as in the expansion dynamics. The plots show the
correlation function at three different times: $t=0.0$, $t=50.0$, and
$t=100.0$ (top to bottom).

To assess the extent of spatial second-order coherence, we present the
normalized two-body correlation function $g^{(2)}(x_{1},x_{2})$ during the
expansion in (Fig.12). Figs.12(a), (c), and (e) represent the correlation
function for contact interactions with $\lambda=25$, while Figs. 12(b), (d),
and (f) correspond to dipolar interactions with $g_{d}=25$. The plots are
presented at three distinct time points: $t=0.0$, $t=50.0$, and $t=100.0$. For
contact interaction (Fig. 12(a)), coherence is maintained at the off-diagonal
elements ($g^{(2)}(x_{1},x_{2})\approx 1$), indicating that there is second-
order coherence between different wells. However, along the diagonal, the
correlation function exhibits a vanishing behavior, known as the correlation
hole. In contrast, for dipolar interaction (Fig. 12(b)), the correlation hole
is broader, and no distinct square-like structure is formed. This behavior
arises due to the long-range nature of the interaction. During the expansion,
the correlation function $g^{(2)}(x_{1},x_{2})$ expands in an organized manner
maintaining its gross features for contact interaction. However, for dipolar
interaction, the two-body correlation as a whole presents restricted but
gradual loss of second-order coherence within and between wells.

Figure 13: Dynamics of the one-body Glauber correlation function
$g^{(1)}(x_{1},x_{2})$ for lattice depth $V=10.0E_{r}$. Figs. (a), (c), and
(e) are for contact interaction with $\lambda=25$, while Figs. (b), (d), and
(f) are for dipolar interaction with $g_{d}=25$. The prequench and post quench
parameters remain same as in the expansion dynamics. The plots show the
correlation functions at three different times: $t=0.0$, $t=200.0$, and
$t=500.0$ (top to bottom).

Fig.13 represents the one-body correlation in the deep lattice, $V=10.0E_{r}$
for selected times $t=0.0$, $200.0$ and $500.0$ to include the long time
scenario. The diagonal structure in Figs.11(a) (contact interaction) and
Figs.11(b) (dipolar interaction) characterize the initial state correlation
(Figs.1). On switching of the trap, the correlation initially arrested but
then expands diffusively for contact interaction whereas the correlation for
dipolar interaction is completely arrested even for very long time. The
corresponding two-body correlation in Figs.14 also concludes the same physics.

Figure 14: Dynamics of the two-body Glauber correlation function
$g^{(2)}(x_{1},x_{2})$ for lattice depth $V=10.0E_{r}$. Figs. (a), (c), and
(e) are for contact interaction with $\lambda=25$, while Figs. (b), (d), and
(f) are for dipolar interaction with $g_{d}=25$. The pre-quench and post
quench parameters remain same as in the expansion dynamics. The plots show the
correlation function at three different times: $t=0.0$, $t=200.0$, and
$t=500.0$ (top to bottom).

## References

  * Bloch  _et al._ [2008] I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80, 885 (2008).
  * Polkovnikov  _et al._ [2011] A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Colloquium: Nonequilibrium dynamics of closed interacting quantum systems, Rev. Mod. Phys. 83, 863 (2011).
  * Schneider  _et al._ [2012] U. Schneider, L. Hackermüller, and J. e. a. Ronzheimer, Fermionic transport in a homogeneous hubbard model: Out-of-equilibrium dynamics with ultracold atoms, Nature Phys 8, 213 (2012).
  * Ronzheimer  _et al._ [2013] J. P. Ronzheimer, M. Schreiber, S. Braun, S. S. Hodgman, S. Langer, I. P. McCulloch, F. Heidrich-Meisner, I. Bloch, and U. Schneider, Expansion dynamics of interacting bosons in homogeneous lattices in one and two dimensions, Phys. Rev. Lett. 110, 205301 (2013).
  * Collura  _et al._ [2013] M. Collura, S. Sotiriadis, and P. Calabrese, Quench dynamics of a tonks–girardeau gas released from a harmonic trap, J. Stat. Mech. P09025 (2013).
  * [6] T. Kinoshita, T. Wenger, and D. S. Weiss, Observation of a one-dimensional tonks-girardeau gas, Science .
  * Wang  _et al._ [2006] B. Wang, P. Fu, J. Liu, and B. Wu, Self-trapping of bose-einstein condensates in optical lattices, Phys. Rev. A 74, 063610 (2006).
  * Koutentakis  _et al._ [2017] G. M. Koutentakis, S. I. Mistakidis, and P. Schmelcher, Quench-induced resonant tunneling mechanisms of bosons in an optical lattice with harmonic confinement, Phys. Rev. A 95, 013617 (2017).
