Thermodynamics of a Tunable Fermi Gas
C. Salomon
Saclay, June 2, 2010
The ENS Fermi Gas Team
S. Nascimbène, N. Navon,
K. Jiang, F. Chevy, C. S.
L. Tarruell, M. Teichmann,
J. McKeever, K. Magalhães,
A. Ridinger, T. Salez, S. Chaudhuri,
U. Eismann, D. Wilkowski, F. Chevy,
Y. Castin, M. Antezza, C. Salomon
Theory collaborators: D. Petrov, G. Shlyapnikov , R. 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
The Equation of State of a Fermi
Gas with Tunable Interactions
Cold atoms, Spin ½
Dilute gas : 1014 at/cm3, T=100nK
BEC-BCS crossover
Spin imbalance, exotic phases
Neutron star, Spin ½
G. Baym, J. Carlson, G. Bertsch,…
a0
EB=-h2/ma2
a
scattering length [nm]
200
100
0
-100
a0
-200
0,0
0,5
1,0
1,5
Magnetic field [kG]
2,0
Lithium 6 Feshbach resonance
Tuning interactions in Fermi gases
Lithium 6
scattering length [nm]
200
a>0
100
BEC-BCS
Crossover
0
a<0
-100
-200
0,0
0,5
1,0
1,5
Magnetic field [kG]
Bound state
2
EB  
ma2
condensate of molecules
2,0
No bound state
BCS phase
Experimental sequence
- Loading of 6Li in the optical trap
- Tune magnetic field to Feshbach
resonance
- Evaporation of 6Li
- Image of 6Li in-situ
Spin balanced Unitary Fermi Gas
a
Optical
Density
(a.u)
Pixels
Direct proof of superfluidity
Critical SF temperature = 0.19 TF
MIT 2006
Thermodynamics of a Fermi gas
We have measured the EoS of the homogeneous Fermi gas
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 the EoS of the Homogeneous Gas
Local density approximation:
gas locally homogeneous at
i=1, spin up
i=2, spin down
Measuring the EoS of the Homogeneous Gas
Extraction of the pressure from in situ images
Ho, T.L. & Zhou, Q.,
Nature Physics, 09
•
doubly-integrated density profiles
equation of state measured for
all values of
The Equation of State at unitarity
1/ kF a  0
Thermodynamics is universal
J. Ho, E. Mueller, ‘04
S. Nascimbene et al., Nature, 463, 1057, (Feb. 2010)
Phase diagram:
exploring two fundamental sectors
Inverse of the fugacity
: balanced gas at finite T, m1=m2=m
: imbalanced gas at T=0
Equation of state of balanced gas
Low temperature
Superfluid region
Accuracy: 6%
High temperature
High T : virial expansion
X. Liu et al., PRL 102, 160401 (2009)
SF
G. Rupak, PRL 98, 90403 (2007)
b4  0.096(15)
No theoretical prediction
 4-body problem
Comparison with Many-Body Theories (1)
Diagram. MC
E. Burovski et. al., PRL 96, 160402 (2006)
QMC
A. Bulgac et al., PRL 99, 120401 (2006)
Diagram.+analytic
R. Haussmann. et al., PRA 75, 023610 (2007)
Comparison with Many-Body Theories (2)
Diagram. MC
E. Burovski et. al., PRL 96, 160402 (2006)
QMC
A. Bulgac et al., PRL 99, 120401 (2006)
Diagram.+analytic
R. Haussmann. et al., PRA 75, 023610 (2007)
New Q. MC
by Amherst
in good
agreement
for   0.2
R. Combescot, Alzetta,
Leyronas, PRA, 09
Low Temperature
Exp. data
B. Svistunov, Prokofiev, 2006
Normal phase
A. Bulgac et al., PRL 99, 120401 (2006)
R. Haussmann. et al., PRA 75,
023610 (2007)
Superfluid at T = 0
Normal phase : Landau theory of the Fermi liquid ?
