Abstract
Quantized vortices are a prototypical feature of superfluidity that have been observed in multiple quantum gas experiments. But the occurrence of vortices in dipolar quantum gases—a class of ultracold gases characterized by longrange anisotropic interactions—has not been reported yet. 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 pancakeshaped trap. Tilting the magnetic field towards the radial plane deforms the cloud into an ellipsoid, 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 agreement with numerical simulations.
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Main
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 2π phase winding that preserves the singlevaluedness of the superfluid wave function and the irrotational nature of its velocity field. In contactinteracting 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} using a number of different techniques. Moreover, vortices play a fundamental role in the description of the Berezinskii–Kosterlitz–Thouless transition in twodimensional (2D) systems^{12}, as well as in the evolution of quantum turbulence^{13,14}, and have been observed in interacting Fermi gases along the BoseEinstein condensate to BardeenCooperSchrieffer crossover^{8,15}.
Recently, a new class of ultracold quantum gases are being created in various laboratories around the world, using strongly magnetic lanthanide atoms^{16,17}. Such a system, providing a quantum analogue of classical ferrofluids, enables access to the physics of dipolar BECs, in which atoms feature a strong longrange anisotropic dipole–dipole interaction (DDI)^{18,19} on top of the traditional contacttype isotropic one. This intriguing platform provided the key to observe, for example, extended Bose–Hubbard dynamics^{20}, roton excitations^{21,22,23}, the quantum version of the Rosensweig instability^{24} and supersolid states of matter^{25,26,27,28}, and is foreseen to host novel phenomena for quantum simulation and metrology^{18,19}.
The dipolar interaction is predicted to also intimately change the properties of vortices in quantum gases^{29}. For instance, theoretical works predict single vortices to exhibit an ellipticshaped core for a quasi2D setting with inplane dipole orientation^{30,31,32,33} or the presence of density oscillations around the vortex core induced by the roton minimum in the dispersion relation^{30,31,32,33,34}. For vortex pairs, the anisotropic DDI is expected to alter the lifetime and dynamics^{33,35} and can even suppress vortex–antivortex annihilation^{33}. These interaction properties are predicted to give rise to a vortex lattice structure that can follow a triangular pattern^{30,34}, as is typical for nondipolar BECs^{11}, or a square lattice for attractive or zero contact interactions^{36,37,38} when the DDI is isotropic (dipoles aligned with the rotation axis). A very striking consequence of the dipoles tilted towards the plane is the formation of vortex stripes^{30,39,40}. Moreover, vortices could provide an unambiguous smoking gun of superfluidity in supersolid states^{41,42,43}. However, despite these intriguing predictions, vortices in dipolar quantum gases have not been observed until now.
This Article presents the experimental realization of quantized vortices in a dipolar BEC of highly magnetic dysprosium (Dy) atoms. Following a method proposed in ref. ^{40}, extended to arbitrary magneticfield angles in ref. ^{44}, we show that the manybody phenomenon of magnetostriction^{45}, 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 vortices in dipolar BECs: the formation of vortex stripes in the system.
In nondipolar gases, quantized vortices have been produced using several conceptually different techniques, for instance, by rotating nonsymmetric optical^{6,11} or magnetic^{46} potentials, by rapidly shaking the gas^{14}, by traversing it with obstacles with large enough velocity^{7,47}, by rapidly cooling the gas across the BEC phase transition^{48,49} or by directly imprinting the vortex phase pattern^{50}. Dipolar quantum gases, while able to form vortices with these same standard procedures^{29}, also offer unique opportunities that have no counterpart in contactinteracting gases. Crucially, the DDI gives rise to the phenomenon of magnetostriction in position space^{45}. When dipolar BECs are polarized by an external magnetic field B—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 favour headtotail dipole configurations, which effectively reduces the interaction energy^{19}.
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 nondipolar BEC takes the same shape as the confining trap (Fig. 1a(i)), introducing dipolar interactions with polarization axis along z stretches the cloud along this axis yet maintains cylindrical symmetry (Fig. 1a(ii)). Tilting the magnetic 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 (Fig. 1a(iii)). Finally, under continuous rotation of the magnetic field, which we coin ‘magnetostirring’, the condensate is predicted to rotate (Fig. 1a(iv)). This unique approach to stir a dipolar condensate can eventually lead to the nucleation of vortices^{40,44}, realizing genuinely interactiondriven vorticity through manybody phenomena.
