landau-zener physics in quantum...

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)


Fast, perturbative:


Slow, adiabatic:

The particle followsthe instantaneous ground state

(Landau-Zener)(Stückelberg, Majorana)

2

the two pathsinterfere

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)


2Δ

V biasempty level



2Δ

V bias

tunnel in



2Δ

V bias


V g


2Δ

V bias

tunnel out



2Δ

V bias

V g

Pick up another electronand repeat the procedure

Transfer one electron per cycleMetrological application:current standard


Electron ejection

2Δ

V bias

tunnel out

Electron ejection

tunnel outLarge ∆

a discrete occupied levelcoupled to an empty continuum

A single-particle problem!(forget the superconductivity)

Electron ejection

tunnel outLarge ∆



need time >> 1/Γto tunnel out

The slower, the better?

rate Γ

Electron ejection

tunnel outLarge ∆



rate Γ

from the Golden Rule(perturbation theory):

normalizedDOS in thesuperconductortunneling rate into

the normal electrode

Electron ejection

tunnel outLarge ∆



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


1D representation of the continuumto reproduce the DOS singularity:

eliminate

inside the continuum

The overlap with the bare levelis small

barelevel

LDOS


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

Dynamical problem

Arbitrary dependence : numerical solution

Special case : analytical

Demkov & OsherovJETP 26, 916 (1968)

Shrödinger equationwith a complex potential

Three asymptotic regimes

~ 1

natural energy scale

two independent parameters


~ 1

perturbative(too fast)



1. Perturbative regime: or ,


~ 1 adiabatic(too slow)





2. Adiabatic regime: ,


~ 1

Markovian

adiabatic(too slow)






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



3. Markovian regime: instantaneous Golden Rule

the two pathsinterfere

A singularity in the DOS produces Landau-Zener physics

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


experiment

analyticaltheory

& numerics

dI/dVb peak @ constant amplitude

(vertical section of the color plot)

experiment

analyticaltheory

compareleft half-width


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!

landau-zener physics in quantum...

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