Observing hyperfine interactions of NV^- centers in diamond in an advanced quantum teaching lab

We start by measuring the ODMR spectrum of the diamond sample as a function of magnetic field $B_{0}$. The faces of the CVD diamond chip are oriented along the $\langle 100\rangle$ directions and the four orientations of NV${}^{-}\>$centers are along the [111], [1$\bar{1}\bar{1}$], [$\bar{1}1\bar{1}$], and [$\bar{1}\bar{1}1$] directions (see Fig. 3). The $B_{1}$ field created by the printed-circuit-board antenna is applied along the [001] direction, and therefore perpendicular to the sample surface. The $B_{0}$ field created by the coils of the electromagnet is applied parallel to the sample surface. By rotating the CVD sample on the sample holder, different $B_{0}$ directions with respect to the crystallographic axes can be created. More specifically, when the sample is mounted in a square orientation [i.e., like □, as shown in Fig. 3(a)] the $B_{0}$ field acts along the [100] direction. This results in all four NV${}^{-}\>$orientations experiencing the same effective angle between their NV${}^{-}\>$axes and $B_{0}$ and having the same two ODMR frequencies. However, when the sample is mounted in a diamond orientation [i.e., like ◇, as shown in Fig. 3(b)] the $B_{0}$ field acts along the [110] direction, resulting in the same effective angle for the [111] and [$\bar{1}\bar{1}1$], and the [1$\bar{1}\bar{1}$] and [$\bar{1}1\bar{1}$] orientations, respectively, and a total of four ODMR frequencies. In order to lift the degeneracy of all four NV${}^{-}\>$orientations and see eight different transitions appear, $B_{0}$ needs to be additionally rotated out of the sample plane.

We mount our sample roughly in the diamond orientation (i.e., like ◇), i.e., with $B_{0}$ directed along the [110] direction, however, we introduce a slight misalignment and let the students work out the exact orientation of the magnetic field. They can achieve this by recording $B_{0}$-dependent ODMR spectra and mapping the observed splittings onto the calculated transition energies from Hamiltonian ${\cal H_{\rm S}}$ (Eq. 2) using Matlab.

We plot such a magnetic field map in Fig. 1(d). Here, the magnitude of $B_{0}$ can be increased without altering the magnetic field direction by increasing the current through the electromagnet (see top x-axis) from 0 A to 1.0 A in 21 steps. For each $B_{0}$ magnitude an ODMR spectra was recorded to build up the 2D ODMR map. The students then fit the simple electron spin Hamiltonian ${\cal H_{\rm S}}$ to the data to work out the coil constant that describes the relationship between current and $B_{0}$ field. The dashed lines in Fig. 1(d) show the best fit for this data. The fit indicates that $B_{0}$ is applied along the [0.56 0.83 0.03] direction and the coil constant is 11.9 mT/A.

Note that the resistance of the electromagnetic coils is about 15 $\Omega$ and that it increases when the coils heat up at larger driving currents. It is therefore important to use a current source to control the magnetic field. Furthermore, the coils will also heat up the diamond sample and therefore the NV${}^{-}\>$center environment. At a high current, the ODMR spectrum will shift according to the temperature dependency of the zero-field-splitting $dD/dT=-74.2$ kHz/K Acosta2010 . For our experimental setup a current of 1 A heats up the diamond sample by $\sim 40$ K and shifts the zero-field splitting by $\sim 3$ MHz.

In the teaching labs at UNSW Sydney, the students recorded eleven ODMR spectra for electromagnet currents between 0 A and 0.5 A in 0.05 A steps. Each spectrum was measured from 2.65 GHz to 3.09 GHz in 221 steps with a wait time per point of 600 ms using the photodiode. This resulted in a total measurement time of $\sim 25$ minutes. While the measurements were running, the students were given a prepared Matlab script to examine the relationship between the coil constant and the magnetic field orientation. The students were then able to iteratively extract the relevant $B_{0}$ parameters by fitting their values to the measurement data, comfortably within the allocated time frame of the lab. This exercise addresses the first learning outcome in Section II.

