Download Lecture 3 : ultracold Fermi Gases Lecture 3 : ultracold Fermi Gases

Lecture
Lecture 33 :: ultracold
ultracold Fermi
Fermi Gases
Gases
The ideal Fermi gas: a reminder
Interacting Fermions
BCS theory in a nutshell
The BCS-BEC crossover and quantum simulation
Many-Body
Many-Body Physics
Physics with
with Cold
Cold Gases
Gases
Diluteness: atom-atom interactions described by 2-body
(and 3 body) physics. At low energy: a single parameter,
the scattering length a
Control of the sign and magnitude of interaction
Control of trapping parameters:
access to time dependent phenomena,
out of equilibrium situations, 1D, 2D, 3D
n(k)
Simplicity of detection
Sherson et al., MPQ 2010
n(k)
Comparison
with quantum Many-Body theories:
1
Gross-Pitaevskii,
Bose and Fermi Hubbard models,
1
search for exotic phases, dipolar gases
disorder effects, Anderson localization, …
Link with condensed matter (high Tc superconductors),
astrophysics (neutron stars), Nuclear physics,
k/kF
high energy physics (quark-gluon
plasma),
1
k/kF
1
Quantum simulation with cold atoms « a la Feynman »
The
The ideal
ideal Fermi
Fermi gas:
gas: aa reminder
reminder
Zero temperature Fermi sea:
E(p)
 2 kF2
EF 
 kBTF
2m
EF
k F  (6 2 n)1/ 3 ~ (particle distance) 1
p
1  2m 
Fermi pressure:


15 2   2 
3/ 2
EF5/ 2
Approximation valid as long as T<<TF
Electrons
Electrons vs
vs cold
cold atoms
atoms
Electrons in metal
Ultra cold atoms
Density
1030 /m3
1020 /m3
Mass
10-30 kg
10-26 kg
Fermi temperature
104 K
1µK
T/TF
<10-5
~ 10-2
Lifetime
Infinite
~10s
Size
1cm
10 µm
Particle number
1024
105
Pauli
Pauli Exclusion
Exclusion Principle
Principle and
and evaporative
evaporative
cooling
cooling of
of ultra-cold
ultra-cold Fermi
Fermi gases
gases
Collision between two atoms. Effective potential in the l-wave:
 (  1)
Veff (r )  V (r ) 
2mr 2
2
Interatomic potential
(long range~-1/r6)
l >0
l=0
centrifugal potential
At low temperature, atoms cannot
cross the centrifugal
barrier: only s-wave collisions.
Symmetrization for identical
particles: even l-wave collisions
forbidden for polarized fermions.
Suppression
Suppression of
of elastic
elastic collisions
collisions
in
in aa spin
spin polarized
polarized Fermi
Fermi gas
gas
Sp
i
n
po
la
riz
ed
(p
-w
av
e)
Spin mixture (s-wave)
B. DeMarco, J. L. Bohn, J.P. Burke, Jr., M. Holland, and D.S. Jin, Phys.
Rev. Lett. 82, 4208 (1999).
Use spin mixtures or several atomic species (eg 6Li-7Li, K-Rb,
different spin states…)
Quantum
Quantum gases
gases in
in harmonic
harmonic traps
traps
Bose-Einstein statistics (1924)
Bose-Einstein condensate
Fermi-Dirac statistics (1926)
Fermi sea
EF
Bose enhancement
Pauli Exclusion
h
TC =
(0.83 N)1/3
kB
h
1/3
T << TF =
(6N)
kB
Dilute gases: 1995, JILA, MIT
Dilute gases: 1999, JILA
Absorption
Absorption imaging
imaging
in situ: cloud size
Bose
-Einstein condensate
Bose-Einstein
condensate and
and Fermi
Fermisea
sea
Lithium 7
2001
ENS
Lithium 6
104 Li 7 atoms, in thermal equilibrium with
104 Li 6 atoms in a Fermi sea.
Quantum degeneracy: T= 0.28 K = 0.2(1) TC= 0.2 TF
Now:
T=0.03 TF
The
The non-interacting
non-interacting Fermi
Fermi gas
gas
Gaussian Fit
Fermi-Dirac
T/TF<0.05
Atom number~105
Quantum
Quantum simulation
simulation of
of strongly
strongly
interacting
interacting Fermions
Fermions
Quantum
Quantum fluids
fluids
Bose Einstein condensates
Superconductivity and helium 3
4He
High Tc and 3He
Dilute gas BEC
BEC
BECof
ofstrongly
stronglybound
boundfermions
fermions
Connecting the two regimes ?
Theory since 80’: Leggett, Randéria, Nozières,Schmidt-Rink, Holland, Kokkelmans,
Levin, Ohashi, Griffin, Strinati, Falco, Stoof, Bruun, Pethick, Combescot, Giorgini….
intermediate
intermediateregime
regime
Near Feshbach resonance,
interactions in cold Fermi gas can be enhanced
Tc ~TF/5
Fermions
Fermions with
with two
two spin
spin states
states
with
with attractive
attractive interaction
interaction
Tc  TF e
BEC of molecules
Bound state
 / 2 k F a
BCS fermionic superfluid
Interaction strength
No bound state
Dilute gases: Feshbach resonance
Cooper
Cooper Pairing
Pairing
Bardeen, Cooper, Schrieffer, 1957
100 years of supraconductivity.
Naive interpretation: Take a homogeneous T=0 Fermi gas, kF, EF
Add two fermions, 1 et 2, which display attractive interaction
 
