Broadband Excitation by Chirped Pulses: Application to Single Electron Spins in Diamond

Pulsed excitation of broad spectra requires very high field strengths if monochromatic pulses are used. If the corresponding high power is not available or not desirable, the pulses can be replaced by suitable low-power pulses that distribute the power over a wider bandwidth. As a simple case, we use microwave pulses with a linear frequency chirp. We use these pulses to excite spectra of single NV-centers in a Ramsey experiment. Compared to the conventional Ramsey experiment, our approach increases the bandwidth by at least an order of magnitude. Compared to the conventional ODMR experiment, the chirped Ramsey experiment does not suffer from power broadening and increases the resolution by at least an order of magnitude. As an additional benefit, the chirped Ramsey spectrum contains not only `allowed' single quantum transitions, but also `forbidden' zero- and double quantum transitions, which can be distinguished from the single quantum transitions by phase-shifting the readout pulse with respect to the excitation pulse or by variation of the external magnetic field strength.

different types of transitions can be distinguished by appropriate shifts in the relative phases of the excitation and readout pulses.

Spin S=1/2 System
We use chirped excitation pulses to excite transitions in a large frequency range. Figure  1 shows the basic idea: Assuming that we want to excite the transition between the |m S = 0 and the |m S = 1 state and that the system is initially in the ground state, we scan the frequency through resonance in such a way that the system has a 50 % transition probability to the |m S = 1 state and ends up in the superposition state which maximizes the coherence between the two levels. The relative phase ϕ depends on the phase, amplitude and scan rate of the microwave. The effect of the chirped pulse can thus be described by a unitary operator [16] U 1 = e −iϕ 1 Sz e −i π 2 Sy . Pulse sequence for broadband Ramsey experiment with chirped excitation pulses. ω start defines the start frequency of the scan and ω bdw the width of the scan. ω 0 is the reference frequency that relates the phase of the two pulses; for details see text. τ p is the pulse duration and t 1 the free evolution time which is incremented between experiments.
As shown in figure 2, the system is then allowed to evolve freely for a time t 1 . If Ω 0 is the Larmor frequency of the system, the superposition state acquires an additional phase Ω 0 t 1 during this time. The resulting state is At this point, a second chirped pulse generates another transformation that we write as thus converting the system into the final state The population of the ground/bright state |0 is thus Clearly, this corresponds to a Ramsey-fringe pattern, which can be Fourier-transformed to obtain the spectrum (a single line at Ω 0 in this case).

Spin S=1 System
The NV-center in diamond is a spin S = 1 system. We write the relevant Hamiltonian Here, D = 2.8 GHz is the zero-field splitting and Ω 0 the Larmor frequency due to the interaction with the magnetic field. Figure 3 shows the resulting level structure, together with the allowed magnetic dipole transitions, marked by arrows. We write |m S for the eigenstates of the Hamiltonian, where m S is the eigenvalue of S z . In the following, we assume that the Rabi frequency is small compared with the frequency separation of the relevant transitions. We therefore can assume that the microwave field drives only one transition at a time [14,15,16]. If we scan from low to high frequency, we first excite the transition |0 ↔ | + 1 in the system shown in figure  3. Starting from the initial state Ψ 0 = |0 , the first passage through resonance converts it into where θ is the effective flip-angle of the pulse. Passing through the second resonance, we obtain Here, we have assumed that the effect of the pulse on both transitions is the same. This is a good approximation if the scan rate and the transition strengths are the same. During the subsequent free evolution period, the system evolves to with Ω ±1 = D ∓ Ω 0 representing the resonance frequencies of the two transitions. This free precession period is terminated by the readout pulse, which is identical to the excitation pulse (apart from an overall phase). It converts part of the coherences back to populations. Here, we are interested only in the population P 0 = P (|0 ) of the bright state |0 : with the amplitudes The first term in this expression is a constant offset. The second term oscillates at the frequency 2Ω 0 = Ω −1 − Ω +1 of the | − 1 ↔ | + 1 transition, while the third term contains the two single quantum transition frequencies. Fourier transformation of this will therefore yield a spectrum with the two allowed single quantum transition and the 'forbidden' double quantum transition frequency, as shown in figure 3. Note that the frequencies in the figure are not the true resonance frequencies. The relation between the apparent and the real frequencies will be discussed in the following section.