Having mapped out the Zeeman splitting of the $S=1$ NV${}^{-}\>$center electron spin in Fig. 1(d), we can now take a closer look at the ODMR spectrum to examine the spectral signature of the hyperfine coupled ¹⁴N nuclear spins. Owing to the optical properties of the NV${}^{-}\>$center, the electron-nuclear spin interactions for the native ¹⁴N nucleus can be seen directly in the ODMR spectrum. The ¹⁴N hyperfine interaction tensor (see Eq. 4) splits each electron energy level [see Fig. 1(a)], causing the electron spin transitions to become nuclear spin dependent, with a $2.16$ MHz splitting in between the different $I_{z}$ projections. The different colored arrows in Fig. 1(a) represent the allowed electron spin transitions for the different nuclear spin orientations $m_{I}=-1$ (yellow), $m_{I}=0$ (orange) and $m_{I}=+1$ (red). These splittings can be resolved with a high-quality, low concentration NV${}^{-}\>$center ensemble sample ($\leq 1$ppm van1997dependences ), a high ODMR frequency resolution and a low microwave power Smeltzer_2011 .

In Fig. 3(c) we show the ODMR spectrum at a magnetic field of $B_{0}=1.3$ mT along the [0.67, 0.74, 0.02] direction (as estimated from the spectrum). The field is only slightly misaligned from the [110] crystallographic axis [see also Fig. 3(c) – inset], leading to similar Zeeman splittings and overlapping transitions for the $[111]$ and $[\bar{1}\bar{1}1]$, and the $[1\bar{1}\bar{1}]$ and $[\bar{1}1\bar{1}]$ NV${}^{-}\>$orientations, respectively. All the coherent control measurement results in the following sections (i.e., Figs. 4, 6, 7) are taken based on this field strength and orientation. Each peak in Fig. 3(c) displays a triplet structure that corresponds to the three different nuclear spin orientations. In the following sections, we will examine closely what effect the presence of the hyperfine lines has in coherent control experiments such as Rabi oscillations, Ramsey oscillations and Hahn echoes.

From the Rabi formula (see Section IV), we know that a lower microwave power results in a narrower HWHM (Half Width at Half Maximum) of the peaks in the spectrum. We map out this dependency in Fig. 4(a), where we plot five ODMR spectra recorded for different MW powers (open circles), together with a fit to the sum of three Lorentzian peaks with equal width and area (solid lines). At the lowest MW power, the three hyperfine peaks are clearly separated, while the power broadening at the higher powers leads to an overlap of the individual peaks.

We therefore need to work with low MW powers if we want to resolve the hyperfine splitting without power broadening. This requires a balance between resolution and signal strength, and we noticed that it is advantageous to work at low magnetic fields where the quenching of the photoluminescence is insignificant and the electron spin transitions from two different NV^- orientations with slightly different angles overlap. For all following experiments, however, we choose a MW power that leads to some overlap between peaks [similar to the $+3$ dBm curve in Fig. 4(a)] in order to observe effects of simultaneously driving neighboring transitions.

In principle, it is also possible to work with NV^- centres of a single orientation. To achieve this, a slightly misaligned and higher magnetic field is needed to split the electron spin transitions associated with the four different NV^- orientations. ³³3As this results in a lower photoluminescence signal, we use continuous ODMR to compensate for the decrease of signal strength, where the laser is on throughout the whole lock-in cycle, while the MW is pulsed on only during the first half. Fig. 4(d) shows an example spectrum of the electron spin transitions between $m_{s}=0$ and $m_{s}=+1$ at $B_{0}=14.3$ mT. The three different nuclear spin states can be clearly resolved and are labeled accordingly. We fit the spectrum with the sum of three Lorentzian peaks with equal amplitude and FWHM shown as black line, with the individual Lorentzian peaks show as red lines.

Applying a microwave pulse in resonance with one of the three hyperfine lines, caused by the ¹⁴N nuclear spin, induces coherent electron spin oscillations between the $m_{s}=0$ and $m_{s}=\pm 1$ spin eigenstates, known as Rabi oscillations. However, it is important to realize that the microwaves are simultaneously driving the detuned hyperfine transitions, due to the finite linewidth of the individual lines as well as the power broadening as described by the Rabi formula [e.g. see overlap of peaks in Fig. 4(a,c,d)].

Figure 4(b) shows a Rabi oscillation experiment, driven with the microwave frequency in resonance with the $m_{I}=0$ transition [middle peak of the left triplet in Fig. 4(a)], and the data is fitted to the sum of two exponentially decaying sinusoids $A_{\alpha}\sin(2\pi f_{\alpha})\exp(-\tau/T_{\alpha})+A_{\beta}\sin(2\pi f_{\beta})\exp(-\tau/T_{\beta})$ (black line). The frequencies for the two sinusoids correspond to the on-resonance drive of the $m_{I}=0$ transition and the detuned drive of the $m_{I}=\pm 1$ transitions. The two oscillation frequencies become apparent when taking the fast Fourier transform (FFT) of the Rabi oscillation experiment data [see Fig. 4(b) – inset], showing peaks at $\Omega_{\rm R}^{\rm res}\approx 0.7$ MHz and $\Omega_{\rm R}^{\rm det}\approx 2.3$ MHz.

