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Collective Modes and Sound Velocity in a Strongly Interacting Fermi Gas
Students: Joe Kinast, Bason Clancy,
Le Luo, James Joseph
Post Doc: Andrey Turlapov
Supported by: DOE, NSF, ARO, NASA
John E. Thomas
Theory: Jelena Stajic, Qijin Chen, Kathy Levin
Strongly- Interacting Fermi Gases as a Paradigm
• Fermions are the building blocks of matter
• Link to other interacting Fermi systems:– High-TC superconductors – Neutron stars
• Strongly-interacting Fermi gases are stable
– Effective Field Theory, Lattice Field Theory
– String theory!Duke, Science 2002
– Quark-gluon plasma of Big Bang - Elliptic flow
- Quantum Viscosity
MITJILA Innsbruck RiceENSDuke
Degeneracy in Fermi Gases
Trap Fermi Temperature Scale:
TF = 2.4 K
5102NHz600)( 3/1 zyx
Optical Trap Parameters:
3/1)3( NhTk FB
Zero Temperature
FBF Tk
hnnn zyx )( Harmonic Potential:
Our atom: Fermionic
1,2
1= 0,
2
1=
spinnuclear spin,electron
Tunable Interactions: Feshbach Resonance
*generated using formula published in Bartenstein, et al, PRL 94 103201 (2005)
aScattering length
840 G
0a @ 528 G
02000L
:Spacing cleInterparti
a
Universal Strong Interactions at T = 0
m
kF
2
22 1
3/1)3( NhTk FB 1
George Bertsch’s problem: (Unitary gas) 0 RLa
L
Ground State:
Trap Fermi Temperature:
1*
mmEffective mass:
5.0Cloud size:
Baker, Heiselberg
Lk
1F
Outline
• All-optical trapping and evaporative cooling
• Experiments– Virial Theorem (universal energy measurement)
– Thermodynamics: Heat capacity (transition energy)
– Oscillations and Damping (superfluid hydrodynamics)
– Quantum Viscosity
– Sound Waves in Bose and Fermi Superfluids
2 MW/cm2
U0=0.7 mK
Preparation of Degenerate 6Li gas
Atoms precooled
in a magneto-optical trap
to 150 K
Forced Evaporation in an Optical Trap
High-Field Imaging
Experimental Apparatus
Experimental Apparatus
Energy input
R
ItRIE 2
0
Temperature
Tools for Thermodynamic Measurements
Temperature from Thomas-Fermi fit
Integrate
x
From Thomas – Fit: FT
T “true” temperature for
non-interacting gas
empirical temperature for
strongly-interacting gas
fitFT
T
Fermi Radius: F Shape Parameter: (T/TF)fit
Zero TempT-F
Maxwell-Boltzmann
(T/TF)fit
0
Calibrating the Empirical temperaturefit
FTT
1fit FF T
T
T
TConjecture:
Calibration using
theoretical density
profiles:
Stajic, Chen, Levin
PRL (2005)
FT
T
1/ fitFTT
S/F transition
predicted
Precision energy input
Trap ON again,
gas rethermalises
heatt time
Trap
ON
Final Energy E(theat)
3
)(
3
2)( heat
2
0heat
tbEtE
Initial energy E0
state Ground0 E
)( heattbExpansion factor:
Virial Theorem
(Strongly-interacting Fermi gas obeys the Virial theorem for an Ideal gas!)
Virial Theorem in a Unitary Gas
),( TnPPressure:
x
U
Trap potential
tot2 Ex Test!
0 UnPForce Balance:
tot2
1tot EU Virial Theorem:
),(3
2Tn
Local energy density (interaction and kinetic)
Ho, PRL (2004)
Verification of the Virial Theorem
Fermi Gas at 840 G
1)0(2
2
x
x
02
2
)02.0(03.1)0( E
E
x
x
Linear Scaling Confirms
Virial Theorem
Fixedexpansiontime
E(theat) calculated assuming hydrodynamic expansion
Consistent with hydrodynamicexpansion over wide range of T!
