Abstract
Superfluidity in its various forms has been of interest since the observation of frictionless flow in liquid helium II1,2. In three spatial dimensions it is conceptually associated with the emergence of long-range order at a critical temperature. One of the hallmarks of superfluidity, as predicted by the two-fluid model3,4 and observed in both liquid helium5 and in ultracold atomic gases6,7, is the existence of two kinds of sound excitation—the first and second sound. In two-dimensional systems, thermal fluctuations preclude long-range order8,9; however, superfluidity nevertheless emerges at a non-zero critical temperature through the infinite-order Berezinskii–Kosterlitz–Thouless (BKT) transition10,11, which is associated with a universal jump12 in the superfluid density without any discontinuities in the thermodynamic properties of the fluid. BKT superfluids are also predicted to support two sounds, but so far this has not been observed experimentally. Here we observe first and second sound in a homogeneous two-dimensional atomic Bose gas, and use the two temperature-dependent sound speeds to determine the superfluid density of the gas13,14,15,16. Our results agree with the predictions of BKT theory, including the prediction of a universal jump in the superfluid density at the critical temperature.
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The hydrodynamic two-fluid theory4 models a fluid below the critical temperature, Tc, as a mixture of a superfluid component and a viscous normal component that carries all the entropy, and assumes that the two are in local thermodynamic equilibrium. The two sounds then correspond to different variations of the total density and the entropy per particle. In three dimensions in the nearly incompressible liquid helium, the higher-speed first sound is a pure density wave and the lower-speed second sound is a pure entropy wave; more generally, both sounds can involve variations in both density and entropy17. At temperatures greater than Tc, the normal fluid supports just the first-sound density wave, and so the appearance of the second sound mode is a clear manifestation of superfluidity.
The two-sound phenomenology is also expected to hold for two-dimensional superfluids, despite the unconventional nature of the infinite-order BKT phase transition. However, in liquid-helium films, in which the BKT transition was first observed18, the propagation of both first and second sound is inhibited because the viscous normal component is pinned by the substrate. In two-dimensional atomic gases, in which many complementary BKT experiments have been performed19,20,21,22,23,24,25,26,27,28,29,30,31, only one sound mode has been seen so far. In a weakly interacting Bose gas, collisionless sound was observed31,32,33 (see also ref. 34) and showed no discontinuity at Tc, whereas in a strongly interacting Fermi gas, one pure density mode was observed35 at temperatures well below Tc.
Here we observe both first and second sound in the long-wavelength density response of a homogeneous two-dimensional Bose gas to an external driving force (Fig. 1a). In our 39K gas, which has a surface density n ≈ 3 μm−2 and is characterized by a dimensionless interaction strength24,36 \(\tilde{g}\) = 0.64(3) (all uncertainties correspond to one standard deviation), the elastic collision rate is sufficiently high for collisional hydrodynamic behaviour. Specifically, near Tc it is about four times larger than the observed first-sound (angular) frequency ω1; for comparison, in the experiment of ref. 31, for an 87Rb gas with a similar geometry and larger n (~50 μm−2) but smaller \(\tilde{g}\) (0.16), the elastic collision rate and the expected16 ω1 were approximately equal near the critical point. At the same time, the compressibility of our gas near Tc is sufficiently high for our driving force to excite both sounds effectively16,17.
Our homogeneous two-dimensional gases are prepared in a node of a vertical one-dimensional optical lattice (Fig. 1b, green) with harmonic trap frequency ωz/(2π) = 5.5(1) kHz. They are deep in the two-dimensional regime; both the interaction and thermal energy per particle are below 0.3ħωz, where ħ is the reduced Planck’s constant. In the x–y plane, we confine the atoms to a rectangular box of size Lx × Ly and potential-energy wall height U0 using a hollow laser beam (Fig. 1b, red); we tune U0/kB, where kB is the Boltzmann constant, to between 100 and 300 nK to vary the gas temperature T (Methods). We control the interaction strength \(\tilde{g}=\sqrt{8{\rm{\pi }}m{\omega }_{z}/\hbar }a\), where m is the atom mass and a the scattering length24, via a magnetic Feshbach resonance37 at 402.7 G. Our value of \(\tilde{g}\) corresponds to a relatively high a = 522(23)a0, where a0 is the Bohr radius, which results in noticeable three-body losses; during the measurements (such as those in Fig. 1c) the sample slowly decays (without heating), but n stays within 15% of its average value and the data are fitted well by assuming a steady-state oscillation.
