Optimal Control for One-Qubit Quantum Sensing

Quantum systems can be exquisite sensors thanks to their sensitivity to external perturbations. This same characteristic also makes them fragile to external noise. Quantum control can tackle the challenge of protecting a quantum sensor from environmental noise, while strongly coupling the sensor with the field to be measured. As the compromise between these two conflicting requirements does not always have an intuitive solution, optimal control based on a numerical search could prove very effective. Here, we adapt optimal control theory to the quantum-sensing scenario by introducing a cost function that, unlike the usual fidelity of operation, correctly takes into account both the field to be measured and the environmental noise. We experimentally implement this novel control paradigm using a nitrogen vacancy center in diamond, finding improved sensitivity to a broad set of time-varying fields. The demonstrated robustness and efficiency of the numerical optimization, as well as the sensitivity advantage it bestows, will prove beneficial to many quantum-sensing applications.

Popular Summary

Quantum systems are sensitive to external perturbations, which makes them precise sensors but also leaves them vulnerable to undesired noise. Optimal control—a mathematical approach to finding the most favorable control algorithm—has great potential for tackling the challenge of quantum sensing. However, it needs to be redefined to strongly couple the sensor to the field to be measured while addressing the conflicting task of noise protection. We have designed a new optimal control algorithm for quantum sensing—based on the unconventional metric of sensitivity—and we experimentally demonstrate that it improves the performance of a nitrogen-vacancy (NV) spin sensor.

We address the problem of detecting and measuring the amplitude of ultraweak, time-varying magnetic signals in noisy environments. This challenging task is critical in many applications, such as investigations of molecular nanomagnetism, as well as medical diagnostics, including monitoring neuron or cardiac activity. The intrinsic frequency selectivity of standard control methods entails significant signal loss. We show that optimal control significantly boosts sensitivity, enabling larger accumulation of the spin phase in the NV sensor that encodes the field information as well as improving shielding to environment-induced decoherence.

Our strategy may find broad applications in the face of ever-more-demanding requirements for NV spin sensors, and it can also be applied to other physical platforms. The enhanced protection of the qubit against decoherence also makes optimal control a strategic tool for solid-state memories.

Figure 1

One-qubit optimization strategy. (a) The electronic spin of a single NV center is optically initialized in the $|0⟩$ state and read out after the sensing period by means of a confocal microscope. An antenna delivers both the resonant control field (mw) and the target magnetic field $b(t)$ to be measured, in the proximity of the spin qubit. Each measurement shot is repeated $N=2\times {10}^5$ times (see also Appendix pp1-s1). (b) Illustration of the optimization protocol. The starting point is an initial guess for the control sequence described by a modulation function $y_n^0(t)$, which depends on a given number of parameters. While in the paper we consider more general cases, for the sake of simplicity, the central panel shows a search in a two-parameter space (sensing time $T$ and phase shift $\alpha$) for a Carr-Purcell ($\text{CP}$) control sequence used to detect a monochromatic ac field $b(t)=b\cos (2\pi {\nu }_{0}t+\alpha )$, with frequency ${\nu }_0=20.5\mathrm{kHz}$ and unknown amplitude $b$ to be measured. The map represents the experimentally measured inverse sensitivity $\mathcal{E}=C/\eta$ (see text). The algorithm computes the sensitivity $\eta$ under the initial control sequence and then produces and evaluates a number of other trial points $y_n^{(i)}(t)$ and moves in the multidimensional parameter space until global convergence is reached. The final point, described by $y_n^{\mathrm{opt}}$, represents the optimal control sequence.

Figure 2

Optimized sensing of monochromatic fields. (a) Experimental signal measured in the presence of a monochromatic target ac magnetic field, $b(t)=b\cos (2\pi \nu t+\alpha )$, with $\nu =9.24\mathrm{kHz}$ and $\alpha =0$, as a function of the target magnetic field amplitude $b$. Here, the spin sensor is controlled with a Carr-Purcell sequence of $n=4$ equidistant $\pi$ pulses. Dots are the experimental data, the curve is a cosinusoidal fit. Error bars are the statistical errors over $2\times {10}^5$ repeated measurements. (b) Theoretical prediction of the optimal sensing time of a Carr-Purcell sequence of $n=8$ equidistant $\pi$ pulses ($\text{CP}\text{−}8$), calculated for a target ac magnetic field $b(t)=b\cos (2\pi \nu t+\alpha )$, as a function of the ac frequency $\nu$. The black empty squares are the optimized solutions of the sensing problem when neglecting the presence of the noisy environment, and the black curve represents the expected optimal time, $T_{\mathrm{opt}}=n/(2\nu )$, with no fitting parameters. Purple dots are optimized solutions including the noise-induced decoherence of the spin qubit (the line is a guide to the eye). (c) Inverse of sensitivity, in the presence of a cosinusoidal field $b(t)$ of frequency ${\nu }_0=20.5\mathrm{kHz}$ under $\text{CP}$ control, with $\alpha =0$ (in red), and with $\alpha =102°$ (that is, an initial delay time $t_0=0.28/{\nu }_0$) resulting from optimization (in blue). Dots are experimental value of $\mathcal{E}$ (left side vertical axis—see text), lines are theoretical $1/\eta$ (right side vertical axis). The experimental error bars come from the slope uncertainty of the experimental signal $s$.

