Two-dimensional atomic Fermi gases Michael Khl University of - PowerPoint PPT Presentation

Two-dimensional atomic Fermi gases • Michael Köhl • University of Bonn

Two-dimensional Fermi gases Two- dimensional gases: “the grand challenge” of condensed matter physics High-T c superconductors: • After 25 years of research still many open questions • Material is too complicated to understand even the basic mechanism How can cold atoms help ? …. Cleanliness, tunability, testing models.

Cold atoms meets condensed matter • Quasiparticle spectroscopy by momentum-resolved photoemission (aka ARPES) • Spin transport and spin diffusion • 2D Hubbard model

Quasi-2D geometry Strong axial harmonic confinement Conditions for 2D: ħ w z E F , k B T < < Spin ½ Fermi gas with contact interaction  2  2 1   g Mean-field coupling constant in 2D 2 D m ln( k a ) (Bloom 1975) F 2 D   ln k F a -   2 D -1 0 1 Bose gas strongly interacting Fermi non-interacting of dimers 2D Fermi gas liquid 2D Fermi gas • 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) Theory: Bloom, P.W. Anderson, Randeria, Shlyapnikov, Devreese, Julienne, Duan, Zwerger, Giorgini, Sa de Melo, ... Experiment : B. Fröhlich et al., PRL 106, 105301 (2011), Inguscio, Grimm, Esslinger, Jochim, Moritz, Turlapov, Vale, Zwierlein

Quasiparticle spectroscopy

Momentum-resolved RF spectroscopy ħ w RF |3> |3>= |-5/2> (E,k) ħ w RF |↑> = |-7/2> final state |3> (weak interactions) Energy E  2 2  k E f k ( ) 2 m |↓> = |-9/2> ħ w RF initial state, e.g. BCS-like 202G Feshbach resonance dispersion    2        2 2  2 k E i k ( ) 2 m momentum k ARPES in 3D Experiment: Jin Theory: Georges, Strinati, Levin, Ohashi, Zwerger, Drummond, ...

Experimental realization |↓> |↑> |3> k y k x RF pulse   2 2 k k k x y

Spin-balanced Fermi liquid • Landau-Fermi liquid quasi-particles are fermionic finite lifetime 1/t ~ (k-k F ) 2 (long- • lived near the Fermi surface) • effective mass: m*/m > 1, depending on interaction strength Fermi liquid: E F , k B T < ħ w (two-dimensional)   ln k F a -   2 D -1 0 1 Bose gas strongly interacting Fermi non-interacting E B < k B T (no pairing) of dimers 2D Fermi gas liquid 2D Fermi gas g=1/ln(k F a 2D ) < 1 (weak interactions)

Comparison with theory Single-particle spectral function Effective mass parameter Experiment Theory T/T F =0.3 B. Fröhlich et al., Phys. Rev. Lett. 109, 130403 (2012)

Strong interactions: Pairing pseudogap Single-particle spectral function E th E F , k B T < ħ w (two-dimensional)   ln k F a 2 D -   -1 0 1 Bose gas strongly interacting Fermi non-interacting E B > k B T (pairing) of dimers 2D Fermi gas liquid 2D Fermi gas g=1/ln(k F a 2D ) > 1 (strong interactions) M. Feld et al., Nature 480, 75 (2011) Observation of polaron quasiparticles: M. Koschorreck et al., Nature 485, 619 (2012)

Spin transport

Spin diffusion Fermi gas at unitarity:   v k F / m v D  Spin diffusion   n  n 2  Quantum limit k D F m of diffusivity 1   k F Semiconductor 3D Fermi gases at unitarity nanostructures   D 6 . 3   2 m D 10 m Weber et al., Nature (2005) Zwierlein group, Nature (2011)

Spin dynamics transversely polarized Fermi gas

Spin-spin interaction Spin exchange / Spin-rotation Spin relaxation e.g. spin-orbit coupling breaks symmetry underlying spin conservation absent in cold atom systems Strength determined by interaction constant  2  2 1   g 2 D m ln( k a ) F 2 D Many-body effects in Fermi liquid (Leggett-Rice effect)

Longitudinal vs. transverse diffusion Magnetisation: longitudinal ~1/T 2 transverse Mullin & Jeon (1992)

Spin-echo technique  /2   /2 0 t 2t time Eliminates effect of magnetic field gradient M z ( t ) = characteristic exponent Theory: Hahn, Purcell, Leggett, Mullin, Dobbs, Lhuillier, Laloe, ... Experiment in 3He: Osheroff M. Koschorreck et al., Nature Physics 9, 405 (2013).

Spin diffusion in the strongly interacting regime Smallest spin diffusion constant ever measured: 0.07(1) ħ/m. M. Koschorreck et al., Nature Physics 9, 405 (2013).

Implications of D< ħ/m ?   v k F / m v   Spin diffusion D v l  MFP n  n Particle separation 2 k F Spin diffusivity D < ħ/m implies l MFP < n 1/D =d Mean-free path for collisions with Resistivity of metals (semiclassically): phonons or electrons MFP As function of temperature: l MFP ~ 1/T BUT: Ioffe-Regel criterion l MFP > d → Saturation of resistivity

Strongly correlated materials Possible ideas for resistivity in solids: • Violation of quasiparticle picture [Nature 405, 1027-1030 (2000)] • Modification of kinetic theory by correlation effects du to stong interactions [PRB 66, 205105 (2002)] For cold atoms → ?

Hubbard model

Hubbard model in two dimensions Simplest interacting lattice model High-resolution imaging: Experimental realization Diffraction limit ~ 2 lattice sites a)

With atoms: Excellent tunability Depends on Inhomogeneity of the trap lattice depth -> convenient access to (~ laser intensity) phase diagram V(x)  x

RF spectroscopy in the lattice • resolve single 2D layer • spectroscopically separate singly and doubly occupied lattice sites Singly Doubly occupied sites occupied sites

Two-dimensional Mott insulator Density vs. chemical potential Singly occupied lattice sites Doubly occupied lattice sites T/t = 3, U/t ~ 30

Thermodynamic quantities 𝜈 Compressibility 𝜆 = 1 𝜖𝑜 Pressure 𝑄 = 𝑜(𝜈′) 𝑒𝜈′ 𝑜 2 𝜖𝜈 −∞ Theory curves: High-temperature series expansion (2nd order) T/t = 3, U/t ~ 30

Summary • Quasiparticle spectroscopy of 2D Fermi gases • Very low spin diffusion D ~ 0.07 ħ/m in a strongly interacting 2D Fermi gas • In-situ measurement of thermodynamics properties of 2D Hubbard model (-> equation of state)

Thanks Fermi gases J. Bernardoff, F. Brennecke, E. Cocchi, J. Drewes, M. Koschorreck, L. Miller, D. Pertot, A. Behrle, K. Gao, T. Harrison, J. Andrijauskas Trapped ions T. Ballance, L. Carcagni, M. Link, H.-M. Meyer, R. Maiwald, J. Silver www.quantumoptics.eu €€€ : Alexander-von-Humboldt Foundation, DFG, EPSRC, ERC, ITN Comiq