ABSTRACT
Quasiparticle Trapping in Andreev Bound States
Maciej Zgirski*, L. Bretheau, Q. Le Masne, H. Pothier, C. Urbina, D. Esteve
Quantronics Group, SPEC, CEA Saclay, France
*presently: Institute of Physics, PAN, Warsaw
Electron transport through superconducting weak links can be understood in
terms of Andreev bound states. They originate from conduction channels with each
conduction channel giving rise to two Andreev bound states. In order to get access to single
Andreev bound states we have used a system with a few conduction channels at most –
quantum point contact. We have studied supercurrent across such a phase-biased atomic
size contacts. For broad phase interval around p we have found suppresion of supercurrent –
effect attributed to quasiparticle trapping in one of the discrete subgap Andreev bound states
formed at the contact. Since single Andreev bound state can sustain supercurrent up to 50nA,
such a trapping has a sound influence on the response of the atomic contact. Next to single
Cooper-pair devices in which parity of the total number of electrons matters, it is another
demonstration of a situation, when a single quasiparticle leaves a macroscopic trace.
However, unlike a single Cooper device, atomic contact contains no island at all. The trapped
quasiparticles are long-lived, with lifetimes up to hundreds of ms. Trapping occurs essentially
when the Andreev energy is smaller than half the superconducting gap D. The origin of this
sharp energy threshold is presently not understood.
PRL ,106, 257003 (2011)
Quasiparticle Trapping in Andreev Bound States
Maciej Zgirski*, L. Bretheau, Q. Le Masne, H. Pothier, C. Urbina, D. Esteve
Quantronics Group, SPEC, CEA Saclay, France
*presently: Institute of Physics, PAN, Warsaw
L. Bretheau
H. Pothier
Q. Le Masne
D. Esteve
C. Urbina
PRL ,106, 257003 (2011)
MOTIVATION
• Josephson effect in superconducting weak
links – unified approach
L
L
S
S I S R
• Spectroscopy
of Andreev Levels
t
S
 L  R
E()
+D
+EA
0
• Andreev Qubit
R
-D

-EA
ANDREEV REFLECTION
COUPLING OF eh AND h$
N-S interface
a  E,  
S
N
for E  D
a E,    1
 D
arg a E,       arccos E
PHASE-BIASED SHORT, BALLISTIC
SINGLE CHANNEL
L x
  L  R
L
t 1
R
a  E, L 
a  E, R 
arg a  E, R   arg a  E,  L   0 mod 2p
Fabry-Perot resonator
ANDREEV BOUND STATES
in a short ballistic channel (t 1 )
E
t= 1
L
R
E()
+D
Andreev spectrum
0
-D
2 resonances
E
 D cos   2
+D
E→
p
E←
2p

0
-D
 p 4
ANDREEV BOUND STATES
in a short reflective channel (t 1 )
t< 1
E
Andreev spectrum
+D
E()
+D
+EA
2D 1  t
0
-D
E A  D 1  t sin   2 
2
0
2p

-D
-EA
Furusaki, Tsukada
C.W.J. Beenakker (1991)
Central prediction of the mesoscopic theory
of the Josephson effect
 p 4
SUPERCONDUCTING WEAK LINKS
Weak link = ensamble of independent transmitting channels,
each characterized by transmission t(Landauer picture)
N – number of transmission channels
t- transmission
Atomic contact:
Tunnel junction:
N~1
N infinity
0t< 1
t->0
L
S
L
S I S
t
S
 L  R
R
= L - R
IJ  I0 sin 
Current phaserelation
Iac() = ?
R
FROM ANDREEV BOUND STATES
TO SUPERCURRENT
E()
+D
Ground state :
+EA
0
-D
2p

-EA
IA   
1   E A 
I A ( ) 
0 
p

Current-phase relation
Current – phase relation…
1   E A 
I A ( ) 
0 
E A  D 1  t sin2   2 
E()
+D
0

