Superconducting flux qubits

Hans MooijDelft University of Technology

International Conference on Nano-ElectronicsLancaster, January 2003

Yasunobu Nakamura (visitor from NEC)Irinel Chiorescu

Adrian Lupascu Kouichi Semba (visit. NTT)Kees Harmans Alexander ter Haar Hannes Majer Jelle PlantenbergPatrice Bertet Erwin HeeresFloor Pauw Paul Scheffers

theoryMilena Grifoni Frank Wilhelm (München)

MIT/Lincoln Labs/HarvardTerry Orlando Seth LloydLeonid Levitov Karl BerggrenMike Tinkham

mesoscopic Josephson junction circuits

Josephson coupling energy

UJ = EJ ( 1 - cosφ )

EJ = 2∆ (h/e2) / (8Rn)

Coulomb charging energy

UC = EC 4n2

EC = e2 / 2C

n , φδnδφ>1

EJ/EC>>1phase excitationsfluxons

EC/EJ>>1charge excitations

Φm

Vg

decoherence spin-oscillator bath Grifoni et al.

spin H = εσz+∆σx δE = 2(∆2+ε2)1/2 (∆ tunnel, ε field energy)

oscillator spectral densityJ(ω) = π/2 Σ ci

2/C1ωi2 δ(ω-ωi)

relaxation rateΓr = ½ (∆/δE)2 J(δE/ħ) coth(δE/2kBT)

dephasing rateΓφ = Γr/2 + (ε/δE)2 α 2πkBT/h

charge noise:

1/f noise

- charge noise: charged defects in barrier,substrate or surface lead to non-integerinduced charge. Static offset, 1/f noise.

- flux noise: trapped vortices, magnetic domains,magnetic impurities.

- critical current noise: neutral defects in barrier.

Ibias

Φ / Φo

0

-1

0

1

0.5

Icirc

E

3 µmIcirc = ∂E/∂(Φ/Φo)LIcirc = 10-3 Φo

0.498 0.500 0.502

~

9.711 GHz

8.650 GHz

6.985 GHz

5.895 GHz

4.344 GHz

3.208 GHz

2.013 GHz

1.437 GHz

1.120 GHz

0.850 GHz

Ι SW

(0.

4 nA

per

div

isio

n)

Φext (Φ0)0 1 2 3

0

5

10

0

2

0

f

(GH

z)∆Φres (10-3 Φ0)

0.8

Φ/Φo

switchingcurrent

Φ/Φo-0.5x10-3

fGHz

Caspar van der Wal

also SUNY

quantum bit: two level quantum system

I0>

I1>∆E = hωo

Ψ = α I0> + β I1>IαI2 + IβI2 = 1

α = cos θ β = eiφ sin θ

dφ/dt = ωο

2θ φ

I0>

I1>

zPauli spin matrices

σx σy σzyx

h set by external fluxt set by fabrication H = hσz + tσx

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

E2t(here t=1)

hH ≈ hσz H ≈ hσz

H = tσx

SQUID readout:

timereference trigger

quantum operations

pulsed bias current~ 50 ns rise/fall time

only two possible outputs:SQUID switched to gap voltageSQUID still at V=0

pulse height adjusted to give ~50% switching

1.0 1.5 2.0 2.5

-20

-15

-10

-5

0

5

18.96 GHz14.34 GHz11.39 GHz10.09 GHz 8.42 GHz 5.00 GHz

<Isw

> - l

inea

ir cu

rve

(nA

*con

v fa

ct)

Vmagnet (V)

high power:

harmonics

subharmonics

Irinel Chiorescu and Yasu Nakamura

4.40 4.42 4.44 4.46 4.48 4.50 4.52 4.54 4.560

2

4

6

8

10

12

14

16

18

20

∆ = 3.365 GHzIp = 350 nA

frequ

ency

(GH

z)

magnetic field (Gs)

pH 5 pH 5

A 20 µ=cI

exI

qqI γ,

0~ Φ

05.0~ Φ

- qubit flux-biased in symmetry point- ramp of SQUID bias current Ιex changes circulating current in SQUID- SQUID-qubit coupling: qubit adiabatically driven from symmetry

trigger

RF

Ib pulse~50ns rise/fall time

time

Rabi: microwave pulse with varying length

pulse length

0 20 40 60 80 100 120 140 160 18030

60

90

30

60

90

30

60

90

switc

hing

pro

babi

lity

(%)

