Dissipative systems, whose linear Hamiltonians obey parity-time ( PT ) symmetry are known to share properties of Hermitian systems; indeed that was a central original motivation for the proposal of such systems in connection to the foundations of quantum mechanics [1, 2]. More recently, this proposal found a fertile ground for experimental realization in a diverse array of other fields, including in optical media (where loss and controllable gain are ubiquitous) [3–6], electronic circuits [7–9] and even mechanical systems [10]. A key feature that most of the above systems share is the possibility to straightforwardly include nonlinearity in the dynamics; e.g., in optical media, this can be achieved via increase of the optical intensity. This rendered the study of these nonlinear systems and of their nonlinear modes/waveforms a canonical next step within such studies.

Nonlinear dimers (two-site-systems) [11–16] and quadrimers [17–19] are among the simplest systems allowing one to observe the above mentioned features. At this point in time, many of the relevant observations have been summarized in comprehensive reviews [20, 21] and books [22].

As is known from the above settings, specific symmetries of the underlying linear system impose constraints on the existence as well as on the types of nonlinear modes sustained by the system of interest. The literature mentioned above was mainly concerned with parity ( P ) - time ( T ) symmetric systems, whose linear Hamiltonians commute with the PT -operator. In this work we address the possibility of anti- PT symmetry of the linear Hamiltonian, as concerns the existence and stability of the associated nonlinear modes. Such dissipative systems in the linear setting were suggested in [23], and since then their experimental feasibility has been argued in linear dissipatively coupled optical systems [24] and illustrated in the context of a warm atomic-vapour cell [25]. More recently, they have been experimentally realized in a dimer of resistively coupled amplifying RLC-circuits, where various intriguing features such as corresponding exceptional points (EPs) and energy difference conserving dynamics were identified [26]. An anti- PT symmetric system was recently proposed in the context of quantum computing [27], where it was shown that the anti- PT symmetric qubit has superior decoherence properties compared to its PT -symmetric and Hermitian counterparts. Further attention to anti- PT symmetric systems stemmed from the possibility of their usage for generating more sophisticated Hamiltonians, for example odd- PT -symmetric systems [28, 29], or for the simulation of anti-parity-time symmetric Lorentz dynamics [30]. A relatively recent summary of the relevant activity can be found in [31].

While a systematic effort has been made to explore anti- PT symmetric linear media, to the best of our knowledge, far less of an effort has been invested in nonlinear analogues thereof. It is toward that latter vein that our effort herein is geared. Specifically, we revisit the prototypical linear anti- PT model motivated from the experimental realization of [26]. We endow the relevant model with nonlinearity which is straightforward in the optical realm, as well as the atomic-vapour setting [25], but also genuinely feasible in the electrical circuit realm as well; e.g., via the dependence of capacitances on the voltage that has been used as a source of numerous nonlinear features in such settings [32]. For this nonlinear anti- PT -symmetric dimer, our aim is to explore the prototypical nonlinear states thereof, as well as their spectral stability features and nonlinear dynamical properties. The algebraic nature of the system permits us to identify the associated nonlinear modes in an exact analytical form. We indeed find two symmetric and two asymmetric branches of solutions. Nevertheless, the corresponding stability matrices cannot be diagonalized to yield the relevant eigenvalues in a simple, explicit closed form. We thus compute the relevant spectrum numerically. We find that out of the four branches of solutions only one symmetric state is stable. Nevertheless, we also elucidate the complex bifurcation structure of the model. Indeed, the two symmetric branches emerge through a saddle-node (SN) bifurcation. The lower (unstable) symmetric branch is also involved in a transcritical bifurcation with the asymmetric branches, with the latter also terminating in a separate SN bifurcation. We then go on to examine the dynamical evolution of both stable and unstable states, corroborating the spectral results, but also illustrating the fate of the unstable waveforms.

Our presentation is structured as follows. In Section 2, we briefly present and explain the relevant mathematical model. In Section 3, we analyze the existence of its nonlinear solutions. In Section 4, we again briefly discuss the spectral linearization around such waveforms. In Section 5, we present our numerical stability and dynamical results. Finally, in Section 6 we summarize our findings and present our conclusions as well as some directions for future study.

