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

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, $i\partial_{t}|\psi(t)\rangle=H_{\mathrm{eff}}[t;\psi(t)]|\psi(t)\rangle$, ensures that the rate of change of $|\psi(t)\rangle$ 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 $H_{\mathrm{eff}}$ depends on $\psi(t)$ [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 $\mathcal{APT}$-symmetric system where a single time-delay $\tau$ 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, $\mathcal{APT}$-symmetric model. Experimental observations of the steady-state laser intensities $I_{1,2}$ as a function of bidirectional feedback strength $\kappa$, individual laser frequencies $\omega_{1,2}$, and the time delay $\tau$ match our (analytical/numerical) predictions qualitatively, but not quantitatively. 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 new dimension for non-Hermitian systems.

When $\tau>0$, 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 $\omega_{0}$ at $e^{i\omega_{0}\tau}=\pm 1$, the Liouvillian commutes with $\mathcal{PT}$ where the $\mathcal{T}$-operator also takes $\tau$ to $-\tau$. 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 $L_{\tau}$ 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 $\omega_{1}$ while keeping $\omega_{2}$ constant, which changes $\omega_{0}$ and $\Delta\omega$ in a correlated manner. We call the corresponding Liouvillian $L_{\mathrm{exp}}$, and it is given by

$$L_{\mathrm{exp}}=i\Delta\omega\sigma_{z}+\kappa e^{-i(\omega_{1}-\Delta\omega)\tau}e^{-\tau\partial_{t}}\sigma_{x}.$$

(4)

Since this Liouvillain does not commute with the $\mathcal{PT}$ operator, its eigenvalues $\lambda_{m}$ are neither complex-conjugate pairs nor symmetric in $\Delta\omega\leftrightarrow-\Delta\omega$. The long-time, steady-state dynamics of the system are determined by the effective amplification rate $U\equiv\max\real\lambda_{m}$. 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 $U_{\tau}$ from $L_{\tau}$ (solid black line) and $U_{\mathrm{exp}}$ from $L_{\mathrm{exp}}$ (dashed blue line) as a function of dimensionless detuning $\Delta\omega/\kappa$ when the time delay is $\kappa\tau$=2. Apart from the central dome present in the $\tau$=0 limit (Fig. 1b inset), both show time-delay induced sideband oscillations whose width, SOW, is constant at large $\Delta\omega$. The SOW for $L_{\tau}$ is twice as large as it is for $L_{\mathrm{exp}}$. 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 $\Delta\omega/\kappa$ 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 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, delay-coupled, nonlinear rate equations. Figure 1d shows exemplary experimental traces for normalized intensity $I_{1}(\Delta\omega)$ at $\tau$=1.3 ns (red) and $\tau$=0.75 ns (blue), at $\kappa$=3.1 GHz. Clear sidebands are visible with an SOW that decreases with increasing delay-time. Conversely, experimental traces in Fig. 1e for normalized $I_{1}(\Delta\omega)$ at $\kappa$=1.9 GHz (red) and $\kappa$=1.1 GHz (blue) at a fixed time delay $\tau$=0.75 ns show that the SOW is largely insensitive to the coupling. Experimental data over a wide range of $\kappa$ and $\tau$ [37] indicate that, while the central dome width $\Delta\omega_{c}$ and sideband oscillation amplitudes depend on both, the SOW is solely determined by the time delay.

The $U_{\tau}>0\leftrightarrow U_{\tau}<0$ 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 $\Delta\omega_{n}$ at which they occur the $\mathrm{SOW}_{n}\equiv(\Delta\omega_{n+2}-\Delta\omega_{n})$ is obtained (Fig. 1b). Since the contours of $G=0$ are independent of $\Delta\omega$, we characterize them first (Fig. 2 solid red lines). When $-u\tau\gg 1$, due to the divergent exponential factor, lines $v_{m}=m\pi/2\tau$ ($m\in\mathbb{Z}$), 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, $\partial_{v}G(u,0)=0$ determines the double zeros along the $u$-axis. They are given by values of $z=2u\tau$ that satisfy the equation $ze^{z}=-2(\kappa\tau)^{2}$. Thus, for $\kappa\tau<1/\sqrt{2e}\approx 0.43$, there are two negative solutions $u_{0,1}=W_{0,1}(-2\kappa^{2}\tau^{2})/2\tau$ where $W_{m}(x)$ is the Lambert $W$ function [38, 39]. We note that $u_{1}$ is the intersection of the $v_{\pm 1}=\pm\pi/2\tau$ 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.

