Spontaneous emission noise resilience of coupled nanolasers

1 Introduction

Over the recent years, coupled nanophotonic laser networks have found a variety of applications, e.g., in photonic integrated circuits, on-chip optical computing, and optical communication (Heil et al., 2001; Zapf et al., 2017; Ma and Oulton, 2019; Ning, 2019; Lingnau et al., 2020; Robertson et al., 2020). A wide bandwidth of new and strongly varying implementations has led to great advantages, such as a small footprint, high speed, and room temperature compatibility (Zhang et al., 2014; Deka et al., 2020). However, new challenges to the theoretical treatment of such systems have arisen from these developments. The small size of nanolasers pushes their dynamical timescales into regimes for which a description via conventional semiconductor laser models [class-B laser models (Arecchi et al., 1984; Erneux and Glorieux, 2010)] is no longer sufficient (van Tartwijk and Agrawal, 1998; Lingnau et al., 2019). Due to device miniaturization, the cavity photon lifetime becomes comparable to the microscopic polarization lifetime. In particular, nanolasers made of metal nanocavities (Ning, 2010), metal-coated semiconductor nanolasers (Koulas-Simos et al., 2022), or 2D materials (Du et al., 2020) exhibit high photon losses and, therefore, have shortened cavity photon lifetime (Neogi et al., 2002; Li and Ning, 2012; Zhang et al., 2014). As a consequence, the dynamic degree of freedom of microscopic polarization is important for the theoretical description in addition to the electric field and charge carrier dynamics (Lorke et al., 2013; Aust et al., 2016).

We use a Maxwell–Bloch-type laser model to describe the three aforementioned dynamical quantities in the coupled setup of two nanolasers [two delay-coupled class-C laser models (Haken, 1975; Haken, 1985; van Tartwijk and Agrawal, 1998; Ohtsubo, 2008)]. The dynamical behavior of such lasers has been investigated previously in various setups, e.g., in feedback and coupling settings (Lingnau et al., 2019; Roos et al., 2021; Wang et al., 2021), and in direct comparison with class-B lasers (Hohl et al., 1997; Javaloyes et al., 2003; Erzgräber et al., 2006; Bonatto et al., 2012; Junges and Gallas, 2015; Seifikar et al., 2018). Here, we focus on the effect of spontaneous emission noise and derive insights about optimal conditions for the noise-resilient phase-locked operation of the two delay-coupled lasers. For our purpose, the exact noise distribution, e.g., as can be obtained via a stochastic simulator approach (Puccioni and Lippi, 2015; Wang et al., 2021), is less important. Hence, we use a simplified Langevin stochastic noise source. To quantify whether the lasers are phase-locked, we compute the phase correlation. By performing multiple parameter studies, we aim to unravel how the polarization timescale influences the noise resilience of synchronized laser emission. On that account, we limit our investigation to parameter settings that permit deterministic (i.e., without the stochastic noise source) phase locking among the two lasers. We quantify the tolerance to noise in parameter space by defining a critical noise strength which we can directly translate into the experimentally accessible intensity auto-correlation (g⁽²⁾ function) of one laser.

2 Materials and methods

2.1 Coupled nanolaser model

The delay-coupled nanolasers are shown in Figure 1 and are described by the following set of macroscopic Maxwell–Bloch equations [class-C laser model, also used in (Roos et al., 2021)]:

$\begin{array}{ll}\hfill \dot{E_j}& =c{P}_j-\frac{E_j}{2}+{\left(-1\right)}^{j}i\delta \omega E_j+K{e}^{i\theta }{E}_k\left(t-\tau \right)+\sqrt{\beta }{\xi }_j,\hfill \end{array}(\mathrm{1a})$

$\dot{P_j}=\frac{\left(i\mathrm{\Delta }\omega -1\right)P_j+E_j{N}_j}{T_2}+{\left(-1\right)}^{j}i\delta \omega P_j,(\mathrm{1b})$

$\dot{N_j}=\frac{1}{T}\left[p-N_j-2c\mathrm{R}\mathrm{e}\left(P_j{E}_j^{\ast }\right)\right].(\mathrm{1c})$

FIGURE 1

FIGURE 1. Sketch of the two coupled nanolasers described by the Maxwell–Bloch class-C equations given in Eq. 1. The solitary frequencies of the lasers are detuned by 2δω = ω₂ − ω₁.

