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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

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
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
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
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
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
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
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

A b flux in an isolated ring
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

`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
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 processes between two leads
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?

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.