Quantum simulator theory This talk: Harvard, Innsbruck-Stuttgart, Michigan, + Stanford experiments Low dimensional systems 1d: spin dynamics of two component Bose mixture Harvard, Mainz collaboration 1d: dynamics of spin chains Harvard, Mainz collaboration (+Weizmann, Munich, Fribourg) 2d: interference of weakly coupled pancakes Harvard, Stanford collaboration Microscopic parameters of low-D systems (Michigan) Dipolar interactions (Harvard, Innsbruck, Stuttgart) Probing fermionic Hubbard model with spin polarization (Harvard)

Experiments with 2D Bose gas z Time of flight Experiments with 2D Bose gas Hadzibabic, Dalibard et al., Nature 441:1118 (2006) Experiments with 1D Bose gas Hofferberth et al. Nature Physics (2008)

Interference of independent 1d condensates S. Hofferberth, I. Lesanovsky, T. Schumm, J. Schmiedmayer, A. Imambekov, V. Gritsev, E. Demler, Nature Physics (2008) Higher order correlation functions probed by noise in interference

Non-equilibrium spin dynamics in one dimensional systems Ramsey interferometry and many-body decoherence Mainz, Harvard collaboration Widera, Trotzky, Cheinet, Foelling, Gerbier, Bloch, Gritsev, Lukin, Demler, PRL (2008) + Kitagawa, Pielawa, Imambekov, Demler, unpublished

Ramsey interference Atomic clocks and Ramsey interference: 1 Ramsey interference is basically Larmor precession for spin ½ particles. If we take a state which is a superposition of spin up and spin down state, the two states will evolve with different energies. So the spin rotates in the plane with the frequency determined by the energy difference. Ramsey precession underlies atomic clocks and is done not with one atoms but with many. Atomic clocks and Ramsey interference: Working with N atoms improves the precision by .

Interaction induced collapse of Ramsey fringes Two component BEC. Single mode approximation Ramsey fringe visibility time Experiments in 1d tubes: A. Widera et al. PRL 100:140401 (2008) What is the role of interaction on Ramsey precession?

Spin echo. Time reversal experiments 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?

Interaction induced collapse of Ramsey fringes in one dimensional systems Low energy effective theory in 1D: Luttinger liquid approach 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 How to distinquish decoherence due to many-body dynamics?

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, et al.

Lattice models Nonequilibrium dynamics in 1d anisotropic Heisenberg spin systems Barmettler, Punk, Altman, Gritsev, Demler, arXiv:0810:4845

Superexchange in Mott state. Spin dynamics in double well systems Jex Mainz, Harvard collaboration (+BU) A.M. Rey et al., PRL (2007) S. Trotzky et al., Science (2008) Experimental measurements of superexchange Jex. Comparison to first principle calculations

Nonequilibrium spin dynamics in 1d. Lattice Spin dynamics in 1D starting from the classical Neel state Coherent time evolution starting with Y(t=0) = Equilibrium phase diagram QLRO D Time, Jt Expected: critical slowdown near quantum critical point at D=1 Observed: fast decay at D=1

Experiment: 1D AF isotropic model prepared in the Neel state: decay of staggered magnetization S. Trotzky et al. (group of I. Bloch)

Quasi 2D condensates: From 2D BKT to 3D Theory: Pekker, Gritsev, Demler (Harvard) B. Clark (UIUC) Experiment: Kasevich et al. (Stanford)

Quasi 2D condensates at Stanford Optical lattice array ~20 disks ~100 87Rb atoms/disk each disk ~60 nm x 4 mm 10 mm kbT/h ~ 1 kHz m/h ~ 200 Hz J/h ~ 5-500 Hz N ~ 100 Interlayer tunneling is a tunable parameter (with lattice depth). 15

Berezinskii–Kosterlitz–Thouless T=TBKT temperature Fisher & Hohenberg, PRB (1988) T=0

Modifications for multiple pancakes 3D Phonons T=TC T=TBKT 3D XY temperature T=2t T=0

Comparison theory and experiment, 12 Er lattice Theory: Classical Monte-Carlo of XY model (also RG analysis) Temperature (nK) 12 ER Experiment RF cut frequency (kHz)

Response vs. Correlations TKT What is typically being measured? Condensed matter response function (e.g. superfluid density) Cold atoms correlations (peak shapes and heights) also possible to do response now

Ulrtacold atoms in low dimensions Luming Duan Realization of Low dimensions: atoms in strong transverse traps Weakly interacting atoms: Projection to the transverse ground state Strongly interacting atoms near Feshbach resonance Simple Projection does not work! Multi-level effects

