Feshbach induced strong interactions in fermionic lithium

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Feshbach induced strong interactions in fermionic
lithium

• Collaboration
• ENS JILA
• Servaas Kokkelmans Murray Holland
• Thomas Bourdel Josh Milstein
• Julien Cubizolles
• Christophe Salomon
• Gora Shlyapnikov

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Outline

• Three subjects
• Atom-molecule model to study superfluidity
• Detailed look at 40K and 6Li
• Signatures of superfluidity
• Validity of theory?
• Include second order fluctuations
• Cross over BEC-BCS
• Study of mean field around Feshbach resonance
• Mean field does not blow up
• Possible to locate the resonance?

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A lot of progress in last months

• Several groups are now in for quest of fermi
  superfluidity
• Some remarkable results
• Measurement of field-dependent decay MIT group,
  Duke group, ENS group
• Zero crossing of scattering length Duke group,
  Innsbruck group
• Asymmetric expansion Duke group, ENS group
• Superfluidity or hydrodynamic regime?
• Need good understanding of resonance in many body
  picture, and mechanisms for decay, superfluidity!

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What is Feshbach resonance?

• Coupling between open and closed channels
• Separate out bound state and treat explicitly

closed channel
Ekin
open channel
abg
a
B

• Resonance short-range molecular state
• Relatively long-lived molecules
• Scattering becomes strongly energy-dependent

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Get int. params from scattering length

• Describe two-body interaction with few parameters

abg
Scattering length
Detuning v
Width g
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Feshbach theory

• Shows that only few parameters needed to describe
  full energy-dependent scattering
• Coupling open en closed channels
• Resonant S-matrix
• T-matrix
• Zero limit
• scattering length

closed channel
open channel
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Double resonance 6Li

• Two lowest hyperfine states (1/2,1/2)(1/2,-1/2)
• Double resonance!
• Double-resonance S-matrix
• With ,
• And coupling strengths g1 and g2

Real background
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Contact scattering - renormalization

• Limiting case R 0
• Cut-off gives renormalization!
• Define parameters

Solve Lippmann-Schwinger equation with contact
potentials and contact coupling
Relation between real and cut-off parameters
(for single resonance)
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Problems with BCS theory

• BCS treatment has several problems close to
  Feshbach resonance
• Scattering independent of energy
• Field theory not convergent when
• Pairing assumed only at Fermi surface
• Solve with explicit description of bound state
• Formulate resonance pairing model valid close to
  resonance

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Many-body physics

• Two different spin states and
• Interaction Hamiltonian
• Large detuning converges to BCS mode
  elimination of molecules
• fi ltbk0gt essentially k0 state classical
  field (or condensate
• Renormalization cut-off K

Theory has build-in scattering equations!
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Hartree-Fock-Bogoliubov
Molecular fields
Mean fields Other fields not relevant (i.e.
) Diagonalize Hamiltonian
(Bogoliubov quasiparticles) Single-particle
energy Upgrade to T-matrix Energy-averaged
T-matrix enters! (is this sufficient?) Gap
The angles are
Normal densities
pairing field
(Cooper pairs)
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Diagonal Hamiltonian
Quasiparticle spectrum Spectrum
independent of cutoff Diagonal
Hamiltonian Next task
thermodynamics
Quasi-particle Energy (µK)
Single-particle Energy (µK)
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Thermodynamics

• Self-consistency
• Given mean-fields, chem. pot., temp.
• Find Ek
• Populate quasiparticles with Fermi-Dirac
  distribution
• Sum over k to recover n and p fields
• Iterate to self-consistency
• Thermodynamic quantities
• Minimum Gibbs free energy Thermodynamic
  solutions Lee et al
• Adjust to give the correct density at each T

(Grand partition function)
(grand potential)
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Homogeneous solutions
Solution for detuning B-B00.15 Gauss
(40K)
Resonance pairing and BCS connect at large
detuning
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Field-dependence Tc
Dependence critical temperature on magnetic field
40K
6Li
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Superfluidity in a trap
Local density approximation Solve for local
chemical potential Density bulge in the
middle!
T0.2 TF
Density (cm-3)
Density (cm-3)
N500000
Temperature (TF)
Radial position (µm)
Radial position (µm)
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Compressiblity
Definition , with P the
pressure Change in C-1 causes
density-bulge C-1 0 Gas becomes unstable!
C-1 / (2nEF)
Radial position (µm)
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Connection BEC-BCS
High Tc places resonance pairing system in
cross-over regime BCS-BEC Gap defined as minimal
energy to break up composite boson into pair of
fermions Magnetic field is
control parameter
Tc/TF
BCS of Fermions BEC of molecules
2?/(kBTF)
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Resonant action

How to include fluctuations in order to properly
describe the superfluid behavior at all detunings
from resonance? Follow method of Nozières and
Schmitt-Rink, 1985, Randeria et al., 1993)
BCS Action
Molecular Action
See also Y. Ohashi and A. Griffin, PRL 13,
130402 (2002)
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Reduction of partition function
Full Partition Function
Integrate Out the Molecular Fields
Bose Partition Function
BCS Action with energy and frequency dependent
potential
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Fluctuations
Introduce complex auxiliary fields c(q),
c(q). Expand the action
Pairing Fluctuations
(Nozières and Schmitt-Rink, 1985)

(Randeria et al., 1993)
Defining the pairing propagator
Fermi distribution
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Gap and Number Equation at TC
Number Equation
Molecules
Atom Pairs / Molecules
Fermions
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Pay attention to molecular states
Desired molecular state
Unwanted molecular state
Asymptotes linear in ?
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Numerical Results 40K
BCS Theory
Smoothly connects the BCS regime to the BEC
regime. V gtgt 0, asymptotes to BCS V ltlt 0,
asymptotes to BEC A maximum of .26TF in the
crossover regime.
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More results
Free Fermions
EF
2µ ? EB
Atom-Pairs
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Further improvements?

• Maximum Tc 0.26 TF
• What about higher-order fluctuations?
• Other interactions?