A Mode-Selective Control in Two-Mode Superradiance from Lambda Three-Level Atoms

Abstract

Dicke superradiance, a single-mode burst of radiation emitted by an ensemble of two-level atoms, has garnered tremendous attention within the physics community. Its extension to multi-level systems introduces additional degrees of freedom, such as mode-selective control over well-known Dicke superradiant behaviors. However, previous work on the extension to two-mode superradiance in three-level atoms has been largely overlooked for over five decades. In this study, we revisit the two-mode superradiance model for a Λ-type three-level system, where two modes couple to a common excited state and two separate lower levels, offering new insights. For the first time, we obtain exact numerical solutions of the two-mode rate equations for this model. We analyze the temporal evolution of two-mode intensities, superradiance time delays, and quantum noise in the time domain as the number of atoms varies. We believe this work will enable external mode-selective control over superradiance processes—a capability unattainable in the single-mode case.

1. Introduction

Open quantum systems are intrinsically fragile, and protecting them against decoherence under practical environmental conditions remains a formidable challenge [1,2,3]. Traditionally, avoiding decoherence has required operating such systems under highly controlled—and often impractical—conditions [1,2,3]. Counterintuitively, recent studies have shown that quantum interactions can persist in quantum materials even at ambient conditions [4,5,6,7]. However, the underlying mechanisms remain unknown [8]. In this context, the understanding of light interactions with the collective two-level [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27] and multi-level atoms is critically important [28,29,30,31,32,33,34,35,36,37]. Particularly, we focus on those interactions that are governed by rate equations [4,9,38,39,40,41,42,43,44,45,46,47,48,49,50] because these systems exhibit behavior with no classical analog [51]. For example, the quantum rate equation theory of the semiconductor lasers implies an adiabatic elimination of the polarization of resonant transitions. Such adiabatic elimination can be performed if the polarization relaxation (i.e., decoherence) rate is much larger than the other relaxation rates in the system (see Ref. [15] and the references therein). It is also known that superradiant states tend to shield quantum systems from specific types of environmental noise (as decoherence) [13].

The original 1954 Dicke superradiance model describes a single-mode light and inter-particle quantum interactions with a net zero dipole moment [9]. Recently, this model was experimentally realized in single diamond nanocrystals at room temperature [4]. Timing in superradiance is essential. For example, the superradiance delay time is scaled approximately by the time for the first photon to be emitted [13]. As the number of emitters increases, the burst of radiation emerges sooner. The superradiance noise fluctuations in time delay are also important quantities because they have a quantum origin [19]. When all $N$ atoms are prepared initially on their excited states, Dicke superradiance exhibits at least five distinct characteristics: (i) the maximum of time-dependent intensity, as defined in Ref. [48], is scaled as $N^2$, rather than $N$ [4,9,28,29,30,31,32,33,34,35,36,37]; (ii) this maximum peak occurs at the superradiance delay given by $(E_0+\mathrm{l}\mathrm{o}\mathrm{g}(N\left)\right)/N$ [41,42], where $E_0=0.57721$ is Euler’s constant; (iii) the time-dependent delay, as defined in Ref. [48], converges to $(E_0+\mathrm{l}\mathrm{o}\mathrm{g}(N\left)\right)/N$; (iv) the time-domain noise fluctuations are given by $\pi /\left[\sqrt{6}\right(E_0+\mathrm{l}\mathrm{o}\mathrm{g}\left(N\right)\left)\right]$ [41,42]; and (v) the time-dependent noise fluctuations, as defined in Ref. [48], converge to $\pi /\left[\sqrt{6}\right(E_0+\mathrm{l}\mathrm{o}\mathrm{g}\left(N\right)\left)\right]$.

The Dicke model has been extended to three-level atoms, enabling two-mode superradiance. This extension was first explored in pioneering works from the 1970s by Agarwal [44,47], Cho et al. [45], and Gilmore [46]. However, these early studies were largely overlooked until their recent revival in 2022 [48].

