1. Quantum Monte Carlo methods applied to ultracold gases Stefano Giorgini Istituto Nazionale per la Fisica della Materia Research and Development Center on Bose-Einstein Condensation Dipartimento di Fisica – Università di Trento BEC CNR-INFM meeting 2-3 May 2006

2. QMC simulations have become an important tool in the study of dilute ultracold gases Critical phenomena Shift of T c in 3D Grüter et al. (´97), Holzmann and Krauth (´99), Kashurnikov et al. (´01) Kosterlitz-Thouless T c in 2D Prokof’ev et al. (´01) Low dimensions Large scattering length in 1D and 2D Trento (´04 - ´05) Quantum phase transitions in optical lattices Bose-Hubbard model in harmonic traps Batrouni et al. (´02) Strongly correlated fermions BCS-BEC crossover Carlson et al. (´03), Trento (´04 - ´05) Thermodynamics and T c at unitarity Bulgac et al. (´06), Burovski et al. (´06)

3. Continuous-space QMC methods Zero temperature Solution of the many-body Schrödinger equation Variational Monte Carlo Based on variational principle energy upper bound Diffusion Monte Carlo exact method for the ground state of Bose systems Fixed-node Diffusion Monte Carlo (fermions and excited states) exact for a given nodal surface  energy upper bound Finite temperature Partition function of quantum many-body system Path Integral Monte Carlo exact method for Bose systems

4. Low dimensions + large scattering length

5. 1D Hamiltonian if g 1D large and negative (na 1D <<1) metastable gas-like state of hard-rods of size a 1D at na 1D  0.35 the inverse compressibility vanishes gas-like state rapidly disappears forming clusters g 1D >0 Lieb-Liniger Hamiltonian (1963) g 1D <0 ground-state is a cluster state (McGuire 1964) Olshanii (1998)

6. Correlations are stronger than in the Tonks-Girardeau gas (Super-Tonks regime) Peak in static structure factor Power-law decay in OBDM Breathing mode in harmonic traps mean field TG

7. Equation of state of a 2D Bose gas Universality and beyond mean-field effects hard disk soft disk zero-range for zero-range potential mc 2 =0 at na 2D 2  0.04 onset of instability for cluster formation

8. BCS-BEC crossover in a Fermi gas at T=0 -1/k F a BCSBEC

9. BEC regime: gas of molecules [mass 2m - density n/2 – scattering length a m ] a m =0.6 a (four-body calculation of Petrov et al.) a m =0.62(1) a (best fit to FN-DMC) Equation of state beyond mean-field effects confirmed by study of collective modes (Grimm)

10. Frequency of radial mode (Innsbruck) Mean-field equation of state QMC equation of state

11. Momentum distribution Condensate fraction JILA in traps

12. Static structure factor (Trento + Paris ENS collaboration) ( can be measured in Bragg scattering experiments) at large momentum transfer k F  k  1/a crossover from S(k)=2 free molecules to S(k)=1 free atoms

13. New projects: Unitary Fermi gas in an optical lattice (G. Astrakharchik + Barcelona) d=1/q= /2  lattice spacing Filling 1: one fermion of each spin component per site (Zürich) Superfluid-insulator transition single-band Hubbard Hamiltonian is inadequate

15. S=1 S=20

16. Bose gas at finite temperature (S. Pilati + Barcelona) Equation of state and universality T  T c T  T c

17. Pair-correlation function and bunching effect Temperature dependence of condensate fraction and superfluid density (+ N. Prokof’ev’s help on implemention of worm-algorithm) T = 0.5 T c