# Vortices in dipolar Bose-Einstein condensates

Thomas Bland  Institut für Experimentalphysik, Universität Innsbruck,
Technikerstraße 25, 6020 Innsbruck, Austria  Giacomo Lamporesi  Pitaevskii BEC
Center, CNR-INO and Dipartimento di Fisica, Università di Trento, I-38123,
Trento, Italy  Manfred J. Mark  Institut für Experimentalphysik, Universität
Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria  Institut für
Quantenoptik und Quanteninformation, Österreichische Akademie der
Wissenschaften, Technikerstraße 21a, 6020 Innsbruck, Austria  Francesca
Ferlaino<EMAIL_ADDRESS>Institut für Experimentalphysik,
Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria  Institut
für Quantenoptik und Quanteninformation, Österreichische Akademie der
Wissenschaften, Technikerstraße 21a, 6020 Innsbruck, Austria

###### Abstract

Quantized vortices are the hallmark of superfluidity, and are often sought out
as the first observable feature in new superfluid systems. Following the
recent experimental observation of vortices in Bose-Einstein condensates
comprised of atoms with inherent long-range dipole-dipole interactions [Nat.
Phys. 18, 1453-1458 (2022)], we thoroughly investigate vortex properties in
the three-dimensional dominantly dipolar regime, where beyond-mean-field
effects are crucial for stability, and investigate the interplay between trap
geometry and magnetic field tilt angle.

##  I Quantum vortices in ultracold gases

Every time a fluid starts to rotate, a vortex can form. This phenomenon is a
universal property of fluid dynamics, and has been observed in many different
systems on all possible lengthscales, from the spiral motion of galaxies, to
the tornadoes in water sinks or in the atmosphere, from superconductors
immersed in a magnetic field to rotating superfluids, such as Helium or
quantum gases. The nature of the fluid strongly affects the motion of the
particles and the properties of the supported vortices. A special case is, in
fact, represented by superfluids, which are characterized by the absence of
viscosity and are described by a continuous, single-valued wavefunction. The
latter feature leads to the quantization of circulation and to quantum
vortices.

First theoretically predicted by Onsager and Feynman at the second half of the
last century [1, 2], quantum vortices were indirectly seen in superfluid
liquid helium [3, 4, 5, 6]. The advent of experiments on dilute, weakly
interacting quantum gases opened the door to an intense series of direct
observations of quantum vortices, providing proof of superfluidity in
ultracold gases. The existence of a quantum vortex in a superfluid requires it
to accumulate a 2$\pi$ phase winding around a line if the fluid is three-
dimensional (3D) or around a point if it is confined in a two-dimensional (2D)
geometry. Typically, the gas density vanishes approaching the vortex core
following a $1/r$-law on a characteristic lengthscale given by the local
healing length. While the ground state of a superfluid is described by a
uniform phase, leading to the absence of superfluid motion, any phase-
fluctuating excited configuration is accompanied by a local superfluid motion,
according to the relation $\mathbf{v_{s}}=\frac{\hbar}{m}\nabla\phi$.

Phase curls, and consequently vortices, can be generated in a superfluid in
many ways. The seminal experimental work of E. Cornell and co-workers at JILA
[7], J. Dalibard and co-workers at LKB [8], and that of W. Ketterle and co-
workers at MIT [9] observed the first vortices in BECs, opening up the field
of quantum vortices in ultracold atoms. Since then, many experimental
techniques have been developed, as illustrated in Fig. 1. In a nutshell, such
techniques can be divided into the following categories:

Figure 1:  Examples of vortices observed in ultracold gases and created with
different methods, as listed in Section I. a) Three-vortex sample [8] and
large vortex lattice (from [9], reprinted with permission from AAAS) obtained
by stirring the BEC with optical potentials at different angular velocities.
Vortex lattice obtained by rotating the magnetic trap and showing Tkachenko
waves [10]. b) Images of topological defects resulting from the decay of an
imprinted planar soliton [11, 12]. c) Interference between expanding 2D
systems containing thermally activated vortices [13] near the BKT transition.
d) Interference pattern between three [14] or many [15] BECs on a ring. e)
Signature of a solitonic vortex in an elongated gas [16] and interference
pattern between a ring containing a vortex and a reference BEC [17], both
created through the Kibble-Zurek mechanism. f) Vortices resulting from
artificial gauge fields [18]. g) Pair of vortices [19] and turbulent vortex
ensemble [20] generated by moving a barrier through the BEC at high speed. h)
Spiral interference signal between a vortexless and a 6-vortex phase pattern
imprinted on an annular Fermi gas [21]. Reproduced figures with permission
from [16, 10, 17, 19, 14, 15, 20, 11, 12, 8]. Copyright by the American
Physical Society. Reproduced figures from [13, 18], with permission from
Springer Nature.

  * •

Rotation. The most intuitive and elegant way to introduce vorticity in a
quantum gas, consists in making the gas rotate. This can be implemented by
introducing an asymmetry in the trapping potential through off-center optical
beams [8, 9, 22] or anisotropic magnetic confinement [23, 24] and having them
rotate about the symmetry axis. For increasing rotation rates, more and more
same-sign vortices enter the system and align their core lines to the rotation
axis. The formation of regular triangular Abrikosov lattices, with a density
proportional to the rotation rate, was observed in Refs. [8, 24, 9]. Such a
vortex crystal can be perturbed by exciting Tkachenko modes [10] and can turn
into a square lattice in the presence of two spin components in the trap [25].

