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

Kaiserslautern, April 2006 quantum 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) Lorentz-force on electron: stationary current: Hall resistance: Dirac flux quantum 2

Kaiserslautern, April 2006 Landau levels

Kaiserslautern, April D electrons in magnetic fields: Landau levels coordinate transformation: Hamiltonian: R X electron center of cyclotron motion radial vector of cyclotron motion commutation relations:

Kaiserslautern, April D electrons in magnetic fields: Landau levels mapping to oscillator: H = h  R² / 2 l² = h  ( a a + ½ ) cc m † Landau levels

Kaiserslautern, April D electrons in magnetic fields: Landau levels typical scales: length magnetic length energy cyclotron frequency

Kaiserslautern, April D electrons in magnetic fields: Landau levels 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 D electrons in magnetic fields: Landau levels wavefunction of lowest Landau level (LLL) in symmetric gauge symmetric gauge Landau gauge introduce complex coordinate LLL analytic b

Kaiserslautern, April D electrons in magnetic fields: Landau levels angular momentum of Landau levels: eigenstates of n´ th Landau level: angular momentum states of LLL:

Kaiserslautern, April D electrons in magnetic fields: Landau levels j wavefunction:

Kaiserslautern, April 2006 Integer Quantum Hall effect

Kaiserslautern, April 2006 Integer Quantum Hall effect spinless (for simplicity) and noninteracting electrons: Pauli principle Slater determinant:

Kaiserslautern, April 2006 Integer Quantum Hall effect compressibility: at integer fillings:

Kaiserslautern, April 2006 Integer Quantum Hall effect Hall current: Heisenberg drift equations of cycoltron center no plateaus ?!

Kaiserslautern, April 2006 Integer Quantum Hall effect Hall plateaus: impurities gap ! impurities pin electrons to localized states  electrons in impurity states do not contribute to current gap  impurity states fill first

Kaiserslautern, April 2006 Fractional Quantum Hall effect

Kaiserslautern, April 2006 Fractional 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

Kaiserslautern, April 2006 Fractional 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 effect fractional Hall plateaus: fractional Hall states are gapped = 1 = 1/3 = 1/5 = 1/7

Kaiserslautern, April 2006 composite particle picture of FQHE

Kaiserslautern, April 2006 composite particle = electron + m magnetic flux quanta composite 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 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 AB AB

Kaiserslautern, April 2006 composite 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 

Kaiserslautern, April 2006 composite particle picture of FQHE Jain hierarchy: experiment: FQHE also for composite fermion picture: since 

Kaiserslautern, April 2006 FQHE for interacting bosons FQHE for interacting bosons

Kaiserslautern, April 2006 FQHE for interacting bosons exact diagonalization  FQH effect for Laughlin state for point interaction composite fermions: boson + single flux quantum + = IQHE for composite fermions

Kaiserslautern, April 2006

effective magnetic fields in rotating traps

Kaiserslautern, April 2006 atoms in dark states |1>|2> |0> γ Ω Ω s p 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

Kaiserslautern, April 2006 R. Dum & M. Olshanii, PRL 76, 1788 (1996) transformation to local adiabatic basis:  gauge potential A + scalar potential |1>|2> |0> ΩΩ s p center of mass motion of atoms in dark states space-dependent dark states & atomic motion:

Kaiserslautern, April 2006 effective vector potential & magnetic field relative momentum vector difference of „center of mass“ of light beams relative orbital angular momentum needed ! (i) magnetic fields ΩΩ s p

Kaiserslautern, April 2006 magnetic fields: (a) vortex light beams G. Juzeliūnas and P.Öhberg, PRL 93, (2004) P. Öhberg, J. Ruseckas, G. Juzeliunas, M.F. PRA 73, (2006) external trap B V ratio of fields eff

Kaiserslautern, April 2006 magnetic fields: (b) shifted light beams x y z Quantum-Hall effect in non-cylindrical systems non-stationary situation possible (current in z) B V eff =    x xx

Kaiserslautern, April 2006 (ii) non-Abelian gauge fields J. Ruseckas, G. Juzeliunas, P. Öhberg, M.F. Phys.Rev.Lett (2005) more than one relevant adiabatic state ! TRIPOD scheme DD 12 Ω 2 x 2 vector matrix

Kaiserslautern, April 2006 magnetic monopole field Ω Ω Ω singularity lines  point singularity at the origin

Kaiserslautern, April 2006 summary 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)

Kaiserslautern, April 2006 quantum gases as many-body model systems 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 systems quantum-Hall physics: rotating traps vortices, vortex lattices; lowest Landau level

Kaiserslautern, April 2006 external trap B V magnetic fields: (a) vortex light beams ratio of fields eff

Kaiserslautern, April 2006 ultra-cold atoms & molecules many-body & solid-state physics instruments of quantum optics & coherent control

Kaiserslautern, April 2006 quantum-Hall physics Ф filling factor quantum effects: ~ 1  =  N # flux quanta ~ N # atoms (R / l ) m 2 hydrodynamics: >> 1 0