PHYSICAL REVIEW A 96, 043836 (2017)

Controllable asymmetric phase-locked states of the fundamental active photonic dimer

Yannis Kominis,1 Vassilios Kovanis,2 and Tassos Bountis3

1
School of Applied Mathematical and Physical Science, National Technical University of Athens, Athens, Greece
2
Department of Physics, School of Science and Technology, Nazarbayev University, Astana, Republic of Kazakhstan
3
Department of Mathematics, School of Science and Technology, Nazarbayev University, Astana, Republic of Kazakhstan
(Received 30 June 2017; published 16 October 2017)

Coupled semiconductor lasers are systems possessing complex dynamics that are interesting for numerous
applications in photonics. In this work, we investigate the existence and the stability of asymmetric phase-locked
states of the fundamental active photonic dimer consisting of two coupled lasers. We show that stable phase-locked
states of arbitrary asymmetry exist for extended regions of the parameter space of the system and that their field
amplitude ratio and phase difference can be dynamically controlled by appropriate current injection. The model
includes the important role of carrier density dynamics and shows that the phase-locked state asymmetry is
related to operation conditions providing, respectively, gain and loss in the two lasers.

DOI: 10.1103/PhysRevA.96.043836

I. INTRODUCTION field amplitudes have small values at or below the threshold

or when we take a constant gain coefficient at its saturated
Coupled laser arrays are photonic structures with great
values, given the knowledge of the field amplitudes [14]. More
potential for a large variety of applications in optical commu-
importantly, this approximation excludes some important
nications, sensing, and imaging. One of the main features that features of the complex dynamics of the system that can be
allows such applications is their electronically controlled op- interesting with respect to photonics applications, such as the
eration and tunability, suggesting their functionality as active existence of symmetric and asymmetric phase-locked states
metasurfaces for the transformation of appropriately designed and limit cycles [29,30] as well as localized synchronization
spatially inhomogeneous current distributions to desirable effects [31,32], which can be described only when carrier
field patterns. In that sense, a pair of coupled lasers can be density dynamics is taken into account [14,29]. The latter
considered as a fundamental element (a photonic “molecule”) allows for nonfixed but dynamically evolving gain and loss
from which larger and more complicated structures can be that enter the coupled field equations and introduce multiscale
built. The properties of such a pair of coupled lasers are characteristics of the system due to the significant difference
determined mostly by its stationary states and their stability, between carrier and photon lifetimes, resulting in dynamical
which can be controlled by the current injection in the two features that have no counterpart in standard coupled oscillator
lasers. The existence of stable asymmetric phase-locked states systems [33]. Moreover, carrier density dynamics introduces
with unequal field amplitudes and phase differences for the pair the role of current injection as a control mechanism for
of coupled lasers crucially determines its far field patterns [1,2] determining the dynamics of the system and its stationary
and the capabilities of such a system as a building block for states.
synthesizing larger controllable active structures characterized In this work, we investigate coupled laser dynamics in
by complex dynamics [3–10]. In addition to beam shaping terms of a model taking into account both laser coupling
applications [11], a pair of coupled lasers can be considered and carrier density dynamics. More specifically, we study the
as an element of a “photonic processor” [12,13]. existence and stability of asymmetric phase-locked modes
A pair of coupled lasers is also a fundamental element and investigate the role of detuning and inhomogeneous
for non-Hermitian optics that have been recently the subject pumping between the lasers. For the case of zero detuning,
of intense research interest. In this context, laser dynamics we show the existence of stable asymmetric modes even
is commonly described by coupled mode equations for the when the two lasers are homogeneously pumped; i.e., the
complex field amplitudes [14–18] and cases of P T -symmetric lasers are absolutely symmetric. These modes bifurcate to
configurations have been considered [19–24]. The essential stable limit cycles in an “oscillation death” scenario (Bar-
condition for P T -symmetry in a linear system is that there is no Eli effect) although an analogous requirement for dissimilar
detuning between the two lasers; in most cases balanced gain oscillators is not fullfiled [33]. For the case of nonzero
and loss are considered, however, this condition can be relaxed detuning, we show the existence of phase-locked modes with
to an arbitrary gain or loss contrast [14,17]. Similar studies arbitrary power, amplitude ratio and phase difference for
have been reported for coupled microcavities [25] and coupled appropriate selection of pumping and detuning values. These
waveguides [26]. Deviation from exact P T -symmetry can be asymmetric states are shown to be stable for large regions of
either necessitated by practical reasons [17] or intentionally the parameter space, in contrast to common coupled oscillators
designed due to advantages related to the existence of stable where asymmetric states are usually unstable [33]. Clearly,
Nonlinear Supermodes [27,28]. the above differences between coupled laser dynamics and
The coupled mode equations, considered in the study of common coupled oscillator systems are a consequence of the
P T -symmetric lasers commonly ignore the nonlinearity of the inclusion of carrier density dynamics in the model. In all cases,
system due to the dynamic coupling between field amplitudes the asymmetric states have carrier densities corresponding
and carrier densities. This approximation is valid only when the to values that are above and below threshold resulting in