-D
p

…is a probe of a configuration
of Andreev bound states

TOWARDS ANDREEV QUBITS
E()
+D
+EA
0
-D
2p

-EA
Use even states
Zazunov, Shumeiko,Bratus’,
Lantz and Wendin, PRL (2003)
Use quasiparticle (spin ½) states
Chtchelkatchev and Nazarov, PRL (2003)
ATOMIC CONTACT = SIMPLEST WEAK LINK
fabrication & characterization
V
I
S
S
1 atom contact = few conduction channels (Al: 3)
Stable system
Can be completely characterized
MICROFABRICATED BREAK-JUNCTIONS
2 µm
metallic film
insulating
layer
pushing
rods
Flexible
substrate
countersupport
PIN code of the atomic contact
Scheer et al. PRL 1997
Current bias in not enough…
Atomic Squid…
Ib  IJJ  IAC
IAC
IJJ
or

V

      2p
0
…allows to determine channels transmissions…
I

Ib
(V ) measurement

OPEN
V
I
transmissions {ti}
…and impose phase on atomic contact
I
() measurement


   

Ib
IJJ >> IAC
“Strength” of the weak link
~ critical current

SHORT
Switching of the Atomic Squid
Ib
switching
V
retrapping
Ib  IJJ  IAC
   

  2p
0
IAC

IJJ
or


I sw  I
JJ
0
p
 I AC (   )
2
SWITCHING MEASUREMENTS
Ib (nA)
1000
Supercurrent
branch
500
<Isw>
0
-500
-1000
-400
-200
0
Ib Pulse height
tp
Tr
time
N
Psw 
V
n
usually
Tr=20µs
tp=1µs
N=5000
time
n
N
Switching probability
P
200
400
V (µV)
« s curve »
Ib (nA)
Switching current [nA]
Flux Modulation pattern for ATOMIC SQUID
= I() of the atomic contact
560
I0-switching current
of junction alone
520
I sw  I
480
0.0
0.5
magnetic flux [/0]
1.0
The ground Andreev state
is well-known…
Theses in
Quantronics:
M. Chauvin,
B. Huard,
Q. Le Masne
Della Rocca et al., PRL 2007
JJ
0
p
 I AC (   )
2
When SQUID switches, phase across
JJ is approx. the same independently of
applied magnetic flux => interference
pattern is current-phase relation of
atomic contact
Switching probability map with normal leads
P (Ib,)
1
P=1
A vertical cut is an s-curve
s = Ib/I0
I0 - critical current of JJ alone
P=0
0
P
SAMPLE
Sample design
bias
line
e-beam
lithography
designed to be 50W
at T < 1K
antenn
a
Switching probability map
with superconducting electrodes
T=40mK,
Period= 20µs
tp
Tr
time
N
j1
= 
2 = 3.05 rad
= 
0.6
Psw
1 = 0.62 rad
1.0
0.8
0.4
0.2
0.0
0.80
0.85
0.90
t={0.95, 0.445, 0.097}
0.95
1.00
s
Height of plateau is period dependent => some
relaxation going on in the system
j2
Switching curve with prepulse
1,0
1.3
1
Psw
Erase memory of the
previous history before
each measurement:
~ 0.1µs
1ms
0,8
{0.95, 0.45 , 0.10}
P1(Ib)
0,6
pP1(Ib)+(1-p)P2(Ib)
0,4
0,2
P2(Ib)
0
{0.45 , 0.10}
delay
« prepulse »
0,0
0,80
0,85
s
0,90
After switching, system is where we
expect it to be with probability p
Blocking the most transmitting channel
1.0
0.8
Psw
{0.45 , 0.10}
0.6
0.4
0.2
{0.95 , 0.45 , 0.10}
0.0
0.80
0.85
s
0.90
QUASIPARTICLES IN A SUPERCONDUCTING
POINT CONTACT
E
D
EA
0
-EA
-D
Ground state
energy  E A
E()
+D
0
-D
EA
E A 
1-qp states
energy  0
energy > -E A +D
2 qps
energy  E A
Excitation picture
All electrons paired
The smallest excitation
breaking parity
= one unpaired
quasiparticle
Excited Cooper pair
Two scenarios
Initial state
QP
nQP
1.
Weight = p
nQP
E
2.
Channel switched on
