Towards Probing Homogeneous Strongly Interacting Fermi...

PROBING LOCAL QUANTITIES IN A STRONGLY

INTERACTING FERMI GAS

Yoav Sagi, JILA/CU, Boulder

Tara Drake, Rabin Paudel, Roman Chapurin and Deborah Jin

Their interaction give rise to complex matter.

Fermions: the building blocks of nature

Conventional BCS superconductivity

Hydrogen

atom

H2 molecule

K. Onnes discovery, 1911

T [K]

Resis

tance

Strongly interacting fermions

• Pairing can happen on a length scale comparable to the

interparticle separation (BCS-BEC crossover)

Quark-Gluon plasma

Neutron stars

Degenerate Fermi gases

High-Tc superconductors

20 orders of magnitude

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

MOT Evaporation in Cloverleaf

magnetic trap

Evaporation in a

Crossed dipole trap

The interaction

energy

dominates the

dynamics !

~𝑘𝐹𝑎

Our Fano - Feshbach

s-wave resonance:

-1 0 1

SuperfluidTe

mp

era

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)

Normal

Fermi

liquid

Molecular

Bose gas

𝑇𝐶 ∼ 0.2𝑇𝐹

-1 0 1

Normal

Fermi

liquidPG?

Superfluid

Molecular

Bose gasT*

Te

mp

era

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,…

𝑇𝐶 ∼ 0.2𝑇𝐹

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 kF

0.2

0.4

0.6

n k

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:

Probing local information

donut beam

transition

mf = -9/2 -7/2 -5/2 …

4S1/2

4P3/2

imaging

transition

f = 7/2

f = 9/2

p-pulse

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

40K

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

Probing a homogeneous gas

The emergence of a sharp Fermi surface !

Getting from trapped-averaged to homogeneous

• When does 𝑛 𝑘 → 𝑛(𝑘) for a gas with density 𝑛 𝑟 ?

𝑛𝐹𝐷 𝑘 =1

𝑒(ℏ2𝑘2

2𝑚 −𝜇)/𝑘𝐵𝑇 + 1𝑇/𝑇𝐹 at < 𝑛 𝑟 >

using a model

of 𝑛(𝑟)

Outline


mitigate it.


interacting gas.


Fermi gas.



regime.


What is the contact?

• For a grand canonical ensemble:

𝑑𝐸 = 𝑇𝑑𝑆 − 𝑃𝑑𝑉 + 𝜇𝑑𝑁

• For a Fermi gas there is an additional variable: the

inverse scattering length 𝑎−1.

• Its conjugate (up to a constant) is the contact:

𝑑𝐸 = 𝑇𝑑𝑆 − 𝑃𝑑𝑉 + 𝜇𝑑𝑁 − 𝐶ℏ2

4𝜋𝑚𝑑𝑎−1

The contact is a fundamental thermodynamic

parameter in a system with controllable interactions.

S. Tan, Annals of Physics 323, 2952 (2008); Ibid., p. 2971; Ibid., p. 2987

E. 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

• VirialTheorem

• RF Lineshape

4)(

k

Ckn 1

0

1, rkka F

p4)(

CssrN pair

ma

Ckd

k

Ckn

m

kUT

p4)(

2

23

4

22

ma

CVUT

p8

2

m

C

ad

dE

Sp4/1

2

m

C 2/324

)(p

S. Tan, Annals of Physics 323, 2952 (2008); Ibid., p. 2971; Ibid., p. 2987

E. 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).

The contact and pair correlations

Naively: the number of pairs, N1N2, should scale as the

volume squared 𝑁𝑝𝑎𝑖𝑟𝑠(𝑟 < 𝑠)~𝑠6

Surprisingly: 𝑁𝑝𝑎𝑖𝑟𝑠(𝑟 < 𝑠) =𝑠4

4ℂ

s

N1 – number of spin up particles

N2 – number of spin down particles

How many pairs are there?

Contact density

𝐶 = ∫ 𝑑3𝑟 ℂ(𝑟)

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 !

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

• We extract the contact from the high frequency tail of an

RF line-shape: Γ(𝜈)~ 𝐶 𝜈3/2

Measuring the homogeneous contact

ℏ𝜈

𝑁 𝜈

Photoemission spectroscopy (PES)

ℏ𝜈| ↑⟩| ↓⟩

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

𝑁 𝜈 ~Γ(𝜈)

Contact vs T

0 1 20

1

2

3

4 Data

C

/(N

kF)

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

Virial 2, Virial 3

C

/(N

kF)

T/TF

Tc


Contact vs T

0 1 20

1

2

3

4 Data

G0G

0, GPF, GG

Virial 2, Virial 3

QMC, ENS

C

/(N

kF)

T/TF

Tc


Contact vs T

0.0 0.2 0.4 0.62

3

4

Data

G0G

0, GPF, GG

Virial 2, Virial 3

QMC, ENS

C

/(N

kF)

T/TF

Tc


Criterion for homogeneity:

𝐶 𝑇/𝑇𝐹 ≈ 𝐶 𝑇/𝑇𝐹

For the theories mentioned before, when the fraction

probed is ≤ 30%, this holds to better than 2%.

