Self-induced superradiant masing

MainSuperradiance, first predicted by Dicke in 1954 (ref. 1), describes an enhanced collective emission of light exhibiting high coherence and a nonlinear scaling of intensity with the number of emitters2,3. It has since been observed in many systems4,5,6,7,8 including solid-state realizations with quantum dots9,10 and the negatively charged nitrogen-vacancy (NV−) centres in diamond11,12. This collective effect lies at the heart of cavity quantum electrodynamics13, where emitters are typically treated as non-interacting and their mutual couplings are mediated solely through the shared cavity field. These light-mediated interactions give rise to a variety of collective phenomena and phase transitions14,15,16,17.Building on these fundamental explorations of collective light–matter interactions, recent advances have shifted the focus towards their applications in quantum technologies18,19. Among other solid-state platforms20,21,22, diamond-based systems stand out due to the exceptional quantum coherence and optical controllability of NV− centres23. Key examples include diamond-based microwave amplifiers24, quantum sensors25, mode cooling platforms26,27 and room-temperature diamond masers28. Operating a diamond maser in a superradiant regime could further narrow its linewidth, with coherence maintained by the high cooperativity of the cavity–spin system itself29, rather than being dictated by the intracavity photon count, as that in conventional masers30. This could pave the way for ultranarrow-linewidth sources, enabling high-precision frequency generation and quantum-limited microwave amplification. However, achieving a continuous-wave superradiant diamond maser remains an open challenge due to the need for both strong and uniform collective spin–photon coupling along with efficient optical pumping31. Although increasing the density of the spin ensemble enhances coupling, it also reduces optical transparency, complicating efficient optical pumping; moreover, it introduces greater dipole–dipole interactions, which can lead to decoherence32,33. In this work, we show that such interactions can take on a more constructive role in dense spin systems like NV− centres. Our measurements reveal that dipole–dipole interactions can actively drive superradiant masing, which opens up a new self-induced regime of coherent collective emission in solid-state systems. We perform a comprehensive experimental characterization of the effect and conclusively trace it back to dipole–dipole interactions by numerically simulating the microscopic dynamics of up to one million NV− centres.In our experiment, we investigate an inhomogeneously broadened ensemble of NV− centres strongly coupled to a microwave cavity. Our system (Fig. 1a) allows for the generation of an inverted spin ensemble by applying strong microwave pulses, and for the controlled release of this inversion in the form of superradiant emission. The cavity, made of two parallel sapphire chips with superconducting split-ring structures, has a resonance frequency of around ωc/2π = 3.1 GHz. The roughly 200-μm-sized diamond sample is positioned between the chips. A wire loop wrapped directly around the chips is used for the rapid switching of the external magnetic field, allowing us to activate or suppress the cavity–spin interaction on demand. This assembly is housed in a copper box inside a dilution refrigerator, cooled to 25 mK and connected to a heterodyne detection scheme. The hybrid system of cavity and spin ensemble, with N ≈ 9 × 1012 NV− centres, is well within the collective strong-coupling regime, having a cooperativity C = 14.6. Uniform cavity–spin coupling over the sample volume guarantees the required spin permutation symmetry to enable the superradiant dynamics we observe. In this dense spin ensemble, the typical nearest-neighbour distance between the NV− spins is r = (N/V)−1/3 ≈ 8 nm. This results in a typical spin–spin coupling strength of about 100 kHz between neighbouring NV−s. Further details on the diamond sample, cavity and setup are provided in the Methods.Fig. 1: Experimental setup, self-induced superradiant masing dynamics and emission spectrum.a, Schematic of the superconducting microwave cavity strongly coupled to the NV− diamond. b, Zoomed-in plot of the cavity amplitude ∣a∣ during the initial superradiant (SR) decay, which is triggered on tuning the inverted spin ensemble back into resonance and is well described by a semiclassical model. c, Expanding the time axis after the initial superradiant decay, we observe a series of narrow masing pulses evolving into a quasi-continuous cavity emission. Note the different y-axis scalings for ∣a∣ compared with that in b. d, Long tail of the quasi-continuous masing emission, showing the quadratures I and Q of the cavity amplitude, digitally demodulated in the rotating frame of the cavity resonance frequency for visual clarity. e, Fourier analysis (interval marked by the purple-shaded area in d), where the emission has a linewidth much smaller than the cavity (inset). The frequency difference Δω/2π is measured relative to the cavity frequency of 3.1 GHz. f, Frequency and linewidth of the emission change over time when the window of the Fourier analysis is shifted (see the main text for details). FFT, fast Fourier transform.Full size imageResultsInitial superradiant decay and spectral holeFor all experiments, we use the previously established protocol34 for generating a uniformly inverted spin state. All NV− spins are tuned into resonance with the cavity, using a static magnetic field with equal projections along the four diamond axes. We apply a microwave inversion pulse to homogeneously invert all spins from a relaxed initial state of the effective two-level systems. Subsequently, we rapidly detune the spin ensemble from the cavity resonance and store the inversion for a set hold time. This procedure allows for the preparation of states with uniform initial spin inversion \({p}_{0}=\langle {\sigma }_{j}^{z}\rangle\) and almost-zero transversal spin components \(\langle {\sigma }_{j}^{-}\rangle \approx 0\). The values for p0, bounded by ±1, are tunable within the range of 0.1–0.4 by modifying the hold time on the order of milliseconds.On tuning the spins back into resonance with the cavity using the detuning loop, the inverted spin state is free to interact with the cavity mode. If the stored inversion exceeds the threshold p0C > 1, the system enters a metastable state35. Here \(C={g}_{{\rm{coll}}}^{2}/\kappa \varGamma \approx 14.6\) is the cooperativity, a dimensionless parameter combining the collective coupling strength gcoll/2π = 4.53 MHz, the cavity linewidth κ/2π = 418 kHz (half-width at half-maximum) and the effective ensemble dephasing rate Γ/2π = 3.36 MHz. The latter accounts for both inhomogeneously broadened spin frequency distribution ρ(ω), with a full-width at half-maximum (FWHM) of W/2π = 8.65 MHz, and the spin linewidths modelled by γ⊥/2π = 179 kHz (equations (9) and (10)).In this metastable inverted state with p0C > 1, any fluctuation will stimulate a collective emission process known as a superradiant decay36. Conversely, if the inversion is below the instability threshold, dephasing due to inhomogeneous broadening becomes dominant and prevents this avalanche process35. In our case, the superradiant decay is triggered by noise photons from the input line. As the spin decay accelerates, the cavity amplitude ∣a∣ increases. It reaches its maximum as the collective spin vector points towards the equator of the Bloch sphere, where the cavity amplitude \(\max (| a| )\propto {p}_{0}-1/C\) serves as a measure of the initial inversion above threshold (equation (12)), and the emitted intensity \(| a{| }^{2} \approx {({p}_{0}N)}^{2}\) exhibits the characteristic quadratic scaling of superradiance with the number of effectively participating spins2,34. Subsequently, energy is coherently exchanged between cavity and spins, visible as damped Rabi oscillations (Fig. 1b). This coherent exchange is eventually stopped by coherence-limiting processes in the system, mainly the dephasing of inhomogeneously broadened spins.Crucially, the cavity-resonant spins—which dominate