Abstract

Superradiance is a fundamental collective effect where radiation is amplified by the coherence of multiple emitters1. Superradiance plays a prominent role in optics (where it enables the design of lasers with substantially reduced linewidths2,3) and quantum mechanics4, and is even used to explain cosmological observations such as Hawking radiation from black holes5. Resonators coupled to spin ensembles6,7,8 are promising future building blocks of integrated quantum devices that will involve superradiance. As such, it is important to study its fundamental properties within such devices. Although experiments in the strong-coupling regime have shown oscillatory behaviour in these systems9,10, a clear signature of Dicke superradiance has so far been missing. Here we explore superradiance in a system composed of a three-dimensional lumped element resonator in the fast cavity limit inductively coupled to an inhomogeneously broadened ensemble of nitrogen–vacancy centres. We observe a superradiant pulse being emitted a trillion times faster than the decay for an individual nitrogen–vacancy centre. This is further confirmed by the nonlinear scaling of the emitted radiation intensity with respect to the ensemble size. Our work provides the foundation for future quantum technologies including solid-state superradiant masers2.

Main

Proposed by Dicke in 19541, superradiance is a collective effect enhancing the radiative decay dynamics of multiple excited emitters such that their decay is much faster than the individual emission rates—hence the name superradiance. Correlations that build up during the decay lead to a nonlinear scaling of the emitted radiation intensity with respect to the number of excited emitters11. The effect of decoherence and dephasing in the system have to occur on a timescale longer than the relevant system dynamics such that coherence can be maintained throughout the decay12, which has made superradiance difficult to observe. Experiments in a number of different systems have been carried out in a regime where dephasing effects are suppressed using a few qubits13,14,15,16, by taking advantage of special symmetries in the sample geometry17,18, or in Bose–Einstein condensates19 and optical lattices3,20,21. A different approach is to use a resonator to decrease the mode volume of the electromagnetic field9,10,13,22,23, which enhances the coupling when compared with dephasing mechanisms such as broadening induced by dipolar interaction. To realize conditions close to those seen for superradiance in vacuum, the system needs to operate in the fast cavity limit, where the cavity decay rate is larger than all other time constants in the system24,25,26, or in the dispersive regime27,28.

A suitable model system is an ensemble of negatively charged nitrogen–vacancy (NV) centres in diamond, coupled to a microwave resonator operating in the fast cavity limit. Our measurements described below show both the nonlinear emission of radiation from the fast cavity, while optical readout of the spin system confirms an enhanced decay rate a trillion times faster compared to the lifetime of a single NV centre. These observations are clear signatures of superradiance.

In our experiment, as illustrated in Fig. 1, we use a dense ensemble of NV centres with a narrow spectral linewidth of \(\gamma _ \bot ^ \ast {\mathrm{/}}2{\rm{\pi}} \approx 4.7\,{\mathrm{MHz}}\) (full-width at half-maximum, FWHM) to reduce dephasing effects originating from inhomogeneous broadening29. The sample is placed in a three-dimensional (3D) lumped element resonator30 with a fundamental frequency of 3.18 GHz and a cavity linewidth of κ/2π = 13.8 MHz (FWHM) corresponding to a quality factor of Q = 230. The resonator focuses the magnetic field such that the spins are homogeneously coupled to the cavity mode with almost no spatial dependence on the coupling rate. This allows us to perform coherent operations on the entire spin ensemble (~1016 spins) and achieve inversion using short microwave pulses, as illustrated in Fig. 1. Operating in the fast cavity limit ensures that the cavity increases the effective coupling to the spin system, but at the same time realizes conditions similar to superradiance in vacuum24. To meet the requirements necessary for the fast cavity limit, we ensure that the photon lifetime in the cavity of τc ≈ 11 ns is shorter than any of the occurring system dynamics.

