Abstract
Classical open systems with balanced gain and loss, i.e. parity-time ($\mathcal{PT}$) symmetric systems, have attracted tremendous attention over the past decade. Their exotic properties arise from exceptional point (EP) degeneracies of non-Hermitian Hamiltonians that govern their dynamics. In recent years, increasingly sophisticated models of $\mathcal{PT}$-symmetric systems with time-periodic (Floquet) driving, time-periodic gain and loss, and time-delayed coupling have been investigated, and such systems have been realized across numerous platforms comprising optics, acoustics, mechanical oscillators, optomechanics, and electrical circuits. Here, we introduce a $\mathcal{PT}$-symmetric (balanced gain and loss) system with memory, and investigate its dynamics analytically and numerically. Our model consists of two coupled $LC$ oscillators with positive and negative resistance, respectively. We introduce memory by replacing either the resistor with a memristor, or the coupling inductor with a meminductor, and investigate the circuit energy dynamics as characterized by $\mathcal{PT}$-symmetric or $\mathcal{PT}$-symmetry broken phases. Due to the resulting nonlinearity, we find that energy dynamics depend on the sign and strength of initial voltages and currents, as well as the distribution of initial circuit energy across its different components. Surprisingly, at strong inputs, the system exhibits self-organized Floquet dynamics, including $\mathcal{PT}$-symmetry broken phase at vanishingly small dissipation strength. Our results indicate that $\mathcal{PT}$-symmetric systems with memory show a rich landscape.
Highlights
Over the past decade, open systems with balanced, spatially separated gain and loss have become a rich area of research
Open, classical systems with balanced gain and loss have seen an explosion of interest
We have presented a new paradigm for PT symmetric systems
Summary
Open systems with balanced, spatially separated gain and loss have become a rich area of research. [25], a single nitrogen-vacancy (NV) center in diamond [26], and a single superconducting qubit [27] Most of these systems are modeled with a static, PT symmetric Hamiltonian whose eigenvalues and eigenvectors determine the PT phase diagram of the system. The eigenvalues α (t ) of the instantaneous Hamiltonian do not govern the system dynamics; instead, the PT phase diagram is determined by an equivalent static Hamiltonian HF called the Floquet Hamiltonian [25,30,31] Another level of complexity is added when we consider systems described by a nonlinear, PT symmetric Schrödinger equation [32].
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