††thanks: These two authors contributed equally to this work.††thanks: These
two authors contributed equally to this work.††thanks: Correspondence should
be addressed to<EMAIL_ADDRESS>
# Observation of vortices and vortex stripes in a dipolar Bose-Einstein
condensate

Lauritz Klaus  Thomas Bland  Institut für Quantenoptik und Quanteninformation,
Österreichische Akademie der Wissenschaften, Innsbruck, Austria  Institut für
Experimentalphysik, Universität Innsbruck, Austria  Elena Poli  Institut für
Experimentalphysik, Universität Innsbruck, Austria  Claudia Politi  Institut
für Quantenoptik und Quanteninformation, Österreichische Akademie der
Wissenschaften, Innsbruck, Austria  Institut für Experimentalphysik,
Universität Innsbruck, Austria  Giacomo Lamporesi  INO-CNR BEC Center and
Dipartimento di Fisica, Università di Trento, 38123, Trento, Italy  Eva
Casotti  Institut für Quantenoptik und Quanteninformation, Österreichische
Akademie der Wissenschaften, Innsbruck, Austria  Institut für
Experimentalphysik, Universität Innsbruck, Austria  Russell N. Bisset
Institut für Experimentalphysik, Universität Innsbruck, Austria  Manfred J.
Mark  Francesca Ferlaino  Institut für Quantenoptik und Quanteninformation,
Österreichische Akademie der Wissenschaften, Innsbruck, Austria  Institut für
Experimentalphysik, Universität Innsbruck, Austria

###### Abstract

Quantized vortices are the prototypical feature of superfluidity. Pervasive in
all natural systems, vortices are yet to be observed in dipolar quantum gases.
Here, we exploit the anisotropic nature of the dipole-dipole interaction of a
dysprosium Bose-Einstein condensate to induce angular symmetry breaking in an
otherwise cylindrically symmetric pancake-shaped trap. Tilting the magnetic
field towards the radial plane deforms the cloud into an ellipsoid through
magnetostriction, which is then set into rotation. At stirring frequencies
approaching the radial trap frequency, we observe the generation of
dynamically unstable surface excitations, which cause angular momentum to be
pumped into the system through vortices. Under continuous rotation, the
vortices arrange into a stripe configuration along the field–in close
corroboration with simulations–realizing a long sought-after prediction for
dipolar vortices.

Since the first experiments on gaseous Bose-Einstein condensates (BECs), the
observation of quantized vortices has been considered the most fundamental and
defining signature of the superfluid nature of such systems. Their very
existence sets a unifying concept encompassing a variety of quantum fluids
from liquid helium 1 to the core of neutron stars 2, and from superconductors
3 to quantum fluids of light 4. Their classical counterparts have as well
fascinated scientists from different epochs and fields as vortices are found
in many scales of physical systems, from tornadoes in the atmosphere to
ferrohydrodynamics.

In the quantum realm, a quantized vortex may emerge as a unique response of a
superfluid to rotation. It can be understood as a type of topologically
protected singularity with a $2\pi$ phase winding that preserves the single-
valuedness of the superfluid wavefunction and the irrotational nature of its
velocity field. In contact-interacting BECs, vortical singularities have been
observed experimentally in the form of single vortices 5, 6, vortex-antivortex
pairs 7, solitonic vortices 8, 9, vortex rings 10 and vortex lattices 6, 11,
12 using a number of different techniques. Moreover, vortices play a
fundamental role in the description of the Berezinskii-Kosterlitz-Thouless
transition in two-dimensional systems 13, as well as in the evolution of
quantum turbulence 14, 15, 16, and have been observed in interacting Fermi
gases along the BEC-to-BCS crossover 17, 8.

Recently, a new class of ultracold quantum gases are being created in various
laboratories around the world, using strongly magnetic lanthanide atoms 18,
19. Such a system, providing a quantum analogue of classical ferrofluids,
enables access to the physics of dipolar BECs, in which atoms feature a strong
long-range anisotropic dipole-dipole interaction (DDI) 20, 21 on top of the
traditional contact-type isotropic one. This intriguing platform provided the
key to observe, e.g., extended Bose-Hubbard dynamics 22, roton excitations 23,
24, 25, the quantum version of the Rosensweig instability 26, supersolid
states of matter 27, 28, 29, 30, and is foreseen to host novel phenomena for
quantum simulation and metrology 20, 21.

The dipolar interaction is predicted to also intimately change the properties
of vortices in quantum gases 31. For instance, theoretical works predict
single vortices to exhibit an elliptic-shaped core for a quasi-two-dimensional
setting with in-plane dipole orientation 32, 33, 34, 35, or the presence of
density oscillations around the vortex core induced by the roton minimum in
the dispersion relation 32, 33, 34, 35, 36. For vortex pairs, the anisotropic
DDI is expected to alter the lifetime and dynamics 35, 37, and can even
suppress vortex-antivortex annihilation 35. These interaction properties are
predicted to give rise to a vortex lattice structure that can follow a
triangular pattern 32, 36, as is typical for non-dipolar BECs 11, or a square
lattice for attractive or zero contact interactions 38, 39 when the DDI is
isotropic (dipoles aligned with the rotation axis). A very striking
consequence of the dipoles tilted into the plane is the formation of vortex
stripes 32, 40, 41. Moreover, vortices could provide an unambiguous smoking
gun of superfluidity in supersolid states 42, 43, 44, 45, 46. However, despite
these intriguing predictions, vortices in dipolar quantum gases have yet to be
observed.

This paper presents the experimental realization of quantized vortices in a
dipolar BEC of highly magnetic dysprosium atoms. Following a method proposed
in Ref. 41, we show that the many-body phenomenon of magnetostriction 47,
genuinely arising from the anisotropic DDI among atoms, provides a natural
route to rotate the systems and nucleate vortices in a dipolar BEC. We carry
out studies on the dynamics of the vortex formation, which agree very well
with our theoretical predictions. Finally, we observe one of the earliest
predictions for dipolar vortices: the formation of vortex stripes in the
system.

In non-dipolar gases, quantized vortices have been produced using several
conceptually different techniques. For instance by rotating non-symmetric
optical 6, 11 or magnetic 48 potentials, by rapidly shaking the gas 15, by
traversing it with obstacles with large enough velocity 7, 49, by rapidly
cooling the gas across the BEC phase transition 50, 51, or directly imprinting
the vortex phase pattern 52. Dipolar quantum gases, while able to form
vortices with these same standard procedures 31, also offer unique
opportunities that have no counterpart in contact-interacting gases.
Crucially, the DDI gives rise to the phenomenon of magnetostriction in
position space 47. When dipolar BECs are polarized by an external magnetic
field–defining the dipole orientation–the DDI causes an elongation of the
cloud along the polarization direction. This is a direct consequence of the
system tendency to favor head-to-tail dipole configurations, which effectively
reduces the interaction energy 21.

Such a magnetostrictive effect provides a simple method to induce an elliptic
effective potential and drive rotation with a single control parameter. This
modification of the effective potential is shown in Fig. 1a for a BEC in an
oblate trap with cylindrical symmetry about the $z$-axis. While a non-dipolar
BEC takes the same shape as the confining trap (a1), introducing dipolar
interactions with polarization axis along $z$ stretches the cloud along this
axis, yet maintains cylindrical symmetry (a2). Tilting the
${\overrightarrow{{\bf\it\rm B}}}$-field leads to a breaking of the
cylindrical symmetry, resulting in an ellipsoidal deformation of the cloud
shape, as seen from the density projection onto the $x$-$y$ plane (a3).
Finally, under continuous rotation of the magnetic-field, which we coin
“magnetostirring”, the condensate is predicted to rotate (a4). This unique
approach to stir a dipolar condensate can eventually lead to the nucleation of
vortices 41, 53, realizing genuinely interaction-driven vorticity through
many-body phenomena.

