Efficient and robust signal sensing by sequences of adiabatic chirped pulses

We propose a scheme for sensing of an oscillating field in systems with large inhomogeneous broadening and driving field variation by applying sequences of phased, adiabatic, chirped pulses. These act as a double filter for dynamical decoupling, where the adiabatic changes of the mixing angle during the pulses rectify the signal and partially remove frequency noise. The sudden changes between the pulses act as instantaneous $\pi$ pulses in the adiabatic basis for additional noise suppression. We also use the pulses' phases to correct for other errors, e.g., due to nonadiabatic couplings. Our technique improves significantly the coherence time in comparison to standard XY8 dynamical decoupling in realistic simulations in NV centers with large inhomogeneous broadening. Beyond the theoretical proposal, we also present proof-of-principle experimental results for quantum sensing of an oscillating field in NV centers in diamond, demonstrating superior performance compared to the standard technique.

Figure 1

Scheme for AC magnetometry with RAP pulses, following the Allen-Eberly (AE) model [57]. The pulses can be phase-shifted for improved performance. Our goal is to sense an oscillating signal with ${\omega }_s=\pi /(T_{\text{pulse}}+\tau )$. The RAP pulses perform population transfer and modulate the signal. The transition time $T_{\text{tr}}$ characterizes the time scale of population transfer. When $T_{\text{tr}}\ll T_{\text{pulse}}+\tau$, its effect can be neglected and the rectified signal takes the bottom shape. Usually, it is advantageous to take $\tau =0$ and increase $T_{\text{pulse}}$ to match ${\omega }_s$, keeping $T_{\text{tr}}$ constant, as this improves adiabaticity without affecting the modulation function.

Figure 2

(a) RAP mechanism. (Top) Rabi frequency and detuning for the AE model with ${\mathrm{\Omega }}_0=2\pi \times 10\mathrm{MHz}$, chirp range $R=2\pi \times 50\mathrm{MHz}$, and $T_{\text{pulse}}=10T=5\mu \mathrm{s}$. (Middle) Eigenenergies in the bare (dashed lines) and adiabatic (solid lines) bases. The composition of the adiabatic states changes, leading to population transfer in the bare basis due to a level crossing. (Bottom) Simulation of population transfer, characterized by the transition time $T_{\text{tr}}=1/\left|{\nu }^{\prime }\left(t_c\right)\right|=4T{\mathrm{\Omega }}_0/R=0.4\mu \mathrm{s}$. Scheme for sensing of the amplitude of an AC field with (b) standard XY8 and (c) RAP-XY8. In both, the atoms are prepared initially in $|1_y\rangle$. Corresponding numerical simulations of the population in the $|1_y\rangle$, observed directly in the bare basis at time intervals of $8\mu \mathrm{s}$ with DD by (d) XY8 with rectangular pulses with $\mathrm{\Omega }\left(t\right)={\mathrm{\Omega }}_0=2\pi \times 10\mathrm{MHz}$, duration $T_{\text{pulse}}=50$ ns and pulse separation $\tau =0.95\mu \mathrm{s}$, and (e) RAP-XY8 with ${\mathrm{\Omega }}_0=2\pi \times 10\mathrm{MHz}$, a target chirp range $R=2\pi \times 95\mathrm{MHz}$, characteristic time $T=0.2\mu \mathrm{s}$, pulse duration $T_{\text{pulse}}=1\mu \mathrm{s}$, and pulse separation $\tau =0$. The sensed field has an amplitude of $g=2\pi \times 4.34\mathrm{kHz}$, initial phase $\xi =0$ and angular frequency ${\omega }_s=2\pi \times 0.5\mathrm{MHz}$. The peak Rabi frequency is the same in both protocols and $T_{\text{pulse}}+\tau =\pi /{\omega }_s$. The slight delay in the ideal, theoretical gray curve with RAP from $p=cos{\left(\chi \left(t\right)\right)}^2$ is mainly due to the noninstantaneous transition time, which is taken into account in the simulation.

