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DARPA OLE

DARPA OLE. Phase 1A Rice Collaboration. Accomplishments. All Phase 1A milestones successfully completed on time 4 theory papers submitted for publication Parish, Baur, Mueller, and Huse, Phys. Rev. Lett. 99, 250403 (2007) Mathy and Huse, arXiv:0805.1507 Zhao and Liu, arXiv:0804.4461

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DARPA OLE

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  1. DARPA OLE Phase 1A Rice Collaboration

  2. Accomplishments • All Phase 1A milestones successfully completed on time • 4 theory papers submitted for publication • Parish, Baur, Mueller, and Huse, Phys. Rev. Lett. 99, 250403 (2007) • Mathy and Huse, arXiv:0805.1507 • Zhao and Liu, arXiv:0804.4461 • Casula, Ceperley, and Mueller • Developed 3 new formal collaborations (joint pubs), and many more informal ones

  3. Quantum Simulation Paradigm Experiment(many details) Effective Model Reduce to relevantdegrees of freedom Theoretical tools(finite size scaling,LDA, extrapolation…) Ideal Model As with digital simulation: Goal is not direct implementation of ideal model Goal is to extract info about ideal model from a simulation (finite size, finite T,…) Ex: Infinite, homogeneous, point interactions, 1D

  4. Goals Ultimate goal (phase 2): Superfluidity in repulsive U Hubbard model • Technically challenging • Extremely important Immediate goal (phase 1A): • Use atoms in optical lattice to map out phase diagram of homogeneous 1D Fermi gas • 2 components -- More up-spins than down-spins • Short range attractive interactions

  5. Phase 1A physics Outline • BCS-BEC Crossover • Polarized Gas • FFLO • 1D Physics • Physical picture + Motivation • 1D scattering • Technical tools • Bethe Ansatz • Luttinger Liquids • Quantum Monte-Carlo • Mean Field Theory • Phase Diagram Where we are coming from

  6. BCS-BEC Crossover Leggett Weak attractive interactions BCS No bound state in free space Pairing is many-body effect(Fermi surface reduces dimensionality) V r V0 Pairing and superfluidity occur simultaneously r0 Continuously connected (Experiment: tune interactions with magnetic field) Strong attractive interactions Pairing (crossover) precedes superfluidity (phase transition) V r BEC

  7. ky kx x FFLO --Polarize Gas (Fulde-Ferrel [1964], Larkin and Ovchinnikov [1965]) Polarize gas -- Shift Fermi surfaces -- suppress pairing Pair atoms at from different Fermi surfaces -- finite center of mass momentum Superimpose different momenta -- spatially modulated order parameter Microscale phase separation Best evidence: heavy fermion superconductors + layered superconductors Short range interactions in 3D: bulk phase separation is typically better than FFLO

  8. ky kx Reducing dimensionalitystabilizes FFLO Increases “nesting” ky kx Alternative picture: Domain walls are “cheap” excitations in 1D p D

  9. Phase Diagram 1D 3D FFLO S N N S This diagram is Phase 1A goal

  10. Theoretical issues • 3D exp to determine 1D phase diagram • Tube-tube coupling • Finite T • Finite System Size • Detection • Optimize parameters    

  11. r(E) E Physics of 1D -- Generic 1D is special (and exciting) Noninteracting system has large density of states at low energies Interactions resolve degeneracy:- strong correlation physics • Typical Features of 1D systems: • Low energy excitations exhausted by spin waves and phonons • Bosonization • Spin-charge separation possible • No true long-range order • Correlation functions fall off algebraically • Typically have multiple orders • Transition temperature to ordering typically zero • Restrictive kinematics Fermi surface topology is simple k

  12. Attractive 2 comp Fermions in 1D High Density = Weak interactions Independent gas of up-spins and down-spins Exact Solution:Bethe Ansatz Power-law pair correlations Power-law CDW correlations Power-law SDW correlations “Guess” for many-body wavefunctionSolves Schrodinger equation if parametersobey a set of coupled integral equations Generic: essentially follows from symmetries/dimensional analysis Low Density = Strong interactions Gas of superfluid pairs + excess up-spin fermions (domain walls) Bose-Bose interactions are short-ranged -- hard core at sufficiently weak interactions Bose-Fermi interactions havelonger range

  13. Our theoretical approaches • Bethe Ansatz • Mueller • Effective Field Theory • Liu • Quantum Monte-Carlo • Ceperley • Mean Field Theory • Bolech, Huse, Mueller, Pu Questions: Observables Parameters 3D-1D Temperature

