 Martin Bruderer
 Wolfgang Belzig
 Quantum Transport Group
 University of Konstanz
 http://qt.uni.kn
Short overview  Motivation: Quantum Simulation of Quantum Transport
 Quantum transport
 Landauer formula, full counting statistics
 Quantum shot noise
 Transport of ultracold fermions
 Modified Landauer approach
 RC circuit analogy
 First experiments
 Quantum engineering & quantum pumping
Conventional mesoscopic physics  Transport through quantum structures (e.g molecules)
 Electrodes are infinite electron reservoirs
 Transport through quantum system is a coherent process
 Measure electronic current, fluctuations etc.
Quantum transport of electrons  Two fermionic reservoirs connected by a tiny channel, chemical potentials are fixed, difference = applied voltage
 Average current given by Landauer formula
 Conductance = Transmission
 Transmissionprobability T
 Electrons arriving in
 the energy window eV
Full counting statistics of quantum transport  Full statistical properties of charge transfer (LevitovLesovik formula)
 Current and current noise
 At zero temperature and bias voltage V
 For T=1 the resistance is finite and quantized, but the fluctuations vanish!
 Quantum shot noise varies between particlelike and wavelike
Quantum simulator for mesoscopic transport  Setup suggested for ultracold atoms
 Mesoscopic transport with ultracold fermions
 Reservoirs are finite size
 + Dynamics is slow (~ milliseconds)
 + System is perfectly clean and versatile
 + Optical observation of the entire system
 R.A. Pepino, J. Cooper, D.Z. Anderson, M.J. Holland, Phys. Rev. Lett. 103, 140405 (2009)
Quantum simulator for transport  Setup suggested for ultracold atoms
 Mesoscopic transport with ultracold fermions
 Reservoirs are finite size
 + Dynamics is slow (~ milliseconds)
 + System is perfectly clean and versatile
 + Optical observation of the entire system
 R.A. Pepino, J. Cooper, D.Z. Anderson, M.J. Holland, Phys. Rev. Lett. 103, 140405 (2009)
 Use the theoretical tools
 of quantum transport…
Tight binding model  Extra term HS describes incoherent and dissipative processes
Modified Landauer approach  Finite reservoirs of size M
 Martin Bruderer, W. Belzig, Phys. Rev. A 85, 013623 (2012)
 Chemical potential µ(t) changes with particle number N(t)
 Landauer formula becomes integrodifferential equation for N(t)
Modified Landauer approach  Finite reservoirs of size M
 M. Bruderer, W. Belzig, Phys. Rev. A 85, 013623 (2012)
 Chemical potential µ(t) changes with particle number N(t)
 Landauer formula becomes integrodifferential equation for N(t)
 invert relation to find
 µ(t) as a function of N(t)
 Valid if μ(t) varies slowly compared
 to microscopic time scale ~ h/J0
Transmission T (ε)  C. Caroli, R. Combescot, P. Nozieres, and D. SaintJames, J. Phys. C 4, 916 (1971)
 Standard machinery for calculating transmission
 System with constant hopping J0
Strong coupling regime  System with constant J0 and strong coupling
 Constant transmission T0 over bandwidth 4J0
Equilibration for constant T 0  Zero temperature and left reservoir initially full
 Equilibration on time scale
Thermal noise in RC model  Fluctuationdissipation theorem at equilibrium
Thermal noise in RC model  Langevin equation for small fluctuations
 Fluctuationdissipation theorem at equilibrium
 Damping term
 (constant Transmission)
Transmission engineering Engineering Transmission and Quantum Noise  Current and current fluctuations for different potential strengths
Quantum simulation  Engineer hopping Ji,j and site energies i
 Hückel method (LCAO) for calculating parameters
 Transport with interactions on/off
Quantum simulation  Engineer hoppings Ji,j and site energies i
 Hückel method (LCAO) for calculating parameters
 Transport with interactions on/off
 M. Bruderer, K. Franke, S. Ragg, W. Belzig, D. Obreschkow, Phys. Rev. A 85, 022312 (2012)
 (on perfect state tansfer through a spin chain, but same algorithm)
 Examples of engineered conductance properties for 1D chains:
First experiments  RC model has been implemented at ETH Zürich
 Weakly interacting fermions
 Interaction in reservoirs
 No interaction in channel
 Number imbalance ~ 20%
 Absorption image of atoms
 (A) Equilibrium
 (B) Different filling levels
 J.P. Brantut, J. Meineke, D. Stadler, S. Krinner and T. Esslinger, arXiv:1203.1927 (accept. at Science)
First experiments  Observe ohmic conductance

 Resistance ~ 1/ width of channel
 “Voltage” falls off exponentially
 RC circuit model works
 Time scale of equilibration ~ 0.2s
Outlook: Quantum simulation of molecules  Create arbitrarily shaped lattices (Esslinger)
 B. Zimmermann et al., New J. Phys. 13, 043007 (2011)
 Simulation of transport through complicated molecules
Outlook: Quantum pumping  Possible scheme for quantum pump
 Nonadiabatic limit ω >>J(preliminary results)
 Sizeable difference in chemical potential Δμ
Main points  Landauer formalism applicable to atomic systems
 Consistent description of finite reservoirs
 RC model in agreement with first experiments
 Quantum simulation of transport through molecules
 (Non)adiabatic quantum pumping
 Local interactions, 1Dwires, spinorbit
 M. Bruderer, W. Belzig, Phys. Rev. A 85, 013623 (2012)
 M. Bruderer, K. Franke, S. Ragg, W. Belzig, D. Obreschkow, Phys. Rev. A 85, 022312 (2012)
Blank  Tune structure and shape of reservoirs
 DOS affects broadening, level shift and compressibility
Modern atomic physics Implementation of reservoirs  How to introduce incoherent and dissipative processes
 Laser excites high energy fermions into second band
 Excited states decay by emission of phonons into superfluid
 Iterative application results in stable cold Fermi distribution
 A. Griessner, A.J. Daley, S.R. Clark, D. Jaksch, P. Zoller, Phys. Rev. Lett 97, 220403 (2006)
Summary Quantum simulator  Many body quantum systems are difficult to simulate
 Simulation of N spins using classical computers
 State described by 2^N amplitudes
 Need 64 × 2^N ~ 10^(0.3 × N) bits
 100 spins ~ ridiculous amount of memory
 Simulate solid state systems with ultracold atoms
 Same Hamiltonian
 Controllable parameters
 Simple preparation and measurment
Tuning interactions  Carlos A. R. Sa de Melo, Physics Today, October 2008, p.45
 Atoms neutral => no Coulomb interaction

 Interaction determined by swave scattering
 Tune interaction via Feshbach resonance
 bound state close
 to scattering energy
Example  Fermions in 3d optical lattice
 Lowest Bloch band is occupied
 Release atoms from trap
 Take picture of first Brillouin zone
 Ultracold fermions are
 warmer than electrons
 Tilman Esslinger, Ann. Rev. Cond. Mat. Phys. 1, 129 (2010)
 Immanuel Bloch, Nature Physics 1, 23 (2005).
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