Local spin transfers with a momentum kick 

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The experimental set-up has been designed and built to produce ultracold bidimensional uniform samples of rubidium atoms. This chapter aims at giving an overview of this set-up.
When I arrived in the group, the experiment was being built for several months, and I participated to the end of this building. Thorough descriptions of the structure of the experiment have been delivered by previous Master and Ph.D. students, and the reader can refer to these works to have detailed information:
The Masterarbeit of Katharina Kleinlein who worked on the early laser system, the vacuum system and the first cooling steps [95].
The Ph.D. thesis of Laura Corman, who detailed the whole scheme to create a uniform bidimensional (2D) sample [64].
The Ph.D. thesis of Jean-Loup Ville, who detailed the manipulation of the internal state, as well as important calibrations of our 2D gas [65].
I develop here all the elements that are useful in order to understand the experiments presented in this thesis. In particular I detail some features that were not discussed in these works. In a first part, the basic functioning of the experiment is presented without going too much into the technicalities. The second part details how the cloud is confined in two dimensions and how a uniform density is maintained. Finally, I explain how the internal state and the temperature of the cloud are prepared prior to investigating its properties.

overview of a cold atom experiment

General features

Electronic structure of 87Rb

Rubidium belongs to the alkali group and has a simple electronic structure. Lasers addressing its electronic transitions are easy to access, enabling cooling schemes as well as optical trapping methods. All these advantages make this atom a widely-spread tool in the cold atoms community. The electronic ground state of the atom is 2S1=2 and its first excited state has two fine levels: 2P1=2 and 2P3=2 . The transition 2S1=2 ! 2P1=2 (resp. 2S1=2 ! 2P3=2) is called the D1 (resp. D2) line and has a wavelength of 795 nm (resp. 780 nm).
We are only interested here in the 2S1=2 and 2P3=2 states, that we use for our cooling scheme. Each of these levels experiences hyperfine splitting due to the coupling between the electron and the spin of the nucleus: the 2S1=2 level splits into the F = 1 and F = 2 level, separated by 6:834 GHz, and the 2P3=2 level splits into four levels labelled from F 0 = 0 to F 0 = 3.
The two hyperfine states of 2S1=2 are the ones in which we perform the interesting physics. They experience Zeeman splitting in the presence of a magnetic field B: the F = 1 state splits into three levels, labelled with mF = 1; 0; 1, and the F = 2 state splits into five levels, labelled with mF = 2; 1; 0; 1; 2. In the low magnetic field regime where the quadratic Zeeman shift is negligible, the displacement of the energy of each level is E = BgF mF B, with B the Bohr magneton, and gF the Landé factor of the hyperfine state of interest. For F = 1 (resp. F = 2), we have gF = 1=2 (resp. 1=2). For magnetic fields around a few gauss, these displacements correspond to frequencies on the order of a few megahertz.
All these features of the electronic structure of rubidium are summarized on Fig. 2.1, where only the relevant states are depicted.


We use lasers with four different optical wavelengths to reach the quantum regime from a metallic sample, to trap them in a controlled geometry and to probe the properties of 2D samples.:
Two lasers at 780 nm to address the D2 line. They are used for the cooling schemes summarized in 2.2.1, and for the imaging of the cloud explained in 2.1.2. We use saturated absorption on a vapour of rubidium to lock the frequency of these lasers with a precision of a few hundreds of kilohertz.
Two lasers at 1064 nm, red-detuned with respect to the D1 and D2 lines, to create conservative attractive potentials (see 2.2.1), referred to as optical dipole traps.
A laser at 532 nm, blue-detuned with respect to the D1 and D2 lines, to create conservative repulsive potentials and shape the final geometry of the cloud (see 2.2.2 and 2.2.3).
A laser at 790 nm, between the D1 and D2 lines, to perform Raman transfers between the two lowest hyperfine states of the atom, as developed in Chapter 3.

Vacuum system

The experiments are performed in a compact vacuum system where a high vacuum is maintained. All experimental steps are performed in a single rect-angular glass cell with high optical access and dimensions 25 25 105 mm. In particular, the early cooling steps to prepare the atomic sample and the experiments performed on it are done at the same position, which prevents any technical difficulty due to the transport of the cloud between different regions of space.
Using a glass cell also allows to have coils and optical elements very near the atoms and outside the cell. Having these tools near the atoms permits to reach higher magnetic fields and optical numerical apertures, and not having them in the vacuum cell is technically easier to develop.
The glass cell is represented seen from three sides on Fig. 2.2, and as many elements as possible that are described in this chapter are depicted on these drawings.

