Introduction

The investigation of topological phases has traditionally been rooted in electronic systems1, while analogous behaviours have been identified across various physical domains, including but not limited to photonic2, acoustic3, and mechanical systems4. Currently, the understanding of topological phases has primarily centred on the global behaviours of wavefunctions within the energy (frequency) bands of materials or artificial structures, spanning diverse space groups and symmetries in real or synthetic dimensions5,6,7. The development of topological phases within frequency bands has led to the discovery of numerous captivating physical phenomena, as exemplified by the topological boundary states that are immune to scattering from disorders or imperfections, and promising applications in quantum computing, lasers, and communications8,9,10,11,12,13.

Recently, it has been noted that the bandstructures can be gapped in momentum (or Bloch momentum), with the gaps known as momentum bandgaps or k gaps14,15,16,17,18,19. This surge in interest in k gaps is largely stimulated by the advent of photonic time crystals14,15,16,20, which modulate permittivity periodically and abruptly in time. The k gaps differ from the frequency bandgaps in fundamental aspects. Firstly, unlike the frequency bandgaps that emerge from spatial reflection and refraction in spatially periodic materials21, the k gaps typically arise from interference between time-reflected22,23,24,25 and time-refracted waves in temporally periodic materials. Secondly, while frequency-gap modes always decay in space and cannot propagate, k-gap modes exhibit exponential growth or decay in time while retaining the ability to propagate in space14,26. These k gaps manifest diverse physical phenomena14,19,26,27,28,29, such as enhanced light-electron interactions26 and superluminal solitons29, offering the potential for various device applications, notably non-resonant tuneable lasers27.

Much akin to frequency bandgaps, k gaps also exhibit topological properties defined by momentum bands lower than the k gaps14,30. However, the fundamental discrepancies between these two categories of bandgaps yield distinctive topological phenomena. For instance, the topological boundary state within the frequency bandgap emerges at a spatial interface (see Fig. 1a) where the frequency is conserved, but momentum is not, while the topological boundary state within the k gap arises at a temporal interface (see Fig. 1b) conserving momentum but not frequency. Intriguingly, the spatial topological boundary state confines itself to an interface between two “insulators,” whereas its temporal counterpart localises at an interface between two media supporting exponentially amplifying k-gap modes. Moreover, the existence of temporal topological boundary states correlates with the principle of causality. Gaining insights into k-gap topology holds immense potential for exploring and harnessing various topological phases defined within momentum bands and temporal topological effects across quantum and classical systems. To date, the k-gap topology has been theoretically proposed in photonic time crystals14. However, its experimental demonstration remains elusive, largely due to the difficulty in periodically modulating refractive indices of photonic materials within extremely short timescales and comparable amplitudes to the materials, particularly at optical frequencies.

Fig. 1: Topological boundary state in the k gap and the experimental setup.
Fig. 1: Topological boundary state in the k gap and the experimental setup.
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a Topological boundary states in a frequency bandgap. Topological boundary states are localised at a spatial interface between two insulators with topologically distinct frequency bandgaps, with the frequency conserved. Topological number w switches in the spatial dimension. b Topological boundary states in a k gap. Topological boundary states are localised at a temporal interface between two media with topologically distinct k gaps, with the momentum conserved. Topological number w switches in the temporal dimension. c Schematic diagram of the experimental setup. The system consists of two fibre loops with different lengths. PM: phase modulator, MZM: Mach–Zehnder modulators, OC: optical coupler, VOC: variable optical coupler, SMF: single-mode fibre, PD: photodetector, EDFA: erbium-doped fibre amplifier. d Temporal synthetic lattice mapped from the setup in (c). The orange and green shadows represent gain (+γ0) and loss (−γ0), respectively, controlled by MZM. e Complex bandstructure of the lattice when γ0 = 0.3. A complete k gap opens near Q = π.

