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Quantum Hall effectsQuantum Hall effects- an introduction -- an introduction -
AvH workshop, Vilnius, 03.09.2006
M. Fleischhauer
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quantum Hall historyquantum Hall history
discovery: 1980
Nobel prize: 1985
K. v. Klitzing
H. Störmer R. Laughlin D. Tsui
discovery: 1982
Nobel prize: 1998
IQHE
FQHE
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classical Hall effect (1880 E.H. Hall)classical Hall effect (1880 E.H. Hall)
Lorentz-force on electron:
stationary current:
Hall resistance:
Dirac flux quantum
2
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Landau levelsLandau levels
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2D electrons in magnetic fields: Landau 2D electrons in magnetic fields: Landau levelslevels
coordinate transformation:
Hamiltonian:
R
X
electroncenter of
cyclotron motionradial vector of cyclotron motion
commutation relations:
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2D electrons in magnetic fields: Landau 2D electrons in magnetic fields: Landau levelslevels
mapping to oscillator:
H = h R² / 2 l² = h ( a a + ½ )c cm†
Landau levels
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2D electrons in magnetic fields: Landau 2D electrons in magnetic fields: Landau levelslevels
typical scales:
• length
BB BB
magnetic length
• energy
cyclotron frequency
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Kaiserslautern, April 2006
2D electrons in magnetic fields: Landau 2D electrons in magnetic fields: Landau levelslevels
degeneracy of Landau levels:
center of cyclotron motion (X,Y) arbitrary degeneracy
• 2D density of states (DOS)
• filling factor
one state per area of cyclotron orbit
# atoms / # flux quanta
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Kaiserslautern, April 2006
2D electrons in magnetic fields: Landau 2D electrons in magnetic fields: Landau levelslevels
wavefunction of lowest Landau level (LLL) in symmetric gauge
symmetric gauge
Landau gauge
introduce complex coordinate
LLL
analytic
b
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2D electrons in magnetic fields: Landau 2D electrons in magnetic fields: Landau levelslevels
angular momentum of Landau levels:
eigenstates of n´th Landau level:
angular momentum states of LLL:
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2D electrons in magnetic fields: Landau 2D electrons in magnetic fields: Landau levelslevels
j
wavefunction:
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Integer Quantum Hall Integer Quantum Hall effecteffect
Integer Quantum Hall Integer Quantum Hall effecteffect
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Integer Quantum Hall effectInteger Quantum Hall effect
spinless (for simplicity) and noninteracting electrons: Pauli principle
Slater determinant:
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Integer Quantum Hall effectInteger Quantum Hall effect
compressibility:
at integer fillings:
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Integer Quantum Hall effectInteger Quantum Hall effect
Hall current:
Heisenberg drift equations of cycoltron center
no plateaus ?!
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Integer Quantum Hall effectInteger Quantum Hall effect
Hall plateaus: impurities
gap !
impurities pin electrons to localized states electrons in impurity states do not contribute to currentgap impurity states fill first
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Fractional Quantum Hall Fractional Quantum Hall effecteffect
Fractional Quantum Hall Fractional Quantum Hall effecteffect
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Fractional Quantum Hall effectFractional Quantum Hall effect
Laughlin state:
• take e-e interaction into account
• generic wavefunction
• requirements
• wave function anstisymmetric• eigenstate of angular momentum• Coulomb repulsion Jastrow-type of wave function
Laughlin wave function
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Fractional Quantum Hall effectFractional Quantum Hall effect
angular momentum of Laughlin wave function and filling factor
maximum single-particle angular momentum
filling factor of Laughlin state
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Kaiserslautern, April 2006
Fractional Quantum Hall effectFractional Quantum Hall effect
fractional Hall plateaus:
fractional Hall states are gapped
= 1
= 1/3 = 1/5 = 1/7
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composite particle picture of composite particle picture of FQHEFQHE
composite particle picture of composite particle picture of FQHEFQHE
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composite particle = electron + m magnetic flux quanta
composite particle picture of FQHEcomposite particle picture of FQHE
+ =
composite fermion
composite boson
