Kaiserslautern, April 2006 Quantum Hall effects - an introduction - AvH workshop, Vilnius, 03.09.2006 M. Fleischhauer

Kaiserslautern, April 2006

Quantum Hall effectsQuantum Hall effects- an introduction -- an introduction -

AvH workshop, Vilnius, 03.09.2006

M. Fleischhauer


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


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


Landau levelsLandau levels


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:



mapping to oscillator:

H = h R² / 2 l² = h ( a a + ½ )c cm†

Landau levels



typical scales:

• length

BB BB

magnetic length

• energy

cyclotron frequency



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



wavefunction of lowest Landau level (LLL) in symmetric gauge

symmetric gauge

Landau gauge

introduce complex coordinate

LLL

analytic

b



angular momentum of Landau levels:

eigenstates of n´th Landau level:

angular momentum states of LLL:



j

wavefunction:


Integer Quantum Hall Integer Quantum Hall effecteffect

Integer Quantum Hall Integer Quantum Hall effecteffect


Integer Quantum Hall effectInteger Quantum Hall effect

spinless (for simplicity) and noninteracting electrons: Pauli principle

Slater determinant:



compressibility:

at integer fillings:



Hall current:

Heisenberg drift equations of cycoltron center

no plateaus ?!



Hall plateaus: impurities

gap !

impurities pin electrons to localized states electrons in impurity states do not contribute to currentgap impurity states fill first


Fractional Quantum Hall Fractional Quantum Hall effecteffect

Fractional Quantum Hall Fractional Quantum Hall effecteffect


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



angular momentum of Laughlin wave function and filling factor

maximum single-particle angular momentum

filling factor of Laughlin state



fractional Hall plateaus:

fractional Hall states are gapped

= 1

= 1/3 = 1/5 = 1/7


composite particle picture of composite particle picture of FQHEFQHE

composite particle picture of composite particle picture of FQHEFQHE


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



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



= 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



Jain hierarchy:

• experiment: FQHE also for

composite fermion picture:

since


FQHE for interacting FQHE for interacting bosons bosons

FQHE for interacting FQHE for interacting bosons bosons


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



effective magnetic fields in rotating trapseffective magnetic fields in rotating traps


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


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:


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


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


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


(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


magnetic monopole field magnetic monopole field

Ω 1 2

3Ω

Ω

singularity lines

point singularity at the origin


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)


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


quantum gases as many-body model quantum gases as many-body model systemssystems

• quantum-Hall physics: rotating traps vortices, vortex lattices; lowest Landau level


external trapB

V

magnetic fields: (a) vortex light beamsmagnetic fields: (a) vortex light beams

ratio of fields

eff


ultra-cold atoms & molecules

many-body & solid-state physics

instruments of quantum optics &coherent control


quantum-Hall physicsquantum-Hall physics

Ф

filling factor

• quantum effects: ~ 1 =

N # flux quanta ~N # atoms

(R / l )m 2

• hydrodynamics: >> 1

0

Documents

Kaiserslautern, April 2006 Quantum Hall effects - an introduction - AvH workshop, Vilnius, 03.09.2006 M. Fleischhauer