Photonic Time Crystals: Temporal Topology

• Photonic Time Crystals are electromagnetic media with refractive indices modulated periodically in time, resulting in distinct time-refraction and time-reflection phenomena.
• They support Floquet–Bloch states with momentum bandgaps where the quantized Zak phase delineates trivial and nontrivial topological regimes.
• These temporal modulations enable robust edge states and nonreciprocal phase control, paving the way for advanced applications in ultrafast optical switching and topological photonics.

Photonic Time Crystals (PTCs) are electromagnetic media whose refractive index varies periodically—and often abruptly—in time, rather than space. This temporal periodicity creates profound new physics: propagating light interacts with time interfaces, experiencing unique wave-mixing phenomena, the emergence of Floquet–Bloch states, and, most notably, the formation of momentum (k) bandgaps hosting topologically nontrivial phases and robust localized states. The interplay among time-refraction, time-reflection, band topology, and edge localization makes PTCs the temporal analog of spatial photonic crystals but with distinct and fundamentally new dynamical and topological properties (Lustig et al., 2018).

1. Temporal Modulation and Wave Dynamics

PTCs are defined by a refractive index profile periodic in time. The canonical model is a step-wise (binary) modulation: $$n(t) = \begin{cases} n_1, & 0 < t < t_1 \ n_2, & t_1 < t < T \end{cases}$$ where $T$ is the modulation period. This abrupt time variation leads to two fundamental phenomena when an optical field traverses such a medium:

• Time-refraction: A part of the field continues to propagate in time with a frequency determined by the new refractive index, conserving spatial momentum ($k$).
• Time-reflection: A second component is "reflected" in time, creating a time-reversed wave (analogous to backward propagation).

At each temporal interface, the continuity conditions for the electromagnetic field and its derivatives split the incident wave into these components. Since the medium is uniform in space, momentum $k$ is strictly conserved, but the frequency $\omega$ can shift at each interface according to the instantaneous refractive index: $\omega_a = k n_a c \quad (a=1,2)$.

2. Floquet–Bloch States and Momentum Bandgaps

Due to the periodic time dependence, PTCs support solutions of Maxwell’s equations in the Floquet–Bloch form: $D(x, t) = D_0(t) e^{-i\omega t - i k z},$ with $D_0(t + T) = D_0(t)$. This imposes a discrete structure for the "quasifrequency" (Floquet frequency) $\phi(k)$, yielding dispersion relations: $\phi(k) = \cos^{-1}(W + X),$ where $W$ and $X$ are determined by the modulation (e.g., segment durations $t_1, t_2$ and indices $n_1, n_2$). When $\phi(k)$ is real, states are "propagating" and energy oscillates. When $\phi(k)$ acquires an imaginary part (i.e., $|\cos \phi(k)| > 1$), modes experience exponential temporal amplification or decay. These bands and gaps appear not in energy, but in $k$-space—an inversion of the spatial photonic crystal paradigm.

A summary of PTC band structure:

Regime	$\phi(k)$	Physical Effect
Propagation band	Real	Stable oscillatory solution
Momentum (k) bandgap	Complex ($\phi = \phi_r + i\phi_i$)	Exponential growth (amplification) or decay

3. Topological Invariants: Zak Phase

A principal discovery is the identification of band topology in PTCs via the Zak phase, a topological invariant associated with each Floquet–Bloch band. For the $m$-th band: $$\varphi_{\text{Zak}}^{(m)} = \int_{0}^{T} i \langle u_m(t) | \partial_t u_m(t) \rangle\, dt,$$ where $u_m(t)$ is the periodic part of the solution. Due to the temporal mirror symmetry (e.g., $t=0$ at the midpoint of a segment), the Zak phase is quantized to $0$ or $\pi$. This quantization distinguishes trivial from nontrivial topological phases as a function of $k$.

Explicit calculations confirm that

• $\varphi_{\text{Zak}} = 0$: topologically trivial band.
• $\varphi_{\text{Zak}} = \pi$: topologically nontrivial band, which manifests in robust phase relationships and protected edge states.

4. Relation Between Forward/Backward Waves and Topology

PTCs produce, for a given $k$, time-refracted (forward) and time-reflected (backward, time-reversed) Floquet modes. For momenta in the bandgap, the outgoing pulses (at the end of the crystal) appear as two equal-magnitude waves. However, their relative phase difference,

$\frac{t}{r} = e^{i\delta\varphi},$

is not arbitrary. It is analytically linked to the Zak phase via: $$\operatorname{sgn}(\delta\varphi_s) = \mathcal{S}(-1)^l \exp(i\varphi_{\text{Zak}}^{(m)}),$$ where $s$ denotes the bandgap, $\mathcal{S}$ is a sign determined by permittivity ratios, and $l$ is the number of lower band crossings. The phase relation is not simply a dynamical quantity but is fundamentally determined by the topological invariant of the occupied band. This phase dictates observable interference phenomena and can directly affect measurable quantities such as carrier-envelope phase shifts.

5. Temporal Topological Edge States

Beyond bulk band topology, the concatenation of two PTCs with identical band-gap structure—but different Zak phases—establishes a "temporal interface" at time $t_{\text{edge}}$. At this interface, robust edge states emerge: they are sharply localized in time, displaying exponential decay both before and after $t_{\text{edge}}$, yet highly concentrated at the interface. This is the temporal analog of spatial edge states found at the boundaries of topologically distinct spatial phases. Notably, even if both sides are in exponentially amplifying regimes, the edge state at the interface remains a robust, localized excitation immune to temporal disorder.

Illustratively:

Structure	Edge State Localization
Spatial topological insulator	Confined at spatial boundary
PTC with temporal topology	Confined at temporal interface $t_{\text{edge}}$

6. Implications and Experimental Relevance

The theoretical analysis establishes:

• The periodic modulation of refractive index in time creates a k-bandgap structure with complex Floquet eigenfrequencies, enabling exponential field amplification not possible in stationary (spatially modulated) media.
• Each Floquet band inherits a quantized Zak phase; consequently, temporal analogs to spatial topological phenomena—such as robust mid-gap edge states—are possible.
• The relative phase of emergent, amplified pulses is dictated by the Zak phase, providing a signature of the underlying band topology.
• Temporally localized edge states can be constructed by engineering sequential PTC segments with different topological invariants. These states are robust against temporal imperfections—a direct consequence of their topological origin.

This analysis clarifies the essential analogy and crucial differences between temporal and spatial photonic crystals. Unlike spatial lattices, where the energy bandgap prohibits propagation, in PTCs the momentum bandgap enables both nontrivial amplification dynamics and topological protection in the time domain. The experimental observability of these effects is intimately linked to the ability to engineer abrupt, periodic changes in refractive index realized via ultrafast optical switching or nonlinear modulation schemes.

The topological classification of PTC momentum bands and the associated temporal edge phenomena provide a powerful platform for nonreciprocal phase control, robust temporal filtering, and the realization of time-domain analogs to topological insulators, as established in the foundational work (Lustig et al., 2018).