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Transmission phase in quantum transport: disorder, chaos and correlation effects. Rodolfo A. Jalabert. Institut de Physique et Chimie des Matériaux de Strasbourg. Philippe Jacquod (Arizona) Rafael A. Molina (Madrid) Peter Schmitteckert (Karlsruhe) Dietmar Weinmann (Strasbourg). - PowerPoint PPT Presentation
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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