Quantum Monte Carlo methodsapplied to ultracold gases

Stefano Giorgini

Istituto Nazionale per la Fisica della Materia Research and Development Center on

Bose-Einstein CondensationDipartimento di Fisica – Università di Trento

BEC CNR-INFM meeting 2-3 May 2006

QMC simulations have become an important tool in the study of dilute ultracold gases

• Critical phenomena

Shift of Tc in 3D Grüter et al. (´97), Holzmann and Krauth (´99), Kashurnikov et al. (´01)

Kosterlitz-Thouless Tc in 2D Prokof’ev et al. (´01)

• Low dimensions

Large scattering length in 1D and 2D Trento (´04 - ´05)

• Quantum phase transitions in optical latticesBose-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

function trial where TTT

TT HE

Low dimensions + large scattering length

1D Hamiltonian

if g1D large and negative (na1D<<1) metastable gas-like state of hard-rods of size a1D

N

i jiijD

iD zg

zmH

112

22

1 )(2

21

222

)1(1

6 D

HR

namn

NE

at na1D 0.35 the inverse compressibility vanishesgas-like state rapidly disappearsforming clusters

1

323

2

1

2

1 03.1122

aa

maa

mag DD

DD

g1D>0 Lieb-Liniger Hamiltonian (1963)

g1D<0 ground-state is a cluster state

(McGuire 1964)

Olshanii (1998)

Correlations are stronger than in the Tonks-Girardeau gas

(Super-Tonks regime)

Peak in staticstructure factor

Power-law decay in OBDM

Breathing mode inharmonic traps

mean field

TG

Equation of state of a 2D Bose gas

)/1ln(2

22

2

D

MF

nan

mNE

Universality and beyond mean-field effects

• hard disk• soft disk• zero-range

for zero-range potential mc2=0 at na2D

20.04onset of instability for cluster formation

BCS-BEC crossover in a Fermi gas at T=0

-1/kFa

BCSBEC

...)(

615

1281

18

52// 2/3

3 mFmFF

b akakNE

BEC regime: gas of molecules [mass 2m - 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)

Equation of state

beyond mean-field effects

confirmed by study of collective modes (Grimm)

Frequency of radial mode (Innsbruck)

Mean-field equation of state

QMCequation of state

Momentum distribution

Condensate fraction

JILA in traps

2/130 )(

38

1 mmann

Static structure factor (Trento + Paris ENS collaboration)

( can be measured in Bragg scattering experiments)

at large momentumtransfer

kF k 1/acrossover from S(k)=2 free moleculestoS(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

)(sin)(sin)(sin2

)( 22222

qzqyqxmq

sVext r

0 1 2 30.0

0.2

0.4

0.6

0.8

1.0

superfluid fraction condensate fraction

s

S=1 S=20

• Bose gas at finite temperature (S. Pilati + Barcelona)

Equation of state and universality

T Tc 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