Exceptional Points in a Non-Markovian Anti-Parity-Time Symmetric System

Abstract

By studying the eigenvalues and eigenvectors of a non-Markovian anti parity-time (APT) symmetric system, we investigate the possibility of exceptional points (EPs) that may arise within it. Our work is motivated by a recently studied APT-symmetric experimental configuration consisting of a pair of time-delay coupled semiconductor lasers (SCLs). In such a system, a single time-delay represents the memory. The time-delayed coupling makes the system’s effective Hamiltonian infinite-dimensional and leads to novel features in the corresponding eigenvalues and eigenvectors. In particular, we demonstrate analytically and numerically that unlike a typical PT-symmetric dimer with zero time-delay, which has one second-order EP, our time-delayed system has parameter regimes that give rise to either one, two, or zero second-order EPs and one third-order EP, and one can select these regimes though a judicious choice of the time-delay and coupling.

1. Introduction

Recent years have seen an explosion of interest in parity-time symmetric (PT) systems which are described by a class of non-hermitian Hamiltonians that may yield real eigenvalues under some conditions [1,2,3,4,5,6,7,8,9,10]. PT-symmetric Hamiltonians are invariant under simultaneous application of parity ($x\to -x$, $p\to -p$) and time-reversal ($x\to x$, $t\to -t$, $i\to -i$) operators [1,2], and a system described by such a Hamiltonian can be tuned via a non-Hermiticity parameter such that the eigenvalues undergo a transition from real to complex. The fundamental interest in PT-symmetric Hamiltonians arises from the possibility of extending quantum mechanics to open systems and complex space, where non-Hermiticity is associated with energy or probability dissipation. PT-symmetry has also garnered much interest in the field of optics, where the index of refraction plays the role of a complex potential. Such PT-symmetric optical systems have been a fertile playground for the experimental implementation of PT-symmetry, and various configurations with balanced gain and loss have been proposed and demonstrated [11,12,13,14,15]. On the applied front, PT-symmetric systems have demonstrated a rich variety of novel phenomena, which include unidirectional light propagation and single-mode lasing, among others [16,17,18,19].

In addition to PT-symmetry, open systems with anti-PT symmetry have begun to surface. Such systems are described by Hamiltonians which anticommute with the PT operator; in other words, the system’s Liouvillian $L=-iH$ commutes with the PT operator. A property of these APT systems is that the eigenvalues of the Liouvillian are either purely real or come in complex conjugate pairs [20]. APT-symmetric systems have been experimentally realized in various settings [9,21], including active electrical circuits [22], rotating disks with thermal gradients [23], and other optical systems. A recent work has proposed algorithms for mimicing an $anti$-PT system in a quantum circuit model [24].

One common realization of a PT-symmetric dimer consists of two evansecently coupled optical resonators, one of which experiences net gain and the other an equal amount of loss [11]. Denoting the gain/loss by $\pm \gamma$ and the mutual coupling by $\kappa$, the resulting Hamiltonian is

$$\begin{array}{c}H=\left(\begin{array}{cc}\mathit{i}\gamma & \kappa \\ \kappa & -\mathit{i}\gamma \end{array}\right).\end{array}$$

(1)

The eigenvalues of this matrix are given by $\lambda =\pm \sqrt{{\kappa }^2-{\gamma }^2}$, which are real for $\gamma <\kappa$, complex for $\gamma >\kappa$. For $\gamma =\kappa$ the two eigenvalues become degenerate and the corresponding eigenvectors coalesce. The regime with real eigenvalues is referred to as the PT-unbroken phase, and the regime with complex eigenvalues is the PT-broken phase. In the unbroken regime the wavefunction norm is bounded, but in the PT-broken regime the norm grows or decays exponentially. The transition from one phase to the other is marked by a sharp threshold, $\gamma =\kappa$ in this case, and this point is called an exceptional point (EP), defined by degenerate eigenvalues and parallel eigenvectors. EPs are generic properties of non-hermitian matrices and as such appear in classical and quantum systems that can be characterized as eigenvalue problems. They were extensively discussed in nuclear physics [25], predicted in quantum chaotic systems [26], and shown to occur in coupled microwave cavities [27]. Recent review articles discussing EPs in photonic systems can be found in Refs. [28,29]. The EP boundary between the PT-broken and unbroken regions is a key feature of PT-symmetric and APT-symmetric systems, and there has been considerable effort devoted to investigating the properties of EPs [30,31,32,33,34,35,36,37,38,39,40,41,42,43]. Indeed, very recently the importance of EPs in PT-symmetric photonic systems has received much attention (see e.g., Ref. [30]). In particular, because the variation across an EP leads to an abrupt increase in the norm of the wavefunction, there has been an interest in exploiting EPs for enhanced sensing applications. It has been shown that when operating a non-Hermitian system near an EP with n degeneracies, a small perturbation $ϵ$ produces an eigenvalue response of order ${ϵ}^{1/n}$, as opposed to a linear response in Hermitian systems [43,44,45]. For $ϵ<1$ this can enormously increase sensing resolution.

