Note: Intro -- Indice -- Preface -- Gruppo fotografico dei partecipanti al Corso -- Fermi gas experiments -- 1. Introduction -- 1.1. Why study ultracold Fermi gases? -- 1.2. Superfluidity -- 1.3. Pairing of fermions -- 1.4. BCS-BEC crossover physics -- 1.5. Status of field -- 2. Weakly interacting Fermi gas -- 2.1. Creating a Fermi gas of atoms -- 2.2. Thermodynamics -- 2.3. Thermometry using the momentum distribution -- 2.4. Thermometry using an impurity spin state -- 3. Feshbach resonance -- 3.1. Predictions -- 3.2. Collisions -- 3.3. Anisotropic expansion -- 3.4. Interaction energy -- 4. Feshbach molecules -- 4.1. Molecule creation -- 4.2. Molecule binding energy -- 4.3. Molecule conversion efficiency -- 4.4. Long-lived molecules -- 5. Condensates in a Fermi gas -- 5.1. Molecular condensates -- 5.2. Fermi condensates -- 5.3. Measurement of a phase diagram -- 6. Exploring the BCS-BEC crossover -- 6.1. Excitations -- 6.2. Atom noise -- 6.3. Thermodynamics -- 7. Conclusion -- Dynamics and superfluidity of an ultracold Fermi gas -- 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 -- Making, probing and understanding ultracold Fermi gases -- 1. Introduction -- 1.1. State of the field -- 1.2. Strongly correlated fermions-a gift of nature? -- 1.3. Some remarks on the history of fermionic superfluidity -- 1.4. Realizing model systems with ultracold atoms -- 1.5. Overview over the sections -- 2. Experimental techniques -- 2.1. The atoms -- 2.2. Cooling and trapping techniques -- 2.3. RF spectroscopy -- 2.4. Using and characterizing Feshbach resonances -- 2.5. Techniques to observe cold atoms and molecules -- 3. Quantitative analysis of density distributions. , 3.1. Trapped atomic gases -- 3.2. Expansion of strongly interacting Fermi mixtures -- 3.3. Fitting functions for trapped and expanded Fermi gases -- 4. Theory of the BEC-BCS crossover -- 4.1. Elastic collisions -- 4.2. Pseudo-potentials -- 4.3. Cooper instability in a Fermi gas with attractive interactions -- 4.4. Crossover wave function -- 4.5. Gap and number equation -- 4.6. Discussion of the three regimes-BCS, BEC and crossover -- 4.7. Single-particle and collective excitations -- 4.8. Finite temperatures -- 4.9. Long-range order and condensate fraction -- 4.10. Superfluid density -- 4.11. Order parameter and Ginzburg-Landau equation -- 4.12. Crossing over from BEC to BCS -- 5. Feshbach resonances -- 5.1. History and experimental summary -- 5.2. Scattering resonances -- 5.3. Feshbach resonances -- 5.4. Broad versus narrow Feshbach resonances -- 5.5. Open channel resonance and the case of [sup(6)]Li -- 6. Condensation and superfluidity across the BEC-BCS crossover -- 6.1. Bose-Einstein condensation and superfluidity -- 6.2. Signatures for superfluidity in quantum gases -- 6.3. Pair condensation below the Feshbach resonance -- 6.4. Pair condensation above the Feshbach resonance -- 6.5. Direct observation of condensation in the density profiles -- 6.6. Observation of vortex lattices -- 7. BEC-BCS crossover: Energetics, excitations, and new systems -- 7.1. Characterization of the equilibrium state -- 7.2. Studies of excitations -- 7.3. New systems with BEC-BCS crossover -- 8. Conclusion -- Basic theory tools for degenerate Fermi gases -- 1. The ideal Fermi gas -- 1.1. Basic facts -- 1.2. Coherence and correlation functions of the homogeneous gas -- 1.3. Fluctuations of the number of fermions in a given spatial zone -- 1.4. Application to the 1D gas of impenetrable bosons -- 1.5. In a harmonic trap -- 2. Two-body aspects of the interaction potential. , 2.1. Which model for the interaction potential? -- 2.2. Reminder of scattering theory -- 2.3. Effective-range expansion and various physical regimes -- 2.4. A two-channel model -- 2.5. The Bethe-Peierls model -- 2.6. The lattice model -- 2.7. Application of Bethe-Peierls to a toy model: two macroscopic branches -- 3. Zero-temperature BCS theory: Study of the ground branch -- 3.1. The BCS ansatz -- 3.2. Energy minimization within the BCS family -- 3.3. Reminder on diagonalization of quadratic Hamiltonians -- 3.4. Summary of BCS results for the homogeneous system -- 3.5. Derivation of superfluid hydrodynamic equations from BCS theory -- Two-channel models of the BCS/BEC crossover -- 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 models -- 11.1. Single-channel BCS theory -- 11.2. Imaginary-time propagation for bosons -- 11.3. Imaginary-time propagation for fermions -- 11.4. Imaginary-time algorithm for the single-channel model -- 11.5. Imaginary-time propagation for the two-channel model -- 12. A mean-field description for the crossover problem -- 12.1. Boson scattering length -- 12.2. Beyond pair correlations -- 13. Summary -- Molecular regimes in ultracold Fermi gases -- Introduction -- 1. Lecture 1. Diatomic molecules in a two-component Fermi gas -- 1.1. Feshbach resonances and diatomic molecules -- 1.2. Weakly interacting gas of bosonic molecules. Molecule-molecule elastic interaction. , 1.3. Suppression of collisional relaxation -- 1.4. Prospects for manipulations with weakly bound molecules -- 2. Lecture 2. Molecular regimes in Fermi-Fermi mixtures -- 2.1. Influence of the mass ratio on the elastic intermolecular interaction -- 2.2. Collisional relaxation. Exact results and qualitative analysis -- 2.3. Molecules of heavy and light fermionic atoms -- 2.4. Crystalline molecular phase -- Ultracold Fermi gases in the BEC-BCS crossover: A review