Here, the indices j, k ∈ {1, 2} enumerate the two lasers. The equations describe three dynamical quantities for each laser: The inversion N of a two-level system, the complex electric field amplitude E, and the microscopic polarization P. Each of these dynamical quantities is subject to internal loss processes and thus connected to a characteristic timescale on which the corresponding dynamics evolve: T is the inversion lifetime, T₂ is the polarization lifetime, and T_ph is the photon lifetime. The latter has been used for transforming Eq. 1 into a non-dimensionless form, e.g., all times and frequencies are given in units of T_ph and $T_{ph}^{-1}$, respectively. A typical value for T_ph is on the order of 1ps for nanolasers (Ding and Ning, 2013). p is the pump supplying the lasers with energy, and Δω is the difference between the transition frequency of the active laser medium and the cavity mode. In our equations, the dynamical impact of Δω resembles the linewidth enhancement factor α (Schunk and Petermann, 1986; Lingnau et al., 2013). For all parameter choices, the first laser threshold is at $p_{\mathit{thr}}^{(1)}=0.5$ due to the scaling with $c={\left((2\mathrm{\Delta }\omega )/(T_2+2)\right)}^2+1$. The detuning between the two lasers is 2δω, and it is implemented such that the rotating frame used for the fields is located equidistantly between the two laser frequencies (Roos et al., 2021). In addition to detuning, the lasers are modeled with identical parameters. Tuning the other parameters typically affects the laser intensity and, thereby, also the emission frequency via the non-zero amplitude-phase coupling. Hence, parameter changes always produce a frequency detuning, which then dominates all other effects. To the best of our knowledge, this observation is similar to experimental observations, where other parameters, e.g., the pump or the crystal temperature, are employed to match the emission frequency of multiple lasers in coupling setups. We, therefore, assume that the frequency detuning is the most relevant parameter and omit the other parameters for the sake of simplicity.

The coupling setup of the lasers is characterized by three parameters: the coupling strength K [we use a value of K = 0.1 corresponding to an intensity reflectivity of $\approx 20\%$ (Roos et al., 2021), where effects due to back-reflected light are negligible (Schelte et al., 2019)], the coupling phase θ describing an additional phase shift between the interfering fields, and the time delay τ of the two laser fields due to the finite propagation speed of the signals [we consider a time delay of τ = 10 that refers to an on-chip coupling distance of 800 μm and can experimentally be realized; see e.g., (Deka et al., 2020)]. Spontaneous emission is included via a stochastic Langevin term added to the field equation, given in Eq. 1a, where ξ_j denotes a delta-correlated Gaussian white noise source and β denotes the spontaneous emission strength. It should be noted that the observed auto- and cross-correlation properties of the laser intensities not only depend on the value of β but also on all other parameters due to the coupled dynamics. The parameters are given in Table 1.

TABLE 1

TABLE 1. Parameters used for simulations.

2.2 Locked emission states

A solitary class-C laser operated below its second laser threshold has a stable emission state and will show only damped relaxation oscillations toward a steady-state solution after a perturbation. Above the second threshold, periodic oscillations or deterministic chaotic behavior occurs (Lingnau et al., 2019). We will operate our laser below the onset of instabilities at four times the first threshold; however, the coupling of two class-C lasers introduces additional emission states with complex dynamics even below the second threshold. Recently, the polarization lifetime T₂ was found to fundamentally influence the locking structure of two coupled class-C lasers (Roos et al., 2021). A scan of the dynamics found in the parameter plane of the coupling phase and the detuning is shown in Figure 2A for a dephasing time of T₂ = 1. The steady-state solutions, commonly referred to as compound laser modes (CLMs) (Erzgräber et al., 2006; Roos et al., 2021), can be followed in parameter space and are plotted in yellow/pink colors, as shown in Figure 2A. The color encodes the constant phase shift Δϕ between the emission of the two lasers, $E_j(t)={\mathcal{E}}_j(t)e^{i{\varphi }_j(t)}$, which is defined as follows (j = 1, 2 enumerates the lasers):

$\mathrm{\Delta }\varphi \left(t\right)={\varphi }_1\left(t\right)-{\varphi }_2\left(t\right)=\text{const.}(2)$

FIGURE 2

FIGURE 2. Deterministic dynamics (A) and phase correlation χ (B–D) of the two lasers in (θ, δω)-parameter space; (B–D) are obtained for noise strengths β (β and the corresponding $g_{\mathit{sol}}^{(2)}$ are indicated in the figure). The purple line in (B) shows the locking range δω. Parameters as given in Table 1 dashed pink line indicates 0 detuning.