 3 / 2 5 2 1/ 2 m  k T  2 
 B  
P( m , T )  2 P1 ( m ,0)  n 
n


8
m
m




we find :
m / m  1.13(3)
C. Lobo et al., PRL 97, 200403 (2006)
Normal-Superfluid phase transition
We find the critical parameters
Fermi liquid
of quasiparticles
Superfluid
transition
E. Burovski et. al., PRL 96, 160402 (2006)
K.B. Gubbels and H.T.C Stoof, PRL 100, 140407 (2008)
A. Bulgac et al., PRL 99, 120401 (2006)
R. Haussmann. et al., PRA 75, 023610 (2007)
(m/EF)c= 0.49 (2)
also
Good agreement with theory, with Riedl et al.,
and with M. Horikoshi, et al. Science 327, 442 (2010);
What happens to superfluidity with
imbalanced Fermi Spheres ?
Superconductors: apply an external magnetic field
but Meissner effect
Cold Atoms: change spin populations
A question discussed extensively since the BCS theory
and more than 30 papers in the last 3 years
Definition :
dmm1m2
P: spin polarization
MIT ’06,: 3 phases, RICE ’06: 2 phases, ENS ’09: 3 phases
Exploring the spin imbalanced gas
at zero temperature
: balanced gas at finite T
Inverse of the fugacity
: imbalanced gas at T=0
Equation of state h(h, 0) i.e.(T=0)
SF
Mixed normal phase
Deviation from hs at
Ideal gas
T=0 SF-Normal Phase Transition
Fixed-Node
h  m2 / m1
MIT: Y. Shin, PRA 08
An interesting limit: the Fermi Polaron
Partially polarized normal phase
- Easier to understand in the limiting case of a
single minority atom immersed in a majority
Fermi sea : the Fermi polaron
- Proposed by Trento, Amherst, Paris
Schirotzek et. al, PRL 101 (2009)
- Observed at MIT by RF spectroscopy
- Can we describe the normal phase as a Fermi liquid of
polarons ?
binding energy of a polaron in the Fermi sea
effective mass
Ideal gas of polarons
Mixed normal phase:
Ideal gas + ideal gas of polarons
Fixed Node (Lobo et. al.)
EOS for a Fermi liquid of polarons
ENS
MIT
Using :
Pilati & Giorgini
Lobo et. al.
Combescot & Giraud
The Equation of State
in the BEC-BCS crossover
1/ kF a  0
The ground state: T=0
N. Navon, S. Nascimbene, F. Chevy and C. Salomon,
Science 328, 729 (2010)
Ground state of a tunable Fermi gas
• Single-component Fermi gas:
• Two-component Fermi gas
δ1: grand-canonical analog of
η: chemical potential imbalance
Equations of State in the Crossover
Paired
SF
Paired
SF
Ideal F Gas
polarized
Normal phase
 m2 / m1
First –order phase transitions: slope of h is discontinuous
Phase diagram
BCS
BEC
Comparison with the two
ideal gases model
We use the most advanced calculations for
Prokof’ev et al., R. Combescot et al,
SF
BEC side, unitarity:
excellent agreement
SF
BCS side: deviation close to
ηc
Superfluid Equation of State
Full pairing:
Symmetric parametrization:
Superfluid Equation of State in the Crossover
BEC
BCS
1/a= 0
Asymptotic behaviors
BCS limit:
mean-field
BEC limit
molecular
binding mean-field
energy
Unitary limit
Lee-Yang
correction
Lee-Huang-Yang
correction
We get: s= 0.41(1)
contact coefficient
z= 0.93(5)
Measurement of the Lee-Huang-Yang correction
mean-field
Fit of the LHY coefficient: 4.4(5)
LHY
theory:
No effect of the composite nature of
the dimers
BCS
BEC
X. Leyronas et al, PRL 99, 170402 (2007)
Beyond the Lee-Huang-Yang correction
For elementary bosons:
• B is not universal for elementary
bosons (Efimov physics)
Λ*: three-body parameter
E. Braaten et al, PRL 88, 040401 (2002)
• Here: universal value
(using an appropriate Padé fit function)
Direct Comparison to Many-Body Theories
Grand-Canonical – Canonical Ensemble
Exp.
