Ebook: Ultra-cold Fermi Gases

All elementary constituents of everyday matter (electrons, protons and neutrons) are fermions, named after Enrico Fermi who introduced, in 1925 in Florence, the quantum statistics of half-integer spin particles. The Varenna school, which carries the name of Enrico Fermi, has witnessed all major advances in physics since 1953. It has been a special honour for us to organize an Enrico Fermi school on Ultracold Fermi Gases, yet another striking development that even the great scientist could not anticipate.

The list of Varenna schools includes cornerstone courses in atomic physics. After the milestones of laser spectroscopy, the fast advances in the field of cold atoms were timely covered by the 1991 School on Laser Manipulation of Atoms and the 1998 School on Bose-Einstein Condensation in Atomic Gases. Following this tradition, the School on Ultracold Fermi Gases highlighted new developments and discussed exciting new directions. These three summer schools on cold atomic gases mark three distinct periods in the exploration of the ultralow temperature regime.

The field of cold atomic gases faced a revolution in 1995 when Bose-Einstein condensation was achieved. Since then, there has been an impressive progress, both experimental and theoretical. The quest for ultra-cold Fermi gases started shortly after the 1995 discovery, and quantum degeneracy in a gas of fermionic atoms was obtained in 1999. The Pauli exclusion principle plays a crucial role in many aspects of ultracold Fermi gases, including inhibited interactions with applications to precision measurements, and strong correlations. The path towards strong interactions and pairing of fermions opened up with the discovery in 2003 that molecules formed by fermions near a Feshbach resonance were surprisingly stable against inelastic decay, but featured strong elastic interactions. This remarkable combination was explained by the Pauli exclusion principle and the fact that only inelastic collisions require three fermions to come close to each other. The unexpected stability of strongly interacting fermions and fermion pairs triggered most of the research which was presented at this summer school. It is remarkable foresight (or good luck) that the first steps to organize the summer school were already taken before this discovery. It speaks for the dynamics of the field, how dramatically it can change course when new insight is obtained.

This summer school took place after the quest for fermionic superfluidity with ultracold atoms has reached its goal, and high-temperature superfluidity was established in ultracold and ultradilute gases. These new superfluid atomic systems provide an ideal laboratory for investigating quantum many-body phenomena. Atomic physics brings to many-body physics the remarkable control and tunability of interactions, as well as of the spatial order provided by atom traps and optical lattices. This approach has stimulated an explosion of theoretical and experimental advances in the quantum physics of many-body systems. We are witnessing an important convergence of research efforts dealing with open problems in many-body physics, covering fields as diverse as high-energy physics, condensed matter, astrophysics, quantum information, and of course quantum gases.

This school brought together many leaders in both the theory and experiments on ultracold Fermi gases as well as a very large number of enthusiastic students from all over the world and from different fields of research. The lectures, which are written up in this volume, provided a detailed coverage of the experimental techniques for the creation and study of Fermi quantum gases, as well as the theoretical foundation for understanding the properties of these novel systems. Many exciting aspects were presented, including basic static and dynamical properties, molecule formation, superfluid behaviour and BEC-BCS crossover, fermions in optical lattices, and Fermi-Bose mixtures. The timing of the school was excellent since the field is still small enough to be fully covered, but it is also undergoing a major expansion.

This volume provides the first systematic review of the many developments that have taken place since the early beginnings of the field less than a decade ago.

The exciting scientific program of the School was enhanced by the special atmosphere of Lake Como combining in a unique blend water and mountains with historical tradition and culture. We warmly thank our scientific secretary, Francesca Ferlaino for her enthusiastic support, and Barbara Alzani for the professional organisation and her dedication which were crucial to the success of the school.

