Transport of ultracold fermions through a mesoscopic channel

1. Transport of ultracold fermions through a mesoscopic channel Martin Bruderer Wolfgang Belzig Quantum Transport Group University of Konstanz http://qt.uni.kn

2. 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

3. Conventional mesoscopic physics • Transport through quantum structures (e.g molecules) classical electrode classical electrode quantum system • Electrodes are infinite electron reservoirs • Transport through quantum system is a coherent process • Measure electronic current, fluctuations etc.

4. Quantum transport of electrons • Two fermionic reservoirs connected by a tiny channel, chemical potentials are fixed, difference = applied voltage 1 transverse channel Energy Electrons arriving in the energy window eV Transmissionprobability T • Average current given by Landauer formula Fermi-Dirac distribution Conductance = Transmission

5. 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

6. Quantum simulator for mesoscopic transport

7. Quantum simulator for transport • Setup suggested for ultracold atoms Use the theoretical tools of quantum transport… • 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)

8. Tight binding model • Lowest band of optical lattice, tight binding regime • Extra term HS describes incoherent and dissipative processes

9. Modified Landauer approach • Finite reservoirs of size M Equilibrium state • Chemical potential µ(t) changes with particle number N(t) invert relation to find µ(t) as a function of N(t) • Landauer formula becomes integro-differential equation for N(t) Martin Bruderer, W. Belzig, Phys. Rev. A 85, 013623 (2012)

10. Modified Landauer approach • Finite reservoirs of size M Equilibrium state • Chemical potential µ(t) changes with particle number N(t) Valid if μ(t) varies slowly compared to microscopic time scale ~ h/J0 invert relation to find µ(t) as a function of N(t) • Landauer formula becomes integro-differential equation for N(t) M. Bruderer, W. Belzig, Phys. Rev. A 85, 013623 (2012)

11. Transmission T(ε) • Standard machinery for calculating transmission Transmission Full Green’s function Coupling to reservoirs • System with constant hopping J0 C. Caroli, R. Combescot, P. Nozieres, and D. Saint-James, J. Phys. C 4, 916 (1971)

12. Strong coupling regime • System with constant J0and strong coupling • Constant transmission T0over bandwidth 4J0

13. Equilibration for constant T0 • Zero temperature and left reservoir initially full Ohm’s 2nd law I = U/R nonlinear in NR • Equilibration on time scale

14. Thermal noise in RC model • Langevin equation for small fluctuations Thermal noise Damping term (constant Transmission) infinite reservoir Damping term (general) ü Fluctuation-dissipation theorem at equilibrium

15. Transmission engineering

16. Engineering Transmission and Quantum Noise Current and current fluctuations for different potential strengths

17. Quantum simulation • Engineer hopping Ji,j and site energies ei • Hückel method (LCAO) for calculating parameters • Transport with interactions on/off

18. Quantum simulation • Engineer hoppings Ji,j and site energies ei • Hückel method (LCAO) for calculating parameters • Transport with interactions on/off Examples of engineered conductance properties for 1D chains: Ji,j Ji,j ei ei Transmission resonances M. Bruderer, K. Franke, S. Ragg, W. Belzig, D. Obreschkow, Phys. Rev. A 85, 022312 (2012) (on perfectstatetansferthrough a spinchain, but same algorithm)

19. 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)

20. First experiments • Observe ohmic conductance • Resistance ~ 1/width of channel • “Voltage” falls off exponentially RC circuit model works Time scale of equilibration ~ 0.2s

21. Outlook: Quantum simulation of molecules • Create arbitrarily shaped lattices (Esslinger) Simulation of transport through complicated molecules B. Zimmermann et al., New J. Phys. 13, 043007 (2011)

22. Outlook: Quantum pumping Possible scheme for quantum pump • Need asymmetry for pumping • Left half is driven • Right half is shifted Non-adiabatic limit ω >>J0 (preliminary results) Sizeable difference in chemical potential Δμ

23. 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)

24. Blank

25. Effect of reservoirs • Tune structure and shape of reservoirs • DOS affects broadening, level shift and compressibility

26. Modern atomic physics

27. 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)

28. Summary Quantum Transport Atomic Physics There is some overlap.

29. 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

30. Tuning interactions • Atoms neutral => no Coulomb interaction • Interaction determined by s-wave scattering • Tune interaction via Feshbach resonance bound state close to scattering energy Carlos A. R. Sa de Melo, Physics Today, October 2008, p.45

31. Example • Fermions in 3d optical lattice • Lowest Bloch band is occupied • Release atoms from trap • Take picture of first Brillouin zone Tilman Esslinger, Ann. Rev. Cond. Mat. Phys. 1, 129 (2010) Immanuel Bloch, Nature Physics 1, 23 (2005). Ultracold fermions are warmer than electrons