collective modes and sound velocity in a strongly interacting fermi gas
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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
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 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!
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
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
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)
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 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
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