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Superfluid behaviour of a two-dimensional Bose gas

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Abstract

Owing to thermal fluctuations, two-dimensional (2D) systems cannot undergo a conventional phase transition associated with the breaking of a continuous symmetry1. Nevertheless they may exhibit a phase transition to a state with quasi-long-range order via the Berezinskii–Kosterlitz–Thouless (BKT) mechanism2. A paradigm example is the 2D Bose fluid, such as a liquid helium film3, which cannot condense at non-zero temperature although it becomes superfluid above a critical phase space density. The quasi-long-range coherence and the microscopic nature of the BKT transition were recently explored with ultracold atomic gases4,5,6. However, a direct observation of superfluidity in terms of frictionless flow is still missing for these systems. Here we probe the superfluidity of a 2D trapped Bose gas using a moving obstacle formed by a micrometre-sized laser beam. We find a dramatic variation of the response of the fluid, depending on its degree of degeneracy at the obstacle location.

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‘Flow without friction’ is a hallmark of superfluidity7. It corresponds to a metastable state in which the fluid has a non-zero relative velocity v with respect to an external body such as the wall of the container or an impurity. This metastable state is separated from the equilibrium state of the system (v = 0) by a large energy barrier, so that the flow can persist for a macroscopic time. The height of the barrier decreases as v increases, and eventually passes below a threshold (proportional to the thermal energy) for a critical velocity vc. The microscopic mechanism limiting the barrier height depends on the nature of the defect and is associated with the creation of phonons and/or vortices7. Whereas the quantitative comparison between experiments and theory is complicated for liquid 4He, cold atomic gases in the weakly interacting regime are well suited for precise tests of many-body physics. In particular, superfluidity was observed in 3D atomic gases by stirring a laser beam or an optical lattice through bosonic8,9,10,11,12 or fermionic13 fluids and by observing the resulting heating or excitations. Here we transpose this search for dissipation-less motion to a disc-shaped, non-homogeneous 2D Bose gas. We use a small obstacle to locally perturb the system. The obstacle moves at constant velocity on a circle centred on the cloud, allowing us to probe the gas at a fixed density. We repeat the experiment for various atom numbers, temperatures and stirring radii and identify a critical point for superfluid behaviour.

Our experiments are performed with 2D Bose gases of N = 35,000–95,000 87Rb atoms confined in the vertical direction by the harmonic potential W (z) and in the horizontal plane by the radially symmetric harmonic potential V (r) (see ref. 14). The trap frequencies are ωr/2π = 25.0(5) Hzin the horizontal plane and ωz/2π = 1.4(1) kHz in the vertical direction. We use gases with temperature T and central chemical potential μ in the range 65–120 nK and kB×(35–60) nK, respectively, where kB is the Boltzmann constant. The interaction energy per particle is given by , where n is the 2D spatial density (typically 100 atoms μm−2 in the centre), m the atomic mass, the dimensionless interaction strength and is Planck’s constant divided by 2π. Here , where a = 5.3 nm is the 3D scattering length and . The energy ωz (kB×70 nK) is comparable to kBT and Uint (kB×40 nK at the trap centre). Thanks to Bose statistics, which limits to typically 10% the fractional atomic density in the axially excited states at the obstacle position, our gas is well described by the quasi-2D fluid model (see Supplementary material of ref. 14).

We stir the cloud with a laser beam which creates a repulsive potential with height V stirkB×80 nK. This is at least twice the local chemical potential μloc(r) = μV (r). The beam has a Gaussian profile with a waist of w0 = 2.0(5) μm, which is larger than the local healing length  (≈0.3 μm at the trap centre), but small compared to the size of the cloud (full width at half maximum ≈25 μm) (see Fig. 1). We stir for typically tstir = 0.2 s at constant velocity v in a circle of radius r centred on the cloud. The intensity of the stirring beam is ramped on and off in ≈5 ms without any significant additional heating. Once the stirring beam is switched off, we let the cloud relax for 0.1 s and measure the temperature Tf.

Figure 1: Stirring a 2D Bose gas.
figure1

a, A trapped 2D gas of 87Rb atoms is perturbed by a focussed laser beam, which moves at constant velocity on a circle centred on the cloud. The stirring beam has a frequency greater than the 87Rb resonance frequency (‘blue detuning’ of ≈2 nm) and thus creates a repulsive potential which causes a dip in the density profile. b, The stirring beam is focussed onto the 2D cloud via a microscope objective of numerical aperture 0.45, which is also used for imaging. We overlap the two beam paths with a polarizing beam splitter (PBS) cube. The position of the stirring beam is controlled by a two-axis piezo-driven mirror. c, In situ false-colour image of the 2D cloud in the presence of the laser beam (average over six images). From the dip in the density we deduce the waist of the laser beam as w0 = 2.0(5) μm. In this image, the intensity of the beam is chosen three times higher than in the stirring experiment to make the hole clearly visible even in the centre of the cloud. We use similar images, but with the stirring beam switched off, to determine the temperature T and the chemical potential μ from a fit of the Hartree–Fock prediction to the wings of the cloud14.

