Continuous dynamical decoupling magnetometry

Solid-state qubits hold the promise to achieve an unmatched combination of sensitivity and spatial resolution. To achieve their potential, the qubits need, however, to be shielded from the deleterious effects of the environment. While dynamical decoupling techniques can improve the coherence time, they impose a compromise between sensitivity and the frequency range of the field to be measured. Moreover, the performance of pulse sequences is ultimately limited by control bounds and errors. Here we analyze a versatile alternative based on continuous driving. We find that continuous dynamical decoupling schemes can be used for ac magnetometry, providing similar frequency constraints on the ac field and improved sensitivity for some noise regimes. In addition, the flexibility of phase and amplitude modulation could yield superior robustness to driving errors and a better adaptability to external experimental scenarios.

Figure 1

Pulse sequences for four ac magnetometry schemes: PDD (P), constant driving (C), RE with optimal frequency (R${}_k^{\text{opt}}$), and spin locking (S). Blue boxes represent microwave driving, with phase ($x$ and $y$) as indicated.

Figure 2

Bandwidth for ac magnetometry. We plot the weight functions $W\left(\omega \right)$ that scale the phase acquired during DD magnetometry for ac fields of frequency $\omega$. Left: $W\left(\omega \right)$ for PDD (blue dotted), RE ($k=1$, red thick), and constant driving (green, dashed line) for $n=2$ cycles, expressing the frequency in terms of the sequence period. We also plot the envelope of the pass-band decay for PDD (blue thin) and RE (red thin), given by $\sim 1/\omega$ and $\sim {\Omega }^2/|{\omega }^2-{\Omega }^2|$ respectively. In the inset: we compare the main peak for $n=1$ (red, thick) and $n=10$ (gray) for RE ($k=1$) showing the reduction in bandwidth. Right: we compare $W\left(\omega \right)$ for continuous driving (green, dotted) and for RE with $k=1$ (red, thick) and $k=4$ (gray, dashed), plotting as a function of $\omega$ in units of the Rabi frequency $\Omega$. The thin lines represent the pass-band decay envelopes for RE.

Figure 3

Sensitivity for ac magnetometry. We compare the magnetic field sensitivity of a single NV center for PDD (left) and RE ($k=1$ center; $k=4$ right). We assumed $T_2=500\mu$s under OU noise (comparable to a ${}^{13}$C bath), yielding a decay $\propto e^{-T^3/\left(n^2{T}_2^3\right)}$, and a single readout with $\mathcal{C}=0.03$. A larger number of refocusing cycles (with shorter periods) achieves better sensitivity but can only detect higher frequencies, as shown by the color of the curves (right bar, MHz).

Figure 4

Sensitivity for ac magnetometry. We compare the achievable sensitivity for constant driving (green, dash-dotted), PDD of $n=50$ echoes (blue, dotted) and RE ($2\pi$-RE –red, dashed– achieving the same sensitivity of PDD at a lower frequency and $8\pi$ RE –black– achieving better sensitivity than PDD at the same frequency). The chosen cycle number is experimentally achievable in the NV center system [18, 39]. The period $T$ is adjusted to match the bandwidth with the frequency of the fields. We assumed $T_2=500\mu$s under OU noise, yielding a superexponential decay $\propto e^{-T^3/\left(n^2{T}_2^3\right)}$, and a readout with $\mathcal{C}=0.03$. The decay of the constant (Rabi) driving was calculated following Ref. [36] for long ${\tau }_c$. In addition, the dashed, thin lines correspond to the optimal sensitivity at each frequency for each one of the sequences considered, obtained by optimizing the cycle time $n$ (here we assumed no driving or pulse errors).