Introduction

Since the seminal work of Bender and co-workers1,2, the field of non-Hermitian Hamiltonians with parity-time (\({{{{{{{\mathcal{PT}}}}}}}}\)) symmetry has diversified and matured over the past two decades3,4,5,6. \({{{{{{{\mathcal{PT}}}}}}}}\)-symmetric Hamiltonians represent open classical systems with balanced gain and loss7, and have been experimentally realized in diverse platforms comprising optics8,9,10,11,12, electrical circuits13,14,15, mechanical oscillators16, acoustics17, and viscous fluids18. Post selection over no-quantum-jump trajectories has further enabled their realizations in minimal quantum systems such as an NV center19, a superconducting qubit20, ultracold atoms21, or correlated photons22. Concurrently, open systems with anti-parity-time (\({{{{{{{\mathcal{APT}}}}}}}}\)) symmetry have emerged. A system has \({{{{{{{\mathcal{APT}}}}}}}}\)-symmetry if its Hamiltonian Hanticommutes or its Liouvillian L ≡ −iH commutes with the \({{{{{{{\mathcal{PT}}}}}}}}\) operator. As a result, the eigenvalues \(({\lambda }_{m},{\lambda }_{m}^{* })\) of the Liouvillian L are purely real or complex conjugates23. Thus, while the symmetry-breaking transition in a \({{{{{{{\mathcal{PT}}}}}}}}\)-symmetric system is marked by the emergence of amplifying and decaying eigenmode pairs, modes of an \({{{{{{{\mathcal{APT}}}}}}}}\)-symmetric system amplify or decay independent of each other. \({{{{{{{\mathcal{APT}}}}}}}}\)-symmetric systems have been realized in atomic vapor and cold atoms24,25, active electrical circuits26, disks with thermal gradients27, and diverse optical setups28,29,30,31.

Markovianity is a key feature of all deeply investigated \({{{{{{{\mathcal{PT}}}}}}}}\)-symmetric, \({{{{{{{\mathcal{APT}}}}}}}}\)-symmetric, or non-Hermitian systems. The first-order differential equation governing the state of such a system, \(i{\partial }_{t}\left\vert \psi (t)\right\rangle ={H}_{{{{{{{{\rm{eff}}}}}}}}}[t;\psi (t)]\left\vert \psi (t)\right\rangle\), ensures that the rate of change of \(\left\vert \psi (t)\right\rangle\) 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 Heff depends on ψ(t)32. The 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 \({{{{{{{\mathcal{APT}}}}}}}}\)-symmetric system where a single time delay τ encodes the memory, and experimentally demonstrates its consequences in a system of two semiconductor lasers with bidirectional, time-delayed feedback33,34. 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 match the results from numerical simulations of time-delayed, nonlinear, modified Lang–Kobayashi (LK) equations for the two electric fields E1,2(t) and the corresponding excess carrier inversions N1,2(t) in the two lasers34,35,36, and thereby validate our minimal, non-Markovian, \({{{{{{{\mathcal{APT}}}}}}}}\)-symmetric model. Experimental observations of the steady-state laser intensities I1,2 as a function of bidirectional feedback strength κ, individual laser frequencies ω1,2, and the time delay τ match our (analytical/numerical) predictions. These robust signatures, clearly observed in experiments with off-the-shelf equipment and no custom fabrications, indicate that non-Markovianity (or time delay) opens up a hitherto-unexplored dimension for non-Hermitian systems.

