Self-induced superradiant masing

Nat. Phys. 22, 158-163 (2026) · DOI: 10.1038/s41567-025-03123-0 · arXiv: 2402.08537

License: CC BY 4.0.

Abstract

In cavity quantum electrodynamics and particularly superradiance, emitters are typically assumed to be independent, interacting only through light shared via a common mode. Although such photon-mediated interactions lead to a wide range of collective optical effects, direct dipole–dipole interactions within the emitter ensemble are generally viewed as a source of decoherence. Here we report the role of direct spin–spin interactions as a drive for the superradiant dynamics of a hybrid system of nitrogen-vacancy centre spins in a diamond coupled to a superconducting microwave cavity. After an initial fast superradiant burst, we observe a train of subsequent emission pulses followed by quasi-continuous masing for up to one millisecond. We show that this behaviour arises from spectral hole refilling, where spin inversion is redistributed into the superradiant window of spins resonant with the cavity. We report measurements that exclude other cavity-related effects and perform microscopic simulations that confirm that the observed behaviour is driven by dipole–dipole interactions between the spins. These findings open pathways for exploring complex spin–spin interactions in dense disordered systems and offer possibilities for ultranarrow-linewidth solid-state superradiant masers powered purely by microwave-driven spin control.

Figures

24 panels with data across 7 figures. Each panel page shows the plot, its columns and its files; each data.csv begins with a header naming the paper, the panel, the source, the license and the provenance route.

Fig. 1

• panel (b): Cavity amplitude $|a|$ (arb. units) against time ($\mu$s) during the initial superradiant decay: measured (dark grey) and simulated with the semiclassical model (blue). data.csv
• panel (c): Cavity amplitude $|a|$ (arb. units) against time ($\mu$s) over the train of masing pulses and the onset of quasi-continuous emission; the initial burst runs off the top of the axis as printed. The green and grey shadings and the pulse arrows are not drawn; the line at 0 $\mu$s marks the end of the detuning. data.csv
• panel (d): Quadratures $I$ (blue) and $Q$ (orange) and the amplitude $|a|$ (grey) of the cavity field, in arb. units, against time ($\mu$s) over the long tail of quasi-continuous masing; values beyond the printed axis range are not drawn. Two purple lines bound the Fourier window starting at $t_{\mathrm{F}}$, shaded in print. data.csv
• panel (e): Fourier spectrum $|\mathrm{FFT}(I + iQ)|^2$ ($\times 10^6$ arb. units) of the emission in the first window of (d) against detuning $\Delta\omega/2\pi$ (kHz) from the cavity frequency: FFT points (open circles) and Lorentzian fit (line). The FWHM arrows are not drawn; the inset is panel e-inset. data.csv
• panel (e-inset): Inset of panel (e): the Lorentzian fit of the masing emission (purple) beside the cavity resonance (black), each normalized, over $\pm 1$ MHz around the cavity frequency. The print draws the inset without axes; the axis scales here are added. data.csv
• panel (f): Peak frequency $\Delta\omega/2\pi$ (kHz) of the masing emission against the start $t_{\mathrm{F}}$ ($\mu$s) of the 200 $\mu$s Fourier window (line), with the FWHM of the Lorentzian fit as a shaded band around it. data.csv

Fig. 2

• panel (a): Cavity amplitude $|a|$ (arb. units) against time ($\mu$s) for three runs with no second hold time and second hold times of 6 and 45 $\mu$s, stacked with vertical offsets of 0.5 and 1.1 as printed; coloured squares mark each revival maximum. The green shading of the hold times is not drawn; the vertical line marks the start of the second hold at 10 $\mu$s. data.csv
• panel (b): Revival amplitude $|a|_{\mathrm{rev}}$ against second hold time ($\mu$s): measured revival amplitudes (open squares, with the three runs of (a) as filled coloured squares), the stretched exponential fit $\propto -\exp(-\sqrt{t/T_{\mathrm{r}}}) + \mathrm{const}$ (magenta dash-dot) and the simulated on-resonance inversion (grey, arb. units). The inset is panel b-inset. data.csv
• panel (b-inset): Inset of panel (b): revival amplitude $|a|_{\mathrm{rev}}$ (log scale) against second hold time (ms) for hold times above 200 $\mu$s, with the exponential decay fit (orange) labelled $\tau = 8.1$ ms. data.csv

Fig. 3

• panel (a): Cavity amplitude $|a|$ (arb. units) against time ($\mu$s): microscopic spin-model simulation (blue) and measurement (grey, offset vertically by 0.14 for clarity, as printed). Vertical lines mark the four key times (i) to (iv) of panel (e) at 0, 5.5, 16 and 100 $\mu$s; the green shading before 0 $\mu$s is not drawn. data.csv
• panel (d): Linewidths of the hybrid system against spin detuning $\Delta/2\pi$ (MHz), each curve normalized and offset as printed (no vertical scale): the cooperativity function $C(\Delta)$, the spin distribution $\rho(\Delta)$ of width $W$, the single-spin line of width $\gamma_\perp$ and the cavity resonance. The width arrows are not drawn; $\rho(\Delta)$ is one colour here, colour-graded in print. data.csv
• panel (e): Simulated spin inversion against spin detuning $\Delta/2\pi$ (MHz) at the four key times $t = 0$, 5.5, 16 and 100 $\mu$s, with the threshold $p = 1/C$ (dotted magenta). data.csv

