P. Bertet

Quantum Transport Group, Kavli Institute for Nanoscience,TU Delft, Lorentzweg 1, 2628CJ Delft, The Netherlands

A. ter Haar A. Lupascu

J. PlantenbergF. PaauwJ. Eroms

C.J.P.M. HarmansJ.E. MooijI. Chiorescu

Y. Nakamura

Photon-noise induced dephasingin a flux-qubit

G. BurkardD. DiVicenzo

+

Introduction

zq

QH 2

nI

kH

2

2)( kS

Dephasing ? 2/)0(21 SnT z (weak coupling)

Very slow and strongly coupled fluctuators

Underdamped modes strongly coupled to qubit

E. Paladino et al., Phys. Rev. Lett. 88, 228304 (2002)

M. Thorwart et al., Chem. Phys. 296, 333 (2004)

Qubit dephased by photon noise

zq

QH 2

aahH pp

ig Temperature T

Dispersive regime : ipq g

Shift of oscillator frequency z

20

Shift of qubit frequency 0n

Coupling

Quality factor Q

Qubit dephased by photon noise

Photon fluctuations )(tn

Qubit frequency

)()( 00 tnt qq

Dephasing factor

Phase shift t

dtntnt0

0 '))'((2)(

2/)(exp)(exp 2tti

t t

dtdtttC0 0

20 ''')'','()(2exp with 2)''()'()'','( ntntnttC

around n

A. Blais et al., PRA 69, 062320 (2004)Dephasing time T

2) Thermal fluctuations in the non-driven oscillator

Thermal field :

n

n

n

nnp

11

1)( 11)/exp( kThn p

)exp()1()0,( tnntC )1()2( 2

0

nnT

D. Schuster et al., PRL 94, 123602 (2005)

Cf also M. Brune et al., PRL 76, 1800 (1996)

Qubit dephased by photon noise

1) Oscillator driven by a coherent field

nn

en

!

2

2

)2/exp()0,( tntC

nT

20 )2(2

2n

Measurement induced dephasing

Photon shot noise

Q/0

Flux-qubit coupled to SQUID plasma mode

Our circuit :Flux-qubit

DC-SQUIDplasma mode

)( bi Ig

Optimal points (with respect to photon noise) whenever

Our measurements : qubit coherence limited by thermal fluctuations in plasma mode

)1()2( 20

nn

T

1) Quantitative agreement with formula

2) Thanks to our circuit geometry, coupling constants

0),(0 xbI

The flux-qubit

Q

2/0 x

0

2 xQ

Josephsonjunctions

elQJ EEH )(

1 control parameter

Al/AlOx/Al junctions by shadow evaporation + e-beam lithography

0.48 0.50 0.52

0

40

80F

requ

enc

y(G

Hz)

|0>

|1>

|2>

|3>

Q/2

qubit

Qubit energy levels

EJ=225GHz

EC=7.2GHz

=0.76

Persistent-current

Property of states |0> and |1> : iH

iIx

i

Useful to measure the qubit state

-0.02 0.00 0.02-300

0

300I (

nA)

I0

I1

Q-

-Ip

+Ip

|0>

|1>|1>|0>

Fre

quen

cy

(

GH

z)

0.50

5

100

1

Q/2

Two-level approximation

Flux-noise optimal point

0Q

Q

d

df

In the 0 1 basis, xzq

hH

2

2/))(/( Qp eI

(cf Saclay)

Control of the qubit state

Arbitrary state

Rabi 01tan

h

I xpRabi

sin2

Rotation axis : ()

Angle : tRabi 201

2 )(

x

x+xcos(2t+)

Microwave pulse t

Our detector : a hysteretic DC-SQUID as on-chip comparator

0 10

4

Ic ( A

)

Sq /0

Persistent-current and detection of the qubit state

Qubit inductively coupled to SQUID

ix

CxC

iC MI

d

dIII

)()(

IC depends on qubit state (i)

Persistent-current and detection of the qubit state

P(1)Psw

5.4 5.6

0

50

100

Psw

itch

(%

)