Results

Time-delayed \({{{{{{{\mathcal{APT}}}}}}}}\)-symmetric model

For a system of two modes E1,2(t) with free-running frequencies ω1,2 = ω0 ± Δω and time-delayed coupling (Fig. 1a), in a frame rotating at the center frequency ω0, the dynamics are described by

$${\partial }_{t}{E}_{1} = i{{\Delta }}\omega {E}_{1}(t)+\kappa {e}^{-i{\omega }_{0}\tau }{E}_{2}(t-\tau ),\\ {\partial }_{t}{E}_{2} = -i{{\Delta }}\omega {E}_{2}(t)+\kappa {e}^{-i{\omega }_{0}\tau }{E}_{1}(t-\tau ).$$
(1)

This model emerges from the microscopic rate equations for full dynamics of two nominally identical, bidirectionally delay-coupled semiconductor lasers34,35,36 operating in the single-mode regime with vanishing excess carrier densities N1,2 (see “Methods”). At zero delay, Eq. (1) reduces to \({\partial }_{t}\vec{E}(t)=L\vec{E}(t)\) where \(\vec{E}(t)={[{E}_{1}(t),{E}_{2}(t)]}^{T}\), the Liouvillian is given by L(Δω, κ) = iΔωσz + κσx, and σz, σx are standard Pauli matrices. It describes a Markovian \({{{{{{{\mathcal{APT}}}}}}}}\)-symmetric system where \({{{{{{{\mathcal{P}}}}}}}}={\sigma }_{x}\) and \({{{{{{{\mathcal{T}}}}}}}}\) is complex conjugation. When the detuning Δω is increased, the eigenvalues \({\lambda }_{\pm }=\pm \sqrt{{\kappa }^{2}-{{\Delta }}{\omega }^{2}}\) 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 I1,2(t) = ∣E1,2(t)∣2 into steady-state values that monotonically decrease with Δω and become constant when Δω ≥ κ (Fig. 1b, c, insets). Although the experimentally accessible steady-state intensities I1,2 scale monotonically with the analytically derived amplification rate \(| {{{{{{{\rm{Re}}}}}}}}{\lambda }_{\pm }|\), their functional dependence is unknown.

Fig. 1: Non-Markovian \({{{{{{{\mathcal{APT}}}}}}}}\)-symmetric system.
figure 1

a Two modes E1,2(t) evolve with opposite phases ± Δωt 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. 4. b Amplification rate Uτ shows sideband oscillations with a constant width (SOW) (solid black traces). Results for \({U}_{\exp }\) 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 \({{{{{{{\mathcal{A}}}}}}}}{{{{{{{\mathcal{P}}}}}}}}{{{{{{{\mathcal{T}}}}}}}}\) transition from U > 0 to U = 0 occurs at Δω = κ. c Steady-state intensity I1(Δω) obtained from four, coupled, nonlinear rate equations shows sideband oscillations whose constant width is halved when ω1 is varied (\({L}_{\exp }\); dashed blue traces) instead of varying Δω while keeping ω0 constant (Lτ; solid black traces). Despite obvious similarities, explicit mapping from U(Δω) to the steady-state I1,2(Δω) is unknown. Inset: At τ = 0, a central dome at small detuning changes into a flat intensity profile for Δω ≥ κ. d Exemplary traces of experimentally measured intensity I1(Δω) 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(Δω) 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 Δω is present in the Markovian limit (τ = 0) and signals the standard \({{{{{{{\mathcal{A}}}}}}}}{{{{{{{\mathcal{P}}}}}}}}{{{{{{{\mathcal{T}}}}}}}}\)-transition. We analytically determine the behavior of the key non-Markovian signature SOWn(κ, τ) for Lτ and \({L}_{\exp }\).

Full size image

When τ > 0, Eq. (1) becomes \({\partial }_{t}\vec{E}={{{{{{{\mathcal{L}}}}}}}}\vec{E}\) where the non-local Liouvillian contains the time-delay operator,

$${{{{{{{\mathcal{L}}}}}}}}({{\Delta }}\omega ,\kappa ,{\omega }_{0},\tau )=i{{\Delta }}\omega {\sigma }_{z}+\kappa {e}^{-i{\omega }_{0}\tau }{e}^{-\tau {\partial }_{t}}{\sigma }_{x}.$$
(2)