  * •

Soliton decay. Standard solitons are usually stable in one-dimensional
systems, while they are subject to decay into other topological excitations in
higher dimensions. The snake instability is a typical example of such a decay
mechanism that shows how the solitonic plane (in 3D systems) or the soliton
line (in 2D systems) breaks into vortex ring pairs or vortex dipoles. These
vortical structures are, in fact, less energetic than solitons and hence more
robust. The decay mechanism and the vortices produced were studied in 3D
isotropic [11], in elongated geometries [26, 16, 12], or in flat potentials
[27].

  * •

Thermal activation in 2D systems. Systems with reduced dimensionalities are
characterized by enhanced fluctuations. In 2D, interactions drive a continuous
phase transition from a superfluid (at low temperatures) to a normal gas (at
high temperatures), called Berezinskii-Kosterlitz-Thouless transition. As the
temperature is increased, the spatial correlations turn from algebraically to
exponentially decaying and initially low-energy vortex-antivortex pairs unbind
toward a proliferation of free vortices. This was observed in continuous
systems [13, 28] and in lattice configurations [29].

  * •

Merging isolated condensates. When independent condensates are released from
their traps and start to spatially overlap, they interfere, as demonstrates
with pairs [30] or linear arrays of BECs [31]. Similarly, initially isolated
condensates with independent phases, arranged in on a ring, interfere and
depending on their phase distributions, they can also show vortices in the
final merged condensate. Such a mechanism was first demonstrated using three
BECs on a ring [14] and then extended to many [15].

  * •

Quenches across phase transitions. A phase transition is usually associated to
a symmetry breaking. When the critical point of the transition is rapidly
crossed, it is likely to create domains with different phases in the system.
Through such a mechanism, turbulent phase patterns form, including the
possibility to have vortices. Spin vortices have been observed by quenching a
spinor BEC across a quantum phase transition [32] and also across the normal
to BEC phase transition [33]. This lead to an intense investigation of the
Kibble-Zurek mechanism in quantum gases [34, 17, 35, 36, 37, 38].

  * •

Artificial gauge fields. Vortices can also be introduced through the
application of synthetic magnetic fields. This possibility has been
demonstrated [18] by introducing a spin-orbit coupling potential made of a
pair of Raman beams propagating along different directions and having a
frequency difference that matches an atomic two-level transition. These
artificial fields do not possess the limitations of real rotating systems and
can potentially lead to the observation of quantum Hall effects [39].
Furthermore, due to the analogy of rotating neutral atoms and charged
particles in a magnetic field, mechanical rotation at the trap frequency maps
the Bose-Einstein condensate onto a single Landau gauge wavefunction, allowing
access to the lowest Landau levels, with the rotating system behaving like a
strongly correlated quantum Hall fluid [40, 41].

  * •

Strong mechanical perturbations. By moving an obstacle such as a focused laser
beam across the superfluid [19, 42, 20, 43, 44], by strongly shaking the
confining potential [45, 46], or by letting two condensates collide with a
large relative momentum [47], the gas can be moved at velocities well above
the local speed of sound, turning it into a turbulent system with strong phase
fluctuations and many vortices. In this case, the net vorticity is on average
zero.

  * •

Optical phase imprinting. The spatial phase pattern characterizing vortices
can be reproduced optically through digital micromirror devices or spatial
light modulators. Illuminating the gas with such a pattern, the wavefunction
acquires a local phase, proportional to the light intensity. This method was
proposed in the past [48, 49], but only recently implemented to imprint
vortices in a controllable way [21]. Optical imprinting techniques with
dynamic beams configurations [7] or static ones using Laguerre-Gauss beams
[50] successfully produced quantum vortices.

These methods have all been observed in gases with short-ranged contact
interactions. In this work, we focus on the recent developments in the vortex
generation in dipolar gases, in particular with the novel magnetostirring
technique. We study the stability of the dipolar condensate for increasing
rotation frequencies, then focus on the vortex core shape as a function of the
relative angle between the rotation axis and the magnetic field, and finally
extend from a single vortex to many, with the formation of exotic ordered
vortex structures, typical of dipolar gases.

##  II Dipolar gases and magnetostirring

Dipolar BECs comprised of highly-magnetic lanthanide atoms [51, 52] are taking
the lead as a promising platform to investigate complex quantum matter
phenomena. Such systems can be viewed as the quantum version of classical
ferrofluids, where the physics is dominated by the interplay of long-range
anisotropic dipole-dipole interactions (DDIs) [53, 54] with short-range
contact interactions and trap potentials. The balance of these contributions
leads to excitations known as rotons. These low energy excitations occur at a
local minimum in the dispersion relation. First predicted in liquid helium
[55], they are considered as the precursor to crystallization due to their
appearance at a finite momentum, and therefore of a fixed wavelength.
Predicted in dipolar systems in 2003 [56, 57], they were observed in dipolar
gases first in cigar [58, 59] then pancake [60, 61] geometries. The transition
to a stable crystal was achieved later, where triggering a roton instability
[58] leads to an appearance of a density modulated state with global phase
coherence, known as a supersolid [62], with translational symmetry broken
along one [63, 64, 65] or two [66, 67] axes. Other observed phenomena in
ultracold dipolar gases thus far include, but are not limited to, lattice spin
models [68, 69], extended Bose-Hubbard dynamics [70], quantum droplets [71,
72], and the quantum Rosensweig instability [73]; constituting only the tip of
the iceberg of possibilities during this second quantum revolution [53, 54].
Equivalent phenomena are also predicted in other systems with long-range
interactions, such as ground-state heteronuclear molecules [74, 75] and
Rydberg atoms [76], and light-atom coupled systems [77, 78, 79].