For each configuration (N, T, r), we repeat this experiment for various v from 0 to 2 mm s−1 and a fixed stirring time tstir. We find two different regimes for the response and we show an example of each in Fig. 2. In Fig. 2a, there is a clear threshold behaviour with no discernable dissipation below a critical velocity. In contrast, in Fig. 2b, the temperature increases without a threshold. We identify these behaviours as the superfluid and normal response, respectively. To model these data we choose for a given configuration the fit function

which describes the heating of a 2D superfluid in the presence of a moving point-like defect15. In equation (1) the three fit parameters are the temperature at zero velocity Tf,0, the heating coefficient κ, and the critical velocity vc. In the normal state, the fit gives vc0 and the according quadratic heating stems from the linear scaling of the drag force. In the absence of the stirring beam, there is no significant heating and we measure the temperature Ti. The presence of the stirring beam at zero velocity leads to a ‘background heating’ Tf,0Ti10 nK, which we attribute to photon scattering. In the following, we use the mean temperature to characterize the cloud.

Figure 2: Evidence for a critical velocity.
figure2

Two typical curves of the temperature after stirring the laser beam at varying velocities. a, In the superfluid regime, we observe a critical velocity (here vc = 0.87(9) mm s−1), below which there is no dissipation. b, In the normal regime, the heating is quadratic in the velocity. The fitted heating coefficients are κ = 18(3) nK s mm−2 and κ = 26(3) nK s mm−2 in a and b, respectively. The experimental parameters are , kB×59 nK, 14.4 μm) and (38,000, 67 nK, kB×39 nK, 16.6 μm) for a and b, respectively, yielding and . The data points are the average of typically ten shots. The y error bars show the standard deviation. The x error bars denote the spread of velocities along the size of the stirring beam ( radius). The solid line is a fit to the data according to equation (1). The stirring time is 0.2 s for all data points. Note that the three low-lying data points in a correspond to the completion of an odd number of half turns. For these data points, where we see a downshift of the temperature by approximately 1.5 nK, we also observe a displacement of the centre of mass of the cloud by a few micrometres. c,d, Calculated radial density distribution for the clouds in a and b, respectively. The dashed blue curve shows the superfluid density, the solid red curve shows the normal density. The stirring beam potential is indicated by the grey shaded area (in arbitrary units). The densities are calculated via the local density approximation from the prediction for an infinite uniform system16. The jump of the superfluid density from zero to a universal value of 4/λdB2 (where λdB is the thermal de Broglie wavelength) is a prominent feature of the BKT transition. The normal density makes a corresponding jump to keep the total density continuous.

In Fig. 3, we summarize our data obtained for different configurations (N, , r). We show in Fig. 3a the fitted critical velocities versus the single parameter . The relevance of this parameter results from two points. First, because of the local character of the excitation, the response of the fluid to the moving perturbation is expected to be similar to that of a uniform gas with the same temperature and the chemical potential μloc. Second, the scale invariance of the weakly-interacting 2D Bose gas implies that the thermodynamic properties do not depend separately on μ and T, but only on the ratio μ/kBT (see refs 14, 16, 17). In particular, this ratio is univocally related to the phase space density, and thus characterizes the degree of degeneracy of the cloud.

Figure 3: Superfluid behaviour across the BKT transition.
figure3

a, The critical velocities vc obtained from the curves as in Fig. 2 plotted versus the single parameter , which is the relevant quantity due to the scale invariance of the weakly-interacting 2D Bose gas. Our data show a threshold between critical velocities compatible with zero and clearly non-zero critical velocities. It is located at (dashed line), somewhat above the prediction for the BKT transition in an infinite uniform system16 (the grey shaded area indicates the normal state by this prediction). The inset to a shows the critical velocity plotted versus the stirring radius r. Owing to the different atom numbers and temperatures of the clouds, we can find superfluid or normal behaviour for the same radius. b, The heating coefficient κ as a function of for the normal data (red circles) and the superfluid data (blue circles). The red solid line shows a fit of κ linear in the normal density, as expected from a single-particle model. The blue dashed line shows an empirical fit quadratic in the superfluid density. The calculation for the densities assumes and the densities are averaged over the size of the stirring beam. The x error bars indicate the region of that is traced by the stirring beam due to its size (using the width of the beam) and due to the ‘background heating’. The y error bars are fitting errors.

