Download - C. Salomon
C. Salomon
Saclay, June 2, 2010
Thermodynamics of a Tunable Fermi Gas Thermodynamics of a Tunable Fermi Gas
The ENS Fermi Gas TeamThe ENS Fermi Gas TeamS. 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. MoraC. 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
The Equation of State of a Fermi Gas with Tunable Interactions
Cold atoms, Spin ½Dilute gas : 1014 at/cm3, T=100nK
BEC-BCS crossoverSpin imbalance, exotic phases
Neutron star, Spin ½G. Baym, J. Carlson, G. Bertsch,…
0a
EB=-h2/ma2
0,0 0,5 1,0 1,5 2,0
-200
-100
0
100
200
scattering le
ngth
[nm
]
Magnetic field [kG]
0a
a
Lithium 6 Feshbach resonanceLithium 6 Feshbach resonance
BCS phase condensate of molecules
BEC-BCSCrossover
0,0 0,5 1,0 1,5 2,0
-200
-100
0
100
200sc
attering le
ngth
[nm
]
Magnetic field [kG]
No bound stateBound state
2
2
maEB
Tuning interactions in Fermi gases Tuning interactions in Fermi gases Lithium 6Lithium 6
Tuning interactions in Fermi gases Tuning interactions in Fermi gases Lithium 6Lithium 6
a>0
a<0
Experimental sequenceExperimental 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 Spin balanced Unitary Fermi Gas
Pixels
Optical Density
(a.u)
a
Direct proof of superfluidity Direct proof of superfluidity
MIT 2006Critical SF temperature = 0.19 TF
Thermodynamics of a Fermi gasThermodynamics of a Fermi gas
Variables : scattering length atemperature Tchemical potential µ
We build the dimensionless parameters :
We have measured the EoS of the homogeneous Fermi gas
±= ¹p2m¹ a
Interaction parameter
Fugacity (inverse)
Canonical analogs
Pressure contains all the thermodynamic information
Local density approximation:
gas locally homogeneous at
Measuring the EoS of the Homogeneous GasMeasuring the EoS of the Homogeneous Gas
i=1, spin up
i=2, spin down
Extraction of the pressure from in situ images
•
doubly-integrated density profiles
equation of state measured for
all values of
Ho, T.L. & Zhou, Q.,Nature Physics, 09
Measuring the EoS of the Homogeneous Gas Measuring the EoS of the Homogeneous Gas
The Equation of State at unitarity The Equation of State at unitarity
0/1 akFThermodynamics is universal
J. Ho, E. Mueller, ‘04
S. Nascimbene et al., Nature, 463, 1057, (Feb. 2010)
Phase diagram: exploring two fundamental sectors
Phase diagram: exploring two fundamental sectors
: balanced gas at finite T, 1=2=
: imbalanced gas at T=0
Inverse of the fugacity
Equation of state of balanced gas Equation of state of balanced gas
Accuracy: 6%
High temperature
Low temperature
Superfluid region
High T : virial expansionHigh T : virial expansion
X. Liu et al., PRL 102, 160401 (2009)
G. Rupak, PRL 98, 90403 (2007)
No theoretical prediction 4-body problem
)15(096.04 b
SF
Comparison with Many-Body Theories (1)Comparison with Many-Body Theories (1)
E. Burovski et. al., PRL 96, 160402 (2006)
Diagram. MCA. Bulgac et al., PRL 99, 120401 (2006)
QMCR. Haussmann. et al., PRA 75, 023610 (2007)
Diagram.+analytic
Comparison with Many-Body Theories (2)Comparison with Many-Body Theories (2)
E. Burovski et. al., PRL 96, 160402 (2006)
Diagram. MCA. Bulgac et al., PRL 99, 120401 (2006)
QMCR. Haussmann. et al., PRA 75, 023610 (2007)
Diagram.+analytic
New Q. MCby Amherstin good agreementfor 2.0
R. Combescot, Alzetta, Leyronas, PRA, 09
Low TemperatureLow Temperature
A. Bulgac et al., PRL 99, 120401 (2006)
R. Haussmann. et al., PRA 75, 023610 (2007)
Normal phase : Landau theory of the Fermi liquid ?
