Efficient continuous-wave noise spectroscopy beyond weak coupling

Kyle Willick, Daniel K. Park, and Jonathan Baugh

Phys. Rev. A 98, 013414 – Published 18 July 2018

Abstract

The optimization of quantum control for physical qubits relies on accurate noise characterization. Probing the spectral density $S (ω)$ of semiclassical phase noise using a spin interacting with a continuous-wave (CW) resonant excitation field has recently gained attention. CW noise spectroscopy protocols have been based on the generalized Bloch equations (GBE) or the filter function formalism, assuming weak coupling to a Markovian bath. However, this standard protocol can substantially underestimate $S (ω)$ at low frequencies when the CW pulse amplitude becomes comparable to $S (ω)$ . Here we derive the coherence decay function more generally by extending it to higher orders in the noise strength and discarding the Markov approximation. Numerical simulations show that this provides a more accurate description of the spin dynamics compared to a simple exponential decay, especially on short timescales. Exploiting these results, we devise a protocol that uses an experiment at a single CW pulse amplitude to extend the spectral range over which $S (ω)$ can be reliably determined to $ω = 0$ .

Received 29 March 2018

DOI:https://doi.org/10.1103/PhysRevA.98.013414

Physics Subject Headings (PhySH)

Coherent control Nuclear & electron resonance Quantum control

Electron spin resonance Nuclear magnetic resonance Perturbative methods

Quantum Information, Science & TechnologyAtomic, Molecular & Optical

Authors & Affiliations

Kyle Willick^1,2,3, Daniel K. Park⁴, and Jonathan Baugh^1,2,5,*

¹Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
²Waterloo Institute for Nanotechnology, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
³Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
⁴Natural Science Research Institute, Korea Advanced Institute of Science and Technology, Daejeon, Republic of Korea 34141
⁵Department of Chemistry, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

^*Corresponding author: baugh@uwaterloo.ca

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Vol. 98, Iss. 1 — July 2018

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Images

Figure 1
The real parts of $F_{2} (ω, Ω, T = 2 π l / Ω)$ , where $l$ is the number of Rabi cycles, for (a) $Ω / 2 π = 1000$ Hz and $l = 1, 2, 3$ , and (b) for $l = 4$ and $Ω / 2 π = 1000, 2000, 3000$ Hz.
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Figure 2
(a) Determination of $S (ω)$ using simple exponential fitting in simulated CW noise spectroscopy experiments. The signal decay is simulated using an input $S_{in} (ω) \approx 30 {Hz}^{2} / ω$ , and least-squares fitting is used to calculate the noise spectrum $S_{0} (ω)$ . At small pulse amplitudes (low Rabi frequencies), the exponential fitting returns inaccurate spectral information. The insets show the signal decays at the indicated pulse amplitudes, illustrating the nonexponential signal decay behavior at low $Ω$ . (b) and (c) Examples of the same procedure applied to different noise spectra, with (b) $S_{in} (ω) \approx 30 {Hz}^{3} / ω^{2}$ , and (c) $S_{in} (ω) \approx 3000 {Hz}^{2} / ω$ . (d) A signal decay taken at $Ω = 16 rad / s$ from the simulated experiment used in (a), illustrating nonexponential decay. The insets show $F_{2} (ω, Ω, T)$ and $S (ω)$ at two different time points, highlighting the importance of the low frequency component of $F_{2} (ω, Ω, T)$ which overlaps with a large spectral noise density.
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Figure 3
The error between the simulated “experimental” signal decays and theoretical decays calculated by standard [i.e., Eq. (19)] and cumulant expansion methods. (a) Using the exponential form $exp [- S (Ω) t / 2]$ results in large errors at short time scales ( $t < 0.5 s$ ), i.e., the region indicated by the dashed box. (b) and (c) Using the cumulant expansion method reduces the error. By using $χ_{2} + χ_{4}$ , the experimental decays are better matched at all times, down to low drive amplitudes ( $Ω \approx 20 rad / s$ for the parameters chosen here).
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Figure 4
Comparison of the true signal decay, calculated by unitary time evolution ( $H$ ), and decays calculated by the cumulant expansion up to $χ_{4}$ , when $S (ω)$ is varied in a region away from the driving amplitude. The correct $S (ω) = 30 {Hz}^{2} / ω$ (with a plateau as $ω \to 0$ ) is used to calculate the red curve. For the green curve, all values of $S (ω)$ for $ω < 10 rad / s$ are reduced by half. The probe frequency, or driving amplitude, is $Ω = 32 rad / s$ .
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Figure 5
$S (ω)$ estimates obtained from the cumulant expansion based noise spectroscopy protocol described in the main text. Results shown are for a simulated experiment with $S_{in} = 30 {Hz}^{2} / ω$ . The final spectrum estimates $S_{final} (ω)$ are shown for three different ( $Ω_{P}$ , $T$ ) conditions, where $T$ is the total pulse duration. The initial estimate $S_{0} (ω)$ is determined using the standard method of exponential fits to Eq. (19). The oscillations that appear in the range of 8–30 rad/s are artifacts (discussed in the text) and can be removed by fitting $S_{final} (ω)$ to a functional form such as $1 / ω^{k}$ .
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Figure 6
The result of applying the cumulant expansion noise spectroscopy protocol to three different noise spectra. The noise spectrum used to simulate each experiment is shown in blue $[S_{in} (ω)]$ . The spectrum obtained from the standard exponential fitting $[S_{0} (ω)]$ is shown in red, and the result of the cumulant expansion protocol $[S_{final} (ω)]$ is shown in green. A fitting of $S_{final}$ to a general power-law form $C ω^{α}$ is shown in the lower panels. The input noise spectrum, pulse amplitude $Ω_{P}$ , and total pulse duration are (a) $S (ω) = 30 {Hz}^{2} / ω$ , $Ω_{P} = 35 rad / s$ , $T = 1 s$ , (b) $S (ω) = 30 {Hz}^{3} / ω^{2}$ , $Ω_{P} = 35 rad / s$ , $T = 5 s$ , and (c) $S (ω) = 3000 {Hz}^{2} / ω$ , $Ω_{P} = 640 rad / s$ , $T = 0.2 s$ .
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