depts. of applied physics & physics yale university expt. k. lehnert l. spietz d. schuster b....
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Depts. of Applied Physics & PhysicsYale University
expt.K. LehnertL. Spietz
D. SchusterB. Turek
Chalmers UniversityK.Bladh
D. GunnarssonP. Delsing
The Cooper-pair Box as a Quantum Spectrum Analyzer
Rob Schoelkopf
The David and LucilePackard FoundationFunding:
And discussions w/:M. Devoret & J. Martinis
theoryA. ClerkS. GirvinD. Stone
Yale
Cooper-pair Box Coupled to an SET
Box
SET
Vg Vge
Cg Cc Cge
Vds
Box SET Electrometer
Superconducting tunnel junction
Qubit Quantum state readout
Quantum spectrum analyzer
Cooper-pair Box SET Transistor
Nonequilibrium noise source
or
Cooper-pair Box
2(2 )4 4 K
2c
eE
C
20.5 K
4 JJ e R
E
Vg
Vg
n
1 fFg jC C C
14
2 2g
elg
c
CE
e
VE
2ˆ ˆ
2ˆz x
e JlHE E
1
0
z
z
n
n
(e.g. Bouchiat et al., 98)
effB
effB
2ggC V
e0.5
0
1
1
01
0E
2ggC V
e
2JE
Cooper-pair Box as Quasi-spin 1/2
14
2 2g
elg
c
CE
e
VE
1 / 2Zn
Measure charge2
ˆ ˆ ˆ2z x
e JlHE E
2elE
zExcited state
Ground state
2JE
2JE
x
z
x
z
x
n
effB
2elE
a b ca b c
a b c
Continuous Measurement of a Single Spin
Measured continuously by SET
Theory: Cooper-pair box ground state
2e
1e
n
0
0.5
0 1 0.5
Measurement must cause additional dephasinguncertainty principle
Measurement may also mix states, drive transitions from ground state
2ggC V
e
1
1
0 1
0E
2ggC V
e
n
Cooper-Pair Resonance Spectroscopy
2ggC V
e
Vapp
0 1 0.5
=38 GHz
Cg
Vapp=Vg+Vacsint
1
0
38 GHz
2-photonPeak
Peak location
2ggC V
e
0
0-1-2 1 2
0.29
0.25
0 1
E 0 / 2eff
JE
2ggC V
e
0effJE
B
0cos( / )effJ JE E
“SQUID box” to vary EJ
Fit parameters:
1.79 0.004
0.626 0.008C
J
E K
E K
Determination of Box Hamiltonian
Vapp
32 GHz
35 GHz
38 GHz
01
Effects of Voltage Noise on Pseudo-Spin
sinelB E
coselB E
slow fluctuations of B dephasing
resonant fluctuations of B mixing
2
201
1sin
boxmix Vmix
eS
T
01 effB
221
0 cosboxV
eS
T
2el boxeE V effB
2elE
z
x
2JE
01
sin JE
01VS 01VS
0T 0T
0101
absorption emission
Emission and Absorption due to Environment
g
e
01
Box envR /
2 R
1env
envV kT
Se
kT
VS
0
Cg Box
Spontaneous Emission into Environment
2
2
01
1
21sin
envV
eS
T
12%gC
C
01 01( ) 2 (50 )envVS
1 0.1 1 sT
50 envR
01kT 01( ) 0envVS
estimate:
Excited-statelifetime, T1
Vg
Pea
k he
ight
(e)
0 time 10 s
Excited-state Lifetime Measurement of Box
1 1.3 sT 2
ggC V
e
n
0 1 0.5
1
0
0.3e
follow peak height after shift
with continuous measurement
910( ) 5 10 pairs /
boxnS Hz (@ 76 GHz)
Relaxation by Electrometer?
Pea
k H
eigh
t (e
)
Electrometer Operating Point (Vge)
0
0.6 1 1.3 sT
1Peak Height T
0.3
Cg
2e SET
Vg
Cc
e-
Vge
1 2
1 Rabi
TT
Peaks saturate when
Vds
Charging Diagram of SET Electrometer
CgVge /e
eVds
Vge
Electrometer SET:
R = 150 kEC ~ ~ 2.4 K
4
2Ec
0
1e
Electrometer operating pt.on “DJQP” feature
Quantum Shot Noise of DJQP* Process
Excitation Relaxation
Sharp thresholds due to opening & closing of transport channels
Γ
*Double Josephson-quasiparticle cycle: (A. Clerk et al. cond-mat/0203338)
0
log VS
2 2small: ( )
4J
VJ
S
Predicted Effects of DJQP on Box Charge
Qubit acts like a spectrum analyzer of the SET quantum noise!