Lines: theory for

homogeneous gas

Symbols: averaging over

the remaining density

inhomogeneity

Determining 𝑇 and 𝑘𝐹

•𝒌𝑭: We reconstruct the density distribution in

trap by using hydrodynamic expansion. We

then model the donut propagation and

calculate the density distribution of the probed

atoms.

•𝑻: We determine the temperature by

measuring the release energy at unitarity, and

using the recently measured EOS (Ku et al.

Science 2012).

Intermediate summary:

Homogenous contact:

• Good agreement with theories at high T.

• None of the theories fully account for the data.

• There is an observable drop in the contact which may be

consistent with Tc.

• There is no observable cusp as predicted by some many-

body theories.

What can we say about the

normal state?

Outline


mitigate it.


interacting gas.


Fermi gas.



regime.


Fermi liquid theory

• Experiments on 3He revealed that some quantities have a

scaling similar to that of an ideal Fermi gas, e.g. the

specific heat is linear in temperature.

• Landau’s idea (1956): low energy excitations may be

regarded as fermionic quasi-particles with well-defined

momentum, 𝑝, and energy 𝜖(𝑝).

• For this to hold, the uncertainty, or width, of 𝜖(𝑝) should

be much smaller than the width of the Fermi surface.

Δ𝐸 ≪ 𝐸𝐹 , 𝑘𝑇

• In the vicinity of the Fermi surface, the dispersion 𝜖(𝑝)follows that of a free particle, with a renormalized effective

mass 𝑚∗.

Probing the many-body wavefunction

ℏ𝜈| ↑⟩| ↓⟩

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

𝑁 𝜈, 𝑘

ℏ𝜈

𝑁 𝜈, 𝑘

Angle-Resolved PES (ARPES)

ℏ𝜈

𝑁 𝜈

Photoemission spectroscopy (PES)

Imaging

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

𝑁(𝜈, 𝑘)~𝐴 𝐸𝑠, 𝑘 𝑓(𝐸𝑠)

The spectral function

Fermi function

ℏ2𝑘2

2𝑚

𝐸𝑠

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

kFkF

k h2k 2

2m*mm

2D

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

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

At what 𝑘𝐹𝑎−1 the pseudogap phase

exist at an appreciable range of

temperatures ?

Homogeneous ARPES

ℏ𝜈| ↑⟩| ↓⟩

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

𝑁 𝜈, 𝑘Imaging

𝑇

𝑇𝑐∼ 0.8

𝑘𝐹𝑎−1 = 0

Homogeneous ARPES on the BEC side

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

1

𝑘𝐹𝑎= 0.3

There is a clear back-bending around kF

ARPES results around Tc


EDCs:


Width is limited by our

resolution of ∼ 0.25𝐸𝐹 Larger width of ∼ 0.5𝐸𝐹 Very large width (∼ 𝐸F)and asymmetric shape

Outline


mitigate it.


interacting gas.


Fermi gas.



regime.


Is the normal state a Fermi liquid?

• Assumption: Long lived quasi-particles with well defined

momentum near the Fermi surface => EDC width Δ𝐸 ≪ 𝐸𝐹 , 𝑘𝑇

For 𝑎 < 0, the data suggests this condition is fulfilled. At unitarity we

observe a width comparable to 𝐸𝐹, and for 𝑎 > 0 , the assumption

does not hold.

• Assumption: The quasi-particles follow a free particle dispersion,

with a renormalized effective mass 𝑚∗

For 𝑎 < 0 and unitarity the dispersion is quadratic, but for 𝑎 > 0 it is

not, and we observe a back-bending.

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=1.1

1/kFa=-0.3

m/m=1.1650.006

Fitting range

Summary

• Contact measurements:

• Sharp drop consistent with 𝑇𝑐• Good agreement with theory only above 0.4𝑇𝐹• Future: lower 𝑇/𝑇𝐹, other values of 1/𝑘𝐹𝑎

• ARPES measurements:

• At unitarity and for positive scattering lengths there is no well-

defined fermionic quasi-particle (i.e., not a Fermi liquid).

• Surprisingly, in the strongly interacting regime on the a<0 side,

we find a well-defined fermionic quasi-particle with a quadratic

dispersion (i.e., a Fermi liquid)

• Future: map width with 𝑇/𝑇𝐹, measurement of the unoccupied

upper branch of A(E,k)

The degenerate Fermi gas team…

Tara Drake, Rabin Paudel , Yoav Sagi

and Roman Chapurin

Deborah Jin

…and lots of fruitful discussions with the greater JILA BEC group!

Towards Probing Homogeneous Strongly Interacting Fermi...

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