PROBING HOMOGENEOUS QUANTITIES IN A TRAPPED INHOMOGENEOUS FERMI GAS FERMI SURFACE, TAN’S CONTACT AND THE SPECTRAL

FUNCTION

Yoav Sagi, JILA/CU, Boulder (soon, Technion ,Israel)

Tara Drake, Rabin Paudel, Roman Chapurin and Deborah Jin

The goal: •Establishing a better understanding of quantum phases of interacting fermions

Superfluidity, magnetic ordering, topological states, glassy phases,…

The mean: ultracold Fermi gas• Clean and controllable system: interactions, potential, spin composition,…

• Unique measurement techniques: spectroscopy, in situ imaging, momentum resolution, transport, thermodynamic, …

Fermionic superfluidity

Fermions at two spin states: electrons, neutrons, holes, Zeeman sublevels of a fermionic isotope (40K, 6Li),…

What happen when the temperature is reduced ?

Weakly interacting: BCS superconductivity

• Below Tc: momentum space pairing around the Fermi surface. Real space pair size is very large. Pairs condense and for long range order.

• Above Tc: normal gapless Fermi liquid.

K. Onnes discovery, 1911

T [K]

Res

ista

nce

Strongly interacting: unconventional superconductivity

Quark-Gluon plasma

Neutron stars

Degenerate Fermi gases

High-Tc superconductors

20 orders of magnitude

Universality

Credit: NASA/CXC/xx;NASA/STScI;M.Weiss Credit: D. Parker, IMI, U. Birmingham 

Credit: Brookhaven National Laboratory Credit: D. Jin group, JILA

JILA’s 40K Fermi gas machine

MOTMOT Evaporation in Cloverleaf Evaporation in Cloverleaf magnetic trapmagnetic trap

Evaporation in a Evaporation in a Crossed dipole trap Crossed dipole trap

The interaction The interaction energy energy dominates the dominates the dynamics !dynamics !

Our Fano - Feshbach s-wave resonance:

-1 0 1

SuperfluidTem

pera

ture

1/kFaBCS limit BEC limit

C. A. Regal, M. Greiner, D. S. Jin, PRL. 92, 040403 (2004)

M. Greiner, C. A. Regal, and D. S. Jin, Nature 426, 537 (2003)

NormalFermi liquid

Molecular Bose gas

-1 0 1

NormalFermi liquid

PG?

Superfluid

Molecular Bose gas

T*

Tem

pera

ture

1/kFaBCS limit BEC limit

What is the nature of the normal state in the BCS – BEC crossover regime ?

Theory Eagles, Leggett,Nozieres and Schmitt-Rink, Holland, Levin, Randeria, Strinati, Ohashi, Zwerger, Haussman, Hu, Griffin,…

Outline• The effect of density inhomogeneity and our way to

mitigate it.• Observation of a sharp Fermi surface for a weakly

interacting gas. • Measurements of the Contact of a homogeneous unitary

Fermi gas.• Measurements of the occupied spectral function of a

homogeneous Fermi gas in the BEC-BCS crossover regime.

• Is the normal state a Fermi liquid?

Outline• The effect of density inhomogeneity and our way to

mitigate it.• Observation of a sharp Fermi surface for a weakly

interacting gas. • Measurements of the Contact of a homogeneous unitary

Fermi gas.• Measurements of the occupied spectral function of a

homogeneous Fermi gas in the BEC-BCS crossover regime.

• Is the normal state a Fermi liquid?

• Sharp features are washed out when averaging over an inhomogeneous density.

• Solutions: “Box” traps (Weizmann, UT at Austin, Cambridge,…), in-situ imaging (Harvard, MIT, ENS, Chicago, MPQ,…), spatial selectivity when probing.

The effect of the trapping potential

0 .5 1 .0 1 .5 2 .0k k F 0 .2

0 .4

0 .6

nk Trapped

Homogeneous

Probing local information• We optically pump the atoms in the outer parts of the

cloud to a dark state.

