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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