Phase 1A Rice Collaboration 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 Casula, Ceperley, and Mueller Developed 3 new formal collaborations (joint pubs), and many more informal ones

Quantum Simulation Paradigm Experiment (many details) Effective Model Reduce to relevant degrees 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

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

Phase 1A physics Outline BCS-BEC Crossover 1D Physics 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

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

Microscale phase separation FFLO --Polarize Gas kx ky (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 x Microscale phase separation Best evidence: heavy fermion superconductors + layered superconductors Short range interactions in 3D: bulk phase separation is typically better than FFLO

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

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

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

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 r(E) E 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

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 wavefunction Solves Schrodinger equation if parameters obey 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 have longer range

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

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

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

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

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

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

LDA works Non-interacting fermions -- very small particle numbers n 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)

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

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

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!!

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

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

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) Pu and Bolech -- multiband effects -- in progress Without sophisticated data analysis: need T<0.08 TF  Adding tunneling Provides robustness against temperature Phase 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

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=0.08 T/TF=0.07 T/TF=0.05

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

Parameters for array of traps FFLO center polarized 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)

Preliminary Data n Fully polarized wing n x

Summary of Theory Tasks 1D Fermi Gas Solve Model Find optimal experimental parameters Develop novel experimental protocols Determine Observables density Exact Solution of ideal model   cooling interference  Quantify corrections from real-world influences Study Thermodynamics  polarization Tube-tube coupling pressure   Finite T Develop theoretical tools to efficiently calculate properties of ideal/nonideal models   expansion  Multiple methods: built in redundency