  * Siegl  _et al._ [2018] P. Siegl, S. I. Mistakidis, and P. Schmelcher, Many-body expansion dynamics of a bose-fermi mixture confined in an optical lattice, Phys. Rev. A 97, 053626 (2018).
  * Vidmar  _et al._ [2015] L. Vidmar, J. Ronzheimer, M. Schreiber, S. Braun, S. H. S. Langer, F. H.-M. I. Bloch, and U. Schneider, Dynamical quasicondensation of hard-core bosons at finite momenta, Phys. Rev. Lett. 115, 175301 (2015).
  * Reinhard  _et al._ [2013] A. Reinhard, J.-F. Riou, L. A. Zundel, D. S. Weiss, S. Li, A. M. Rey, and R. Hipolito, Self-trapping in an array of coupled 1d bose gases, Phys. Rev. Lett. 110, 033001 (2013).
  * Xia  _et al._ [2015] L. Xia, L. A. Zundel, J. Carrasquilla, A. Reinhard, J. M. Wilson, M. Rigol, and D. S. Weiss, Quantum distillation and confinement of vacancies in a doublon sea, Nature Phys 11, 316 (2015).
  * Xu and Rigol [2017] W. Xu and M. Rigol, Expansion of one-dimensional lattice hard-core bosons at finite temperature, Phys. Rev. A 95, 033617 (2017).
  * Trujillo-Martinez  _et al._ [2021] M. Trujillo-Martinez, A. Posazhennikova, and J. Kroha, Expansion dynamics in two-dimensional bose-hubbard lattices: Bose-einstein condensate and thermal cloud, Phys. Rev. A 103, 033311 (2021).
  * Boschi  _et al._ [2014] C. D. E. Boschi, E. Ercolessi, L. Ferrari, P. Naldesi, F. Ortolani, and L. Taddia, Bound states and expansion dynamics of interacting bosons on a one-dimensional lattice, Phys. Rev. A 90, 043606 (2014).
  * Mark Jreissaty and Rigol [2011] F. A. W. Mark Jreissaty, Juan Carrasquilla and M. Rigol, Expansion of bose-hubbard mott insulators in optical lattices, Phys. Rev. A 84, 043610 (2011).
  * Jreissaty  _et al._ [2013] A. Jreissaty, J. Carrasquilla, and M. Rigol, Self-trapping in the two-dimensional bose-hubbard model, Phys. Rev. A 88, 031606(R) (2013).
  * L. Vidmar  _et al._ [2013] S. L. L. Vidmar, I. P. McCulloch, U. Schneider, U. Schollwöck, and F. Heidrich-Meisner, Sudden expansion of mott insulators in one dimension, Phys. Rev. B 88, 235117 (2013).
  * Hauschild  _et al._ [2015] J. Hauschild, F. Pollmann, and F. Heidrich-Meisner, Sudden expansion and domain-wall melting of strongly interacting bosons in two-dimensional optical lattices and on multileg ladders, Phys. Rev. A 92, 053629 (2015).
  * Anker  _et al._ [2005] T. Anker, M. Albiez, R. Gati, S. Hunsmann, B. Eiermann, A. Trombettoni, and M. K. Oberthaler, Nonlinear self-trapping of matter waves in periodic potentials, Phys. Rev. Lett. 94, 020403 (2005).
  * Albiez  _et al._ [2005] M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, Direct observation of tunneling and nonlinear self-trapping in a single bosonic josephson junction, Phys. Rev. Lett. 95, 010402 (2005).
  * Smerzi  _et al._ [1997] A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy, Quantum coherent atomic tunneling between two trapped bose-einstein condensates, Phys. Rev. Lett. 79, 4950 (1997).
  * Raghavan  _et al._ [1999] S. Raghavan, A. Smerzi, S. Fantoni, and S. R. Shenoy, Coherent oscillations between two weakly coupled bose-einstein condensates: Josephson effects, $\pi$ oscillations, and macroscopic quantum self-trapping, Phys. Rev. A 59, 620 (1999).
  * Trombettoni and Smerzi [2001a] A. Trombettoni and A. Smerzi, Discrete solitons and breathers with dilute bose-einstein condensates, Phys. Rev. Lett. 86, 2353 (2001a).
  * Trombettoni and Smerzi [2001b] A. Trombettoni and A. Smerzi, Variational dynamics of bose-einstein condensates in deep optical lattices, Journal of Physics B: Atomic, Molecular and Optical Physics 34, 4711 (2001b).
  * Creffield [2007] C. E. Creffield, Coherent control of self-trapping of cold bosonic atoms, Phys. Rev. A 75, 031607(R) (2007).
  * Rosenkranz  _et al._ [2007] M. Rosenkranz, D. Jaksch, F. Y. Lim, and W. Ba, Self-trapping of bose-einstein condensates expanding into shallow optical lattices, Phys. Rev. A 77, 063607 (2007).
