Optimizing phase-estimation algorithms for diamond spin magnetometry

We present a detailed theoretical and numerical study discussing the application and optimization of phase-estimation algorithms (PEAs) to diamond spin magnetometry. We compare standard Ramsey magnetometry, the nonadaptive PEA (NAPEA), and quantum PEA (QPEA) incorporating error checking. Our results show that the NAPEA requires lower measurement fidelity, has better dynamic range, and greater consistency in sensitivity. We elucidate the importance of dynamic range to Ramsey magnetic imaging with diamond spins, and introduce the application of PEAs to time-dependent magnetometry.

Figure 1

Quantum metrology: (a) Mach-Zehnder interferometer senses the relative phase shift between two beams paths. (b) Quantum circuit representation of the process. (c) The analogous realization via Ramsey interferometry. The first $\pi /2$ pulse brings the spin vector from $|0\rangle$ to $\left(\right|0\rangle +|1\rangle )/\sqrt{2}$. The free precession interval introduces a phase $\varphi$ on $|1\rangle$ relative to $|0\rangle$. Measurement after the second $\pi /2$ pulse senses this phase with respect to the control phase $\Phi$.

Figure 2

Ramsey simulations: (a) Measurement fidelity: The histogram obtained with $N=10000$ measurement for both $|0\rangle$ and $|1\rangle$ states is shown. The lesser the overlap, the higher the fidelity in distinguishing the states. A sample number $\kappa =5{\kappa }_{th}$ leads to a fidelity $\approx 98.8\%$. (Inset) Fidelity improves with $\kappa$. (b) Simulation of Ramsey experiment: Ramsey signal after 50 averages when free time $t$ is varied while $B_{\mathrm{ext}}=142.9\mu$T is fixed (top) and as $B_{\mathrm{ext}}$ varied while $t=640$ ns is fixed (bottom). Here, $\kappa =5{\kappa }_{th}$ was used.

Figure 3

Ramsey imaging: (a)–(f) Calculated resonant imaging contours by fixing the detuning of the Ramsey sequence at a value expected from the field of a single electron (el), proton (p), or carbon-13 (C) spin. The linewidth of the sensor is set to be 30 pT which could be achieved with coherence time $T_2=100$ ms and averaging time $T=14$ s. The sample-probe height is 10 nm. The resonance condition will be met when the field from the target spins projected on the NV axis is within one linewidth $\delta B/{\gamma }_e$ of the contour value (Ramsey detuning), and the corresponding pixel in the image is shaded red to represent a dip in the fluorescence level. The contour images do not sufficiently reveal the existence of different spins. For instance, existence of carbon-13 is not revealed in (b) and (c), whereas the existence of proton is not revealed in (f).

Figure 4

Quantum circuit diagrams: (a) QPEA, (b) NAPEA for the case $K=4$. The dark lines represent the classical bits. The measured bit string in QPEA itself gives an estimate ${\varphi }_{\mathrm{est}}$ for the unknown phase with $u_1$ being the least significant bit. The control rotations $R\left(\Phi \right)$ in QPEA depend on preceding measurement results. In contrast, measurements in NAPEA are performed with a predefined set of control phases $\left\{\Phi \right\}$ and Bayes' method is used to obtain ${\varphi }_{\text{MLE}}$.

Figure 5

Phase likelihood distribution: (a) Phase likelihood distribution. (b) Distribution of MLE's. (c) Magnetic field readout of QPEA (right) with $\kappa =10{\kappa }_{th}$, $M_K=1$, $F=0$, $K=6$ and NAPEA (left) $\kappa =5{\kappa }_{th}$, $M_K=F=4$, $K=6$.

Figure 6

Control phases in NAPEA: (a) A simplified form of PEA with a single control phase $\Phi =\left\{0\right\}$ leads to a final phase likelihood distribution which is symmetric around “0” and consequently gives two possible MLE's. (b) An additional control phase can solve this problem by suppressing the wrong MLE. Red, green, and blue lines give the corresponding individual probability distributions for $k=1$, 2, and 3 measurement outcomes, respectively. Dashed lines correspond to the case of $\Phi ={90}^{\circ }$. (c) The red circle, green square, blue triangle, and black star represent the DUAL, QUAD, OCT, and VAR sets of control phases, respectively. DUAL and QUAD lead to similar results, and have large variance of the phase readout ${\left(\delta {\varphi }_{\text{MLE}}\right)}^2$ at points corresponding to $\varphi \sim 0$ or $\pm \pi /2$. The OCT suppresses this effect and thus will maintain almost a constant sensitivity over the full field range. The VAR leads to further improvement in consistency because of the rapid increment in the number of control phases according to the weighting scheme. The standard deviation of the phase readout variances at all points ${\sigma }_{{\left(\delta {\varphi }_{\text{MLE}}\right)}^2}$, corresponding to DUAL, QUAD, OCT, and VAR sets are 4.13, 4.28, 1.80, and 0.71 10${}^{-3}$ rad${}^2$, respectively.

Figure 7

Precision scaling of the phase measurement. (a) The NAPEA sensitivity ${\left(\delta {\varphi }_{\text{MLE}}\right)}^{2}N$ obtained as resource is increased by increment of $K$ for different system phases. (Inset) The average obtained for the above system phases is shown. (b)–(f) The precision scaling of average NAPEA (QPEA) phase sensitivity under different conditions of fidelity and PEA parameters are shown in (b) and (c) [(d)–(f)]. $F=0$ implies no weighting. The best results in QPEA require extremely high fidelity $>99\%$, whereas for NAPEA a fidelity $\approx 90\%$ is sufficient. The black dashed (solid) line shows standard-quantum (Heisenberg) limit for phase measurement.

Figure 8

Magnetic field sensitivity. (a) Field sensitivity ${\eta }^2={\left(\delta B\right)}^{2}T$ averaged over many different fields with the number of resources. The resources are increased by the increment of $K$ and given in units of time. The best cases obtained from QPEA and NAPEA are compared with the optimum Ramsey sensitivity averaged over its full field range. (b) Field sensitivity with unitless number of resources $N=T/t_{min}$. The simulation here was performed with the exact experimental conditions and is in good agreement with experimental results. (c) The NAPEA sensitivity is roughly constant throughout the full field range, whereas QPEA sensitivity is not. The red (blue) curves show theoretical limits of Ramsey sensitivity for $t=20$ ns $(t=600$ ns).

Figure 9

Dynamic range and time constants: (a) The standard Ramsey approach shows a tradeoff between the sensitivity and $B_{max}$ whereas PEA can maintain the same sensitivity up to a wide field range. (b) (Top) The total time for the NAPEA as a function of minimum evolution time interval $t_{min}$. The parameter $K$ is chosen such that the longest evolution time interval is always the same, 1280 ns. (Bottom) The corresponding dynamic range $\text{DR}=B_{max}/\delta B$ is shown, where $\delta B$ is the minimum detectable field change. The black dashed line is obtained from the Ramsey theoretical limit ${\eta }_{th}/\sqrt{T}$ for the same experimental conditions. Here, the total time for PEA with $M_K=F=8$ is used for $T$.

Figure 10

ac magnetometry with PEA: (a) Type-I (left) and type-Q (right) MW pulse sequence used for necessary phase accumulations from the ac magnetic field. (b) PEA readout with type-I (left) and type-Q (right) sequences. (c) Detection of the ac magnetic field phase.