Low frequency noise in superconducting qubits

Lara Faoro and Lev Ioffe

Rutgers University (USA)

Exp. Collaborators: Oleg Astafiev (NEC, Tsukuba) , Ray Simmonds (NIST, Boulder),Dale Van Harlingen (UIUC, Urbana Champaign) and Fred Wellstood (MD)

Outline

• Studies of decoherence in superconducting qubits (almost complete phenomenology of the noise) :

• low frequency noise (1/f noise in charge, critical currents, flux)• high frequency noise ( f noise for charge qubits but ... for the other devices ??)

• Recent developments in Fault-Tolerant QEC show that proofs and estimates of the error thresholds strongly depend on the physical characteristics of the noise: i.e. temporal (memory effects) and spatial (inter-qubits) correlations.

• It is essential to achieve the complete phenomenological characterization of the noise in superconducting qubit in order to design “realistic” strategies for QEC.

• We need to understand the microscopic origin of the charge/flux sources of noise :

• weakly interacting quantum Two Level Systems (TLSs)• environment made by Kondo-like traps

Novel ideas on charge noise

State of the field

Motivation

Problem

Josephson junction qubitsphase flux

charge - flux

charge ˆcos2

ENN̂

C2

eH J2

gtot

2

ElectrostaticJosephson energy

CPB in a cavity

NECIBM

2m 01.0~size

2m 1.0~size 2m 10~size Josephson junction

e21

e00

i]N̂,ˆ[

Where are we?

10

DESIGN GROUP T2 T1

Phase qubit NIST - UCSB-MD 85 nsec 25 nsec

Flux qubit Delft - NEC- IBM 1.5 sec 4 sec

Charge qubit NEC - Chalmers 6 nsec 100 nsec

Hybrid charge-flux qubit Saclay - Yale 0.5 sec 4 sec

CPB in a cavity Yale 0.5 sec 7 sec

GHz 10520

Relaxation time Dephasing time

Error Per Gate

4320 1010TQ

Characterization of the noise Too short due to 1/f noise

idealideal1

1T 2T

Sources of noise

• external circuit, quasi particle measurement• motion of trapped vortices in superconductor• motion of charges in associated dielectrics and oxides (responsible for 1/f noise in metallic junction)

A strategy to identify the sources of noise

Level II : Fingerprint experiments in order to infer spectralproprieties of the charge noise (correlated or uncorrelated noise? Use of dynamical decoupling schemes?)

Zorin et al. 1996

Level I : Complete Phenomenological model of the noise.Proper model of dephasing [fluctuator model]Non-Markovian bath, non gaussian noise.

Level III : Novel ideas on microscopic origin of 1/f charge noise Experiments in progress at NEC, NIST, UIUC, MD

Analysis oferror threshold

for fault-tolerantQC.

Phenomenological model of decoherence

ESsinT X 2

11 2

11

02

1

4

11 22

22 XX ScosESsin

T

Relaxation rate

Dephasing rate

Charged defects in barrier, substrate or surface lead to fluctuating induced charge

bathzq HX̂HH Longitudinal couplingto the charge degree of freedom

Golden Rule:

ti

X e0X,tXdt2

1S Noise power spectrum

Pure dephasing

1*T

But 1/f noise is special...

2

X2* cosA0Scos

2

1

0for A

SX

Golden Rule fails for 1/f noise, where

Cottet et al. (01)

Non-exponential decay of coherence

cosAT 1*

tlnt

2

cosAexpdcosiexpt ir

22t

001

-0.3 -0.2 -0.1 0.0

10

100

500

Coh

eren

ce t

imes

(ns

)

|/2|

0.05 0.10

10

100

500Free decaySpin echo

|Ng-1/2|

-4.0x10-4 -2.0x10-4 0.0 2.0x10-4 4.0x10-40.0

5.0x106

1.0x107

1.5x107

2.0x107

2.5x107

3.0x107

1/T

2(s-1

)

qb [

0]

G. Ithier et al. PRB 05

Saclay, Charge – Flux Qubit

Y. Nakamura et al. 2006 NEC, Flux Qubit

K. Kakuyanagi et al. 2006 NTT, Flux Qubit

cosAT 1*Robustness of

From Random Telegraph Signals to1/f Noise: the role of classical fluctuators

A superposition of many RTSs with a distribution of switching rates exponentially broad gives a 1/f noise spectrum

c

22

22

E c12

v)(S

Random Telegraph Signal (RTS)

Switching rate:

Noise power spectrum:

0.1 1 100.1

1

10

S

f

1/f

1P

APdSS

22j

j

2d vnA

dn

v

Number of fluctuators/decade

Average coupling to the qubit

Falci et al., PRL 2005

Interplay of several energy scales

Non gaussian effects are relevant for

initial decoherence

(inhomogeneous broadening)

and crucial for error correction!