Two-mode superradiance emerges in three main three-level configurations: (i) the ladder or ‘cascade’ system, (ii) the Λ or ‘Lambda’ system, and the V or ‘Vee’ system. While the rate equations for these systems were formally derived in Refs. [44,45,46,47], exact numerical solutions and detailed analyses of their properties have only recently become available. For instance, the cascade superradiance model was reintroduced for the ladder three-level system was reported in Ref. [48], while the quantum statistical theory of the two-mode superradiance from the V-three-level system is presented in separate work.

In this work, we study the remaining two-mode superradiance model for a Λ-type three-level system, where two modes couple to a common excited state and two separate lower levels, offering new insights. Our results demonstrate that one mode exhibits modified Dicke superradiance whose properties can be tuned through the parameters of the other mode. This finding offers new possibilities towards engineering quantum emission processes. The next section introduces the Lambda three-level model and presents exact numerical solutions for the two-mode rate equations. Data interpretations and detailed discussions of the results, comparing them with the characteristic quantities of Dicke superradiance, are presented in the following section. The final section concludes the study with a summary of our findings.

2. Results

We consider an ensemble of Λ-type three-level N atoms confined to a volume that is small compared to the wavelengths of both radiation modes. In this configuration, direct one-photon transitions between the two lower states are forbidden, while one-photon transitions between the common excited state and each lower state are allowed. The dipole–dipole interactions between atoms and the effect associated with the collective frequency shifts are ignored in this model. Thus, the system dynamics are governed by the two-mode rate equation [44,45,46,47]:

$$\begin{array}{ll}{\displaystyle \frac{\mathit{d}{\mathit{P}}_{\mathit{n},\mathit{m}}\left(\mathit{t}\right)}{\mathit{d}\mathit{t}}}=& \left[{\mathit{I}}_1\left(\mathit{n}+1,\mathit{m}\right){\mathit{P}}_{\mathit{n}+1,\mathit{m}}\left(\mathit{t}\right)-{\mathit{I}}_1\left(\mathit{n},\mathit{m}\right){\mathit{P}}_{\mathit{n},\mathit{m}}\left(\mathit{t}\right)\right]\\ & +\mathit{\alpha }\left[{\mathit{I}}_2\left(\mathit{n}+1,\mathit{m}+1\right){\mathit{P}}_{\mathit{n}+1,\mathit{m}+1}\left(\mathit{t}\right)-{\mathit{I}}_2\left(\mathit{n},\mathit{m}\right){\mathit{P}}_{\mathit{n},\mathit{m}}\left(\mathit{t}\right)\right]\end{array}$$

(1)

where $P_{n,m}\left(t\right)$ represents the probability of finding the system with $N$ atoms in a state of $|n,m-n,N-m⟩$ at a given time $t$, with the normalization condition that all probabilities sum to unity. As mentioned in Refs. [44,47], the cooperative decay rates are given by $I_1\left(n,m\right)=n\left(m-n+1\right)$ and $I_2\left(n,m\right)=n\left(N-m+1\right)$, the two distinct transitions, respectively. Here, $n$ denotes the atoms in the common upper state, and $m-n$ and $N-m$ represent the atoms in the two lower states; see an inset in Figure 1.

The parameter $\alpha ={\Gamma }_2/{\Gamma }_1$ (assuming ${\Gamma }_1\ge {\Gamma }_2$ and $\alpha \le 1$) represents the ratio of single-atom spontaneous decay rates ${\Gamma }_2$ and ${\Gamma }_1$ for the two mode transitions [48]. Time is expressed in dimensionless units as $t\to {\Gamma }_{1}t$. For extended atomic media, this scaling parameter generalizes to $\alpha ~{\mu }_2{\Gamma }_2/{\mu }_1{\Gamma }_1~{\lambda }_2^2{\Gamma }_2/{\lambda }_1^2{\Gamma }_1$ [48], where λ₁,₂ (λ₁ ≥ λ₂) are the mode wavelengths and ${\mu }_{1,2} ({\mu }_1\ge {\mu }_2)$ are geometric factors [10,11,12,13,45,48], with the dimensionless time becoming $t\to {{\mu }_1\Gamma }_{1}t$ [48]. For small samples, the parameter $\alpha$ is determined by a class of three-level energy configurations where the ratio of the spontaneous decay rates is identical in the same and/or different atomic species. For large samples, this parameter also depends on the ratio of the geometrical factors, which can be experimentally controlled within the same atomic species as well as among different species.