If the two mode-frequencies are swept antisymmetrically while maintaing ω0 at \({e}^{-i{\omega }_{0}\tau }=\pm \!1\), the Liouvillian commutes with \({{{{{{{\mathcal{PT}}}}}}}}\) where the \({{{{{{{\mathcal{T}}}}}}}}\)-operator also takes τ to − τ. Then this non-Markovian system has \({{{{{{{\mathcal{APT}}}}}}}}\) symmetry. We will denote this Liouvillian as \({L}_{\tau }\equiv i{{\Delta }}\omega {\sigma }_{z}+\kappa {e}^{-\tau {\partial }_{t}}{\sigma }_{x}\). The characteristic equation for the eigenmodes \(\vec{E}(t)=\exp (\lambda t)\vec{E}(0)\) of Lτ is given by

$${\lambda }^{2}+{{\Delta }}{\omega }^{2}-{\kappa }^{2}{e}^{-2\lambda \tau }=0.$$
(3)

This transcendental equation has infinitely many eigenvalue pairs \(({\lambda }_{m},{\lambda }_{m}^{* })\). Experimentally it is easier to sweep ω1 while keeping ω2 constant, which changes ω0 and Δω in a correlated manner. We call the corresponding Liouvillian \({L}_{\exp }\), and it is given by

$${L}_{\exp }=i{{\Delta }}\omega {\sigma }_{z}+\kappa {e}^{-i({\omega }_{1}-{{\Delta }}\omega )\tau }{e}^{-\tau {\partial }_{t}}{\sigma }_{x}.$$
(4)

Since this Liouvillain does not commute with the \({{{{{{{\mathcal{PT}}}}}}}}\) operator, its eigenvalues λm are neither complex-conjugate pairs nor symmetric in Δω ↔ −Δω. The long-time, steady-state dynamics of the system are determined by the effective amplification rate \(U\equiv \max {{{{{\mathrm{Re}}}}}} {\lambda }_{m}\). 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 Uτ from Lτ (solid black line) and \({U}_{\exp }\) from \({L}_{\exp }\) (dashed blue line) as a function of dimensionless detuning Δω/κ when the time delay is κτ = 2. Apart from the central dome present in the τ = 0 limit (Fig. 1b, inset), both show time-delay induced sideband oscillations whose width, SOW, is constant at large Δω. The SOW for Lτ 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 (see “Methods”). The results for the intensity of the first laser, normalized to its large Δω/κ value (Fig. 1c), also show sidebands with an SOW that is twice as large for the \({{{{{{{\mathcal{APT}}}}}}}}\)-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 obvious similarity of results in (b, c), occurring over a wide range of time delays and feedback37, indicates that our minimal model captures key signatures of non-Markovianity that emerge from four, delay-coupled, nonlinear rate equations. Figure 1d shows exemplary experimental traces for normalized intensity I1(Δω) 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 I1(Δω) 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 τ37 indicate that, while the central dome width Δωc and sideband oscillation amplitudes depend on both, the SOW is solely determined by the time delay (Supplementary Note 1).

SOW theory and experimental results

The emergence of constant-width oscillations in the steady-state intensity is the key signature of non-Markovianity on an \({{{{{{{\mathcal{APT}}}}}}}}\)-symmetric system. To analytically determine SOW(κ, τ), we investigate the flow of eigenvalues λ = u + iv of Lτ. It is best understood via common zeros of two real functions comprising Eq. (3),

$$F(u,v)={u}^{2}-{v}^{2}+{{\Delta }}{\omega }^{2}-{\kappa }^{2}{e}^{-2u\tau }\cos (2v\tau ),$$
(5)
$$G(u,v)=2uv+{\kappa }^{2}{e}^{-2u\tau }\sin (2v\tau ).$$
(6)