Current Ib ( A)

|0>

Theoretical |1>

rela

xatio

n

P(1)=Psw

Persistent-current and detection of the qubit state

SQUID shunted by a capacitor

PLASMA MODE

shJ

pCLL

2

1

)Re(/2 1 ZCQ shp

)2/1( aahH pp

)(0 aaaa

),( bSqJ IL

Coupling of the qubit and the plasma mode

Complex : qubit Circ current J Plasma mode currentM dJ/dIb(Ib)

2 different effects :

a) Effective inductive coupling with tunable mutual inductance

b) Flux dependent SQUID Josephson inductance

SQUID circulating current

-0.5 0.0 0.5

Circ

. cur

rent

JIb/2IC

dJ/dIb=0

dJ/dIb(Ib)

Ideal symmetric SQUID : dJ/dIb(0)=0

Including asymmetries : 0*)( bb

IdI

dJDecoupling current

Coupling of the qubit and the plasma mode

1) Measurement shift

-0.02 0.00 0.02

Ene

rgy

(GH

z)

21

Cur

rent

(A

)

(e/Ip)

221 )()()()( aaIhgaaIhgH xbxbI

bJb dI

d

LLIg

01 2

1)(

2

22

02 4

1)(

bJb

dI

d

LLIg

2) Coupling hamiltonian

NON RESONANT

inductive Flux-dependent Josephson inductance

)()( bx I

The sample

Ib

V

Microwave antenna

Csh

G. Burkard et al., cond-mat/0405273

1k 3k

The setup

Qubit spectroscopy

timetrig

ger

Ib pulse

read-outt

Microwave pulseat frequency f

Parameters : =5.85GHz, Iq=270nA

5.7 5.8 5.9 6.0

Psw

itch

Frequency (GHz)

B

-0.01 0.00 0.010

5

10

15

20

25

Larm

or fr

eque

ncy

(GH

z)

(x-0/2)/0

Plasma mode spectroscopy

time

Microwave pulseat frequency f

Ib

Switching probability enhancement if f=p : resonant activation

4.4 4.5 4.6 4.7 4.8 4.9

2.6

2.8

3.0

2.4 2.6 2.80

50

100

0

2

4

Fre

qu

en

cy (

GH

z)

Magnetic field (Gs)

Psw

itch

(%)

Frequency (GHz)

Sw

itch

ing

cu

rre

nt

(A

) Resonant activation peak :

Typical width : 20-50MHz

15050

Q

shxJ

pCLL ))'((2

1

Csh=12pF, L=170pH (design)

P. Bertet et al., Phys. Rev. B 70, 100501 (2004)

bJb dI

d

LLIg

01 2

1)(

2

22

02 4

1)(

bJb

dI

d

LLIg

Evaluating the coupling constantsbJ

b dI

d

LLIg

01 2

1)(

2

22

02 4

1)(

bJb

dI

d

LLIg

Measure (Ib)

Spectroscopy

0.0 0.3 0.6

-1

0

(G

Hz)

Ib (A)

Ib*

-0.001 0.000 0.001

5.5

6.0

6.5

Fre

quen

cy (

GH

z)

(x-0/2)/0

Ib=0A

Ib=0.6A

Evaluating the coupling constantsbJ

b dI

d

LLIg

01 2

1)(

2

22

02 4

1)(

bJb

dI

d

LLIg

Measure (Ib)

Spectroscopy

0.0 0.3 0.6

-1

0

(G

Hz)

Ib (A)

Ib*

g1

g2

Ib*0.0 0.3 0.6-0.2

0.0

Cou

plin

g (G

Hz)

Ib (A)

Frequency shift

xzq

hH

22

21 )()()()( aaIhgaaIhgH xbxbI

)2/1( aahH pp

ac-Zeeman shift. Always >0

Frequency shift 0 due to g1

0.0 0.3-2

0

2

Ib (A)

epsi

lon

(GH

z)

Ib*

-20MHz

+26MHz

0MHz

Frequency shift 0 due to g2

Shift has same sign as epsilon

0.0 0.3-2

0

2

Ib (A)

epsi

lon

(GH

z)

=0

Frequency shift

0.0 0.3-2

0

2

epsi

lon

(GH

z)

