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Interacting Bosons and Fermions in 3D Optical Lattice Potentials

Interacting Bosons and Fermions in 3D Optical Lattice Potentials. Sebastian Will, Thorsten Best, Simon Braun, Ulrich Schneider, KC Fong, Lucia Hackermüller, Stefan Trotzky, Yuao Chen, Ute Schnorrberger, Stefan Kuhr, Jacob Sherson, Christof Weitenberg, Manuel Endres,

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Interacting Bosons and Fermions in 3D Optical Lattice Potentials

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  1. Interacting Bosons and Fermions in 3D Optical Lattice Potentials Sebastian Will, Thorsten Best, Simon Braun, Ulrich Schneider, KC Fong, Lucia Hackermüller,Stefan Trotzky, Yuao Chen, Ute Schnorrberger, Stefan Kuhr, Jacob Sherson,Christof Weitenberg, Manuel Endres, Theory: Belén Paredes, Mariona Moreno Immanuel Bloch funding by€ DFG, European Union,$ AFOSR, DARPA (OLE) www.quantum.physik.uni-mainz.de Johannes Gutenberg-Universität, Mainz

  2. Fermions in a 3D Lattice with repulsive interactions Quantum Phase Diffusion and Bose-Fermi Mixtures

  3. Our starting point: Ultracold Quantum Gases Parameters:Densities: 1015 cm-3Temperatures: nanoKelvinNumber of Atoms: about 106 ground states at T=0 Bose-Einstein condensate e.g. 87Rb atoms Degenerate Fermi gas e.g. 40K atoms

  4. Optical Lattice in 3D 3D lattice: array ofquantum dots

  5. Hubbard Hamiltonian Restriction to single (lowest) band and expansion in localized wannier functions yields: Bose- Hubbard Hamiltonian Tunneling matrix element: Onsite interaction matrix element:

  6. Fermionswith repulsive interactions U. Schneider, L. Hackermüller, S. Will, Th. Best, I.Bloch & A. Rosch, Th. Costi, D. Rasch, R. Helmes (Science, 322, 1520 (2008))

  7. Strongly Interacting Fermions in Optical Lattices • Phases predicted at half filling for strong interactions U/12J > 1: maximal entropy: S/N = kB 2 ln(2) Related experimental work at ETHZ (T. Esslinger) e.g. M. Köhl et al., PRL 94, 080403 (2005), R. Jördens et al. Nature 455, 204 (2008)

  8. Hubbard Model and High-Tc Can we help identifying the phase diagram of the Hubbard model? W. Hofstetter, J.I. Cirac, P. Zoller, E. Demler, M.D. Lukin, PRL 89, 220407 (2002), P. A. Lee, N. Nagosa, X. G. Wen, Rev. Mod. Phys. 78 , 17 (2008)

  9. Experimental Setup: Fermions in the Optical Lattice • Crossed Dipole Trap 1030nm (elliptical beams) Spin mixture of K atoms in F=9/2, mF=-9/2 and F=9/2, mF=-7/2: • Blue Detuned Lattice Beams 738nm (160 µm waist) T=0.06 to 0.13 TF with about 3 x 105 atoms!

  10. Independent control of Lattice Depth and Dipole Trap Depth Compression of the Quantum Gas Total Potential for Atoms: Optical Latticecombined withDipole Trap! + Compression Range:

  11. Hubbard Hamiltonian: All Parameters Tunable! 40K Feshbach resonance: + (JILA parametrization)

  12. Experimental Observables: • Global Observable: Compressibility 2R For example: in-situ cloud size with phase-constrast imaging • Local Observable: For example: pair fraction with Feshbach ramp orcentral occupation (see L. De Leo et al., 2008,alternative method: see Zürich experiment R. Jördens et al., Nature 455, 204 (2008)) U. Schneider, L. Hackermüller, S. Will, Th. Best, I.Bloch & A. Rosch, Th. Costi, D. Rasch, R. Helmes (Science, 322, 1520 (2008))

  13. Quantum Phases of Repulsive Fermions in Trap compressible! incompressible! incompressible!

  14. Comparison with Theory (I) Dynamical Mean Field Theory (DMFT) Metzner, Vollhardt, Georges, Kotliar e.g. A. Georges et al. Rev. Mod. Phys.68, 13 (1996) Real Space Adaptation (Inhomogeneous Systems) Achim Rosch, Theo Costi (here LDA + DMFT) see also: L. De Leo et al. PRL, 101, 210403 (2008) and work by W. Hofstetter Calculations at Forschungszentrum Jülich: JUGENE, IBM Blue-Gene Supercomputer # 1 in Germany # 6 on TOP 500 list worldwide First test bed for DMFT in 3D!

