Figure 1

(a) Schematic of the hybrid resonator-spin system. A single-crystal diamond piece ($1.7\mathrm{mm}\times 1.7\mathrm{mm}\times 1.1\mathrm{mm}$, type-Ib Sumicrystal, $\sim 200\mathrm{ppm}$ N content) is placed on top of one of four coplanar-waveguide resonators capacitively coupled to a common feed line. The resonators (with $10\mu \mathrm{m}$ central conductor and $1.5\mu \mathrm{m}$ gap widths) are patterned on a NbTiN film (70 nm thick, critical temperature $T_c=12.5\mathrm{K}$) on sapphire ($\mathrm{C}$-plane, $430\mu \mathrm{m}$ thick). An external magnetic field $B_{\parallel }$ is applied parallel to the film, along the diamond [100] direction. (b) Hyperfine interaction $A\approx 94\mathrm{MHz}$ with the N host nuclear spin splits each electron-spin level into a triplet. Only electron-spin transitions that preserve nuclear spin (solid arrows) are allowed. (c) $B_{\parallel }$ tunes the electron-spin energy levels through resonance with the $\lambda /4$ or $3\lambda /4$ modes of the resonator. The dashed line represents the thermal energy $k_{B}T/h\approx 5\mathrm{GHz}$ at $T=0.25\mathrm{K}$.Reuse & Permissions

Figure 2

Transmission spectroscopy. Image plots of normalized feed line transmission as a function of $B_{\parallel }$ and frequency near the $\lambda /4$ and $3\lambda /4$ mode resonances at $T=0.25\mathrm{K}$ (a), (b) and 1.2 K (c), (d). Each plot reveals three avoided crossings, corresponding to allowed hyperfine-split electron-spin transitions. Note that the frequency span in (b), (d) is 10 times larger than in (a), (c). The arrow in (a) points to a flux jump shifting the resonator frequency. All other image plots shown are corrected for these rare events.Reuse & Permissions

Figure 3

Determination of the collective coupling strength $g_{\mathrm{ens}}$. (a) A vertical cut of Fig. 2b at $B_{\parallel }=272.8\mathrm{mT}$ (dashed arrow) shows Rabi-split transmission dips. The best-fit to Eq. (1) gives $g_{\mathrm{ens}}/2\pi =17.0\mathrm{MHz}$. (b) Measured loaded quality factor of the $\lambda /4$ mode as a function of $B_{\parallel }$ at $T=0.25\mathrm{K}$. The best fit of Eq. (1) away from the avoided crossings gives $g_{\mathrm{ens}}/2\pi =3.9\mathrm{MHz}$. Arrows point to satellites resulting from the hyperfine coupling of the electron spin to the nuclear spin of 13C atoms adjacent to some $P1$ centers (see Ref. 23 for further discussion). (c) Best fit $g_{\mathrm{ens}}$ to the $\lambda /4$ (circles) and $3\lambda /4$ (squares) modes as a function of temperature. Solid curves are the best fits of Eq. (2). Error bars are smaller than the symbol size.Reuse & Permissions

Figure 4

Measurement of the spin linewidth and relaxation times using dispersive spin-resonator interactions. (a) Scheme (not to scale) for (b) the measurement of the spin linewidth ($T=0.25\mathrm{K}$, $B_{\parallel }=86\mathrm{mT}$) by probing the frequency shift of the $\lambda /4$ mode after applying a pump pulse ($0.4\mathrm{s}$ duration, $-50\mathrm{dBm}$ incident power) through resonance with the $m_I=+1$ transition (60 s wait between successive measurements) [23]. A similar measurement of $\gamma$ at $B_{\parallel }=263\mathrm{mT}$ is obtained using the $3\lambda /4$ mode. (c) $\gamma$ at $B_{\parallel }=86\mathrm{mT}$ (circles) and $B_{\parallel }=263\mathrm{mT}$ (squares) as a function of temperature. (d) Measurement of the spin relaxation time $T_1$ by probing the resonator shift as a function of time after the pump pulse is switched off. A biexponential decay is observed. (e) Temperature dependence of the two time constants, extracted by probing with the $\lambda /4$ (circles) and $3\lambda /4$ (squares) modes. Error bars, unless shown, are smaller than the symbol size.Reuse & Permissions

Figure 5

(a) Transmission spectroscopy similar to Fig. 2b, with an additional pump pulse resonant with the $m_I=0$ transition (incident pump power $P_p=-50\mathrm{dBm}$, 100 ms duration) prior to $|S_{21}|$ measurement. Complete disappearance of the $m_I=0$ avoided crossing and reduction in the coupling strength of the undriven transitions are observed. Color scale identical to Fig. 2b. (b) Inset: Vacuum-Rabi-split dips at $B_{\parallel }=269.1\mathrm{mT}$ and pump frequency of 7.54 GHz as a function of $P_p$. The merging of dips with increasing $P_p$ indicates cross relaxation between the subensembles. Main panel: Extracted polarization $\overline{P}$ (normalized to the value without pump) for the undriven $m_I=+1$ subensemble. The curve corresponds to the steady-state solution of a rate equation modeling fast equilibration between the subensembles compared to $T_1$ (see text and Ref. 23 for details).Reuse & Permissions