When all $N$ atoms are prepared initially on their common upper states, we numerically solve Equation (1) using the following parameters throughout our analysis: atom numbers $N$ ranging from 10 to 300, time intervals $t\in [0,1]$, and three decay rate ratios $\alpha =0.1$ (panels A, B), $0.5$ (panels C, D), and 1 (panels E, F), with ${\Gamma }_1\equiv {\mu }_1={\mu }_2=1$. The results for the first and second modes are labeled (A, C, E) and (B, D, F), respectively. The two-mode intensities as functions of the number of atoms $N$ and time $t$ are depicted in Figure 1. Two-mode time-dependent intensities are given by $I_{1,2}\left(t\right)={\sum }_{n,m=0}^N{I}_{1,2}\left(n,m\right)P_{n,m}\left(t\right)$ [48]. Note that these intensities include constant factors involving the transition wavelengths ${\lambda }_{1,2}$ [47], which we omit here as they only affect absolute scaling without altering the qualitative behavior.

As seen from the intensity profiles in Figure 1, both modes exhibit pulsed behavior, with the pulses forming at different delays and showing increasingly rapid buildup and higher peak intensities as $N$ increases. Following Ref. [48], we define the partial two-mode pulse areas as $〈A_{1,2}\left(t\right)〉={\sum }_{n,m=0}^N{I}_{1,2}\left(n,m\right){\int }_0^t{P}_{n,m}\left(t^{\prime }\right)d{t}^{\prime }$, where the corresponding total pulse areas are $〈A_{1,2}\left(\infty \right)〉={\sum }_{n,m=0}^N{I}_{1,2}\left(n,m\right){\int }_0^{\infty }{P}_{n,m}\left(t^{\prime }\right)d{t}^{\prime }$. The detailed temporal characteristics of these pulsed modes will be discussed in the following section.

To characterize the temporal properties of the pulse pairs, we calculate the time-dependent delay times using the partial pulse areas $〈A_{1,2}\left(t\right)〉$. The delays are given by $〈{\tau }_{1,2}\left(t\right)〉={\sum }_{n,m=0}^N{I}_{1,2}\left(n,m\right){\int }_0^t{t}^{\prime }{P}_{n,m}\left(t^{\prime }\right)d{t}^{\prime }/〈A_{1,2}\left(t\right)〉$ [48]. In the long-time limit ($t\to \infty$), these converge to $〈{\tau }_{1,2}(\infty )〉={\sum }_{n,m=0}^N{I}_{1,2}\left(n,m\right){\int }_0^{\infty }{t}^{\prime }{P}_{n,m}\left(t^{\prime }\right)d{t}^{\prime }/〈A_{1,2}(\infty )〉$, where the normalization uses the total pulse areas. Figure 2 presents our numerical results for $〈{\tau }_{1,2}\left(t\right)〉$ as functions of both time and atom number $N$.

As evident from the delay profiles in Figure 2, both modes exhibit qualitatively similar behavior. The delays decrease sharply with the increasing atom number before transitioning to a more gradual decline, eventually reaching a steady-state value at long times ($t\to \infty$). Our analysis focuses particularly on this stationary regime, which will be examined in detail in the following section.

The intrinsic noise fluctuations in the system originate from quantum interactions between atoms and vacuum fluctuations (in the case of two-level atoms, see Refs. [19,20]; for three-level atoms, see Refs. [21,22,26,36,48]). For the three-level configuration studied here, the time-dependent two-mode noise in pulse delays is quantified by $〈{\sigma }_{1,2}\left(t\right)〉=\sqrt{〈{\tau }_{1,2}{\left(t\right)}^2〉-{〈{\tau }_{1,2}\left(t\right)〉}^2}/〈{\tau }_{1,2}\left(t\right)〉$ [48], where the second moment is calculated as $〈{\tau }_{1,2}{\left(t\right)}^2〉={\sum }_{n,m=0}^N{I}_{1,2}\left(n,m\right){\int }_0^t{t}^{\prime 2}{P}_{n,m}\left(t^{\prime }\right)d{t}^{\prime }/〈A_{1,2}\left(t\right)〉$. Figure 3 presents our numerical results for $⟨{\sigma }_{1,2}\left(t\right)⟩$ as functions of both time and atom number. The noise fluctuations exhibit distinct dynamical features, an initial rapid decrease followed by pronounced minima (‘valleys’), before stabilizing at longer times. Our subsequent discussion will focus on both the minimum delays observed at these valleys and the asymptotic behavior in the $t\to \infty$ limit.