According to the Rabi formula (Eq. 5), the Rabi oscillation frequency is given by:

where $\Omega_{\rm R}^{\rm res}$ is the on-resonance Rabi oscillation frequency and $\Delta$ is the detuning. $\Omega_{\rm R}^{\rm res}$ is given by:

where $B_{1}$ is the oscillating magnetic field component perpendicular to the NV^- axis delivered by the antenna Pham2013MagneticFS . Therefore, the Rabi frequency of $\Omega_{\rm R}=0.667\pm 0.005$ MHz in Fig. 4(b) corresponds to a field amplitude of $B_{1}=0.034$ mT, and we keep it fixed to this value for all coherent control experiments. The faster frequency of $\Omega_{\rm R}^{\rm det}=2.35\pm 0.06$ MHz in Fig. 4(b) corresponds to Rabi oscillations on the detuned hyperfine transition and agrees well with the one calculated using Eq. 6, where $\Delta_{A}=2.16$ MHz is the detuning between adjacent hyperfine peaks: $\Omega_{\rm R}^{\rm det}=\sqrt{(\Omega_{\rm R}^{\rm res})^{2}+\Delta_{A}^{2}}=\sqrt{(0.667~{}{\rm MHz})^{2}+(2.16~{}{\rm MHz})^{2}}=2.26$ MHz. The Rabi coherence time extracted from the on-resonance component is $T_{2}^{\rm R}=3.96$ $\mu$s, which is most likely limited by the inhomogeneity of the $B_{1}$ field over the sample area Vallabhapurapu2021 .

The effect of power broadening and cross-talk between the different hyperfine lines can be well visualized in a so-called Rabi chevron experiment, where the MW frequency and the pulse length are both swept – or in other words, a Rabi oscillation measurement is performed for various MW frequencies. Note that due the large number of data points, a Rabi chevron is usually outside of the scope of a teaching lab. Fig. 4(c), shows such a measurement for the same triplet of peaks as in Fig. 4(a,b). Each of the three hyperfine lines shows its own chevron fringes, overlapping with the neighbouring lines, and leading to the deviation from a single, decaying sinusoid in Fig. 4(b).

In principle, a simultaneous drive of all three hyperfine lines with similar strength can be achieved by applying a MW drive with stronger power to increase the Rabi oscillation frequency beyond the hyperfine coupling ($2.16$ MHz) Vallabhapurapu2021 . However, in our experiments the MW power was purposefully kept low, so that the effect of hyperfine transitions can be observed. This allows the students to develop an intuitive understanding of the consequences and origin of power broadening, fulfilling the second learning outcome in section II.

The Rabi experiment discussed in this section is the gateway between performing spin resonance spectroscopy and applying discrete gate operations in pulse sequences. Extracting the Rabi frequency $\Omega_{\rm R}=0.667$ MHz allows us to calibrate the lengths of our MW pulses to perform controlled rotations around the Bloch sphere. Here a $\pi$-pulse corresponds to a MW pulse of correct length $\tau_{\pi}=0.75$ $\mu$s to rotate the electron spin $180^{\circ}$ around the Bloch sphere, e.g. from $m_{s}=0$ to $m_{s}=-1$. A $\pi/2$-pulse of length $\tau_{\pi/2}=0.38$ $\mu$s rotates the electron spin by $90^{\circ}$, e.g. from the $m_{s}=0$ state to the $|{+y}\rangle=\tfrac{1}{\sqrt{2}}({|0\rangle}+{i|{-1}\rangle})$ superposition state at the equator of the Bloch sphere. Once the pulse lengths have been calibrated, experiments using pulse sequences, such as the coherence times measurements in the next sections can be implemented. See also Ref. Sewani2020, for a more detailed discussion and visualizations on the spin rotations.

A Ramsey experiment measures the free-induction decay or $T_{2}^{*}$ coherence time of a spin or a spin ensemble. The $T_{2}^{*}$ time is affected not only by noise during the pulse sequence, but also by detunings between the transition and the MW frequency. This is because the sequence does not contain any refocussing or echo pulses, in contrast to the Hahn echo and Carr-Purcell-Meiboom-Gill (CPMG) sequences discussed in Section VI.5.

