Two-dimensional Fermi gases Two dimensional Fermi gases Michael Khl - - PowerPoint PPT Presentation

Two-dimensional Fermi gases Two dimensional Fermi gases

Michael Köhl

BEC-BCS crossover

What happens in reduced dimensions?

S d M l Ph i T d (2008) Sa de Melo, Physics Today (2008)

A very short theory overview for 2D

)

kHz) 8 / (  h EF

shift of resonance Mean-field coupling in 2D

(Bloom 1975)

) ln( 1 2

2 2 2 D F D

a k m g    

n(kF a2D)

        

D D

a m g

3 2 3

4  

l B [G] Hz] 2D

3D

B [G]

 

D Fa

k

2

ln

1 1   EB [kH confinement-induced bound state

Fermi liquid ? strongly interacting 2D Fermi gas non-interacting 2D Fermi gas Bose gas

• f dimers
• 1

1

• 

 B [G] bou d state

• BKT transition at TBKT ≈ 0.1 TF in the strongly interacting regime
• TBKT decays exponentially towards weak attractive interactions (as in 3D)

Theory: Bloom, P.W. Anderson, Randeria, Shlyapnikov, Petrov,

[ ]

y , , , y p , , Devreese, Julienne, Duan, Zwerger, Giorgini, Sa de Melo, ...

Quasi-2D geometry

kHz 80 π 2    kHz 8   h EF

Conditions for 2D

E k T << ħ

Hz 130 π 2  



 kHz 80 π 2  

z



EF , kBT << ħz

Strong axial confinement required

ħz Optical lattice: array of 2D quantum gases

• lattice depth 83 Erec
• hopping rate 0 002 Hz

hopping rate 0.002 Hz

• ~ 2000 Fermions per spin state
• ~ 30 "pancakes" / layers

y x

• B. Fröhlich, M. Feld, E. Vogt, M. Koschorreck, W. Zwerger, MK, PRL 106, 105301 (2011)

p y

l0

 60 nm

L/2

= 532 nm

x z

, , g , , g , , , ( )

• ther 2D Fermi gases: Inguscio, Grimm, Esslinger, Turlapov, Vale, Zwierlein

Preparing strongly interacting 2D systems

Feshbach resonances in 40K

|1> & |2> |1> & |3> |3> = |-5/2> a3D / a0 |2> = |-7/2> a |1> = |-9/2> B [G] |2> & |3> B [G]

174a aBG 

|2> & |3>

Radio-frequency spectroscopy

RF pulse (association/dissociation)

|1> & |3> |3>

224G FB res.

hyperfine transition frequency

|3>

ħRF,0 ħRF

|2>

frequency

|2> |1>

e.g. repulsively interacting gas

|1

3D: Regal et al., Nature (2003) g , ( ) 2D: B. Fröhlich, M. Feld, E. Vogt, M. Koschorreck, W. Zwerger, MK, PRL 106, 105301 (2011)

Momentum-resolved RF spectroscopy ("ARPES")

ħRF

|3>= |-5/2>

ħ

|3> (E,k)

|2>= |-7/2>

ħRF E

final state |3> (weak interactions)

|1>= |-9/2>

Energy E ħRF

m k f k

E

2

2 2

) (





initial state, e.g. BCS-like dispersion

 

2 2 2

2 2

) (       

m k i k

E





202G FB res.

momentum k

 

2

) (  

m i

ARPES in 3D Experiment: Jin Theory: Georges, Strinati, Levin, Ohashi, Zwerger, Drummond, ...

Population of the spin states

|-9/2> |-7/2> |-5/2> RF pulse

2 2 y x

k k k  

Single-particle spectral function

atoms

“BCS” side ln(kFa2D) > 0 , EB < EF ln(kFa2D) 0 , EB EF no isolated dimers, attractive interactions, pairs are huge compared to inter-particle “Condensation energy” of Cooper pairs p g p p spacing

pairs

B th

E E k E     2 ) (

2

gy p p

(MF theory, T=0, Randeria 1989)

pairs

ln(kFa2D) = 0.19

F th

E 2

Only the case in 2D

(in 3D: E 0 on the BCS side) (in 3D: EB=0 on the BCS side)

Pairing pseudogap phenomenon

) , (

) , ( ) , (

x T i

e x T x T



  

complex order parameter  





 =0 but  > 0

spatial average

Pairing pseudogap phenomenon

) , (

) , ( ) , (

x T i

e x T x T



  

complex order parameter  =0  =0  0

(condensed Cooper pairs) (phase-fluctuating

• rder parameter)

Temperature

In 3D weak coupling BCS: T

c ≈ T* (pairs condense as they form)

Theory: Randeria, Levin, Sa De Melo, Kleinert, ...

Spectra at k=0: Determining the pseudogap

Eth

th

E k E 2 ) (

2

  

F

E 2

• M. Feld et al., Nature 480, 75 (2011)

Temperature dependence

Finite temperature mean field theory mean field theory, no free parameters

• M. Feld et al., Nature 480, 75 (2011)

Pairing pseudogap

th/EB

ln(kF a2D)

0 2

Et

0.2 0.5 0.8

T/T* T/T

In 3D: Observation of Fermion condensates below T/TF ≈ 0.15 [b j ti t l l ] [by projection onto molecules] In 2D: No condensation observed.

