Non-Markovian protection of states from decay in quasi-PT-symmetric systems

Abstract

We consider a quasi-PT-symmetric system of two resonators, one of which interacts with a finite-size environment. The interaction with the environment leads to energy losses in the resonators, and the finite size of the environment leads to a non-Markovian dynamics of the relaxation process. We demonstrate that non-Markovian processes in the quasi-PT-symmetric system can make the states of the system infinitely living, loss-protected states, even in the absence of gain. There is a critical value of the interaction between the resonator and the environment below which any state of the system is loss-protected. When the interaction magnitude is greater than the critical value, depending on the coupling strength between the resonators, either one or both states are unprotected. We show that the boundaries of regions with different numbers of protected states are determined by the relaxation rates in the quasi-PT-symmetric system, calculated in the Markovian approximation. By changing the coupling strength between the resonators and the interaction magnitude between the resonator and the environment, the system switches between modes with two, one, or no loss-protected states. This makes it possible to realize stable PT-symmetric devices based on purely dissipative systems. The obtained results are applicable to quantum systems with single excitations, allowing the concept of PT symmetry to be extended to such systems.

Figures (4)

• Figure 1: Dependence of the first (blue solid line) and second (red solid line) resonator's probability amplitudes $\vert a_{1,2}(t)\vert^{2}$ on time calculated by the Eqns. (\ref{['eq:3']})-(\ref{['eq:5']}) in the case $\gamma>>\delta\omega$ and $\Omega \sim \gamma$. The black dashed lines corresponds to the $\vert a_{1,2}(t)\vert^{2}$ calculated by non-Hermitian equations (\ref{['eq:6']}). Here, $T_R=2\pi/\delta\omega$ is the time of first revival. We consider $N=100$, $\delta\omega = 2 \times 10^{-3} \omega_0$, $g=3\times 10^{-3} \omega_0$, $\gamma=\pi g^2/\delta\omega \approx 1.4\times 10^{-2} \omega_0>>\delta\omega$, $\Omega=6 \times 10^{-3} \omega_0$.
• Figure 2: Dependence of the first (blue solid line) and second (red solid line) resonator's probability amplitudes $\vert a_{1,2}(t)\vert^{2}$ on time calculated by the Eqns. (\ref{['eq:3']})-(\ref{['eq:5']}) in the case $\gamma \lesssim \delta\omega$. Here, $T_R=2\pi/\delta\omega$ is the time of first revival. We consider $N=100$, $\delta\omega = 2 \times 10^{-3} \omega_0$, $g=7.5 \times 10^{-4} \omega_0$, $\gamma=\pi g^2/\delta\omega \approx 8.8\times 10^{-4} \omega_0<\delta\omega$, $\Omega=5 \times 10^{-4} \omega_0$.
• Figure 3: Dependence of the system's memory $M$ on the coupling strengths $g$ and $\Omega$ with the initial condition $a_1(t=0)=1$, $a_2(t=0)=0$ (a) and $a_1(t=0)=0$, $a_2(t=0)=1$ (b). Here, $N=50$, $\delta\omega = 2 \times 10^{-3} \omega_0$. Vertical red line corresponds to the condition $\gamma =\delta\omega$ (or $g=\delta\omega/\sqrt{\pi}$). The inclined red line corresponds to the condition $\Gamma_1\ \approx 2\Omega^2/\gamma =\delta\omega$ (or $\Omega=\sqrt{\frac{\pi}{2}}g$).
• Figure 4: Phase diagram of the states in which $\left| a_1\right| \ge \left| a_1\right|$ (a) and the state in which $\left| a_1\right| \le \left| a_1\right|$ (b). The orange color indicates areas where states are loss-protected by non-Markovian effects. The blue color indicates areas where all states are unprotected. The numbers indicate areas whose boundaries are determined by the conditions discussed in the section the influence of PT-symmetry transition on the time of life of states. The boundaries of first and second areas are determined by conditions $\Omega>\Omega_{EP}$ (see (\ref{['eq:11']})) and (\ref{['eq:12']}) and (\ref{['eq:13']}, respectively. The boundaries of third and fourth areas are determined by conditions $\Omega<\Omega_{EP}$, and (\ref{['eq:14']}), (\ref{['eq:15']}), respectively.