Ext. Data Fig. 1

• panel (a): Cavity amplitude $|a|$ (arb. units) against time ($\mu$s) for runs with a resonant hole-burning pulse (bottom), no pulse (middle, offset 1.2) and an off-resonant pulse (top, offset 2.4), as printed; squares mark each superradiant decay maximum. The red pulse and green detuning shadings are not drawn; vertical lines mark their edges. data.csv
• panel (b): Maximum cavity amplitude $\max(|a|)$ of the superradiant decay against hole-burning pulse detuning (MHz): measured points joined by a line, a Gaussian fit (orange), the level without a pulse (dashed orange, "No pulse") and the two pulsed runs of (a) as coloured squares. data.csv

Ext. Data Fig. 2

• panel (a): Time delay $\Delta t$ ($\mu$s) between the initial superradiant decay and the first revival against the reduced initial inversion $p_0 - 1/C$: measured (grey circles) and estimated from the stretched exponential hole refilling of (d) (squares joined by a line, coloured blue to purple by $p_0$ in print, one colour here). data.csv
• panel (b): Cavity amplitude $|a|$ (arb. units) against the time $\Delta t$ ($\mu$s) after the initial superradiant decay for three runs at initial inversions $p_0 = 0.110$, 0.191 and 0.288, measured (grey) and Maxwell-Bloch simulation of the initial decay (blue), as a 3D waterfall along $p_0$. The green detuning bars, the grey $\Delta t$ plane and the $1/C$ line of the print are not drawn. data.csv
• panel (c): Simulated spin inversion profiles $p(\Delta, t = 0)$ against spin detuning $\Delta/2\pi$ (MHz) from Maxwell-Bloch simulations for ten initial inversions $p_0$ (blue to purple, legend), with the threshold $1/C$ (dotted magenta). data.csv
• panel (d): Hole-centre inversion $p(\Delta = 0, t)$ against time ($\mu$s) from the stretched exponential refilling $p(\Delta = 0) - \bar{p} \propto \exp(-\sqrt{t/T_{\mathrm{r}}})$ for the ten simulated holes of (c), with the threshold $1/C$ (dotted magenta); short vertical marks give the delay $\Delta t$ at which each curve crosses $1/C$ (coloured per curve in print, grey here). data.csv

Ext. Data Fig. 3

• panel (a): The data of Fig. 2(b): revival amplitude $|a|_{\mathrm{rev}}$ (arb. units, left axis, increasing downwards as printed) against 2$^{\mathrm{nd}}$ hold time ($\mu$s), read on the right axis as the relaxation function $y(t)$ that decays from 1 to 0; the horizontal line marks $y(t) = 0$. The printed label $y(t) \sim \exp(-(t/T_{\mathrm{r}})^\alpha)$ is not drawn. data.csv
• panel (b): Double-logarithmic plot of $\ln(-\ln y(t))$ against $\ln t$ for the data of (a) (open squares), with a line of slope $\alpha = 1/2$ (magenta). data.csv

Ext. Data Fig. 4

• panel (a): Dispersive shift $\chi$ (kHz) of the cavity resonance against the detuning (MHz) of a long weak microwave drive, measured (points joined by a line) with a q-Gaussian fit (orange); the printed FWHM arrows and label are not drawn. data.csv
• panel (b): Hahn echo amplitude (arb. units) against the total free evolution time $2\tau$ ($\mu$s), measured (open circles) with an exponential decay fit (blue) labelled $T_2 = 0.89\,\mu$s. The inset is panel b-inset. data.csv
• panel (b-inset): Inset of panel (b): cavity amplitude $|a|$ against time ($\mu$s) during one Hahn echo sequence, with the $(\pi/2)_x$ and $(\pi)_y$ pulses (orange, shaded in print, edges marked here) separated by $\tau$ and the echo after a further $\tau$. The print gives no $|a|$ scale; the hosted values are as stored. The echo marker is not drawn. data.csv
• panel (c): Transmission $S_{21}$ (dB) against frequency (MHz) for the coupled system at low power (grey), the cavity with the spins saturated at high power (blue) and the detuned spins (green): measured points (dark blue for the cavity, lying mostly under its fit) with their fitted curves; the thin vertical line is drawn as printed. data.csv

Cite

Wenzel Kersten et al. (12 authors). Self-induced superradiant masing. Nat. Phys. 22, 158-163 (2026). https://doi.org/10.1038/s41567-025-03123-0

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