Dynamical decoupling noise spectroscopy at an optimal working point of a qubit

Abstract

I present a theory of environmental noise spectroscopy via dynamical decoupling of a qubit at an optimal working point. Considering a sequence of $n$ pulses and pure dephasing due to quadratic coupling to Gaussian distributed noise $\xi (t)$, I use the linked-cluster (cumulant) expansion to calculate the coherence decay. Solutions allowing for reconstruction of spectral density of noise are given. For noise with correlation time shorter than the timescale on which coherence decays, the noise filtered by the dynamical decoupling procedure can be treated as effectively Gaussian at large $n$, and well-established methods of noise spectroscopy can be used to reconstruct the spectrum of ${\xi }^2(t)$ noise. On the other hand, for noise of dominant low-frequency character ($1/f^{\beta }$ noise with $\beta >1$), an infinite-order resummation of the cumulant expansion is necessary, and it leads to an analytical formula for coherence decay having a power-law tail at long times. In this case, the coherence at time $t$ depends both on spectral density of $\xi (t)$ noise at $\omega =n\pi /t$, and on the effective low-frequency cutoff of the noise spectrum, which is typically given by the inverse of the data acquisition time. Simulations of decoherence due to purely transverse noise show that the analytical formulas derived in this paper apply in this often encountered case of an optimal working point, provided that the number of pulses is not very large, and the longitudinal qubit splitting is much larger than the transverse noise amplitude.

Substantially expanded final version, includes discussion of dynamical decoupling from transverse noise

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• Spectroscopy of classical environmental noise with a qubit subjected to projective measurements
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• High-fidelity two-qubit gates via dynamical decoupling of local 1/f noise at optimal point
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• Dynamical Correlation Functions of the Quadratic Coupling Spin-Boson Model
• Decoherence-assisted detection of entanglement of two qubit states
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• A generalized model of the noise spectrum of a two-level fluctuator in the presence of an electron subbath
• Quantifying electron-nuclear spin entanglement dynamics in central-spin systems using one-tangles
• Qubit decoherence under two-axis coupling to low-frequency noises