What can we learn about quantum gases from 2- and 3-atom problems?

Fei Zhou

University of British Columbia, Vancouver

at Institute for Nuclear Theory, University of Washington,March 31, 2011

Special Thanks to Canadian Institute for Advanced Research Izzak Walton Killam Foundation Center for Ultracold Atoms, Harvard and MIT

Cold-Atom Theory group at UBC:

Junliang Song (Currently at Innsbruck, PDF)

Mohammad Mashayekhi (PhD student, UBC)

Demitri Borzov (MSc student, UBC)

Majid Hasan (MSc student, UBC)

David MacNeill (Currently at Cornell, PhD student)

Xiaoling Cui (Currently at IASTU, Bejing)

BCS

BEC

Unitary Gas )1,1( akF

0a

0a

Outline

1) Nearly Fermionized 3D Bose Gases near Feshbach Resonance. Comparison to recent experiments.

2) Anomalous Dimers and Trimers

in a Quantum Mixture near Resonance

---- the role of multiple scattering, Fermi surface and Fermi surface dynamics.

Bose Gas near Feshbach Resonance (Upper Branch)

a

Binding energy

B

Bose Gases:

Rb: JILA (Jin & Cornell’s group , 2008);

Li: Rice Hulet’s group, 2009.

Li: Salamon’s group, 2011

Na: MIT Ketterle’s group, 1998. Fermi Gases MIT Ketterle’s group, 2009.

Possible approaches to a Bose gas at large scattering lengths

1) Diagrammatic and similar approaches

(developed in 1957-58; 1960s for small scattering lengths or hardcore bosons)

2) Monte Carlo simulations of a quantum Bose gas with attractive interactions in the upper branch haven’t been successful.

3) Monte Carlo simulations of the ground state of hardcore bosons (?).

Dilute Bose Gases

Lee-Yang-Huang (56; 57-58) and Beliaev (58)

And is valid for small scattering lengths.

There have been efforts to improve LYH-Beliaev theory by taking into the higher order contributions. (Wu, Sawada, 59;….. Braatten, 02…) At infinity a, each term diverges.

13 a

na

......)]272.4)[ln(3/)334(8

15

1281(

2

33

322

naBna

nam

anE

Lee-Yang-Huang effect:An alternative view based on two-body physics

...)6

1281(

4)( 3

na

m

na

N

NE

.2

......)()(2

12

LifLNLE

mb

2body spectrum in 3D Trap

a=0

Resonance

a=0

Scaling dimensions of interaction energy for two atoms in a trap of size L (kinetic energy ~ 1/L^2.)

.,1

~)(

;...),2

1(~)(

22

32

LamL

LE

LaL

aC

mL

aLE

b

b

(Near Resonances)

Renormalization-group-equation approach

RG flow

dV (L)

dL 1 m

2 V2 . V ()

4am

a(L) a(14a

L ...), a(L)

mV (L)

4

Near resonance

a , 1

2m 2 1

m 2 (n 3 )

n 1/ 3 or n2 / 3

2m F

Comparison of 2-body wavefunctions1) dashed Blue---fermion; 2) Thin Red---far away resonance; 3) Thick red---at resonance.

Constrained variation approach

General features

1) At long wavelengths there should be gapless excitations of Bogoliubov type;

2) At short distance, the two-particle scattering state is universal.

2

22

1;1)0()2

0..|)11

1)(1

kg

r

ar

sgbg

gb

g

k

k

k

kk

k

Wavefunction

Bkgk

Ag

gg

sConstra

g

gncN

bbgbcsg

kk

kmk

k

k

k

kkkk

1,

0

:int

1,||

0|)exp()exp(..|

02

2

2

200

00

Song and FZ, Phys Rev Lett.103, 025302 ( 2009)

Song and FZ, Phys Rev Lett.103, 025302( 2009)

)(),,( *

d

afrad F

80%

Nearly Fermionized 3D Bose gases near Resonance

Density Profile and Size of Condensates

Blue Data from Pollack et al., Phys. Rev.Lett. 102, 90402 (2010)

Navon et al., ArXiv. 1103. 4449 (2011). (7Li, ENS) a=2500Bohr. Chemical potential about 44% of the Fermi energy.

Previous Results Jastrow-wavefunction approach (Cowell et al., 2002)

indicates a quantum phase transition at a finite scattering length and the chemical potential is 2.93 times of the Fermi energy.

Quantum Mont Carlo (Giorgini et al, 1999) had a convergence problem near resonance (due to instability).

Relation to the estimate of liquid Helium condensation fraction (C.F.)

R/d

C.F.=50%

a/d

~1

~1

C.F.=8 % (Onsager and Penrose, 56)

??

V<0

V>0

Validity/Relevance to experiments

Although the role of Efimov physics remains to be understood, our results appear to suggest a reasonable qualitative picture

and even a quantitative estimate of the sizes of condensates near resonances.

Conjecture: In some cold Bose gases studied in labs, the atoms which

have developed non-trivial three-body correlations and which are not captured by our descriptions might have recombined quickly and left the trap before the rest of atoms become self- thermalized and probed in measurements?

Multiple Scattering Near Resonance:From 2 to N ----A Quiz for MacNeill

Scattering Cross-section is

24

).,*,;()4;4)3;2)2;0)1

a

Ddrf

Effective scattering scattering length ~ d

“Three-body” bound States

1/a

E3

1/d^2

1/d

1/a 1/a

3D 2D 4D (E3=E2 when r*=0)

E3 E3

1/a d 1/d^4

r*/d^2

Efimov Trimers in a Quantum Gas:Effect of Pauli Blocking

Energy versus density at resonance (BBF in a FS) (MacNeill and Zhou, 2010; To appear in Phys. Rev. Lett.)

Vertex correction related toAnderson Infrared catastrophe

Particle-Hole fluctuations

Efimov Trimers (Efimov, 1970-73)

Discovery of Tetramers: Hammer and Platter Eur.Phys. 2007;Stecher, D’Incao and C.Greene, Nature Phys 2009.

529

Loss spectrum: Observation of 22.7

Kramer et al. (Grimm’s group at Innsbruck), Nature 440, 315 (2006);Zaccanti et al. (Lorence group), Nature Physics 5, 568 (2009) Pollack et al. (Hulet’s group at Rice), Science 326, 1683 (2009)….

Anomalous Dimers in Quantum Mixtures(Song, Mashayekhi and FZ, PRL, 2010)

40K

87Rb

k-K K

F.S.

E(k)

.0)(,2

2

QM

QW BB

tot

FBB

B

FFB m

k

2

2

Mass Ratio= 0.05

Song and FZ, 2011, to appear.

Conclusions

1) Cold gases near resonances have fascinating/surprising properties. 3D Bose gases near resonance are nearly Fermionized analogous to 1D dilute Tonks-Girardeau gas.

2) Anomalous dimers and trimers in a quantum gas can have very distinct structures and dynamics which have yet to be better understood. ----work in progress.

B: Resonant Fermi-Bose mixtures

C. A. Stan et al., ( Ketterle’s group), Phys. Rev. Lett. 93, 143001 (2004). S. Inouye et al., (J. Bohn, and D. Jin’s group), Phys. Rev. Lett. 93, 183201 (2004). Ferlaino et al., (Inguscio, Modugno) Phys. Rev. A 73, 040702 (2006).Ospelkaus et al., ( K. Sengstock’s group), Phys Rev Lett. 97, 20403 (2006). B. Deh et al., (P. Courteille’s group), Phys. Rev. A 77, 010701(R) (2008).