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Decoherence at optimal point: beyond the Bloch equations

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)

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Decoherence at optimal point: beyond the Bloch equations

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  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 noisy environment qubit 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. Power spectrum: , Models for noise and classification longitudinal – transverse – quadratic (longitudinal) … 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. S(w) 1/c 1/c c¿ T1, T2 works for 1/T1 1/T2* weak short-correlated noise w 0 DE/~ Bloch equations, applicability Bloch (46,57) Redfield (57) perturbation theory 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. -1 Fx/F0 0.5 1 Vg Quantronium Charge-phase qubit Vion et al. (Saclay) E1 operation at saddle point E0

  10. p/2 p/2 Decay of Ramsey fringes at optimal point Vion et al., Science 02, …

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

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

  13. similar: Rabenstein et al. 04 Paladino et al. 04 in general even more general 1/f spectrum „quasistatic“ for linear coupling For optimal decoupled pointX2 X4:

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

  15. High frequency contribution w f

  16. Gaussian approximation and accurate calculation

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

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

  23. Many noise sources 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 point or

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

  27. Summary • 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

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