per delsing lecture at the international summer school on quantum information, dresden 2005...
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Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Superconducting qubitsSuperconducting qubits
1. Overview of Solid state qubits
2. Superconducting qubits
3. Detailed example: The Cooper pair box
4. Manipulation methods
5. Read-out methods
6. Relaxation and dephasing
7. Single qubits: Experimental status
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
1. Overview of solid state qubits1. Overview of solid state qubits
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Solid State QubitsSolid State Qubits
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Energy scalesEnergy scales
Atoms and ionsSingle system
Identical systems possible
∆E=> ~500 THz
optical frequencies
High temperature
Hard to scale
Solid state systemsSingle system
System taylored
∆E=> ~10 GHz
microwave frequencies
Low temperature 20mK
Potentially scalable
NMR-systemsEnsemble system
Identical systems possible
∆E=> ~100 MHz
radio frequencies
High temperature
Impossible to scale ?
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Things to keep in mindThings to keep in mind
• These is a macroscopic system, they typically contains 109 atoms
• Thus coherence is hard to keep, relatively short decoherence times
• Nanolithography makes the system (relatively) easy to scale
• Experimental research on solid state qubits started late compared
to other types of qubits
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Requirements for read-out systemsRequirements for read-out systems
• Photons Photon detectors (hard for IR and mw)
• Magnetic flux SQUIDs
• Charge Single Electron Transistors
• Single spin (Convert to charge)
Two different strategiesTwo different strategies::Single quantum systems Single quantum systems quantum limited detectorsquantum limited detectors
Ensembles of systemsEnsembles of systems normal detectorsnormal detectors
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
2. Superconducting qubits2. Superconducting qubits
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Advantages and drawbacks of Advantages and drawbacks of superconducting systemssuperconducting systems
AdvantagesEnergy gap Protects against low energy exitationsGood detectors SQUIDs, SETsNano lithography Relatively easy scaling
Drawbacks”Large systems” Relatively short decoherence timesCooling The low energies require cooling to <<1K
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Superconducting qubit characteristicsSuperconducting qubit characteristics
Which degree of freedom charge, flux, phaseTrue 2-level system quasi 2-level systemRepresentation of one qubit single systemManipulation Microwave pulses, rectangular pulsesLevel splitting ~10GHzType of read-out Squid, SET, dispersiveBack action of read out Single shot possibleOperation time 100 ps to 1nsDecoherence time 4 µs obtained Scalability potentially goodCoupling between qubits static, tunable, via cavity
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Carge versus phase/fluxCarge versus phase/fluxFluxoid Quantisation in a
superconducting ringCharge Quantisation on a metalic
island
€
dϕ = n ⋅2π∫ =2e
hLI + δ
I = Ic sinδ
Φ = LI =h
2en ⋅2π −δ( ) = n ⋅Φ0 −
h
2eδ
€
ϕ
€
ϕ + n ⋅2π
€
Q = n ⋅e
Q = n ⋅2e
Reservoir
Island
Reservoir
Island
€
Q,ϕ[ ] = ieFlux Charge
Inductance CapacitanceJosephson coupling energy EJ Charging energy EC
Current VoltageConductance Resistance
€
Q,Φ[ ] = ih€
EC =e2
2C, EQ =
2e( )2
2C
€
EJ =RQ
R
Δ
2=
h
2eIC
€
δJosephsonjunction
(Josephson)junction
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
charge qubit
EJ ~ 0.1 EC
Quantronium
EJ ~ EC
flux qubit
EJ ~ 30 EC
phase qubit
EJ ~ 100000 EC
NIST/Santa Barbara
Superconducting qubitsSuperconducting qubits
Delft, NTTSaclay, YaleNEC, Chalmers, Yale
€
EJ
EC
small
€
E J
EC
Large
Q well defined well defined
€
Q,ϕ[ ] = ieQ and j are conjugate variables which obey the commutation relation:
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
ProbeProbe
TunnelTunnel junctionjunction
SQUIDSQUID looploop
Box Box GateGate
SingleSingleCooper-pair Cooper-pair tunnelingtunneling ReservoirReservoir
Ωk 10~1R
ΩM 30~2R
Y. Nakamura et al., Nature 398, 786 (1999).
0.5 1 1.5
-1
0
1
0 1
( )102
1+
1 0
( )102
1−
JE
eQ /0
CEE /
ttΔ
I0 〉
I1 〉superposition state
The NEC qubitThe NEC qubit
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
The quantronium, charge-phase (Saclay)The quantronium, charge-phase (Saclay)
D. Vion et al., Science 296, 286 (2002) €
EJ ≈ EC
Level splitting ≈ EJ
T2≈0.5 µs
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Coupling a qubit to a resonator (Yale)Coupling a qubit to a resonator (Yale)
Vacuum Rabi splitting, A. Wallraff, Nature 431 162 (2004)
Jaynes-Cummings Hamiltonian
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Persistent-current qubit (Delft)Persistent-current qubit (Delft)flux qubit with three junctions flux qubit with three junctions && small geometric loop inductance small geometric loop inductance
Science 285, 1036 (1999)
H = hz + Δx
with h=( o-0.5) oIp
0
-1
0
1
0.5
Icirc
E
/o
2Δ
+Ip
-Ip
0Ibias
€
Δ ≈1.3 EJ EC e−0.64
E J
EC
⎛
⎝ ⎜
⎞
⎠ ⎟
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
The phase qubit (NIST)The phase qubit (NIST)
McDermott et al. Science 307, 1299 (2005)
€
H = −EJ cosδ − 12π Φ0Ibδ +
q2
2CJ
€
EJ >> EC
€
Δ ≈ 8EJ EC 1−I
IC
⎛
⎝ ⎜
⎞
⎠ ⎟
2 ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
1
4
Large size~100µm
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
3. Detailed example:3. Detailed example:The Cooper pair boxThe Cooper pair box
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
The Single Cooper-pair box (SCB)The Single Cooper-pair box (SCB)
€
H = 4EC (n − ng )2 + EC
C
Cg
−1 ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟ng
2 − EJ cosθ
EC =e2
2C, n =
Q
2e, ng =
C1Vg
2e
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2
<n>
ng
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2
E /
EC
n g
EJ
∆ > EC > EJ > T2.5K ~1.5K ~0.5K 20mK
C
Cg
Vg
Q=n2e
Likharev and Zorin LT17 (84)
Charge statesdegenerate
€
0
€
1
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
The Single Cooper-pair box (SCB)The Single Cooper-pair box (SCB)
Which degree of freedom ChargeRepresentation of qubit single circuitLevels multi but uses only the two lowestManipulation mw- or rectangular- pulsesType of read-out Single Electron Transistor, Josephson junction,
dispersiveBack action of read out single shot possible, very low for dispersive ROOperation time 100 psDecoherence time 1µsScalability YesCoupling capacitive, or via resonator
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
How does the SCB work as a QubitHow does the SCB work as a Qubit
€
H = 4EC (n − ng )2 − EJ cosθ
i.e. the same Hamiltonian as for Bloch electrons
using ket representation we get
H = 4EC n − ng( )2
nn
∑ n − 12 EJ n n +1 + n +1 n( )
n
∑
If we assume that EJ << EC and stay close to the degeneracy point,
only two states, 0 and 1 , matters. Thus we get
H = 2EC (1− 2ng )1 0
0 −1
⎛
⎝ ⎜
⎞
⎠ ⎟− 1
2 EJ (B)0 1
1 0
⎛
⎝ ⎜
⎞
⎠ ⎟= 2EC (1− 2ng )σ z − 1
2 EJ (B)σ x
Analogy to a single spin in a magnetic fieldShnirman, Makhlin, Schön, PRL, RMP, Nature
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Eigenstates and EigenvaluesEigenstates and Eigenvalues
€
g = cosθ
2
⎛
⎝ ⎜
⎞
⎠ ⎟0 + sin
θ
2
⎛
⎝ ⎜
⎞
⎠ ⎟1
e = −sinθ
2
⎛
⎝ ⎜
⎞
⎠ ⎟0 + cos
θ
2
⎛
⎝ ⎜
⎞
⎠ ⎟1
θ = arctanEJ
4EC (1− 2ng )
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
Eigen statesEigen states
Eigen values for two state systemEigen values for two state system
€
Eg,e = m 16EC 1− 2ng( )2
+ EJ2
€
n = ng −1
2EC
∂E
∂ng
Expectation value for nExpectation value for n
E/4EC
<n>
ng
ng
The Coulombstaircase
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Energies, and the optimal pointEnergies, and the optimal point
E/4EC
<n>
ng=CgVg/2e
Optimal point
€
∂E
∂ng
= 0
Optimal point
Energy levels
Level splitting
€
∂E
∂ng
= 0
∂E
∂δ= 0
At the optimal point the system is insensitive (to first order) to fluctuations in the control parameters ng and
The Coulombstaircase
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Representation of the QubitRepresentation of the Qubit
• The basic building block of a quantum computer is called a Qubit
• Any two level system which acts quantum mechanically (having quantum coherence) can in principle be used as a Qubit.