Figure 1: Magnetostirring of a Dy dipolar BEC and evolution of the cloud
aspect ratio. a, Illustration of a non-dipolar (a1) and dipolar BEC with ${\rm
B}\neq 0$ (a2-a4) in a cylindrically symmetric, oblate trap. The magnetic-
field angle with respect to the $z$ axis varies from $\theta=0$° (a2) to
$\theta=35^{\circ}$ (a3) and rotating at $\theta=35^{\circ}$ around $z$ (a4).
b, Representative axial absorption images showing the dipolar BEC while
spinning up the magnetic field for
$t_{\dot{\Omega}}=[140,430,627,692]\,$\mathrm{m}\mathrm{s}$$ (b1) and
subsequent constant rotation at $\Omega=2\pi\times 36\,$\mathrm{H}\mathrm{z}$$
for $t_{\Omega}=[0,6,11,17]\,$\mathrm{m}\mathrm{s}$$ (b2). c, Cloud aspect
ratio AR for different final rotation frequencies. $\Omega$ is linearly
increased to its final value at a speed of $\dot{\Omega}=2\pi\times
50\,$\mathrm{H}\mathrm{z}$/$\mathrm{s}$$. To mitigate influences of trap
anisotropies on the AR, a full period at the final rotation frequency is
probed. The error bars are smaller than the symbol size. The solid (dashed)
black line shows the corresponding eGPE simulations with a $2\,$\mathrm{s}$$
($1\,$\mathrm{s}$$) ramp and $a_{s}=110\,a_{0}$,
$(\omega_{\perp},\omega_{z})=2\pi\times[50,130]\,$\mathrm{H}\mathrm{z}$$, and
$N=15000$.  Figure 2: Observation of vortices in a dipolar BEC. Each column
shows the simulated (upper) and experimental (lower) images for various
rotation times $t_{\Omega}$. For the experiment, the atoms are imaged along
the $z$ direction. In each experimental run, we rotate the magnetic field
counter-clockwise at $\Omega=0.74\,\omega_{\perp}$ for different rotation
times $t_{\Omega}$. The magnetic field value is kept to
$|{\overrightarrow{{\bf\it\rm B}}}|=5.333(5)\,$\mathrm{G}$$. The initial
condensed atom number is $N=15000$. The decreasing size of the cloud suggests
a decrease in atom number. However, for states with vortices or spiral shapes,
appearing at large $t_{\Omega}$, our bimodal fit to extract the atom number
breaks down. For the corresponding simulations, the parameters are
$a_{s}=112\,$\mathrm{a}_{\mathrm{0}}$$, trap frequencies
$(\omega_{\perp},\omega_{z})=2\pi\times[50,150]\,$\mathrm{H}\mathrm{z}$$,
$N=8000$, and $\Omega=0.75\,\omega_{\perp}$.

We explore this protocol using a dipolar BEC of 162Dy atoms. We create the BEC
similarly to our previous work 45 with the distinction that here the magnetic
field unit vector, $\hat{{\bf\it\rm B}}$, is kept tilted at an angle of
$\theta=35$° with respect to the $z$-axis both during evaporative cooling and
magnetostirring (see Fig. 1a3 and Methods). After preparation, the sample
contains about $2\times 10^{4}$ condensed atoms confined within a
cylindrically symmetric optical dipole trap (ODT) with typical radial and
axial trap frequencies
$(\omega_{\perp},\omega_{z})=2\pi\times[50.8(2),140(1)]\,$\mathrm{H}\mathrm{z}$$.
Here, prior to stirring, the magnetostriction is expected from simulations to
increase the cloud aspect ratio in the horizontal plane from 1 up to 1.03,
whereas the trap anisotropy is negligible. We use a vertical ($z$) absorption
imaging to probe the radial ($x$,$y$) atomic distribution after a short time-
of-flight (TOF) expansion of $3\,$\mathrm{m}\mathrm{s}$$. The atom number is
instead measured using horizontal absorption imaging with a TOF of
$26\,$\mathrm{m}\mathrm{s}$$.

Similarly to a rotation of a bucket containing superfluid helium or of a
smoothly deformed ODT for non-dipolar BECs, magnetostirring is predicted to
transfer angular momentum into a dipolar BEC 41, 53. In response to such an
imposed rotation, the shape of an irrotational cloud is expected to elongate
with an amplitude that increases with the rotation frequency $\Omega$. This
phenomenon is clearly visible in our experiments, as shown in Fig. 1b. Here,
we first revolve the tilted $\hat{{\bf\it\rm B}}$ around the $z$-axis with a
linearly increasing rotation frequency ($\dot{\Omega}=2\pi\times
50\,$\mathrm{H}\mathrm{z}\mathrm{/}\mathrm{s}$$) and observe that the dipolar
BEC starts to rotate at the same angular speed as the field and deforms with
increasing elongation (b1). We then stop the adiabatic ramp at a given value
of $\Omega$ and probe the system under continuous rotation. We now find that
the cloud continues rotating in the radial plane with an almost constant shape
(b2). Note that $|{\overrightarrow{{\bf\it\rm B}}}|$ is held constant at
$5.333(5)\,$\mathrm{G}$$, where we estimate a contact scattering length of
about $a_{s}=111\,a_{0}$ (Methods).

We further explore the response of our dipolar BEC to magnetostirring by
repeating the measurements in Fig. 1b1, but stopping the ramp at different
final values of $\Omega$. The maximum value used for $\Omega$ approaches
$\omega_{\perp}$, corresponding to a ramp duration of $1\,$\mathrm{s}$$. We
quantify the cloud elongation in terms of the aspect ratio
AR$=\sigma_{\text{max}}/\sigma_{\text{min}}$, where the cloud widths
$\sigma_{\text{max}}$ and $\sigma_{\text{min}}$ are extracted by fitting a
rotated two-dimensional Gaussian function to the density profiles. Figure 1c
summarizes our results. We observe that initially the AR slightly deviates
from one due to magnetostriction. It then slowly grows with increasing
$\Omega$, until a rapid increase at around 0.6$\,\omega_{\perp}$ occurs, as
this allows the angular momentum to increase, which decreases the energy in
the rotating frame54. Suddenly, at $\Omega_{c}\approx 0.74\,\omega_{\perp}$,
the AR abruptly collapses back to $\mathrm{AR}\approx 1$, showing how the
superfluid irrotational nature competes with the imposed rotation.

To substantiate our observation, we perform numerical simulations of the zero-
temperature extended Gross-Pitaevskii equation (eGPE) 55, 56, 57, 58 (see
Methods). Quantum and thermal fluctuations are added to the initial states,
which are important to seed the dynamic instabilities once they emerge at
large enough $\Omega$; see later discussion. The lines in Fig. 1c show our
results. The dashed line is obtained through the same procedure as the
experiment, whereas for the solid line we halve the ramp rate, spending more
time at each frequency. Both ramp procedures show quantitatively the same
behaviour up to $\Omega=0.8\,\omega_{\perp}$, and are in excellent agreement
with the experimental results. The stability of the $1$\mathrm{s}$$ ramp
exceeds the experimentally observed critical frequency. We partly attribute
this discrepancy to asymmetries of the rotation in the experiment that are not
present in the simulations, which may lead to an effective speedup of the
dynamical instabilities. However, in all cases, the AR rapidly decreases to
about one.

The growing AR and subsequent collapse to one is a signature of the dynamical
instability of surface modes, known for being an important mechanism for
seeding vortices and allowing them to penetrate into the high-density regions
of rotated BECs 54, 59, 60, as also predicted for our dipolar system 41. To
search for quantum vortices in our system, we perform a new investigation
where we directly set $\Omega$ close to $\Omega_{c}$. We then hold the
magnetic-field rotation fixed at this constant frequency for a time
$t_{\Omega}$. As shown in Fig. 2 (lower row), the cloud rapidly elongates, and
the density starts to exhibit a spiral pattern, emanating from the tips of the
ellipsoid. As early as $t_{\Omega}=314\,$\mathrm{m}\mathrm{s}$$ clear holes
are observed in the density profile, forming in the density halo around the
center, the first clear indication of vortices in a dipolar gas. These
vortices, initially nucleated at the edge of the sample, persist as we
continue to stir, and eventually migrate towards the central (high-density)
region. Owing to their topological protection, vortices are still visible in
the experiment after one second of magnetostirring, although our atom number
decreases throughout this procedure. Our observations bear a remarkable
resemblance to the simulations; Fig. 2 (upper row) showing the in situ column
densities. Taking a fixed atom number of $N=8000$, but otherwise repeating the
experimental sequence, we observe many similar features. First, the spiral
density pattern appears before the instability, forming two arms that are
filled with vortices close to the central density. Next, turbulent dynamics
ensue as the density surface goes unstable and vortices emerge in the central
high density region. This turbulence, however, inhibits the creation of a
vortex lattice on these timescales.

