Pulse-phase control for spectral disambiguation in quantum sensing protocols

We present a method to identify spurious signals generated by finite-width pulses in quantum sensing experiments and apply it to recently proposed dynamical decoupling sequences for accurate spectral interpretation. We first study the origin of these fake resonances and quantify their behavior in a situation that involves the measurement of a classical magnetic field. Here we show that a change of the initial phase of the sensor or, equivalently, of the decoupling pulses leads to oscillations in the spurious signal intensity while the real resonances remain intact. Finally we extend our results to the quantum regime for the unambiguous detection of remote nuclear spins by utilization of a nitrogen vacancy sensor in diamond.

Figure 1

(a) Visualization of the actual control axis at the present of signal fields on the Bloch sphere. The driving field $\mathrm{\Omega }$ in the $x-y$ plane and the ac field ${\mathrm{\Omega }}_{ac}^j$ parallel to the $z$ axis add to a total driving field along ${\widehat{n}}_j$, which set the angle ${\beta }_j$ out of the $x-y$ plane. (b) Locations of the $X$ [blue (dark grey)] and $Y$ [orange (light grey)] pulses with respect to the ac field for $\alpha =1$ and $\theta =0$. The height of each square pulse is proportional to the tilting angle $\beta$. Panels (c) and (d) are the illustrations similar to (b) but for $\alpha =2$ and $\alpha =4$, respectively.

Figure 2

(a) Impact of the second order in the tilting angle for the XY-8 sequence as a function of $1/\alpha$ and the phase angle $\varphi$. The cuts indicated by the dashed lines have different values of $\alpha$ and are shown in (b), where we compare the analytic results (solid lines) of Eqs. (7), (9), and (10) with a numerical simulation (dots) of the behavior of the transition probability $P$ under the Hamiltonian given in Eq. (1) for ${\omega }_{\text{ac}}=2\pi \times 1\mathrm{MHz}, {\gamma }_{n}B=2\pi \times 0.12\mathrm{MHz}, \theta =0$, and ${\beta }_{\text{max}}=0.012$.

Figure 3

Simulation of $P_{\varphi }$ for a NV center coupled to a single spin employing the XY-8 [blue (solid dark grey)] and AXY-8 [orange (solid light grey)] sequences and different initial phases of $\pi /4,0,-\pi /4$ in each row from top to bottom. The graphs in column (a) show the large resonance for $\alpha =1$ and the smaller $\alpha =4/5$ resonance. In columns (b)–(d) the spurious peaks for $\alpha =4/3,2,4$ are displayed respectively. The $\alpha =1$ resonance corresponds to a sensing time of $T\approx 2.4\mathrm{ms}$. The green (dark grey) dashed curves represent an XY-8 sequence but with no error in the Rabi frequency.

Figure 4

Simulation of $P_{\varphi }$ for an NV center with a hydrogen and a strongly coupled carbon spin under the control of AXY sequences. Panels (a) and (b) show results for $f_1=4/\left(1.2\pi \right)$ and $f_3=4/\left(1.2\pi \right)$ respectively. The arrow in (b) indicates the position of the hydrogen resonance, while the arrow in (c) marks the ${}^{13}\mathrm{C}$ resonance. Panels (b) and (c) show the effect when changing the initial phase from 0 to $\pi /4$. All calculations are performed with $\mathrm{\Delta }=2\pi \times 1\mathrm{MHz}, \mathrm{\Omega }=2\pi \times 20\mathrm{MHz}$, and $\delta =0.03\times \mathrm{\Omega }{t}_{\text{flip}}$. Here, 1920 decoupling pulses are used which corresponds to a sensing time $T\approx 23.9\mu \mathrm{s}$ for the $\alpha =1$ and $T\approx 71.7\mu \mathrm{s}$ for the $\alpha =1/3$ resonance of the hydrogen atom.

Figure 5

Impact of the second order after three applications of the XY-8 sequence. The values for the figure are calculated using Eq. (B2)