Fractional Quantum Hall states in optical lattices

Eugene Demler

Collaborators:

Anders SorensenMikhail Lukin

Bose-Einstein Condensation

Cornell et al., Science 269, 198 (1995)

Ultralow density condensed matter system

Interactions are weak and can be described theoretically from first principles

New Era in Cold Atoms Research

• Optical lattices

• Feshbach resonances

• Rotating condensates

• One dimensional systems

• Systems with long range dipolar interactions

Focus on systems with strong interactions

http://jilawww.colorado.edu/bec

Vortex lattice in rotating condensatesPictures courtesy of JILA

Lindeman criterion suggests that the vortexlattice melts when . Cooper et al., Sinova et al.

QH states in rotating condensatesFractional quantum Hall states have beenpredicted at fast rotation frequencies:Wilkin and Gunn, Ho, Paredes et.al., Cooper et al,…

Corriolis force : F = 2m

v

Lorentz force : F = q

v

B

Read-RezayiMoore-Read

Composite fermions

Laughlin

Vortexlattice

QHE in rotating BEC

It is difficult to reach small filling factors

- scattering length

Current experiments: Schweikhard et al., PRL 92:40404 (2004)

Small energies in the QH regime require very low temperatures

This work: Use optical lattices

Atoms in optical lattices

Theory: Jaksch et al. PRL 81:3108(1998)

Experiment: Kasevich et al., Science (2001); Greiner et al., Nature (2001); Phillips et al., J. Physics B (2002) Esslinger et al., PRL (2004);

4

Bose Hubbard Model. Mean-field Phase Diagram.

1n

U

02

0

M.P.A. Fisher et al.,PRB40:546 (1989)

MottN=1

N=2

N=3

Superfluid

Superfluid phase

Mott insulator phase

Weak interactions

Strong interactions

Mott

Mott

Superfluid to Insulator TransitionGreiner et al., Nature 415 (02)

U

1n

t/U

SuperfluidMott insulator

Outline

1. How to get an effective magnetic field for neutral atoms

2. Fractional Quantum Hall states of bosons on a lattice

3. How to detect the FQH state of cold atoms

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:

U t n2

U t 2

n

e iTx / 2e 2iAxy /e iTy / e2iAxy /e iTx / 2 n

e-i Heff t /

Heff J x x1 x1 x x J y y 1e 2ix e2ix y 1 y

y

: Flux per unit cell 0≤ ≤1

Lattice: Hofstadter Butterfly

E/J

~B

Particles in magnetic fieldContinuum: Landau levels

En eBmc

(n1/ 2)

B

E

Similar « 1

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

periodic boundary conditions)

Ground Laughlin

2

99.98%

95%

Dim(H)=8.5·105

?

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

N=2N

Energy gap

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

EJ

N=2N

> 0.1ascatterring

l ?

E ~ 0.25 J

DetectionIdeally: Hall conductance, excitations

Realistically: expansion image

HallSuperfluid Mott

Conclusions•Effective magnetic field can be created for cold neutral atoms in an optical lattice•Fractional Quantum Hall states can be realized with atoms in optical lattices•Detection remains an interesting open problem

Future

- Quasi particles - Exotic states- Magnetic field generation

Two dimensional rotating BEC

xy

Single particle Hamiltonian

E

01 2-1-2 0 1 2-1-2

System at rest System rotating at frequency

E’