Figure 5 shows the pulse sequence that we use in our experiments and the outputs of the individual channels (CH0 - CH3) of the Pulse Blaster. The spins are initialized into the ${|0\rangle}$ state by the first laser pulse and read out by the second laser pulse (see CH1). The MW pulse sequence (See CH2 and CH3) is applied only during the first half-cycle for the lock-in amplifier (see CH0), so that the lock-in detection scheme is sensitive to changes in the photoluminescence caused by the MW pulse sequence (see also Ref. Sewani2020, ). The first microwave $\pi/2$-pulse, applied along the $X$-axis, rotates the NV^- electron spins from $|0\rangle$ to the $Y$-axis of the Bloch sphere, corresponding to the ${|{+y}\rangle}=\tfrac{1}{\sqrt{2}}({|0\rangle}+i{|{-1}\rangle})$ superposition state in the rotating frame. We then let the spins dephase freely for a duration $\tau$, before applying a second microwave $\pi/2$-pulse along the $X$-axis to transfer the NV^- spin polarization from ${|{+y}\rangle}$ to ${|{-1}\rangle}$ for readout. A free induction decay curve can be recorded by varying the free precession time $\tau$ and the resulting data shows an exponential decay in ${|{-1}\rangle}$ population as a function of $\tau$ with decay time $T_{2}^{*}$ caused by decoherence and inhomogeneities Pham2013MagneticFS . For NV^-, the main inhomogeneities come from the variations in local strain and electric fields, as well as temporal and spatial fluctuations of the surrounding ¹³C nuclear spin bath Doherty2013 . In addition, a constant frequency detuning between the spins and the MW drive will lead to a sinusoidal modulation of the data. This is because the spins precess at the speed of the detuning in the reference frame of the MW source, causing them to rotate from ${|{+y}\rangle}$ towards ${|{+x}\rangle}$ or ${|{-x}\rangle}$ and onward, depending on the sign of the detuning. As the second $\pi/2$-pulse in the sequence maps ${|{+y}\rangle}\rightarrow{|{-1}\rangle}$, ${|{+x}\rangle}\rightarrow{|{+x}\rangle}$, ${|{-y}\rangle}\rightarrow{|{0}\rangle}$, and ${|{-x}\rangle}\rightarrow{|{-x}\rangle}$, the constant detuning manifests as a sinusoidal oscillation in the ${|{-1}\rangle}$ population at the end of the sequence. As a consequence, a Ramsey sequence is often used for the accurate estimation of shifts in the spin qubit resonance frequency, as is important in magnetometry experiments Barry2020 .

In Fig. 6(a) we show in blue the Ramsey data obtained for driving the NV^- in resonance with the $m_{I}=0$ transition [same as the Rabi oscillations in Fig. 4(b)]. The data shows the exponential decay superimposed with a sinusoidal modulation. We fit the data to the function $\exp(-\tau/T_{2}^{*})\sum_{i=1}^{n}[A_{i}\sin(2\pi f_{i})]$, where $n$ corresponds to the number of expected transitions Grezes2016 . Looking at the corresponding FFT of the data in Fig. 6(c), we can clearly see two peaks. The peak at low frequencies indicates a slight detuning of our MW source to the $m_{I}=0$ transition frequency. The smaller peak at $\sim 2$ MHz corresponds to the signal from the detuned $m_{I}=\pm 1$ hyperfine lines, which are detuned from the $m_{I}=0$ transition by $\Delta_{A}=2.16$ MHz.

In Fig. 6(b), we show very similar data with the Ramsey experiment performed in resonance with the $m_{I}=-1$ transition. The data now shows two discrete modulation frequencies, one for the $m_{I}=0$ electron spin subsystem that is detuned by $\Delta_{A}$ and a second one for the $m_{I}=+1$ subsystem that is detuned by $2\Delta_{A}$. The two frequency components can be clearly seen as peaks in the Fourier-transform of the decay data in Fig. 6(d).