2469-9926/2017/96(4)/043836(9) 043836-1 ©2017 American Physical Society

YANNIS KOMINIS, VASSILIOS KOVANIS, AND TASSOS BOUNTIS PHYSICAL REVIEW A 96, 043836 (2017)

gain coefficients of opposite signs in each laser, so that the introducing the amplitude and phase of the complex electric
respective electric fields experience gain and loss, as in the field amplitude in each laser as Ei = Xi eiθi , Eqs. (4) for M = 2
case of P T -symmetric configurations. However, it is shown are written as
that deviation from P T -symmetry, expressed by a nonzero
detuning, enables the existence of asymmetric phase-locked
dX1
states of arbitrary field amplitude ratio and phase difference, = X1 Z1 − X2 sin θ,
that are most promising for applications. dτ
This paper is organized as follows: The rate equations dX2
= X2 Z2 + X1 sin θ,
model for coupled diode lasers is described in Sec. II. In dτ
Secs. III and IV we analytically solve the inverse problem  
dθ X1 X2
of determining the phase-locked states of the system, that = − α(Z2 − Z1 ) +  − cos θ, (6)
dτ X2 X1
is, we obtain the appropriate selection of the parameters
of the system to have a given asymmetric phase-locked dZ1
2P = P1 − Z1 − (1 + 2Z1 )X12 ,
state and investigate their stability, for the case of zero and dτ
nonzero detuning, respectively. In Sec. V we numerically dZ2
investigate the deformation of the symmetric phase-locked 2P = P2 − Z2 − (1 + 2Z2 )X22 ,
dτ
states of the system under nonzero detuning and asymmetric
pumping. In Sec. VI the main conclusions of this work are
summarized. where = 2 − 1 is the detuning, θ = θ2 − θ1 is the phase
difference of the electric fields, and we have used a reference
II. RATE EQUATIONS MODEL FOR COUPLED value P = (P1 + P2 )/2 to define  as in Eq. (3). As a reference
DIODE LASERS case, we consider a pair of lasers with α = 5, T = 400, which
is a typical configuration relevant to experiments and we
The dynamics of an array of M evanescently coupled take P = 0.5. For these values we have  = 5 × 10−2 and a
semiconductor lasers is governed by the following equations coupling constant η in the range of 10−5 ÷ 100 corresponds to
for the slowly varying complex amplitude of the normalized a  in the range 0.5 × 10−3 ÷ 0.5 × 102 . The phase-locked
electric field Ei and the normalized excess carrier density Ni states are the equilibria of the dynamical system Eq. (6),
of each laser: given as the solutions of the algebraic system obtained by
dEi setting the time derivatives of the system equal to zero and
= (1 − iα)Ei Ni + iη(Ei+1 + Ei−1 ) + iωi Ei ,
dt their linear stability is determined by the eigenvalues of
dNi the Jacobian of the system. For the case of zero detuning
T = Pi − Ni − (1 + 2Ni )|Ei |2 , i = 1...M, (1) ( = 0) and symmetric electrical pumping (P1 = P2 = P0 ),
dt
two phase-locked states are known analytically: X1 = X2 =
√
where α is the linewidth enhancement factor, η is the P0 , Z1 = Z2 = 0 and θ = 0,π . The in-phase state (θ = 0) is
normalized coupling constant, Pi is the normalized excess stable for η > αP0 /(1 + 2P0 ), whereas the out-of-phase state
electrical pumping rate, ωi is the normalized optical frequency (θ = π ) is stable for η < (1 + 2P0 )/2αT [29].
detuning from a common reference, T is the ratio of carrier The phase difference θ and the electric field amplitude
to photon lifetimes, and t is the normalized time [29]. When ratio ρ ≡ X2 /X1 of a phase-locked state crucially determine
the lasers are uncoupled (η = 0), they exhibit free running the intensity response of the system. The incoherent intensity
relaxation with frequencies is defined as the sum of the individual laser intensities S ≡

