S. Stenger, M. R. Inouye, H. Andrews, D. M. Miesner, W. Stamper-kurn et al., Strongly Enhanced Inelastic Collisions in a Bose-Einstein Condensate near Feshbach Resonances, Physical Review Letters, vol.82, issue.12, p.2422
DOI : 10.1103/PhysRevLett.82.2422

M. Kraemer, P. Mark, J. G. Waldburger, C. Danzl, B. Chin et al., Evidence for Efimov quantum states in an ultracold gas of caesium atoms, Nature, vol.47, issue.7082, p.315, 2006.
DOI : 10.1038/nature04626

S. Petrov, C. Salomon, G. V. Shlyapnikov, O. Sørensen, D. V. Fedorov et al., 17 For N spin-1 / 2 fermions, the spin configuration can be chosen as ? N ? ? N ? where the first N ? factors of the tensorial product are the spin-up state and the N ? = N ? N ? remaining factors are the spin-down state. In this case, the wave function r 1 ,...,r N is antisymmetric for the permutation of the positions of the N ? first particles and also antisymmetric for the permutation of the last N ? particles. 18 In the context of cold atoms, hyperspherical coordinates were used, e.g, Phys. Rev. A Phys. Rev. Lett.D.S. Petrov, Phys. Rev. A Phys. Rev. A, vol.71, issue.584, 1998.

F. Albeverio, R. Gesztesy, H. Hoegh-krohn, and . Holden, Solvable Models in Quantum Mechanics, 1988.

. Ultra-cold-fermi and . Gases, Proceedings of the International School of Physics " Enrico Fermi, Course CLXIV, 2006.

. Blume, 26 L. Pricoupenko and Y. Castin, Phys. Rev. Lett. J. Phys. A Phys. Rev. Lett, vol.99, issue.91, 2003.

E. Astrakharchik, 29 O. Juillet, Ultra-cold Fermi Gases 3. 31 L. Tarruell et al.,i nUltra-cold Fermi Gases 3, 2004.

L. Platter, 47 For details, see F. Werner, e-print arXiv:0803, Phys. Rev. A Europhys. Lett. Rev. Mod. Phys, vol.70, issue.463, 1999.

D. Carr, 56 In low dimensions there is a caveat: the thermalization time may diverge with the system size, see A. Polkovnikov and V. Gritsev, Phys. Rev. Lett. Nat. Phys, vol.92, issue.4, 2008.

N. Bogoliubov and Y. Shin, 60 If the expansion dynamics is known theoretically, then other, Lectures on quantum statistics Gordon & Breach, pp.2179-2226, 1967.

J. Carlson, Superfluid Fermi Gases with Large Scattering Length, Physical Review Letters, vol.91, issue.5, p.50401, 2003.
DOI : 10.1103/PhysRevLett.91.050401

URL : http://arxiv.org/abs/physics/0303094

J. Stenger, Strongly Enhanced Inelastic Collisions in a Bose-Einstein Condensate near Feshbach Resonances, Physical Review Letters, vol.82, issue.12, p.2422, 1999.
DOI : 10.1103/PhysRevLett.82.2422

T. Kraemer, Evidence for Efimov quantum states in an ultracold gas of caesium atoms, Nature, vol.47, issue.7082, p.315, 2006.
DOI : 10.1038/nature04626

E. Braaten and H. Hammer, Universality in few-body systems with large scattering length, Physics Reports, vol.428, issue.5-6, p.259, 2006.
DOI : 10.1016/j.physrep.2006.03.001

E. Braaten, Dilute Bose-Einstein Condensate with Large Scattering Length, Physical Review Letters, vol.88, issue.4, p.40401, 2002.
DOI : 10.1103/PhysRevLett.88.040401

URL : http://arxiv.org/abs/cond-mat/0108380

S. Jonsell, H. Heiselberg, and C. J. Pethick, Universal Behavior of the Energy of Trapped Few-Boson Systems with Large Scattering Length, Physical Review Letters, vol.89, issue.25, p.250401, 2002.
DOI : 10.1103/PhysRevLett.89.250401

M. Stoll and T. Köhler, Production of three-body Efimov molecules in an optical lattice, Physical Review A, vol.72, issue.2, p.22714, 2005.
DOI : 10.1103/PhysRevA.72.022714

R. 1. O-'hara, S. L. Hemmer, M. E. Gehm, S. Granade, and J. E. Thomas, Observation of a Strongly Interacting Degenerate Fermi Gas of Atoms, Science, vol.298, issue.5601, p.2179, 2002.
DOI : 10.1126/science.1079107

T. Bourdel, J. Cubizolles, L. Khaykovich, K. M. Magalhães, S. J. Kokkelmans et al., Fermi Gas, Physical Review Letters, vol.91, issue.2, p.20402, 2003.
DOI : 10.1103/PhysRevLett.91.020402

URL : https://hal.archives-ouvertes.fr/hal-00001552

M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. Raupach, S. Gupta et al., Observation of Bose-Einstein Condensation of Molecules, Physical Review Letters, vol.91, issue.25, p.250401, 2003.
DOI : 10.1103/PhysRevLett.91.250401

C. A. Regal, M. Greiner, and D. S. Jin, Observation of Resonance Condensation of Fermionic Atom Pairs, Physical Review Letters, vol.92, issue.4, pp.40403-120403, 2004.
DOI : 10.1103/PhysRevLett.92.040403

C. Chin, M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim et al., Observation of the Pairing Gap in a Strongly Interacting Fermi Gas, Science, vol.305, issue.5687, p.1128, 2004.
DOI : 10.1126/science.1100818

P. Törmä and P. Zoller, Laser Probing of Atomic Cooper Pairs, Physical Review Letters, vol.85, issue.3, p.487, 2000.
DOI : 10.1103/PhysRevLett.85.487