The Bloch Sphere
Ψ =α 0 +β1 =α
β⎛ ⎝ ⎜ ⎞
⎠
α and β are complex numbers
0 =1
0
⎛ ⎝ ⎜ ⎞
⎠ , 1 =
0
1
⎛ ⎝ ⎜ ⎞
⎠ €
θ
€
g =cos θ /2( )
sin θ /2( )
⎛
⎝ ⎜
⎞
⎠ ⎟
e =−sin θ /2( )
cos θ /2( )
⎛
⎝ ⎜
⎞
⎠ ⎟
Ground state
Exited state
€
θ =arctanEJ
4 EC (1− 2ng )
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟= arctan
Bx
Bz
⎛
⎝ ⎜
⎞
⎠ ⎟
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
4. Manipulation methods4. Manipulation methods
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Manipulation with rectangular pulses Manipulation with rectangular pulses
-2.0
-1.0
0.0
1.0
2.0
3.0
0 0.5 1 1.5 2
0 0.5 1 1.5 2
ng
∆t
t<0 Starting at ng0
t=0 Go to ng0+∆ng
t=∆t Go back to ng0
Nakamura et al. Nature (99)
The left sphere with two adjacent pure charge states at the north and south poles corresponds to a CPB with EJ /EC << 1, which is driven to the charge degeneracy point with a fast dc gate pulse.
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Microwave pulses (NMR-style)Microwave pulses (NMR-style)
NMR-like Control of a Quantum Bit Superconducting CircuitE. Collin, et al. Phys. Rev. Lett., 93, 15 (2004).
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0 0.5 1 1.5 2ng
The energy eigenstates at the poles described within the rotating wave approximation. The spin is represented by a thin arrow whereas fields are represented by bold arrows. The dotted lines show the spin trajectory, starting from the ground state.
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Comparing the two methodsComparing the two methods
Microwave pulses
Slower, typical π pulse
Timing is easier, NMR techniques can be used
Smaller amplitude and monocromatcMore gentle on the environment
Works at the optimal point
€
τ π ≈10h
EJ
≈1 ns
Rectangular pulses
Faster, typical π pulse
More accurate timing required
Large amplitude and wide frequency contentShakes up the environment
Can not stay at optimal point
€
τ π ≈1
2
h
EJ
≈ 50 ps
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
5. Read-out methods5. Read-out methods
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
• Josephson-quasiparticle cycle Fulton et al. PRL ’89
-2e
Cooper-pair box
JE qp1Γ>> hJE
• detect the state as current • initialize the system to
1
0
CC EeVE 322 +Δ<<+Δ
+ probe
-e-e
qp1Γ
qp2Γ
Repeted measurement gives a current
Read-out with a probing junction (NEC)Read-out with a probing junction (NEC)
