Experiments with ultracold atomic
gases
Andrey Turlapov
Institute of Applied Physics, Russian Academy of Sciences
Nizhniy Novgorod
How ultracold Fermi atoms are related to
nuclear physics ?
The atoms are fermions
With the atoms, one may see major Fermi phenomena (as in other Fermi
systems):
Fermi statistics;
Cooper pairing and superfluidity
+ strong interactions, i.e. Uint ~ EF
One may see even more with the atoms (the phenomena unobserved in the
other Fermi systems)
BEC-to-BCS crossover, i.e. crossover between a gas of Fermi atoms
and a gas of diatomic Bose molecules;
stability of a resonantly interacting matter;
resonant superfluidity;
viscosity at the lowest quantum bound (???);
itinerant ferromagnetism (???).
Good about atoms:
Fundamentally no impurities
Control over interactions:
tunable s-wave collisions
somewhat tunable p-wave collisions
dipole-dipole collisions (perspective)
Tunable spin composition, more than 2 spins
Tunable energy, temperature, density
Tunable dimensionality (2D – at Nizhniy Novgorod)
Direct imaging
Bad about atoms:
Small particle number (N = 102 – 106 << NAvogadro)
Non-uniform matter (in parabolic potential)
Coarse temperature tuning (dT > EF/20 as opposed to dT ~ EF/105 in solid-statephysics experiments)
No p-wave (and higher) collisions in thermal equilibrium
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
Fermi degeneracy
Fermi energy: EF   w (6N )1/ 3
At T=0:
Optical dipole trap:
EF  TF
Focus of a CO2
laser:
700 x 50 x 50 mm3
Phase space density:
r = Natoms / Nstates = 1
w /2p =(wx wy wz)1/3 /2p ~few kHz
Natoms=200 000
EF ~ 100 nK - 10 mK
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
Scattering in 1-channel model
V(r)
V(r)
r
r
R
R
a>0
(a > >R)
Repulsive mean field
The mean field (for weak interactions):
a<0
( | a| > >R )
Attractive mean field
4p  2 a
U
n
m
Fano – Feshbach resonance
1
valenceelectrons  (...) triplet  (...) singlet
Singlet 2-body potential:
2
electron spins ↑↓
 
U  m  B
Triplet 2-body potential:
electron spins ↓↓
On resonance
a
Fano – Feshbach resonance: Zero-energy scattering
length a vs magnetic field B
5000
834 gauss
a 
2500
a, bohr
200
400
600
800
1000
0
-2500
528 gauss
a0
-5000
-7500
В, gauss
1200
1400
1600
Instability of the a>0 region towards molecular
formation
5000
Singlet 2-body potential:
electron spins ↑↓
2500
a, bohr
0
200
400
600
800
1000
1200
1400
1600
-2500
-5000
Triplet 2-body potential:
electron spins ↓↓
-7500
В, gauss
BCS-to-BEC crossover
5000
Singlet 2-body potential:
electron spins ↑↓
2500
a, bohr
0
200
400
600
800
1000
1200
1400
1600
-2500
-5000
-7500
BEC
of Li2
В, gauss
BCS
s/fluid
Triplet 2-body potential:
electron spins ↓↓
Resonant s-wave interactions (a → ± ∞)
Is the mean field U int
4p  2 a

n
m
?
?
2
4p  2 a
2 2/3
Energy balance at a → - ∞:
6p 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
4p  2 aeff
1

n, where aeff ~ 
m
kF
n~k
3
F
U int


 2 k F2
2
2 2/3
~
  F (n)  
6p n
2m
2m
Universality
L
R
-V0
Strong interactions: |a|>L>>R
At a→∞, the system is universal, i.e.,
L is the only length scale:
- No dependence on microscopic details of binary interactions
- All local properties depend only on n and T


At T  0,
  4 p a 2 ,   4p / k F2
U int
 const  b
 F ( n)


 2 3p 2 n
 F ( n) 
2m
local Fermi energy
23
Experiment (sound propagation, Duke, 2007): b = - 0.565(.015)
Theory: Carlson (2003) b = - 0.560, Strinati (2004) b = - 0.545
Compare with neutron matter:
a = –18 fm, R = 2 fm
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
wlaser
|g
pphoton=hk
patom atom
|g
photon
|e
patom-hk
1st stage of cooling: Magneto-optical trap
mj = –1
|e
wlaser
|g
|g>
Energy
mj= -1
mj=+1
wlaser
+

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
mw  T 


n1 ( x)  
2p   w 
3/ 2
 m  mw x
Li 3 / 2   e T 2T


2 2




Linear density, mm-1
Temperature measurement
from transverse density profile
Gaussian fit
mw  T 


2D Thomas-Fermi profile: n1 ( x)  
2p   w 
3/ 2
T=(0.10 ± 0.03) EF
=20 nK
 m  mw x
Li 3 / 2   e T 2T


2 2




The apparatus (main vacuum chamber)
Superfluid and normal hydrodynamics of
a strongly-interacting Fermi gas
a 
T < 0.1 EF
Superfluidity ?
Duke,
Science (2002)
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:

p 

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wt   
w = frequency
t = damping time
Breathing mode frequency w
Transverse frequencies of the trap:
wx , w y
Trap
wz
wx wy 1.107
w  w xw y
Prediction of universal isentropic hydrodynamics
(either s/fluid or normal gas with many collisions):
w  1.84w
at any T
Prediction for normal collisionless gas:
w  2 wx  2.11w
Frequency ( ww )
Frequency w 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/w)
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/w)
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 ( ww )
Damping rate (1/w)
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
amping rate 1/ vs temperature
Damping rate (1/w)
or 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
Maksim Kuplyanin, A.T., Tatyana Barmashova,
Kirill Martiyanov, Vasiliy Makhalov