Next, let us consider the evolution of $F=0$ contours when $\Delta\omega$ is varied at a fixed $\tau$ (Fig. 2 dashed blue lines). When $-u\tau\gg 1$, parallel lines at $v_{m^{\prime}}=(2m^{\prime}+1)\pi/4\tau$ ($m^{\prime}\in\mathbb{Z}$) are solutions of $F=0$. At $u\tau\gg 1$, they are given by hyperbolas $v=\pm\sqrt{u^{2}+\Delta\omega^{2}}$. For small $\Delta\omega$, 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 $u_{+}=W_{0}(\kappa\tau)/\tau$ when $\Delta\omega=0$ to zero as $\Delta\omega\rightarrow\Delta\omega_{c}\sim\kappa$. It corresponds to the amplifying mode underlying the central dome that persists in the Markovian limit (Fig. 1b-e).

At larger $\Delta\omega$, the $F=0$ contours intersect the $v$-axis, at two, mirror-symmetric intersections $(0,\pm\bar{v})$. As the zeros of $G$ are at $n\pi/2\tau$, 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

$$\mathrm{SOW}_{n}(\kappa,\tau)=\frac{\pi}{\tau}-\frac{\kappa^{2}\tau}{\pi n(n+2)}\xrightarrow[n\gg 1]{}\frac{\pi}{\tau}.$$

(7)

With a similar analysis for eigenvalues of $L_{\mathrm{exp}}$, Eq.(4), we find that the SOW is reduced by a factor of two, i.e. $\mathrm{SOW}(\kappa,\tau)=\pi/2\tau$, because $\Delta\omega$ generated by varying $\omega_{1}$ with a fixed $\omega_{2}$ is half of what is generated when $\omega_{1}$ and $\omega_{2}$ are varied antisymmetrically. Figure 3 shows these predictions for $L_{\tau}$ (solid gray) and $L_{\mathrm{exp}}$ (dot-dashed gray) as lines with slopes $\pi$ and $\pi/2$ 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 $I_{1,2}(\Delta\omega|\kappa,\tau)$ are obtained over a range of $\Delta\omega$ such that $\gtrsim 20$ sidebands are present away from the central dome. For a given $\kappa$ and $\tau$, 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 $\tau$ ranging from 0.6 ns to 2.5 ns are plotted in Fig. 3. They are obtained for $\kappa$=0.4 GHz (open circles) and $\kappa$=2 GHz (filled squares), and yet SOWs derived from the full LK simulations do not depend on $\kappa$. Their striking agreement with the analytical predictions shows that our minimal models, defined by $L_{\tau}$ and $L_{\mathrm{exp}}$, capture the key consequences of introducing non-Markovianity in non-Hermitian ($\mathcal{APT}$-symmetric) systems.

We obtain experimental laser intensity profiles $I_{1,2}(\Delta\omega)$ by changing the temperature and consequently the frequency $\omega_{1}$ 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 $\sim 20$ oscillations away from the central dome to obtain the SOW. Figure 3 shows that the experimentally obtained $\mathrm{SOW}(\tau)$ varies inversely with delay time $\tau$, and essentially remains unchanged when the feedback strength $\kappa$ is varied over a factor of six. The slope of the experimental data for $\mathrm{SOW}$ vs. $1/\tau$ best-fit line (dotted gray) is halfway between the predictions for the $\mathcal{APT}$-symmetric $L_{\tau}$ model and non-Hermitian $L_{\mathrm{exp}}$ model, but two key features of Eq.(7)—namely, $1/\tau$ variation and vanishing $\kappa$ dependence—are robustly retained.

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 environment [19, 20, 22, 21]. 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.

Supplementary Material