€
I ∝ P 1( )€
1 ⇒ 2e
0 ⇒ 0e
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
ProbeProbe
TunnelTunnel junctionjunction
SQUIDSQUID looploop
Box Box GateGate
SingleSingleCooper-pair Cooper-pair tunnelingtunneling ReservoirReservoir
Ωk 10~1R
ΩM 30~2R
Y. Nakamura et al., Nature 398, 786 (1999).
0.5 1 1.5
-1
0
1
0 1
( )102
1+
1 0
( )102
1−
JE
eQ /0
CEE /
ttΔ
I0 〉
I1 〉superposition state
The NEC qubitThe NEC qubit
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
C
C
C
C C
st
bt
s
b t
SET
TrapReservoir
Readout gate
SET gate
Control gate
Box
1 mμ
Trap gate
Box gate
1. Measurement circuit is electrostatically decoupled from the qubit
2. Final states are read out after termination coherent state manipulation
Read-out with SET sample and hold (NEC)Read-out with SET sample and hold (NEC)
1. Manipulation2. Trap quasi particles3. Measure trap with SET
Single shot, but still fairly slow
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Lecture at the international summer school on Quantum Information, Dresden 2005
each readout only two possible outputs pulse height adjusted for ~50% switchingSQUID switched to gap voltage probability measured with ~ 5000 readoutsSQUID still at V=0
Switching SQUID readout scheme (Delft)
time
Vthr
V
2.3 2.4 2.5 2.6 2.70000000000
123456789
100
switc
hing
pro
babi
lity
(%)
pulse height @ AW generator (V)pulse height
pulsed bias current~ 5 ns rise/fall time
τmeas~5 ns, τtrail~500 ns
time quantum
operationsreference
trigger
Room temp. output signal
IV
Switching
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Switching Junction readout scheme (Saclay)
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Read out with the RF-SETRead out with the RF-SET
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
A single Cooper-pair box qubit (Chalmers)A single Cooper-pair box qubit (Chalmers)integrated with an RF-SET Read-out systemintegrated with an RF-SET Read-out system
RF-SET
V
singleCooper-pairbox
Büttiker, PRB (86) Bouchiat et al. Physica Scripta (99)Nakamura et al., Nature (99)Makhlin et al. Rev. Mod. Phys. (01)Aassime, PD et al., PRL (01)Vion et al. Nature (02)
A two level systen based on the charge states|1> = One extra Cooper-pair in the box|0> = No extra Cooper-pair in the box
∆ >> EC >> EJ(B) >> T2.5K 0.5-1.5K 0.05-1K 20mK
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
The Single Electron TransistorThe Single Electron Transistor
ICg CC
Vg = Qg Cg-30
-20
-10
0
10
20
30
-200 -100 0 100 200
Id
(pA)
Vds (µV)
Current-Voltage Characteristics for a Single Electron Transistor
ICg C
C
Vg = Qg Cg
Qg=0
Qg=e/2
drain
gate
source
-100-90-80-70-60-50-40-30
-400-300-200-100 0 100 200 300400
V (µV)
Vg (µV)
e/Cg
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
The Radio-Frequency Single Electron Transistor (RF-SET)The Radio-Frequency Single Electron Transistor (RF-SET)
L
C
DirectionalCoupler
Mixer
SET-bias
Tank circuit
ColdAmplifier
WarmAmplifier
~RF-sourceOutput
RFLO
Bias-Tee
C
SingleElectron
Transistor
∑
Modulation of conductanceand reflection
-20
-15
-10
-5
0
5
10
15
20
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
V (mV)
RSET=44.1kΩ
C∑=370 aF
Cg≈20aF
Vdc
Vac
Very high speed: 137 MHzR. Schoelkopf, et al. Science 280 1238 (98)
Charge sensitivity: ∂Q= 3.2 µe/√HzA. Aassime et al. APL 79, 4031 (2001)
Current Voltagecharacteristics
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Performance of the RF-SETPerformance of the RF-SET
Time domain:
∆Q=0.2e, inset 0.05e
Results:Results: Aassime et al.Charge sensitivity: ∂Q= 3.2 µe/√Hz APL 79, 4031 (2001)Energy sensitivity: ∂ = 4.8 h PRL 86, 3376 (2001)
-800
-600
-400
-200
0
200
400
600
800
-100 -50 0 50 100
Vdet (mV)
t (µs)
Bw=1 MHz0.2 epp
-0.04
0.00
0.04
0.08
-80 -40 0 40 80
Step= 0.05 eBw= 1MHz
Frequency domain:∂Q=0.035erms, fg=2MHz
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Dispersive read-outs, parametric Dispersive read-outs, parametric capacitance or inductancecapacitance or inductance
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Coupling a qubit to a resonator (Yale)Coupling a qubit to a resonator (Yale)
A. Wallraff, Nature 431 162 (2004)
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
The Cooper-pair box as a Parametric CapacitanceThe Cooper-pair box as a Parametric Capacitance
€
CQ = ±Cg
2
Cg + CJ( )2
2e2
E J
We define an effective capacitance which contains two parts
€
Ceff =∂ Qg
∂Vg
= Cgeom + CQ =
CQ =Cg
CΣ
2e∂ n
∂Vg
= Cgeom −∂ 2E
∂Vg2 = ±
Cg2
Cg + CJ( )
