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).