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3D Hydrodynamics and Quasi-2D Thermodynamics
in Strongly Correlated Fermi Gases
John E. ThomasNC State University
J. E. Thomas
Research Scholars:James JosephIlya Arakelian
Ethan ElliotWillie OngChingyun ChengArun JaganathanNithya ArunkumarJayampathi KangaraLorin Baird
Graduate Students:
JETLab Group
Support:
NSFARODOE
AFOSR
Outline
• Scale Invariance in Expanding Fermi gases:– Ballistic
expansion of a hydrodynamic
gas
• Shear Viscosity measurement– Local viscosity from trap-averaged viscosity
Topics
• Spin-imbalanced Quasi-two-dimensional Fermi gas– Failure of true 2D-BCS theory– 2D-polaron model of the thermodynamics– Phase-transition to a balanced core
Strongly Interacting Fermionic
Systems
Neutron Star
Quark Gluon Plasma Ultra‐Cold Fermi Gas
Why Study Strongly InteractingFermi Gases?
Layered High Temperature Superconductors
Optically Trapped Fermi gas
agnet coils
Our atom: Fermionic
Feshbach
Resonance
g1
Singlet Diatomic Potential: Electron Spins Anti-Parallel
832 G-Resonant Scattering! 2coll 4 dB
B
u3
Triplet Diatomic Potential: Electron Spins Parallel
B
u3
B
u3
SCALE INVARIANT!
Really strong interactions!
ShockFronts
• Trapped gas is divided into two
clouds with a repulsive optical potential.
• The repulsive potential is extinguished, the two clouds accelerate towards each other and collide.
Strong Interactions: Shock waves in Fermi gases
Scale Invariance in Expanding Resonant Fermi Gases
Compressed “Balloons”
Expanded “Balloons”
2coll 4 dB
Scale-invariance: Connecting Strongly to Weakly Interacting
• Can we connect elliptic flow
of a resonant
gas to the ballistic flow of an ideal gas in 3D?
• Anti-de Sitter-Conformal Field TheoryCorrespondence:
Connects strongly
interacting
fields in 4-dimensions to weakly
interacting gravity in 5-dimensions.
For both, the pressure is 2/3 of the energy density:
032 pp Scale Invariant?
Elliptic Flow: Observe 2 dimensions + time
Measuring expanding clouds in 3D
• Measure all three cloud radii using two cameras.
Trap with 1:3 transverse (x-y)
aspect ratio
E/EF
=0.52E/EF
=0.75E/EF
=1.22E/EF
=1.69Ballistic
832 G
527.5 G
Transverse Aspect Ratio versus Timefor a Unitary Fermi gas
Energy Dependent!
x
y
Scale Invariance: Ideal Gas
0
2
0
22 Urrr mt Ballistic Flow
0
22
0
22 vrr t
Cloud average
Virial
Theorem:
00
2 Urv m
tvrr 0 Ballistic flow
t2r
How does the mean square radius evolve in time? 2222 zyx r
Ideal gas:
20r
)(rU trap potential
Defining Scale Invariant Flow
Cloud average
20r
t2r
2222 zyx r
2
0
0
22
tm
Ur
rrDefines Scale Invariant Flow!
t
= expansion time
Scale Invariance: Resonant Gas
Hydrodynamic gas:
20r
Elliptic flow
20r
t2r ?
How does the mean square radius evolve in time? 2222 zyx r
)(rU trap potential
Hydrodynamic Expansion
From the Navier-Stokes
and continuity
equations, it is easy to show that a single component fluid obeys:
vr Biiiiii pd
NUxm
xmdtd S
322
2
2 1v2
Shear and Bulk Viscosity
PressureTrap potential
Stream KE
Need to find the volume integral of the pressure:
Equilibrium: 0033 Upd
N rr Measured
from the cloud profile and trap parameters
pp 32
Global Energy Conservation
Internal Energy (t)
Stream KE (t)
Internal Energy (t = 0+)
0312
231 εε dd N
mN rvr
Just after
the optical trap is abruptly extinguished*: t = 0+ :
Scale Invariant Expansion
vrpprUrr
BdN
dN
mdtd 3
03
0
2
2
2 332
Using the hydrodynamic equations and energy conservation it is easy to show that
Initial trap potential Bulk viscosityConformal symmetrybreaking Dp
32 pp
Scale Invariance!
032 ppResonant gas
0
2
0
22 Urrr mt
The bulk viscosity also vanishes so
Can we observe ballistic flow of an elliptically expanding gas?
Ballistic Flow!
