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## Detection of the order parameter variation: topological defects

In this section, we will focus on the case of the order parameter of a BoseEinstein condensate; in particular, since this order parameter is a complex number, we will be interested in its phase, which is the symmetry-breaking parameter.

After a temperature quench, the distribution of the phase of the condensate isnot uniform as it would be after an infinitely slow cooling procedure: it consistsof areas of typical size ξˆ where the phase of two points is correlated, while the phase of points whose distance is larger than ξˆ is uncorrelated. In a simplified picture, we can picture the phase distribution consisting of “patches” of size ξˆ. If we can measure directly the phase of the system, we can access the freezingout size ξˆ. A recent experiment using ultracold atoms has been probing directly the first-order correlation function after a quench cooling [68]. However, this is usually difficult to do, which is why the Kibble-Zurek prediction was originally based on the observation of topological defects in the phase. A topological defect is a perturbation of the phase from the uniform distribution that cannot be transformed continuously back into a uniform phase distribution. A typical example of a topological phase defect is a vortex, that is a phase winding of 2π around a point in a 2D system. Other examples of topological defects include vortex lines in 3D systems or solitons, i. e. a phase change of π between two parts of the system. One advantage of topological defects is that they can have stronger signatures than phase domains. For example, in a 2D degenerate gas, a vortex can be detected in Time-of-Flight (ToF) experiment as an expanding density hole. Another advantage is that topological defects do not disappear easily, so they might live for a long time after the quench has been performed. Zurek used the prediction on the typical domain size ξˆ to predict the density of topological defects [107, 118]. In the naive phase patch picture for a system in d dimensions, a patch has a volume V µ ξˆd. Depending on the dimension D of the defect (a point vortex has D = 0, a vortex line D = 1, a soliton D = d − 1), the scaling of the density of defects can be computed. We will consider two cases: the case of a vortex in a ring (1D system) and the case of a vortex in a 2D system.

### limitations to the observation of the kibble-zurek prediction

Let us now state some of the obstacles that must be overcome when trying to verify experimentally the Kibble-Zurek scaling.

First, experimental verification of a power-law scaling requires in principle to probe several orders of magnitude in the different parameters, that is in the topological defect density and in the quench duration, which is a challenging experimental task. In the case of the BKT transition, the orders of magnitude that have to be spanned for an accurate determination of the scaling behaviour are for the moment far out of reach of atomic physics experiments [117]. In addition, in the case of a power-law behaviour, the values of the exponents to be measured are small: 1/8 or 1/6 for the phase winding in the ring, depending on the model (mean-field or model F), and 1/2 to 2/3 in the case of vortices in a plane, also depending on the model; this makes their accurate determination more difficult. Second, the finite size L of the experimental systems can also limit the range of quench durations to be tested. The scaling behaviour is indeed valid only in the limits L ≫ ξˆ or L ≪ ξˆ: the intermediate regime L ∼ ξˆ does not follow such simple laws [118].

Last, the picture of two extremely well separated regimes t < ˆt where the system follows adiabatically the parameter ε and t > ˆt where the order parameter is completely frozen is not completely valid [120–122]. Even if the dynamic is slow below the transition point ε > 0, the system can still evolve. This was important in the demonstration of Kibble-Zurek scaling in the experiments performed in [68]. Even the density of topological defects, which cannot easily decay and whose lifetime is therefore long, can evolve during the quench, for example in the case of annihilation of vortices with opposite charge in a 2D system [90].

#### Blue-detuned laser for 2D and box confinement

2D vertical confinement as well as in-plane, horizontal, uniform trapping is provided by a 10W laser at 532 nm, a Verdi V10. The light is brought to the experiment via optical fibers with length up to 4 m, which limits the maximum power available due to Brillouin scattering. The maximum power of 1W at the output of the fibers is obtained using the laser at a restrained operating point of 4W and coupling the light into end-capped fibers from Schäfter-Kirchhoff.

