1. Slava Kashcheyevs Colloquium at Physikalisch-Technische Bundesanstalt (Braunschweig, Germany) November 13 th, 2007 Converging theoretical perspectives on charge pumping

2. Pumping: definitions rectification photovoltaic effect photon-assisted tunneling ratchets Pumping overlaps with:Interested in “small” pumps to witness: quantum interference single-electron charging f I

3. Outline Adiabatic Quantum Pump Thouless pump Brouwer formula Resonances and quantization Beyond the simple picture Non-adiabaticity (driving fast) Rate equations and Coulomb interaction Single-parameter, non-adiabatic, quantized

4. Adiabatic pump by Thouless If the gap remains open at all times, I = e f  (exact integer) Argument is exact for an infinite system DJ Thouless, PRB 27, 6083 (1983)

5. Adiabatic Quantum Pumping a phase-coherent conductor Pump by deforming Change of interference pattern can induce “waves” traveling to infinity

6. Brouwer formula gives I in terms of Brouwer formula: “plug-and-play” Vary shape via X1(t), X2(t),.. Solve for “frozen time” scattering matrix

7. Brouwer formula gives I in terms of Brouwer formula: “plug-and-play”

8. Brouwer formula gives I in terms of Brouwer formula: “plug-and-play” Brouwer formula gives I = Depends on a phase Allows for a geometric interpretation Need  2 parameters! “B”“B”

9. Outline Adiabatic Quantum Pump Thouless pump Brouwer formula Resonances and quantization Beyond the simple picture Non-adiabaticity (driving fast) Rate equations and Coulomb interaction Single-parameter, non-adiabatic, quantized

10. Idealized double-barrier resonator Tuning X1 and X2 to match a resonance I  e f, if the whole resonance line encircled Resonances and quantization Y Levinson, O Entin-Wohlman, P Wölfle Physica A 302, 335 (2001) X1 X2

11. How can interference lead to quantization? Resonances correspond to quasi-bound states Proper loading/unloading gives quantization X1 X2 Resonances and quantization V Kashcheyevs, A Aharony, O Entin-Wohlman, PRB 69, 195301 (2004)

12. Outline Adiabatic Quantum Pump Thouless pump Brouwer formula Resonances and quantization Beyond the simple picture Non-adiabaticity (driving fast) Rate equations and Coulomb interaction Single-parameter, non-adiabatic, quantized

13. Driving too fast: non-adiabaticity What is the meaning of “adiabatic”? Can develop a series: Q: What is the small parameter? Thouless: staying in the ground state Brouwer: a gapless system! O Entin, A Aharony, Y Levinson PRB 65, 195411 (2002)

14. M Moskalets, M Büttiker PRB 66, 205320 (2002) Floquet scattering for pumps Adiabatic scattering matrix S(E; t) is “quasi-classical” Exact description by Typical matrix dimension (# space pts)  (# side-bands) LARGE! h f

15. M Moskalets, M Büttiker PRB 66, 205320 (2002) Adiabaticity criteria Adiabatic scattering matrix S(E; t) Floquet matrix Adiabatic approximation is OK as long as ≈ Fourier T.[ S(E; t)] For a quantized adiabatic pump, the breakdown scale is f ~ Γ (level width) h f

16. Outline Adiabatic Quantum Pump Thouless pump Brouwer formula Resonances and quantization Beyond the simple picture Non-adiabaticity (driving fast) Rate equations and Coulomb interaction Single-parameter, non-adiabatic, quantized

17. A different starting point Consider states of an isolated, finite device Tunneling to/from leads as a perturbation! Rate equations: concept ΓRΓR ΓLΓL

18. Rate equations: an example For open systems & Thouless pump, see GM Graf, G Ortelli arXiv:0709.3033 ε 0 - i (Γ L +Γ R ) Loading/unloading of a quasi-bound state Rate equation for the occupation probability Interference in an almost closed system just creates the discrete states!

19. Backbone of Single Electron Transistor theory Conditions to work: Tunneling is weak: Γ << Δε or Ec No coherence between multiple tunneling events:Γ << k B T Systematic inclusion of charging effects! Rate equations are useful! DV Averin, KK Likharev “Single Electronics” (1991) CWJ Beenakker PRB 44, 1646 (1991)

20. Outline Adiabatic Quantum Pump Thouless pump Brouwer formula Resonances and quantization Beyond the simple picture Non-adiabaticity (driving fast) Rate equations and Coulomb interaction Single-parameter, non-adiabatic, quantized

21. Single-parameter non-adiabatic quantized pumping B. Kaestner, VK, S. Amakawa, L. Li, M. D. Blumenthal, T. J. B. M. Janssen, G. Hein, K. Pierz, T. Weimann, U. Siegner, and H. W. Schumacher, arXiv:0707.0993

22. “Roll-over-the-hill”

23. V 2 (mV) Fix V 1 and V 2 Apply V ac on top of V 1 Measure the current I(V 2 ) V1V1 V2V2 V1V1 V2V2 Experimental results

24. A simple theory ε0ε0 Given V 1 (t) and V 2, solve the scattering problem Identify the resonance ε 0 (t), Γ L (t) and Γ R (t) Rate equation for the occupation probability P(t)

25. A: Too slow (almost adiabatic) Enough time to equilibrate Charge re-fluxes back to where it came from → I ≈ 0 ω<<Γ

26. B: Balanced for quantization Tunneling is blocked, while the left-right symmetry switches to opposite Loading from the left, unloading to the right → I ≈ e f ω>>Γ

27. C: Too fast Tunneling is too slow to catch up with energy level switching The chrage is “stuck” → I ≈ 0 ω

28. A general outlook I / (ef)