Nearly all PT-symmetric and APT-symmetric systems to date have been Markovian, i.e., memoryless. In other words, the rate equation governing the system’s evolution is local in time and does not explicitly on the system’s history: $i{\partial }_t|\psi \left(t\right)\rangle =H[t;\psi \left(t\right)]|\psi \left(t\right)\rangle$. This definition includes even non-Hermitian and non-linear systems. Despite not being fundamentally prohibited, the Markovian property is typically treated as a basic system feature.

Here we aim to investigate the eigenvalues and exceptional points that arise in a system subject to non-markovianity and APT-symmetry, and discuss how the results differ from a typical PT-symmetric system. This paper is motivated by a recent experimental implementation [46] of classical APT-symmetry in a pair of mutually delay-coupled semiconductor lasers (SCLs), wherein light from one SCL is injected into the active region of the other SCL and vice-versa, as shown schematically in Figure 1. A time-delayed coupling arose naturally in this system due to the physical separation between the lasers, which was comparable to the characteristic time scales of laser’s photon and carrier inversion lifetimes. This time delay thereby introduces a novel element into the study of experimental PT-symmetry [47]. It is important to emphasize that even though there are only two optical resonators, this dimer configuration becomes infinite-dimensional due to the delayed coupling. Without the time-delay the system has, under certain approximations, an effective Hamiltonian reminiscent of Equation (1), with the diagonal elements now being the relative detuning between the lasers ($\pm \mathsf{\Delta }\omega$) and the off-diagonal elements being the coupling strength of light from one laser into the other ($\kappa$) [45,46]. The relative detuning $\mathsf{\Delta }\omega$ is therefore the non-hermiticity parameter for the coupled SCLs. As we have shown in prior work [46,48], despite a time-delayed coupling the SCLs continue to show remnants of PT-symmetric behavior commonly seen for zero delay, i.e., an abrupt phase transition across a boundary determined by the EP at $\mathsf{\Delta }\omega =\kappa$. The delay also introduces new features into the behavior of the system, e.g., additional transitions which manifest themselves as oscillations in a laser’s intensity as the relative detuning is varied. These additional transitions were observed experimentally and faithfully reproduced in our numerical analysis based on a model that is commonly used to study coupled SCLs [47,49]. In Ref. [46] we characterized the dependence of these additional transitions on coupling strength and time-delay through experiments, simulations, and analytic calculations.

The infinite-dimensional nature of the system provided the stimuli for our examination of the EPs. Are there EPs in an APT-symmetric dimer if a time-delay is introduced in the coupling term? If so, how many are permitted, and under what conditions do these EPs appear? To address these questions, the emphasis of this paper is on an analysis of the eigenvalues (and eigenvectors) of the infinite-dimensional effective Liouvillian that arises when a time-delayed coupling is introduced into an otherwise conventional APT-symmetric system. We begin our investigation by presenting an effective Liouvillian equation model that is used to describe a pair of mutually coupled SCLs, which we have shown in a previous work reduces from the lasers’ microscopic rate equations under certain reasonable approximations [48]. We then use the Lambert W function, which we have shown recently can be used to study the dynamics of a laser with time-delayed coupling [50], to obtain analytical results which are in excellent agreement with numerical results [46]. The analytic results allow us to establish constraints on parameters of the system which lead to EPs. In addition to revealing the overall behavior of the eigenvalues and eigenvectors in a time-delayed PT-symmetric system, our work establishes, numerically and analytically, that depending on parameter choices this system can have one, two, or zero second order EPs and one third-order EP, in contrast to the zero-delay configuration wherein there is only one EP at $\mathsf{\Delta }\omega =\kappa$.

