On Decoherence in Solid-State Qubits - Capri SchoolOn Decoherence in Solid-State Qubits • Josephson charge qubits • Classification of noise, relaxation/decoherence • Josephson

On Decoherence in Solid-State Qubits

• Josephson charge qubits• Classification of noise, relaxation/decoherence• Josephson qubits as noise spectrometers• 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)

http://www.tfp.uni-karlsruhe.de/

2 energy scales EC , EJcharging energy, Josephson coupling

2 degrees of freedomcharge and phase[ ]θ, n i= −

2 control fields: Vg and Φxgate voltage, flux

Vg

Φxn

tunable JE

2 states only, e.g. for EC » EJ

z xh xJgc1

2

1

2σ) ( ) σ(E EH V= − Φ−

0

g xJ

gC 2 θcos(π ) cos

eE

CH n

VE= − −

ΦΦ

2 ( )

Vgg

Φx /Φ0 Cg Vg/2e

Shnirman, G.S., Hermon (PRL 97)Makhlin, G.S., Shnirman (Nature 99)

1. Josephson charge qubits

Observation of coherent oscillationsNakamura, Pashkin, and Tsai (Nature 99)

τop ≈ 100 psec, τϕ ≈ 5 nsec

z xg Jch11

2 2( )σ σE VH E= − −

( ) 0 1/ /e 0 e 1iE t iE tt a bψ − −= +h h

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

2ch J

2 x0g0g x

1 1 2x z

1

2 4g xz

2δ δ V

E EV

H VEτ ττ Φ∂ ∂

∂ ∂Φ− ∆ Φ= − −

Charge-phase qubit EC ≈ EJ

0

g xJ

gC 2 θcos(π ) cos

eE

CH n

VE= − −

ΦΦ

2 ( )gate

Cg Vg/2eΦx /Φ0

x y

z

x y

z

π2( )

xtd

ϕ= ∆ Eh dt

ϕ

ϕ

x y

z

ϕ

gatevoltage

time

π2( )

xσz final< > =cos

Ramsey fringes

Tool box:

1 1

2 2(cos sin )z z x yRH B t tσ ω σ ω σ= − − Ω +

1

2' xRH σ= − Ωin rotating frame

(unitary transformation)

operate at resonance zBω =

in lab frame

Free decay (Ramsey fringes)

Echo signal

π/2 π/2

π/2 π π/2

0

0

t

tt/2

τ

Echo experiment

Rabi oscillations

0 200 400 600 800

25

30

35

40

45

50

55detuning=50MHz

T2 = 300 ns

switc

hing

pro

babi

lity

(%)

Delay between π/2 pulses (ns)

Decay of Ramsey fringes at optimal point

π/2 π/2

Vion et al. (Science 02)

Experiments Vion et al.

Gaussian noiseSδ

ω1/ω

4MHz

SNg

ω

1/ω

0.5MHz

-0.3 -0.2 -0.1 0.0

10

100

500

Coh

eren

ce ti

mes

(ns)

Φx/Φ0

0.05 0.10

10

100

500Free decaySpin echo

|Ng-1/2|

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

Properties of noise- spectrum: Ohmic (white), 1/f, ….- Gaussian or non-Gaussian

coupling:

longitudinal – transverse – quadratic (longitudinal) …

zz bathxz22

11 11

2 422 = H E XX HX ττ ττ ⊥− ∆ − − − +

B

1

2

1

( ) ( ), (0)

coth , / , ...2

Xi tS dt X t X

k T

e ωω

ωω ω

+=

∝

∫h

2. Noise and Decoherence

Ohmic

Spin bath

1/f(Gaussian)

model

noise

Bosonic bath

Quantum Baths

Bloch equations, relaxation (Γrel = 1/T1) and dephasing (Γϕ = 1/τϕ =1/T2)

( )1 2

01 1 ( )z z x x y y

d M M M Mdt T T

= × − − − +M B M e e eBloch (46,57)Redfield (57)

[ ]Trσ σρ= =M

00 01

10 11

ρ ρρ

ρ ρ

=

00 00 11

11 00 11

01 01 01zBi ϕ

ρ ρ ρρ ρ ρρ ρ ρ

↑ ↓

↑ ↓

= −Γ +Γ

= Γ −Γ

= − −Γ

&

&

&

0

rel ( ) /( )M

↑ ↓

↓ ↑ ↑ ↓

Γ = Γ + Γ

= Γ − Γ Γ + Γ

Relaxation (T1) and Dephasing (T2)

