1. Optical lattices for ultracold atomic gases Sestri Levante, 9 June 2009 Andrea Trombettoni (SISSA, Trieste)

2. Outlook A brief introduction on ultracold atoms Why using optical lattices? Effective tuning of the interactions Experimental realization of interacting lattice Hamiltonians Ultracold bosons on a disordered lattice: the shift of the critical temperature

3. Trapped ultracold atoms: Bosons Bose-Einstein condensation of a dilute bosonic gas Probe of superfluidity: vortices System: - typically alkali gases (e.g., Rb or Li) - temperature order of 10-100 nK - number of particles: 10 3 -10 6 - size order of 1-100  m

4. Trapped ultracold atoms: Fermions Tuning the interactions… … and inducing a fermionic “condensate” A non-interacting Fermi gas

5. Ultracold atoms in an optical lattice a 3D lattice It is possible to control: - barrier height - interaction term - the shape of the network - the dimensionality (1D, 2D, …) - the tunneling among planes or among tubes (in order to have a layered structure) …

6. Tuning the interactions with optical lattices bosonic field s-wave scattering length For large enough barrier height tight-binding Ansatz [Jaksch et al. PRL (1998)] increasing the scattering length or increasing the barrier height  the ratio U/t increases Bose-Hubbard Hamiltonian Ultracold fermions in an optical lattice  (Fermi-)Hubbard Hamiltonian [Hofstetter et al., PRL (2002) – Chin et al., Nature (2006)]

7. Quantum phase transitions in bosonic arrays Increasing V, one passes from a superfluid to a Mott insulator [Greiner et al., Nature (2001)] Similar phase transitions studied in superconducting arrays [see Fazio and van der Zant, Phys. Rep. 2001]: Why using optical lattices? Effective tuning of the interactions Nonlinear discrete dynamics: negative mass, solitons, dynamical instabilities Experimental realization of interacting lattice Hamiltonians: Study of quantum & finite temperature phase transitions

8. [A. Trombettoni, A. Smerzi and P. Sodano, New J. Phys. (2005)] central peak of the momentum distribution: Good description at finite T by an XY model thermally driven vortex proliferation [Schweikhard et al., PRL (2007)] In the continuous 2D Bose gas BKT transition observed in the Dalibard group in Paris, see Hadzibabibc et al., Nature (2006) Finite temperature Berezinskii-Kosterlitz- Thouless transition in a 2D lattice

9. 2D optical lattices “simulating” graphene With three lasers suitably placed: Zhu, Wang and Duan, PRL (2007)

10. Ultracold bosons and/or fermions in trapping potentials provide new experimentally realizable interacting systems on which to test well-known paradigms of the statistical mechanics: -) in a periodic potential -> strongly interacting lattice systems -) interaction can be enhanced/tuned through Feshbach resonances (BEC-BCS crossover – unitary limit) -) inhomogeneity can be tailored – defects/impurities can be added -) effects of the nonlinear interactions on the dynamics -) strong analogies with superconducting and superfluid systems -) used to study 2D physics -) predicted a Laughlin ground-state for 2D bosons in rotation: anyionic excitations … Trapped ultracold atoms

11. Outlook A brief introduction on ultracold atoms Why using optical lattices? Effective tuning of the interactions Experimental realization of interacting lattice Hamiltonians Ultracold bosons on a disordered lattice: the shift of the critical temperature Infinite-range model:  T c <0, and vanishing  T c for large filling f 3D lattice: ordered limit & connection with the spherical model 3D lattice with disorder:  T c >0 for large f -  T c <0 for small f with: L. Dell’Anna, S. Fantoni (SISSA), P. Sodano (Perugia) [J. Stat. Mech. P11012 (2008)]

12. Bosons on a lattice with disorder filling total number of particles number of sites random variables: produced by a speckle or by an incommensurate bichromatic lattice From the replicated action  disorder is similar to an attractive interaction

13. Replicated action Introducing N replicas  =1,…,N  effective attraction

14. Shift of the critical temperature in a continuous Bose gas due to the repulsion For an ideal Bose gas, the Bose-Einstein critical temperature is What happens if a repulsive interaction is present? The critical temperature increases for a small (repulsive) interaction… …and finally decreases [see Blaizot, arXiv:0801.0009]

15. Long-range limit (I) Without random-bond disorder The relation between the number of particles and the chemical potential is The critical temperature is then

16. Long-range limit (II) With random-bond disorder Using results from the theory of random matrices [in agreement with the results for the spherical spin glass by Kosterlitz, Thouless, and Jones, PRL (1976)]

17. 3D lattice without disorder The relation between the number of particles and the chemical potential is single particle energies For large filling

18. 3D lattice, with random-bond and on-site disorder: 3D lattice with disorder Introducing N replicas of the system and computing the effective replicated action Disorder (both on links and on-sites) is equivalent to an effective attraction among replicas Diagram expansion for the Green’s functions for N  0 Computing the self-energy New chemical potential (effective t larger, larger density of states)

19. 3D lattice with disorder: Results for random-bond disorder For large filling When both random-bond and random on-site disorder are present

20. 3D lattice with disorder: numerical results results for the continuous (i.e., no optical lattice) Bose gas [Vinokur & Lopatin, PRL (2002)]

21. A (very) qualitative explanation Continuous Bose gas: Repulsion  critical temp. T c increases Disorder  “attraction”  T c decreases Lattice Bose gas: Disorder  “attraction” Small filling  continuous limit  T c decreases Large filling  all the band is occupied  effective “repulsion”  T c increases

22. Thank you!

23. Some details on the diagrammatic expansion (I) Green’s functions: N -> 0 At first order in v 0 2

24. Some details on the diagrammatic expansion (II)

25. For large filling, the critical temperature coincides with the critical temperature of the spherical model Connection with the spherical model The ideal Bose gas is in the same universality class of the spherical model [Gunton-Buckingham, PRL (1968)] with the (generalized) constraint

26. Long-range limit (I) Without random-bond disorder The matrix to diagonalize iswhere The relation between the number of particles and the chemical potential is The critical temperature is then

27. 3D lattice with disorder: Results for an incommensurate potential Two lattices:

28. Stabilization of solitons by an optical lattice (I) Recent proposals to engineer 3-body interactions [Paredes et al., PRA 2007 -Buchler et al., Nature Pysics 2007] In 1D with attractive 3-body contact interactions: no Bethe solution is available – in mean-field [Fersino et al., PRA 2008] : in order to have a finite energy per particle

29. Stabilization of solitons by an optical lattice (II) Problem: a small (residual) 2-body interaction make unstable such soliton solutions Adding an optical lattice : Soliton solutions stable for for small q

30. 2-Body Contact Interactions N=2 Lieb-Liniger modelit is integrable and the ground-state energy E can be determined by Bethe ansatz: Mean-field works for[3]: [3] F. Calogero and A. Degasperis, Phys. Rev. A 11, 265 (1975) is the ground-state of the nonlinear Schrodinger equation with energy in order to have a finite energy per particle

31. N-Body Attractive Contact Interactions We consider an effective attractive 3-body contact interaction and, more generally, an N-body contact interaction: With contact interaction N-body attractive (c>0)