Relations to the pair and triplet distributions at short distances

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The atomic beam source

The atomic beam is produced by heating a reservoir containing several grams of lithium (with natural isotopic abundance). The reservoir is connected to the rest of the vacuum chamber via a small tube that collimates the jet of atoms. The bottom of the reservoir is heated up to 400C, the entrance of the collimation tube to 510 C and its end to 190 C. In this configuration, lithium is liquid at the bottom of the reservoir and a sufficient atomic flux leaves the oven. In principle, lithium cannot solidify in the tube and the temperature gradient allows lithium droplets to get back to the reservoir thanks to a temperature dependent surface tension. From time to time, the tube gets clogged and further heating (up to 600 C ) of the tube allows us to evaporate the undesired liquid. In order to guarantee an ultra-high vacuum in the main chamber, the atomic beam goes through two differential pumping stages before reaching the Zeeman slower, see Fig. 2.5.

The Zeeman slower

The oven produces a jet of atoms with a mean velocity of 1700m:s􀀀1. The MOT will only capture atoms with velocities less than the capture velocity 50m:s􀀀1. Therefore, we use a Zeeman slower which is a specific combination of counter-propagating bi-chromatic beam and a designed magnetic field in order to reduce the average atomic velocity. After each photon absorption event, the atom gets decelerated and quickly moves out of resonance due to the Doppler effect. This effect is compensated by the spatially-dependent magnetic field along the trajectory of the atoms provided by the coils. Indeed, the energy levels of the atoms are shifted by the Zeeman effect and the magnetic field is finely tailored so that the Zeeman shift constantly compensates the Doppler shift. Our Zeeman slower is in a spin-flip configuration consisting of two coils with opposite sign current, see Fig. 2.5, so that in between the two the magnetic field becomes zero and reversed, which avoids to have resonant light with the cold trapped atoms. The magnetic field between both ends goes from 800G to -200G and the Zeeman slower has a capture velocity of about 1100m:s􀀀1 [129].
Two laser beams are used to slow down 6Li and 7Li atoms (called principal beams). They are tuned to the D2 : F = 3=2; mF = 3=2 ! F0 = 5=2; mF = 5=2 and D2 : F = 2; mF = 2 ! F0 = 3; mF = 3 transitions respectively and shifted by 􀀀400MHz to compensate the Doppler effect at the magnetic field zero crossing (v ‘ 250m:s􀀀1).
The 6Li principal beam slightly deteriorates the slowing of 7Li due to the 7Li-D1 6Li-D2 coincidence, which finally forces a trade-off between the 6Li and 7Li flux. Two repumper beams are also necessary for the zero crossing region for the two following reasons: First, due to the narrow hyperfine structure of the 22P3=2 manifold, the open transitions D2 : F = 3=2 ! F0 = 3=2 and D2 : F = 2 ! F0 = 2 are excited and the atoms can fall in the 22S1=2 F = 1=2 and F = 1 states. Second, there is no adiabatic following near the zero crossing and atoms can flip to the “wrong” mF0 states. We use the transitions D1 : F = 1=2 ! F0 = 3=2 and D2 : F = 1 ! F0 = 2 for 6Li and 7Li respectively, also shifted by -400MHz in order to repump these depolarized atoms.

