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Effective Hamiltonians and quantum magnetism of ultra cold atoms. Ehud Altman Harvard university. Collaborators: E. Demler, M. Lukin, W. Hofstetter, A. Polkovnikov, A. Auerbach. Overview. Anderson etal, Science (95). 1 trap. Lattice.

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## Effective Hamiltonians and quantum magnetism of ultra cold atoms

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**Effective Hamiltonians and quantum magnetism of ultra cold**atoms Ehud Altman Harvard university Collaborators: E. Demler, M. Lukin, W. Hofstetter, A. Polkovnikov, A. Auerbach**Anderson etal, Science (95)**1 trap Lattice Macroscopic occupation of a single particle wave function. Greiner etal., Nature (02) Cloud density momentum distribution in trap BEC and Matter Waves**From matter waves to particles**Proposal: D. Jaksch, et al. PRL (98) Greiner etal., Nature 415 (02) Mott insulator Superfluid Adiabatic increase of interaction t/U No description in terms of a single particle wavefunction**Nevertheless these systems exhibit very long coherence times**Playground for non equilibrium quantum dynamics. Greiner et. Al., Nature 419 (2002) Collapse - revival 0.5 0 - 0.5 Time evolution of a Mott state when the lattice is rapidly changed to the supefluid regime?**0.5**Mott insulator 0 - 0.5 Dynamics Weak interactions: ? Gross-Pitaevskii (GP) Non linear single particle wave equation ? ?**We understand the plainvanilla Mott transition.**What aboutchocolate vanilla? Time of flight image is not sensitive to many body correlations. How to detect them ?**Outline**• Superfluid-Mott insulator transition – plain vanilla:mean field theory, collective modes, dynamics • Chocolate vanilla Mott insulator – “spin ½” bosonsspin ordered phases How the ice-cream melts • Detecting correlations via noise in time of flight imageWhat does time of flight imaging measure?How boring is the vanilla Mott state?Detecting spin correlations.Pairing correlations.**V(x)**Standing light waves U J Bose Hubbard Model: Projection to lowest Bloch band Atoms in an optical lattice and the BHM D. Jaksch, et al. PRL (98) Enhanced interaction effects !**0.5**Mott insulator 0 - 0.5 Phase diagram Mean field theory: Fisher et. al. PRB (1989) Dynamics: ? GP + Bogoliubov ? ? How to calculate excitations and dynamics in the vicinity of the insulating phase?**Breakdown of the Gross-Pitaevskii description**Variational state: Amounts to replacing the operator aiby the c-number yi: Uniform solution: Classical equation of motion: Cannot describe Mott transition: • Particle number fluctuation • Nothing special at integer filling • Quadratic fluctuations (Bogoliubov theory) don’t help.**n**-1 0 1 Variational mean field theory (I) Factorizable state: Rokhsar & Kotliar PRB (91) Near the transition Keep 3 states Constraint:**-1**0 1 n Variational mean field theory (II) Superposition on every site: 0 +1 -1 ( ) ( ) … Easy to calculate expectation values: Order parameter (commensurate filling):**0.5**Mott insulator 0 - 0.5 1 Variational mean field theory (III) Minimize variational energy: Expression for the Mott lobes: Is the dimensionless interaction**Fluctuations (I)**Rewrite H in terms of the new operators Spin - 1 n -1 0 1 Can also be written as the pseudo spin – 1 model (n>>1):**Example: Mott state**Fluctuations: Fluctuations (II) Mean field state is a Fock state in terms of a new operator Orthogonal operators create fluctuations: Constraint:**Fluctuations (III)**Use constraint to eliminate : Rewrite H in terms of fluctuation operators: Regime of validity: Diagonalize with Bogoliubov transformation**Amplitude mode**Superfluid: w w Mott: Phase fluctuations k k Collective modes Gapped particle/hole**Canonical conjugate pairs**Dynamics Use as (over-complete) basis to construct a path integral for the time evolution: Close to the transition@integer filling, integrate over h to obtain: Recall effective action from Sachdev’s talk**Superfluid**Mott J/U E Application: sudden quench E. Altman & A. Auerbach, PRL (2002) Different regime: Polkovnikov et. al. PRA (2002) Saddle point equation of motion: Uniform order parameter evolution: Frequency scale: Non uniform configurations?**OP**Which way will they fall?