Bearing in mind optical applications to a two-waveguide geometry [24], atomic ones for a pair of two collective spin-wave excitations [25], or a pair of RLC circuits per the experiment of [26], we chose, arguably, the simplest model of an anti- PT -symmetric dimer ψ = ( ψ 1 , ψ 2 ) T (T stands for transpose, and ψ is associated with voltage in the electric circuit scenario, while |ψ|² constitutes the observable for the optical intensity or atomic density scenarios) governed by the equation: i d ψ d z = H 0 ψ + F ψ ψ , H 0 = − δ + i γ i C i C δ + i γ (1) where C, δ and γ are real parameters describing non-conservative coupling between the waveguides (or circuits), difference of the propagation constants (it will be assumed without loss of generality that δ > 0), and gain (if γ > 0) or loss (if γ < 0) in the waveguides (or circuits), respectively. We notice, that while one of the parameters, say δ, in H ₀ can be scaled out, we keep all of them since they correspond to different physical processes, and thus facilitate interpretation of the results. The non-conservative nonlinearity in Eq. 1 is given by the diagonal matrix F ψ = g | ψ 1 | 2 + g ̃ | ψ 2 | 2 0 0 g | ψ 2 | 2 + g ̃ | ψ 1 | 2 (2) with g = g ₁ − ig ₂ and g ₂ > 0 describes the nonlinear absorption (g _1,2 and g ̃ are real). Defining the parity P = σ 3 and time-reversal (anti-linear) operator T as a complex conjugation, T ψ = ψ * one can verify that PT H 0 + H 0 PT = 0 (3) We notice that the introduced system is characterized by the active (non-Hermitian) coupling which was previously addressed in a number of publications without [24–26, 30] and with conservative and non-conservative nonlinear contributions [33]. It is relevant to mention in passing that some of these works, including experimental ones such as [25] indicate how nonlinearity can be incorporated in the relevant considerations even though they do not study it in detail. Within the model (1), nonlinearity stems from self- and cross-phase modulation, characterized by the strengths g ₁ and g ̃ , respectively, as well as from nonlinear absorption of strength g ₂.

At the linear level, the eigenvalue problem for H ₀ H 0 φ j = b j φ j (4) is readily solved b + − = i γ + − Λ , φ 1 = i C δ + − Λ (5) where Λ = C 2 − δ 2 (6) describes the deviation from the EP δ = |C| of the linear Hamiltonian.

Turning to the nonlinear problem we start with steady state solutions of Eq. 1 and employing the ansatz ψ = e i b z ψ 10 e i ϕ / 2 ψ 20 e − i ϕ / 2 , (7) where b is a real spectral parameter and ψ ₁₀ and ψ ₂₀ are real, we obtain the system 0 = b − δ + i γ + g | ψ 10 | 2 + g ̃ | ψ 20 | 2 ψ 10 + i C ψ 20 e − i ϕ (8) 0 = b + δ + i γ + g | ψ 20 | 2 + g ̃ | ψ 10 | 2 ψ 20 + i C ψ 10 e i ϕ (9)

The solutions found above need to be analyzed for their stability, in order to assess their potential dynamical robustness. This is achieved by studying the eigenvalues of the stability matrix, given by M ̄ ̄ = ∇ u ⃗ F ⃗ ( u ⃗ ) , where u ⃗ is a steady state solution in the form u ⃗ = ( ψ 1 , ψ 2 , ψ 1 * , ψ 2 * ) T , and F ⃗ = ( F ψ 1 , F ψ 2 , F ψ 1 * , F ψ 2 * ) = ( F ψ 1 , F ψ 2 , − F ψ 1 * , − F ψ 2 * ) .

For the anti- PT symmetric equations this has the form: M ̄ ̄ = A ̄ ̄ B ̄ ̄ − B ̄ ̄ * − A ̄ ̄ * where: A ̄ ̄ = − δ + i γ + b + 2 g | ψ 1 | 2 + g ̃ | ψ 2 | 2 g ̃ ψ 1 ψ 2 * + i C g ̃ ψ 1 * ψ 2 + i C δ + i γ + b + 2 g | ψ 2 | 2 + g ̃ | ψ 1 | 2 and B ̄ ̄ = g ψ 1 2 g ̃ ψ 1 ψ 2 g ̃ ψ 1 ψ 2 g ψ 2 2

Using the ansatz u ⃗ = e ̂ λ e λ z in the equation for the stability i d u ⃗ d z = ∇ u ⃗ F ⃗ ( u ⃗ ) u ⃗ we obtain the eigenvalue problem as follows λ e ̂ λ = − i M ̄ ̄ e ̂ λ ≡ M ̄ ̄ ′ e ̂ λ

Thus the eigenvalues of M ̄ ̄ are the eigenfrequencies of the problem (ω), while those of M ̄ ̄ ′ = − i M ̄ ̄ are its eigenvalues (λ). [The two are connected via λ = −iω].

We look for solutions of the asymmetric form by performing a Newton method search of the algebraic Eq. 23, followed by continuation in the parameter C of any solution thus found. For the symmetric solutions, we did the same, although we could simply use our explicit analytical expressions within the stability matrix (in order to identify their spectral stability properties).

