1508/1508.05007.md

Observing the Rosensweig instability of a quantum ferrofluid

Holger Kadau¹, Matthias Schmitt¹, Matthias Wenzel¹, Clarissa Wink¹, Thomas Maier¹, Igor Ferrier-Barbut¹ & Tilman Pfau¹

¹5. Physikalisches Institut and Center for Integrated Quantum Science and Technology, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany

Ferrofluids show unusual hydrodynamic effects due to the magnetic nature of their constituents. For increasing magnetization a classical ferrofluid undergoes a Rosensweig instability¹ and creates self-organized ordered surface structures² or droplet crystals³. A Bose-Einstein condensate with strong dipolar interactions is a quantum ferrofluid that also shows superfluidity⁴. The field of dipolar quantum gases is motivated by the search for new phases that break continuous symmetries^5,6. The simultaneous breaking of continuous symmetries like the phase invariance for the superfluid state and the translational symmetry for a crystal provides the basis of novel states of matter. However, interaction-induced crystallization in a superfluid has not been observed. Here we use in situ imaging to directly observe the spontaneous transition from an unstructured superfluid to an ordered arrangement of droplets in an atomic dysprosium Bose-Einstein condensate⁷. By utilizing a Feshbach resonance to control the interparticle interactions, we induce a finite-wavelength instability⁸ and observe discrete droplets in a triangular structure, growing with increasing atom number. We find that these states are surprisingly long-lived and measure a hysteretic behaviour, which is typical for a crystallization process and in close analogy to the Rosensweig instability. Our system can show both superfluidity and, as shown here, spontaneous translational symmetry breaking. The presented observations do not probe superfluidity in the structured states, but if the droplets establish a common phase via weak links, this system is a very good candidate for a supersolid ground state^9-11.

Research in condensed matter physics is driven by the discovery of novel phases of matter, in particular phases simultaneously displaying different types of order. A prime example is the supersolid state featuring simultaneously crystalline order and superfluidity^9-11. This state has been elusive and claims of its discovery in helium^12,13 have been withdrawn recently¹⁴. One of the requirements for a spontaneous spatially ordered structure formation in the ground state of a many-body system are long-range interactions which are present in ferrofluids. As a consequence, a magnetized ferrofluid forms stationary surface waves due to a competition between gravitational or magnetic trapping, dipolar interaction and surface tension. This effect is known as the normal-field instability or Rosensweig instability² and leads to stable droplet patterns on a superhydrophobic surface³. For ferrofluids the dispersion relation of surface excitations displays a minimum at finite momentum, resembling the well-studied

roton spectrum in liquid helium¹⁵. However, for a ferrofluid the physical interpretation of this minimum is very different than in helium. Its origin is an energy gain due to the attractive part of the dipolar interaction for clustering polarized dipoles in a head-to-tail configuration in periodic structures. Such roton-induced structures have also been discussed for quantum ferrofluids^8,16. In close similarity with a classical ferrofluid, a competition exists between the harmonic trapping, dipolar interaction and contact interaction. For increasing relative dipolar interaction, the roton instability can lead to a periodic perturbation of the atomic density distribution, closely connected to the Rosensweig instability¹⁷. However, it was believed that these rotonic structures would be unstable due to subsequent instabilities of the forming droplets¹⁸.

Here we use the most magnetic element dysprosium (Dy) with a magnetic moment of $\mu = 9.93 \mu_B$ , where $\mu_B$ is the Bohr magneton, and generate a Bose-Einstein condensate (BEC)⁷. We observe an angular roton instability^16,19 and find subsequent droplet formation to triangular structures with surprisingly long lifetimes. We use two key tools to study these self-organized structures. First, we use a magnetic Feshbach resonance²⁰ to tune the contact interaction and to induce the droplet formation. Second, we use a microscope with high spatial resolution to detect the atomic density distribution in situ.

