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
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