2-level system: relaxation of density matrix

↓Γ0

1

Relaxation

2

2

00 + 1

1a

b

p aa b

p b

=

=→

probability

ϕΓ

Dephasing

Transverse coupling ⇒ relaxation

1 12 2z x BathH E X Hτ τ= − ∆ − +

Golden Rule:

( )

( )

[ ]

2

,

,

2

2

2

/

/

2 1 0, | |1,4

2 1 1| | | | exp /4 2

1 | ( ) (0) | exp /4

1 ( ) (0)41 ( ) (0)

4

Bath

Bath

Bath

i fi f

i fi f

i

xii

ii

ii

E

E

i X f E E E

i X f f X i dt i E E E t

dt i X t X i i Et

X t X

X t X

ω

ω

π ρ σ δ

π ρπ

ρ

↑

↑

↓

=∆

=−∆

Γ = + ∆ −

= + ∆ −

= ∆

Γ =

Γ =

∑

∑ ∫

∑∫

h

h

h

hh h

hh

h

h

21

rel1 1 ( / )

2 XS ET

ω↑ ↓≡ Γ = Γ + Γ = = ∆ hh

compare “P(E)-theory”

Longitudinal coupling ⇒ pure dephasing

1 12 2z z BathH E X Hτ τ= − ∆ − +

X(t) treated as classical, Gaussian random field

0

1 2 1 220 0

01 exp ( )1

( ) ( )2

( ) expt t tiX d d d X Xt τ τ τ τ τ τρ −

∝ = ∫ ∫ ∫

h h

2

2 2 2

1 sin ( / 2) 1exp ( ) exp ( 0)2 2 ( / 2) 2X X

d tS S tω ωω ωπ ω

= − ≈ − ≈

∫h h

2

2

sin ( / 2) 2 ( )( / 2)

t tω πδ ωω

≈

2* 1 ( 0)

2 XSϕ ωΓ = ≈h

“Golden-rule” approximation:

0 0

01 ( ) ( )exp exp( ) 0 (0) 1t ti iH d H dt T Tτ τ τ τρ ρ− =

∫ ∫h h

off-diagonal comp. of density matrix

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

rel1

21

2s n( i1 )XS E

Tω η= Γ = = ∆

1

2

2

1 1

2 2co1 1 ( 0) sXST Tϕ ω η= Γ = + ≈

exponential decay law

pure dephasing: *ϕΓ

1 1 1

2 2 2co ss i n z z x BathH E X X Hητ τ η τ= − ∆ − − +

General linear coupling

Golden rulete−Γ∝

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

⇒

( ) coth2VB

S Rk Tδωω ω=h

h

relB

2 coth/ 2R E Eh e k T

∆ ∆Γ ∝

h

* B2/k TR

h eϕΓ ∝h

1

2( ) z BathH E X Hτ= − ∆ + +

Golden rule* 1

2( 0)XSϕ ωΓ = =

( )2

1/ for 0| |

fX

ES ω ω

ω= →∞ →

fails for 1/f noise,

where

2

01 20

21/ 2

1

2

sin ( / 2)( ) exp ( ) exp ( )2 ( / 2)

exp ln | |2

t

X

fir

d tt i X d S

Et t

ω ωρ τ τ ωπ ω

ωπ

= = −

= −

∫ ∫

2

2

sin ( / 2)( ) regular 2 ( )

( / 2)X

tS t

ωω π δ ω

ω⇒ = ⇒

Cottet et al. (01)

Non-exponential decay of coherence

Golden rule, exponential decay

1/f noise, longitudinal linear coupling

At symmetry point: Quadratic longitudinal 1/f noise

Shnirman, Makhlin (PRL 03)

E. Paladino et al. 04D. Averin et al. 03

static noise (random distribution of value X)

long t:

1/f spectrum ‘‘quasi-static”

short t:

Fitting the experiment

G. Ithier, E. Collin, P. Joyez, P.J. Meeson, D. Vion, D. Esteve, F. Chiarello, A. Shnirman, Y. Makhlin, J. Schriefl, GS, PRB 2005

Longitudinal coupling: exact quantum mechanical solutionreduced density matrix

Low-Temperature Dephasing 1

2( ) z BathH E X Hτ= − ∆ + +

Factorized initial conditions:

‘Keldysh’-contour

σ = +1

σ = -1

Longitudinal coupling: exact quantum mechanical solution, ctd.

