Non-Markovian anti-parity-time symmetric systems: theory and experiment

Open systems with anti parity-time (anti $\mathcal{PT}$-) or $\mathcal{PT}$ symmetry exhibit a rich phenomenology absent in their Hermitian counterparts. To date all model systems and their diverse realizations across classical and quantum platforms have been local in time, i.e. Markovian. Here we propose a non-Markovian system with anti-$\mathcal{PT}$-symmetry where a single time-delay encodes the memory, and experimentally demonstrate its consequences with two time-delay coupled semiconductor lasers. A transcendental characteristic equation with infinitely many eigenvalue pairs sets our model apart. We show that a sequence of amplifying-to-decaying dominant mode transitions is induced by the time delay in our minimal model. The signatures of these transitions quantitatively match results obtained from four, coupled, nonlinear rate equations for laser dynamics, and are experimentally observed as constant-width sideband oscillations in the laser intensity profiles. Our work introduces a new paradigm of non-Hermitian systems with memory, paves the way for their realization in classical systems, and may apply to time-delayed feedback-control for quantum systems.

Markovianity is a key feature of all deeply investigated PT -symmetric, APT -symmetric, or non-Hermitian systems.The first-order differential equation governing the state of such a system, i∂ t |ψ(t) = H eff [t; ψ(t)]|ψ(t) , ensures that the rate of change of |ψ(t) depends only on the system's properties at time t and not on its history.This includes cases with a nonlinearity where the effective Hamiltonian H eff depends on ψ(t) [32].Markovian (or memoryless) nature of such effective non-Hermitian dynamics is considered inviolate, although no fundamental principles prohibit it.
Here, we propose a non-Markovian APT -symmetric system where a single time-delay τ encodes the memory, and experimentally demonstrate its consequences in a system of two semiconductor lasers with bidirectional, time-delayed feedback [33,34].
A transcendental equation with infinitely many eigenvalues, that results from the non-local-in-time nature of a delay-differential equation, distinguishes our model from its Markovian counterpart with a quadratic eigenvalue equation.
We analytically obtain predictions for the key features of steady-state intensity profiles as a function of non-Markovianity, i.e. the time delay.These predictions coincide exceptionally well with results from numerical simulations of time-delayed, nonlinear, modified Lang-Kobayashi (LK) equations for the two electric fields E 1,2 (t) and the corresponding excess carrier inversions N 1,2 (t) in the two lasers [34][35][36], and thereby validate our minimal, non-Markovian, APT -symmetric model.Experimental observations of the steady-state laser intensities I 1,2 as a function of bidirectional feedback strength κ, individual laser frequencies ω 1,2 , and the time delay τ match our (analytical/numerical) predictions qualitatively, but not quantitatively.
These robust signatures, clearly observed in experiments with off-theshelf equipment and no custom fabrications, indicate that non-Markovianity (or time delay) opens up a new dimension for non-Hermitian systems.Time-delayed APT -symmetric model.For a system of two modes E 1,2 (t) with free-running frequencies ω 1,2 = ω 0 ± ∆ω and time-delayed coupling (Fig. 1a), in a frame rotating at the center frequency ω 0 , the dynamics are described by ( This model emerges from the microscopic rate equations (Supplementary Materials) for full dynamics of two, nominally identical, bidirectionally delay-coupled semiconductor lasers [34][35][36] operating in the single- mode regime with with vanishing excess carrier densities N 1,2 .At zero delay, Eq.( 1) reduces to T , the Liouvillian is given by L(∆ω, κ) = i∆ωσ z + κσ x , and σ z , σ x are standard Pauli matrices.It describes a Markovian APT -symmetric system where P = σ x and T is complex conjugation.When the detuning ∆ω is increased, the eigenvalues λ ± = ± √ κ 2 − ∆ω 2 of the Liouvillian change from real to complex conjugates, and the amplifying/decaying modes change into oscillatory ones with constant intensity.