3. Discussions

We begin our analysis by examining whether the two-mode emission maintains the characteristic $N^2$ scaling of conventional Dicke superradiance. Figure 4 presents the peak intensities extracted from Figure 1 plotted against $N^2$. The first mode (panels A, C, E) exhibits excellent linear scaling across the full range of atom numbers ($N$ = 10–300) for all three decay rate ratios ($\alpha =0.1,0.5,1$). This clear $N^2$ dependence confirms that the first mode satisfies one of the fundamental criteria for Dicke superradiance, as observed in collective two-level systems.

The linear fits in Figure 4 reveal an inverse relationship between slope magnitude and decay rate ratio $\alpha$, indicating that the first mode’s superradiant intensity can be effectively controlled through the spontaneous decay rate of the second mode in the Λ configuration. This control mechanism represents a unique feature of the two-mode system that does not present in conventional single-mode superradiance.

For the second mode (Figure 4B,D,F), we observe significant deviation from ideal superradiant behavior, particularly at lower $\alpha$ values. A quantitative analysis of the linear fits shows progressively deteriorating $R^2$ values; here, $R^2$ is a goodness-of-fit measure for the linear regression model: $1.00 (\alpha =1), 0.99 (\alpha =0.5)$, and $0.96 (\alpha =0.1)$. This degradation in linearity suggests that only the symmetric case ($\alpha =1$) maintains proper superradiant characteristics in the second mode, while asymmetric configurations ($\alpha <1$) show a marked departure from ideal Dicke superradiance.

To examine the temporal characteristics, we normalized the intensity maxima from Figure 1 and plotted them against the theoretical superradiance delay formula $(E_0+\mathrm{l}\mathrm{o}\mathrm{g}(N\left)\right)/N$. Figure 5 presents these normalized intensity profiles in a perspective view, revealing that the first mode with $\alpha =0.1$ (Figure 1A) shows the most linear relationship with this nonlinear delay function. This agreement with theoretical prediction provides additional evidence for superradiant behavior in the first mode across all α values, while further confirming the breakdown of superradiance in the second mode for $\alpha <1$. As seen from Figure 5B, an efficient emission in the first mode disrupts the condition for the second mode emission. The number of the excited atoms becomes much less than $N$. This updated variable condition prevents any efficient burst of radiation in the second mode from occurring.

For quantitative analysis, we focus on the delay time data from Figure 2 rather than the intensity profiles of Figure 1 or Figure 5. The temporal evolution in Figure 2 reveals two distinct dynamical regimes: (i) an initial rapid transient stage followed by (ii) asymptotic convergence to steady values in the $t\to \infty$ limit. This characteristic bistage relaxation behavior is a significance of the two-mode superradiant transitions. The normalized representation in Figure 6 provides a clearer visualization of the characteristic dynamics. Two distinct features emerge: (i) well-defined plateaus in the steady-state region (yellow regions), and (ii) significantly improved linear correlation with the theoretical delay formula at plateau boundaries compared to the intensity profiles in Figure 5.

Figure 7 presents the absolute steady-state delays $⟨{\tau }_{1,2}(\infty )⟩$ extracted from Figure 2. Notably, the data for mode 1 with $\alpha =0.1$ (Figure 1A) show exact correspondence with the theoretical prediction $(E_0+\mathrm{l}\mathrm{o}\mathrm{g}(N\left)\right)/N$. This agreement demonstrates that in the limit $\alpha \ll 1$, the Λ-type three-level system effectively reduces to a two-level system, recovering the conventional Dicke superradiance behavior.