The observed evolution of the system under constant rotation shows some
concurrence between the appearance of vortices in the absorption images and
the formation of a round density pattern in the radial plane with
$\mathrm{AR}\approx 1$ (Fig. 2). To study this dynamical evolution in more
detail, we adopt an analysis protocol for both the experiment and theory that
allows us to quantitatively track the evolution of the average number of
vortices, $\mathcal{N}_{v}$ (Methods). The result is shown in Fig. 3a. In
brief, for each single image (a1) we create a blurred reference image by
applying a 2D Gaussian filter 61, 62. We then calculate the difference between
each single image (a1) and the corresponding reference (a2) to obtain the
residual image (a3), from which we count $\mathcal{N}_{v}$ by finding local
minima below a certain threshold.

For the experimental density profiles, which are affected by both the limited
resolution of the imaging system and the weak contrast in the low-density
zones (halo) where the vortices initially nest, we expect $\mathcal{N}_{v}$ to
be underestimated relative to the number expected by theory. However, in order
to carry out a quantitative comparison with the simulations, we apply a
blurring filter and add noise to the latter that mimics the actual resolution
in the experiment (Methods).

Figure 3b shows both the evolution of $\mathcal{N}_{v}$ and cloud AR as a
function of rotation time, $t_{\Omega}$. Solid lines are the results from the
eGPE simulations without any adjustable parameters. For
$t_{\Omega}<200\,$\mathrm{m}\mathrm{s}$$, $\mathcal{N}_{v}$ is below 1, where
vortices, if present, are at the edge of the cloud. For longer times,
$\mathcal{N}_{v}$ increases and saturates to an average value of about three
and a maximum of six vortices (see Fig. 3a for an example of five vortices).
The observed saturation might be due to the decreased visibility and to the
atom-loss-induced shrinking of the BEC size, which is not accounted for in the
theory. We also compare the course of the average vortex number with the AR of
the cloud. After initial large oscillations, due to the sudden jump in
rotation frequency, the AR declines towards $\approx$ 1 60. This happens as
the vortex number simultaneously increases.

Figure 3: Time evolution of the average vortex number, $\mathcal{N}_{v}$, and
cloud aspect ratio AR. a1, Sample image after rotating for
$t_{\Omega}=474\,$\mathrm{m}\mathrm{s}$$. a2, Blurred reference image
($\sigma=2.1\,\mu$m). a3, Residuals with markers (black circles) indicating
the identified vortices. b, The detected vortex number $\mathcal{N}_{v}$ (top
panel) and the AR of the cloud (bottom panel) after the rotation time
$t_{\Omega}$. Data points and error bars show the mean and standard error from
about 10 experimental runs. Solid lines indicate the averaged results from 10
corresponding simulations with different initial noise for parameters
$a_{s}=110\,a_{0}$, $(\omega_{\perp},\omega_{z})=2\pi\times[50,130]$ Hz,
$N=10000$ and $\Omega=0.75\,\omega_{\perp}$, the shaded area gives its
standard error.

One fascinating prediction with dipolar vortices under the influence of a
rotating magnetic field relates to the structure of the resulting vortex
lattice. Due to magnetostriction and the anisotropic vortex cores, the
resulting vortex configuration is also anisotropic, producing a stripe phase
in the strongly dipolar regime 32, 31, instead of the usual triangular lattice
in non-dipolar BECs 6. The ground state stripe lattice solution for our
parameters is shown in Fig. 4a, with a cloud ${\text{AR}=1.08}$. In the vortex
stripe phase, vertical planes of high density regions, parallel to the
magnetic field, alternate with low density ones, that host rows of vertical
vortex filaments. Such a configuration promotes head-to-tail dipolar
attraction within the high density ridges, and this acts to lower the energy.
It should be noted that these states are distinct from the oscillating vortex
sheets states, which appear after squeezing a triangular vortex lattice 63.

Figure 4:  Stripe nature of dipolar vortices. a1, Ground state stripe lattice
solution for our experimental parameters $a_{s}=109\,a_{0}$, trap frequencies
$(\omega_{\perp},\omega_{z})=2\pi\times[50,130]\,$\mathrm{H}\mathrm{z}$$,
$N$=10000 and $\Omega=0.75\,\omega_{\perp}$. a2, Corresponding residual image,
found by subtracting the ground state from the blurred image, with circles
showing the detected vortices. a3, Fourier transform of the residual image.
b1, Single experimental image after $500\,$\mathrm{m}\mathrm{s}$$ of
continuous rotation at $\Omega=0.75\,\omega_{\perp}$. b2, The corresponding
residual image. b3, Fourier transform of the residual images, averaged over 49
runs, with example shots shown to the right. c1, Simulation result for the
dynamic experimental procedure in b. c2-c3, Residuals and FT analysis (115
temporal images) as in b2-b3. d1-d3, Same as b1-b3 for 121 runs, but we rotate
for an additional $100\,$\mathrm{m}\mathrm{s}$$ and then spiral the magnetic
field to $\theta=0^{\circ}$ over a further $100\,$\mathrm{m}\mathrm{s}$$
before imaging. e1-e3, Simulation result for procedure in d. All simulation
images are rotated to have the same magnetic field direction as the
experiment.

To explore this prediction, we perform two new surveys. First, we slightly
reduce the magnetic field value, reducing the scattering length to
$a_{s}\approx 109\,a_{0}$ and hence making the system relatively more dipolar,
while still avoiding the droplet regime. We magnetostir the BEC at a constant
rotation frequency $\Omega=0.75\,\omega_{\perp}$ for
$500\,$\mathrm{m}\mathrm{s}$$, but during TOF we stop the magnetic-field
rotation and keep it in place at $\theta=35^{\circ}$. The stripe structure is
revealed in Fig. 4b1 for a single experimental run, and is clearly visible in
the residual image, Fig. 4b2, where the vortices align along three stripes.
The spatial structure of the residual image can be assessed through the
absolute value of two-dimensional Fourier transform (FT). After taking the FT
of each residual image, we then average the result, see Fig. 4b3, finding a
clear peak at the $k$ of the inter-stripe spacing. This shows that the stripe
spatial structure survives the averaging, implying that the majority of images
show stripes with the same spacing, and they also have the same orientation as
set by the magnetic field, as evidenced by the example images shown in the
right of Fig. 4. Note that these observations do not rely on our ability to
resolve individual vortices, as the stripes are an ensemble effect of many
aligned vortices. In fact, by comparing with the numerical simulations of the
dynamical procedure (Fig. 4c1-c3), we expect there are more vortices than
detected here that fill in the stripes, forging out this structure. In
general, our simulations show that the stripes appear faster when the
scattering length is lower, and when the atom number is larger. In the long
time limit of the scenario presented in Fig. 2, we expect the stationary
solution to also be the stripe state, but this is not observable on our
timescales.

Remarkably, the stripe structure washes out when we subsequently tilt the
magnetic-field orientation to $\theta=0^{\circ}$ (parallel to the trap
symmetry axis), as shown in Fig. 4d1. Here, after
$600\,$\mathrm{m}\mathrm{s}$$ of magnetostirring, we add another step in which
we spiral up the magnetic field to ${\theta=0^{\circ}}$ (with $\Omega$ fixed)
over $100\,$\mathrm{m}\mathrm{s}$$, before imaging. Under these conditions,
all vortex properties are again isotropic within the plane. The final non-
equilibrium positioning of the vortices is arbitrary, and if we average the FT
of the residuals directly, we observe a homogeneous ring in the average FT
(Fig. 4d3). Also, this behavior is confirmed by the simulations, as shown in
Fig. 4e1-e3.

By exploiting magnetostirring — a novel, robust, method of generating angular
momentum — we have observed quantized vortices in a dipolar quantum gas, and
the appearance of the vortex stripe configuration. Future works will focus on
investigations of the individual vortex shape and behaviour, such as the
anisotropic nature of the vortex cores for in-plane magnetic fields 32, 33,
34, 35, the interplay between the vortex and roton excitations 32, 33, 34, 35,
36, exotic vortex patterns such as square lattices 31, and investigations into
anisotropic turbulence 64. This work also opens the door to studying more
complex matter under rotation, such as dipolar droplets 65, 66, 67 and
supersolid states 43, 42, 44, 45. Such proposals will be challenging due to
the intricate density patterns 68, however such observations would provide
conclusive evidence of superfluidity in supersolids. Rotating the magnetic
field at frequencies far larger than the radial trap frequencies, but smaller
than the Larmor frequency, has been observed to tune the sign and magnitude of
the dipole-dipole interaction 69, 70–a method also employed in Nuclear
Magnetic Resonance spectroscopy 71–but there remain open questions on the
stability of this procedure 72, 73, which if rectifiable would unlock new
research directions 69. Other vortex generation methods, such as thermally
activated pairs in quasi-2D to assess the Berezinskii-Kosterlitz-Thouless
transition, and stochastically generated vortex tangles through temperature
quenches to assess the Kibble-Zurek mechanism, remain unexplored in dipolar
gases 31. The technique introduced here is also applicable to a wide range of
systems governed by long-range interactions through the manipulation of
magnetic or electric fields.