Expansion time
Scale-invariant “Ballistic”
Expansionof a Resonant Fermi gas
0
02 )(Ur
rr 22
mt
2t
The aspect ratio exhibits elliptic flow, but the mean-square cloud radius expands Ballistically: SCALE-INVARIANT!
x
y
Shear and Bulk Viscosity: Unitary Fermi Gas
Shear viscosityBulk viscosity nBB
nSS
The bulk viscosity vanishes!
Quantum Viscosity
n = density ( particles/cc)n Viscosity:
dimensionless shear viscosity coefficient
Water:
n = 3.3 x 1022 n300
Air:
n = 2.7 x 1019 n6000
Fermi gas:
n = 3.0 x 1013 n4.0
Nuclear Matter:
n = 3.0 x 1038 n?
Measuring Shear Viscosity: Scaling Approximation
)(2
0
22 tbxx iii
)(v 2
0
22 tbx iii
zyx bbbVolume scale factor
)/,/,/(
),,,( 0 zyx bzbybxntxyxn
iiii bbx /v Velocity field is linear
in the spatial coordinates
322
i
iii b
b
imagiii
iiSpQ
i
ii b
bxmtCtC
bb 2
0
23/2
2
)()(1
0
202
3 ii xm
U
r
Cloud-averaged shear viscosity coefficient
Aspect Ratio versus Expansion Time
E/EF
=0.52E/EF
=0.75E/EF
=1.22E/EF
=1.69
Ballistic
x
y
Shear Viscosity
only Fit Parameter)()(
tbtb
y
x
x
y
y
x
527.5 G
832 G
Shear Viscosity at Resonanceversus Reduced Temperature
Transition to
Superfluid
High Temperature Scaling of the Cloud-Averaged Viscosity
3.60 03/2
Red: For trap that is 50
times deeperScience 331, 58 (2011)
Blue: New Data
Universal Scaling!
Cloud Averaged Shear Viscosity versus Reduced Temperature
Reduced temperatureat the trap center
Transition to
Superfluid
*EoS
from Ku et al., Science, 2012
Local
Shear Viscosity: Linear Inverse Problem
Equivalent matrix
equation: Cαα
mmmm Ψββ αCαCααα Τ )1(1
Can be solved iteratively
For each trap averaged shear viscosity measured at reduced temperature 0j , we create a linear
equation
i
ijijC
1 ii Discrete Local a
0
3 )( )( 1 rnrrdNS
Determining the Local Shear Viscosity: Finite Volume
Average
CR
S rnrrdN 0
3 )( )( 1 3
41 30
2/302/3
2/301 CRn
Nc
2/1298.0 rRC
Fit from our datac1
= 3.60from theory3/2
= 2.77
Volume
2/32
0 32
rNn For a Gaussian
density
rnr
Divergent!
Image Processing Technique Iterative Matrix Inversion
mSmmm Ψββ αCαCααα Τ )1(1
Iterative Solution for Test Function
mSmmm Ψββ αCαCααα Τ )1(1
0
q
Local Shear Viscosity versus Reduced Temperature
Structure appears atlow temperature
High Temperature Limit
= 2.77 θ3/2
Local Shear Viscosity (Comparison to Theory)
“Measured”
Kinetic theory
= 2.77 θ3/2
Guo
et al., PRL 2011
Enss
et al., Annals 2011(Diagrammatic Kubo formula)
Wlazłowski
et al., PRL 2012 (Monte Carlo)
Derivative of the Local Shear Viscosity with respect to Reduced Temperature
*EoS
from Ku et al., Science, 2012
Ratio of the Local Shear Viscosity to the Entropy Density*
String theory limitKovtun,Son,StarinetsPRL 2005
•
Determined Local Shear Viscosity
-
Image processing techniques
-
Tests of non-perturbative
many-body theory
Summary: 3D Hydrodynamics
• Scale-Invariant Strongly Interacting Fermi Gas-
Tests theories of scale-invariant hydrodynamics
• Measured Cloud-Averaged Shear Viscosity-
Single-shot method to find initial cloud size(temperature) and cloud-averaged shear viscosity self consistently
Quasi-2D
Fermi Gases
Enhancement
of the superfluid
transition temperature compared to true 2D
materials:
• In copper oxide and organic films, electrons are confined in a quasi-two-dimensional geometry
• Complex, strongly interacting many-body systems• Phase diagrams are not well understood• Exotic superfluids
Search for high temperature superconductivity in layered
materials:
• Heterostructures
and inverse layers• Quasi-2D organic superconductors• Intercalated structures and films of transition metals
Creating a Quasi-2D Fermi gas
Transverse Density Profiles: )(2 Dn
)()( 222 yxndyxn Dc
Measure Column
Density:N1
=
Majority
#(800 per site)
N2
= Minority #
Two-lowest hyperfine states of fermionic
6Li
in CO2
laser trap at T< 0.2 TF
:x
z
5.3 m
Two-Dimensional Gas
r
z nz
2D Density of States
0
20 )(
hd
sN
Two-dimensional
n = 0
n = 1
n = 2
0
zh
Ns
z
z
sF NhE 20 2D TransverseFermi Energy
zF hE True
2D if:
Two-dimensional
n = 0
n = 1
n = 2
0
zh
Ns
Quasi-two-dimensional
01
Ns0
Ns1
Quasi-Two-Dimensional Gas
What is the ground state of a quasi-2D strongly-interacting Fermi gas?