Fig. 1: Experimental setup.
Fig. 1

a, Our hybrid system consists of a 3D lumped element resonator (rendering without the top lid and the side walls) bonded with a diamond sample in combination with the laser readout scheme and the microwave setup operating at 25 mK in a dilution refrigerator. A 3D Helmholtz coil configuration provides magnetic fields in arbitrary directions with field strengths up to 200 mT. A fibre is glued to one side of the diamond sample to apply optical readout pulses and collect the scattered fluorescence, with a dichroic mirror to separate them. The microwave setup consists of a standard autodyne detection scheme with two mixers for (de-)modulation of the I/Q-signal. b, The diamond lattice shows four different orientations for a given vacancy, resulting in four possible different NV directions (labelled I–IV). c, The Zeeman tuning of the NV levels by an external magnetic field is given by the projection of the magnetic field on the spins’ principal axis. This allows us to bring either one, two, three or four NV subensembles into resonance with the cavity mode (ωc), by applying appropriate magnetic fields in the [1, 1, 1], [1, 1, 0] and [1, 0, 0] directions.

We begin our exploration of this hybrid system by driving the spins with a 50 ns long pulse and measure the emitted radiation intensity from the cavity as a function of the input power as presented in Fig. 2. For low power levels the system primarily remains in the ground state, corresponding to the behaviour in the low-excitation (boson-like) regime. For a certain threshold power we achieve maximum inversion, indicated by the black dashed line in Fig. 2. At this point the system is in a metastable state—comparable to a classical pendulum turned upside down. The coupling of the spin system to the cavity mode is suppressed and it remains in this state for an extended period of time. In this metastable state the excitations reside in the spin system with the cavity mode empty. Approximately 300 ns after we switch off the microwave excitation, we observe a burst of photons exiting the cavity, depicted as red measurement data in Fig. 3. If we further increase the input power, the delay of this photon burst becomes shorter, because excess photons from the drive pulse keep interacting with the spin ensemble. This stimulates the emission of photons and drives the spins out of their metastable state earlier. The dependence of the delay on this tipping angle θ is given by td ≈ Γsrlog[tan2(θ/2)] as derived in Methods and shown in Fig. 2 as a green dashed line. For smaller drive powers (<−5 db attenuation of the maximum input power) we observe a similar behaviour, as a pulse exits the cavity earlier than for full inversion. In this case the number of photons in the excitation pulse is insufficient to invert the spin ensemble, again leading to an earlier emission. Due to inhomogeneous broadening, the intensity of these emitted pulses is smaller than for large drive powers. After driving the system and while emptying the cavity mode, the spins remain in a state with a high polarization (S_) value for an extended period of time. Dephasing and decoherence are most prominent here and reduce the signal intensity.

Fig. 2: Cavity response under varying drive powers.
Fig. 2

The response after applying a coherent microwave pulse of 50 ns with three NV subensembles in resonance with the cavity. The colour bar shows the emitted intensity \(|A|^2\). At a power level approximately equal to −5 db attenuation of the maximum power (indicated with a black dashed line) we achieve maximum inversion and observe a pulse emitted from the cavity (green dotted line). For higher powers the delay of this emitted pulse becomes shorter, since excess photons in the cavity lead to stimulated emission. The dependence of the delay on this tipping angle is fitted by equation (9) as derived in Methods. The inset shows the fluorescence collected from the scattered light of the NV ensemble 50 ns after the drive pulse is switched off. This driven Rabi-like measurement confirms that we reach maximum inversion (lowest fluorescence) for −5 dB attenuation of the maximum input power. The error bars are the s.d. of the measured fluorescence.

Fig. 3: Dynamics of the superradiant decay.
Fig. 3

a, Close to the region where the inversion Sz is zero and the spin quantum number is maximum, correlations in the spin system lead to an enhanced photon emission rate ΓN2 (note that we assume an S = 1/2 system here). Uncorrelated emission ΓN governs the decay for the excited spin system. Dephasing decreases the spin quantum number and evolves the system out of the purely symmetric subspace. Superradiance also occurs in these partly dephased inner shells, but the fully symmetric ground state is no longer reached after the decay. Further, the number of photons emitted during the superradiant decay becomes smaller. b, The red measurement curve shows a detailed view of the trace for the emitted photon intensity with three NV subensembles in resonance with the cavity and where the inversion of the spin ensemble is maximum (depicted as a black dashed line in Fig. 2). The shaded area is the time for which the excitation drive is turned on. Shown in blue is the dynamics of the spin inversion, measured using the optical transition of the NV centre and the inversion polarization normalized to the number of spins. The error bars are the s.d. of the measured fluorescence. The inset shows the measurement sequence with microwave (MW) excitation and optical readout pulses. After maximum inversion is reached the spins remain in a metastable state until fluctuations lead to a stimulated superradiant decay. This is accompanied by a burst of photons that builds up in the cavity mode. The blue solid line represents a fit of the fluorescence data according to a hyperbolic tangent11,43 as derived in Methods.