For both Ramsey experiments in Figs. 6(a) and (b) we also show data that is obtained by applying the second microwave $\pi/2$-pulse along the $Y$-axis (purple data). This modifies the mapping of the spin before readout to ${|y\rangle}\rightarrow{|{y}\rangle}$, ${|x\rangle}\rightarrow{|{0}\rangle}$, ${|{-y}\rangle}\rightarrow{|{-y}\rangle}$, and ${|{-x}\rangle}\rightarrow{|{-1}\rangle}$. As the projection pulse along the $Y$-axis (purple data) is shifted by $\pi/2$ in phase with respect to the projection pulse along the $X$-axis (blue date), the sinusoidal modulations are also shifted in phase by $\pi/2$ as seen in Figs. 6(a) and (b). The possibility to modify the projection axis by adjusting the phase of the MW source forms the basis for quantum state tomography Nielsen2002 and allows the extension of the teaching labs to include further experiments.

The Hahn echo experiment is designed to cancel out constant detunings and frequency offsets by adding an extra $\pi$-pulse to the sequence. The $\pi$-pulse inverts the spins halfway through the precession and as such refocusses variations in their Larmor precession frequency Hahn1950 . We have already introduced this experiment in Ref. Sewani2020, for the purpose of dynamical decoupling, where it helps to improve the transverse relaxation time of the relevant spins, allowing the coherence time to be extended beyond the free induction decay time.

The MW pulse sequence we implement for the Hahn echo measurement is $X_{\pi/2}-\tau-X_{\pi}-\tau-X_{\pi/2}$. In this sequence the electron spins are initialised in the $\ket{0}$ state and rotated into the superposition state $|y\rangle=\frac{1}{\sqrt{2}}(\ket{0}+i\ket{-1})$ using a MW $\frac{\pi}{2}$-pulse. The spins are then left to freely precess for a length of time $\tau$, after which they are inverted on the Bloch sphere with a $\pi$-pulse. The spins are again left to precess for the same amount of time, causing them to refocus before being rotated back into the $\ket{0}$ state with another $\frac{\pi}{2}$-pulse for optical readout. The spin signal is then measured as a function of the free precession time $\tau$, and the $T_{2}^{\rm Hahn}$ coherence time is given as the time constant of the exponential decay of the signal.

The Hahn echo can also be used to reveal interactions between spins – in this case the hyperfine coupling to the ¹³C nuclei in proximity to the NV spins that have a natural abundance of $1.1$%. The randomly distributed ¹³C nuclei of the diamond lattice produce a local magnetic field that is modulated by their Larmor precession frequencies, effectively resulting in an oscillating background magnetic field. Since the Hahn echo sequence can only remove slow fluctuations in detuning, the Larmor precession of the nuclei will lead to gradual decoherence of the electron spin. However, if the refocussing pulse occurs after exactly one period of nuclear spin precession, the background field modulation in both free precession intervals is exactly the same and the coherence reappears.

In Fig. 7(a), we present the Hahn echo signal measured at a magnetic field of $B_{0}=1.3$ mT. From the exponential decay, we extract $T_{2}^{\rm Hahn}=11.7\pm 0.2$ $\mu$s. Upon further investigation, a measurement at a higher magnetic field of $B_{0}=5.84$ mT reveals the presence of the above-mentioned coherence revival within the Hahn echo signal, as shown in Fig. 7(b). The decay and revival in Fig. 7(b) is attributed to the weak hyperfine coupling to the ¹³C spins near the NV center lattice sites Childress281 . The signal revival at $2\tau\sim 30$ $\mu$s corresponds to a frequency of $\sim 66.7$ kHz, which is consistent with the $f_{\rm C}=\gamma_{\rm C}B_{0}=62.5$ kHz Larmor precession frequency of the spin-1/2 nuclei with $\gamma_{\rm C}=10.705$ MHz/T and $B_{0}=5.84$ mT Childress281 ; PhysRevB.82.201201 . The black line is a fit of the spin signal to two inverted Gaussian peaks whose amplitude follows an exponential decay $\propto\exp[-(2\tau/T_{2}^{\rm Hahn})]$, which gives $T_{2}^{\rm Hahn}=63\pm 10$ $\mu$s. According to Ref. PhysRevB.82.201201, , the $T_{2}^{\rm Hahn}$ of an NV^- ensemble can be as long as $600$ $\mu$s at room temperature. However, the coherence time reduces when the magnetic field is not parallel to the NV^- axis, as is the case in our samples and experiments.

As the coherence revival rate is linked to the Larmor precession frequency of the ¹³C nuclei and therefore proportional to the applied magnetic field, this revival is not present in Fig. 7(a) where the lower magnetic field of $B_{0}=1.3$ mT would have led to a revival at $2\tau=144$ $\mu$s, longer than the $T_{2}^{\rm Hahn}$ coherence time.