Bayesian estimation for quantum sensing in the absence of single-shot detection

Quantum information protocols, such as quantum error correction and quantum phase estimation, have been widely used to enhance the performance of quantum sensors. While these protocols have relied on single-shot detection, in most practical applications only an averaged readout is available, as in the case of room-temperature sensing with the electron spin associated with a nitrogen-vacancy center in diamond. Here, we theoretically investigate the application of the quantum phase estimation algorithm for high dynamic-range magnetometry, when single-shot readout is not available. We show that, even in this case, Bayesian estimation provides a natural way to efficiently use the available information. We apply Bayesian analysis to achieve an optimized sensing protocol for estimating a time-independent magnetic field with a single electron spin associated to a nitrogen-vacancy center at room temperature and show that this protocol improves the sensitivity over previous protocols by more than a factor of 3. Moreover, we show that an extra enhancement can be achieved by considering the timing information in the detector clicks.

Figure 1

A sketch of the estimation protocols. Previous work addressed implementations of the quantum phase estimation algorithm for magnetometry when single-shot readout is available, i.e., the measurements are close to projective quantum measurements (a). In this case, the outcome ($u=0,1$) of the qubit readout after each Ramsey experiment is used to update the current probability distribution $P\left(f_B\right)$ for the Larmor frequency $f_B$ through Bayes' rule. This scheme allows for real-time choice of the optimal setting for the readout basis, through the choice of $\theta$, based on the current $P\left(f_B\right)$. When single-shot readout is not available [case (b)], each individual interrogation of the qubit does not provide sufficient information to discriminate between the spin states $\left|0\right.〉$ and $\left|1\right.〉$. However, a Bayesian approach, compatible with real-time adaptation of the measurement basis is still possible.

Figure 2

Sensitivity $\eta$ as a function of the total measurement time, for different protocols, in the case where only “averaged” qubit readout is available. We consider the qubit dephasing time $T_2^{*}=1.3\mu \mathrm{s}$, and we set $K=6$, i.e., the interrogation time is changed between $2^6{\tau }_0$ and ${\tau }_0$, with ${\tau }_0=12.5\mathrm{ns}$. Blue lines with $\blacksquare$ and $\square$ correspond to the threshold protocols for $G=F=9$ and $G=15,F=1$, respectively. Orange line with $▵$ shows the Bayesian batching protocol with $G=15,F=1$, while green line with $\circ$ is the Bayesian protocol with single-measurement updating (SMU) for $R=700$ and $F=1$. The data points for the threshold and batching approaches are obtained by increasing the number of repetitions $R$ of each batch of Ramsey experiments. The data points of the Bayesian with single-measurement updating is obtained by keeping $R=700$ fixed and changing $G$.

Figure 3

(a) Optically detected magnetic-resonance experiments provide more information than the number of detected photons. When considering the arrival times of the detected photons, photon clicks corresponding to arrival times between 100 and 200 ns, for example, are more likely to be associated with a spin prepared in the state $\left|0\right.〉$. The photoluminescence signal is simulated based on a five-level model [37]. The blue line with $•$ (red line with $▴$) corresponds to the spin prepared in $\left|0\right.〉$ ($\left|1\right.〉$) state. (b) This additional information can also be included in Bayesian estimation by updating the probability distribution for the magnetic field $P\left(f_B\right)$ from the distribution of photon arrival times.

Figure 4

Sensitivity $\eta$ vs total measurement time. The orange lines with $▴$ and $▵$ show the batching approach with and without taking the arrival time information (t-info) of photons into account, respectively. For these lines we have set $K=6$ and $G=15,F=1$ while varying $R$, the number of repetitions. The green lines with $•$ and $\circ$ show the single-measurement updating (SMU) approach with and without the arrival time information, respectively. In this case $R=700$ and $F=1$ are kept fixed and $G$ is varied.