Nozières-Schmitt-Rink approximation
Hu et al, EPL 74, 574 (2006)
Diagrammatic theory
Haussmann et al, PRA 75, 23610 (2007)
Quantum Monte Carlo
Bulgac et al, PRA 78, 23625 (2008)
Fixed-Node Monte-Carlo theories
Chang et al, PRA 70, 43602 (2004)
Astrakharchik et al, PRL 93, 200404 (2004)
Pilati et al, PRL 100, 030401 (2008)
Conclusion - Perspectives
- EOS of a uniform Fermi gas at unitarity in two sectors
1) balanced gas at finite T
2) T = 0 imbalanced gas
- Precision Test of Many-body Theories
- EoS in the BEC-BCS crossover at T=0
- First quantitative measurement of LeeHuang-Yang quantum corrections and
Lee-Yang on BCS side
- Simple description of the normal phase
as two ideal gases on BEC and unitary;
breakdown on BCS side
- Next: Mapping the EOS in the complete
 imbalanced gas at finite T , mass imbalance
- Lattice experiments
space
Thermodynamics of a unitary Fermi gas
- Equation of state of a homogeneous two-component Fermi gas
(appropriate variables experimentally)
Two problems
1) Thermometry of a strongly correlated system : difficult !
 immersing weakly interacting bosonic 7Li (« ideal thermometer »)
2) Trapping potential  inhomogeneous gas
Measured values averaged over the trap
Conclusion
Measurement of the EoS :
- Unitary gas at finite temperature
- Fermi gas T=0 in the BEC-BCS crossover
Quantitative many-body physics – benchmark for theories
Comparison with Tokyo group
M. Horikoshi, et al. Science 327, 442 (2010);
Tokyo
Disagrees with
Viriel 2 expansion
?
Viriel 2
ENS
Grand-canonical
Canonical ensemble
Typical images
- Image of 6Li in-situ
Image of 7Li in TOF
One experimental run : density profile + temperature
Conclusion
Measurement of the EoS :
- Unitary gas at finite temperature
- Fermi gas T=0 in the BEC-BCS crossover
Quantitative many-body physics – benchmark for theories
Trapped gas : Results
We find :
Boulder
Duke
Tokyo
Innsbruck
Perspectives – Open Questions
Superfluid
Pseudo-gap
Fermi liquid
- Critical temperature in the BEC-BCS crossover
- Low-lying excitations of the superfluid
- Nature of the Normal phase in the crossover
Simulation of Neutron Matter
Neutron characteristics
• spin ½
• scattering length
• effective range
Universal regime:
•
• « dilute » limit
(mean density
• Tc=1010 K =1 MeV, T=TF/100 kFa ~ -4,-10,…
)
Thermodynamics
PV  NkBT
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  mN
Pressure
Volume
Temperature
Entropy
Chemical potential
Atom number
Internal energy
Equ. of state useful for engines, chemistry, phase transitions,….
We have measured the grand potential
of a tunable Fermi gas
k/k
F
1
k/k
F
1 Nature, 463, 1057, (Feb.
S. Nascimbene et al.,
2010), arxiv 0911.0747
N. Navon et al., Science 328, 729 (2010)
MIT and Rice experiments
Difference
of optical
density of
the two spin
populations
MIT: M. Zwierlein, A. Schirotzek, C. Schunck,
and W. Ketterle, Science 311, 492 (2006).
RICE: G. Partridge, W. Li, R. Kamar, Y. Liao,
and R. Hulet, Science 311, 503 (2006).