In magnetic atoms, these phenomena arise due to the long-range nature of the
interaction, which for a polarized gas of ultracold atoms presents as a
pseudo-potential

$\displaystyle
U_{\text{dd}}(\textbf{r})=\frac{C_{\text{dd}}}{4\pi}\frac{1-3\left(\hat{\mathbf{e}}\cdot\hat{\mathbf{r}}\right)^{2}}{r^{3}}\,,$
(1)

characterized by the dipole strength $C_{\text{dd}}=\mu_{0}\mu_{m}^{2}$ for
atoms with a permanent dipole moment with magnetic moment $\mu_{m}$ and the
vacuum permeability $\mu_{0}$, dipoles polarized along $\hat{\textbf{e}}$, and
$\hat{\textbf{r}}=\textbf{r}/r$ with $r=|\textbf{r}|$.

Figure 2: Illustration of magnetostirring. Dipolar BEC (red) in an azimuthally
symmetric pancake trap (gray mesh) stretches along the B-field, at an angle
$\theta$, and follows the polarization orientation as it is stirred at a
frequency $\Omega$.

The anisotropic nature of Eq. (1) reveals a novel method of vortex generation.
Here, two notions are important. First, the atomic magnets always align along
the magnetic-field vector, which sets the polarization axis. Second, a
polarized gas stretches along the polarization axis in a process known as
magnetostriction [80], where dipoles align head-to-tail to maximize their
attraction. This can be exploited to induce asymmetry into the dipolar BEC
density profile, in an otherwise cylindrically symmetric trap, by tilting the
magnetic field into the plane. Once tilted, the magnetic field can be rotated
around the trap centre producing vortices in an analogous method to mechanical
trap rotation of a non-dipolar BEC [81]. See Fig. 2 for a schematic of this
process. Recently, this method was employed to produce the first observed
vortices in a dipolar gas [82], in which the vortices exhibit remarkable
properties due to the dipole-dipole interaction, to be explored in the next
section. Rapid magnetostirring is known to produce a different effect, where
the atoms remain stationary but the dipole moments rapidly rotate [83]. This
leads to a time-averaged dipole-dipole interaction that can be reversed from
the bare DDI (i.e. introducing a negative sign to Eq. (1)), known as an anti-
dipolar interaction, where head-to-tail anti-dipoles attract and side-by-side
anti-dipoles repel. This was observed experimentally in 2018 [84], though the
condensate had a reduced lifetime due to destructive dynamical instabilities
[85]. There remains hope that these can be suppressed by rotating at much
larger frequencies [86], and the impact of changing the trap geometry is
unexplored. Furthermore, slow magnetostirring is also predicted to be a good
probe of the stability of two-dimensional bright solitons [87] and droplets
[88], and may even produce vortices inside these intriguing self-bound objects
[89, 90].

##  III Quantum vortices in dipolar gases

The long-range and anisotropic DDI also drastically modifies the now well-
established superfluid vortex properties [91]. At the single vortex level, it
is the anisotropy of the interaction that in turn produces elliptical vortex
cores, observable when the vortex line and magnetic field vector are non-
parallel [92, 93, 94, 95, 96]. Instead, under the condition of vortex lines
parallel to the external magnetic field polarization, the presence of the
roton minimum in the dispersion relation can produce concentric alternating
rings of high and low density surrounding the vortex core [92, 97, 94, 95, 96,
98]. These effects can be understood from a simple toy model introduced in a
review by Martin et al. [91]: decomposing the condensate density for a vortex
into $n(\textbf{r})=n_{0}-n_{v}(\textbf{r})$, with the vortex-free solution
$n_{0}$ and the vortex density $n_{v}$, and calculating the dipolar energy
contribution from this solution provides the necessary insight. The dipolar
interaction energy is then given by

$\displaystyle 2E_{\text{dd}}$
$\displaystyle=\int\text{d}^{3}\textbf{r}\int\text{d}^{3}\textbf{r}^{\prime}\,n(\textbf{r})U_{\text{dd}}(\textbf{r}-\textbf{r}^{\prime})n(\textbf{r}^{\prime})$
$\displaystyle 2E_{\text{dd}}$
$\displaystyle=\int\text{d}^{3}\textbf{r}\int\text{d}^{3}\textbf{r}^{\prime}\,n_{0}U_{\text{dd}}(\textbf{r}-\textbf{r}^{\prime})n_{0}$
$\displaystyle~{}+\int\text{d}^{3}\textbf{r}\int\text{d}^{3}\textbf{r}^{\prime}\,n_{v}(\textbf{r})U_{\text{dd}}(\textbf{r}-\textbf{r}^{\prime})n_{v}(\textbf{r}^{\prime})$
$\displaystyle~{}-2\int\text{d}^{3}\textbf{r}\int\text{d}^{3}\textbf{r}^{\prime}\,n_{0}U_{\text{dd}}(\textbf{r}-\textbf{r}^{\prime})n_{v}(\textbf{r}^{\prime})\,.$
(2)

The negative sign of the last term in this expansion can be interpreted such
that the vortex acts as a giant anti-dipole sitting inside the (normal)
dipolar BEC. For dipoles aligned parallel to the vortex line, this means that
the density is pulled towards the vortex core, causing the ripple structure.
For dipoles tilted into the plane, the antidipolar interaction causes head-to-
tail repulsion, hence the vortex core repels atoms along the B-field and
stretches elliptically, analogously to magnetostriction. Anisotropic vortex
cores have also been observed in vortices in superconductors due to the same
effect [99].

