On Decoherence in Solid-State Qubits
Gerd Schön
Karlsruhe
work with:
Alexander Shnirman
Yuriy Makhlin
Pablo San José
Gergely Zarand
Karlsruhe
Landau Institute
Karlsruhe
Budapest and Karlsruhe
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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
Universität
Karlsruhe (TH)
1. Josephson charge qubits
Fx
n
g
Vg
H  EC ( 2n 
2 degrees of freedom
 n, θ   i
charge and phase
2 control fields: Vg and Fx
Shnirman, G.S., Hermon (97)
2
2 states only, e.g. for EC » EJ
charging energy, Josephson coupling
flux
Cg Vg/2e
Fx
)  EJ cos(π
) cos θ
e
F0
CgVg
tunable EJ
2 energy scales EC , EJ
gate voltage,
F x /F 0
Vg
1
1
2
2
H   Ech (Vg ) σ z  EJ (F x ) σ x
Observation of coherent oscillations
Nakamura, Pashkin, and Tsai, ‘99
1
1
H   Ech (Vg ) σ z  EJ σ x
2
2
1
1
  t   ae
iE0 t /
Qg/e
0  be
iE1 t /
top ≈ 100 psec, tj ≈ 5 nsec
1
major source of decoherence:
background charge fluctuations
EC ≈ EJ
Charge-phase qubit
H  EC ( 2n 
CgVg
e
)2  EJ cos(π
Fx
) cos θ
F0
gate
Quantronium (Saclay)
Operation at saddle point:
to minimize noise effects
- voltage fluctuations couple transverse
- flux fluctuations couple quadratically
F x /F 0
Cg Vg/2e
1
H   E t z 
2
1
2
Ech
Vg
Vg0
δVg t x 
1
 2 EJ
2 F x0
4 F x
δF 2x t z
Decay of Ramsey fringes at optimal point
switching probability (%)
55
detuning=50MHz
/2
50
/2
45
40
35
T2 = 300 ns
30
25
0
200
400
600
800
Delay between /2 pulses (ns)
Vion et al., Science 02, …
Coherence times (ns)
500
Spin echo
Free decay
500
Gaussian noise
Sd
100
SNg
100
1/w
1/w
w
w
4MHz
0.5MHz
10
10
-0.3
Experiments Vion et al.
-0.2
Fx/F0
-0.1
0.0
0.05 0.10
|Ng-1/2|
2. Models for noise and classification
Sources of noise
- noise from control and measurement circuit, Z(w)
- background charge fluctuations
-…
Properties of noise
S X (w ) 
- spectrum: Ohmic (white), 1/f, ….
- Gaussian or non-Gaussian
1
2
 dt
 w coth
coupling:
1
1
1
1
2
2
2
4
 X (t ), X (0) eiw t
w
2kBT
,
1 / w , ...
H =  E t z  X t z  X  t x  X 22 t z  H bath
longitudinal
– transverse
– quadratic (longitudinal) …
model
noise
Ohmic
1/f
(Gaussian)
Bosonic bath
Spin bath
Relaxation (T1) and dephasing (T2)


d
1
1
M  B  M  M z  M 0 z  (M x x  M y y )
dt
T1
T2
Bloch (46,57), Redfield (57)
For linear coupling, regular spectra, T ≠ 0
Golden rule: exponential decay law
1
1
  rel      S X  (w  ΔE )
T1
2
1
1 1 1
 j 
 S X (w  0)
T2
2 T1 2
pure dephasing:
00   00   11
11   00   11
01  i Bz 01  j 01
Example: Nyquist noise due to R
(fluctuation-dissipation theorem)
SdV (w )  w R coth
  rel 
j* 
j*
w
2kBT
R E
E
coth
h / e2
2kBT
R kBT
h / e2
Dephasing due to 1/f noise, T=0, nonlinear coupling ?
1/f noise,
longitudinal linear coupling
1
H   (ΔE  X )t z + H bath
2
S X w  =
E1/2 f
|w|
  for w  0
 t

 1 dw
sin 2 (wt / 2) 
01 (t )  exp  i  X (t ) dt   exp   
S X (w )

2
2
2π
(
w
/
2
)


 0

 E1/2 f 2

 exp  
t ln | wir t | 
 2π



non-exponential decay of coherence
 time scale for decay
1/ T2  j*  E1/ f
Cottet et al. (01)
3. Noise Spectroscopy via JJ Qubits
Astafiev et al. (NEC)
Martinis et al., …
Josephson qubit + dominant background charge fluctuations
1
1
1
2
2
2
H   Ech (Vg )  z  EJ (F x )  x  X (t )  z
1
1
1
2
E  Ech
(Vg )  EJ2 (F x )
2
2
2
tan  EJ (F x ) / Ech (Vg )
H   E t z  X  t  cos t z  X  t  sin t x
eigenbasis of qubit
transverse component
of noise  relaxation
1
1
  rel  S X (w  E ) sin 2
T1
2
longitudinal component
of noise  dephasing
1/f noise S X w  
E1/2 f
|w |
probed in exp’s
 E1/2 f 2
t cos 2  ln wir t
 01 (t )  exp  
 2
1
*


j  E1/ f cos
*
T2



Relaxation (Astafiev et al. 04)
 rel 
1
2
S X (w  E ) sin 2
data confirm expected
dependence on
2
E
J (F x )
sin 2 
2
Ech
(Vg )  EJ2 (F x )
 extract
S X (w  E )
w
Low-frequency noise and dephasing
S X w  
E1/f
w
(e)
1E-5
E
Dephasing
low frequency 1/f noise
0.015
/
1E-6
1
*