Figure 3

Optimized sensing of multitone ac fields. (a) Upper panel: Sample multitone target field, $b(t)=b\sum _i^3{w}_i\cos (2\pi {\nu }_{i}t+{\alpha }_i)$, with ${\alpha }_i=0$, frequencies ${\nu }_i=(77;96;141)\mathrm{kHz}$, and amplitudes $w_i=(0.12;0.43;0.45)$, respectively. Bottom panel: In green, position of the first 19 $\pi$ pulses of a Carr-Purcell sequence of 50 equidistant $\pi$ pulses ($\text{CP}\text{−}50$) with optimized sensing time ($T=260\mu \mathrm{s}$); in blue, position of the first 27 $\pi$ pulses of an optimal control sequence of 50 non-equally-distributed $\pi$ pulses with optimized time intervals and optimized initial phase ($T=187\mu \mathrm{s}$, ${\alpha }_i=0.3$). (b) Phase ${\varphi }_n(T)={\phi }_n(T)/b$ accumulated by the spin qubit sensor during the sensing time $T$ in the presence of the field $b(t)$, under a control field of $n=50$ $\pi$ pulses, in the cases of $\text{CP}\text{−}50$ (solid line) and optimized control (blue squares). (c) Experimentally measured $\mathcal{E}=C/\eta$ in the presence of the field $b(t)$ under $\text{CP}\text{−}50$ (dots) and optimized control (diamonds). The curves represent the theoretical prediction for $\mathcal{E}$, for $\text{CP}\text{−}50$ (solid green line) and optimized control (blue line), respectively, obtained by rescaling $1/\eta$ (right-hand side, vertical scale) with the unique factor $C$. The shaded area takes into account the experimental uncertainty due to $C$ (see Appendix pp1-s4).

Figure 4

Optimal sensing of Gaussian impulses. (a) Upper panel: Target signal made of a train of Gaussian impulses of width $\sigma$ and repetition rate $1/\mathrm{\Delta }t$. Bottom panel: Position of the first $10$ pulses of a $\text{CP}\text{−}50$, with total sensing time $T=280\mu \mathrm{s}$ optimized to sense a train of Gaussian impulses with $\sigma =2\mu \mathrm{s}$, $\mathrm{\Delta }t=11.2\mu \mathrm{s}$ (in green), and optimal position of the first $10$ pulses of a control sequence of $50$ pulses (in blue), optimized to sense the same target field (50 optimization parameters). (b)–(i) Modulus of the phase accumulated by the spin qubit sensor in the presence of the target field (left panels), and the inverse of sensitivity $1/\eta$ (right panels), as a function of the sensing time $T$, under $\text{CP}\text{−}50$ control (green solid curves) and under optimized control (blue lines with squares) with bounds $\tau \in (0.6–10)\mu \mathrm{s}$ (see Appendix pp1-s3). (c) Measurement of the experimental observable $\mathcal{E}$ as resulting from $\text{CP}$ experiments (yellow dots) and from optimized control (blue diamonds), scaling as indicated on the right-hand side, vertical axis. The target field parameters of panels (b)–(i) are as follows: (b,c) $\sigma =1.0\mu \mathrm{s}$, $\mathrm{\Delta }t=10\mu \mathrm{s}$; (d,e) $\sigma =0.5\mu \mathrm{s}$, $\mathrm{\Delta }t=11.2\mu \mathrm{s}$; (f,g) $\sigma =1.0\mu \mathrm{s}$, $\mathrm{\Delta }t=11.2\mu \mathrm{s}$; (h,i) $\sigma =2.0\mu \mathrm{s}$, $\mathrm{\Delta }t=11.2\mu \mathrm{s}$.

Figure 5

NV spin coherence under the $\text{CP}$ sequence. Measured signal $s(T)$ (gray dots) after a $\text{CP}$ decoupling sequence with $n=50$, as a function of time $T=n\tau$. Here, the sensor is not irradiated with any external target field $b(t)$ to be measured. The solid red line is a fit of the signal based on a classical model of the bath (see text).

Figure 6

Time interval ${\tau }_j$ engineering. Optimization scheme of the $n$ time intervals ${\tau }_j$ of a measurement sequence with $n$ pulses. Here, ${\tau }_j=(t_j+t_{j+1})/2$, where $t_j$ are the $n+1$ time intervals between the $\pi$ pulses, with $j=0,\dots ,n$, and $t_0=t_{n+1}=0$.