2Pi |E1 |2 + |E2 |2 = (1 + ρ 2 )X12 and can be measured by placing
i = . (2) a broad detector next to the output face of the system. The
T
coherent intensity corresponds to a coherent superposition
Since we consider inhomogeneously pumped (Pi = Pj ) lasers, of the individual electric fields I ≡ |E1 + E2 |2 = (1 + ρ 2 +
we use a reference value P to define a frequency 2ρ cos θ )X12 and can be measured by placing a detector at the
 focal plane of an external lens. The coherent intensity depends
2P
= , (3) on the phase difference θ ; however, it does not take into account
T the spatial distribution of the lasers, i.e., the distance between
which is further untilized to rescale Eqs. (1) as them. The latter determines the far-field pattern of the intensity
dEi resulting from constructive and destructive interference at
= (1 − iα)Ei Zi + i(Ei+1 + Ei−1 ) + ii Ei , different directions in space. By considering, for the sake of
dτ
simplicity, the lasers as two point sources at a distance d, the
dZi coherent intensity is given by Iφ = |E1 + ei2π(d/λ) sin φ E2 |2 =
2P = Pi − Zi − (1 + 2Zi )|Ei |2 , i = 1...M, (4)
dτ E2 | = [1 + ρ 2 + 2ρ cos(θ + 2π (d/λ) sin φ)]X12 , where φ is
where the azimuthal angle measured from the direction normal to
the distance between the lasers and λ is the wavelength
τ ≡ t, Zi ≡ Ni / ,  ≡ η/, i ≡ ωi / . (5)
[34]. It is clear that the asymmetry of the phase-locked state
In the following, we investigate the existence and stability described by ρ and θ along with the geometric parameter of the
of asymmetric phase-locked states for a pair of coupled system d/λ define a specific far-field pattern Iφ with desirable
lasers under symmetric or asymmetric electrical pumping. By characteristics.

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FIG. 1. Stability regions of asymmetric phase-locked states in a symmetric configuration with = 0 and P1 = P2 = P0 in the (log10 ,ρ)
parameter space. Dark blue and light yellow areas correspond to stability and instability, respectively. (a) α = 5 and T = 400, (b) α = 1.5 and
T = 400, (c) α = 5 and T = 2000 (case of Ref. [29]).

III. ASYMMETRIC PHASE-LOCKED STATES UNDER amplitude X0 given by

ZERO DETUNING
 sin θ (ρ 2 + 1)
Although we cannot find analytical solutions of the system X02 = , (10)
ρ[(ρ 2 − 1) − 4ρ sin θ ]
of equations that provide the field amplitudes and phase
difference for a given set of laser parameters, we can solve and the common pumping is
 
explicitly the reverse problem: for a given phase-locked state P0 = X02 + 1 + 2X02 ρ sin θ. (11)
with field amplitude ratio ρ ≡ X2 /X1 and phase difference (θ )
we can analytically solve the algebraic system of equations It is quite remarkable that an asymmetric phase-locked
obtained by setting the right-hand side of Eq. (6) equal to state, with arbitrary amplitude ratio (ρ) exists even for the case
zero to determine the steady-state carrier densities (Z1,2 ), of identical coupled lasers. This asymmetric state describes a
and the appropriate detuning ( ) and pumping rates (P1,2 ), localized synchronization effect [31,32], with the degree of
in terms of ρ and θ . In this section, we consider the case localization determined by the field amplitude ratio ρ. More
of zero detuning ( = 0) between the coupled lasers. It is interestingly, this state is stable within a large area of the
straightforward to verify that for every ρ there exists an parameter space as shown in Fig. 1, in contrast to what is
equilibrium of the dynamical system Eq. (6) with a fixed phase expected [30]. In fact, there exist areas of the parameter space
difference θ , where the previously studied [29] symmetric, in-phase and
out-of-phase, states are unstable, so that this asymmetric state
1 ρ2 − 1 is the only stable phase-locked state of the system. The stability
tan θ = , (7) of this state depends strongly on the parameters α and T , as
α ρ2 + 1
shown in Fig. 1. The asymmetric phase-locked state undergoes
and Hopf-bifurcations giving rise to stable limit cycles where
the fields oscillate around the respective phase-locked values,
similarly to the case of symmetric phase-locked states [29] but
Z1 = ρ sin θ, with different oscillation amplitudes in general [32]. It is worth
(8) mentioning, that this “oscillation death” (or Bar-Eli) effect