Figure 3

(a) Energy level diagram of the NV center. The NV is excited with off-resonant green light (532 nm), coherently manipulated using microwave fields, and optically read-out through spin-dependent fluorescence between states $\left|0\right.〉$ and $\left|\pm 1\right.〉$. (b) Experimental setup scheme. (c) Measurement of population transfer by a RAP pulse with ${\mathrm{\Omega }}_0=2\pi \times 5\mathrm{MHz}$, duration $T_{\text{pulse}}=11.4\mu \text{s}$, a target chirp range $R=2\pi \times 40\mathrm{MHz}$, and characteristic time $T/T_{\text{pulse}}=0.17$. The transition time is short compared to the pulse duration ($T_{tr}\approx 0.07T_{\text{pulse}}$), as expected from theory. The time between the dashed lines depicts the theoretical transition time $4{\mathrm{\Omega }}_{0}T/R$ (see Appendix pp1-s7). Coherence decay curves in presence of an artificial external magnetic AC field with amplitude of 52 nT and frequency of $14.5\mathrm{kHz}$ for the low Rabi frequency experiment ${\mathrm{\Omega }}_0=2\pi \times 1.7\mathrm{MHz}$ (see Appendix pp8) for (d) XY8 and (e) for RAP-XY8. The estimate of the oscillation period allows for sensing of the field amplitude (see text). The error terms for the estimates of $T_2$ and $\eta$ are their standard deviations. (f) Sensitivity vs. number of cycles, corresponding to the number of pulses, (g) Direct measurement of sensitivity experiment: variation of fluorescence vs. amplitude of the sensed magnetic field for a direct measurement of the sensitivity at its optimal points, marked with a circle in (f), i.e., for 16 (40) cycles for XY8 (RAP-XY8). We fit to a $sin$ function with the steeper curve slope at zero field with RAP-XY8 showing improved sensitivity; the slightly lower contrast with RAP-XY8 is due to the higher pulse number.

Figure 4

Numerical simulation of an example for population transfer with RAP for three different detuning errors ${\mathrm{\Delta }}_{\epsilon }\left(t\right)=2\pi \mathrm{\Delta }\epsilon$ (blue), 0 (green), $-\mathrm{\Delta }\epsilon$ (red) with $\mathrm{\Delta }\epsilon =2\pi \times 15\mathrm{MHz}$. (a) Time dependence of the respective parameters, used in the simulation: Rabi frequency $\tilde{\mathrm{\Omega }}\left(t\right)={\mathrm{\Omega }}_0\text{sech}(t/T)$, where the peak Rabi frequency ${\mathrm{\Omega }}_0=2\pi \times 10\mathrm{MHz}$, detuning $\tilde{\mathrm{\Delta }}\left(t\right)=\mathrm{\Delta }\left(t\right)+{\mathrm{\Delta }}_{\epsilon }\left(t\right)$ with $\mathrm{\Delta }\left(t\right)=(R/2)tanh(t/T)$, where $R=2\pi \times 50\mathrm{MHz}$ is the target chirp range, $T=0.5\mu \mathrm{s}$. Time evolution of (b) the mixing angle $\tilde{\nu }\left(t\right)$, (c) the population of the bare state $|1\rangle$, (d) the population of the bare state $|2\rangle$ for the respective values of ${\mathrm{\Delta }}_{\epsilon }\left(t\right)$. The population transfer efficiency is very robust to detuning errors even though they are greater than the peak Rabi frequency but the flip of the quantum state happens at different times for each value of ${\mathrm{\Delta }}_{\epsilon }\left(t\right)$, i.e., the transfer process is centered at the time when the respective mixing angle is $\tilde{\nu }\left(t\right)=\pi /4$.