In reality, the nonlinearity of the gain medium saturates the exponentially amplifying mode intensities I 1,2 (t) = |E 1,2 (t)| 2 into steady-state values that monotonically decrease with ∆ω and become constant when ∆ω ≥ κ (Fig. 1b,c If the two mode-frequencies are swept antisymmetrically while maintaing ω 0 at e iω0τ = ±1, the Liouvillian commutes with PT where the T -operator also takes τ to −τ .Then this non-Markovian system has APT symmetry.We will denote this Liouvillian as L τ ≡ i∆ωσ z + κe −τ ∂t σ x .The characteristic equation for the eigenmodes E(t) = exp(λt) E(0) of L τ is given by This transcendental equation has infinitely many eigenvalue pairs (λ m , λ * m ).Experimentally it is easier to sweep ω 1 while keeping ω 2 constant, which changes ω 0 and ∆ω in a correlated manner.We call the corresponding Liouvillian L exp , and it is given by Since this Liouvillain does not commute with the PT operator, its eigenvalues λ m are neither complexconjugate pairs nor symmetric in ∆ω ↔ −∆ω.The longtime, steady-state dynamics of the system are determined by the effective amplification rate U ≡ max Re λ m .A positive U means that there is an amplifying mode, while U < 0 means all modes are below the lasing threshold.
Figure 1b shows the numerically obtained U τ from L τ (solid black line) and U exp from L exp (dashed blue line) as a function of dimensionless detuning ∆ω/κ when the time delay is κτ =2.Apart from the central dome present in the τ =0 limit (Fig. 1b inset), both show timedelay induced sideband oscillations whose width, SOW, is constant at large ∆ω.The SOW for L τ is twice as large as it is for L exp .For these two system configurations, we obtain the steady-state intensities by solving four modified LK equations (Supplementary Material).The results for the intensity of the first laser, normalized to its large ∆ω/κ value (Fig. 1c), also show sidebands with an SOW that is twice as large for the APT -symmetric model (solid black line) as it is for the experimental setup (dashed blue line); these sidebands are absent in the Markovian limit (Fig. 1c inset).
The striking similarity of results in (b)-(c), occurring over a wide range of time delays and feedback [37], indicates that our minimal model captures key signatures of non-Markovianity that emerge from four, delaycoupled, nonlinear rate equations.Figure 1d shows exemplary experimental traces for normalized intensity I 1 (∆ω) at τ =1.3 ns (red) and τ =0.75 ns (blue), at κ=3.1 GHz.Clear sidebands are visible with an SOW that decreases with increasing delay-time.Conversely, experimental traces in Fig. 1e for normalized I 1 (∆ω) at κ=1.9 GHz (red) and κ=1.1 GHz (blue) at a fixed time delay τ =0.75 ns show that the SOW is largely insensitive to the coupling.Experimental data over a wide range of κ and τ [37] indicate that, while the central dome width ∆ω c and sideband oscillation amplitudes depend on both, the SOW is solely determined by the time delay.SOW theory and experimental results.Emergence of constant-width oscillations in the steady-state intensity is the key signature of non-Markovianity on an APT -symmetric system.To analytically determine SOW(κ, τ ), we investigate the flow of eigenvalues λ = u + iv of L τ .It is best understood via common zeros of two real functions comprising Eq.(3), The U τ > 0 ↔ U τ < 0 transitions that lead to the sidebands occur when G = 0 and F = 0 contours intersect in the vicinity of the vertical v-axis (Fig. 2).By determining the detunings ∆ω n at which they occur the SOW n ≡ (∆ω n+2 −∆ω n ) is obtained (Fig. 1b).Since the contours of G = 0 are independent of ∆ω, we characterize 2. Eigenvalues λ = u + iv of Liouvillian Lτ occur at the intersections of F (u, v) = 0 and G(u, v) = 0 contours, shown here for κτ = 1.5 and ∆ω/κ = 1.Properties of G(u, v) = 0 contours and their intersections with the two axes are analytically determined by the Lambert W function [38,39].At small detuning, the hyperbolic F (u, v) = 0 contour always intersects the u > 0 axis and gives the central dome that survives in the Markovian limit.At large ∆ω, intersections of the G = 0 and F = 0 contours on the vertical axis (u = 0) give an infinite sequence of Uτ (∆ω) > 0 ↔ Uτ (∆ω) < 0 transitions that manifest as sideband oscillations seen in Fig. 1b-e.