Since superradiance is the enhanced coherent decay of the inverted spin system, it is instructive to measure the polarization inversion of the ensemble during this decay. The NV centre possesses an optical transition, which enables a direct measurement of the spin polarization by optically detected magnetic resonance31,32. We implement optically detected magnetic resonance in our experiment at ultra-low (mK) temperatures by illuminating parts of the sample using 20 ns long optical pulses delivered through an optical (multimode) fibre and collect the scattered fluorescence with the same fibre (see Methods for details). The fluorescence level then gives a direct measurement for Sz. By varying the time delay of the laser readout pulse with respect to the microwave pulse set for maximum inversion of the spin system, a time-resolved measurement of the inversion is obtained. We measure the microwave cavity output and the scattered fluorescence simultaneously (details of the measurement are presented in Methods). As can be seen from the blue measurement data in Fig. 3, the inverted spin ensemble stays in its metastable state for approximately 200 ns before thermal and vacuum fluctuations stimulate the decay of the spins and the emission of microwave photons in a characteristic superradiant burst, as depicted in red in Fig. 3b. During drive and decay, dephasing evolves some spins into a subradiant state (see Fig. 3a). This has the effect that we do not reach full inversion after the drive pulse and the spins do not decay back into the fully symmetric ground state after the superradiant decay, with 5% incoherent inversion remaining. Dephasing also reduces the number of photons emitted during the superradiant burst. These dephasing effects are evident from the fluorescence level, which does not reach the initial level after the superradiant burst (Fig. 3b). The remaining incoherent excitation decays either through single-spin Purcell-enhanced spontaneous emission33 (ΓP/2π ≈ 8 × 10−10 Hz) or by longitudinal relaxation mediated by spin–phonon interactions34 (γ||/2π ≈ 3 × 10−5 Hz), with the latter dominating in our solid-state spin system.

We can study the nonlinear scaling of the emitted photon radiation intensity, as expected from Dicke superradiance, by tuning either one, two, three or four NV subensembles in resonance with the cavity mode and thus changing the number of spins coupled from about 0.38 to 1.5 × 1016. In Fig. 4a we show the corresponding measurement data for the emitted intensity where the delay of the superradiant burst is maximum for each of these four cases. By determining the maximum values of the intensities of the superradiant bursts and plotting them as a function of the number of spins coupled (Fig. 4b), we observe the dependence IN1.52. This nonlinear scaling, as expected from a superradiantly enhanced decay11,35, clearly demonstrates the superradiant nature of the emitted photon burst.

Fig. 4: Nonlinear scaling of the emitted radiation intensity.
Fig. 4

a, By bringing either N, 2N, 3N or 4N spins into resonance with the cavity mode we are able to measure the dependence of the emitted photon intensity on the number of spins. Depicted here are the traces for each of these cases where maximum inversion is reached, and a microwave drive duration of 50 ns, with the blue dots indicating the maximum power of the emitted radiation intensity. b, By measuring the maximum value of the emitted intensity we observe the nonlinear scaling \(|A|^2 \propto N^{1.52}\). The dashed grey lines show N and N2 scaling. The error bars are the standard errors of the mean on the emitted power, with the red shaded area the 95% confidence interval of the fit. c, The delay and the width of the superradiant burst as functions of the number of spins. Error bars, s.d. of amplitude and delay. The width of the pulse can be modelled by equation (12) in Methods, and shows the expected 1/N dependence11 (see the red dashed line). The delay of the burst does not closely follow equation (9), since it is very sensitive to small deviations in the inclination angle, which are experimentally hard to prevent. Further, for smaller number of spins the theoretically predicted delay is much longer (>5) than other decay mechanisms driving the spins out of the metastable state earlier.