  14. Effective Low Energy Theory Erhai Zhao and W. Vincent Liu, arXiv:0804.4461 Low energy excitations: two linear modes -- density and spin are coupled Allows studying stability of ideal system to perturbations(Temperature, intertube coupling…) Results: Consistency among all our methods

  15. Quantum Monte-Carlo Casula, Ceperely, and Mueller, to be submitted Variational Inhomogeneous Jastrow (2+3 particle correlations) Free particles ~30 parameters Diffusion Imaginary time propagation of variational wavefunction Can be made arbitrarily accurate in 1D Path integrals Perform trace in position basis-- expand exponential as a product -- insert position basis resolution of identity Maps onto classical polymer problem Can be made arbitrarily accurate in 1D

  16. Quasi-1D to 1D Olshanii -- Harmonic transverse confinement V 3D scattering length a3D 1D scattering length a1D r Our collaboration: Mean field and effective field theory Sufficiently weak intertube coupling leads to 1D physics Quantification: in progress

  17. Measuring phase diagram laser Image(can not resolve tubes) cloud Integrate over one direction to improve signal-to-noise(“axial density”) Each image gives one point on phase diagram boundary -- EOS on fixed line

  18. FFLO S N Theoretical axial densities (LDA) LocalDensityApproximation (exact EOS) n+n n-n n-n n+n These points mark phase boundaries   Interpretation requires LDA Each point in trap assumed locally homogeneous

  19. LDA works Non-interacting fermions -- very small particle numbers n N=20 z z Wiggles -- discreteness of particles Scale as 1/N (Small numbers of particles shown to illustrate agreement) Erich J. Mueller, Density profile of a harmonically trapped ideal Fermi gas in arbitrary dimension, Phys .Rev. Lett.93,190404 (2004)

  20. Also works in interacting system 20 particles g=2 d/a = 20 (very strong interactions) n (pairs) 9 pairs2 unpaired fermions n-n (unpairedfermions) Dots: Monte-Carlo z Lines: TF Smeared out by interparticle spacing We understand wiggles Even better agreement for larger N

  21. Wiggles at strong interaction Dots: QMC n Solid: 9 non-interacting fermions (mass m)-- lengths scaled by 0.795 (pairs -- hard-core bosons) z Dots: QMC n-n Solid: 2 non-interacting fermions (mass m) -- lengths scaled by 1.35 (excess fermions/domain walls) z

  22. Axial Density “averages out” finite size wiggles Ex: 20 noninteracting particles in one trap na n z z Array of traps with 20 noninteracting particles in central Do not need to worry about corrections to LDA -- even for very small particle numbers!!

  23. LDA+Harmonic trap:Axial density gives local pressure Derivative of axial density gives 3D density

  24. If LDA fails -- can still reconstruct Reconstructed density contours Empirical (no modelling)fit to column density data Inverse Abel

  25. Resolved technical issues • Temperature • Zhao and Liu, arXiv:0804.4461 • Casula, Ceperley, and Mueller • Tunneling between wires • Parish, Baur, Mueller, and Huse, Phys. Rev. Lett. 99, 250403 (2007) • Zhao and Liu, arXiv:0804.4461 • Pu and Bolech -- multiband effects -- in progress Without sophisticated data analysis: need T<0.08 TF  Adding tunneling Provides robustness against temperaturePhase diagram/Density profiles distort continuously   In progress: quantify distortion(or how to extract zero coupling result from finite coupling data) -have mean-field results -tools in place for extend to strong coupling

  26. Temperature Monte-Carlo: T effects are small if T/Tf<0.08 If necessary: can do scaling T/TF=0.4 T/TF=0.2 T/TF=0.2 T/TF=0.08 T/TF=0.07 T/TF=0.05

  27. Other open questions • Kinetics Equilibration is inhibited in 1D • find equilibrium? • Monte Carlo algorithm slowed down • Leave “hopping” on as long as possible 1D bosons -- Weiss

  28. P Parameters for array of traps FFLO centerpolarized wings FFLO center Paired wings • a -- 1D scattering length • l -- spacing between tubes • d -- Harmonic confinement length along each tube • w -- Harmonic confinement length transverse to tubes • N -- Total number of particles • P = (Nu-Nd)/(Nu+Nd)

  29. Preliminary Data n Fully polarizedwing n x

  30. 1D Fermi Gas Summary of Theory Tasks Find optimal experimental parameters Develop novelexperimentalprotocols Solve Model DetermineObservables density  Exact Solution ofideal model  cooling interference  Quantify correctionsfrom real-world influences StudyThermodynamics  polarization Tube-tubecoupling pressure   Finite T Develop theoretical toolsto efficiently calculate propertiesof ideal/nonideal models   expansion  Multiple methods:built in redundency

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