High-Dynamic-Range Imaging of Nanoscale Magnetic Fields Using Optimal Control of a Single Qubit

Abstract

We present a novel spectroscopy protocol based on optimal control of a single quantum system. It enables measurements with quantum-limited sensitivity ( √ denoting the system's coherence time) but has an orders of magnitude larger dynamic range than pulsed spectroscopy methods previously employed for this task. We employ this protocol to image nanoscale magnetic fields with a single scanning NV center in diamond. Here, our scheme enables quantitative imaging of a strongly inhomogeneous field in a single scan without closed-loop control, which has previously been necessary to achieve this goal.Optimal control of quantum systems is an experimental technique that has evolved over the two past decades [1-3] as a generalization of related techniques like composite pulses [4] or adiabatic control [5]. It implements unitary operations ("quantum gates") of very high fidelity by irradiating a quantum system with numerically optimized excitation pulses. Amplitude and phase of this pulse are an arbitrary function of time, which is tailored such as to result in a specific unitary operation.Numerical optimization can generate pulses that achieve near-perfect operation (i.e. high fidelity) over a wide range of experimental parameters, such as excitation power or detuning, rather than a single specific set. This is in contrast to simple (e.g. rectangular) pulses and arises from the fact that optimization has access to the much larger space of arbitrary amplitude and phase profiles. Thanks to these additional degrees of freedom, the resulting pulse can satisfy a larger number of constraints. In practice, this has been used to generate "robust" pulses which are immune against fluctuations of the excitation power, or pulses that implement a specific operation within a large bandwidth of different system frequencies [3], as they may arise e.g. by inhomogeneous broadening.Here we show that optimal control can be used to achieve an opposite goal, a pulse that is maximally sensitive to fluctuations of one experimental parameter (in our case the static magnetic field) while it preserves robustness against fluctuations of all other parameters and, in particular, a large operating bandwidth. With these properties, such a pulse enables sensitive spectroscopy of a system even in the presence of large unknown frequency offsets. The concept is illustrated in more detail in Fig. 1(a-b). Sensitive spectroscopy classically relies on sharp selective excitation ( Fig. 1(a)), realized for instance by a long low-power excitation pulse (Rabi spectroscopy) or a suitable pulse sequence (Ramsey spectroscopy). We extend these schemes by designing an optimal control pulse which generates a grating of equally sharp excitation lines, evenly spaced over a large bandwidth (Fig. 1(b)). With this protocol, small changes of the system's resonance frequency can be tracked without tuning the excitation pulse to the system's frequency.

Training Schrödinger’s cat: quantum optimal control

“…of NV centers in diamond, with many potential applications in sensing and quantum information processing [294][295][296]. Hyperpolarization techniques (also known as spin cooling), [297][298][299] can generate highly polarized non-thermal spin states.…”

Controlling open quantum systems: tools, achievements, and limitations

“…Protocols derived with optimal control have by now reached a maturity that allows them to be tested in experiments. Examples include the crossing of a phase transition studied with trapped, cold atoms [49,50]; the improvement of the imaging capabilities of a single NV center [51]; and the creation of spin entanglement [52], quantum error correction [37] and matter wave interferometry [50,53].…”

Optimizing phase-estimation algorithms for diamond spin magnetometry

“…However, a quantitative map of the magnetic field is impossible in a single run due to the restriction of a single contour 27 . In order to further illustrate the importance of dynamic range, we show in Fig.…”