The first prediction of structured ground states in a dipolar BEC dates back to the early days of quantum gases²¹ and the first mechanical effects have been seen with chromium atoms²². There the dipolar attraction deforms the compressible gas and its shape is balanced by a repulsive contact interaction, described with the scattering length $a$ . To compare the strengths of the contact and dipolar interaction, we introduce a length scale characterizing the magnetic dipole-dipole interaction strength $a_{dd} = \mu_0 \mu^2 m / 12\pi \hbar^2$ . By tuning the scattering length $a$ with a Feshbach resonance such that $a < a_{dd}$ , the dipolar attraction dominates the repulsive contact interaction and an instability of a dipolar gas can occur^4,23. However, in a pancake-shaped trap the dipoles sit mainly side-by-side and predominantly repel each other due to the anisotropy of the dipole-dipole interaction, and hence the dipolar BEC is stabilized. In such a pancake-shaped configuration the roton instability at finite wavelength is predicted^8,16.

For our experiments we used the isotope ¹⁶⁴Dy with a dipolar length of $a_{dd} = 132 a_0$ , where $a_0$ is the Bohr radius. This dipolar length is already higher than the background scattering length $a_{bg} = 92(8) a_0$ , which is the value far from Feshbach resonances^24,25. To obtain a stable BEC, we tuned the scattering length to $a \approx a_{dd}$ using a mag-netic field of $B = 6.962(3)$ G in the vicinity of a Feshbach resonance located at $B_0 = 7.117(3)$ G. We then obtained typically 15,000 atoms in nearly-pure Dy BECs (see Methods section). The atoms were trapped in a radially symmetric pancake-shaped trap with harmonic frequencies of $(\nu_x, \nu_y, \nu_z) = (46, 44, 133)$ Hz and the external magnetic field aligned the magnetic dipoles in axial $z$ -direction. Subsequently, we tuned the magnetic field $B \lesssim 6.9$ G, which reduces $a$ to $a_{bg} < a_{dd}$ resulting in an angular roton instability¹⁶ that triggered the transition to ordered states (Fig. 1a). We observed then the formation of droplets that arrange in ordered structures by in situ phase-contrast polarization imaging along the $z$ -direction with a spatial resolution of $1 \mu\text{m}$ .

In Fig. 1b we show typical in situ images of the resulting triangular patterns for the quantum ferrofluid with different number of droplets $N_d$ ranging from two to ten. In order to analyze the average atom number per droplet, we count the number of droplets $N_d$ in relation to the total atom number. Fig. 1c indicates a linear dependence between $N_d$ and atom number with a slope of $1750(300)$ atoms per droplet. In the case of $N_d = 2$ we observe a droplet distance of $d = 3.0(4) \mu\text{m}$ . The droplets, having a large effective dipole moment of $N_d\mu$ , strongly repel each other while the radial trapping applies a restoring force. Hence, this length distance $d$ can be calculated in a simplified one-dimensional classical system by minimizing its energy. We assume two strongly dipolar particles with 1750 times the mass and magnetic moment of a dysprosium atom that are confined in a harmonic trap. For our experimental parameters they minimize their energy with a distance of $d = 3.3 \mu\text{m}$ , in agreement with the observed distance. For $N_d > 2$ the droplets arrange mostly in a triangular structures and form a microscopic crystal with a droplet distance of $d = 2 - 3 \mu\text{m}$ . Owing to the isotropy of the repulsion between droplets in the radial plane, we expect the triangular configuration to have the lowest energy. Because of the repelling dipolar force between the droplets, we observe in the radial direction nearly round discrete droplets with possible weak overlap to neighbouring ones. Comparing our quantum ferrofluid with a classical ferrofluid, very similar behaviour and patterns have been observed on a superhydrophobic surface³. In this system a single droplet first deforms for increasing external magnetic field and divides above a critical field into two droplets. However, for a quantum ferrofluid a single droplet should be unstable for $a < a_{dd}$ due to the attractive part of the dipolar interaction and collapse. In addition the counteracting quantum pressure, the zero-point energy existing due to an external trapping potential, can compensate attraction and prevent collapse²⁶. It was however predicted not to be the case by mean field calculations^18,19. Our observation of stable droplet ensembles is thus striking and further work is needed to understand their stability. A possible stabilizing effect are quantum fluctuations leading to beyond mean-field effects²⁷. Such a stabilization has been suggested in a similar situation of competing attraction and repulsion²⁸ and an increased effect of quantum fluctuations has been calculated