The electric fields for laser emission on a CLM are given by a constant intensity emission of each laser with an optical frequency Ω and a constant phase shift ${\varphi }_j^c$:

$E_j\left(t\right)=E_j^0{e}^{i\left(\mathrm{\Omega }t+{\varphi }_j^c\right)}.(3)$

In order to quantify the range for which locking is observed, the locking range δω_lock has been introduced by Roos et al. (2021). It denotes the maximum possible detuning for which a stable CLM exists. In the deterministic case without noise (Figure 2A), the locking range is given by the extent of the yellow/pink areas in the y-direction. As can be seen in Figure 2A, the coupling phase is an important parameter for the dynamics of the system, and not every phase permits locking (there are phases where no CLM solutions exist). For our analysis, we are interested in the maximum locking range reached for the case of an optimally adjusted phase. We, therefore, define $\delta {\omega }_{\text{max}}^{\theta }$, as in the study by Roos et al. (2021) via

$\delta {\omega }_{\text{max}}^{\theta }=\underset{\theta \in \left[\mathrm{0,2}\pi \right)}{\max }\delta {\omega }_{\text{lock}}\left(\theta \right),(4)$

which then allows us to study its dependence on system timescales as, e.g., the polarization lifetime T₂. The maximum locking range is indicated in Figure 2B by the pink-dashed horizontal line.

For the stochastic case, where the Langevin noise source is applied, another measure needs to be defined to determine the transition to unlocked emission. For that, we define several tools. One quantity to identify whether the lasers remain on a CLM for a given noise strength is to look for the phase correlation of the electric fields and determine the noise strength where it overcomes a threshold (see the pink line in Figure 2B). The phase correlation χ is defined via the time average ⟨⋅⟩,

$\chi =|⟨e^{i{\varphi }_1\left(t\right)}{e}^{-i{\varphi }_2\left(t\right)}⟩|=|⟨e^{i\mathrm{\Delta }\varphi \left(t\right)}⟩|.(5)$

The phase correlation resembles the electric field correlation $⟨E_1{E}_2^{*}⟩=⟨{\mathcal{E}}_1(t)e^{i{\varphi }_1(t)}{\mathcal{E}}_2(t)e^{-i{\varphi }_2(t)}⟩$. Since we are interested in phase synchronization among the lasers, the phase correlation is deprived of the amplitudes of the fields. If the lasers are entirely locked, the phase shift Δϕ stays at a constant value and the phase correlation amounts to χ = 1. On the other hand, if the lasers emit independently, the phase shift e^iΔϕ(t) exhibits a random walk on the complex unit circle. Therefore, the correlation averages to χ = 0. In general, only phase-synchronized emission states exhibit a phase correlation of χ = 1, while desynchronized lasing modes exhibit a phase correlation of χ ≠ 1.

The intensity auto-correlation function $g_{jj}^{(2)}(0)$ of one laser within the coupled setup is defined as

$g_{jj}^{\left(2\right)}\left(\mathrm{\Delta }t\right)=⟨|E_j\left(t+\mathrm{\Delta }t\right){|}^2|E_j\left(t\right){|}^2⟩/⟨|E_j\left(t\right){|}^4⟩.(6)$

It is also a good measure for describing the stochastic emission and will be used later on. It provides a value of 2 for noise-dominated emission and a value of 1 for coherent laser emission (Redlich et al., 2016).

To complete the picture and better understand the nature of the unlocking transition, we additionally consider the ensemble variance of the phase difference in order to define a diffusion coefficient D of the random walk-like dynamics of the phase (Otto et al., 2012; Redlich et al., 2017). D then also measures the state of synchronization as it is 0 for locked operation (i.e., no random walk) and larger than 0 otherwise.

$\begin{array}{ll}\hfill \text{Var}\left\{\mathrm{\Delta }\varphi \right\}\left(t\right)& ={⟨\mathrm{\Delta }{\varphi }^2\left(t\right)⟩}_{\mathit{ens}}-{⟨\mathrm{\Delta }\varphi \left(t\right)⟩}_{\mathit{ens}}^2\hfill \\ \hfill & =2D\left(\beta \right)\cdot t.\hfill \end{array}(7)$

The expectation value ⟨⋅⟩_ens is with respect to an ensemble of time series with different stochastic realizations. We find that the variance evolves linearly with time for sufficiently large times. The last equality implies that the time evolution of the variance can be modeled as an uncorrelated random walk. This only holds true for times t larger than the characteristic system timescales such that all correlations have decayed.

