High resolution quantum sensing with shaped control pulses

TL;DR

This paper demonstrates that amplitude-shaped control pulses significantly improve the temporal and spectral resolution of quantum sensing using nitrogen vacancy centers, enabling detection of weak magnetic signals with unprecedented precision.

Contribution

It introduces a simple method using cosine square envelope pulses to enhance quantum sensing resolution beyond hardware limitations.

Findings

Achieved 0.6 ps timing resolution for interpulse delay.

Enhanced spectral resolution in magnetic field and NMR detection.

Method is straightforward and applicable to fast quantum phase gates.

Abstract

We investigate the application of amplitude-shaped control pulses for enhancing the time and frequency resolution of multipulse quantum sensing sequences. Using the electronic spin of a single nitrogen vacancy center in diamond and up to 10,000 coherent microwave pulses with a cosine square envelope, we demonstrate 0.6 ps timing resolution for the interpulse delay. This represents a refinement by over 3 orders of magnitude compared to the 2 ns hardware sampling. We apply the method for the detection of external AC magnetic fields and nuclear magnetic resonance signals of carbon-13 spins with high spectral resolution. Our method is simple to implement and especially useful for quantum applications that require fast phase gates, many control pulses, and high fidelity.

Figures

Equations

$$\delta f=\frac{1}{2\tau }-\frac{1}{2\tau +2t_s}\approx 2t_s{f}_{\mathrm{ac}}^2.$$

$$p(t,\tau )$$

$$W_{N,\tau }(f_{\mathrm{ac}})$$

Full text

Pulse shaping is a well-established method in many areas of physics including magnetic resonance Bauer et al. (1984); Murdoch et al. (1987); Baum et al. (1985), trapped ion physics Choi et al. (2014); Timoney et al. (2008) and superconducting electronics Kelly et al. (2014) for improving the coherent response of quantum systems. Introduced to the field of nuclear magnetic resonance (NMR) spectroscopy in the 1980s, shaped pulses enable selective excitation of nuclear spins with uniform response over the desired bandwidth, which led to more selective and more sensitive measurement techniques. More recently, with the advent of fast arbitrary waveform generators (AWG), pulse shaping techniques also entered the field of electron paramagnetic resonance (EPR), thereby improving spectrometer performance via chirped pulses for broadband excitation Segawa et al. (2015); Doll and Jeschke (2016). In quantum information processing applications, shaped microwave pulses are utilized to optimize the fidelity and stability of gate operations against environmental perturbations that cause detuning of transition frequencies or fluctuations in the driving frequency Chow et al. (2010); Cross and Gambetta (2015).

In this Letter we investigate the application of pulse shaping to enhance the timing resolution in the emerging field of dynamical decoupling spectroscopy Cywinski et al. (2008); Alvarez and Suter (2011). Dynamical decoupling is a quantum control method developed to protect a quantum system against dephasing by environmental noise Suter and Alvarez (2016). More recently dynamical decoupling sequences have also been applied to map out noise spectra and to detect time-varying signals with high signal-to-noise ratio Alvarez and Suter (2011); Kotler et al. (2011); Lange et al. (2011); Bylander et al. (2011). In their simplest implementation, dynamical decoupling sequences consist of a periodic series of $\pi$ pulses with repetition time $\tau$ Carr and Purcell (1954) (see Figure 1(a)). For large numbers of pulses $N$, the spectral response of these sequences resembles that of a narrowband frequency filter, with center frequency $1/(2\tau)$ and bandwidth $1/(N\tau)$, that rejects noise at all frequencies except for those commensurate with the repetition time $\tau$. By tuning $\tau$ into resonance with a signal at a particular frequency $f_{\mathrm{ac}}\approx 1/(2\tau)$, the sensitivity to the signal can be enhanced while suppressing the influence of noise, thereby resembling the properties of a classical lock-in amplifier in the quantum regime Kotler et al. (2011).

When using many control pulses, the filter bandwidth becomes very narrow and the repetition time $\tau$ must be precisely tuned to the signal frequency $f_{\mathrm{ac}}\approx 1/(2\tau)$. On any controller hardware, however, $\tau$ can only be adjusted in increments of the sampling time $t_{s}$. This limits in practice the frequency resolution of the technique. Specifically, when detecting a signal with frequency $f_{\mathrm{ac}}$, the minimum frequency increment is given by

[TABLE]

Arbitrary waveform generators (AWGs) employed for the control of superconducting and spin qubits have sampling rates of typically $1\,\mathrm{GS/s}$, corresponding to a time resolution of $t_{s}=1\,\mathrm{ns}$. When operating at high frequencies $f_{\mathrm{ac}}$ this timing resolution quickly becomes prohibitive. For example, at a signal frequency of $f_{\mathrm{ac}}=5\,\mathrm{MHz}$, the minimum frequency increment is $\delta f=50\,\mathrm{kHz}$, which precludes the detection of weak signals with sharply defined spectra. Although AWGs with faster sampling rates exist, they are extremely costly and barely reach adequate timing resolution. Hardware sampling therefore is a severe limitation for dynamical decoupling spectroscopy.