Our system can be described by the driven Tavis–Cummings Hamiltonian36, which gives the dynamics of N spins coupled to a quantized field mode. In the fast cavity limit it is straightforward (details in Methods) to derive an equation of motion for the inversion of the spins as

$$\left\langle {\dot S_z} \right\rangle = - \frac{{4g\eta }}{\kappa }\left\langle {S_y} \right\rangle - \gamma _{||}\left( {\left\langle {S_z} \right\rangle + N} \right) - \frac{{4g^2}}{\kappa }\left\langle {S_ + S_ - } \right\rangle$$
(1)

where η is the drive amplitude, κ the decay rate of the cavity intensity, g the coupling rate of a single spin to the cavity mode, γ|| the longitudinal decay rate and Sx,y,z, S± the collective spin operators.

Analysing equation (1), it is apparent that the cavity provides an enhancement of the spontaneous emission for the undriven system (η = 0), known as the Purcell factor37. This Purcell factor 4g2/κ is increased by emerging correlations in the spin system during the decay of the inverted spin ensemble with a maximum proportionality of S+SN2 (derived in Methods). For our typical system parameters this enhances the small Purcell factor of ΓP/2π ≈ 8 × 10−10 Hz by about 16 orders of magnitude to a value of several megahertz. The enhancement is clearly visible from the measurement data as displayed in blue in Fig. 3b, where we observe a spin decay with a maximum rate of Γsr/2π ≈ 17 MHz, much faster than the Purcell-enhanced decay of a single spin.

These emerging correlations also account for the fact that the intensity of the superradiant radiation emitted by the decaying spin system \(I \propto \frac{{g^2}}{\kappa }S_ + S_ -\) scales nonlinearly as the number of spins coupled to the cavity mode changes24,26. The nonlinear scaling of the emitted radiation intensity as illustrated in Fig. 4b, together with the enhanced decay of the spins as derived in equation (1) and shown in Fig. 3, are both indicators of the superradiant nature of the emitted light.

Pure N2 scaling is only expected for the limit of emitters without a cavity mode. However, as we are working in a cavity the scaling of the emitted radiation intensity becomes smaller the narrower the cavity linewidth becomes35. Further, this is enhanced by experimental imperfections, such as slightly misaligned magnetic fields for more than one subensemble in resonance with the cavity mode, which leads to more dephasing for more than one subensemble coupled. By ensuring that the cavity linewidth surpasses all other time constants and minimizing experimental imperfections, we still observe a nonlinear scaling as expected from Dicke superradiance. The fact that the fast cavity limit is only a first-order approximation also accounts for the occurrence of a faint second pulse in Fig. 2 for high drive powers. Photons that are emitted during the first superradiant pulse do not exit the cavity immediately and instead are partially reabsorbed by the spin ensemble, creating a superradiant burst.

Our studies described above clearly demonstrate Dicke superradiance in a hybrid quantum system embedding a macroscopic solid-state spin ensemble, by measuring both the nonlinear scaling of the emitted radiation and the enhanced decay of the spins. In particular, the implementation of optically detected magnetic resonance techniques shows that protocols that rely on the optical transition in the NV centre are feasible even at low temperatures38. This opens up new perspectives for solid-state hybrid quantum systems employing both the full coherent microwave and optical control, and shows the versatility of these types of system. Our experiments show that the NV centre is a suitable candidate for quantum technologies that rely on a superradiant enhancement, such as the NV-based superradiant maser by pumping of the optical transition instead of driving the microwave transition directly2,39. The superradiantly enhanced decay of the spins back into the ground state increases the repetition rate of the experiment by many orders of magnitude, which allows us to perform high-sensitivity measurements40,41,42 without relying on the small spontaneous emission rate at low temperatures to polarize the spins back into the ground state.