3 Results

Figures 2B–D show the phase correlation, as defined in Eq. 5, plotted color-coded in the two-dimensional (θ, δω) parameter space for three different noise strengths. For a noise strength of β = 10^–4 [see (b)], the dynamical locking structure of the deterministic system can still be seen [compare to (a)]. Here, it can be confirmed that the locked modes exhibit a phase correlation of χ = 1 (light blue colors). However, if the applied noise strength β is further increased [see (c) and (d)], the phase correlation strongly decreases for each parameter set. In addition to this unlocking transition that we quantify further in the following section, the noise does not induce any new features in the investigated parameter space.

To elaborate on the physical context of the unlocking process induced by the noise, we plot the phase correlation χ as a function of the noise strength β in Figure 3 (blue line) for zero detuning and zero phase (δω = 0, θ = 0). As expected, the phase correlation transitions from 1 to 0 as β increases. The noise strengths for which the scans of Figures 2B–D are performed can be seen as vertical dashed lines. The transition of the phase correlation gives rise to a practical possibility to quantify the unlocking of the lasers. We define the critical noise strength β_lock in Eq. 8 as the maximum noise strength for which the phase correlation does not exceed a certain threshold of 1 − ϵ (we set ϵ = 0.05).

${\beta }_{\mathit{lock}}=\max \left\{\beta |\chi \left(\beta \right)>1-ϵ\right\}.(8)$

FIGURE 3

FIGURE 3. Statistical measures of the laser emission as functions of the noise strength β. The phase correlation χ is shown as a solid blue line (blue shading indicates ϵ defining the critical noise strength β_lock). The green and black lines show the intensity auto-correlation (Eq. 6) and the diffusion coefficient (Eq. 7). Dashed vertical lines reference panels (B–D) of Figure 2. The top x-axis indicates the auto-correlation of a solitary laser (Eq. 9). Parameters as in Table 1, δω = 0 and θ = 0.

Figure 3 also shows the intensity auto-correlation $g_{11}^{(2)}(0)$ of one of the lasers (green line) calculated from Eq. 6 where the transition from 1 to 2 can be detected. As long as $g_{jj}^{(2)}(0)<1.1$, we are below the locking threshold.

The critical noise strength β_lock that we use to define the unlocking of the lasers cannot be measured and also changes with the set of equations that is used. In order to overcome this issue, we determine the intensity auto-correlation of a single laser without coupling.

$g_{\mathit{sol}}^{\left(2\right)}\left(0\right)=g_{11}^{\left(2\right)}\left(0\right)\left(K=0,\beta \right),(9)$

which can directly be measured (Koulas-Simos et al., 2022) and is thus better suited for quantifying the amount of noise in the system. Thus, we always plot both quantities, β and $g_{\mathit{sol}}^{(2)}(0)$, when discussing the results (see, e.g., upper x-axis in Figure 3).

The last quantity that we evaluate to characterize the transition to unlocking is the diffusion coefficient D (black line in Figure 3). It increases from a value of 0 for low noise strengths to a maximum of D = 15 × 10⁻³ for β = 0.05. It should be noted that we excluded the region of high noise strength due to convergence problems (hashed area in Figure 3). As can be seen in Figure 3, the unlocking transition seen in quantity D is closely related to the change observed in the phase correlation χ with increasing noise strength. A diffusion coefficient of zero is found when e^iΔϕ is entirely locked to a certain point on the unit circle, thus leading to χ = 1. Once the lasers unlock, the diffusion coefficient D increases linearly with the noise strength. This resembles a characteristic feature of a random walk.

To understand how the noise resilience of the synchronized state changes with the coupling phase θ, we plot the phase correlation (Eq. 5) as a function of θ and β in Figure 4 (blue color code). The critical noise strength β_lock is printed as a blue line. As defined previously, it is the contour line that satisfies χ = 1 − ϵ and separates regions corresponding to unlocked and locked emission states. We see that the critical noise strength strongly depends on the coupling phase θ. Within the white areas, the deterministic system does not have a stable CLM solution that can be unlocked by noise (see Figure 2A). To exclude the details of the phase effects and to be able to study the maximum possible noise resilience of our laser system, we introduce the maximum critical noise strength.

