Fast Coherent Control of a Nitrogen-Vacancy-Center Spin Ensemble Using a ${\mathrm{K}\mathrm{Ta}\mathrm{O}}_3$ Dielectric Resonator at Cryogenic Temperatures

Microwave delivery to samples in a cryogenic environment can pose experimental challenges such as restricting optical access, space constraints, and heat generation. Moreover, existing solutions that overcome various experimental restrictions do not necessarily provide a large homogeneous oscillating magnetic field over macroscopic length scales, which is required for control of spin ensembles or fast gate operations in scaled-up quantum computing implementations. Here, we show fast and coherent control of a negatively charged nitrogen-vacancy spin ensemble by taking advantage of the high permittivity of a ${\mathrm{K}\mathrm{Ta}\mathrm{O}}_3$ dielectric resonator at cryogenic temperatures. We achieve Rabi frequencies of up to 48 MHz, with a total power-to-field conversion factor $C_{\mathrm{P}}=9.66\mathrm{mT}/\sqrt{\mathrm{W}}$ (approximately $191{\mathrm{MHz}}_{\mathrm{Rabi}}/\sqrt{\mathrm{W}}$). We use the nitrogen-vacancy-center spin ensemble to probe the quality factor, the coherent enhancement, and the spatial distribution of the magnetic field inside the diamond sample. The key advantages of the dielectric resonator utilized in this work are ease of assembly, in situ tunability, a high magnetic field conversion efficiency, a low-volume footprint, and optical transparency. This makes ${\mathrm{K}\mathrm{Ta}\mathrm{O}}_3$ dielectric resonators a promising platform for the delivery of microwave fields for the control of spins in various materials at cryogenic temperatures.

Figure 1

The setup and the ODMR response. (a) An overview of the measurement setup, showing the schematic of the microwave circuit, an exploded view of the optical setup, and the sample mounting. The samples are mounted on an Attocube piezoelectric stage, allowing the samples to be repositioned inside the cryostat. (b) A sketch of the magnetic (red) and electric (blue) field lines corresponding to the ${\mathrm{TE}}_{11\delta }$ excitation mode of the dielectric resonator. (c) The reflection magnitude and phase signals of the overcoupled ${\mathrm{K}\mathrm{Ta}\mathrm{O}}_3$ dielectric resonator, when tuned through a fixed microwave reference frequency at $2.967$ GHz. (d) The ODMR signal as a function of the microwave frequency and the magnetic field $B_0$. The white dashed line indicates the dielectric resonator frequency. (e) The measured temperature dependence of the dielectric resonator (DR) frequency in the cryostat.

Figure 2

The $B_1$ conversion factor and the Rabi decay rate. (a) Selected Rabi oscillation measurements at three microwave powers, specified at the loop coupler. The red lines are fits to the measured data with a decaying sinusoidal function including an exponent $n$. (b) The measured Rabi frequency is shown to double for +6-dB increments of microwave power, as expected (the red fit excludes the last three data points due to amplifier saturation). The trend of the $1/{T_2}^{\mathrm{Rabi}}$ decay rate (blue line) can be explained by power broadening. This leads, in combination with the spectral distribution of the ensemble (green line) and the $B_1$ inhomogeneity within the detection volume (magenta line), to the observed S bend.

Figure 3

The detuning dependence of the Rabi oscillation enhancement. (a) The chevron pattern of the Rabi oscillations plotted against a range of $\mathrm{N}$-$V^{-}$–resonator detunings. The microwave source is kept in resonance with the $\mathrm{N}$-$V^{-}$ ODMR frequency throughout this measurement. (b) The corresponding Rabi frequencies fitted to a Lorentzian function. The on-resonance enhancement is calculated as approximately $7.1\times$.

Figure 4

The position dependence and homogeneity of $B_1$ enhancement. (a) The measured and appropriately compensated Rabi frequency ${\mathrm{\Omega }}_{\mathrm{R}}$ along the $x$ and $y$ axes of the diamond sample [see Eq. (A1) in the Appendix]. The black line corresponds to predicted ${\mathrm{\Omega }}_{\mathrm{R}}$ values, obtained from modeling the trend of the $B_1$ field of the dielectric resonator in cst microwave studio. We find good agreement between the experimental data and the cst simulation model. (b) $1/{T_2}^{\mathrm{Rabi}}$ decay rates measured and plotted across both axes of the sample. We observe a general increase in ${T_2}^{\mathrm{Rabi}}$ coherence times toward the center of the resonator, due to higher $B_1$ homogeneity. (c) ${T_2}^{\mathrm{Hahn}}$ decay times for positions along the $x$ axis of the resonator. We do not observe significant variations of the ${T_2}^{\mathrm{Hahn}}$ decay times across the sample. The inset shows a sample Hahn-echo measurement for a position close to the center of the resonator, corresponding to the data point highlighted in red in the main panel. The error bars represent 95% confidence intervals for fits.

