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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