Virtual Parity-Time Symmetry

Parity-time ($PT$) symmetry has recently been opening exciting directions in photonics, yet the required careful balance of loss and gain has been hindering its widespread applicability. Here, we propose a gain-free route to $PT$ symmetry by extending it to complex-frequency excitations that can mimic gain in passive systems. Based on the concept of virtual absorption, extended here to implement also virtual gain, we implement $PT$ symmetry in the complex-frequency plane and realize its landmark effects, such as broken phase transitions, anisotropic transmission resonances, and laser-absorber pairs, in a fully passive, hence inherently stable, system. These results open a path to establish $PT$ symmetry and non-Hermitian physics in passive platforms.

Figure 1

Virtual gain: instantaneous signals in the quasistationary regime versus position $z$ in a TL with uniform loss ${\gamma }_0$, when the signal decay rate (a) $\sigma =0$, (b) $\sigma ={\gamma }_0$, and (c) $\sigma >{\gamma }_0$.

Figure 2

(a) Schematic of a virtual $PT$-symmetric circuit. In its quasistationary state, the system is mapped onto a $PT$-symmetric circuit (inside the dashed circle) consisting of gain and loss balanced parallel $\tilde{L}RC$ resonators, with resistance $\pm R=\pm 1/G$ and renormalized inductance $\tilde{L}=L(1+{\gamma }_0^2/{\omega }_{PT}^2).$ (b) The eigenvalue spectrum of the effective Hamiltonian of the system in (a) as a function of the Hermiticity parameter $G$. In the quasistationary state, these curves (solid green lines) are mapped onto the dashed lines, corresponding to a $PT$-symmetric transition. Here, ${\widehat{\gamma }}_0=\widehat{c}=0.01$.

Figure 3

Demonstration of virtual ATR. Time-dependent signals $v_L^{+}(t)=v_R^{+}(t)$ incident from (a) the left and (b) right port. (a),(b) We plot incident (red), reflected (green), and transmitted signals (blue). (Insets) Operation sketches in the quasistationary state. Here, ${\omega }_0=2\pi \times 50\mathrm{MHz}$, $L=10\mathrm{nH}$, and ${\widehat{\gamma }}_0=0.01$. Thus, $C=1/(L{\omega }_0^2)$ and ${\widehat{\gamma }}_R=0$, $\widehat{ϵ}\approx {\widehat{\gamma }}_L+\widehat{c}$, ${\widehat{\gamma }}_L={\widehat{\gamma }}_0\{1+1/{[(1+\widehat{\delta })}^2+{\widehat{\gamma }}_0^2]\}$, $\widehat{\delta }=-\frac{1}{2}\widehat{c}$, and $\widehat{c}\approx 4{\widehat{\gamma }}_0$ for ATR with $|S_{22}|=1$. In addition, $G_1=2{\widehat{\gamma }}_L\sqrt{C/L}$, $G_2=0$, $C_c=\widehat{c}C$, $G_c={\widehat{\gamma }}_0\widehat{c}\sqrt{C/L}$, $Z_0=\sqrt{L/C}/\widehat{ϵ}$, and the input signals have ${\gamma }_0={\widehat{\gamma }}_0{\omega }_0$ and ${\omega }_{PT}=(1+\widehat{\delta }){\omega }_0$.

Figure 4

Demonstration of absorption and amplification in a virtual CPA laser around its threshold. The initial state $|\mathrm{\Psi }(t=0)⟩$ is prepared by monochromatic excitation before $t=0$, injected from left (red curve) and right ports (blue). Time-domain reflected waves at the left (green) and right (black) ports are also shown. (a) Virtual CPA after $t=0$. (b) Virtual lasing. Here, $\widehat{\delta }=-\frac{1}{2}\widehat{c}$, ${\widehat{\gamma }}_{L/R}\approx {\widehat{\gamma }}_0\pm ({\gamma }_{\mathrm{th}}/{\omega }_0)(1-{\widehat{\delta }}_{\gamma })$ with detuning ${\widehat{\delta }}_{\gamma }=0.01$, and $\widehat{c}=\widehat{ϵ}={\widehat{\gamma }}_0=0.01$. We choose ${\omega }_0=2\pi \times 50\mathrm{MHz}$, $L=10\mathrm{nH}$, and other circuit elements are determined as in Fig. 3. (Insets) The overall output power $\mathrm{\Theta }$ versus ${\widehat{\delta }}_{\gamma }$, confirming lasing and antilasing after $t=0$. Red dots indicate the result in the main panels.