Heat Capacity
Energy versus empirical temperature(Superfluid transition)
Input Energy vs Measured Temperature
Noninteracting Gas (B=528 G)
Ideal Fermi Gas Theory
0E
E
FF T
T
T
T
fit
Strongly-Interacting Gas at 840 G
Ideal Fermi Gas Theorywith scaled Fermi temperature
0E
E
FF T
T
T
T
fit
Input Energy vs Measured Temperature
Low temperature region
Strongly-Interacting Gas (B=840 G)
fit
FT
T
Ideal Fermi gas theorywith scaled temperature
0E
EPower law fit
Energy vs on log-log scale
Transition!
fit
FT
T
10
E
E
33.0fit FTT
fit
FTT
Blue – strongly-int. gasGreen – non-int. gas
Ideal Fermi gas theory
Fit
58.10
E
E
Energy vs FT
T
FTT
10
E
E
Theory for Strongly-interacting gas (Chicago, 2005)
Oscillation ofa trapped Fermi gas
Study same system (strongly-interacting Fermi gas)by different method
Breathing mode in a trapped Fermi gas
Trap ON again,
oscillation for variable
offtholdt
Image
1 ms
Releasetime
Trap
ON
Excitation &
observation:
Breathing Mode Frequency and Damping
528 GNoninteracting Gas
840 G Strongly- Interacting Gas
tAxtx t cose)( /0rms
frequency damping time
Radial Breathing Mode: Frequency vs Magnetic Field
Hu et al.
Radial Breathing Mode: Damping Rate vs Magnetic Field
Pair Breaking
Frequency versus temperature for strongly-interacting gas (B=840 G)
Hydrodynamicfrequency, 1.84
Collisionless gasfrequency, 2.10
2.1
2.0
1.9
1.8
1.7
Fre
quen
cy (
/
trap
)
1.51.00.50.0( T/TF )fit
0.10
0.05
0.00
Dam
ping
rat
e (1
/)
1.51.00.50.0( T/TF )fit
Damping 1/ versus temperature for strongly-interacting gas (B=840 G)
Transition!
Transition in damping:
35.0or5.0fit
FF TT
TT
Transition in heat capacity:
27.0or33.0fit
FF TT
TT
S/F transition (theory):
Levin:
Strinati:
Bruun:
29.0FTT
31.0FTT30.0FTT
Superfluid behavior: Hydrodynamic damping 0 as T 0
Quantum Viscosity?
1 z
1)3(3
413/1N
Radial mode:
1)3(5
1613/1Nzz
Axial mode:
Innsbruck Axial: = 0.4 Duke Radial: = 0.2
nL
L
2
/Viscosity:section cross
momentum n
Shuryak (2005)
Wires!