Acknowledgements:
We are grateful to S. B. Prasad, M. Norcia, R. M. W. van Bijnen, and L. Santos
for helpful discussions. We acknowledge M. Norcia and A. Patscheider for
experimental contributions. This study received support from the European
Research Council through the Consolidator Grant RARE (No. 681432), the
QuantERA grant MAQS by the Austrian Science Fund FWF No. I4391-N, the DFG/FWF
via FOR 2247/PI2790 a joint-project grant from the FWF (Grant No. I4426,
RSF/Russland 2019). M.J.M. acknowledges support through an ESQ Discovery Grant
by the Austrian Academy of Sciences. We also acknowledge the Innsbruck Laser
Core Facility, financed by the Austrian Federal Ministry of Science, Research
and Economy. 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.

Author contributions:
L.K., C.P., G.L., E.C., M.J.M., and F.F. performed the experimental work and
data analysis. E.P., T.B., and R.B. performed the theoretical work. All
authors contributed to the interpretation of the results and the preparation
of the manuscript.

## References

  * Donnelly [1991] R. J. Donnelly, Quantized vortices in Helium II (Cambridge University Press, Cambridge, 1991).
  * Pines and Alpar [1985] D. Pines and M. A. Alpar, Superfluidity in neutron stars, Nature 316, 27 (1985).
  * Abrikosov [2004] A. A. Abrikosov, Nobel Lecture: Type-II superconductors and the vortex lattice, Rev. Mod. Phys. 76, 975 (2004).
  * Lagoudakis  _et al._ [2008] K. G. Lagoudakis, M. Wouters, M. Richard, A. Baas, I. Carusotto, R. André, L. S. Dang, and B. Deveaud-Plédran, Quantized vortices in an exciton–polariton condensate, Nat. Phys. 4, 706 (2008).
  * Matthews  _et al._ [1999] M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell, Vortices in a Bose-Einstein condensate, Phys. Rev. Lett. 83, 2498 (1999).
  * Madison  _et al._ [2000] K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, Vortex formation in a stirred Bose-Einstein condensate, Phys. Rev. Lett. 84, 806 (2000).
  * Neely  _et al._ [2010] T. W. Neely, E. C. Samson, A. S. Bradley, M. J. Davis, and B. P. Anderson, Observation of vortex dipoles in an oblate Bose-Einstein condensate, Phys. Rev. Lett. 104, 160401 (2010).
  * Ku  _et al._ [2014] M. J. H. Ku, W. Ji, B. Mukherjee, E. Guardado-Sanchez, L. W. Cheuk, T. Yefsah, and M. W. Zwierlein, Motion of a solitonic vortex in the BEC-BCS crossover, Phys. Rev. Lett. 113, 065301 (2014).
  * Donadello  _et al._ [2014] S. Donadello, S. Serafini, M. Tylutki, L. P. Pitaevskii, F. Dalfovo, G. Lamporesi, and G. Ferrari, Observation of solitonic vortices in bose-einstein condensates, Phys. Rev. Lett. 113, 065302 (2014).
  * Anderson  _et al._ [2001] B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark, and E. A. Cornell, Watching dark solitons decay into vortex rings in a bose-einstein condensate, Phys. Rev. Lett. 86, 2926 (2001).
  * Abo-Shaeer  _et al._ [2001] J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, Observation of vortex lattices in Bose-Einstein condensates, Science 292, 476 (2001).
  * Coddington  _et al._ [2003] I. Coddington, P. Engels, V. Schweikhard, and E. A. Cornell, Observation of Tkachenko oscillations in rapidly rotating Bose-Einstein condensates, Phys. Rev. Lett. 91, 100402 (2003).
  * Hadzibabic  _et al._ [2006] Z. Hadzibabic, P. Krüger, M. Cheneau, B. Battelier, and J. Dalibard, Berezinskii–Kosterlitz–Thouless crossover in a trapped atomic gas, Nature (London) 441, 1118 (2006).
  * Neely  _et al._ [2013] T. W. Neely, A. S. Bradley, E. C. Samson, S. J. Rooney, E. M. Wright, K. J. H. Law, R. Carretero-González, P. G. Kevrekidis, M. J. Davis, and B. P. Anderson, Characteristics of two-dimensional quantum turbulence in a compressible superfluid, Phys. Rev. Lett. 111, 235301 (2013).
  * Navon  _et al._ [2016] N. Navon, A. L. Gaunt, R. P. Smith, and Z. Hadzibabic, Emergence of a turbulent cascade in a quantum gas, Nature (London) 539, 72 (2016).
  * Tsatsos  _et al._ [2016] M. C. Tsatsos, P. E. Tavares, A. Cidrim, A. R. Fritsch, M. A. Caracanhas, F. E. A. dos Santos, C. F. Barenghi, and V. S. Bagnato, Quantum turbulence in trapped atomic Bose–Einstein condensates, Physics Reports 622, 1 (2016).
  * Zwierlein  _et al._ [2005] M. W. Zwierlein, J. R. Abo-Shaeer, A. Schirotzek, C. H. Schunck, and W. Ketterle, Vortices and superfluidity in a strongly interacting Fermi gas, Nature (London) 435, 1047 (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).
  * Norcia and Ferlaino [2021] M. A. Norcia and F. Ferlaino, Developments in atomic control using ultracold magnetic lanthanides, Nat. Phys. 17, 1349 (2021).
  * Chomaz  _et al._ [2022] 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, arXiv:2201.02672 (2022).
  * Baier  _et al._ [2016] S. Baier, M. J. Mark, D. Petter, K. Aikawa, L. Chomaz, Z. Cai, M. Baranov, P. Zoller, and F. Ferlaino, Extended Bose-Hubbard models with ultracold magnetic atoms, Science 352, 201 (2016).
  * Landau [1941] L. Landau, Theory of the superfluidity of Helium II, Phys. Rev. 60, 356 (1941).
  * Chomaz  _et al._ [2018] L. Chomaz, R. M. W. van Bijnen, D. Petter, G. Faraoni, S. Baier, J. H. Becher, M. J. Mark, F. Wächtler, L. Santos, and F. Ferlaino, Observation of roton mode population in a dipolar quantum gas, Nat. Phys. 14, 442 (2018).
  * Schmidt  _et al._ [2021] J.-N. Schmidt, J. Hertkorn, M. Guo, F. Böttcher, M. Schmidt, K. S. H. Ng, S. D. Graham, T. Langen, M. Zwierlein, and T. Pfau, Roton excitations in an oblate dipolar quantum gas, Phys. Rev. Lett. 126, 193002 (2021).
  * Kadau  _et al._ [2016] H. Kadau, M. Schmitt, M. Wenzel, C. Wink, T. Maier, I. Ferrier-Barbut, and T. Pfau, Observing the Rosensweig instability of a quantum ferrofluid, Nature 530, 194 (2016).
  * Tanzi  _et al._ [2019] L. Tanzi, E. Lucioni, F. Famà, J. Catani, A. Fioretti, C. Gabbanini, R. N. Bisset, L. Santos, and G. Modugno, Observation of a dipolar quantum gas with metastable supersolid properties, Phys. Rev. Lett. 122, 130405 (2019).
  * Böttcher  _et al._ [2019a] F. Böttcher, J.-N. Schmidt, M. Wenzel, J. Hertkorn, M. Guo, T. Langen, and T. Pfau, Transient supersolid properties in an array of dipolar quantum droplets, Phys. Rev. X 9, 011051 (2019a).
  * Chomaz  _et al._ [2019] L. Chomaz, D. Petter, P. Ilzhöfer, G. Natale, A. Trautmann, C. Politi, G. Durastante, R. M. W. van Bijnen, A. Patscheider, M. Sohmen, M. J. Mark, and F. Ferlaino, Long-lived and transient supersolid behaviors in dipolar quantum gases, Phys. Rev. X 9, 021012 (2019).
  * Norcia  _et al._ [2021a] M. A. Norcia, C. Politi, L. Klaus, E. Poli, M. Sohmen, M. J. Mark, R. Bisset, L. Santos, and F. Ferlaino, Two-dimensional supersolidity in a dipolar quantum gas, Nature 596, 357 (2021a).
  * Martin  _et al._ [2017] A. M. Martin, N. G. Marchant, D. H. J. O’Dell, and N. G. Parker, Vortices and vortex lattices in quantum ferrofluids, J. Phys. Condens. Matter 29, 103004 (2017).
  * Yi and Pu [2006] S. Yi and H. Pu, Vortex structures in dipolar condensates, Phys. Rev. A 73, 061602 (2006).
  * Ticknor  _et al._ [2011] C. Ticknor, R. M. Wilson, and J. L. Bohn, Anisotropic superfluidity in a dipolar Bose gas, Phys. Rev. Lett. 106, 065301 (2011).
  * Mulkerin  _et al._ [2013] B. C. Mulkerin, R. M. W. van Bijnen, D. H. J. O’Dell, A. M. Martin, and N. G. Parker, Anisotropic and long-range vortex interactions in two-dimensional dipolar Bose gases, Phys. Rev. Lett. 111, 170402 (2013).
  * Mulkerin  _et al._ [2014] B. C. Mulkerin, D. H. J. O'Dell, A. M. Martin, and N. G. Parker, Vortices in the two-dimensional dipolar bose gas, J. Phys. Conf. Ser. 497, 012025 (2014).
  * Jona-Lasinio  _et al._ [2013] M. Jona-Lasinio, K. Łakomy, and L. Santos, Roton confinement in trapped dipolar Bose-Einstein condensates, Phys. Rev. A 88, 013619 (2013).
  * Gautam [2014] S. Gautam, Dynamics of the corotating vortices in dipolar Bose-Einstein condensates in the presence of dissipation, J. Phys. B: At. Mol. Opt. Phys. 47, 165301 (2014).
  * Cooper  _et al._ [2005] N. Cooper, E. Rezayi, and S. Simon, Vortex lattices in rotating atomic bose gases with dipolar interactions, Phys. Rev. Lett. 95, 200402 (2005).
  * Kumar  _et al._ [2016] R. K. Kumar, T. Sriraman, H. Fabrelli, P. Muruganandam, and A. Gammal, Three-dimensional vortex structures in a rotating dipolar bose–einstein condensate, J. Phys. B: At. Mol. Opt. Phys 49, 155301 (2016).
