Reconstructing the profile of time-varying magnetic fields with quantum sensors

Quantum systems have shown great promise for precision metrology thanks to advances in their control. This has allowed not only the sensitive estimation of external parameters but also the reconstruction of their temporal profile. In particular, quantum control techniques and orthogonal function theory have been applied to the reconstruction of the complete profiles of time-varying magnetic fields. Here, we provide a detailed theoretical analysis of the reconstruction method based on the Walsh functions, highlighting the relationship between the orthonormal Walsh basis, sensitivity of field reconstructions, data compression techniques, and dynamical decoupling theory. Specifically, we show how properties of the Walsh basis and a detailed sensitivity analysis of the reconstruction protocol provide a method to characterize the error between the reconstructed and true fields. In addition, we prove various results about the negligibility function on binary sequences which lead to data compression techniques in the Walsh basis and a more resource-efficient reconstruction protocol. The negligibility proves a fruitful concept to unify the information content of Walsh functions and their dynamical decoupling power, which makes the reconstruction method robust against noise.

Figure 1

Control sequences derived from Walsh functions (in sequency ordering). The first sequence (Ramsey) describes the simplest parameter estimation protocol with no modulating $\pi$ pulses. The second sequence corresponds to the Hahn spin echo and, in general, the $m$th Walsh function requires $m$ $\pi$ pulses in the control sequence.

Figure 2

Negligibility [$p\left(m\right)$] and logarithm of the bound factor [$1-p\left(m\right)$] for the first 32 Walsh sequences in Paley ordering. PDD are in red and CPMG in green.

Figure 3

Negligibility for first $2^8$ natural numbers (red) compared with a constant negligibility of 9 (blue). Clearly most natural numbers have large negligibility and can potentially be ignored.

Figure 4

Bar plot of MSQE for the five different functions $f_1$ through $f_5$.

Figure 5

Log plot of first 32 Walsh coefficients for exp($-t$).

Figure 6

(a) Fifth-order reconstruction (partial sum with 32 terms) and exact function exp($-t$). (b) Threshold-sampled reconstruction (threshold negligibility = 6) and exact function. (c) Subsampled reconstruction and exact function.

Figure 7

First 32 Walsh coefficients for $f_4\left(t\right)=2+3\cos \left(2\pi t\right)+4\cos \left(4\pi t\right)+6\sin \left(2\pi t\right)+2\sin \left(4\pi t\right)$.

Figure 8

(a) Plot of the complete fifth-order reconstruction (partial sum with 32 terms) and exact function $f_4\left(t\right)=2+3\cos \left(2\pi t\right)+4\cos \left(4\pi t\right)+6\sin \left(2\pi t\right)+2\sin \left(4\pi t\right)$. (b) Plot of the threshold-sampled reconstruction (threshold negligibility = 6). (c) Plot of the subsampled reconstruction.

Figure 9

First 32 Walsh coefficients for $\sin \left(2\pi t\right)$.

Figure 10

(a) Plot of the complete Fifth-order reconstruction (partial sum with 32 terms) and exact function $\sin \left(2\pi t\right)$. (b) Plot of the threshold-sampled reconstruction (threshold negligibility = 6). (c) Plot of the subsampled reconstruction.