1. Introduction

Superradiance, first predicted by Dicke [1], is the collective spontaneous emission of highly condensed emitters in a dimension smaller than a transition wavelength [1]. To date, superradiance has been realized in various platforms, including cold atoms [2–5], trapped ions [6], Bose-Einstein condensates [7–10], semiconductors [11–14], free-electron lasers [15–18], Josephson junctions [19,20], quantum dots [21,22], nano-diamonds with nitrogen-vacancy centers [23], etc. In fact, superradiance occurs by strong collective coupling between optical fields and dipole momenta of emitters [1,2,24]. When this strong collective coupling is obtained, it synchronously modulates the radiation of each emitter and eventually leads to frequency synchronization as well as resulting in phase locking amongst them [2,23–26], thereby exhibiting their collective behavior, i.e., the collective spontaneous emission with high coherence. Indeed, this self-constructed high coherence has a potential to break through the coherence limit of cavity-based lasers [2,3,27]. To exploit this fascinating characteristic and potential, the subject on superradiant synchronization of emitters has intensely been studied in various fields [6,24,28–30].

In particular, two mesoscopic atomic ensembles have recently been investigated for superradiant synchronization [24,29], the key findings of which were two folds: (1) The synchronization occurs only when the transition frequency difference between the atomic ensembles is below a certain limit. (2) Strong enough coupling capable of modulating frequencies of the atomic ensembles can still exist even when they are unsynchronized. It is, in fact, noteworthy that the influence of collective coupling becomes significant as long as the transition frequency difference is sufficiently small even when the frequencies fail to converge completely. Unlike a perfectly synchronized superradiant state, this frequency difference offers a beating mechanism, so that the resultant oscillation of the total field in turn gives rise to a periodic influence on the two atomic ensembles.

Inspired by these recent studies [24,29], which mainly focused on the perfectly synchronized state of two mesoscopic ensembles of atoms in the context of steady-state superradiance, we aim to investigate a general unsynchronized superradiant state that can collectively give rise to higher-order frequency components of superradiant emission in the context of non-steady-state superradiance. In fact, the experimental work reported in [29] could not be explained to a satisfactory level under the constant steady-state approximation. Therefore, in order to deal with the transient behavior of an unsynchronized superradiant state of two mesoscopic ensembles of atoms in a rigorous manner, it is necessary to give up the constant steady-state condition out of the numerical modeling [31–35], so that it can eventually unveil the specific unsynchronized superradiant state and its detailed dynamics.

We here construct a full numerical model on non-steady-state, multi-frequency superradiance, building upon our preliminary work presented in [34,35] that was only applicable to a specific unsynchronized superradiant state in a limited condition with limited accuracy. We then numerically analyze the superradiance in two mesoscopic ensembles of atoms having different transition frequencies particularly when they are in unsynchronized superradiant states. We show that such states can lead to frequency mixing of superradiant emissions, i.e., superradiant mixing. We discuss the detailed behavior of the superradiant mixing in comparison with both synchronized superradiant and non-superradiant states, also overviewing the subsequent changes that follow with increase of the number of individual atomic ensembles with detuned frequencies.

2. Numerical model

In our numerical model, we make the following basic assumptions: The superradiant system is based on two mesoscopic atomic ensembles very similar to the systems reported in [24,29]. Each atom is a two-level system having its own transition frequency. They form two individual atomic ensembles according to their own transition frequencies. These atomic ensembles are confined in a one-dimensional waveguide system, which can be an optical fiber or a photonic crystal waveguide, for example [36,37], and interact with one waveguide mode. The system’s intrinsic nonlinearity and dispersion are small enough to ignore. Repumping and cooling processes are constantly applied to the atomic ensembles. The schematic of the superradiant system with the two atomic ensembles and the corresponding energy transitions taking place inside them are illustrated in Fig. 1 [34,35]. Two different transition frequencies for the two ensembles can be prepared by using two different dressing lasers, for example. Other methods, such as the Zeeman splitting method and the use of quantum dots of different sizes, can also be exploited for the preparation of such ensembles [3,24]. The repumping process supplies sufficient energy for the atomic ensembles, so that they can continuously radiate superradiant emissions. The cooling process is well preventing the atomic ensembles from interacting with environmental or thermal noise that may disturb the construction of superradiance, so that dephasing by such noise remains in an ignorable level. Thus, repumping is mainly responsible for the total dephasing of the superradiant emissions.

