On Decoherence in Solid-State Qubits - Capri SchoolOn Decoherence in Solid-State Qubits •...
Transcript of On Decoherence in Solid-State Qubits - Capri SchoolOn Decoherence in Solid-State Qubits •...
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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/
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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
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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
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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
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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
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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)
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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|
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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
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Ohmic
Spin bath
1/f(Gaussian)
model
noise
Bosonic bath
Quantum Baths
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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
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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”
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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
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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
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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
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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:
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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
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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
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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
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Longitudinal coupling: Ohmic spectrum
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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))
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( ) ( )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., …
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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
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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
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• 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
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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
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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
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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
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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
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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
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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
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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
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( )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
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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
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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
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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
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c) Free evolution, dephasing
d) pulse
e) Measurement of
Slow oscillators ⇒ dephasing
Fast oscillators ⇒ renormalization
Appropriate basis: renormalized (dressed) spin