Figure 1: Microscopic droplet crystal growth. a, Scheme of the experimental procedure: We prepared a stable strongly dipolar Dy BEC at $a \approx a_{dd}$ in a pancake-shaped trap. By decreasing the scattering length $a$ , we induced an instability close to $a \approx a_{bg}$ . Followed by this instability the atoms clustered to droplets in a triangular pattern. b, Representative single-shot in situ images of droplet patterns with droplet numbers $N_d$ ranging from two to ten. c, We used a set of 112 realizations with different droplet and atom numbers for a statistical analysis. The plot shows the mean number of atoms in dependence of visible droplets $N_d$ , with the standard deviation as error bars. We fitted a linear relation (grey dashed line) and extracted a slope of $1750(300)$ atoms per droplet. This shows the growth of the microscopic droplet crystal by increasing the atom number.Figure 2: Evaluation of the structures and lifetime analysis. a-c, This part presents our statistical evaluation procedure. We started with a the spatial density distribution, calculated b the absolute value of the two-dimensional Fourier transform $S(k_x, k_y)$ and radially averaged over $k = (k_x^2 + k_y^2)^{1/2}$ to get the spectrum $S(k)$ . We removed the white noise from the spectrum $S(k)$ such that $S(k) = 0$ for $k > 7 \mu\text{m}^{-1}$ , which are structures below our resolution, and show in c an average of 13 images for BECs and Patterns. For patterns in the spatial density distribution, we observe enhanced signal for $k \approx 2.5 \mu\text{m}^{-1}$ in the spectrum (red line), whereas $S(k)$ of BECs (black line) shows only a monotonic decay for increasing momentum. We define the sum of these spectra over a momentum range as relative spectral weight SW (coloured area, as defined in the text), which is a quantity for the strength of the structured states. d, We performed a sudden quench (0.5 ms) of the BEC below the instability to $B = 6.656(3)$ G for varying wait times. To extract creation and lifetime we give the relative spectral weight SW (red diamonds) against wait time, where each point is an average of 13 realizations with error bars indicating the standard error. The plot shows a fast pattern formation within 7 ms and surprisingly long 1/e-lifetimes of $\sim 300$ ms. The lifetime seems to be limited by losses of the atom number (blue circles). The insets are typical single-shot spatial density distributions before pattern formation (BEC) and at three different wait times.

for strongly dipolar gases²⁹.

For further quantitative statistical analysis, we computed the Fourier spectrum $S(k)$ of the obtained images (Fig. 2a-c). The patterns are visible as a local maximum at finite momentum $k = 2\pi/d \approx 2.5 \mu\text{m}^{-1}$ , whereas the spectrum $S(k)$ of a BEC shows only a monotonic decrease in $k$ . Hence, we define the spectral weight

$$\text{SW}=\sum _{k=1.5\mu {\text{m}}^{-1}}^{5\mu {\text{m}}^{-1}}S(k),(1)\backslash text\{SW\}\; =\; \backslash sum\_\{k=1.5\; \backslash mu\backslash text\{m\}^\{-1\}\}^\{5\; \backslash mu\backslash text\{m\}^\{-1\}\}\; S(k),\; \backslash quad\; (1)$$

which is a quantity for the strength of the structured states and give it in relative units such that a BEC has $\text{SW}{\text{BEC}} = 1$ . After a quench of the interactions from $a \approx a{\text{dd}}$ to $a \approx a_{\text{bg}}$ , we investigated statistically the pattern formation time and the lifetime of these patterns (Fig. 2d). We repeated this measurement 13 times and found statistically that after 7 ms the pattern formation is fully developed and we observed a