the collective emission dynamics—become de-excited through superradiant emission, whereas the off-resonant spins maintain their inversion. This leaves behind a spectral hole37, a region of depleted inversion centred at the cavity frequency. This initial superradiant decay dynamics is well captured by the Maxwell–Bloch equations38 that provide a semiclassical description of the collective coupling between a non-interacting spin ensemble and the cavity mode. This standard picture does not predict further dynamics beyond the initial emission pulse, once the resonant spin inversion has been sufficiently depleted below the threshold p(Δ = 0) \ j}\,\frac{{J}_{0}}{| {{\bf{r}}}_{jk}{| }^{3}}\frac{1}{{\hslash }^{2}}\left[3({{\bf{S}}}_{j}\cdot {\hat{{\bf{u}}}}_{jk})({{\bf{S}}}_{k}\cdot {\hat{{\bf{u}}}}_{jk})-{{\bf{S}}}_{j}\cdot {{\bf{S}}}_{k}\right],$$ (4) where J0/2π = 51.9 MHz nm3, rjk is the vector connecting the spins and \({\hat{{\bf{u}}}}_{jk}={{\bf{r}}}_{jk}/| {{\bf{r}}}_{jk}|\). In our experiment, only the transition between the m = 0 and m = +1 magnetic sublevels is of relevance. Taking into account the four different orientations of NV−s in our system, we can write the Hamiltonian as$${{\mathcal{H}}}_{{\rm{dd}}}=\hslash{\sum}_{{j,k}\atop{k> j}}\left[\left({J}_{jk}{\sigma }_{j}^{+}{\sigma }_{k}^{-}+{J}_{jk}^{* }{\sigma }_{k}^{+}{\sigma }_{j}^{-}\right)+{Q}_{jk}{\sigma }_{j}^{\;\rm{ee}}{\sigma }_{k}^{\;\rm{ee}}\right],$$ (5) $${\rm{with}}\quad {J}_{jk}=-\frac{{J}_{0}}{| {r}_{jk}{| }^{3}}({g}_{jk}+{\rm{i}}{h}_{jk}),\quad {Q}_{jk}=-\frac{{J}_{0}}{| {r}_{jk}{| }^{3}},$$ (6) where \({g}_{jk}=\frac{1}{2}({T}_{jk}^{\;xx}+{T}_{jk}^{yy})\), \({h}_{jk}=\frac{1}{2}({T}_{jk}^{xy}-{T}_{jk}^{yx})\) and \({q}_{jk}={T}_{jk}^{zz}\), with \({T}_{jk}^{\alpha \beta }\)\(=3({\hat{{\bf{e}}}}_{{O}_{j}}^{\alpha }\cdot {\hat{{\bf{u}}}}_{jk})({\hat{{\bf{e}}}}_{{O}_{k}}^{\beta }\cdot {\hat{{\bf{u}}}}_{jk})-{\hat{{\bf{e}}}}_{{O}_{j}}^{\alpha }\cdot {\hat{{\bf{e}}}}_{{O}_{k}}^{\beta }\). Here \({\hat{{\bf{e}}}}_{{O}_{j}}^{\alpha }\) denotes the α-oriented unit vector of the local coordinate system at site j, with α, β ∈ {x, y, z} (Supplementary Section 1).The total Hamiltonian is given by$${\mathcal{H}}=\hslash \sum _{j}{{\mathit\varDelta} }_{j}{\sigma }_{j}^{\rm{ee}}+\hslash {g}_{0}\sum _{j}\left({a}^{\dagger }{\sigma }_{j}^{-}+{\sigma }_{j}^{+}a\right)+{\rm{i}}\hslash \eta \left({a}^{\dagger }-a\right)+{{\mathcal{H}}}_{{\rm{dd}}},$$ (7) with a† (a) being the creation (annihilation) operator of the cavity mode and \({\sigma }_{j}^{\rm{ee}},{\sigma }_{j}^{\pm }\) being the projection on the excited state and raising/lowering operators for the jth spin, respectively. The cavity losses are accounted for by a rate κ. We disregard T1 processes as the spin-lattice relaxation rate is negligible compared with all other dynamical timescales of our system. In our simulations, we also consider a weak external driving field η to model the triggering of superradiant pulses by technical noise. It mainly affects the precise timing of the first revival pulse as well as its shape.We neglect three-spin and cavity-induced contributions in the dynamics of the spin–spin coherences, \(\langle {\sigma }_{j}^{+}{\sigma }_{k}^{-}\rangle\), as they have negligible impact on the dynamics. Since the (single-spin) decoherence rate γ⊥ is the dominant timescale, we adiabatically eliminate the dynamics of these correlations, \({\partial }_{t}\langle {\sigma }_{j}^{+}{\sigma }_{k}^{-}\rangle \approx 0\). Factorizing spin–cavity, \(\langle a{\sigma }_{j}^{+}\rangle \approx \langle a\rangle \,\langle {\sigma }_{j}^{+}\rangle\), and two-spin expectation values, \(\langle {\sigma }_{j}^{\;{\rm{ee}}}{\sigma }_{k}^{-}\rangle \approx \langle {\sigma }_{j}^{\;\rm{ee}}\rangle \,\langle {\sigma }_{k}^{-}\rangle\), we arrive at the dynamical equations of our system:$${\partial }_{t}\langle a\rangle =-\kappa \langle a\rangle -{\rm{i}}{g}_{0}\sum _{j}\langle {\sigma }_{j}^{-}\rangle +\eta \,,$$ (8a) $$\begin{array}{ll}{\partial }_{t}\langle {\sigma }_{j}^{-}\rangle=-({\gamma }_{\perp }+{\rm{i}}{{\varDelta} }_{j})\langle {\sigma }_{j}^{-}\rangle +{\rm{i}}{g}_{0}\langle a\rangle {p}_{j}\\\qquad\qquad+\,{\rm{i}}{p}_{j}\mathop{\sum}\limits_{{k}\atop{k\,\ne\,j}}{J}_{jk}\langle {\sigma }_{k}^{-}\rangle -{\rm{i}}\langle {\sigma }_{j}^{-}\rangle\mathop{\sum}\limits_{{k}\atop{k\,\ne\,j}}{Q}_{jk}({p}_{k}+1)/2\,,\end{array}$$ (8b) $${\partial}_{t}{p}_{j}=-4{g}_{0}{\rm{Im}}(\langle {a}^{\dagger}\rangle \langle {\sigma }_{j}^{-}\rangle )-\mathop{\sum}\limits_{{k}\atop{k\,\ne\,j}}\frac{4{\gamma }_{\perp }\vert {J}_{jk}{\vert}^{2}\left({p}_{j}-{p}_{k}\right)}{{({{\varDelta} }_{j}-{{\varDelta} }_{k})}^{2}+4{\gamma }_{\perp}^{2}}\,.