zF hE 0True
2D if:zF hE Quasi-2D if:
zF hE 03D if:
Column Densities versus N2
/N1
6.6b
F
EE• 832 G 75.0
b
F
EE• 775 G
siteper 8001 NMajority state: 12 NN Minority state:Eb
= 2D dimer
binding energy EF
= 2D ideal gas Fermi energy
Majority
and Minority
Radii
6.6b
F
EE
75.0b
F
EE
Eb
= 2D dimer
binding energy EF
= 2D ideal gas Fermi energyIdeal gas Thomas-Fermi radius -
Majority
Majority
and Minority
Radii
6.6b
F
EE
75.0b
F
EEIdeal gas
2D-BCS—Balanced
2b
FE
2D-BCS
2D-BCS Theory Fails!
Polaron
Gas Model
q
k
FS
Fermi energy
20
1q 1k
2q-k
2-Collision with the 1-Fermi sea
12 1
21 21 2 1 2
1 21 2 12 1 11
qkkq k-qpkq
01;,0,0polaronFSFS
single spin down cloud of particle-hole pairs
• Chevy’s Fermi-polaron
2D-Polaron Thermodynamics
• Free energy density-imbalanced gas: )2(22221
1121
pFF Ennnf
11)()2( Fmp qyE • Polaron energy:
Klawunn and Recati 2011)21log(
2)(1
1 qqym
11 n
f
2
2 nf
• Chemical potentials:
fnnp 2211 • Pressure:
b
F
Eq 1
1
1
2
12 n
mF
Minority PolaronEnergy
Ideal Fermi gas
2D Fermi-Polaron
Analytic Approx.
Shahin
Bour, Dean Lee, H.-W. Hammer, and Ulf-G. MeinerarXiv:1412.8175v2 (2014)
Meera
M. Parish and Jesper
LevinsonPhys. Rev. A 87, 033616 (2013)
Polaron
Energy: 2D-Fermi Polaronvs
Analytic Approximation
Lianyi
He, Haifeng
Lu, Gaoqing
Cao, Hui
Hu, Xia-Ji
LiuarXiv:1506.07156v1 (2015)
)/2ln( F21
bE
Majority
and Minority
Radii
6.6b
F
EE
75.0b
F
EE
Ideal gas Polaron model 2D-BCS—balanced
Measuring the 2D Pressure
0)(2
UnPDForce balance:
Harmonic Trap: 2221)( mU
NmP
2)0(
2
Pressure at trap center
FDnP 221
ideal Ideal 2D Pressure (for density n2D
)
)0()0(
)0()0(
22
2ideal
ideal Dnn
PP
= 1
for an Ideal
Fermi gas:
Determine n2D
(0)
from the column densities.
2D-Pressure Balanced Gas
ideal
)0(pp
2D-polaron model
2D-BCS
2D-Pressure—Wide Range
A. Turlapov, Phys. Rev. Lett. 112, 045301 (2014).
-
PolaronModel
2D-Central Density Ratio
• Transition to balanced core:• Not predicted!
Polaron model
Ideal gas
6.6b
F
EE
1.2b
F
EE 75.0
b
F
EE
Summary: Quasi-2D Fermi Gas
• 2D Polaron model explains several features of the density profiles in the quasi-2D regime.
• 2D Polaron model with the analytic approximation is too crude to predict the transition to a balanced core.
• Measurements with imbalanced mixtures providethe first benchmarks for predictions of the phase diagram for quasi-2D Fermi gases.
• 2D BCS theory for a True 2D system fails
in the Quasi-2D regime.
Birthday Party Dec-2014