Here, we report the experimental observation of temporal topological boundary states stemming from the k-gap topology of light. Our experimental framework is based on an optical system comprising two fibre loops31,32,33,34,35,36,37, which is mapped into a large-scale temporal synthetic lattice featuring a Bloch-momentum bandstructure inclusive of a complete k gap. Using this setup, we directly observe the emergence of topological boundary states at temporal interfaces between two subsystems displaying distinct k-gap topology. Intriguingly, these boundary states demonstrate a counterintuitive behaviour, with energy decaying away from the interface in both temporal directions [i.e., with energy growing (decaying) before (after) the temporal interfaces], despite both subsystems supporting exponential growth of k-gap modes. Furthermore, we observe the distinctive behaviour of optical pulses experiencing time-refraction and time-reflection within the k gap and the momentum bands. Specifically, when traversing a temporal slab, pulses within the momentum band bifurcate into four distinct pulses, whereas their k-gap counterparts diverge into two amplified pulses.

Results

Momentum gap in a time-synthetic lattice

In our experiment, optical pulses propagate in two fibre loops with slightly different lengths connected by a variable optical coupler (VOC), as shown in Fig. 1c. This coupler facilitates a tuneable coupling ratio β, controlled by an electric signal sourced from an arbitrary waveform generator (AWG). A 50 ns rectangular-shaped pulse is injected into the longer loop via a 50:50 optical coupler (OC). The pulse is generated by a 1550 nm distributed-feedback laser, with its output beam modulated by an acousto-optic modulator (AOM). Utilising a Mach–Zehnder modulator (MZM) and the erbium-doped fibre amplifiers (EDFAs), we can amplify or attenuate the signal amplitude within the loops (see details in Supplementary Notes 1). Besides, a phase modulator (PM) is integrated into the loops to manipulate the phase of the optical signal, which is used to generate a pulse with predesigned Bloch momenta (see Supplementary Note 3). As the optical pulses propagate, the power in each loop is monitored using photodetectors (PDs).

In our fibre-loop system, pulses in the shorter loop experience a temporal advancement, whereas those in the longer loop undergo a delay. Such a pulse delay or advancement in the time domain, within a round-trip, can be conceptually analogous to a distance in the space dimension. Concurrently, the count of round-trips inherently offers a temporal degree of freedom, which can be viewed as a time dimension. As a result, this fibre-loop system can be equivalently mapped into a spatiotemporal lattice network, as illustrated in Fig. 1d. The pulse evolution in the lattice is governed by the following equations that describe light dynamics in a temporal unit comprising two successive time steps,

$$\begin{array}{c}{u}_{n}^{m+1}=[\cos ({{\beta }}){u}_{n+1}^{m}+i\,\sin ({{\beta }}){v}_{n+1}^{m}]{e}^{{{\gamma }}(m)}\\ {v}_{n}^{m+1}=i\,\sin ({{\beta }}){u}_{n-1}^{m}+\,\cos ({{\beta }}){v}_{n-1}^{m}\end{array}$$
(1)

where \({u}_{n}^{m}\) denotes the amplitude at lattice position n and time step m on left-moving paths, and \({v}_{n}^{m}\) denotes the corresponding amplitude on right-moving paths. γ(m) is a temporally amplitude-modulated signal applied to the left-moving paths, satisfying

$${{\gamma }}(m)=\Bigg\{\begin{array}{ccc}+{{{\gamma }}}_{0} & {{{\rm{for}}}} & {{{\rm{mod}}}}(m;2)=1\\ -{{{\gamma }}}_{0} & {{{\rm{f}}}}{{{\rm{o}}}}{{{\rm{r}}}} & {{{\rm{mod}}}}(m;2)=0\end{array}$$
(2)

By employing the Floquet-Bloch ansatz, we obtain the system’s bandstructure, showcased in Fig. 1e. The relation between quasienergy, θ (or the longitudinal propagation constant), and Bloch momentum (or the transverse wavenumber) along the 1D lattice, Q, is given by

$$\cos ({{\theta }})={\cos }^{2}({{\beta }})\cos (Q)-{\sin }^{2}({{\beta }})\cosh ({{{\gamma }}}_{0})$$
(3)

