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Landau-Zener physicsin quantum turnstiles
D. M. Basko
Laboratoire de Physique et Modélisation des Milieux Condensés
CNRS & Université Grenoble Alpes
D. van Zanten, C. Winkelmann, H. Courtois
Institut Néel
CNRS, Université Grenoble Alpes, Institut polytechnique de Grenoble
Thanks to: M. Houzet, I. Khaymovich, M. Kiselev, J. Pekola, X. Waintal, R. Whitney
Landau-Zener(Stückelberg, Majorana)
Slow, adiabatic:
The particle followsthe instantaneous ground state
Quantum SINIS turnstile
SS SN
Vbias
gate
V (t)
gate voltage
ground
Small metallic island- large level spacing- strong Coulomb blockade
single leveloccupation = 0,1
g
superconductingelectrodes
Pekola et al., Nature Phys. 4, 120 (2008)van Zanten et al., PRL 116, 166801 (2016)
Quantum SINIS turnstile
2Δ
V biasempty level
Pekola et al., Nature Phys. 4, 120 (2008)van Zanten et al., PRL 116, 166801 (2016)
Quantum SINIS turnstile
2Δ
V bias
tunnel in
Pekola et al., Nature Phys. 4, 120 (2008)van Zanten et al., PRL 116, 166801 (2016)
Quantum SINIS turnstile
2Δ
V bias
Pekola et al., Nature Phys. 4, 120 (2008)van Zanten et al., PRL 116, 166801 (2016)
V g
Quantum SINIS turnstile
2Δ
V bias
tunnel out
Pekola et al., Nature Phys. 4, 120 (2008)van Zanten et al., PRL 116, 166801 (2016)
Quantum SINIS turnstile
2Δ
V bias
V g
Pick up another electronand repeat the procedure
Transfer one electron per cycleMetrological application:current standard
Pekola et al., Nature Phys. 4, 120 (2008)van Zanten et al., PRL 116, 166801 (2016)
Electron ejection
tunnel outLarge ∆
a discrete occupied levelcoupled to an empty continuum
A single-particle problem!(forget the superconductivity)
Electron ejection
tunnel outLarge ∆
a discrete occupied levelcoupled to an empty continuum
A single-particle problem!(forget the superconductivity)
need time >> 1/Γto tunnel out
The slower, the better?
rate Γ
Electron ejection
tunnel outLarge ∆
a discrete occupied levelcoupled to an empty continuum
A single-particle problem!(forget the superconductivity)
rate Γ
from the Golden Rule(perturbation theory):
normalizedDOS in thesuperconductortunneling rate into
the normal electrode
Electron ejection
tunnel outLarge ∆
a discrete occupied levelcoupled to an empty continuum
A single-particle problem!(forget the superconductivity)
rate Γ
from the Golden Rule(perturbation theory):
normalizedDOS in thesuperconductortunneling rate into
the normal electrode
divergence!Golden Rule breaks down
Exact solution for a fixed level
discretelevel quasiparticle
continuum
tunnelingmatrix
element
Self-energy for the discrete level:
near thesingularity
Exact solution for a fixed level
discretelevel quasiparticle
continuum
tunnelingmatrix
element
Self-energy for the discrete level:
near thesingularity
count from the singularity
real poles of the Green's function
energies of the eigenstates
E
E−Ed
Ed
they always cross
Singularity in the DOS
Divergence of
Existence of the bound state for any
Kramers-Kronig
(quantum-mechanical level repulsion)
L. Yu, Acta Physica Sinica 21, 75 (1965)H. Shiba, Prog. Theor. Phys. 40, 435 (1968)A. I. Rusinov, JETP Letters 9, 85 (1969).
Singularity in the DOS
Divergence of
Existence of the bound state for any
Kramers-Kronig
(quantum-mechanical level repulsion)
If the levelis moved adiabatically,
the electron will never escape
The bound state for a fixed level
1D representation of the continuumto reproduce the DOS singularity:
eliminate
The bound state for a fixed level
1D representation of the continuumto reproduce the DOS singularity:
eliminate
inside the continuum
The overlap with the bare levelis small
barelevel
LDOS
The bound state for a fixed level
barelevel
LDOS
Ed
E
(γ Δ/2)0
2 1/3
bare levelLDOS
E
Z
d(γ Δ/2)0
2 1/3
1
The overlapwith the bare level
is small
Dynamical problem
Arbitrary dependence : numerical solution
Special case : analytical
Demkov & OsherovJETP 26, 916 (1968)
Shrödinger equationwith a complex potential
Three asymptotic regimes
~ 1
perturbative(too fast)
natural energy scale
two independent parameters
1. Perturbative regime: or ,
Three asymptotic regimes
~ 1 adiabatic(too slow)
perturbative(too fast)
natural energy scale
two independent parameters
1. Perturbative regime: or ,
2. Adiabatic regime: ,
Three asymptotic regimes
~ 1
Markovian
adiabatic(too slow)
perturbative(too fast)
natural energy scale
two independent parameters
1. Perturbative regime: or ,
2. Adiabatic regime: ,
3. Markovian regime: instantaneous Golden Rule
From Markovian to adiabatic
The first correction to the Markovian result:
decayphase
The result looks like two-path interference
1. Perturbative regime: or ,
2. Adiabatic regime: ,
3. Markovian regime: instantaneous Golden Rule
Experiment
Dynes paramaterfor aluminum: 10‒4‒10‒5
due to noise from the circuit
Pekola et al., PRL 105, 026803 (2010)
electrodes
grain
The DOS singularity is very pronounced
Device parameters:
Δ = 260 µeV
EC > 100 Δ
δ > 10 Δ
T = 0.05 Δ
γ ≈ (0.005 – 0.1) Δ
Experiment
current = 0
current = ef
overshoot(escape back
to the same electrode)
Dynes paramaterfor aluminum: 10‒4‒10‒5
due to noise from the circuit
Pekola et al., PRL 105, 026803 (2010)
electrodes
grain
The DOS singularity is very pronounced
Device parameters:
Δ = 260 µeV
EC > 100 Δ
δ > 10 Δ
T = 0.05 Δ
γ ≈ (0.005 – 0.1) Δ
Simulating the experiment
1. Ejection probability on each half-cycle (from numerics or from the parabolic solution)
2. Rate equations for the occupation probabilities (assuming no coherence between different events)
3. Current
Theory and Experiment(no fitting parameters)
Gaps in dI/dVb: the ejection onset crosses the overshooting
theory
experiment
Theory and Experiment(no fitting parameters)
experiment
analyticaltheory
& numerics
dI/dVb peak @ constant amplitude
(vertical section of the color plot)
experiment
analyticaltheory
compareleft half-width
Theory and Experiment(no fitting parameters)
experiment
analyticaltheory
& numerics
dI/dVb peak @ constant amplitude
(vertical section of the color plot)
experiment
analyticaltheory
compareleft half-width
Poor agreement @ large Vb:
the bare level is deep in the continuum (Ed ~ Δ/2)
the bound state is very shallow
(E* ~ 30 neV ~ 10
-4Δ)
- Dynes smearing of the DOS singularity- noise of the gate voltage - absorption of photons (24 MHz = 100 neV) coherence between tunneling events
Conclusions and outlook
1. DOS singularity Landau-Zener-like behavior fundamental limitation on quantum turnstile operation
2. Qualitative agreement between experiment and theory
3. Need to improve the theory:- include the noise in the gate voltage- include coherence between successive tunneling events
Thank you for your attention!
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