(A. Clerk et al. cond-mat/0203338)
(see also Aguado & Kouwenhoven, 2000 for double dot)
01
2ggC V
e
n
1
0
01 0
Average box charge
Log
[SV(
)]
SET noise spectrum
on resonanceoff resonance
• Inelastic lifetime is long > 1 s : (and electrometer affects T1 !)
• Cooper-pair box as a “quantum spectrum analyzer”
• RF-SET a good probe of the charge states of box
• Spectroscopic determination of Hamiltonian of box
Conclusions
1 1 01 500,000Q T
( )VS ( )VS
Measures all Noise Classical (symmetric) Quantum (asymmetric)
0
( )VS
p
Gap rise (Vds= 1200 V)
JQP (Vds= 800 V)
Supercurrent (Vds=0)
dseV
2ggC V
e
gg eC V
e
Coulomb Staircase vs. Electrometer Bias
T=20 mK
n
Back-action increases with
electrometer bias
DJQP (Vds= 400 V)
Cooper-pair Staircase vs. Electrometer Bias
Theory: Cooper-pair box ground state
2e
1e
n
0
0.5
0 1 0.5 2ggC V
e
1
sweep gate @ 2e per 100 s
Data:Vds= 350 VVds= 275 VVds= 250 V
Cooper-pair Staircase vs. Josephson CouplingTheory: Cooper-pair box w/ max EJ
2e
1e
n
0
0.5
0 1 0.5 2ggC V
e
1
Data: maximum EJ
minimum EJ
Pea
k he
ight
0/ 0-1-2 1 2
B
0cos( / )effJ JE E
“SQUID box” to vary EJ
Charge States Coupled by EJ
Vapp
2elE
z
2JE
x
Peak location 2
ggC V
e32 GHz
35 GHz
38 GHz
ne=-1 ne=0 ne=1
Ec
E
ggC V
e0-0.5 0.5
/q e
Single-electron Box: Coulomb Staircase
ee
Coulomb Staircase
Thermally broadened
kT/Ec
-1 1
500 mK200 mK 50 mK0
-1
1
First demonstrated by Lafarge et al, ’91
(CEA Saclay)
B 1 T
Ec/4 1.6 KCE
Temperature Dependence in Normal State
2
*2
1
T
0.235 2HWHM
0
2ggC V
e
n
Peak width
Decoherence Time of Box
*2 500 psT
2Power (arb) R
*2 1 nsT
0.265
74 GHz 78 GHz0.2
01 /E h
DJQP Noise, Off-resonance
0 0.5 1 1.5 20
2
-1 -0.5 0 0.5 110
-15
10-10
Her
tz-1
Γ > Γ Population inversion in the qubit.
Ω / ECSA
vg. Q
ubit
Cha
rge
•Move away from the center of the resonance by increasing VDS…
NB
Pea
k he
ight
0/
0-1-2 1 2
0 1
E 0 / 2eff
JE
2ggC V
e
0effJE
B
0cos( / )effJ JE E
“SQUID box” to vary EJ
Charge States Coupled by EJ
Vapp
01
cos( )2
elEt
z
2JE
x
Effects of Voltage Noise on Qubit
xelE
effB
sinelB E
coselB E
2JE
slow fluct. of Bdephasing
resonant fluct. of B mixing
2
201
1sin
boxmix Vmix
eS
T
01 effB
2
210 cos
boxV
eS
T
z 2el boxeE V
2ggC V
e
n
Box State Depends on Electrometer Bias
Vds (V)
250290
1200
0
420470760
Cg
2e SET
SET Box Environment
Spontaneous Emission
Vg
50 envR
CcVds
E
Relaxatione
g
Backaction of SET on Box
boxV
SETV
t
SET
eC
Cm
Cg
Zenv
mSEox
oxb T
b
CV
CV
SETV
effB
2elE
z
2JE
x
Who’s measuring whom?
Measured continuously by SET
Theory: Cooper-pair box ground state
2e1e
m r
g
e
n
0
1
Can Electrical Circuits be ‘Quantum?’