We explore this protocol using a dipolar BEC of ^{162}Dy atoms. We create the BEC similar to our previous work^{51} with the distinction that here the magneticfield unit vector, \(\hat{{{{\rm{B}}}}}\), is kept tilted at an angle of θ = 35° with respect to the z axis both during evaporative cooling and magnetostirring (Fig. 1a(iii) and Methods). After preparation, the sample contains about 2 × 10^{4} condensed atoms confined within a cylindrically symmetric optical dipole trap (ODT) with typical radial and axial trap frequencies (ω_{⊥}, ω_{z}) = 2π × [50.8(2), 140(1)] Hz. Here, before stirring, the magnetostriction is expected from simulations to increase the cloud aspect ratio (AR) 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 timeofflight (TOF) expansion of 3 ms. The atom number is instead measured using horizontal absorption imaging with a TOF of 26 ms.
Similarly to a rotation of a bucket containing superfluid helium or of a smoothly deformed ODT for nondipolar BECs, magnetostirring is predicted to transfer angular momentum into a dipolar BEC^{40,44}. 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 Ω. This phenomenon is clearly visible in our experiments, as shown in Fig. 1b. Here we first revolve the tilted \(\hat{{{{\rm{B}}}}}\) around the z axis with a linearly increasing rotation frequency (\({{\dot{{{\Omega }}}}}={2}{{\uppi}} \times 50\,{{{\rm{Hz}}}}\,{{{\rm{{s}^{1}}}}}\)) and observe that the dipolar BEC starts to rotate at the same angular speed as the field and deforms with increasing elongation (Fig. 1b, top). We then stop the adiabatic ramp at a given value of Ω and probe the system under continuous rotation. We now find that the cloud continues rotating in the radial plane with an almost constant shape (Fig. 1b, bottom). Note that B is held constant at 5.333(5) G, where we estimate a contact scattering length of about a_{s} = 111a_{0}, where a_{0} is the Bohr radius (Methods).
We further explore the response of our dipolar BEC to magnetostirring by repeating the measurements in Fig. 1b (top), but stopping the ramp at different final values of Ω. The maximum value used for Ω approaches ω_{⊥}, corresponding to a ramp duration of 1 s. We quantify the cloud elongation in terms of the aspect ratio AR = σ_{max}/σ_{min}, where the cloud widths σ_{max} and σ_{min} are extracted by fitting a rotated 2D Gaussian function to the density profiles. Figure 1c summarizes our results. We observe that initially the AR slightly deviates from 1 due to magnetostriction. It then slowly grows with increasing Ω, until a rapid increase at around 0.6ω_{⊥} occurs, as this allows the angular momentum to increase, which decreases the energy in the rotating frame^{52}. Suddenly, at a critical rotation frequency Ω_{c} ≈ 0.74ω_{⊥}, the AR abruptly collapses back to AR ≈ 1, showing how the superfluid irrotational nature competes with the imposed rotation. This critical frequency is close to the value found in nondipolar gases with a rotating elliptical harmonic trap, associated with a resonance at the quadrupole frequency^{53}.
To substantiate our observation, we perform numerical simulations of the zerotemperature extended Gross–Pitaevskii equation (eGPE)^{54} (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 Ω; 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 Ω = 0.8ω_{⊥} and are in excellent agreement with the experimental results. The stability of the 1 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 1.
The growing AR and subsequent collapse to 1 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 highdensity regions of rotated BECs^{52,53,55}, as also predicted for our dipolar system^{40}. To search for quantum vortices in our system, we perform a new investigation where we directly set Ω close to Ω_{c}, aiming to trigger the instability at an earlier time when more atoms are condensed. We then hold the magneticfield rotation fixed at this constant frequency for a time t_{Ω}. As shown in Fig. 2 (bottom), the cloud rapidly elongates, and the density starts to exhibit a spiral pattern, emanating from the tips of the ellipsoid. As early as t_{Ω} = 314 ms, clear holes are observed in the density profile, forming in the density halo around the centre, 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 (highdensity) region. Vortices are still visible in the experiment after 1 s of magnetostirring, although our atom number decreases throughout this procedure. Our observations bear a remarkable resemblance to the simulations; Fig. 2 (top) shows the in situ column densities. Taking a fixed atom number of N = 8,000, but otherwise repeating the experimental sequence, we observe many similar features. In the first 100 ms, the system elongates, consistent with Fig. 1, and a 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 highdensity region. For this scattering length and atom number, the relaxation timescale to a stable vortex lattice is longer than the experimentally available (see Extended Data Fig. 3 for more images from this dataset). Note that at angles θ deeper into the plane, more atoms align headtotail in the loose radial confinement direction. Thus, when performing the rotation procedure, we find that the BEC is resilient to instability on the timescales of the experiment.