M. Inguscio, W. Ketterle and C. Salomon

1. Introduction; 2. Weakly interacting Fermi gas; 3. Feshbach resonance; 4. Feshbach molecules; 5. Condensates in a Fermi gas; 6. Exploring the BCS-BEC crossover; 7. Conclusion

1. Introduction; 2. Ideal Fermi gas in harmonic trap; 3. Role of interactions: The BEC-BCS crossover; 4. Equilibrium properties of a trapped gas; 5. Dynamics and superfluidity; 6. Rotating Fermi gases and superfluidity; 7. Conclusions

1. Introduction; 2. Experimental techniques; 3. Quantitative analysis of density distributions; 4. Theory of the BEC-BCS crossover; 5. Feshbach resonances; 6. Condensation and superfluidity across the BEC-BCS crossover; 7. BEC-BCS crossover: Energetics, excitations, and new systems; 8. Conclusion

1. The ideal Fermi gas; 2. Two-body aspects of the interaction potential; 3. Zero-temperature BCS theory: Study of the ground branch

1. Introduction; 2. Bose-Einstein condensation and superfluidity; 3. Description of a superfluid in a dilute atomic gas; 4. Breakdown of the mean-field picture—resonance superfluids; 5. Single-channel vs. two-channel approaches; 6. Poles of the molecular propagator; 7. The equivalent single-channel theory; 8. Connection with the theory of Feshbach resonances; 9. The BCS/BEC crossover; 10. Momentum distribution in a dilute Fermi gas; 11. Imaginary-time methods for single- and two-channel BCS; 12. A mean-field description for the crossover problem; 13. Summary

The use of Feshbach resonances for tuning the interparticle interaction in ultracold Fermi gases has led to remarkable developments, in particular to the creation and Bose-Einstein condensation of weakly bound diatomic molecules of fermionic atoms. These are the largest diatomic molecules obtained so far, with a size of the order of thousands of angstroms. They represent novel composite bosons, which exhibit features of Fermi statistics at short intermolecular distances. Being highly excited, these molecules are remarkably stable with respect to collisional relaxation, which is a consequence of the Pauli exclusion principle for identical fermionic atoms. The purpose of these lectures is to describe molecular regimes in two-component Fermi gases and Fermi-Fermi mixtures, focusing attention on quantum statistical effects.

1. Introduction; 2. Brief history of experiments on strongly interacting Fermi gases; 3. Interactions in a ⁶Li spin mixture; 4. The molecular route into Fermi degeneracy: creation of a molecular Bose-Einstein condensate; 5. Crossover from mBEC to a fermionic superfluid; 6. Collective excitations in the BEC-BCS crossover; 7. Pairing gap spectroscopy in the BEC-BCS crossover; 8. Conclusion and outlook

1. Introduction; 2. Optical lattices; 3. Concept of the experiment; 4. Imaging Fermi surfaces; 5. Interacting fermionic atoms in an optical lattice: the Hubbard model and beyond; 6. Weakly bound molecules in an optical lattice; 7. Bose-Fermi mixtures in a three-dimensional optical lattice; 8. Outlook

Various topics at the interface between condensed-matter physics and the physics of ultra-cold fermionic atoms in optical lattices are discussed. This article starts with basic considerations on energy scales, and on the regimes in which a description by an effective Hubbard model is valid. Qualitative ideas about the Mott transition are then presented, both for bosons and fermions, as well as mean-field theories of this phenomenon. Antiferromagnetism of the fermionic Hubbard model at half-filling is briefly reviewed. The possibility that interaction effects facilitate adiabatic cooling is discussed, and the importance of using entropy as a thermometer is emphasized. Geometrical frustration of the lattice, by suppressing spin long-range order, helps revealing genuine Mott physics and exploring unconventional quantum magnetism. The importance of measurement techniques to probe quasi-particle excitations in cold fermionic systems is emphasized, and a recent proposal based on stimulated Raman scattering briefly reviewed. The unconventional nature of these excitations in cuprate superconductors is emphasized.

We review some of the basic concepts in quantum information processing, including both the gate-based and the measurement-based quantum computation set-ups. We also show how one can perform quantum simulations using a quantum computer. Finally, we review how one can implement some of these ideas using neutral atoms and trapped ions.

These lecture notes discuss two effects which contribute to the reduction of the interference fringe contrast in matter interferometers. The first effect is the shot noise arising from a finite number of atoms used in experiments. Focusing on a single-shot measurement, we provide explicit calculations of the full distribution functions of the fringe contrast for the interference of either the coherent or the number states of atoms. Another mechanism of the suppression of the amplitude of interference fringes discussed in these lecture notes is the quantum and thermal fluctuations of the order parameter in low-dimensional condensates. We summarize recent theoretical and experimental studies demonstrating that suppression of the interference fringe contrast and its shot to shot variations can be used to study correlation functions within individual condensates. We also discuss full distribution functions of the fringe amplitudes for one and two-dimensional condensates and review their connection to high-order correlation functions. We point out intriguing mathematical connections between the distribution functions of interference fringe amplitudes and several other problems in field theory, systems of correlated electrons, and statistical physics.

