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Phase measurements and Persistent Currents in A-B interferometers

Phase measurements and Persistent Currents in A-B interferometers. Yoseph Imry The Weizmann Institute In collaboration with Amnon Aharony , Ora Entin-Wohlman (TAU), Bertrand I. Halperin (HU), Yehoshua Levinson (WIS)

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Phase measurements and Persistent Currents in A-B interferometers

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  1. Phase measurements and Persistent Currents in A-B interferometers Yoseph Imry The Weizmann Institute In collaboration with Amnon Aharony, Ora Entin-Wohlman (TAU), Bertrand I. Halperin (HU), Yehoshua Levinson (WIS) Peter Silvestrov (Leiden) and Avraham Schiller (HUJ). Inspired by results of A. Jacoby, M. Heiblum et al. Discussions with: J. Kotthaus, A. stern, J. von Delft, and The late A. Aronov.

  2. Outline • The Aharonov-Bohm (AB) interferometer, with a Quantum dot (QD) • Experiment: Open vs closed ABI. • Theory: Intrinsic QD, (Fano) ,Closed ABI+ QD, Open ABI + QD • (The sensitivity of the phase to Kondo correlations.) • Mesoscopic Persistent Currents • The Holstein Process • Phonon/photon induced persistent current • Conclusions PRL 88, 166801 (2002); PRB 66, 115311 (2002); PRL 90, 106602 , 156802 (2003), 91, 046802, (2003), cond-mat/0308382, 0311609

  3. Two-slit interference--a quintessential QM example: “Two slit formula” When is it valid???

  4. A. Tonomura: Electron phase microscopy Each electron produces a seemingly random spot, but: Single electron events build up to from an interference pattern in the double-slit experiments.

  5. Closed system! scatterer scatterer h/e osc. –mesoscopic fluctuation. Compare: h/2e osc. – impurity-ensemble average, Altshuler, Aronov, Spivak, Sharvin2

  6. The AB interferometer Use 2-slit formula: AB phase shift 2 Measure aa-ab(e.g. of a QD) from f dependence of I?

  7. Semiconducting Quantum Dots Red=semiconducting 2D electron gas White=insulating Blue=metal

  8. Model for Quantum Dot: • Basic model for “intrinsic” QD: • On QD: single electron states plus interactions. • QD connected to 2 reservoirs via leads. • No interactions on the leads. QD S D Transmission:

  9. Transmission through a “QD” Landauer conductance: How to measure the “intrinsic” phase a? ??? ??

  10. Solid-State Aharonov-Bohm interferometers (interference effects in electronic conduction) Landauer formula

  11. ? Higher harmonics?

  12. The Onsager (Casimir)(1931) relations: Time reversal symmetry + Unitarity (conservation of Electron number) Phase rigidity holds for CLOSED Systems! (e.g. M. Buttiker and Y.I., J. Phys.C18, L467 (1985), for 2-terminal Landauer) 2-slit formula no good??

  13. For 2-slit formula, must use (HOW?) OPEN (non-unitary) interferometer! Nature 385, 417 (1997) See: Hackenbroich and Weidenmuller

  14. 8.5 8.0 Collector Voltage (a.u.) 7.5 V 7.0 -0.58 -0.56 Plunger Gate Voltage [V] -15 -10 -5 0 5 10 15 I A Magnetic Field [mT] V C C E B F P B E AB-oscillations along a resonance peak Collector Voltage (a.u)

  15. G(f) A B What is b??

  16. What is the difference between 2-slit and the AB interferometer? D S 2-slit: NO reflections From S or D: Waves MUST be Reflected from S and D K real

  17. Theory, Three results: * “Intrinsic” QD transmission: can deduce a! * Closed AB interferometer: one can measure the intrinsic phase a, without violating Onsager! * Open AB interferometer: the phase shift bdepends on how one opens the system, but there exist openings that give a! PRL 88, 166801 (2002); PRB 66, 115311 (2002); PRL 90, 156802 (2003); cond-mat/0308382

  18. Example: No interactions V f f

  19. f 8p

  20. Phase increases by  around the Kondo resonance, sticks at /2 on the resonance

  21. SCIENCE 290, 79 2000

  22. A-B Flux in an isolated ring • A-B flux equivalent to boundary condition. • Physics periodic in flux, period h/e (Byers-Yang). • “Persistent currents”exist due to flux (which modifies the energy-levels). • They do not(!!!) decay by impurity scattering (BIL).