The same line of thought elucidates the properties of vortex-vortex
interactions. In a non-dipolar condensate confined to a quasi-2D geometry,
vortex-vortex interactions are also long-ranged (scaling as $1/r$) but are
isotropic, and the nature of the interaction (attractive/repulsive) is set by
the sign of the circulation. Same sign vortices repel and rotate around one
another at a fixed distance, whereas opposite sign vortices attract one
another and annihilate. In 3D, there are also interesting recombination and
rebounding effects between vortex filaments, depending also on the relative
angle of approach and speed [100]. The isotropy of the vortex-vortex
interaction leads to the triangular Abrikosov vortex lattice. In the dipolar
system, we can see from Eq. (2) that the inter-vortex interaction behaves like
a bare dipole-dipole interaction. Two dipolar vortices sat in a side-by-side
configuration (with respect to the B-field angle) will repel one another, and
attract head-to-tail. Thus, there are predictions that there can be
suppression of vortex-antivortex annihilation [96] and an elliptic precession
of two same-sign vortices [96, 101]. Beyond vortex pairs, long-range dipolar
interactions also alter the bulk lattice structure. Dipoles pointing out of
the plane–i.e. interacting isotropically within the plane–give rise to a
triangular lattice structure when the contact interaction is repulsive [92,
98], but this can be square or rectangular for attractive or zero contact
interactions [102, 103, 104]. Breaking the isotropy by tilting the dipoles,
breaks the symmetry of the lattice, where instead the vortices align in
stripes, preferring to maximize head-to-tail configurations as much as
possible [92, 105, 81]. There are open questions pertaining to finite
temperature effects on dipolar vortices, such as Kibble-Zurek physics and the
Berezinskii-Kosterlitz-Thouless transition, though in the absence of external
driving there is evidence of anisotropic and polarized turbulent decay [106].
To our knowledge, the impact of quantum fluctuations on these predictions is
unknown.

In spite of 20 years of theoretical and experimental endeavour, the first
experimental evidence of vortices in a dipolar Bose-Einstein condensate was
only observed in an experiment last year [82]. Vortices in a dipolar BEC are
predicted to be generated through all the aforementioned methods at the
beginning of this article [91]. However, Klaus et al. utilized the
magnetostirring protocol of Fig. 2. Turbulent clouds of vortices were
generated, and ordering into a stripe configuration was observed. This
experiment was inspired by earlier theoretical works that predicted the
formation of vortex lattices under continuous magnetostirring at magnetic
fields tilted at an arbitrary angle from the trap geometry [81, 107]. In
future experiments, it will be interesting to see single vortex properties,
and to generate ground state lattices under continuous rotation [81].

There are open questions on how quantum fluctuations change the predictions
made thus far, though in the superfluid phase their effect is expected to be
minimal. In the supersolid phase, however, they play the crucial role of
stabilizing the gas against collapse, and the observation of vortices could
provide an unambiguous smoking gun of superfluidity in supersolid states [108,
109, 110, 111, 112]. In this special issue, we analyze the effect of varying
the scattering length and dipolar angle on the predicted vortex
properties–including single vortex properties and the resulting lattice
formation–focussing on the experimentally relevant regime, and including
beyond-mean-field effects.

##  IV Theoretical model

We focus our investigations on parameter regimes relevant to the recent
experimental observation of vortices in a dipolar BEC [82]. We use an extended
Gross-Pitaevskii formalism for the wavefunction
$\Psi\equiv\Psi(\textbf{r},t)$, where the time-dependent equation reads [113,
72, 71, 114]

$\displaystyle i\hbar\frac{\partial\Psi}{\partial t}$
$\displaystyle=\bigg{[}-\frac{\hbar^{2}\nabla^{2}}{2m}+\frac{1}{2}m\left[\omega_{r}^{2}(x^{2}+y^{2})+\omega_{z}^{2}z^{2}\right]$
$\displaystyle~{}+\int\text{d}^{3}\textbf{r}^{\prime}\,U(\textbf{r}-\textbf{r}^{\prime})|\Psi(\textbf{r}^{\prime},t)|^{2}+\gamma_{\text{QF}}|\Psi|^{3}\bigg{]}\Psi(\textbf{r},t)\,,$
(3)

for atoms of particle mass $m$ confined in a harmonic trap with frequency
$f_{j}=\omega_{j}/2\pi$ and aspect ratio $\gamma=\omega_{z}/\omega_{r}$,
reduced Planck’s constant $\hbar$, and the wavefunction $\Psi$ is normalized
to the total atom number $N=\int{\rm d}^{3}\mathbf{x}|\Psi|^{2}$. The last two
terms of Eq. (3) describes the interparticle interactions, determined by the
pseudo-potential,

$\displaystyle U(\textbf{r})=\frac{4\pi\hbar^{2}a_{\rm
s}}{m}\delta(\textbf{r})+\frac{3\hbar^{2}a_{\text{dd}}}{m}\frac{1-3\left(\hat{\mathbf{e}}(t)\cdot\hat{\mathbf{r}}\right)^{2}}{r^{3}}\,,$
(4)

where the first term derives from short-range interactions with strength given
by the s-wave scattering length $a_{s}$. The second term is the anisotropic
and long-ranged dipole-dipole interaction given by Eq. (1), characterized by
dipole length $a_{\text{dd}}=C_{\text{dd}}m/12\pi\hbar^{2}$. We always
consider 162Dy, such that $a_{\text{dd}}=129.2\,a_{0}$, where $a_{0}$ is the
Bohr radius. The time dependent polarization $\hat{\mathbf{e}}(t)$ gives us
the ability to model magnetostirring, with the dipoles polarized
instantaneously along a time-dependent axis, given by

