1. Ana Maria Rey March Meeting Tutorial May 1, 2014

2. Brief overview of Bose Einstein condensation in dilute ultracold gases What do we mean by quantum simulations and why are ultra cold gases useful The Bose Hubbard model and the superfluid to Mott insulator quantum phase transition Exploring quantum magnetisms with ultra-cold bosons

3. High temperature T: Thermal velocity v Density d -3 “billiard balls” Ketterle Low temperature T: De Broglie wavelength DB =h/mv~T -1/2 “Wave packets” T=T crit : Bose Einstein Condensation De Broglie wavelength DB =d “Matter wave overlap” T= 0 : Pure Bose Condensate “Giant matter wave ”

4. In 1995 (70 years after Einstein’s prediction) teams in Colorado and Massachusetts achieved BEC in super-cold gas.This feat earned those scientists the 2001 Nobel Prize in physics. S. Bose, 1924 Light A. Einstein, 1925 Atoms E. Cornell W. Ketterle C. Wieman Using Rb and Na atoms

5. One way to achieve 2. is with an optical lattice A BEC opened the possibility of studying quantum phenomena on a macroscopic scale. Ultra cold gases are dilute How can increase interactions in cold atom systems? 1.increase a s : Using Feshbach resonances 2. Increase the effective mass m  m* a s : Scattering Length n: Density a s : Scattering Length n: Density * * Cold gases have almost 100% condensate fraction: allow for mean field description

6. Periodic light shift potentials for atoms created by the interference of multiple laser beams. Two counter-propagating beams Standing wave |e|e h |g|g  a= /2 ~Intensity

7. Perfect Crystals Quantum Simulators Quantum Information Precision Spectroscopy Polar Molecules Scattering Physics e.g. Feshbach resonances Bose Hubbard and Hubbard models Quantum magnetism Many-body dynamics Quantum gates Robust entanglement generation Reduce Decoherence AMO Physics

8. Single particle in an Optical lattice q: Quasi-momentum –k/2≤ |q| ≤ k/2 n: Band Index Solved by Bloch Waves Effective mass m* grows with lattice depth k=2  /a Reciprocal lattice vector Recoil Energy: ћ 2 k 2 /(2m)

9. Single particle in an Optical lattice Wannier Functions localized wave functions: Bloch Functions V=0V=0.5 Er V=4 Er V=20 Er

10. And expand  in lowest band Wannier states Assuming: Lowest band, Nearest neighbor hopping We start with the Schrodinger Equation If V=0 Cosine spectrum Band width = 4 J

11. M. Greiner

12. Idea: Use one physical system to model the behavior of another with nearly identical mathematical description. Important: Establish the connection between the physical properties of the systems

13. Richard Feynman We want to design artificial fully controllable quantum systems and use them to simulate complex quantum, many-body behavior What can we simulate with cold atoms? Bose Hubbard models Quantum phase transitions Fermi Hubbard models Cuprates, high temperature superconductors, Quantum magnetism …

14. We start with the full many-body Hamiltonian and expand the field operator  in Wannier states Assuming: Lowest band, Short -range interactions, Nearest neighbor hopping H= - J  â i † â j External potential Interaction Energy Hopping Energy + U/2  j â j † â † j â j â j +  j ( V j –  â j † â j J j+1 j U w 0 (x) V D. Jacksh et al, PRL, 81, 3108 (1998)

15. M.P.A. Fisher et al., PRB40:546 (1989) Superfluid phase Mott insulator phase Weak interactions Strong interactions 4 2 0 Mottn=1 n=2 n=3 Superfluid Mott n=1

16. Superfluid – Mott Insulator Superfluid Mott Insulator Quantum phase transition: Competition between kinetic and interaction energy Shallow potential: U<<J Deep potential: U>>J Weakly interacting gas Strongly interacting gas Superfluid Mott insulator

17. Superfluid Mott Insulator Poissonian Statistics Condensate order parameter Off diagonal long Range Order Atom number Statistics No condensate order parameter Short Range correlations Gapless excitations Energy gap ~ U

18. Step 1: Use the decoupling approximation Step 2: Replace it in the Hamiltonian z: # of nearest neighbor sites Step 3: Compute the energy using  as a perturbation parameter and minimize respect to  Mott: E (2) >0 Energy  SF: E (2) < 0 Energy  E (2) = 0 Critical point Van Oosten et al, PRA 63, 053601 (2001)

19. t=0 Turn off trapping potentials Imaging the expanding atom cloud gives important information about the properties of the cloud at t=0: Spatial distribution -> Momentum distribution after time of flight at t=0

20. In the lattice at t=0 After time of flight σ(t)= tħ/(mσ o ) a  |G|=

22. Lattice depth : Laser Intensity Superfluid Mott insulator Quantum Phase transition Markus Greiner et al. Nature 415, (2002); shallowdeepshallow The loss of the interference pattern demonstrates the loss of quantum phase coherence.

