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

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

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

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

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

14. April 2003 Quantum Mechanics on the Large Scale Banff, Alberta 6 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 7 Schmid model: particle in a tilted cosine potential TB limit  Current-biased Josephson junction (charge-phase duality)  Impurity scattering in 1d quantum wire  Point contact tunneling between quantum Hall edges  Boundary sine-Gordon model  Exact selfduality in the Ohmic scaling limit  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 8 Density matrix: Global system: Reduced description: partial trace time-local interactionstime-nonlocal interactions reduced dynamics: full dynamics: W(t)

14. April 2003 Quantum Mechanics on the Large Scale Banff, Alberta 9 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 10 Keldysh contour Laplace representation in the limit :

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

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

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

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

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

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

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

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

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

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

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

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