Frank Hekking - Physics@Technion

Superconducting charge pumps
Frank Hekking
Université Joseph Fourier
& Institut Universitaire de France
Laboratoire de Physique et Modélisation
des Milieux Condensés
Maison des Magistères Jean Perrin
CNRS-Grenoble, France
Together with V. Brosco, R. Fazio, F. Taddei, Pisa (Italy)
M. Möttönen, J. Pekola, J. Vartiainen, Helsinki (Finland)
M. Governale, Bochum (Germany)
Acknowledgment: D. Greenbaum, Y. Gefen
Quantum Pumping, January 7-12, 2007
The Lewiner Institute for Theoretical Physics, Technion, Haifa, Israel

Introduction & Motivation
Outline
Adiabatic charge transfer in Cooper pair pumps
- Relation between pumped charge and geometric phase
Open Cooper pump
-Adiabatic manipulation of Andreev bound states
Closed Cooper pump
-Effect of measuring device on pumped charge
Conclusion

Introduction & Motivation

Parametric pumping in the normal state
« Open » systems « Closed » systems
(Brouwer PRB98)
- modulation of transmission phase
- electronic phase coherence crucial
- interactions weak
- non-integer number of electrons per cycle
(Thouless PRB83, Brouwer PRB98
Switkes et al. Science99)
(Zorin JAP00)
- modulation of Coulomb blockade
- interactions crucial (charge quantization)
- interference effects weak
- integer number of electrons per cycle
(Delft-Saclay, NIST Boulder
PTB Braunschweig, TKK Helsinki)

Normal three-junction pump
Central idea: controlled charge transfer
upon gate modulation
I = e f

Experiment (Delft-Saclay)
(Pothier et al. EPL92)

Single Cooper pair pump (Delft-Saclay)
(Geerligs et al., ZPB ’91)

Revived interest in superconducting pumps
Metrology:
operation of normal state multijunction pumps limited by RC-time
- Pekola et al. PRB99: theory of coherent charge transfer
- Zorin et al. cond-mat00: experiment on a resistive pump
- Vartiainen et al, cond-mat 06, nanoampere pumping of Cooper pairs
Quantum information processing:
adiabatic manipulation of individual Cooper pairs
- Averin cond-mat97: adiabatic quantum computing with Cooper pairs
- Falci et al., Science 00: geometricJosephson qubits
- Ioffe & Feigelman PRB02: protected Josephson qubits
- Bibow et al . PRL02: experiment on resonant tunnelling through coherent charge states

Adiabatic charge transfer in Cooper pair pumps
Relation between pumped charge and geometric phase
Möttönen et al, PRB06
Brosco et al., unpublished

Adiabatic charge transfer in Cooper pair pumps
H(t) = H(Φ, {λ i(t)})
Phase difference across pump
Control parameters
Adiabatic transfer of Cooper pairs is obtained by slow periodic modulation of
control parameters

Adiabatic expansion
Solution of Schrödinger equation up to first
order in 1/T
T denotes the duration of a
pumping cycle
Dynamical contribution, O(T) Geometrical
contribution, O(0)

Instantaneous eigenstates
Initial condition
Instantaneous eigenstates
Expansion in terms of instantaneous eigenstates
Dynamical phase Berry phase

Charge pumped in a cycle of duration T
Dynamical
contribution, O(T)
Geometrical
contribution, O(0)
Berry phase

First example: « open » Cooper pump
Adiabatic manipulation of Andreev bound states
Governale et al, PRL05

−φ/2
Δ0 Hamiltonian
0
0
Open SNS pump
φ(r)
Δ(r)
φ/2
Δ 0

Andreev bound states in the absence of normal
scattering
Plane-wave solutions for particles and holes
Matching yields discrete spectrum
(Kulik, Sov. Phys. JETP70)
NS interface:
Andreev reflection process
χ
Scattering amplitude: α e
±i χ

Andreev bound states in the presence of normal
scattering
(Beenakker, Les Houches 94)
Andreev reflection
Phase factor
Normal scattering
Normal state scattering matrix

Andreev bound states in a short single channel pump
Schrödinger equation
Instantaneous eigenstates
If W < ξ 0
States above the gap do not contribute to current

Dynamical contribution, O(T)
Pumped charge
Geometrical contribution, O(0)

Results
Difference in scales
Q Jos(φ) is O(T), Q p(φ) is O(1)
Parity:
Q Jos(φ) odd, Q p(φ) even
Measure Q(φ)+Q(−φ)
Adiabaticity condition

Second example: « closed » Cooper pair pump
Effect of measuring device
on geometrically pumped charge
Fazio et al, PRB03

Superconducting three-junction pump
Charging energy dominates!

Charging energy: stability diagram

Josephson coupling: coherent mixing of charge states

Results

(Pekola et al. PRB99)
Coherent Cooper pair pump
Triangular gating sequence
incoherent classical
contribution
coherent correction
Interference due to higher order process

Pump: current generates
flux through SQUID
SQUID: critical current
depends on flux
Two possible measuring circuits
Pump: current generates extra
current through escape junction
Escape junction in quantum limit:
voltage sensitive to small current changes
I (μA)
15
10
5
0
-5
-10
-15
V=0
V~2Δ/e
-0.6 -0.4 -0.2 0 0.2 0.4 0.6
Uech (mV)

Measurement circuit: the model
Phase across junction fluctuates
Physics governed by partition function

Integrate out circuit degrees of freedom
Here
Second order: interaction term
Effective theory
First order: renormalized Josephson coupling
where

up to a frequency ω c
RG-analysis:
Resistive environment
Effect on pumped charge
where
logarithmic divergence!
low energy cut-off ω 0 = max (f, eV, …)

Coulomb blockade
lifted at finite bias
Resistive Cooper pair pump
(Zorin, cond-mat00)
Coulomb blockade at
zero bias

α = ω
LCL LCL
s/R s/R
Measurement with a SQUID loop
Frequency-dependent impedance
γ = M 2 /(L s L)
Renormalized Josephson coupling
independent of low energy cut-off!

η 0 R K/ω LCL J
0.8
0.7
0.6
0.5
0.4
0.3
0.2
Measurement with an escape junction
2 4 6 8 10 12 14
α = ωLCLJ/R Value of R: trade-off hysteresis-fluctuations
Frequency-dependent
impedance
Renormalized Josephson coupling
independent of low energy cut-off!

Conclusions
Adiabatic charge transfer in Cooper pair pumps
- Relation between pumped charge and geometric phase
“Open” Cooper pump
-Adiabatic manipulation of Andreev bound states
“Closed” Cooper pump
-Effect of measuring device on pumped charge:
suppression of Josephson energy
Examples
-Resistive environment: efficiently suppresses coherent part
-Inductive environment: coherent part survives
-SQUID
-Escape junction