them first (Fig. 2 solid red lines).When −uτ 1, due to the divergent exponential factor, lines v m = mπ/2τ (m ∈ Z), parallel to the u-axis, are solutions of G = 0.These points, i.e. 0 + iv m , also satisfy G = 0 along at v-axis, as does the entire u-axis (v = 0).In addition to these simple zeros, ∂ v G(u, 0) = 0 determines the double zeros along the u-axis.They are given by values of z = 2uτ that satisfy the equation ze z = −2(κτ ) 2 .Thus, for κτ < 1/ √ 2e ≈ 0.43, there are two negative solutions u 0,1 = W 0,1 (−2κ 2 τ 2 )/2τ where W m (x) is the Lambert W function [38,39].We note that u 1 is the intersection of the v ±1 = ±π/2τ branches with the u-axis, while u 0 is the intersection of the deformation of the v-axis, which is a solution of G = 0 at zero delay.
At larger ∆ω, the F = 0 contours intersect the v-axis, at two, mirror-symmetric intersections (0, ±v).As the zeros of G are at nπ/2τ , the most dominant eigenvalue λ = 0 ± + iv changes from positive to negative when v traverses "even n" branches of G = 0 contours.When v = v n , this leads to ∆ω n = v 2 n + κ 2 .Therefore, we predict that With a similar analysis for eigenvalues of L exp , Eq.( 4), we find that the SOW is reduced by a factor of two, i.e.SOW(κ, τ ) = π/2τ , because ∆ω generated by varying ω 1 with a fixed ω 2 is half of what is generated when ω 1 and ω 2 are varied antisymmetrically.Figure 3 shows these predictions for L τ (solid gray) and L exp (dot-dashed gray) as lines with slopes π and π/2 respectively.To validate the eigenvalue-analysis predictions, we obtain the SOWs from full laser-dynamics simulations for the two cases [35,36].Steady state intensities I 1,2 (∆ω|κ, τ ) are obtained over a range of ∆ω such that 20 sidebands are present away from the central dome.For a given κ and τ , SOW is obtained by Fourier transform of the sideband data; the error-bars indicate full width at half maximum (FWHM) of the single peak that is present in the Fourier transform.The results from such analysis carried out for delay times τ ranging from 0.6 ns to 2.5 ns are plotted in Fig. 3.They are obtained for κ=0. 4 GHz (open circles) and κ=2 GHz (filled squares), and yet SOWs derived from the full LK simulations do not depend on κ.Their striking agreement with the analytical predictions shows that our minimal models, defined by L τ and L exp , capture the key consequences of introducing non-Markovianity in non-Hermitian (APTsymmetric) systems.
We obtain experimental laser intensity profiles I 1,2 (∆ω) by changing the temperature and consequently the frequency ω 1 of the first laser, and normalize each by the minimum recorded intensity at large detuning.Since the amplitude of sideband oscillations is small, we average ∼ 20 oscillations away from the central dome to obtain the SOW. Figure 3 shows that the experimentally obtained SOW(τ ) varies inversely with delay time τ , and essentially remains unchanged the feedback strength κ is varied over a factor of six.The slope of the experimental data for SOW vs. 1/τ best-fit line (dotted gray) is halfway between the predictions for the APTsymmetric L τ model and non-Hermitian L exp model, but two key features of Eq.( 7)-namely, 1/τ variation and vanishing κ dependence-are robustly retained.Discussion.
They have long been used for classical random-number generation and control [46][47][48][49] APT -symmetric model is motivated by a standard setup of two, bidirectionally coupled semiconductor lasers.