T. E. Drake, Y. Sagi, R. Paudel, J. T. Stewart, J. P. Gaebler, and D. S. Jin, PRA 86, 031601(R) (2012)

hollow beam:

donut beamtransition

mf = -9/2 -7/2 -5/2 …

4S1/2

4P3/2

imagingtransition

f = 7/2

f = 9/2

-pulse

|9/2,-5/2>|11/2,-11/2>

40K

Probing a homogeneous non-interacting gas

The emergence of a sharp Fermi surface !The emergence of a sharp Fermi surface !

T. E. Drake, Y. Sagi, R. Paudel, J. T. Stewart, J. P. Gaebler, and D. S. Jin, PRA 86, 031601(R) (2012)

Outline• The effect of density inhomogeneity and our way to

mitigate it.• Observation of a sharp Fermi surface for a weakly

interacting gas. • Measurements of the Contact of a homogeneous unitary

Fermi gas.• Measurements of the occupied spectral function of a

homogeneous Fermi gas in the BEC-BCS crossover regime.

• Is the normal state a Fermi liquid?

What is the contact?

S. Tan, Annals of Physics 323, 2952 (2008); Ibid., p. 2971; Ibid., p. 2987E. Braaten and L. Platter, Phys. Rev. Lett. 100, 205301 (2008); S. Zhang and A. J. Leggett, Phys. Rev. A 79, 023601 (2009).

Universal relations with the contact

• Momentum Distribution

• Energy

• Local Pair Size

• Adiabatic Sweep

• Virial Theorem

• RF Lineshape

4)(

k

Ckn 1

01, rkka F

4)(

CssrN pair

ma

Ckd

k

Ckn

m

kUT

4)(

2

23

4

22

ma

CVUT

8

2

m

C

ad

dE

S4/1

2

m

C 2/324

)(

S. Tan, Annals of Physics 323, 2952 (2008); Ibid., p. 2971; Ibid., p. 2987E. Braaten and L. Platter, PRL 100, 205301 (2008); S. Zhang and A. J. Leggett, PRA 79, 023601 (2009).J. T. Stewart, J. P. Gaebler, T. E. Drake, D. S. Jin, PRL 104, 235301 (2010); E. D. Kuhnle et al. PRL 105, 070402 (2010).G. B. Partridge et al., PRL 95, 020404 (2005); F. Werner et al., EPJ B 68, 401 (2009).

Temperature dependence of the contact

The homogeneous contact is an excellent benchmark for many-body theories !

E. D. Kuhnle et al. PRL 106, 170402 (2011) Hui Hu et al., NJP 13, 035007 (2011)

Trap average Homogeneous

Measuring the homogeneous contact

Photoemission spectroscopy (PES)

mf = -9/2 -7/2 -5/2

Contact vs T

0 1 20

1

2

3

4 Data

C

/(N

k F)

T/TF

Tc

Y. Sagi, T. E. Drake, R. Paudel, and D. S. Jin, PRL 109, 220402 (2012)

Contact vs T

0 1 20

1

2

3

4 Data G

0G

0, GPF, GG

Virial 2, Virial 3 QMC, ENS

C

/(N

k F)

T/TF

Tc

Y. Sagi, T. E. Drake, R. Paudel, and D. S. Jin, PRL 109, 220402 (2012)

Contact vs T

0.0 0.2 0.4 0.62

3

4

Data G

0G

0, GPF, GG

Virial 2, Virial 3 QMC, ENS

C

/(N

k F)

T/TF

Tc

Y. Sagi, T. E. Drake, R. Paudel, and D. S. Jin, PRL 109, 220402 (2012)

Outline• The effect of density inhomogeneity and our way to

mitigate it.• Observation of a sharp Fermi surface for a weakly

interacting gas. • Measurements of the Contact of a homogeneous unitary

Fermi gas.• Measurements of the occupied spectral function of a

homogeneous Fermi gas in the BEC-BCS crossover regime.

• Is the normal state a Fermi liquid?

