Multifrequency spin resonance in diamond

Magnetic resonance techniques provide a powerful tool for controlling spin systems, with applications ranging from quantum information processing to medical imaging. Nevertheless, the behavior of a spin system under strong excitation remains a rich dynamical problem. In this paper, we examine spin resonance of the nitrogen-vacancy center in diamond under conditions outside the regime where the usual rotating-wave approximation applies, focusing on effects of multifrequency excitation and excitation with orientation parallel to the spin quantization axis. Strong-field phenomena such as multiphoton transitions and coherent destruction of tunneling are observed in the spectra and analyzed via numerical and analytic theory. In addition to illustrating the response of a spin system to strong multifrequency excitation, these observations may inform techniques for manipulating electron-nuclear spin quantum registers.

Figure 1

(a) Spin levels of the electronic ground state and optically excited state of the N-$V$ center. Note that the orbital doublet in the excited state is averaged at room temperature, where we conduct our experiments [26]. The excited-state level anticrossing (ESLAC) near 510 G is indicated by the dashed oval. (b) Geometry of our experiments, illustrating the orientation of oscillating magnetic fields. (c) Multifrequency continuous-wave excitation in a low magnetic field. Inset: zoom of boxed region. By extrapolating from Rabi nutation frequencies at higher MW intensity, we estimate the MW electron spin Rabi frequency for these data to be ${\Omega }_{\mathrm{MW}}=(2\pi )0.46$ MHz. The rf intensity is significantly higher; based on quasistatic analysis (below), we estimate the rf longitudinal coupling to be ${\Omega }_{\mathrm{rf}}\approx (2\pi )6$ MHz. Each horizontal sweep in MW frequency is normalized by its average value, compensating for slow fluctuations in fluorescence intensity owing to sample drift, as the data are collected over several days.

Figure 2

(a) Multifrequency spectrum at the ESLAC. ${\Omega }_{\mathrm{rf}}\approx (2\pi )15$ MHz and ${\Omega }_{\mathrm{MW}}\approx (2\pi )0.43$ MHz are pulsed for 5 $\mu$s prior to fluorescence detection (sequence shown). Each horizontal sweep in MW frequency is normalized by its average value and shifted in frequency (using ESR data collected at regular intervals during data acquisition) to compensate for drifts in fluorescence intensity and magnetic field. Inset: Data superposed with the lines $\delta =n{\omega }_{\mathrm{rf}}$ (red) and zeros of $J_{\delta /{\omega }_{\mathrm{rf}}}(2{\Omega }_{\mathrm{rf}}/{\omega }_{\mathrm{rf}})$ (white) for $\delta ={\omega }_{\mathrm{MW}}-1.4506$ GHz and ${\Omega }_{\mathrm{RF}}=15.4$ MHz. (b) NMR and ENDOR at the ESLAC. Higher intensities, ${\Omega }_{\mathrm{MW}}\approx (2\pi )2$ MHz and ${\Omega }_{\mathrm{rf}}\approx (2\pi )20$ MHz, and longer pulses (10 $\mu$s) allow NMR and ENDOR via the mechanism illustrated in the inset. Examples of weak ENDOR resonances are marked with arrows. Note that the ENI 525LA rf amplifier used in this experiment does not have a flat response below 2 MHz.

Figure 3

Numerical simulation of continous-wave multifrequency spectroscopy. All simulations are performed using a fourth-order Runge-Kutta integration with a time step of 0.1 ns over a total time of 100 $\mu$s [32]. In each case, ${\Omega }_{\mathrm{MW}}=0.5$ MHz, ${\Omega }_{\mathrm{rf}}=3$ MHz, and $\Delta =100$ MHz, while the simulations average over time and $\varphi$. The microwave (MW) frequency is indicated as a detuning from the resonant frequency of $100$ MHz; this resonance frequency is far lower than the experimental resonance at approximately a few gigahertz, but the simulations are not sensitive to this parameter because the RWA is valid for the MW excitation. The average population in $m_s=0$ (for a spin initially polarized into $m_s=0$) is shown on a color scale, with a darker color indicating a lower population. (a) ${\mathit{\Omega }}_{\mathbf{MW}}\left|\right|\mathrm{x̂}$ and ${\mathit{\Omega }}_{\mathbf{rf}}\left|\right|\mathrm{ẑ}$. (b) ${\mathit{\Omega }}_{\mathbf{MW}}\left|\right|\mathrm{ẑ}$ and ${\mathit{\Omega }}_{\mathbf{rf}}\left|\right|\mathrm{x̂}$. (c) ${\mathit{\Omega }}_{\mathbf{MW}}\left|\right|\mathrm{x̂}$ and ${\mathit{\Omega }}_{\mathbf{rf}}\left|\right|\mathrm{x̂}$. Note that no spin transitions occur if both oscillating magnetic fields are oriented along $\mathrm{ẑ}$.

Figure 4

(a) A single trace from Fig. 2a at ${\omega }_{\mathrm{rf}}=300$ kHz, shown with the expected quasistatic line shape (red line) for ${\Omega }_{\mathrm{rf}}=(2\pi )15$ MHz convolved with a Lorentzian of HWHM 0.5 MHz. (b) MW spectrum vs rf power. Solid lines illustrate functional dependence of Rabi frequency ${\Omega }_{\mathrm{rf}}$ on rf power. (c) Resonant MW excitation (${\omega }_{\mathrm{MW}}\approx \Delta$) vs rf frequency, illustrating CDT. Experimental data are obtained by averaging over the central 1-MHz line in data from Fig. 2a. Theory shows the population in $m_s=0$ after 5-$\mu$s numerical integration of the Bloch equations for Eq. (2) using ${\Omega }_{\mathrm{MW}}=(2\pi )0.43$ MHz and ${\Omega }_{\mathrm{rf}}=(2\pi )15.4$ MHz, incorporating 0.5-MHz pure dephasing (which has the effect of slightly broadening the CDT resonances and suppressing small oscillations between them). Upper curve indicates corresponding zeros in the Bessel function $J_0(2{\Omega }_{\mathrm{rf}}/{\omega }_{\mathrm{rf}})$.