Parity-time-symmetric coupled asymmetric dimers

We investigate a parity-time ($\mathcal{PT}$)-symmetric system that consists of two symmetrically coupled asymmetric dimers. The enclosed magnetic flux controls the $\mathcal{PT}$ phase transition. The system can reenter the exact $\mathcal{PT}$-symmetric phase from a broken $\mathcal{PT}$-symmetric phase with large non-Hermiticity. Two-state coalescence may have one or two defective eigenstates. The topology of exceptional points is reflected by the phase rigidity scaling exponents. The topology changes when exceptional points coincide. The geometric phases accumulate when encircling the exceptional points and vary as the magnetic flux. The magnetic flux is favorable for the realization of high-order exceptional points. A triple point of different quantum phases has an order of 4. The perturbation around the four-state coalescence leads to a fourth-root mode frequency splitting; the sensing sensitivity is significantly enhanced.

Figure 1

Schematic of four coupled resonators consisting of two asymmetric dimers. One dimer has loss (red) and the other has gain (green). The coupling strengths are $\kappa$ and $J$. The dotted blue line is the $\mathcal{PT}$-symmetric axis. The primary (auxiliary) resonators are round (elliptical). The blue arrows indicate the counterclockwise mode supported by the primary resonators. The yellow (magenta) arrows illustrate the path lengths of counterclockwise mode photons traveling in counterclockwise (clockwise) direction between primary resonators. The length difference induces a directional phase factor in the coupling, which results in the nonreciprocity and induces the synthetic magnetic flux $\mathrm{\Phi }$.

Figure 2

(a,d) $\mathrm{\Phi }=0$, (b,e) $\mathrm{\Phi }=\pi /2$, and (c,f) $\mathrm{\Phi }=\pi$. $\mathcal{PT}$ phase diagrams are shown in the ${\gamma }^2-J^2$ plane for (a–c). I is the exact $\mathcal{PT}$-symmetric phase; II and III are the broken $\mathcal{PT}$-symmetric phases with one and two pairs of conjugate energy levels, respectively. The critical lines of ${\gamma }_{\mathrm{c},1}$ and ${\gamma }_{\mathrm{c},2,\pm }$ are shown in the $\kappa -J$ plane for (d–f); the couplings are in the unit of a decay rate, $\gamma /2=1$. The dashed green curve divides the pure imaginary and complex energy levels in region III. The blue solid curves (${\gamma }_{\mathrm{c},1}$) represent the EP2 with one defective eigenstate; the cyan (${\gamma }_{\mathrm{c},2,-}$) and green curves (${\gamma }_{\mathrm{c},2,+}$), both the solid and dashed ones, indicate the EP2 with two defective eigenstates; black crosses mark the high-order EP4s where two types of EP2s coincide.

Figure 3

Energy levels as functions of gain and loss $\gamma$. (a) $J=0.5$, (b) $J=1.0$, (c) $J=1.5$, (d) $J=J_{\mathrm{c},2}=2.0$, (e) $J=2.01$, (f) $J=J_{\mathrm{c},1}=\sqrt{2+\sqrt{5}}\approx 2.058$, (g) $J=2.5$, and (h) $J=3.0$. Other parameters are $\kappa =1,\mathrm{\Phi }=0$.

Figure 4

Energy levels as functions of gain and loss $\gamma$. (a) $J=1$, (b) $J=\sqrt{2}$, (c) $J=\sqrt{1+\sqrt{2}}\approx 1.554$, and (d) $J=2$. Other parameters are $\kappa =1,\mathrm{\Phi }=\pi /2$.

Figure 5

(a,c,e) Phase rigidity and (b,d,f) the scaling law when EPs coincide for $\kappa =1$. (a,b) $J=J_{\mathrm{c},2}=2,\mathrm{\Phi }=0$; (c,d) $J=J_{\mathrm{c},1}=\sqrt{2+\sqrt{5}},\mathrm{\Phi }=0$; and (e,f) $J=J_{\mathrm{c},2}=\sqrt{2},\mathrm{\Phi }=\pi /2$; corresponding spectra are in Figs. 3 and 4. The fitting exponent approaches 1.0 (0.5) for diamonds (circles) for ${\gamma }_{\mathrm{c},2,\pm }$ (${\gamma }_{\mathrm{c},1}$) in (b,f) and approaches 0.75 (0.5) for the diamonds (circles) for ${\gamma }_{\mathrm{c},2,+}$ (${\gamma }_{\mathrm{c},2,-}$) in (d). Energy levels that coalesced at EPs exhibit identical scaling law in (b,d). (f) The green square energy level in (e); the other levels that coalesced at EPs possess the identical scaling exponent.

Figure 6

Encircling ${\gamma }_{\mathrm{c},1}$ at the $\delta -\kappa$ plane, $\delta =\rho cos\theta$ and $\kappa =1+\rho sin\theta$. (a,c) Energy levels and (b,d) accumulated geometric phase. The upper panel is for $J=0.5,\gamma =3/4$; the lower panel is for $J=1.5,\gamma =5/4$. Other parameters are $\mathrm{\Phi }=0,\rho =0.1$.

Figure 7

Encircling ${\gamma }_{\mathrm{c},2,+}$ at the $\delta -\gamma$ plane, $\delta =\rho cos\theta$ and $\gamma ={\gamma }_{\mathrm{c}}+\rho sin\theta$. (a,c,e) Energy levels and (b,d,f) accumulated geometric phase. The upper panel is for $\mathrm{\Phi }=0,J=2,{\gamma }_{\mathrm{c}}=2\sqrt{2}$; the middle panel is for $\mathrm{\Phi }=\pi /2,J=\sqrt{2},{\gamma }_{\mathrm{c}}=2$; the lower panel is for $J=1,\mathrm{\Phi }=\pi ,{\gamma }_{\mathrm{c}}=2$. Other parameters are $\kappa =1,\rho =0.1$.