Atoms in optical lattices and the Quantum Hall effect Anders S. Sørensen Niels Bohr Institute,...
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Transcript of Atoms in optical lattices and the Quantum Hall effect Anders S. Sørensen Niels Bohr Institute,...
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
€
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
€
Corriolis force : r F = 2m
r v ×
r Ω
€
Lorentz force : r F = q
r v ×
r B
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
€
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
€
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
€
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
€
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
€
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:
€
H = gr J i ⋅
r J j
{i,j}
∑
€
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
€
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:
€
U t =n2πω
⎛ ⎝ ⎜
⎞ ⎠ ⎟= U t =
2πω
⎛ ⎝ ⎜
⎞ ⎠ ⎟n
= e−iβTx / 2he−2iAxy / ωhe−iβTy / he2iAxy / ωhe−iβTx / 2h( )
n
= e-i Heff t / h
€
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
€
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)
€
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
€
EJ
N=2NΦ
€
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
€
EJ
€
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