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## Scaling laws for the emergence of coherence

We have plotted in figure 3.7 the ensemble of our results for the threshold value of the total 2D phase-space-density Dc ≡ Nc λ2 dB/A as a function of ζ = kBT/hνz, determined both from the onset of bimodality as in figure 3.5 (closed symbols) or from the onset of visible interference as in figure 3.6c) (open symbols). Two trapping configurations have been used along the z direction, ωz/2π = 1460 Hz and ωz/2π = 365 Hz. In the first case, the z direction is nearly frozen for the temperatures studied here (ζ . 2). In the second one, the z direction is thermally unfrozen (ζ & 8). All points approximately fall on a common curve, independent of the shape and the size of the gas: Dc varies approximately linearly with ζ with the fitted slope 1.4 (3) for ζ & 8 and approaches a finite value ∼ 4 for ζ . 2.

In the frozen case, a majority of atoms occupy the vibrational ground state jz = 0 of the motion along the z direction, so that D essentially represents the 2D phase-space-density associated to this single transverse quantum state (see chapter 1). Then for D ≥ 1, we know from Equation 1.47 and the associated discussion that a broad component arises in G1 with a characteristic length ℓ that increases exponentially with the phase-space-density. The observed onset of extended coherence around D ∼ 4 can be understood as the place where ℓ starts to exceed significantly λdB. The regime around D ∼ 4 is reminiscent of the presuperfluid state identified in [45, 86]. It is different from the truly superfluid phase, which is expected at a higher phase-space-density (D ∼ 8) for our parameters [99]. Therefore the threshold Dc is not associated to a true phase transition, but to a crossover where the spatial coherence of the gas increases rapidly with the control parameter N.

### creation of topological defects by quench cooling the gas

As explained in chapter 2, for a system undergoing a phase transition, a lot of information can be retrieved from studying the freezing out dynamics during a quench cooling as described by the Kibble-Zurek (KZ) mechanism. The density and scaling of topological defects with the quench duration can indicate which transition is being crossed and leading to the appearance of phase domains. The Kibble-Zurek mechanism has already been experimentally studied in a variety of systems, such as liquid crystals[139], helium [140, 141], ion chains [142, 143], superconducting loops [144], hydrodynamic systems [145] and Bose- Einstein condensates [67, 68, 146, 147].

Here, we use the flexibility of our method to produce box potentials of different shapes to study this mechanism. In particular, having a uniform system brings us closer to the original proposal and scalings by Kibble and Zurek. However, note that recent experimental and theoretical studies have investigated the influence of introducing a harmonic trap on the Kibble-Zurek scaling [67, 142, 146, 148, 149].

We study the appearance of topological defects by quench cooling a gas of atoms in two different configurations. First, point vortices are revealed in short time-of-flight experiments. Second, supercurrents (i. e. phase windings) created in a ring of atoms are studied interferometrically by using a trap potential shaped as a “target” with a disk of atoms surrounded by a ring (see figure 3.3d). The preparation of the gas is similar for both cases and the scaling of the number of topological defects is recorded as a function of the quench duration to compare it to the predictions of the KZ mechanism (see chapter 2).

#### Observation of topological defects

From now on we use the weak trap along z (ωz/2π = 365 Hz) so that the onset of extended coherence is obtained thanks to the transverse condensation phenomenon. We are interested in the regime of strongly degenerate, interacting gases, which is obtained by pushing the evaporation down to a point where the residual thermal energy kBT becomes lower than the chemical potential μ (see 3.1.3.4). The final box potential is ∼ kB × 40 nK, leading to an estimated temperature of ∼ 10 nK, whereas the final density (∼ 50 μm−2) leads to μ ≈ kB × 14 nK. In these conditions, for most realizations of the experiment, defects are present in the gas. They appear as randomly located density holes after a short 3D ToF (figure 3.8), with a number fluctuating between 0 and 5. To identify the nature of these defects, we have performed a statistical analysis of their size and contrast, as a function of their location and of the ToF duration τ (figure 3.8 c and d, and next subsection). For a given τ, all observed holes have similar sizes and contrasts. The core size increases approximately linearly with τ, with a nearly 100% contrast. This favours the interpretation of these density holes as single vortices, for which the 2π phase winding around the core provides a topological protection during the ToF. This would be the case neither for vortex–antivortex pairs nor phonons, for which one would expect large fluctuations in the defect sizes and lower contrasts.

**Analysis of the density holes created by the vortices**

We first calculate the normalized density profile n/¯n where the average ¯n is taken over the set of images with the same ToF duration τ. Then we look for density minima with a significant contrast and size. Finally for each significant density hole, we select a square region centered on it with a size that is ∼ 3 times larger than the average hole size for this τ. In this region, we fit the function A0 1 − c + c tanh q x2 + y2/ξ (3.14) to the normalized density profile, where A0 accounts for density fluctuations. We also correct for imaging imperfections (finite imaging resolution and finite depth of field) by performing a convolution of the function defined in equation 3.14 by a Gaussian of width 1 μm, which we determined from a preliminary analysis.

**Dynamical origin of the topological defects**

In principle the vortices observed in the gas could be due to steady-state thermal fluctuations. BKT theory indeed predicts that vortices should be present in an interacting 2D Bose gas around the superfluid transition point [85]. Such “thermal” vortices have been observed in non-homogeneous atomic gases, either interferometrically [44] or as density holes in the trap region corresponding to the critical region [133]. However, for the large and uniform phase-space densities that we obtain at the end of the cooling process (nλ2 dB ≥ 100), Ref. [152] predicts a vanishingly small probability of occurrence for such thermal excitations. This supports a dynamical origin for the observed defects.