**Method for producing a hollow beam**

Some tests to produce a hollow beam were performed using a phase-plate producing a Laguerre-Gauss beam of order six [90, 123]. The quality of the potential was not sufficient for our applications, since the light potential has defects whose peak intensity can reach 40% of the maximum intensity. We therefore turned to direct imaging of a mask onto the atoms to produced dark regions in the blue-detuned box potential beam with sufficient quality. However, note that a Laguerre-Gauss beam has been successfully used to produced three-dimensional box potentials for ultracold atoms [55, 130].

We create the box-like potential in the xy plane using a laser beam that is blue-detuned with respect to the 87Rb resonance. At the position of the atomic sample, we image a dark mask placed on the path of the laser beam. This mask is realized by a metallic deposit on a wedged, anti-reflective coated glass plate2.

The xy confinement of the atoms is provided by a hollow, blue-detuned beam trapping the atoms in its dark regions. We characterize the box-like character of the resulting trap in two ways. (i) The flatness of the domain where the atoms are confined is characterized by the root mean square intensity fluctuations of the inner dark region of the beam profile. The resulting variations of the dipolar potential are δU/Ubox ∼ 3%, where Ubox is the potential height on the edges of the box. The ratio δU/kBT varies from ∼ 40% at the loading temperature to ∼ 10% at the end of the evaporative cooling. In particular, it is of ∼ 20% at the transverse condensation point for the largest square pattern. (ii) The sharp spatial variation of the potential at the edges of the box-like trapping region is characterized by the exponent α of a power-law fit U(r) µ rα along a radial cut. We restrict the fitting domain to the central region where U(r) < Ubox/4 and find α ∼ 10–15, depending on the size and the shape of the box.1

**Temperature measurement in the box potentials**

Due to the spatial extension of the condensate in a box potential, it is necessary to perform long ToF in order to separate the thermal from the degenerate part in very cold samples. Therefore another method for thermometry has been developed.

All temperatures indicated in this chapter are deduced from the value of the box potential, assuming that the evaporation barrier provided by Ubox sets the thermal equilibrium state of the gas. This hypothesis was tested, and the relation between T and Ubox calibrated, using atomic assemblies with a negligible interaction energy. For these assemblies, we compared the variance of their velocity distribution Dv2 obtained from a ToF measurement to the prediction for the ideal gas description of trapped atomic samples as in Equation 1.6. The calibration obtained from this set of measurements can be empirically written asT(Ubox) = T0 1 − e−Ubox/(η kBT0) 0.

**Chemical potential in the degenerate interacting regime**

To compute the chemical potential μ of highly degenerate interacting gases, we perform a T = 0 mean-field analysis. We solve numerically the 3D Gross– Pitaevskii equation in imaginary time using a split–step method, and we obtain the macroscopic ground state wave-function ψ(r). Then we calculate the different energy contributions at T = 0 for N atoms with mass M – namely the potential energy Epot, the kinetic energy Ekin and interaction energy Eint – by integrating over space: Epot = N 2 Mω2 z Z z2 |ψ (r)|2 d3r.

**Time-of-flight measurements of gases in a box potential**

To characterize the coherence of the gas, we study the velocity distribution, i. e., the Fourier transform of the G1(r) function. We approach this velocity distribution in the xy plane by performing a 3D ToF: we suddenly switch off the trapping potentials along the three directions of space, let the gas expand for a duration τ, and finally image the gas along the z axis. In such a 3D ToF, the gas first expands very fast along the initially strongly confined direction z. Thanks to this fast density drop, the interparticle interactions play nearly no role during the ToF and the slower evolution in the xy plane is governed essentially by the initial velocity distribution of the atoms. The ToF duration τ is chosen so that the size expected for a Boltzmann distribution τ√kBT/m is at least twice the initial extent of the cloud. Typical examples of ToF images are given in figure 3.4. Whereas for the hottest and less dense configurations, the spatial distribution after ToF has a quasi-pure Gaussian shape, a clear non- Gaussian structure appears for larger N or smaller T. A sharp peak emerges at the center of the cloud of the ToF picture, signaling an increased occupation of the low-momentum states with respect to Boltzmann statistics, or equivalently a coherence length significantly larger than λdB the thermal wavelength (λdB = h/√2πMkBT).

**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 physics **

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 cloud 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**