On Decoherence in Solid-State Qubits

• Josephson charge qubits• Classification of noise, relaxation/decoherence• Josephson qubits as noise spectrometers• Decoherence due to quadratic 1/f noise• Decoherence of spin qubits due to spin-orbit coupling

Gerd Schön Karlsruhe

work with:Alexander Shnirman Karlsruhe Yuriy Makhlin Landau InstitutePablo San José KarlsruheGergely Zarand Budapest and Karlsruhe

UniversitätKarlsruhe (TH)

2 energy scales EC , EJ

charging energy, Josephson coupling

2 degrees of freedomcharge and phase θ, n i

2 control fields: Vg and x

gate voltage, flux

Vg

x

n

tunable JE

2 states only, e.g. for EC » EJ

z xh xJgc1

2

1

2σ) ( ) σ(E EH V

0

g xJ

gC 2 θcos(π ) cos

eE

CH n

VE

2 ( )

Vg

gx /0 Cg Vg/2e

Shnirman, G.S., Hermon (97)

1. Josephson charge qubits

Observation of coherent oscillations Nakamura, Pashkin, and Tsai, ‘99

op ≈ 100 psec, ≈ 5 nsec

z xg Jch11

2 2( )σ σE VH E

0 1/ /e 0 e 1iE t iE tt a b

Qg/e

1

1

major source of decoherence:background charge fluctuations

Quantronium (Saclay)

Operation at saddle point: to minimize noise effects- voltage fluctuations couple transverse- flux fluctuations couple quadratically

2

ch J2 x0g0

g x

1 1 2x z

1

2 4g xz

2δ δ V

E E

VH VE

Charge-phase qubit EC ≈ EJ

0

g xJ

gC 2 θcos(π ) cos

eE

CH n

VE

2 ( )gate

Cg Vg/2ex /0

0 200 400 600 800

25

30

35

40

45

50

55detuning=50MHz

T2 = 300 ns

switc

hing

pro

babi

lity

(%)

Delay between /2 pulses (ns)

Decay of Ramsey fringes at optimal point

/2 /2

Vion et al., Science 02, …

Experiments Vion et al.

Gaussian noise

S

1/

4MHz

SNg

1/

0.5MHz

-0.3 -0.2 -0.1 0.0

10

100

500

Coh

eren

ce t

imes

(ns

)

x

0.05 0.10

10

100

500Free decaySpin echo

|Ng-1/2|

Sources of noise

- noise from control and measurement circuit, Z()- background charge fluctuations - …

Properties of noise

- spectrum: Ohmic (white), 1/f, ….

- Gaussian or non-Gaussian

coupling:

2. Models for noise and classification

longitudinal – transverse – quadratic (longitudinal) …

zz bathxz22

11 11

2 422 = H E XX HX

B

1

2

1

( ) ( ), (0)

coth , / , ...2

Xi tS dt X t X

k T

e

Ohmic

Spin bath

1/f(Gaussian)

model

noise

Bosonic bath

Relaxation (T1) and dephasing (T2)

1 2

z x y01 1

( )d

M M M Mdt T T

z x yM B M

Bloch (46,57), Redfield (57)

Dephasing due to 1/f noise, T=0, nonlinear coupling ?

rel1

1

2( Δ )

1XT

S E

2 1

1 1

22

1( )

10XT T

S

Golden rule: exponential decay law

For linear coupling, regular spectra, T ≠ 0

pure dephasing: *

Example: Nyquist noise due to R(fluctuation-dissipation theorem)

( ) coth2V

B

S Rk T

relB

2 coth/ 2

R E E

h e k T

* B2/

k TR

h e

00 00 11

11 00 11

01 z 01 01Bi

1/f noise, longitudinal linear coupling

z bath

1

2(Δ ) +H E X H

21/= for 0

| |f

X

ES

21/

20

2

2

01

1

2

sin ( / 2)( ) exp ( ) exp

exp ln | |2

( )2π ( / )

π

2

fir

t

X

d tt i X d

Et

S

t

Cottet et al. (01)

non-exponential decay of coherence*

2 1/1/ fT E

time scale for decay

1 1

2

1

2 2cos sinzz xtH XE X t

2 2Jch ( ) ( )g xE E V E

J chtan ( ) / ( )x gE E V eigenbasis of qubit

Josephson qubit + dominant background charge fluctuations

Jch1 1 1

2 2 2( ) ( ) ( )g xz x zH E V E X t

3. Noise Spectroscopy via JJ Qubits

probed in exp’s

transverse component of noise relaxation

2

1rel

1

2

1( )sinXS E

T

*1/*

2

1cosfE

T

longitudinal componentof noise dephasing

2

1/

| |f

X

ES

1/f noise

21/ 2 2

01( ) exp cos ln2

fir

Et t t

Astafiev et al. (NEC)Martinis et al., …

Relaxation (Astafiev et al. 04)2

rel1

2( )sinXS E

data confirm expecteddependence on

22

xJ2 2

g xJch

( )sin

( ) ( )

E

E V E

extract ( )XS E

1 10 100

1E-8

1E-7

1E-6

1E-5

1E-4

Sq (

arb

.u.)

f (Hz)

1/f

2

1/ fX

ES

T 2 dependence of 1/f spectrum observed earlier by F. Wellstood, J. Clarke et al.