$\displaystyle\hat{\mathbf{e}}(t)=(\sin\theta(t)\cos\phi(t),\sin\theta(t)\sin\phi(t),\cos\theta(t))\,,$
(5)

with time dependent polarization angle $\theta(t)$ and
$\phi(t)=\int_{0}^{t}\text{d}t^{\prime}\Omega(t^{\prime})$, for rotation
frequency protocol $\Omega(t)$. Finally, beyond-mean-field effects are treated
through the inclusion of a Lee–Huang–Yang correction term [115, 116, 117]

$\displaystyle\gamma_{\text{QF}}=\frac{128\hbar^{2}}{3m}\sqrt{\pi
a_{s}^{5}}\,\text{Re}\left\\{\mathcal{Q}_{5}(\epsilon_{\text{dd}})\right\\}\,,$
(6)

with the auxiliary function
$\mathcal{Q}_{5}(\epsilon_{\text{dd}})=\int_{0}^{1}\text{d}u\,(1-\epsilon_{\text{dd}}+3u^{2}\epsilon_{\text{dd}})^{5/2}$,
and the relative dipole strength $\epsilon_{\text{dd}}=a_{\rm dd}/a_{\rm s}$.
We identify two regimes that are used throughout this work: the contact
dominated regime $\epsilon_{\text{dd}}<1$ and the dipole dominated regime
$\epsilon_{\text{dd}}>1$. The initial state $\Psi(\textbf{r},0)$ of the real-
time simulations is obtained by first solving Eq. (3) in imaginary time,
giving $\Psi_{0}(\textbf{r})$, and then adding non-interacting noise to this
state. This noise is generated through single-particle eigenstates $\phi_{n}$
and the complex Gaussian random variables $\alpha_{n}$ sampled with
$\langle|\alpha_{n}|^{2}\rangle=(e^{\epsilon_{n}/k_{B}T}-1)^{-1}+\frac{1}{2}$
for a temperature $T=20$ nK, such that the total initial state can be
described as
$\Psi(\textbf{r},0)=\Psi_{0}(\textbf{r})+\sum_{n}^{\prime}\alpha_{n}\phi_{n}(\textbf{r})$,
where the sum is restricted only to the modes with $\epsilon_{n}\leq 2k_{B}T$
[118]. The simulations are performed with a split-step Fourier method, in a
$192^{3}$ box of equal lengths $L=40\,\mu$m, imposing a spherical cut-off for
the dipolar potential to avoid contamination of our results with phantom
simulation copies [119].

##  V Stability of the condensate under magnetostirring

The stability of a non-dipolar condensate under mechanical rotation is well
understood from a hydrodynamic formalism [120, 121]. This approach tells us
that, in spite of the energetic favorability of generating vortices for
rotation frequencies exceeding $\Omega\sim 0.3\,\omega_{r}$, with symmetric
radial trap frequency $\omega_{r}$, a dynamical instability is required to
nucleate the vortices into the system. This phenomenon was first observed by
the LKB group led by J. Dabilard in a non-dipolar BEC [8]. Typically, this
dynamical instability occurs at $\Omega\sim 0.7\,\omega_{r}$, where low-lying
collective modes are seeded and become unstable [8, 122]–associated with a
resonance with the quadrupole mode at
$\Omega\sim\omega_{r}/\sqrt{2}$–providing the necessary avenue for vortex
creation.

This same phenomenon has been thoroughly investigated in dipolar gases, where
in spite of the complexity of the underlying integro-differential equation
there are analytic solutions in the Thomas-Fermi regime [123, 124]. A key
feature of the non-dipolar analysis is that there exists three solutions for
the stationary elliptic parabolic density profile at large rotation
frequencies: one stretched parallel to the trap deformation, and two stretched
perpendicularly. Of the solutions aligned perpendicularly to the trap
deformation, one is weakly deformed and highly stable, and the other is highly
unstable and strongly deformed. In the dipolar system, under weak trap
ellipticity and dipoles pointing out of the plane, all three solutions have
been shown to exist, and possess similar features [125, 126, 127]. For
stronger dipole-dipole interactions, both the critical value of $\Omega$ at
which these three solutions exist can be reduced from the non-dipolar result,
and the critical frequency of the dynamical instability is similarly decreased
[127]. In these works, ellipticity is induced by breaking the radial trap
symmetry, however when dipoles are pointing into the plane elliptic
deformation is instead provided by magnetostriction, and angular momentum is
induced through magnetostirring. Crucially, this method exhibits the same
features from the mechanically rotating Thomas-Fermi analysis, following a
series of earlier theoretical predictions from Prasad et al. [85, 81], work
that was also extended to arbitrary rotation angles [107]. Rotation at an
arbitrary angle is known to present unique features due to centrifugal forces
pushing the condensate orientation towards the plane, tilted with respect to
the trap or magnetic field angle, and can facilitate stable solutions for
rotation frequencies greater than the radial frequency [107, 128].

Figure 3: Aspect ratio of a 162Dy dipolar gas under accelerated
magnetostirring. Each curve shows the aspect ratio evolution for increasing
B-field tilt $\theta$, in steps of 10∘. Colors indicate which category the
solutions fall under with (i) green, (ii) orange, and (iii) magenta (see
text). The orange circles are the experimental data of Ref. [82] (Figure 1(c))
taken with the same system parameters at $\theta=35^{\circ}$. Insets show
column densities for fixed time and increasing $\theta$, indicated by border
color, and rotated to the same B-field angle given by the arrow. Each panel is
$20\mu\text{m}\times 20\mu\text{m}$. Parameters: $N=15000$,
$a_{s}=110\,a_{0}$, $\vec{f}=(50,130)$ Hz, i.e. $\gamma=2.6$.

