Cyclic permutation-time symmetric structure with coupled gain-loss microcavities

We study the coupled even number of microcavities with the balanced gain and loss between any pair of their neighboring components. The effective non-Hermitian Hamiltonian for such a structure has the cyclic permutation-time symmetry with respect to the cavity modes, and this symmetry determines the patterns of the dynamical evolutions of the cavity modes. The systems also have multiple exceptional points for the degeneracy of the existing supermodes, exhibiting the “phase transition” of system dynamics across these exceptional points. We illustrate the quantum dynamical properties of the systems with the evolutions of cavity photon numbers and correlation functions. Moreover, we demonstrate the effects of the quantum noises accompanying the amplification and dissipation of the cavity modes. The reciprocal light transportation predicted with the effective non-Hermitian models for the similar couplers is violated by the unavoidable quantum noises.

Figure 1

A schematic of CPT systems with even number of microcavities, which carry the balanced gain and loss (at the rate $\kappa$) for each pair of the neighboring components. These cavities are coupled with each other by an adjustable strength $J$. In the non-Hermitian model for the system, the light transported from fiber 1 to fiber 2 has the same intensity as in the reverse process. Here we also consider the quantum noises as the random driving fields into the cavities.

Figure 2

Evolution of photon numbers and average contrast in the regime $\kappa <J$. Here the plots are obtained with $J/\kappa =2.5,E/\kappa =20$, and $\Delta =0$. The initial quantum state of the system is assumed to be a vacuum state. (a) illustrates the evolution of $\langle {\widehat{a}}_1^{†}{\widehat{a}}_1\rangle$ (the thinner purple curve) and $\langle {\widehat{b}}_1^{†}{\widehat{b}}_1\rangle$ (the thicker orange curve), (b) is about the evolution of the average contrast ${\mathcal{C}}_{a_1{b}_1}$ between the two modes, and (c) and (d) illustrate the corresponding quantities for modes ${\widehat{a}}_2$ and ${\widehat{b}}_2$.

Figure 3

Evolution of photon numbers and average contrast in the regime $J<\kappa <2J$. In (a) the thinner (purple) solid curve stands for the evolution of $\langle {\widehat{b}}_1^{†}{\widehat{b}}_1\rangle$ with $J/\kappa =0.6$ and the thicker (orange) one stands for that of $\langle {\widehat{b}}_1^{†}{\widehat{b}}_1\rangle$ with $J/\kappa =0.7$. In (b) the curves stand for the evolution of the average contrast between the modes ${\widehat{a}}_1$ and ${\widehat{b}}_1$ for the ratios in (a), respectively. Correspondingly, the plots in (c) and (d) are about the evolution of $\langle {\widehat{a}}_2^{†}{\widehat{a}}_2\rangle$ [with the same $J/\kappa$ ratios as in (a)] and the average contrast between the modes ${\widehat{a}}_2$ and ${\widehat{b}}_2$, respectively. All other parameters are the same as in Fig. 2.

Figure 4

Evolution of photon numbers and average contrast in the regime $\kappa >2J$. All plots here describe the same quantities as in Fig. 3, except that the thinner (purple) curves now stand for the situation with $J/\kappa =0.4$ and the thicker (orange) ones for $J/\kappa =0.2$.

Figure 5

Comparison between the different dynamical regimes for a CPT system of $N=6$. (a) shows the photon number $\langle {\widehat{b}}_1^{†}{\widehat{b}}_1\rangle$ distributing with the parameter $J/\kappa$ at the time $\kappa t=20$, when the supermodes of the largest eigenvalue have dominated the system evolution. (b) is the corresponding distributions of the average contrast ${\mathcal{C}}_{a_1{b}_1}$ at two different moments, under a driving field of $E/\kappa =20$. Here we mark the two boundaries of the three regimes with the dashed vertical lines. The photon numbers in other cavities and the average contrasts between other modes have the similar distributions.

Figure 6

Difference between transported photon numbers along the mutually reverse paths. The solid curves stand for the transported photon number differences, which are proportional to the difference in the measured light powers. Here we illustrate the quantity $|{\langle {\widehat{b}}_3^{†}{\widehat{b}}_3\rangle }_F-{\langle {\widehat{a}}_1^{†}{\widehat{a}}_1\rangle }_B|$, where ${\langle {\widehat{b}}_3^{†}{\widehat{b}}_3\rangle }_F$ is the photon number of the third dissipation cavity, which is found in the forward transmission pattern with the coherent drive on the first amplification cavity, and ${\langle {\widehat{a}}_1^{†}{\widehat{a}}_1\rangle }_B$ is the photon number in the first amplification cavity due to the backward transmission pattern of driving the third dissipation cavity. Such differences come from the amplification noises. The dashed curves represent the constant forwardly and backwardly transported photon numbers purely due to a coherent driving field with the intensity $E/\kappa =5$, which are found with the non-Hermitian model. (a)–(c) are about the ratio $J/\kappa =1.2,0.6$, and $0.4$, respectively.