E J2EQ
ΔE 3
At the degeneracypoint we get:
Büttiker, PRB (86) Likharev Zorin, JLTP (85)c.f. the parametric Josephson inductance
C, EJ
Cg
VgQ=n2e
C
Cg
Vg=>CQ[ng,α]
€
E / EQ
€
n
€
CQ /Cgeom
€
ng = CgVg /2e
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Quadrature measurements with the RF-SETQuadrature measurements with the RF-SET
DirectionalCoupler
SET-biasColdAmplifier
~RF-source
C
SingleElectron
Transistor
∑
WarmAmplifier
QuadratureMixer
Inphase
Bias-Tee
L
C
Tank circuit
Out ofphase
RFLO
RFLO
90°
€
Im Vr[ ] = V0 Im Γ[ ]
Reactive part, C (or L)
€
Re Vr[ ] = V0 Re Γ[ ]
Disspative part, R
High speed: 137 MHzSchoelkopf, et al. Science (98)
Charge sensitivity: ∂Q= 3.2 µe/√HzAassime et al. APL 79, 4031 (2001)
Cooper-pair transistor similarto Cooper-pair box
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Lecture at the international summer school on Quantum Information, Dresden 2005
The Josephson Quasiparticel cycle: JQPThe Josephson Quasiparticel cycle: JQP
J2
J1
One junction isresonant part of the time
Quasi parttransition
Cooper-pairresonance
Cooper-pairresonance
Quasi parttransition
DJQP
Drain-3, -1
Drain-1, 1
Source-1, 1
Source0, 2
Phase response
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Lecture at the international summer school on Quantum Information, Dresden 2005
The Double Josephson Quasiparticle cycle: DJQPThe Double Josephson Quasiparticle cycle: DJQP
One junction is always resonantbut only one at the time
This results in an average CQ
Drain-3, 1
Drain-1, 1
Source-1, 1
Source0, 2 Drain jcn in
resonance
Source jcn inresonance
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Quantitative Comparison of the Quantum CapacitanceQuantitative Comparison of the Quantum Capacitance
€
θ ≈−2QCQ
CT
=CJ
CT'
EJ2EQ
ΔE 3 ,
If the phase shift is small it can be approximated by:
Assuming equal capacitances we can calculate CQ for a two level system and compare with the data.
From spectroscopy EJ/EC=0.12
Temperature adds to the FWHM€
FWHM =22 / 3 −1
4
EJ
EC
= 0.191EJ
EC
, T = 0
T=140 mK
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Spectroscopy on the quantum capacitanceSpectroscopy on the quantum capacitance
Junction 1
Junction 2
By tuning only one junction into resonance we can excite the system to the exited state, which has a capacitance of the opposite sign
€
CQ = 1− P( )CQ 0 + P −CQ 0( )
P =1
2 ⇒ CQ = 0
EJ1=3.0 GHzEJ2=2.8 GHz
P= Probability to be in the exited state
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Quantum Cap QubitQuantum Cap Qubit
• Operate and readout at optimal point• No intrinsic dissipation• Tank circuit protects qubit by filtering environment• Lumped element version of Yale cavity experiment, Wallraff et al. Nature (04)
CT
Cc
Cg
Cin
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Measuring the charge on the boxMeasuring the charge on the boxThe Coulomb StaircaseThe Coulomb Staircase
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Lecture at the international summer school on Quantum Information, Dresden 2005
The Coulomb Staircase ComparingThe Coulomb Staircase Comparingthe Normal and the Superconducting Statethe Normal and the Superconducting State
2e periodicity is achievedif Ec is sufficiently small
<1K
Bouchiat et al. Physica Scripta (98)
Aumentado et al. PRL (04)
0
1
2
3
4
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
Normal state
Superconducting state
CgVg [e]
Gunnarsson et al. PRB (04)
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Lecture at the international summer school on Quantum Information, Dresden 2005
The Coulomb staircase comparingThe Coulomb staircase comparingthe normal and the superconducting statethe normal and the superconducting state
0
1
2
3
4
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
Normal state
Superconducting state: Sample 1
Superconducting state: Sample 2
CgVg [e]
Small step occures due to quasi particle poisoning
Bouchiat et al. Physica Scripta (98)
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Lecture at the international summer school on Quantum Information, Dresden 2005
What would you expectWhat would you expectin the superconducting statein the superconducting state
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0 0.5 1 1.5 2ng
∆~
Using EC1.0K pure 2e periodicity is obtained
Tuominen et al. PRL (93)Lafarge et al. Nature (93)
€
˜ Δ ≈ Δ0 − kBT ln N( )
Δ0 ≈ 2.4 K for Al
˜ Δ ≈L − S
L + S,
L = size of long step
S = size of short step