  * Cai  _et al._ [2018] Y. Cai, Y. Yuan, M. Rosenkranz, H. Pu, and W. Bao, Vortex patterns and the critical rotational frequency in rotating dipolar Bose-Einstein condensates, Phys. Rev. A 98, 023610 (2018).
  * Prasad  _et al._ [2019a] S. B. Prasad, T. Bland, B. C. Mulkerin, N. G. Parker, and A. M. Martin, Vortex lattice formation in dipolar Bose-Einstein condensates via rotation of the polarization, Phys. Rev. A 100, 023625 (2019a).
  * Roccuzzo  _et al._ [2020] S. M. Roccuzzo, A. Gallemí, A. Recati, and S. Stringari, Rotating a supersolid dipolar gas, Phys. Rev. Lett. 124, 045702 (2020).
  * Gallemí  _et al._ [2020] A. Gallemí, S. M. Roccuzzo, S. Stringari, and A. Recati, Quantized vortices in dipolar supersolid Bose-Einstein-condensed gases, Phys. Rev. A 102, 023322 (2020).
  * Ancilotto  _et al._ [2021] F. Ancilotto, M. Barranco, M. Pi, and L. Reatto, Vortex properties in the extended supersolid phase of dipolar Bose-Einstein condensates, Phys. Rev. A 103, 033314 (2021).
  * Norcia  _et al._ [2021b] M. A. Norcia, E. Poli, C. Politi, L. Klaus, T. Bland, M. J. Mark, L. Santos, R. N. Bisset, and F. Ferlaino, Can angular oscillations probe superfluidity in dipolar supersolids?, arXiv:2111.07768 (2021b).
  * Roccuzzo  _et al._ [2022] S. M. Roccuzzo, A. Recati, and S. Stringari, Moment of inertia and dynamical rotational response of a supersolid dipolar gas, Phys. Rev. A 105, 023316 (2022).
  * Stuhler  _et al._ [2005] J. Stuhler, A. Griesmaier, T. Koch, M. Fattori, T. Pfau, S. Giovanazzi, P. Pedri, and L. Santos, Observation of dipole-dipole interaction in a degenerate quantum gas, Phys. Rev. Lett. 95, 150406 (2005).
  * Haljan  _et al._ [2001] P. C. Haljan, I. Coddington, P. Engels, and E. A. Cornell, Driving Bose-Einstein-Condensate Vorticity with a Rotating Normal Cloud, Phys. Rev. Lett. 87, 210403 (2001).
  * Kwon  _et al._ [2021] W. J. Kwon, G. Del Pace, K. Xhani, L. Galantucci, A. Muzi Falconi, M. Inguscio, F. Scazza, and G. Roati, Sound emission and annihilations in a programmable quantum vortex collider, Nature (London) 600, 64 (2021).
  * Corman  _et al._ [2014] L. Corman, L. Chomaz, T. Bienaimé, R. Desbuquois, C. Weitenberg, S. Nascimbène, J. Dalibard, and J. Beugnon, Quench-induced supercurrents in an annular bose gas, Phys. Rev. Lett. 113, 135302 (2014).
  * Liu  _et al._ [2018] I. K. Liu, S. Donadello, G. Lamporesi, G. Ferrari, S. C. Gou, F. Dalfovo, and N. P. Proukakis, Dynamical equilibration across a quenched phase transition in a trapped quantum gas, Commun. Phys. 1, 24 (2018).
  * Del Pace  _et al._ [2022] G. Del Pace, K. Xhani, A. M. Falconi, M. Fedrizzi, N. Grani, D. H. Rajkov, M. Inguscio, F. Scazza, W. J. Kwon, and G. Roati, Imprinting persistent currents in tunable fermionic rings (2022), arXiv:2204.06542 .
  * Prasad  _et al._ [2021] S. B. Prasad, B. C. Mulkerin, and A. M. Martin, Arbitrary-angle rotation of the polarization of a dipolar Bose-Einstein condensate, Phys. Rev. A 103, 033322 (2021).
  * Recati  _et al._ [2001] A. Recati, F. Zambelli, and S. Stringari, Overcritical rotation of a trapped Bose-Einstein condensate, Phys. Rev. Lett. 86, 377 (2001).
  * Wächtler and Santos [2016] F. Wächtler and L. Santos, Quantum filaments in dipolar Bose-Einstein condensates, Phys. Rev. A 93, 061603 (2016).
  * Ferrier-Barbut  _et al._ [2016] I. Ferrier-Barbut, H. Kadau, M. Schmitt, M. Wenzel, and T. Pfau, Observation of Quantum Droplets in a Strongly Dipolar Bose Gas, Phys. Rev. Lett. 116, 215301 (2016).
  * Chomaz  _et al._ [2016] L. Chomaz, S. Baier, D. Petter, M. J. Mark, F. Wächtler, L. Santos, and F. Ferlaino, Quantum-fluctuation-driven crossover from a dilute Bose-Einstein condensate to a macrodroplet in a dipolar quantum fluid, Phys. Rev. X 6, 041039 (2016).
  * Bisset  _et al._ [2016] R. N. Bisset, R. M. Wilson, D. Baillie, and P. B. Blakie, Ground-state phase diagram of a dipolar condensate with quantum fluctuations, Phys. Rev. A 94, 033619 (2016).
  * Sinha and Castin [2001] S. Sinha and Y. Castin, Dynamic instability of a rotating Bose-Einstein condensate, Phys. Rev. Lett. 87, 190402 (2001).
  * Madison  _et al._ [2001] K. W. Madison, F. Chevy, V. Bretin, and J. Dalibard, Stationary states of a rotating Bose-Einstein condensate: Routes to vortex nucleation, Phys. Rev. Lett. 86, 4443 (2001).
  * Abo-Shaeer  _et al._ [2002] J. R. Abo-Shaeer, C. Raman, and W. Ketterle, Formation and decay of vortex lattices in Bose-Einstein condensates at finite temperatures, Phys. Rev. Lett. 88, 070409 (2002).
  * Kwon  _et al._ [2014] W. J. Kwon, G. Moon, J.-y. Choi, S. W. Seo, and Y.-i. Shin, Relaxation of superfluid turbulence in highly oblate Bose-Einstein condensates, Phys. Rev. A 90, 063627 (2014).
  * Engels  _et al._ [2002] P. Engels, I. Coddington, P. C. Haljan, and E. A. Cornell, Nonequilibrium effects of anisotropic compression applied to vortex lattices in Bose-Einstein condensates, Phys. Rev. Lett. 89, 100403 (2002).
  * Bland  _et al._ [2018] T. Bland, G. W. Stagg, L. Galantucci, A. W. Baggaley, and N. G. Parker, Quantum ferrofluid turbulence, Phys. Rev. Lett. 121, 174501 (2018).
  * Cidrim  _et al._ [2018] A. Cidrim, F. E. dos Santos, E. A. Henn, and T. Macrì, Vortices in self-bound dipolar droplets, Phys. Rev. A 98, 023618 (2018).
  * Lee  _et al._ [2018] A.-C. Lee, D. Baillie, R. N. Bisset, and P. B. Blakie, Excitations of a vortex line in an elongated dipolar condensate, Phys. Rev. A 98, 063620 (2018).
  * Ferrier-Barbut  _et al._ [2018] I. Ferrier-Barbut, M. Wenzel, F. Böttcher, T. Langen, M. Isoard, S. Stringari, and T. Pfau, Scissors Mode of Dipolar Quantum Droplets of Dysprosium Atoms, Phys. Rev. Lett. 120, 160402 (2018).
  * Hertkorn  _et al._ [2021] J. Hertkorn, J.-N. Schmidt, M. Guo, F. Böttcher, K. Ng, S. Graham, P. Uerlings, T. Langen, M. Zwierlein, and T. Pfau, Pattern formation in quantum ferrofluids: From supersolids to superglasses, Phys. Rev. Res. 3, 033125 (2021).
  * Tang  _et al._ [2018] Y. Tang, W. Kao, K.-Y. Li, and B. L. Lev, Tuning the Dipole-Dipole Interaction in a Quantum Gas with a Rotating Magnetic Field, Phys. Rev. Lett. 120, 230401 (2018).
  * Giovanazzi  _et al._ [2002] S. Giovanazzi, A. Görlitz, and T. Pfau, Tuning the Dipolar Interaction in Quantum Gases, Phys. Rev. Lett. 89, 130401 (2002).
  * Maricq and Waugh [1979] M. Maricq and J. S. Waugh, NMR in rotating solids, The Journal of Chemical Physics 70, 3300 (1979).
  * Prasad  _et al._ [2019b] S. B. Prasad, T. Bland, B. C. Mulkerin, N. G. Parker, and A. Martin, Instability of rotationally tuned dipolar bose-einstein condensates, Phys. Rev. Lett. 122, 050401 (2019b).
  * Baillie and Blakie [2020] D. Baillie and P. B. Blakie, Rotational tuning of the dipole-dipole interaction in a Bose gas of magnetic atoms, Phys. Rev. A 101, 043606 (2020).
  * Tang  _et al._ [2015] Y. Tang, A. Sykes, N. Q. Burdick, J. L. Bohn, and B. L. Lev, $s$-wave scattering lengths of the strongly dipolar bosons 162Dy and 164Dy, Phys. Rev. A 92, 022703 (2015).
  * Tang  _et al._ [2016] Y. Tang, A. G. Sykes, N. Q. Burdick, J. M. DiSciacca, D. S. Petrov, and B. L. Lev, Anisotropic Expansion of a Thermal Dipolar Bose Gas, Phys. Rev. Lett. 117, 155301 (2016).
  * Lucioni  _et al._ [2018] E. Lucioni, L. Tanzi, A. Fregosi, J. Catani, S. Gozzini, M. Inguscio, A. Fioretti, C. Gabbanini, and G. Modugno, Dysprosium dipolar Bose-Einstein condensate with broad Feshbach resonances, Phys. Rev. A 97, 060701 (2018).
  * Böttcher  _et al._ [2019b] F. Böttcher, M. Wenzel, J.-N. Schmidt, M. Guo, T. Langen, I. Ferrier-Barbut, T. Pfau, R. Bombín, J. Sánchez-Baena, J. Boronat, and F. Mazzanti, Dilute dipolar quantum droplets beyond the extended Gross-Pitaevskii equation, Phys. Rev. Res. 1, 033088 (2019b).
  * Chin  _et al._ [2010] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Feshbach resonances in ultracold gases, Rev. Mod. Phys. 82, 1225 (2010).
  * Patscheider  _et al._ [2022] A. Patscheider, L. Chomaz, G. Natale, D. Petter, M. J. Mark, S. Baier, B. Yang, R. R. W. Wang, J. L. Bohn, and F. Ferlaino, Determination of the scattering length of erbium atoms, Phys. Rev. A 105, 063307 (2022).
  * Lima and Pelster [2011] A. R. P. Lima and A. Pelster, Quantum fluctuations in dipolar Bose gases, Phys. Rev. A 84, 041604 (2011).
  * Blakie  _et al._ [2008] P. Blakie, A. Bradley, M. Davis, R. Ballagh, and C. Gardiner, Dynamics and statistical mechanics of ultra-cold Bose gases using c-field techniques, Adv. Phys. 57, 363 (2008).