M

v

dn

??? MHz (indirect echo)

???

???

Noise in superconducting qubits

qS

2T

mK 100T

Small Josephson charge qubits Critical current fluctuations for all other qubits

Same origin of the noise at low and high frequency?

t iNNq e0QtQdtS

L

deQ 1

substrate nm 500 L

barrier nm3L

0J00 Ie2

E IA

AI

20

N

0I2

01

0I IN~S IS

A

TS K2.4TK09.0

1

Hz

pA

mA

AI144K2.4,Hz1S

2

0I

2

2

20

0I

O. Astafiev et al. 2004

A. Shnirman et al. 2005

F. C. Wellstood et al. 2004

D. Van Harlingen et a. 2004

Dephasing by TLSs

zii

xii

zii

i

TLS

E

tEH

4322

1010 de

t

t,EP

eVcm

103

20

i

zii

i ij3ij

iijiijjiint pp̂

r 4

p̂r̂ p̂r̂3p̂p̂H

dep

Mechanisms of relaxation for TLSs

A common belief: charged impurities are TLSs in the surrounding insulators

J. L. Black and B. I. Halperin, (1977)L. Levitov (1991)A. L. Burin (1995)

• interaction with low energy phonons T >100 mK• many TLSs interact via dipole-dipole interactions:

Fundamental Problem!!

Faoro & Ioffe, 2006

The effective strength of the interactions is controlled by and it is always very weak.

2p

Quantum coherent TLSEach TLS is coupled weakly to a dissipative bath

?

Some notations.

L

pVCQ

Q

EpV g

nmL 3

nmL 300

tiggq eQtQdtS 0

iiizii deppp̂

i

tiiz

iz etdtG 0

ixi

izi

iz sincos 22

iii

i

ii

i

ii

tEE

E

tsinθ

E

Eθcos

Each dipole induces a change in the island potential or in the gate charge

i.e.

barrier

substrate

Charge Noise Power Spectrum:

Rotated basis:

Dephasing rates for the dipoles

i

ixi

izii

effint sincosthH

ij j

jijci Ecoshcoskh

222 1

Tp

2

pure dephasing:

The weak interaction between dipoles causes:• a width in each TLS• at low frequency some of the TLSs become classical

Effective electric field

34

3

ij

jijiijjiij r

pr̂pr̂ppk

N.B: density of thermally activated TLSs enough (Continuum)

T 310

An important limit of this analysis: we neglect the interaction with the qubit, but it might be important ! (future research work...)

Relaxation rates for the dipoles

Tp

sin ii

2221

jiijj ji

i sinsinkEE

222

221

From Fermi Golden Rule we can calculate the relaxation rates:

0 ji EE

But in presence of large disorder, some of TLSs:

These dipoles become classical and will be responsible for 1/f noise:i.e. how classical fluctuators emerge from an ensemble of quantum TLSs

Charge noise power spectrum

tiggq eQtQdtS 0

iiizii deppp̂

i

tiiz

iz etdtG 0

ixi

izi

iz sincos 22

iii

i

ii

i

ii

tEE

E

tsinθ

E

Eθcos

Rotated basis:

22

2

20

i

iii

tiiz

iz

EsinetdtG

2

12

12 20

i

i

iii

tiiz

iz cosetdtG

Low frequency

High frequency

T10T p 3

2

222

2

q eT

Le

V pS

222

2

q eLe

VpS

qS

T

Because of the qualitatively disagreement: search for fluctuators of different nature !!