The Uτ > 0 ↔ Uτ < 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 Δωn at which they occur the SOWn ≡ (Δωn+2 − Δωn) is obtained (Fig. 1b). Since the contours of G = 0 are independent of Δω, we characterize them first (Fig. 2 solid red lines). When − uτ ≫ 1, due to the divergent exponential factor, lines vm = mπ/2τ (\(m\in {\mathbb{Z}}\)), parallel to the u-axis, are solutions of G = 0. These points, i.e., 0 + ivm, also satisfy G = 0 along at v-axis, as does the entire u-axis (v = 0). In addition to these simple zeros, ∂vG(u, 0) = 0 determines the double zeros along the u-axis. They are given by values of z = 2uτ that satisfy the equation zez = −2(κτ)2. Thus, for \(\kappa \tau \, < \, 1/\sqrt{2e}\approx 0.43\), there are two negative solutions u0,1 = W0,1( −2κ2τ2)/2τ where Wm(x) is the Lambert W function38,39. We note that u1 is the intersection of the v±1 = ±π/2τ branches with the u-axis, while u0 is the intersection of the deformation of the v-axis, which is a solution of G = 0 at zero delay.

Fig. 2: Eigenvalue analysis of time-delayed \({{{{{{{\mathcal{A}}}}}}}}{{{{{{{\mathcal{P}}}}}}}}{{{{{{{\mathcal{T}}}}}}}}\) system.
figure 2

Eigenvalues λ = u + iv of Liouvillian Lτ occur at the intersections of F(u, v) = 0 (blue dot–dash) and G(u, v) = 0 (red solid) contours, shown here for κτ = 1.5 and Δω/κ = 1. Properties of G(u, v) = 0 contours and their intersections with the two axes are analytically determined by the Lambert W function38,39. 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 Δω, intersections of the G = 0 and F = 0 contours on the vertical axis (u = 0) give an infinite sequence of Uτ(Δω) > 0 ↔ Uτ(Δω) < 0 transitions that manifest as sideband oscillations seen in Fig. 1b–e.

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Next, let us consider the evolution of F = 0 contours when Δω is varied at a fixed τ (Fig. 2, dashed blue lines). When −uτ ≫ 1, parallel lines at \({v}_{{m}^{{\prime} }}=(2{m}^{{\prime} }+1)\pi /4\tau\) (\({m}^{{\prime} }\in {\mathbb{Z}}\)) are solutions of F = 0. At uτ ≫ 1, they are given by hyperbolas \(v=\pm \sqrt{{u}^{2}+{{\Delta }}{\omega }^{2}}\). For small Δω, the F = 0 contour intersects with the positive u-axis at \({z}^{{\prime} }=u\tau\) that satisfies \({z}^{{\prime} }{e}^{{z}^{{\prime} }}=\kappa \tau {[1-{({{\Delta }}\omega {e}^{{z}^{{\prime} }}/\kappa )}^{2}]}^{1/2}\). The solution reduces from u+ = W0(κτ)/τ when Δω = 0 to zero as Δω → Δωc ~ κ. It corresponds to the amplifying mode underlying the central dome that persists in the Markovian limit (Fig. 1b–e).

At larger Δω, the F = 0 contours intersect the v-axis, at two, mirror-symmetric intersections \((0,\pm \bar{v})\). As the zeros of G are at nπ/2τ, the most dominant eigenvalue \(\lambda ={0}^{\pm }+i\bar{v}\) changes from positive to negative when \(\bar{v}\) traverses “even n” branches of G = 0 contours. When \(\bar{v}={v}_{n}\), this leads to \({{\Delta }}{\omega }_{n}=\sqrt{{v}_{n}^{2}+{\kappa }^{2}}\). Therefore, we predict that

$${{{{{{{{\rm{SOW}}}}}}}}}_{n}(\kappa ,\tau )=\frac{\pi }{\tau }-\frac{{\kappa }^{2}\tau }{\pi n(n+2)}\mathop{\to }\limits_{n\gg 1}\frac{\pi }{\tau }.$$
(7)

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 Δω generated by varying ω1 with a fixed ω2 is half of what is generated when ω1 and ω2 are varied antisymmetrically. Figure 3 shows these predictions for Lτ (solid gray) and \({L}_{\exp }\) (dot-dashed gray) as lines with slopes π and π/2, respectively.