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 AR ≈ 1 (Fig. 2). Note that the drop in AR observed in Fig. 1 is concurrent with the creation of vortices, but they reside in the lowdensity regions at this time, and we do not see them. 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 (Fig. 3a, left), we create a blurred reference image by applying a 2D Gaussian filter^{56,57}. We then calculate the difference between each single image (Fig. 3a, left) and the corresponding reference (Fig. 3a, middle) to obtain the residual image (Fig. 3a, right), 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 lowdensity zones (halo) where the vortices initially nest, we expect \({{{{\mathcal{N}}}}}_{v}\) to be underestimated relative to the true value and the number expected by theory. However, 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_{Ω}. Solid lines are the results from the eGPE simulations without any adjustable parameters. For t_{Ω} < 200 ms, \({{{{\mathcal{N}}}}}_{v}\) 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 atomlossinduced 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 ~1 (ref. ^{53}). This happens as the vortex number simultaneously increases.
One fascinating prediction with vortices in a strongly dipolar gas 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^{29,30}, instead of the usual triangular lattice in nondipolar BECs^{6}. The ground state stripe lattice solution for our parameters is shown in Fig. 4a, with a cloud AR = 1.08. In the vortex stripe phase, vertical planes of highdensity regions, parallel to the magnetic field, alternate with lowdensity ones, which host rows of vertical vortex filaments. Such a configuration promotes headtotail dipolar attraction within the highdensity 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^{58}.
To explore this prediction, we perform two new surveys. First, we slightly reduce the magneticfield value, reducing the scattering length to a_{s} ≈ 109a_{0} and hence making the system relatively more dipolar. We magnetostir the BEC at a constant rotation frequency Ω = 0.75ω_{⊥} for 500 ms, but during TOF, we stop the magneticfield rotation and keep it in place at θ = 35°. The stripe structure is revealed in Fig. 4b (left) for a single experimental run and is clearly visible in the residual image (Fig. 4b, middle) where the vortices align along three stripes. The spatial structure of the residual image can be assessed through the absolute value of 2D Fourier transform (FT). After taking the FT of each residual image, we then average the result (Fig. 4b, right), finding a clear peak at the wave number k of the interstripe 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. 4c), 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 magneticfield orientation to θ = 0° (parallel to the trap symmetry axis), as shown in Fig. 4d (left). Here, after 600 ms of magnetostirring, we add another step in which we spiral up the magnetic field to θ = 0° (with Ω fixed) over 100 ms, before imaging. Under these conditions, all vortex properties are again isotropic within the plane. The nonequilibrium 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. 4d, right). Also, this behaviour is confirmed by the simulations, as shown in Fig. 4e. The vortices survive long after the magnetostirring has stopped (not shown), due to their topological protection.
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 inplane magnetic fields^{30,31,32,33}, the interplay between the vortex and roton excitations^{30,31,32,33,34} and exotic vortex patterns such as square lattices^{29}, and investigations into anisotropic turbulence^{59}. This work also opens the door to studying more complex matter under rotation, such as dipolar droplets^{60,61,62} and supersolid states^{41,42,43,51}. Such proposals will be challenging due to the intricate density patterns^{63}; 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^{64}—a method also employed in nuclear magnetic resonance spectroscopy—but there remain open questions on the stability of this procedure^{65,66}, which if rectifiable would unlock new research directions^{64}. Other vortex generation methods, such as thermally activated pairs in quasitwo dimensions 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^{29}. The technique introduced here is also applicable to a wide range of systems governed by longrange interactions through the manipulation of magnetic or electric fields.
Methods
Experimental procedure
We prepare an ultracold gas of ^{162}Dy atoms in an ODT. Three 1,064 nm laser beams, overlapping at their foci, form the ODT. The experimental procedure to BEC is similar to the one followed in our previous work^{51}, but the magneticfield unit vector, \(\hat{{{{\rm{B}}}}}\), is tilted by an angle of θ = 35° with respect to the zaxis during the whole sequence. After preparation, the sample contains about 2 × 10^{4} condensed atoms. The corresponding trap frequencies are typically (ω_{⊥}, ω_{z}) = 2π × [50.8(2), 140(1)] Hz. For all our measurements, the deviation of the trap AR in the x–y plane AR_{trap} = ω_{y}/ω_{x} from 1 is always smaller than 0.6%. We evaporate the atoms at B = 5.423(5) G and jump to the final magneticfield 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 ms and a pulse duration of 3–4 μs. For the data in Figs. 1–3, we let the magneticfield spinning during TOF, whereas for Fig. 4, we use a static field orientation.