When $\alpha \ll 1$ (e.g., $\alpha =0.1$), the system exhibits pure Dicke superradiance through mode 1 exclusively. In this regime, all atoms decay via mode 1 before the second mode’s significantly delayed buildup (ten times longer in this case) can occur. This results in the complete depletion of the upper state population by the time mode 2 would begin emitting, preserving the ideal $N^2$ scaling and the temporal characteristics of single-mode Dicke superradiance.

For larger decay rate ratios ($\alpha =0.5$ or $1$), the residual population remains in the upper state during mode 2’s emission phase. As evident from Figure 1B–F, this leads to measurable deviations (solid curves) from the ideal linear Dicke superradiance relationship (dotted lines). Especially for $\alpha =1$, the observed deformations arise from intermodal competition—the upper state population leaks through both decay channels simultaneously, disrupting the conditions required for pure superradiance in either mode.

This competition between modes, while disruptive to ideal superradiance, provides a novel mechanism for controlling cooperative emission. By tuning the decay rate ratio $\alpha$, one mode can be selectively enhanced or suppressed—a degree of control absent in traditional two-level superradiance. Notably, this effect is more pronounced in the Λ configuration compared to the ladder three-level systems [48], where mode competition is inherently weaker due to the sequential (i.e., cascade) nature of the transitions. For $\alpha \ll 1,$ a sequential pair of pure Dicke superradiant pulses in two modes can be produced in the ladder three-level system, while in the Λ three-level system, there is a pure Dicke superradiance in one mode and a deformed one in the other. For $\alpha =1$, the sequential (simultaneous) pair of deformed superradiant pulses can occur in the ladder three-level system (Λ three-level system).

The temporal evolution of two-mode quantum noise fluctuations was analyzed in comparison with the theoretical prediction $\pi /\left[\sqrt{6}\right(E_0+\mathrm{l}\mathrm{o}\mathrm{g}\left(N\right)\left)\right]$. From the data in Figure 3, we extracted both the asymptotic noise values $⟨{\sigma }_{1,2}(\infty )⟩$ and their temporal minima, then plotted these quantities against the analytical expression. The numerical results demonstrate consistent agreement with theory across the entire parameter space, including all examined atom numbers ($N$ = 10 to 300) and decay rate ratios ($\alpha =0.1,0.5,1$). This agreement persists despite the system’s inherent dynamic evolution of fluctuations, confirming that the fundamental noise characteristics of Dicke superradiance remain preserved in this Λ configuration system. Both the asymptotic values and fluctuation minima (except for the second mode for $\alpha =0.1,0.5$) follow the predicted theoretical scaling, with the minimum values (represented by dotted curves in Figure 8) particularly highlighting this correspondence during the most stable emission periods. The consistent validation of the noise scaling relationship holds true for both radiation modes, even when accounting for their competitive interaction (except for the second mode for $\alpha =0.1,0.5$), suggesting the universal nature of quantum fluctuations in collective atomic systems.

4. Conclusions

This work demonstrates that multi-mode extensions of Dicke superradiance provide new opportunities for mode-selective control of cooperative emission processes. We have investigated the Λ-type three-level system, where two optical modes interact through a common excited state, obtaining the first exact numerical solutions to the two-mode rate equations originally proposed in the 1970s. Our comprehensive analysis has characterized the complete temporal dynamics of the system, including the evolution of two-mode intensities, superradiant delay times, and quantum noise fluctuations in the time domain, all carefully benchmarked against standard Dicke superradiance characteristics.

The results reveal a remarkable control mechanism: when the spontaneous decay rates are asymmetric (${\Gamma }_1\ne {\Gamma }_2$), one mode exhibits modified Dicke superradiance whose properties can be precisely tuned through the parameters of the other mode. This finding represents an important advance beyond conventional single-mode superradiance, offering new possibilities for engineering quantum emission processes. The demonstrated ability to selectively control one mode’s superradiant behavior via another mode’s characteristics opens promising avenues for applications in quantum information processing and controlled light–matter interactions.

This work provides both novel insights into multi-mode superradiant systems and practical foundations for developing quantum technologies with tailored emission properties. Future research directions include experimental implementations of this control scheme and investigations of more complex multi-mode quantum systems.