##  Appendix A Methods

###  A.1 Experimental Procedure

We prepare an ultracold gas of 162Dy atoms in an optical dipole trap (ODT).
Three 1064 nm laser beams, overlapping at their foci, form the ODT. The
experimental procedure to Bose-Einstein condensation is similar to the one
followed in our previous work 45, but the magnetic field unit vector,
$\hat{{\bf\it\rm B}}$, is tilted by an angle of $\theta=35$° with respect to
the $z$-axis during the whole sequence. After preparation, the sample contains
about $2\times 10^{4}$ condensed atoms. The corresponding trap frequencies are
typically
$(\omega_{\perp},\omega_{z})=2\pi\times[50.8(2),140(1)]\,$\mathrm{H}\mathrm{z}$$.
For all our measurements, the deviation of the trap aspect ratio in the
$xy$-plane AR${}_{\text{trap}}=\omega_{y}/\omega_{x}$ from 1 is always smaller
than $0.6\%$. We evaporate the atoms at $|{\overrightarrow{{\bf\it\rm
B}}}|=5.423(5)\,$\mathrm{G}$$ and jump to the final magnetic-field value
during the last evaporation ramp. After the preparation of the BEC, the
magnetic field is rotated as described in the next section. We use standard
absorption imaging to record the atomic distribution. We probe the vortices
using the vertical imaging taken along the axis of rotation ($z$), for which
the dark spots within the condensate correspond to the cores of individual
vortices. The vertical images are taken with a short TOF of
$3\,$\mathrm{m}\mathrm{s}$$ and a pulse duration of 3-4 $\mu$s. For the data
in Fig. 1-3, we let the magnetic field spinning during TOF, whereas for Fig. 4
we use a static field orientation.

###  A.2 Control of the magnetic field

Calibration: Three pairs of coils – each oriented along a primary axis in the
laboratory frame – enable the creation of a homogeneous field with arbitrary
orientations. The absolute magnetic field value $|\overrightarrow{{\bf\it\rm
B}}|$ of each pair of coils is independently calibrated using radio frequency
(RF) spectroscopy. The RF drives transitions to excited Zeeman states, leading
to a resonant dip in the atom number. The long-term stability – measured via
the peak position of the RF resonance over the course of several days – is on
the order of $\Delta B=\pm 1\,$\mathrm{m}\mathrm{G}$$, while shot-to-shot
fluctuations – measured via the width of the RF resonance for a single
calibration set – is $\Delta B=\pm 5\,$\mathrm{m}\mathrm{G}$$.
Rotation: We drive the rotation of the magnetic field by sinusoidally
modulating the magnetic-field value components $B_{x}$ and $B_{y}$ with a
phase difference of $90^{\circ}$ between them. Since we want to keep the
absolute magnetic field value $|{\overrightarrow{{\bf\it\rm B}}}|$ constant
during rotation, we measure it for various values of the azimuthal angle
$\phi$ and fixed $\theta=35^{\circ}$ by performing Feshbach loss spectroscopy
around $5.1\,$\mathrm{G}$$. We find an average shift of
$|{\overrightarrow{{\bf\it\rm B}}}|$ of about $10\,$\mathrm{m}\mathrm{G}$$
from the $\theta=0^{\circ}$ case, which we take into account. We also find
small deviations as a function of $\phi$ of
$\Delta|{\overrightarrow{{\bf\it\rm B}}}|<20\,$\mathrm{m}\mathrm{G}$$, which
might appear due to slightly non-orthogonal alignment of the magnetic fields.
We did not correct these deviations for the sake of simplicity.

