Residual radial confinements for a decentered cloud compared to the light-sheet beam focus 

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Overall description of the experimental setup

As in any cold atoms experiment our setup consists of a vacuum system in which we can trap and cool atomic sample at the nano-Kelvin scale. The specificity of our own setup is that it is constituted of two separated chambers with two different vacuum pressures:
— a steel «MOT chamber» with moderate vacuum in which we trap a sample of 87Rb atoms from the background vapor and perform preliminary cooling on it.
— a glass «science cell», in which cooling is carried further to reach degeneracy and in which the 2D gas is produced and studied.
These two chambers are linked by a differential pumping stage along which atoms are magnetically transported. To reach the desired regime of degenerate 2D Bose gas within this global setup, a sequence of tens of events mainly consisting in modifying either the strength of the magnetic fields or the intensity of the optical beams is necessary. These are controlled and synchronized at the microsecond scale. The control is realized via a computer based program developed in MIT by Aviv Keshet [130] communicating with nine analog or digital National Instruments cards and two GPIB devices. These tens of unit events can be decomposed in typically ten global phases. I will describe them, distinguishing the creation of a 3D BEC and the further study of a 2D degenerate Bose gas.

Possible Time–of–Flight (some ms)

Depending on the quantities we want to measure, we have the possibility to let the cloud expand freely along certain directions of space for some milliseconds before imaging it. This technique is usually named Time–of–Flight (ToF) measurement. More precisely, we may switch off:
— The light-sheet confinement that tightly confines the atoms in the vertical direction. This switching off can be done by fast electronic control of Acousto Optic Modulator (AOM) on the order of the microsecond. When we only switch off this trap, the gas expands along the vertical direction but not in the xy plane. Such an expansion is called «one–dimensional Time–of–Flight » which we abbreviate in 1D ToF 8.
— The box–trap confinement that keeps the atoms in a restricted region of space in the 2D plane. This switching off can be done in 10 ms. When we only switch this trap, the gas expands in the xy plane but not along z. Such an expansion is called «two–dimensional Time–of–Flight » (2D ToF). To perform such a Time–of–Flight we usually ramp down the light sheet power to reduce the effect of possible defects and rugosities of the light-sheet potential (see 2.2.1). We typically choose a remaining confinement of 350 Hz.
— When switching off simultaneously both traps, the gas is expanding in the three-dimensional free-space and this is conventional «three-dimensional time of flight» (3D ToF).
We implement these Time–of–Flights without modifying the magnetic fields. Such changes would induce strong Eddy currents (especially in the translation stage holding the microscope objective for vertical imaging setup) and thus strong perturbations to the later imaging process. We choose the value of this gradient to b02D z = b0mag z so that it exactly compensates the effect of gravity. This is highly convenient as it enables to keep the cloud center of mass in place for all ToF durations. Hence, we can produce long free–expansion without being limited by the gas hitting the glass cell and the cloud stays advantageously at the imaging focus position. The remaining gradients do not imply any important modification to ToF dynamics as the resulting frequencies are small enough (some Hz).

Current realization of tight transverse confinement for 2D trapping: the Hermite-Gaussian beam

To experimentally realize quasi–two–dimensional confinement of our gas of cold atoms, we need to generate a trap with very tight confinement in one direction. Such a tight confinement must freeze out atomic motion in the considered direction (see 1.1.2). The freezing is realized when the energy gap between the ground state and the first excited state is much smaller than both the thermal energy kBT and the interaction energy per particle eint. If the z confinement is harmonic with pulsation wz, we enter the quasi-2D regime when ¯hwz & kBT, eint. In our typical 2D experiments, the temperature T is of the order of T 100 nK and the chemical potential m kB 50 nK. Then the quasi-2D regime necessitates wz & 2p 2 kHz.To implement this tight trapping, we use an optical dipole trap generated via a specially shaped blue–detuned laser beam (typically l = 532 nm) showing an anisotropic geometry.

