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arXiv:1708.05535 · Physical Review A

Accuracy of dynamical-decoupling-based spectroscopy of Gaussian noise

Piotr Szańkowski, Łukasz Cywiński

Published Mar 5, 2018• 22 citations
arXivHTMLPDF

Abstract

The fundamental assumption of dynamical-decoupling-based noise spectroscopy is that the coherence decay rate of qubit (or qubits) driven with a sequence of many pulses, is well approximated by the environmental noise spectrum spanned on frequency comb defined by the sequence. Here we investigate the precise conditions under which this commonly used spectroscopic approach is quantitatively correct. To this end we focus on two representative examples of spectral densities: the long-tailed Lorentzian, and finite-ranged Gaussian---both expected to be encountered when using the qubit for nanoscale nuclear resonance imaging. We have found that, in contrast to Lorentz spectrum, for which the corrections to the standard spectroscopic formulas can easily be made negligible, the spectra with finite range are more challenging to reconstruct accurately. For Gaussian line shape of environmental spectral density, direct application of the standard dynamical-decoupling-based spectroscopy leads to erroneous attribution of long-tail behavior to the reconstructed spectrum. Fortunately, artifacts such as this, can be completely avoided with the simple extension to standard reconstruction method.

DOI
10.1103/physreva.97.032101

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The fundamental assumption of dynamical-decoupling-based noise spectroscopy is that the coherence decay rate of qubit (or qubits) driven with a sequence of many pulses, is well approximated by the environmental noise spectrum spanned on frequency comb defined by the sequence. Here we investigate the precise conditions under which this commonly used spectroscopic approach is quantitatively correct. To this end we focus on two representative examples of spectral densities: the long-tailed Lorentzian, and finite-ranged Gaussian---both expected to be encountered when using the qubit for nanoscale nuclear resonance imaging. We have found that, in contrast to Lorentz spectrum, for which the corrections to the standard spectroscopic formulas can easily be made negligible, the spectra with finite range are more challenging to reconstruct accurately. For Gaussian line shape of environmental spectral density, direct application of the standard dynamical-decoupling-based spectroscopy leads to erroneous attribution of long-tail behavior to the reconstructed spectrum. Fortunately, artifacts such as this, can be completely avoided with the simple extension to standard reconstruction method.
I present a theory of environmental noise spectroscopy via dynamical decoupling of a qubit at an optimal working point. Considering a sequence of nnn pulses and pure dephasing due to quadratic coupling to Gaussian distributed noise $\ensuremath{\xi}(t)$, 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 time scale on which coherence decays, the noise filtered by the dynamical decoupling procedure can be treated as effectively Gaussian at large nnn, and well-established methods of noise spectroscopy can be used to reconstruct the spectrum of ${\ensuremath{\xi}}^{2}(t)$ noise. On the other hand, for noise of dominant low-frequency character $(1/{f}^{\ensuremath{\beta}}$ noise with $\ensuremath{\beta}>1)$, 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 ttt depends both on spectral density of $\ensuremath{\xi}(t)$ noise at $\ensuremath{\omega}=n\ensuremath{\pi}/t$, 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 that the longitudinal qubit splitting is much larger than the transverse noise amplitude.
Here, we explore the prospects of carrying out the single qubit spectroscopy of environmental noise when the resolution of the frequency filters produced by the dynamical decoupling pulse sequences can be set high enough to reveal the discrete structure of the noise spectral density. The standard form of spectroscopy is applicable when the filter is unable to resolve the discrete spectral lines and only coarse-grained approximation of the spectrum is recorded in the qubit's decoherence rate. When the discrete structure becomes accessible, the qubit probe evolves in a qualitatively different manner, and the procedure for recovering spectral density has to be redesigned.
Here we demonstrate how the standard, temporal-only, dynamical-decoupling-based noise spectroscopy method can be extended to also encompass the spatial degree of freedom. This spatiotemporal spectroscopy utilizes a system of multiple qubits arranged in a line that are undergoing pure dephasing due to environmental noise. When the qubits are driven by appropriately coordinated sequences of π pulses the multi-qubit register becomes decoupled from all components of the noise, except for those characterized by frequencies and wavelengths specified by the pulse sequences. This allows for employment of the procedure for reconstruction of the two-dimensional spectral density that quantifies the power distribution among spatial and temporal harmonic components of the noise.
A qubit subjected to pure dephasing due to classical Gaussian noise can be turned into a spectrometer of this noise by utilizing its readout under properly chosen dynamical decoupling (DD) sequences to reconstruct the power spectral density of the noise. We review the theory behind this DD-based noise spectroscopy technique, paying special attention to issues that arise when the environmental noise is non-Gaussian and/or it has truly quantum properties. While we focus on the theoretical basis of the method, we connect the discussed concepts with specific experiments, and provide an overview of environmental noise models relevant for solid-state based qubits, including quantum-dot based spin qubits, superconducting qubits, and NV centers in diamond.
Here we explore the prospects of carrying out the single qubit dynamical decoupling spectroscopy of environmental noise characterized by discrete spectral density. We show that the approach to the spectroscopy and spectrum reconstruction has to be adjusted to the resolution of frequency filters produced by the dynamical decoupling pulse sequences. The standard form of spectroscopy is applicable when the filter is unable to resolve discrete spectral lines and only coarse grained approximation of the spectrum is recorded in the qubit's decoherence rate. When the filter's resolution is high enough the qubit probe evolves in a qualitatively different manner and the procedure for recovering spectral density has to be redesigned.
Detecting and characterizing decoherence-inducing noise sources is critical for developing robust quantum technologies and deploying quantum sensors operating at molecular scales. However, current noise spectroscopies rely on severe approximations that sacrifice accuracy and precision. We propose a novel approach to overcome these limitations. It self-consistently extracts noise spectra that characterize the interactions between a quantum sensor and its environment from commonly performed dynamical decoupling-based coherence measurements. Our approach adopts minimal assumptions and is resilient to measurement errors. We quantify confidence intervals and sensitivity measures to identify experiments that improve spectral reconstruction. We employ our method to reconstruct the noise spectrum of a nitrogen-vacancy sensor in diamond, resolving previously undetected nuclear species at the diamond surface and revealing that previous measurements had overestimated the strength of low-frequency noise by an order of magnitude. Our method uncovers previously hidden structure with unprecedented accuracy, setting the stage for precision noise spectroscopy-based quantum metrology.