Recently, an elaborate experimental scheme termed quantum interpolation has been devised and demonstrated to overcome this issue Ajoy et al. (2017). It enables a frequency sampling beyond the hardware limit of the control electronics by varying the interpulse delay between subblocks of the sensing sequence. This leads to an interpolation of the spin evolution at a more precisely controlled interpulse delay.

Here, we study the complementary and conceptually simpler approach of utilizing shaped control pulses to interpolate the pulse timing. The concept of our method is illustrated in Figure 1. Instead of modulating the high frequency control field by the common square pulse profile (Fig. 1(b)), we shape the envelope of the pulses by a smooth function. In this study we use a cosine-square profile (Fig. 1(c)), although any other smooth profile could be applied Ernst et al. ; Steffen et al. (2003). The pulse shape can be computed numerically before uploading the waveform data to the AWG and is therefore exceedingly simple to implement. (Numerical code is given in the supplemental material sup ). Because the AWG has a high vertical resolution, we can interpolate the center position of a pulse with a timing resolution $\delta t$ that is far better than sampling time $t_{s}$. The interpolated timing resolution $\delta t$ is approximately given by the slope of the pulse envelope, $\delta t=(\partial y/\partial t)^{-1}\delta y\sim t_{\pi}\delta y$, where $t_{\pi}$ is the duration of the pulse and $\delta y$ the minimum amplitude increment. Specifically, for a cosine-square shaped pulse of duration $t_{\pi}=25\,\mathrm{ns}$ implemented on an AWG with 14 bits of vertical resolution ($\delta y=2^{-14}$), an interpolated timing resolution of $\delta t\sim 1\,\mathrm{ps}$ can be expected.

To experimentally demonstrate the shaped-pulse interpolation method we study the coherent response of the electronic spin associated with a single nitrogen-vacancy (NV) center in a diamond single crystal. Due to their excellent coherence properties, even at room temperature, NV centers have pioneered the area of applied quantum sensing, with applications in nanoscale magnetometry of condensed matter systems Rondin et al. (2014), imaging of current distributions in nanostructures Chang et al. (2017) or structural magnetic resonance imaging of single proteins Kong et al. (2015); Ajoy et al. (2015). In particular for structural NMR imaging, very high frequency resolutions combined with MHz detection frequencies are required, providing a demanding test case for dynamical decoupling spectroscopy. In our experiments, control pulses are generated on an AWG operating at 500 MS/s with 14 bits of vertical resolution (Tektronix AWG5002C), and upconverted to the $\sim 2.2\,\mathrm{GHz}$ qubit resonance by frequency mixing with a local oscillator (Fig. 1(a)). Additional amplification is used to reach Rabi frequencies of $\sim 20\,\mathrm{MHz}$ corresponding to pulse durations of $t_{\pi}\sim 25\,\mathrm{ns}$. Microwave pulses are applied to the NV center by a coplanar waveguide structure, and the NV spin state is initialized and read out by optical means Loretz et al. (2013). All experiments are performed under ambient conditions.

Figure 2 shows a first set of measurements, in which we directly compare the timing resolution of dynamical decoupling sequences with and without shaped pulses. For this purpose we combine the control field with a sinusoidal AC test signal ($f_{\mathrm{ac}}=9.746969\,\mathrm{MHz},B_{\mathrm{ac}}=7.15\,\mathrm{\,\mu{\rm T}}$) before delivering it to the coplanar waveguide. In this case the sensing sequence becomes resonant with the AC field for a pulse repetition time of $\tau=51.298\,\mathrm{ns}$. When utilizing square pulses, the repetition time can only be stepped in increments of $t_{s}=2\,\mathrm{ns}$ and the sampling of the AC signal spectrum is very coarse (Fig. 2(a)). In stark contrast, we can finely sample the spectrum using shaped control pulses and clearly augment the hardware-limited time resolution (Fig. 2(b)).