1/e-lifetime of $\sim 300$ ms. The decay of the droplet structure is accompanied by an atom loss showing a 1/e-lifetime of $\sim 130$ ms, while the residual thermal cloud is constant. Due to the decreasing atom number the structures melt back to lower number of droplets $N_d$ until they merge back to one droplet (insets of Fig 2d). In comparison, as we measured lifetimes of a non-structured BEC of more than 5 s, we assume increased three-body losses as a reason for the reduced lifetime. One indication is the measured atomic peak density for droplets of $n \gtrsim 5 \cdot 10^{20} \text{ m}^{-3}$ , which is larger than the density of a BEC with $n \approx 10^{20} \text{ m}^{-3}$ . To explore the nature of this instability, we performed the following experimental sequence, depicted in Fig. 3a. We prepared the BEC close to the Feshbach resonance with $a \approx a_{\text{dd}}$ and ramped linearly to varying magnetic field values around the instability point. We ensured that the structures are formed within 10 ms also for magnetic field values close to the stability threshold, and waited here for twice this time. Fig. 3b shows a clear hysteresis for the way down in magnetic field compared to the return. For the way back we have the same spectral weightFigure 3: Hysteresis of pattern formation. a, A time line of the experiment where we observed hysteresis. We prepared the Dy BEC close to the Feshbach resonance at 6.962(3) G and ramped linearly to different magnetic field values with a constant ramp speed down to a lowest value of 6.860(3) G. To ensure that the structures have enough time to form, we waited afterwards for 20 ms and in situ imaged the atomic sample. For the way back, we waited first at the lowest field value for 20 ms and then increased with the same ramp speed to different higher field values and held again for 20 ms. b, Hysteresis plot for the structured patterns. It shows the spectral weight against magnetic field for the way down (red diamonds and line) and the way back (green squares and line), each point is an average of 14 realizations with error bars indicating standard errors. Our long-term field stability was 3 mG shown with horizontal error bars for selected points. A clear hysteresis is visible comparing the way down and the return, although the total wait time is twice longer for the way back. A few points are marked in a and b with numbers to help the understandability and the lines serve as a guide to the eye.

for $\sim 20$ mG higher magnetic field values. This demonstrates that our system features bistability in the transition region. We expect that the transition from one state to the other state is driven by thermal excitations or weak currents due to overlapping droplets. In the thermodynamic limit such behaviour is a clear signature of a first-order phase transition and a latent heat in the crystallization process.

To verify that we are dealing with a transition to a ground state and not a metastable state resulting from quench dynamics, we performed forced evaporative cooling at a constant magnetic field far away from any Feshbach resonance with $a \approx a_{\text{bg}}$ . We observed very similar self-organized structures, which were visible for temperatures near the expected critical temperature for the phase transition to a BEC.

As structures can melt back into a BEC and we observed evaporative cooling to patterns, it is quite plausible that the droplets are superfluid individually. Whether they are sharing the same phase via weak links or lose their mutual phase coherence will have to be investigated in future experiments. Another open question is the creation dynamics of a self-organized structure. It will be interesting to study phonons in such a droplet crystal which we expect to have eigenfrequencies on the order of many inverse lifetimes. We expect that these phonons will be coupled to collective Josephson oscillations via weak links³⁰.

Methods

Preparation of a Dy BEC Bosonic $^{164}\text{Dy}$ atoms are first cooled in a narrow-line magneto-optical trap, operating on the 626 nm optical cycling transition, which polarizes the atoms to the lowest Zeeman state $J = 8, m_J = -8$ , and subsequently directly loaded into a single-beam optical dipole trap (ODT)³¹ created by a broadband fiber laser operating at 1070 nm with a power of 72 W. By moving the focusing lens of the transport ODT over a range of 375 mm we transport the atoms into a glass cell. After the transport, we load the atoms from the transport ODT into a crossed optical dipole trap created by a single-mode laser operating at 1064 nm. Following this, the atoms are Doppler cooled with 626 nm light and we utilize forced evaporative cooling by ramping down the power of both trapping laser beams at a magnetic field of $B = 1.012$ G far away from any resonance. When close to degeneracy we tune the magnetic field close to a Feshbach resonance and achieve there a quasi-pure BEC with typically 15,000 atoms and a temperature of $T = 70$ nK. Before shaping the final trap with harmonic frequencies of $(\nu_x, \nu_y, \nu_z) = (46, 44, 133)$ Hz, we apply an additional magnetic gradient of 1.1 G/cm to partially compensate gravity.