We have mapped the complex, nonlinear system into simple, analytically tractable non-Markovian models.Their multifarious dynamics contain robust signatures of transitions that occur solely due to the non-Markovianity.We find that predictions from the minimal models quantitatively capture those from the full laser dynamics model.Their variance from the experimental data is likely due to the failure of the single-mode approximation or the weak coupling approximation, and possible variation of the second-laser frequency when the frequency of the first laser is varied.
We have considered a system with APT -symmetry.Its Wick-rotated counterpart, i.e. a PT -symmetric system with time delay, can naturally arise in electrical oscillator circuits and classical wave systems.In the quantum domain, PT -symmetric systems have been realized through post-selection on a minimal quantum system coupled to an environment [19][20][21][22].Coherent feedback with time-delay has been proposed as a control mechanism for precisely such open quantum systems [50,51].Investigation of non-Markovianity induced phenomena in such systems remains an open question.

FIG. 1 .
FIG. 1. Non-Markovian APT -symmetric system.(a) Two modes E1,2(t) evolve with opposite phases ±∆ωt in frame rotating with frequency ω0.Due to finite speed of light, each mode at time t (filled circles) couples to the other at an earlier time t − τ (open circles).This non-Markovian coupling κ is shown along the (shaded) past light-cones.This model, described by Eq.(1), is experimentally realized with two semiconductor lasers with bidirectional, time-delayed feedback; see Fig. S1.(b) Amplification rate Uτ shows sideband oscillations with a constant width (SOW) (solid black traces).Results for Uexp show that the SOW is halved (dashed blue traces).U > 0 region (pink) denotes amplifying modes, while U < 0 region (violet) denotes decaying modes.Inset: in the Markovian limit τ = 0, the APT transition from U > 0 to U = 0 occurs at ∆ω = κ.(c) Steady state intensity I1(∆ω) obtained from four, coupled, nonlinear rate equations shows sideband oscillations whose constant width is halved when ω1 is varied (Lexp; dashed blue traces) instead of varying ∆ω while keeping ω0 constant (Lτ ; solid black traces).Despite obvious similarities, explicit mapping from U (∆ω) to the steady-state I1,2(∆ω) is unknown.Inset: At τ = 0, a central dome at small detuning changes into a flat intensity profile for ∆ω ≥ κ.(d) Exemplary traces of experimentally measured intensity I1(∆ω) obtained by sweeping ω1 at τ =0.75 ns (blue) and τ =1.3 ns (red) show that observed SOW is reduced with increasing τ .Their features are consistent with our model and full laser dynamics simulations.(e) Exemplary traces of intensity I1(∆ω) obtained at κ=1.1 GHz (blue) and κ=1.9 GHz (red) show that the observed SOW is insensitive to the coupling κ.The central dome in (b)-(e) at small ∆ω is present in the Markovian limit (τ = 0) and signals the standard APT -transition.We analytically determine the behavior of the key non-Markovian signature SOWn(κ, τ ) for Lτ and Lexp.
insets).Although the experimentally accessible steady-state intensities I 1,2 scale monotonically with the analytically derived amplification rate |Reλ ± |, their functional dependence is unknown.When τ > 0, Eq.(1) becomes ∂ t E = L E where the non-local Liouvillian contains the time-delay operator, . We have shown that non-Markovianity via time delay adds a novel dimension to the verdant field of non-Hermitian open systems.Our choice of Time-delay induced transitions.Eigenvalue analysis predicts SOW ∝ 1/τ with κ-independent prefactor of π for Lτ (gray solid curve) and π/2 for Lexp (gray dot-dashed curve).SOWs extracted from steady-state intensity sidebands obtained from the full LK simulations match the eigenvalue predictions exceptionally well; error bars, obtained from FWHM of the Fourier transform, are smaller than symbols when not shown.κ-independence of the prefactor in full LK simulations is clear ( squares: κ=2 GHz; circles: κ=0.4 GHz).SOW(τ ), obtained from experimental data for κ that varies by a factor of six, clearly show a 1/τ behavior.Vertical errorbars are FWHM of the sideband Fourier transform; horizontal error-bars in time-delay estimate are from a fixed uncertainty ∆l=1 cm in the optical path length.