A few-fermion BCS superconductor
J. Lofthouse & G.J. Conduit
Theory of Condensed Matter Group, Department of Physics, Cambridge
Superconductivity and Scale in Quantum Systems
Experimental Realization of a Cold Atom Gas
6Li atom
Up spin electron
Down spin electron
A Model of a Cold Atom Gas
• Optical trap modeled by 3-D QHO
3-D QHO Orbitals
x
y
3-D QHO Orbitals
x
y
3-D QHO Orbitals
x
y
3-D QHO Orbitals
Non-Interacting System Ground State
Non-Interacting System Ground State
Non-Interacting System Ground State
Non-Interacting System Ground State
Non-Interacting System Ground State
Quantum Monte-Carlo
• Non-interacting system orbitals suitable basis for weak interactions
• G.S. Energy Monte-Carlo integration
Superfluid Hamiltonian in External Trap
• Pairwise attractive interactions between up and down spins
Effect of Interactions on Degeneracy
Effect of Interactions on Degeneracy
Effect of Interactions on Degeneracy
Interaction Energy
Interaction Energy
Interaction Energy
System Geometry
System Geometry
System Geometry
Pairing Effect
Pairing Effect
Asymmetric Trap
Non-interacting energy of N = 8 state Kept constant for
Asymmetric Trap
Summary
• Unique opportunity to realize a few body interacting system
• Link between microscopic physics and macroscopic phenomenology
• DMC simulations allowed us to probe attractive interactions
BCS state
BCS state
BCS state
BCS FFLO state
BCS FFLO state
BCS FFLO state
BCS FFLO state
Δ(r )=Δ0 Δ(r )=Δ0ei q⋅r
q
nx
ny
Available states
0,0
0,1 1,1
1,0
nx
ny
Available states
0,0
0,1 1,1
1,0
1,1
1,0
1,1 1,10,1
nx
ny
2 atoms in a spherical trap
nx
ny
2 atoms in a spherical trap
nx
ny
3 atoms in a spherical trap
nx
ny
3 atoms in a spherical trap
nx
ny
3 atoms in a spherical trap
3 atoms in a spherical trap
nx
ny
2 atoms in a spherical trap
Δ(r)=〈ψ∣c↑(r)c↓(r)∣ψ〉
2 atoms in a spherical trap
Δ̄(r )Δ(0)=〈ψ∣c↓† (r)c↑
† (r)c↑(0)c↓(0)∣ψ〉
2 atoms in a spherical trap
θ
3 atoms in a spherical trap
3 atoms in a spherical trap
Where does the extra majority spin reside?
x
Δ
Where does the extra majority spin reside?
x
Δ
3 atoms in a spherical trap
3 atoms in a spherical trap
θ
ρ↑-ρ↑
Phase diagram
ω∥=2ω⊥
Phase diagram
nx=1 ny=1
nz=1
or
Phase diagram
Phase diagram
Phase diagram
Doubly degenerate
Singly degenerate
Few trapped fermions offers chance to observe spatiallymodulated pairing
Trap ellipticity and central barrier are experimentalprobes of the pairing state
Summary
Appendix: System Geometry
Appendix: System Geometry
Hamiltonian for outermost electron
Appendix: System Geometry
Hamiltonian for outermost electron
Appendix: System Geometry
Appendix: System Geometry
Appendix: Magnetised Fermionic Gases
Appendix: Magnestised fermionic gases
• N = 14
M = 0 M = 6/2 =3
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.417
18
19
20
21
Appendix: Asymmetric Trap
Trapping potential
ω⊥
ω∥V=1
2 [ω⊥2 (x2+ y2)+ω∥
2 z2 ]
One trapped atom
E=12ω∥+ω⊥
Two trapped atoms
E=ω∥+2ω⊥
Three trapped atoms
E=52ω∥+3ω⊥
E=32ω∥+4ω⊥
Three trapped atoms
E=52ω∥+3ω⊥+√ ω∥ω∥+ωBV B(2+ ω∥ω∥+ωB )
E=32ω∥+4ω⊥+3√ ω∥ω∥+ωBV B
Three trapped atoms
E=52ω∥+3ω⊥+√ ω∥ω∥+ωBV B(2+ ω∥ω∥+ωB )+ aa∥ω⊥√
2π(32−4√2π V Bω∥+ωB )
E=32ω∥+4ω⊥+3√ ω∥ω∥+ωBV B+ aa∥ω⊥√
2π(32−6√2π V Bω∥+ωB )
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