Strong magnetic coupling of an inhomogeneous nitrogen-vacancy ensemble to a cavity

We study experimentally and theoretically a dense ensemble of negatively charged nitrogen-vacancy centers in diamond coupled to a high-$Q$ superconducting coplanar waveguide cavity mode at low temperature. The nitrogen-vacancy centers are modeled as effective spin-one defects with inhomogeneous frequency distribution. For a large enough ensemble the effective magnetic coupling of the collective spin dominates the mode losses and inhomogeneous broadening of the ensemble and the system exhibits well-resolved normal-mode splitting in probe transmission spectra. We use several theoretical approaches to model the probe spectra and the number and frequency distribution of the spins. This analysis reveals an only slowly temperature-dependent $q$-Gaussian energy distribution of the defects with a yet unexplained decrease of effectively coupled spins at very low temperatures below $100\mathrm{m}\mathrm{K}$. Based on the system parameters we predict the possibility to implement an extremely stable maser by adding an external pump to the system.

Figure 1

(a) Measured transmission as a function of $f_{-}^{\text{I}}={\omega }_{-}^{\text{I}}/\left(2\pi \right)$ tuned by the magnetic field and pump frequency $f_p={\omega }_p/\left(2\pi \right)$. (b) Transition frequencies $f_{\pm }^{\text{I,II}}={\omega }_{\pm }^{\text{I,II}}/\left(2\pi \right)$ as a function of the magnetic field for a field angle of $\varphi =22.5^{{}^{\circ }}$. The frequency of the cavity $f_c={\omega }_c/\left(2\pi \right)$ is denoted by the (red) dashed line. (c) Fit of $|{\langle a_c\rangle }_{\text{st}}{|}^2$ to the transmitted signal close to the resonance. We obtain the decay rate $\gamma =10.92\mathrm{M}\mathrm{Hz}$. (d) $|{\langle a_c\rangle }_{\text{st}}{|}^2$ as a function of $f_{a_1}$ and $f_p$ incorporating the parameters obtained from the fit.

Figure 2

Simulation of the normal-mode splitting at resonance (${\omega }_c-U_a={\omega }_{a1}$) for different values of $\gamma$. Finally, for large values of $\gamma$ the doublet is not resolved any more and the avoided crossing (sketched on the right) becomes blurred.

Figure 3

(a) Collective coupling strength obtained from our measurements for different temperatures (black markers) and $\Omega \left(T\right)=a_0{\left\{\tanh [\hslash {\omega }_{-}^{\text{I}}/\left(2k_{B}T\right)]\right\}}^{1/2}$ (red dashed line), where $a_0$ was chosen to match the high-temperature behavior. Including the $m_S=+1$ state via Eq. (16) gives the result denoted by the blue dotted line. (b) Collective coupling strength determined from the results of the numerical integration of the coupled equations with $g\sqrt{N}=a_0$ [$a_0$ determined from (a), black markers] and $\Omega \left(T\right)$ (red dashed line).

Figure 4

Coupling density of the ensemble determined from an exemplary transmission measurement at $T=20\mathrm{m}\mathrm{K}$. The error bars are calculated from the uncertainties in $R_{\text{cc}}^{+}\left(\omega \right)$ which we obtain from fitting Eq. (27) to the data. We fit the $q$ Gaussian (red) in Eq. (28) to the coupling density to determine the width and the behavior of the tails of the distribution.

Figure 5

(a) Coupling density for different temperatures. The $q$ parameter and width ${\gamma }_q$ of the coupling density do not change significantly with increasing temperature. The coupling $g\sqrt{N}={[\int \rho \left(\omega \right)d\omega ]}^{1/2}$ is reduced, as can be seen in (b). The red line shows $\Omega \left(T\right)=a_0{\left\{\tanh [\hslash {\omega }_{-}^{\text{I}}/\left(2k_{B}T\right)]\right\}}^{1/2}$ with $a_0$ chosen to match the high-temperature limit of the data (black markers). To compare $\Omega \left(T\right)$ with our previous results determined directly from the splitting we plot the data already shown in Fig. 3 using gray markers.

Figure 6

$|G_{\text{cc}}{\left(\omega \right)|}^2$ as a function of the collective coupling strength $g\sqrt{N}$ and for three different values of the $q$-Gaussian parameter $q$ (grayscale figures on the left are plotted on a logarithmic scale). For $g\sqrt{N}=4,8,\text{and}16\mathrm{M}\mathrm{Hz}$, marked by $\text{(d})$, $\text{(e})$, and $\text{(f})$, we show the transmission for all three values of $q$ together. The transmissions for the Gaussian coupling density with $q\to 1$ (dotted black line), for the coupling density of our ensemble with $q=1.39$ (dashed red line), and for the Lorentzian coupling density with $q\to 2$ (solid black line) are shown on the right. The parameters were chosen to be ${\gamma }_q/\left(2\pi \right)=10\mathrm{M}\mathrm{Hz}$, $\kappa /\left(2\pi \right)=0.4\mathrm{M}\mathrm{Hz}$, and ${\gamma }_{\text{hom}}/\left(2\pi \right)=1\mathrm{Hz}$.

Figure 7

(a) Real part and (b) modulus of the imaginary part of one of the complex poles of $G_{\text{cc}}^{+}\left(\omega \right)$ in the regime where $g\sqrt{N}/\left(2\pi \right)>{\gamma }_q/\left(2\pi \right)=10\mathrm{M}\mathrm{Hz}$. The collective coupling strength $g\sqrt{N}$ is shown as a dotted line in (a). With increasing $g\sqrt{N}$ the modulus of the imaginary part, proportional to the width of the Rabi peaks, is reduced.

Figure 8

$|G_{\text{cc}}{\left(\omega \right)|}^2$ on resonance as a function of the inhomogeneous width ${\gamma }_q$ for an ensemble with (a) Gaussian coupling density, (b) $q$-Gaussian $(q=1.39)$ coupling density, and (c) Lorentzian coupling density. To keep the transmission visible for large values of ${\gamma }_q$ the range of the color coding is limited to $[0,0.1]$. The parameters were chosen to be $g\sqrt{N}/\left(2\pi \right)=10\mathrm{M}\mathrm{Hz}$, $\kappa /\left(2\pi \right)=0.4\mathrm{M}\mathrm{Hz}$, and ${\gamma }_{\text{hom}}/\left(2\pi \right)=1\mathrm{Hz}$.

Figure 9

(a) Steady-state number of photons in the cavity ${\langle a_c^{†}{a}_c\rangle }_{\text{st}}$ as a function of the incoherent pump rate $w$ and the inhomogeneous width ${\gamma }_p$. The resulting linewidth $\Delta f$ is shown in (b). For small ${\gamma }_p$ both $\Delta f$ and ${\langle a_c^{†}{a}_c\rangle }_{\text{st}}$ show rapid changes as $w$ becomes larger than ${\gamma }_{\text{hom}}$. The critical width of the inhomogeneity is given by ${\gamma }_{p,\text{crit}}=g^{2}N/\kappa$. The parameters were chosen to be $N={10}^{12}$, $g/\left(2\pi \right)=10\mathrm{Hz}$, $\kappa /\left(2\pi \right)=1\mathrm{M}\mathrm{Hz}$, and ${\gamma }_{\text{hom}}/\left(2\pi \right)=1\mathrm{Hz}$.