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Size of the odd step versus magnetic fieldSize of the odd step versus magnetic field
•Lower Ec is better
•∆ is suppressed by parallell magntic field
Faster suppression in the reservoir than in the box, due to film thickness
Reservoir 40 nmBox 25 nm
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
SpectroscopySpectroscopy
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Energy Levels of the Cooper-Pair BoxEnergy Levels of the Cooper-Pair Box
0 1 20
0.5
1
1.5
2
2.5
3
3.5
4
Q0
Qb
ox
[e]
EJ>EC
EJ<E
C
2/)1(2 0 xJzC EQEH −−=
0 1 2
0
1
2
3
4
E [
Ec]
0 1 20
1
2
Gate Charge Q0 [e]
Box C
harg
e N
box [
e]
| 0 >
| 1 >
01
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Spectroscopy of the Cooper-Pair BoxSpectroscopy of the Cooper-Pair Box
0 1 20
0.5
1
1.5
2
2.5
3
3.5
4
Q0
Qbo
x [e
]
EJ>EC
EJ<EC 0 0.2 0.4 0.6 0.8
15
20
25
30
35
40
ng [e]
f HF
[GH
z] data
EC=42.0GHz, E
J=20.2GHz
SpectroscopySpectroscopy
16
12
0
8
4 Spectroscopy
perpendicular B-field (/)
Level splitting ∆E(nLevel splitting ∆E(ngg))B-field dependenceB-field dependence
of Eof EJJ
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Relaxation and dephasingRelaxation and dephasing
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Dephasing and mixingDephasing and mixing
Relaxation or mixing• The environment can exchange energy with
the qubit, mixing the two states by stimulated emission or absorption. This has the characteristic time T1
• Describes the diagonal elements in the density matrix
• Fluctuations at resonance, S(
• Important during read-out
T1 T2
Dephasing• The environment can create loss of phase memory
by smearing the energy levels, thus changing the phase velocity. This process requires no energy exchange, and it has the characteristic time T
• Describes the decay of the off-diagonal elements in the density matrix
• Fluctuations at low frequencies, S(0)
• Important during “computation”
The qubit can be disturbed in two different ways.
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Decoherence sourcesDecoherence sourcesWhat are the major decoherence sources in your system?Active sources (stimulated) absorption and emision heat, noise,….
Passive sources (spontaneous) emission only quantum fluctuations
external degrees of freedom
photons, phonons, quasiparticles...
Can they be controlled ?Cooling, shielding, filtering, tailoring the environment
Read-outQubit
50Ω
Manipulation
Material dependentMicroscopic fluctuators
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
The spin boson modelThe spin boson model
A single two level system (weakly) coupled to an environment described as a bath of harmonic oscillators.
€
θ =arctanE j
EC 1− 2ng( )
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
€
H = 2EC 1− 2ng( )σ z −1
2EJσ x
H0 Two level system1 2 4 4 4 4 3 4 4 4 4
− 4ECδng t( )σ z
δH t( ) Perturbation1 2 4 4 3 4 4
€
θ€
z
€
x
€
δH t( )
€
δH|| t( )
€
δH⊥ t( )
€
H0
The effect of the harmonic oscillators can be described as a fluctuating gate voltage, or a fluctuating ng
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Decoherence ratesDecoherence rates
The slow longitudinal fluctuations leads to dephasing =>
The transversal fluctuations which are resonant with the levelsplitting causes mixing (relaxation) =>
The rates are directly proportional to the spectral densities
€
Γ2 =1
T2
=1
2Γ1 + Γϕ
€
Γ1 =1
T1
=e2
h2sin2 θ SV ω =
ΔE
h
⎛
⎝ ⎜
⎞
⎠ ⎟
€
Γϕ =1
Tϕ
=e2
h2cos2 θ SV ω = 0( )
Positive frequenciesrelaxation
Negative frequenciesexitation€
SV ω( )
€
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Examples for spectral densitiesExamples for spectral densities
Environmental circuit impedance 50Ω
1/f like charge fluctuations
Shot noise from SET read-out
Log S(f)
Log f [Hz]
If these are the only contributions1/f noise will be important for dephasingand environment will be important for relaxation (if SET is switched off)
€
SV ω( ) = hω ⋅Re Z ω( )[ ] 1+ cothhω
2kBT
⎛
⎝ ⎜
⎞
⎠ ⎟
⎛
⎝ ⎜
⎞
⎠ ⎟
€
SV ω( ) = 4ESET
2
e2
4I /e
ω2 +16I2 /e2€
SV ω( ) =α
ω
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
MeasurementsMeasurements of T of T1 1 and Tand T22
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Manipulation with dc-pulses Manipulation with dc-pulses
-2.0
-1.0
0.0
1.0
2.0
3.0
0 0.5 1 1.5 2
0 0.5 1 1.5 2
ng
∆t
t<0 Starting at ng0
t=0 Go to ng0+∆ng
t=∆t Go back to ng0
The probability to find the qubit in the exited state oscillates as a function of ∆t.