from the Innsbruck perspective -- 1. Introduction -- 2. Brief history of experiments on strongly interacting Fermi gases -- 3. Interactions in a [sup(6)]Li spin mixture -- 3.1. Energy levels of [sup(6)]Li atoms in a magnetic field -- 3.2. Tunability at the marvelous 834G Feshbach resonance -- 3.3. Weakly bound dimers -- 4. The molecular route into Fermi degeneracy: creation of a molecular Bose-Einstein condensate -- 4.1. A brief review of different approaches -- 4.2. The all-optical Innsbruck approach -- 4.3. Formation of weakly bound molecules -- 4.4. Evaporative cooling of an atom-molecule mixture -- 4.5. The appearance of mBEC -- 5. Crossover from mBEC to a fermionic superfluid -- 5.1. BEC-BCS crossover physics: a brief introduction -- 5.2. Basic definitions, typical experimental parameters -- 5.3. Universal Fermi gas in the unitarity limit -- 5.4. Equation of state -- 5.5. Phase diagram, relevant temperatures and energies -- 5.6. First Innsbruck crossover experiments: conservation of entropy, spatial profiles, and potential energy of the trapped gas -- 6. Collective excitations in the BEC-BCS crossover -- 6.1. Basics of collective modes -- 6.2. Overview of recent experiments -- 6.3. Axial mode -- 6.4. Radial breathing mode: breakdown of hydrodynamics -- 6.5. Precision test of the equation of state -- 6.6. Other modes of interest -- 7. Pairing gap spectroscopy in the BEC-BCS crossover. , 7.1. Basics of radio-frequency spectroscopy -- 7.2. RF spectroscopy on weakly bound molecules -- 7.3. Observation of the pairing gap in the crossover -- 8. Conclusion and outlook -- A lab in a trap: Fermionic quantum gases, Bose-Fermi mixtures and molecules in optical lattices -- 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 -- Condensed-matter physics with light and atoms: Strongly correlated cold fermions in optical lattices -- 1. Introduction: A novel condensed-matter physics -- 2. Considerations on energy scales -- 3. When do we have a Hubbard model? -- 4. The Mott phenomenon -- 4.1. Mean-field theory of the bosonic Hubbard model -- 4.2. Incompressibility of the Mott phase and "wedding-cake" structure of the density profile in the trap -- 4.3. Fermionic Mott insulators and the Mott transition in condensedmatter physics -- 4.4. (Dynamical) mean-field theory for fermionic systems -- 5. Ground state of the 2-component Mott insulator: Antiferromagnetism -- 6. Adiabatic cooling: Entropy as a thermometer -- 7. The key role of frustration -- 7.1. Frustration can reveal "genuine" Mott physics -- 7.2. Frustration can lead to exotic quantum magnetism -- 8. Quasi-particle excitations in strongly correlated fermion systems, and how to measure them -- 8.1. Response functions and their relation to the spectrum of excitations -- 8.2. Measuring one-particle excitations by stimulated Raman scattering -- 8.3. Excitations in interacting Fermi systems: A crash course -- 8.4. Elusive quasi-particles and nodal-antinodal dichotomy: The puzzles of cuprate superconductors. , Quantum information processing: Basic concepts and implementations with atoms.

Note: Intro -- The Hydrogen Atom -- Foreword -- Preface -- Table of Contents -- List of Contributing Participants -- Introduction to Simple Atoms -- 1 Historica Remarks -- 2 Precision Physics of Simple Atoms -- 3 Studying the Simple Atoms -- 3.1 Hydrogen and deuterium (see [6,7] and Part VI) -- 3.2 Muonium and Positronium (see [12,13] and Part VII) -- 3.3 Muonic Atoms and Nuclear Structure (see Part VIII) -- 3.4 Nuclear-Structure Independent Differences -- 3.5 Fine Structure in Helium (see [23] and Part VI) -- 3.6 Few-Electron Ions (see [26,27] and Part XI) -- 3.7 Medium-Z Physics (see [26,27] and Part XI) -- 3.8 Higher-Order Corrections -- 3.9 Bound State QED -- 3.10 Exotic Atoms (see [35,36] and Part IX) -- 3.11 Antihydrogen and CPT Violation (Part IX) -- 3.12 Exotic Events -- 3.13 Variation of Constants (Part X) -- 3.14 Precision Frequency Metrology ([38] and Part X) -- 3.15 Determination of Fundamental Constants ([39,40] and Part X) -- 4 About This Publication -- Acknowledgements -- References -- Precision Spectroscopy of Atomic Hydrogen -- 1 Introduction -- 2 The Hydrogen 1S -2S Transition -- 2.1 Hydrogen 1S -2S Two-Photon Spectroscopy in an Atomic Beam -- 2.2 Theoretical Line Shape Model -- 2.3 Optical Lamb Shift Measurements -- 2.4 Absolute Measurements of the 1S -2S Transition Frequency in Atomic Hydrogen -- 2.5 1S-2S Isotope Shift and the Deuteron Structure Radius -- 3 Spectroscopy of the2S-nS and 2S-nD Transitions -- 3.1 Method -- 3.2 Optical Frequency Measurements in Paris -- 3.3 Comparison of the 1S-3S and 2S-6 S/D Transitions -- 4 Determination of the Rydberg Constant and Lamb Shifts -- 4.1 Rydberg Constant -- 4.2 Lamb Shifts -- 5 Conclusion and Prospects -- References -- Ultracold Hydrogen -- 1 Introduction -- 2 Ultracold Hydrogen Research at MIT -- 2.1 The Road to Bose-Einstein Condensation -- 2.2 Two-Photon 1S-2S Spectroscopy. , 2.3 Bose-Einstein Condensation -- 2.4 High Resolution Spectroscopy in Ultracold Hydrogen -- Acknowledgments -- References -- Review of High Precision Theory and Experiment for Helium -- 1 Introduction -- 2 Principal Effects -- 3 Nonrelativistic Wave Functions -- 3.1 Recent Advances -- 4 Asymptotic Expansions -- 5 Relativistic Corrections -- 6 Quantum Electrodynamic Corrections -- 6.1 Electron-Nucleus Terms -- 6.2 Electron-Electron Terms -- 6.3 Higher Order Terms -- 7 Comparison with