per delsing lecture at the international summer school on quantum information, dresden 2005...

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

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1. Overview of solid state qubits1. Overview of solid state qubits

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Solid State QubitsSolid State Qubits

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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 ?

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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

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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

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2. Superconducting qubits2. Superconducting qubits

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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

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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

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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

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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:

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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

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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

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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

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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

⎛

⎝ ⎜

⎞

⎠ ⎟

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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

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3. Detailed example:3. Detailed example:The Cooper pair boxThe Cooper pair box

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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

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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

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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

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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

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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

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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

⎛

⎝ ⎜

⎞

⎠ ⎟

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4. Manipulation methods4. Manipulation methods

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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.

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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.

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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

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5. Read-out methods5. Read-out methods

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• 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

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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

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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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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

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Switching Junction readout scheme (Saclay)

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Read out with the RF-SETRead out with the RF-SET

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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

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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

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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

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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

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Dispersive read-outs, parametric Dispersive read-outs, parametric capacitance or inductancecapacitance or inductance

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Coupling a qubit to a resonator (Yale)Coupling a qubit to a resonator (Yale)

A. Wallraff, Nature 431 162 (2004)

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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

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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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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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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

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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

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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

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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

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Measuring the charge on the boxMeasuring the charge on the boxThe Coulomb StaircaseThe Coulomb Staircase

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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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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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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

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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

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SpectroscopySpectroscopy

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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

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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

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Relaxation and dephasingRelaxation and dephasing

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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.

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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

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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


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( )

€

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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


MeasurementsMeasurements of T of T1 1 and Tand T22

Per Delsing


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


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


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


Rabi and Ramsey in the Quantronium (Saclay)Rabi and Ramsey in the Quantronium (Saclay)

Ramsey fringes

Vion, Esteve, et al. Science 2002

Per Delsing


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


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


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


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


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


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


Back-actionBack-action

Per Delsing


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


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


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


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


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

per delsing lecture at the international summer school on quantum information, dresden 2005...

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