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

TL;DR

This paper develops a theoretical framework for noise spectroscopy of qubits at optimal working points using dynamical decoupling, addressing both short- and long-range correlated noise and providing formulas for coherence decay and spectral density reconstruction.

Contribution

It introduces a comprehensive theory for environmental noise spectroscopy at optimal qubit points, including analytical formulas for coherence decay under various noise conditions.

Findings

Effective Gaussian approximation for short correlation time noise with large number of pulses.

Analytical power-law decay formula for dominant low-frequency noise.

Simulation results confirm applicability of formulas for transverse noise decoherence.

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.