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Connecting Berezinskii-Kosterlitz-Thouless and BEC phase transitions by tuning interactions in a trapped gas

We study the critical point for the emergence of coherence in a harmonically trapped two-dimensional Bose gas with tuneable interactions. Over a wide range of interaction strengths we find excellent agreement with the classical-field predictions for the critical point of the Berezinskii-Kosterlitz-Thouless (BKT) superfluid transition. This allows us to quantitatively show, without any free parameters, that the interaction-driven BKT transition smoothly converges onto the purely quantum-statistical Bose-Einstein condensation (BEC) transition in the limit of vanishing interactions.

(1) This ideal-gas BEC transition is similar to the familiar condensation in 3d; it is purely statistical and follows from Einstein's standard argument of the saturation of the excited states. However, the importance of dimensionality emerges if interactions are introduced. In 3d, ideal-gas BEC occurs when in the trap centre D ≈ 2.612; weak interactions slightly shift the critical atom number [29][30][31], but do not alter the BEClike nature of the transition. In 2d, while N 0 c is finite, the phase-space density required in the trap centre for the excited states to saturate is infinite, just as in a uniform system [13]. For anyg > 0 this is unattainable and the excited states can accommodate any number of particles [32]. The BEC transition is thus suppressed and one expects it to be replaced by the BKT transition with non-infinite D BKT [9].
These arguments suggest that the two conceptually very different phase transitions, the interaction-driven BKT and the saturation-driven BEC, are in fact continuously connected as g → 0 [9,33]. The harmonic trapping potential offers the opportunity to experimentally observe this unification of BKT and BEC physics. While in an infinite uniform gas no transition occurs forg = 0, in a harmonic trap a transition always occurs at a finite N c and always results in significantly extended coherence of the gas. The nature of this transition is quantitatively encoded in the value of N c . This picture is supported by theoretical calculations of the critical atom number N BKT c for a harmonically trapped gas [9,34], which suggest that N BKT c smoothly connects to N 0 c in the limitg = 0. Various signatures of a phase transition, including emergence of extended coherence [14,[16][17][18]20] and superfluidity [23], have been observed in trapped 2d gases with different specificg values. On the other hand, systematic studies with tuneableg have focussed on in-situ measurements of the equation of state [21,26], which do not directly reveal any striking signatures of the infinite-order BKT transition.
In this Letter, we systematically study the critical point for the emergence of extended coherence in a harmonically trapped 2d Bose gas over a wide range of interaction strengths, 0.05 <g < 0.5. We show, without any free parameters, that N c generally agrees very well with the beyond-mean-field calculation of N BKT c [34], and converges onto N 0 c of Eq. (1) as g → 0. The critical chemical potential, µ c , which directly reveals uniform-system conditions for a phase transition to occur in the trap centre, also agrees with the BKT theory and converges onto the BEC value, µ c = 0, forg → 0. Our measurements also reiterate the importance of the suppression of density fluctuations in the normal state near the BKT critical point, previously observed in [17,18,[20][21][22].
The experiment was carried out using a 39 K gas, in the apparatus described in [35]. For 2d trapping, the tight axial (vertical) confinement is provided by two repulsive "blades" of blue-detuned light, formed by passing a 532-nm gaussian beam through a 0-π phase-plate [19], while a red-detuned 1064-nm dipole trap provides the in-plane (horizontal) con-finement. The radial and axial trapping frequencies are (ω r , ω z ) ≈ 2π × (38, 4100) Hz. For all our measurements T ∈ [140 nK, 190 nK] and µ/k B < 100 nK, resulting in a small (< 30%) occupation of the excited axial states. The interaction strength,g = √ 8πa/ z [13], where a is the s-wave scattering length and z = h/(mω z ), is controlled via a Feshbach resonance centred at 402.5 G [35,36].
To characterise long-range coherence of a gas we study its (in-plane) momentum distribution n(k) [18]. A change in the functional form of g 1 (r) leads to a dramatic change in its values at distances much larger than the thermal wavelength λ = h/ √ 2πmk B T [13], and an increase of coherence over some large distance L manifests itself in enhanced population of the low-momentum states k < ∼ 2π/L. Thus, unlike the intrap density distribution, which varies very smoothly through the BKT critical point [17,18,21,22], n(k) can provide a dramatic signature of the phase transition [18].
