1. Bose-Fermi mixtures in random optical lattices: From Fermi glass to fermionic spin glass and quantum percolation Anna Sanpera. University Hannover Cozumel 2004

2. 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

3. Atomic physics meets condensed matter physics or Atomic physics beats condensed matter physics ????

4. Outline OUTlLINE 1 Bose-Fermi (BF) mixtures in optical lattices 2 Disorder and frustration in BF mixtures

5. Bose gas in an optical lattice Idea: 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 Superfluid Mott insulator

6. Before talking about disorder, let us define order: an optical lattice with atoms loaded on it. First band Tunneling On site interactions Bose-Hubbard model

7. Some facts about Fermi-Bose Mixtures  Fermions and bosons on equal footing in a lattice:  Atomic physics “beats” condensed matter physics!!!  Novel quantum phases and novel kinds of pairing: Fermion-boson pairing!!!  Novel possibilites of control of the system  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) Fermi-Bose mixtures in optical lattices: Some of the people working on the subject (theory):

8. Quantum phases of the Bose-Fermi Hubbard model Description: i) Bose-Fermi Hubbard model i)Phase I – Mott (n) plus Fermi gas of fermions with NN interactions ii)Phase II – Interacting composite fermions (fermion + bosonic hole) iii)Phase III – Interacting composite fermions (fermion + 2 bosonic holes) Phase I – Mott(2)+ Fermi seaPhase II– Fermion-hole pairingPhase III– Fermion-2 holes pairing Lewenstein et al. PRL (2003), Ferhman et al. Optics Express (2004)

9. Low tunneling J<< Ubb,Ubf Effective Fermi-Hubbard Hamiltonian Lattice gases: Bose-Fermi mixtures U bf /U bb 0 1 2 -2 II AD II AS II RF I RF I RD II RD II AS II RF II RD II RF 01  b  U bb.... Composite interactions Different quantum phases Attractive: Superfluid fermionic Domains Repulsive: Fermi liquid Density modulations

10. |f 0f0b | 2 |f 0f1b | 2 |f 1f0b | 2 |f 1f1b | 2  = 0 No composites U bb =1 J b =J f =0.02  b =10 -7  f =5x10 -7 N f =40 N b =60 No composites  |f 0f1b | 2 |f 1f0b | 2 Composites Fermi liquidFermionic domain

11. 2. DISORDER AND FRUSTRATION IN ULTRACOLD ATOMIC GASES 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)

12. What are spin glasses? Spin glasses are disordered systems with competing ferromagnetic ( )and antiferromagnetic ( ) interactions, which generates FRUSTRATION.

13. 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) ?

14. 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. h External magnetic Field Spin Glasses

15. Spin glasses= quenched disorder + frustration. Mean Field Theory: Sherrington-Kirkpatrick model 75 PARAMAG. FERROMAG SPIN GLASS KT/J 1 10 J

16. 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“

17. 1.How many pure states are in a spin glass at low temperature? 2.Which is the nature and complexity of the glassy phase? 3.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 ! REAL SYSTEMS (short range interactions) 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.

18. From Bose-Fermi mixtures in optical lattices to spin glasses: From disordered Bose-Fermi Hubbard Hamiltonian : to spin glass Hamiltonian

19. Low tunneling Low Temperature DISORDER (chemical potential varies site to site) Effective Fermi-Hubbard Hamiltonian (second order perturbation theory) HOPPING of COMPOSITES INTERACTIONS between COMPOSITES

20. 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

21. Speckle radiation or supperlattices or… Disorder: Tunneling On site interactions Damski et al. PRL 2003

22. How to make a quantum SG with atomic lattice gas?  There can be N i = 1 or = 0 atoms at a site!!  We can define Ising spins s i = 2 M i – 1.  What we need are:  RANDOM NEXT NEIGHBOUR INTERACTIONS, H QSG = 1/4  K ij s i s j + quantum tunneling terms +... 1. Use spinless fermions or bosons with strong repulsive interactions: Composites

23. Effective n.n. coulings in FB mixture in a random optical lattice Here  ij =  i -  j SPIN GLASS !

24. Physics of Fermi-Bose mixtures in random optical lattices  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“)  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 Regime of small disorder (weak randomness of on-site potential) Regime of strong disorder

25. SUMMARY OF Bose-Fermi Mixtures  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 Fermionic spin glasses in optical lattices:

26. nFjnFj y x nFjnFj y x Transition from Fermi liquid to Fermi glass “in vivo” Here composite fermions = a fermion + a bosonic hole

27. Question: Can AMO physics help? 2. Can cold atoms and ions be used as quantum simulators of complex systems? 3. Can cold atoms and ions be used for quantum information processing in complex systems? 1.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 YES! YES?