Magnetic trapping and radio-frequency evaporation

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Superfluidity of Bose and Fermi gases

After giving an account of the thermodynamical properties of interacting Bose and Fermi gases, we turn to the description of superfluidity and its manifestations in theses systems. Both interacting Bose-Einstein condensate and Fermi gases in the BEC-BCS crossover are superfluid at zero temperature, and a description of the superfluid state will be useful later in the study of a mixture of such superfluids and their relative motion.
Superfluidity in general refers to an ensemble of phenomena related to flow properties. Its most famous manifestation is ‘flow without friction’. The first experimental signatures of frictionless flow were seen as the drop of resistance in a piece of superconducting metal [Kamerlingh Onnes, 1913], and in liquid helium, Kapitza, Allen and Misener observed a drop of viscosity below 2.18 K [Kapitza, 1938, Allen and Misener, 1938]. It refers to a state in which the (super)fluid has a non-zero velocity with respect to an external body in contact with it, without any dissipation. This state is not the ground state as demonstrated in [Leggett, 2001], it is actually metastable, with a macroscopically long life-time. A spectacular consequence of flow without friction is the existence of persistent currents in a torroidal (ring shaped) geometry. Such long-lived currents were measured in superconducting rings, where their lifetime could be inferred to be longer than 105 years [File and Mills, 1963]. In ultracold atoms, persistent currents were observed in single-component BECs contained in a ring trap, and the life-time of the current was shown to be limited only by the lifetime of sample itself, extending up to two minutes [Ryu et al., 2007, Beattie et al., 2013].

Lithium isotopes, atomic structure

Lithium has two stable isotopes 7Li, a boson and 6Li, a fermion. The natural abundances are 92.5% for 7Li and 7.5% for 6Li, their masses m6 = 6.015 u and m7 = 7.016 u with u the atomic mass unit u = 1.661 × 10−27 kg [Heavner et al., 2001, Nagy et al., 2006].
Lithium is the third element in the periodic table, and the lightest of the alkali metals (considering that Hydrogen is not an alkali). Alkalis have a single electron in their outermost electronic shell (a s-shell), so they have a simple atomic structure. 6Li has a nuclear spin of I = 1, 7Li of I = 3/2. The energy levels at zero magnetic field of 6Li and 7Li are represented in figure 2.1, the optical transition from the 2s orbital to the 2p is at 671 nm. The fine structure splitting between the 22P1/2 (a) and the 22P3/2 state is about 10 GHz. Incidentally, the isotope shift of the 2S → 2P transition is also of 10 GHz, and the D1 line (22S1/2 → 22P1/2) of 7Li is almost tuned with the D2 line (22S1/2 → 22P3/2) of 6Li. The natural linewidth of the optical transitions is 􀀀 = 2π × 5.9 MHz for both isotopes, from which we can conclude that the hyperfine levels of the 22P3/2 are not resolved since their splitting is smaller than the linewidth 􀀀 (see fig. 2.1).

Double Magneto-Op􀆟cal trap

At the output of the tube we use a spin-flip Zeeman slower, with cooling light on the D2 line of both isotopes. At the end of the Zeeman slower, the atoms have a mean velocity of about 50 m/s. From this jet we load a magneto-optical trap (MOT) of both isotopes. By turning on or off their respective cooling light, we can also selectively trap only 6Li or 7Li in the MOT. The MOT is performed in a glass cell, by three pairs of counter-propagating beams, represented in figure 2.3. The cooling light for 7Li and 6Li is tuned to the D2 line and we use the cycling transitions F = 2 → F = 3 and F = 3/2 → F = 5/2 for 7Li and 6Li respectively (these transitions are used in the Zeeman slower as well). Due to the rather narrow hyperfine structure of the excited state, the open transitions F = 2 → F′ = 2 and F = 3/2 → F′ = 3/2 are also excited and a strong repumper is needed to pump atoms fallen into the lower hyperfine manifolds (F = 1, F = 1/2) back in the cooling cycle.
For 6Li, this repumping is done using the D1 transition F = 1/2 → F′ = 3/2. To avoid having near resonant light for 6Li, we repump 7Li using the D2 F = 1, F′ = 2 transition, as indicated in figure 2.1(e). The cooling beam is detuned by δ = −5 􀀀 for 7Li and −4􀀀 for 6Li, the repumpers are detuned respectively by −4 􀀀 and −2 􀀀. All frequencies are mixed in each beam, which have a 1/e2 diameter of ≃ 1.5 cm. The maximum intensity per frequency is I ≃ 3.5 mW/cm2, except for the 7Li cooling frequency which is amplified by a TA and is about I ≃ 5.5 mW/cm2. This corresponds to I ≃ I′sat/50 for the cooling beams, where I′sat = Isat(1 + (2δ/􀀀)2) is the saturation intensity at the beams detuning and Isat = 2.4 mW/cm2.

