Experiments with ultracold atomic
gases
Andrey Turlapov
Institute of Applied Physics, Russian Academy of Sciences
Nizhniy Novgorod
Fermions: 6Li atoms
Ground state splitting in high B
2p
4
670 nm
3
2s
Electronic ground
state: 1s22s1
Nuclear spin: I=1
 10 7
2
State 1 : spin 1 up
2
5
6
1
1 
State 2 : spin 1 down 2 
2
1
  ,1
2
1
  ,0
2
Optical dipole trap
Trapping potential of a focused laser beam:
 
2
U  d  E   E
Laser:
P = 100 W
llaser=10.6 mm
Trap:
U ~ 0 – 1 mK
The dipole potential is nearly conservative: 1 photon absorbed per 30 min
b/c llaser=10.6 mm >> llithium=0.67 mm
2-body strong interactions in a dilute gas (3D)
1
V (r )
L = 10 000 bohr
2
R=10 bohr ~ 0.5 nm
l (l  1) 2
Veff (r )  V (r ) 
2m r 2
At low kinetic energy, only s-wave scattering (l=0).
For l=1, the centrifugal barrier ~ 1 mK >> typical energy ~ 1 mK
s-wave scattering length a is the only interaction parameter (for R<< a)
Physically, only a/L
matters
Feshbach resonance. BCS-to-BEC crossover
Singlet 2-body potential:
5000
electron spins ↑↓
2500
a, bohr
0
200
400
600
800
1000
1200
1400
1600
-2500
-5000
-7500
BEC
of Li2
BCS
s/fluid
В, gauss
On resonance :
Triplet 2-body potential:
electron spins ↓↓
  4  a 2 ,   4 / k F2
b/c s-wave scattering amplitude: f l 0  
1
1

ik  1 / a
ik F
Superfluid and normal hydrodynamics of
a strongly-interacting Fermi gas (a → ∞)
M. Gyulassy: “Elliptic flow is everywhere”
Crab nebula
[Duke,
Science
(2002)]
Elliptic, accelerated expansion
Superfluid and normal hydrodynamics of
a strongly-interacting Fermi gas (a → ∞)
[Duke,
Science
(2002)]
T < 0.1 EF
Superfluidity ?
Superfluidity
1. Bardeen – Cooper – Schreifer hamiltonian
on the far Fermi side of the Feshbach resonance
p2 
H   a p a p  U 0  a p ' a p ' a p a p
p , 2
p ', p
2. Bogolyubov hamiltonian
on the far Bose side of the Feshbach resonance
H 
p
p2 
apap U0
2

a p1 ' a p2 ' a p2 a p1
p1 , p2 , p1 ', p2 '|
p1  p2  p1 '  p2 '
High-temperature superfluidity in the unitary limit
(a → ∞)
Bardeen – Cooper – Schrieffer:

 

