David Pekker, Rajdeep Sensarma, Ehud Altman, Takuya Kitagawa, Susanne Pielawa, Adilet Imambekov,

1. Nonequilibrium dynamics of ultracold atoms in optical lattices.Lattice modulation experiments and more David Pekker, Rajdeep Sensarma, Ehud Altman, Takuya Kitagawa, Susanne Pielawa, Adilet Imambekov, Mikhail Lukin, Eugene Demler Harvard University Collaboration with experimental groups of I. Bloch, T. Esslinger, J. Schmiedmayer $$ NSF, AFOSR, MURI, DARPA,

2. Outline Lattice modulation experiments Probe of antiferromagnetic order Fermions in optical lattice. Decay of repulsively bound pairs. Nonequilibrium dynamics of Hubbard model Ramsey interferometry and many-body decoherence. Quantum noise as a probe of dynamics

3. Lattice modulation experiments with fermions in optical lattice. Probing the Mott state of fermions Sensarma, Pekker, Lukin, Demler, arXiv:0902.2586 Related theory work: Kollath et al., PRA 74:416049R (2006) Huber, Ruegg, arXiv:0808:2350

4. 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

5. Fermions in optical lattice Electrons in solids • System size, number of doublons • as a function of entropy, U/t, w0 • Thermodynamic probes • i.e. specific heat • X-Ray and neutron • scattering • Bragg spectroscopy, • TOF noise correlations • Lattice • modulation • experiments • Optical conductivity • STM Probing many-body states Beyond analogies with condensed matter systems: Far from equilibrium quantum many-body dynamics

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

8. Lattice modulation experiments Probing dynamics of the Hubbard model R. Joerdens et al., Nature 455:204 (2008)

9. Mott gap for the charge forms at Antiferromagnetic ordering at “Low” temperature regime Mott state Regime of strong interactionsU>>t. “High” temperature regime All spin configurations are equally likely. Can neglect spin dynamics. Spins are antiferromagnetically ordered or have strong correlations

10. Bosons Fermions Constraint : Singlet Creation Boson Hopping Schwinger bosons and Slave Fermions

11. Schwinger bosons and slave fermions Fermion hopping Propagation of holes and doublons is coupled to spin excitations. Neglect spontaneous doublon production and relaxation. Doublon production due to lattice modulation perturbation Second order perturbation theory. Number of doublons

12. “Low” Temperature d h = + Incoherent part: dispersion Schwinger bosons Bose condensed Propagation of holes and doublons strongly affected by interaction with spin waves Assume independent propagation of hole and doublon (neglect vertex corrections) Self-consistent Born approximation Schmitt-Rink et al (1988), Kane et al. (1989) Spectral function for hole or doublon Sharp coherent part: dispersion set by J, weight by J/t

13. Propogation of doublons and holes Spectral function: Oscillations reflect shake-off processes of spin waves Comparison of Born approximation and exact diagonalization: Dagotto et al. Hopping creates string of altered spins: bound states

14. “Low” Temperature Rate of doublon production • Sharp absorption edge due to coherent quasiparticles • Broad continuum due to incoherent part • Spin wave shake-off peaks

15. “High” Temperature Atomic limit. Neglect spin dynamics. All spin configurations are equally likely. Aij (t’) replaced by probability of having a singlet Assume independent propagation of doublons and holes. Rate of doublon production Ad(h) is the spectral function of a single doublon (holon)

16. Propogation of doublons and holes Hopping creates string of altered spins Retraceable Path Approximation Brinkmann & Rice, 1970 Consider the paths with no closed loops Spectral Fn. of single hole Doublon Production Rate Experiments

17. Lattice modulation experiments. Sum rule Ad(h) is the spectral function of a single doublon (holon) Sum Rule : Experiments: Most likely reason for sum rule violation: nonlinearity The total weight does not scale quadratically with t

18. Fermions in optical lattice.Decay of repulsively bound pairs Experiment: ETH Zurich, Esslinger et al., Theory: Sensarma, Pekker, Altman, Demler

19. Fermions in optical lattice.Decay of repulsively bound pairs Experiments: T. Esslinger et. al.

20. 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 important for quantum simulations, adiabatic preparation

