Eugene Demler Harvard University

1. Strongly correlated many-body systems: from electronic materials to ultracold atomsto photons Eugene DemlerHarvard University

2. “Conventional” solid state materials Bloch theorem for non-interacting electrons in a periodic potential

3. EF Consequences of the Bloch theorem B VH d Metals I Insulators and Semiconductors EF First semiconductor transistor

4. “Conventional” solid state materials Electron-phonon and electron-electron interactions are irrelevant at low temperatures ky kx Landau Fermi liquid theory: when frequency and temperature are smaller than EF electron systems are equivalent to systems of non-interacting fermions kF Ag Ag Ag

5. UCu3.5Pd1.5 CeCu2Si2 Strongly correlated electron systems Quantum Hall systems kinetic energy suppressed by magnetic field Heavy fermion materials many puzzling non-Fermi liquid properties High temperature superconductors Unusual “normal” state, Controversial mechanism of superconductivity, Several competing orders

6. What is the connection between strongly correlated electron systems and ultracold atoms?

7. Bose-Einstein condensation of weakly interacting atoms Density 1013 cm-1 Typical distance between atoms 300 nm Typical scattering length 10 nm Scattering length is much smaller than characteristic interparticle distances. Interactions are weak

8. Feshbach resonances • Rotating systems • Low dimensional systems • Atoms in optical lattices • Systems with long range dipolar interactions New Era in Cold Atoms Research Focus on Systems with Strong Interactions

9. Feshbach resonance and fermionic condensates Greiner et al., Nature (2003); Ketterle et al., (2003) Ketterle et al., Nature 435, 1047-1051 (2005)

10. One dimensional systems 1D confinement in optical potential Weiss et al., Science (05); Bloch et al., Esslinger et al., One dimensional systems in microtraps. Thywissen et al., Eur. J. Phys. D. (99); Hansel et al., Nature (01); Folman et al., Adv. At. Mol. Opt. Phys. (02) Strongly interacting regime can be reached for low densities

11. Atoms in optical lattices Theory: Jaksch et al. PRL (1998) Experiment: Kasevich et al., Science (2001); Greiner et al., Nature (2001); Phillips et al., J. Physics B (2002) Esslinger et al., PRL (2004); and many more …

12. Atoms in optical lattices Electrons in Solids Strongly correlated systems Simple metals Perturbation theory in Coulomb interaction applies. Band structure methods wotk Strongly Correlated Electron Systems Band structure methods fail. Novel phenomena in strongly correlated electron systems: Quantum magnetism, phase separation, unconventional superconductivity, high temperature superconductivity, fractionalization of electrons …

13. By studying strongly interacting systems of cold atoms we expect to get insights into the mysterious properties of novel quantum materials: Quantum Simulators BUT Strongly interacting systems of ultracold atoms and photons: are NOT direct analogues of condensed matter systems These are independent physical systems with their own “personalities”, physical properties, and theoretical challenges Strongly correlated systems of ultracold atoms should also be useful for applications in quantum information, high precision spectroscopy, metrology

14. New Phenomena in quantum many-body systems of ultracold atoms Long intrinsic time scales- Interaction energy and bandwidth ~ 1kHz- System parameters can be changed over this time scale Decoupling from external environment- Long coherence times Can achieve highly non equilibrium quantum many-body states New detection methods Interference, higher order correlations

15. Strongly correlated many-body systems of photons

16. Linear geometrical optics

17. Strongly correlated systems of photons Strongly interacting polaritons in coupled arrays of cavities M. Hartmann et al., Nature Physics (2006) Strong optical nonlinearities in nanoscale surface plasmons Akimov et al., Nature (2007) Crystallization (fermionization) of photons in one dimensional optical waveguides D. Chang et al., Nature Physics (2008)

18. Outline of these lectures • Introduction. Systems of ultracold atoms. • Bogoliubov theory. Spinor condensates. • Cold atoms in optical lattices. • Bose Hubbard model and extensions • Bose mixtures in optical lattices Quantum magnetism of ultracold atoms. Current experiments: observation of superexchange • Fermions in optical lattices Magnetism and pairing in systems with repulsive interactions. Current experiments: Mott state • Detection of many-body phases using noise correlations • Experiments with low dimensional systems Interference experiments. Analysis of high order correlations • Non-equilibrium dynamics • Emphasis of these lectures: • Detection of many-body phases • Dynamics

19. Ultracold atoms

20. Ultracold atoms Most common bosonic atoms: alkali 87Rb and 23Na Most common fermionic atoms: alkali 40K and 6Li Other systems: BEC of133Cs (e.g. Grimm et al.) BEC of52Cr (Pfau et al.) BEC of 84Sr (e.g. Grimm et al.),87Srand88Sr(e.g. Ye et al.) BEC of 168Yb, 170Yb, 172Yb, 174Yb, 176Yb Quantum degenerate fermions 171Yb 173Yb (Takahashi et al.)

