Transport of ultracold fermions through a mesoscopic channel


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Transport of ultracold fermions through a mesoscopic channel

  • 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.
  • classical
  • electrode
  • classical
  • electrode
  • quantum system

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
  • Fermi-Dirac distribution
  • 1 transverse channel
  • Transmissionprobability T
  • Energy
  • Electrons arriving in
  • the energy window eV

Full counting statistics of quantum transport

  • Full statistical properties of charge transfer (Levitov-Lesovik 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 particle-like and wave-like

Quantum simulator for mesoscopic transport

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)

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)
  • Equilibrium state
  • Chemical potential µ(t) changes with particle number N(t)
  • Landauer formula becomes integro-differential equation for N(t)

Modified Landauer approach

  • Finite reservoirs of size M
  • M. Bruderer, W. Belzig, Phys. Rev. A 85, 013623 (2012)
  • Equilibrium state
  • Chemical potential µ(t) changes with particle number N(t)
  • Landauer formula becomes integro-differential 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. Saint-James, J. Phys. C 4, 916 (1971)
  • Standard machinery for calculating transmission
  • Transmission
  • Coupling to reservoirs
  • Full Green’s function
  • 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
  • Ohm’s 2nd law I = U/R
  • nonlinear in NR
  • Equilibration on time scale

Thermal noise in RC model

  • Fluctuation-dissipation theorem at equilibrium
  • Damping term
  • Thermal noise

Thermal noise in RC model

  • Langevin equation for small fluctuations
  • Fluctuation-dissipation theorem at equilibrium
  • Damping term
  • (constant Transmission)
  • Thermal noise
  • Damping term (general)
  • infinite reservoir

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)
  • Ji,j
  • i
  • Ji,j
  • i
  • Transmission resonances
  • 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
  • Non-adiabatic 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, 1D-wires, spin-orbit
  • 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

Effect of reservoirs

  • 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 Transport
  • Atomic Physics
  • There is some overlap.

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
  • Solid State System
  • Atomic System
  • Electrons
  • Fermionic Atoms
  • Spin
  • Internal States
  • Crystal Lattice
  • Optical Lattice
  • Coulomb Interaction
  • Contact Interaction
  • 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 s-wave 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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