Fractional Quantum Hall states in optical lattices

Anders Sorensen

Ehud Altman

Mikhail Lukin

Eugene Demler

Physics Department, Harvard University

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 BECPictures courtesy of JILA

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

QH states in rotating BECFractional 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 (1998)

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

Bose Hubbard Model

tunneling of atoms between neighboring wells

repulsion of atoms sitting in the same well

U

t

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 (2002)

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 states 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

Quantum 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

(r1,.....,rN)exp z2/ 4 (zk zl)

m

kl

z = x + i yLaughlin:

Energy gap

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

N =2N

DetectionIdeally: Hall conductance, excitations

Realistically: expansion image

Quantum HallSuperfluid Mott

Time of flight experiments

Quantum noise interferometry of atoms in optical lattices

Second order coherence

Altman et al., PRA(2004); Read and Cooper, PRA (2004)

Second order coherence in the insulating state of bosons.Hanburry-Brown-Twiss experiment

Theory: Altman et al., PRA 70:13603 (2004)

Experiment: Folling et al., Nature 434:481 (2005)

Hanburry-Brown-Twiss stellar interferometer

Second order coherence in the insulating state of bosons

Bosons at quasimomentum expand as plane waves

with wavevectors

First order coherence:

Oscillations in density disappear after summing over

Second order coherence:

Correlation function acquires oscillations at reciprocal lattice vectors

Second order coherence in the FQH state

In the Landau gauge for states in the LLL, momentum corresponds to the guiding center coordinate. From one can calculate

Read and Cooper, PRA (2004)

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