Self-Induced Superradiant Masing

Abstract

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

Figures (8)

• Figure 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 upon 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 $|a|$ as compared to panel 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. The purple shaded area marks the interval for the Fourier analysis, which is plotted in e, where the emission has a linewidth much smaller than the cavity, see inset. The frequency difference $\Delta \omega /2\pi$ is measured relative to the cavity frequency of $3.1G Hz$. f, The frequency and linewidth of the emission changes over time when the window of the Fourier analysis is shifted (see text for details).
• Figure 2: Influence of second hold time on revival dynamics.a, Stacked cavity signals of the superradiant dynamics with a second stabilization sequence (hold time), represented by light green shading. Increasing the duration of this second hold time extends the spectral hole-filling process, influencing the amplitude of the superradiant masing pulse revival. b, Revival amplitude $|a|_\mathrm{rev}$ for varying second hold times. An initial stretched exponential increase is followed by an exponential decrease for longer timescales (see inset). The data points are well described by simulating the on-resonance inversion using the parameters of Fig. \ref{['fig3:spectralholesim']}.
• Figure 3: Simulation of superradiant dynamics driven by spin-spin interactions.a, Simulation of the cavity amplitude $|a|$ based on the microscopic spin model compared to measurement (offset vertically for clarity), with four key timesteps marked (cf. e). b, NV– center in the diamond unit cell, illustrating one of four possible alignments. c, Schematic of the simulated spin network: NV– centers are randomly distributed in space, orientation, and resonance frequency, and interact via dipole-dipole couplings. d, Relevant linewidths in the hybrid system: cavity linewidth, inhomogeneous spin distribution of width $W$, single-spin dephasing rate $\gamma_\perp$, and the frequency-dependent cooperativity function $C(\Delta)$. e, Simulated spin inversion profiles at four key times: (i) at $t=0$: uniform inversion, (ii) at $t=5.5µ s$: deep spectral hole, (iii) at $t=16µ s$: refilled above threshold, (iv) at $t=100µ s$: broad, depleted quasi-steady state.
• Figure 4: Probing superradiant dynamics after a hole-burning pulse.a, Cavity dynamics when applying a strong microwave pulse (red shading) to the spins while detuned (green shading) and subsequently triggering the superradiant decay and revival pulses. This will create a spectral hole in the ensemble, resulting in a slice of decreased spin inversion at varying frequencies for different runs. A resonant hole-burning pulse reproduces a spectral hole similar to the one created by the initial superradiant decay when no hole-burning pulse is applied. When the hole-burning pulse is off-resonant at the sides of the spin distribution, it still leads to an earlier triggering of the initial superradiant decay, but will otherwise recreate similar dynamics as when no pulse is applied. b, Maximum amplitude of the superradiant decay pulse for different hole-burning frequencies, revealing the frequency distribution of the detuned spins, although appearing with a larger width due to power broadening.
• Figure 5: Influence of initial inversion on refilling time.a, Measured time delay $\Delta t$ between the initial superradiant decay and the first revival pulse (circles), plotted as a function of the reduced initial inversion $p_0 - 1/C$. This reduced inversion corresponds to the depth of the spectral hole created by the initial emission, since the resonant spins involved effectively undergo a $\pi$-rotation (cf. panel c). b, Experimental runs for three different initial inversion levels $p_0$, illustrating how $\Delta t$ is extracted from the cavity signal $|a|$. The values of $p_0$ are obtained by fitting the initial superradiant decay using Maxwell-Bloch equations [Eqs. (\ref{['eq:MBE_all']}) without spin-spin interactions]. c, From such Maxwell-Bloch simulations, we obtain access to the inversion profile. These profiles are shown for different initial $p_0$, evaluated at the time when the cavity amplitude $|a|$ reaches its peak value during the initial superradiant pulse. In all cases, the initial inversion $p_0$ is assumed to be uniform across the spin ensemble. d, Using the stretched exponential relaxation of the on-resonance inversion, Eq. \ref{['eq:strechedExp']}, we can estimate the refilling dynamics with the experimentally obtained time constant $T_\mathrm{r} = 11.6µs$ from the fit of Fig. \ref{['fig2:secondholdtime']}b. We extract from these curves the time $\Delta t$ when the on-resonance inversion reaches $1/C$ (squares in panel a). This time delay $\Delta t$ serves as a measure for how long it takes to refill the spectral hole above the instability threshold $p(\Delta\,{=}\,0) > 1/C$, enabling the revival pulse. We find good agreement between the experiment and the estimated time delay.