Z2 = − sin θ, [33], commonly occurring for coupled dissimilar oscillators,
ρ is taking place even for the case of identical lasers due to the
  consideration of the role of carrier density dynamics.
P1 = X0 + 1 + 2X02 ρ sin θ,
2

(9) The amplitude ratio ρ determines the phase difference θ

   as shown in Eq. (7) and Fig. 2(a). The extreme values of the
P2 = ρ 2
X02 − 1 + 2ρ 2 X02 sin θ,
ρ phase difference are determined by the linewidth enhancement
factor α with smaller values of α allowing for larger phase
where X0 ≡ X1 . For the case of symmetrically pumped lasers differences. Moreover, the amplitude of the field depends
(P1 = P2 = P0 ) the phase-locked states have a fixed field strongly on the amplitude ratio ρ and the normalized coupling

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FIG. 2. Steady-state phase difference θ (a) and field amplitude X0 (in logarithmic scale; logarithms are to base 10) (b) of the stable
asymmetric phase-locked state in the (log10 ,ρ) parameter space. Parameter values correspond to Fig. 1(a).

constant  as shown in Eq. (10) and Fig. 2(b). The appropriate as the appropriate detuning ( ) and pumping rates (P1,2 ) for
symmetric pumping P0 in order to have an asymmetric an arbitrary field amplitude ratio (ρ) and phase difference (θ ),
phase-locked state is given by Eq. (11). It is obvious from as follows:
the above equations that for ρ = 1 the well-known symmetric
states are obtained [29]. Z1 = ρ sin θ,
(12)
Phase-locked states with fixed phase difference θ but 
arbitrary electric field amplitude X0 exist for different pumping Z2 = −
sin θ,
ρ
between the two lasers, given by Eq. (9). The stability of these    
states depends crucially on the electric field amplitude X0 as 1 1
= −α sin θ + ρ −  cos θ −ρ , (13)
shown in Fig. 3. In comparison to Fig. 1(a) corresponding to ρ ρ
the same parameter set but with P1 = P2 = P0 the extent of  
the stability region is significantly reduced. P1 = X02 + 1 + 2X02 ρ sin θ,
(14)
  
P2 = ρ 2
X02 − 1 + 2ρ 2 X02 sin θ.
IV. PHASED-LOCKED STATES WITH ARBITRARY ρ
ASYMMETRY UNDER NONZERO DETUNING
Therefore, there always exists a phase-locked state with
Analogously to the case of zero detuning we can find arbitrary field amplitude asymmetry and phase difference,
analytically the steady-state carrier densities (Z1,2 ) as well provided that the detuning and the pumping rates P1,2 have

FIG. 3. Stability regions of asymmetric phase-locked states in an asymmetric configuration with = 0 and P1 = P2 in the (log10 ,ρ)
parameter space. Dark blue and light yellow areas correspond to stability and instability, respectively. α = 5, T = 400, and (a) log10 X0 = −3,
(b) log10 X0 = −1.5, (c) log10 X0 = −1.

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FIG. 4. Stability regions

√ of phase-locked states of arbitrary asymmetry, characterized by the steady-state field amplitude ratio ρ, phase
difference θ and X0 = 0.5. Dark blue and light yellow areas correspond to stability and instability, respectively. The parameters of the coupled
lasers are α = 5, T = 400 and log10  = −2.1,−1.9,−1.7,0,0.5,2 (a)–(f). The respective detuning and pumping P1,2 values are given by
Eqs. (13) and (14). The stability regions are symmetric with respect to the transformation ρ → 1/ρ and θ → 2π − θ. The topology and the
extent of the stability region depends crucially on the coupling coefficient .

