Exploring the nonlinear regime of light-matter interaction using electronic spins in diamond

The coupling between defects in diamond and a superconducting microwave resonator is studied in the nonlinear regime. Both negatively charged nitrogen-vacancy and P1 defects are explored. The measured cavity mode response exhibits strong nonlinearity near a spin resonance. Data is compared with theoretical predictions and a good agreement is obtained in a wide range of externally controlled parameters. The nonlinear effect under study in the current paper is expected to play a role in any cavity-based magnetic resonance imaging technique and to impose a fundamental limit upon its sensitivity.

Figure 1

The experimental setup. (a) A loop antenna (LA) is coupled to the spiral resonator. Two multimode optical fibers are coupled to the diamond wafer. Fiber F1 is employed for delivering laser light of wavelength ${\lambda }_{\mathrm{L}}=532$ nm, and fiber F2 probes the emitted photoluminescence (PL). (b) The spiral resonator has three turns, an inner radius of 0.59 mm, and an outer radius of 0.79 mm. (c) The resonance line shape of the spiral's fundamental mode vs temperature. (d) The magnetic induction magnitude $\left|{\mathbf{B}}_{\mathrm{c}}\left(\mathbf{r}\right)\right|$ of the fundamental mode vs position $\mathbf{r}$ in a plane perpendicular to both wafers that contains the center of the spiral.

Figure 2

ODMR. (a) The measured emitted PL spectrum. (b),(c) ODMR spectrum vs driving frequency ${\omega }_{\mathrm{p}}/2\pi$ and magnetic field $\left|\mathbf{B}\right|$. The white dotted lines represent the frequencies ${\omega }_{\pm }/2\pi$ calculated using Eq. (1). The fitting procedure yields the direction of the magnetic field $\widehat{\mathbf{b}}$, which is expressed as $\widehat{\mathbf{b}}=T_{\widehat{\mathbf{z}}}\left({\theta }_z\right)T_{\widehat{\mathbf{y}}}\left({\theta }_y\right)T_{\widehat{\mathbf{x}}}\left({\theta }_x\right)\widehat{\mathbf{z}}$, where $\widehat{\mathbf{x}}$, $\widehat{\mathbf{y}}$, and $\widehat{\mathbf{z}}$ are unit vectors in the crystal directions $\left[100\right]$, $\left[010\right]$, and $\left[001\right]$, respectively, and $T_{\widehat{\mathbf{s}}}\left({\theta }_s\right)$ is a rotation operator around the axis $\widehat{\mathbf{s}}$, where $s\in \left\{x,y,z\right\}$. The rotation angles found from the fitting procedure are ${\theta }_x/\pi =-0.02$, ${\theta }_y/\pi =0.002$, and ${\theta }_z/\pi =0.05$.

Figure 3

Cavity mode reflectivity $R_{\mathrm{c}}$ with ${\mathrm{NV}}^{-}$ defects for various values of injected microwave power $P_{\mathrm{p}}$ ($P1=-90$, $P2=-70$, $P3=-60$, and $P4=-50$ dBm) and laser power $I_{\mathrm{L}}$ ($L0=0$, $L1=5.6$, $L2=12.8$, and $L3=30$ mW mm${}^{-2}$). For each pair, the top plot is experimental data (labeled D) and the bottom is the theoretical prediction (labeled F). The bottom row shows the effective normalized resonance frequency (${\omega }_{\text{eff}}/{\omega }_{\text{c}}$), which is obtained from the minimum reflectivity signal for each magnetic field, as a function of magnetic field. Each column corresponds to a single microwave power (DF; P1–P4) and each plot shows all four laser powers (L3 in red, L2 in black, L1 in blue, and L0 in green). Cross markers denote experimental data and solid lines represent theoretical predictions. Plots are vertically shifted for clarity. The parameters used for the calculation for the case of laser on (off) are $P_{z\mathrm{ST}}=-0.035$, $P_{z\mathrm{SO}}=-0.55$, ${\rho }_{\mathrm{s}}=1.23\times {10}^{17}{\mathrm{cm}}^{-3}$, $T_2=219\mathrm{ns}$, $T_{1\mathrm{T}}=23\mathrm{ms}$ ($T_{1\mathrm{T}}=565\mathrm{ms}$) [23] and $g_{\mathrm{s}}/2\pi =5.05\mathrm{Hz}$ ($g_{\mathrm{s}}/2\pi =2.72\mathrm{Hz}$). The rate $T_{1\mathrm{O}}^{-1}$ of OISP is taken to be given by $T_{1\mathrm{O}}^{-1}=0.16\times {\gamma }_{\mathrm{O}}$, where ${\gamma }_{\mathrm{O}}=I_{\mathrm{L}}\sigma {\lambda }_{\mathrm{L}}/hc$ is the rate of optical absorption, where $\sigma =3\times {10}^{-17}{\mathrm{cm}}^2$ [24] is the optical cross section, $h$ is the Plank's constant, and $c$ is the speed of light in vacuum. The effective coupling coefficient $g_{\mathrm{s}}$ for both cases of laser on and off is calculated using Eq. (5) and the numerically calculated mode shape [see Fig. 1]. The volume inside the diamond wafer illuminated by the laser is $0.76{\mathrm{mm}}^3$.

Figure 4

Cavity mode reflectivity $R_{\mathrm{c}}$ with P1 defects for various values of injected microwave power $P_{\mathrm{p}}$ ($P1=-90$, $P2=-80$, and $P3=-70$ dBm) with laser off. For each pair, the top plot is experimental data (labeled D) and the bottom is the theoretical prediction (labeled F). The parameters used for the calculation are ${\gamma }_{\mathrm{c}}/2\pi =0.304$ MHz, ${\gamma }_{\mathrm{f}}/2\pi =0.349$ MHz, ${\rho }_{\mathrm{s}}=1\times {10}^{18}$ cm${}^{-3}$, $T_2=438$ ns, and $T_{1\mathrm{T}}=470$ ms [36]. The values of $P_{z\mathrm{ST}}$ and $g_{\mathrm{s}}$ are the same as in Fig. 3 for the case of laser off.