Magneto-Optical trap

The next step is to trap the slow atoms of the beam and further cool them down in a magneto-optical trap (MOT). It consists of three pairs of circularly polarized counterpropagating laser beams as depicted in Fig. 2.1 and a magnetic field gradient provided by one pair of coils. As a result, the atoms feel a combination of restoring and friction forces that trap them in the vicinity of the zero magnetic field region.
We use the D2 : F = 3=2 ! F0 = 5=2 and D2 : F = 2 ! F0 = 3 lines as cooling transitions for the 6Li and 7Li MOT respectively, similarly to the zeeman slower. We also need strong repumping beams to avoid loosing atoms that fall in the 22S1=2 F = 1=2 and F = 1 states. For this purpose we utilize the lines D1 : F = 1=2 ! F0 = 3=2 and D2 : F = 1 ! F0 = 2 of 6Li and 7Li respectively. The cooling beams are red-detuned by 6P ‘ 􀀀6􀀀 for 6Li and 7P ‘ 􀀀7􀀀 for 7Li, and the repumpers beams by 6R ‘ 􀀀3􀀀 for 6Li and 7R ‘ 􀀀5:5􀀀 for 7Li. Each beam has a 1=e2 diameter of 1:5cm and a peak intensity of about 2 mW/cm2 (The on-resonance saturation intensity of the D1 and D2 lines are Isat = 7:6 mW/cm2 and Isat = 2:5 mW/cm2 respectively). The applied magnetic field gradient at the MOT is 25 G/cm. In these conditions, we are able to load the dual-MOT in 40 s and it typically contains several 109 7Li atoms and 108 6Li atoms at a temperature of approximately 3 mK. In com38 parison, a single-isotope MOT can have twice as many atoms (by switching off the other MOT beams). In particular, the 6Li cooling beams enhance the light-assisted inelastic collisions in the 7Li MOT, and we tune its power to have a good balance between the two isotope numbers. Once the dual-MOT is fully loaded, we perform a compressed-MOT phase. The cooling beams are brought closer to resonance (6P = 􀀀1:5 􀀀 and 7P = 􀀀5 􀀀) and the repumping light intensities are ramped to zero in 8 ms. This results in all atoms being pumped in the lowest hyperfine manifold and a reduced temperature of T ‘ 600 K. We do not reach a temperature as low as the Doppler temperature TD = ~􀀀=2kB = 140 K mainly because of the unresolved hyperfine structure of the 22P3=2 states and multiple photon scattering processes.

Lower magnetic trap and transfer to the appendage

After the optical pumping, we quickly turn on a quadrupole trap (within 2 ms) using the MOT coils (resulting in a magnetic field gradient of 200 G.cm􀀀1 at 300 A). The atoms not in states j6fi and j8bi are lost by spin-exchange collisions and/or expelled by the magnetic trap. The overall efficiency of the optical pumping and magnetic trapping loading is typically 50% for 7Li atoms and 30% for 6Li atoms.
Next, we transport the cloud in the appendage of the cell: We simultaneously ramp up the current in the Feshbach coils and ramp down the current in the MOT coils in 500 ms so that the magnetic trap center is progressively elevated to the appendage. The cloud is transported over 6 cm and during the transfer approximately 50% of the atoms are lost, mainly due to collisions with the walls of the appendage.
Atoms are then transferred from the quadrupole trap to a Ioffe-Pritchard trap [133]: A strong magnetic radial confinement is provided by four bars placed near the appendage, a pair of coils (the pinch-curve coils) create a curvature along the axial direction, and finally another pair of coils (Feshbach coils) create an adjustable bias field (and only a small curvature). This creates a cigar-shaped harmonic trap with a non-zero minimum which avoid Majorana losses [134].

Doppler Cooling

In the Ioffe trap, the cloud has an initial temperature of T ‘ 3 mK. The collisional crosssection is cancelled at momenta corresponding to a temperature of 6mK and is still weak at 3mK[128]. Hence, to start evaporative cooling with better conditions (i.e. a larger collisional rate) we perform an additional Doppler cooling on 7Li atoms. We send on the cloud a circularly polarized light beam red-detuned to the transition D2 : F = 2;mF = 2 ! F0 = 3;mF0 = 3 at the corresponding bias field felt by the atoms (500 G). The beam will solely cool the atoms lying at the center of the trap and in the axial direction. The anharmonicity of the trap and at some point the elastic collisions will lead to a thermalization of the cloud in all spatial directions. This cooling procedure is repeated twice, with a stronger confinement and a smaller detuning the second time. As a result, the cloud has a temperature of 300 K, with a loss of 25% 7Li atoms, and the collisional rate is increased by a factor of 16.

