Noise spectroscopy of a quantum-classical environment with a diamond qubit

Knowing a quantum system's environment is critical for its practical use as a quantum device. Qubit sensors can reconstruct the noise spectral density of a classical bath, provided long enough coherence time. Here, we present a protocol that can unravel the characteristics of a more complex environment, comprising both unknown coherently coupled quantum systems, and a larger quantum bath that can be modeled as a classical stochastic field. We exploit the rich environment of a nitrogen-vacancy center in diamond, tuning the environment behavior with a bias magnetic field, to experimentally demonstrate our method. We show how to reconstruct the noise spectral density even when limited by relatively short coherence times, and identify the local spin environment. Importantly, we demonstrate that the reconstructed model can have predictive power, describing the spin qubit dynamics under control sequences not used for noise spectroscopy, a feature critical for building robust quantum devices. At lower bias fields, where the effects of the quantum nature of the bath are more pronounced, we find that more than a single classical noise model are needed to properly describe the spin coherence under different controls, due to the back action of the qubit onto the bath.

Figure 1

System-environment model and experimental protocol. The NV electronic spin (blue sphere) is sensitive to ${}^{13}\mathrm{C}$ impurities in the diamond (brown spheres), including isolated nearby spins and a larger ensemble spin bath. The NV is addressed and manipulated via optically detected magnetic resonance in the presence of an external bias magnetic field, aligned with the NV axis. After optical initialization in the $m_s=0$ spin state, the NV is manipulated with resonant microwave pulses (in blue), and read out optically (red). A detailed description of our experimental setup can be found in Refs. [1, 30, 31].

Figure 2

(a)–(c) Spin coherence, mapped onto population $P_n$, as a function of $n$. The interpulse delay time $t_1$ is fixed: (a) $t_1=242$ ns, $B=635$ G; (b) $t_1=456$ ns, $B=528$ G; and (c) $t_1=585$ ns, $B=208$ G. (d) NSD and coupling with nearby nuclei. We report $1/T_2^L$ (blue, left-hand side vertical scale) and the amplitude of the observed coherent modulations (red, right-hand side vertical scale), for different magnetic field strengths ($B=208–635$ G). Blue lines are the fit of high-order harmonics to extract the NSD, and red lines are the simulations of the completely resolved nearby carbons (see text). The red shadow describes the uncertainty on the estimation of the coupling strength components. The coupling strength to the third carbon is too weak to be distinguishable from the zeroth-order collapse, thus it has been extracted from higher-order harmonics (see Fig. III.S in Ref. [31]).

Figure 3

$1/T_2^L$ and $S\left(\omega \right)$. [(a) and (b)] Dots are experimental $1/T_2^L$ values peaked (a) at ${\omega }_L/5$ ($l=1$) and ${\omega }_L/3$ ($l=2$), and (b) at ${\omega }_L$ ($l=0$), with $B=635\left(1\right)$ G. The blue line is a Gaussian fit of harmonics $l=1,2$, from which we extract $S\left(\omega \right)$ (red dashed line, peaked at ${\omega }_L$). [(c) and (d)] Reconstruction of a model NSD, used to proof self-consistency of the method. The blue squares are $1/T_2^L$ values resulting from the fit of the simulated coherence. The solid lines are the Gaussian fits to the harmonics of orders $l=1,2$ (blue) and $l=0$ (green). The red dashed line is the original model NSD.

Figure 4

NSD peak across the quantum-to-classical spin bath transition. Peak position (a), width (b), and area (c), are obtained from a Gaussian fit of the measured NSD, and reported as a function of the magnetic field strength. The blue line in (a) is a linear fit, consistent with the expected coupling frequency for a ${}^{13}\mathrm{C}$ spin bath. The vertical dashed line represents a guide for the eye indicating a sudden change of the bath behavior—where the condition $R\gg 1$ is no longer fulfilled. The gray horizontal region in (c) denotes the mean value $\overline{A}$ and standard deviation $\sigma$ of the NSD area for $B>150$ G. Values at field $B\le 150$ G deviate from $\overline{A}$ by $>6\sigma$. Inset: mean squared residuals of the experimentally observed coherence with the simulation obtained from the measured NSD (squares) and with the two-model simulation (triangles). Each point results from several datasets collected under different controls [31].

Figure 5

Time evolution of spin coherence under different DD sequences. Dots are experimental data with statistical error. Red lines are the predicted coherence simulated using the measured environment, with no free parameters. (a) Four repetitions of XY-8 ($n=32$). (b) UDD with $n=32$. (c) AXY-8, with $n=40$ pulses at positions $t_{ij}=\frac{T}{80}(10i+j-8)$, with $i=1,\cdots ,8$ and $j=1,\cdots ,5$ [31]. (d-e) Low-field results for (d) spin-echo and (e) Uhrig $n=32$. The blue dashed line is obtained from the NSD best fit to a set of multipulse sequences (Uhrig, AXY-4, and AXY-8). Mean squared residuals of the experiments with simulation are summarized in the inset of Fig. 4.