${\beta }_{\max }^{\theta }=\underset{\theta \in \left[\mathrm{0,2}\pi \right)}{\max }{\beta }_{\mathit{lock}}\left(\theta \right).(10)$

This quantity is used to study the impact of the internal laser timescales on noise resilience as it describes the maximum noise that can be tolerated within one laser for the case where the phase is adjusted to its optimal value. As mentioned previously, the value of the noise strength is strongly model-dependent, and we rather determine the experimentally accessible intensity auto-correlation of a solitary laser at this maximum noise strength $g_{\mathit{sol}}^{(2)}{(0)}_{{\beta }_{max}^{\theta }}$ (see Figure 4).

FIGURE 4

FIGURE 4. Phase correlation χ as a function of noise strength and coupling phase θ. The critical noise strength β_lock, as determined from Eq. 8, is shown as a blue line, and the maximum noise strength is indicated by the horizontal pink-dashed line. The right y-axis gives the intensity auto-correlation of one solitary laser. Parameters as in Table 1, δω = 0.

One internal timescale that is specifically important for the dynamics of the nanolasers described with our class-C laser equations is the polarization lifetime T₂. We aim to determine if there is an optimal value $T_2^{\text{opt}}$ that maximizes the noise tolerance of the coupled system and its dependence on the frequency detuning δω between the two lasers. In order to determine the value, we evaluate and plot the correlation $g_{\mathit{sol}}^{(2)}{(0)}_{{\beta }_{max}^{\theta }}$ of the solitary laser at the maximum critical noise strength in (T₂, δω)-parameter space, as shown in Figure 5. The largest relevant detuning is the deterministic maximum locking range $\delta {\omega }_{\text{max}}^{\theta }$ (black line) because above that line, even without noise, there is no stable locking (no stable CLM exists). We see that $g_{\mathit{sol}}^{(2)}{(0)}_{{\beta }_{max}^{\theta }}$ decreases with the detuning of the lasers (dark orange). The maximum locking range $\delta {\omega }_{\text{max}}^{\theta }$ (black line) is maximized by a polarization lifetime of T₂ ≈ 1 as an optimal condition between gain bandwidth and linear system stability is reached (Roos et al., 2021). The best parameter for a large noise tolerance (large $g_{\mathit{sol}}^{(2)}{(0)}_{{\beta }_{max}^{\theta }}$), however, seems to be found for a T₂ between 2 and 3 (see bright regions in Figure 5). For this value of the dephasing time, a strong incoherence, i.e., noisiness, is needed to evoke a desynchronization, while a much smaller noise is sufficient to bring the laser out of synchronization at T₂ ≈ 1. The optimal noise resistance of the coupled laser system is thus predicted to be found for a polarization lifetime of $T_2^{\text{opt}}\approx 2.5$ as long as the detuning does not exceed δω = 0.06.

FIGURE 5

FIGURE 5. Maximum tolerated noise as a function of T₂ and detuning δω color coded via the intensity auto-correlation $g_{\mathit{sol}}^{(2)}{(0)}_{{\beta }_{max}^{\theta }}$. The black line shows the maximum locking range $\delta {\omega }_{\text{max}}^{\theta }$, which is ∝ 1/T₂ for large T₂. Parameters as in Table 1.

4 Conclusion

We investigated the impact of spontaneous emission noise on delay-coupled prototypical micro/nanolasers, which we describe using a Maxwell–Bloch type class-C laser model. This approach, despite its simplifications regarding the gain medium, provides valuable qualitative insights. In particular, it unravels the interplay among the different dynamical quantities and thereby highlights relevant trends. In order to quantify the noise-induced unlocking of the lasers, we have discussed the maximum tolerated noise strength in parameter space. The results for the noise strength are converted to the intensity auto-correlation of a single laser on the verge of unlocking. This enables easy comparison with experiments and other laser models. Our results revealed that the noise resilience of coupled lasers is, for small detunings, optimized by a polarization lifetime of two to three times the photon lifetime. Comparing the results for spontaneous emission noise with the behavior of the system under frequency detunings reveals an interesting trend: the lasers show a fundamentally different response to the two perturbations. The detuning locking range is maximized by a polarization lifetime of the order of the photon lifetime, which, however, leads to a drastic decrease in the noise resilience of the lasers.

Data availability statement

The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.

Author contributions

The project was initiated and lead by SM and KL. Numerical simulations were performed by AR. All authors discussed the results and wrote the manuscript.

Funding

The work was funded by the German Research Foundation within SFB910 project B9.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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