Noise and decoherence in the Josephson Charge Qubits

Oleg Astafiev, Yuri Pashkin, Tsuyoshi Yamamoto,Yasunobu Nakamura, Jaw-Shen Tsai

RIKEN Frontier Research SystemNEC Fundamental Research Laboratories

Outline

• The Josephson charge qubit

• Single-shot readout with charge trap

• Measurements of energy relaxation

• Charge fluctuators and energy relaxation

The Josephson Charge Qubit

b

g

C

ennU

2

2 22

Charging energy (for Cooper pair):

Josephson energy: EJ

EJ

Cb >> Cg

E

ng =Vg Cg /e

Reservoir

BoxCb

Cg

2e2

Cb

Control gate

ng =VgCg

e

Degeneracy

>> kT

2e2

Cb

>> EJ

0 2 31 4

The Hamiltonian

sincos2 xz

EH

C

neU g

2222

JEUE E

EJ

tan

2/sin

2/cos

2

E

2/cos

2/sin

2

E

Eigenstates

Eigenenergies

|0

|1 2

10t = 0:

t > 0: 2/2/

2

1 titi JJ ee

2/sin12/cos0 tit JJ

2

cos11

tP J

Coherent Oscillations

J

J

E

P

t

1

0

E

t

|12

|12

-pulse: J t =

q

EJ

0

1

2

3

4

01234

0

1

2

3

4

0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

4

0 10 20 30

I 1

(pA

) I 2

(pA

)

I 1 (p

A)

I 2 (p

A)

t (ns)

13.4 GHz

9.1 GHz

f (GHz)

Cooper-Pair BoxCooper-Pair BoxCooper-Pair BoxCooper-Pair Box

GateGateGateGate

ＳＱＵＩＤＳＱＵＩＤＳＱＵＩＤＳＱＵＩＤ

Probe Probe JunctionJunctionProbe Probe

JunctionJunction

1m

Probe Probe JunctionJunctionProbe Probe

JunctionJunction

Cooper-Cooper-pair Boxpair BoxCooper-Cooper-pair Boxpair Box

GateGateGateGate

ＳＱＵＩＤＳＱＵＩＤＳＱＵＩＤＳＱＵＩＤ

Al/AlOAl/AlOxx/Al tunnel junctions/Al tunnel junctionsAl/AlOAl/AlOxx/Al tunnel junctions/Al tunnel junctions

Pulse induced current in SQUID – box – probe junction

circuit is measured

0+1 I = 2e ||2/Tr

Tr

Control pulse sequence

t (ns)0 1

2e

Cooper-pair box

JE

qp1 JE

Final state read-out

A pair of qusiparticles tunnels through the probe junction biased to Vb 2/e

CC EeVE 322

ee

qp1

qp2

+ probe

Single-shot Readout

Coherent oscillations

Quasiparticle tanneling (when the trap is biased to 2/e)

Reservoir

Box

Control gate

qubit

CbCts

SET

Readout circuit

Cs

I

Cbt

Trap

Readout gate

Ct

t

readout:

control:

Pulses

C

C

C

C C

s t

b t

s

b t

S E T

Tr a pR e s e r v o ir

R ea dou t g a te

S E T g a te

C on tro l ga te

B o x

1 m

Tra

p ga

te

Bo

x ga

te

• Measurement circuit is electrostatically decoupled from the qubit

• Final states are read out after termination coherent state manipulation

Reservoir

Box

SET

Trap

gate

Readout with control -pulses

0 5 10 15 200

100

2000

4

I SE

T (

pA

)

t (ms)

11 Q

t(e)

1 11 1 0 1 0 1

tot

switch

N

NP 1

Readout pulse

Control -pulse

ng

I

0.0 0.2 0.40.00.20.40.60.81.0

P

tc (ns)

0 .8

0 .1

0.2 0.4 0.6 0.8 1

0.75

0.80

0.85

0.90

0.95

t (ns)

b

P

Quantum Oscillations

q

(e)

= /

-pulse

0 2 4 6 80.2

0.4

0.6

0.8

P

tc (ns)

Degeneracy

Crossectiont

q

0 500 1000 1500 20000

50

100

150

200

250

N (

coun

ts)

t delay (ns)