C. M-e-n-o-t-t-i and P. , P e d r i ,S .S t r i n g a r i ,P h y s .R e v .L e t t, p.250402, 2002.

M. Cozzini and S. Stringari, Fermi Gases in Slowly Rotating Traps: Superfluid versus Collisional Hydrodynamics, Physical Review Letters, vol.91, issue.7, p.70401, 2003.
DOI : 10.1103/PhysRevLett.91.070401

D. L. Feder, Vortex Arrays in a Rotating Superfluid Fermi Gas, Physical Review Letters, vol.93, issue.20, p.200406, 2004.
DOI : 10.1103/PhysRevLett.93.200406

N. Nygaard, G. M. Bruun, C. W. Clark, and D. L. Feder, Microscopic Structure of a Vortex Line in a Dilute Superfluid Fermi Gas, Physical Review Letters, vol.90, issue.21, p.210402, 2003.
DOI : 10.1103/PhysRevLett.90.210402

T. Isoshima and K. Machida, Instability of the nonvortex state toward a quantized vortex in a Bose-Einstein condensate under external rotation, Physical Review A, vol.60, issue.4, p.3313, 1999.
DOI : 10.1103/PhysRevA.60.3313

F. Dalfovo and S. Stringari, Shape deformations and angular-momentum transfer in trapped Bose-Einstein condensates, Physical Review A, vol.63, issue.1, p.11601, 2000.
DOI : 10.1103/PhysRevA.63.011601

K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, P h y s .R e v .L e t t, p.806, 2000.

S. Sinha and Y. Castin, Dynamic Instability of a Rotating Bose-Einstein Condensate, Physical Review Letters, vol.87, issue.19, p.190402, 2001.
DOI : 10.1103/PhysRevLett.87.190402

URL : https://hal.archives-ouvertes.fr/hal-00002093

K. Madison, F. Chevy, V. Bretin, and J. Dalibard, Stationary States of a Rotating Bose-Einstein Condensate: Routes to Vortex Nucleation, Physical Review Letters, vol.86, issue.20, p.4443, 2001.
DOI : 10.1103/PhysRevLett.86.4443

URL : https://hal.archives-ouvertes.fr/hal-00002550

E. Hodby, G. Hechenblaikner, S. A. Hopkins, O. M. Maragò, and C. , Vortex Nucleation in Bose-Einstein Condensates in an Oblate, Purely Magnetic Potential, Physical Review Letters, vol.88, issue.1, p.10405, 2002.
DOI : 10.1103/PhysRevLett.88.010405

D. L. Feder, A. A. Svidzinsky, A. L. Fetter, and C. W. Clark, P h y s .R e v .L e t t, p.564, 2001.

C. L-o-b-o and A. , C a s t i n ,P h y s .R e v .L e t t, p.20403, 2004.

D. S. Petrov, M. Holzmann, G. Shlyapnikov, D. S. Petrov, G. V. Shlyapnikov et al., Bose-Einstein Condensation in Quasi-2D Trapped Gases, Physical Review Letters, vol.84, issue.12, pp.2551-012706, 2000.
DOI : 10.1103/PhysRevLett.84.2551

URL : https://hal.archives-ouvertes.fr/hal-00001658

M. Randeria, J. Duan, and L. Shieh, Superconductivity in a two-dimensional Fermi gas: Evolution from Cooper pairing to Bose condensation, Physical Review B, vol.41, issue.1, p.327, 1990.
DOI : 10.1103/PhysRevB.41.327

C. Mora and Y. Castin, Extension of Bogoliubov theory to quasicondensates, Physical Review A, vol.67, issue.5, p.53615, 2003.
DOI : 10.1103/PhysRevA.67.053615

URL : https://hal.archives-ouvertes.fr/hal-00157445

G. Tonini, Study of Ultracold Fermi gases: crystalline LOFF phase and nucleation of vortices, Phys. Rev. A, vol.72, 2005.

K. Köhl, T. Günter, H. Stöferle, T. Moritz, and . Esslinger, magnetic moment of the closed channel with respect to the two-atom open channel. For K-Rb in their ground state, J. Phys. B, vol.39, issue.6, 2006.

G. Winkler, F. Thalhammer, R. Lang, J. Grimm, A. J. Hecker-denschlag et al., In the other regimes attractively and repulsively interacting atoms, we have 1 / k typ a rel In the harmonic approximation for the experimental lattice depth a rel = 103 nm. Thus, the above inequality is fairly well satisfied. 15 Y. Castin, e-print arXiv:cond-mat/0612613 17 A. Simoni private communication Hecker Denschlag, e-print arXiv:cond-mat/ 0608721v2. 19 We note that we neglect many low-energy basis states N , L ,0,0,0,0 with 2N + L max when the scattering length is small and positive a s 0. These molecule states with highly excited center-of-mass energy have nevertheless low total energy since the energy of the relative motion is large and negative . We have included the states max to allow for the flattening of the center of mass wave function due to the anharmonicity of the lattice site potential. However, we have neglected the states with 2N + L max since they are only weakly coupled to the states 0 repulsively interacting atoms of the decoupled problem, nm 16. Using E res ?4.6 nm. It remains to estimate k typ . In the molecule regime These matrix elements become even zero if 2N + L p p =1,...,8 since in this case the matrix elements 0 are exactly zero. 20 C. Ospelkaus, S. Ospelkaus, K. Sengstock, and K. Bongs, 2001.