2. Model

In the general case with non-zero delay, the evolution of the electric fields in the system is represented by ${\partial }_t\overrightarrow{E}\left(t\right)=\mathcal{L}\overrightarrow{E}\left(t\right)$, with eigenmodes $\overrightarrow{E}\left(t\right)={[E_1\left(t\right),E_2\left(t\right)]}^T$ and the non-local Liouvillian given by:

$$\mathcal{L}(\mathsf{\Delta }\omega ,\kappa ,\tau ,\theta )=i\mathsf{\Delta }\omega {\sigma }_z+\kappa e^{-i\theta \tau }{e}^{-\tau {\partial }_t}{\sigma }_x$$

(2)

In the context of a delay-coupled laser system [51], $\mathsf{\Delta }\omega$ is the frequency detuning between the resonators, $\kappa$ is the coupling between them, $\tau$ is the delay time, and $\theta$ is the resonator phase difference. All these values represent physical quantities, and are therefore assumed to be real. In addition, the quantities $\kappa$ and $\tau$ are positive by definition. The product $\theta \tau$ is referred to as the “phase accumulation”, and allows for greater tuning of the system’s non-Hermiticity. If $\theta \tau \ne n\pi$ (where n is an integer), the system’s APT-symmetry is broken.

Assuming solutions of the form $\overrightarrow{E}\left(t\right)=\exp \left(\lambda t\right)\overrightarrow{E}\left(0\right)$, the Liouvillian’s characteristic equation is

$${\lambda }^2+\mathsf{\Delta }{\omega }^2-{\kappa }^2{e}^{-2\lambda \tau }{e}^{-2i\theta \tau }=0.$$

(3)

It is important to note that when $\tau \ne 0$ the transcendental characteristic equation is not a 2 × 2 eigenvlaue problem; rather, it is infinite dimensional due to the time delay, resulting in infinitely many eigenvalues. Separating the real and imaginary components with the substitution $\lambda ={\lambda }_R+\mathit{i}{\lambda }_I$, where ${\lambda }_R$ and ${\lambda }_I$ are real, a pair of coupled equations is obtained. Here $F({\lambda }_R,{\lambda }_I)$ contains the real parts and $G({\lambda }_R,{\lambda }_I)$ contains the imaginary parts:

$$\begin{array}{cc}\hfill F({\lambda }_R,{\lambda }_I)& =\mathsf{\Delta }{\omega }^2+{\lambda }_R^2-{\lambda }_I^2\hfill \\ \hfill & -{\kappa }^2{e}^{-2{\lambda }_R\tau }\cos \left(2\tau ({\lambda }_I+\theta )\right)=0\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill G({\lambda }_R,{\lambda }_I)& =2{\lambda }_R{\lambda }_I\hfill \\ \hfill & +{\kappa }^2{e}^{-2{\lambda }_R\tau }\sin \left(2\tau ({\lambda }_I+\theta )\right)=0\hfill \end{array}$$

(5)

The G equation is independent of $\mathsf{\Delta }\omega$, an observation that will be important later in the discussion. We find solutions for the eigenvalues, $\lambda$, by solving the coupled F and G equations.

The eigenvalues are then used to generate corresponding eigenvectors. Assuming a generic eigenvector ${(\alpha ,\beta )}^T$ and substituting it into equation Equation (2) results in coupled equations for $\alpha$ and $\beta$:

$$\lambda \alpha =\mathit{i}\mathsf{\Delta }\omega \alpha +\kappa e^{-\lambda \tau }{e}^{-\mathit{i}\theta \tau }\beta$$

(6)

$$\lambda \beta =\kappa e^{-\lambda \tau }{e}^{-\mathit{i}\theta \tau }\alpha -\mathit{i}\mathsf{\Delta }\omega \beta .$$

(7)

Without loss of generality we can set $\alpha$ =1:

$$\beta =\frac{\lambda -\mathit{i}\mathsf{\Delta }\omega }{\kappa e^{-\lambda \tau }{e}^{-\mathit{i}\theta \tau }}.$$

(8)

The evolution of the eigenvectors (1, ${\beta )}^T$ can therefore be tracked by following the evolution of $\beta$. From the degeneracy of eigenvalues and eigenvectors we will obtain information about the EPs of the time-delayed system.