Obvious phase separation seen on the in situ optical density difference
Clogston limit at MIT is P= 0.75 and close to 1 at Rice
Direct comparison to many-body theories
Grand-Canonical – Canonical Ensemble
Nozières-Schmitt-Rink approximation
Hu et al, EPL 74, 574 (2006)
Diagrammatic theory
Haussmann et al, PRA 75, 23610 (2007)
Quantum Monte Carlo
Bulgac et al, PRA 78, 23625 (2008)
Fixed-Node Monte-Carlo theories
Chang et al, PRA 70, 43602 (2004)
Astrakharchik et al, PRL 93, 200404 (2004)
Pilati et al, PRL 100, 030401 (2008)
Polaron effective mass
We fit our data in the region
Simple analytical theory
R. Combescot et al, PRL 98, 180402 (2007)
Fixed Node Monte Carlo
S. Pilati et al, PRL 100, 030401 (2008)
Diagrammatic Monte Carlo
N. Prokof’ev et al, PRB 77, 020408 (2008)
Most advanced analytical theory
R. Combescot et al, EPL 88, 60007 (2009)
MIT measurement
Collective modes
Y. Shin, PRA 77, 041603 (2008)
S.Nascimbene. et al, PRL 103, 170304 (2009)
Determination of m0
Density profile  segment
of
Unknown !
7Li
No model-independent determination of m0 !
Progressive construction of P by connecting adjacent segments
From high temperature side assuming viriel 2 coefficient
Free parameter for each image m0
 Y axis value fixed, X axis rescaled by a free factor
Construction of the EOS
Application: Critical polarization for superfluidity
superfluid core
normal phase
Superfluid core for
We calculate
using fits of our EoS
MIT
EOS
Partially polarized phase
Simple physical picture:
one builds a Fermi sea of fermionic polarons
• Single-polaron ground state:
• Excited states:
Then
Fermi pressure of
majority atoms
Fermi pressure of
polarons
Using 7Li as a thermometer
7Li
6Li
- Feshbach resonance @ 834 G
a76 = 2 nm  Good ! Weakly interacting
a77 = -3 nm  Not so good… BEC instability
 Low temperature limited by collapse :
- With typical atom numbers
close to expected Tc for SF transition…
Clogston-Chandrasekhar limit
- Naive argument using BCS picture :
the energy of excess particles must be compared with
« robustness » of the fermion pairs :
: SF is stable with equal densities
?
- Relation predicted by BCS theory :
Density Profiles
- In-situ imaging at high field of 2 spin states
SF core
Fully
polarized
shell
Vertical width = 40 µm
- Data consistent with 3 phases + LDA as MIT
Partially polarized
normal phase
- In agreement with theory (3 phases)
-- Solid line: model with approximate eq of state ( Recati et al., 2008)
Previous works:
superfluid equation of state
Radial breathing mode
• experiments
Altmeyer et al, PRL 98, 040401 (2006)
• theory
Mean field
QMC calculations +
hydrodynamics +
scaling ansatz
Astrakharchik et al, PRL 95, 030404
(2005)
Extracting the BCS asymptotic behavior
On the BCS side: Pade approximant
mean-field value used as a constraint
From
mean-field
Lee-Yang coefficient: 0.18(2)
theory:
mean-field + LY
:
Calculation of the density ratio
Using
and
, one calculates
• at
: jump of n2/n1
• low imbalance phase:
n2=n1
• large imbalance phase:
0<n2<n1
Thermodynamics of the unitary gas
From
:
•
agrees with previous measurements
•
Related to short-range pair correlations
Two weeks ago (Hu et al, arXiv:1001.3200): Dynamic structure factor measurement
Prospects
-- One example of a cold atom system at the interface
with condensed matter: surprising richness of the simplest of
strongly correlated systems. Precision measurement of the EOS.
Useful to understand other more complex quantum many-body
systems.
-- In progress:
General equation of state at finite temperature (ongoing !)
Superfluid temperature = f(kFa), polaron oscillation (PRL’ 09)
-- FFLO phase,…
-- Lattice experiments and low D systems
A good reading !
Enrico Fermi on lake Como
Experimental approach
Cooling of 7Li and 6Li
1000 K: oven
1 mK: laser cooling
10 mK: evaporative cooling
in magnetic trap
E= - m.B
Ioffe-Pritchard
trap
Quantum simulation with cold atoms
Other many-body Hamiltonians
• Bosons, fermions, and mixtures, 6Li—40K
• Periodic potential or disordered (Anderson loc.)