M. Arimondo, P. Inguscio, and . Violino, 27 For the scattering length in the initial 87 Rb1, Rev. Mod. Phys, vol.4931, issue.2 2, pp.11977-12017

K. Thalhammer, F. Winkler, S. Lang, R. Schmid, J. Grimm et al., Long-Lived Feshbach Molecules in a Three-Dimensional Optical Lattice, Physical Review Letters, vol.96, issue.5, p.50402, 2006.
DOI : 10.1103/PhysRevLett.96.050402

P. S. Petrov, E. Naidon, W. Tiesinga, P. Mitchell, and . Julienne, 92007, and references therein. 47 For an equivalent energy-independent formulation of such a model, see L. Pricoupenko, Phys. Rev. Lett. New J. Phys. Phys. Rev. A, vol.931, issue.73, 2006.

T. Stoll and . Köhler, 51 F. Werner and Y. Castin 52 VFG-150, a FPGA-driven fast versatile frequency generator, controlled via an USB2.0 interface and developed by Th, Phys. Rev. A Phys. Rev. Lett, vol.72, issue.97, 2006.

D. Jaksch, Cold Bosonic Atoms in Optical Lattices, Physical Review Letters, vol.81, issue.15, p.3108, 1998.
DOI : 10.1103/PhysRevLett.81.3108

S. Jochim, Bose-Einstein Condensation of Molecules, Science, vol.302, issue.5653, p.2101, 2003.
DOI : 10.1126/science.1093280

M. W. Zwierlein, Observation of Bose-Einstein Condensation of Molecules, Physical Review Letters, vol.91, issue.25, p.250401, 2003.
DOI : 10.1103/PhysRevLett.91.250401

T. Bourdel, Experimental Study of the BEC-BCS Crossover Region in Lithium 6, Physical Review Letters, vol.93, issue.5, p.50401, 2004.
DOI : 10.1103/PhysRevLett.93.050401

URL : https://hal.archives-ouvertes.fr/hal-00001237

P. B. Blakie and A. Bezett, Adiabatic cooling of fermions in an optical lattice, Physical Review A, vol.71, issue.3, p.33616, 2005.
DOI : 10.1103/PhysRevA.71.033616

W. Hofstetter, High-Temperature Superfluidity of Fermionic Atoms in Optical Lattices, Physical Review Letters, vol.89, issue.22, p.220407, 2002.
DOI : 10.1103/PhysRevLett.89.220407

A. Georges, Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions, Reviews of Modern Physics, vol.68, issue.1, p.13, 1996.
DOI : 10.1103/RevModPhys.68.13

R. Staudt, Phase diagram of the three-dimensional Hubbard model at half filling, The European Physical Journal B, vol.17, issue.3, p.411, 2000.
DOI : 10.1007/s100510070120

A. Georges and W. Krauth, =??? Hubbard model: Evidence for a Mott transition, Physical Review Letters, vol.69, issue.8, p.1240, 1992.
DOI : 10.1103/PhysRevLett.69.1240

A. Georges and W. Krauth, Physical properties of the half-filled Hubbard model in infinite dimensions, Physical Review B, vol.48, issue.10, p.7167, 1993.
DOI : 10.1103/PhysRevB.48.7167

M. J. Rozenberg, Finite Temperature Mott Transition in the Hubbard Model in Infinite Dimensions, Physical Review Letters, vol.83, issue.17, p.3498, 1999.
DOI : 10.1103/PhysRevLett.83.3498

A. Daré and G. Albinet, Magnetic properties of the three-dimensional Hubbard model at half filling, Physical Review B, vol.61, issue.7, p.4567, 2000.
DOI : 10.1103/PhysRevB.61.4567

D. Arovas and A. Auerbach, Functional integral theories of low-dimensional quantum Heisenberg models, Physical Review B, vol.38, issue.1, p.316, 1988.
DOI : 10.1103/PhysRevB.38.316

E. Altman, Probing many-body states of ultracold atoms via noise correlations, Physical Review A, vol.70, issue.1, p.13603, 2004.
DOI : 10.1103/PhysRevA.70.013603

S. Fölling, Spatial quantum noise interferometry in expanding ultracold atom clouds, Nature, vol.22, issue.7032, p.481, 2005.
DOI : 10.1103/PhysRevA.59.4595

. Du-piège, Nous comptons également généraliser l'étude des résonances à N corps présentée ici Une autre voie est l'étude du problème à N corps unitaire dans un réseau optique profond

]. M. Bibliographie1, T. Stoll, and . Köhler, Production of three-body Efimov molecules in an optical lattice, Phys. Rev. A, vol.72, p.22714, 2005.

K. Lange, A. Pilch, H. Jaakkola, R. Naegerl, and . Grimm, Evidence for Efimov quantum states in an ultracold gas of caesium atoms, Nature, vol.440, p.315, 2006.

J. Von-stecher, C. H. Greene, and D. Blume, Energetics and structural properties of trapped two-component Fermi gases, Physical Review A, vol.77, issue.4, p.43619, 2008.
DOI : 10.1103/PhysRevA.77.043619

B. Bernu and G. Misguich, Specific heat and high-temperature series of lattice models: Interpolation scheme and examples on quantum spin systems in one and two dimensions, Physical Review B, vol.63, issue.13, p.134409, 2001.
DOI : 10.1103/PhysRevB.63.134409

E. Tiesinga, B. J. Verhaar, and H. T. Stoof, Threshold and resonance phenomena in ultracold ground-state collisions, Physical Review A, vol.47, issue.5, pp.4114-4122, 1993.
DOI : 10.1103/PhysRevA.47.4114

S. Inouye, M. R. Andrews, J. Stenger, H. Miesner, D. M. Stamper-kurn et al., Observation of Feshbach resonances in a Bose???Einstein condensate, Nature, vol.77, issue.6672, p.151, 1998.
DOI : 10.1038/32354

K. M. O-'hara, S. L. Hemmer, M. E. Gehm, S. R. Granade, and J. E. Thomas, Observation of a Strongly Interacting Degenerate Fermi Gas of Atoms, Science, vol.298, issue.5601, p.2179, 2002.
DOI : 10.1126/science.1079107

