We report on the anomalous behavior of control pulses for spins under spin-spin relaxation and subject to classical noise with a singular autocorrelation function. This behavior is not detected for noise with analytic autocorrelation functions. The effect is manifest in the different scaling behavior of the deviation of a real pulse to the ideal, instantaneous one. While a standard pulse displays scaling ∝τp(1), a first-order refocusing pulse normally shows scaling ∝τp(2). But in presence of cusps in the noise autocorrelation the scaling ∝τp(3/2) occurs. Cusps in the autocorrelation are characteristic for fast fluctuations in the noise with a spectral density of Lorentzian shape. We prove that the anomalous exponent cannot be avoided; it represents a fundamental limit. On the one hand, this redefines the strategies one has to adopt to design refocusing pulses. On the other hand, the anomalous exponent, if found in experiment, provides important information on the noise properties.
We consider pulses of finite duration for coherent control in the presence of classical noise. We derive the corrections to ideal, instantaneous pulses for the case of general decoherence (spin-spin relaxation and spinlattice relaxation) up to and including the third order in the duration τ p of the pulses. For pure dephasing (spin-spin relaxation only), we design π and π/2 pulses with amplitude and/or frequency modulation which resemble the ideal ones up to and including the second order in τ p . For completely general decoherence including spin-lattice relaxation the corrections are computed up to and including the second order in τ p as well. Frequency modulated pulses are determined which resemble the ideal ones. They are used to design a low-amplitude replacement for XY8 cycles. In comparison with pulses designed to compensate quantum noise less conditions have to be fulfilled. Consequently, we find that the classical pulses can be weaker and simpler than the corresponding pulses in the quantum case.
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