In a recent experiment, Klaus et al. [82] fixed the tilt angle of the magnetic
field at $\theta=35^{\circ}$ and investigated the consequences of
magnetostirring a dipolar BEC. In what follows, we investigate the
significance of this choice of angle. We explore the stability of a dipolar
BEC in the dominantly dipolar regime ($\epsilon_{\text{dd}}>1$) with fixed
scattering length $a_{s}=110\,a_{0}$ under increased rotation frequencies,
increasing at a rate of $2\pi\times 50$ Hz s-1, and different B-field tilt
angles. The measured aspect ratio from the column density is shown in Fig. 3.
As expected, increasing the initial tilt angle the state is stretched due to
magnetostriction, and we clearly observe the extension of the aspect ratio at
low rotation frequencies. However, the stability of the state at fast
rotations is angle-dependent and can be categorized into one of three groups:

  1. (i)

$0\leq\theta<20^{\circ}$ (green): At shallow tilt angles the condensate
density remains parabolic throughout the simulation, and although the aspect
ratio initially increases, it finally decreases to $\sim 0.85$, indicating
that the stationary solution follows the middle branch from the Thomas-Fermi
analysis. This branch is known to be highly stable, hence, no vortices are
formed here.

  2. (ii)

$20^{\circ}\leq\theta<50^{\circ}$ (orange): At intermediate angles the
condensate density undergoes a dynamical instability (at $\Omega\sim
0.8\,\omega_{r}$), and vortices begin to penetrate the condensate surface.

  3. (iii)

$50^{\circ}\leq\theta\leq 90^{\circ}$ (magenta): At deeper angles, as the
condensate elongates, more atoms align in a head-to-tail configuration, and
the attractive dipolar interaction holds the atoms together, suppressing a
dynamical instability.

The demarcating value of $\theta$ between each one of these categories will
depend on the choice of scattering length and trap geometry. Remarkably, the
solution for $\theta=90^{\circ}$ is a BEC for $\Omega=0$, but due to
elongation of the state and increased head-to-tail attraction becomes a
rotating trap-bound droplet state as $\Omega\to\omega_{r}$, stable due to the
quantum fluctuations. We have verified that after another second of continuous
driving at $\Omega=\omega_{r}$ the droplet continues to rotate without a
growing collective mode amplitude. As the scattering length is increased, and
the relative dipolar effects are reduced, the range of stable solutions for
large $\theta$ (iii) will be decreased. Furthermore, the effect of
magnetostriction will be diminished, such that increasing the scattering
length plays a similar role as reducing the trap ellipticity in non-dipolar
condensates, where this is known to aid stability. This, in turn, will also
increase the $\theta$ range of stable solutions at small tilt angles (i).
There, the ramp procedure is non-adiabatic compared to the timescales required
to generate a dynamical instability, hence it is possible for the condensate
to remain stable on the central stable branch predicted from Thomas-Fermi
theory [81, 107].

For this ramp procedure, however, there exists the Goldilocks range of angles
(ii) where a dynamical instability occurs on a short timescale observable in
experiments. After a collective mode goes unstable, vortices flood the
condensate surface and the aspect ratio (obtained by a fit to a 2D Gaussian
profile) no longer accurately captures the condensate density. Also overlaid
on Fig. 3 are the data from Figure 1 of Ref. [82], obtained with the same
protocol but with $\theta=35^{\circ}$ (orange points). At short times the data
agree perfectly, it is only at late times, where the condensate undergoes a
much earlier dynamical instability, that the results diverge. The time at
which the instability happens will be sensitive to finite temperature effects
and small inhomogeneities in the rotation procedure, which we attribute to the
earlier onset of instability. The precise role of the trap aspect ratio
$\gamma$ in the dipole dominated regime remains an open question, noting that
previous studies have shown in the contact dominated regime that there exist
trap aspect ratios where the dipolar interaction can fully suppress dynamical
instabilities [124].

##  VI Single vortex in an oblate geometry

Vortices in a dipolar BEC are known to have unique features compared to their
non-dipolar counterparts [129]. For $\theta=0$ and scattering lengths close to
a roton instability, the vortex core exhibits a high density ring around the
core, and the core size tends to the inter-droplet distance as the scattering
length is crossed into the supersolid phase. In tighter quasi-2D geometries,
roton effects are more prominent prior to mean-field instability, and instead
there are many concentric rings decreasing in amplitude radially. Tilting the
B-field into the plane causes magnetostriction of the vortex core itself, and
this has intriguing consequences on the resulting vortex lattice structure,
that we will explore later.

Figure 4: Column densities of a single vortex in a dipolar BEC, for varying
tilt angle $\theta$ and scattering length $a_{s}$. For 90$a_{0}$ the
condensate undergoes a roton instability at 25° (red crosses), and the
stationary state is instead two tilted droplets. The core eccentricity,
measured at an arbitrary low isodensity $3\times 10^{19}$m-3, tends to 1 for
increasing $\theta$ in all cases. The inset shows the width of the minor axis.
Other parameters the same as Fig. 3.

For our system parameters we assess how the ellipticity and core size depend
on both the scattering length and dipole angle, paying particular heed to the
results in the range $20^{\circ}<\theta<50^{\circ}$, which we have seen can
generate vortices within a reasonable experimentally feasible timescale
($<1\,$s). In Fig. 4, we show the column densities for a single vortex, for
three tilt angles and three values of the scattering length. These stationary
solutions are obtained through generating an initial condition with a 2$\pi$
phase winding in the $xy$-plane and evolving in imaginary time. For
$\theta=0^{\circ}$, the core is cylindrically symmetric, increasing in radial
size as the scattering length is lowered. At first glance, this contrasts the
energy argument given around Eq. (2). However, as the scattering length is
reduced the healing length increases, setting a wider vortex core size. In
this trap geometry, the effect of the roton is present, but subtle, appearing
as a slightly increased density around the core, and also widening the core
size to be comparable to the roton wavelength [109]. For any non-zero tilt
angle the vortex core is stretched into an ellipse. We measure the core
eccentricity as $\epsilon=\sqrt{1-R_{x}^{2}/R_{y}^{2}}$, for major axis
$R_{y}$ along the B-field and minor axis $R_{x}$ perpendicular to the B-field,
taken at an arbitrary low isodensity $3\times 10^{19}$ m-3. We see that the
core eccentricity exceeds 0.5 for all considered scattering lengths when the
tilt angle exceeds $\sim 20^{\circ}$. Therefore, in the angle regime of
interest ($20^{\circ}<\theta<50^{\circ}$) the effect of the dipolar
interactions should be visible from the column density.

