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