 
V ( r1  r2 )  V  ( r1  r2 )
a< 0
avec V < 0
Then these particles will always form at state with an energy lower than
EF , a bound state.
Pairs
 
k, k
at Fermi surface:
| k | kF
These pairs form a superfluid phase
TBCS  0.3 TF e


2 k F |a | 
k F a  1
Mean-field
Mean-field Theory
Theory (BCS)
(BCS) at
at T=0
T=0
Bardeen, Cooper, Schrieffer, 1957
Many-body hamiltonian:
Hˆ



3

3
3


ˆ
ˆ
ˆ
ˆ
ˆ (r ') 
ˆ (r )
d
(
)
h
(
)
d
d
'
(
)
(r ')V (r  r ')
r

r

r

r
r

r





   0 



ˆ



 d r

3


ˆ (r ) g d 3r
ˆ  (r )
ˆ  (r )
ˆ (r )
ˆ (r )
(r )h0 

b




2

h0  
2m
ˆ (r )  
In momentum space: 

k
gb

ˆ
H   k aˆk aˆk  3
L
k
 aˆ
eik .r
3
L
aˆk ,


k  q  k ' q  k ' k 
k ,k ',q
aˆ
aˆ aˆ
2k 2
k 

2m
Mean-field
Mean-field Theory
Theory (2)
(2)
Search for a solution in the form:  N  N    ck ak† a† k  
ck  c k

k
 (ri  rj )
Fourier components of spatial function
N /2

| vac 
Assumption: zero momentum for BCS pairs:
 k  aˆk  aˆk  ~ order parameter for a gas of bosonic
molecules with zero center of mass momentum.
BCS order parameter
Hˆ  cte   k aˆk aˆk  *  aˆ k  aˆk     aˆ k  aˆ  k   ...
k
k
gb
 3
L
gb
k k  L3
k

k
aˆ k  aˆk 
Gap equations
Excitation
Gap
Excitation
Gap
In the ground state, no excitation of the Bogoliubov modes
Ek  k2  |  |2
Ek
|D|
BCS wavefunction: |
xk
2k 2
k 