Back-bending of dispersion relation

• M. Feld et al., Nature 480, 75 (2011)

Back-bending of dispersion relation

log(kFa2D) = 0.2 T/TF = 0.65 T/TF = 0.45 BCS-like dispersion T/TF = 0.27 BCS like dispersion at low T

• M. Feld et al., Nature 480, 75 (2011)

Strongly imbalanced Fermi gases in 2D

The “N+1” problem: one |↓> impurity in a large |↑> Fermi sea

Energy

• Mobile impurity

ln(k a ) repulsive polaron

Metastability

• Mobile impurity

interacting with a Fermi sea of atoms Tunable interactions

ln(kFa2D) attractive polaron

• Tunable interactions
• Polaron properties

determine phase

molecule

diagram of imbalanced Fermi mixtures

3D Theory: Bruun, Bulgac, Chevy, Giorgini, Lobo, Prokofiev, Stringari, Svistunov, ... 3D E t Z i l i S l G i

It’s disputed whether there is a transition in 2D

2D Theory: Bruun, Demler, Enss, Parish, Pethick, Recati, ... 3D Expmt: Zwierlein, Salomon, Grimm

Characterizing the attractive polaron

l (k ) Energy

repulsive polaron

ln(kFa2D)

repulsive polaron molecule attractive polaron molecule Theory from: R. Schmidt, T. Enss, ( ) free particle dispersion s btracted

• V. Pietilä, E. Demler, PRA 85, 021602 (2012)

0 Signal 1

free-particle dispersion subtracted

• M. Koschorreck et al., Nature (2012), Advanced Online Publication 23/5/2012

Coherence of the polaron

Rabi oscillations between polaron and free particle

E [kHz]

• 10
• 20
• 30

4 8

k[μm-1]

Incoherent transfer: rate ~ amplitude

• M. Koschorreck et al., Nature (2012), Advanced Online Publication 23/5/2012

Repulsive polaron

Theory: V N tik t l Energy repulsive polaron

• V. Ngampruetikorn et al.,

EPL 98 30005 (2012). Energies comparable to ln(kFa2D) repulsive polaron l l attractive polaron Energies comparable to:

• R. Schmidt, T. Enss,
• V. Pietilä, E. Demler,

PRA 85, 021602 (2012) molecule Similar experiments in 3D: Ketterle & Grimm groups

• M. Koschorreck et al., Nature (2012), Advanced Online Publication 23/5/2012

Scale invariance and viscosity of a 2D Fermi gas

Quadrupole mode

• insensitive to EoS

h i it Breathing mode

• measures compressibility

b lk i it

• measures shear viscosity
• measures bulk viscosity

Equation of state and scale invariance

• Scale invariance in a homogeneous system: H(x)= H(x)/
• Scale invariance in a homogeneous system: H(x)= H(x)/

 Simple equation of state P=2/D

I li d i ll t i t l i i i l d b

• In cylindrically symmetric trap: scale invariance is replaced by a

SO(2,1) Lorentz symmetry (Pitaevskii/Rosch, 1997) Two remarkable predictions: Two remarkable predictions:

• 1. Breathing mode: B=2 (independent of interaction strength!)
• 2. bulk viscosity is zero
• Quantum anomaly due to log-dependence of coupling strength?

(Olshanii 2010)

For bosons: Pitaevskii/Rosch, Perrin/Olshanii, Chin, Dalibard; for fermions: Schaefer, Hofmann, Randeria/Taylor, Bruun,...

Collective modes

Breathing mode Quadrupole mode

collisionless (zeroth sound) hydrodynamic (f ) (first sound)

• E. Vogt et al, PRL 108, 070404 (2012).

Temperature dependent damping

Shear viscosity

 

F

T T n /    

   



 

F F

T T T T / /

0 



dimensionless function

   

F F

• E. Vogt et al, PRL 108, 070404 (2012).

Summary

• ARPES measurements in 2D

ARPES measurements in 2D

• Pairing pseudogap
• Pairing pseudogap
• 2D Fermi polaron
• 2D Fermi polaron
• Collective modes to determine equation of state
• Collective modes to determine equation of state

Thanks

Theory discussions/ collaboration: collaboration:

• S. Baur,
• C. Berthod
• N. Cooper
• N. Cooper
• E. Demler
• A. Georges
• T. Giamarchi
• C. Kollath
• C. Kollath
• J. Levinsen
• M. Parish
• W. Zwerger

Fermi gases

• E. Vogt, M. Koschorreck, D. Pertot, L. Miller, E. Cocchi, J. Bohn

Ion & BEC

• C. Zipkes, L. Ratschbacher, C. Sias, J. Silver, L. Carcagni

Ion trap QIP H.-M. Meyer, M. Steiner www.quantumoptics.eu Funding: EPSRC, ERC, Leverhulme Trust