Setup and Samples
The experiments were performed with a home-built confocal microscope. A diodepumped solid-state laser with an emission wavelength of 532 nm was used. The cw laser beam was sent through an acousto-optical modulator to generate laser pulses for excitation and readout. We used an oil immersion microscope objective (with NA = 1.4) mounted on a nano-positioning system to focus the laser light to single NV-centers. The microscope objective also collects light emitted by the NV-centers during readout. For electronic excitation we used a setup consisting of a microwave synthesizer and an arbitrary waveform generator, which were connected to a mixer and up-converted. Here the synthesizer was used as local oscillator and the arbitrary waveform generator, which had a sampling frequency of 4 GS/s, delivered the intermediate frequency. We were able to control the phase as well as the frequency of the up-converted signal by changing the phase and the frequency of the arbitrary waveform generator. The controllable frequency bandwidth was < 2 GHz. The microwaves were guided through a Cu wire mounted on the surface of the diamond. The maximal excitation power was 8 W. We used a permanent magnet to apply a magnetic field to the sample. We applied the chirped Ramsey sequence shown in figure 2 to two different diamond samples both of type IIa. One is a 12 C enriched (concentration of 99.995 %) diamond with a relaxation time of T * 2 > 200 µs the other a natural abundance diamond with The enriched sample is a diamond single crystal grown at 5.5 GPa and 1400°C from Co-Ti-Cu alloy by using a temperature gradient method. As a solid carbon source, polycrystalline diamond plates synthesized by chemical vapor deposition (CVD) utilizing 12 C enriched methane were used. Secondary ion mass spectrometry (SIMS) analysis has shown that typically a 12 C concentration of 99.995 % in the grown crystals was achieved. The crystal was irradiated at room temperature with 2 MeV electrons and a total flux intensity of 10 11 /cm 2 . Subsequently it was annealed at 1000°C for 2 hours in vacuum.
We first present measurements of the enriched sample to illustrate different features of this experiment, in particular how the phases of the excitation pulses affect the observed frequency and phase of the different types of resonance lines.

Reference Frequency
In the experiments, we are not interested in the dc component 2A 2 1 + A 2 2 , which we omit in the following. We now compare experiments where we change the phase of the second pulse with respect to that of the first one by an angle α. The resulting signal is then In the experiments, we use this additional phase for two purposes: we increment it linearly with the free precession period t 1 to shift the effective precession frequency, and we use it to distinguish the double quantum transition, which does not depend on α, from the single quantum transitions. Looking first at the linear phase increments, we set α = ω 0 t 1 . The resulting signal is then We therefore expect that the single quantum transitions appear shifted to the frequencies (Ω ±1 − ω 0 ), while the double quantum transition remains at the natural frequency 2Ω 0 = Ω +1 − Ω −1 . This is clearly borne out in figure 4, where we compare spectra obtained with the same excitation scheme, but different reference frequencies.
The three groups of lines appear centered around Ω +1 − ω 0 , 2Ω 0 = Ω +1 − Ω −1 , and Ω −1 − ω 0 . For these experiments, we chose ω 0 such that the resulting frequencies fall into a frequency window that is easily accessible. In the case of the spectra shown here, we incremented t 1 by 2 ns between scans, which yields, according to the Nyquist theorem a 250 MHz frequency window. The maximum value of t 1 was 5 µs. The data were recorded in the same magnetic field, which splits the |m s = ±1 lines by 146 MHz. All measurements were done with frequency chirps starting at 2770 MHz and the pulse lengths were τ p = 120 ns. It is clearly seen that the single quantum transitions are shifted in the opposite direction from the reference frequency, while the double quantum transitions (at 146 MHz) are not affected by the detuning.