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Quadrupole trap, magne􀆟c transport and transfer to the Ioffe- Pritchard Trap

Once the atoms are optically pumped, a quadrupole magnetic trap is turned-on in 2 ms using the MOT coils with opposite currents between the coils. The atoms are then transported in the vertical direction to a small appendage of the glass cell see figure 2.3. This transport is done with two sets of coils: the MOT coils and the Feshbach coils. The zero of the magnetic field is moved by decreasing current in the MOT coils to zero and simultaneously ramping up current in the Feshbach coils, in 500ms. At the end of transport the atoms lie in the center of the Feshbach coils pair, the overall efficiency of this transport is about 40 %, most of the losses are due to the cutting of the density distribution by the appendage walls.
From the quadrupole trap created by the Feshbach coils, the atoms are transferred to a Ioffe-Pritchard trap. This trap is realized by four Ioffe bars which generate a strong field gradient confining the atoms in the radial direction. The axial confinement is provided by the Pinch coils that create a curvature of the magnetic field along the trap symmetry axis.

The hybrid op􀆟cal dipole – magne􀆟c trap (ODT)

The phase space densities at the end of evaporation in the Ioffe-Pritchard trap are close to quantum degeneracy, and evaporating more we could reach the Bose-Einstein condensation threshold for 7Li. But since the background scattering length in the |2, 2⟩ state is negative : abg = −27 a0, reaching this threshold would result in a collapse. Furthermore, in the Ioffe trap, the magnetic field offset and the trap confinement cannot be adjusted independently. For these reasons, we transfer the atomic clouds to a hybrid optical – magnetic trap (dipole trap or ODT). The optical trapping relies on the dipole force [Grimm et al., 2000], which dominates at large detunings over the radiation pressure force used for laser cooling. The dipole potential takes the form Udip = 3πc2 2ω3 0 􀀀 I(r),

Table of contents :

Introduction
􀍙 Bose-Einstein condensates and Fermi superfluids 
1.1 Ideal Quantum Gases
1.2 The Local Density Approximation
1.3 S-wave Interactions and Feshbach Resonances
1.3.1 S-wave interactions
1.3.2 Feshbach resonances
1.3.3 Feshbach resonances in lithium
1.3.4 6Li-7Li interactions and Feshbach resonances
1.4 Interacting Bose-Einstein Condensates
1.4.1 Weakly interacting Bose-Einstein condensates
1.4.2 Approaching the unitary Bose gas
1.5 Fermi Superfluids in the BEC-BCS crossover
1.5.1 Stability of Fermi gases on Feshbach resonances
1.5.2 The BEC-BCS crossover
1.5.3 The equation of state
1.6 Superfluidity of Bose and Fermi gases
1.6.1 Landau’s criterion for superfluidity
1.6.2 Landau’s criterion for a mixture of Bose and Fermi superfluids
1.6.3 Beyond Landau’s criterion
1.6.4 Some experiments on a critical velocity in superfluid dilute gase
1.6.5 Other hallmarks of superfluidity
􀍚 Experimental set-up 
2.1 Lithium isotopes, atomic structure
2.2 Laser cooling
2.2.1 Laser system
2.2.2 The lithium source
2.2.3 Double Magneto-Optical trap
2.2.4 Optical pumping
2.3 Magnetic trapping and radio-frequency evaporation
2.3.1 Quadrupole trap, magnetic transport and transfer to the Ioffe- Pritchard Trap
2.3.2 Doppler cooling of 7Li
2.3.3 Radio-frequency evaporative cooling of 7Li
2.4 The hybrid optical dipole – magnetic trap (ODT)
2.5 Preparation of strongly interacting degenerate gases
2.5.1 Preparing a resonantly interacting Bose gas
2.5.2 Preparing a mixture of Bose and Fermi superfluids
2.6 Imaging
2.7 Calibrations
2.7.1 Imaging Calibration
2.7.2 Frequencies measurement
2.7.3 Magnetic field calibration
􀍛 D􀍙 sub-Doppler cooling of 7Li 
3.1 Grey molasses cooling in a nutshell
3.2 Implementation on 7Li
3.3 The Λ model
3.4 The perturbative approach
3.5 The continued fractions approach
􀍜 Three-body losses in strongly interacting Bose gases 
4.1 Three-body losses
4.2 A glance at Efimov Physics
4.3 Results on the stability of a unitary Bose gas
4.4 Conclusion: stability domain of the strongly interacting Bose gas
􀍝 Mixtures of Bose and Fermi superfluids 
5.1 Two-body interactions in the |1f⟩ , |2f⟩ , |2b⟩ mixture
5.2 Mean-field interactions and phase separation
5.3 Evidences for superfluidity
5.3.1 Thermometry of the mixture
5.3.2 Degeneracy points
5.3.3 Superfluidity of 7Li
5.3.4 Superfluidity of 6Li
5.3.5 Frictionless counter-flow
5.4 Coupled dipole modes, theory
5.4.1 Simple model for the BEC dipole frequency shift
5.4.2 Sum rules and coupled 6Li-7Li dipole oscillations
5.5 Coupled dipole modes, experiments
5.5.1 Dipole modes excitation
5.5.2 Uncoupled oscillations
5.5.3 Coupled oscillations
5.5.4 Discussion of the model
5.6 Damping of the dipole modes and critical velocity
5.6.1 Damping at unitarity
5.6.2 Damping in the BEC-BCS crossover
5.6.3 Friction at finite temperature
5.7 Concluding remarks and prospects with Bose-Fermi superfluid mixtures
Conclusion
Appendix 
References

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