quantum monte carlo methods applied to ultracold gases stefano giorgini

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