The driving force Fy = F0 sin(ωt) is aligned along the y direction, spatially uniform, and oscillating sinusoidally in time, t. It excites the longest-wavelength sound mode(s)38, with wavevector q = π/Ly. The resulting density perturbation δn(y, t) is proportional to sin(πy/Ly), with y = 0 in the box centre, and we define its oscillating amplitude b(t) such that δn(y, t)/n = b(t) sin(πy/Ly). The corresponding displacement of the centre of mass of the cloud is then d(t) = 2b(t)Ly/π2. We choose the driving-force amplitude F0 so that even for resonant ω the maximal d(t) is a few per cent of Ly (Fig. 1c) and fit d(t) = R(ω)sin(ωt) − A(ω)cos(ωt), which gives the reactive (R) and absorptive (A) response.
We focus on A(ω), which is proportional to the imaginary part of the density response function (Methods) and reveals the phase transition. From energy conservation, namely the equality of the excitation energy and the energy pumped into the system, it follows that the response spectrum A(ω) must satisfy the f sum rule39
with different excitation modes contributing to fsum according to their oscillator strengths. Experimentally verifying that fsum = 1 ensures that all modes have been observed. In our case, below Tc the excitation spectrum should consist of first and second sound, with contributions f1 and f2, respectively, to fsum. Above Tc the second-sound mode vanishes and a diffusive heat mode should appear; this mode also couples to the density and contributes fdiff to fsum (Methods). From A(ω) we also obtain the dynamical structure factor17 S(q, ω) = πq2kBTA(ω)/(8ωF0), which reveals the qualitative difference between propagating sound modes and the diffusive mode40,41—for the former, S(ω) has a maximum at non-zero ω, whereas for the latter, the maximum occurs at ω = 0.
Figure 2a, b shows the different responses of the system below and above the critical temperature. Here we express all results in dimensionless form, using the Bogoliubov frequency ωB = cBq, where \({c}_{{\rm{B}}}=\hbar \sqrt{n\tilde{g}}/m\) ≈ 2.3 mm s−1 is the Bogoliubov sound speed. Specifically, we define \(\tilde{A}={\rm{\pi }}m{\omega }_{{\rm{B}}}^{2}A/(8{F}_{0})\), so \({f}_{{\rm{sum}}}=\int {\rm{d}}\omega \,\omega \tilde{A}/{\omega }_{{\rm{B}}}^{2}\) and \(S={k}_{{\rm{B}}}T\tilde{A}/(m{c}_{{\rm{B}}}^{2}\omega )\); for the fitting procedure, see Methods. In Fig. 2a, the two resolved resonances observed below Tc correspond to the first and second sound, and the resonance frequencies ω1 and ω2 < ω1 give the sound speeds c1,2 = ω1,2/q. Above Tc, the sole resonance is attributed to the first sound, whereas the low-frequency ‘shoulder’ is attributed to the diffusive mode that replaces the second sound. Figure 2b shows the corresponding S(ω), and the inset highlights the difference between second sound (c2 > 0) and the diffusive mode corresponding to c2 = 0. The width of the diffusive mode gives the thermal diffusivity DT = 5(2)ħ/m. With a caveat that our sound resonances might be broadened by the loss-induced density drift, their widths imply sound diffusivities Ds,1 = 7(1)ħ/m and 10(2)ħ/m for the first sound below and above Tc, respectively, and Ds,2 = 6(1)ħ/m for the second sound below Tc. For comparison, sound diffusivities that are several times lower, at approximately ħ/m, were observed in strongly interacting two-dimensional35 and three-dimensional42 Fermi gases.