###  A.3 Scattering length

The scattering length in 162Dy is currently not known with large accuracy 74,
75, 76, 77. To estimate the scattering length in the small magnetic-field
range around $B=5.3\,$\mathrm{G}$$, relevant to this work, we use the well-
known relation $a_{s}=a_{\text{bg}}\prod_{i}[1-\Delta B_{i}/(B-B_{0,i})]$ 78,
where $B_{0,i}$ and $\Delta B_{i}$ are the center position and the width of
the Feshbach loss features reported in Ref. 77, respectively.

The value of the background scattering length, $a_{\text{bg}}$, is empirically
fixed by measuring the magnetic field value at which the supersolid transition
occurs and comparing it with the corresponding critical $a_{s}$ predicted from
simulations. Such an approach leads to
$a_{s}=111(9)\,$\mathrm{a}_{\mathrm{0}}$$ at $B=5.333\,$\mathrm{G}$$. Extended
Data Fig. 1 shows the resulting scattering lengths for the relevant magnetic
fields. Although such an approach gives very good agreement between theory and
experiments, future works on a precise determination of $a_{s}$, similar to
the one achieved with erbium 79, would be desirable.

###  A.4 Magnetostirring

Tilting the magnetic field vector ${\overrightarrow{{\bf\it\rm B}}}$ away from
the symmetry axis of our cylindrical trap leads to an ellipsoidal deformation
of the cloud 47 and therefore to a breaking of the cylindrical symmetry. This
allows for the transfer of angular momentum to the sample by rotating the
magnetic field (magnetostirring). In all our measurements we use a
${\overrightarrow{{\bf\it\rm B}}}$ tilted with respect to the $z$-axis by 35°
and a constant value $|{\overrightarrow{{\bf\it\rm B}}}|$. That value is
$|{\overrightarrow{{\bf\it\rm B}}}|=5.333(5)\,$\mathrm{G}$$ for the surveys in
Figs. 1-3 and $|{\overrightarrow{{\bf\it\rm B}}}|=5.323(5)\,$\mathrm{G}$$ for
Fig. 4. For these parameters, the magnetostricted aspect ratio of the cloud is
$\mathrm{AR}=1.03$. For all our measurements, the measured trap
$\mathrm{AR}_{\text{trap}}<1.006$ is much smaller than the deformation due to
magnetostriction. Additionally, we have confirmed with simulations that even
with trap asymmetries of up to 10%, e.g. $(\omega_{x},\omega_{y})=(55,50)$Hz,
this procedure can still generate vortices in a lattice configuration.
Adiabatic frequency ramp: We employ different magnetic-field rotation
sequences for the different data sets. For the data set of Fig. 1c, the
rotation frequency of the magnetic field is linearly increased to different
final values at a speed of $\dot{\Omega}=2\pi\times
50\,$\mathrm{H}\mathrm{z}$/$\mathrm{s}$$ and for a duration of
${t_{\dot{\Omega}}=0\text{-}1\,$\mathrm{s}$}$. The ramp time is much longer
than the period of the rotation $\Omega^{-1}$ for higher rotation frequencies
$\Omega\gtrsim\Omega_{c}$, and therefore the ramp is adiabatic for the regimes
considered, until the onset of dynamical instabilities. After the ramp, the
magnetic field direction is rotated at the target rotation frequency $\Omega$
for one final period (as shown in Fig. 1b2). We sample 10 different final
magnetic field angles during this last rotation, measuring the corresponding
aspect ratio and averaging the result in order to remove any potential biases
due to latent trap anisotropies. Each data point is then obtained with 8-10
experimental runs.
Constant rotation frequency: For the dataset of Fig. 2, Fig. 3 and Fig. 4b we
directly start to rotate at the final rotation frequency $\Omega$ without any
acceleration phase. The magnetic field is then rotated for a variable time
$t_{\Omega}$, after which the atoms are released from the trap and a vertical
image is taken.
Spiral up magnetic field: For the dataset of Fig. 4d we employ a similar
sequence as described above. However, after constantly rotating the magnetic
field at $\Omega=0.75\,\omega_{\perp}$, the magnetic field is spiraled up in
$100\,$\mathrm{m}\mathrm{s}$$ to $\theta=0$° by linearly reducing $\theta$
while continuing rotating. Afterwards, the atoms are released from the trap
and a vertical image is taken.

###  A.5 Theoretical model

We employ an extended Gross-Pitaevskii formalism to model our experimental
setup. In this scheme, the inter-particle interactions are described by the
two-body pseudo-potential,

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

with the first term describing short-range interactions governed by the s-wave
scattering length $a_{s}$, with Planck’s constant $\hbar$ and particle mass
$m$. The second term represents the anisotropic and long-ranged dipole-dipole
interactions, characterized by dipole length
$a_{\text{dd}}=\mu_{0}\mu_{m}^{2}m/12\pi\hbar^{2}$, with magnetic moment
$\mu_{m}$ and vacuum permeability $\mu_{0}$. We always consider 162Dy, such
that $a_{\text{dd}}=129.2\,a_{0}$, where $a_{0}$ is the Bohr radius. The
dipoles are polarized uniformly along a time-dependent axis, given by

$\displaystyle\hat{\mathbf{e}}(t)=(\sin\theta(t)\cos\phi(t),\sin\theta(t)\sin\phi(t),\cos\theta(t))\,$
(2)

with time dependent polarization angle $\theta(t)$ and
$\phi(t)=\int_{0}^{t}\text{d}t^{\prime}\Omega(t^{\prime})$, for rotation
frequency protocol $\Omega(t)$.

Beyond-mean-field effects are treated through the inclusion of a
Lee–Huang–Yang correction term 80

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

with
$\mathcal{Q}_{5}(\epsilon_{\text{dd}})=\int_{0}^{1}\text{d}u\,(1-\epsilon_{\text{dd}}+3u^{2}\epsilon_{\text{dd}})^{5/2}$
being the auxiliary function, and the relative dipole strength is given by
$\epsilon_{\text{dd}}=a_{\rm dd}/a_{\rm s}$. Finally, the full extended Gross-
Pitaevskii equation (eGPE) then reads 55, 56, 57, 58

$\displaystyle i\hbar\frac{\partial\Psi(\textbf{x},t)}{\partial
t}=\bigg{[}-\frac{\hbar^{2}\nabla^{2}}{2m}+\frac{1}{2}m\left(\omega_{x}^{2}x^{2}+\omega_{y}^{2}y^{2}+\omega_{z}^{2}z^{2}\right)$
$\displaystyle+\int\text{d}^{3}\textbf{x}^{\prime}\,U(\textbf{x}-\textbf{x}^{\prime})|\Psi(\textbf{x}^{\prime},t)|^{2}+\gamma_{\text{QF}}|\Psi(\textbf{x},t)|^{3}\bigg{]}\Psi(\textbf{x},t)\,,$
(4)

where $\omega_{x,y,z}$ are the harmonic trap frequencies. The wavefunction
$\Psi$ is normalized to the total atom number $N=\int{\rm
d}^{3}\mathbf{x}|\Psi|^{2}$. The stationary solution for Fig. 4a from the main
text is found through the imaginary time procedure in the rotating frame,
introducing the usual angular momentum operator $\Omega\hat{L}_{z}$ into Eq.
(4). The initial state $\Psi(\textbf{x},0)$ of the real-time simulations is
always obtained by adding non-interacting noise to the ground state
$\Psi_{0}(\textbf{x})$. Given the 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 $T=20\,$\mathrm{n}\mathrm{K}$$, the initial state can be
described as
$\Psi(\textbf{x},0)=\Psi_{0}(\textbf{x})+\sum_{n}^{\prime}\alpha_{n}\phi_{n}(\textbf{x})$,
where the sum is restricted only to the modes with $\epsilon_{n}\leq 2k_{B}T$
81. Throughout, the density images are presented in situ, with a scaling
factor to account for the 3ms TOF for the experimental images.