 

Fig. 1. Schematics of superradiant system with two atomic ensembles. Two ensembles of atoms are pumped at repumping rate ${\mathrm{\Gamma }_R}$, having their own transition frequencies of ${\omega _1}$ and ${\omega _2}$, respectively [34,35].

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The numerical model of the two atomic ensembles is constructed based on the Maxwell-Bloch equations under the slowly varying envelope approximation [5,13,24,38]. It is assumed that both ensembles can be built upon their own synchronized superradiant states, and that the transition frequency difference between them, namely initial detuning, is small enough in comparison with their own transition frequencies. The mean value of the two transition frequencies can be regarded as a carrier frequency. In such a state, atoms within the same ensemble share nearly the same quantum state [1–3,24]. This feature is also reflected on the induced dipole fields. Therefore, the dipole fields share the equal amplitude and phase amongst them.

Upon this background, Maxwell’s wave equation for the electric field of light can be written as

In addition, the Maxwell-Bloch equations derived above can be reformulated in terms of the normalized parameters via the initial detuning $\delta \equiv {\omega _{20}} - {\omega _{10}} = \frac{1}{2}({\omega - {\omega_{10}}} )= \frac{1}{2}({{\omega_{20}} - \omega } )$ as the following:

The variables ${P_j}$, ${Q_j}$, and ${R_j}$ form the Bloch vector of the $j$-th ensemble, and their initial state can be represented by $({{P_j},{Q_j},{R_j}} )= ({\cos {\phi_j}\sin {\theta_j},\sin {\phi_j}\sin {\theta_j},\cos {\theta_j}} )$. We can assume that the dynamics starts from an excited state with small noise due to the uncertainty principle [40]. The uncertainty principle requires ${\theta _j}$ to be a random variable with Gaussian distribution of standard deviation $2/\sqrt {{N_a}} $, where ${N_a}$ is the total number of atoms. Thus, we can choose the angular values for each ensemble randomly in the uniform distribution of $0 < {\phi _j} \le 2\pi $ and in a Gaussian distribution of ${\theta _j}$ centered at zero with standard deviation of $0.01547$ (from the Ref. [40]), together with ${a_k}$’s initially set to zeros. This small noise can trigger collective coupling and superradiant emission. With this initial condition, we numerically solve the Maxwell-Bloch equations, using the Runge-Kutta method until the solution converges to a specific state.

3. Numerical simulation results

Based on the numerical model constructed in the preceding section, we conduct numerical simulations for a system of two atomic ensembles, varying both the collective decay rate and the repumping rate. In particular, we investigate and analyze an unsynchronized superradiant state generally defined in terms of its phase difference, polarization amplitude, and normalized intensity.

First of all, we discuss typical characteristics of an unsynchronized superradiant state via some simulation results obtained for ${N_0}{\gamma _a}/\delta = 2$ and ${\mathrm{\Gamma }_R}/\delta = 0.5$, which are shown in Fig. 2. In this unsynchronized superradiant state, the transition frequencies are shifted from their initial values, undergoing modulated detuning. Since the polarizations of the individual ensembles alternate at the modulation frequencies, they become in-phase and out-of-phase periodically as shown in Fig. 2(a). Whilst the relative phases of the polarizations change continually, their evolutions are relatively slow in terms of the time scale of the fast oscillations with the transition frequencies. Thus, when the polarizations remain relatively in-phase for a certain period time, the polarizations converge as if they formed a synchronized superradiant system, resulting in the maximized amplitude for the total polarization as depicted in Fig. 2(b). This subsequently leads to the maximized collective coupling, which yields superradiant emission with a high collective decay rate. In contrast, when the polarizations remain relatively out-of-phase for a certain period time, superradiant emission is suppressed, because the dipole fields in opposite polarities cancel out each other, thereby being unable to build up strong collective coupling. In fact, these procedures repeat and give rise to periodic oscillation of the radiation field, generating a periodic, superradiant pulse train, i.e., coherent mode-locked radiation based on the modulated detuning. This superradiant pulse train is composed of numerous different frequency components with equal mode spacing, which can be analyzed in more detail in frequency domain represented by the normalized detuned frequency $\omega ^{\prime}/\delta $ as shown in Fig. 2(d). We note that if the radiation were based on a synchronized superradiant state, there had to be a single peak at the center frequency of zero detuning or just two peaks at the modulated transition frequencies for too weak collective coupling. However, one can see that there exist a number of side peaks with considerable heights, which can be regarded as an evidence of frequency mixing resulted from the strong collective coupling of superradiant emissions. Apparently, this outcome is similar to that of degenerated four-wave mixing; however, one should bear in mind that the frequency components here are based on superradiant emissions. That is, the frequency mixing is not caused by the Kerr nonlinear effect but by the coupling of superradiant emissions. For this reason, we call this frequency mixing of superradiant emissions as “superradiant mixing”.