At deeper angles than 25∘, and small scattering lengths, the increased head-
to-tail attraction triggers a roton instability, and the stationary state is a
pair of droplets. Note that the ground state with these parameters is a single
droplet with no vortex. Here, a vortex is imprinted between two droplets,
though not visible in the density profile, highlighting the challenge of
observing vortices in the density profile in the supersolid phase [108, 109,
110, 111, 112]. One recent proposal [111] has suggested quenching the
scattering length into the BEC phase after creating the vortex through
rotation of the supersolid state. In this way, the vortex remains after the
quench and would be visible in the density profile. One other method could be
to create a persistent current in a ring supersolid, and observe the
circulation through the interference pattern in time-of-flight imaging [130].

Figure 5: Generation of vortex stripes in a dipolar BEC. Top row: Time series
of real-time simulation snapshots at (16, 164, 542, 1200)ms using the
experimental parameters. Bottom row: Time series of the corresponding
experimental snapshots at (20, 127, 314, 447)ms. For the last image of each
series we applied the vortex detection algorithm [82] and show the calculated
residuals, with the detected vortices marked with crosses. Used parameters:
$\theta=35^{\circ}$, $\Omega=0.7\,\omega_{r}$, $a_{s}=109a_{0}$, $N=10000$,
$\vec{f}=(50,130)$ Hz.

##  VII From one to many vortices in a dipolar gas

We have seen that the recipe to generate vortices through magnetostirring in a
dipolar gas requires careful tuning of the dipolar parameters. Choosing an
angle in the intermediate regime ($20^{\circ}<\theta<50^{\circ}$) is ideal for
triggering the dynamical instability required to induce vortices. Moreover,
individual vortices induced have an eccentricity $>$0.5, and dipolar effects
are observable in the in situ density.

Rather than increasing $\Omega$ linearly from 0, as in Fig. 3, previous
experiments have investigated the formation of vortex lattices either through
jumping to the target value of $\Omega\sim 0.7\,\omega_{r}$ after generation
of the BEC, or by cooling across the thermal cloud-to-BEC phase transition
under continuous rotation at this value [23]. Here, we continuously stir a
dipolar BEC jumping from $\Omega=0$ to $\Omega=0.7\,\omega_{r}$ at $t=0$ with
fixed $\theta=35^{\circ}$ to observe rapid dynamical instability and vortex
lattice nucleation. In Fig. 5 (top row) we show four simulation snapshots
taken at; $t=0$, a static BEC; after 164 ms showing the formation of spiral
arms, a hallmark feature of the growing dynamical instability; post
instability at 542 ms with vortices flooding the condensate density; after
1200 ms of continuous rotation where the system is settling towards a static
vortex stripe lattice. Vortex stripes—rows of vortices aligned along the
magnetic field direction—were one of the first predicted consequences of the
dipole-dipole interaction on vortex structure [92]. We also show snapshots of
the same process taken from the experimental data from Ref. [82] [Fig. 5
(bottom row)]. Remarkably, similar features are observed, such as the
appearance of spiral arms and the eventual production of vortices inside the
density profile. In this experiment, due to atom losses, it was not possible
to continuously stir until a lattice has formed, however vortex stripes were
observed for the first time. These effects are readily observed in the
residual images, where the average density from many experimental runs–which
is smooth due to the random distribution of vortices–is subtracted from the
single experimental image (see Ref. [82] for more details) leaving behind
peaks at the vortex positions and troughs along the high density stripes. From
our simulations, we have found that larger atom numbers facilitate a faster
reorganization into the lattice, and there is hope that future experiments
will be able to observe this exciting prediction.

##  VIII Vortex lattices

Finally, for our system parameters, we investigate the transition from
triangular to stripe vortex lattice, and the fate of the vortices across the
BEC-to-supersolid transition. In Fig. 6, we present the stationary state
lattice solutions found through imaginary time propagation of Eq. (3) with
$\Omega=0.7\,\omega_{r}$. When $\theta=0^{\circ}$ the lattice is triangular,
as in non-dipolar BECs. We note, however, that a square lattice here is only
0.03% larger in energy, and we do not rule out that a square lattice would be
possible in this geometry, similar to the results of Refs. [102, 103, 104] but
with positive scattering length. Already at $\theta=30^{\circ}$ we find that
the vortex lattice takes on the stripe structure, with the stripes more
prominent at lower scattering lengths. This is consistent with the findings of
Klaus et al., who observed vortex stripes at smaller scattering lengths
(approx. $105\,a_{0}$). At deeper angles still, the number of stripes is
reduced as the magnetostriction stretches the condensate along the magnetic
field direction, until the condensate becomes a single rotating droplet and
the energy cost of adding a vortex exceeds that obtained from maintaining an
irrotational flow. The critical rotation frequency to induce a vortex across
the BEC-to-droplet transition is an open question, and would be an interesting
future study.