Tc ~ E F exp  
 2 kF | a | 
Theories appropriate for strong interactions
Levin et al. (Chicago):
Burovsky, Prokofiev, Svistunov, Troyer
(Amherst, Moscow, Zurich):
Tc EF  0.29
Tc EF  0.22
The Duke group has observed signatures of phase transition in different
experiments at T/EF = 0.21 – 0.27
High-temperature superfluidity in the unitary limit
(a → ∞)
Group of John Thomas
[Duke, Science 2002]
Superfluidity ?
vortices
Group of Wolfgang Ketterle
[MIT, Nature 2005]
Superfluidity !!
Breathing mode in a trapped Fermi gas
Excitation &
observation:
Trap
ON
Trap ON again,
oscillation for variable t hold
toff
Image
1 ms
time
Release
300 mm
[Duke, PRL 2004, 2005]
Breathing Mode in a Trapped Fermi Gas
840 G Strongly-interacting Gas ( kF a = 30 )
Fit: xrms (t )  x0  A e
t / 
cost   
w = frequency
t = damping time
Breathing mode frequency 
Transverse frequencies of the trap:
x ,  y
Trap
z
x y 1.107
   x y
Prediction of universal isentropic hydrodynamics
(either s/fluid or normal gas with many collisions):
  1.84
at any T
Prediction for normal collisionless gas:
  2 x  2.11
Frequency ( )
Frequency  vs temperature
for strongly-interacting gas (B=840 G)
Collisionless gas
frequency, 2.11
2.0
Tc
Hydrodynamic
frequency, 1.84
at all T/EF !!
1.8
0.0
0.2
0.4
0.6
T/EF
0.8
1.0
1.2
Damping rate 1/ vs temperature
Damping rate (1/)
for strongly-interacting gas (B=840 G)
0.10
0.05
0.00
0.0
0.2
0.4
0.6
T/EF
0.8
1.0
1.2
Hydrodynamic oscillations.
Damping vs T/EF
Superfluid
hydrodynamics
As T  0 :
Bigger superfluid
fraction.
Collisional hydrodynamics
of Fermi gas
In general,
more collisions
longer damping.
As T  0 :
Collisions are Pauli blocked b/c
final states are occupied.
coll  T / EF   0
2
Slower damping
Oscillations damp faster !!
Damping rate 1/ vs temperature
Damping rate (1/)
for strongly-interacting gas (B=840 G)
0.10
0.05
0.00
0.0
0.2
0.4
0.6
T/EF
0.8
1.0
1.2
Black curve – modeling by kinetic equation
0.10
0.05
0.00
0.0
Frequency ( )
Damping rate (1/)
f
f 1 U trap  U mean field  f
T
v 
  coll ( )( f  f localequil.)
t
r m
r
v
EF
0.2
0.4
0.6
0.8
1.0
1.2
0.8
1.0
1.2
T/EF
2.0
1.8
0.0
0.2
0.4
0.6
T/EF
Damping rate (1/)
Damping rate 1/ vs temperature
for strongly-interacting gas (B=840 G)
0.10
Phase
transition
Phase transition
Tc
0.05
0.00
0.0
0.2
0.4
0.6
T/EF
0.8
1.0
1.2
TF
 0.27
Shear viscosity bound
d

v
Shear viscosity coefficient  :

A
F  v
d
Kovtun, Son, Starinets (PRL, 2005):
In a strongly-interacting quantum system
s – entropy density



s 4
Strongly-interacting atomic Fermi gas –
fluid with min shear viscosity ?!!
Quantum Viscosity?
momentum
 L
Viscosity:  
 2  n
cross section
L
   (...)  n
Calculate viscosity from breathing mode
Assumption: Universal isentropic hydrodynamics
One eq.: normal & s/f
u
m 2
 P
m
   u  U  
component flow together
t
2
 n
  ui 2 ul 
1
0 

  xi
  ik
  2
n i ,k ,l xk   xk 3 xl 
T
   
 F

 T 
  n    2 / 3   n
n 

Viscosity / Entropy density
for a universal isentropic fluid
T
   
 F

  n

1
E
 
  
3
d
 x

1







s  d 3x s
S/N
where S N -
E
 
1
S/N
entropy per particle
Viscosity / Entropy density

1 E
1

s
  S / N
 

s 4
3He
& 4He
near l-point
1.5
/s

s
?
s/f
transition
1.0
Quark-gluon plasma,
S. Bass, Duke, priv.
0.5
0.0
0.5
T 0
1.0
1.5
2.0
E/EF
2.5
String theory
limit 1/4
T  1.1 EF
Ferromagnetism: An open problems
Itinerant ferromagnetism in 2D
Ferromagnet
Normal
phase
2D at T=0:
Enorm
2
2
 4
n2  g
n2 ,
2m
2m
Eferro
2
 4
n2  2
2m
Eferro < Enorm at g > 4
a
g~
lz
2D Fermi gas in a harmonic trap
m z2 z 2
2
z


 m2 x 2  y 2 m z2 z 2
V ( x) 

2
2
z  
EF   2 N
where N = # of atoms
EF  z – condition of 2D in ideal gas at T=0
Open problems
2. Superfluidity in 2D
Berezinskii – Kosterlitz – Thouless transition
BKT transition not yet observed directly in Fermi systems.
Indirect observations in s/c films questioned [Kogan, PRB (2007)]
3. 3-body bound states
2D and quasi-2D analogs of the 3D Efimov states ?
How to parameterize
a universal Fermi gas ?
Temperature (T) or Total energy per particle (E)
Temperature:
1 S