Methods

Spin system

The negatively charged nitrogen–vacancy centre is a paramagnetic point defect centre in diamond, consisting of a nitrogen atom replacing a carbon atom and an adjacent lattice vacancy. Two unpaired electrons form a spin S = 1 system, which can be described by a simplified Hamiltonian of the form \(H = \hbar DS_z^2 + \hbar \mu B_zS_z\) with a zero-field splitting parameter D/2π = 2.878 GHz, an axial magnetic field strength Bz and a gyromagnetic ratio of μ/2π = 28 MHz mT−1. The splitting corresponds to a temperature of D/ħkb = 138 mK, which allows us to thermally polarize the spins to the ground state at the refrigerator base temperature of ~25 mK with more than 99% fidelity. The diamond lattice possesses four different crystallographic orientations (shown in Fig. 1b), which results in four different possible directions for the NV centre in the \(\langle 1,\,1,\,1\rangle\) directions. Each of these four vectors shows the same abundance of NV centres all exhibiting the same properties. By applying magnetic fields in the [1, 0, 0] direction, the external magnetic field projection onto the NV axis is equal for all four subensembles, which allows us to tune all of them into resonance with the cavity mode. By applying appropriate magnetic fields in the [1, 1, 1] direction, we can bring either only one or three subensembles into resonance with the cavity depending on the magnetic field strength. Two subensembles can be brought into resonance by applying magnetic fields in the [1, 1, 0] direction.

NV sample

The sample is a type-Ib high-pressure, high-temperature diamond crystal with an initial nitrogen concentration of 50 parts per million (ppm). To create lattice vacancies, the sample is irradiated with electrons of energy 2 MeV at 800 °C and is subsequently annealed multiple times at 1,000 °C. The total electron dose was 5.6 × 1018 cm−2. This gives a total NV density of 13 ppm and an inhomogeneously broadened linewidth of γinh/2π ≈ 4.7 MHz (FWHM). This value is fundamentally limited by the hyperfine coupling to the nuclear spin of nitrogen (2.3 MHz). The small spectral broadening allows us to increase our signal-to-noise ratio and number of photons emitted during the superradiant decay. Further, excess nitrogen P1 centres (S = 1/2), uncharged NV0 and naturally abundant 13C nuclear spins serve as a source of decoherence and inhomogeneous broadening. The electron irradiation was performed using a Cockcroft–Walton accelerator in QST, Takasaki (Japan). The linewidth was measured using a dispersive measurement scheme relying on the dispersive shift of the microwave resonator coupled to a polarized spin ensemble8.

Spin decay

The longitudinal decay of the NV ensemble γ|| is quantified using a dispersive measurement scheme, and exhibits the small value of 3 × 10−5, mediated by spin–phonon interactions34. This value is many orders of magnitude larger than the Purcell factor, which on resonance is given by

$$\Gamma _{\mathrm{P}} = 4\frac{{g^2}}{\kappa }$$
(2)

With an estimated single-spin coupling strength of g/2π = 72 and a cavity linewidth of κ/2π = 13.8, this computes to a value of ΓP/2π = 7.5 × 10−10 Hz.

Hybrid system

We use a 3D lumped element resonator machined from oxygen-free copper that allows us to couple homogeneously to the whole diamond sample with a root mean square deviation of the coupling rate of approximately 1.5%30. Our hybrid system is formed by bonding the NV sample within the 3D lumped element resonator. The collective coupling strength to the entire spin ensemble changes from 3.1 to 6.2 MHz depending on the number of subensembles coupled to the cavity mode. We operate the resonator in the largely overcoupled regime \(\kappa _1 \gg \kappa _2 \gg \kappa _{\mathrm{int}}\) with the input coupling much larger than the output coupling. Because the linewidth is governed by external (coupling) losses instead of internal (ohmic) losses, we refer to this as the ‘fast’ cavity limit. This increases the number of photons in the cavity and consequently reduces the time it takes to perform a π-pulse. Further, it allows us to perform the experiments in the fast cavity regime, such that the cavity linewidth surpasses all other time constants in the system (\(\kappa > \Delta _j,\gamma _ \bot ,\gamma _{||},\sqrt N g\)).

