# The weakly interacting 2D Bose gas: fundamental equations

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## One way to implement the classical field Monte Carlo approach

The idea of a Monte Carlo simulation in statistical physics is to compute a lot of microscopical states and evaluate thermodynamic quantities on these states. If the states are well sampled, an average of these values constitutes a thermal average of the thermodynamic quantity. A good sampling requires ergodicity, meaning that the process used for computing the microscopical states should be able to create all the possible states.
When the work is done in the canonical ensemble, an efficient and often used way of sampling the states has been introduced by Metropolis, Rosenbluth and Teller [112] and consists in sampling the system by modifying it slightly around equilibrium and attributing different weights to the obtained states. The weights are simply the Boltzmann’s factor between the computed energy of the states. In the simulations described in this section, we choose a certain energy of the system which we keep constant along the evolution to get the different microscopical states. We are thus working in the microcanonical ensemble, and all the states have equal probabilities. The bench of states obtained this way corresponds to a sampling of all the microscopical states which we hope is representative.

### Principle of computation of the microscopical states

We do our simulations on a grid of size NS × NS. We compute the microscopical states from a state (r) with white noise, corresponding to the energy put into the system, with Dirichlet boundary conditions. We thus are simulating a finite size system. We then do a time evolution of this system by the splitting method, using (r) for the evolution with the interaction energy, and its discrete sine transform (q) for the kinetic energy evolution. We let this state evolve so that it “thermalizes” thanks to the non-linearity in the interaction energy evolution. We then compute the phase-space density D and post select the states so that the D found after evolution is compatible with the initial D used for the evolution with a relative error of less than 10%. The full process is described with more details in the Annexe A.

#### Monte Carlo simulations of Prokof’ev and Svitsunov

We will not try here to explain fully their simulations. Further explanations are given in their book [114]. We will just try to stress the main differences with our simulations. This is important because we will use the values they found for the equation of state of the 2D Bose gas later in this thesis. We said that we compute states all with the same energy, and that we are working in the microcanonical ensemble. They on the contrary use states with different energies, and are thus in the grand canonical ensemble. They then need a powerful process to sample their states, that they call Worm algorithm and which they describe in [102]. As discussed section 1.5.1, the results of our simulations are cutoff dependent. To have cutoff independent results, Prokof’ev and Svitsunov use an other type of universality that we did not describe yet. It can be shown that close to the critical point, all the differences between a physical quantity and its value at the critical point, forexample D − Dc, can be expressed as a function of one variable: 1 ˜g  » μ kBT − μ kBT c # .

Reaching the degenerate regime

When I arrived in the lab, the vacuum system was already prepared and the laser systems with their locks were mounted but nothing was aligned on the experiment. I thus will not describe in details all apparatus, beam paths, AOM frequencies, etc here, but these can be found in Laura’s thesis [117], who took care to write most of the information in her chapter 4. Further information can also be found in Katharina’s report [118] who worked on the laser systems and on early stages of the experiment such as the computer control. I will nevertheless briefly explain the sequence used to reach the degenerate regime, as some of the tools will be reused later in the sequence for other purposes.

Imaging axes of the setup

We built several imaging paths on the experiment to be able to probe the cloud in different directions. I will describe only the horizontal and vertical ones which are the most used now on the experiment. Two additional Lumenera cameras LM135M are mainly used to image the atoms in the MOT or in the quadrupole.
• The horizontal axis corresponds to the imaging light propagating along the x direction. The absorption picture is taken by a Pixelfly PF-M-QE-PIV with a magnification 3 used in interframed mode. This axis allows us to check the vertical confinement of the atoms. An example of picture is shown in figure 2.4 (a).
• For the vertical axis, the imaging light propagates in the z direction (downwards). The absorption picture is taken by a Princeton Instrument PIXIS 1024 Excelon with a magnification of 11 used in frame-transfer mode. Its quantum efficiency is of 0.95. This axis has a high numerical aperture of 0.45 thanks to the use of a microscope objective described in the following. The imaging light is linearly polarized and examples of pictures are shown in figure 2.6. We have in addition 4 Chameleon cameras from Point Grey allowing to monitor the drifts of the different beams. The most used is the one monitoring the position of the accordion fringes which is used for a feedback as explained section 2.3.1. In the next paragraph we focus on the vertical imaging axis which provides the highest quality pictures for the experiment.