Although recent theoretical and experimental progress has considerably clarified pairing mechanisms in spin-(1/2) fermionic superfluid with equally populated internal states, many open questions remain when the two spin populations are mismatched. We show here that, taking advantage of the universal behavior characterizing the regime of infinite scattering length, the macroscopic properties of these systems can be simply and quantitatively understood in the regime of strong interactions.

1. Introduction; 2. Mean-field treatment for the homogeneous case; 3. Mean-field treatment for the trapped case; 4. Exact equations in the dilute case; 5. Numerical results and comparison with experiments; 6. Extension to vortices (rotating frame); 7. Perspectives and open problems

1. Introduction; 2. Review: functional integral crossover theory; 3. Application to optical lattices; 4. Application to multi-species Fermi mixtures; 5. Conclusion

1. Introduction; 2. Feshbach resonances in the K-Rb mixture; 3. Feshbach spectroscopy; 4. Three-body losses at a Feshbach resonance; 5. Tuning of the interaction in the quantum degenerate regime; 6. Formation of dimers; 7. Outlook

1. Introduction; 2. Repulsively bound pairs; 3. Analytical solution of two-particle problem in an optical lattice; 4. Numerical approach for repulsively bound pairs; 5. Experimental realization; 6. Experiments; 7. Repulsively bound pairs of fermions; 8. Other related physical systems; 9. Conclusion

1. Introduction; 2. The Clogston-Chandrasekhar limit; 3. The Fulde-Ferrell-Larkin-Ovchinnikov phases; 4. Vicinity of the tricritical point; 5. Collective mode in the BEC-BCS crossover

We review several experimental aspects of ultracold bosonic and fermionic quantum gases in optical lattices. After introducing fundamental aspects of optical lattices, we use the superfluid-Mott insulator transition of ultracold bosonic quantum gases in optical lattices, to highlight the physics of strongly correlated quantum systems. We discuss the coherence properties and recent measurements of the shell structure of in the Mott insulating phase. Furthermore, we show how quantum noise correlations can be used to detect quantum phases of strongly correlated bosonic and fermionic quantum gases in optical lattices.

1. BCS-BEC crossover theory and the physical effects of temperature; 2. Theory outline; 3. Behavior of T_c and trap effects; 4. Experimental evidence for a pseudogap in cold gases; 5. Establishing superfluidity in cold Fermi gases; 6. Fermi gases with imbalanced spin population; 7. Conclusions

1. Introduction; 2. Numerical method; 3. Extrapolation towards continuum system in the thermodynamic limit; 4. Simulation results; 5. Trapped Fermi gas; 6. Conclusions

1. Nuclear matter and QCD; 2. The true vacuum; 3. Homogeneous color superconductivity; 4. Color superconductivity and compact stars; 5. Inhomogeneous color superconductivity: LOFF phase; 6. LOFF phase of QCD with three flavors in the Ginzburg-Landau approximation; 7. Stability of the LOFF phase of QCD with three flavors; 8. Neutrino emission by pulsars and the LOFF state

1. Introduction; 2. Relevant atomic physics; 3. Molecular and collision physics; 4. MOT results; 5. Magnetic trapping and one-dimensional Doppler cooling; 6. Bose-Einstein condensation of helium-4; 7. Fermi degeneracy of helium-3; 8. Mixtures; 9. Prospects

1. Introduction; 2. Description of atomic lattice excitons; 3. Probing atomic lattice excitons; 4. Exciton formation on an optical lattice; 5. Summary and outlook

We present an experimental study of the time-of-flight properties of a gas of ultracold fermions in the BEC-BCS crossover. Since interactions can be tuned by changing the value of the magnetic field, we are able to probe both non-interacting and strongly interacting behaviors. These measurements allow us to characterize the momentum distribution of the system as well as its equation of state. We also demonstrate the breakdown of superfluid hydrodynamics in the weakly attractive region of the phase diagram, probably caused by pair breaking taking place during the expansion.