It is elucidating to begin with a discussion of the F and G equations, as they underlie the behavior and evolution of the system’s eigenvalues and eigenvectors. Typical plots of $F({\lambda }_R,{\lambda }_I)$ (blue) and $G({\lambda }_R,{\lambda }_I)$ (red) are given in Figure 2. To illustrate the important features, we will, for now, set the phase accumulation term $\theta \tau$ to be $2n\pi$ (where n is an integer). Each F and G plot has “fingers” which extend from Re($\lambda$) = $-\infty$ and curve back around near Re(λ) = 0, and the spacing of each “finger” corresponds to the period of the sine and cosine terms in Equations (4) and (5), $\pi /2\tau$. We label these “fingers” with indices 0, ±1, ±2, etc., with the 0th finger corresponding to the one that straddles the Re($\lambda )$-axis and the index labels increasing (decreasing) as the fingers shift up (down) along the Im($\lambda )$-axis. Refer to Figure 2 for an example of this labeling. The location of the turning point for each finger (i.e., the rightmost point the finger reaches on the Re($\lambda )$-axis) depends on both $\kappa$ and $\tau$. Only if $\kappa \tau >1/\sqrt{2e}$ does the turning point lie beyond Re($\lambda$) = 0, shown with the black arrows in Figure 2b. As the detuning ($\mathsf{\Delta }\omega$) approaches zero, the F plot’s “antennae”, the two branches that extend outward toward Re($\lambda$) $=\pm \infty$, move closer to the origin (result not shown). The G plot, as stated earlier, does not depend on $\mathsf{\Delta }\omega$, and the line Im($\lambda$) = 0 is always a solution to the G equation. One particular branch of G is important in the eigenvalue dynamics, namely the branch that straddles the Re($\lambda$) = 0 axis (pictured in Figure 2a insert). This branch is continuous for $\kappa \tau$ < $1/\sqrt{2e}$ and is broken otherwise—as we will see in the next section, this breaking condition is critical to the existence of exceptional points. Finally, it is clear from these plots that besides the solutions on the Im($\lambda$) = 0 axis, all the intersections of F and G come in complex conjugate pairs (as long as $\theta \tau =n\pi$). How these eigenvalue solutions evolve with $\mathsf{\Delta }\omega$, $\tau$, $\kappa$, and $\theta$ is discussed in the next section, with an emphasis on eigenvalue degeneracies. Using these degeneracies we can map out the system’s exceptional point landscape.

3. Analytic Results

Time-delayed differential equations are often intractable, but the Lambert W function can be used under some conditions to extract analytical solutions. In the following, the W function is used to obtain closed-form solutions for the system’s eigenvalues and exceptional points. We refer the reader to Ref. [52] for an excellent description of the Lambert W function.

The Liouvillian’s characteristic equation, Equation (3), can be factored as

$$\begin{array}{cc}\hfill {\lambda }^2+\mathsf{\Delta }{\omega }^2-& {\kappa }^2{e}^{-2\tau (\lambda +i\theta )}\hfill \\ \hfill & =(\lambda -x_1)(\lambda -x_2)(\lambda -x_3)\dots =0,\hfill \end{array}$$

(9)

where $x_n$ are solutions to Equation (3). In the case of an eigenvalue degeneracy of order N, Equation (7) can be written as

$${(\lambda -x_1)}^N(\lambda -x_2)(\lambda -x_3)\dots =0,$$

(10)

where $x_1$ is the degenerate eigenvalue. A consequence of this degeneracy is that the 1st to Nth derivatives of Equation (8) with respect to $\lambda$ also vanish at $x_1$, since the term $(\lambda -x_1)$ is present in all terms after differentiation. In the following sections, we begin by analyzing the exceptional point landscape of 2nd order degeneracies, and then extend the analysis to higher-order EPs.