23. Optical lattice and parabolic potential 4 2 0 Mottn o =1 n o =2 n o =3 Superfluid Mott ultracold.uchicago.edu

24. Observing the Shell structure Spatially selective microwave transitions and spin changing collisions S. Foelling et al., PRL 97:060403 (2006) G. Campbell et al, Science 313,649 2006 S. Waseems et al Science, 2010 J. Sherson et al : Nature 467, 68 (2010). Also N. Gemelke et al Nature 460, 995 (2009)

25. Why are some materials ferro or anti- ferromagnetic A fundamental question is whether spin-independent interactions e.g. Coulomb fources, can be the origin of the magnetic ordering observed in some materials. Study role of many-body interactions in quantum systems: Non-interacting electron systems universally exhibit paramagnetism Useful applications Ferromagnetic RAMMagnetic Heads High Tc Superconductivity

26. Exchange interactions Basic Idea Singlet Triplet Effective spin-spin interactions can arise due to the interplay between the SPIN-INDEPENDENT forces and EXCHANGE SYMMETRY Energy Exchange Direct overlap  

27. Experimental Control of Exchange Interactions M. Anderlini et al. Nature 448, 452 (2007) Spin : |0  =|F=1,m F =0  |1  =|F=1,m F =-1  Singlet < Triplet Orbitals: Two bands g and e w0w0 w1w1

28. Superimpose two lattices: one with twice the periodicity of the other Adjustable bias and barrier depth by changing laser intensity and phase

29. Experimental Control of Exchange Interactions Measured spin exchange: using band-mapping techniques and Stern- Gerlach filtering Prepare a superposition of singlet and triplet

30. Spin dynamics Experimental Control of Exchange Interactions

31. Super-Exchange Interactions Spin order can arise even though the wave function overlap is practically zero. Super- Exchange Virtual processes E.g. Two electrons in a hydrogen molecule, MnO Singlet Triplet Energy   P.W. Anderson, Phys. Rev. 79, 350 (1950) Mn O

32. J lifts the degeneracy: An effective Hamiltonian can be derived using second order perturbation theory via virtual particle hole excitations Consider a double well with two atoms At zero order in J, the ground state is Mot insulator with one atom per site and all spin configurations are degenerated JJ Super-exchange in optical lattices JJ

33. For spin independent parameters - Bosons, + Fermions

34. Reversing the sign of super-exchange Add a bias: 2  >U implies J ex <0 S. Trotzky et. al, Science, 319,295(2008)

35. Two bosons in a Double Well with S z =0 2 singly occupied configurations: 2 doubly occupied configurations: (2,0)|t , (0,2)| t  (1,1)| s , (1,1)| t  (0,2)(2,0) (1,1) Singlet Triplet Only 4 states: (0,2)| S   Vibrational spacing  o >>U,J

36. Energy levels in symmetric DW:   » U Good basis:  |s , |-  are not coupled by J. They have E=0,U for any J.  |t , |+  are coupled by J: Form a 2 level system In the U>>J limit ħ  1 ~ 4J 2 /U: Super-exchange ħ  2 ~ U ħħ ħħ |-|- | t  +  ’|+  U |+  +  | t  |s|s

37. Magnetic field gradient In the limit U>>J, only the singly occupied states are populated and they form a two level system: |s  = |   and |t  =|   If |  B | « J ex then | s  and | t  are the eigenstates If |  B | » J ex then | ↓↑  and | ↑↓  are the eigenstates A Magnetic field gradient couples |   and |  

38. Experimental Observation t M Measure spin imbalance Prepare| ↑↓  |  B|>0 Turn of  B Evolve N z : # atoms | ↑↓  - # atoms | ↓↑  S. Trotzky et. al, Science, 319,295(2008) |s  |↑↓  |t z  In the limit J <<U, Simple Rabi oscillations

39. Measuring Super-exchange V=6Er V=11 Er V=17 Er Two frequencies Almost one frequency

40. Comparisons with B. H. Model 2J ex Shadow regions: 2% experimental lattice uncertainty

41. Extended B. H. Model

42. Real Materials: Complicated Disorder Condensed matter models: Difficult to calculate ? Cold atoms in optical lattice: Clean realization of CM models Direct experimental test of condensed matter models: Great success and a lot of new challenges