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.
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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
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Measuring phase Quantum
Interference
4
Brick or wood
armor
5
Two-slit interference--a quintessential QM
example
Two slit formula
When is it valid???
6
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.
7

Closed system!
scatterer
scatterer
h/e osc. mesoscopic fluctuation.
Compare h/2e osc.
impurity-ensemble average, Altshuler, Aronov,
Spivak, Sharvin2
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The AB interferometer
Use 2-slit formula
AB phase shift
2
Measure aa- ab (e.g. of a QD) from f dependence
of I?
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Semiconducting Quantum Dots
Redsemiconducting
2D electron gas
Whiteinsulating
Bluemetal
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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
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Transmission through a QD
Landauer conductance
How to measure the intrinsic phase a?
???
??
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Interferometry on Quantum Dot
QD on one path in the AB interferometer
G(f)
Resonance
13
Solid-State Aharonov-Bohm interferometers (interfe
rence effects in electronic conduction)
Landauer formula
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?
Higher harmonics?
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Phase rigidity? Higher harmonics?
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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??
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For 2-slit formula, must use (HOW?) OPEN
(non-unitary) interferometer!
Nature 385, 417 (1997)
See Hackenbroich and Weidenmuller
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AB-oscillations along a resonance peak
Collector Voltage (a.u)
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G(f)
A
B
What is b??
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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
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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 b depends
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
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Theory I Intrinsic case
V
S
D
x
J
J
S
D
L
R
R
L
J
J
Ng and Lee, PRL 61, 1768 (1988)
a phase of dots Green function (Friedel)
MEASURE PHASE a!
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t
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Alternative Equations of Motion method
g
Solve for Gs
D
Self energy
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Example
JL QD JR
J
J
V
x
Tight binding, QD with 1 bound state, with
energy V Gate Voltage (no interactions)
Self energy
Gate Voltage VeD
2
Tsin a!
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V
x
Tight binding, QD with 4 bound states, whose
energy V depends linearly on the Gate Voltage
T
Hartree for Coulomb Blockade VV0Ultngt
a
2
Tsin a!
Fano interference
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Another example Fano resonances
PRB 62, 2188 (2000)
N5
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Theory II Closed AB interferometer
f
f
f
Renormalized self-energy
Real
Contain effects of reference path
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Theory II Closed AB interferometer

f
f
f
From clock counterclockwise contributions
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Theory II Closed AB interferometer

f
f
Depends explicitly on a and I (and on
interactions)
Measurements at different small I allow
extraction of a!
PRL 90, 156802 (2003) cond-mat/0308382
Kondo?
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Example No interactions
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Example No interactions
V
f
f
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f
8p
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Theory OPEN AB interferometer
A possible unitarity breaking, or opening
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Theory III OPEN AB interferometer
A possible unitarity breaking
a
b
Measured phase b depends on strength of coupling
to loss channel!
NOT GOOD
PRL 88, 166801 (2002)
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Try
?
R
T
R
T
Jx hopping to lossy channel, cf Hackenbroich
and Weidenmuller (6-terminal calc.)
R
T
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Jx
E
K is complex!
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A
B
b, a
Jx 0 .15 .5
.9 1.5
ba ONLY FOR INTERMEDIATE Jx!
PRB 66, 115311 (2002)
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Lossy channels
QD
Large Jx Large R Back and forth
rattling of electron narrower
resonances on the QD!
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Add leak from dot
Opening of QD itself Eliminates the Fano
zeroes In the transmission!
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Theory
?
Experiment
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Theory (with phase choice)?
Experiment
?
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Phase increases by ? around the Kondo resonance,
sticks at ?/2 on the resonance
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SCIENCE 290, 79 2000
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A-B Flux in an isolated ring

• A-B flux equivalent to boundary condition.
• Physics periodic in flux, period h/e
  (Byers-Yang).
• Persistent currentsexist due to flux (which
  modifies
• the energy-levels).
• They do not(!!!) decay by impurity scattering
  (BIL).

