Tunneling in Complex Systems: From Semiclassical Methods to Monte Carlo simulations Joachim Ankerhold Theoretical condensed matter physics University of Freiburg Germany „Challenges in Material Sciences“ Hanse-Kolleg, February 16/17, 2006

Barrier transmission: Scattering

Semiclassics (WKB): Action of a periodic path in the inverted barrier with Energy -E Equivalent:

Alpha-Decay (Gamow)

Tunneling rate: Density of states Probability distribution Incoherent tunneling from a reservoir Total rate:

Scanning tunneling microscope SiC (0001) 3  3 surface Tip Sample

x 0 d Tunneling current (Temperature = 0) Tunneling resistance: Tunneling resistance Exponential sensitivity

Tunneling in NH 3 x Friedrich Hund 1926:

Coherent tunneling H N H H E[1/cm] Energy doublets

Incoherent tunneling in presence of a dissipative environment

Example: Josephson-junction phase difference V( j ) Applied current: Potential energy: (Josephson 1961) Particle in a periodic potential

Macroscopic quantum tunneling phase difference Tunneling of a collective degree of freedom Squids Vortices Nanomagnets Superfluids Bose-Einstein Condensates potential energy

1 m m Environment: Electromagnetic modes Groupe Quantronique, CEA Saclay

Decay rate of metastable systems Tunneling rate in presence of thermal environment: (Leggett et al) Decay channels: thermal activation quantum tunneling

Open quantum systems ++ System + reservoir: reduced density

Path integrals Feynman: “Sum over all paths“

Path integrals Feynman: “Sum over all paths“ Density matrix:

Influence functional Influence functional: describes interaction with environment Path integral in imaginary time:

Semiclassics: Periodic orbits in the inverted barrier with period | wellbarrier Thermal activation

Semiclassics: Periodic orbits in the inverted barrier with period | wellbarrier | wellbarrier Quantum tunnelingThermal activation

Devoret et al, 1988 Experiment

Thermal activation Quantum tunneling Experiment

Rate processes Rate theory in JJ equivalent to rate theory for  chemical reactions  diffusion of interstitials in metals  collaps of BECs with attractive interactions  proton transfer  JJ as detectors for: read-out in quantum bit devices measurement of non-Gaussian electrical noise

Tunneling of a qubit: Crossing of surfaces ? Flip: Smaller barrier larger rate ? Landau-Zener transitions „under“ the barrier: MQT of a Spin JA et al, PRL 91, (2003) Vion et al & JA, PRL 94, (2005)

Tunneling in the system and Tunneling in the phonon environment

Large Molecules: Photosynthesis 2 nm

Photosynthesis: Reaction center 2 nm

Photosynthesis: Reaction center Electron transfer fast: ~ 3ps efficient: 95% 2 nm

„Bottom up“ instead of „top down“: Molecular electronics Reed et al, 2002

Classical Marcus theory Polar environment: Fluctuating polarization electronic tunnelingactivation energy Marcus et al, 1985

Classical Marcus theory Polar environment: Fluctuating polarization electronic couplingactivation energy Low T: Nuclear tunneling

Open quantum systems: Nonequilibrium dynamics ++ System + reservoir: reduced dynamics

Reduced dynamics paths Path integrals: Paths in real and imaginary time

Reduced dynamics paths Influence functional: self-interactions non-local in time In general no simple equation of motion ! Mak, Egger, JCP 1995; Mühlbacher & JA, JCP 2004, 2005

Redfield-Equation 2. order perturbation theory in coupling  powerful method for many chemical systems  numerically efficient  weak friction, higher temperatures  sufficiently fast bath modes

How to evaluate high-dimensional integrals? Monte Carlo: Stochastic evaluation (numerically exact) MC weight Distributed according to MC weight ( K >> 1 )

Electron transfer along molecular wires: Tight binding system Davis, Ratner et al, Nature 1998 D A In general: d localized states

Real-time Quantum Monte Carlo Dicretization of time (Trotter)

Real-time Quantum Monte Carlo System: d orthonormal states At each time step: d different configurations possible d-possible orientations at each time step= configurations

Real-time Quantum Monte Carlo System: d orthonormal states At each time step: d different configurations possible Important sampling over spin chains Convergence:

Real-time Quantum Monte Carlo Integrand oscillates: Dynamical sign problem Treat subspace exactly: Reduction of Hilbert space to be sampled Mak et al, PRB 50, (1994); Mühlbacher & JA, JCP 121, (2004); ibid 122, (2005) Quantum mechanicslives from interferences ! Wave mechanics lives from interferences

Coherent / Incoherent dynamics

Assembling of molecular wires Davis, Ratner et al, Nature 1998 D A Not an ab initio method: Structure Dynamics

Population dynamics:

Molecular wire: Diffusion versus Superexchange qm class

Molecular wire: Phonon tunneling vs. Superexchange Mayor et al, Angew. Chemie 2002 Mühlbacher & JA, JCP 122, (2005) qm class

Park et al, Science 2002 Tunneling in presence of Charging effects: Coulomb-blockade

Quantum dots: artificial molecules

Dissipative Hubbard system Two charges with opposite spin: Polarization operator

Non-Boltzmann equilibrium Charges on same site U > 0 Charges on different sites ???

Non-Boltzmann equilibrium Mühlbacher, JA, Komnik, PRL 95, (2005)

Non-Boltzmann equilibrium Mühlbacher, JA, Komnik, PRL 95, (2005) Invariant subspace bosons „Coherent“ channels for faster transfer

Summary and Conclusions Nanosystems show a variety of tunneling phenomena Strongly influenced by the surrounding Semiclassics: very successful for mesoscopics Exact reduced dynamics: Real-time Monte Carlo L. Mühlbacher M. Duckheim H. Lehle M. Saltzer Thanks