1. Decoherence at optimal point: beyond the Bloch equations
Yuriy Makhlin
Landau Institute
A. Shnirman (Karlsruhe) R. Whitney (Geneva)
G. Schön (Karlsruhe) Y. Gefen (Rekhovot)
J. Schriefl (Karlsruhe / Lyon)
S. Syzranov (Moscow)
Quantronics group: G. Ithier, E. Collin, P. Joyez, P. Meeson,
D. Vion, D. Esteve (Saclay)

2. Outline Josephson qubits & decoherence sources
Bloch equations, applicability
Beyond Bloch equations:
1/f noise
optimal points, higher-order terms: X2, X4
FID and echo at optimal point
- time-dependent field, Berry phase

3. Coherent oscillations and decoherence
qubit
noisy
environment
Pashkin et al. `03
random phase, decoherence
P=|s+ |
Analyze decay of coherence:
dephasing time
short-time asymptotics
(important for QEC)
decay law t
qubits as spectrometers of quantum noise

4. Models for noise and classification
longitudinal – transverse – quadratic (longitudinal) …
Power spectrum: ,
X – fluctuating field , e.g.:
Gaussian noise with given spectrum
collection of incoherent fluctuators
bosonic bath
spin bath
non-gaussian effects –
cf. Paladino et al. ´02
Galperin et al. ´03

5. Longitudinal coupling  “pure” dephasing
X – classical or quantum field
for regular spectrum
for 1/f noise
e.g., Cottet et al. 01

6. Transverse coupling  relaxation
Golden Rule:

7. c¿ T1, T2 Bloch equations, applicability works for 1/c 1/T1 1/T2* w
Redfield (57)
perturbation theory
DE/~
S(w) w
1/c
1/T2*
1/T1
c¿ T1, T2
works for
weak short-correlated noise Dt Dt Dt
at c¿  t ¿ T1, T2: weak and uncorrelated
on neighboring  t‘s t

8. Beyond Bloch equations
1/f noise – long correlation time c
optimal operation points: X2 or higher powers
sharp pulses (state preparation)
time-dependent field, Berry phase
Study: decay of coherence P(t)=h+(t)i

9. Charge-phase qubit E1 E0 Fx/F0 Vg Quantronium
Vion et al. (Saclay) Vg
Fx/F0
0.5 1 -1 E1
operation at saddle point E0

10. Decay of Ramsey fringes at optimal point
Vion et al., Science 02, …

11. H= - (DE+lX )s 1 Free induction decay (Ramsey) Echo signal
Longitudinal quadratic coupling, non-Gaussian effects
H= - (DE+lX )s 2 z 1
Free induction decay (Ramsey)
p/2
Echo signal
p/2 p

12. Line shape: Linked cluster expansion for X21/f
Determinant regularization

13. for linear coupling in general even more general
For optimal decoupled point X2  X4:
even more general
in general
similar:
Rabenstein et al. 04
Paladino et al. 04
1/f spectrum „quasistatic“
for linear coupling

14. Fitting the experiment
G. Ithier et al. (Saclay)

15. High frequency contributionw f

16. Gaussian approximation and accurate calculation

17. transverse X -> longitudinal X^2
adiabaticity +
transverse X -> longitudinal X^2
YM, Shnirman `03
X(t) B0
DE ¼ B0 + X2/2B0 DE
useful:
- consider transverse and longitudinal noise together
- treat several independent noise sources
- explains „non-Gaussian“ effects of bistable fluctuators
(reduces to Gaussian in X^2; cf. Paladino et al. `02)

18. Example: TLF‘s and the qubit
transverse coupling to TLF‘s (Paladino et al. ‘02):
(t) = 0,v ) (2(t)/2 ) = 0, v´ v´= v2/2
(t)
P(t) = e-t/(2T1)¢ P‘(t) 
for symmetric coupling = v/2, -v/2 - no dephasing
for one strong TLF (vÀ) or many weak TLFs:
it is enough to use the Golden rule
=h X2i/2 1/T2=v‘2/(4)

19. Echo for 2
1/t
1/t 1
e-f t /2

20. Quasi-static environment

21. Multi-pulse echo for quadratic 1/f noise
cf.: NMR; Viola, Lloyd `98; Uchiyama, Aihara `02
p/2 p
N=4 periods
between -pulses
contributions of frequency ranges:
f t<<1
1¿ft¿ N
N¿ft

22. Multi-pulse echo for quadratic 1/f noise
cf.: NMR; Viola, Lloyd `98; Uchiyama, Aihara `02
p/2 p
N=4
P(t)
=10-6
N=30
N=20
N=10
FID
analyt
=104
all perturbative orders involved smooth decay with N

23. Many noise sources E=  V2 +  2 +  V  - cross-term in general,
E(V,) has extrema w.r.t. parameters (V,)
2e- and 0-periodic => torus => 1min, 1max, 2 saddles
E=  V2 +  2 +  V  - cross-term
But: no cross-term for H = H(V) + H()
still, cross-terms for flux qubits

24. Many noise sources two uncorrelated comparable noise sources:
reduces to the problem for one source:
(for similar noise spectra of X and Y)
Remark: subtleties for 1/f noises w/ different infrared cutoffs

25. Decoherence close to optimal pointor

26. Berry phase in dissipative environment
Whitney, YM, Shnirman, Gefen ‘04
Geometric complex correction
BP - monopole
correction - quadrupole

27. Summary - describes large number of weak fluctuators
decoherence beyond the Bloch equations
effects of Gaussian 1/f noise:
- describes large number of weak fluctuators
- non-Gaussian effects for (effective) quadratic coupling
- single- and multi-pulse echo experiments
- non-trivial decay laws
- benchmark!
decoherence in the vicinty of optimal point
effect of several noise sources
geometric phase and noise