  15. Measuring the Cloud Size…

  16. Measuring the Compressibility (I) U. Schneider, et al. (Science, 322, 1520 (2008)) Theory: R. W. Helmes et al. (PRL 100, 056403(2008))

  17. Measuring the Compressibility (II)

  18. Pair Fraction versus Compression

  19. Entropy Distibution in the Trap Entropy of non-interacting gas in harmonic trap T/TF = 0.15 S/N > kB 2 ln(2) While entropy of MI is only S/N = kB ln(2) ! U/12J = 1.5

  20. Summary: Pair Fraction & Compression Measurements In-situ cloud size / Compression Measurements: • Very good quantitative agreement with ab-initio DMFT forweakandstrong compressions! • Direct measurement of the (in-)compressibility of the many-body system. • Deviations beyond U/6J = 4 in low compression regime Pair fraction measurements: • Good agreement with ab-initio DMFT theory (T approx. 0.15 TF) • But note: Melted MI and strongly interacting metallic phases can also show suppression of pairs!

  21. Multi-Orbital Quantum Phase Diffusion Sebastian Will, Thorsten Best, Simon Braun, Ulrich Schneider, KC Fong, Lucia Hackermüller, Dirk-Sören Lühmann, Immanuel Bloch

  22. From BEC to a Superfluid in an Optical Lattice… BEC in a harmonic trap… …plus a weak lattice Onsite picture: Non-interacting, homogeneous case: Coherent State  Poisson distribution

  23. Dynamics of a coherent state: In the limit of zero tunneling (J = 0) evolution is determined by: time-evolution of coherent state The matterwave field on a lattice site… experimentally observable as Visibility

  24. Phase Diffusion Dynamics: Collapse and Revival • Matterwave field collapses and revives after multiple times of h/U • Collapse time depends on the variance of the atom number distribution Theory: Yurke & Stoler, 1986, F. Sols 1994; Wright et al. 1997; Imamoglu, Lewenstein & You et al. 1997, Castin & Dalibard 1997, E. Altman & A. Auerbach 2002, Exp: M. Greiner et al. 2002, G.-B. Jo et al. 2006, J. Sebby-Strabley et al. 2007, A. Widera et al., 2007, M. Oberthaler et al. 2008

  25. Dynamical Evolution of the Interference Pattern t=50µs t=150µs t=200µs t=300µs t=400µs t=450µs t=600µs Dynamics after potential jump from 8Erec to 22Erec!

  26. Collapse & Revival under Optimal Harmonic Confinement • Up to 70 revivals can be detected! • And: Multiple frequency components!

  27. n = 2 n = 3 n = 4 U(2) U(3) U(4) Why Multiple Frequencies? Here U is assumed to be constant, independent of filling… Breakdown of the single-band approximation! Admixture of higher-band orbitals! for a differential measurement, see also: G. Campbell et. al., Science (2006)

  28. Fourier Spectrum c2 · c32 · c4 E(2) + E(4) – 2E(3) c1 · c22 · c3 2 E(2) - E(3) c0 · c12 · c2 E(2) Strong signal of small contributions due to heterodyning effect!

  29. Comparison with Exact Diagonalization of order 2∙U of order U Theory: D.-S. Lühmann, Hamburg University

  30. Atom distribution along the SF to MI transition:

  31. BOSE-FERMI Mixtures in the Optical Lattice Lattice site with 1 Fermion Effective Onsite Potential for Boson at Ubf < 0: Theory: D.-S. Lühmann et al., PRL 2008 R. Lutchyn, S. Tewari, S. Das Sarma, arxiv:0812.0815v2 nboson= 1 nboson= 2

  32. Self-Trapping of 87Rb due to 40K Increasing Boson Filling due to Bose-Fermi Attraction! Increasing Boson Repulsion due to Self-Trapping of Fermions!

  33. Shift of SF-MI Transition in Bose-Fermi Mixtures Experiment: Th. Best, S. Will et al., arXiv:0807.4504 (in press) S. Ospelkaus et al., PRL 2006, K. Günter et al. PRL 2006, J. Catani et al. PRA 2008 Theory: D.-S. Lühmann et al., PRL 2008

  34. Conclusion Global Compressibility Measurements on Repulsively Interacting Fermi Gases in a 3D Optical Lattice • Evidence for Incompressible Mott Core • Good Agreement with ab-initioDMFT calculations Quantum Phase Diffusion as a Probe in Strongly Interacting Quantum Gas Mixtures • Quantum Phase Diffusion with Fock State Resolution • Renormalized Hubbard Parameters • Self-Trapping in Bose-Fermi Mixtures (Multi-Band Physics) THANK YOU!

  35. Useful Variables: Interactions versus Kinetic Energy Confinement versus Kinetic Energy, where characteristic trap energy = Fermi energy at T=0, J=0 and no interaction trap aspect ratio: Initial Temperature (Entropy) Doubly occupied sites, compressibility, , …

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