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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
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
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!
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)
*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
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-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.