RF evaporation

The last step performed in the Ioffe-Pritchard trap is a radio-frequency evaporation of the 7Li atoms. Evaporative cooling consists in removing the hottest atoms of the cloud so that the mean energy per particle is reduced and the phase-space density is increased [135]. The evaporation cannot be done with 6Li atoms as they can neither collide via s-wave channel since they are spin-polarized in the j6fi state, nor via higher partial wave channels as they are already inhibited by the low temperature of the sample (The p-wave threshold for Li is mK). 6Li atoms are instead cooled by thermal contact with the 7Li atoms that can be efficiently evaporated.
Here, the Ioffe-Pritchard is highly compressed and the bias field is maintained to a low value (5 G). A radio-frequency field blue-detuned with respect to the transition j1bi􀀀j8bi (803.5MHz at B = 0) is sent on the cloud. As a result energetic 7Li atoms are transferred to the high-field seeking state j1bi and expelled from the trap. Thanks to the smaller isotope hyperfine splitting of the 22S3=2 state of 6Li (228.2 MHz), the RF-knife essentially only affect 7Li atoms.
The RF-knife frequency is ramped from 1050MHz down to 840MHz in 22s. Typical final numbers are N6 = 2:5 106 and N7 = 0:5 106 at a temperature T = 10 K which correspond to a peak phase-space density of 10􀀀1 for 6Li atoms. These numbers strongly depends on the initial numbers of both species and in practice we adjust them byvarying the population balance between the two MOT clouds.

Hybrid magnetic-optical trap

The Ioffe-Pritchard trap does not allow for an independent adjustment of the magnetic bias field and trap confinement. Hence, in order to profit from the 832 G Feshbach resonance of 6Li where we can perform a second evaporation to quantum degeneracy, we load the atoms in a hybrid-magnetic optical trap resulting from the combination of a focussed 1073 nm beam and a axial magnetic curvature + bias field created by the pinch-curve and the Feshbach coils.

Trap description

The dominant effect of a far-detuned light beam on an atom is a shift of its energy levels proportional to the beam intensity (known as light shift or Stark shift) [136]. If the beam is red-detuned, the shift is negative for atoms in their ground state and they will be attracted by maxima of intensity. For lithium atoms, if the fine structure splitting of the 22P state is negligible with respect to the laser detuning, the dipole potential created by a beam is well described by the equation : Udip(r) = 3c2 2!3 0 􀀀 I(r).

Table of contents :