In Fig. 2c we plot the fitted fsum for a range of temperatures, showing that it always satisfies the f sum rule. We also show how f2 vanishes with increasing T, a finding that we used to experimentally identify Tc. In absolute terms, including our systematic uncertainties in n and T (Methods), Tc = 42(4) nK for n = 3.0(5) μm−2, which is compatible with the BKT prediction13 \({T}_{{\rm{c}}}=2{\rm{\pi }}n{\hbar }^{2}/[m{k}_{{\rm{B}}}\,\mathrm{ln}(380/\mathop{g}\limits^{ \sim })]\) = 37(6) nK. In a finite-size system the transition is rounded-off into a crossover19,43,44,45 (see also refs. 18,34 for non-zero-ω effects), and for our system parameters this could indeed increase Tc by approximately 10%; however, within our errors this shift is not conclusive.
Figure 3a displays the temperature-dependent speeds of the first and second sound, normalized to cB. The solid lines represent values predicted according to infinite-system theory15,16 without any free parameters, and are generally in good agreement with the data. Crucially, these theoretical results rely on both thermodynamic calculations and the predictions of the superfluid density11,12,14. The relevant thermodynamic calculations were performed in ref. 14 assuming \(\tilde{g}\) ≪ 1; however, previous experiments have verified their applicability for a wide range of interaction strengths22,23,26, including the one used in this work. Regarding the superfluid density ns, a key prediction of the infinite-system BKT theory is that it exhibits a jump12 at Tc from 0 to 4/λ2, where λ is the thermal wavelength. For our value of \(\tilde{g}\), the most pronounced effect of this jump is a discontinuity in c2 of about 0.45cB, with less pronounced discontinuities in c1 and f2 (see Fig. 2c)15. For our data at T/Tc < 0.75, where the expected superfluid fraction is close to 100% and c2 ≈ cB, the predictions for c1 are not reliable because they are very sensitive to the exact value of the vanishingly small normal-component density16.
Finally, instead of comparing the measured and predicted sound speeds, we can combine our measurements of c1,2 and the previously verified22,23,26 thermodynamic calculations14 to deduce ns and compare it directly with theory12,14 (Methods); a similar procedure was previously used for a strongly interacting three-dimensional Fermi superfluid7. In Fig. 3b, we plot the deduced superfluid phase-space density \({{\mathcal{D}}}_{{\rm{s}}}={n}_{{\rm{s}}}{\lambda }^{2}\) versus \({\mathcal{D}}/{{\mathcal{D}}}_{{\rm{c}}}\), where \({\mathcal{D}}=n{\lambda }^{2}\) is the total phase-space density and \({{\mathcal{D}}}_{{\rm{c}}}\) is its critical value. The solid line is the theoretical prediction14 without any free parameters, which shows the universal, \(\tilde{g}\)-independent jump of \({{\mathcal{D}}}_{{\rm{s}}}\) from 0 to 4 at the critical point.
Our experiments establish the applicability of the two-fluid model to unconventional BKT superfluids. We provide the first—to our knowledge—measurement of the superfluid density in an atomic two-dimensional gas, and experimentally demonstrate the predicted universal jump at the critical point. Our measurements also extend into the low-temperature regime, in which a complete theoretical picture is not yet available. An experimental challenge for future work is to explore even lower temperatures, at which hybridization of the first and second sound is expected46,47. More generally, establishing measurements of the superfluid density in two-dimensional quantum gases provides an invaluable diagnostic tool for many future studies, including exploration of non-equilibrium phenomena and of the effects of disorder on superfluidity.
