Temporal Parity-Time Symmetry for Extreme Energy Transformations

Temporal interfaces introduced by abrupt switching of the constitutive parameters of unbounded media enable unusual wave phenomena. So far, their explorations have been mostly limited to lossless media. Yet, non-Hermitian phenomena leveraging material loss and gain, and their balanced combination in parity-time (PT)-symmetric systems, have been opening new vistas in photonics. Here, we unveil the role that temporal interfaces offer in non-Hermitian physics, introducing the dual of PT symmetry for temporal boundaries. Our findings reveal unexplored interference mechanisms enabling extreme energy manipulation, and open new scenarios for time-switched metamaterials, connecting them with the broad opportunities offered by non-Hermitian phenomena.

Figure 1

(a) Schematic of TPT-symmetric temporal slabs. (b) ${\widehat{P}}_{\mathrm{tot}}^{\min }$ [solid (or dashed) blue line] and ${\widehat{P}}_{\mathrm{tot}}^{\max }$ [solid (or dashed) red line] in the case of fixed ${\widehat{n}}_2^{\prime }=2$ (or varying ${\widehat{n}}_2^{\prime }=\sqrt{1-{\widehat{n}}_2^{\prime \prime 2}}$) as a function of ${\widehat{n}}_2^{\prime \prime }$. Blue down-pointing (red up-pointing) triangles represent the perturbative result for ${\widehat{P}}_{\mathrm{tot}}^{\min }$ (${\widehat{P}}_{\mathrm{tot}}^{\max }$) in Eq. (7).

Figure 2

Normalized energy ${\widehat{w}}_{em}(t)$ (solid lines), ${\widehat{U}}_i(t)$ (dashed lines) and ${\widehat{U}}_c(t)$ (empty circles) as a function of time when (a) ${\varphi }_{\min }=0.13$, ${\mathrm{\Delta }}_f^{\min }=3.5$ and (b) ${\varphi }_{\max }=3.27$, ${\mathrm{\Delta }}_f^{\max }={\mathrm{\Delta }}_f^{\min }$ for the extreme values of ${\widehat{P}}_{\mathrm{tot}}$ [see Fig. 1] in the case of ${\widehat{n}}_2^{\prime }=2$ and ${\widehat{n}}_2^{\prime \prime }=0.2$. Insets: snapshots obtained with COMSOL of $E_x(z,t)$ versus position $z$ (in units of $2\pi /k_z$) at time $t=-16\pi /{\omega }_1$ (1st row), $\mathrm{\Delta }t/2$ (2nd row), $3\mathrm{\Delta }t/2$ (3rd row), and $16\pi /{\omega }_1$ (4th row) for finite pulses traveling in the medium. Other free parameters are $n_1=1$ and ${\omega }_1/(2\pi )=1\mathrm{GHz}$, and the colored backgrounds indicate the material properties over time. The amplitude of the two input waves is $\sqrt{2}\mathrm{V}/\mathrm{m}$.

Figure 3

Same as Fig. 2 for the minimum (a) and the maximum (b) of ${\widehat{P}}_{\mathrm{tot}}$ when ${\widehat{n}}_2^{\prime \prime }=0.2$ and ${\widehat{n}}_2^{\prime }\approx 0.98$, such that we switch only the material conductivity.