Measuring Environmental Quantum Noise Exhibiting a Nonmonotonic Spectral Shape

Understanding the physical origin of noise affecting quantum systems is important for nearly every quantum application. Quantum-noise spectroscopy has been used in various quantum systems, such as superconducting qubits, nitrogen-vacancy centers, and trapped ions. Traditional spectroscopy methods are usually efficient in measuring noise spectra with mostly monotonically decaying contributions. However, there are important scenarios in which the noise spectrum is broadband and nonmonotonous, thus posing a challenge to existing noise-spectroscopy schemes. Here we compare several methods for noise spectroscopy: spectral decomposition based on the Carr-Purcell-Meiboom-Gill sequence, the recently presented dynamic sensitivity control (DYSCO) sequence, and a modified DYSCO sequence with a Gaussian envelope (gDYSCO). The performance of the sequences is quantified by analytic and numeric determination of the frequency resolution, bandwidth, and sensitivity, revealing a supremacy of gDYSCO to reconstruct nontrivial features. Using an ensemble of nitrogen-vacancy centers in diamond coupled to a high-density ${}^{13}\mathrm{C}$-nuclear-spin environment, we experimentally confirm our findings. The combination of the schemes presented offers potential to record high-quality noise spectra as a prerequisite to generate quantum systems unlimited by their spin-bath environment.

Figure 1

The schematic for the CPMG and DYSCO experiments: green indicates a laser pulse, orange indicates a $\pi$ or $\pi /2$ pulse, while yellow and blue indicate microwave-pulse blocks composed of $4\pi$ pulses with the denoted phases (in which $\phi$ is a function of the block number).

Figure 2

The time-domain sensitivity functions for CPMG-16, DYSCO, and gDYSCO experiments (a) and the respective FFs (b). The higher harmonics in the CPMG FF do not appear for DYSCO FF at the cost of a smaller maximum filter-function height. When a Gaussian envelope is added, the side lobes disappear at the cost of an even further reduced maximum filter-function height and an increased width of the main peak.

Figure 3

Simulations of the various reconstructions methods. (a) A noise spectrum is defined as a dc-centered Lorentzian (blue). On the basis of the noise spectrum, CPMG-$N$ decoherence curves are generated (in the inset the numbers correspond to $N$, the $X$ axis is time in milliseconds, and the $Y$ axis is coherence in arbitrary units). Normally distributed noise added to the coherence curves simulates experimental uncertainties. Application of the decomposition code to the coherence curves reconstructs the noise spectrum (red). Equivalently, the reconstruction of the noise spectrum for the DYSCO sequence is simulated (green). (b)–(d) Extension of the analysis to noise spectra featuring a pronounced peak for CPMG-$N$, DYSCO, and gDYSCO schemes, respectively. The spectra obtained from a noise-free coherence curve are represented by the red data points, whereas the data represented by the green points incorporate the simulated experimental uncertainty. In general, the consideration of experimental uncertainties blurs the noise peak. Higher harmonics of the CPMG-$N$ FF cause artifacts in the reconstructed spectrum at roughly 30 and 15 kHz, which do not appear in the DYSCO and gDYSCO measurements. The gDYSCO scheme, which avoids the side lobes in the FF, reconstructs the noise spectrum with highest precision.

Figure 4

(a) Experimental coherence curves for CPMG-$N$ experiments. Short timescales are densely sampled to reveal the ${}^{13}\mathrm{C}$ collapse. The revival points are also sampled to extract the decoherence curve. The envelope functions $\exp [-(x/T{)}^p]$ fitted to the revival points are shown as solid lines. (b) Coherence curve obtained from a gDYSCO experiment as a function of the modulation frequency. (c) The reconstructed spectrum obtained with CPMG-N experiments, with N ranging from 2 to 128, and the spectrum obtained from the gDYSCO experiment. Both spectra are fitted with a Gaussian; the fitted parameters are given in Table 1. The ${}^{13}\mathrm{C}$ Larmor peak reconstructed by gDYSCO is narrower than the one obtained by CPMG SD, as predicted from the simulation results [Fig. 3]. In addition, it can be clearly seen that the limited sensitivity of gDYSCO does not reveal the noise spectrum when its power is below approximately ${10}^5(\mathrm{rad}/\mathrm{s})$ but results in a higher maximum detected noise spectrum (see the Appendix for more information).

Figure 5

(a) The coherence curve obtained for a Hahn-echo ($T_2(1)$) experiment. The function is fitted to a sum of Gaussians centered around the Larmor revivals multiplied by an envelope $e^{-(t/T_2{)}^p}\left[\sum _i{e}^{-(t-i{t}_l{)}^2}/2{\sigma }^2\right]$, where $i$ ranges from 0 to 6, and the decay time $T_2=488\mu \mathrm{s}$. (b) The coherence curve obtained for a DYSCO experiment with zero sensitivity at all times as a function of the experiment time. The measurements reported in the main text are done at $200\mu \mathrm{s}$, near the first peak in the curve. The occurrence of collapses and revivals is not well understood and will be the subject of further investigations.

Figure 6

The experimental results for CPMG-8 experiments (blue) and the simulated CPMG-8 data (red).

Figure 7

Coherence decay curve. Superimposed are the relevant magnitude of DYSCO maximum contrast, $a_{\mathrm{DYSCO}}$, and uncertainty, $ϵ$.

Figure 8

(a) The coherence curve obtained from a DYSCO measurement as a function of the DYSCO modulation frequency. (b) The spectrum obtained from the DYSCO measurements with a fitted Gaussian centered around $67.1(2)\mathrm{kHz}$ with a width of 17.3(2) kHz.