Fig. 3: Dependence of sideband oscillations width (SOW) on the time delay.
figure 3

Eigenvalue analysis predicts SOW ∝ 1/τ with κ-independent prefactor of π for Lτ (gray solid curve) and π/2 for \({L}_{\exp }\) (gray dot-dashed curve). SOWs extracted from steady-state intensity sidebands obtained from the full LK simulations quantitatively match the eigenvalue predictions; 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 error bars are FWHM of the sideband Fourier transform; horizontal error bars in time-delay estimate are from a fixed uncertainty Δl = 1 cm in the optical path length.

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To validate the eigenvalue-analysis predictions, we obtain the SOWs from full laser dynamics simulations for the two cases35,36. Steady-state intensities I1,2(Δω∣κ, τ) are obtained over a range of Δω 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 quantitative agreement with the analytical predictions shows that our minimal models, defined by Lτ and \({L}_{\exp }\), capture the key consequences of introducing non-Markovianity in non-Hermitian (\({{{{{{{\mathcal{APT}}}}}}}}\)-symmetric) systems.

We obtain experimental laser intensity profiles I1,2(Δω) by changing the temperature and consequently the frequency ω1 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 when 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 \({{{{{{{\mathcal{APT}}}}}}}}\)-symmetric Lτ 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

Delay-differential equations model systems from engineering, physics, chemistry, biology, and epidemiology40,41,42,43,44, and exhibit synchronization, bifurcation, and chaos41,45. They have long been used for classical random-number generation and control46,47,48,49. We have shown that non-Markovianity via time delay enriches the verdant field of non-Hermitian open systems. Our choice of \({{{{{{{\mathcal{APT}}}}}}}}\)-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.

For a larger network of coupled lasers with a single time delay, the characteristic equation Eq. (3) will have polynomial terms of higher order with no analytical solution. However, determining the transition from decaying modes to amplifying modes requires only locating the intersection with the u = 0 axis, and our approach with Lambert W function remains applicable.

Opto-electronic oscillators (OEOs) are another classical realization of non-Markovian, non-Hermitian system50. A proven test-bed showing quantitative agreement between the model and experiments51,52, they have been used to study time-delayed, coupled, nonlinear, and chaotic dynamics53,54,55. Introducing delay in stand-alone classical, non-Hermitian, chaotic wave platforms56 will require further ingenuity.

We have considered a system with \({{{{{{{\mathcal{APT}}}}}}}}\)-symmetry. Its Wick-rotated counterpart, i.e., a \({{{{{{{\mathcal{PT}}}}}}}}\)-symmetric system with time delay, can naturally arise in electrical oscillator circuits and classical wave systems. In the quantum domain, \({{{{{{{\mathcal{PT}}}}}}}}\)-symmetric systems have been realized through post selection on a minimal quantum system coupled to an environment19,20,21,22. Coherent feedback with time delay has been proposed as a control mechanism for precisely such open quantum systems57,58. Investigation of non-Markovianity-induced phenomena in such systems remains an open question.

Methods

Numerical model

Our system of two, coupled semiconductor lasers is described by a modified version of the well-known Lang–Kobayashi model34,35,36 for a solitary, single-mode laser with weak, time-delayed feedback. Two identical, single-mode lasers are operating at free-running frequencies ω1,2 = ω0 ± Δω. The slowly varying envelopes E1,2(t) of the electric fields inside the rectangular laser cavities are defined in a reference frame rotating at the average ω0 of the two optical frequencies. The rate equations describing the complex electric fields and excess carrier densities N1,2(t) can be written as follows35,36