3.1. Second-Order EPs

For $N=2$, the first derivative of Equation (7) with respect to $\lambda$ is

$$2\lambda +2\tau {\kappa }^2{e}^{-2\tau (\lambda +i\theta )}=0,$$

(11)

and combining Equation (7) with Equation (9) we solve for the detuning $\mathsf{\Delta }\omega$ for which there is a degeneracy. After eliminating $\lambda$ we arrive at

$$\begin{array}{cc}\hfill 1\mp & \sqrt{1-{\left(2\mathsf{\Delta }\omega \tau \right)}^2}\hfill \\ \hfill & -2{\kappa }^2{\tau }^2{e}^{-2i\theta \tau }{e}^{1\mp \sqrt{1-{\left(2\mathsf{\Delta }\omega \tau \right)}^2}}=0,\hfill \end{array}$$

(12)

which can be written as

$$-z{e}^{-z}=-2{\kappa }^2{\tau }^2{e}^{-2i\theta \tau },$$

(13)

where $z=1\mp \sqrt{1-{\left(2\mathsf{\Delta }\omega \tau \right)}^2}$. The solutions of Equation (11) are given by the Lambert W function:

$$z=-{\mathrm{W}}_n(-2{\kappa }^2{\tau }^2{e}^{-2i\theta \tau })$$

(14)

where n is an integer. Equation (11) is written in the standard form of the Lambert W function, which is defined as the inverse of $f\left(\omega \right)=\omega e^{\omega }$. The Lambert W function is commonly found in spectral analyses of systems with time delay due to the inclusion of terms like $e^{\lambda \tau }$ in the characteristic equations [52]. It is real-valued only when $f\left(\omega \right)$ is real and greater than $-1/e$ (Figure 3). From Equation (12) we derive an expression for the critical detuning $\mathsf{\Delta }{\omega }_c$ at which a degeneracy occurs:

$$\mathsf{\Delta }{\omega }_c=\frac{\pm 1}{2\tau }\sqrt{1-{[{\mathrm{W}}_n(-2{\kappa }^2{\tau }^2{e}^{-2i\theta \tau })+1]}^2}.$$

(15)

We require that $\mathsf{\Delta }{\omega }_c$ be a real frequency, which implies that the term under the radical is positive. Assuming a general solution ${\mathrm{W}}_n(-2{\kappa }^2{\tau }^2{e}^{-2i\theta \tau })=a+ib$ results in the requirement that ${\mathrm{W}}_n$ have the form

$${\mathrm{W}}_n(-2{\kappa }^2{\tau }^2{e}^{-2i\theta \tau })=a$$

(16)

or

$${\mathrm{W}}_n(-2{\kappa }^2{\tau }^2{e}^{-2i\theta \tau })=-1+ib,$$

(17)

where a and b are real. This requirement places constraints on the argument of ${\mathrm{W}}_n$, and therefore the parameter space swept by $\kappa ,\tau$, and $\theta$ is narrowed.

When substituted into Equation (13), the first solution ${\mathrm{W}}_n(-2{\kappa }^2{\tau }^2{e}^{-2i\theta \tau })=a$ gives

$$\mathsf{\Delta }{\omega }_c=\frac{\pm 1}{2\tau }\sqrt{-(a^2+2a)},$$

(18)