Remarkably, the ensemble of our data for vc when plotted as a function of shows a threshold between values compatible with zero and clearly non-zero values. This threshold is located at , above the prediction for the superfluid phase transition in a uniform system16 with . If we assume that the stirrer must stand entirely in the superfluid core to yield a non-zero critical velocity, then the deviation can be attributed to the non-zero width of the stirring beam. The range of corresponding to the extent of this beam is indicated by the horizontal error bars in Fig. 3a. Note that the finite size of our trapped atomic clouds might also shift the BKT transition, but the effect is expected to be small (a few per cent) and in the opposite direction16.

We limit the presented stirring radii to r≥10 μm such that the stirring frequencies ω = v/r for the relevant velocities vvc are well below ωr. Indeed, smaller radii correspond to a larger centripetal acceleration. This could lead to additional heating via the phonon analog of synchrotron radiation, as observed in the formally similar context of capillary waves generated by a rotating object18.

For a homogeneous system, the value of the critical velocity is limited by two dissipation mechanisms, the excitation of phonons or vortices. For a point-like obstacle15, phonon excitation dominates and vc is equal to the speed of sound, given in the zero-temperature limit by (≈1.6 mm s−1 for n = 50 atoms μm−2) (this situation is described by the celebrated Landau criterion7). When the obstacle size w0 increases and becomes comparable to ξ, dissipation via the nucleation of vortex–antivortex pairs (vortex rings in 3D) becomes significant19,20,21. The corresponding vc is then notably reduced with respect to cs. In the limit of very large obstacles (w0ξ), an analytical analysis of the superfluid flow stability yields vc/m w0cs (see refs 22, 23). With an obstacle size w0ξ, our experimental situation is intermediate between these two asymptotic regimes. For a non-homogeneous system such as ours with the stirring obstacle close to the border of the expected superfluid regime, one can also excite surface modes24,25, which constitute a further dissipation mechanism.

Our measured critical velocities are in the range 0.5–1.0 mm s−1, that is, vc/cs = 0.3–0.6. By contrast, previous experiments in 3D clouds found lower fractions, vc/cs0.1 (see ref. 9). The difference may be due to the larger size of the obstacles that were used, and to the average along the axis of the stirring beam of the density distribution in the 3D gas26. The dominant dissipation mechanism could be revealed, for example, by directly observing the created vortex pairs as in ref. 12 or interferometrically detecting the Cerenkov-like wave pattern for v>cs as in experiments with a non-equilibrium 2D superfluid of exciton–polariton quasi-particles27.

Figure 3b shows the fitted heating coefficients κ for the normal (red circles) and superfluid data (blue circles). In the normal region, we expect the heating to scale linearly with the normal density nno (see ref. 10). Using the prediction of ref. 16 for (averaged over the size of the stirring beam) we fit and obtain a1≈3×10−6 nK s. This value is in reasonable agreement with the prediction of a model10 of a single particle with a thermal velocity distribution of mean colliding with a moving hard wall of width L = w0, yielding (for N = 65,000 and T = 90 nK). In particular our data nicely reproduce the maximum of around the expected superfluid transition point. In the superfluid case and v>vc, we empirically fit a quadratic scaling of the heating with density κ = a2· nSF2 and find a2 = 8×10−9 nK s μm2. In principle, one could develop a more refined model to describe the superfluid region, by taking into account the coexistence of the normal and superfluid states via the sum of two heating terms. However, within the accuracy of our data, we did not find any evidence for the need of such a more refined description.

We have presented a direct proof of the superfluid character of a trapped 2D Bose gas. An interesting extension of our work would be the study of superfluidity from the complementary point of view of persistent currents, by adapting to 2D the pioneering experiments performed in 3D toroidal traps28,29,30.

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Acknowledgements

We would like to thank I. Carusotto, M. Holzmann, S. Nascimbéne, H. Perrin, L. Pitaevskii and W. Zwerger for helpful discussions. We acknowledge funding by IFRAF, ANR (project BOFL), the Alexander von Humboldt foundation (C.W.) and DGA (L.C.). Laboratoire Kastler Brossel is a research unit of École normale supérieure and Université Pierre et Marie Curie, associated with CNRS.

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The experiments were carried out by R.D., L.C., T.Y. and C.W.; the stirring laser set-up was developed by J.L. and J.B.; C.W., R.D. and L.C. analysed the data; C.W. wrote the manuscript with the input from all coauthors; J.B. and J.D. planned and supervised the project.

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Correspondence to Christof Weitenberg.

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Desbuquois, R., Chomaz, L., Yefsah, T. et al. Superfluid behaviour of a two-dimensional Bose gas. Nature Phys 8, 645–648 (2012). https://doi.org/10.1038/nphys2378

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