Figure 5

Numerical simulation of single RAP pulse transition probability vs. peak Rabi frequency ${\mathrm{\Omega }}_0$ and target chirp range $R$. We assume a two-state quantum system interacting with a driving field with a Rabi frequency $\tilde{\mathrm{\Omega }}\left(t\right)={\mathrm{\Omega }}_0\text{sech}\left(\frac{t-t_c}{T}\right)$ and detuning $\tilde{\mathrm{\Delta }}\left(t\right)={\mathrm{\Delta }}_{\epsilon }+(R/2)tanh\left(\frac{t-t_c}{T}\right)$ with pulse duration $T_{\text{pulse}}=1\mu \mathrm{s}$ and characteristic time $T=0.1\mu \mathrm{s}$ for (left) zero static detuning ${\mathrm{\Delta }}_{\epsilon }=0$ and (right) ${\mathrm{\Delta }}_{\epsilon }={\mathrm{\Omega }}_0$. The black dotted/dashed/solid lines correspond to $p$ of $0.5/0.95/0.99$, respectively. The vertical, white, dashed lines correspond to pulse areas of $\pi$, $3\pi$, and $5\pi$. The top solid white line corresponds to the adiabaticity parameter $\tilde{\mathrm{\Omega }}{\left(t_{\text{c}}\right)}^2/\left|\dot{\stackrel{̃}{\mathrm{\Delta }}}\left(t_{\text{c}}\right)\right|=3.33$ [see Eq. (A27)]. The bottom solid line corresponds to (a) the chirp range condition $Rsinh(T_{\text{pulse}}/2T)={\mathrm{\Omega }}_0\sqrt{\frac{2}{{\epsilon }_{\text{max}}}-4}$ for ${\epsilon }_{\text{max}}=0.01$ [see Eq. (A22)] and (b) $R=2{\mathrm{\Delta }}_{\epsilon }$. As expected from theory, the good performance of RAP pulses takes place in the region between the white solid lines and can be achieved even for moderate pulse areas.

Figure 6

Example of transition time $T_{\text{tr}}$ in RAP. We perform a numerical simulation of a two-state quantum system interacting with a driving field with a Rabi frequency $\tilde{\mathrm{\Omega }}\left(t\right)={\mathrm{\Omega }}_0\text{sech}\left(\frac{t-t_c}{T}\right)$, where the peak Rabi frequency ${\mathrm{\Omega }}_0=2\pi \times 10\mathrm{MHz}$ and $T=0.5\mu \mathrm{s}$, detuning $\tilde{\mathrm{\Delta }}\left(t\right)=(R/2)tanh\left(\frac{t-t_c}{T}\right)$, where $R=2\pi \times 50\mathrm{MHz}$ is the target chirp range, the initial and end times are $t=0$ and $t=T_{\text{pulse}}=5\mu \mathrm{s}$. We show the time evolution of (green) the transition probability $p\left(t\right)$ in the bare states basis, (red) the function $|{\tilde{\nu }}^{\prime }\left(t_c\right)|(t-t_c)$, where $t_c=2.5\mu \mathrm{s}$ is the center of the pulse. The transition time is $T_{\text{tr}}=4{\mathrm{\Omega }}_{0}T/R=0.4\mu \mathrm{s}$. As expected from theory, the transition probability at $t=t_c+T_{\text{tr}}/2$ is $p=(2+\sqrt{2})/4\approx 0.854$.

Figure 7

Scheme for sensing with (a) RAP-X and (b) RAP-XY8. In both, the atoms are prepared initially in $|1_y\rangle$. Corresponding numerical simulations of the population in the $|1_y\rangle$, observed directly in the bare basis at time intervals of $8\mu \mathrm{s}$ with DD by (c) RAP-X and (d) RAP-XY8, where the only difference between the two are the phases of the poulses, which have the following characteristics for both simulations: ${\mathrm{\Omega }}_0=2\pi \times 10\mathrm{MHz}$, a target chirp range $R=2\pi 95\mathrm{MHz}$, characteristic time $T=0.2\mu \mathrm{s}$, pulse duration $T_{\text{pulse}}=1\mu \mathrm{s}$, and pulse separation $\tau =0$. The sensed field has an amplitude of $g=2\pi \times 4.34\mathrm{kHz}$, initial phase $\xi =0$ and angular frequency ${\omega }_s=2\pi \times 0.5\mathrm{MHz}$. The peak Rabi frequency is the same in both protocols and $T_{\text{pulse}}+\tau =\pi /{\omega }_s$.

