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##### Anna Sanpera. University Hannover Cozumel 2004

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**Bose-Fermi mixtures in random optical lattices:From**Fermi glass to fermionic spin glassand quantum percolation Anna Sanpera. University Hannover Cozumel 2004**Theoretical Quantum Optics**at the University of Hannover • Cold atoms and cold gases: • Weakly interacting Bose and Fermi gases (solitons, vortices, phase fluctuations, atom optics, quantum engineering) • Dipolar Bose and Fermi gases • Collective cooling, CW atom laser, quantum master equation • Strongly correlated systems in AMO physics • Quantum Information: • Quantification and classification of entanglement • Quantum cryptography and communications • Implementations in quantum optics V. Ahufinger, B. Damski, L. Sanchez-Palencia, A. Kantian, A. Sanpera M. Lewenstein**Atomic physics meets condensed matter physicsor**Atomic physics beats condensed matter physics ????**Outline**OUTlLINE 1 Bose-Fermi (BF) mixtures in optical lattices 2 Disorder and frustration in BF mixtures**Mott insulator**Superfluid Bose gas in an optical latticeIdea: D. Jaksch, C. Bruder, J.I. Cirac, C.W. Gardiner and P. Zoller By courtesy of M. Greiner, I. Bloch, O. Mandel, and T. Hänsch**Bose-Hubbard model**On site interactions Tunneling Before talking about disorder, let us define order: an optical lattice with atoms loaded on it. First band**Some facts about Fermi-Bose Mixtures**Fermi-Bose mixtures in optical lattices: • Fermions and bosons on equal footing in alattice: • Atomic physics “beats” condensed matter physics!!! • Novel quantum phases and novel kinds of pairing: Fermion-boson pairing!!! • Novel possibilites of control of the system Some of the people working on the subject (theory): • A. Albus, J. Eisert (Potsdam), F. Illuminati (Salerno), H.P. Büchler (Innsbruck), G. Blatter (ETH), A.B. Kuklov, B.V. Svistunov (Amherst/Kurchatov), M.Yu. Kagan, D.V. Efremov, A.V. Klaptsov (Kapitza), M.-A. Cazalilla (Donostia), A.F. Ho (Birmingham)**Phase I – Mott(2)+ Fermi sea**Phase II– Fermion-hole pairing Phase III– Fermion-2 holes pairing Quantum phases of the Bose-Fermi Hubbard model • Description: • i) Bose-Fermi Hubbard model • Phase I – Mott (n) plus Fermi gas of fermions with NN interactions • Phase II – Interacting composite fermions (fermion + bosonic hole) • Phase III – Interacting composite fermions (fermion + 2 bosonic holes) Lewenstein et al. PRL (2003), Ferhman et al. Optics Express (2004)**Lattice gases: Bose-Fermi mixtures**IIAD IIAS IIRF • Low tunneling J<< Ubb,Ubf • Effective Fermi-Hubbard Hamiltonian Ubf/Ubb Composite interactions Different quantum phases 2 Attractive: Superfluid fermionic Domains Repulsive: Fermi liquid Density modulations 1 IRD 0 IRF IIRD IRD IIRF -1 IIAS IIRF . . IIRD -2 0 1 mb/Ubb****|f0f1b |2 = 0 No composites |f1f0b |2 |f0f0b |2 Composites |f0f1b |2 Fermi liquid Fermionic domain No composites |f1f0b |2 |f1f1b |2 Ubb=1 Jb=Jf=0.02 b=10-7 f=5x10-7 Nf=40 Nb=60**2. DISORDER AND FRUSTRATIONIN ULTRACOLD ATOMICGASES**B. Damski, et al. Phys. Rev. Lett. 91, 080403 (2003) A. Sanpera, et al.. cond-mat/0402375, Phys. Rev. Lett. 93, 040401 (2004) V. Ahufinger et al. (a review of AMO disordered systems – work in progress)**What are spin glasses?**Spin glasses are disordered systems with competing ferromagnetic ( )and antiferromagnetic ( ) interactions, which generates FRUSTRATION.**?**Frustration: if we only have 2 possible spin orientations and the interactions are random, no spin configuration can simultaneously satisfy all couplings. Ferromagentic (J=1) Antiferromagnetic (J=-1)**Spin Glasses**70‘s Edwards & Anderson: Essential physics of spin glasses lay not in the details of their microscopic interaction but rather in the competetion between quenched ferromagentic and antiferromagentic interactions. It is enough to study: i- site of a d-dimensional lattice Ising classical spins Independent Gaussian random variables with zero mean and variance 1. External magnetic Field h**Spin glasses= quenched disorder + frustration.