C. A. Regal and D. S. Jin, Measurement of Positive and Negative Scattering Lengths in a Fermi Gas of Atoms, Physical Review Letters, vol.90, issue.23, p.230404, 2003.
DOI : 10.1103/PhysRevLett.90.230404

T. Bourdel, J. Cubizolles, L. Khaykovich, K. M. Magalhaes, S. J. Kokkelmans et al., Fermi Gas, Physical Review Letters, vol.91, issue.2, p.20402, 2003.
DOI : 10.1103/PhysRevLett.91.020402

URL : https://hal.archives-ouvertes.fr/hal-00001552

I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Reviews of Modern Physics, vol.80, issue.3, p.885, 2008.
DOI : 10.1103/RevModPhys.80.885

URL : https://hal.archives-ouvertes.fr/hal-00195515

S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory of ultracold atomic Fermi gases, Reviews of Modern Physics, vol.80, issue.4, p.1215, 2008.
DOI : 10.1103/RevModPhys.80.1215

E. Wigner, ???ber die Streuung von Neutronen an Protonen, Zeitschrift f???r Physik, vol.83, issue.3-4, p.253, 1933.
DOI : 10.1007/BF01331145

H. Bethe and R. Peierls, Quantum Theory of the Diplon, Proc. R. Soc, p.146, 1935.

E. Braaten and H. Hammer, Universality in few-body systems with large scattering length, Physics Reports, vol.428, issue.5-6, p.259, 2006.
DOI : 10.1016/j.physrep.2006.03.001

V. N. Efimov, Weakly-bound states of three resonantly interating particles, Yad. Fiz.Sov. J. Nucl. Phys, vol.12, issue.12, pp.1080-589, 1970.

J. Stenger, S. Inouye, M. R. Andrews, H. Miesner, D. M. Stamper-kurn et al., Strongly Enhanced Inelastic Collisions in a Bose-Einstein Condensate near Feshbach Resonances, Physical Review Letters, vol.82, issue.12, pp.2422-2425, 1999.
DOI : 10.1103/PhysRevLett.82.2422

J. L. Roberts, N. R. Claussen, S. L. Cornish, and C. E. Wieman, Magnetic Field Dependence of Ultracold Inelastic Collisions near a Feshbach Resonance, Physical Review Letters, vol.85, issue.4, pp.728-731, 2000.
DOI : 10.1103/PhysRevLett.85.728

A. Marte, T. Volz, J. Schuster, S. Dürr, G. Rempe et al., Precision Measurement and Analysis, Feshbach Resonances in Rubidium Phys. Rev. Lett, vol.87, issue.89, p.283202, 2002.

T. Weber, J. Herbig, M. Mark, H. Nägerl, and R. Grimm, Three-Body Recombination at Large Scattering Lengths in an Ultracold Atomic Gas, Physical Review Letters, vol.91, issue.12, p.123201, 2003.
DOI : 10.1103/PhysRevLett.91.123201

D. S. Petrov, C. Salomon, and G. V. Shlyapnikov, Weakly Bound Dimers of Fermionic Atoms, Physical Review Letters, vol.93, issue.9, p.90404, 2004.
DOI : 10.1103/PhysRevLett.93.090404

URL : https://hal.archives-ouvertes.fr/hal-00002933

D. S. Petrov, C. Salomon, and G. V. Shlyapnikov, Diatomic molecules in ultracold Fermi gases???novel composite bosons, Journal of Physics B: Atomic, Molecular and Optical Physics, vol.38, issue.9, p.645, 2005.
DOI : 10.1088/0953-4075/38/9/014

URL : https://hal.archives-ouvertes.fr/hal-00004896

S. B. Papp, J. M. Pino, R. J. Wild, S. Ronen, C. E. Wieman et al., Bose-Einstein Condensate, Physical Review Letters, vol.101, issue.13, p.135301, 2008.
DOI : 10.1103/PhysRevLett.101.135301

E. Braaten and H. Hammer, Efimov physics in cold atoms, Annals of Physics, vol.322, issue.1, p.120, 2007.
DOI : 10.1016/j.aop.2006.10.011

E. Braaten, H. Hammer, and T. Mehen, Dilute Bose-Einstein Condensate with Large Scattering Length, Physical Review Letters, vol.88, issue.4, p.40401, 2002.
DOI : 10.1103/PhysRevLett.88.040401

D. Blume, J. Stecher, and C. H. Greene, Universal Properties of a Trapped Two-Component Fermi Gas at Unitarity, Physical Review Letters, vol.99, issue.23, p.233201, 2007.
DOI : 10.1103/PhysRevLett.99.233201

T. Stöferle, H. Moritz, K. Günter, M. Köhl, and T. Esslinger, Molecules of Fermionic Atoms in an Optical Lattice, Physical Review Letters, vol.96, issue.3, p.30401, 2006.
DOI : 10.1103/PhysRevLett.96.030401

C. Ospelkaus, S. Ospelkaus, L. Humbert, P. Ernst, K. Sengstock et al., Ultracold Heteronuclear Molecules in a 3D Optical Lattice, Physical Review Letters, vol.97, issue.12, p.120402, 2006.
DOI : 10.1103/PhysRevLett.97.120402

Y. Castin, Exact scaling transform for a unitary quantum gas in a time dependent harmonic potential, Comptes Rendus Physique, vol.5, issue.3, p.407, 2004.
DOI : 10.1016/j.crhy.2004.03.017