Photon intensity measurements

We measure the emitted photon intensity by performing transmission measurements using a autodyne detection scheme. The carrier frequency is split into two paths, with one serving as the signal going into the cryostat. It is modulated using a fast arbitrary waveform generator (Tabor WW2182B) with 2 GS s−1 sampling frequency to produce short microwave pulses. The other path is used as a reference signal to demodulate the signal exiting the crysotat. The demodulated signal, with the intensity proportional to the emitted photon flux, is then recorded by a fast digitizer card (Acqiris U1082A) with 5 GS s−1 sampling frequency.

Optical fluorescence measurement

To obtain a relative measure for the inversion of the spin system, we make use of the optical transition of the NV centre, which gives us a direct way to measure the expectation value for the Sz component. We glue a multimode fibre to one side of the diamond sample and illuminate it with short (20 ns) green 526 nm laser pulses using a high-bandwidth laser diode (LD-520). The scattered red light is collected using the same fibre, and filtered using a dichroic mirror reflecting the ingoing green light and transmitting the scattered red light. An optical longpass filter further filters the scattered red light, which is then measured by an avalanche photodiode (Thorlabs ADP110A). Because the coupling rate to the cavity mode shows almost no spatial dependence, we can assume that optical readout of a small part of the sample is equivalent to the readout of the whole ensemble. We record the fluorescence for several hundred repetitions for different time delays of the readout pulse with respect to the microwave excitation pulse. For a spin system that is fully polarized in the ground state this gives the highest fluorescence, while the maximally inverted spin system exhibits the lowest fluorescence. By comparing the fluorescence values of this scattered light we obtain a relative measure for the Sz component.

Equations of motion and the adiabatic elimination in the fast cavity limit

To derive the equation of motions in the fast cavity limit we begin with a driven Tavis–Cummings Hamiltonian for N spins in the rotating frame

$$\begin{array}{ccccc}\cr {\cal H} = & \hbar \Delta _{\mathrm{c}}a^{\mathrm{\dagger }}a + \frac{\hbar }{2}\mathop {\sum}\limits_{j = 1}^N {\Delta _{\mathrm{s}}^j\sigma _z^j} &\\\cr& + {\mathrm{i}}\hbar g\mathop {\sum}\limits_{j = 1}^N {\left( {a^\dagger \sigma _ - ^j - \sigma _ + ^ja} \right) + {\mathrm{i}}\hbar \eta \left( {a^\dagger - a} \right)} \cr \end{array}$$
(3)

with \(\sigma _z^j\), \(\sigma _ \pm ^j\) the Pauli-z and raising/lowering operators for the jth spin and a, a the bosonic creation and annihilation operators. Note that the collective spin operators in the main text are defined as \(S_{x,y,z} = \frac{1}{2}\mathop {\sum}\nolimits_{j = 1}^N \sigma _{x,y,z}^j\) and \(S_ \pm = \mathop {\sum}\nolimits_{j = 1}^N \sigma _ \pm ^j\). Further, \(\Delta _{\mathrm{s}}^j\) is the detuning of the jth spin with respect to the cavity resonance frequency accounting for the inhomogeneous broadening of the spectral spin linewidth and Δc is the detuning of the cavity mode with respect to the drive frequency. In the fast cavity limit we adiabatically eliminate the cavity mode, and arrive at a Hamiltonian that only contains spin operators:

$${\cal H}_{\mathrm{at}} = \frac{\hbar }{2}\mathop {\sum}\limits_{j = 1}^N \Delta _{\mathrm{s}}^j\sigma _z^j - 2\hbar \frac{{g\eta }}{\kappa }\mathop {\sum}\limits_{j = 1}^N \left( {\sigma _ + ^j + \sigma _ - ^j} \right)$$
(4)