Methods
Optical confinement of the two-dimensional gas
The one-dimensional optical lattice and the rectangular hollow beam are blue-detuned from the atomic resonance, and create repulsive potentials for the atoms. Both are shaped using digital micromirror devices (DMDs). The hollow beam providing the in-plane confinement has a wavelength of 760 nm and is created by direct imaging of a DMD pattern. The vertical one-dimensional lattice is made of 532-nm light and is created by Fourier imaging of a DMD pattern, which allows dynamical tuning of the lattice period Δz. Specifically, using a DMD we create two horizontal light strips, each of width corresponding to 50 micromirror pixels, separated vertically by ΔZ, so their interference in the Fourier plane creates a lattice with Δz ∝ 1/ΔZ. We additionally impose a phase shift of π between the two interfering beams, which places the central node of the symmetric interference pattern at z = 0 independently of the varying Δz. To dynamically change Δz we shift the DMD pattern pixel by pixel (moving the two light strips symmetrically in opposite directions) in 25-ms steps. We start with a large Δz = 18.5 μm to load a three-dimensional gas, pre-cooled as in ref. 48, into a single lattice node. We then reduce Δz over a period of 1.5 s to 3.3 μm in order to compress the gas into the two-dimensional geometry. In the final two-dimensional configuration, the lattice depth around the central node is Uz ≈ 3.0 μK, giving the trap frequency \({\omega }_{z}/(2{\rm{\pi }})={(\Delta z)}^{-1}\sqrt{{U}_{z}/(2m)}\) = 5.5 kHz.
Calibration of the experimental parameters
Our absorption imaging system, used to measure the cloud density n, is calibrated with a systematic uncertainty of 15% using measurements of the critical temperature for Bose–Einstein condensation in a three-dimensional harmonic trap49; this calibration also agrees with an independent one that is based on the rates of the density-dependent three-body decay50. We assess the absolute gas temperature with a systematic uncertainty of 10% using measurements of the scale-invariant two-dimensional equation of state14,22,23,26, as in ref. 31; we have made equation of state measurements for several trap depths U0 and also different trap dimensions Lx and Ly, which show linear dependence of T on U0. The wavevector q = π/Ly is determined using in situ absorption images (such as the one shown in Fig. 1b), with a systematic 5% error due to the fact that the cloud edges are not infinitely sharp; the half-wavelength of the density modulation closely corresponds to the length of the region in which the density is greater than 90% of its value in the bulk. The driving-force amplitude F0 is calibrated with an error of 5% by applying a constant force on a cloud released from the trap and measuring the resulting centre-of-mass acceleration.
Response function, f sum rule and S(ω)
The density response function is defined in Fourier space as χnn(q, ω) = δn(q, ω)/δU(q, ω), where δU(q, ω) is the driving potential. Our monochromatic and spatially uniform driving force corresponds to a potential −F0 y sin(ωt) for −Ly/2 ≤ y ≤ Ly/2, and Fourier decomposition of this gives δU(q = π/Ly, ω) = −4F0Ly/π2. Following our definition of A(ω) gives Im[χnn(q = π/Ly, ω)] = −π2nq2A(ω)/(8F0). Inserting this into the conventional form of the f sum rule39,
gives the dimensionless sum rule in equation (1), which is insensitive to uncertainties and variations in n and q. Our dimensionless results in Fig. 2c (and also Fig. 3) are also not affected by changes in F0; for these measurements we have varied F0 by a factor of 3. We have also varied q by about 50%, by using two different box sizes of (Lx, Ly) ≈ (21 μm, 33 μm) and (56 μm, 23 μm). The dynamical structure factor is (for kBT ≫ ħω, which is always satisfied in our experiments) given by S(q, ω) = −kBT Im[χnn(q, ω)]/(πnω), which is equivalent to the form given in the text in terms of A(ω).