 

Fig. 2. Oscillating dynamics of an unsynchronized superradiant state. (a) The phase difference between the two ensembles; (b) the average polarization amplitude of the two ensembles; (c) the normalized intensity of the superradiant emission in time domain; and (d) the normalized intensity of the superradiant emission in frequency domain represented by the normalized detuned frequency $\omega ^{\prime}/\delta $. Dashed red line in (d) is a Gaussian-shape envelope line that connects all the individual peaks as closely as possible.

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In some sense the formation of the superradiant pulse train or superradiant mixing, is similar to that of photon echo [41–44]: In photon-echo processes, emitters are first excited, and their polarizations disperse by inhomogeneous broadening effects. This inhomogeneous dephasing can be recovered by inverting the optical Bloch vector with a π-inversion pulse. Subsequently, when the phases of the emitters become fully converged, they radiate an echo pulse of the initial excitation. We note that contrast to such a photon echo system, the two atomic ensembles under consideration here are constantly pumped, thereby undergoing in-phase and out-of-phase intervals periodically. That is, an unsynchronized superradiant state radiates a mode-locked pulse train continuously. In fact, it is noteworthy that the numerical observation of this superradiant mixing is largely thanks to the rigorous numerical approach not relying on the constant steady-state approximation [31–35].

To elucidate the overall influence of parameters on the superradiant characteristics of the system of two atomic ensembles, we choose the full normalized spectral bandwidth $\Delta \omega /\delta $ as the order parameter and plot a phase diagram with respect to ${N_0}{\gamma _a}/\delta $ and ${\mathrm{\Gamma }_R}/\delta $ in Fig. 3. We note that the full normalized spectral bandwidth is determined at the level 10-dB down from its peak as specified in Fig. 3(a) for three representative cases presented with respect to the normalized detuned frequency $\omega ^{\prime}/\delta $. In Fig. 3(a), the synchronized regime shows the case of perfectly synchronized frequency. Thus, its spectral bandwidth is the same as that of superradiant emission. In the other cases, one can find distinct frequency components, which are slightly shifted from the initial detuning frequencies owing to collective coupling. In the zero-quantum-correlation regime, the total emission is just the sum of two very weak and broad emissions, and thus, the overall spectral bandwidth is just determined by that of the overlapped spectrum of the two broad emissions. In contrast, the spectral bandwidth of the unsynchronized regime is determined by how far and high the side peaks are. It is noteworthy that we take the full normalized spectral bandwidth $\Delta \omega /\delta $ as the order parameter, because the side peaks are a clear evidence of superradiant mixing and the level of their relative magnitudes indicates how high the nonlinear polarization is in the given superradiant system. Moreover, the distinct characteristics of the spectral bandwidth in different regimes make it a decent criterion for distinguishing amongst regimes. On the phase diagram shown in Fig. 3(b), the overall superradiant states are now classified into three constituent regimes: the unsynchronized, synchronized, and zero-quantum-correlation regimes [24,35].

 

Fig. 3. (a) Spectral profile of the normalized superradiant emission intensity for each different regime with respect to the normalized detuned frequency $\omega ^{\prime}/\delta $: ${\mathrm{\Gamma }_\textrm{R}}/\delta = 0.8$ (fixed) and ${N_0}{\gamma _a}/\delta = 0.6,\; 1.2,\; \textrm{and}\; 1.8$ for the zero-quantum-correlation, unsynchronized, and synchronized regimes, respectively. The dashed red line is an envelope connecting local peak points, and the spectral bandwidth $\Delta \omega $ at 10 dB level is indicated by the arrows. (b) Phase diagram of the spectral bandwidth $\Delta \omega $. The grey dashed lines denote the boundaries of the three different regimes.