Producing an Hermite–Gauss beam

The shaping of this beam is realized in the current setup via the use of a phase plate as previously implemented in [134]. This phase plate shows a step in phase along one direction and it imprints a phase of p on the upper half of the beam (z > 0) with respect to the other half (z 0): f (z) = p if z > 0 , f (z) = 0 if z 0 (2.11).
The phase plate is placed on the path of elliptic collimated beam (see below for the choice of the yz aspect ratio). We refocus this beam after the phase plate at the atom position by a converging lens of focal f = 100mm so that the phase imprinted translates into a specific Hermite–Gaussian intensity profile. To quantitatively describe the effect of this far blue detuned (l = 532 nm) beam on the atoms, we need to calculate the spatial dependance of the dipolar force. Hence, we are interested in the intensity profile IHG at the position of the atoms. The resulting dipole potential is U = aIHG where a
can be computed as previously done in 2.1.1. As w = 2p 5.631014 Hz, we take into account the beyond Rotating Wave Approximation term and neglect the detuning difference to the fine structure. Thus a = 􀀀3pc2 2w3 0 G w0􀀀w + G 􀀀w and numerical application gives a = kB 59mK mm2/W.We then need to calculate the IHG.

Table of contents :

Detailed Outline
1. Elements of theory on quasi- two-dimensional Bose gases 
1.1. Specificity of two-dimensional physics
1.1.1. Absence of true long-range order in 2D systems
1.1.2. Realizing a 2D Bose gas in a 3D world: the deep thermal 2D regime
1.1.3. The ideal two-dimensional Bose gas The uniform 2D gas The harmonic 2D gas
1.1.4. The interacting quasi-two-dimensional gas The «collisionnally» quasi-2D regime and scale invariance Reduction of the density fluctuations and quasi-condensation Equation of state of an interacting 2D gas in two limiting cases The superfluid state at low T and Berezinskii–Kosterlitz– Thouless mechanism:
1.2. 3D-2D crossover in a uniform trap: transverse condensation phenomenon
1.2.1. Relevance of the uniform case
1.2.2. General description of an non-interacting gas in an oblate confinement uniform in-plane
1.2.3. Transverse condensation phenomenon of a in-plane uniform gas in an oblate confinement Saturation of z excitation populations and transverse condensation phenomenon. Population of z ground-state at transverse condensation point Link to 3D Bose–Einstein condensation phenomenon Influence of transverse condensation on in-plane coherence
1.3. Conclusion
2. Experimental methods for producing two–dimensional Bose gases 
2.1. Experimental setup
2.1.1. Overall description of the experimental setup
2.1.2. Experimental sequence toward tri–dimensional Bose–Einstein Condensation
2.1.3. Experimental sequence from 3D BEC to the study of two–dimensional gases
2.2. Tight transverse confinement for 2D trapping: the Hermite-Gaussian beam
2.2.1. Producing an Hermite–Gauss beam
2.2.2. Focusing the beam on the atoms by frequency measurement: Finding the focus Choosing the right position for loading the trap
2.2.3. Characterizing the resulting potential in the radial plane . Deconfining Potentials Defects on the beam profile
2.3. A new setup for tight transverse confinement: the accordion setup
2.3.1. The accordion scheme of principle
2.3.2. Choices for experimental implementation
2.3.3. Characterizing the transverse confinement (toward tighter trapping)
2.3.4. Dynamical variation of the confinement strength (toward optimal loading of the 2D-trap)
2.4. Conclusion
3. Imaging of our experimental gases 
3.1. Non-saturating and saturating absorption imaging
3.1.1. Principle of saturating absorption imaging
3.1.2. Absorption imaging at arbitrary intensity, some theoretical analysis
3.1.3. Constraints on the imaging parameters Doppler effect Atomic displacement Summary
3.2. Imaging setup
3.2.1. Horizontal Imaging
3.2.2. Vertical Imaging
3.3. Calibrating absorption imaging coefficients.
3.3.1. Frame Transfer and additional intensity per frame exposure: introducing a new coefficient g
3.3.2. Calibrating the number of counts to the actual intensity seen by the atoms: efficiency h