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 magneticfield value 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 longterm stability—measured via the peak position of the RF resonance over the course of several days—is on the order of ΔB = ±1 mG, while shottoshot fluctuations, measured via the width of the RF resonance for a single calibration set, is ΔB = ±5 mG.
Rotation
We drive the rotation of the magnetic field by sinusoidally modulating the magneticfield value components B_{x} and B_{y} with a phase difference of 90° between them. As we want to keep the absolute magneticfield value B constant during rotation, we measure it for various values of the azimuthal angle ϕ and fixed θ = 35° by performing Feshbach loss spectroscopy around 5.1 G. We find an average shift of B of about 10 mG from the θ = 0° case, which we take into account. We also find small deviations as a function of ϕ of ΔB < 20 mG, which might appear due to slightly nonorthogonal alignment of the magnetic fields. We did not correct these deviations for the sake of simplicity.
Scattering length
The scattering length in ^{162}Dy is currently not known with large accuracy^{67,68,69,70}. To estimate the scattering length in the small magneticfield range around B = 5.3 G, relevant to this work, we use the wellknown relation a_{s} = a_{bg}∏_{i}[1 − ΔB_{i}/(B − B_{0,i})] (ref. ^{71}), where B_{0,i} and ΔB_{i} are the centre position and the width of the ith feature of the Feshbach loss measurement reported in ref. ^{70}, respectively. The value of the background scattering length, a_{bg}, is empirically fixed by measuring the magneticfield 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)a_{0} at B = 5.333 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^{72}, would be desirable.
Magnetostirring
Tilting the magneticfield vector B away from the symmetry axis of our cylindrical trap leads to an ellipsoidal deformation of the cloud^{45} 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 B tilted with respect to the z axis by 35° and a constant value B. That value is B = 5.333(5) G for the surveys in Figs. 1–3 and B = 5.323(5 )G for Fig. 4. For these parameters, the deformed magnetostricted AR of the cloud is AR − 1 = 0.03. For all our measurements, the measured trap AR_{trap} − 1 < 0.006 is much smaller than the deformation due to magnetostriction. Additionally, we have confirmed with simulations that even with trap asymmetries of up to 10%, for example, (ω_{x}, ω_{y}) = (55, 50) Hz, this procedure can still generate vortices in a lattice configuration.
At the scattering lengths considered in this work, 35° is an optimal choice to see the vortices within ~500 ms of rotation and anisotropic enough to observe the stripe phase. From the simulations, we find that tilt angles smaller than 35° increase the timescale to vortex nucleation. Similarly, tilting the angle further into the plane increases the number of atoms that are aligned head to tail, making the dipolar interaction dominantly attractive. This attractive force holds the condensate together during the rotation, also increasing the time to vortex nucleation. From the experimental side, increasing the tilt angle reduces the contrast of the absorption imaging, since the magnetic field is not parallel to the imaging axis. As the TOF is only 3 ms, we do not rotate up the magnetic field before imaging to avoid undesired effects, such as losing the anisotropy given by θ ≠ 0°.
Adiabatic frequency ramp
We employ different magneticfield rotation sequences for the different datasets. For the dataset of Fig. 1c, the rotation frequency of the magnetic field is linearly increased to different final values at a speed of \(\dot{{{\Omega }}}=2{{\uppi}} \times 50\,{{{\rm{Hz}}}}\,{{{\rm{{s}^{1}}}}}\) and for a duration of \({t}_{{{\dot{{{\Omega }}}}}}=0{{{\rm{}}}}1\,{{{\rm{s}}}}\). The ramp time is much longer than the period of the rotation Ω^{−1} for higher rotation frequencies Ω ≳ Ω_{c}, and, therefore, the ramp is adiabatic for the regimes considered, until the onset of dynamical instabilities. After the ramp, the magneticfield direction is rotated at the target rotation frequency Ω for one final period (as shown in Fig. 1b). We sample ten different final magneticfield angles during this last rotation, measuring the corresponding AR and averaging the result to remove any potential biases due to latent trap anisotropies. Each data point is then obtained with eight to ten experimental runs.
