prl ,106, 257003 (2011)
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ABSTRACTQuasiparticle 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)
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
Q. Le Masne
D. Esteve
C. Urbina
H. Pothier
MOTIVATION• Josephson effect in superconducting weak
links – unified approach
• Spectroscopy of Andreev Levels
• Andreev Qubit
S SIL R
L R
tS SL R
E()
-EA
+EA+D
-D
0
ANDREEV REFLECTION
COUPLING OF eh AND h$
D
D
for E
a E, 1
Earg a E, arccos
SN
N-S interface ,a E
PHASE-BIASED SHORT, BALLISTIC SINGLE CHANNEL
L x
p arg , arg , 0 mod 2R La E a E
Fabry-Perot resonator
L R
L R
, La E , Ra E
t1
ANDREEV BOUND STATESin a short ballistic channel (t1 )
Andreev spectrumE()
2pp
+D
-D
0
E→
E←
D cos 2E
2 resonances
E
+D
-D
0 p 4
t= 1
L R
ANDREEV BOUND STATESin a short reflective channel (t1 )
tD 21 sin 2AEFurusaki, TsukadaC.W.J. Beenakker (1991)
E()
-EA
+EA+D
-D
0tD 2 1
2p
Andreev spectrumt< 1
Central prediction of the mesoscopic theoryof the Josephson effect
E
+D
-D
0 p 4
SUPERCONDUCTING WEAK LINKS
S SIL R
Tunnel junction:N infinityt->0
L R
tS SL R
Atomic contact:N ~ 10t< 1
J 0I I sin
N – number of transmission channelst- transmission
Current phase-relation
= L - R
Weak link = ensamble of independent transmitting channels,each characterized by transmission t(Landauer picture)
Iac() = ?
FROM ANDREEV BOUND STATESTO SUPERCURRENT
E()
-EA
+EA+D
-D
0
AI
p
2p
Current-phase relation
Ground state :
0
1( )AAE
I
Current – phase relation…
0
1( )AAE
I
…is a probe of a configuration of Andreev bound states
p
E()
+D
-D
0
tD 21 sin 2AE
TOWARDS ANDREEV QUBITS
Chtchelkatchev and Nazarov, PRL (2003)
Use quasiparticle (spin ½) states
E()
-EA
+EA+D
-D
02p
Zazunov, Shumeiko,Bratus’, Lantz and Wendin, PRL (2003)
Use even states
ATOMIC CONTACT = SIMPLEST WEAK LINK
fabrication & characterization
IS SV
1 atom contact = few conduction channels (Al: 3)Stable system
Can be completely characterized
2 µm
insulating layer counter-
support
Flexiblesubstrate
metallic film pushing rods
MICROFABRICATED BREAK-JUNCTIONS
PIN code of the atomic contact
Scheer et al. PRL 1997
Current bias in not enough…
Atomic Squid…
b JJ ACI I I
JJIor
V
IAC
p
0
2
…allows to determine channels transmissions…
IbI
V
measurement( )I V
transmissions {ti}
OPEN
…and impose phase on atomic contact
“Strength” of the weak link~ critical current
( )I measurement
IJJ >> IAC
Ib
SHORT
b JJ ACI I I
JJIorIAC
Ib
V
)2
(0 p AC
JJsw III
0
2
p
switching
retrapping
Switching of the Atomic Squid
Ib Pulse height
V
time
time
tp Tr
N
Tr=20µstp=1µsN=5000
usually Sw
itchi
ng
prob
abili
ty
Ib (nA)
P « s curve »
SWITCHING MEASUREMENTS
-400 -200 0 200 400-1000
-500
0
500
1000
Ib (nA)
V (µV)
<Isw>Supercurrent branch
NnPsw
n
Flux Modulation pattern for ATOMIC SQUID= I() of the atomic contact
0.0 0.5 1.0
480
520
560
Sw
itchi
ng c
urre
nt [n
A]
magnetic flux [/0]
I0-switching current of junction alone
When SQUID switches, phase across JJ is approx. the same independently of applied magnetic flux => interference pattern is current-phase relation of atomic contact
)2
(0 p AC
JJsw III