  23. Early history of normal persistent currents L. Pauling: “The diamagneticAnisotropy of Aromatic molecules”, J. Chem. Phys. 4, 673 (1936); F. London: “Theorie Quantique des Courants Interatomiques dans les Combinaisons aromatiques”, J. Phys. Radium 8, 397 (1937); Induced currents in anthracene

  24. Thermodynamic persistent current in one-dimensional ring zero temperature

  25. `normal’ thermodynamic currents in response to a phase I. O. Kulik: “Flux Quantization in Normal Metals”, JETP Lett. 11, 275 (1970); weak-disorder M. Buttiker, Y. Imry, and R. Landauer: “Josephson Behavior in Small Normal One-dimensional Rings”, Phys. Lett. 96A, 365 (1983): ELASTIC SCATTERING IS OK! persistent currents in impure mesoscopic systems (BUT: coherence!!!)

  26. Persistent current induced by a flux of phonons/photons Due to Holstein 2nd order process (boson emission and absorption), generalizing previous work (discrete and equilibrium case) with Entin-Wohlman, Aronov and Levinson.  boson number (if decoherence added, T, DW fixed…)! Leads make it O(2), instead of O(3) for discrete case. Sign opposite to that of electrons only. Process retains coherence!

  27. Persistent currents in Aharonov-Bohm interferometers: Coupling to an incoherent sonic/em source does the electron-phonon interaction have necessarily a detrimental effect on coherence-related phenomena? (as long as the sonic/em source does not destroy coherence) T. Holstein: “Hall Effect in Impurity Conduction”, Phys. Rev. 124, 1329 (1961);

  28. The Holstein process-invoking coupling to phonons (energy conservation with intermediate state!) coupling with a continuum, with exact energy conservation-> the required imaginary (finite!) term

  29. the Holstein process--doubly-resonant transitions For DISCRETE I and j The transition probability through the intermediate site requires two phonons (at least)

  30. The Holstein mechanism-consequences The transition probability—dependence on the magnetic flux result from interference! 1. When used in the rate equations for calculating transport coefficients yields a term odd in the flux, i.e., the Hall coefficient. 2. Coherence is retained.

  31. Violation of detailed balance Persistent current at thermal equilibrium

  32. phonon-assisted transition probabilities charge conservation on the triad- the difference is odd in the AB flux (phonon-assisted) persistent current- does not violate the Onsager-Casimir relations!

  33. Detailed calculation polaron transformation the current: Debye-Waller factor O. Entin-Wohlman, Y. I, and A. Aronov, and Y. Levinson (‘95)

  34. persistent currents and electron-phonon coupling in isolated rings-summary -reduction due to Debye-Waller factor; -counter-current due to doubly-resonant (energy-conserving) transitions, which exist only at T>0. non-monotonic dependence on temperature

  35. manipulating the orbital magnetic moment by an external radiation phonon modes of doubly-resonant transitions all phonon modes O. Entin-Wohlman, YI, and A. Aronov, and Y. Levinson, (‘95)

  36. Using boson-assisted processesbetween two leads • Quantum analogue of “peristaltic pump”, to transfer charge between the leads. • We will discuss the flux-sensitive circulating current produced by the boson (incoherent) source.

  37. `open’ interferometers What is left of the Holstein mechanism? Can the current be manipulated by controlling the radiation?

  38. `open’ interferometers-the model circulating current:

  39. Method of calculation All interactions are confined to the QD Use Keldysh method to find all partial currents Express all partial currents in terms of the exact (generally, un-known) Green fn. on QD Use current conservation to deduce relations on the QD Green fn.

  40. Coupling to a phonon source Debye-Waller factor dot occupation elec.-ph. coupling Bose occupations phonon frequency L. I. Glazman and R. I. Shekhter , JETP 67, 163 (‘88)

  41. Acousto-magnetic effect in open interferometers (as compared to the Holstein process in closed interferometers) Both controllable by boson intensity -reduction due to Debye-Waller factor; -counter-current due to doubly-resonant (energy-conserving) transitions, which exist only at T>0. operative at a specific frequency-band Original Holstein process: One virtual and one real phonon -reduction due to Debye-Waller factor; -no need for exact resonance conditions, exists also at T=0. -no need for 2nd “real” phonon. operative in a wide frequency-band open ring: single (virtual) phonon

  42. Conclusions • Experimentalists and theorists benefit talking to each other! • THREE Ways to determine transmission phase. • Phase measured in the open AB interferometer depends on method of opening; Need experiments which vary the amount of opening; must optimize • One CAN obtain the QD phase from dot’s transmission and from closed interferometers! -- Need new fits to data. • Phase is moresensitive to Kondo correlations than transmission. • Possible to “pump” persistent currents in open and closed ABI’s by phonons/photons. Differences between the two.

  43. the end

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