Relaxation to the reservoir

Readout

td

Control -pulse

220 exp(-t/288)+32

T1res = 288 ns

Ntot = 327

Reservoir

Box

SET

Trap

Relaxation to the Trap

0 100 200 300 400 5000

50

100

150

200

250

300

N (

coun

ts)

t width (ns)

Control -pulse

twidth

Teff = (1/T1res + 1/T1

trap)-1 = 31 ns

ReadoutReservoir

Box

SET

Trap

90.031288

2881

11

11

nsns

ns

TT

TP

restrap

res

Readout efficiency

0 500 1000 1500 20000.0

0.2

0.4

0.6

0.8

P

t delay (ns)

0 2000 4000 6000 80000.2

0.3

0.4

0.5

0.6

0.7

P

t (ps)

N0+ N Exp[-t/] N

0 = 152

= 5380 ps

90.000 P

Reservoir

Box

SET

Trap

Two-level Systemas a Quantum Noise Spectrometer

Two-level systemTLS

Environment

zxz tUE

H sincos2

Electrostatic energy noise

Charge basis:

Eigenbasis:

tan = EJ

E

U

EJ

U

z

x

transitions

dephasing

Dephasing Transitions

U

U22 UEE J

sincos2

xzz tUE

H

E

Charge qubit q charge noise spectral density: Sq()

deqqS iq 0

2

1SU() = (2e/C)2Sq()

1 = 22

SU()Relaxation rate: sin2

Dephasing: 1

0

2

2

2

cos

dSU

SU

Dephasing

RelaxationExcitation

0 100 200 300 400 5000.0

0.2

0.4

0.6

P

ta (ns)

T1 time measurements

ng

E

0

1

0 1

ta

P(1) exp(-ta/T1)

time

timereadout pulse

Control -pulse Adiabatic pulse

0 1 2 3 4 5

10

100

T1 (

ns)

Vp (V)

Degeneracy

T1 time vs Gate Voltage

10

100

100 50 0 -50

0.8 0.6 0.4 0.2 0.0 -0.2 -0.4

T1 (

ns)

B = 8 Gs (Ej = 3.7 GHz) B = 5 Gs (Ej = 6.0 GHz) B = 0 Gs (Ej = 8.1 GHz)

E (GHz)

qg (e)

15 20 25 30 351E-3

0.01

0.1

1

E = 400 eV (off degeneracy)

(

ns-1

)

EJ (eV)

E = EJ (degenercy)

~E2

J

EJ-dependences

Degeneracy

Off degeneracy

C

C

C

C C

s t

b t

s

b t

S E T

Tr a pR e s e r v o ir

R ea dou t g a te

S E T g a te

C on tro l ga te

B o x

1 m

Tra

p ga

te

Bo

x ga

te

Coupling to Environment through Electrical Leads

Coupling to gates:

3107.1600

1 aF

aF

C

C

b

g

Coupling to SET:

31051000600

10030

aFaF

aFaF

CC

CC

tb

tsbt

Measured relaxation time can not be explained by coupling to the external environment through electrical leads

-1.0 -0.5 0.0 0.5

107

108

Readout SET normally in ON state OFF state

(

s-1)

q (e)

Effect of the measurement SET

1 10 100107

108

109

SE/22 (

s-1)

(GHz)

c

1/f

f

The noise derived from 1 time

mKk

T cc 100

1 = 22

SU(0)

sin2

dt

St U

2

2

2 2/sin2cos

0

ExpI

2

2ln 2

2

21 tCe

ExpI

CeT

2ln 12

2

2

2

2T

tExpI

T2-2

0 100 200 300 400 500 600

I

t (ps)

I = I1(1-cos(t)exp(-(t/T

2)2/2))-I

0

T2 = 300 ps

Classical Quantum Noise

Quantum f-noise ( > 0): Classical 1/f-noise:

(kTeff)2

Do low frequency 1/f and high frequency f noises have common origin?