Experiment -- 7.1 Measurement of the Fine Structure Constant -- 7.2 Applications to Lithium -- 8 Concluding Remarks -- Acknowledgements -- References -- Spectroscopy of the Muonium Atom -- 1 Introduction -- 2 Muonium Formation -- 3 Ground State Hyperfine Structure -- 4 1s-2s Energy Interval -- 5 Connection to a New Measurement of the Muon Magnetic Anomaly -- 6 Muonium-Antimuonium Conversion -- 7 Long Term Future Possibilities -- 8 Conclusions -- 9 Acknowledgements -- References -- Experimental Tests of QED in Positronium: Recent Advances -- 1 Introduction -- 2 Decay Rates -- 2.1 Para-Positronium Decay Rate λ (1^1 S_0) -- 2.2 Ortho-Positronium Decay Rate λ (1^3 S_1) -- 3 Energy Level Intervals -- 3.1 Ground State Interval -- 3.2 Rydberg Interval -- 3.3 Intervals in the n=2 and 3 excited states -- 4 Summary and Conclusions -- References -- A New Type of Frequency Chain and Its Application to Fundamental Frequency Metrology -- 1 Introduction -- 2 Kerr-Lens Mode-Locked Lasers -- 3 Femtosecond Frequency Combs -- 4 Spectral Broadening by Self-Phase Modulation -- 5 Photonic Crystal Fibers -- 6 Phase-Locking the Frequency Comb -- 7 Self-calibrated Optical Combs: Absolute Optical Frequencies -- 8 Accuracy Tests of the fs Laser Comb Approach -- 9 The Fine Structure Constant α -- 10 Conclusion -- Acknowledgement -- References -- Fundamental Constants and the Hydrogen Atom*. , 1 Introduction -- 2 1998 Least Squares Adjustment -- 3 Electron Magnetic Moment Anomaly -- 4 Rydberg Constant -- 4.1 Theory Relevant to the Rydberg Constant -- 4.2 Self Energy -- 4.3 Two-Photon Corrections -- 4.4 Finite Nuclear Size -- 4.5 Total Energy and Uncertainty -- 4.6 Result of LSA for the Rydberg Constant -- 5 Conclusion -- References -- Present Status of g − 2 of Electron and Muon -- 1 Introduction -- 2 Electron Magnetic Moment Anomaly -- 3 Muon Magnetic Moment Anomaly -- 4 Improving the alpha^4 Term of the Electron g-2 -- 5 Concluding remarks -- Acknowledgments -- Appendix: VEGAS and Feynman Integral -- References -- Laser Spectroscopy of Hydrogen-Like and Helium-Like Ions -- 1 Introduction -- 2 Fast-Beam Laser Resonance Technique -- 2.1 Signal Formation -- 2.2 Co-linear Geometry and Kinematic Compression -- 2.3 Determination of the Beam Velocity: Doppler-tuned Spectroscopy with Co- and Counter-Propagating Laser Beams -- 2.4 Wavefront Curvature Effects -- 2.5 Alternatives to the Beam-Foil Technique -- 3 Hydrogen-like Ions -- 3.1 Lamb Shift -- 3.2 Experimental Considerations -- 3.3 Lamb Shift Measurements in F^8+, P^14+, S^15+ and Cl^16+ -- 3.4 Future Prospects for Laser Lamb Shift Measurements -- 3.5 Ground-state Hyperfine Structure of High-Z Hydrogen-like Ions -- 4 Helium-like Ions -- 4.1 Experimental Considerations -- 4.2 2^3 S_1 - 2^3 P_J Transitions in Li^+ ,Be^2+ and B^3+ -- 4.3 2^1 S_0 - 2^3 P_1, 2^3 P_0 Intercombination Transitions in N^5+ -- 4.4 2^3 P_J - 2^3 P_J' Fine Structure Transitions in F^7+ and Mg^10+ -- 4.5 Future Prospects -- 5 Conclusions -- Acknowledgments -- References -- The g Factor of Hydrogenic Ions: A Test of Bound State QED -- 1 Introduction -- 2 Summary of Theory -- 3 Experiment -- 4 Results -- 5 Future Prospects -- 5.1 Electron Mass -- 5.2 Fine Structure Constant -- 5.3 Electron Binding Energies. , 5.4 Nuclear Magnetic Moments -- 5.5 Lithium-like Ions -- Acknowledgements -- References -- Elementary Relativistic Atoms -- 1 Introduction -- 2 Atom consisting of π and µ mesons -- 2.1 A_pi mu properties -- 2.2 Observation of A_πµ -- 2.3 Measurement of the A_πµ formation rate in K^0_L decay -- 3 Ultrarelativistic positronium atoms (A_2e) -- 3.1 Source of the ultrarelativistic A_2e and their quantum numbers -- 3.2 Superpenetration of ultrarelativistic atoms -- 3.3 Time-of-formation effects in production of ultrarelativistic A_2e -- 3.4 Observation of ultrarelativistic positronium and measurement of the branching ratio for the pi°-mesons decay... -- 3.5 Measurement of the total cross section for interaction of ultrarelativistic positronium atoms with carbon. -- 4 π^+ π^- atom -- 4.1 A_2π production and lifetime -- 4.2 A_2π detection method and setup description -- 4.3 Data processing -- 4.4 Approximation procedure for the pi^+ pi^ - pair distribution A_2pi number obtaining -- 4.5 Status of A_2π investigation at CERN -- 4.6 A_piK as a source of model-independent data on piK S-wave scattering lengths -- Acknowledgements -- References -- Antiprotonic Helium - An Exotic Hydrogenic Atom -- 1 Introduction -- 2 Unique Facets of Antiprotonic Helium -- 3 Advanced Theories -- 4 Laser Spectroscopy of Antiprotonic Helium -- 5 Precise Determination of Transition Energies -- 6 Chemical Physics Aspects -- 6.1 State Dependent Lifetime Shortening -- 6.2 Pressure Shifts of Resonance Lines -- 6.3 Quenching with H_2 Admixtures -- 6.4 Hydrogen-Assisted Inverse Resonances and Individual Quenching Rates -- 7 Hyperfine Structure -- 8 The Future -- References -- Towards a Precise Measurement of the He+ 2S Lamb Shift -- 1 Introduction -- 2 Overview of Lamb Shift Calculations -- 3 The Experimental Method -- 3.1 The UV Ligh Source -- 3.2 The Interaccion Region. , 3.3 The He^+2S Ion Source and Beam -- 3.4 Metastable Detection -- 3.5 The Reference Laser System -- 3.6 Detection of the Two-Photon Transition -- 4 Conclusion -- 5 Acknowledgments -- References -- High Precision Measurements on Helium at 1083 nm -- 1 Helium and Fundamental Physics -- 2 Fine Structure of the Helium 2^3 P Level: Experiments -- 2.1 The Florence Experiment -- 2.2 State of the Art of the 2^3 P Helium Splittings -- 3 Hyperfine Iodine Transitions at 541 nm: a New Frequency Reference for Helium Spectroscopy -- 4 Conclusions and Final Remarks -- Acknowledgments -- References -- Absolute Frequency Measurement of the 1S-3S Transition in Hydrogen -- Conclusion -- References -- 2s Hyperfine Structure in Hydrogen Atom and Helium-3 Ion -- 1 Introduction -- 2 Theory -- 2.1 Non-recoil limit -- 3 Present status of D21 theory -- 3.1 Old theory and recent progress -- 3.2 Our results -- 4 Present status -- Acknowledgments -- References -- Three-Loop Slope of the Dirac Form Factor and the 1S Lamb Shift in Hydrogen -- 1 Introduction -- 2 Strategy of the calculation -- 3 Basics of the program -- 4 Results -- 5 Acknowledgments -- References -- Radiative Decay of Coupled States in an External dc Field -- 1 Theory and results -- References -- Atomic Interferometer and Coherent Mixing of 2S and 2P States in the Hydrogen Atom -- 1 Atomic interferometer method -- 2 Discussions -- References -- Ground State Energy of the Helium Atom -- 1 Introduction -- 2 Framework of the calculation -- 3 Soft Scale Contributions -- 3.1 Irreducible corrections -- 3.2 Reducible Corrections -- 3.3 Total soft scale contribution -- 4 Hard Scale Contributions -- 4.1 Radiative Recoil Correction -- 4.2 Radiative Corrections -- 4.3 Pure Recoil Correction -- 5 Conclusion -- Acknowledgments -- Appendix -- References -- Two-Loop Corrections to the Decay Rate of Orthopositronium. , 1 Introduction.

Note: Title Page -- Indice -- Preface -- Gruppo fotografico dei partecipanti al Corso -- A brief history of our understanding of BEC: From Bose to Beliaev -- Introduction -- A. Einstein -- F. London -- L. Tisza -- L. D. Landau -- N. N. Bogoliubov -- O. Penrose and L. Onsager -- R. P. Feynman -- Golden era -- Concluding remarks -- Experiments in dilute atomic Bose-Einstein condensation -- Introduction -- Why BEC? -- BEC in an ideal gas -- Some real systems -- Why BEC in dilute BEC is hard -- History -- Conceptual beginnings -- Spin-polarized hydrogen -- Laser cooling and the ascendancy of alkalis -- Cold reality: limits on laser cooling -- Evaporative cooling and the return of hydrogen -- Hybridizing MOT and evaporative cooling techniques -- Collisional concerns -- Increasing elastic collisions per background loss -- Multiple loading -- Adiabatic compression -- Enhancing MOT density -- Reducing background loss -- Forced evaporation -- Magnetic trap improvements -- Imaging techniques -- Survey of BEC technology, present and future -- Magnetic (and other) traps -- Atomic species -- Precooling and compression -- MOT loading -- Chamber design -- Imaging technique -- Effects of interactions -- Energy and size -- Two-species condensates -- Negative scattering length -- Finite temperature -- Critical temperature -- Energy content -- Density distributions -- Future directions -- Finite T -- Feshback resonances -- Vortices -- Many-body effects -- A perspective -- Excitations -- Probing excitations -- Driving the excitations -- Connection to theory -- Finite T -- Nonlinear excitations -- Many-body effects -- Perspective -- Condensates and quantum phase -- A common misconception -- The basic phase thought experiment -- How is a condensate coherent? -- Relative phase of two condensates -- Noncondensate clouds -- Coherence and decoherence. , Mixtures and superpositions -- A final warning on vocabulary -- Stray heating: a pessimistic note -- Introduction to the problem -- Miscellaneous small effects -- Anti-evaporation -- Field noise/shaking trap -- Stray rf and optical fields -- Black-body radiation -- Collisions with background atoms -- Direct creation of "hot" atoms -- Heating from secondary scattering -- Collisions with the products of inelastic decay -- The Oort cloud paradigm -- Collisions and evaporation: an optimistic note -- Collisional scaling predicts success! -- Background loss -- Three-body recombination -- Dipolar relaxation -- Except of course... -- But on the other hand... -- Making, probing and understanding Bose-Einstein condensates -- Introduction -- Basic features of Bose-Einstein condensation -- Length and energy scales -- BEC of composite bosons -- Bose-Einstein condensation as thermal-equilibrium state -- Macroscopic wave function -- BEC 1925-1995 -- BEC and condensed-matter physics -- Spin-polarized hydrogen -- Alkali atoms -- Cooling, trapping, and manipulation techniques -- Pre-cooling -- Standard laser cooling techniques -- Cryogenic pre-cooling -- Conservative atom traps -- Magnetic trapping -- Quadrupole-type traps -- Ioffe-Pritchard traps -- Mode matching -- Adiabatic compression -- Figure of merit for magnetic traps -- Evaporative cooling -- Trap loss and heating -- Manipulation of Bose-Einstein condensates -- Magnetic fields -- Optical dipole forces -- Rf fields -- Bragg and Raman transitions -- Excitation of sound -- Atoms for BEC -- Techniques to probe Bose-Einstein condensates -- Atom-light interactions -- Absorptive and dispersive methods -- Quantum treatment of light scattering -- Non-destructive imaging -- Other aspects of imaging -- Fluorescence imaging -- Maximum light intensity in single-shot imaging -- Optical pumping. , Imaging in inhomogeneous magnetic fields -- Experimental aspects -- Quantitative analysis of images -- Thermal clouds above the BEC transition temperature -- Bose-Einstein condensates