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Early history of normal persistent currents
L. Pauling The diamagnetic Anisotropy 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
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Thermodynamic persistent current in
one-dimensional ring
zero temperature
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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!!!)
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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!
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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) This
is a realistic experiment
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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)
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does the electron-phonon interaction have
necessarily a detrimental effect on
coherence-related phenomena?
T. Holstein Hall Effect in Impurity
Conduction, Phys. Rev. 124, 1329 (1961)
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phonon-assisted hopping conduction
(variable-range conductivity)
transition amplitude
energy levels are random impurity sites are
random
transition probability
---phonon energy
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phonon-assisted hopping conduction in the
presence of a magnetic field---orbital effect
transition amplitude
Aharonov-Bohm flux
transition probability
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transition amplitude (direct)
transition amplitude (indirect)
intermediate
initial
final
reminiscent of the Onsager-Casimir relations!
transition probability
even in the field
-Aharonov-Bohm flux
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the Holstein process
transition amplitude (direct)
transition amplitude (indirect)
transition probability
boson source
To make (at least) one of the amplitudes complex
(GOOD TO GET HALL EFFECT)
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The Holstein process-invoking coupling to phonons
(energy conservation with intermediate state!)
coupling with a continuum, with exact energy
conservation-gt the required imaginary (finite!)
term
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the Holstein process--doubly-resonant
transitions For DISCRETE I and j
The transition probability
through the intermediate site
requires two phonons (at least)
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The Holstein mechanism-consequences
The transition probabilitydependence 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.
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Violation of detailed balance
Persistent current at thermal equilibrium
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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!
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Detailed calculation
polaron transformation
the current
Debye-Waller factor
O. Entin-Wohlman, Y. I, and A. Aronov, and Y.
Levinson (95)
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persistent currents and electron-phonon coupling
in isolated rings-summary
-reduction due to Debye-Waller factor -counter-cu
rrent due to doubly-resonant (energy-conserving)
transitions, which exist only at Tgt0.
non-monotonic dependence on temperature
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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)
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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.

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open interferometers
What is left of the Holstein mechanism? Can the
current be manipulated by controlling the
radiation?
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open interferometers-the model
circulating current
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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.
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Detailed calculation
interacting quantum dot
d states
QD
k states
non-interacting reference site
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Detailed calculation (cont.)
current conservation
determination of the charge on the dot
single-particle property
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Self-energy due to connection of the QD to the
outside
line-width
energy-shift
coupling of the dot to the ring
interference
QD
interacting quantum dot
QD
transmission
coupling of the reference site to the ring
non-interacting reference site
reflection
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-all interactions are confined to the quantum
dot -use Keldysh technique to find all currents
exact (advanced) Green fn. on the dot
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-no interactions
resonance with width-gt
scattering (Friedel) phase shift
lt-dot occupation
E. Akkermans, A. Aurbach, J. E. Avron, and B.
Shapiro, (91)
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-with interactions
exact (advanced) Green fn. on the dot
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-with interactions, no bias
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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)
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Coupling to a phonon source (cont.)
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Putting everything together
total persistent current
the part due to the radiation
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Modification of the persistent current due to
radiation in terms of the Friedel phase
radiation enhanced
Debye-Waller factor
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Zero temperature
Debye-Waller
phonon emission
scattering phase shift
(Friedel phase)
equal chemical potentials
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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-cu
rrent due to doubly-resonant (energy-conserving)
transitions, which exist only at Tgt0.
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
T0. -no need for 2nd real phonon.
operative in a wide frequency-band
open ring
single (virtual) phonon
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Solid-State Aharonov-Bohm interferometers A tool
to probe quantum dots
can be manipulated by coupling to a sonic source
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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 dots
  transmission and from closed interferometers! --
  Need new fits to data.
• Phase is more sensitive to Kondo correlations
  than transmission.
• Possible to pump persistent currents in open
  and closed ABIs by phonons/photons. Differences
  between the two.

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the end