Fits of the response spectra
In the two-fluid model, ignoring dissipation, the density response function is17
with the two poles giving the sound speeds c1,2, and Z1 + Z2 = 1 to satisfy the f sum rule. Including linear damping51, we fit the experimental spectra with A(ω) = A1(ω) + A2(ω), where
Here the amplitudes x1,2, resonance frequencies ω1,2 (with ω1 > ω2) and damping rates Γ1,2, are fit parameters, and the sound diffusivities are then given by Γ1,2/q2. For consistency, we first apply the same fit to the data taken at all temperatures, and find that it always captures the data well and gives fsum ≈ 1 (Fig. 2c). The first sound is always underdamped and the A1 term gives f1, its contribution to fsum. For the spectra identified as being below Tc (as in the top panel of Fig. 2a), the fit gives that the second sound is also underdamped, and its contribution to S(ω) peaks at a non-zero ω. In this case A2 gives the non-zero f2 contribution to fsum. For the data identified as being above Tc (as in the bottom panel of Fig. 2a), the second term in the fit function shows that this mode is overdamped, and its contribution to S(ω) peaks at ω = 0. This demonstrates, in an unbiased way, that the second sound is replaced by the diffusive mode. In this case the A2 term gives fdiff, the diffusive-mode contribution to fsum, with f1 + fdiff ≈ 1, while f2 = 0. Note that approximating the diffusive mode by a δ-function response17 at ω = 0 gives fdiff = 0, but in reality this mode has a non-zero contribution to the experimental fsum shown in Fig. 2c (top). To quantitatively assess the thermal diffusivity DT, following refs. 51,52 we refit the data for T > Tc with A(ω) = A1(ω) + AT(ω), where
corresponds to a contribution to S(ω) that is a Lorentzian centred at ω = 0, and gives DT = ΓT/q2.
Sound speeds and the superfluid density
The theoretical predictions for the two sound speeds, c1 and c2, are the solutions of the quartic equation for c:
where \({c}_{10}^{2}\) = 1/(mnκS) and \({c}_{20}^{2}\) = Ts2ns/[mcV(n − ns)]; here cV is the specific heat per particle at constant volume, s is the entropy per particle and γ = κT/κS ≥ 1 is the ratio of the isothermal and isentropic compressibilites17,51. As detailed in equations (2) and (3) of ref. 15, all the relevant thermodynamic quantities are linked to the phase-space density calculated in ref. 14 and experimentally verified in refs. 22,23,26. Crucially, in addition to these thermodynamic quantities, c20 explicitly depends on the superfluid density ns. For γ → 1, which is the case in nearly incompressible superfluids such as three-dimensional liquid helium and unitary Fermi gases, c1 → c10 and c2 → c20, so to a good approximation only c2 depends on ns. However, when γ is clearly distinct from 1, which is typically the case for a Bose gas near Tc, both sound speeds depend on ns; for our value of \(\tilde{g}\), we find γ ≈ 1.6 at the critical temperature16. The theoretical speeds14,15,16 in Fig. 3a are based on the previously verified thermodynamic predictions and the hitherto unverified predictions for ns. Conversely, we can use our measured c1 and c2 (for T < Tc), which together give c20, and the established thermodynamic calculations to deduce ns; this gives the results shown in Fig. 3b. Note that owing to the scale-invariance in two dimensions22,23, for a given \(\tilde{g}\) both the superfluid fraction and all the thermodynamic quantities depend only on T/Tc.
Data availability
The data that support the findings of this study are available in the Apollo repository (https://doi.org/10.17863/CAM.66056). Any additional information is available from the corresponding authors upon reasonable request. Source data are provided with this paper.
Change history
02 August 2021
A Correction to this paper has been published: https://doi.org/10.1038/s41586-021-03746-2
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Acknowledgements
We thank J. Man for experimental assistance; and R. P. Smith, J. Dalibard, M. Zwierlein, R. J. Fletcher, T. A. Hilker, S. Nascimbene and T. Yefsah for discussions. This work was supported by the EPSRC (grant nos EP/N011759/1 and EP/P009565/1), ERC (QBox) and QuantERA (NAQUAS, EPSRC grant no. EP/R043396/1). J.S. acknowledges support from Churchill College (Cambridge), and Z.H. acknowledges support from the Royal Society Wolfson Fellowship.
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P.C. led the data acquisition and analysis. M.G. and J.S. contributed to the data acquisition. P.C., M.G., R.L. and J.S. contributed to the experimental setup. P.C., N.D. and J.S. produced the figures. Z.H. supervised the project. All authors contributed to the data analysis, interpretation of the results and writing of the manuscript.
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Christodoulou, P., Gałka, M., Dogra, N. et al. Observation of first and second sound in a BKT superfluid. Nature 594, 191–194 (2021). https://doi.org/10.1038/s41586-021-03537-9
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