1/f

f

SU()

kT/ emissionabsorption

Relaxation through Fluctuators

• Dephasing is caused by 1/f noise of charge fluctuators with activation energy less than kT

• Fluctuators with activation energy of ( >> kT) accept qubit excess energy

kTE

fSq 2

1 10 10010-5

10-4

10-3

10-2

Si (

pA

2 /Hz)

f (Hz)

3x10-2pA2/Hz

Low frequency 1/f noise

Temperature dependences of the 1/f noise

1 10 10010-5

10-4

10-3

10-2

Si (

pA

2 /Hz)

f (Hz)

3x10-2pA2/Hz

T=0.055K

1 10 100

10-3

10-2

10-1

Si (

pA

2 /Hz)

f (Hz)

8x10-2pA2/Hz

T=0.5K

1 10 10010-4

10-3

10-2

10-1

Si (

pA

2 /Hz)

f (Hz)

3x10-2pA2/Hz

T=0.9K

-8 -6 -4 -2 0 2

0

5

10

15

20

25

30

35

40

45

I (p

A)

Vg (V)

0.90 K 0.85 K 0.80 K 0.75 K 0.70 K 0.65 K 0.60 K 0.55 K 0.50 K 0.45 K 0.40 K 0.35 K 0.30 K 0.25 K 0.20 K 0.15 K 0.10 K 0.055 K

0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

4

5

6

71/

2 (1

0-3e)

T (K)

EC = 110 eV

1/2 = 6x10-3eT

Standard qubit on 400 nm thick Si3N4

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

12 current measurements

T2

*

= 13x10-3 eT

(

10-3

e)

T (K)

EC = 50 eV

CeT

2ln2

0 100 200 300 400 500 600

I

t (ps)

I = I1(1-cos(t)exp(-(t/T

2)2/2))-I

0

T2 = 300 ps

1/f noise in superconducting – normal SETs

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

12

14

B = 0T B = 0.2T (normal)

= 17x10-3 eT

EC = 125 eV

(

10-3

e)

T (K)

0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

41/

2 (1

0-3 e

)

T (K)

EC = 110 eV

1/2=4x10-3 eT

GaAs

SET on GaAs substrate

SET on Al2O3

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

(

10-3

e)

T (K)

B = 0 (superconductivity) B = 0.4T (normal)

EC = 320 eV

1/2 = 7x10-3 eT

Si

Al Al2O3

SET island

Large area SETs

0.0 0.1 0.2 0.3 0.4 0.50

5

10

15

20

1/2 (

10

-3 e

)

T (K)

EC = 80 eV

Box: 3x2.3 m2

1/2= 43x10-3 eT

1/f noise properties from experiments

• does not depend on substrate type• noise appears in oxide of Al(?)• scales with SET size (area?)• saturation level at low temperatures depends on current

Basic properties of the 1/f noise caused by bistable fluctuators

22

,

s

0.1 1 100.1

1

10

S

2S

dsPS ,

kT

exp0

dPkT

dP

1P

S()

Qubit

TLS(fluctuators)

Environment at T > 0

Qubit island TLSfluctuators

C

eUq

2

The qubit is coupled to environment through charge degree of freedom

1/f noise

2313

12

3

Environment at T > 0

kT13

013 exp 031

kTDV ij

ijenijij 2coth1

2

12 2

022 2PkTqSq

kT13

012 exp

kT23

021 exp

high frequency cutoffof the 1/f noise

01312 , PP If , then

31

2

130

2 enDV

2221122112

21122112

14,,

s

0

23132112

0

2313 ,,, dsPS

k

kkV 1113

2

13

2

12

3

13231

31

11

ee

e

130

2

13

2 PV

130

2

13

2 ePV

Qubit relaxation (excitation)

22

22

qC

eV

2

22

C

eSq

02 40 PqSq

022 2PkTqSq

02 40 PqSq

kT

2

1/f low frequency noise: f high frequency noise:

Crossover frequency:

Same fluctuators contribute in the 1/f noise and the quantum f noise

Constant distribution of two energy parameters for the fluctuators is required

kT

exp0 d

kTd

021, PP 21

210

2210

ddPkTddP

02

0

23132112

0

0

2,, PkTdsPS

20

2, kT

AdsAS

AP

Two energy parameters:

Single energy:

Single energy (TLS)

12

1

2

Environment at T > 0

enDV 22

21 sin2

22sin

2coth1

2

kTDV ij

ijenij

0

High frequency cutoff

12

tan

C

eqV

2

A

P

1/f noise: << kT f noise:

< 105 Hz 1010 1011 Hz

Different TLS contribute in 1/f and f noises

Conclusion

We have demonstrated single-shot readout using charge trap

Energy relaxation of the qubit has been measured

The energy relaxation is caused by quantum f noise which has crossover frequency with 1/f noise at kT/

Nearly T2 dependence of the 1/f noise has been observed