Phase Shifts
Instead of incrementing the phase proportionally with t 1 , we can also compare two spectra with different constant phase shifts of the readout pulse. The two traces of figure 5 (b) show an example: the spectra were obtained with phase shifts of 0 and π between the two pulses; only expanded regions of the full spectrum shown in figure 5 (a) are shown. These data were recorded with a different NV-center in a higher magnetic field strength. The chirp bandwidth was 500 MHz, the pulse length τ p = 50 ns and the maximum value of t 1 was 5 µs. According to equation (2), we expect that the phase of the single quantum transitions |0 ↔ | ± 1 should change with α, while the double quantum transition | + 1 ↔ | − 1 should not change. Inspection of the experimental data shows that the spectral lines close to 60 and 375 MHz are inverted between the two spectra, while the signals close to 315 MHz do not change. We therefore interpret the outer lines as single quantum transitions, the inner ones as double quantum transitions. This assignment is also consistent with the splittings due to the hyperfine interaction with the 14 N nuclear spin, which is 2.15 MHz for the single quantum transitions and 4.3 MHz for the double quantum transition.
Using this phase dependence, we can also separate the two types of transitions by calculating the sum and difference of the two spectra. According to equation (2), the difference of the two spectra should be and the sum s α=0°+ s α=180°= 4A 2 1 · cos (2Ω 0 t 1 ) .
The lower part of figure 5 shows the result of this operation: The sum (upper trace) contains mostly the double quantum signals, while the difference is dominated by the single quantum transitions which corresponds to the results of equation (3) and (4). The incomplete suppression of the other signals can be attributed to instabilities in the experimental setup, which result in thermal frequency shifts and changing amplitudes. Figure 6 shows spectra of the 12 C enriched crystal for different magnetic field strengths. For these measurements the reference frequency was ω 0 = 2670.8 MHz. The chirp pulses had a bandwidth of 500 MHz and a duration of τ p = 50 ns. The start frequency of the chirp was ω start = 2650.8 MHz and the bandwidth ω bdw = 500 MHz. The sampling interval of 1 ns results in a bandwidth of 500 MHz and maximum value of t 1 of 5 µs yields a digital frequency resolution of 100 kHz. In each spectrum of the figure, we list the splitting between the single quantum transitions, which corresponds to the magnetic field component along the symmetry axis of the center, measured in frequency units. The outer triplets correspond to the single quantum transitions (|0 ↔ |±1 ), the inner lines to the double quantum transition (|+1 ↔ |−1 ). With increasing magnetic field strength, the splitting between the single quantum transitions increases proportionally and is always equal to the frequency of the double quantum transition. The frequency changes for the left and right triplets are not the same, this can be explained by transversal components in the Zeeman interaction which we have neglected in the Hamiltonian equation (1).

Multi-Line Broadband Spectrum
The chirped excitation scheme is particularly useful when the spectra cover a broad frequency range with many resonance lines. Such a situation exists in NV-centers with a 13 C nuclear spin in the first coordination shell. figure 7 shows the spectrum of such a center. In this particular center, the electron spin is coupled to a nearest-neighbor 13 C nuclear spin with a hyperfine coupling Figure 7. Spectra of NV-center in natural abundance diamond with two adjacent 13 C nuclear spins. One strongly coupled with A ≈ 126.5 MHz (nearest-neighbor) and one with A ≈ 6.55 MHz [9]. Ω 0 ≈ 10 MHz is the Zeeman interaction, D the zero-field splitting and ω 0 the reference frequency. (a) Absolute value spectrum. (b) sum and (c) difference of the spectra obtained with phase shifts α = 0 • and α = 180 • .
constant A ≈ 126.5 MHz as well as to an additional 13 C with a coupling constant of A ≈ 6.55 MHz. For this measurement we used a type IIa natural abundance diamond and applied a magnetic field strength of approximately 9 G. The field was not aligned and had an angle of ≈ 65°with respect to the symmetry axis of the NV-center, which corresponded to a projected field strength of 3.7 G. The chirp bandwidth was 250 MHz, starting from 2750.3 MHz and the pulse-duration was τ p = 60 ns.
The top graph of figure 7 shows the absolute value of a chirped Ramsey spectrum. The center graph shows the sum and the lower the difference of two phase-shifted spectra, which correspond to the double-and single quantum transitions, respectively. The line at 126.5 MHz in b) is a zero-quantum transition. Its transition frequency matches the hyperfine coupling constant of the nearest-neighbor 13 C. In the spectra, we also indicate how the spectral lines can be assigned to transitions of the electron spin with different configurations of the three coupled nuclear spins. If we consider only the Hamiltonian of equation (1) for the electron spin and the hyperfine interactions with the nuclear spins, the single quantum spectrum (bottom of figure 7) should consist of 4 groups of six lines. In the experimental spectrum, the four groups contain more than six lines. This difference can be attributed to the splitting of the |m S = 0 ground state due to the interaction with the transverse components of the magnetic field and the nonsecular hyperfine interaction. [17]

Conclusions
We have introduced a new experimental technique for measuring broad spectra of single electron spins. This approach does not require high microwave power. The precession frequency of the spins is measured in the absence of microwave irradiation, in the form of Ramsey fringes, which results in high resolution spectra. The resulting spectra contain not only the dipole-allowed single quantum transitions, but also multiple quantum transitions that can only be excited by multiple absorption/emission processes. This technique is particularly useful in the case of electron spins coupled to multiple nuclear spins. Such clusters of spins may be useful tools for quantum computing applications [5,6,7,8]. We have demonstrated the technique on the example of single electron spins in the diamond NV-center, but the same approach should also be applicable to other systems, where the excitation bandwidth can be sufficiently large.