Superfluid at T = 0
2
2/12
2/31 8
5)0,(2),(
Tk
m
mPTP B
nn
B. Svistunov, Prokofiev, 2006
we find :
C. Lobo et al., PRL 97, 200403 (2006))3(13.1/ mm
Normal phase
Exp. data
Normal-Superfluid phase transitionNormal-Superfluid phase transition
We find the critical parameters
E. Burovski et. al., PRL 96, 160402 (2006)
R. Haussmann. et al., PRA 75, 023610 (2007)
K.B. Gubbels and H.T.C Stoof, PRL 100, 140407 (2008)
A. Bulgac et al., PRL 99, 120401 (2006)
Fermi liquidof quasiparticles
Superfluidtransition
also
Good agreement with theory, with Riedl et al., and with M. Horikoshi, et al. Science 327, 442 (2010);
/EFc= 0.49 (2)
What happens to superfluidity with imbalanced Fermi Spheres ?
What happens to superfluidity with imbalanced Fermi Spheres ?
Superconductors: apply an external magnetic fieldbut Meissner effect
Cold Atoms: change spin populations
Definition :
P: spin polarization
A question discussed extensively since the BCS theory and more than 30 papers in the last 3 years
MIT ’06,: 3 phases, RICE ’06: 2 phases, ENS ’09: 3 phases
Exploring the spin imbalanced gas at zero temperature
Exploring the spin imbalanced gas at zero temperature
: balanced gas at finite T
: imbalanced gas at T=0
Inverse of the fugacity
Equation of state h(, 0) i.e.(T=0)Equation of state h(, 0) i.e.(T=0)
Deviation from hs at
T=0 SF-Normal Phase Transition
MIT: Y. Shin, PRA 08
Fixed-Node
SF
Mixed normal phase
Ideal gas
12 /
An interesting limit: the Fermi PolaronAn 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
- 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
Schirotzek et. al, PRL 101 (2009)
Ideal gas of polaronsIdeal gas of polarons
EOS for a Fermi liquid of polarons
Using :
ENS MIT
Pilati & Giorgini
Lobo et. al.Combescot & Giraud
Fixed Node (Lobo et. al.)Mixed normal phase:Ideal gas + ideal gas of polarons
The Equation of Statein the BEC-BCS crossover
The Equation of Statein the BEC-BCS crossover
0/1 akF
The ground state: T=0
N. Navon, S. Nascimbene, F. Chevy and C. Salomon, Science 328, 729 (2010)
• Single-component Fermi gas:
• Two-component Fermi gas
δ1: grand-canonical analog of
η: chemical potential imbalance
Ground state of a tunable Fermi gas Ground state of a tunable Fermi gas
Equations of State in the Crossover Equations of State in the Crossover
12 /
First –order phase transitions: slope of h is discontinuous
PairedSF
polarizedNormal phase
Ideal F Gas
PairedSF
Phase diagram Phase diagram
BCS BEC
SFSF
We use the most advanced calculations forProkof’ev et al., R. Combescot et al,
BEC side, unitarity: excellent agreement
BCS side: deviation close to ηc
Comparison with the two ideal gases model
Comparison with the two ideal gases model
Full pairing:
Symmetric parametrization:
Superfluid Equation of State Superfluid Equation of State
Superfluid Equation of State in the Crossover
Superfluid Equation of State in the Crossover
BCS BEC
1/a= 0
BCS limit:
BEC limit
Unitary limit
Lee-Yang correction
mean-field
molecularbinding energy
mean-field Lee-Huang-Yang correction
We get: s= 0.41(1)contact coefficient = 0.93(5)
Asymptotic behaviorsAsymptotic behaviors
Fit of the LHY coefficient: 4.4(5)
theory:
No effect of the composite nature of the dimers
X. Leyronas et al, PRL 99, 170402 (2007)
mean-field
LHY
Measurement of the Lee-Huang-Yang correctionMeasurement of the Lee-Huang-Yang correction
BCS BEC
For elementary bosons:
• B is not universal for elementary bosons (Efimov physics)
Λ*: three-body parameterE. Braaten et al, PRL 88, 040401 (2002)
• Here: universal value (using an appropriate Padé fit
function)
Beyond the Lee-Huang-Yang correctionBeyond the Lee-Huang-Yang correction
Direct Comparison to Many-Body TheoriesGrand-Canonical – Canonical Ensemble
Direct Comparison to Many-Body TheoriesGrand-Canonical – Canonical Ensemble
Chang et al, PRA 70, 43602 (2004)Astrakharchik et al, PRL 93, 200404 (2004)
Pilati et al, PRL 100, 030401 (2008)
Fixed-Node Monte-Carlo theories
Nozières-Schmitt-Rink approximation Hu et al, EPL 74, 574 (2006)
Diagrammatic theoryHaussmann et al, PRA 75, 23610 (2007)
Quantum Monte CarloBulgac et al, PRA 78, 23625 (2008)
Exp.