Weight = 1 - p
E
Channel switched off
Switching probability is the weighted average of these 2 scenarios.
Modulation curves on different contacts
{1,0.072,0.072}
AC1
{0.998,0.56,0.124}
AC2
{1,0.7,0.24,0.24,0.06}
AC3
The most transmitting channel is sometimes switched off
1QP STATE RELAXATION MEASUREMENTS
Ib
Current
line
Flux
line
0
waiting time
w
across
i contact
i
1,0
0,85
TR()
Pinf()
0,75
0,6
Psw
P=0.815
0,80
0,8
0,70
0,4
1 µs
580 µs
0,65
0
0,2
0,0
0,76
 Phase
0,78
0,80
0,82
s
0,84
0,86
0,88
P0
100
TR = 172 µs
200
300
400
500
waiting time (µs)
600
A few 100ms relaxation time
{1,0.07,0.07}
T=29mK
TR(µs)
100
10
1
-0.6p
0
 phase across atomic contact
Symmetry around p
Monotonous behaviour
0.6p
Relaxation as a function of phase across Atomic
Contact for different transmissions
200
{0.85,0.22,0.22}
TR(µs)
100
{0.96,0.03,0.03}
{1,0.07,0.07}
10
{0.74,0.01}
1
1.00
T=29mK
{0.74,0.01}
0.95
0.90
p
{0.85,0.22,0.22}
0.85
{0.96,0.03,0.03}
0.80
{1,0.07,0.07}
0.75
-0.9p
-0.6p
-0.3p
0.0p
p
0.3p
0.6p
0.9p
Energy threshold for relaxation
200
{0.85,0.22,0.22}
TR(µs)
100
E()
+D
0
-D
p
{0.96,0.03,0.03}
{0.74,0.01}
{1,0.07,0.07}
10
2p 
E1
1,00
E
  1  t sin2  2 
D
0,95
{1,0.07,0.07}
0,90
p
Relaxation
instantaneous only for
Andreev Bound states
with energies bigger
than
0.5 D~25GHz ~1K
{0.74,0.01}
{0.85,0.22,0.22}
0,85
{0.96,0.03,0.03}
0,80
0,75
-1,00
-0,75
-0,50
-0,25
Eground / D
0,00
Energy threshold for relaxation
200
nQP
{0.85,0.22,0.22}
TR(µs)
100
{0.96,0.03,0.03}
{0.74,0.01}
{1,0.07,0.07}
10
E
1
nQP
1,00
{0.74,0.01}
0,95
{1,0.07,0.07}
D
D/2
p
0,90
E
{0.85,0.22,0.22}
0,85
{0.96,0.03,0.03}
0,80
WHY?
0,75
-1,00
-0,75
-0,50
-0,25
Eground / D
0,00
Possible explanation
hn
nQP
E
hn ~ D/2
nQP
D
D/2
E
Conclusions
• Atomic contacts with tunable transmissions
• Atomic Squid to measure current-phase relation of atomic contact
with switching measurements - for ground Andreev bound states
excellent agreement with theory
• Quasiparticle poisoning => disappearence of the most transmitting
channel;
• long relaxation for Andreev Cooper pair binding energies smaller
than 0.5D, sharp cut off for binding energies bigger than 0.5D ?
• Dispersive measurements of resonant frequency of resonator +
atomic squid
• Trials to observe avoided level crossing (atomic contact embedded in
resonator)
• No evidence of excited Andreev state in 2 different experiments
(switching measurements, coupling to resonator )
• Current Status: Josephson Junction spectroscopy of Atomic Squid –
observed avoided level crossing
PLASMA FREQUENCY – ANDREEV GAP
Temperature dependence
TR(µs)
100
10
29mK
66.5mK
101mK
129mK
150mK
168mK
184mK
202mK
214mK
1
1.00
0.95
p
0.90
0.85
0.80
0.75
-0.6p
-0.3p
0.0p
p
0.3p
0.6p
{1,0.07,0.07}
Does excited Andreev state exist?
(OPTIONAL)
Sample design
bias
line
e-beam
lithography
designed to be 50W
at T < 1K
antenn
a
Andreevmon (or Andreevnium)
Capacitor + inductive lines
140µ
m
10µm gap
Capacitor
C = 60 pF
680µm
inductive lines, 900nm
wide, 70 + 54 nm thick Al
Ltotal = 1.8nH
antenna (5µm wide short of
CPW)
Electromagnetic environment is important
RF
line