at zero temperature -- Ideal-gas limit -- Thomas-Fermi limit -- Partly condensed clouds -- Column densities -- Extracting static quantities -- Extracting dynamic properties -- Comparison of time-of-flight and in situ images -- Static properties -- Zero-temperature condensates -- Condensate density profile -- Mean-field energy -- Bose condensates at non-zero temperatures: thermodynamics -- The BEC transition temperature -- Condensate fraction -- Density profiles -- Specific heat -- Bose condensates with a negative scattering length -- Sound and other dynamic properties -- Collisionless excitations in a homogeneous Bose gas -- Collisionless excitations in an inhomogeneous, trapped Bose gas -- Experiments on collective excitations near T = 0 -- Measurements of the speed of Bogoliubov sound -- Collective excitations at non-zero temperature -- Oscillations of the thermal cloud -- Frequency shifts of condensate oscillations -- Damping of condensate oscillations -- First and second sound in a Bose gas -- Challenges ahead -- Other dynamic properties -- Free expansion and large-amplitude oscillations of a Bose-Einstein condensate -- The search for quantized circulation -- Collapse of a negative scattering length Bose condensate -- Formation and decay of the condensate -- Coherence properties and the atom laser -- The atom laser and bosonic stimulation -- Derivation of Bose-Einstein statistics from bosonic stimulation -- Formation of the condensate in a magnetic trap -- Interference between two condensates -- Condensate interferometry -- Higher-order coherence -- Output couplers for an atom laser -- Optically confined Bose-Einstein condensates. , Optical confinement of a Bose-Einstein condensate -- Reversible formation of a Bose-Einstein condensate -- Observation of Feshbach resonances in a Bose-Einstein condensate -- Spinor Bose-Einstein condensates -- Miscibility and phase separation of spinor condensate components -- Conclusion -- Appendix A -- Image Processing -- Absorption image processing -- Phase-contrast image processing -- Bose-Einstein condensation of atomic hydrogen -- Introduction -- Origins of the search for BEC in an atomic gas -- Hydrogen trapping and cooling -- Optical detection of trapped hydrogen -- Evaporative cooling -- Evaporation techniques -- Cold-collision frequency shift -- Observation of the condensate -- Properties of the condensate -- Prospects -- Theory of a dilute low-temperature trapped Bose condensate -- Uniform dilute Bose gas -- Brief review of a uniform ideal Bose gas -- Effect of weak repulsive interactions -- Review of scattering theory -- Bogoliubov quasiparticles -- Macroscopic properties -- Moving condensate -- Dilute Bose gas in a harmonic trap -- Ideal Bose gas in a harmonic trap -- Effect of interactions on a trapped Bose condensate -- Basic physics of the Gross-Pitaevskii equation -- The behavior of the condensate for large N -- Effect of an attractive interaction -- Surface region of Thomas-Fermi condensate -- Excited states of a trapped Bose condensate -- Hydrodynamic description of a dilute trapped bose gas -- Uniform dilute Bose gas -- Sum rules for a uniform dilute Bose gas -- Hydrodynamic description of non-uniform dilute Bose gas -- Hydrodynamics in the Thomas-Fermi limit -- Spherical trap -- Anisotropic axisymmetric trap -- Sum-rules for a non-uniform Bose condensate -- Surface modes -- Lowest compressional mode -- Vortices in a dilute trapped Bose gas -- Review of classical vortices -- Quantized circulation -- Classical vortex dynamics. , Effect of rigid boundaries in classical hydrodynamics -- Effect of rotation -- Vortices in a dilute uniform Bose gas -- A vortex in a dilute trapped Bose condensate -- Critical angular velocity Omega_{c1} -- Possible scenario for creation of a vortex -- Possible detection of a vortex line -- Stability of a vortex in a trapped Bose condensate -- Ultracold interactions and mean-field theory of Bose-Einstein condensates -- Introduction -- Low-energy interactions -- Mean-field theory of condensed dilute gases -- Excitations for a trapped gas -- The effect of the medium on collisions -- Gapless mean-field theories and the pair correlation function -- The many-body T-matrix in the homogeneous limit -- Predictions for excitation frequencies -- So when is the effective interaction really important? -- Discussion -- Trapped Bose-Einstein condensed gas: mean-field approximation and beyond -- Introduction -- Mean-field approximation -- Basic equations -- Hydrodynamics and Thomas-Fermi approximation -- Quantum effects in the mean-field equation -- Elementary excitations -- Linearized equations -- Semiclassical approximation -- Thermodynamic scaling -- Landau damping of collective modes -- Quantum fluctuations -- Corrections to the density distribution -- Corrections to the collective mode frequencies -- Fluctuations of phase -- Quantum collapse of collective modes -- Collisional dynamics of ultra-cold atomic gases -- Scattering theory: a brief reminder -- Collision between two particles -- The scattering amplitude -- The low-energy limit -- The Born approximation -- Radial potentials and partial wave expansion -- Scattering states and phase shifts -- The 1D radial Schrodinger equation -- Identical particles -- The low-energy limit -- Scattering length and mean-field energy -- The scattering length for some simple potentials -- The square potential barrier. , The square potential well.

