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Fermi Superfluids in the BEC-BCS crossover
We continue our description of interacting dilute quantum gases by reviewing interacting fermions. We have discussed earlier the case of an ideal Fermi gas, which is realized in ultracold gases with a single spin component. The addition of a second component allows for s-wave collisions, and many-body effects can play a major role. The use of Feshbach resonances is crucial for that matter. We will thus review in this section the physics of atomic Fermi gases containing two spin states (|↑⟩ , |↓⟩)(i) in equal population, with a Feshbach resonance between them.
Stability of Fermi gases on Feshbach resonances
Before entering into details of Fermi gases on a Feshbach resonance, let us consider their stability. As we discussed in 1.4, this issue is crucial for zero-temperature Bose gases. First, homogeneous BECs are mechanically unstable when a < 0 eliminating half of the accessible range. Second, 3-body recombination dramatically shortens their lifetime when approaching the strongly-interacting regime at the center of the resonance. On the opposite, Fermi gases exhibit a precious stability over most of the range of interactions accessible. The fundamental difference with bosons arises from fermionic statistics with the Pauli exclusion principle. On the macroscopic level it creates the so-called Fermi pressure, famous for preventing neutron stars from gravitational collapse, which protects ultracold Fermi gases from a mechanical collapse.
On the microscopic level, Fermi statistics reduces relaxation processes to bound states which would imply association of two atoms with the same spin. Thus in a gas containing two spin states, dimers can be formed but three-body processes are strongly hampered. The main loss mechanism for a > 0 has been shown to be dimer-dimer collisions with a relaxation rate γ decreasing with increasing interactions: γ ∝ a−2.55 [Petrov et al., 2004]. As a result the lifetime of Fermi gases on Feshbach resonances is long compared to equilibration as was demonstrated experimentally in [Bourdel et al., 2003, Regal et al., 2004a]. This confirms that Fermi gases can span the different regimes of interactions accessible on a Feshbach resonance, we will qualitatively describe these regimes in the following.
The BEC-BCS crossover
The zero-temperature Fermi gas on a Feshbach resonance realizes the so-called BEC-BCS crossover [Zwerger, 2012]. The experimental realization of such a crossover, proposed early by Leggett, and Nozières and Schmitt-Rink [Leggett, 1980, Nozières and Schmitt-Rink, 1985], has been extremely valuable for the advance of the fermionic many-body problem. It first makes the connection between Bardeen-Cooper-Schrieffer superfluidity of conventional superconductors on the one end, and Bose-Einstein condensation at the other end. Second, in the midst of the resonance, the regime where the scattering length becomes larger than the inter-particle distance (na3 > 1) can be studied, contrarily to the actual situation on Bose gases. At its center lies the unitary Fermi gas. Ultracold Fermi gases are thus used as a quantitative test bench for many-body theories benefitting from their high controllability and the – apparent – simplicity of their constituents. At zero temperature when the scattering length describes fully the two-body interactions, the only additional parameter describing the gas is the Fermi energy EF = ℏ2k2F/2m. kF is the Fermi wave vector proportional to the inverse of the inter-particle distance (kF = (3π2n)1/3) in a homogeneous system. All measurable quantities are function of one universal dimensionless parameter 1/kFa which is formally equivalent to the parameter na3 introduced for bosons(j).
For small and positive scattering length (1/kFa > 1), fermions pair-up into the bosonic universal shallow dimer with binding energy ℏ2/ma2. The gas is thus in the strongly attractive limit(k), but due to the Pauli exclusion principle the interaction between dimers is repulsive, with effective dimer-dimer scattering length ad = 0.6a [Petrov et al., 2004]. These dimers then form a Bose-Einstein condensate at zero temperature as was observed experimentally by [Greiner et al., 2003, Jochim et al., 2003b, Zwierlein et al., 2003, Regal et al., 2004b, Bourdel et al., 2004]. It was shown in [Leyronas and Combescot, 2007] that the equation of state of the dimer gas is that of a BEC, including the Lee-Huang-Yang correction. This has been demonstrated experimentally on the lithium experiment [Navon et al., 2010], in a study of the thermodynamics of the Fermi gas through the BEC-BCS crossover. This BEC limit is asymptotically valid when the dimers are effectively tightly bound, that is for 1/kFa ≫ 1. When the size of the dimers become of the order of the inter-particle distance 1/kFa ∼ 1, the repulsive Bose gas picture does not hold anymore.
Superfluidity of Bose and Fermi gases
After giving an account of the thermodynamical properties of interacting Bose and Ferm 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].