Cooper-pair boxY. Nakamura et al, Nature 1999
New Challenges:
•Understand and minimize decoherence
•Develop efficient quantum readout
New Opportunities:
•Create artificial atoms
•Quantum computation
Macroscopic Quantum Coherence:
( , )E f Q ( , )?QH f
Quantum Circuits for Quantum Computing
Classical bit
values 0 or 1
Information as state of a two-level quantum system
orvalues ,0 10 1
Prediction: a 2,000 bit quantum computer = a conventional computer the size of universe.
Quantum bit (or “qubit”)
superposition:
0
( )VS
The Quantum Spectrum Analyzer
( )VS
( )VS
pCmeas
?
Vbias
Measures all Noise Classical (symmetric) Quantum (asymmetric)
Quantum Computing
Scalable
Coherent
ControllableMeasurable
Cooper-pair boxSQUID’s
Ion TrapsLiquid State NMRNuclear Spins in
Semiconductors
How coherent is a Cooper-pair box?
Outline
•Charge quantization on a normal-metal islandSingle-electron Box
•Superconducting island as quantum two-level systemCooper-pair Box
•Spectroscopy of the Cooper-pair boxSingle-electron Tranistor (SET) measures box
•Box Measures SET Quantum Spectrum Analyzer
Microwaves
Small, Cold and Fast
1 m
Dilution refrigeratorT = 15 mK
MillikelvinsNanometers
Experiment Diagram
Quantum Shot Noise of DJQP* Process
0 0.5 1 1.5 20
2
Avg
. Qub
it C
harg
e
-1 -0.5 0 0.5 110
-15
10-10
Her
tz-1
NBΩ / ECS
Excitation RelaxationΩ= -ECS Ω= ECS
Qubit acts like a spectrum analyzer of the SET quantum noise!
*Double Josephson-quasiparticle cycle: (A. Clerk et al. cond-mat 0203338)
(see also Aguado & Kouwenhoven, 2000 for double dot)
Single Spin ½ Quantum Measurement
NMR of a Single Spin
Box
SET
Vgb Vge
Cgb Cc Cge
Vds
e
ne
The Single-Electron Box
2/c g geE E n C V e
ne to ne+1 electrons
Vg
island
Cj Rj
1 fFg jC C C Cg
2
1 K2c
eE
C
ne=-1 ne=0 ne=1
Ec
E
ggC V
e
Ec/4
Normal tunnel junction
ne=-1 ne=0 ne=1
Ec
E
ggC V
e0-0.5 0.5
/q e
Single-electron Box: Coulomb Staircase
ee
Coulomb Staircase
Thermally broadened
kT/Ec
-1 1
500 mK200 mK 50 mK0
-1
1
First demonstrated by Lafarge et al, ’91
(CEA Saclay)
B 1 T
Ec/4 1.6 KCE
CgeVge
Vds
Ids
Vds
0ge eg VC
e
1
2gegeVC
e
10 nA
1 mV
Single-electron Transistor: Electrometer
Electrometerinput gate
drain
source
SET
Quantum Shot Noise of DJQP* Process
-1 -0.5 0 0.5 110
-15
10-10
Her
tz-1
Ω / ECS
Excitation RelaxationSharp thresholds due to opening & closing of transport channels
Γ
*Double Josephson-quasiparticle cycle: (A. Clerk et al. cond-mat/0203338)
10-5 e/Hz1/2 charge noise
Sub-electron sensitivity for > 100 MHz bandwidth
-10 0 10Time ( s )
0.2 electrons
Electrometer input gate
TransformerSET
RF
Ref
lect
ed p
ower
Measure RF power reflected from LC transformer
Schoelkopf et al., (Science 1998)
Radio-Frequency Single Electron Transistor (RF-SET)
Response to step in Vge
single time trace
e
ne
The Single-Electron Box
2/c g geE E n C V e
ne to ne+1 electrons
Vg
island
Cj Rj
1 fFg jC C C Cg
2
1 K2c
eE
C
ne=-1 ne=0 ne=1
Ec
E
ggC V
e
Ec/4
Normal tunnel junction
Conclusions
•Cooper-pair Box: A quantum two-level systemworst-case coherence
•Box Hamiltonian determined with spectroscopy
•Long excited-state lifetime while continuously measured.
*2 01 100Q T
1 1 01 100,000Q T
1.79 0.004CE 0.626 0.008JE
Gap rise 500 pA
Eye 100 pA
Weak JQP 100 pAStrong JQP 150 pADouble JQP 300 pA
dseV
2ggC V
e
gg eC V
e
Coulomb Staircase vs. Electrometer Bias
T=20 mK
n
Back-action increases with
electrometer current
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