values given by Eqs. (13) and (14), respectively, while the bifurcations giving rise to stable limit cycles characterized by
steady-state carrier densities (Z1,2 ) are given by Eqs. (12). asymmetric synchronized oscillations of the electric fields,
These phase-locked states exist in the whole parameter space that can have different mean values and amplitudes of
and can have an arbitrary power X0 . However, their stability oscillation.
depends strongly on the coupling () as well as the power Characteristic cases for the time evolution of the electric
(X0 ) and the degree of their asymmetry, characterized by ρ field amplitudes X1,2 and the phase difference θ are depicted
and θ , as shown in Figs. 4(a)–4(f). It is worth mentioning in Fig. 5 for various degrees of asymmetry. The parameters
that there is enough freedom in parameter selection to have of the system correspond to those of Fig. 4(a), with phase
a controllable configuration that supports a large variety of difference θ = 0.9π  2.83 and various values of ρ. For ρ =
stable asymmetric phase-locked states with unequal field 0.75 [Fig. 5(a)] the asymmetric phase-locked state is stable
amplitudes and phase differences, with the latter crucially and perturbed initial conditions evolve to the stable state. As ρ
determining the far field patterns of the pair of coupled decreases to ρ = 0.5 [Fig. 5(b)] and 0.25 [Fig. 5(c)], the phase-
lasers. The asymmetric states are characterized by carrier locked states become unstable and the system evolves to stable
densities having opposite signs Z1 /Z2 = −ρ 2 < 0 so that the limit cycles of increasing period. Close to the center of the
electric fields of the two lasers experience gain and loss, unstable region the system evolves to chaotic states [ρ = 0.15,
respectively. For ρ = 1 we have equal gain and loss and Fig. 5(d)]. Further decreasing ρ results in stable limit cycles
a phase-locked state with equal field amplitude and phase [ρ = 0.10, Fig. 5(e)] and stable phase-locked states [ρ = 0.05,
difference given by Eq. (13) as sin θ = − /2α. At the Fig. 5(f)] corresponding to the stability region of lower ρ
boundaries of the stability regions, the system undergoes Hopf shown in Fig. 4(a).

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YANNIS KOMINIS, VASSILIOS KOVANIS, AND TASSOS BOUNTIS PHYSICAL REVIEW A 96, 043836 (2017)

FIG. 5. Time evolution of the electric field amplitudes and phase difference for parameters corresponding to Fig. 4(a). The initial conditions
correspond to asymmetric phase-locked states with θ = 0.9π  2.83 and ρ = 0.75 (a), 0.50 (b), 0.25 (c), 0.15 (d), 0.10 (e), 0.05 (f), perturbed
by random noise. In accordance to Fig. 4(a), the phase-locked states are unstable for 0.06 < ρ < 0.54. Cases of stable phase-locked states
are shown in (a) and (f). In the case of unstable phase-locked states the system evolves either to stale limit cycles [(b), (c), (e)] or to chaotic
states (d).

FIG. 6. Existence and stability regions of phase-locked states in the (log10 , P ) parameter space for α = 5 and T = 400 when P0 = 0.5.
Dark blue and light yellow areas correspond to stability and instability, respectively. The green area corresponds to nonexistence of a
phased-locked state due to nonzero detuning. (a) = 0, (b) = 0.05, (c) = 0.1.

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FIG. 7. Steady-state phase difference (θ ) of the stable phase-locked states in the (log10 , P ) parameter space for α = 5, T = 400 and
P0 = 0.5. (a) = 0, (b) = 0.05 correspond to the cases of Figs. 6(a) and 6(b), respectively.