I Theory and experimental setup
1 Overview of the two-dimensional Bose gas
1.1 The ideal uniform Bose gas in three and two dimensions
1.1.1 Does condensation occur?
1.1.2 Coherence length
1.2 The weakly interacting 2D Bose gas: fundamental equations
1.2.2 From the quantum Hamiltonian to classical field
1.2.3 Classical field formalism
1.2.4 Factorization of the “frozen direction”
1.2.5 The quasi-2D regime
1.2.6 Scale invariance
1.3 The weakly interacting 2D Bose gas: some properties
1.3.1 Reduction of density fluctuations and quasi-condensate
1.3.2 Bogoliubov analysis
1.3.3 Algebraic decay of correlations
1.4 Superfluid transition in the 2D Bose gas
1.4.1 Definition of superfluidity and superfluid fraction
1.4.2 The superfluid universal jump
1.4.3 Connexion to the Berezinskii-Kosterlitz-Thouless transition
1.4.4 Critical temperature of the transition
1.5 Classical field Monte Carlo simulations
1.5.1 One way to implement the classical field Monte Carlo approach
1.5.2 Result on correlation functions
1.5.3 Monte Carlo simulations of Prokof’ev and Svitsunov
1.6 Equation of state
1.6.1 Limiting cases of equation of state
1.6.2 Quasi-condensate and superfluid fractions
1.6.3 Summary of transitions crossed by a finite-size 2D uniform Bose gas
2 Experimental Setup
2.1 Reaching the degenerate regime
2.1.1 Experimental sequence to load the final trap
2.1.2 Computer control
2.1.3 Laser system
2.1.4 Magnetic field control
2.2 Absorption imaging of the cloud
2.2.1 Brief description
2.2.2 Imaging axes of the setup
2.2.3 Optical resolution of the vertical axis
2.2.4 Calibration of intensity on the atoms
2.3 Description of the final trap
2.3.2 Compression in the accordion
2.3.3 In-plane confinement: Potentials of arbitrary shape
2.4 Manipulation of hyperfine states
2.4.1 Hyperfine structure and Breit-Rabi formula
2.4.2 Stabilisation of the magnetic field
2.4.4 Micro-wave transfers
2.5 Crossing the condensation threshold of the gas
2.5.1 Blasting
2.5.2 Making a thermal 2D Bose gas
2.5.3 Condensate focussing
2.6 Thermometry of the gas
2.6.1 Difficulty for the uniform 2D Bose gas
2.6.2 Improved thermometry
II Light scattering
3 About light scattering by atomic clouds
3.1 Introduction
3.2 Modelling light scattering
3.2.1 Response of a single dipole
3.2.2 Coherent and incoherent parts
3.2.3 Modelling of the coherent part: dielectric approach
3.2.4 Modelling of the incoherent part: random walk approach
3.2.5 “Full” Modelling: coupled dipoles approach
3.2.6 Definition of “cooperative effects”
3.3 Experimental investigations and interpretations
3.3.1 Different experimental investigations
3.3.2 About the shifts of the resonance
3.3.3 Finite-size scaling for coupled dipoles
3.3.4 About “cooperative effects” in MOTs
3.4 Conclusion
4 Study of light transmission
4.1 Experimental methods
4.1.1 Cloud preparation
4.1.2 Transmission measurement
4.1.3 Computation of the optical depth
4.1.4 Atom number calibration
4.1.5 Experimental protocol
4.2 Theoretical description
4.2.1 Perturbative approach
4.2.2 Coupled dipole simulations
4.3 Experimental results
4.4 Conclusion
4.5 Additional remark: reflection coefficient of a 2D gas
5 Study of light diffusion
5.1 Results
5.1.1 Light spreading in a dense atomic cloud
5.1.2 Diffusion model
5.1.3 Variation of the decay length with density at resonance
5.2 Discussion
5.3 Methods
5.3.1 Preparation of the atomic cloud
5.3.2 Imaging system
5.3.3 Coupled dipole simulations
5.3.4 Semi-analytical model of light guiding
III Out-of-equilibrium and superfluid studies
6 Merging of N independent condensates
6.1 Motivation for this work
6.1.1 Kibble-Zurek mechanism and its study in bulk systems
6.1.2 Disentangling the Kibble-Zurek mechanism: study of coarsening dynamics .
6.2 Study of the relaxation dynamics in the merging of N independent condensates
6.2.1 Experimental protocol
6.2.2 Principle of the measurements
6.2.3 Results for the number of segments
6.2.4 Results on time evolution
6.2.5 Conclusion
6.3 Supplementary information
6.3.2 Phase reference of the inner ring
6.3.3 Independence of BECs
6.3.4 Width of the distribution for different merging times
6.3.6 Center values for the relaxation measurements
6.3.7 Experimental sequence and data analysis for the merging of two condensates
7 Further characterization of the setup
7.1 Calibration of the magnifications
7.1.1 Lattice in situ: measure of M1 ×M2
7.1.2 Bragg diffraction through a lattice: measure of M2/M1
7.1.3 Conclusion of the calibration
7.2 Calibration of the effective atomic cross-section with light
7.2.1 Principle of the calibration
7.2.2 Experimental sequence
7.2.3 Taking shot noise and read-out noise into account for one image
7.2.4 Taking shot noise and read-out noise into account for the difference of two images
7.2.5 A first method of calibration
7.3 Calibration by modeling the optical response
7.3.1 A coarse-grain approximation
7.3.2 Extracting the matrix using correlations
7.3.3 Robustness using subsets of images
8 Study of Sound propagation
8.1 Motivation for this project
8.1.1 Hydrodynamics and two-fluid model
8.1.2 Reminder of thermodynamics
8.1.3 Prediction
8.2 Experimental study
8.2.1 Protocol for travelling waves
8.2.2 Analysis of the data of travelling waves
8.2.3 Results
8.2.4 Standing waves
8.2.5 Conclusion
8.3 Supplementary information
8.4 Experimental setup
8.4.1 Protocol for characterizing standing waves
8.4.2 Determination of the cloud’s degeneracy
8.4.3 Density calibration
8.5 Summary of the measurements
8.5.1 Complementary results
8.5.2 Landau damping
8.5.3 Influence of the excitation
9 Conclusion and perspectives
9.1 Conclusion of the thesis
9.2 Remaining experimental challenges of the 2D Bose gas
9.3 Perspectives
9.3.1 First-order correlation function measurements
9.3.2 Demixing experiments
IV Appendices
A Implementation of Monte Carlo simulations
A.1 Modelisation
A.2 Space discretization
A.2.1 Implementation of the ultraviolet cutoff
A.2.2 Normalization of the fields
A.3 Field evolution
A.3.1 Algorithm used
A.4 Useful variables of the model
B PHYSICAL REVIEW A 95, 013632 (2017)
C Clebsch-Gordan coefficients
C.1 Electric dipole
C.2 Magnetic dipole
D Appendix of the atom number calibration
D.1 Monte Carlo simulations for checking validity of linearisation of noise
D.2 Modelling the optical response
D.2.1 Correlations between pixels
D.2.2 Computation of spatial correlations of the differences of ODs
E List of publications
Bibliography

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