which, since $\mathsf{\Delta }{\omega }_c$ must be real, produces an additional requirement that a must lie between $-2$ and 0. Now, because the Lambert W function is only real-valued when the argument is purely real and is greater than $-1/e$, $-2{\kappa }^2{\tau }^2{e}^{-2i\theta \tau }$ must be real and greater than $-1/e$, which implies that $\theta \tau =n\pi$ (where n is an integer). When evaluating the Lambert W function with a real argument, there are two branches that can produce purely real outputs, namely the 0 and $-1$ branches. A plot of these real-valued solutions is given in Figure 3. The ${\mathrm{W}}_0\left(x\right)$ branch (blue) is bounded on the left by the point ($-1/e$, $-1$) and goes to infinity as x increases, while the ${\mathrm{W}}_{-1}\left(x\right)$ branch (red) is bounded on the left by ($-1/e$, $-1$) and travels down asymptotically to the y-axis as x approaches zero. ${\mathrm{W}}_{-1}\left(x\right)$ has no real solutions for $x>0$. It’s clear from the graph that ${\mathrm{W}}_0\left(x\right)$ satisfies $-2\le {\mathrm{W}}_0\left(x\right)\le 0$ for all $-1/e\le x\le 0$; however, ${\mathrm{W}}_{-1}\left(x\right)$ satisfies $-2\le {\mathrm{W}}_{-1}\left(x\right)\le 0$ only for $-1/e\le x\le 2/e^2$. This translates into the following constraints on $\kappa \tau$: for the ${\mathrm{W}}_0\left(x\right)$ branch $\kappa \tau$ must satisfy $0\le 2{\kappa }^2{\tau }^2\le 1/e$, and for the ${\mathrm{W}}_{-1}\left(x\right)$ branch $\kappa \tau$ must satisfy $2/e^2\le 2{\kappa }^2{\tau }^2\le 1/e$.

In the context of the F and G equations, the existence of the ${\mathrm{W}}_0$ solution depends on the continuity of the G branch that straddles the Re($\lambda$) = 0 axis—when $2{\kappa }^2{\tau }^2>1/e$, that branch becomes discontinuous near the origin and the two “antennae” branches of F will never merge (Figure 4a,b). The existence of the ${\mathrm{W}}_{-1}$ solution is related to the merging of the first two tongues of the F plot—when $2/e^2\le 2{\kappa }^2{\tau }^2\le 1/e$, there exists a detuning for which the $\pm 1$ tongues merge into the 0th tongue, and the merging location on the Im(λ) = 0 axis is determined by where the 0th tongue of G crosses the Im($\lambda$) = 0 axis (Figure 5a,b).

The second solution, ${\mathrm{W}}_n=-1+ib$, is more complicated. Starting from Equation (15),

$$-1+ib={\mathrm{W}}_n(-2{\kappa }^2{\tau }^2{e}^{-2i\theta \tau })$$

(19)

we take the inverse and solve for $\theta$:

$$e^{-2i\theta \tau }=\frac{(-1+ib)}{-2e{\kappa }^2{\tau }^2}{e}^{ib}.$$

(20)

Taking the complex logarithm of both sides and separating the real and imaginary parts allows us to solve for b:

$$b^2={\left(2e{\kappa }^2{\tau }^2\right)}^2-1.$$

(21)

When substituted, the resulting equation provides solutions for ${\theta }_c$, the phase for which, given both $\kappa$ and $\tau$, there exists an eigenvalue degeneracy:

$$\begin{array}{cc}\hfill {\theta }_c\tau =\frac{1}{2}[& \pm \sqrt{{\left(2e{\kappa }^2{\tau }^2\right)}^2-1}\hfill \\ \hfill & \mp arctan(\sqrt{{\left(2e{\kappa }^2{\tau }^2\right)}^2-1}\left)\right]+n\pi ,\hfill \end{array}$$

(22)

where n is an integer. A plot of ${\theta }_c\tau$ vs. $\kappa \tau$ is shown in Figure 6. For $\kappa \tau \le 1/e$ there are no values of $\theta$ that produce a degeneracy, which agrees with our previous conclusion (i.e., ${\mathrm{W}}_n=a$), and is a consequence of requiring that $\theta$ be real. When $\kappa \tau >1/e$ there are two supplementary solutions for ${\theta }_c\tau$ in the range $0\ge \theta \tau \ge \pi$, their sum being $\pi$.

Graphically, the solutions provided by Equation (20) come from the separation of tongues in the G plot as $\kappa \tau$ increases. The proper index in Equation (13) is chosen based on the observation of which tongue m is being created—explicitly, the index is $n=m\pm 1$ (positive for tongues above the Im($\lambda$) = 0 axis and vice versa). For example, when $\kappa \tau =2.5$ and $\theta \tau =0.51$ we see that the −5th tongue is separating, and therefore we use $n=-6$ in Equation (13), resulting in the proper detuning $\mathsf{\Delta }\omega /\kappa =6.8$ (see Figure 7 for a visualization of this merging process).

