1. Nonequilibrium dynamics of ultracold fermions Theoretical work: Mehrtash Babadi, David Pekker, Rajdeep Sensarma, Ehud Altman, Eugene Demler $$ NSF, MURI, DARPA, AFOSR Experiments: T. Esslinger‘s group at ETH W. Ketterle’s group at MIT Harvard-MIT

2. Antiferromagnetic and superconducting Tc of the order of 100 K Atoms in optical lattice Antiferromagnetism and pairing at sub-micro Kelvin temperatures Same microscopic model

3. 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

4. Outline Relaxation of doublons in Hubbard model Expts: Strohmaier et al., arXiv:0905.2963 Quench dynamics across Stoner instability Expts: Ketterle et al.,

5. Fermions in optical lattice. Decay of repulsively bound pairs Ref: N. Strohmaier et al., arXiv:0905.2963 Experiment: T. Esslinger’s group at ETH Theory: Sensarma, Pekker, Altman, Demler

6. Signatures of incompressible Mott state of fermions in optical lattice Suppression of double occupancies Jordens et al., Nature 455:204 (2008) Compressibility measurements Schneider et al., Science 5:1520 (2008)

7. Lattice modulation experiments Probing dynamics of the Hubbard model Measure number of doubly occupied sites Main effect of shaking: modulation of tunneling Modulate lattice potential Doubly occupied sites created when frequency w matches Hubbard U

8. Lattice modulation experiments R. Joerdens et al., Nature 455:204 (2008)

9. Fermions in optical lattice. Decay of repulsively bound pairs Experiments: N. Strohmaier et. al.

10. Relaxation of repulsively bound pairs in the Fermionic Hubbard model U >> t For a repulsive bound pair to decay, energy U needs to be absorbed by other degrees of freedom in the system Relaxation timescale is determined by many-body dynamics of strongly correlated system of interacting fermions

11.  Energy carried by spin excitations ~ J =4t 2 /U  Relaxation requires creation of ~U 2 /t 2 spin excitations Relaxation of doublon hole pairs in the Mott state Relaxation rate Very slow Relaxation Energy U needs to be absorbed by spin excitations

12. Doublon decay in a compressible state Excess energy U is converted to kinetic energy of single atoms Compressible state: Fermi liquid description Doublon can decay into a pair of quasiparticles with many particle-hole pairs U p-pp-p p-hp-h p-hp-h p-hp-h

13. Doublon decay in a compressible state To calculate the rate: consider processes which maximize the number of particle-hole excitations Perturbation theory to order n=U/6t Decay probability

14. Doublon decay in a compressible state Doublon decay Doublon-fermion scattering Doublon Single fermion hopping Fermion-fermion scattering due to projected hopping

15. Fermi’s golden rule Neglect fermion-fermion scattering + other spin combinations + 2 G =  k1  k2  k = cos k x + cos k y + cos k z Particle-hole emission is incoherent: Crossed diagrams unimportant

16. Comparison of Fermi’s Golden rule and self-consistent diagrams Need to include fermion-fermion scattering Self-consistent diagrammatics Calculate doublon lifetime from Im S Neglect fermion-fermion scattering

17. Self-consistent diagrammatics Including fermion-fermion scattering Treat emission of particle-hole pairs as incoherent include only non-crossing diagrams Analyzing particle-hole emission as coherent process requires adding decay amplitudes and then calculating net decay rate. Additional diagrams in self-energy need to be included No vertex functions to justify neglecting crossed diagrams

18. Correcting for missing diagrams type presenttype missing Including fermion-fermion scattering Assume all amplitudes for particle-hole pair production are the same. Assume constructive interference between all decay amplitudes For a given energy diagrams of a certain order dominate. Lower order diagrams do not have enough p-h pairs to absorb energy Higher order diagrams suppressed by additional powers of (t/U) 2 For each energy count number of missing crossed diagrams R[n 0 ( w )] is renormalization of the number of diagrams

19. Doublon decay in a compressible state Comparison of approximationsChanges of density around 30%

20. Why understanding doublon decay rate is important Prototype of decay processes with emission of many interacting particles. Example: resonance in nuclear physics: (i.e. delta-isobar) Analogy to pump and probe experiments in condensed matter systems Response functions of strongly correlated systems at high frequencies. Important for numerical analysis. Important for adiabatic preparation of strongly correlated systems in optical lattices

21. Quench dynamics across Stoner instability

22. Stoner model of ferromagnetism Spontaneous spin polarization decreases interaction energy but increases kinetic energy of electrons Mean-field criterion U N(0) = 1 U – interaction strength N(0) – density of states at Fermi level Does Stoner ferromagnetism really exist ? Counterexample: 1d systems. Lieb-Mattis proof of singlet ground state Kanamori’s argument: renormalization of U then

23. Magnetic domains could not be resolved. Why?

24. Stoner Instability New feature of cold atoms systems: non-adiabatic crossing of U c Quench dynamics: change U instantaneously. Fermi liquid state for U>Uc. Unstable collective modes

25. Outline Relaxation of doublons in Hubbard model Expts: Strohmaier et al., arXiv:0905.2963 Quench dynamics across Stoner instability Expts: Ketterle et al.,

26. Quench dynamics across Stoner instability

27. Stoner model of ferromagnetism Spontaneous spin polarization decreases interaction energy but increases kinetic energy of electrons Mean-field criterion U N(0) = 1 U – interaction strength N(0) – density of states at Fermi level Does Stoner ferromagnetism really exist ? Counterexample: 1d systems. Lieb-Mattis proof of singlet ground state Kanamori’s argument: renormalization of U then

28. Magnetic domains could not be resolved. Why?

29. Stoner Instability New feature of cold atoms systems: non-adiabatic crossing of U c Quench dynamics: change U instantaneously. Fermi liquid state for U>Uc. Unstable collective modes

30. Quench dynamics across Stoner instability Find collective modes Unstable modes determine characteristic lengthscale of magnetic domains For U<U c damped collective modes w q = w ’- i w ” For U>U c unstable collective modes w q = + i w ”

31. Quench dynamics across Stoner instability For MIT experiments domain sizes of the order of a few l F D=3D=2 When

32. Quench dynamics across Stoner instability Open questions: Interaction between modes. Ordering kinetics. Scaling? Classical ordering kinetics: Brey, Adv. Phys. 51:481 Stoner Instability in the Hubbard model?

33. Relaxation of doublons in Hubbard model Expts: Strohmaier et al., arXiv:0905.2963 Quench dynamics across Stoner instability Expts: Ketterle et al., Conclusions Experiments with ultracold atoms open interesting questions of nonequilibrium many-body dynamics