Low-frequency noise and dephasing

0 100 200 300 400 500 600 700 800 900 10000.000

0.005

0.010

0.015 Dephasinglow frequency 1/f noise

(

e)

T (mK)

21/

2fE a T

*1/*

2

1fE

T

E1/f

same strength for low- and high-frequency noise

a BB

B

2

( ) for

o

f r

XSa

kk

T

k

T

a T

Astafiev et al. (PRL 04)

1 10 10010

7

108

109

2e2Rћ

S

X(

)/2ћ2 (

s)

(GHz)c

ћ2E1/f2

Relation between high- and low-frequency noise

• Qubit used to probe fluctuations X(t)

• each TLS is coupled (weakly) to thermal bath Hbath.j at T and/or other TLS

weak relaxation and decoherence2 2

,rel, , j jj jj E

• Source of X(t): ensemble of ‘coherent’ two-level systems (TLS)

High- and low-frequency noise from coherent two-level systems

qubit

TLS

TLS

TLS

TLS

TLS

,rel, , jj bath

inter-action

Spectrum of noise felt by qubit

distribution of TLS-parameters, choose

exponential dependence on barrier height for 1/f

for linear -dependence

overall factor

• One ensemble of ‘coherent’ TLS

• Plausible distribution of parameters produces:

- Ohmic high-frequency (f) noise → relaxation

- 1/f noise → decoherence

- both with same strength a

- strength of 1/f noise scaling as T2

- upper frequency cut-off for 1/f noise

Shnirman, GS, Martin, Makhlin (PRL 05)

low : random telegraph noiselarge : absorption and emission

4. At symmetry point: Quadratic longitudinal 1/f noise

Paladino et al., 04Averin et al., 03

static noise

1/f spectrum “quasi-static”

Shnirman, Makhlin (PRL 03)

Fitting the experiment

G. Ithier, E. Collin, P. Joyez, P.J. Meeson, D. Vion, D. Esteve, F. Chiarello, A. Shnirman, Y. Makhlin, J. Schriefl, G.S., Phys. Rev. B 2005

5. Decoherence of Spin Qubits in Quantum Dots or Donor Levels with Spin-Orbit Coupling

Coherent Manipulation of Coupled Electron Spins in Semiconductor Quantum DotsPetta et al., Science, 2005

Generic Hamiltonian

y bath

1 1 1 1

2 2 2 2 = ( ) ( , )z x zH B b XZ ZX H

������������������������������������������

= strength of s-o interaction direction depends on asymmetries

b

published work concerned with large ,

→ vanishing decoherence for

(Nazarov et al., Loss et al., Fabian et al., …)

We find: the combination of s-o

and Xx and Zz leads to decoherence,

based on a random Berry phase.

0B ��������������

spin + ≥ 2 orbital states + spin-orbit couplingnoise coupling to orbital degrees of freedom

dot2 orbital

states

noise 2 independent fluct. fieldscoupling to orbital degrees of freedom

spin-orbitspin

B��������������

dot noise

1

2s-o = ( , , , ) ( , , , )x yH XB H x y p p H H x Zy

����������������������������

2 2s-o ( ) ( ) ( )y x x y x x y y x y x y x yH p p p p p p p p

Rashba + Dresselhaus + cubic Dresselhaus

Specific physical system: Electron spin in double quantum dot

• Phonons with 2 indep. polarizations

• Charge fluctuators near quantum dot

,( () )X t Z tSpectrum:

, 3s s / ( )X ZS

1 and/or/

+ Z(t)

X(t)

noise

1

2( ) = ( )( )x zZ tX tH

2-state approximation:

Fluctuations

20 1

...

0

y x x yx

y

z

b

b

p p

b

i p p

y

1

2s-o = bH

��������������

1 1 1 1z x z y

2 2 2 2( = [ ( )) ] ( )Z tX t hbH t

= natural quantization axis for spin b

,x

,y

,z

( ) sin ( ) co( ) s ( )

( ) sin ( ) sin ( )

( ) cos (

( )

( )

( )

( ) )

h t t t

h

X t

Z t

t t t

h t

h t

h t

h t t

b

1 1 1z x z y

2 2 2 = ( )XH bZ

��������������0B

��������������

For two projections ± of the spin along b

For each spin projection ± we consider orbital ground state

Ground (and excited) states 2-fold degenerate due to spin (Kramers’ degeneracy)

0 0

1

2( )E h t E

x

y

z

( )h t

( )h t

b-b

x

y

z

( )h t

( )h t

In subspace of 2 orbital ground states for + and - spin state:

+eff

2 = cos

bH i U U

��������������

Instantaneous diagonalization introduces extra term in Hamiltonian

+ += H U H U i U U

Gives rise to Berry phase

+ eff,+

1

21

2

1 = d ( ) d cos

2

d cos

t H t t

, , ( ( )) Z tX t

random Berry phase dephasing

bounded 3/ 22 2cos ( ) ( )

bdt dt X dt Z t X t

b

X(t) and Z(t) small, independent, Gaussian distributed

effective power spectrum and dephasing rate

2

32 2

2

0

( ( )) ZX

Tbd

bSS

Small for phonons (high power of and T)

Estimate for 1/f– noise or 1/f ↔ f noise

2

2 232 2

9 5 4(10 ...10 ) 1...10 HzT

bTX Z

b

• Nonvanishing dephasing for zero magnetic field• due to geometric origin (random Berry phase)• measurable by comparing 1 and for different initial spins

4( 0) 1...10 HzB

Conclusions

• Progress with solid-state qubits

Josephson junction qubits

spins in quantum dots

• Crucial: understanding and control of decoherence

optimum point strategy for JJ qubits: 1 sec >> op ≈ 1…10 nsec

origin and properties of noise sources (1/f, …)

mechanisms for decoherence of spin qubits

• Application of Josephson qubits:

as spectrum analyzer of noise