Conclusion - PerspectivesConclusion - 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
- Next: Mapping the EOS in the complete space
imbalanced gas at finite T , mass imbalance
- Lattice experiments
- EoS in the BEC-BCS crossover at T=0
- First quantitative measurement of Lee- Huang-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
Thermodynamics of a unitary Fermi gasThermodynamics of a unitary Fermi gas
- Equation of state of a homogeneous two-component Fermi gas
(appropriate variables experimentally)
2) Trapping potential inhomogeneous gas
Measured values averaged over the trap
1) Thermometry of a strongly correlated system : difficult !
Two problemsTwo problems
immersing weakly interacting bosonic 7Li (« ideal thermometer »)
ConclusionConclusion
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
Grand-canonical Canonical ensemble
Viriel 2
?
Disagrees with Viriel 2 expansion
M. Horikoshi, et al. Science 327, 442 (2010);
Tokyo
ENS
Typical imagesTypical images
One experimental run : density profile + temperature
- Image of 6Li in-situ Image of 7Li in TOF
ConclusionConclusion
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 : ResultsTrapped gas : Results
We find :
Boulder
Duke
Tokyo
Innsbruck
Perspectives – Open QuestionsPerspectives – Open Questions
- Critical temperature in the BEC-BCS crossover
- Nature of the Normal phase in the crossover
- Low-lying excitations of the superfluid
Fermi liquid
Pseudo-gap
Superfluid
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,…
6Li
Simulation of Neutron Matter Simulation of Neutron Matter
k/kF1
n(k)
1
k/kF1
n(k)
1
Thermodynamics Thermodynamics
Is a useful but incomplete equation of state !
Complete information is given by thermodynamic potentials:
TNkPV B
Equ. of state useful for engines, chemistry, phase transitions,….
NTSEPV Grand potential
Entropy
Pressure TemperatureAtom number
Chemical potential
Volume
Internal energy
We have measured the grand potential of a tunable Fermi gas
S. Nascimbene et al., Nature, 463, 1057, (Feb. 2010), arxiv 0911.0747N. Navon et al., Science 328, 729 (2010)
MIT and Rice experimentsMIT and Rice experiments
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
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).
Differenceof optical density of the two spinpopulations
Direct comparison to many-body theoriesGrand-Canonical – Canonical Ensemble
Direct comparison to many-body theoriesGrand-Canonical – Canonical Ensemble
Chang et al, PRA 70, 43602 (2004)Astrakharchik et al, PRL 93, 200404 (2004)
Pilati et al, PRL 100, 030401 (2008)
Fixed-Node Monte-Carlo theories
Nozières-Schmitt-Rink approximation Hu et al, EPL 74, 574 (2006)
Diagrammatic theoryHaussmann et al, PRA 75, 23610 (2007)
Quantum Monte CarloBulgac et al, PRA 78, 23625 (2008)
We fit our data in the region
Simple analytical theoryR. Combescot et al, PRL 98, 180402 (2007)
Fixed Node Monte CarloS. Pilati et al, PRL 100, 030401 (2008)
Diagrammatic Monte CarloN. Prokof’ev et al, PRB 77, 020408 (2008)
Most advanced analytical theoryR. Combescot et al, EPL 88, 60007 (2009)
Polaron effective mass
Collective modesS.Nascimbene. et al, PRL 103, 170304 (2009)
MIT measurementY. Shin, PRA 77, 041603 (2008)
Determination of 0Determination of 0
Density profile segment
7Li
Unknown !
