Temporal Interfaces in Non-Hermitian Systems

• Temporal interfaces in non-Hermitian systems are abrupt time boundaries where rapid gain/loss and parameter switching drive asymmetric wave scattering and topological transitions.
• The methodologies reveal that sudden shifts in metrics such as permittivity and conductivity yield nonreciprocal energy transfer, enabling extreme amplification and perfect absorption.
• Experimental and theoretical studies probe these interfaces to uncover topological temporal boundary states and unidirectional quantum dynamics, offering new routes for dynamic control.

Temporal interfaces in non-Hermitian systems are abrupt or smooth time boundaries where system Hamiltonians, constitutive parameters, or gain/loss profiles are discontinuously or rapidly switched, generically leading to novel dynamical, topological, and energetic phenomena fundamentally unattainable in either standard Hermitian or purely spatial analogues. Central to their analysis are the consequences of non-Hermitian physics—such as asymmetric (chiral) wave propagation, nonorthogonal mode interference, temporal parity-time (PT) symmetry, topological braiding, and the emergence of mid-gap temporal boundary states—at the moment of switching. Applications span ultrafast wave amplification/absorption, topological control of transient states, and nonreciprocal time-dependent metamaterials.

1. Formulation of Temporal Interfaces in Non-Hermitian Media

Temporal interfaces are realized by abrupt or rapid time variations of system parameters. In electromagnetic systems, the constitutive relations are expressed via time-dependent permittivity $\varepsilon(t)$ and conductivity $\sigma(t)$, which together define a complex, time-dependent refractive index $n(t) = n'(t) - j n''(t)$ with $\varepsilon(t) = \varepsilon_0 [n(t)]^2$, $\sigma(t) = 2 \omega \varepsilon_0 n''(t)$ (Li et al., 2021). A temporal interface at $t = t_s$ is an abrupt change in these parameters: $$(\varepsilon, \sigma)(t) = \begin{cases} (\varepsilon_1, \sigma_1) & t < t_s \ (\varepsilon_2, \sigma_2) & t > t_s \ \end{cases}$$ with matching conditions dictated by Maxwell's equations: continuity of electric displacement and magnetic field. The propagation problem reduces to a wave equation with time-dependent coefficients for, e.g., a transverse electric field, leading to discontinuities in generalized frequency and nontrivial mode coupling across the time boundary.

Non-Hermiticity arises from gain/loss: positive (negative) $\sigma$ corresponds to loss (gain), i.e., $\mathrm{Im}\,n < 0$ ($\mathrm{Im}\,n > 0$). Non-Hermitian dynamical models generalize this framework to quantum (Schrödinger) and classical systems, introducing intuitive analogs such as time-dependent non-Hermitian Hamiltonians $H(t)$ and their associated time-evolution operators (Longhi et al., 2017, Frith, 2020).

2. Temporal Scattering, Asymmetric Transitions, and Transitionless Dynamics

The instantaneous switching implements a “temporal scattering” problem, with transfer (matching) matrices relating incoming and outgoing mode amplitudes. For electromagnetic waves, the “scattering” matrix $S$ that connects amplitudes across the jump is (Li et al., 2021): $$S = \begin{pmatrix} \tau & \rho \ \rho & \tau \end{pmatrix},$$ where, for a jump from refractive index $n_1$ to $n_2$,

$$\tau = \frac{n_1}{n_2}, \quad \rho = \frac{n_1(n_1-n_2)}{n_2(n_2+n_1)}.$$

These expressions reduce to the Hermitian results for purely real $n$ but, more generally, encode the gain/loss asymmetric energy transfer inherent to non-Hermitian parameter sets.

In quantum mechanics, non-Hermitian time-dependent perturbation theory exhibits fundamentally asymmetric transition probabilities $P_{n\to m} \neq P_{m\to n}$ due to the complex and, potentially, one-sided nature of the perturbation’s temporal Fourier spectrum. For strictly one-sided $f(t)$ (e.g., analytic in upper/lower half-plane), transitions can be fully unidirectional—either upward or downward in energy are forbidden—depending on the analyticity domain of $f(t)$. For certain pulse profiles $f(t)$, all transitions can be suppressed entirely, yielding transitionless evolution despite a temporally strong perturbation (Longhi et al., 2017). These dynamics are unique to non-Hermitian temporal interfaces and have no analog in conventional Hermitian time-dependent perturbation theory.

3. Temporal Parity-Time Symmetry and Topological Properties

A central breakthrough is the identification of temporal PT symmetry (TPT), the time-domain analog of spatial PT symmetry, defined via P$_t$ (time-parity: $t \mapsto -t$) and T$_t$ (time-reversal: complex conjugation and $z \mapsto -z$). A TPT-symmetric temporal slab consists of matched gain/loss regions, with complementary refractive indices $n_2 = n'^{} - j n''$, $n_3 = n'^{} + j n''$. The resulting temporal switching protocol is invariant under P$_t$T$_t$, and the scattering matrix $S$ satisfies the constraint $P_t T_t S P_t T_t = S^{-1}$.

Such temporal interfaces support new forms of “topological” responses. In non-Hermitian SSH-type (Su-Schrieffer-Heeger) lattices, an abrupt temporal jump of Hamiltonian parameters imprints reflected and refracted components whose amplitude structure encodes a braiding number $\nu$, equal to the difference of winding numbers (topological invariants) before and after the interface: $\nu = |w_f - w_i|$ This bulk-boundary correspondence establishes that topological transitions in non-Hermitian bands have direct consequences for measurable temporal scattering, i.e., the evolution of amplitudes at the moment of switching (Jiang et al., 9 Dec 2025).