The frequency detunings associated with these degeneracies can be found using Equation (13) with the proper values of $\kappa \tau$ and $\theta \tau$; however, the proper branch of ${\mathrm{W}}_n$ must be used as well. Figure 8a is a summary plot of the second-order EPs that arise on the $\mathsf{\Delta }\omega /\kappa$ vs. $\kappa \tau$ plane for $\theta \tau =n\pi$. Curve A shows the branch of EPs between $0\le \kappa \tau \le 1/\sqrt{2e}$ which corresponds to solutions with the ${\mathrm{W}}_0$ branch. This EP curve lies very close to the parameters that give rise to an EP for the zero delay case, i.e., $\mathsf{\Delta }\omega /\kappa =1$. Similarly, curve B represents solutions with the ${\mathrm{W}}_{-1}$ branch for $1/e\le \kappa \tau \le \sqrt{2e}$.

For $\kappa \tau >1/\sqrt{2e}$ there are broadly spaced points of EPs, which arise from the $\theta \tau =0$ solutions to Equation (20). These point-EPs asymptotically follow a linear trajectory and increase in frequency as $\kappa \tau$ grows. The slope of this asymptote is e and the period between EP points decreases as $\sqrt{\frac{\pi }{4ne}}$, where n is a positive integer. This can be proven by setting $\theta \tau =0$ in Equation (20). In the limit of large $\kappa \tau$, the equation becomes

$$0=\pm 2e{\kappa }^2{\tau }^2\mp \pi /2+2\pi n,$$

(23)

which when rearranged gives the value of $\kappa \tau$ for which there exists an EP:

$${\left({\kappa }^2{\tau }^2\right)}_n=\frac{\pi }{4e}(1+4n),$$

(24)

The choice of sign in Equation (21) ensures that the value of (${\kappa }^2{\tau }^2{)}_n$ in Equation (22) is positive and therefore physically meaningful. The period between successive EPs can be evaluated by taking the difference ${\left({\kappa }^2{\tau }^2\right)}_{n+1}-{\left({\kappa }^2{\tau }^2\right)}_n$ and using the binomial approximation for large n:

$$\begin{array}{cc}\hfill {\left({\kappa }^2{\tau }^2\right)}_{n+1}& -{\left({\kappa }^2{\tau }^2\right)}_n\hfill \\ \hfill & =\sqrt{\frac{\pi }{4e}}[\sqrt{4(n+1)+1}-\sqrt{4n+1}]\hfill \\ \hfill & =\sqrt{\frac{\pi }{4ne}}\hfill \end{array}$$

(25)

The asymptotic EP slope can be evaluated by substituting Equation (17) into Equation (13), setting $\theta \tau =0$, and remembering that b is given in Equation (19):

$$\begin{array}{cc}\hfill \frac{\mathsf{\Delta }{\omega }_c}{\kappa }& =\frac{\pm 1}{2\kappa \tau }\sqrt{1-{\left(ib\right)}^2}\hfill \\ \hfill & =\frac{\pm 1}{2\kappa \tau }\sqrt{{\left(2e{\kappa }^2{\tau }^2\right)}^2}\hfill \\ \hfill & =\pm e\kappa \tau \hfill \end{array}$$

(26)

We conclude from these features that there is a continuous range of $\kappa \tau$ values for which there can exist one or two second-order EPs. The existence of this latter regime is a direct consequence of the time-delayed coupling — that is, for zero time-delay there is only one EP at $\mathsf{\Delta }\omega =\kappa$.

To verify the existence of the aforementioned EPs, the corresponding eigenvectors were examined as well. If the system’s eigenvectors coalesce for the same parameters as the degenerate eigenvalues, then the EPs truly exist. As an example, Figure 8b displays the magnitude of the complex inner-product of the two eigenvectors which produce the exceptional points near $\mathsf{\Delta }\omega /\kappa =1$ (Figure 8a, curve A) — not all eigenvector pairs are shown. When the magnitude of the inner-product is 1 the two eigenvectors are degenerate, and from the plot it’s clear that this curve matches the same curve in Figure 8a (curve A). Beyond the breaking point $\kappa \tau =1/\sqrt{2e}$ there is a clear phase transition, and the inner-product displays vastly different behavior, instantly jumping to completely different values. Similar eigenvector plots were obtained for curve B and the isolated EP points.