Below we present some representative results for the parameter values γ = 1.0, δ = 0.1, g ₁ = 0.3, g ₂ = 0.4 and g ̃ = 0.5 . C is scanned from δ up to δ 2 + γ 2 , which are the limits for a real amplitude for the “negative” branch of the symmetric solution, as can be seen in Eq. 12. Given our analytical formulae and numerical setup, similar findings can be obtained for other parameter values.

For the symmetric case, Figures 1, 2 show the results for the “positive” branch (hereafter termed “upper”). Represented are the amplitudes of the two nodes (|A|² and |B|²), the phase difference between them (ϕ) (left panel), and the complex plane representation of the eigenvalues for two values of the scanned parameter, C (right panel). Then, in the second figure we show the dependence on C of the real and imaginary parts of the eigenvalues. One can observe that the amplitude grows with C, while the phase difference (right axis) varies from π/2 to a little above zero. Superposed to the numerical results are those of the analytic expressions found above, and we can see that the two are essentially identical, as is of course expected. The right panel of Figure 1 shows that the eigenvalues are purely real and indeed, as shown in Figure 2, they remain real throughout.

Figure 2 shows the real and imaginary parts of the eigenvalues as a function of C. Given that the largest value of the real part is zero, this upper branch is spectrally stable. That is, all the relevant eigendirections are associated with decay, aside from a neutral one (associated with an overall phase freedom). Recall that this is a non-conservative system, hence the relevant eigenvalues have to be in the left-half of the spectral plane (or on the imaginary axis thereof) for stability, as is the case for this branch. Indeed, we will see below that this is the only spectrally stable branch of this nonlinear anti- PT -symmetric dimer.

In Figures 3, 4 we illustrate the corresponding results for the “negative sign” solution in Eq. 11, i.e., the hereafter termed lower branch. This time the amplitude decreases with increasing C, and the phase difference increases from just over π/2 to π. The continuation was started a little above C = δ; for C = δ we would expect both solutions to have ϕ = π/2. It is relevant to also note that the branches of Figures 1–3 coincide at the critical point of C = δ at which the relevant SN bifurcation arises with the upper branch corresponding to the node, while the lower one to the saddle. In accordance with this picture the spectra show again a purely real set of eigenvalues and in Figure 4 with one of them being positive and hence corroborating the instability of the saddle (−) symmetric configuration of the lower branch.

This is confirmed systematically also in Figure 4, where the relevant unstable eigenvalue is seen to grow from 0 beyond the bifurcation point. Interestingly, an additional unstable eigendirection arises at some intermediate value of C as well, rendering the relevant branch more unstable. We will return to the latter more elaborate bifurcation shortly. Nevertheless, for the interval of values of C considered, the former instability is always stronger (i.e., has a higher growth rate) than the latter one.

Now we turn to the results obtained for the two branches of asymmetric solutions. Here, the bifurcation picture is far more elaborate. The bifurcation diagram as a function of the parameter C is shown in Figure 5. Let us note that in this diagram the symmetric (node upper and saddle lower) branches are also shown and their SN bifurcations are shown via the green (solid) curves, while the asymmetric branches are shown with blue (dashed) lines. The main feature of the latter is that there is a 3-way collision between the 2 asymmetric branches and the lower symmetric one, close to C c r ( 1 ) = 0.51 . This is easily seen in the detailed (right) panel for the amplitude dependence. The upper asymmetric branch goes past a fold en route to that collision, existing as a solution up to C c r ( 2 ) = 0.5128 . Indeed, the latter point is associated with a SN bifurcation corresponding to the termination (in terms of values of C) of the upper asymmetric branch. That is, the relevant branch does not exist for higher C values. Interestingly, the algebraic picture is somewhat more complicated in that when solving Eq. 22, the upper branch goes past the turning point of C c r ( 2 ) and upon turning around collides with the lower branch C c r ( 1 ) . Nevertheless, this is, in a sense, an “artifact” of the closed form formulae of the analytical solutions of Eq. 22. Observing the curves in the bifurcation diagram of Figure 5, one can see that the “inner” curves (the ones closer to the green line before C c r ( 1 ) = 0.51 at this critical point) collide between them and therefore become instantaneously symmetric before smoothly continuing en route to the collision with the “outer” (top and bottom) curves at C c r ( 2 ) = 0.5128 . That is to say, the former critical point signals a transcritical bifurcation, between the asymmetric and the symmetric branch, while the latter critical point signals a SN bifurcation leading to the termination of asymmetric branches.