Moreover, in non-Hermitian spatial crystals with spatially periodic gain/loss, a sudden flip in the sign of gain/loss at a temporal interface induces a topological transition, leading to the emergence of topological temporal boundary states (TTBSs)—localized analogs of Jackiw-Rebbi zero modes but in the time domain (Li et al., 2023).

4. Extreme Energy Transformations and Nonorthogonal Wave Interference

Temporal interfaces in non-Hermitian systems enable extreme modulation of stored and transmitted energy. The core mechanism is the interference of nonorthogonal forward and backward waves due to non-Hermitian-induced nonorthogonality. The total electromagnetic energy is

$W_{em}(t) = U_i(t) + U_c(t)$

where $U_i \propto |E^+|^2 + |E^-|^2$ is the incoherent part and $U_c \propto \mathrm{Re}[n(n^* - n)E^+ E^{-*}]$ is the nonorthogonal interference term, vanishing in Hermitian media (Li et al., 2021).

In TPT bilayer slabs, the outgoing power $P_{tot}$ after the temporal sequence can be tuned from extreme amplification to perfect absorption by scanning the relative incoming phase $\varphi$, with

$$P_{max} \sim \frac{n'^4 + 2 n''^2}{(n'^2 - n''^2)^2}, \quad P_{min} \sim 1/P_{max}$$

For systems that realize “pure conductivity switching,” $P_{max}$ diverges and $P_{min}$ vanishes as $n'' \to 1$. This corresponds to “temporal laser” or “temporal perfect absorber” operation, unattainable in Hermitian or even spatial (exact-phase) PT-symmetric systems.

5. Topological Temporal Boundary States and Phase-Transition Probing

Temporal interfaces that result in topological transitions produce temporally localized boundary modes. In the non-Hermitian spatial crystal, a temporal flip in the non-Hermitian Dirac-mass term ($m_r(t) = m_0 \operatorname{sgn}(-t)$) results in a TTBS for the mode with Bloch momentum $k = \pi/d$: $$|\Psi_{TTBS}(t)\rangle \propto \begin{cases} e^{+m_0 t} |+\rangle & t < 0 \ e^{-m_0 t} |-\rangle & t > 0 \ \end{cases}$$ This state is sharply peaked at the switch ($t=0$) and is mathematically equivalent to the Jackiw-Rebbi zero mode, but now localized in the time domain rather than space (Li et al., 2023).

Dynamical signatures of topological transitions at the temporal interface can also be probed via geometric similarity measures—overlaps between pre- and post-interface (biorthogonal) eigenstates—whose crossing signals critical momenta where phase transitions occur in the complex spectrum (Jiang et al., 9 Dec 2025).

6. Quantum Dynamics, Entropy, and Correlation Effects at Temporal Interfaces

Time-dependent non-Hermitian quantum systems require careful treatment of state evolution, metric, and observable definition. The time-dependent Dyson map $\eta(t)$ provides a metric $\rho(t) = \eta^\dagger \eta$ ensuring the generalized norm and observability. Interface matching requires continuity of the metric, normed state, and energy observable $E(t) = H(t) + i \eta^{-1} \partial_t \eta$ (Frith, 2020).

Notably, temporal boundaries across PT symmetry regimes result in distinct entropic behaviors. Across an unbroken-to-broken PT boundary, the von Neumann entropy $S_A(t)$ can tend to a nonzero plateau (“eternal life of entropy”), a property unattainable in conventional Hermitian quantum systems. These entropic signatures are robust to the precise nature (sharp or smooth) of temporal switching and persist in higher-dimensional and oscillator-model extensions.

Temporal interfaces enable quantum dynamics with temporal Leggett-Garg correlations that exceed Hermitian quantum bounds, reaching the algebraic maximum $K_3 = 3$ even with a two-level system, due to nonlinear Bloch-sphere evolution induced by non-Hermitian terms (Varma et al., 2019).

7. Experimental Realizations and Applications

Experimental implementation of temporal interfaces in non-Hermitian systems span multiple physical platforms:

• Active mechanical lattices employing spatially periodic gain/loss with a temporal flip realize TTBSs that are observed as mid-gap temporal-localized modes (Li et al., 2023).
• Time-switched metamaterials enable tunable absorption/amplification, ultrafast power modulation, and nonreciprocal temporal metasurfaces (Li et al., 2021).
• Quantum simulation protocols embedding non-Hermitian dynamics via ancillary coupling in finite Hilbert spaces directly manifest maximal temporal correlations (Varma et al., 2019).

Potential technological directions include robust quantum information storage exploiting persistent entropy plateaus, topologically protected delay lines, momentum-selective filtering, and ultrafast control of photonic, acoustic, or mechanical wave packets.

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In summary, temporal interfaces in non-Hermitian systems define a multi-faceted theoretical and experimental arena in which abrupt temporal changes in gain/loss, Hamiltonian structure, or underlying symmetry drive dynamically asymmetric, topologically distinct, and energetically extreme phenomena. Robust connection to topological invariants, unique interference effects, and new directions for quantum and classical control distinguish this subject as a major research frontier (Li et al., 2021, Longhi et al., 2017, Jiang et al., 9 Dec 2025, Li et al., 2023, Frith, 2020, Varma et al., 2019).