• Polarized bath (bath relaxed to state with spin pointing up)

• Unpolarized bath (no interaction between spin and bath before t=0)

compare P(E) theory

Longitudinal coupling: Ohmic spectrum

A. Shnirman, G.S., NATO ARW "Quantum Noise in Mesoscopic Physics", Delft, 2002cond-mat/0210023

• in general no exponential decay• dephasing in finite time even at T = 0• decay may depend on cutoff ωc (due to factorization of ρ(0))

( ) ( )1 1

2

1

2 2cos sinzz xtH XE X t η ττ η τ= − ∆ − −

2 2Jch ( ) ( )g xE E V E∆ = ∆ + Φ

J chtan ( ) / ( )x gE E Vη = Φ ∆eigenbasis of qubit

Josephson qubit + dominant background charge fluctuations

Jch1 1 1

2 2 2( ) ( ) ( )g xz x zH E V E X tσ σ σ= − ∆ − Φ −

3. Noise Spectroscopy via JJ Qubits

probed in exp’s

transverse componentof noise ⇒ relaxation

2

1rel

1

2

1 ( ) sinXS ET

ω η≡ Γ = = ∆

*1/*

2

1 cosfET ϕ η≡ Γ ∝

longitudinal componentof noise ⇒ dephasing

( )2

1/

| |f

X

ES ω

ω=1/f noise

21/ 2 2

01( ) exp cos ln2

fir

Et t tρ η ω

π

= −

⇒

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

Relaxation (Astafiev et al. 04)2

rel1

2( ) sinXS Eω ηΓ = = ∆

data confirm expecteddependence on

22

xJ2 2

g xJch

( )sin( ) ( )E

E V Eη Φ=∆ + Φ

⇒ extract ( )XS Eωω= ∆

∝

1 10 100

1E-8

1E-7

1E-6

1E-5

1E-4

Sq (a

rb.u

.)

f (Hz)

1/f

( )2

1/ fX

ES ω

ω=

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

α1/2 (e

)

T (mK)

21/

2fE a T=

*1/*

2

1fE

T ϕ≡ Γ ∝

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

ω ωω

ωω

→

→

h

h

h

h

Astafiev et al. (PRL 04)

1 10 100107

108

109

2e2Rω/ћ

πS X

(ω)/2ћ2

(s)

ω/2π(GHz)ωc

/ћ2ωE1/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 decoherence 2 2,rel, , j jj jj Eϕ ε→ Γ Γ << = + ∆

• 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 ϕΓ Γ bath

inter-action

Spectrum of noise felt by qubit

distribution of TLS-parameters, choose

exponential dependence on barrier height for 1/ffor linear ω-dependence

overall factor

• One ensemble of ‘coherent’ TLS

• Plausible distribution of parameters produces:~ ε→ Ohmic high-frequency (f) noise ~ 1/∆ → 1/f noise - 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 ω: random telegraph noiselarge ω: absorption and emission

4. Decoherence of Spin Qubits in Quantum Dotswith Spin-Orbit Coupling

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

What is spin decoherence at ?

Spin Decoherence

Published work concerned with large ,fluctuations due to piezoelectric phononscouple via spin-orbit interaction to spin need breaking of time reversal symmetry → vanishing decoherence for

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

0B =ur

Bur

0B =ur

P. San-Jose, G. Zarand, A. Shnirman, and G. Schön,Geometrical spin dephasing in quantum dots, cond-mat/0603847

The combination of two independent fluctuating field and spin-orbit interaction leads to decoherence of spin at

based on a random Berry phase.0B =

ur

Model Hamiltonian

bath1 1 1 1

2 2 2 2 = ( ) ( , )z y x zH Hb ZB ZX Xµ σ ετ τ σ τ τ− ⋅ − − ⋅ − + +

rur ur ur

= strength of s-o interactiondirection depends on asymmetries

br

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

dot2 orbital

states

noise2 independent fluct. fieldscoupling to orbital degrees of freedom

spin-orbitspin

dot noise1

2s-o = ( , , , ) ( , , , )x yH XB H x y p p H H x Zyµ σ− ⋅ + + +

ur ur

2 2s-o ( ) ( ) ( )y x x y x x y y x y x y x yH p p p p p p p pα σ σ β σ σ γ σ σ= − + − + + −

Rashba + Dresselhaus + cubic Dresselhaus

Specific physical system: Electron spin in double quantum dot

ε + Z(t)

X(t)

2 orbital states:

20 1

...