Constant rotation frequency
For the dataset of Figs. 2, 3 and 4b, we directly start to rotate at the final rotation frequency Ω without any acceleration phase. The magnetic field is then rotated for a variable time t_{Ω}, 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 Ω = 0.75ω_{⊥}, the magnetic field is spiralled up in 100 ms to θ = 0° by linearly reducing θ while continuing rotating. Afterwards, the atoms are released from the trap and a vertical image is taken.
Theoretical model
We employ an eGP formalism to model our experimental setup. In this scheme, the interparticle interactions are described by the twobody pseudopotential
with \(\delta ({\bf{r}})\) being the Kronecker delta function and \({\bf{r}}=(x,y,z)\). The first term describes the shortrange interactions governed by the swave scattering length a_{s}, with Planck’s constant ℏ and particle mass m. The second term represents the anisotropic and longrange dipole–dipole interactions, characterized by dipole length \({a}_{{{{\rm{dd}}}}}={\mu }_{0}{\mu }_{{\mathrm{m}}}^{2}m/12{\mathrm{\pi}} {\hslash }^{2}\), with magnetic moment μ_{m} and vacuum permeability μ_{0}. We always consider ^{162}Dy, such that a_{dd} = 129.2a_{0}, where a_{0} is the Bohr radius. The dipoles are polarized uniformly along a timedependent axis, given by
with timedependent polarization angle θ(t) and \(\phi (t)=\int\nolimits_{0}^{t}{{{\rm{d}}}}t^{\prime} {{\Omega }}(t^{\prime} )\), for rotation frequency protocol Ω(t).
Beyondmeanfield effects are treated through the inclusion of a Lee–Huang–Yang correction term^{73}
with \({{{{\mathcal{Q}}}}}_{5}({\epsilon }_{{{{\rm{dd}}}}})=\int\nolimits_{0}^{1}{{{\rm{d}}}}u\,{(1{\epsilon }_{{{{\rm{dd}}}}}+3{u}^{2}{\epsilon }_{{{{\rm{dd}}}}})}^{5/2}\) being the auxiliary function, and the relative dipole strength is given by ϵ_{dd} = a_{dd}/a_{s}. Finally, the full eGPE then reads^{54}^{,}^{74,75,76}
where ω_{x,y,z} are the harmonic trap frequencies. The wave function Ψ is normalized to the total atom number N = ∫d^{3}x∣Ψ∣^{2}. The stationary solution for Fig. 4a is found through the imaginary time procedure in the rotating frame, introducing the usual angular momentum operator \({{{\varOmega }}}{\hat{L}}_{z}\) into equation (4). The initial state Ψ(x, 0) of the realtime simulations is always obtained by adding noninteracting noise to the ground state Ψ_{0}(x). Given the singleparticle eigenstates ϕ_{n} and the complex Gaussian random variables α_{n} sampled with \(\langle {\alpha _n}{^2}\rangle = {({e^{{_n}/{k_{\rm{B}}}T}}  1)^{  1}} + \frac{1}{2}\) for a temperature T = 20 nK and Boltzmann’s constant \(k_{\mathrm{B}}\), the initial state can be described as \({\varPsi }\left({\mathbf{x}},0\right)={\varPsi }_{0}\left({\mathbf{x}}\right)+{\mathop{\sum}\limits_{n}}\,^{\prime} {\alpha }_{n}{\phi }_{n}\left({\mathbf{x}}\right)\), where the sum is restricted only to the modes with ϵ_{n} ≤ 2k_{B}T (ref. ^{77}). Throughout, the density images are presented in situ, with a scaling factor to account for the 3 ms TOF for the experimental images.
To obtain the average residual FT images for Fig. 4c,e, we first Fourier transform 115 frames from the simulation between 700 ms and 1.1 s in the rotating frame before averaging the result.
Atom number
Extended Data Fig. 2 shows the condensed atom number N_{c} for the measurement with an adiabatic ramp of the magneticfield rotational velocity (\({{\dot{{{\Omega }}}}}=2{{\uppi}} \times 50\,{{{\rm{Hz}}}}\,{{{\rm{{s}^{1}}}}}\)), corresponding to the data of Fig. 1c. Threebody losses are negligible in the lowdensity BEC phase, with losses probably coming from imperfections in the rotation procedure and heating. To extract the atom number, we use the horizontal imaging with 26 ms of TOF. About 3 ms 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 ms. 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} = 15,000. Note that in all following datasets, in which we abruptly accelerate the magneticfield rotation to the desired final velocity, we observe a faster decay, and our simulations are performed with either N_{c} = 8,000 or N_{c} = 10,000.