2m
 BCS   (| uk | | vk | ei ak† a† k  ) | vac 
k
| uk |2  | vk |2  1
Tuning
Tuning interactions
interactions in
in Fermi
Fermi gases
gases
Lithium
Lithium 66
scattering length [nm]
200
a>0
100
BEC-BCS
Crossover
0
a<0
-100
-200
0,0
Bound state
2
EB  
ma 2
0,5
1,0
1,5
Magnetic field [kG]
condensate of molecules
2,0
No bound state
BCS phase
The
The Equation
Equation of
of State
State of
of aa Fermi
Fermi
Gas
Gas with
with Tunable
Tunable Interactions
Interactions
Cold atoms, Spin ½
Dilute gas : 1014 at/cm3, T=100nK
BEC-BCS crossover
Spin imbalance, exotic phases
Neutron star, Spin ½
a = -18.6 fm, n~2 1036cm-3
•Tc=1010 K , T=TF/100
• kFa ~ -4,-10,…
• kFre<<1
Experimental
Experimental sequence
sequence
- Loading of 6Li in the optical trap
- Tune magnetic field to Feshbach
resonance
- Evaporation of 6Li
- Image of 6Li in-situ
Suppression
Suppression of
of vibrational
vibrational relaxation
relaxation for
for fermions
fermions
W(r)
a
a
EB
r
Re
 C6 / r 6
Pauli exclusion principle
Inhibition by factor (a/Re)2 >>1
Re
h2/ma2
Binding energy: EB=
Momentum of each atom: h/a
G~ 1/as
with s = 2.55 for dimer-dimer coll.
3.33 for dimer-atom coll.
D. Petrov, C. Salomon, G. Shlyapnikov, ‘04
Unitary
Unitary Fermi
Fermi Gas
Gas ::
a
Optical
Density
(a.u)
Pixels
Direct
Direct proof
proof of
of superfluidity:
superfluidity:
classical
classical vs.
vs. quantum
quantum rotation
rotation
Rotating classical gas
velocity field of a rigid body
Rotating a quantum macroscopic object
macroscopic wave function:
In a place where
, irrotational velocity field:
The only possibility to generate a non-trivial rotating motion is to
nucleate quantized vortices (points in 2D or lines in 3D) with quantized
circulation around vortex core. Feynman, Onsager
to keep
single-valued
Vortices now all have the same sign, imposed by the external rotation
MIT 2005: Vortex lattices in the BEC-BCS Crossover
Energy [MHz]
1
Direct proof of superfluidity
0
BEC side
BCS side
M. Zwierlein
A. . Schirotzek
C. Stan
C. Schunk
P. Zarth
W. Ketterle
Science 05
-1
650
834.15
Magnetic Field [G]
Direct
Direct proof
proof of
of superfluidity
superfluidity
Critical SF temperature = 0.19 TF
MIT 2006
Superflow
Superflow in
in fermionic
fermionic superfluids
superfluids
Landau criterion: dissipation for
V>wk/k
Theory: Combescot, Kagan and
Stringari.
Experiment: MIT, 2008.
strongly
strongly interacting
interacting Fermions
Fermions
thermodynamic
thermodynamic properties
properties
Thermodynamics
Thermodynamics
PV  Nk BT
Is a useful but incomplete equation of state !
Complete information is given by thermodynamic potentials:
Grand potential
n(k)
n(k)
1
1
   PV  E  TS  N
Pressure
Volume
Temperature
Entropy
Chemical potential
Atom number
Internal energy
We have measured the grand potential of a tunable Fermi gas
S. Nascimbène et al., Nature, 463, 1057, (2010), arxiv 0911.0747
N. Navon et al., Science 328, 729 (2010)
1 NJP (2010) k/kk/k
F
S. Nascimbène et al.,
F
S. Nascimbène et al.,1arXiv 1012.4664: normal
phase, to appear PRL 2011
M. Horikoshi et al., Science, 327, 442 (2010)
The ENS Fermi Gas Team
S. Nascimbène, N. Navon, F. Chevy,
A. Ridinger, T. Salez, S. Chaudhuri,
K. Gunther, B. Rem, A. Grier, I Ferrier-barbut, D. Wilkowski, F. Chevy, F. Werner,
U. Eismann, K. Magalhães, K. Jiang
Y. Castin, M. Antezza, D. Fernandes
Theory collaborators: D. Petrov, G. Shlyapnikov , D. Papoular,
J. Dalibard, R. Combescot, C. Mora
C. Lobo, S. Stringari, I. Carusotto,
L. Dao, O. Parcollet, C. Kollath, J.S. Bernier, L. De Leo, M. Köhl, A. Georges
Useful
Useful thermodynamic
thermodynamic quantities
quantities
•
Gibbs –Duhem relation:
•
Density:
VdP  SdT  Nd 
P
n

P
T
• Free energy in canonical ensemble:
•
Entropy density:
S /V 
P
F (T ,  ,V )
V
 2 F 
• Isothermal Compressibility: 1/  T  V 
2 

V

T , N
 2 F 
• Specific heat: CV  T  2 
 T  N ,V
•
Legendre transform between grand canonical and
canonical variables
Thermodynamics
Thermodynamics of
of aa Fermi
Fermi gas
gas
The Equation of State of a uniform Fermi gas can be written as:
(  , T ; a )  E  TS  N   P (  , T ; a ) V
Pressure contains all the thermodynamic information
Variables :
scattering length
temperature
chemical potential
a
T
µ
We build the dimensionless parameters :
Interaction parameter
Fugacity (inverse)
Canonical analogs
Measuring
Measuring the
the EoS
EoS of
of the
the Homogeneous
Homogeneous Gas
Gas
Local density approximation:
gas locally homogeneous at
Measuring
Measuring the
the EoS
EoS of
of the
the Homogeneous
Homogeneous Gas
Gas
Extraction of the pressure from in situ images
Ho, T.L. & Zhou, Q.,
Nature Physics, 09
S. K Yip, 07
•
doubly-integrated density profiles
equation of state measured for
all values of
Derivation
Derivation
LDA:
1
1
2
2
2
 ( x, y, z )    mr ( x  y )  m z 2 z 2
2
2
0
Cylindrical symmetry
Gibbs-Duhem:
dP   SdT  nd   nd 
Integrating over x and y between 0 and +

at constant temperature

1
P(  z ) 
mr2  n( x, y, z )2 rdr
2
0
2 
2
mr
mr
P(  z , T ) 
n( x, y, z )dxdy 
n ( z)

2 0
2
Directly relates the pressure at abscissa z
to the doubly integrated absorption signal at z
Unitary
Unitary Fermi
Fermi Gas
Gas
a
Doubly integrated
Density
Pressure of the
locally
homogeneous gas
Position / Chemical potential z
S. Nascimbène et al., Nature, 463, 1057, (2010)
Next
Next lecture:
lecture:
measurement
measurement of
of the
the
Equation
Equation of
of State
State P(  , T )