  * Xiong  _et al._ [2009] B. Xiong, J. Gong, H. Pu, W. Bao, and B. Li, Symmetry breaking and self-trapping of a dipolar bose-einstein condensate in a double-well potential, Phys. Rev. A 79, 013626 (2009).
  * Roy  _et al._ [2022] R. Roy, B. Chakrabarti, and A. Trombettoni, Quantum dynamics of few dipolar bosons in a double-well potential., European Physical Journal D 76, 215303 (2022).
  * Morsch and Oberthaler [2006] O. Morsch and M. Oberthaler, Dynamics of bose-einstein condensates in optical lattices, Rev. Mod. Phys. 78, 179 (2006).
  * Duan [2005] L. M. Duan, Effective hamiltonian for fermions in an optical lattice across a feshbach resonance, Phys. Rev. Lett. 95, 243202 (2005).
  * Li  _et al._ [2012] W. Li, L. Hamadeh, and I. Lesanovsky, Probing the interaction between rydberg-dressed atoms through interference, Phys. Rev. A 85, 053615 (2012).
  * Defenu  _et al._ [2023] N. Defenu, T. Donner, T. Macrì, G. Pagano, S. Ruffo, and A. Trombettoni, Long-range interacting quantum systems, Rev. Mod. Phys. 95, 035002 (2023).
  * Hazzard  _et al._ [2014] K. R. Hazzard, B. Gadway, and e. a. Michael Foss-Feig, Many-body dynamics of dipolar molecules in an optical lattice, Phys. Rev. Lett. 113, 195302 (2014).
  * T.  _et al._ [2007] L. T., T. Koch, and F. B. et al., Strong dipolar effects in a quantum ferrofluid., Nature 448, 672 (2007).
  * Lahaye  _et al._ [2008] T. Lahaye, J. Metz, B. F. ans T. Koch, M. Meister, A. Griesmaier, T. Pfau, Y. K. H. Saito, and M. Ueda, d-wave collapse and explosion of a dipolar bose-einstein condensate, Phys. Rev. Lett. 101, 080401 (2008).
  * Koch  _et al._ [2008] T. Koch, T. Lahaye, and M. J. et al., Stabilization of a purely dipolar quantum gas against collapse., Nature Phys 4, 218 (2008).
  * Griesmaier  _et al._ [2005] A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau, Bose-einstein condensation of chromium, Phys. Rev. Lett. 94, 160401 (2005).
  * Lu  _et al._ [2011] M. Lu, N. Q. Burdick, S. H. Youn, and B. L. Lev, Strongly dipolar bose-einstein condensate of dysprosium, Phys. Rev. Lett. 107, 190401 (2011).
  * Aikawa  _et al._ [2012] K. Aikawa, A. Frisch, M. Mark, S. Baier, A. Rietzler, R. Grimm, and F. Ferlaino, Bose-einstein condensation of erbium, Phys. Rev. Lett. 108, 210401 (2012).
  * Böttcher  _et al._ [2021] F. Böttcher, J.-N. Schmidt, J. Hertkorn, K. S. H. Ng, S. D. Graham1, M. Guo, T. Langen, and T. Pfau, New states of matter with fine-tuned interactions: quantum droplets and dipolar supersolids, Rep. Prog. Phys. 84, 012403 (2021).
  * Chomaz  _et al._ [2023] L. Chomaz, I. Ferrier-Barbut, F. Ferlaino, B. Laburthe-Tolra, B. L. Lev, and T. Pfau, Dipolar physics: a review of experiments with magnetic quantum gases, Rep. Prog. Phys. 86, 026401 (2023).
  * Su  _et al._ [2023] L. Su, A. Douglas, and M. e. a. Szurek, Dipolar quantum solids emerging in a hubbard quantum simulator, Nature 622, 724 (2023).
  * Zöllner  _et al._ [2011] S. Zöllner, G. M. Bruun, C. J. Pethick, and S. M. Reimann, Bosonic and fermionic dipoles on a ring, Phys. Rev. Lett. 107, 035301 (2011).
  * Zöllner [2011] S. Zöllner, Ground states of dipolar gases in quasi-one-dimensional ring traps, Phys. Rev. A 84, 063619 (2011).
  * Astrakharchik  _et al._ [2008] G. E. Astrakharchik, G. Morigi, G. D. Chiara, and J. Boronat, Ground state of low-dimensional dipolar gases: Linear and zigzag chains, Phys. Rev. A 78, 063622 (2008).
  * Bera  _et al._ [2019] S. Bera, B. Chakrabarti, A. Gammal, M. C. Tsatsos, M. L. Lekala, B. Chatterjee, C. Lévêque, and A. U. J. Lode, Sorting fermionization from crystallization in many-boson wavefunctions, Scientific Reports 9, 17873 (2019).