The charge is measured continuously by the RF-SET
Difference between these two curves = excess charge ∆Qbox
c.f. Nakamura et al. (99)
0 1 2 30
0.5
1
1.5
2
2.5
3
gate charge Q0 [e]
Qbox
[e]
pulse train offpulse train on
0
1
2
3
0 1 2Q
g [e]
Pulse train offPulse train on
T ≈100 ns r
∆t≈100ps trise≈30ps
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Oscillations at the charge degeneracyOscillations at the charge degeneracy
Good news: • We observe oscillations in 6 samples• A high fidelity! >70%Deviation from 1.0 e due to finite risetime (~30ps) of pulses, i.e. no missing amplitude
Bad news:T2 only ~10 ns
0 0.5 1 1.50
0.20.40.60.8
1
?t [ns]
.5 .5 2
.2 .4 .6 .8
gate charge Q
[e]
16
12
0
8
4 SpectroscopyCoherent oscillation
perpendicular B-field
EJ [
GH
z]
Oscillation frequency = EJ/hagrees well with EJ from spectroscopy.
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
0 20 40 60 80 100 120 140 160 18030
60
90
30
60
90
30
60
90
switching probability (%)
A = -3 dB
RF pulse length (ns)
A = 3 dB
A = 9 dB
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00
100
200
300
400
500
600
Rab
i fre
quen
cy (
MH
z)
RF amplitude, 10A/20
(a.u.)
Flux qubit : Rabi oscillationsFlux qubit : Rabi oscillations
Chuorescu, Bertet
time
trigger
Ib pulse
read-outoperation
AMW
τMW
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Rabi and Ramsey in the Quantronium (Saclay)Rabi and Ramsey in the Quantronium (Saclay)
Ramsey fringes
Vion, Esteve, et al. Science 2002
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Extracting T2 from free precession oscillationsExtracting T2 from free precession oscillations
0 1 2 3 4 5 6
- 0 . 3
- 0 . 2
- 0 . 1
0
0 . 1
0 . 2
0 . 3
∆ t [ n s ]
Q
b
o
x
[
e
]
0 . 4 0 . 5
2
4
6
8
1 0
1 2
g a t e c h a r g e n
g
T
2
[
n
s
]
Note: relatively large visibility, here ~60%
Oscillation period agrees well with level splitting
T2 decreases rapidly as the gate charge is detuned from the degeneracy point.
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Measurements of TMeasurements of T22 vs. Gate Charge n vs. Gate Charge ngg
•Double pulse data agrees with single pulse data•Q0 dependence coupling to charge•T2 limited by relaxation at the degeneracy point•T2 limited by 1/f-niose away from the degeneracy point
0.2 0.4 0.6 0.8 1 1.210
8
109
1010
gate charge ng [e]
EJ/h=3.6GHz
EJ/h=9.4GHz
T2-1
T. Duty et al., J. Low Temp. Phys. (04)T. Duty et al., J. Low Temp. Phys. (04)
•Very similar to data from NEC and JPL
€
Using Sq ω( ) =α
ω
for longitudinal
Gaussian fluctuations
Γ*2 =
E1/ f
h1π ln
E1/ f
hωir
where
E1/ f = 4EC α
€
We find α = 4 ×10−3e
This is higher than
values from low
frequency measurements
(0.3−1.0 ×10−3e).