Indeed, this picture is corroborated by the relevant eigenvalue plots. Figure 6 illustrates some prototypical examples of the spectral plane of the upper and lower asymmetric solutions. Both of them bear a complex eigenvalue pair (i.e., are associated with an oscillatory instability featuring both growth and oscillation, as we will also see below). However, in the case of the upper branch this instability is persistent up to C ≈ 0.430, while in the lower branch, it splits into two real eigenvalues earlier (parametrically), i.e., for C ≈ 0.287. Notice, accordingly, the difference for the red points of C = 0.4 in the right panel of Figure 6.

The conversion of these complex pairs into real ones is also manifest explicitly in the top panels of the detailed Figure 7, which constitutes a central set of our numerical findings. Indeed, the top right panel shows how the complex pairs collide at the above critical points, thereafter splitting into two real eigenvalues for the respective branches, as shown in the top left panel of Figure 7. The left panel, admittedly, becomes rather complicated as we approach the critical points C c r ( 1 ) and C c r ( 2 ) although a collision with the green (lower symmetric) branch is apparent. To that effect, we provide further details in the middle and bottom row panels of this Figure, where the detail of each of the relevant eigenvalues is shown in the vicinity of C c r ( 1 ) and C c r ( 2 ) , i.e., close to the transcritical and the SN bifurcation points, respectively. The “collision” of the asymmetric lower and symmetric branch is evident in these three panels at C c r ( 1 ) . Especially telling within the right panel of the middle row is the exchange of stability between the symmetric (lower) green branch and the asymmetric branch. Notice that both branches already bear a positive real eigenvalue (hence are unstable). However, the asymmetric branch has a second eigenvalue crossing from positive to negative, while the symmetric one goes in the opposite direction, with the two exchanging their stability in the aforementioned transcritical bifurcation event. Lastly, the asymmetric branch terminates at C c r ( 2 ) through a SN bifurcation featured in the middle right panel through two eigenvalues colliding at 0. We believe that this description offers a comprehensive understanding of the bifurcation phenomenology present in the system.

In the present work, we have explored a nonlinear variant of the anti- PT symmetric dimer problem. The linear version of this setup has already been explored in a variety of settings, including optical waveguides [23, 24, 31], coupled electrical circuit resonators [26] and atomic vapour cells [25]. Some of these works have already proposed variants of the relevant settings that would involve nonlinearity [25], while for others we argued about the fact that nonlinearity inclusion would be natural on the basis of the nature of the response of such systems at larger amplitudes. We have explored the most prototypical nonlinear dimer setting and were able, given the few-degrees-of-freedom nature of the setting, to obtain solutions analytically for the stationary states of the system. We found, in particular, two symmetric solutions arising via a saddle-node bifurcation and also identified two asymmetric solutions which are involved in a transcritical bifurcation with the lower symmetric branch, as well as in a saddle-node bifurcation leading to the termination of the asymmetric solutions. Out of these four solution branches, only one was found to be spectrally stable and indeed was identified as a generic attractor of the dynamics of the system, even when starting from the unstable symmetric or asymmetric solutions. Our spectral analysis was straightforwardly corroborated via direct numerical simulations of the evolution dynamics which showed the growth along the predicted unstable eigendirections of unstable stationary states with the appropriate rates, and the eventual approach to the sole dynamical attractor of this system, namely the stable (upper) symmetric branch.

Naturally, these results pave the way for numerous further studies of anti- PT symmetric systems along a similar vein to what was done in the PT -symmetric case [20–22]. In particular, one can examine so-called anti- PT symmetric oligomers ( PT -symmetric ones were explored, e.g., in [12, 17, 18]), as well as lattices of such elements (again, corresponding PT -symmetric explorations could be found in [34, 35]).

Given that the bifurcation picture for the anti- PT -symmetric dimer is far more complex (and involving multiple bifurcations), as shown herein, in comparison to the corresponding PT -symmetric dimer, it is expected that the situation with anti- PT -symmetric oligomers (trimer, quadrimer, etc.) will be significantly more complex than the regular PT -symmetric ones considered earlier and summarized, e.g., in [20–22]. This is a topic particularly relevant for future studies, and the results/methods proposed herein as well as the differences between the PT and anti- PT systems for the dimer could be a useful guide towards such efforts in the near future. These extensions can be considered not only in one- but also in higher dimensions, with the latter implying a different coupling between the nodes constituting the oligomer. Furthermore, here, we have concerned ourselves with cubic Kerr-type nonlinearities, yet some of the above settings seem to be well-suited for different types of nonlinear terms, including four-wave-mixing ones [25], with the latter being another topic worthwhile of further study. Such considerations are currently in progress and will be reported in future publications.