  * Streltsov  _et al._ [2006] A. I. Streltsov, O. E. Alon, and L. S. Cederbaum, General variational many-body theory with complete self-consistency for trapped bosonic systems, Phys. Rev. A 73, 063626 (2006).
  * Streltsov  _et al._ [2007] A. I. Streltsov, O. E. Alon, and L. S. Cederbaum, Role of excited states in the splitting of a trapped interacting bose-einstein condensate by a time-dependent barrier, Phys. Rev. Lett. 99, 030402 (2007).
  * Alon  _et al._ [2007] O. E. Alon, A. I. Streltsov, and L. S. Cederbaum, Unified view on multiconfigurational time propagation for systems consisting of identical particles, J. Chem. Phys. 127, 154103 (2007).
  * Alon  _et al._ [2008] O. E. Alon, A. I. Streltsov, and L. S. Cederbaum, Multiconfigurational time-dependent hartree method for bosons: Many-body dynamics of bosonic systems, Phys. Rev. A 77, 033613 (2008).
  * Lode [2016] A. U. J. Lode, Multiconfigurational time-dependent hartree method for bosons with internal degrees of freedom: Theory and composite fragmentation of multicomponent bose-einstein condensates, Phys. Rev. A 93, 063601 (2016).
  * Fasshauer and Lode [2016] E. Fasshauer and A. U. J. Lode, Multiconfigurational time-dependent hartree method for fermions: Implementation, exactness, and few-fermion tunneling to open space, Phys. Rev. A 93, 033635 (2016).
  * Lode  _et al._ [2020] A. U. J. Lode, C. Lévêque, L. B. Madsen, A. I. Streltsov, and O. E. Alon, Colloquium: Multiconfigurational time-dependent hartree approaches for indistinguishable particles, Rev. Mod. Phys 92, 011001 (2020).
  * Lin  _et al._ [2020] R. Lin, P. Molignini, L. Papariello, M. C. Tsatsos, C. Lévêque, S. E. Weiner, E. Fasshauer, and R. Chitra, Mctdh-x: The multiconfigurational time-dependent hartree method for indistinguishable particles software, Quantum Sci. Technol. 5, 024004 (2020).
  * Lode  _et al._ [2024] A. U. J. Lode, M. C. Tsatsos, E. Fasshauer, S. E. Weiner, R. Lin, L. Papariello, P. Molignini, C. Lévêque, M. Büttner, J. Xiang, S. Dutta, and Y. Bilinskaya, Mctdh-x: The multiconfigurational time-dependent hartree method for indistinguishable particles software (2024).
  * Sinha and Santos [2007] S. Sinha and L. Santos, Cold dipolar gases in quasi-one-dimensional geometries, Phys. Rev. Lett. 99, 140406 (2007).
  * Deuretzbacher  _et al._ [2010] F. Deuretzbacher, J. C. Cremon, and S. M. Reimann, Ground-state properties of few dipolar bosons in a quasi-one-dimensional harmonic trap, Phys. Rev. A 81, 063616 (2010).
  * Chatterjee and Lode [2018] B. Chatterjee and A. U. J. Lode, Order parameter and detection for a finite ensemble of crystallized one-dimensional dipolar bosons in optical lattices, Phys. Rev. A 98, 053624 (2018).
  * Molignini  _et al._ [2022] P. Molignini, C. Lévêque, H. Keßler, D. Jaksch, R. Chitra, and A. U. J. Lode, Crystallization via cavity-assisted infinite-range interactions, Phys. Rev. A 106, L011701 (2022).
  * Novotny  _et al._ [2023] L. Novotny, J. Vijayan1, J. Piotrowski, C. Gonzalez-Ballestero, K. Weber, and O. Romero-Isart, Cavity-mediated long-range interactions in levitated optomechanics,  (2023).
  * Kramer and Saraceno [1981] P. Kramer and M. Saraceno,  _Geometry of the Time-Dependent Variational Principle in Quantum Mechanics_ , Lecture Notes in Physics, Vol. 140 (Springer, 1981).
  * Kvaal [2013] S. Kvaal, Variational formulations of the coupled-cluster method in quantum chemistry, Molecular Physics 111, 1100 (2013).
  * McLachlan [1964] A. McLachlan, A variational solution of the time-dependent schrodinger equation, Molecular Physics 8, 39 (1964).
  * Cao  _et al._ [2017] L. Cao, V. Bolsinger, S. I. Mistakidis, G. M. Koutentakis, S. Krönke, J. M. Schurer, and P. Schmelcher, A unified ab initio approach to the correlated quantum dynamics of ultracold fermionic and bosonic mixtures, J. Chem. Phys. 147, 044106 (2017).