**Table of contents :**

Introduction

**1 the bose gas from three to two dimensions **

1.1 Statistics and Bose gases

1.1.1 Non-interacting bosons in the grand-canonical ensemble

1.1.2 Bose-Einstein condensation

1.1.3 Validity of the derivation of Bose-Einstein condensation

1.2 Dimensional crossover from two to three dimensions

1.2.1 Experimental realization of a 2D Bose gas

1.2.2 Transverse condensation

1.2.3 Coherence length at the transverse condensation point

1.3 Behaviour of a 2D plane of atoms

1.3.1 Non-interacting Bose gas

1.3.2 The interacting 2D Bose gas

1.3.3 Superfluid regime: Berezinskii-Kosterlitz-Thouless transition versus Bose-Einstein condensation

1.4 Conclusion

**2 the kibble-zurek mechanism **

2.1 Phase transitions and critical slowing down

2.1.1 Static critical exponents

2.1.2 Dynamical exponent

2.2 The Kibble-Zurek prediction

2.2.1 Correlation length and thermalization time

2.2.2 Freezing out of the system

2.2.3 Detection of the order parameter variation: topological defects

2.3 Limitations to the observation of the Kibble-Zurek prediction

2.4 Conclusion

**3 quenching the bose gas between three and two dimensions**

3.1 Experimental set-up

3.1.1 Laser set-up

3.1.2 Production of degenerate gases

3.1.3 Parameter estimation of Bose gases in box potentials

3.2 Experimental evidence for the dimensional crossover

3.2.1 Phase coherence revealed by velocity distribution measurements

3.2.2 Phase coherence revealed by matter-wave interference

3.2.3 Scaling laws for the emergence of coherence

3.3 Creation of topological defects by quench cooling the gas

3.3.1 Vortices in square geometries

3.3.2 Supercurrents in ring geometries

3.3.3 Discussion on possible improvements on the measurements

3.4 Conclusion

**4 a new experimental set-up for 2d physic**

4.1 Producing degenerate gases of rubidium

4.1.1 Design principle of the experiment

4.1.2 Laser system

4.1.3 Vacuum system

4.1.4 Laser cooling

4.1.5 Quadrupole trap and radio-frequency evaporation

4.1.6 Production of Bose-Einstein condensates in a crossed dipole trap

4.1.7 Imaging the cloud

4.1.8 Obtaining degenerate gases in shaped potentials

4.2 Shaping the cloud

4.2.1 Making box potentials

4.2.2 Confining the gas to two dimensions

4.3 Conclusion

**5 collective effects in light-matter interaction **

5.1 Position of the problem

5.1.1 Importance of collective effects in atom-light interactions

5.1.2 Observation of collective effects

5.1.3 Relevance of collective effects for our systems

5.2 Modelling multiple and recurrent scattering effects

5.2.1 Choice of the model

5.2.2 Coupled classical dipoles

5.2.3 Programs

5.3 Preparing a sample

5.3.1 Calibration of the imaging set-up

5.3.2 Computing the optical density of the cloud

5.3.3 Preparation and properties of the atomic sample

5.4 Resonances

5.4.1 Resonance curves for dilute clouds

5.4.2 Lorentzian fits

5.4.3 Wing fits

5.5 Local excitation of a coud of atom

5.6 Conclusion

**6 prospective experiment: evaporation in a tilted lattice **

6.1 Solving the scattering problem of an atom in a tilted lattice

6.1.1 Position of the problem—Outline of the resolution

6.1.2 Scattering matrix in real and reciprocal space

6.1.3 Definition of the Bloch-Stark states

6.1.4 Expression of the scattering matrix

6.1.5 Results

6.2 Evaporation using particle interactions

6.2.1 Principle

6.2.2 Simulations

6.2.3 Results

6.3 Conclusion

**7 prospective experiment: using magnetic texture to produce supercurrents **

7.1 A ring of atoms in a quadrupole field

7.1.1 A neutral atom in a real magnetic field interpreted as a charge in an artificial magnetic field

7.1.2 Case of a spin 1 atom

7.1.3 Higher order spins

7.1.4 Case of the quadrupole field

7.1.5 Higher order fields

7.1.6 Artificial magnetic field

7.2 Condensation in presence of an artificial gauge field

7.2.1 Computing the ground state

7.2.2 Higher order spins or multipolar fields

7.2.3 Spinor and choice of gauge

7.2.4 Detecting the angular momentum

7.3 Measuring Berry’s phase

7.4 A vortex pump

7.4.1 Basic idea

7.4.2 Topological interpretation of the vortex pump

7.4.3 Simulation of experimentally relevant parameters

7.5 Conclusion

Conclusion

**a fit-free determination of scale invariant equations of state: application to the two-dimensional bose gas **

**b calculation of the probability distribution of the relative velocity **

**c description of the coupled dipole programs**

c.1 drawPositions

c.1.1 Inputs

c.1.2 Output

c.2 Heff

c.2.1 Inputs

c.2.2 Output

c.3 excVector

c.3.1 Inputs

c.3.2 Output

c.4 transmission

c.4.1 Inputs

c.4.2 Output

c.5 doResonancesVaryOD

c.5.1 Inputs

c.5.2 Output

**d estimating confidence intervals using the bootstrapping method **

d.1 Position of the problem

d.2 The bootstrap principle

d.3 Precautions in using the bootstrap

d.3.1 The bootstrap does not provide better estimates of parameters

d.3.2 Bootstrap caveats

d.4 Conclusion

**bibliography **