The ground Andreev state is well-known…
Theses in Quantronics:M. Chauvin,
B. Huard,Q. Le Masne
Della Rocca et al., PRL 2007
P (Ib,)
A vertical cut is an s-curve
P = 0
P = 1
P1
0
s = Ib/I0I0 - critical current of JJ alone
Switching probability map with normal leads
SAMPLE
Sample design
antenna
bias line
designed to be 50W at T < 1K
e-beam lithography
T=40mK,Period= 20µs
t={0.95, 0.445, 0.097}
Switching probability map with superconducting electrodes
0.80 0.85 0.90 0.95 1.000.0
0.2
0.4
0.6
0.8
1.0
= 2
= 3
.05
rad
Psw
s
= 1
= 0
.62
rad
j1 j2timetp Tr
N
Height of plateau is period dependent => some relaxation going on in the system
0,80 0,85 0,900,0
0,2
0,4
0,6
0,8
1,0
P sw
s
P1(Ib)
P2(Ib)
pP1(Ib)+(1-p)P2(Ib)
Switching curve with prepulse
After switching, system is where we expect it to be with probability p
{0.45 , 0.10}
{0.95, 0.45 , 0.10}Erase memory of the previous history before each measurement:
1
0
1.3
delay
~ 0.1µs
1ms
« prepulse »
0.80 0.85 0.900.0
0.2
0.4
0.6
0.8
1.0
Psw
s
Blocking the most transmitting channel
{0.95 , 0.45 , 0.10}
{0.45 , 0.10}
QUASIPARTICLES IN A SUPERCONDUCTINGPOINT CONTACT
E
D
-D
0
Ground state 1-qp states
energy 0 DAenergy > -E + Aenergy E
Aenergy E
EA
-EA
2 qps
E()
+D
-D
0 AE
AE
Excitation picture
All electrons paired The smallest excitationbreaking parity= one unpaired quasiparticle
Excited Cooper pair
Two scenariosQP
E
nQP
1.
2.E
nQP
Weight = p
Weight = 1 - p
Channel switched off
Channel switched on
Switching probability is the weighted average of these 2 scenarios.
Initial state
Modulation curves on different contacts
{1,0.7,0.24,0.24,0.06}AC3
{1,0.072,0.072}AC1
{0.998,0.56,0.124}AC2
The most transmitting channel is sometimes switched off
1QP STATE RELAXATION MEASUREMENTS
0,76 0,78 0,80 0,82 0,84 0,86 0,880,0
0,2
0,4
0,6
0,8
1,0
P sw
1 µs 580 µs
s
0 100 200 300 400 500 6000,65
0,70
0,75
0,80
0,85
P0
waiting time (µs)
TR = 172 µs
P=0.815
Current line
waiting time0
Ib
TR() Pinf()
Flux line
i
w Phase across contacti
A few 100ms relaxation time
T=29mK
Symmetry around pMonotonous behaviour
{1,0.07,0.07}
1
10
100
T R(
µs)
-0.6p 0 0.6p phase across atomic contact
T=29mK
Relaxation as a function of phase across Atomic Contact for different transmissions
-0.9p -0.6p -0.3p 0.0p 0.3p 0.6p 0.9p0.75
0.80
0.85
0.90
0.95
1.00
{0.85,0.22,0.22}
p
p
{0.74,0.01}
{1,0.07,0.07}
{0.96,0.03,0.03}
1
10
100
{0.85,0.22,0.22}
{0.96,0.03,0.03}{1,0.07,0.07}
{0.74,0.01}
T R(µ
s)
200
-1,00 -0,75 -0,50 -0,25 0,000,75
0,80
0,85
0,90
0,95
1,00
{0.96,0.03,0.03}
{1,0.07,0.07}
{0.74,0.01}
p
Eground / D
{0.85,0.22,0.22}
Energy threshold for relaxation
21 sin 2 DE t
1
10
100
200
{0.74,0.01}
T R(µ
s)
{0.85,0.22,0.22}
{1,0.07,0.07}
{0.96,0.03,0.03}
E()
E-
2pp
+D
-D
0
Relaxation instantaneous only for Andreev Bound states with energies bigger than 0.5 D~25GHz ~1K
-1,00 -0,75 -0,50 -0,25 0,000,75
0,80
0,85
0,90
0,95
1,00
{0.96,0.03,0.03}
{1,0.07,0.07}
{0.74,0.01}
p
Eground / D
{0.85,0.22,0.22}
Energy threshold for relaxation
1
10
100
200
{0.74,0.01}
T R(µ
s)
{0.85,0.22,0.22}
{1,0.07,0.07}
{0.96,0.03,0.03}
E
nQP
E
nQP
D/2D
WHY?