Beyond Landau’s criterion
The critical velocity calculated from Landau’s criterion must be handled with care. First it describes a homogeneous superfluid. Already in an inhomogeneous system such as a trapped gas, the geometry of the superfluid modifies the dispersion relation of elementary excitations with respect to the homogeneous case [Zaremba, 1998, Stringari, 1998].
In turn this reduces strongly the critical velocity compared to the naive expectation of the sound velocity given the density n at the center of the cloud: vc ̸= √ gn(0)/m [Fedichev and Shlyapnikov, 2001]. So already, taking finite-size and geometry effects into account changes the expected value of the critical velocity, however this is still within the scope of Landau’s criterion.
As said before, Landau’s criterion is valid in the weak coupling limit. Experimentally, the ‘impurities’ are rarely weakly coupled. For a strong coupling, higher-order excitations can be created. Notoriously ‘hydrodynamic’ excitations such as vortices and solitons can appear. Creation of vortices or solitons is dependent on the geometry of the problem, if we consider a superfluid moving inside a container, its proportions and form will influence the creation of defects. Identically the shape and size of a moving defect will determine its critical velocity [Frisch et al., 1992]. It is known for instance that in most cases critical velocities measured in liquid helium cannot be described by Landau’s criterion and that turbulence sets in, see for example [Wilks and Betts, 1987]. Only in experiments performed with moving ions inside 4He, the critical velocity measured is in agreement with Landau’s criterion, where the dispersion relation presents a roton minimum [Allum et al., 1976, Allum et al., 1977]. An argument originally formulated by Feynman treats a vortex or a vortex ring as an elementary excitation (replacing the momentum p by the moment of inertia I) in order to salvage Landau’s criterion [Feynman, 1955]. It is valid only in a cylindrical geometry and in general this cannot be applied and a specific hydrodynamic analysis is required.
In dilute ultracold gases, homogeneous systems have been created only very recently [Gaunt et al., 2013] and so far all experiments studying superfluid flow have been performed on inhomogeneous gases. Most searches for superfluid flow have been performed by moving an impurity created by a laser beam (with infinite effective mass) focused at the position of the ultracold gas. The laser creates a potential usually large with respect to inter-particle distance, and the impurity created this is not necessarily weakly coupled. The local density approximation can be used to calculate the local density in the region where the laser creates an additional potential and then extract the local critical velocity [Watanabe et al., 2009] from Landau’s criterion. A possible instability due to vortex creation can be investigated analytically in simple cases [Frisch et al., 1992] or using for instance hydrodynamic numerical simulations [Stießberger and Zwerger, 2000]. We review the experimental results obtained until present in the following.
Some experiments on a crical velocity in superfluid dilute gases
On dilute Bose-Einstein condensates, a critical velocity for the motion of a laser impurity has been first observed in [Raman et al., 1999, Onofrio et al., 2000, Raman et al., 2001]. The observed critical velocity lies around 0.3 cb,0 where cb,0 is the sound velocity at the center of the cloud. Considering the inhomogeneity of the gas perturbed by the laser, this value of the critical velocity did not allow for a comparison with theory and so could not determine the dissipation mechanism. Other measurements realized on a superfluid ring stirred by a laser showed the influence of the barrier height on the critical velocity in good agreement with a model using Landau’s criterion and accounting for a reduced density at the barrier position [Wright et al., 2013]. The creation of vortices by a moving laser beam was first observed in [Inouye et al., 2001]. In an elegant experiment, Neely et al. demonstrated the formation of vortices by a moving laser impurity in an oblate geometry [Neely et al., 2010] with a critical velocity in very good agreement with numerical simulations (vc ≃ 0.1c). The creation of solitons by a moving elongated potential barrier was observed in [Engels and Atherton, 2007].
Finally, in a two-dimensional Bose gas the existence of a critical velocity has been experimentally proved by stirring a laser at constant radius, the two-dimensional geometry allowing to probe a constant density [Desbuquois et al., 2012].
It is clear in the experiments on Bose-Einstein Condensates listed above that the usage of a laser as an impurity does not realize the weak-coupling limit required for Landau’s criterion to be applicable. To make a more weakly coupled impurity, [Chikkatur et al., 2000] have used a stimulated Raman process to turn condensate atoms into impurities expelled by the trapping potential. They have measured the energy dissipated by the impurities as a function of their velocity, and observed a dramatic reduction for velocities under the speed of sound of the condensate. The critical velocity of microscopic, weakly-coupled impurities is thus in agreement with Landau’s criterion, identically to the case of liquid 4He where the motion of ions agreed with this criterion.
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 gases
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