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
t
Qb
ox(t
)
1
2rT
Determining a TDetermining a T11 that is smaller than T that is smaller than Tmeasmeas
€
< n > (tR ) = 2n0
t1tR
1− e−t R / t1
1+ e−t R / t1
The average charge The average charge <Q<Qboxbox> depends both > depends both
on Ton T11 and T and TRR
0 500 1000
0.2
0.4
0.6
0.8
1
TR [ns]
T1=72ns
T1=87ns
<Q
box
> [
e] n0 depends on the pulse rise time
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
TT11 Measurements vs n Measurements vs ngg and E and EJJ
provide info on S(provide info on S() and form of coupling) and form of coupling
€
Γrelax ≡ T1−1 =
e2
h2κ 2 sin2η ⋅S ω = ΔE( )
0
10
20
30
40
50
60
0 0.2 0.4 0.6 0.8 1
gate charge ng [e]
T1 [
ns]
EJ
5GHz
EJ
8GHz
EJ
9GHz
The dependenceindicates that the qubit is coupling to charge.
We can extrapolate the measurement to the degeneracy point and compare with T2 measurements
€
sin2η
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Measured samplesMeasured samples
K. Bladh, T. Duty, D. Gunnarsson, P. DelsingNew Journal of Physics, 7, 180 (2005)Focussed issue on: Solid State Quantum InformationSolid State Quantum Information
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Summarizing T1 and T2Summarizing T1 and T2
T2 away from degeneracy pointT2 seems to be limited by charge noise whan the qubit is tuned away from Degeneracy point. The extracted value for the 1/f noise is almost an order of magnitude worse than standard values for SETs
T2 at degeneracy pointAt degeneract T2 seems to be limited by relaxation, Best value 10ns.
T1T1 seems to be due to charge noise. Possibly due to the back ground charges
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Back-actionBack-action
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
The needed measurement timeThe needed measurement time
€
tm =δQ
eκ
⎛
⎝ ⎜
⎞
⎠ ⎟2
δQ is the charge sensitivity of the SET, κ is the coupling coeficient CC
Cqb
€
tmix−1 = κ 2 e2
h2
EJ2
ΔE 2SV
ΔEh( )
The mixing time depends on the shot noise in the The mixing time depends on the shot noise in the SETSET
Signal to noise ratio (SNR)Signal to noise ratio (SNR)
€
SNR =tmix
tm
= hΔE
E J
1
δQ SVΔEh( )
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Spectral density of the voltage fluctuationsSpectral density of the voltage fluctuationsof the SET island for our best SET of the SET island for our best SET
10-2
10-1
100
0.1 1 10 100 1000
Full Expression [nV2/Hz]
Shot-noise [nV2/Hz]
Nyquist [nV2/Hz]
SV
[nV2
/Hz]
Frequency [GHz]
RF-carrier
∆EAl/h
G. Johansson et al.PRL 2001
Assuming readout at ∆E=2.4K
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
Summarizing results for two samplesSummarizing results for two samplesAssume =1% , EJ=0.1K, ∆EAl=2.4K, ∆Enb=15Kand that the SET dominates the mixing
Sample 1 : I= 6.7 nA, ∂Q= 6.3µe/√Hz [A.Aassime et al. PRL 86, 3376 (2001)]
tm tmix SNRAl-qubit 0.40 µs 8.6 µs 4.6Nb-qubit 0.40 µs 1.9 ms 68
Sample 2: I= 8 nA, ∂Q= 3.2µe/√Hz [A.Aassime et al. APL 79, 4031 (2001)]
tm tmix SNRAl-qubit 0.10 µs 6.4 µs 8.0Nb-qubit 0.10 µs 1.3 ms 114
Summarized in: K. Bladh et al. Physica Scripta T102, 167 (2002)
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
0 100 200 300 400 500 600 700 800 9000
50
100
150
Ibias [pA]
T1
[ns]
We find TWe find T11 short and independent of short and independent of
SET bias in 6 different samples.SET bias in 6 different samples.
DJQP bias
JQP bias
Per Delsing
Lecture at the international summer school on Quantum Information, Dresden 2005
SummarySummary
• Macroscopic systems that allow tailoring and scaling.
• Energy gap protects against low energy excitations.
• Different flavors depending on EJ/EC ratio
• Optimal point important to avoid decoherence
• T2/Top ≈ 1 µs / 1 ns = 1000
• T1 and T2 can be estimated from spectral densities
• Dispersive read-out schemes promising -> QND
• Coupling to cavities allow coupling and cavity QED on-chip