Figure 8

Numerical simulations of the population in the $|1_y\rangle$, observed directly in the bare basis at time intervals of $8\mu \mathrm{s}$ with DD by (a) KDD in XY8 pulse sequence with rectangular pulses with $\mathrm{\Omega }\left(t\right)={\mathrm{\Omega }}_0=2\pi \times 10\mathrm{MHz}$, duration $T_{\text{pulse}}=50$ ns and pulse separation $\tau =0.95\mu \mathrm{s}$, and (b) RAP-XY8 with ${\mathrm{\Omega }}_0=2\pi \times 10\mathrm{MHz}$, a target chirp range $R=2\pi \times 95\mathrm{MHz}$, characteristic time $T=0.2\mu \mathrm{s}$, pulse duration $T_{\text{pulse}}=1\mu \mathrm{s}$, and pulse separation $\tau =0$. The sensed field has an amplitude of $g=2\pi \times 4.34\mathrm{kHz}$, initial phase $\xi =0$ and angular frequency ${\omega }_s=2\pi \times 0.5\mathrm{MHz}$. The peak Rabi frequency is the same in both protocols and $T_{\text{pulse}}+\tau =\pi /{\omega }_s$.

Figure 9

Numerical simulations of the population (a) in the $|1_z\rangle$ state with double drive continuous dynamical decoupling with the first drive with a Rabi frequency ${\mathrm{\Omega }}_1\left(t\right)={\mathrm{\Omega }}_0=2\pi \times 10\mathrm{MHz}$, and second drive ${\mathrm{\Omega }}_2\left(t\right)=0.1{\mathrm{\Omega }}_1\left(t\right)$, observed stroboscopically at time intervals of $2\mu \mathrm{s}$. The red (gray) curve shows the evolution with (without) inhomogeneous broadening, frequency and amplitude noise. The lower oscillating frequency in comparison to RAP-XY8 without noise is expected from theory as we sense the amplitude of the oscillating field in the second order interaction basis, where it is reduced [19]. (b) Sensing with RAP-XY8 with parameters described in Fig. 8.

Figure 10

(a) Scheme for sensing with preparation and readout for (a) the standard XY8 sequence of rectangular pulses and (b) the RAP-XY8 sequence of phase shifted, chirped, adiabatic pulses. In both schemes we assume that the atoms are initially in the ground state $|1_z\rangle$, so we need to prepare the system in a coherent superposition state, e.g., by a $\pi /2$ pulse, perform sensing and then readout. Corresponding numerical simulations of the population of the excited state $|1_{-z}\rangle$ in the bare basis for quantum sensing, observed stroboscopically directly in the bare basis at time intervals of $8\mu \mathrm{s}$ with dynamical decoupling with (c) XY8 with rectangular pulses, and (d) RAP-XY8 with chirped pulses. The experimental parameters are the same as in Fig. 3 in the main text with the only difference that we apply $\pi /2$ and half-RAP pulses for preparation and readout, respectively. The light gray curve shows the respective theoretical evolution in an ideal system without inhomogeneous broadening, frequency and amplitude noise. The red curve shows the evolution with inhomogeneous broadening, frequency and amplitude noise. We note that the slight delay in the ideal, theoretical curve with RAP pulses from $p=cos{\left(\chi \left(t\right)\right)}^2$, defined in Eq. (C9), is expected and due mainly to the noninstantaneous transition time, which is taken into account in the simulation. Both the coherence time and the contrast are much higher with the RAP-XY8 protocol.

Figure 11

RAP experimental optimization for the high Rabi experiment with peak Rabi frequency ${\mathrm{\Omega }}_0=2\pi \times 5\mathrm{MHz}$ and pulse duration $T_{\text{pulse}}=11.4\mu \text{s}$. (a) Fluorescence vs. ratio characteristic time $T/T_{\text{pulse}}$ for a single RAP pulse. The lower the fluorescence the more efficient is the population from $|0\rangle$ to $|1\rangle$ (see Fig. 3 in the main text). The chirp range is taken $R=2\pi \times 40\mathrm{MHz}$. (b) Optimization of the target chirp range $R$ for a characteristic time $T/T_{\text{pulse}}=0.17$ for the RAP-XY8 DD sequence, repeated once.

