Nonlinear Single-Spin Spectrum Analyzer

Qubits have been used as linear spectrum analyzers of their environments. Here we solve the problem of nonlinear spectral analysis, required for discrete noise induced by a strongly coupled environment. Our nonperturbative analytical model shows a nonlinear signal dependence on noise power, resulting in a spectral resolution beyond the Fourier limit as well as frequency mixing. We develop a noise characterization scheme adapted to this nonlinearity. We then apply it using a single trapped ion as a sensitive probe of strong, non-Gaussian, discrete magnetic field noise. Finally, we experimentally compared the performance of equidistant vs Uhrig modulation schemes for spectral analysis.

Figure 1

(a) ${}^{88}\mathrm{Sr}^{+}$ ground state manifold comprising the Zeeman sensitive quantum probe. (b) Typical experimental sequence with $n$ modulation pulses sandwiched between two $\pi /2$ pulses with a relative ${\varphi }_{rf}$ phase. (c), (d) Probability of finding the probe in the $\mid \uparrow ⟩$ state at the end of a sequence vs the rf relative phase. Red line is a best fit to $P_{\uparrow }=1/2-A/2\cos ({\varphi }_{rf})$, $A$ is the coherence. (e) Nonlinear response of the coherence with respect to the number of equidistant pulses $n$. Interpulse distance was 5 ms. The first and last points correspond to the fringes in 1(c) and 1(d), respectively. Red line is a single parameter best fit to Eq. (2). Inversion of the fringe contrast in (d) corresponds to negative coherence. (f) Nonlinear response of the coherence with respect to modulation frequency, with $n=11$ equidistant pulses. Spectrum corresponds to a single, highly nonperturbative noise component. Spectral features, narrower than the Fourier resolution, are highlighted by the shaded regions. These features are 4 Hz FWHM whereas the Fourier limit is 17 Hz. Red line is a single parameter best fit to Eq. (2).

Figure 2

Spectral peak identification adapted to the nonlinear regime. Two scans taken several months apart. Each scan is a color map of coherence vs the number of pulses $n$ and the modulation frequency. The upper scan shows clear features at 100 and 150 Hz. The latter nearly vanishes in the lower scan and a new 200 Hz component appears. The crossing from positive to negative coherence marked by the dashed line in (b) can be used to extract the noise magnitude at 100 Hz as described in the text.

Figure 3

Fine-tuning of noise magnitudes in the nonlinear regime. Once noise components have been identified in frequency and magnitude, a fine-tuning estimate is obtained by scanning the modulation frequency and fitting to the full spectrum to Eq. (2) (red line). Here we use an $n=11$ equidistant pulse sequence. Nonlinearity is well pronounced around $f=100\mathrm{Hz}$ where it renders spectral features narrower than the Fourier limit with almost a tenfold improvement in spectral resolution, as discussed in the text.

Figure 4

Comparison of equidistant vs Uhrig modulation. (a) Each column is a scan of ${\varphi }_{rf}$ at $n=19$ equidistant pulses and a fixed $T$. The total experiment time $T$ is varied from column to column and the corresponding contrast and modulation frequency is shown in (c). (b) Same as (a), for an Uhrig scheme of 20 pulses. (c) Red line is a fit to Eq. (2). (d) Same as (c), for the Uhrig scheme. The fit indicated by the red line yields field magnitudes consistent with the equidistant modulation fit in (c). Inset in (c) shows the spectrum of equidistant modulation with 19 pulses and a total duration of 66.7 ms. Inset in (d) is the same as the inset in (c) for Uhrig modulation with 20 pulses.