3.3.3. Calibration of the a coefficient Need for a new calibration of the a coefficient Principle of the calibration procedure for a coefficient . A new calibration procedure for a coefficient Description of the new calibration procedure: direct fit of the optical density terms. Qualitatively (possible) physical explanation to the a dependency on the atomic density n: Conclusion on a analysis:
3.3.4. Calibrating the global detectivity factor b
3.4. Conclusion
4. Fit-free determination of the equation of state and scale invariance 
4.1. General « fit free » formalism for determination of a scale-invariant EoS .
4.1.1. A set of new dimensionless variables Xn
4.1.2. Determination of the EoS by mean of the Xn
4.1.3. Some examples The non interacting 2D gas The interacting 2D gases
4.2. Application to the 2D Bose gas across BKT transition
4.2.1. Acquiring Experimental profile n(U) Experimental sequence Characterization of the in-plane trapping potential . Imaging procedure
4.2.2. Implementation of the « fit-free » method EoS in terms of the new variables Xn EoS for the conventional variables m/kBT and D . Removing the contribution for thermally populated transverse states Comparison to theory predictions EoS in other variables
4.3. Conclusion
5. Superfluidity in two dimensions 
5.1. Data acquisition
5.1.1. Experimental scheme
5.1.2. Heating response and fit
5.2. Observation of a critical velocity
5.3. Comparison with theory and previous measurements
5.4. Heating coeficient
5.5. conclusion
6. The uniform two-dimensional Bose gas 
6.1. Experimental specifications for our 2D Uniform trap
6.1.1. Physical properties of the gas under study
6.1.2. Deduced constraints on the trapping potential The maximal energy barrier Ubarrier confining the atoms in the central region The uniformity of the trap center: The steepness of the walls:
6.1.3. Choice on experimental technique to realize the uniform trap
6.2. Uniform trap via holographic shaping
6.2.1. Principle
6.2.2. Theoretical expectations Radius of the box-trap Barrier height of the box-trap Steepness of the edges of the box-trap
6.2.3. Experimental results and reasons for abandon Radius of the box-trap and trap bottom Steepness of the edges of the box-trap Uniformity of the trap bottom
6.3. The dark spot method
6.3.1. Principle and theoretical expectations. Barrier height and optimal waist Steepness of the edges and flatness of the bottom of the trap: Aperture Effects on the trap quality
6.3.2. Experimental setup
6.3.3. Experimental results and reason for the choice Potential Barrier and Radius of the box-trap Uniformity of the trap bottom Steepness of the edges of the box-trap: Various shapes of the potential
6.4. Conclusion
7. Coherence of the Uniform 2D Bose gas 
7.1. Thermometry of 2D uniform Bose gases
7.1.1. Measuring Velocity Distribution Variance
7.1.2. Estimate of temperature T for the less interacting clouds Self-consistent validation of the estimate Fit via Bose law computation
7.1.3. Empirical model for temperature dependency Justification of the temperature dependency at loading . Choice of the empirical model Generalization of the temperature relation
7.2. Emergence of Coherence seen in the momentum distribution
7.2.1. Estimating the population of the low excited states
7.2.2. Phase Diagram for D Measurements Comparing to theoretical predictions
7.2.3. Critical Atom Numbers for D
7.3. Matter-wave interferences
7.3.1. Detecting matter-wave interference
7.3.2. Characterizing fringes pattern
7.3.3. Phase Diagram and critical atom number for G Phase Diagram Critical atom number for G
7.4. Scaling Laws for emergence of Coherence
7.5. Conclusion
8. Kibble–Zurek mechanism at the dimensional crossover 
8.1. Kibble–Zurek Mechanism
8.1.1. General description of the mechanism
8.1.2. Specific description of the mechanism in our experiment.
8.1.3. Homogeneous Kibble–Zurek mechanism Linear ramp model Some insights on non–linear ramps and application to our experimental ramp:
8.1.4. Specificity of transition crossing in homogeneous systems: some insights on the inhomogeneous KZ phenomenon
8.1.5. Some insights on finite size effects on scaling properties .
8.2. Nucleation of vortices in a 2D uniform gas
8.2.1. Characterizing vortices via Time–of–Flight measurements . Experimental sequence Identifying density holes to vortex cores.
8.2.2. Justifying dynamical origin of the vortices
8.2.3. Studying the quench dynamics Fit of critical exponent for the quench dynamics . Description of the plateau at longer quench times