A = -3 dB

RF pulse length (ns)

A = 3 dB

A = 9 dB

-20

-15

-10

-5

0

10 20 30 40 50 60 70 80 90 100

RF

pow

er (d

B)

RF pulse length (ns)0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0

100

200

300

400

500

600

Rab

i fre

quen

cy (M

Hz)

RF amplitude, 10A/20 (a.u.)

2.70 2.75 2.80 2.85 2.90 2.95 3.000

20

40

60

80

100

switc

hing

pro

babi

lity

(%)

bias pulse height (V)bias pulse height →

switching ↑probability

0 5 10 15 20 25 30 35 40 45 50

50

60

70

80

90

A = 9 dBF0 = 6.607 GHz∆f = 250 MHz

Switc

hing

pro

babi

lity

(%)

time between two π/2 pulses (ns)

Ramseyfree induction decay

π/2π/2 Ib pulsetrigger

result:dephasing time T2= 20 ns

time between π/2 pulses →

switching ↑probability

0 50 100 150 200 250 300 350 400 450 50020

30

40

50

60

70

80

90

100

0 10 20 3030

60

90

500 510 520 530

58

60

62

Switc

hing

pro

babi

lity

(%)

RF pulse length (ns)

Psw

(%)

RF pulse length (ns)

Psw

(%)

RF pulse length (ns)

switching ↑probability(%)

pulse length (ns) →

dephasing: flux noise leads to δE=ħδω=π/τφ

τφ=20 ns corresponds to δf=25 MHz

Rabi frequency off-resonanceωR’ = ωR (1 + δω2/ωR

2)1/2 = ωR (1+π2/ωR2τφ

2)1/2

10 20 30 40 50

50

100

150↑Rabi

dephasingtime (ns)

for δω↔25 MHz

Rabi period (ns) →

0 10 20 30 40 50 60 70 80

0

20

40

60

80

100F=6.512 GHzA=-3 dBm

N- number of averaged events N=1 N=10 N=10000

switc

hing

pro

babi

lity

(%)

RF pulse length (ns)

Rabi with low number of measurements,single shot contains some information

-25 -20 -15 -10 -5 0 5 10 15 20 25

52

56

60

64

68

π π/2π/2

switc

hing

pro

babi

lity

(%)

position of π pulse (ns)

closed superconducting wire loop, w<λ

-2 -1.5 -1 -0.5 0.5 1 1.5 2

0.2

0.4

0.6

0.8

1

f=Φ/Φο

E

n=1 n=0 n=-1 Φ ⊗

I

γqubit

A

B

n=1

standard SQUID

Π/2-SQUID

Π-SQUID

-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5

0.0

0.2

0.4

0.6

0.8

1.0

1.2

standard SQUID Π SQUID Π/2 SQUID

I c (µA

)

f

Hannes Majer, Jeremy Butcher

Φ1 Φ2

trapped fluxoidgradiometer qubit

response only to Φ1-Φ2

∆E = Ip(Φ1-Φ2) - ζIp(Φo+Φ1+Φ2)

asymmetry parameter ζ ≈ 0.01 (fabrication)reduction flux noise by this factor

two coupled qubitsHannes Majer, Floor Pauw

-8 -6 -4 -2 0 2 4 6 8-5

-3

0

3

5Fitted Step RF 14 GHz 11dbm 15 aug 2002

Sign

al (V

)Qubit Flux (mΦ0)

conclusions:

single qubits at present: decoherence time / operation time 20-50modulation range maximum 60%

gradiometer qubit, soon:determine origin flux noiseimprove coherenceimprove modulation range

two-qubit systems starting