As illustrated in Fig. 1, to identify the critical point for a giveng, we start with a highly coherent 2d gas and measure n(k) after holding the cloud in the trap for a variable time t. During the hold time, the atom number N slowly decays through various inelastic processes [37], while the elastic-collision rate (≈ 0.1 Ng 2 s −1 ) remains sufficiently high to ensure that the gas is in quasi-static equilibrium. To measure n(k), we employ the "momentum focussing" technique [18,28,38,39]: we turn off just the tight z confinement, so the rapid vertical expansion removes all the interaction energy on a timescale 1/ω z 1/ω r , and the subsequent horizontal evolution in the remaining in-plane harmonic potential reveals n(k) as the spatial distribution after a quarter of the trap period. We probe this distribution by absorption imaging along z [see Fig. 1(a)].
Our k-space imaging resolution, ∆k ≈ 0.4 µm −1 , sets the largest distance over which we can probe coherence to L = 2π/∆k ≈ 15 µm, which is much larger than λ ≈ 0.7 µm. To probe coherence on this lengthscale, we simply monitor the peak value of the momentum distribution, P 0 , without making any theoretical assumptions about the exact shape of n(k) at low k. To get the corresponding atom number N we do a simple summation over the image. Importantly, we eliminate the systematic error due to the uncertainty in the absorptionimaging cross section by independently calibrating our imaging system through measurements of the BEC critical point in a 3d gas [40].
In Fig. 1(b) we show a typical evolution of P 0 and N (herẽ g = 0.28). While N decays smoothly, P 0 shows two distinct regimes, which allows us to identify the critical hold time t c and the corresponding N c . We note that n(k) is narrower than a Gaussian even for N significantly below N c , indicating some coherence on a lengthscale > λ [13,17,20]. The smooth evolution of such non-Gaussian "peakiness" of n(k) does not reveal a phase transition [20], and only P 0 corresponding to L λ shows a clear change in behaviour at a well-defined N c [41]. Our large L is still small compared to the thermal diameter of the cloud, 2 k B T / (mω 2 r ) ≈ 50 µm, so the observed N c is closely linked to the occurrence of a . Extended coherence is revealed as a sharp peak in n(k). Each image is an average of three experimental realisations. (b) Evolution of the momentum-distribution peak, P0, and the smoothly decaying total atom number N . We associate the threshold-like behaviour of P0 with the critical time tc and deduce the corresponding Nc. The solid line is a heuristic piecewise fit function used to determine tc [40]. phase transition in the centre of the trap [40].
For comparisons with theory, we also fit µ and T to each n(k) image. Unlike in 3d, in 2d interactions affect n(k) appreciably even in the normal state, and near the critical point it is in general insufficient to treat them at a mean-field (MF) level. However, beyond-MF correlations primarily affect the highly-populated low-k states [34]. We restrict our fits to the high-k wings of the distribution (h 2 k 2 > 2gmk B T ), where we expect the beyond-MF effects to be small, and still carefully include the effects of interactions at a MF level [40]. Following [33], we also account for the residual thermal occupation of the axial excited states and the interaction-induced deformation of the axial eigenstates.
In Fig. 2 we summarise our measurements of the critical atom number for a wide range of interaction strengths. To compare our data with the strictly-2d theoretical calculations, we correct the observed "raw" N c by subtracting the calculated population of the excited axial states [40]. We scale this corrected critical number,N c , to the BEC critical atom number, N 0 c of Eq. (1), and plot it versusg. Our ∆klimited value of L imposes a lower bound ong for which we can reliably identify the critical point. In the absence of any phase transition, in a weakly-interacting degenerate gas We thus do not expect our measurements to reliably identify N c if 0 > L for some D < D BKT . This occurs for g < 380 λ 2 /(4πL 2 ) ≈ 0.06, indicated by the shaded area in Fig. 2 [42]. Our measurements also stop being reliable for g > ∼ 0.5; in that regime our MF temperature fits are restricted to very high k values, which are affected by the anharmonicity of the optical trap. The error bars in Fig. 2 are statistical, while the systematic uncertainty inN c /N 0 c is < ∼ 0.2 [40]. Without any free parameters, we find generally excellent agreement with the BKT prediction of Ref. [34]: which is based on fixing the phase-space density in the trap centre to D BKT and integrating a uniform-system equation of state over the trap, using classical-field results of Ref. [44]. This agreement strongly supports the interpretation that the emergence of extended coherence corresponds to the BKT transition occurring in the central region of the trap. Moreover, the agreement with Eq. (2) over a very broad range of interaction strengths, and the proximity of our lowest reliablẽ g values to zero, allow us to conclude that the critical atom number, without any free parameters, indeed smoothly converges onto the BEC result of Eq. (1).