URL : https://hal.archives-ouvertes.fr/hal-00001653

Y. Castin, Proceedings of the International School of Physics "Enrico Fermi" on Ultra- Cold Fermi gases, 2007.

T. Busch, B. G. Englert, K. Rzazewski, and M. Wilkens, Two Cold Atoms in a Harmonic Trap, Foundations of Physics, vol.28, issue.4, p.549, 1998.
DOI : 10.1023/A:1018705520999

S. Albeverio, F. Gesztesy, R. Hoegh-krohn, and H. Holden, Solvable models in quantum mechanics, 1988.

M. Jona-lasinio, L. Pricoupenko, and Y. Castin, -wave Feshbach resonance, Physical Review A, vol.77, issue.4, p.43611, 2008.
DOI : 10.1103/PhysRevA.77.043611

URL : https://hal.archives-ouvertes.fr/hal-00623276

M. Reed and B. Simon, Methods of modern mathematical physics 1 : functional analysis, 1980.

S. Y. Chang and G. F. Bertsch, Unitary Fermi gas in a harmonic trap, Physical Review A, vol.76, issue.2, p.21603, 2007.
DOI : 10.1103/PhysRevA.76.021603

T. Köhler, K. Góral, and P. S. Julienne, Production of cold molecules via magnetically tunable Feshbach resonances, Reviews of Modern Physics, vol.78, issue.4, p.1311, 2006.
DOI : 10.1103/RevModPhys.78.1311

J. Carlson, S. Chang, V. R. Pandharipande, and K. E. Schmidt, Superfluid Fermi Gases with Large Scattering Length, Physical Review Letters, vol.91, issue.5, p.50401, 2003.
DOI : 10.1103/PhysRevLett.91.050401

G. E. Astrakharchik, J. Boronat, J. Casulleras, and S. Giorgini, Equation of State of a Fermi Gas in the BEC-BCS Crossover: A Quantum Monte Carlo Study, Physical Review Letters, vol.93, issue.20, p.200404, 2004.
DOI : 10.1103/PhysRevLett.93.200404

J. R. Engelbrecht, M. Randeria, and C. A. Sá-de-melo, BCS to Bose crossover: Broken-symmetry state, Physical Review B, vol.55, issue.22, pp.15153-15156, 1997.
DOI : 10.1103/PhysRevB.55.15153

O. Juillet, Sign-free stochastic mean-field approach to strongly correlated phases of ultracold fermions, New Journal of Physics, vol.9, issue.6, p.163, 2007.
DOI : 10.1088/1367-2630/9/6/163

URL : https://hal.archives-ouvertes.fr/hal-00090944

L. Tarruell, M. Teichmann, J. Mckeever, T. Bourdel, J. Cubizolles et al., Expansion of an ultra-cold lithium gas in the BEC-BCS crossover, Proceedings of the International School of Physics "Enrico Fermi" on Ultra-Cold Fermi gases, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00123470

J. Kinast, A. Turlapov, J. Thomas, Q. Chen, J. Stajic et al., Heat Capacity of a Strongly Interacting Fermi Gas, Science, vol.307, issue.5713, p.1296, 2005.
DOI : 10.1126/science.1109220

G. B. Partridge, W. Li, R. I. Kamar, Y. Liao, and R. G. Hulet, Pairing and Phase Separation in a Polarized Fermi Gas, Science, vol.311, issue.5760, p.503, 2006.
DOI : 10.1126/science.1122876

J. T. Stewart, J. P. Gaebler, C. A. Regal, and D. S. Jin, Fermi Gas in the BCS-BEC Crossover, Physical Review Letters, vol.97, issue.22, p.220406, 2006.
DOI : 10.1103/PhysRevLett.97.220406

S. Y. Chang, V. R. Pandharipande, J. Carlson, and K. E. Schmidt, Quantum Monte Carlo studies of superfluid Fermi gases, Physical Review A, vol.70, issue.4, p.43602, 2004.
DOI : 10.1103/PhysRevA.70.043602

B. M. Fregoso and G. Baym, Stability of trapped fermionic gases with attractive interactions, Physical Review A, vol.73, issue.4, p.43616, 2006.
DOI : 10.1103/PhysRevA.73.043616

E. Burovski, N. Prokof-'ev, B. Svistunov, and M. Troyer, Critical Temperature and Thermodynamics of Attractive Fermions at Unitarity, Physical Review Letters, vol.96, issue.16, p.160402, 2006.
DOI : 10.1103/PhysRevLett.96.160402

E. Burovski, N. Prokof-'ev, B. Svistunov, and M. Troyer, Critical Temperature and Thermodynamics of Attractive Fermions at Unitarity, Physical Review Letters, vol.96, issue.16, pp.160402-239902, 2006.
DOI : 10.1103/PhysRevLett.96.160402

E. Burovski, N. Prokof-'ev, B. Svistunov, and M. Troyer, The Fermi???Hubbard model at unitarity, New Journal of Physics, vol.8, issue.8, p.153, 2006.
DOI : 10.1088/1367-2630/8/8/153

A. Bulgac, J. E. Drut, and P. Magierski, Fermions in the Unitary Regime: A Superfluid of a New Type, Physical Review Letters, vol.96, issue.9, p.90404, 2006.
DOI : 10.1103/PhysRevLett.96.090404

Y. Castin, Simple theoretical tools for low dimension Bose gases, Journal de Physique IV (Proceedings), vol.116, 2003.
DOI : 10.1051/jp4:2004116004

URL : https://hal.archives-ouvertes.fr/hal-00002178

L. Houches and S. School, Quantum Gases in Low Dimensions, J. Phys. IV France, vol.116, pp.89-132, 2004.

C. Mora and Y. Castin, Extension of Bogoliubov theory to quasicondensates, Physical Review A, vol.67, issue.5, p.53615, 2003.
DOI : 10.1103/PhysRevA.67.053615

