Quantum Monte Carlo methods applied to ultracold gases

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

QMC simulations have become an important tool in the study of dilute ultracold gases • Critical phenomena Shift of Tc in 3 D Grüter et al. (´ 97), Holzmann and Krauth (´ 99), Kashurnikov et al. (´ 01) Kosterlitz-Thouless Tc in 2 D Prokof’ev et al. (´ 01) • Low dimensions Large scattering length in 1 D and 2 D 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 Tc at unitarity Bulgac et al. (´ 06), Burovski et al. (´ 06)

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

Low dimensions + large scattering length

1 D Hamiltonian g 1 D>0 Lieb-Liniger Hamiltonian (1963) g 1 D<0 ground-state is a cluster state (Mc. Guire 1964) Olshanii (1998) if g 1 D large and negative (na 1 D<<1) metastable gas-like state of hard-rods of size a 1 D at na 1 D 0. 35 the inverse compressibility vanishes gas-like state rapidly disappears forming clusters

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

Equation of state of a 2 D Bose gas Universality and beyond mean-field effects • hard disk • soft disk • zero-range for zero-range potential mc 2=0 at na 2 D 2 0. 04 onset of instability for cluster formation

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

Equation of state beyond mean-field effects confirmed by study of collective modes (Grimm) BEC regime: gas of molecules [mass 2 m - density n/2 – scattering length am] am=0. 6 a (four-body calculation of Petrov et al. ) am=0. 62(1) a (best fit to FN-DMC)

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

Momentum distribution JILA in traps Condensate fraction

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

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

S=1 S=20

• Bose gas at finite temperature (S. Pilati + Barcelona) Equation of state and universality T Tc

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 Tc