C. Salomon Saclay, June 2, 2010 Thermodynamics of a Tunable Fermi Gas.

transcript

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

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,…

EB=-h2/ma2

0,0 0,5 1,0 1,5 2,0

scattering le

Magnetic field [kG]

Lithium 6 Feshbach resonanceLithium 6 Feshbach resonance

BCS phase condensate of molecules

BEC-BCSCrossover

0,0 0,5 1,0 1,5 2,0

attering le

Magnetic field [kG]

No bound stateBound state

Tuning interactions in Fermi gases Tuning interactions in Fermi gases Lithium 6Lithium 6

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

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

: 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

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)

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

2/31 8

5)0,(2),(

mPTP B

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

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

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 ?

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

: 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

Mixed normal phase

Ideal gas

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

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

First –order phase transitions: slope of h is discontinuous

PairedSF

polarizedNormal phase

Ideal F Gas

PairedSF

Phase diagram Phase diagram

BCS BEC

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

Full pairing:

Symmetric parametrization:

Superfluid Equation of State Superfluid Equation of State

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

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

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)

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);

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

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,…

Simulation of Neutron Matter Simulation of Neutron Matter

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

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

Unknown !

No model-independent determination of0 !Progressive construction of P by connecting adjacent segments

From high temperature side assuming viriel 2 coefficient

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

Partially polarized phase

Simple physical picture:

one builds a Fermi sea of fermionic polarons• Single-polaron ground state:• Excited states:

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

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

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

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

2/32(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

is a universal number

Double-integrated profiles of density differenceDouble-integrated profiles of density difference

Flat top = hollow core with LDA = paired core

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

Unknown !

No model-independent determination of0 !Progressive construction of P by connecting adjacent segment

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

Innsbruck

Imbalanced gas preparationImbalanced gas preparation

- RF evaporation of 7Li cooling of 6Li

- Loading of 6Li alone in the optical trap

10 µs

|1> |2>...

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

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

C. Salomon Saclay, June 2, 2010 Thermodynamics of a Tunable Fermi Gas.

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