URL : https://hal.archives-ouvertes.fr/hal-00157445

L. Pricoupenko and Y. Castin, Three fermions in a box at the unitary limit: universality in a lattice model, Journal of Physics A: Mathematical and Theoretical, vol.40, issue.43, p.12863, 2007.
DOI : 10.1088/1751-8113/40/43/003

URL : https://hal.archives-ouvertes.fr/hal-00145589

S. Tan, Short Range Scaling Laws of Quantum Gases With Contact Interactions

R. L. Pease and H. Feshbach, Binding Energy of the Triton, Physical Review, vol.81, issue.1, pp.142-143, 1951.
DOI : 10.1103/PhysRev.81.142

G. S. Danilov, On the three-body problem with short-range forces, JETPSov. Phys. JETP, vol.40, issue.13, p.349, 1961.

R. A. Minlos and L. D. Faddeev, Comment on the problem of three particles with point interactions, JETPSov. Phys. JETP, vol.41, issue.14, pp.1850-1315, 1961.

R. A. Minlos and L. D. Faddeev, On the point interaction for a three-particle system in quantum mechanics, Sov. Phys. Dokl, vol.6, pp.1335-1072, 1961.

S. Albeverio, R. Høegh-krohn, and T. T. Wu, A class of exactly solvable three-body quantum mechanical problems and the universal low energy behavior, Physics Letters A, vol.83, issue.3, p.105, 1981.
DOI : 10.1016/0375-9601(81)90507-7

S. Jonsell, H. Heiselberg, and C. J. Pethick, Universal Behavior of the Energy of Trapped Few-Boson Systems with Large Scattering Length, Physical Review Letters, vol.89, issue.25, p.250401, 2002.
DOI : 10.1103/PhysRevLett.89.250401

D. S. Petrov, Three-Boson Problem near a Narrow Feshbach Resonance, Physical Review Letters, vol.93, issue.14, p.143201, 2004.
DOI : 10.1103/PhysRevLett.93.143201

L. Pricoupenko, Pseudopotential in resonant regimes, Physical Review A, vol.73, issue.1, p.12701, 2006.
DOI : 10.1103/PhysRevA.73.012701

URL : https://hal.archives-ouvertes.fr/hal-00007776

A. O. Gogolin, C. Mora, and R. Egger, Analytical Solution of the Bosonic Three-Body Problem, Physical Review Letters, vol.100, issue.14, p.140404, 2008.
DOI : 10.1103/PhysRevLett.100.140404

URL : https://hal.archives-ouvertes.fr/hal-00272446

L. Pricoupenko, -Wave Scattering, Physical Review Letters, vol.96, issue.5, p.50401, 2006.
DOI : 10.1103/PhysRevLett.96.050401

URL : https://hal.archives-ouvertes.fr/hal-01355490

V. Efimov, Energy levels of three resonantly interacting particles, Nuclear Physics A, vol.210, issue.1, p.157, 1973.
DOI : 10.1016/0375-9474(73)90510-1

I. S. Gradshteyn and I. M. Ryzhik, Tables of integrals, series, and products, 1994.

F. Werner and Y. Castin, A unitary quantum gas in a harmonic trap : exact mapping to free space zero energy eigenstates and application to the three-body problem

K. M. Case, Singular Potentials, Physical Review, vol.80, issue.5, p.797, 1950.
DOI : 10.1103/PhysRev.80.797

P. Morse and H. Feshbach, Methods of Theoretical Physics, tome II, p.1665, 1953.

M. D. Lee, T. Köhler, and P. S. Julienne, Excited Thomas-Efimov levels in ultracold gases, Physical Review A, vol.76, issue.1, p.12720, 2007.
DOI : 10.1103/PhysRevA.76.012720

G. V. Skorniakov and K. A. Ter-martirosian, Three Body Problem for Short Range Forces. I. Scattering of Low Energy Neutrons dy Deuterons, JETPSov. Phys. JETP, vol.31, issue.4, p.648, 1956.

B. Marcelis, S. J. Kokkelmans, G. V. Shlyapnikov, and D. S. Petrov, Collisional properties of weakly bound heteronuclear dimers, Physical Review A, vol.77, issue.3, p.32707, 2008.
DOI : 10.1103/PhysRevA.77.032707

URL : https://hal.archives-ouvertes.fr/hal-00266160

I. Stetcu, B. R. Barrett, U. Van-kolck, and J. P. Vary, Effective theory for trapped few-fermion systems, Physical Review A, vol.76, issue.6, p.63613, 2007.
DOI : 10.1103/PhysRevA.76.063613

G. Gasaneo and J. H. Macek, Hyperspherical adiabatic eigenvalues for zero-range potentials, Journal of Physics B: Atomic, Molecular and Optical Physics, vol.35, issue.10, p.2239, 2002.
DOI : 10.1088/0953-4075/35/10/304

M. C. Birse, Renormalization-group analysis of repulsive three-body systems, Journal of Physics A: Mathematical and General, vol.39, issue.2, p.49, 2006.
DOI : 10.1088/0305-4470/39/2/L01

D. S. Petrov, Three-body problem in Fermi gases with short-range interparticle interaction, Physical Review A, vol.67, issue.1, p.10703, 2003.
DOI : 10.1103/PhysRevA.67.010703

R. B. Laughlin, Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations, Physical Review Letters, vol.50, issue.18, p.1395, 1983.
DOI : 10.1103/PhysRevLett.50.1395

V. Efimov, Effective interaction of three resonantly interacting particles and the force range, Physical Review C, vol.47, issue.5, p.1876, 1993.
DOI : 10.1103/PhysRevC.47.1876

A. Comtet, P. Leboeuf, and S. N. Majumdar, Level Density of a Bose Gas and Extreme Value Statistics, Physical Review Letters, vol.98, issue.7, p.70404, 2007.
DOI : 10.1103/PhysRevLett.98.070404