Using the Lindblad master equation, introducing loss channels for the cavity mode and the spins as

$$\begin{array}{*{20}{l}} {\dot \rho = } \hfill & { - \frac{\mathrm{i}}{\hbar }\left[ {{\cal H},\rho } \right] } \hfill \cr + \hfill & {\kappa \left( {a\rho a^{\mathrm{\dagger }} - \frac{1}{2}\left( {a^\dagger a\rho - \rho a^\dagger a} \right)} \right)} \hfill \cr + \hfill & {\gamma _{||}\mathop {\sum}\limits_j^N \left( {2\sigma _ - ^j\rho \sigma _ + ^j - \sigma _ + ^j\sigma _ - ^j\rho - \rho \sigma _ + ^j\sigma _ - ^j} \right)} \hfill \cr + \hfill & {\gamma _ \bot \mathop {\sum}\limits_j^N \left( {\sigma _z^j\rho \sigma _z^j - \rho } \right)} \hfill \end{array}$$
(5)

we derive an equation of motion for the cavity mode operator as

$$\dot a = - {\mathrm{i}}\Delta _{\mathrm{c}}a + g\mathop {\sum}\limits_{j = 1}^N \sigma _ - ^j + \eta - \frac{\kappa }{2}a$$
(6)

In the fast cavity limit we assume \(\dot a = 0\), since the presence of the coupled spin system in this limit does not alter the cavity amplitude which remains unchanged. The resulting expression for the mode operator, assuming the cavity mode in resonance with the probe frequency (c = 0), can be resubstituted into equation (3), which then leads to a Hamiltonian only for the atomic operators, as given in equation (4). The losses in the cavity in the Lindblad superoperator from equation (5) can then be expressed by the atomic operators:

$${\cal L}_{{\mathrm{cav}}} = \frac{{2g^2}}{\kappa }\mathop {\sum}\limits_{j,k}^N \left( {2\sigma _ - ^j\rho \sigma _ + ^k - \sigma _ + ^j\sigma _ - ^k\rho - \rho \sigma _ + ^j\sigma _ - ^k} \right)$$
(7)

From this we can derive the equation of motions for the spin inversion as given in equation (1).

Delay time and width of the superradiant pulse

The delay of the superradiant pulse can be modelled by equation (1) with η = 0 and neglecting longitudinal relaxation. The resulting differential equation,

$$\left\langle {\dot S_z} \right\rangle = - \frac{{4g^2}}{\kappa }\left( {\frac{N}{2} + S_z} \right)\left( {\frac{N}{2} - S_z + 1} \right)$$
(8)

for a state S, M with S = N/2 and M = Sz = −N/2 cos(θ) can be solved for an arbitrary tipping angle θ. Solving for Sz = 0 gives the time delay

$$\begin{array}{ccccc}\cr t_{\mathrm{d}} = & \frac{\kappa }{{4g^2N}}{\mathrm{log}}\left( { - \frac{{\left( {2 + N} \right)\left( {{\mathrm{cos}}(\theta ) - 1} \right)}}{{2 + N + N\,{\mathrm{cos}}(\theta )}}} \right)&\\\cr& \approx \frac{\kappa }{{4g^2N}}{\mathrm{log}}\left( {\mathop {{\mathrm{tan}}}\nolimits^2 \left( {\frac{\theta }{2}} \right)} \right)\cr \end{array}$$
(9)

for the superradiant burst. The lineshape and the dynamics of the spin decay is obtained from the solution of equation (8) and reads11

$$\left\langle {S_z} \right\rangle = - \frac{N}{2}{\mathrm{tanh}}\left( {\frac{{2g^2N}}{\kappa }\left( {t - t_{\mathrm{d}}} \right)} \right)$$
(10)
$$I \propto - \frac{{{\mathrm{d}}S_z}}{{{\mathrm{d}}t}} = \frac{{g^2N^2}}{\kappa }{\mathrm{sech}}^2\left( {\frac{{2g^2N}}{\kappa }\left( {t - t_{\mathrm{d}}} \right)} \right)$$
(11)

with an FWHM for the burst duration of

$$\tau _{{\mathrm{sr}}} = \frac{{\kappa \,\mathop {{\mathrm{cosh}}}\nolimits^{ - 1} \left( {\sqrt 2 } \right)}}{{Ng^2}}$$
(12)