**Mean Field Theory: Sherrington-Kirkpatrick model 75 KT/J PARAMAG. FERROMAG 1 SPIN GLASS J 0 1**Mean Field (infinite range)Sherrington-Kirkpatrick model:**Use replicas: ……. n-replicas Solution: Parisi 80‘s: Breaking the replica symmetry. The spin glass phase is characterized by an infinite number of pure states organized in an ultrametric structure, and a phase transition occurring in a magnetic field. Order parameter= „overlap between replicas“**REAL SYSTEMS (short range interactions)**• How many pure states are in a spin glass at low temperature? • Which is the nature and complexity of the glassy phase? • Does exist a transition in a non zero magnetic external field? Despite 30 years of effort on the subject No consensus has been reached for real systems ! Alternative: Droplet model: phase glass consist in two pure states related by global inversion of the spins and no phase transition occurring in a magnetic field.**From Bose-Fermi mixtures in optical lattices to spin**glasses: From disordered Bose-Fermi Hubbard Hamiltonian: to spin glass Hamiltonian**Low tunneling**• Low Temperature • DISORDER (chemical potential varies site to site) • EffectiveFermi-Hubbard Hamiltonian (second order perturbation theory) INTERACTIONS between COMPOSITES HOPPING of COMPOSITES**J=0, no tunneling of fermions or bosons**Depending on the disorder 2 types of lattice sites: A-sites n=1,m=1 B-sites n=0,m=1**Tunneling**On site interactions Disorder: Speckle radiation or supperlattices or… Damski et al. PRL 2003**How to make a quantum SG with atomiclattice gas?**1. Use spinless fermions or bosons with strong repulsive interactions: • There can be Ni= 1 or = 0 atoms at a site!! • We can define Ising spins si = 2 Mi– 1. • What we need are: • RANDOM NEXTNEIGHBOUR INTERACTIONS, HQSG = 1/4 Kijsisj+ quantum tunneling terms + ... Composites**Effective n.n. coulings in FB mixture in a random**optical lattice Here ij = i - j SPIN GLASS !**Physics of Fermi-Bose mixtures in random optical lattices**Regime of small disorder (weak randomness of on-site potential) • With weak repulsive interactions we deal essentially with a Fermi glass (i.e. an analog of Fermi liquid, but with Anderson localized quasi-particle states) • With attractive interactions we deal with the interplay of superfluidity and disorder • Both situations might occur simultaneously with quantum site percolation (some sites might be „blocked“) Regime of strong disorder • Using the superlattices method we may make local potential to fluctuate on n.n. sites strongly, being zero on the mean. This leads to quantum fermionic spin glass • There is a possibility of novel metallic phases at the interplay between disorder, hopping and n.n. interactions**SUMMARY OF Bose-Fermi Mixtures**Fermionic spin glasses in optical lattices: • Spin glasses (SG) are spin systems with random (disoredered) interactions: equally probable to be ferro- or antiferromagnetic. The spin behaviour is dominated by frustration!!! The nature of ordering in SG poses one the most outstanding open questions of classical (sic!) and quantum statistical mechanics. • COLD ATOMIC BOSE-FERMI (BOSE-BOSE) MIXTURES in optical lattices with disorder can be used to study in “vivo” the nature of short range spin glasses. (real replicas) - Many novel phases related to composite fermions in disorder lattices are expected! NEW & RICH PHYSICS**y**y nFj nFj x x Transition from Fermi liquid to Fermi glass “in vivo” Here composite fermions = a fermion + a bosonic hole**Question: Can AMO physics help?**YES! • Can cold atoms or ions be used to model complex systems? • Bose gas in a disordered optical lattice: From Anderson to Bose glass • Fermi-Bose mixtures in random lattices: From Fermi glass to • fermionic spin glass and quantum percolation • Trapped ions with engineered interactions: Spin chains with • long range interactions and neural networks • Atomic lattice gases in non-abelian gauge fields: From Hofstadter • butterfly to Osterloh cheese 2. Can cold atoms and ions be used as quantum simulators of complex systems? YES! 3. Can cold atoms and ions be used for quantum information processing in complex systems? YES?