Relaxation and Decoherence in Quantum Impurity Models From

Relaxation and Decoherence in Quantum Impurity Models: From Weak to Strong Tunneling Ulrich Weiss Institute for Theoretical Physics University of Stuttgart Ø Quantum impurity models (spin-boson, Kondo, Schmid, BSG, . . ) Ø Dynamics Ø From weak to strong tunneling § Quantum relaxation § Decoherence H. Saleur (USCLA) A. Fubini (Florence) H. Baur (Stuttgart) 14. April 2003 Quantum Mechanics on the Large Scale Banff, Alberta 1

Electron transfer (ET): solvent tunneling donor acceptor biological electron transport molecular electronics quantum dots molecular wires charge transport in nanotubes 14. April 2003 Quantum Mechanics on the Large Scale Banff, Alberta bath dynamics dissipation decoherence classical rate theory Marcus theory of ET activationless ET inverted regime nonadiabatic ET 2

Spin-boson model with ultracold atoms: Recati et al. 2002 a 14. April 2003 Quantum Mechanics on the Large Scale Banff, Alberta b 3

Global system: System Heat bath T Physical baths: Phonons Conduction electrons (Fermi liquid) 1 d electrons (Luttinger liquid) BCS quasiparticles Electromagn env. (circuits, leads) Nuclear spins Solvent Electromagnetic modes Spectral density of the coupling: s 14. April 2003 >1 super-Ohmic =1 Ohmic <1 sub-Ohmic Quantum Mechanics on the Large Scale Banff, Alberta phonons (d > 1) e-h excitations RC transmission line 4

Truncated double well: TSS: T stochastic force: driven TSS: T Spin-boson Hamiltonian: stochastic force 14. April 2003 Quantum Mechanics on the Large Scale Banff, Alberta 5

Anisotropic Kondo model conduction band spin polarization conserved spin flip scattering Correspondence with spin-boson model: universal in the regime ferromagnetic Kondo regime antiferromagn. Kondo regime 14. April 2003 Quantum Mechanics on the Large Scale Banff, Alberta 6

Schmid model: particle in a tilted cosine potential TB limit v Current-biased Josephson junction (charge-phase duality) v Impurity scattering in 1 d quantum wire v Point contact tunneling between quantum Hall edges Ø Boundary sine-Gordon model Ø Exact selfduality in the Ohmic scaling limit Ø Scaling function for transport and noise at T=0 is known in analytic form A. Schmid, Phys. Rev. Lett. 51, 1506 (1983) P. Fendley, A. W. W. Ludwig, and H. Saleur, Phys. Rev. B 52, 8934 (1995) 14. April 2003 Quantum Mechanics on the Large Scale Banff, Alberta 7

Density matrix: Global system: Reduced description: full dynamics: W(t) time-local interactions 14. April 2003 partial trace reduced dynamics: time-nonlocal interactions Quantum Mechanics on the Large Scale Banff, Alberta 8

Tight-binding model: charges Influence functional: Absorption and emission of energy according to detailed balance 14. April 2003 Quantum Mechanics on the Large Scale Banff, Alberta 9

Keldysh contour Laplace representation in the limit 14. April 2003 : Quantum Mechanics on the Large Scale Banff, Alberta 10

Ohmic scaling limit: Spectral density: Pair interaction between tunneling transitions: Kondo scale: TSS model Schmid model at fixed Kondo scale 14. April 2003 Quantum Mechanics on the Large Scale Banff, Alberta 11

N=2: N=5: charges: scaling limit: 14. April 2003 friction noise (Gaussian filter) phase factor noise integral Quantum Mechanics on the Large Scale Banff, Alberta 12

Incoherent tunneling: golden rule limit: is probability for transfer of energy phase factor 14. April 2003 noise integral { to from phase factor Quantum Mechanics on the Large Scale Banff, Alberta }the bath noise integral 13

Order (1) + c. c. = + c. c. (2) + c. c. = + c. c. (3) + c. c. = + c. c. (4) = 14. April 2003 Quantum Mechanics on the Large Scale Banff, Alberta 14

Noise integrals: __ Ø Up-hill partial rates are zero Ø Scaling property general! particular! Formidable relations between the various noise integrals of same order l 14. April 2003 Quantum Mechanics on the Large Scale Banff, Alberta 15

Results: Schmid model: Ø Only minimal number of transitions contribute to the rate cancelled contributes Ø All rates can be reconstructed from the known mobility Ø Knowledge of all statistical fluctuations (full probability distribution) H. Saleur and U. Weiss, Phys. Rev. B 63, 201302(R) (2001) TSS model: Ø Exact relations between rates of the Schmid and TSS model 14. April 2003 Quantum Mechanics on the Large Scale Banff, Alberta 16

Weak-tunneling expansion Integral representation Im(z) C Re(z) H. Baur, A. Fubini, and U. Weiss, cond-mat/0211046 14. April 2003 Quantum Mechanics on the Large Scale Banff, Alberta 17

Strong-tunneling expansion The case K<1: Leading asymptotic term: 14. April 2003 Quantum Mechanics on the Large Scale Banff, Alberta 18

Strong-tunneling expansion The case K>1: 14. April 2003 Quantum Mechanics on the Large Scale Banff, Alberta 19

weak tunneling large bias 14. April 2003 Quantum Mechanics on the Large Scale Banff, Alberta strong tunneling small bias 20

weak tunneling small bias 14. April 2003 Quantum Mechanics on the Large Scale Banff, Alberta strong tunneling large bias 21

Decoherence Conjecture: holds in all known special cases Strong-tunneling expansion: 14. April 2003 Quantum Mechanics on the Large Scale Banff, Alberta 22