Note: Title Page -- Contents -- Preface -- Course group shot -- Introduction to the physics of artificial gauge fields -- Magnetism and quantum physics -- Gauge invariance -- Cyclotron motion and Landau levels -- The Aharonov-Bohm effect -- Rotating gases -- Geometric phases and gauge fields for free atoms -- Berry's phase -- Adiabatic following of a dressed state -- The two-level case -- Validity of the adiabatic approximation -- Spontaneous emission and recoil heating -- Non-Abelian potentials and spin-orbit coupling -- Non-Abelian potentials in quantum optics -- Tripod configuration and 2D spin-orbit coupling -- 1D version of spin-orbit coupling -- Gauge fields on a lattice -- Tight-binding model -- Hofstadter butterfly -- Chern number for an energy band -- Generation of lattice gauge fields via shaking or modulation -- Rapid shaking of a lattice -- Resonant shaking/modulation -- Generation of lattice gauge fields via internal atomic transitions -- Laser-assisted tunneling in a 1D ladder -- Lattice with artificial dimension -- Laser-induced tunneling in a 2D lattice -- Optical flux lattices -- Conclusion -- Appendix A. Landau levels -- Eigenstates with the Landau gauge -- Probability current in a Landau state -- Eigenstates with the symmetric gauge -- Appendix B. Topology in the square lattice -- Band structure and periodicity in reciprocal space -- Constant force and unitary transformation -- Bloch oscillations and adiabatic following -- The velocity operator and its matrix elements -- The Berry curvature -- Conduction from a filled band and Chern number -- The Chern number is an integer -- Strongly interacting Fermi gases -- Feshbach resonances -- Two-body scattering -- Feshbach resonances -- Three-body losses -- Unitary bosons and the Efimov effect -- Tan relations -- Thermodynamic relations -- Quantitative results for the contact. , Closed-channel fraction -- Single-channel model and zero-range limit -- Short-distance correlations -- Unitary fermions: universality and scale invariance -- Quantum critical point and universality -- Thermodynamics of the unitary Fermi gas -- Luttinger-Ward theory -- Scale invariance -- Broken scale invariance and conformal anomaly in 2D -- RF-spectroscopy and transport -- RF-spectroscopy -- Quantum limited viscosity and spin diffusion -- Thermodynamics of strongly interacting Fermi gases -- Introduction -- Universal thermodynamics -- Thermodynamics of trapped gases -- Zero-temperature equation of state -- Viral theorem for the trapped gas at unitarity -- General thermodynamic relations -- Obtaining the pressure from density profiles -- ``Magic formula'' for harmonic trapping -- Universal thermodynamics of the unitary Fermi gas -- Compressibility equation of state -- Specific heat versus temperature - the Lambda transition in a gas -- Chemical potential, energy and free energy -- Entropy, density and pressure -- Importance of cross-validation with theory -- Further applications of the ``fit-free'' method -- Equation of state in the BEC-BCS crossover - The contact -- Energy of molecular Bose-Einstein condensates -- Energy of weakly interacting Fermi gas -- Near unitarity -- Pressure relation -- General Virial theorem -- Equation of state in the BEC-BCS crossover - Experiments -- Equation of state from density profiles -- Momentum distribution -- Radiofrequency spectroscopy -- Photoassociation -- Bragg spectroscopy -- Temperature dependence of the homogeneous contact -- Collective oscillations -- Condensation energy -- The normal state above Tc: Pseudo-gap phase, Fermi liquid, or Fermi gas? -- Fermionic superfluidity with spin imbalance -- Chandrasekhar-Clogston limit -- Phase separation -- Limit of high imbalance - the Fermi polaron. , Fermi liquid of polarons -- Thermodynamics of spin-imbalanced Fermi mixtures -- Equation of state at unitarity -- Prospects for observing the FFLO state -- Conclusion and perspectives -- Spinor Bose-Einstein gases -- Basic properties -- The quantum fluids landscape -- Atomic species -- Alkali atoms -- High-spin atoms -- Stability against dipolar relaxation -- Rotationally symmetric interactions -- Magnetic order of spinor Bose-Einstein condensates -- Bose-Einstein magnetism in a non-interacting spinor gas -- Spin-dependent s-wave interactions in more recognizable form -- Ground states in the mean-field and single-mode approximations -- Mean-field ground states under applied magnetic fields -- Experimental evidence for magnetic order of ferromagnetic and anti-ferromagnetic F = 1 spinor condensates -- Correlations in the exact many-body ground state of the F = 1 spinor gas -- Imaging spinor condensates -- Stern-Gerlach imaging -- Dispersive birefringent imaging -- Circular birefringent imaging -- Projective imaging -- Absorptive spin-sensitive in situ imaging (ASSISI) -- Noise in dispersive imaging and ASSISI -- Spin-spin correlations and magnetic susceptibility -- Multi-axis imaging and topological invariants -- Multi-axis imaging of ferromagnetic structures -- Magnetization curvature -- Spin dynamics -- Microscopic spin dynamics -- Mean-field picture of collective spin dynamics -- Spin-mixing instability -- Experiments in the single-mode regime -- Quantum quenches in spatially extended spinor Bose-Einstein condensates -- Magnetic excitations -- Quasiparticles of a spin-1 spinor condensate -- Linearized Schrödinger equation -- Ferromagnetic F = 1 condensate -- Polar F = 1 condensate -- Making and detecting magnons -- Magnon propagation -- Magnon contrast interferometry and recoil frequency -- Conclusion. , Probing and controlling quantum many-body systems in optical lattices -- Introduction -- Bose and Fermi Hubbard models -- Bose-Hubbard model -- Fermi-Hubbard model -- Quantum magnetism with ultracold atoms in optical lattices -- Superexchange spin interactions -- Superexchange interactions in a double well -- Superexchange interactions on a lattice -- Resonating valence bond states in a plaquette -- Site-resolved imaging -- Thermometry at the limit of individual thermal excitations -- Single-site-resolved addressing of individual atoms -- Quantum gas microscopy-new possibilities for cold quantum gases -- Using quantum gas microscopes to