It is instructive to also compare our data with the approximation N BKT c /N 0 c = 1 + 3gD 2 BKT /π 3 [9,11], shown by the dashed line in Fig. 2. Here, the critical phase-space density is again set to D BKT , but the suppression of bosonic fluctuations in the normal state is neglected, i.e. the density profile is calculated using MF theory with an interaction potential 2gn(r), where g = (h 2 /m)g. Our data strongly exclude this result, confirming the importance of the suppression of density fluctuations near the critical point even for our lowestg values. For a more direct comparison with the uniform-system BKT theory, we also consider the critical chemical potential for the onset of coherence. Like N c in Fig. 1, µ c is experimentally defined via the critical hold time t c . The classical-field simulations [7] predict D BKT to be reached for µ BKT c = k B T (g/π) ln (13.2/g), which reduces to the BEC prediction, µ c = 0, forg = 0.
In Fig. 3 we plotμ c = µ c /(k B T ) versusg, and again observe generally good agreement with the BKT theory (solid line), all the way down tog ≈ 0.06, i.e. very close to the expected BEC limit. The small systematic difference between the data and the theory is comparable to our systematic uncertainty inμ c , of ∼ 0.05 [40].
Finally, we note that in previous experiments [21,26], on the in-trap equation of state,μ c was deduced by defining it so as to satisfy the theoretical expectation [7,44] that the phasespace density should be a universal function of (μ −μ c )/g. Our measurements ofμ c defined through the emergence of extended coherence show that the two definitions indeed lead to very similar values.
In conclusion, by studying the critical point for the emer-gence of extended coherence in a harmonically trapped 2d gas with tuneable interactions, we have quantitatively confirmed the predictions based on BKT theory and the expected unification of BKT and BEC transitions in the limit of vanishing interactions. The in-plane harmonic potential enables the observation of this unification. However, to quantitatively study the exact functional form of the slowly decaying correlations in a BKT superfluid, in the future it would be very interesting to study coherence of a tuneable 2d gas in a uniform potential [45][46][47]. Just below T c this should reveal an interaction-strength-independent algebraic decay of the firstorder correlation function, corresponding to a universal jump in the superfluid density [48]. We thank Naaman Tammuz and Alex Gaunt for experimental contributions in the early stages of the project, Shannon Whitlock for help with image processing, and Andre Wenz, Markus Holzmann and Jean Dalibard for useful discussions. These two authors contributed equally. † rjf45@cam.ac.uk ‡ mrdsv2@cam.ac.uk Fringe removal: Optical density (OD) images are usually affected by fringes due to the temporal variations in the probe intensity. We eliminate this effect using the fringe-removal algorithm of Ockeloen et al. [1].
Imaging at high magnetic field: Near the Feshbach resonance our atoms are in the electronic ground state approximately described by |4S 1/2 , m J = 1/2, I = 3/2, m I = 3/2i, and are imaged on the transition to the excited state |4P 3/2 , m J = 3/2, I = 3/2, m I = 3/2i. However, due to the residual hyperfine mixing in the ground state manifold, the imaging transition is not perfectly cycling, and the probability of optical pumping into other ground states is ⇡ 3% per cycle. Consequently, the relation between the observed OD and the atomic column density depends on the number of photons scattered per atom, which in turn varies with the inhomogeneities of the probe beam.
To correct for this effect, we experimentally calibrate the relative variation of the observed OD(r) with the camera counts, C(r), in the reference image taken without atoms. For this, we image a 3d thermal cloud after expansion from a dipole trap. Assuming cylindrical symmetry of the cloud, any variations in OD observed along a ring concentric with the cloud centre we attribute to the variations in the probe intensity. Averaging over several images, we build, pixel by pixel, a correlation plot of OD/OD 0 versus C/C 0 , where C 0 is a fixed reference value and OD 0 the corresponding optical density. We construct such plots for various ring radii (i.e. various atomic column densities) and various (mean) probe intensities, and find that to a very good approximation the scaling of the optical density due to the optical pumping depends only on the light intensity and not on the atomic column density. We thus finally create an interpolated rescaling function F (C/C 0 ) = OD 0 /OD, and correct our images according to OD scaled (r) = OD(r) · F (C(r)/C 0 ). This relative calibration corrects for both the optical pumping and for the Doppler shifting of atoms with increasing number of scattering events.
