Phase measurements and persistent currents in a b interferometers
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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

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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.


  • 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!



h/e osc. –mesoscopic fluctuation.


h/2e osc. – impurity-ensemble average,

Altshuler, Aronov, Spivak, Sharvin2

The AB interferometer

Use 2-slit formula:

AB phase shift


Measure aa-ab(e.g. of a QD) from f dependence of I?

Semiconducting Quantum Dots


2D electron gas



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.





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


(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



Collector Voltage (a.u.)






Plunger Gate Voltage [V]










Magnetic Field [mT]










AB-oscillations along a resonance peak

Collector Voltage (a.u)




What is b??

What is the difference between 2-slit and the AB interferometer?



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


* 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


No interactions






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

SCIENCE 290, 79 2000

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);


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:



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



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


  • 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.

the end

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