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
Top Related