Figure 5

Setup photographs. (a) The sample stack with the copper stage and the three sapphire spacers below the ${\mathrm{K}\mathrm{Ta}\mathrm{O}}_3$ dielectric resonator–$\mathrm{N}$-$V^{-}$ diamond sample sandwich. The sample stack is held together with minimal vacuum grease. (b) The sample stack mounted on the dilution refrigerator cold finger atop an Attocube positioner. The coaxial loop coupler is positioned roughly 1.5 mm above the topmost ${\mathrm{K}\mathrm{Ta}\mathrm{O}}_3$ dielectric resonator. (c) A photograph of the coaxial loop coupler, showing the size of the looped inner conductor.

Figure 6

The loop-coupler–resonator coupling. (a) The VNA measurement of the dielectric resonator stack at room temperature, showing critical coupling. (b) VNA measurements at 4 K showing the S11 magnitude (top panel) and phase (bottom panel) for various coupling regimes of the dielectric resonator to the coaxial loop antenna. The coupling strength can be modified by adjusting the sample $z$ position. At critical coupling, $Q_I=1275\pm 12$, $Q_E=1328\pm 5$, $Q_L=650\pm 4$. For the overcoupled case, the values are $Q_I=1414\pm 5$, $Q_E=591\pm 8$, $Q_L=416\pm 7$. The quality factors are extracted using a resonance-fitting script [62]. We attribute the low internal quality factor to the lossy $\mathrm{N}$-$V^{-}$ CVD diamond sample (see also Table 3).

Figure 7

The dielectric resonator-frequency tuning. (a) The dielectric resonator frequency measured as a function of the cryostat temperature. (b) The dielectric resonator frequency as a function of the infrared laser power as measured at the optical window into the cryostat. (c) The temporal step response of the dielectric resonator frequency after infrared laser modulation.

Figure 8

The dielectric resonator-frequency stabilization. (a) The resonator-frequency drift is likely due to heating caused by longer microwave pulses and higher microwave powers when no pulse compensation is employed. All lasers are off. (b) The resonator frequency is stabilized by keeping the total microwave duty cycle (${\mathrm{T}}_1+{\mathrm{T}}_2$) constant for various microwave pulse widths (${\mathrm{T}}_1$). In the first half cycle, a measurable ODMR signal is created, while the second half cycle acts as reference for the lock-in amplifier: channel 1, detector gating; channel 2, 520-nm laser excitation; channel 3, microwave pulse; channel 4, lock-in reference.

Figure 9

The magnetic field simulation. (a) The simulation model, showing the loop coupler and relevant samples stacked on a conductive platform. (b) The simulated ${\mathrm{S}}_{11}$ magnitude, showing fitted quality factors similar to measured values in Fig. 1. (c) The simulated magnetic field distribution within the relevant sample volume with 1 W of applied microwave power at the loop coupler.

Figure 10

Laser excitation and magnetic field simulations. (a) A cross-section plot of the 520-nm laser-intensity model within the signal collection volume, showing a minimum beam waist of $4.6\mu \mathrm{m}$ at the focus. (b) A plot of the simulated $B$-field magnitude within the signal collection volume: the $B$-field intensity is rescaled.

Figure 11

The dielectric resonator-frequency stability. (a) The resonator deviation from 2.967 GHz as recorded while implementing continuous resonator tuning during the ODMR measurement shown in Fig. 1 in the main text. (b) The histogram of the resonator-frequency deviation from the set point, accumulated throughout the measurement corresponding to Fig. 2 in the main text. The corresponding histogram data shows the distribution in terms of percentage change of the Rabi frequencies ($\delta B_1$) as a result of resonator deviation. The calculation of $\delta B_1$ uses measured data shown in Fig. 3 in the main text. (c) The combined histogram of all measured resonator deviation values from the detuning set points used for the measurement shown in Fig. 3 in the main text.

Figure 12

The collection of the measured (red) and simulated (blue) Rabi signals for assorted applied microwave powers. The measured data are fitted to the function $f(t)=e^{-(t/\tau {)}^n}\sin (2\pi {\mathrm{\Omega }}_{\mathrm{R}}t+p)+c$.

Figure 13

The collection of the measured and fitted ${\tau }^2$ Hahn-echo signals for various positions along the $x$ axis of the resonator. The data points at the end of the measured signals are scaled to 0.5 to represent the spins entering into a mixed state. The fit function used is $f(t)=e^{-(t/\tau {)}^n}(a_1+a_2({\sin }^2(0.5\omega t+p)))+c$.