8.2.4. Analyzing dissipation dynamics of the vortices Principle of vortices dissipative dynamics simulations . Results of vortex dissipative dynamics simulations . Further observations on the evolution with thold
8.2.5. Complementary analysis Evolution in occurrence of vortex numbers Vortices location Correlation between the vortices
8.2.6. Conclusion
8.3. Nucleation of supercurrents in an annular Bose gas
8.3.1. Characterizing vortices current by matter-wave interference in a target Experimental sequence Identifying supercurrent from fringe patterns
8.3.2. Studying the supercurrent origin Stochastic Origin Location of the phase winding Dynamical origin
8.3.3. Studying the quench dynamics Fit of the power-law exponent for quench dynamics. Estimation of domain numbers and corrections of the power-law scaling Corrections due to predicted long time plateau for quench through BEC?
8.3.4. Other limitations
8.3.5. Characterizing phonons from fringes patterns
8.4. Conclusion
Concluding remarks 
Toward a characterization of the fluctuations of a 2D uniform gas
Toward tighter confinement and strongly interacting gases
Toward other uniform geometries and transport measurements
Appendix A. Ideal Bose description of the uniform 2D gases. 
A.1. Ideal description of the gas at thermal equilibrium
A.2. Computing predictions
A.2.1. Occupation of the single-particle states
A.2.2. Mean value of one-body observables
A.2.3. Mean value of two-body observables
A.3. self-consistent validity of the non-interacting treatment
Appendix B. Gross-Pitaevskii simulations to estimate the gas parameters in the highly degenerate regime. 
B.1. Principle of GrossPitaevskii simulations
B.1.1. Time dependent Gross-Pitaevskii equations
B.1.1.1. Defining the confining potential V
B.1.1.2. 2D approximation for the Gross-Pitaevskii equation
B.1.2. Method for solving the time-dependent GPE
B.2. Numerical computations
B.2.1. Computing the macroscopic ground state wavefunction for the trap sample
B.2.2. Deducing the gas parameters
B.2.3. Computing 2D and 3D Time-of-Flight evolutions of the wavefunction.
Appendix C. Residual radial confinements for a decentered cloud compared to the light-sheet beam focus 
C.1. Decentering along the LS-beam propagation direction x
C.2. Decentering along the transverse and weakly direction, y
C.3. Decentering along both x and y directions
Appendix D. Conventional imaging of an atomic ensemble, limitations due to bi–dimensionality 
D.1. Theoretical analysis of the multiple scattering effect
D.2. Modelling the atom-light interaction
D.2.1. The electromagnetic field
D.2.2. The atomic medium
D.2.3. The atom-light coupling
D.3. Interaction of a probe laser beam with a dense quasi-2D atomic sample .
D.3.1. Wave propagation in an assembly of driven dipoles.
D.3.2. Absorption signal
D.3.3. Light absorption as a quantum scattering process
D.3.4. Beyond the sparse sample case: 3D vs. 2D
D.4. Absorption of light by a slab of atoms
D.4.1. Reaching the ‘thermodynamic limit’
D.4.2. Measured optical density vs. Beer–Lambert prediction .
D.4.3. Absorption line shape
D.5. Conclusion
Appendix E. Historical procedure and results for the calibration of the a imaging coefficient 
E.1. Principle
E.2. Results
Appendix F. A conventional analysis of equation of state of the two-dimensional Bose gas 
F.1. Analysis of the images
F.2. Thermodynamic analysis
F.2.1. EoS for the pressure
F.2.2. EoS for the phase space density
F.2.3. EoS for the entropy
F.3. Measuring the interaction energy
F.4. Conclusion
Appendix G. Aperture effect on the uniform trap implementation via dark mask imaging. 
G.1. Principle of the simulation
G.1.1. Simulating the beam propagation
G.1.2. Analyzing the simulated intensity profile
G.2. Results on a disk
G.2.1. Intensity Profiles
G.2.2. Characterizing trap properties
G.2.3. auxiliary effects: varying the size and the shape of the mask .
G.3. Conclusion
Appendix H. Procedure for Initialization of fringe pattern fit 
Appendix I. Estimate of the collision time in a thermal cloud 
I.1. Definition
I.2. Calculation using Boltzmann predictions
I.2.1. Average velocity
I.2.2. Average density
I.2.3. Collision time
I.3. Calculation using Bose Law
I.3.1. Approximating spatial density dependency

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