V. DEFORMATION OF SYMMETRIC PHASED-LOCKED space and extends to values P = P0 for which only one of
STATES UNDER NONZERO DETUNING AND PUMPING the lasers is pumped above threshold (P1 = 1, P2 = 0). This
ASYMMETRY area of stability extending from intermediate to high values of
coupling is enabled by the asymmetric pumping and indicates
For the case of a given nonzero detuning and/or asym-
its stabilizing effect. The dependence of the phase difference of
metrically pumped lasers P1 = P0 + P ,P2 = P0 − P , the
the stable states on the coupling  and pumping difference P
equilibria of the system Eq. (6) cannot be analytically obtained
is depicted in Fig. 7(a). In Fig. 8, the electric field amplitudes
for a given set of values ( , P ). The respective algebraic
X1,2 and carrier densities X1,2 of all the stable states √ are
system consists of transcendental equations and is solved
shown. It is clear that for P = 0 we have X1,2 = P 0 and
by utilizing a numerical continuation algorithm, according to
Z1,2 = 0 and for small values of  we have X1,2 = P1,2
which we start from = 0 and P = 0 corresponding to the
and Z1,2 = 0, as the two lasers are essentially uncoupled.
symmetric case with the two known equilibria (θ = 0,π ). For
For P > 0 and finite coupling values the electric field and
each one of them, we increase and/or P in small steps; in
carrier density destributions in the two lasers become highly
each step the solution of the previous step is used as an initial
asymmetric.
guess for the iterative procedure (Newton-Raphson method)
A nonzero detuning ( = 0) between the two lasers
that provides the solution.
strongly affects the existence of a stable out-of-phase state
For the case of zero detuning = 0, the domain of exis-
in the weak coupling regime, as shown in Figs. 6(b) and 6(c)
tence of stable phase-locked states in the (, P ) parameter
for = 0.05,0.1, respectively. The role of detuning in terms
space is shown in Fig. 6(a). For P = 0 the results are similar
to the case considered in [29], with the in-phase state being of the phase of the stable states is clearly presented in Fig. 7(b)
stable for large  and the out-of-phase being stable for small for = 0.05. In comparison to Fig. 7(a), corresponding to
values of . As the pumping difference increases, the stable zero detuning, it is obvious that the phase of the stable
in-phase state extends only over a quite small range of P , states existing for intermediate and strong coupling is hardly
whereas the out-of-phase state extends almost in the entire affected by the detuning, whereas the stable state, existing
range of P . In both cases, as P increases from zero the in the weak coupling regime, has a phase that ranges from
phase difference is slightly differentiated from the values θ = π to 3π/2 depending strongly on  and P . The role of
0,π for P = 0. Surprisingly, another region of stable phase- detuning is also shown in Figs. 9(a) and 9(b) for = 0.05
locked states appears in the strong coupling regime (large ) in the case of strong (log10  = 1) and weak (log10  =
above a threshold of pumping difference P . This is an −2.2) coupling, respectively. The detuning introduces phase
out-of-phase state with phase difference close to π that appears sensitivity to current injection for stable modes of the weak
for values of P for which no stable in-phase state exists. In coupling regime that can be quite interesting for beam-steering
fact, this stable state exists for a large part of the parameter applications [1].

FIG. 8. Steady-state electric field amplitudes X1,2 (a) and carrier densities Z1,2 (b) of the stable phase-locked states in the (log10 , P )
parameter space for α = 5, T = 400 and P0 = 0.5, corresponding to the case of Fig. 6(a).

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FIG. 9. Steady-state phase difference (θ ) of the stable phase-locked states in the ( , P ) parameter space for α = 5, T = 400 and P0 = 0.5.
(a) strong coupling log10  = 1, (b) weak coupling log10  = −2.2.

VI. CONCLUSIONS the pumping profile of the system. The role of the current
injection suggests a dynamic mechanism for the control of
We have investigated the existence of stable asymmetric
the phase-locked states and, therefore, the far-field emission
phase-locked states in a system of two coupled semiconductor
patterns of this fundamental photonic element consisting of
lasers. The asymmetric phase-locked states are characterized
two coupled lasers.
by a nonunitary field amplitude ratio and nontrivial phase
difference. The asymmetry is shown to be directly related to
operation conditions that result in the presence of gain in one
ACKNOWLEDGMENTS
laser and loss in the other. The crucial role of carrier density
dynamics has been taken into account by considering a model Y.K. is grateful to the School of Science and Technology of
where the field equations are dynamically coupled with the Nazarbayev University, Astana, Kazakhstan for its hospitality
carrier density equations, in contrast to standard coupled mode during his visit at NU. This research is partly supported
equations, commonly considered in studies on non-Hermitian by the ORAU grant entitled “Taming Chimeras to Achieve
photonics and PT-symmetric lasers. It has been shown that the Superradiant Emitter,” funded by Nazarbayev University,
stable asymmetric states exist even in absolutely symmetric Republic of Kazakhstan. This work was partially also sup-
configurations and that states of arbitrary asymmetry can ported by the Ministry of Education and Science of the
be supported by appropriate selection of the detuning and Republic of Kazakhstan via Contract No. 339/76-2015.

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