Magnetic fields

The magnetic field in the glass cell has to be well-controlled in order to implement some of the cooling stages (see 2.2.1) and to control the energy splitting between Zeeman states of the atoms (see 2.3.1) and reliably address transitions between these states. There are several pairs of coils to achieve these tasks:
A pair of water-cooled coils in anti-Helmholtz configuration along the vertical (z) axis. They produce a quadrupolar field for our quadrupole magnetic trap with a maximal vertical gradient of 240 G=cm. A pair of water-cooled coils in anti-Helmholtz configuration along the y axis. They produce a quadrupolar field for our magneto-optical trap (MOT) with a maximal gradient of 22 G=cm.
Three pairs of coils in Helmholtz configuration along the three axes, to create bias fields. The pair on the vertical axis (resp. horizontal axes) creates a maximum bias field of 2 G (resp. 1 G). These coils are located around the glass cell, but they are not represented on Fig. 2.2. The intensity is provided by power supplies (Delta Elektronika ES 030-5) with a relative intensity noise of 10 4, which corresponds to a fluctuation of magnetic field of 0:2 (resp. 0:1) mG.

Control of the experiment

The whole experiment is computer-controlled. The free software Cicero-Word generator is used to generate a series of instructions. These instructions are transmitted to several National Instruments cards that send ‘analogical’ and digital signals to all the instruments we use (lasers, power supplies, generator, etc.). The different cards are synchronized by a FPGA module that acts as a clock.
The temporal resolution of all the signals is 1 µs, and their temporal accuracy is below 100 ns. The ‘analogical’ signals are in fact digitized with a resolution of approximately 2 mV, and they are bounded between 10 and 10 V. The digital signals provide 0 or 5 V.

Taking pictures of atoms

The determination of the density distribution of the cloud is realised via absorption imaging on the state F = 2. A laser pulse is tuned on the closed transition from F = 2 to F 0 = 3, which has a frequency !L , a linewidth and a wavelength L = 2 c=!L with c the speed of light. It is sent on the atoms during a few tens of microseconds. The atoms scatter photons in all directions of space and a camera collects the remaining ones in the forward direction.
In the regime of low saturation, where the intensity I of the laser probe is small compared to the saturation intensity Isat = ~!L3 =(12 c2), and in the regime of dilute clouds where the atomic density n3D is small compared to L 3, the intensity collected on the camera is given by the Beer-Lambert law: Iwith = I exp dl n3D + Ibgd; (2.1)
where is the scattering cross-section of the atoms, whose value is discussed in 2.2.3, and the integral is performed along the direction of the probe laser. Ibgd represents any spurious light that can hit the camera and pollute our signal.
Another pulse of light without atoms is then sent to the camera to measure Iwithout = I + Ibgd (2.2) and acts as a reference, and an image is also taken without any light pulse to measure Ibgd.
We then numerically compute the optical density OD of the cloud:
OD = ln Iwith Ibgd ; (2.3)
Iwithout Ibgd
and one should get from equation (2.1)
OD=Z dl n3D: (2.4)
Departing from the low saturation regime I Isat broadens the absorption line, and the number of photons each of the atoms scatter is not proportional to I any more, which is an important hypothesis to get the Beer-Lambert law. Moreover, having a too dense cloud where the condition n3D 3L 1 is not experimental set-up fulfilled can lead the light field to excite collective modes of the cloud mediated by dipole-dipole interactions, and the intensity Iwith that we would measure can be very different from the single-atom picture given by the Beer-Lambert law. These effects have been studied into more details during the beginning of my Ph.D. They have been the subject of two publications [96, 97] and they are discussed in the two previous Ph.D. works [64, 65].
In this thesis, we restrict ourselves to the regime where the Beer-Lambert law is valid. More precisely, we use a probe light with I=Isat < 0:2, and we measure optical densities that are always smaller than 1:5 to avoid leaving this regime due to collective excitations. In order to do that, the experiments are performed with the atoms in the F = 1 state, which is not sensitive to the probe light. A controlled fraction of these atoms is transferred in the F = 2 state to be imaged. This fraction is adjusted to limit the optical density to values lower than 1:5. The detail of how these transfers from F = 1 to F = 2 are done is explained in 2.3.1.
For quantitative measurements, we mainly use a vertical imaging to probe the spatial distribution of the bidimensional cloud in the xy-plane. We use a microscope objective with a numerical aperture of 0.45 to have a diffraction-limited resolution around 1 µm on the atoms. We image the atoms with a magnification of 11 with a low-noise CCD camera (Princeton Instruments, Pixis 1024 Excelon) that has an effective pixel size of 1:15 µm on the atoms. This calibration has been performed by combining the imaging of atoms trapped in a periodic potential and the diffraction of these atoms by a periodic lattice. More details are given in [65].