Introduction
1 From few to many 
1.1 Two-body problem
1.1.1 Interacting potential
1.1.2 Scattering theory
1.1.3 Pseudo-potential
1.1.4 Feshbach resonance
1.2 Three-body problem
1.2.1 Setting up the framework
1.2.2 Zero-range model
1.2.3 Efimov’s ansatz
1.2.4 Hyperangular problem
1.2.5 Hyperradial problem
1.2.6 Finite scattering length
1.2.7 Adding more bodies
1.3 Universal thermodynamics of the many-body problem
1.3.1 Ideal gases
1.3.2 Interacting bosons
1.3.3 Interacting fermions: The BEC-BCS crossover
2 Producing a dual Bose-Fermi superfluid 
2.1 General description
2.2 The lithium atom
2.2.1 Atomic structure
2.2.2 Feshbach resonances
2.3 Laser system
2.4 Loading the dual magneto-optical trap
2.4.1 The atomic beam source
2.4.2 The Zeeman slower
2.4.3 Magneto-Optical trap
2.5 Magnetic trapping
2.5.1 Optical pumping
2.5.2 Lower magnetic trap and transfer to the appendage
2.5.3 Doppler Cooling
2.5.4 RF evaporation
2.6 Hybrid magnetic-optical trap
2.6.1 Trap description
2.6.2 Trap loading
2.6.3 Mixture preparation
2.6.4 Evaporation at 835 G
2.7 Imaging
2.7.1 Absorption imaging
2.7.2 Imaging directions
2.7.3 Imaging transitions
2.7.4 Double and triple imaging sequences at high field
2.8 Evidence for superfluidity
2.8.1 Bose gas
2.8.2 Fermi gas
2.9 Final trap calibrations
2.9.1 Magnetic field calibration
2.9.2 Trap frequency calibration
2.9.3 Number calibration
2.10 Conclusion
3 Counterflowing mixture of Bose and Fermi superfluids 
3.1 Creating a counterflow of Bose and Fermi superfluids
3.2 Low amplitude oscillations: Coherent energy exchange
3.2.1 Frequency shift
3.2.2 Amplitude modulation
3.2.3 Sum-rule approach
3.2.4 Frequency shift in the crossover
3.3 Large amplitude oscillations: Friction and critical velocity
3.3.1 Simple and generalized Landau criterion for superfluidity
3.3.2 Critical velocity in the BEC-BCS crossover
3.4 Conclusion
4 Numerical simulation of counterflowing superfluids 
4.1 Mathematical and numerical settings
4.1.1 Gross-Pitaevskii equations
4.1.2 Dimensionless equations
4.1.3 Numerical methods
4.1.4 Simulation parameters
4.2 Low amplitude oscillations
4.3 Large amplitude oscillations
4.3.1 First observations: The center of mass evolution
4.3.2 Fluctuation analysis using a principal analysis component
4.3.3 PCA’s modes versus collective excitations of the superfluids
4.3.4 Linearly forced modes
4.3.5 Parametric modes
4.4 Conclusion
5 Contact relations 
5.1 General framework
5.2 The two and three-body contact
5.3 Relation to the tail of the momentum distribution
5.4 Relations to the pair and triplet distributions at short distances
5.4.1 Pair distribution
5.4.2 Triplet distribution
5.5 Relation to the energy
5.6 Extension to statistical mixtures
5.7 Conclusion
6 Universal inelastic losses in cold gases 
6.1 Three-body recombination
6.2 General principles
6.2.1 A general statement
6.2.2 A justification using a microscopic model
6.2.3 Application to some generic cases
6.2.4 Inelasticity parameter
6.3 Scalings for the Bose gas
6.4 Conclusion
7 Inelastic losses in a strongly interacting Bose gas 
7.1 Universal loss dynamics
7.1.1 The model
7.1.2 Analysis of the experimental data
7.1.3 Conclusion
7.2 Momentum distribution of a dilute unitary Bose gas with three-body losses
7.2.1 The model
7.2.2 Comment on the depletion time scale
7.2.3 First virial correction
7.2.4 Comparison to the JILA experiment
7.2.5 Conclusion
8 Inelastic losses of a weakly coupled impurity immersed in a resonant Fermi gas 
8.1 Bose-Fermi losses scalings in the BEC-BCS crossover
8.2 Experiments on the BEC side
8.2.1 Nature of the losses
8.2.2 Loss coefficient measurement on the BEC side
8.2.3 Molecule fraction
8.2.4 Bose-fermi losses versus molecular fraction and magnetic field
8.3 Experiment at unitarity
8.3.1 Loss coefficient measurement
8.3.2 Density dependence of the loss rate
8.4 Concluding remarks and perspectives
Conclusion
A Derivation of the coupled oscillator model using the sum-rule approach 
B Instability domains of modified 2D Mathieu’s equations 
C Numerical simulation of counterflowing superfluids: supplemental data 
D Momentum distribution of a dilute unitary Bose gas: Supplemental material
D.1 Derivation of the loss equations
D.2 Decomposition of the solution over the Laguerre Polynomial basis
D.3 Calculation of C
D.4 Momentum distribution in a harmonic trap
E Analysis of impurity losses of the Innsbruck experiment. 
Remerciements 

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