 E1/ f
j
*
T2
0.010

Sq (arb.u.)
1E-4
2
1/ f
0.005
1E-7
E1/2 f  a T 2
1/f
1E-8
0.000
1
10
f (Hz)
100
0
100 200 300 400 500 600 700 800 900 1000
T (mK)
T 2 dependence of 1/f spectrum observed earlier by F. Wellstood, J. Clarke et al.
Relation between high- and low-frequency noise
Astafiev et al. (PRL 04)
9
2
 SX(w)/2ћ (s)
10
8
2
10
2e Rw/ ћ
E1/f/2 ћ
2
w
7
10
1
wc
10
w/ (GHz)
S X (w ) 
a  kBT 
w
 a w
100
2
for
w
kBT
for
w
kBT
same strength a
for low- and
high-frequency noise
High- and low-frequency noise from coherent two-level systems
• Qubit used to probe fluctuations X(t)
• Source of X(t): ensemble of ‘coherent’ two-level systems (TLS)
• each TLS is coupled (weakly) to thermal bath Hbath.j at T and/or other TLS
 weak relaxation and decoherence
 rel, j , j , j  E j   2j   2j
TLS
TLS
qubit
TLS
TLS
TLS
rel, j , j , j
interaction
bath
Spectrum of noise felt by qubit
low w: random telegraph noise
large w: absorption and emission
distribution of TLS-parameters, choose
for linear w-dependence
exponential dependence on barrier height for 1/f
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)
4. At symmetry point: Quadratic longitudinal 1/f noise
static noise
Paladino et al., 04
Averin et al., 03
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 Dots
Petta et al., Science, 2005
spin + ≥ 2 orbital states + spin-orbit coupling
noise coupling to orbital degrees of freedom
Generic Hamiltonian
1
1
1
1
2
2
2
2
H =   B    t z  t y b    ( X t x  Zt z )  H bath ( X , Z )
spin
dot
2 orbital
states
spin-orbit
b
= strength of s-o interaction
direction depends on asymmetries
B,
B0
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 Xtx and Ztz leads to decoherence,
based on a random Berry phase.
noise
2 independent fluct. fields
coupling to orbital degrees of freedom
Specific physical system: Electron spin in double quantum dot
1
H =   B    H dot ( x, y, px , p y )  H s-o  H noise ( x, y, X , Z )
2
Hs-o   ( py  x  px  y )   ( px  x  py  y )   ( px py2 x  py px2 y )
Rashba
+
Dresselhaus
Fluctuations
X (t ), Z (t )
+ cubic Dresselhaus
Spectrum:
• Phonons with 2 indep. polarizations
• Charge fluctuators near quantum dot
2-state approximation:
s
w
,s  3
S X / Z (w ) 
1 / w and/or
1
H s-o =  t y b  
2
bx  i 0  p y   px   px p y2 1
X(t)
 + Z(t)
1
H noise =  ( X (t ) t x  Z (t ) t z )
2
by  ...
bz  0
w
1
1
1
2
2
2
H =  t z  ( X t x  Zt z )  t y b  
B0
b
For two projections ± of the spin along
b
1
1
1
2
2
2
H  =  t z  [ X (t )t x  Z (t )t z ] 
h ,x (t )  X (t )  h(t ) sin  (t ) cos j (t )
h ,y (t )   b   h(t ) sin  (t ) sin j (t )
h ,z (t )    Z (t )  h(t ) cos  (t )
For each spin projection ±
we consider orbital ground state
1
E0    h  (t )  E0 
= natural quantization axis for spin
1
b t y   h  (t ) t
2
z
h (t )

-b
j j
h (t )
b
x
2
Ground (and excited) states 2-fold degenerate due to spin (Kramers’ degeneracy)
y
Instantaneous diagonalization introduces extra term in Hamiltonian
H = U +H U  i U + U
In subspace of 2 orbital ground states for + and - spin state:
H eff =  i U + U  j cos   b
2
Gives rise to Berry phase
z
h (t )

j j
h (t )
y
+ =
1
2

1
 dt H eff,+ (t ) 
1
2
 dt j cos 
dj cos 

2
      j ,  X (t ), Z (t )
random Berry phase  dephasing
x
b


   dtj cos   bounded   dt X    dt


 2  b2


3/ 2
Z (t ) X (t )
X(t) and Z(t) small, independent, Gaussian distributed
 effective power spectrum and dephasing rate
j 
T
b2

2
b
2

3
2
d
w
w
S X (w ) S Z (w )

0
Small for phonons (high power of w and T)
Estimate for 1/f – noise or 1/f ↔ f noise
j 
b2

2
b
2

3
X2
Z2 T
 (109...105 ) T  1...104 Hz
• Nonvanishing dephasing for zero magnetic field
• due to geometric origin (random Berry phase)
• measurable by comparing 1 and j for different
initial spins
j ( B  0)  1...104 Hz
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: tj  1 sec >> top ≈ 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