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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
Kaiserslautern, April 2006
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
Kaiserslautern, April 2006
Landau levelsLandau levels
Kaiserslautern, April 2006
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:
Kaiserslautern, April 2006
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
Kaiserslautern, April 2006
2D electrons in magnetic fields: Landau 2D electrons in magnetic fields: Landau levelslevels
typical scales:
• length
BB BB
magnetic length
• energy
cyclotron frequency
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
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
Kaiserslautern, April 2006
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:
Kaiserslautern, April 2006
2D electrons in magnetic fields: Landau 2D electrons in magnetic fields: Landau levelslevels
j
wavefunction:
Kaiserslautern, April 2006
Integer Quantum Hall Integer Quantum Hall effecteffect
Integer Quantum Hall Integer Quantum Hall effecteffect
Kaiserslautern, April 2006
Integer Quantum Hall effectInteger Quantum Hall effect
spinless (for simplicity) and noninteracting electrons: Pauli principle
Slater determinant:
Kaiserslautern, April 2006
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
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
Kaiserslautern, April 2006
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
Kaiserslautern, April 2006
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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Kaiserslautern, April 2006
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
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
Kaiserslautern, April 2006
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
Kaiserslautern, April 2006
quantum-Hall physicsquantum-Hall physics
Ф
filling factor
• quantum effects: ~ 1 =
N # flux quanta ~N # atoms
(R / l )m 2
• hydrodynamics: >> 1
0