CT

Theory of TLSs NEC Experiments

qS

2T

mK 100TC

3710

AVFor substrate volume

mk 120T~ Ke10S 126q

In the barrier...3

710

AV The density of TLSs ~ too low! K/.10

Astafiev et al. 2004

Edet

EnE

HHHH

zxzxJ

zgC

ITLSQ

22

214

22 tedEE

Strongly coupled TLS

Relaxation in phase qubit, NIST UCSB

… and the solutions?

qubit

v

Faoro, Bergli, Altshuler and Galperin, 2005

ccccTHccH

HT

T~ccvH

TE

zTI

0

0

2

0

2g

dg2T - dependence at low

frequency

T 1V10VW

TN 0

60

mKT 20

Andreev fluctuators

g

Sq1

dididi

i kdikkisd

dii

didii

didd

kkkkk

kkBCS

sddBCS

ccn

.c.hccVH

nnUccH

.c.hccccH

HHHH

0

Uexp

UT d

i

d

ii

iK

00

122

202 ii VN

Kondo Temperature

U

0d

… and the solutions? Faoro & Ioffe, 2006

Kondo-like traps

Properties of the ground state and the localized excited state

“Physics” of the Kondo-like traps

*K

0

0

*KK

K

K0

0KK0K

T

V

TTT

dTV dV

2*K

42*K3

Al

2*K

20 T10T

1TA

r

eAA

0

Slow processes Fast processes

430d

K 1010T

w

barrier

superconductor

Superconductor coherence lenght

Density of states closeto the Fermi energy

bare density

weight of the Kondo resonanceL

HzW

Hz.T *K

14

10

10

1030

ji tjitATransition amplitude:

So far only numerics ...

2g Linear density!!

at low and high frequency qS

2

2

*K

02

q eT

V

L

awS

NB: Andreev fluctuators have the same but … and

1 21 2

22

*K

02

2q

T

T

V

L

rweS

*K

0

T

T V

L

rwe

410w e32 1010

e43 1010

3710

AV

HzA?,d

Ai

iii

80 10 but maxmin

33

*K

0 A10T

V

Agreement with experimental value:

estimates :

In the barrier

High frequency - fast processes

Low frequency - slow processes

1/f noise due to critical current fluctuations:

0IS

Fred Wellstood, Ph. D thesis 1988Wellstood et al, APL 85, 5296 (2004)Van Harlingen et al. PRB (2004)

Hz

pA

mA

AI144K2.4,Hz1S

2

2

20

I0

1

A

TS K2.4TK09.0

2

I0

with the Kondo-like traps model 0IS

2t

0

0 r4I

I

A

A

TK

TtanhT

eR2I

Bn0

221t

22t m103

s

xs4r

1

T

TVr4

A

IS

2

*k

022

t2

20

0I

4.2KTat 85.0T

T

K33.16meV4.1 nm1s

*k

Nb

Nb-Al2O3-Nb

w7.0t

Hz

pA

mA

AI144K2.4,Hz1S

2

2

20

I0

Hz

pA

mA

AI104K2.4,Hz1S

2

2

20

I0

At higher frequency:

2226-22

20

I

HzA106mA

1

A

I

S0

Testing our theoretical ideas...

In collaboration with NEC, Tsukuba:* is superconductivity crucial for 1/f noise in Josephson charge qubits? [magnetic field, SET with very high charging energy]* are the charged fluctuators in the barrier? Is charge noise non-Markovian but local?

In collaboration with NIST, Boulder and UIUC, Urbana-Champaign* is the noise in the phase qubit due to TLSs in the substrate and barrier?* Test T-dependence of the 1/f noise [Van Harlingen, Illinois]* Measurement high frequency critical current fluctuations. [Van Harlingen, Illinois]

Measurement of second spectrum both in chargenoise and critical current fluctuations

Supported by LPS, NSA and ARO

7±3x10-6 [ 0] SQUID~2500-160000 m2 F.C.Wellstood et al. APL50, 772 (1987)

1.5x10-6 [ 0] phase qubit ~10000? m2 (UCSB)

~ 1x10-6 [ 0] flux qubit ~1000 m2 (Berkeley)

~ 1x10-6 [ 0] flux qubit ~ 100 m2 (NTT)

~ 1x10-6 [ 0] flux qubit ~ 3 m2 (NEC)

A re-discovered low frequency noise• Microscopic origin of the excess low frequency noise in dc-SQUIDs

• above 1K (due to critical current fluctuations and/or apparent flux noise)• below 1K (always due to apparent flux noise)

- An “old problem”: is it the ultimate limitation for all superconducting qubit?

• Loop size independent ??• Slope of the noise 2/3 ??• There are no RTSs!

Impressive universality:

(2006)

* in collaboration with Fred Wellstood, MD.

L2

1

V

SS

2V

Hz1,K1TS 2/1