0

y x x yx

y

z

b

b

p p

b

i p pα β γ= − +

=

=

y1

2s-o = bH τ σ− ⋅

r ur

noise1

2( ) = ( )( )x zZ tX tH τ τ− +

• Phonons with 2 indep. polarizations

• Ohmic fluctuations due to circuit

• Charge fluctuators near quantum dot

,( () )X t Z t

FluctuationsSpectrum:

, 3s sω ≥/ ( )X ZS ω ∝ ω

1/ω

1 1 1 1z x z y

2 2 2 2( = [ ( )) ] ( )Z tX t hbH tετ τ τ τ τ±± − − + ± = − ⋅

rr r

= natural quantization axis for spin br

,x

,y

,z

( ) sin ( ) co( ) s ( )( ) sin ( ) sin ( )

( ) cos (

( )

( )

( )

( ) )

h t t th

X t

Z t

t t t

h t

h th t

h t t

bθ ϕ

θ ϕ

ε θ

±

±

±

= =

= ± = ±

= + =

1 1 1z x z y

2 2 2 = ( )XH bZετ τ τ τ σ− − + − ⋅

r ur0B =

ur

For two projections ± of the spin along br

For each spin projection ±we consider orbital ground state

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

0 01

2( )E h t E++ −= − =

r

ϕ−ϕ

θ

x

y

z

( )h t+

r( )h t−

r

b-b

ϕ−ϕ

θ

x

y

z

( )h t+

r( )h t−

r

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

+eff

2 = cos bH i U U ϕ θ σ− = ur

hh

Instantaneous diagonalization introduces extra term in Hamiltonian

+ += H U HU i U U− h

Gives rise to Berry phase

+ eff,+12

12

1= d ( ) d cos

d cos

t H t tφ ϕ θ

ϕ θ

=

→

∫ ∫

∫h

, , ( ( )) Z tX tφ φ φ ϕ θ+ −∆ = − ↔ ↔

random Berry phase ⇒ dephasing

( )bounded 3/ 22 2( ( )cos )bdt dt X dt t

bXZ tφ ϕ θ φ

ε ∆ = = + +

∫ ∫ ∫

X(t) and Z(t) independent⇒ effective power spectrum

and dephasing rate ( )2

32 2

2

0( ( )) ZX

Tb db

SSϕ ω ωωε

ωΓ =+

∫

Estimate for GaMnAs quantum dot

level spacing ω0 = 1 K

T = 100 mK

• Nonvanishing dephasing for zero magnetic field• due to geometric origin (random Berry phase)

4( 0) 1...10 HzBϕΓ = =

P. San-Jose, G. Zarand, A. Shnirman, GS, cond-mat/0603847

Conclusions

• Progress with solid-state qubits

Josephson junction qubitsspins in quantum dots

• Crucial: understanding and control of decoherence

optimum point strategy for JJ qubits: τϕ ≥ 1 µsec >> τop ≈ 1…10 nsecorigin and properties of noise sources (1/f, …)mechanisms for decoherence of spin qubits

• Application of Josephson qubits:

as spectrum analyzer of noise

Selected References

Yu. Makhlin, G. Schön, and A. Shnirman, Quantum-state engineering with Josephson-junction devices, Rev. Mod. Phys. 73, 357 (2001)

A. Shnirman and G. Schön,Dephasing and renormalization of quantum two-state systemsin "Quantum Noise in Mesoscopic Physics", Y.V. Nazarov (ed.), p. 357, Kluwer (2003), Proceedings of NATO ARW "Quantum Noise in Mesoscopic Physics", Delft, 2002cond-mat/0210023

Yu. Makhlin and A. Shnirman, Dephasing of solid-state qubits at optimal points, Phys. Rev. Lett. 92, 178301 (2004)

A. Shnirman, G. Schön, I. Martin, and Yu. Makhlin, Low- and high-frequency noise from coherent two-level systems, Phys. Rev. Lett. 94, 127002 (2005)

P. San-Jose, G. Zarand, A. Shnirman, and G. Schön,Geometrical spin dephasing in quantum dots, cond-mat/0603847

Preparation Effects Introduce frequency scale

Slow modes dephasing, fast modes renormalization

a) Initially

ground state of

b) pulse

implemented as

Slow oscillators do not reactFast oscillators follow adiabatically

BUT

c) Free evolution, dephasing

d) pulse

e) Measurement of

Slow oscillators ⇒ dephasing

Fast oscillators ⇒ renormalization

Appropriate basis: renormalized (dressed) spin