**Top View with fluctuations**How many defects survive? E Growth modes: Vortex trapping (I) Initial state: Fastest growing:k=0 Do not grow:**Vortex trapping (II)**sets initial size of domains Quantum Kibble-Zurek mechanism**w**k Quantum Corrections Damping Emission of phason pairs Damping rate (Q factor) of the oscillations: Over-damped in the critical region!**Optical lattice for two species**Realized in I. Bloch’s group: O. Mandel et. al., PRL (2003) Two hyperfine states trapped by two different laser beams: va(x) vb(x) Va U Ja Jb See also Greiner’s talk**Strong coupling @ integer occupation**ta = tb = 0 All configurations with 1 atom per site are degenerate !**degenerate manifold**U U Lifting the degeneracy Hopping is frozen. Spin is the only remaining degree of freedom. 2 sites (qualitative): competing Va Effective spin Hamiltonian:**degenerate manifold**Effective spin-½ Hamiltonian 1 1 1 1 + + + + 2 2 2 2 + + + Recall:**Effective spin-½ Hamiltonian**Phase diagram ta,b<< U<< Va,b Spin operators: Svistunov and Kuklov (PRL 03), Duan, Demler and Lukin (PRL 03)**+**+ + + + ? ? ? Counter-flow superfluid Treatment valid only deep in the Mott insulating phase Transition to superfluid?**(**) ( ) Variational mean field theory (I) Mott: qi = 0 Order parameters: Note: superfluid of both a and b is necessarily also a paired and a counterflow SF because both total and relative phases are fixed.**independent of local spin orientation.**Massive degeneracy Variational mean field theory (II) Minimize variational energy: Superfluid Mott Where are the spin phases?**Dilema**Effective Hamiltonian in low energy subspace captures spin ordering but cannot access the superfluid phase Variational mean field theory captures transition to superfluid but spin orders degenerate in the Mott phase. Solution Quantum fluctuations about the variational states lift the degeneracy selecting specific ordered states “Order from disorder” mechanism**kBT**F(l,T) Digression: “classical order from disorder” E(l) Similarly, quantum fluctuations can lift degeneracy by the zero point energy.**Rotate the basis:**Fluctuations (I) As before, define second quantized operators that create the local Hilbert space: Constraint: Fluctuation operators**Fluctuations (II)**Eliminate using the constraint: (Precise form of the Hamiltonian depends on the variational state) Diagonalize with Bogoliubov transformation Zero point energy: Depends on the assumed spin ordering**Mott-SF transition of 2 comp. bosons**More detail: E. Altman, W. Hofstetter, E. Demler and M. Lukin, NJP 5 (03) Mean Field Theory Hysteresis + Quantum fluctuations 1st order MFT alone cannot resolve spin order! 2nd order line**Mott-SF transition of 2 comp. bosons**and ordered 2nd order x-y transition. Ordering of ordered 2nd order line How to detect spin order experimentally?**Probing many body states of ultra-cold atoms via noise**correlations III**Anderson etal, Science (95)**1 trap Lattice Macroscopic occupation of a single particle wave function. Greiner etal., Nature (02) BEC and Matter Waves What if the state is not a product of single particle wave functions?**In trap**After expansion But a single shot does not measure an expectation value ! There are fluctuations (shot noise). Time of flight image Average cloud density after long expansion:**In trap**After expansion Proposal: extract information from the noise E. Altman, E. Demler and M. Lukin, PRA (04) Correlations in the noise:**Time of flight image from an optical lattice (I)**Density expectation value after free expansion for time t : Lives in lowest Bloch band Projection on lowest Bloch band: Time evolved Wannier function**Time of flight image from an optical lattice (II)**Assuming Gaussian Wannier function and long time of flight ( r>>Ri ): Gaussian envelope Interesting part: Width of the gaussian envelope: Defines correspondance between position in the cloud and lattice momentum in the trap**What to expect from the expectation value**Superfluid: Mott:**Second order correlations (Noise)**After time of flight t: n r’ r After normal ordering we can replace again: And after long time of flight as before : Bose Fermi**Plain vanilla Mott state**r Sharp Bragg peaks in 2nd order coherence! r’ This is simply bunching/antibunching.

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