Ib (A) 0=0

Quantitative prediction : optimal point for photon noise

Optimal point for flux/current noise

Optimal pointFor flux-noise

Optimal point for photon noise

Characterizing decoherence (1) : spectroscopy

5 types of experiments :

Low-power spectroscopy

Rabi oscillations

T1 measurements

Spin-echo measurements

At decoupled optimal point (Ib=Ib*,=0)

5.52 5.56 5.60

68

78

Psw

itch

(%)

Freq F(GHz)

f1,w1

f2,w2

Strongly coupled 2-level fluctuator

)(/2 212 wwt

Ramsey fringes

Thermal photon noise :« high frequency »

sTRabi 5.1

Psw

itch

(%)

0 1 2

100

200

Pulse duration Dt (s)

Non-exponential because low-frequency noise

Characterizing decoherence (2) : Rabi oscillations

At decoupled optimal point (Ib=Ib*,=0)

Dt

MW=Q

0.00 0.06

60

80

Pulse length (s)

Characterizing decoherence (3) : T1 measurements

Dt

0 10 20

60

80

Psw

itch

(%)

Delay Dt (s)

- Exponential decay

At decoupled optimal point (Ib=Ib*,=0)

sT 41

Characterizing decoherence (4) : Ramsey fringes

0.0 0.1 0.2 0.3 0.4

60

80

100

120

140

160

Psw

itch

(%)

Delay between pulses (microseconds) T/2

/2 /2

T/2

MW-Q

Difficult to extractdephasing time …

At decoupled optimal point (Ib=Ib*,=0)

1.0 1.1 1.2

60

80

Psw

itch

(%)

t (s)

T/2=2.2s

0 1 2 3 4 5

60

70

80

Psw

itch

(%)

t (s)0 4000 8000

0

10

20

Ech

o a

mp

l (%

)

T/2 (ns)

Techo=3.9+-0.1s

Bertet et al., cond-mat/0412485

Characterizing decoherence (5) : spin-echo sequence

t

/2 /2

T/2

T/2/2

T1 dependence on Ib

-0.2 0.0 0.2 0.4

1

T1 (s

)

Ib (A)

Ib*

Away from Ib*, T1 limited by coupling to measuring circuit

Spin-echo and t2 dependence on Ib and

-0.001 0.000 0.001

100

1000

5.5

5.6

5.7

Tim

e (n

s)

(e/Ip)

Fre

quen

cy (

GH

z)

Ib=Ib*

g1=0

Techo

t2=2/(w1+w2)

Best coherence :=0 (optimal point)

Ib=0A

g1=80MHz

-0.001 0.000 0.00110

100

1000

5.6

5.8

Tim

e (n

s)

(e/Ip)

Fre

quen

cy (

GH

z)

Best coherence :

=m<0

NOT LIMITED

by flux-noise

Decoherence due to qubit-plasma mode coupling

-0.001 0.000 0.00110

100

1000

5.6

5.8

Tim

e (n

s)

(e/Ip)

Fre

quen

cy (

GH

z)

m

-0.2 0.0 0.2 0.4

-1

0

m (

GH

z)

Ib (A)

0=0

Dephasing minimum for spin-echo and Ramsey when 0=0

Quantum coherence limited by photon noise

-0.001 0.000 0.001

100

1000

5.5

5.6

5.7

Tim

e (n

s)

(e/Ip)

Fre

quen

cy (

GH

z)

Ib=Ib*

g1=0

T=70mK, Q=150

Ib=0A

g1=80MHz

-0.001 0.000 0.00110

100

1000

5.6

5.8

Tim

e (n

s)

(e/Ip) F

requ

ency

(G

Hz)

Spin-echo and t2 dependence

Quantitative agreement

Conclusion

Long spin-echo time (4s) at optimal bias point

Dephasing due to thermal fluctuations of the photon number in an underdamped resonator coupled to the qubit : very general situation

Case of a flux-qubit coupled to the plasma mode of its SQUID detector

By tuning coupling constants, could decouple qubit from photon noise

Quantitative agreement with simple model for spin-echo time

2 questions :

- mechanism for low-freq noise ? (charge or critical current noise ?)

- effect of dispersive shifts in usual spin-boson model ?