probe quantum magnetism -- Long-range-interacting quantum magnets -- Outlook -- New theoretical approaches to Bose polarons -- Introduction -- Derivation of the Fröhlich Hamiltonian -- Microscopic Hamiltonian: Impurity in a BEC -- Fröhlich Hamiltonian in a BEC -- Microscopic derivation of the Fröhlich model -- Characteristic scales and the polaronic coupling constant -- Lippmann-Schwinger equation -- Overview of common theoretical approaches -- Perturbative approaches -- Rayleigh-Schrödinger perturbation theory -- Green's function perturbation theory and self-consistent Born -- Exact solution for infinite mass -- Lee-Low-Pines treatment -- Weak coupling mean-field theory -- Self-consistency equation -- Polaron energy -- Polaron mass -- Strong coupling Landau-Pekar approach -- Polaron energy -- Polaron mass -- Feynman path integral approach -- Jensen-Feynman variational principle -- Feynman's trial action -- Polaron mass -- Monte Carlo approaches -- Renormalization group approach -- Fröhlich model and renormalized coupling constants -- Renormalization group formalism for the Fröhlich model -- Dimensional analysis -- Formulation of the RG -- RG flow equations -- Solutions of RG flow equations. , Polaron ground state energy in the renormalization group approach -- Logarithmic UV divergence of the polaron energy -- Ground state polaron properties from RG -- Polaron mass -- Phonon number -- Quasiparticle weight -- Gaussian variational approach -- UV regularization and log-divergence -- Regularization of the power-law divergence -- Explanation of the logarithmic divergence -- Results for experimentally relevant parameters -- Experimental considerations -- Conditions for the Fröhlich model -- Experimentally achievable coupling strengths -- RF spectroscopy -- Basic theory of RF spectroscopy -- Basic properties of RF spectra -- Properties of polarons -- Polaronic mass -- Phonon number -- Quasiparticle weight -- Example of a dynamical problem: Bloch oscillations of Bose polarons -- Time-dependent mean-field approach -- Equations of motion-Dirac's time-dependent variational principle -- Bloch oscillations of polarons in lattices -- Model -- Time-dependent mean-field description -- Adiabatic approximation and polaron dynamics -- Polaron transport properties -- Outlook -- Appendix A -- Lee-Low-Pines formalism in a lattice -- Coupling constant and relation to experiments -- Time-dependent Lee-Low-Pines transformation in the lattice -- Renormalized impurity mass -- Polaron properties from the RG-derivations -- Polaron phonon number -- Polaron momentum -- Quasiparticle weight -- Clean and dirty one-dimensional systems -- Introduction -- Why one dimension -- 1D basics -- What are one-dimensional systems? -- Some realizations with cold atoms or CM -- Universal physics in one dimension (Luttinger liquid) -- Fermions and spins -- Luttinger parameters -- Experimental tests of TLL -- Magnetic insulators -- Cold atomic systems -- Other experimental features of 1d: Fractionalization of excitations -- TLL and beyond -- Effect of a lattice: Mott transition. , Disorder.

Note: Title Page -- CONTENTS -- Preface -- Course group shot -- MODULE I. METROLOGY FOR QUALITY OF LIFE -- Reference methods and commutable reference materials for clinical measurements -- 1. Introduction -- 1.1. Importance of medical tests reliability -- 1.2. Role of a national metrology institute in bioanalysis -- 1.3. Examples of projects undertaken in LNE (France) -- 1.4. Regulatory drivers and traceability chains in laboratory medicine -- 1.5. JCTLM -- 1.6. Accreditation according to ISO 15189 -- 2. Importance of reference methods in EQAS -- 3. Importance of commutability -- 3.1. Why commutability matters -- 3.2. Principle of commutability assessment -- 4. Conclusions and perspectives -- Reference measurement systems for biomarkers: Towards biometrology -- 1. Introduction -- 2. Metabolites and small molecules -- 2.1. Kidney disease: creatinine -- 2.2. Diabetes mellitus: glucose -- 3. Peptides and proteins -- 3.1. Absolute quantification of peptides and proteins -- 3.2. Diabetes mellitus: HbA1c -- 3.3. Sepsis and antimicrobial resistance: Procalcitonine -- 3.4. Alzheimer's disease: amyloid beta & -- tau -- 3.5. Iron-related disorders: hepcidin -- 4. Lipids and lipoproteins -- 4.1. Cardiovascular diseases: cholesterol, triglycerides, LDL-C and HDL-C -- 4.2. Lipoprotein particle concentration: beyond LDL-C in CVD risk assessment -- 5. Conclusion -- SI traceable measurements of the Earth from space to monitor and mitigate against climate change -- 1. Introduction -- 1.1. Climate -- 1.2. Earth Observation data quality -- 1.3. Sensor post launch Calibration and Validation (Cal/Val) -- 1.4. Summary -- 2. Key climate parameters -- 2.1. Essential Climate Variables (ECV) -- 2.2. Earth Radiation Budget (ERB) -- 2.3. Solar variability -- 2.3.1. Total Solar Irradiance (TSI) -- 2.3.2. Solar spectral irradiance (SSI) -- 2.4. Climate feedbacks. , 2.4.1. Introduction -- 2.4.2. Cloud feedback on climate -- 3. Establishing SI traceability for the Earth observing system -- 3.1. Introduction -- 3.2. Near simultaneous overpass calibrations (SNO) -- 3.3. Reference standard calibration test sites -- 3.4. Lunar calibration -- 3.5. Dominant sources of uncertainty -- 3.6. Radiometric accuracy and traceability to SI -- 4. Traceable Radiometry Underpinning Terrestrial- and Helio-Studies (TRUTHS): an NMI in space -- 4.1. Mission requirements and objectives -- 4.2. TRUTHS instrumentation -- 4.2.1. Calibration system -- 4.3. On-board calibration methods -- 4.3.1. Overview -- 4.3.2. Step 1: Calibration of TR against CSAR using LDs -- 4.3.3. Step 2: HIS (Earth radiance view) calibrated against the TR at each LD wavelength (radiance mode) -- 4.3.4. Step 3: HIS (Earth radiance view) calibrated at intermediate wavelengths with a lamp -- 4.3.5. Step 4: Measurements of the Earth and Sun -- 4.3.6. Summary -- 5. Conclusions -- Amount of substance - the Avogadro constant and the SI unit "mole -- 1. Introduction -- 2. History -- 3. The mole as an SI unit in chemistry -- 4. Realization and dissemination -- 4.1. Primary standards as reference points -- 4.2. Example: National standards for the determination of