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The unsynchronized regime is specified by the area located on the lower right corner with the collective decay rate higher than the repumping rate, which has a relatively broad normalized spectral bandwidth. The synchronized regime is specified by the dark area located on the upper right corner with both a high collective decay rate and a high re-pumping rate, which has a relatively narrow spectral bandwidth. Finally, the zero-quantum-correlation regime is specified by the area located on the upper left corner with the repumping rate higher than the collective decay rate, which has an unlimited spectral bandwidth. We note that in the unsynchronized regime, the superradiant mixing always takes place. The spectral bandwidth is reduced to a minimum value where ${\mathrm{\Gamma }_R} = 0$ or ${N_0}{\gamma _a} = {\mathrm{\Gamma }_R}$. The minimum value takes place near the initial detuning, which indicates that superradiant mixing requires both collective decay and repumping processes to occur at the same time. A low repumping rate is hardly able to excite atoms fast enough before the next pulse is generated, and a collective decay rate lower than the given repumping rate is not sufficient to yield superradiant emission.

On the other hand, stable periodic oscillation of superradiant emission is greatly favored in the area with either a high collective decay rate or a high repumping rate. In fact, this condition makes the population inversion rise and fall relatively faster, thereby yielding a sharp-pulse train and more vivid superradiant mixing. Whilst it is not depicted in Fig. 3(b), the normalized spectral bandwidth can exceed over $10$ for even higher collective decay rates. Increasing either the collective decay rate or the repumping rate can strengthen superradiant mixing. However, if the collective decay rate and the repumping rate satisfy ${N_0}{\gamma _a}/\delta > \sim 1$ and ${\mathrm{\Gamma }_R}/\delta > \sim 1$, respectively, at the same time, they eventually lead to a well-balanced, constant steady state with nearly all the atoms behaving as a single entity, which is the synchronized regime. The synchronized superradiant state sustains unless either of the conditions is broken.

To explore how the state of the two atomic ensembles evolves through different regimes, we investigate the changes in their total polarization with respect to the collective decay rate for a fixed repumping rate, which is directly related to superradiant emission. In particular, the normalized repumping rate is fixed to ${\mathrm{\Gamma }_\textrm{R}}/\delta = 0.8$, with which one can make the two atomic ensembles operate in all the three regimes, depending on how large the normalized collective decay rate is. In Fig. 4, we illustrate the normalized amplitude of the total polarization of the two atomic ensembles in frequency domain represented by the normalized detuned frequency $\omega ^{\prime}/\delta $. The three different regimes can readily be identified by the three distinct regions divided at ${N_0}{\gamma _a}/\delta ={\sim} \; 0.8$ and ${\sim} \; 1.5$.

 

Fig. 4. Normalized amplitude of the total polarization of the two atomic ensembles in frequency domain with respect to the normalized detuned frequency $\omega ^{\prime}/\delta $ and the collective decay rate when ${\mathrm{\Gamma }_\textrm{R}} = 0.8\; \delta $. The normalized polarization is presented in logarithmic scale. The grey dashed lines denote the boundaries of the three different regimes.

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When the collective decay rate is small, there is no coherence between the two atomic ensembles, so that the total polarization nearly vanishes, which means superradiance does not occur in this zero-quantum-correlation regime. On the other hand, when the collective decay rate is excessively high, the total polarization has only one strong peak at a single frequency, which implies the synchronization of the superradiant emissions by the two atomic ensembles. It is noteworthy that the state of the total polarization remains independent of the collective decay rate as far as the two atomic ensembles operate in the synchronized regime. Between the zero-quantum-correlation regime and the synchronized regime, there is the unsynchronized regime. Right above the boundary between the zero-quantum-correlation regime and the unsynchronized regime, where ${N_0}{\gamma _a}/\delta ={\sim} 0.8$, two strong peaks of the total polarization start to appear with a spectral split similar to the initial detuning. Since there is no sign of other peaks at higher-order frequencies, the two atomic ensembles are presumably in independent, individual superradiant states without undergoing any superradiant mixing process. However, as the collective decay rate increases further, additional spectral peaks at higher-order frequencies start to grow along with slight changes in the individual transition frequencies. The number and strength of the spectral peaks increase with the collective decay rate, and each peak has a narrow Fourier-limited linewidth. The superradiant mixing becomes further significant as the collective decay rate increases until the upper bound of the unsynchronized regime. As the parameter condition tends closer to the boundary of the unsynchronized and synchronized regimes, we do see frequency shift to synchronization, which matches with other’s expectation [24]. Right at the boundary of the two regimes, there is a competition between two different states during the early stage of the numerical iterations: one with main peaks near the detuned frequencies at $\omega ^{\prime} ={\pm} \delta /2$ and the other with the main peak at zero detuning ($\omega ^{\prime} = 0$). However, the competition invariably tends to settle down to either of the unsynchronized or synchronized state after a sufficient number of iterations, depending on the precise conditions of the given parameters.

