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?
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
• 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?
• 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
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
G0G
0, GPF, GG
Virial 2, Virial 3
QMC, ENS
C
/(N
kF)
T/TF
Tc
Y. Sagi, T. E. Drake, R. Paudel, and D. S. Jin, PRL 109, 220402 (2012)
Contact vs T
0.0 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
Y. Sagi, T. E. Drake, R. Paudel, and D. S. Jin, PRL 109, 220402 (2012)
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
• 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
• 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
ARPES results around Tc
EDCs:
ARPES results around Tc
Width is limited by our
resolution of ∼ 0.25𝐸𝐹 Larger width of ∼ 0.5𝐸𝐹 Very large width (∼ 𝐸F)and asymmetric shape
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?
• 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!