Post on 12-Jan-2016
description
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