Feshbach resonance The background scattering length of $^{164}\text{Dy}$ has been measured to be $a_{\text{bg}} = 92(8) a_0$ , where $a_0$ is the Bohr radius^24,25. In the vicinity of a Feshbach resonance the scattering length scales as $a = a_{\text{bg}}(1 - \Delta B/(B - B_0))$ with $B$ the applied magnetic field, $B_0$ the center and $\Delta B$ the width of the Feshbach resonance. To tune the scattering length $a$ , we used a narrow resonance at $B_0 = 7.117(3)$ G with $\Delta B = 51(15)$ mG. We calibrated the magnetic field with a weak radiofrequency field driving transitions between magnetic sublevels. To measure the resonance position we use atom trap-loss spectroscopy and find the maximal loss at the central position $B_0$ of the Feshbach resonance. To estimate their width we did a thermalization experiment at different magnetic field positions and found a maximum temperature at the field position $B_0 + \Delta B$ (Extended Data Fig. 1b). Close to the maximal temperature we observed another narrow resonance that influences our width measurement. Hence, we cannot state precise values for the scattering length. In addition, we did the same investigations for the structured states on a second even narrower resonance located at $B_0 = 1.326(3)$ G with a width of $\Delta B = 8(5)$ mG.Extended Data Figure 1: Atom trap-loss spectroscopy for two Feshbach resonances. Atom-loss spectroscopy mapping Feshbach resonances of $^{164}\text{Dy}$ . The atom number (blue circles) and temperature (red diamonds) is normalized. a, Atom number minimum shows the center of the Feshbach resonance at $B_0 = 1.326(3)$ G, while the temperature is maximized at $B_0 + \Delta B$ , with $\Delta B = 8(5)$ mG. We prepared stable BECs at $B = 1.285(3)$ G (dashed black line) before we induced an instability for lower magnetic field values. b, We used for the investigations in this manuscript the shown resonance at $B_0 = 7.117(3)$ G with $\Delta B = 51(15)$ mG. Stable BECs were created at $B = 6.962(3)$ G (dashed black line).

(Extended Data Fig. 1a). We observed the same qualitative results with this resonance, showing that self-organization is independent of a particular Feshbach resonance.

In situ phase-contrast polarization imaging Phase-contrast polarization imaging was first introduced with lithium atoms²⁶ and relies on the dispersive phase shift instead of direct absorption giving rise to the optical density. We use an off-resonant beam ( $\Delta = -1.1$ GHz = $-35$ $\Gamma$ ) near the strongest optical transition at 421 nm with a linewidth of $\Gamma \approx 32$ MHz to suppress absorption. The imaging beam is linearly polarized and we apply a magnetic field of more than 1 G in the beam propagation direction so that the atoms see a mixture of left- and right-handed circularly polarized light. In our case the dysprosium atoms are fully spin-polarized in the lowest lying Zeeman state $m_J = -8$ . Thus the atoms couple mainly to the $\sigma^-$ optical transition, while the $\sigma^+$ transition is suppressed by a factor of $\sim 150$ due to the difference in the Clebsch-Gordan coefficients. Hence, the atoms show a strong circular birefringence or optical rotation. If both coupled and uncoupled polarizations are combined on a linear polarizer, the transferred intensity distribution depends on the dispersive shift of the atoms. The atomic plane is then imaged through a commercial diffraction limited objective (NA = 0.32), corrected for the upper window of our glass cell, with a resolution of $1 \mu\text{m}$ . With a second commercial objective the image is magnified by a factor of 50 and guided through a polarizer to the camera.

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Acknowledgements We would like to thank Axel Griesmaier for support at the early stage of the experiment and we thank Damir Zajec, David Peter, Hans Peter Büchler and Luis Santos for discussions. This work is supported by the German Research Foundation (DFG) within SFB/TRR21. H.K. acknowledges support by the 'Studienstiftung des deutschen Volkes'.

Author Contributions All authors discussed the results, made critical contributions to the work and contributed to the writing of the manuscript.