Ana Maria Rey

1. Ultra-cold bosonic atoms in optical lattices: An Overview Ana Maria Rey March Meeting Tutorial May 1, 2014

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

4. BEC in dilute ultracold gases 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. A. Einstein, 1925 S. Bose, 1924 W. Ketterle E. Cornell C. Wieman Atoms Using Rb and Na atoms Light

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

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

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

8. Single particle in an Optical lattice Band Structure Solved by Bloch Waves q: Quasi-momentum –k/2≤ |q| ≤ k/2 n: Band Index k=2 p/a Reciprocal lattice vector Recoil Energy: ћ2k2/(2m) Effective mass m* grows with lattice depth

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

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

11. Why we want to have strong interactions? M. Greiner

12. Simulators 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. Quantum Simulations 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 • … Richard Feynman

14. Bose Hubbard Hamiltonian We start with the full many-body Hamiltonianand expand the field operator Y in Wannier states Assuming: Lowest band, Short -range interactions, Nearest neighbor hopping H=-J<i,j>âi† âj + U/2 jâj† â†jâjâj + j(Vj –m)âj† âj External potential Hopping Energy Interaction Energy J w0(x) U V j+1 j D. Jacksh et al, PRL, 81, 3108 (1998)

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

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

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

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

19. Time of flight images 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. so a Superfluid: Interference pattern In the lattice at t=0 After time of flight σ(t)= tħ/(mσo) |G|=

21. Superfluid: Interference pattern

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

23. Optical lattice and parabolic potential no=3 Mott 4 Superfluid no=2 Mott 2 no=1 Mott 0 ultracold.uchicago.edu

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

25. Quantum Magnetism 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 High Tc Superconductivity Ferromagnetic RAM Magnetic Heads

26. Exchange interactions Effective spin-spin interactions can arise due to the interplay between the SPIN-INDEPENDENTforces and EXCHANGE SYMMETRY f2 • Exchange Direct overlap f1 Basic Idea Triplet Energy Singlet

27. Experimental Control of Exchange Interactions M. Anderliniet al. Nature 448, 452 (2007) Spin : |0=|F=1,mF=0  Singlet < Triplet |1=|F=1,mF=-1  w0 Orbitals: Two bands g and e w1

28. Period-two Superlattice 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 Prepare a superposition of singlet and triplet Measured spin exchange: using band-mapping techniques and Stern-Gerlach filtering

30. Experimental Control of Exchange Interactions Spin dynamics

31. Spin order can arise even though the wave function overlap is practically zero. Super-Exchange Interactions Virtual processes Super- Exchange f1 f2 E.g. Two electrons in a hydrogen molecule, MnO Triplet Energy O Mn Singlet P.W. Anderson, Phys. Rev. 79, 350 (1950)

32. Super-exchange in optical lattices 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 lifts the degeneracy: An effective Hamiltonian can be derived using second order perturbation theory via virtual particle hole excitations J J

33. Super-exchange in optical lattices For spin independent parameters - Bosons , + Fermions

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

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

36. Energy levels in symmetric DW: wo»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 |++ a|t |- ħw2 U In the U>>J limit ħw1~ 4J2/U: Super-exchange ħw2~ U |s ħw1 |t + a’|+

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=| A Magnetic field gradient couples | and | • If |DB|« Jexthen| s and | t are the eigenstates • If| DB |» Jex then | ↓↑ and |↑↓ are the eigenstates

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

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

40. Comparisons with B. H. Model ? 2Jex Shadow regions: 2% experimental lattice uncertainty

41. Extended B. H. Model

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