Nonlinear scaling

During our superradiant decay the time evolution for all emitters is identical and thus correlated because all spins are coupled to the cavity mode equally. Therefore, the emitted radiation is coherent and interferes constructively, leading to the typical N2 scaling of the emitted radiation intensity. This can be seen by rewriting the expression for the polarization in terms of the operators for the individual emitters:

$$\left\langle {S_ + S_ - } \right\rangle = \mathop {\sum}\limits_i^N \left\langle {\sigma _ + ^i\sigma _ - ^i} \right\rangle + \mathop {\sum}\limits_{i \ne j}^N \left\langle {\sigma _ + ^i\sigma _ - ^j} \right\rangle$$
(13)

The first sum gives the number of excitations and is proportional to the number of emitters N. The second sum accounts for interference terms and contains N2 elements, responsible for the nonlinear scaling of the emitted radiation intensity as given by \(I \propto \frac{{g^2}}{\kappa }S_ + S_ -\). Dephasing processes lead to differing phases for the \(\sigma _ + ^i\) and \(\sigma _ - ^j\) terms. For strong dephasing the second sum averages to zero and the intensity scales linearly with N as in regular spontaneous emission, with no superradiant burst observable. For an excited ensemble of emitters the coherence effects are negligible because the polarization term \(\sigma _ + ^i\sigma _ - ^j\) is zero. Half way through the decay process these terms become maximal, leading to the typical radiation intensity peak. Due to the fact that the experiment is carried out in a cavity with additional experimental imperfections such as misaligned magnetic fields for more than one subensemble in resonance with the cavity, this nonlinear scaling exponent becomes smaller than two.

Data availability

The data that support the plots within this paper and the findings of this work are available from the corresponding author upon request.

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Acknowledgements

We would like to thank D. Krimer, M. Zens, S. Rotter and H. Ritsch for discussions and G. Wachter for help with the setup of the laser system. The experimental effort has been supported by the Top-/Anschubfinanzierung grant of the TU Wien and the JTF project “The Nature of Quantum Networks” (ID 60478). A.A. and T.A. acknowledge support by the Austrian Science Fund (FWF) in the framework of the Doctoral School “Building Solids for Function” Project W1243. K.N. acknowledges support from the MEXT KAKENHI Grant-in-Aid for Scientific Research on Innovative Areas “Science of Hybrid Quantum Systems” no. 15H05870. J.I. acknowledges support by the Japan Society for the Promotion of Science KAKENHI grant no. 26220903 and grant no. 17H02751. J.S. acknowledges financial support from the Wiener Wissenschafts- und TechnologieFonds (WWTF) project No MA16-066 (“SEQUEX”).

Author information

Author notes

    • Kirill Streltsov

    Present address: Institute for Theoretical Physics, Universität Ulm, Ulm, Germany

    • Stefan Putz

    Present address: Department of Physics, Princeton University, Princeton, NJ, USA

Affiliations

  1. Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, Vienna, Austria

    • Andreas Angerer
    • , Kirill Streltsov
    • , Thomas Astner
    • , Stefan Putz
    • , Jörg Schmiedmayer
    •  & Johannes Majer
  2. Sumitomo Electric Industries Ltd., Itami, Japan

    • Hitoshi Sumiya
  3. National Institutes for Quantum and Radiological Science and Technology, Takasaki, Japan

    • Shinobu Onoda
  4. Research Centre for Knowledge Communities, University of Tsukuba, Tsukuba, Japan

    • Junichi Isoya
  5. NTT Basic Research Laboratories, NTT Corporation, Atsugi, Japan

    • William J. Munro
  6. National Institute of Informatics, Hitotsubashi, Tokyo, Japan

    • William J. Munro
    •  & Kae Nemoto
  7. Wolfgang Pauli Institute c/o Faculty of Mathematics, Universität Wien, Vienna, Austria

    • Johannes Majer

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Contributions

A.A., S.P., T.A., J.S. and J.M designed and set up the experiment. A.A. and K.S. carried out the measurements under the supervision of J.M. W.J.M. and K.N provided the theoretical framework. J.I., S.O. and H.S. characterized and provided the diamond sample. A.A. wrote the manuscript, to which all authors suggested improvements.

Competing interests

The authors declare no competing interests.

Corresponding authors

Correspondence to Andreas Angerer or Johannes Majer.

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DOI

https://doi.org/10.1038/s41567-018-0269-7