Figure 12

Fluorescence vs interaction time for XY8 and RAP-XY8 without an artificial AC field for (a) high Rabi experiment with a peak Rabi frequency ${\mathrm{\Omega }}_0=2\pi \times 5\mathrm{MHz}$ and $T_{\text{pulse}}+\tau =11.4\mu \mathrm{s}$. The rectangular pulses in standard XY8 have a duration $T_{\text{pulse}}=\pi /{\mathrm{\Omega }}_0=100\mathrm{ns}$, $\tau =11.3\mu \mathrm{s}$; for RAP-XY8: $T_{\text{pulse}}=11.4\mu \mathrm{s}$, $\tau =0$, target chirp range $R=2\pi \times 40\mathrm{MHz}$, characteristic time $T/T_{\text{pulse}}=0.17$. (b) high Rabi with noise: the same pulse parameters as in (a) plus added amplitude noise with $\sigma =0.2{\mathrm{\Omega }}_0$. The coherence time $T_2$ of standard XY8 decreases by $\sim 35\%$ in comparison to the high Rabi case while remaining approximately the same with RAP-XY8. (c) low Rabi experiment with peak Rabi frequency ${\mathrm{\Omega }}_0=2\pi \times 1.7\mathrm{MHz}$, $T_{\text{pulse}}+\tau =34.46\mu \mathrm{s}$. The rectangular pulses in standard XY8 have a duration $T_{\text{pulse}}=\pi /{\mathrm{\Omega }}_0=294\mathrm{ns}$, $\tau =34.166\mu \mathrm{s}$; for RAP-XY8: $T_{\text{pulse}}=34.46\mu \mathrm{s}$, $\tau =0$, target chirp range $R=2\pi \times 20\mathrm{MHz}$, characteristic time $T/T_{\text{pulse}}=0.19$. The coherence time $T_2$ of standard XY8 decreases by $\sim 40\%$ in comparison to the high Rabi case while remaining approximately the same with RAP-XY8. All experiments show fluorescence difference from two measurements in which the phase of the last $\pi /2$ or half RAP pulse is $X$ and $-X$. The error terms for the estimates of $T_2$ are given as $\pm$ one standard deviation.

Figure 13

Fluorescence vs interaction time for a quantum sensing experiment with (top) XY8 and (bottom) RAP-XY8 with an artificial AC field with a frequency corresponding to half the pulse repetition rate. The pulse parameters are the same as in Fig. 12. The magnitude and angular frequency of the artificial sensed field are respectively (a) $78\mathrm{nT}$ and ${\omega }_s=2\pi 43.8\mathrm{kHz}$, (b) $52\mathrm{nT}$ and ${\omega }_s=2\pi 43.8\mathrm{kHz}$, and (c) $52\mathrm{nT}$ and ${\omega }_s=2\pi 14.5\mathrm{kHz}$. The lower sensed frequency ${\omega }_s$ in the low Rabi case (c) is due to the slower repetition rate and longer duration of the RAP pulses, needed to maintain some minimum adiabaticity (see the Discussion section in the main text). The slight difference in the artificial field magnitudes does not affect significantly the coherence times and sensitivity. The coherence time $T_2$ of standard XY8 in the high Rabi with noise experiment decreases by $\sim 40\%$ in comparison to the high Rabi case while remaining approximately the same with RAP-XY8. The corresponding drop of $T_2$ of standard XY8 is $\sim 45\%$ for the low Rabi case while again remaining approximately the same with RAP-XY8. All experiments show fluorescence difference from two measurements in which the phase of the last $\pi /2$ or half RAP pulse is $X$ and $-X$. The error terms for the estimates of $T_2$ are given as $\pm$ one standard deviation.

Figure 14

Sensitivity vs number of cycles, where each cycle consists of free evolution for time $\tau /2$ before a pulse—the pulse duration—free evolution for time $\tau /2$ after the pulse. Thus the number of cycles corresponds to the number of applied pulses. The free evolution time is zero for these examples of RAP sensing. The pulse parameters and characteristics of the artificial AC field as in Fig. 13. The error terms for the estimates of $\eta$ are given as $\pm$ one standard deviation.