URL : https://hal.archives-ouvertes.fr/hal-00134522

S. Tan, Large momentum part of a strongly correlated Fermi gas, Annals of Physics, vol.323, issue.12, p.2971, 2008.
DOI : 10.1016/j.aop.2008.03.005

S. Tan, Energetics of a strongly correlated Fermi gas, Annals of Physics, vol.323, issue.12, p.2952, 2008.
DOI : 10.1016/j.aop.2008.03.004

H. Bateman and A. Erdélyi, Tables of integral transforms, 1954.

D. Blume and C. H. Greene, Fermi pseudopotential approximation: Two particles under external confinement, Physical Review A, vol.65, issue.4, p.43613, 2002.
DOI : 10.1103/PhysRevA.65.043613

E. Tiesinga, C. J. Williams, F. H. Mies, and P. S. Julienne, Interacting atoms under strong quantum confinement, Physical Review A, vol.61, issue.6, p.63416, 2002.
DOI : 10.1103/PhysRevA.61.063416

E. Braaten, M. Kusonoki, and D. Zhang, Scattering models for ultracold atoms, Annals of Physics, vol.323, issue.7, p.1770, 2008.
DOI : 10.1016/j.aop.2007.12.004

G. Smirne, R. M. Godun, D. Cassettari, V. Boyer, C. J. Foot et al., Collisional relaxation of Feshbach molecules and three-body recombination, Phys. Rev. A, vol.87, issue.75, p.20702, 2007.

L. P. De-goey, H. T. Stoof, B. J. Verhaar, and W. Glöckle, Role of three-body correlations in recombination of spin-polarized atomic hydrogen, Physical Review B, vol.38, issue.1, pp.646-658, 1988.
DOI : 10.1103/PhysRevB.38.646

H. T. Stoof, B. J. Verhaar, L. P. De-goey, and W. Glöckle, molecular states, Physical Review B, vol.40, issue.13, pp.9176-9182, 1989.
DOI : 10.1103/PhysRevB.40.9176

E. Braaten, H. Hammer, and M. Kusunoki, Universal equation for Efimov states, Physical Review A, vol.67, issue.2, p.22505, 2003.
DOI : 10.1103/PhysRevA.67.022505

E. Braaten and H. Hammer, Enhanced dimer relaxation in an atomic and molecular Bose-Einstein condensate, Physical Review A, vol.70, issue.4, p.42706, 2004.
DOI : 10.1103/PhysRevA.70.042706

D. S. Petrov, C. Salomon, and G. V. Shlyapnikov, Scattering properties of weakly bound dimers of fermionic atoms, Physical Review A, vol.71, issue.1, p.12708, 2005.
DOI : 10.1103/PhysRevA.71.012708

URL : https://hal.archives-ouvertes.fr/hal-00003932

Y. Kagan, B. V. Svistunov, and G. V. Shlyapnikov, Effect of Bose condensation on inelastic processes in gases, JETP Lett, vol.42, p.209, 1985.

J. P. Incao and B. D. Esry, Scattering Length Scaling Laws for Ultracold Three-Body Collisions, Phys. Rev. Lett, vol.94, p.213201, 2005.

E. Braaten and H. Hammer, Three-Body Recombination into Deep Bound States in a Bose Gas with Large Scattering Length, Physical Review Letters, vol.87, issue.16, p.160407, 2001.
DOI : 10.1103/PhysRevLett.87.160407

C. Cohen-tannoudji, J. Dupont-roc, and G. Grynberg, Atom-Photon Interactions, C I .3b, 1992.

G. Cennini, C. Geckeler, G. Ritt, and M. Weitz, Interference of a variable number of coherent atomic sources, Physical Review A, vol.72, issue.5, p.51601, 2005.
DOI : 10.1103/PhysRevA.72.051601

C. Chin and P. S. Julienne, Radio-frequency transitions on weakly bound ultracold molecules, Physical Review A, vol.71, issue.1, p.12713, 2005.
DOI : 10.1103/PhysRevA.71.012713

E. G. Van-kempen, S. J. Kokkelmans, D. J. Heinzen, and B. J. Verhaar, Interisotope Determination of Ultracold Rubidium Interactions from Three High-Precision Experiments, Physical Review Letters, vol.88, issue.9, p.93201, 2002.
DOI : 10.1103/PhysRevLett.88.093201

P. J. Leo, C. J. Williams, and P. S. Julienne, Atoms, Physical Review Letters, vol.85, issue.13, pp.2721-2724, 2000.
DOI : 10.1103/PhysRevLett.85.2721

B. Marcelis, B. Verhaar, and S. Kokkelmans, Total Control over Ultracold Interactions via Electric and Magnetic Fields, Physical Review Letters, vol.100, issue.15, p.153201, 2008.
DOI : 10.1103/PhysRevLett.100.153201

URL : http://arxiv.org/abs/0710.0733

G. K. Campbell, J. Mun, M. Boyd, P. Medley, A. E. Leanhardt et al., Imaging the Mott Insulator Shells by Using Atomic Clock Shifts, Imaging the Mott Insulator Shells By Using Atmoic Clock Shifts, p.649, 2006.
DOI : 10.1126/science.1130365

M. Köhl, K. Günter, T. Stöferle, H. Moritz, and T. Esslinger, Strongly interacting atoms and molecules in a 3D optical lattice, Journal of Physics B: Atomic, Molecular and Optical Physics, vol.39, issue.10, p.47, 2006.
DOI : 10.1088/0953-4075/39/10/S05

T. Müller, S. Fölling, A. Widera, and I. Bloch, State Preparation and Dynamics of Ultracold Atoms in Higher Lattice Orbitals, Physical Review Letters, vol.99, issue.20, p.200405, 2007.
DOI : 10.1103/PhysRevLett.99.200405