$$ (8c) We sample \({n}_{{\rm{sim}}}=1{0}^{6}\) NV− centres distributed randomly in a cube. For this number, we find convergence for the cavity amplitude with respect to the spatial and frequency distributions as well as the orientations of the NV−s. To take into account the actual number N of NV−s in the experiment, we have to consider \(N/{n}_{{\rm{sim}}}\) copies of our box when coupling to the cavity in equation (8a). To avoid stiffness issues, we single out neighbouring spins that equilibrate faster than all other timescales in the system and approximate them to instantaneously equilibrate. We simulate the remaining equations using an explicit Runge–Kutta method. A detailed derivation is provided in Supplementary Section 3.Superradiant instability threshold and cooperativityThe instability criterion p0C > 1 is formally derived35 from equation (8c) as a necessary condition for the growth of the cavity amplitude starting from zero photons 〈a〉 = 0, in the case of uniform initial inversion \({p}_{0}=\langle {\sigma }_{j}^{z}\rangle\) and vanishing \(\langle {\sigma }_{j}^{-}\rangle =0\), assuming no spin–spin interactions (Jjk = Qjk = 0). The frequency-resolved single-spin cooperativity is given by$$C({\varDelta})=\frac{{g}_{0}^{2}}{\kappa}{\left({\gamma}_{\perp}+\frac{{\varDelta}^{2}}{{\gamma}_{\perp}}\right)}^{-1}\,,$$ (9) and defines the total cooperativity as an integral over the spin distribution (∫ρ(Δ) dΔ = N):$$C=\,\int\,C({\varDelta} )\,\rho ({\varDelta} )\,d{{\varDelta}} =\frac{{g}_{{\rm{coll}}}^{2}}{\kappa {\varGamma}}\,,$$ (10) where Γ is the effective ensemble linewidth, incorporating both inhomogeneous broadening and intrinsic spin dephasing. For our system, we find C = 14.6 and Γ/2π = 3.36 MHz. For a non-uniform inversion profile p(Δ), this criterion generalizes in the continuum limit as a weighted integral over spin detunings. The system becomes unstable when$${\overline{pC}}=\,\int\,p({\varDelta} )\,C({\varDelta} )\,\rho ({\varDelta} )\,d{\varDelta} > 1\,.$$ (11) This expression highlights that the instability is dominated by the spin inversion near resonance p(Δ = 0), where C(Δ) is sharply peaked. Although this threshold strictly applies only in the absence of spin–spin interactions, it remains a valuable heuristic for interpreting the self-pulsing behaviour.Accordingly, the peak cavity amplitude during superradiant emission follows the relation$$\max (| a| )\propto {p}_{0}-\frac{1}{C}\,$$ (12) as a consequence of the equations of motion (equation (8c)). To motivate this relation, we consider the moment when the cavity field reaches its maximum: ∂t〈a〉 ≈ 0. Equation (8a) then implies \(\langle a\rangle \propto {\sum }_{j}\langle {\sigma }_{j}^{-}\rangle \equiv {S}_{-}\). The transverse spin component S− = Sx − iSy is built up during the superradiant burst via the collective rotation of an initially +z-oriented ensemble and peaks simultaneously with the cavity amplitude in our superradiant regime. The system starts with Sz ∝ p0N and negligible initial coherence S− ≈ 0, the latter maintained at almost all times by rapid single-spin dephasing γ⊥. Only during the superradiant emission does coherence buildup dynamically. This leads to a linear scaling \(\max (| a| ) \approx {p}_{0}N\). The offset in equation (12) reflects the superradiant threshold: for p0 < 1/C, the system remains stable and does not emit collectively. Since the dynamics are dominated by cavity-resonant spins, this scaling primarily reflects the on-resonance inversion p(Δ = 0).