When $\theta \tau \ne n\pi$ there is no region where the exceptional points continuously change with $\kappa \tau$ (Equation (20)); rather, the degeneracies are isolated points in the $\kappa \tau$ vs. $\frac{\mathsf{\Delta }{\omega }_c}{\kappa }$ plane.

3.2. Higher-Order EPs

All the preceding analysis is for a second order degeneracy; can this system exhibit higher order exceptional points? For a third order eigenvalue degeneracy, we again start from Equation (7). Now both the first and second derivatives of must be zero—these derivatives, along with the original equation, produce a set of three coupled equations that can be solved for the degenerate $\lambda$ and the detunings $\mathsf{\Delta }\omega$ for which they occur:

$$\begin{array}{cc}\hfill {\lambda }^2+\mathsf{\Delta }{\omega }^2-{\kappa }^2{e}^{-2\tau (\lambda +i\theta )}& =0\hfill \end{array}$$

(27a)

$$\begin{array}{cc}\hfill 2\lambda +2\tau {\kappa }^2{e}^{-2\tau (\lambda +i\theta )}& =0\hfill \end{array}$$

(27b)

$$\begin{array}{cc}\hfill 1-2{\left(\kappa \tau \right)}^2{e}^{-2\tau (\lambda +i\theta )}& =0\hfill \end{array}$$

(27c)

By rearranging Equation (27c), it’s clear that the exponential term has to satisfy the relation $e^{-2\tau (\lambda +i\theta )}=1/2{\left(\kappa \tau \right)}^2$. Since the right-hand side of the expression is always real, this constrains $\theta \tau$ to be an integer multiple of $\pi /2$. This expression, when substituted into Equation (27b), gives the singular eigenvalue for which there exists a third-order degeneracy: $\lambda =-\frac{1}{2\tau }$. With this eigenvalue we can then evaluate Equation (27a) and find the detunings for which this degeneracy occurs: $\mathsf{\Delta }\omega =\pm \frac{1}{2\tau }$. Taking the eigenvalue solution and substituting it back into Equation (27c) results in a familiar expression for the constraints on $\kappa$ and $\tau$:

$$\kappa \tau =\frac{1}{\sqrt{2e}}$$

(28)

There is only a single combination of $\kappa \tau$ for which there exists a third-order exceptional point (Figure 9)—the corresponding eigenvectors also coalesce at the specified detuning. This value of $\kappa \tau$ places the third-order EP at the cusp formed by the second-order EP curves A and B in Figure 8a.

For the case of $N\ge 4$ degeneracies, the N-th derivative of Equation (7) is

$$-{\kappa }^2{(-2\tau )}^N{e}^{-2\tau (\lambda +i\theta )}=0.$$

(29)

For any non-zero delay or feedback strength this equation has no solutions for $\lambda$, and thus there are no N-th order eigenvalue degeneracies.

4. Discussion

We have analytically demonstrated that a time-delay coupled, APT-symmetric system displays rich EP behavior when compared with the corresponding zero-delay dimer case. By tuning the system’s delay time, coupling rate, or phase accumulation, we can move the system through three distinct regions in the exceptional point landscape, i.e., regions with 0, 1, or 2 s-order exceptional points and one third-order exceptional point. Recent experimental results have demonstrated the effectiveness of the Liouvillian in predicting and describing the origin of intensity behavior in a system of two delay-coupled semiconductor lasers [46]. The focus of this paper has been on the EPs that arise in the time-delayed effective Liouvillian, and the delay-coupled SCL system may provide a good test bed for verifying the predictions made here. Among the open questions to be addressed in future work are whether the effects of EPs in a time-delayed system produce observable signatures in the intensity, carrier populations, or phase-locking evolution of the coupled-SCL system.