Whilst we have hitherto investigated superradiance of two atomic ensembles represented by two Bloch vectors, our numerical model has, in fact, no specific limitation in terms of the number of Bloch vectors that can be dealt with at the same time. Thus, our investigation can extend to more than two atomic ensembles. In particular, one can intuitively expect that ensembles having their transition frequencies equally spaced, such as atoms with Zeeman splitting or multiple dressing lasers, are likely to generate mode-locked superradiant emission in a similar way as shown in Fig. 2. In such a multiple-ensemble system, the individual polarizations of ensembles can still undergo in-phase and out-of-phase intervals periodically, depending on the detuning condition. The total polarization, however, obtains its maximum value in a very short interval, because the total dispersion of the system becomes significant and larger as the number of individual ensembles with detuned frequencies increases. This shorter in-phase interval produces superradiant pulses with a shorter pulsewidth and generates a broader spectral bandwidth of the resultant superradiant mixing.

For example, numerical results for 2, 4 and 6 atomic ensembles with ${N_0}{\gamma _a}/\delta = 1.5$ and ${\mathrm{\Gamma }_R}/\delta = 0.5$ are presented with respect to the normalized detuned frequency $\omega ^{\prime}/\delta $ in Fig. 5, where we have assumed that the initial detuning $\delta $ between two nearest neighboring transition frequencies of the atomic ensembles remains the same for all the cases, and that the total number of atoms remains the same whilst the number of individual ensembles with detuned frequencies varies. When the number of the individual atomic ensembles is changed from 2 to 4, the mode spacing is a bit reduced, and the overall heights of side peaks increase. We note that the side peaks grow unevenly with the detuning frequency. In particular, the growths of the second side peaks counted from the main peaks in the middle are most significant. In the case of 6 atomic ensembles, the mode spacing is a bit further reduced and the overall heights of the side peaks even further increase, which eventually results in a significant increase in the overall spectral bandwidth. We suspect that this intriguing outcome is thanks to many-body effects. Detailed physical interpretation on this matter would require further extensive investigation with vast computational resources elsewhere.

 

Fig. 5. Normalized intensities of superradiant emissions for 2, 4, and 6 different atomic ensembles with ${N_0}{\gamma _a}/\delta = 1.5$ and ${\mathrm{\Gamma }_R}/\delta = 0.5$ in frequency domain represented by the normalized detuned frequency $\omega ^{\prime}/\delta $. The normalized intensity is presented in logarithmic scale.

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4. Conclusions

We have constructed a numerical model that can rigorously analyze the transient behavior of superradiant dynamics in atomic ensembles confined in a one-dimensional waveguide without relying on the constant steady-state approximation. Based on this model, we have conducted numerical simulations on superradiant atomic ensembles of different transition frequencies, unveiling their detailed dynamics and characteristics, which result from superradiant mixing. We have first analyzed this superradiant mixing phenomenon and its dependency on the collective decay rate and the repumping rate for a system of dual atomic ensembles, and then extended our investigation to systems of 4 and 6 different atomic ensembles.

In particular, the numerical analysis on the superradiant mixing suggests that it can be a novel frequency generation method that does not rely on the conventional Kerr nonlinearity. The high coherence of the superradiance can make frequencies have higher spectral purity than the simple cavity-based methods do. The mode spacing can safely extend up to ∼ 10% of the transition frequency as long as the emitters remain as a two-level system. If the initial detuning is prepared by dressing lasers or Zeeman splitting, the mode spacing and carrier envelope offset can readily be controlled by external light or magnetic fields. Broad spectral bandwidth can be achieved from the multiple atomic ensembles with different transition frequencies. In addition, the superradiant mixing can maintain the carrier envelope offset stable, because the frequencies are locked around the carrier frequency. Furthermore, as long as repumping rate is under control, the superradiance can also maintain high coherence emission with low noise and narrow bandwidth for each frequency component. These suggest that superradiant mixing has great potential for frequency comb and multiple frequency generation.

We expect this work to be an important basis of novel multiple frequency generation methods for a quantum optical system. Further investigation on superradiant mixing based on multiple atomic ensembles and related experimental studies should follow.