how to produce a uniform 2d bose gas

Cooling atoms down to quantum degeneracy

In order to reach the quantum degeneracy from a hot vapour of rubidium, we follow four successive cooling steps briefly detailed here. The geometrical configuration of the various tools are shown on Fig. 2.2.
In a primary glass cell separated from the science cell by approximately 30 cm, a two-dimensional magneto-optical trap (2D MOT) cools the initial vapour of rubidium in two dimensions of space, y and z. A beam resonant with the D2 line pushes the atoms to the science cell. The rest of the experimental steps are performed in this science cell.
A MOT confines and cools around 109 atoms down to approximately 250 µK. The quadrupolar magnetic field is provided by the horizontal anti-Helmholtz coils. A first pair of laser beams propagates along the y axis, and the two others are in the xz-plane, at an angle of 60 with respect to the vertical axis. After this step, followed by a compressed-MOT step and a molasses step, we are left with 6 108 atoms at a temperature of 15 µK, and they are optically pumped in the F = 1 state.
A strong quadrupolar magnetic field provided by the vertical anti-Helmholtz coils is ramped up (vertical gradient of 240 G=cm). Only the atoms in the sub-state mF = 1 are trapped in this magnetic landscape, and their temperature rises to 200 µK. We proceed to evaporative cooling by sending a radio-frequency field to couple this sub-state to the other Zeeman sub-states. We ramp the frequency from 35 to 2:5 MHz during 12 s to transfer the most energetic atoms to the non-trapped Zeeman states, eject them and let the residual ones rethermalize. At the end of this step we have 2:5 107 atoms at a temperature of 20 µK.
A crossed dipolar trap made of two red-detuned lasers located 50 µm below the centre of the quadrupolar field is turned on. The quadrupolar magnetic field is ramped down and the atoms fall into the optical dipolar trap. We proceed to further evaporative cooling by lowering the depth of this trap. Around 3 105 atoms reach quantum degeneracy at a temperature of 200 nK and in the F = 1; mF = 1 state. A vertical magnetic field around 1 G is maintained to keep the atoms polarised in this state.
All these steps last in total 26 s, and a few more seconds are needed to create a 2D gas and probe its properties.

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Confining the gas in two dimensions

The construction of the vertical confinement of the atoms is crucial to have a 2D gas. The goal is to confine the atoms around a minimum of potential energy in the z direction, to freeze the degrees of freedom along this axis and get a system that effectively evolves in the two other dimensions only. This 2D regime is attained when the vertical frequency !z of the confinement corresponds to an energy ~!z much larger than both the interaction energy Eint and the thermal energy kBT of the cloud. In that case, the atoms occupy only the lowest state of the vertical potential. Their dynamics is frozen in this direction of space and the gas is in the quasi-2D regime.

