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

A very short theory overview for 2D  ( E F / h 8 kHz) Mean-field coupling in 2D shift of resonance (Bloom 1975) ) n(k F a 2D )  2 2 1    g 2 D m ln( k a ) F 2 D l    2 4     g a   3 D 3 D m   B [G] B [G] 2D   3D Hz] ln k F a 2 D -     E B [kH -1 1 0 0 1 1 Bose gas strongly interacting Fermi non-interacting confinement-induced of dimers 2D Fermi gas liquid ? 2D Fermi gas bou d state bound state • BKT transition at T BKT ≈ 0.1 T F in the strongly interacting regime • T BKT decays exponentially towards weak attractive interactions (as in 3D) B [G] [ ] Theory: Bloom, P.W. Anderson, Randeria, Shlyapnikov, Petrov, y , , , y p , , Devreese, Julienne, Duan, Zwerger, Giorgini, Sa de Melo, ...

Quasi-2D geometry Conditions for 2D  h  E F 8 kHz E F , k B T << ħ  z k T << ħ      2  2  E π π 80 80 kHz kHz z ħ  z   2  π 130 Hz  Strong axial confinement required Optical lattice: array of 2D quantum gases • lattice depth 83 E rec • hopping rate 0 002 Hz hopping rate 0.002 Hz • ~ 2000 Fermions per spin state y • ~ 30 "pancakes" / layers p y x x l 0 z  L /2  60 nm = 532 nm B. Fröhlich, M. Feld, E. Vogt, M. Koschorreck, W. Zwerger, MK, PRL 106, 105301 (2011) , , g , , g , , , ( ) other 2D Fermi gases: Inguscio, Grimm, Esslinger, Turlapov, Vale, Zwierlein

Preparing strongly interacting 2D systems Feshbach resonances in 40 K |3> = |-5/2> |1> & |2> |1> & |3> a 3D / a 0 |2> = |-7/2> a |1> = |-9/2> |2> & |3> |2> & |3> B [G] B [G] a BG  174 a 0

Radio-frequency spectroscopy RF pulse |1> & |3> (association/dissociation) 224G FB res. |3> |3> ħ  RF,0 ħ  RF hyperfine transition |2> |2> frequency frequency |1> |1 e.g. repulsively interacting gas 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> |3>= |-5/2> (E,k) ħ  RF ħ |2> = |-7/2> final state |3> (weak interactions) E Energy E  2 2 E f k ( ) k  2 m |1> = |-9/2> ħ  RF initial state, e.g. BCS-like 202G FB res. dispersion      2          2 2 2 E i k ( ( ) ) k  i 2 2 m m momentum k 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 k k k x y

Single-particle spectral function “BCS” side atoms ln(k F a 2D ) > 0 , E B < E F ln(k F a 2D ) 0 , E B E F no isolated dimers, attractive interactions, p pairs are huge compared to inter-particle g p p spacing “Condensation energy” of Cooper pairs gy p p (MF theory, T=0, Randeria 1989)  2    E ( k 0 ) E th th B pairs pairs 2 2 E E F Only the case in 2D ln(k F a 2D ) = 0.19 (in 3D: E (in 3D: E B =0 on the BCS side) 0 on the BCS side)

Pairing pseudogap phenomenon     i ( T , x ) ( T , x ) ( T , x ) e complex order parameter         =0 but  > 0 spatial average

Pairing pseudogap phenomenon     i ( T , x ) ( T , x ) ( T , x ) e complex order parameter  =0  =0   0 (condensed (phase-fluctuating Cooper pairs) order 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 E th  2   E ( k 0 ) th 2 2 E E F 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 ln ( k F a 2D ) th /E B 0 0 2 0.2 E t 0.5 0.8 T/T* T/T In 3D: Observation of Fermion condensates below T/T F ≈ 0.15 [b [by projection onto molecules] j ti t l l ] In 2D: No condensation observed.

Back-bending of dispersion relation M. Feld et al., Nature 480, 75 (2011)

Back-bending of dispersion relation log(k F a 2D ) =  0.2 T/T F = 0.65 T/T F = 0.45 BCS-like dispersion BCS like dispersion T/T F = 0.27 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 • Mobile impurity interacting with a Metastability Fermi sea of atoms repulsive polaron ln(k a ln(k F a 2D ) ) • Tunable interactions Tunable interactions • Polaron properties attractive polaron determine phase diagram of imbalanced molecule Fermi mixtures It’s disputed whether there is a transition in 2D 3D Theory: Bruun, Bulgac, Chevy, Giorgini, Lobo, Prokofiev, Stringari, Svistunov, ... 3D E 3D Expmt: Zwierlein, Salomon, Grimm t Z i l i S l G i 2D Theory: Bruun, Demler, Enss, Parish, Pethick, Recati, ...

Characterizing the attractive polaron Energy repulsive polaron repulsive polaron ln(k F a 2D ) l (k ) attractive polaron molecule molecule Theory from: R. Schmidt, T. Enss, free particle dispersion s btracted free-particle dispersion subtracted V. Pietilä, E. Demler, PRA 85, 021602 (2012) ( ) 0 Signal 1 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] 0 -10 -20 -30 0 4 8 k[ μ m -1 ] Incoherent transfer: rate ~ amplitude M. Koschorreck et al., Nature (2012), Advanced Online Publication 23/5/2012

Repulsive polaron Energy Theory: repulsive polaron repulsive polaron V N V. Ngampruetikorn et al., tik t l ln(k F a 2D ) EPL 98 30005 (2012). attractive polaron Energies comparable to Energies comparable to: molecule l l R. Schmidt, T. Enss, V. Pietilä, E. Demler, PRA 85, 021602 (2012) 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 Breathing mode • insensitive to EoS • measures compressibility • measures shear viscosity h i it • measures bulk viscosity b lk i it

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 In cylindrically symmetric trap: scale invariance is replaced by a li d i ll t i t l i i i l d b 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 Quadrupole mode Breathing mode collisionless (zeroth sound) hydrodynamic (f (first sound) ) E. Vogt et al, PRL 108, 070404 (2012).

Temperature dependent damping Shear viscosity      n T / T  F dimensionless function             0  T / T T / T F F 0 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