Possible explanation
E
nQP
E
nQP
D/2D
hn ~ D/2
hn
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
• No evidence of excited Andreev state in 2 different experiments (switching measurements, coupling to resonator )
• 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)
• Current Status: Josephson Junction spectroscopy of Atomic Squid – observed avoided level crossing PLASMA FREQUENCY – ANDREEV GAP
1
10
100 29mK 66.5mK 101mK 129mK 150mK 168mK 184mK 202mK 214mK
T R(µ
s)
{1,0.07,0.07}
Temperature dependence
-0.6p -0.3p 0.0p 0.3p 0.6p0.75
0.80
0.85
0.90
0.95
1.00
p
p
Does excited Andreev state exist?(OPTIONAL)
Sample design
antenna
bias line
designed to be 50W at T < 1K
e-beam lithography
Capacitor + inductive lines
inductive lines, 900nm wide, 70 + 54 nm thick Al
680µm
10µm gap140µm
antenna (5µm wide short of CPW)
Capacitor C = 60 pF
Ltotal = 1.8nH
Andreevmon (or Andreevnium)
Electromagnetic environment is important
C
R IB
bias line
RF line
VB
L
n A1
1Re Z
T
Trials to observe excited Andreev state
Peak position is frequency-dependent
I
10.50/2p
Expected
Andreev Qubit in cavity
Weak coupling
strong coupling regime
VAC in
VAC out
Cavity Quantum Electrodynamics
Let 2 level system interact with resonator
avoided level crossingCoherent exchange of energy between resonator and artificial atom
Andreev Gap
E0=-Ea
E1=Ea
0-D
2pp
0
E()
Bare Resonator eigenfrequency
0|| resonator
1|0| ba
Interaction “on”
Interaction “off”Red – expected position of resonance
{0.95, 0.94, 0.60, 0.34, 0.30, 0.29, 0.27, 0.26, 0.24, 0.2}
2 CHANNELS POISONING
0.60 0.65 0.70 0.75 0.800.0
0.2
0.4
0.6
0.8
1.0
Psw
s
340 360 380 400 420 4400.00.20.40.60.81.0
Ib nAP
{0.957, 0.948, 0.601, 0.344, 0.295, 0.291, 0.27, 0.262, 0.242, 0.2}
Pollution of 2 channels
All channels
2 channels blocked
1 channel blocked
49/19
Atomic SQUID in cavity
460 480 500 5200.0
0.2
0.4
0.6
0.8
1.0
P
Ib (nA)
-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
Flux pulse cleans excited Andreev state
Flux line
Vfluxbig enough
Current line
period
period
delayRF line
MULTIPLE CHARGE TRANSFER PROCESSES
eV
I
Blonder, Tinkham, Klapwijk (‘82)
2D / 12D / 22D / 3
VI
t
S S
52/19
S S
Atomic contact
few channels, {ti} tunableAl film
2 µm
Δx
pushing rod
counter-support
Elastic substrateΔz
i ii
I I{ , ,} t t {ti} measurable
53/19
QUASIPARTICLES IN A BULK SUPERCONDUCTOR
E
D
-D
0
Ground state 1-qp states 2 qps
Denergy > Denergy > 2energy 0
QUASIPARTICLES AND SUPERCURRENT IN A SUPERCONDUCTING POINT CONTACT
E()
+D
-D
0 AE
AE
AI
p
0i
0
1( )AAI
E
Lowest-lying 1-qp excitations
p
( ) 0AI
0
1( )AAI E
AI
p
0i
Aenergy E energy 0 DAenergy > -E + Aenergy E
Ground state 1-qp state Excited singlet
CORRELATED SWITCHING EVENTS
0 20 40 60 80 1001
10
100
# of
occ
uren
ces
D t / Tr
V(t)
0 5000 10 000 15 000 20 000
Need a ‘’reset’’ between pulses
V(t)
MEASURING THE SWITCHING PROBABILITY
1µs
sI0
swPNn
meast
hold
Vb(t)/Rb
pulsesN
eventsn
V(t)
MEASURING THE SWITCHING PROBABILITY
Dt
1µs
1.3 sI0
sI0
prepulse (reset)meast
hold
Vb(t)/Rb
pulsesN
eventsn
Uncorrelated switching events
QP
Reaching 1QP odd state
E1=0
E0=-Ea
E2=E
a
50GHzD
0-D
2pp
0
E()
τD 2 1
Ground state
1QP state (x2)
2QP state
1
0
2 I0=Ia
I2=-I
a
0
2pp
0
I()
I1=0
for Al
0.80 0.85 0.900.0
0.2
0.4
0.6
0.8
1.0
Psw
s
1
0
E
nQP
0
1
2
0.00 0.25 0.50 0.75 1.00
1
10
100
T1 (
µs)
0.0
0.1
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}
p
EA / D
/p
RELAXATION VERSUS ANDREEV ENERGY