We have compared the experimental data to the expected spectral response for the dynamical decoupling sequence. Because the phase of the AC magnetic field is not synchronized to the measurement sequence, we can describe the probability that the sensor qubit maintains its original state by Degen et al. (2016)

[TABLE]

Here, $\gamma=2\pi\times 28\,\mathrm{GHz/T}$ is the gyromagnetic ratio of the electronic sensor spin, $t=N\tau$ is the total duration of the phase acquisition, and $J_{0}$ is the zeroth-order Bessel function of the first kind. Further, $W_{N,\tau}(f_{\mathrm{ac}})$ is the spectral weighting or filter function of the sequence Degen et al. (2016),

[TABLE]

which has a maximum response of $W=2/\pi$ when $f_{\mathrm{ac}}=1/(2\tau)$. We find excellent agreement between the experimental spectra and the analytical filter profile of the dynamical decoupling sequence, but only the interpolated sequence provides the required fine frequency sampling. We have verified that the filter profile does not depend on whether square or shaped control pulses are used sup .

In a second set of experiments, shown in Figure 3, we analyze the sensor response under the action of an increasing number of shaped microwave pulses. Here, we keep the frequency of the test signal unchanged but reduce its amplitude. We first record the response to a sequence with $N=192$ and 672 shaped pulses to calibrate the amplitude of the AC signal. In both cases the sensor is operated in the small signal regime where the argument of the Bessel function is small ($W_{N,\tau}\gamma B_{\mathrm{ac}}t\lesssim 2\gamma B_{\mathrm{ac}}t/\pi\ll\pi$).

Subsequently, we tune the sensor into the strongly non-linear regime by increasing $N$ up to $10,000$ (Fig. 3(c)). For this large number of pulses, the argument of the Bessel function in Eq. 2 is no longer a small quantity because the total duration of the sequence $t=N\tau$ becomes very long. The non-linear regime has recently been explored with trapped ions Kotler et al. (2013) and it has been found that this regime gives rise to spectral features far below the Fourier limit. Here, we exploit these features to precisely characterize and test the frequency response of the sensor. Figure 3(d) shows a zoom into the center region of spectrum (c) that is acquired with a time increment of $\delta t=0.6\,\mathrm{ps}$. We observe that even for this fine time resolution the observed response of the sensor spin agrees well with the model expressed by Eq. 2. The time increment of $0.6\,\mathrm{ps}$ corresponds to a frequency sampling of $114\,\mathrm{Hz}$, which is an improvement by over three orders of magnitude compared to the frequency sampling of $380\,\mathrm{kHz}$ possible without pulse shaping.

To demonstrate the ability of the interpolated dynamical decoupling sequence to spectrally resolve nearby signals, we expose the sensor to a two-tone AC magnetic field composed of two slightly different frequencies. We select equal amplitudes for both frequency components and operated the sensor in the linear regime. As shown in Figure 4(a) both frequency components can be clearly distinguished in the resulting spectrum even though the frequencies are only $3\,\mathrm{kHz}$ apart.

Finally, we demonstrate that the application of shaped pulses also enables the detection of NMR spectra with high frequency resolution. Specifically, we detect the NMR signal of 13C nuclei located in close proximity to the NV center Taminiau et al. (2012); Zhao et al. (2012); Kolkowitz et al. (2012). Figure 4(b) shows the observed spectrum (points) for a sequence with $N=320$ control pulses together with a theoretical model (lines). Because the response of the sensor spin is no longer described by the classical description of Eq. (2), we perform density matrix simulations of the NV-13C system to calculate the $p(t,\tau)$ response curve. The simulations require knowledge of the parallel and perpendicular hyperfine coupling constants $a_{\mathbin{\|}}$ and $a_{\mathbin{\bot}}$, respectively, which we determine in separate high-resolution correlation spectroscopy measurements Boss et al. (2016). Two simulations are shown with Figure 4(b): A first curve (dashed line) plots the expected response from the single 13C nuclear spin. The second curve (solid line) reflects a simulation that includes three additional, more weakly coupled 13C nuclei. The example of Figure 4(b) clearly shows the advantage of a high sampling resolution for detecting NMR spectra.

To conclude, we have presented a simple method for greatly increasing the timing resolution of dynamical decoupling sequences. Using sequences with up to 10,000 coherent, amplitude-shaped control pulses, we were able to improve the effective timing resolution of the interpulse delay by more than 3 orders of magnitude, with time increments as small as $0.6\,\mathrm{ps}$. The resulting high frequency resolution has been demonstrated by sensing AC magnetic fields and NMR signals from individual carbon nuclear spins. The method provides a simple technique to further push the boundaries in frequency resolution and sensitivity in quantum sensing applications and can also be applied to other physical implementations, such as other solid state defect spins, trapped ultracold atoms and ions, or superconducting qubits.