Sound Wave Propagationin Bose and Fermi Superfluids
Magnetic tuning between Bose and Fermi Superfluids
g1
Singlet Diatomic Potential: Electron Spins Anti-parallel
u3
Triplet Diatomic Potential: Electron Spins Parallel
1,2
1= 0,
2
1=
spinnuclear spin,electron Stable molecules
g1
u3
B = 710 GB
g1
u3
B = 834 G
Resonance
g1
u3
B = 900G
Cooper Pairs
Molecular BECs are cold
Lin
ea
r d
en
sity
-150 -100 -50 0 50 100 150
Radial position, m
“Hot” BEC, 710 G(after free expansion)
Lin
ea
r d
en
sity
-150 -100 -50 0 50 100 150
Radial position, m
“Cold” BEC, 710 G(after free expansion,from the same trap)
Sound: Excitation by a pulse of repulsive potential
Trapped atoms
Slice of green
light (pulsed)
Sound excitation:
Observation:
hold, release & image
thold= 0
Sound propagation on resonance (834 G)
Sound propagation at 834 G
200
150
100
50
0
-50
z (m
)
86420 thold (ms)
Forward Moving Notch
Backward Moving Notch
Speed of Sound, u1 in the BEC-BCS Crossover
0.4
0.3
0.2
0.1
0.0
u 1/v F
5 4 3 2 1 0 -1 -2
1/kFa
710 750 780 834 900
B (Gauss)
Sound Velocity in a BEC of Molecules
M
ag mol
24 mM 2
2
2
mol
'1)'(
r
grn 2
2 2
M
molMF ngU
Mean field:
222
1Trap ')'( rmrU
Harmonic Trap:
M
rng
n
P
Mrc
)'(1)'( mol
mol
2
Local Sound Speed c:
5
1
FF0 )(k
128.0
v
Ba
cFull trap average:
vF0= Fermi velocity, trap center, noninteracting gas
2mol2
1 ngP Dalfovo et al, Rev Mod Phys1999
)(6.0)(mol BaBa For (Petrov, Salomon, Shlyapnikov)
0.4
0.3
0.2
0.1
0.0
u 1/v F
5 4 3 2 1 0 -1 -2
1/kFa
710 750 780 834 900
B (Gauss)
Speed of Sound, u1 for a BEC of Molecules
Sound Velocity at Resonance
2
3
2F
2'1)0()'(
r
nrn2
2F
)0(2
m
F222
1Trap ')'( rmrU
Harmonic Trap:
Pressure: nnP )()1(5
2F
Local Sound Speed c:m
n
n
P
mrc
)()1(
3
21)'( F2
41
F0F
1
vv
2
1
2
2
F
'1v
3
1
F
rc
vF0 = Fermi velocity, trap center, noninteracting gas
from the sound velocity at resonance
3
178.0
v
4
1
F0
cFull trap average:
61.0
49.0
54.0Rice, cloud size 06
Duke, cloud size 05
Duke, sound velocity 06
Carlson (2003) = - 0.560Strinati (2004) = - 0.545
Theory:
Experiment:
(Feshbach resonance at 834 G)
Transverse Average—I lied!
2
1
2
2
1)0()(
z
czc z
zc
dzt
0 )'(
'
2)0(
6
11)0(
)0(sin
tc
tctc
z
2)0(
tc %4
2
1
6
12
tcz )0(
More rigorous theory with correct c(0) agrees with trap average to 0.2 %
(Capuzzi, 2006):
0.4
0.3
0.2
0.1
0.0
u 1/v F
5 4 3 2 1 0 -1 -21/kFa
710 750 780 834 900
B (Gauss)
Speed of sound, u1 in the BEC-BCS crossover
Theory: Grigory Astrakharchik (Trento)
Monte-Carlo Theory
Speed of sound, u1 in the BEC-BCS crossover
0.4
0.3
0.2
0.1
0.0
u 1/v F
5 4 3 2 1 0 -1 -2
1/kFa
710 750 780 834 900
B (Gauss) Monte-Carlo Theory
Theory: Grigory Astrakharchik (Trento)
0.4
0.3
0.2
0.1
0.0
u 1/v F
5 4 3 2 1 0 -1 -2
1/kFa
710 750 780 834 900
B (Gauss)
Speed of sound, u1 in the BEC-BCS crossover
Leggett Ground State Theory
Theory: Yan He & Kathy Levin (Chicago)
Monte-Carlo Theory
Theory: Grigory Astrakharchik (Trento)
Summary
• 2 Experiments reveal high Tc transitions in behavior: - Heat capacity - Breathing mode
• Strongly-interacting Fermi gases: - Nuclear Matter – High Tc Superconductors
• Sound-wave measurements: - First Sound from BEC to BCS regime - Very good agreement with QMC calculations
The Team (2005)
Left to Right: Eric Tong, Bason Clancy, Ingrid Kaldre, Andrey Turlapov, John Thomas, Joe Kinast, Le Luo, James Joseph