$$\frac{d{E}_{1}}{dt}=\left[\frac{(1+i\alpha )}{2}G{N}_{1}(t)+i{{\Delta }}\omega \right]{E}_{1}(t)+K{E}_{2}(t-\tau ),$$
(8)
$$\frac{d{E}_{2}}{dt}=\left[\frac{(1+i\alpha )}{2}G{N}_{2}(t)-i{{\Delta }}\omega \right]{E}_{2}(t)+K{E}_{1}(t-\tau ),$$
(9)
$$\frac{d{N}_{1}}{dt}={J}_{1}-\frac{{N}_{{{{{{{{\rm{th}}}}}}}}}}{{\tau }_{s}}-\frac{{N}_{1}(t)}{{\tau }_{s}}-\left[\frac{1}{{\tau }_{p}}+G{N}_{1}(t)\right]| {E}_{1}(t){| }^{2},$$
(10)
$$\frac{d{N}_{2}}{dt}={J}_{1}-\frac{{N}_{{{{{{{{\rm{th}}}}}}}}}}{{\tau }_{s}}-\frac{{N}_{2}(t)}{{\tau }_{s}}-\left[\frac{1}{{\tau }_{p}}+G{N}_{2}(t)\right]| {E}_{2}(t){| }^{2}.$$
(11)

Here, α is the linewidth enhancement factor, G is the gain rate, \(K={{{{{{{\mathcal{K}}}}}}}}{e}^{-i{\omega }_{0}\tau }/{\tau }_{{{{{{{{\rm{in}}}}}}}}}\), \({{{{{{{\mathcal{K}}}}}}}}\) is the dimensionless feedback strength, τin is the internal round-trip time for the laser cavity (<1 ps), τ = l/c is the time delay, and l is the free-space optical path length. In the N1, N2 equations, J1,2 are the injection current above threshold. Nth is the steady-state inversion, τp is the photon lifetime (10 ps) and τs is the carrier lifetime (1 ns). This model has been used with great success to describe the nonlinear l behavior of coupled semiconductor lasers34,35,36.

We numerically solve the four coupled, nonlinear, time-delayed equations by using the fourth-order Runge–Kutta method with 0.1 ps time-step increment. The seed solutions for the history E1,2( −τ ≤ t ≤ 0) are obtained by solving the corresponding τ = 0 equations37. If the excess carrier densities for the steady-state solution are zero, N1,2 = 0, it follows that the electric-field envelope equations, Eqs. (8)–(9), reduce to Eq. (1) in the main text with \(\kappa ={{{{{{{\mathcal{K}}}}}}}}/{\tau }_{{{{{{{{\rm{in}}}}}}}}}\).

These rate equations, (8)–(11), do not account for the microscopic details of a semiconductor, but instead treat quantities like differential gain or loss as globally averaged parameters. This simplicity means one can use it to make qualitative predictions—successfully done in other studies with such lasers—but quantitative agreement is not expected34,59. For example, Eqs. (8)–(9) describe the intracavity field. Since the laser cavity mirrors’ reflections and transmissions are not precisely known, the relevant photon decay rate has significant uncertainty. Similar considerations apply to the carrier decay rate, and linewidth enhancement factor; so we use typical values. Thus, output intensities are not expected to quantitatively match experimentally measured quantities.

Experimental setup

Our experimental system is shown in Fig. 4. It consists of two, identical single-mode (HL7851G) semiconductor lasers (SCL1, SCL2), an external cavity consisting of two beam splitters (BS1 and BS2) which optically coupled the two lasers, and an external control of the coupling strength, \({\kappa }_{\exp }={{{{{{{\mathcal{K}}}}}}}}/{\tau }_{{{{{{{{\rm{in}}}}}}}}}\), via a variable neutral density filter (VND).

Fig. 4: Schematic of the experimental setup.
figure 4

Two nominally identical semiconductor lasers SCL1 and SCL2 are controlled by pump currents J1 and J2 respectively, and independent temperature controllers. The glass slide GS1 (GS2) reflects a small amount (8%) of light to the photodiode PD1 (PD2) to measure steady-state laser intensities I1,2. Mirrors M1 and M2, along with the variable neutral density (VND) filter provide bidirectional, time-delayed feedback. The transmission through the VND, and therefore the coupling κ, is determined by using another laser SLC3 along with a photodiode PD3.