T E
?
Energy measured from the cloud size !!
In a universal Fermi system: pressure
2
P(n, T ) 
3
[Ho, PRL (2004)]
Trap potential
U
Force Balance:
Virial Theorem:
 (n, T )
Local energy density
(interaction + kinetic)
P  nU  0
2U total  Etotal
z
E  Etot / N  3m z2 z 2
Thomas,
PRL (2005)
Resonant s-wave interactions (a → ± ∞)
Is the mean field U int
4  2 a

n
m
?
?
2
4  2 a
2 2/3
Energy balance at a → - ∞:
6 n 
n
2m
m

s-wave scattering amplitude:
f l 0  

1
ik  1 / a
In a Fermi gas k≠0. k~kF. Therefore, at a =∞,
U int
Collapse
f l 0  
1
ik F
4  2 aeff
1

n, where aeff ~ 
m
kF
n~k
3
F
U int


 2 k F2
2
2 2/3
~
  F (n)  
6 n
2m
2m
2 stages of laser cooling
1. Cooling in a magneto-optical trap
Tfinal = 150 mK
Phase-space density ~ 10-6
2. Cooling in an optical dipole trap
Tfinal = 10 nK – 10 mK
Phase-space density ≈ 1
The apparatus
1st stage of cooling: Magneto-optical trap
|e
laser
|g
pphoton=hk
patom atom
|g
photon
|e
patom-hk
1st stage of cooling: Magneto-optical trap
mj = –1
|e
laser
|g
|g>
Energy
mj= -1
mj=+1
laser
+

mj=0
0
z
mj = 0
mj = +1
1st stage of cooling: Magneto-optical trap
N ~ 109
T ≥ 150 mK
n ~ 1011 cm-3
phase space
density ~ 10-6
2nd stage of cooling: Optical dipole trap
Trapping potential of a focused laser beam:
 
2
U  d  E   E
Laser:
P = 100 W
llaser=10.6 mm
Trap:
U ~ 250 mK
The dipole potential is nearly conservative: 1 photon absorbed per 30 min
b/c llaser=10.6 mm >> llithium=0.67 mm
2nd stage of cooling: Optical dipole trap
Evaporative cooling
N
Evaporative cooling:
- Turn on collisions by tuning to the Feshbach resonance
- Evaporate
The Fermi degeneracy is achieved at the cost of loosing 2/3 of atoms.
Nfinal = 103 – 105 atoms, Tfinal = 0.05 EF, T = 10 nK – 1 mK,
n = 1011 – 1014 cm-3
Absorption imaging
Laser beam
l=10.6 mm
CCD matrix
Imaging over few microseconds
Mirror
Trapping atoms in anti-nodes of
a standing optical wave
Laser beam
l=10.6 mm
V(z)
z
Fermions: Atoms of lithium-6 in spin-states |1> and |2>
Mirror
Absorption imaging
Laser beam
l=10.6 mm
CCD matrix
Imaging over few microseconds
Photograph of 2D systems
x, mm
atoms/mm2
Each cloud is
an isolated 2D system
Each cloud ≈ 700 atoms
per spin state
Period = 5.3 mm
T = 0.1 EF = 20 nK
z, mm
[N.Novgorod, PRL 2010]
Linear density, mm-1
Temperature measurement
from transverse density profile
x, mm
Linear density, mm-1
Temperature measurement
from transverse density profile
2D Thomas-Fermi profile:
T=(0.10 ± 0.03) EF
m  T 


n1 ( x)  
2    
3/ 2
 m  m x
Li 3 / 2   e T 2T


2 2




Linear density, mm-1
Temperature measurement
from transverse density profile
Gaussian fit
m  T 


2D Thomas-Fermi profile: n1 ( x)  
2    
3/ 2
T=(0.10 ± 0.03) EF
=20 nK
 m  m x
Li 3 / 2   e T 2T


2 2




The apparatus (main vacuum chamber)
Maksim Kuplyanin, A.T., Tatyana Barmashova,
Kirill Martiyanov, Vasiliy Makhalov