Institut de Physique et Chimie des Matériaux de Strasbourg

Rodolfo A. Jalabert

Transmission phase in quantum transport: disorder, chaos and correlation effects

Philippe Jacquod (Arizona)

Rafael A. Molina (Madrid)

Peter Schmitteckert (Karlsruhe)

Dietmar Weinmann (Strasbourg)

CAN WE MEASURE THE SCATTERING PHASE IN A QUANTUM DOT ?CAN WE MEASURE THE SCATTERING PHASE IN A QUANTUM DOT ?

Conductance of a two-lead Aharonov-Bohm interferometer:

Time reversal: T(Φ)=T(-Φ) the phase β is locked at 0 or π

Conductance of a quantum dotconnected to monochannel leads:

Different peaks are in phase

TWO-LEAD PHASE SENSITIVE MEASUREMENTSTWO-LEAD PHASE SENSITIVE MEASUREMENTS

A. Yacoby et al, PRL’95

trivial !

The phase is locked at 0 or π

mistery !!

Levy-Yeyati, Büttiker, PRB’95

Different peaks are in phase: π lapses when the transmission vanishes !!!

MULTY-LEAD PHASE SENSITIVE MEASUREMENTSMULTY-LEAD PHASE SENSITIVE MEASUREMENTS

R. Schuster et al, Nature’97

The phase increases continuously by π at each resonance (Friedel sum rule)

Universal regime: Large dots N > 14, correlated behavior between phase jumps and lapses

CROSSOVER FROM MESOSCOPIC TO UNIVERSAL PHASECROSSOVER FROM MESOSCOPIC TO UNIVERSAL PHASE

M. Avinum-Kalish et al, Nature’05

Mesoscopic regime: Small dots N < 10, random behavior of phase jumps and lapses

TRANSPORT THROUGH AN INTERACTING QUANTUM TRANSPORT THROUGH AN INTERACTING QUANTUM DOT IN THE REGIME OF COULOMB BLOCKADEDOT IN THE REGIME OF COULOMB BLOCKADE

Charging energy U = e2/C > kBT

Two approaches:

1) Constant interaction model: ΔVp = U + ΔE(1)

reduced to a one-body problem

2) Full many-body approach, include electronic correlations

quantum dot

partial-width amplitude:

PHASE EVOLUTION BETWEEN TWO RESONANCESPHASE EVOLUTION BETWEEN TWO RESONANCES

total-width:

PHASE EVOLUTION IN THE COMPLEX PLANEPHASE EVOLUTION IN THE COMPLEX PLANE same

parity:opposite parity:

ZEROS OF THE TRANSMISSION AND PHASE LAPSESZEROS OF THE TRANSMISSION AND PHASE LAPSES

L

large fluctuations in the partial-width amplitudes, unrestricted off-resonance behavior (UOR): 2 zeros

Dn <

0

Dn >

0

Levy-Yeyati and Büttiker, PRB 2000

Correlations between wave-functions of different eigenstates ?

• If Dn > 0 there is a zero between the n and the n+1 resonance

• If Dn < 0 there is no zero between the n and the n+1 resonance

PARITY RULE FOR THE TRANSMISSION ZEROSPARITY RULE FOR THE TRANSMISSION ZEROS

Experimentally (universal regime)

For a disordered quantum dot

independent random phases

BERRY CONJECTURE FOR WAVE-FUNCTION CORRELATIONSBERRY CONJECTURE FOR WAVE-FUNCTION CORRELATIONS

Random Wave Modelfor a chaotic billiard:

uniformly distributed vectors of magnitude k

for disordered systemsBessel function

transmission zero (and phase lapse) between the n and the n+1 resonance

UNIVERSAL BEHAVIOR IN THE SEMICLASSICAL LIMITUNIVERSAL BEHAVIOR IN THE SEMICLASSICAL LIMIT

Statistical independence of different eigenstates + Berry’s conjecture:

Two-dimensional billiard L

Probability of not havinga phase lapse:

NUMERICAL CALCULATIONS: TRANSMISSION AND PHASESNUMERICAL CALCULATIONS: TRANSMISSION AND PHASES

plateaus with the same number ofresonances and

transmission zeros

L

accumulated phase

transmission

scattering phase

IS THERE ALWAYS A ZERO BETWEEN TWO RESONANCES ?IS THERE ALWAYS A ZERO BETWEEN TWO RESONANCES ?

Nr and Nz grow with

the same rate in thesemiclassical limit !number of resonances

number of zeroes

The probability of observing out-of-phase

resonances vanishes as 1/kL

probability of havingmore resonances than zeroes in a k-interval

VALIDITY OF FRIEDEL SUMMATION RULEVALIDITY OF FRIEDEL SUMMATION RULE

A finite field Blifts the ambiguity inthe definition of theaccumulated phase

accumulated phase

At finite magnetic field the transmission

does not vanish

number of resonances

ARE CORRELATION NEEDED ?ARE CORRELATION NEEDED ?

mesoscopic to universal behavior by changing the

ratio δ/Γ, provided that

U >> Γ

A quantitative description requires correlations to be treated accurately

STRONGLY CORRELATED SCATTERERS: CONDUCTANCE STRONGLY CORRELATED SCATTERERS: CONDUCTANCE AND PHASE FROM THE EMBEDDING METHODAND PHASE FROM THE EMBEDDING METHOD

Scattering phase from the ground state energy

of rings with different number of electrons

Persistent current: ground-state property

DMRG CALCULATION OF CONDUCTANCE AND DMRG CALCULATION OF CONDUCTANCE AND PHASE FOR A SIMPLE QUASI-1D SCATTERERPHASE FOR A SIMPLE QUASI-1D SCATTERER

-

Spinless electrons with nearest neighbor interaction U

U=0

CAN INTERACTIONS INDUCE UOR BEHAVIOR?CAN INTERACTIONS INDUCE UOR BEHAVIOR?

Three-level system with large fluctuations among the level couplings:

U=0

U=2

UOR

U=0

U=2

conductance peaks ≠ resonances

INTERACTIONS &INTERACTIONS &CORRELATIONSCORRELATIONS

UOR behavior extra zeros (in pairs) incomplete filling of resonances

N=6

INTERACTION EFFECTS IN SMALL QUANTUM DOTSINTERACTION EFFECTS IN SMALL QUANTUM DOTS

Disordered quantum dot:random in-site energiesnon arbitrary couplings

N=8

No new zeros induced by the interactions

U=0

U=2

CONCLUSIONSCONCLUSIONS • Indirect determination of the transmission phase by

transport experiments in multilead rings• Phase lapses of π when the transmission vanishesrandom at low N: mesoscopic

regimeregular for higher N: universal regime

• Universal behavior emerges with probability 1-1/kL

• The wave-function correlations

are responsible for the universal behavior, not the electronic correlations R.A. Molina, R.A. Jalabert, D. Weinmann, Ph. Jacquod, Phys. Rev. Lett. 2012 R.A. Molina, P. Schmitteckert, D. Weinmann, R.A. Jalabert, Ph. Jacquod, unpublished 2012

• Difference chaotic vs. disordered

EMBEDDING METHOD: DETAILS AND IMPLEMENTATIONEMBEDDING METHOD: DETAILS AND IMPLEMENTATION

one-particle eigenstates of the ring:transfer matrix

of the sample

transfer matrix of the lead

1/L expansion of the ground state energy

Also, persistent current and scattering phase

EMBEDDING METHOD: EFFECTIVE ONE-PARTICLE SCATTERINGEMBEDDING METHOD: EFFECTIVE ONE-PARTICLE SCATTERING

The interacting region acts as a

local non-interacting scatterer