Fermi liquid theory

Probing the many-body wavefunction

mf = -9/2 -7/2 -5/2

Angle-Resolved PES (ARPES)Photoemission spectroscopy (PES)

Imaging

J. T. Stewart, J. P. Gaebler, and D. S. Jin, Nature 454, 744 (2008)

The spectral functionFermi function

Photoemission Spectroscopy – limiting cases

Weak Interactions

Strong Interactions

Molecular Limit

J. T. Stewart, J. P. Gaebler, and D. S. Jin, Nature 454, 744 (2008)

Molecular branch

k/kF

Superfluid

Evidence of pseudogap with trapped 40K

J. P. Gaebler, J. T. Stewart, T. E. Drake, D. S. Jin, A. Perali, P. Pieri, and G. C. Strinati, Nat. Phys. 6, 569 (2010).

Hotter

• The true width of the dispersion might be obscured by the density inhomogeneity. Can it still be a Fermi liquid?

• The existence of a pseudogap phase in a strongly interacting Fermi gas remains controversial

Homogeneous ARPESmf = -9/2 -7/2 -5/2

Imaging

Homogeneous ARPES on the BEC side

Purple – center of mass of the EDC, White – fit to a Gaussian

There is a clear back-bending around kF

ARPES results around Tc

ARPES results around Tc

EDCs:

ARPES results around Tc

Outline• The effect of density inhomogeneity and our way to

mitigate it.• Observation of a sharp Fermi surface for a weakly

interacting gas. • Measurements of the Contact of a homogeneous unitary

Fermi gas.• Measurements of the occupied spectral function of a

homogeneous Fermi gas in the BEC-BCS crossover regime.

• Is the normal state a Fermi liquid?

Is the normal state a Fermi liquid?

Fermi liquid Non-Fermi liquid

Fermi liquid effective mass (BCS side)

• We fit the dispersion peak to a quadratic function, and extract the effective mass:

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

1.12

1.14

1.16

1.18

1.20

1.22

1.24

1/kFa=-0.3

m*/

m0

T/Tc

0 0.5 1 1.5

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

k/kF

E/E

F

T/Tc=1.11/kFa=-0.3

m/m=1.1650.006

Fitting range

Summary

The degenerate Fermi gas team…

Tara Drake, Rabin Paudel , Yoav Sagiand Roman Chapurin

Deborah Jin

The contact and pair correlations

s

N1 – number of spin up particlesN2 – number of spin down particles

How many pairs are there?

E. Braaten, in The BCS-BEC Crossover and the Unitary Fermi Gas, Lecture Notes in Physics, Vol. 836 (Springer, 2012). ArXiv 1008.2922.

The number of pairs in a small volume is much larger than one would expect by extrapolating from larger volumes !

Lines: theory for homogeneous gasSymbols: averaging over the remaining density inhomogeneity

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4

5

6

7

T/TC

C/N

k F1/k

Fa=0.3, Data from ARPES datasets (lineshape)

Data

Strinati, t-matrixStrinati, popov

Theory: PRA 82, 021605(R) (2010)

Signature of pairing

0

1

2

0 1 0 1 0 1

E/E

F

k/kF k/kFk/kF

Non-interacting gas Normal Fermi liquid BCS superfluid

kF

k h2k 2

2m*mm

Does a Fermi gas has PG phase ?

Experiments:• Thermodynamics : not a sensitive probe - ?• Transport: Duke experiment measures low viscosity -> no well

defined quasi-particles. - YES• RF spectroscopy (JILA): evidence of pairing in the normal state. -

YES

P. Magierski, G. Wlazłowski, A. Bulgac, PRL 107, 145304 (2011).

Theories: most predict a pseudogap at unitarity.

G0G0, GG0, Virial, QMC – YES GG - NO

Width dependence on momentum

Near the phase transition, at different interaction strength

On the BEC side, at different temperatures

In these figures we plot the full width at half the maximum:

Comparison with Fermi liquid theory – averaging over the remaining inhomogeneity

BC

S

Unitarity

BE

C

v v

Looking around the Fermi surface

v

v

Homogeneous condensate fraction at unitarity

High-Tc superconductors versus strongly interacting Fermi gases

Credit: Laboratoire National des Champs Magnétiques Intenses, Toulouse, France

Credit: HIGH ENERGY ACCELERATOR RESEARCH ORGANIZATION, KEK 

Controlling the interaction• Magnetic scattering resonance (Fano-Feshbach)

New molecular bound state leads to a divergence of the scattering properties!