element concentrations in solutions -- 4.3. Examples of primary measurement procedures -- 4.4. International comparability -- 5. Redefinitions -- 5.1. General -- 5.2. Silicon single crystal and the Avogadro constant -- 5.3. Realization and dissemination in accordance with the redefinitions -- 6. Summary -- Comparisons of gas standards for climate change and air quality monitoring -- 1. Introduction to air quality and greenhouse gas monitoring -- 2. Methods for gas standard preparation and verification -- 3. Standards produced by static gravimetric methods (CH4 and NO) -- 3.1. Methane in air standards. , 3.2. Nitrogen monoxide in nitrogen standards -- 4. Dynamic methods (NO2 and HCHO) -- 5. Spectroscopic methods (O3 and FTIR) -- 5.1. Ozone standards -- 5.2. FTIR for the comparison and analysis of gas standards -- 5.3. FTIR measurements of isotope ratios in CO2 -- 6. Manometric methods (CO2 and O3) -- 6.1. Manometric reference method for CO2 mole fraction value assignment -- 6.2. Manometric method for ozone cross-section measurements -- Chemical primary reference materials: From valine to C-peptide -- 1. Introduction: Organic primary reference materials for analytical chemistry -- 2. The Mass balance purity assignment method applied to valine -- 2.1. Measurement of mass fraction of related structure impurities (wRS) in valine -- 2.1.1. Experimental and method description -- 2.1.2. Metrological uncertainty and SI-traceability of mass fraction of related structure impurities -- 2.2. Measurement of mass fraction of water (wW) in valine -- 2.3. Measurement of the mass fraction of residual organic solvent (wOS) for valine -- 2.4. Measurements of the mass fraction of non-volatile materials (wNV ) in valine -- 2.5. Assignment of the mass fraction content of valine -- 3. qNMR applied to pure material standards of folic acid -- 3.1. Introduction to qNMR -- 3.2. Application of qNMR for purity measurements of folic acid -- 4. Methods for large organics: peptides -- 4.1. High-resolution mass spectrometry, identification and quantification of impurities in peptides -- 4.1.1. Typical configuration of a mass spectrometer -- 4.1.2. Electrospray ionization (ESI) -- 4.1.3. Mass analysers and mass resolution -- 4.1.4. The Orbitrap mass analyser -- 4.2. High-resolution mass spectrometry of Angiotensin I -- 4.2.1. Sequencing of Angiotensin I using MS/MS -- 4.2.2. Identification of impurities in ANG I peptide material -- 4.2.3. PICAA analysis of ANG I. , 4.3. Characterization of C-peptide primary reference material -- 4.3.1. Peptide impurity identification and quantification -- 4.3.2. PICCA analysis of C-peptide material -- MODULE II. FUNDAMENTALS OF METROLOGY -- Uncertainty of measurement -- Disclaimer -- 1. Introduction -- 2. Physical quantities and their values -- 2.1. Quantities -- 2.2. Quantity values -- 2.3. Physical constants -- 2.4. Other quantities -- 3. Measurement -- 3.1. Measurement and measurand -- 3.2. Measurement result and measurand estimate -- 4. Measurement uncertainty -- 4.1. General -- 4.2. Definitional uncertainty -- 5. Modelling a measurement -- 6. Propagating uncertainty -- 7. Evaluating uncertainties -- 7.1. General -- 7.2. Random and systematic effects -- 7.3. Frequentist and Bayesian (or subjective) approaches -- 7.4. Change of paradigm -- 8. Coverage intervals - propagation of PDFs -- 9. Bayesian inference -- 10. Conclusion -- International recognition of NMI calibration and measurement capabilities: The CIPM MRA -- 1. Introduction -- 2. The origins of the CIPM MRA -- 3. Launch of the CIPM MRA -- 4. Structure and mechanisms of the CIPM MRA -- 4.1. Key and supplementary comparisons -- 4.2. Calibration and Measurement Capabilities (CMCs) -- 5. The CIPM MRA today -- 6. The CIPM MRA review and the way forward -- 7. In conclusion -- On the proposed re-definition of the SI: "for all people for all time -- 1. Metrology and the role of the SI -- 2. The development of the SI -- 3. Establishing units for electrical quantities -- 4. The 1990 convention for the electrical units -- 5. The Kibble balance and the silicon Avogadro experiment -- 6. A method for establishing a mass unit from the atomic mass unit -- 7. The 2005 proposal to redefine the kilogram -- 8. The 2006 proposals -- 9. Articulating the definitions -- 10. The motivation for re-definition. , 10.1. Eliminating the dependence on artefacts -- 10.2. Widening access to the realization of the base units -- 11. Conclusions -- The Metre Convention and the creation of the BIPM -- 1. Introduction -- 2. The call by geodesists in 1867 for a better standard of the metre and for a European international Bureau of weights and measures -- 3. The International Metre Commission -- 4. The Metre Convention of 1875 -- 5. The creation of the BIPM and early scientific work -- 6. The BIPM since 1921 -- 7. The CIPM MRA -- 8. Conclusions -- Bibliography -- The development of units of measurement from the origin of the metric system in the 18th century to proposals for redefinition of the SI in 2018 -- 1. Introduction -- 2. The origin of the metric system in 1791 -- 3. The Metre and Kilogram of the Archives 1796 -- 4. The Berlin Conference of 1867 and proposals for an international European bureau of weights and measures -- 5. The construction of the new prototypes of the Metre and Kilogram -- 6. Next steps: Maxwell, Michelson and successive definitions of the Metre -- 7. Electrical standards -- 8. Developments in other base units until 1971 -- 9. The changes since 1971 that finally opened the way to units based on constants of physics -- 10. The Kibble balance -- 11. The x-ray crystal density method -- 12. Comparison of the two methods -- 13. Present plans for a fully constants-based SI for 2018 -- 13.1. What does it mean to fix the numerical value of a constant of nature? -- 13.2. The form of the proposed new definition of the SI and its base units -- 13.3. Advantages of units defined in terms of constants of nature: explicit constant definitions rather than an explicit unit definitions -- Bibliography -- Frequency combs applications and optical frequency standards -- 1. Introduction -- 2. Frequency combs from mode locked lasers. , 2.1. Derivation from cavity boundary conditions.