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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)
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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
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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
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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
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Longitudinal coupling “pure” dephasing
X – classical or quantum field for regular spectrum for 1/f noise e.g., Cottet et al. 01
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Transverse coupling relaxation
Golden Rule:
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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
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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
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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
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Decay of Ramsey fringes at optimal point
Vion et al., Science 02, …
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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
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Line shape: Linked cluster expansion for X21/f
Determinant regularization
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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
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Fitting the experiment
G. Ithier et al. (Saclay)
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High frequency contribution
w f
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Gaussian approximation and accurate calculation
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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)
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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)
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Echo for 2 1/t 1/t 1 e-f t /2
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Quasi-static environment
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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
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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
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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
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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
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Decoherence close to optimal point
or
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Berry phase in dissipative environment
Whitney, YM, Shnirman, Gefen ‘04 Geometric complex correction BP - monopole correction - quadrupole
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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
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