Figure 6: Vortex lattice solutions of a dipolar BEC. Light blue points mark
vortex positions. Parameters: $\Omega=0.7\omega_{r}$, other parameters are the
same as Fig. 3.

Vortices in the supersolid regime are known to possess intriguing properties,
taking on the core size of the inter-droplet spacing [109], and snaking
between the droplets in order to interact with one another [110]. Typically,
the crystal structure of two-dimensional supersolids are known to be
triangular lattices, with some evidence of the possibility of square lattices
[131], and predictions for exotic labyrinthine-like and honeycomb crystal
solutions at high densities [132, 133, 134], resembling the nuclear pasta
states of neutron stars. The energy minimum position of a vortex is in the
centre of each triangle, however there is a metastable minimum between each
droplet pair [110], and vortices fill all of these minima forming the
supersolid vortex lattice at fast rotation frequencies [109]. Rotating high
density honeycomb structure supersolids is predicted to break apart the
crystal [112]. In our system, the stationary state with $a_{s}=90a_{0}$ is a
supersolid for all angles and $\Omega=0.7\,\omega_{r}$, however is an
unmodulated BEC at $\Omega=0$, hinting at the intriguing possibility of
triggering the unmodulated-to-modulated transition through rotation. This
behaviour has been predicted previously [130], due to the two competing
avenues for angular momentum production in a supersolid: either a vortex can
enter contributing to the superfluid angular momentum, a contribution
proportional to the superfluid fraction $f_{s}$, or the classical angular
momentum $\Omega m\langle x^{2}+y^{2}\rangle$ can be increased by reducing the
superfluid fraction, i.e. inducing supersolidity. Intermediate tilt angles
display a complex interplay between maintaining a fixed interdroplet spacing
and the vortices attempting to align in stripes, until at deeper tilt angles
where the droplet number reduces to one droplet without vortices.

##  IX Conclusion and outlook

In conclusion, we have shown that rotation at intermediate tilt angles are
ideal for generating vortices in a short timescale, delicately balancing the
increase of ellipticity with deeper tilt angles against the increased head-to-
tail attraction suppressing the necessary dynamical instability. We have shown
that the predictions for the shape and size of dipolar vortices in the absence
of beyond-mean-field corrections remain present in the dominantly dipolar
regime, where these corrections are necessary for stability. Whereas previous
studies, that have neglected beyond-mean-field effects, required
$\theta\approx 70^{\circ}$ in order to see the transition from triangular to
square lattice [105], we are able to access the dominantly dipolar regime
where this transition occurs as low as $30^{\circ}$. Recent experimental
results only present the first steps to observe a large quantity of intriguing
properties. At finite temperatures, the impact of the dipolar interaction on
universal scaling laws for Kibble-Zurek physics and the nature of the BKT
transition in dipolar gases remains unknown. Future research directions may
investigate the properties of dipolar vortices in binary condensates, such as
dipolar spin mixtures between two spin states of a single species,
multispecies mixtures such as erbium-dysprosium [135], or dipolar non-dipolar
mixtures such as dysprosium-potassium [136] and erbium-ytterbium [137].
Predictions thus far have investigated the interplay between roton
immiscibility–the two species separating at the roton wavelength–and vortex
lattices [138, 139, 140, 141, 142]. Recent theoretical investigations into
binary systems have revealed novel interweaving supersolid states [143, 144]
and miscible supersolid states [145]. In liquid helium, a second component was
utilized in order to directly image the vortices from the bulk component
[146], and the same ideas may be brought to binary supersolidity as an ideal
platform to image vortices in supersolids, for example extending the work of
Ref. [147] to dipolar gases. Prospects for rotating an unmodulated BEC into
the supersolid regime [130], and tests of superfluidity of the supersolid, are
currently a hotly debated topic in the field [148, 149, 150].

## Acknowledgments

In December of 2021, Jean Dalibard was awarded with the most prestigious
French prize for physicists, the CNRS Gold medal. This paper is a contribution
to celebrate his numerous achievements throughout his prestigious scientific
carrier. In the last 40 years, Professor Dalibard enormously contributed to
the field of ultracold gases manipulation in several fundamental ways, always
finding the key ingredient to solve problems and elegant solutions to describe
complex phenomena in a beautiful and intuitive manner. His research on laser
cooling and invention of the magneto-optical trap was fundamental for the
achievement of extremely low temperatures towards the successful realization
of Bose-Einstein condensates [151]. He then played a crucial role in the
investigation of the many-body properties of polarized condensates and spinor
systems, from collective modes to topological excitations. Undoubtedly, one of
his major and fundamental contributions is related to vortex physics, with his
experiments on the first rotating condensates and the observation of vortex
lattices, with his studies on quasi two-dimensional gases in harmonic and flat
potentials, from the observation of vortices created via the Kibble-Zurek
mechanism [152, 153] to the characterization of the Berezinskii-Kosterlitz-
Thouless transition [154, 155].

We are grateful to E. Poli and the Innsbruck erbium-dysprosium team for
helpful discussions. This study received support from the European Research
Council through the Advanced Grant DyMETEr (No. 1010545), the QuantERA grant
MAQS by the Austrian Science Fund FWF (No. I4391-N), the DFG/FWF via Dipolare
E2 (No. I4317-N36) and a joint-project grant from the FWF (No. I4426). Part of
the computational results presented have been achieved using the HPC
infrastructure LEO of the University of Innsbruck. G. L. acknowledges
financial support from Provincia Autonoma di Trento. T.B. acknowledges
financial support by the ESQ Discovery programme (Erwin Schrödinger Center for
Quantum Science & Technology), hosted by the Austrian Academy of Sciences
(ÖAW).

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