Atoms in optical lattices and the Quantum Hall effect

Anders S. Sørensen

Niels Bohr Institute, Copenhagen

IntroAtomic physics: simple, well understood

Extremely good experimental control of atoms (lasers)

=> Let us try to use atoms as a tool to solve other peoples problems

BEC with cold atoms

What they do

1. Cool and trap atoms with lasers

2. Atom = magnet => trap with magnetic fields

3. Evaporative cooling => BEC

4. Release from trap; look at velocity distribution.

Features

1. Many body system with well known simple properties Vij= g (ri-rj)

2. Properties highly tunable (in real time)

Optical trapping

Dielectric attracted into electric field

+Q

-Q

F

Laser beam attracts dielectrics, cells, molecules, atoms....

Low D condensates1D condensates

2D condensates

A. Görlitz et al., Phys. Rev. Lett. 87, 130402 (2001)

But...Condensates are simple

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Φ(r1,r2,....rN) = ϕ(r1)ϕ(r2 )......ϕ(rN)

(H0 + gNϕ(r)2)ϕ(r) = μϕ(r)

Mean field theory:

Interactions among particles are weak

Not very challenging theoretically

Strong correlations, strong interactions => challenging

Strong Interactions 1:Rapid rotation

Rotating condensates

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Corriolis force : r F = 2m

r v ×

r Ω

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Lorentz force : r F = q

r v ×

r B

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are needed to see this picture.

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r Ω

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

MIT

Vortices:

Quantum Hall in rotating BECWilikin and Gunn, Ho, Paredes et al., .......

rot~vib => NVotices~NAtoms => Fractional quantum Hall

Mz0 1 2-1-2

E No Rotation

Mz0 1 2-1-2

E With Rotation

Rotation: Many degenerate states => Interactions dominate

Interactions still very weak

Strong Interactions 2:Feshbach Resonances

Feshbach resonancesVij= g (ri-rj) => Change g

ri-rj

E

Bound state

Move bound state up and down => Dramatically change interaction

Feshbach resonances

Bosons: three body loss => no good

Fermions: VERY nice experiments

(cooling harder for fermions)

Strong Interactions 3:Optical lattices

Optical lattices

Two lasers => Standing wave

Atoms trapped in planes

4 lasers = > atoms trapped in tubes

6 lasers => cubic lattice

Optical latticesAtomic potential

J U

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H = Jx,y,z(ai+a j

{i,j}

∑ +H.C.)+U ni(ni −1)i

∑

Tunneling: J~exp(-I....) => can be tuned

V0~I

(Bose-Hubbard model)

Strong interactions: atoms confined to small volume => U Big

State preparation

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H = Jx,y,z(ai+a j

{i,j}

∑ +H.C.)+U ni(ni −1)i

∑ (Bose-Hubbard model)

Load atoms in to lattice, cool, look at ground state => doesn’t work; can’t cool in lattices

E<<V0

V0

Load cold atoms into lattice. Adiabatic loading => Constant Entropy

Mott insulator

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H = Jx,y,z(ai+a j

{i,j}

∑ +H.C.)+U ni(ni −1)i

∑ (Bose-Hubbard model)

J>>U U>>J

Superfluid One atom at each site

J~U

Quantum phase transition

Load BEC

Have been done in 1, 2, and 3D

DetectionVelocity distribution = Fourier transform of density matrix

~ Probes long range order of off-diagonal elements

Superfluid SuperfluidMott

Not the most convincing probe (did also probe excitation spectrum + density correlations)

Tonks Giradeau GasOne dimensional Bose gas, strong interactions

~ non interacting fermions

Tune lattice potential => go from one regime to the other

Achievements - Bosons

• Mott insulator

• Tonks Giradeau

• “Entangling operations”

• Collapse and revival of matter wave field

• Spin dynamics

• Molecule formation

• Several experiments with weaker interactions

FermionsHarder to work with experimentally. Cooling harder (use Bosons to cool).

• Fermi degenerate gas loaded into lattice, observed Fermi surface, dynamics, interactions.• Confinement induced change of collision properties (molecules always bound)• More experiments underway

ExtensionsNow: atoms with a few spin states jumping around in lattice

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H = Jx,y,z(ai+a j

{i,j}

∑ +H.C.)+U ni(ni −1)i

∑

Extensions:

• Magnetism• Bose-Fermi mixtures• Quantum Hall• Three particle interactions• ......