T1 

L
IB
R
VB
C
1
Re  Z  n A  
bias
line
Trials to observe excited Andreev state
Expected
Peak
position is
frequencydependent
I
0
0.5
/2p
1
Andreev Qubit in cavity
Weak coupling
Cavity Quantum Electrodynamics
VAC in
VAC out
strong coupling regime
Let 2 level system interact with resonator
0
E1=Ea
-D
E0=-Ea
E()
Andreev Gap
0
p

Bare Resonator eigenfrequency
Red –
expected
position of
resonance
Interaction “off”
Interaction “on”
 | resonator  | 0 
  a | 0  b | 1 
avoided level crossing
Coherent exchange of energy between resonator and artificial atom
2p
2 CHANNELS POISONING
{0.95, 0.94, 0.60, 0.34, 0.30, 0.29, 0.27, 0.26, 0.24, 0.2}
1.0
0.8
Psw
0.6
0.4
0.2
0.0
0.60
0.65
0.70
s
0.75
0.80
Pollution of 2 channels
49/19
1 channel
blocked
2 channels
blocked
All channels
P
{0.957, 0.948, 0.601, 0.344, 0.295,
0.291, 0.27, 0.262, 0.242, 0.2}
1.0
0.8
0.6
0.4
0.2
0.0
340 360 380 400 420 440
Ib nA
Atomic SQUID in cavity
Flux pulse cleans excited Andreev state
Current
line
RF
line
Flux
line
period
delay
Vflux
big enough
period
1.0
P
0.8
0.6
-0.61 x 2p
-0.53 x 2p
-0.44 x 2p
-0.35 x 2p
-0.26 x 2p
-0.17 x 2p
-0.09 x 2p
no flux pulse
0.4
0.2
0.0
460
480
Ib (nA)
500
520
MULTIPLE CHARGE TRANSFER PROCESSES
V
S
S
I
t
I
Blonder, Tinkham, Klapwijk (‘82)
2D / 3 2D / 2
2D / 1
eV
52/19
Atomic contact
53/19
S
S
few channels, {ti} tunable
2 µm
Al film
I
 {ti },     I  ti,  
i
Δx
pushing
rod
Δz
countersupport
Elastic substrate
{ti} measurable
QUASIPARTICLES IN A BULK
SUPERCONDUCTOR
E
D
0
-D
Ground state
1-qp states
energy  0
energy > D
2 qps
energy > 2D
QUASIPARTICLES AND SUPERCURRENT IN A
SUPERCONDUCTING POINT CONTACT
IA   
i0
i0
p
I A ( ) 

1   E A 
0 
Ground state
energy  E A
E()
+D
0
-D
EA
E A 

IA   
p
I A ( ) 
I A ( )  0
Lowest-lying 1-qp excitations
energy  0
p

1-qp state
energy > -E A +D
1 E A
0 
Excited singlet
energy  E A

CORRELATED SWITCHING EVENTS
V(t)
5000
# of occurences
0
10 000
100
10
1
0
20
40
60
80
100
D t / Tr
 Need a ‘’reset’’ between pulses
MEASURING THE SWITCHING PROBABILITY
meast
1µs
hold
N pulses
Vb(t)/Rb
sI0
n events
V(t)
Psw
n

N
MEASURING THE SWITCHING PROBABILITY
prepulse (reset)
meast
1µs
hold
N pulses
1.3 sI0
sI0
Vb(t)/Rb
Dt
n events
V(t)
 Uncorrelated switching events
Reaching 1QP odd state
D
50GHz
E2=Ea
for Al
2D 1  τ
0
1QP state (x2)
1
Ground state
0
E1=0
-D 0
E0=-Ea
p
2p

2
I2=-Ia
I0=Ia
0
I1=0
0
p
I()
E()
2QP state
2p

1.0
QP
Psw
0.8
0
nQP
0.6
0.4
1
0.2
0.0
E
0.80
0.85
s
0.90
RELAXATION VERSUS ANDREEV ENERGY
10
1
T (µs)
100
1
p
0.2
{0.994, 0.10, 0.10}
{0.96, 0.03, 0.03}
{0.91, 0.62, 0.15}
{0.85, 0.22, 0.22}
{0.74, 0.01}
0.1
0.0
/p
2
1
0
0.00
0.25
0.50
EA / D
0.75
1.00