atoms in optical lattices and the quantum hall effect anders s. sørensen niels bohr institute,...
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Atoms in optical lattices and the Quantum Hall effect
Anders S. Sørensen
Niels Bohr Institute, Copenhagen
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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
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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.
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Features
1. Many body system with well known simple properties Vij= g (ri-rj)
2. Properties highly tunable (in real time)
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Optical trapping
Dielectric attracted into electric field
+Q
-Q
F
Laser beam attracts dielectrics, cells, molecules, atoms....
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Low D condensates1D condensates
2D condensates
A. Görlitz et al., Phys. Rev. Lett. 87, 130402 (2001)
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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
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Strong Interactions 1:Rapid rotation
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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:
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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
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Strong Interactions 2:Feshbach Resonances
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Feshbach resonancesVij= g (ri-rj) => Change g
ri-rj
E
Bound state
Move bound state up and down => Dramatically change interaction
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Feshbach resonances
Bosons: three body loss => no good
Fermions: VERY nice experiments
(cooling harder for fermions)
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Strong Interactions 3:Optical lattices
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Optical lattices
Two lasers => Standing wave
Atoms trapped in planes
4 lasers = > atoms trapped in tubes
6 lasers => cubic lattice
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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
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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
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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
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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)
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Tonks Giradeau GasOne dimensional Bose gas, strong interactions
~ non interacting fermions
Tune lattice potential => go from one regime to the other
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Achievements - Bosons
• Mott insulator
• Tonks Giradeau
• “Entangling operations”
• Collapse and revival of matter wave field
• Spin dynamics
• Molecule formation
• Several experiments with weaker interactions
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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
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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
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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
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Fractional Quantum Hall states in optical lattices
Collaborators: Harvard PhysicsEugene DemlerMikhail D. LukinMohammad HafeziMartin Knudsen (NBI)
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Fractional quantum Hall effect
Tsui, Störmer, and Gossard, PRL 48, 1561 (1982)
V/I
=
(2D)
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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
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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?
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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
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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
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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
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Lattice: Hofstadter Butterfly
E/J
~B
Particles in magnetic fieldContinuum: Landau levels
€
En = heBmc
(n+1/ 2)
B
E
Similar « 1
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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
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Energy gap
N=2 N=3 N=4 N=5
€
EJ
N=2NΦ
€
E ~ 0.25 J
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Making the stateAdiabatically connect to a BEC
Quantum HallBEC
Mott-insulator
?
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Making the state
U0
4 Atoms, 66 lattice, =2/9=0.222
U0/J
Overlap 98%
U0/J
€
EJ
€
ΨGround ΨLaughlin
2
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DetectionIdeally: Hall conductance, excitations
Realistically: expansion image
HallSuperfluid Mott
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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
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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