21. Relaxation of doublon hole pairs in the Mott state Energy U needs to be absorbed by spin excitations • Relaxation requires • creation of ~U2/t2 • spin excitations • Energy carried by spin excitations ~ J =4t2/U Relaxation rate Very slow Relaxation

22. p-h p-h U p-h p-p 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

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

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

25. + gk1 gk2 Fermi’s golden rule Neglect fermion-fermion scattering 2 G = + other spin combinations Particle-hole emission is incoherent: Crossed diagrams unimportant gk = cos kx + cos ky + cos kz

26. Self-consistent diagrammatics Calculate doublon lifetime from Im S Neglect fermion-fermion scattering Emission of particle-hole pairs is incoherent: Crossed diagrams are not important Suppressed by vertex functions

27. Self-consistent diagrammatics Neglect fermion-fermion scattering Comparison of Fermi’s Golden rule and self-consistent diagrams Need to include fermion-fermion scattering

28. 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

29. type present type missing Including fermion-fermion scattering Correcting for missing diagrams 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[n0(w)] is renormalization of the number of diagrams

30. Doublon decay in a compressible state Doublon decay with generation of particle-hole pairs

31. Ramsey interferometry and many-body decoherence Quantum noise as a probe of non-equilibrium dynamics

32. 0.4 low T z middle T 0.2 high T 0 x 0 30 10 20 L [pixels] Interference between fluctuating condensates high T BKT Time of flight low T 2d BKT transition: Hadzibabic et al, Claude et al 1d: Luttinger liquid, Hofferberth et al., 2008

33. Distribution function of interference fringe contrast Hofferberth, Gritsev, et al., Nature Physics 4:489 (2008) Quantum fluctuations dominate: asymetric Gumbel distribution (low temp. T or short length L) Thermal fluctuations dominate: broad Poissonian distribution (high temp. T or long length L) Intermediate regime: double peak structure • First demonstration of quantum fluctuations for 1d ultracold atoms • First measurements of distribution function of quantum • noise in interacting many-body system

34. Can we use quantum noise as a probe of dynamics? Example: Interaction induced collapse of Ramsey fringes A. Widera, V. Gritsev, et al. PRL 100:140401 (2008), T. Kitagawa, S. Pielawa, A. Imambekov, J. Schmiedmayer, E. Demler

35. 1 Working with N atoms improves the precision by . t 0 Ramsey interference Atomic clocks and Ramsey interference:

36. Interaction induced collapse of Ramsey fringes Two component BEC. Single mode approximation. Kitagawa, Ueda, PRA 47:5138 (1993) Ramsey fringe visibility time Experiments in 1d tubes: A. Widera et al. PRL 100:140401 (2008)

37. Spin echo. Time reversal experiments Single mode analysis The Hamiltonian can be reversed by changing a12 Predicts perfect spin echo

38. Spin echo. Time reversal experiments Expts:A. Widera et al., PRL (2008) Experiments done in array of tubes. Strong fluctuations in 1d systems. Single mode approximation does not apply. Need to analyze the full model No revival?

39. Interaction induced collapse of Ramsey fringes.Multimode analysis Low energy effective theory: Luttinger liquid approach Luttinger model Changing the sign of the interaction reverses the interaction part of the Hamiltonian but not the kinetic energy Time dependent harmonic oscillators can be analyzed exactly

40. Interaction induced collapse of Ramsey fringesin one dimensional systems Only q=0 mode shows complete spin echo Finite q modes continue decay The net visibility is a result of competition between q=0 and other modes Decoherence due to many-body dynamics of low dimensional systems Fundamental limit on Ramsey interferometry How to distinquish decoherence due to many-body dynamics?

41. Interaction induced collapse of Ramsey fringes Single mode analysis Kitagawa, Ueda, PRA 47:5138 (1993) Multimode analysis evolution of spin distribution functions T. Kitagawa, S. Pielawa, A. Imambekov, J. Schmiedmayer, E. Demler

42. Summary Lattice modulation experiments as a probe of AF order Fermions in optical lattice. Decay of repulsively bound pairs Ramsey interferometry in 1d. Luttinger liquid approach to many-body decoherence

43. Harvard-MIT Thanks to