Imaging-intensity optimisation: In our main experimental sequences we take two sets of interlaced images, under identical conditions, but with two different (mean) imagingbeam intensities: a "low" intensity I 0 ⇡ 0.08 mW/cm 2 and a "high" intensity I 1 ⇡ 0.3 mW/cm 2 (which is still well below the saturation intensity I sat ⇡ 1.8 mW/cm 2 ), for which the scaling function F is larger by a factor of 2.7. The lowintensity images are used for determining N , T and µ, while the high-intensity images are used just to determine, with a larger dynamic range, the high-OD momentum peak P 0 .

Extraction of thermodynamic parameters
Model functions: To extract µ and T from our images, we fit a model to the azimuthally-averaged momentum distributions. By constraining the fit to the wings where~2k 2 > 2mk B Tg, we ensure that interaction effects are described well within a mean-field (MF) framework [4].
The model we use includes the thermal occupation of the excited axial states and the interactions between atoms in all axial states, as in [5,6], and also the axial deconfinement of the cloud due to the repulsive interactions, as in [7]. We construct the model 2d momentum distribution for an in-plane potential V (x, y) within the 2d local density approximation. Defining the local chemical potential µ L (x, y) = µ V (x, y), and writingr = (x, y), we have where {" j }(r) are the eigenenergies of the axial states of motion, {| j (z)i}(r). Due to interactions, they are not simplỹ ! z (j + 1/2), but depend on µ L (r). and estimated systematic shifts affecting the measuredNc/N 0 c and µc. Statistical errors are shown as error bars in the main paper. All values are typical for the experimentally explored range ofg. The positive (negative) systematic shifts correspond to the measured values being higher (lower) than the true ones.
To evaluate the set {" j }(µ L ) we solve the 1d axial problem defined by two coupled equations: where n 3 is 3d density, and where g 3 = 4⇡~2a/m. We iteratively find a self-consistent solution to Eqs. (3)(4), and then compute n(k) in Eq. (2). Critical parameters: To identify the critical time t c , we fit the following heuristic function to P 0 (t): where ⇥(t) is the Heaviside function. In the limit b ⌧ 1/|t t c |, this is simply a piecewise linear function, while the nonzero b allows for the saturation of the peak OD at short t.
To determine N (t c ), T (t c ) and µ(t c ) we fit the smooth time evolutions of N , T and µ by third-order polynomials.
Finally, we use the above MF model to compute the occupations of the j > 0 axial states for (T (t c ), µ(t c )), and correct N c to getN c . This operation assumes that beyond-MF effects affect only the axial ground state. This approximation is justified by the fact that~! z /k B T >g [4].

Momentum-space resolution
Our k-space resolution is limited by the camera pixel size and by the slight anisotropy of the in-plane harmonic potential in which we perform the momentum focusing. After removing the green "blade" beams, we have ! x,y =! ± !, with!/(2⇡) = 36.3(1) Hz and !/(2⇡) = 2.5(1) Hz. This means that the focusing is not obtained simultaneously in both directions, but at the focusing time t f = ⇡/(2!) ⇡ 7 ms the blurring of the momentum distribution is isotropic, with k ⇡ p mk B T /~2 sin( ! t f ) ⇡ 0.4 µm 1 . This happens to closely coincide with the resolution set by the pixel size. To determine the height of the momentum peak, P 0 , we average the image within one pixel of the fitted cloud centre.

Confidence intervals
In Table I, we summarise the main uncertainties in the critical parametersN c /N 0 c andμ c in Figs. 2-3 in the main paper. The statistical errors, defined as 68% confidence interval and arising from experimental noise and fitting errors, are shown by the error bars in the main paper. Here they are averaged over the experimental points. Table I fall into three categories: (i) calibration errors and drifts in the trapping frequencies, imaging magnification and absolute detectivity, (ii) finite-size (non-zero ! r ) effects, and (iii) anharmonicity of the optical trapping potentials. These affect the calculated N 0 c and also the model functions used for extracting µ and T . The calibration errors we calculate by propagating errors, the finite-size effects are calculated following [3], and we assess the typical magnitude of the anharmonicity shifts by numerical simulations in the ideal-gas limit.

Systematic uncertainties in
Additionally, two more fundamental systematic effects could shift the apparent critical point. If we assume that the emergence of coherence over lengthscale L requires the whole region near the cloud centre, of radius L/2, to be superfluid, then the trap inhomogeneity leads to apparent positive shifts ofN c /N 0 c andμ c , of . 0.2 and ⇡ 0.05, respectively. On the other hand, the critical divergence of the correlation length above T c has an opposite effect, which we estimate to be of comparable absolute size [8]. We thus expect these two effects to partially cancel, and lead to overall shifts that do not exceed our statistical and systematic errors (see also [9]).