M. Anderlini, P. J. Lee, B. L. Brown, J. Sebby-strabley, W. D. Phillips et al., Controlled exchange interaction between pairs of neutral atoms in an optical lattice, Nature, vol.83, issue.7152, p.452, 2007.
DOI : 10.1038/nature06011

G. Bruun, Y. Castin, R. Dum, and K. Burnett, BCS theory for trapped ultracold fermions, The European Physical Journal D, vol.7, issue.3, p.433, 1999.
DOI : 10.1007/s100530050587

S. Stringari, Proceedings of the International School of Physics "Enrico Fermi" on Ultra-Cold Fermi gases, 2007.

A. Collin, P. Massignan, and C. J. Pethick, Energy-dependent effective interactions for dilute many-body systems, Physical Review A, vol.75, issue.1, p.13615, 2007.
DOI : 10.1103/PhysRevA.75.013615

A. Derevianko, J. F. Babb, and A. Dalgarno, High-precision calculations of van der Waals coefficients for heteronuclear alkali-metal dimers, Physical Review A, vol.63, issue.5, p.52704, 2001.
DOI : 10.1103/PhysRevA.63.052704

E. L. Bolda, E. Tiesinga, and P. S. Julienne, Effective-scattering-length model of ultracold atomic collisions and Feshbach resonances in tight harmonic traps, Physical Review A, vol.66, issue.1, p.13403, 2002.
DOI : 10.1103/PhysRevA.66.013403

Z. Idziaszek and T. Calarco, Analytical solutions for the dynamics of two trapped interacting ultracold atoms, Physical Review A, vol.74, issue.2, p.22712, 2006.
DOI : 10.1103/PhysRevA.74.022712

R. B. Diener and T. Ho, Fermions in Optical Lattices Swept across Feshbach Resonances, Physical Review Letters, vol.96, issue.1, p.10402, 2006.
DOI : 10.1103/PhysRevLett.96.010402

E. Arimondo, M. Inguscio, and P. Violino, Experimental determinations of the hyperfine structure in the alkali atoms, Reviews of Modern Physics, vol.49, issue.1, p.31, 1977.
DOI : 10.1103/RevModPhys.49.31

E. Altman, W. Hofstetter, E. Demler, and M. D. Lukin, Phase diagram of two-component bosons on an optical lattice, New Journal of Physics, vol.5, p.113, 2003.
DOI : 10.1088/1367-2630/5/1/113

S. Trotzky, P. Cheinet, S. F?lling, M. Feld, U. Schnorrberger et al., Time-Resolved Observation and Control of Superexchange Interactions with Ultracold Atoms in Optical Lattices, Science, vol.319, issue.5861, p.295, 2008.
DOI : 10.1126/science.1150841

L. D. Carr, G. V. Shlyapnikov, and Y. Castin, Achieving a BCS Transition in an Atomic Fermi Gas, Physical Review Letters, vol.92, issue.15, p.150404, 2004.
DOI : 10.1103/PhysRevLett.92.150404

URL : https://hal.archives-ouvertes.fr/hal-00002433

L. Luo, B. Clancy, J. Joseph, J. Kinast, and J. E. Thomas, Measurement of the Entropy and Critical Temperature of a Strongly Interacting Fermi Gas, Physical Review Letters, vol.98, issue.8, p.80402, 2007.
DOI : 10.1103/PhysRevLett.98.080402

T. Gericke, F. Gerbier, A. Widera, S. Foelling, O. Mandel et al., Adiabatic loading of a Bose???Einstein condensate in a 3D optical lattice, Journal of Modern Optics, vol.54, issue.5, p.735, 2007.
DOI : 10.1103/PhysRevLett.89.250404

URL : https://hal.archives-ouvertes.fr/hal-00116297

E. Altman, E. Demler, and M. D. Lukin, Probing many-body states of ultracold atoms via noise correlations, Physical Review A, vol.70, issue.1, p.13603, 2004.
DOI : 10.1103/PhysRevA.70.013603

S. Fölling, F. Gerbier, A. Widera, O. Mandel, T. Gericke et al., Spatial quantum noise interferometry in expanding ultracold atom clouds, Nature, vol.22, issue.7032, p.481, 2005.
DOI : 10.1103/PhysRevA.59.4595

T. Rom, T. Best, D. Van-ooosten, U. Schneider, S. Fölling et al., Free fermion antibunching in a degenerate atomic Fermi gas released from an optical lattice, Nature, vol.73, issue.7120, p.733, 2006.
DOI : 10.1038/nature05319

I. Carusotto, Bragg scattering and the spin structure factor of two-component atomic gases, Journal of Physics B: Atomic, Molecular and Optical Physics, vol.39, issue.10, p.211, 2006.
DOI : 10.1088/0953-4075/39/10/S20

D. S. Petrov, M. Holzmann, and G. V. Shlyapnikov, Bose-Einstein Condensation in Quasi-2D Trapped Gases, Physical Review Letters, vol.84, issue.12, pp.2551-2555, 2000.
DOI : 10.1103/PhysRevLett.84.2551

URL : https://hal.archives-ouvertes.fr/hal-00001658

M. S. Makivi? and H. Ding, Two-dimensional spin-1/2 Heisenberg antiferromagnet: A quantum Monte Carlo study, Physical Review B, vol.43, issue.4, p.3562, 1991.
DOI : 10.1103/PhysRevB.43.3562

J. Kim and M. Troyer, Low Temperature Behavior and Crossovers of the Square Lattice Quantum Heisenberg Antiferromagnet, Physical Review Letters, vol.80, issue.12, p.2705, 1998.
DOI : 10.1103/PhysRevLett.80.2705