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)

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

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

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

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

Transverse coupling  relaxation Golden Rule:

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

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

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

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

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

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

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

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

High frequency contribution w f

Gaussian approximation and accurate calculation

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)

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)

Echo for 2 1/t 1/t 1 e-f t /2

Quasi-static environment

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

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

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

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

Decoherence close to optimal point or

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

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