May or may not be feasible

Magnetism

• Mott regime U >> J• Atoms have spin (several internal states)• Interaction dependent on internal state (or use spin dependent tunneling)

Include virtual processes:

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H = gr J i ⋅

r J j

{i,j}

∑

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g ~J2

UδUU

<< J <<U

Fractional Quantum Hall states in optical lattices

Collaborators: Harvard PhysicsEugene DemlerMikhail D. LukinMohammad HafeziMartin Knudsen (NBI)

Fractional quantum Hall effect

Tsui, Störmer, and Gossard, PRL 48, 1561 (1982)

V/I

=

(2D)

Theory

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Ψ(r1,.....,rN) = exp − z2/ 4∑( ) (zk − zl)

m

k<l

∏ z = x + i y

Magnetic flux: Φ = B · A = NΦ · Φ0 Φ0 = h/e

Laughlin: if NΦ=m ·N incompressible quantum fluid

Quasi particles: charge e/m, anyons

Particle+m fluxes composite particle (boson) condenses

Goal: produce these states for cold bosonic atoms (m=2)

Energy gap to excited state ∆E. Phase transition kBT~ ∆E

Requirements/outline

1. Effective magnetic field

2. What does the lattice do?

3. How do we get to the state?

4. How do we detect it?

Magnetic field

See also Jaksch and Zoller, New J. Phys. 5, 56 (2003)

1. Oscillating quadropole potential: V= A ·x·y ·sin(t)2. Modulate tunneling

x

y

Magnetic field

See also Jaksch and Zoller, New J. Phys. 5, 56 (2003)

1. Oscillating quadropole potential: V= A ·x·y ·sin(t)2. Modulate tunneling

x

y

Magnetic field

See also Jaksch and Zoller, New J. Phys. 5, 56 (2003)

1. Oscillating quadropole potential: V= A ·x·y ·sin(t)2. Modulate tunneling

Proof:

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U t =n2πω

⎛ ⎝ ⎜

⎞ ⎠ ⎟= U t =

2πω

⎛ ⎝ ⎜

⎞ ⎠ ⎟n

= e−iβTx / 2he−2iAxy / ωhe−iβTy / he2iAxy / ωhe−iβTx / 2h( )

n

= e-i Heff t / h

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Heff ≈ J x x +1 + x +1 x +x

∑ J y y +1e−2iπαx +e2iπαx y +1 yy

∑

: Flux per unit cell 0≤ ≤1

Lattice: Hofstadter Butterfly

E/J

~B

Particles in magnetic fieldContinuum: Landau levels

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En = heBmc

(n+1/ 2)

B

E

Similar « 1

Hall states in a latticeIs the state there? Diagonalize H (assume J « U = ∞,

periodic boundary conditions)

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ΨGround ΨLaughlin

2

99.98%

95%

Dim(H)=8.5·105

?

N=2 N=3 N=4 N=5

N=2NΦ

Energy gap

N=2 N=3 N=4 N=5

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EJ

N=2NΦ

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E ~ 0.25 J

Making the stateAdiabatically connect to a BEC

Quantum HallBEC

Mott-insulator

?

Making the state

U0

4 Atoms, 66 lattice, =2/9=0.222

U0/J

Overlap 98%

U0/J

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EJ

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ΨGround ΨLaughlin

2

DetectionIdeally: Hall conductance, excitations

Realistically: expansion image

HallSuperfluid Mott

Requirements/outline1. Effective magnetic field

2. What does the lattice do?

3. How do we get to the state?

4. How do we detect it?

Conclusion (1)

Future- Quasi particles - Exotic states- Magnetic field generation

Conclusion• Ultra cold atoms: Flexible many body system with well

understood and controllable parameters

• Beginning to enter into the regime of strong

coupling strong correlations: lattices, Feshbach resonances

• More complex system can be engineered

• Open question how much is feasible

• Quantum Hall: tunneling only turned on at short instances

=> reduced energy gap, super lattice hard. Not very near

future