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_{img}, the reference image n_{ref} and the residual image n_{res}. The image is normalized such that the maximum density \(\max ({n}_{{{{\rm{img}}}}})=1\). We create the reference image by blurring the image via applying a 2D Gaussian filter with σ = 5 pixel, corresponding to about 2.1 μm. 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_{ref} = ∫∫n_{ref} ≐ N_{img} = ∫∫n_{img}. The residual image is calculated as the difference between the image and the reference \({n}_{\mathrm{res}}={n}_{\mathrm{img}}{n}_{\mathrm{ref}}\). We additionally mask the region where the density of the reference is below a certain threshold (\({n}_{{{{\rm{res}}}}}=0\), where n_{ref} ≤ 0.1).
Second, we identify local minima in the residual image and determine whether they are connected to vortices. For this, we create a list of local minima \(({x}_{\min },{y}_{\min })\), defined by the condition that the pixel density \({n}_{{{{\rm{res}}}}}({x}_{\min },{y}_{\min })\) is lower than of all surrounding pixels. Then we remove minima with density values above zero \({n}_{{{{\rm{res}}}}}({x}_{\min },{y}_{\min })\ge 0\) 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 ± 2 pixel values away from it \({n}_{{{{\rm{con}}}}}({x}_{\min },{y}_{\min })={n}_{{{{\rm{res}}}}}({x}_{\min },{y}_{\min }){{{\rm{mean}}}}({n}_{{{{\rm{res}}}}}({x}_{\min }\pm 2\,{{{\rm{px}}}},{y}_{\min }\pm 2\,{{{\rm{px}}}}))\), and remove minima above a certain threshold n_{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 pixel, the minimum with the higher residual density value n_{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 × 2 binning to make the pixel size of the theory density profiles \({n}_{{{{\rm{img}}}}}^{{{{\rm{theo}}}}}\) essentially the same as for the experimental images (sizes are within 5%). After normalizing to \(\max ({n}_{{{{\rm{img}}}}}^{{{{{\rm{theo}}}}}})=1\), 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 σ = 1 pixel (~0.42 μm); 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 ms, 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 lowdensity 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 that 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_{con} > −0.08 and decreased the minimum distance between vortices d < 3 pixel, 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.
Data availability
Data pertaining to this work can be found at https://doi.org/10.5281/zenodo.7019859.
Code availability
The codes that support the findings of this study are available from the corresponding author upon reasonable request.
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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. I4391N), and the DFG/FWF via FOR 2247/PI2790 a jointproject grant from the FWF (No. I4426). 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.
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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.N.B. performed the theoretical work. All authors contributed to the interpretation of the results and the preparation of the manuscript.
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Extended data
Extended Data Fig. 1 Calculated Btoa_{s} conversion for ^{162}Dy.
Scattering length as a function of the magneticfield value with the background scattering length a_{bg} = 129(9) a_{0}. We find a_{s} = 111(9) a_{0} at B = 5.333 G.
Extended Data Fig. 2 Condensed atom number N_{c} during magnetostirring (Fig. 1c).
Condensed atom number as a function of spinup time \({t}_{\dot{\Omega }}\) for the same sequence as in Fig. 1c. The condensed atom number is extracted by fitting a twodimensional bimodal distribution of ThomasFermi 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 (a_{i}, b_{i}, c_{i}) for a different rotation time: t_{a} = 127 ms, t_{b} = 207 ms, and t_{c} = 741 ms. The rotation frequency is Ω = 0.74ω_{⊥} with the trap frequencies being ω_{t} = 2π × [50.7(1), 50.8(1), 129(1)] Hz. The magnetic field value is \(B =5.333(5)\,{{{\rm{G}}}}\). For the simulation the scattering length used is 112 a_{0}, the trap frequencies are (50, 50, 150) Hz, the condensed atom number is N = 8000 and the rotation frequency is 0.75ω_{⊥}.
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Klaus, L., Bland, T., Poli, E. et al. Observation of vortices and vortex stripes in a dipolar condensate. Nat. Phys. 18, 1453–1458 (2022). https://doi.org/10.1038/s41567022017938
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DOI: https://doi.org/10.1038/s41567022017938
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