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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 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.
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
Two-slit interference--a quintessential QM example: “Two slit formula” When is it valid???
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.
Closed system! scatterer scatterer h/e osc. –mesoscopic fluctuation. Compare: h/2e osc. – impurity-ensemble average, Altshuler, Aronov, Spivak, Sharvin2
The AB interferometer Use 2-slit formula: AB phase shift 2 Measure aa-ab(e.g. of a QD) from f dependence of I?
Semiconducting Quantum Dots Red=semiconducting 2D electron gas White=insulating Blue=metal
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:
Transmission through a “QD” Landauer conductance: How to measure the “intrinsic” phase a? ??? ??
Solid-State Aharonov-Bohm interferometers (interference effects in electronic conduction) Landauer formula
? Higher harmonics?
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??
For 2-slit formula, must use (HOW?) OPEN (non-unitary) interferometer! Nature 385, 417 (1997) See: Hackenbroich and Weidenmuller
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)
G(f) A B What is b??
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
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
Example: No interactions V f f
f 8p
Phase increases by around the Kondo resonance, sticks at /2 on the resonance
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).
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
Thermodynamic persistent current in one-dimensional ring zero temperature
`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!!!)
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!
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);
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
the Holstein process--doubly-resonant transitions For DISCRETE I and j The transition probability through the intermediate site requires two phonons (at least)
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.
Violation of detailed balance Persistent current at thermal equilibrium
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!
Detailed calculation polaron transformation the current: Debye-Waller factor O. Entin-Wohlman, Y. I, and A. Aronov, and Y. Levinson (‘95)
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
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)
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.
`open’ interferometers What is left of the Holstein mechanism? Can the current be manipulated by controlling the radiation?
`open’ interferometers-the model circulating current:
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.
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)
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
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.