Instability domains of modified 2D Mathieu’s equations 

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Trap frequency calibration

In order to characterize the trapping potential we measure the associated frequencies by exciting the dipole mode of a cloud. This also allows us to check the harmonicity of the trap which is essential for the counterflow experiment described in the next chapter. In the axial direction, we profit from the small position difference between the two curvature minima of the optical potential and the magnetic field. By slowly increasing the optical power, the cloud is adiabatically displaced toward the optical potential minimum and a sudden decrease of the power to its initial value will then make the cloud oscillate in the trap. In virtue of the Kohn’s theorem, if the trap is harmonic the oscillations should last forever. In Fig. 2.10, we show two examples of dipole-mode oscillations. Measuring the oscillations over numerous periods allows for the extraction of the trap frequency with a high precision. Furthermore even for amplitude larger than the cloud size, no damping is visible over 1 s which validates the harmonic approximation in the axial direction.
In the radial direction, we excite the dipole mode by abruptly turning off the optical dipole trap for a short period of time (500 s), the slight displacement induced by the anti-trapping magnetic curvature and the gravity is enough to initiate radial oscillations of the cloud. In order to amplify the oscillations amplitude, we image the cloud after a time-of-flight (which typically increase the amplitude by a factor of 3). Depending on the isotope used and the temperature of the cloud, the oscillations can exhibit a fast damping or long-lived oscillations. In the case of a pure 7Li BEC, there is no damping for amplitudes similar to the cloud sizes of the two isotopes (8 m amplitude while the fermions radial size is 10 m). In contrast, the oscillations of a fermionic superfluid show a strong damping rate as it experiences more easily the anharmonicities of the trap due to its larger radial extent (10 m while the Bose gas has a typical radial extent of 3 m).

Creating a counterflow of Bose and Fermi superfluids

The full experimental sequence to cool our 6Li-7Li mixture down to double degeneracy is described in chapter 2. For the final cooling step, we start with 6 Li atoms in a balanced mixture of their two lowest hyperfine states j1fi & j2fi and 7Li atoms spin polarized in the second lowest state j2bi. The clouds are confined in a strong optical dipole trap, and by lowering the trap depth in the vicinity of the 6Li Feshbach resonance we perform an extremely efficient evaporation and the clouds reach the dual degenerate regime. Typical final atoms numbers are Nf = 3 105 and Nb = 4 104 at a temperature T ‘ 80 nK which is significantly below the critical temperatures for superfluidity of both gases, see section 2.8.
Contrary to liquid 3He-4He experiments, the Bose-Fermi interaction in our system is weak (abf = 40:8a0) and ensures that the superfluid mixture is stable with respect to phase separation. However, this also means that the effect of the interspecies coupling is difficult to see directly on the density profiles (and even more if the profiles are doubly integrated)1.
As we will see, the small coupling between the two superfluids can be actually probed by counterflow experiment.
We create a relative motion between the two superfluids by exciting the center of mass oscillations of the two clouds, a scheme used previously for the study of mixtures of Bose-Einstein condensates [69, 156], mixtures of Bose-Einstein condensates and spinpolarized Fermi seas [157], spin diffusion in Fermi gases [158], or integrability in onedimensional systems [159]. The oscillations are initiated by displacing the clouds from the trap center in the axial direction where the confinement is mostly magnetic and highly harmonic. At high field, both isotopes are in the Paschen-Back regime and have the same energy dependence with magnetic field so that both clouds feel the same trapping potential in the axial direction. As a consequence, due to the different atomic masses of the two clouds, the trapping frequencies are related by2 !f=!b = p 7=6 ‘ 1:08. Hence, as depicted in Fig. 3.1, after being displaced by the same quantity, the clouds oscillate in the trap and progressively acquire a relative motion. The relative velocity reaches its maximal value after ‘ 4:5 periods. For an initial displacement of 100 m, we have typically vrel;max = 1:8 cm/s. The cloud positions are monitored during up to 4s, which represents 60 periods of oscillations and we can determine the oscillations frequency with a typical precision of !=! . 2 10􀀀2.

Low amplitude oscillations: Coherent energy exchange

For a small initial displacement (typically 100 m), contrary to the large damping observed in the Bose-Bose mixtures [69], we observe long-lived oscillations of the Bose- Fermi superfluid mixture. They can extend up to 4 s with no visible damping, see fig (3.2) and our measurement is solely limited by the lifetime of the mixture. Furthermore, the oscillations of the mixture exhibits two features not observed in absence of one of the two isotopes. First, the 7Li oscillation frequency ~!b is downshifted by several percent, while the 6Li frequency ~!f is almost unchanged. At unitarity (832 G), we measure ~!b = 15:00(2) and ~!f = 16:80(2), while the associated bare trapping frequencies are !b = 15:27(1) and !f = 16:80(50). Second, we observe an amplitude modulation on the bosons oscillations at a frequency ‘ (~!f 􀀀 ~!b)=2, which implies a coherent energy exchange between the two superfluids.

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Large amplitude oscillations: Friction and critical velocity

For small initial displacements of the clouds, we observed long-lived oscillations of their center of mass as expected in virtue of the frictionless flow property of superfluids. In sharp contrast, for amplitudes larger than a critical value the BEC oscillations are rapidly damped until a steady state regime is reached, as shown in Fig. 3.4. To extract the damping rate, we fit the data using equations (3.7, 3.8) with phenomenological time-dependent amplitudes d given by:
db(t) = (1 + be􀀀 bt)d; (3.21).
df(t) = (1 + fe􀀀 f t)d.

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. 
List of publications
Remerciements

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