21. Single valence electron in the s-orbital and Nuclear spin Hyperfine coupling mixes nuclear and electron spins Total angular momentum (hyperfine spin) Zero field splitting between and states Magnetic properties of individual alkali atoms For 23Na AHFS = 1.8 GHz and for 87Rb AHFS = 6.8 GHz

22. Magnetic properties of individual alkali atoms Effect of magnetic field comes from electron spin gs=2 and mB=1.4 MHz/G When fields are not too large one can use (assuming field along z) The last term describes quadratic Zeeman effect q=h 390 Hz/G2

23. Magnetic trapping of alkali atoms Magnetic trapping of neutral atoms is due to the Zeeman effect. The energy of an atomic state depends on the magnetic field. In an inhomogeneous field an atom experiences a spatially varying potential. Example: Potential: Magnetic trapping is limited by the requirement that the trapped atoms remain in weak field seeking states. For 23Na and87Rb there are three states

24. - polarizability Optical trapping of alkali atoms Based on AC Stark effect Dipolar moment induced by the electric field Typically optical frequencies. Potential: Far-off-resonant optical trap confines atoms regardless of their hyperfine state

25. Bogoliubov theory of weakly interacting BEC. Collective modes

26. - boson annihilation operator at momentum p, - strength of contact s-wave interaction - volume of the system - chemical potential BEC of spinless bosons. Bogoliubov theory We consider a uniform system first For non-interacting atoms at T=0 all atoms are in k=0 state. Mean field equations Minimizing with respect to N0 we find

27. When N0>>1 we can treat b0 as a c-number and replace by - means that p,-p pairs should be counted only once BEC of spinless bosons. Bogoliubov theory . We now expand around the nointeracting solution for small U0 From the definition of Bose operators Mean-field Hamiltonian n0 = N0/V - density and m = n0 U0

28. Bogoliubov transformation Bosonic commutation relations are preserved when

29. Bogoliubov transformation Mean-field Hamiltonian becomes

30. Cancellation of non-diagonal terms requires To satisfy take Bogoliubov transformation Solution of these equations Mean-field Hamiltonian

31. Define healing length from Long wavelength limit, , sound dispersion Short wavelength limit, , free particles Sound velocity Bogoliubov modes Dispersion of collective modes

32. Probing the dispersion of BEC by off-resonant light scattering Excitation rate out of the ground state |g> Treat optical field as classical Dynamical structure factor For small q wefind For details see cond-mat/0005001

33. PRL 83:2876 (1999)

34. PRL 88:60402 (2002) Ω

36. Rev. Mod. Phys. 77:187 (2005)

38. Gross-Pitaevskii equation

39. Commutation relations This is operator equation. We can take classical limit by assuming that all atoms condense into the same state . Gross-Pitaevskii equation Hamiltonian of interacting bosons Equations of motion The last equation becomes an equation on the wavefunction

40. Gross-Pitaevskii equation Analysis of fluctuations on top of the mean-field GP equations leads to the Bogoliubov modes

41. Two-component mixtures

42. Two-component Bose mixture Consider mean-field (all particles in the condensate) Repulsive interactions. Miscible and immiscible regimes System with a finite density of both species is unstable to phase separation

43. Bose mixture: dynamics of phase separation What happens if we prepare a mixture of two condensates in the immiscible regime? Two component GP equation Assume equal densities Analyze fluctuations

44. Bose mixture: dynamics of phase separation density fluctuation phase fluctuation Define Equations of motion: charge conservation and Josephson relation Equation on collective modes

45. Bose mixture: dynamics of phase separation When we get imaginary frequencies Imaginary frequencies indicate exponential growth of fluctuations, i.e. instability. The most unstable mode sets the length for pattern formation. Note that when . This is required by spin conservation. Here Most unstable mode

46. PRL 82:2228 (1999)

48. Spinor condensates F=1

49. Spinor condensates. F=1 Three component order parameter: mF=-1,0,+1 Contact interaction depends on relative spin orientation When g2>0 interaction is antiferromagnetic. Example 23Na When g2<0 interaction is antiferromagnetic. Example 87Rb

50. - spin operators for F=1 F=1 spinor condensates. Hamiltonian Total Fz is conserved so linear Zeeman term should (usually) be understood as Lagrange multiplier that controls Fz. Quadratic Zeeman effect causes the energy of mF=0 state to be lower than the energy of mF=-1,+1 states. The antiferromagnetic interaction (g2>0) favors the nematic (polar) state (mF=0 and its rotations). The ferromagnetic interaction (g2<0) favors spin polarized state (mF=+1) and its rotations).