• Gauge field with rotation or geometrical phase
• Quantum Hall physics and Laughlin states
• Non abelian Gauge field for simulating the
Hamiltonian of strong interactions in particle physics
D. Weiss et al.,
Nature Phys, 2007
New experimental methods:
• Image a many-body wavefunction with
Period: 4.9 mm
micrometer resolution in optical lattice
• Measure correlation functions
• Photoemission spectroscopy to measure Fermi surface
and single particle excitations, Dao et al., 07
• Cooling in optical lattices: J.S. Bernier et al. 09, J. Ho 09,..
• Time-dependent phenomena in 1, 2, and 3 D
Dispersion relation of polarons
p2
Low energy spectrum of the polaron: E2 (r , p)  AEF 1 (r )  V (r ) 
 ....

2m
A: binding energy of a polaron= - 0.60
F. Chevy `06, Lobo et al., Prokofiev, Svistunov, MC
Within LDA: local Fermi energy of majority species: EF 1 (r )  EF 1 (0)  V (r )
The quasi-particle evolves in an effective potential:
V  (r )  (1  A) V (r )


(1  A) m
Oscillation frequency of the polaron:


m
Measuring the oscillation frequency of polarons
gives access to the effective mass m*
N. Nascimbene, N. Navon, K. Jiang, L. Tarruell, J. Mc Keever,
M. Teichmann, F. Chevy, C. Salomon, to appear in PRL 2009
arXiv:0907.3032
Universal equation of state of the
unitary Fermi gas at zero temperature
For
a   m  (1   )
2
6 n )
(
2m
2
2/3
  EF
Determination of 
Experiment
ENS (6Li)
0.42(15)
Rice (6Li)
Theory
BCS
0.59
0.46(5)
Astrakharchik
0.42(1)
JILA(40K)
0.46(10)
Perali
0.455
Innsbruck (6Li)
0.27(10)
Carlson
0.42(1)
Duke (6Li)
0.51(4)
Haussmann
0.36
 is a universal number
Double-integrated profiles of density difference
MIT
RICE
Flat top = hollow core with
LDA = paired core
Non monotonic behaviour :
LDA violating feature
Effect of the number of atoms, aspect ratio and temperature ?
In-situ density profiles
- easy to interpret at T=0
- Trapping potential is useful (this time !)
- under LDA, the trapping provides us with locally homogeneous
gases with different values of
: in one image, we
have the full T=0 phase diagram of the system !
Local chemical potentials
Column density
Superfluid
Normal partially
polarized
Majority
Fully polarized
Minority
Theory, F. Chevy ‘06, Lobo et al.,
Difference
Typical images
One experimental run :
we get density profile + temperature
Density profile of 6Li
6Li
Temperature
7Li
Mixture remarkably stable !
Determination of m0
Density profile  segment
of
Unknown !
7Li
No model-independent determination of m0 !
Progressive construction of P by connecting adjacent segment
Free parameter for each image m0
 Y axis value fixed, X axis rescaled by a free factor
Construction of the EOS
Using our EOS : Trapped gas
Previous works : EOS of the trapped gas (Duke, Boulder)
L. Luo and J. Thomas, JLTP 154, 1-29 (2009)
Legendre transform
Integrals  Discrete sums
over experimental data for h
 Results independent of fitting or
interpolation functions
Can be computed using our EOS + LDA !
Trapped gas : Results
We find :
Boulder
Duke
Tokyo
Innsbruck
Imbalanced gas preparation
- RF evaporation of 7Li  cooling of 6Li
- Loading of 6Li alone in the optical trap
- Landau-Zener sweep for imbalanced
spin mixture
- Fast imaging of the two spin states
Absorption pictures
Reference pictures
10
µs
|1>
10
µs
10
µs
|2>
...
|1>
10
µs
|2>
Previous episode : Typical density profiles
SF core
Fully
polarized
shell
Vertical width = 40 µm
Partially polarized
normal phase
S. Nascimbène, N.N, K. Jiang, L. Tarruell, M.
Teichmann, J. McKeever, F. Chevy, C. Salomon
PRL 103, 170402 (2009)