effective magnetic field
composite particle are anyons (fractional statistics) exist only in 2D
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composite particle picture of FQHEcomposite particle picture of FQHE
some remarks about anyons:
• two-particle wave function
• exchange particles
• exchange particles a second time
in 3D: Boson
Fermion
3D:no projected area in (xy) 2D always projected area in (xy)
particles can pick up e.g. Aharanov-Bohm phase
A BA B
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composite particle picture of FQHEcomposite particle picture of FQHE
= 1 / m FQE
(A) electron + flux quanta
form composite boson 0
Bose condensation of composite bosons
(B) electron + flux quanta
form composite fermion
IQHE for composite fermions
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composite particle picture of FQHEcomposite particle picture of FQHE
Jain hierarchy:
• experiment: FQHE also for
composite fermion picture:
since
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FQHE for interacting FQHE for interacting bosons bosons
FQHE for interacting FQHE for interacting bosons bosons
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FQHE for interacting bosonsFQHE for interacting bosons
exact diagonalization FQH effect for
Laughlin state for point interaction
composite fermions:
boson + single flux quantum + =
IQHE for composite fermions
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effective magnetic fields in rotating trapseffective magnetic fields in rotating traps
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atoms in dark statesatoms in dark states
|1> |2>
|0> γ
Ω Ωsp
Δ
Ω
-
D
+
adiabatic eigenstates:
γ
γ
for dark states see e.g.: E. Arimondo, Progress in Optics XXXV (1996)
dark state (no fluoresence):p
s
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R. Dum & M. Olshanii, PRL 76, 1788 (1996)
transformation to local adiabatic basis:
gauge potential A + scalar potential
|1> |2>
|0>
Ω Ωsp
center of mass motion of atoms in dark center of mass motion of atoms in dark statesstates
• space-dependent dark states & atomic motion:
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effective vector potential & magnetic field
relative momentum vector
difference of „center of mass“of light beams
relative orbital angular momentum needed !
(i) magnetic fields(i) magnetic fields
Ω Ωsp
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magnetic fields: (a) vortex light beamsmagnetic fields: (a) vortex light beams
G. Juzeliūnas and P.Öhberg, PRL 93, 033602 (2004)P. Öhberg, J. Ruseckas, G. Juzeliunas, M.F. PRA 73, 025602 (2006)
external trapB
Vratio of fields
eff
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magnetic fields: (b) shifted light beamsmagnetic fields: (b) shifted light beams
x
yz
• Quantum-Hall effect in non-cylindrical systems• non-stationary situation possible (current in z)
B
Veff
= x
x
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(ii) non-Abelian gauge fields (ii) non-Abelian gauge fields
J. Ruseckas, G. Juzeliunas, P. Öhberg, M.F. Phys.Rev.Lett 95 010404 (2005)
• more than one relevant adiabatic state ! TRIPOD scheme
D D 1 2
Ω
2 x 2 vector matrix
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magnetic monopole field magnetic monopole field
Ω 1 2
3Ω
Ω
singularity lines
point singularity at the origin
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summarysummary
• motion of atom in space-dependent dark states gauge potential A
• light beams with relative OAM magnetic field B
• degenerate dark states non-Abelian magnetic fields (monopoles,...)
• vortex light beams• displaced beams (non-cylindrical geometry, currents)
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quantum gases as many-body model quantum gases as many-body model systemssystems
• lattice models:
• BCS – BEC crossover:
Bose-Hubbard model;Bose-Fermi-H. model;spin models
Feshbach resonances;fermionic superfluidity
• quantum-Hall physics: rotating traps vortices, vortex lattices; lowest Landau level
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Kaiserslautern, April 2006
quantum gases as many-body model quantum gases as many-body model systemssystems
• quantum-Hall physics: rotating traps vortices, vortex lattices; lowest Landau level
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external trapB
V
magnetic fields: (a) vortex light beamsmagnetic fields: (a) vortex light beams
ratio of fields
eff
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ultra-cold atoms & molecules
many-body & solid-state physics
instruments of quantum optics &coherent control
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quantum-Hall physicsquantum-Hall physics
Ф
filling factor
• quantum effects: ~ 1 =
N # flux quanta ~N # atoms
(R / l )m 2
• hydrodynamics: >> 1
0
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