An optical accordion

We chose to build an ‘optical accordion’ following the scheme published in [98], which is also close to the optical set-up described in [99] and implemented with cold atoms in [100]. It consists in creating a vertical optical lattice with two blue-detuned lasers beams interfering with an angle that can be dynamically varied to change the lattice spacing. The same approach has also been used in [101] by reflecting a beam on a surface with a variable angle.
The building and the optical testing of our accordion was the subject of my master internship in the group, and it was followed by its implementation on the experiment in the very beginning of my Ph.D. To our knowledge, it is the first implementation of an optical accordion of this kind to produce 2D ultracold gases.
I describe here the set-up and give some important technical details that were not discussed in the previous Ph.D. theses.
When using a lattice made of repulsive light (wavelength A = 532 nm), the atoms sit in the dark fringes of the interference pattern. If the lattice is deep enough, the potential that the atoms experience is well approximated by a harmonic potential, with a vertical frequency
!z = r 2U
2md2 ; (2.5)
where U is the maximum potential height of the lattice, m is the mass of an atom and d is the distance between two sites of the lattice. In order to have a high confinement, one therefore needs a lattice with a high potential U and a small lattice spacing d. For a given laser wavelength, the value of U depends only on the intensity we use to create the lattice. In our case we are limited to values of U=kB below 10 µK. If we want to have a vertical confinement of several kilohertz (so that ~!z > kBT ), we need to have a fringe spacing d below 4 µm, so we aim at having d around 1 or 2 µm.
Efficiently loading a large cloud of atoms (size 10 µm) in a single node of a lattice with such a small spacing is a difficult task. The accordion solves this problem with the possibility to dynamically vary the lattice spacing: we load the atoms in a large spacing configuration so that they populate only one site of the lattice, and we adiabatically decrease the spacing to reach a high confinement.
The scheme of the accordion set-up is presented on Fig. 2.3: a single laser beam is separated into two by a first polarising beam splitter. The combination of a mirror, a quarter-wave plate and a second polarising beam splitter makes the two beams parallel. Another polarising beam splitter is added to filter the vertical polarisation and make sure that both beams have the same horizontal polarisation. This element is not represented on Fig. 2.3. The two beams are then sent on an aspherical lens (Asphericon ALL50-100-S-U, labelled ‘L1’ on the figure) of focal length f = 100 mm and they interfere in the glass cell. The waists of the beams at the position of the atoms are wz 40 µm in the vertical direction and wx 90 µm in the horizontal direction. A second aspherical lens, identical to the first one, is used to image the interference pattern on a control camera. The position xb of the incoming beam on the beam splitters can be varied thanks to a mirror mounted on a motorized translation stage (PI miCos LS-100), which changes the distance between the two beams before the aspherical lens and then modifies the lattice spacing on the atoms. The angle 1 between the two interfering beams can be varied between 3 and 15 . The former is limited by the edge of the two polarising beam splitters, and the latter is limited by the aperture of the MOT coils through which the beams pass. The lattice spacing is
A A s f 2
d = = 1 + ; (2.6)
2 sin( 1=2) 2 xb xb;0
which varies between 2 and 10 µ m. In the second expression, the value xb;0 corresponds to the position of the stage where the two beams overlap perfectly. We measured the lattice spacing d on the control camera as a function of xb. The results are presented on Fig. 2.4a.

Limitation due to the quality of the focussing lens

The most important element in the set-up is the quality of the focussing lens. When realising tests with a doublet, we observed that the aberrations of such a experimental set-up lens did not allow to make two beams with a waist smaller than 100 µm cross at the same position when displacing the incoming beam with the translation stage. We could have such beams overlapping and interfering properly on the whole range of the translation stage only with an aspherical lens that corrects for these aberrations.
For this aspherical lens as well, we observe that the centres of the two beams are not exactly at the same vertical position in the focal plane of the lens when varying the lattice spacing. On Fig. 2.4b are reported these vertical positions as a function of the lattice spacing. The biggest distance between them is 20 µm, which is why we chose the vertical waist of the beams wz to be larger than this value.
The way the centres of the two beams are vertically displaced one with respect to the other is very reproducible and is independent of the speed of the translation stage that displaces the light beam. We found that it mostly depends on the surface of the aspherical lens. We have indeed tested several lenses from the same company and with the same specifications, and we measured different relative motions. More precisely, we have purchased successively one pair of lenses, then another pair, and then an additional single lens. These three sets of lenses have been manufactured separately. When testing them, we found good correlations between two lenses belonging to a pair, and much less correlations between the pairs, and between a lens from a pair and the single lens.
We have also performed a simple numerical simulation to check whether defects of the profile of the lens could lead to deflections of the beams on this order of magnitude. The manufacturer provides the equation of the surface of the lens and specifies the tolerance on this surface. The root mean square (RMS) of the irregularities does not exceed 100 nm and the error on the slope of the surface is smaller than 0:06 mrad.
Thanks to Snell’s law, we compute the geometrical path of a ray of light parallel to the optical axis of the lens and hitting it at various positions of the surface and look at the position of the beam in the focal plane of the lens (see Fig. 2.5a). We perform this computation either with the perfect surface profile, or adding to it a small sinusoidal modulation within the specifications of the manufacturer. On Fig 2.5b we show the results of this computation. We find that the displacement of a beam with respect to the optical axis does not exceed 5 µm, which would translate into a distance smaller than 10 µm between two beams symmetric with respect to the optical axis. This simple picture may not grasp all the effects that we experimentally observe. For example a misalignment between the first and the second aspherical lens that we use may increase the measured displacements.
The conclusion is nevertheless that we have good indications that the minute deviations from the optimal surface of the aspherical lens is limiting the accuracy with which we are able to focus the two beams on the same spot. It also constrains the minimal vertical waist of the accordion beams that we can have in order to always maintain a good interference between the accordion beams and keep the atoms in a dark fringe while varying the spacing of the vertical lattice.

