1. Phase 1A Rice Collaboration
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, (2007)
Mathy and Huse, arXiv:
Zhao and Liu, arXiv:
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 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

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 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

6. 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

7. 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

8. Reducing dimensionality stabilizes FFLOkx ky
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 Finite T
Tube-tube coupling
Finite T
Finite System Size
Detection
Optimize parameters    

11. 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

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 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

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:
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. 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

19. LDA works Non-interacting fermions -- very small particle numbers nz 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, (2004)

20. 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

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:
Casula, Ceperley, and Mueller
Tunneling between wires
Parish, Baur, Mueller, and Huse, Phys. Rev. Lett. 99, (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

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=0.08
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. 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)

29. Preliminary Data n
Fully polarized wing n x

30. 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