No model-independent determination of0 !Progressive construction of P by connecting adjacent segments
From high temperature side assuming viriel 2 coefficient
of
Free parameter for each image 0
Y axis value fixed, X axis rescaled by a free factor
Construction of the EOSConstruction of the EOS
superfluid core normal phase
Superfluid core for
We calculate using fits of our EoS
Application: Critical polarization for superfluidity Application: Critical polarization for superfluidity
EOS
MIT
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 Using 7Li as a thermometer
6Li 7Li
- 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 limitClogston-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 ProfilesDensity Profiles
- Data consistent with 3 phases + LDA as MIT
- In agreement with theory (3 phases)
-- Solid line: model with approximate eq of state ( Recati et al., 2008)
- In-situ imaging at high field of 2 spin states
Vertical width = 40 µm
SF core
Partially polarized normal phase
Fully polarizedshell
Previous works: superfluid equation of state
Radial breathing mode
• experimentsAltmeyer et al, PRL 98, 040401 (2006)
• theoryMean field
QMC calculations +
hydrodynamics +
scaling ansatzAstrakharchik 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 :
Lee-Yang coefficient: 0.18(2)
theory:
mean-field
mean-field + LY
Using and , one calculates
• at : jump of n2/n1
• low imbalance phase: n2=n1
• large imbalance phase: 0<n2<n1
Calculation of the density ratio Calculation of the density ratio
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
-- 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-bodysystems. -- 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
ProspectsProspects
A good reading ! A good reading !
Enrico Fermi on lake Como
Experimental approachExperimental approach Experimental approachExperimental approach
Cooling of 7Li and 6Li
1000 K: oven
1 mK: laser cooling
10 K: evaporative cooling in magnetic trap
E= - .B
Ioffe-Pritchardtrap
New experimental methods:
• Image a many-body wavefunction with 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
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
Quantum simulation with cold atomsQuantum simulation with cold atoms
Period: 4.9 m
D. Weiss et al., Nature Phys, 2007
Dispersion relation of polarons Dispersion relation of polarons
Low energy spectrum of the polaron:2
2 1( , ) ( ) ( ) ....2F
pE r p AE r V r
m
1 1( ) (0) ( )F FE r E V r
( ) (1 ) ( )V r A V r
Within LDA: local Fermi energy of majority species:
The quasi-particle evolves in an effective potential:
Oscillation frequency of the polaron: (1 )A m
m
Measuring the oscillation frequency of polarons gives access to the effective mass m*
A: binding energy of a polaron= - 0.60 F. Chevy `06, Lobo et al., Prokofiev, Svistunov, MC
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
2
2/32(1 ) 62 Fn Em
Experiment ENS (6Li) 0.42(15)
Rice (6Li) 0.46(5)
JILA(40K) 0.46(10)
Innsbruck (6Li) 0.27(10)
Duke (6Li) 0.51(4)
Theory BCS 0.59
Astrakharchik 0.42(1)
Perali 0.455
Carlson 0.42(1)
Haussmann 0.36
Determination of
Universal equation of state of the unitary Fermi gas at zero temperature
a For
is a universal number
Double-integrated profiles of density differenceDouble-integrated profiles of density difference
MIT
Flat top = hollow core with LDA = paired core
RICE
Non monotonic behaviour :
LDA violating feature
Effect of the number of atoms, aspect ratio and temperature ?
Local chemical potentials
In-situ density profilesIn-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 !
Superfluid
Normal partially polarized
Fully polarized
Majority
Difference
Minority
Column density
Theory, F. Chevy ‘06, Lobo et al.,
Typical imagesTypical images
One experimental run : we get density profile + temperature
Density profile of 6Li Temperature
6Li 7Li
Mixture remarkably stable !
Determination of 0Determination of 0
Density profile segment
7Li
Unknown !
No model-independent determination of0 !Progressive construction of P by connecting adjacent segment
of
Free parameter for each image 0
Y axis value fixed, X axis rescaled by a free factor
Construction of the EOSConstruction of the EOS
Using our EOS : Trapped gasUsing our EOS : Trapped gas
Previous works : EOS of the trapped gas (Duke, Boulder)
Can be computed using our EOS + LDA !
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
Trapped gas : ResultsTrapped gas : Results
We find :
Boulder
Duke
Tokyo
Innsbruck
Imbalanced gas preparationImbalanced gas preparation
- RF evaporation of 7Li cooling of 6Li
- Loading of 6Li alone in the optical trap
10 µs
10 µs
|1> |2>...
10 µs
10 µs
|1> |2>
Absorption pictures Reference pictures
- Fast imaging of the two spin states
- Landau-Zener sweep for imbalanced spin mixture
Previous episode : Typical density profilesPrevious episode : Typical density profiles
SF core
Partially polarized normal phase
Fully polarizedshell
S. Nascimbène, N.N, K. Jiang, L. Tarruell, M. Teichmann, J. McKeever, F. Chevy, C. Salomon PRL 103, 170402 (2009)
Vertical width = 40 µm