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)
BEC
a0
Unitary Gas
BCS
k F a  (1,1)
a0
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)
Bose Gases:
a
Rb: JILA (Jin & Cornell’s group ,
2008);
Li: Rice Hulet’s group, 2009.
B
Li: Salamon’s group, 2011
Na: MIT Ketterle’s group, 1998.
Binding energy
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.
na 
3
a

 1
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.
2 2 n 2 a
128
E
(1 
m
15 
na3
 8(4  3 3 ) / 3  [ln(na )  4.72  2 B]na  ......)
3
3
Lee-Yang-Huang effect:
An alternative view based on two-body physics
E ( N ) 4na
128


(1 
na3  ...)
N
m
6 
E2  b
1
( L) N ( L)  ......  
2
2m
if
L  .
2body spectrum in 3D Trap
a=0
a=0
Resonance
Scaling dimensions of interaction energy for two
atoms in a trap of size L (kinetic energy ~ 1/L^2.)
a
a
E2 b ( L ) ~
(1  C
 ...),a  L;
3
mL
2L
1
E2 b ( L ) ~
, a  L.
2
mL
(Near Resonances)
Renormalization-group-equation approach
RG flow
4a
dV(L)
m 2
V () 
1   2 V .
dL

m
4a
m V(L)
 a(L)  a(1
 ...), a(L) 
L
4
Near resonance
1
1
3
a  ,  

(n ) 
2
2
2m
m
n
1/ 3
2/ 3
n
or  
 F
2m
Comparison of 2-body wavefunctions
1) dashed Blue---fermion; 2) Thin Red---far away resonance;
3) Thick red---at resonance.
Constrained variation approach
General features
1)
2)
At long wavelengths there should be gapless excitations of
Bogoliubov type;
At short distance, the two-particle scattering state is universal.
1)(
1
1  g k2
bk 
gk
1  g k2

k
b ) | g.s.  0
a
1
2) (r  0)  1  ;  g k  2
r
k
Wavefunction
| g.s.  exp(c b )   exp(g b b ) | 0 

0 0
 
k k k
k
N 0 | c0 |2 , nk 
g k2
1  g k2
Constraint s :
g
m
k
g k 
gk  0
A
 2 , g k 0  1  Bk
k
Song and FZ, Phys Rev Lett.103, 025302 ( 2009)
Nearly Fermionized 3D Bose gases near Resonance
80%
Song and FZ, Phys Rev Lett.103, 025302( 2009)
a
 ( d , a, r * )   F f ( )
d
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.)
V<0
~1
C.F.=50%
a/d
V>0
~1
C.F.=8 %
(Onsager and Penrose, 56)
R/d
??
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 1)0;
2)2 ;
3)4 ;
4) f ( ; r*, d , D).
  4a 2
Effective scattering scattering length ~ d
“Three-body” bound States
E3
E3
E3
1/d
r*/d^2
1/a
1/a
1/a
1/d^4
1/a d
1/d^2
3D
2D
4D (E3=E2 when r*=0)
Efimov Trimers in a Quantum Gas:
Effect of Pauli Blocking
Vertex
correction
related to
Anderson
Infrared
catastrophe
Energy versus density at resonance (BBF in a FS)
(MacNeill and Zhou, 2010; To appear in Phys. Rev. Lett.)
Particle-Hole fluctuations
Efimov Trimers (Efimov, 1970-73)
529
Discovery of Tetramers:
Hammer and Platter Eur.Phys. 2007;
Stecher, D’Incao and C.Greene, Nature Phys 2009.
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)
E(k)
40K
87Rb
F.S.
2
k
 BF  F
2mB
-K
K
k
2
Q
F
WB   B 
 B , B (Q)  0.
2M tot
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).