Description of Strongly interacting atoms in low dimensions Renormalization of atom-atom scattering length (model I) (Olshanni etc., 1D, PRL; Petrov, Shlyapnikov, etc., 2D, PRL) Effective low-D scattering length Not adequate yet near Resonance Reason: Existence of two-body bound-state at any detuning in low-D Effective low-D atomic scattering length does not include this bound state

Effective Hamiltonian for low-D strongly interacting gas Effective interaction between atoms and dressed molecules (model II) (Kestner, Duan, PRA, 06) Atoms in transverse ground level Dressed molecules, accounting for atomic population in excited transverse levels.

Comparison of predictions of model I and model II Comparison of Tomas-Fermi Radius of 2D gas in a weak planar trap Fails to reproduce a shrinking radius at the BEC side BCS side BEC side Zhang, Lin, Duan, PRA 08

Dipolar interactions R. Cherng, E. Demler (Harvard) D.W. Wang (Tsing-Hua Univ) H.P. Buchler (Stuttgart), P. Zoller (Innsbruck)

- + Dipolar interactions in low dimensional systems Attractive interaction head-to-tail Repulsive interaction side-by-side Dipole-Dipole Interactions in 2D pancake Attractive at short distances Repulsive at long distances Roton-maxon spectrum and roton softening Santos, Shlyapnikov, Lewenstein (2000) Fischer (2006)

Enhancement of roton softening in multi-layer systems Amplification of attractive interaction for dipoles in different layers on top of each other Wang, Demler, arXiv:0812.1838 Roton softening for 10, 20, and 100 layers Growth rate of unstable modes Decoherence of Bloch oscillations In agreement with expts on 39K: Fattori et al, PRL (2008) momentum out of plane momentum in the plane

B F Spin-dipolar interactions for ultracold atoms Larmor Precession (100 kHz) dominates over all other energy scales. Effective interaction based on averaging over precession B F Quasi 2D system of 87Rb. Spin-roton softening Wide range of instabilities tuned by quadratic Zeeman, AC Stark shift, initial spiral spin winding

Dipolar instabilities in spinor condensates Fourier spectrum Spontaneously modulated textures in Rb condensates Vengalattore et al., PRL (2008) Dipolar spin instabilities R. Cherng, E. Demler, arXiv:0806.1991 Checkerboard pattern observed in experiments reflects unstable spin modes

Polar molecules Objectives: Polar molecules rotation of the molecule dipole moment rotation of the molecule Objectives: - control and design the interactions potentials - derive extended Hubbard models Polar molecules - permanent dipole moment: - polarizable with static electric field, and microwave fields - interactions are increased by compared to magnetic dipole interactions

Polar molecules Three-body interactions Repulsive shield Spin toolbox - systematic approach to strong many-body interactions (H.P. Büchler, A. Micheli, and P. Zoller, Nature Physics 2007) Repulsive shield - crystalline phases (H.P. Büchler, E. Demler, M. Lukin, A. Micheli, G. Pupillo, P. Zoller, PRL 2007) - design of a repulsive potential between polar molecules - quenches inelastic collisions Spin toolbox - polar molecules with spin - realization of Kitaev model (A. Micheli, G. Brennen, P. Zoller, Nature Physics 2006)

Probing fermionic Hubbard model with spin polarization B. Wunsch, E. Demler (Harvard) E. Manousakis (FSU)

Antiferromagnetic Mott state and spin imbalance Do we have spin separation in parabolic trap? Perfect AF Mott state Spin polarized edges Spin polarized edges Canted antiferromagnetic phase in the Mott plateau W. Hofstetter et al., NJP(2008) Hartree-Fock approximation

Antiferromagnetic Mott state and spin imbalance Both states are self-consistent solutions of the HF equations Canted antiferromagnetic phase is lower in energy

Quantum simulator theory This talk: Harvard, Innsbruck-Stuttgart, Michigan Low dimensional systems 1d: spin dynamics of two component Bose mixture Harvard, Mainz collaboration 1d: dynamics of spin chains Harvard, Mainz collaboration (+Weizmann, Munchen,Fribourg) 2d: interference of weakly coupled pancakes Harvard, Stanford collaboration Microscopic parameters of low-D systems (Michigan) Dipolar interactions (Harvard, Innsbruck, Stuttgart) Probing fermionic Hubbard model with spin polarization (Harvard)