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The transmission through the VND is determined by an independent laser (SL3) and a photodiode (PD3) which allows us to calibrate the coupling strength as \({\kappa }_{\exp }={{{{{{{\mathcal{K}}}}}}}}/{\tau }_{{{{{{{{\rm{in}}}}}}}}}=({r}^{-1}-r)\zeta {\tau }_{p}/{\tau }_{{{{{{{{\rm{in}}}}}}}}}\). Here r < 1 is the reflectivity of the external laser facet, ζ2 is the fraction of optical power transmitted by all the optical elements, τin is the internal round-trip time, and τp is the photon lifetime. Once the transmission through the VND is recorded, ζ2 can be determined since all the other optical elements are fixed. This model assumes that the fractional power is fully coupled into the active region of the semiconductor lasers. However, due to the relative sizes of the beam profile (>100 μm) and the active region (about 10 μm), only a portion of the power is coupled into the active region. Through literature reports and comparison of our experiments to LK simulations, we find that the “effective coupling strength” is reduced by a factor of ten, i.e., \({\kappa }_{{{{{{{{\rm{LK}}}}}}}}}={\kappa }_{\exp }/10\).

The experiment is designed such that the light coupling from the first laser into the second is equal to light coupling from the second into the first. A Faraday rotator is placed in the coupling beam path to eliminate self-coupling. The glass slides GS1 and GS2 independently reflect a small portion of (8%) of the intensity from SCL1 and SCL2, respectively, to corresponding 1 GHz photodiodes (PD1 and PD2) in conjuction with a 1 GHz oscilloscope. The pump currents J1,2 and temperatures T1,2 of the lasers SCL1 and SCL2 are stabilized to 10 μA and 0.01 C, respectively.

After bidirectionally coupling the two lasers, the temperature of SCL1 is slowly scanned (<10 Hz) and the steady-state intensities I1.2 of the two lasers are monitored. The key parameters κ and Δω can be varied via the VND and the temperature of SCL2, respectively. For our room-temperature lasers, the frequencies ω1,2 are proportional to the temperature of the relevant active region. For a small temperature range (<4 C), we use a linear approximation for the laser frequency ω and intensity I,

$$\omega (T)=\omega ({T}_{0})-{A}_{T}(T-{T}_{0}),$$
(12)
$$I(T)=I({T}_{0})+{B}_{T}(T-{T}_{0}),$$
(13)

where the coefficients AT = 20 GHz/C and BT = 0.15 mW/C are experimentally determined for our setup. AT is experimentally determined by scanning the temperature of a laser while monitoring the transmitted intensity through a fixed 2 GHz free spectral range Fabry–Perot etalon. When n peaks are observed through the etalon, we obtain AT = 2n/ΔT GHz/C where ΔT is the range of temperature scanned. To obtain BT, the scanned temperature and the emitted laser light intensity are recorded and fit to Eq. (13).

For detuning beyond the central dome, the analytical models (Lτ and \({L}_{\exp }\)) and simulations of the full laser model, Eqs. (8)–(11), show that time delay causes oscillations in the steady-state intensities as a function of Δω. We assign these oscillations to those in the sign of Uτ or \({U}_{\exp }\). To test this hypothesis, we operate the two lasers at constant injection currents barely (3%) above their stand-alone lasing thresholds. This guarantees that they remain above threshold when the temperature is scanned. The optical spectrum of the uncoupled (free-running) lasers is independently measured by a scanning an optical spectrum analyzer. The temperature to one laser was scanned and recorded. Using Eq. (8) along with temperature and wavelength measurements, we calculated the detuning, Δω. The temperature, coupling strength and the two SCL intensities were simultaneously recorded resulting in the reported intensity profiles of Fig. 1. The photodiodes have a large load resistor that decreases the bandwidth to <1 GHz. This bandwidth, along with the scan rate of the oscilloscope, leads to intensities Ik that are averaged over a time window of 1 ns.