Table of contents :

1 introduction 
i producing and manipulating 2d bose gases
2 experimental set-up 
2.1 Overview of a cold atom experiment
2.1.1 General features
2.1.2 Taking pictures of atoms
2.2 How to produce a uniform 2D Bose gas
2.2.1 Cooling atoms down to quantum degeneracy
2.2.2 Confining the gas in two dimensions
2.2.3 Creating a cloud with a uniform atomic density
2.3 How to control the initial state of the cloud
2.3.1 The internal state of the atoms
2.3.2 The phase-space density of the cloud and its temperature
2.4 Conclusion
3 implementation of spatially-resolved spin transfers
3.1 How to induce Raman processes on cold atoms
3.1.1 Elements of theory about two-photon transitions
3.1.2 Our experimental set-up
3.2 Raman transitions without momentum transfer
3.2.1 Measuring Rabi oscillations
3.2.2 Focus and size of the DMD
3.2.3 Local spin transfers
3.3 Raman transitions with momentum transfer
3.3.1 Calibrating the momentum transfer
3.3.2 Local spin transfers with a momentum kick
3.4 Conclusion
ii measuring the first correlation function of the 2d bose gas
4 theoretical considerations on the first correlation function 
4.1 The first-order correlation function of infinite 2D systems
4.1.1 The XY-model and the BKT transition
4.1.2 An ideal gas of bosons in 2D
4.1.3 Interacting bosons in 2D
4.2 Developments for realistic experimental measurements
4.2.1 Exciton-polaritons and out-of-equilibrium effects
4.2.2 Cold atoms and trapping effects
4.2.3 Finite-size effects
4.2.4 Conclusion
5 probing phase coherence by measuring a momentum distribution 
5.1 Measuring the momentum distribution of our atomic clouds
5.1.1 Creating an harmonic potential with a magnetic field
5.1.2 Evolution of atoms in the harmonic potential
5.2 Investigating the width of the momentum distribution
5.2.1 Influence of the initial size of the cloud
5.2.2 Influence of the temperature of the cloud
5.2.3 Determining the first-order correlation function?
5.3 Conclusion
6 measuring g1 via atomic interferometry 
6.1 Interference between two separated wave packets
6.1.1 Free expansion of two wave packets in one dimension
6.1.2 Free expansion of two wave packets in two dimensions
6.2 Setting up and characterising the experimental scheme
6.2.1 The experimental sequence
6.2.2 Measuring the expansion of one line
6.2.3 Measuring the expansion of two lines
6.3 Measuring the phase ordering across the BKT transition
6.3.1 Extracting the contrast of the averaged interference pattern
6.3.2 Results of the measurements across the critical temperature
6.3.3 Discussion and effects that may affect the measurements
6.4 Conclusion
iii dynamical symmetry of the 2d bose gas
7 elements of theory on dynamical symmetries 
7.1 Symmetries of a physical system
7.1.1 The symmetry group as a Lie group
7.1.2 Linking different solutions of a differential equation
7.1.3 Linking solutions of two differential equations
7.2 Dynamical symmetry of weakly interacting bosons in 2D
7.2.1 Symmetry group of the free Gross-Pitaevskii equation
7.2.2 Symmetry group with a harmonic trap
7.2.3 Link between different trap frequencies
7.3 More symmetries in the hydrodynamic regime
7.4 Conclusion
8 an experimental approach of dynamical symmetries
8.1 Experimental sequence
8.1.1 The course of events
8.1.2 The measured observables
8.1.3 Some calibrations
8.2 Verification of the SO(2,1) symmetry
8.2.1 Evolution of the potential energy
8.2.2 Evolution in traps of different frequency
8.3 Universal dynamics in the hydrodynamic regime
8.3.1 Evolution with different interaction parameters
8.3.2 Evolution with different sizes and atom numbers
8.4 Conclusion
9 breathers of the 2d gross-pitaevski i equation 
9.1 Experimental hints
9.1.1 Initial triangular shape
9.1.2 Initial disk shape
9.2 Numerical simulations
9.2.1 Initially triangular-shaped cloud
9.2.2 Initially disk-shaped cloud
9.2.3 Other initial shapes
9.3 Towards an analytical proof?
9.4 Conclusion
10 conclusion 
a coupling two hyperfine states with raman beams
b correlation function of an ideal 2d bose gas
c details on the interferometric measurements of g1
d details on the scaling laws of the 2d bose gas
d.1 Free Gross-Pitaevskii equation
d.2 Gross-Pitaevskii equation with a harmonic trap
d.2.1 General case: a variable trap frequency
d.2.2 Particular case: a constant trap frequency
d.2.3 Invariant transformations
d.3 Hydrodynamic equations
e publications
f résumé en français


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