1 9 8 2 0 0 2 0 2 2 0 4 2 0 6 2 0 8

M a g n e t ic

F ie ld G 2 0 0 0

1 0 0 0

1 0 0 0

2 0 0 0S c a t te r in g le n g th a 0

Strong interactions• When is the gas strongly interacting?

• Generally, there is no small parameter and the system cannot be described by mean field theories.

The interaction energy dominates the dynamics !

Fermionic condensation

M. Greiner, C. A. Regal, and D. S. Jin, Nature 426, 537 (2003)C. A. Regal, M. Greiner, D. S. Jin, PRL. 92, 040403 (2004)

Probing a homogeneous gas

• We fit to a homogeneous Fermi-Dirac distribution:

0 5 1 0 1 5 2 0 2 5 3 0 3 5T im e

0 .2

0 .4

0 .6

0 .8

1 .0G ro u n d S ta te P o p u la tio n

e 2 1

t

O B E solut ion 4

0.1

The probability to scatter a photon• We model the optical pumping with a two-level open

system:

g

e

)1( - Rabi frequency

1 - Excited state lifetime - Branching ratio

• We solve using the optical Bloch equations:

Hollow beam propagation• Assumption: each scattering event results in the removal

of one photon and one atom:

2),,(1 1),,(

),,( CzyxIezyxCdz

zyxdI

The probability to scatter a photonNumber of atoms

The change in the number of photons:

AtomsHollow beam

Angle Resolved Photo-Emission Spectroscopy (ARPES)

Raw Signal

Conservation of energy and momentum

Measures the occupied part of the single-particle spectral function in the energy-momentum space.

Crossover theories I

…

T~Tc

Crossover theories II

Crossover theories III

NSR BCS-Leggett

NSR BCS-Leggett

Crossover theories IV

T/TF=0.01

0.06 0.14

0.16 (Tc) 0.18 0.3

Luttinger-Ward formalism

Other experiments - thermodynamics

S. Nascimbene et al. (ENS), Nature 463, 1057 (2010)

M. J. H. Ku et al. (MIT), arXiv: 1110.3309 (2011)

Also, spin transport measurements are not conclusive (Sommer et al. Nature 472, 201, 2011).

A tale of two tails…

0 0.5 1.0 1.5 2.0 2.50

2

4

6

8

k4 n(

k)

k [kF]

T/TF » 0.11

0 2 4 6 8 10 120

2

4

6

23

/223

/2

-3 -2 -1 0 1

0

2

4

6

8 momentum tail RF lineshape tail T=0 Quantum Monte Carlo

C [

Nk F

]

1/kFa

J. T. Stewart, J. P. Gaebler, T. E. Drake, D. S. Jin, PRL 104, 235301 (2010)

F. Werner, L. Tarruell, Y. Castin, Euro. Phys. J. B 68, 401 (2009)

Universal energy relations

-2 -1 0

0

2

4 from E derivative from n(k) & I()

C [N

kF]

1/kFa

2/1

C

akd

dE

F

-2 -1 0-0.06

-0.03

0

0.03

0.06

1/kFa

Ene

rgy

[EF]

T+I-V Contact from n(k),I()

ak

CVIT

F4

J. T. Stewart, J. P. Gaebler, T. E. Drake, D. S. Jin, PRL 104, 235301 (2010)

Contact vs. fraction

Symbols=full local density calculation

0.0 0.5 1.01.6

2.0

2.4

2.8

fraction probed

C/(

Npk

F, a

vg )

Pairing pseudogap in high-Tc SC• Suppression of low-energy spectral weight due to

incoherent pairing in the normal state.

Tunneling Spectra (DOS) of underdoped Bi2Sr2CaCu2O8 .Renner et al., PRL 80, 149 (1998).

Tc

DOS

Energy

Mom

ent

um

ARPES spectra of Bi2Sr2CaCu2O8 at 140K>Tc=90K.Kanigel et al., PRL 101, 137002 (2008).