In order to obtain the average residual Fourier Transform images for Figs. 4c3
and e3, we first Fourier Transform 115 frames from the simulation between
700ms and 1.1s in the rotating frame before averaging the result.

###  A.6 Atom number

Extended Data Fig. 2 shows the condensed atom number $N_{c}$ for the
measurement with an adiabatic ramp of the magnetic-field rotational velocity
($\dot{\Omega}=2\pi\times 50\,$\mathrm{H}\mathrm{z}$/$\mathrm{s}$$),
corresponding to the data of Fig.1c of the main text. To extract the atom
number, we use the horizontal imaging with $26\,$\mathrm{m}\mathrm{s}$$ of
TOF. About $3\,$\mathrm{m}\mathrm{s}$$ before flashing the imaging resonant
light to the atoms, we rotate the magnetic field in the imaging plane and
perform standard absorption imaging. From the absorption images, we extract
$N_{c}$ from a bimodal fit up to $700\,$\mathrm{m}\mathrm{s}$$. At later
times, the system undergoes a dynamic instability (see discussion in the main
text) and the density profile deviates from a simple bimodal distribution.
During the observation time, we see a slight decrease of $N_{c}$ and for our
theory simulations we use a constant atom number of $N_{c}=15000$. Note that
in all following datasets, in which we abruptly accelerate the magnetic-field
rotation to the desired final velocity, we observe a faster decay and our
simulations are performed with either $N_{c}=8000$ or $N_{c}=10000$.

###  A.7 Vortex detection

Vortex detection algorithm: Since vortices appear as dark holes in the density
profile of a BEC, which would otherwise have a smooth profile, our approach to
extract the number of vortices is to look at deviations between the image and
an unmodulated reference image. To extract the vortex number from the raw
images, we proceed as follows:

First, we prepare the image $n_{\text{img}}$, the reference image
$n_{\text{ref}}$ and the residual image $n_{\text{res}}$. The image is
normalized such that the maximum density $\max(n_{\text{img}})=1$. We create
the reference image by blurring the image via applying a 2D Gaussian filter
with $\sigma=5\,$pixel, corresponding to about $2.1\,\mu$m. This blurring
smoothens any structure on the lengthscale of the filter width, therefore any
holes in the density profile wash out. We then normalize the atom number of
the reference to be the same as from the image $N_{\text{ref}}=\int\int
n_{\text{ref}}\doteq N_{\text{img}}=\int\int n_{\text{img}}$. The residual
image is calculated as the difference between the image and the reference
$n_{\text{res}}=n_{\text{img}}-n_{\text{ref}}$. We additionally mask the
region where the density of the reference is below a certain threshold
($n_{\text{res}}=0$ where $n_{\text{ref}}\leq 0.1$).

Second, we identify local minima in the residual image and determine if they
are connected to vortices. For this, we create a list of local minima
$(x_{\text{min}},y_{\text{min}})$, defined by the condition that the pixel
density $n_{\text{res}}(x_{\text{min}},y_{\text{min}})$ is lower than of all
surrounding pixels. Then we remove minima with density values above zero
$n_{\text{res}}(x_{\text{min}},y_{\text{min}})\geq 0$ or which are within one
pixel distance of the mask border. Now we determine a local contrast for each
minimum by calculating the difference between its central density value and
the mean of the density values $\pm 2\,$ pixel values away from it
$n_{\text{con}}(x_{\text{min}},y_{\text{min}})=n_{\text{res}}(x_{\text{min}},y_{\text{min}})-\text{mean}(n_{\text{res}}(x_{\text{min}}\pm
2\,$px$,y_{\text{min}}\pm 2\,$px$))$, and remove minima above a certain
threshold $n_{\text{con}}>-0.11$. As a last step we check the distance $d$
between all remaining minima to avoid double counting of minima too close to
each other. In case $d$ is below the threshold $d<5\,$px the minimum with the
higher residual density value $n_{\text{res}}$ is discarded.

Preparation of theory density profiles: For the direct comparison of the
vortex number shown in Fig. 3b, we apply additional steps to the density
profiles obtained from theory. First, we reduce the resolution by a $2\times
2$ binning to make the pixel size of the theory density profiles
$n^{\text{theo}}_{\text{img}}$ essentially the same as for the experimental
images (sizes are within $5\%$). After normalizing to
$\max(n^{\text{theo}}_{\text{img}})=1$ we apply Gaussian white noise with zero
mean and a variance of $0.01$ to each pixel, recreating the noise pattern from
empty regions of experimental images. Then we blur the image using a 2D
Gaussian filter with $\sigma=1\,$pixel ($\sim 0.42\,\mu$m), this recreates the
same resolution condition as our experimental setup. The resulting density
profile is taken as the input image for the vortex detection algorithm
described above.

Benchmarking the vortex detection algorithm: As the vortex positions for the
simulation images are known a priori due to the available phase map, we can
derive the fidelity of the vortex detection algorithm for simulation data. For
the theory data shown in Fig. 3b in the time frame between $600$ and
$700\,$\mathrm{m}\mathrm{s}$$, the average detected vortex number in the
simulated density profiles (applying the preparation scheme described above)
is about 9, while the real number of vortices present in the same area of the
image is about 33 in average. This mismatch is explained by the conservative
choice of the thresholds for vortex detection together with the added noise,
which results in only counting clear density dips as vortices, throwing out
many vortices in the low density region. This conservative choice of
thresholds on the other hand leads to a very high fidelity of $>97\%$, where
we define the fidelity as the percentage of detected vortices which correspond
to actual present vortices in the data. For raw simulation data (without
resolution reduction, added noise and blurring) the vortex detection algorithm
would detect up to $80\%$ of the vortices present with a fidelity of $>95\%$.

Note that for the visualization of the vortex positions for Fig. 4 we slightly
increased the local threshold $n_{\text{con}}>-0.08$ and decreased the minimum
distance between vortices $d<3\,$px, which increases the overall number of
vortices detected. For the density distributions obtained from theory, we
additionally omit the resolution reduction, addition of noise and blurring
steps.

##  Appendix B Extended data figures

Extended Data Fig. 1: Calculated ${\rm B}$-to-$a_{s}$ conversion for 162Dy.
Scattering length as a function of the magnetic-field value with the
background scattering length
$a_{\text{bg}}=129(9)\,$\mathrm{a}_{\mathrm{0}}$$. We find
$a_{s}=111(9)\,$\mathrm{a}_{\mathrm{0}}$$ at $B=5.333\,$\mathrm{G}$$. Extended
Data Fig. 2: Condensed atom number $N_{c}$ during magnetostirring (Fig. 1c).
Condensed atom number as a function of spin-up time $t_{\dot{\Omega}}$ for the
same sequence as in Fig. 1c. The condensed atom number is extracted by fitting
a two-dimensional bimodal distribution of Thomas-Fermi and Gaussian function
to the horizontal density distributions.

Extended Data Fig. 3: Repeatability of the vortex generation protocol. Each
row shows the simulated image (a, b, c) and the corresponding vertical TOF
images from independent experimental runs ($\mathrm{a}_{i}$, $\mathrm{b}_{i}$,
$\mathrm{c}_{i}$) for a different rotation time:
$t_{a}=127\,$\mathrm{m}\mathrm{s}$$, $t_{b}=207\,$\mathrm{m}\mathrm{s}$$, and
$t_{c}=741\,$\mathrm{m}\mathrm{s}$$. The rotation frequency is
$\Omega=0.74\,\omega_{\perp}$ with the trap frequencies being
$\omega_{t}=2\pi\times[50.7(1),50.8(1),129(1)]\,$\mathrm{H}\mathrm{z}$$. The
magnetic field value is $|{\overrightarrow{{\bf\it\rm
B}}}|=5.333(5)\,$\mathrm{G}$$. For the simulation the scattering length used
is $112\,$\mathrm{a}_{\mathrm{0}}$$, the trap frequencies are
$(50,50,150)\,$\mathrm{H}\mathrm{z}$$, the condensed atom number is $N=8000$,
and the rotation frequency is $0.75\,\omega_{\perp}$.