Interestingly, the above bandstructure is gapped in the momentum, around Q = θ, signifying an absence of real-value solutions for quasienergy. We note that our system is “spatially” uniform yet temporally modulated, which preserves the “spatial” translational symmetry but breaks the temporal translational symmetry. At each temporal interface, the Bloch-momentum Q (not the quasienergy θ) is conserved, which is thus a good quantum number. This is fundamentally different from spatially modulated yet stationary systems, such as spatial photonic crystals, in which frequency is conserved while momentum is not. Besides, throughout our experiments (including those of topological boundary states), the “spatial” (temporal) translational symmetry is preserved (broken). In view of the above facts, we consider the bandgap here as a complete k gap. In some sense, our temporal synthetic lattice resembles the photonic time crystal, with the modes propagating in two loops mapped respectively to the forward- and backward-propagating modes in the photonic time crystal (see Supplementary Note 8).

Light propagation in the momentum gap

We now perform experiments to investigate the time-refraction and time-reflection behaviour of light in the momentum bands or within the k gap, respectively. In the first case, the lattice framework has triple time periods, mimicking a temporal slab (Fig. 2a, b). More specifically, both the first and third time periods have β = π/4, while the second time period has β = 0. To better illustrate the light behaviour solely attributed to the bands, all three periods feature no k gaps, and the first and third time periods have distinct bandstructures from the second one (Fig. 2e). At time step m = 0, a Gaussian-shaped wave packet from the upper band is excited, which carries an initial momentum Q = 0.95π and propagates forward. Upon reaching the first temporal interface at m = 21, the momentum conservation principle dictates that the pulse couples to two distinct sets of Bloch modes in the second period: a time-refracted pulse propagating forward and its time-reversed counterpart propagating backwards. At the subsequent temporal interface at m = 41, a similar bifurcation occurs, yielding a total of four pulses out from the temporal slab (Fig. 2b, c). Interestingly, the pulse propagation in the temporal slab markedly differs from that in a spatial counterpart, which undergoes infinite times of refraction and reflection at two interfaces, resulting in superpositions of refracted and reflected waves in the spatial slab. As the bands are purely real, energy is conserved in this scenario, as shown in Fig. 2d.

Fig. 2: Pulse propagation in the k gap.
Fig. 2: Pulse propagation in the k gap.
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a Schematic diagram of the time refraction and time-reflection behaviour in the band. A pulse undergoes a splitting event at each temporal interface. b Parameter β as a function of time (left panel), the measured pulse propagation (middle panel), and the calculated result (right panel). In the first and third time periods (from m = 0 to 21 and from m = 41 to 66), γ0 = 0 and β = π/4, with the bandstructure corresponding to the blue line in (e). In the second time period (from m = 21 to 41), γ0 = 0 and β = 0, with the bandstructure corresponding to the green line in (e). c Pulse intensity distributions at m = 10, 30, and 60, respectively. d Total energy of the pulses. e Bandstructures in different time periods. The Grey dashed line indicates the conservation of Bloch momentum at the temporal interface. f Schematic diagram of the time refraction and time-reflection behaviour in the k gap. g Parameter γ0 as a function of time (left panel), the measured pulse propagation (middle panel), and the calculated result (right panel). In the first and third time periods (from m = 0 to 21 and from m = 41 to 66), γ0 = 0 and β = π/4, with the bandstructure corresponding to the blue line in (j). In the second time period (from m = 21 to 41), γ0 = 0.23 and β = π/4, with the bandstructure corresponding to the orange (real part) and red (imaginary part) lines in (j). h Pulse intensity distributions at m = 10, 28, and 60, respectively. i Total energy of the pulses. j Bandstructures in different time periods. The real and imaginary parts are represented by the solid and dashed lines, respectively.

In the second case, there are also triple time periods (Fig. 2f, g), however, with the second time period featuring a k gap (Fig. 2j). Similarly, at m = 0, a Gaussian-shaped wave packet from the upper band is excited, which carries an initial momentum Q = 0.95π inside the k gap. Upon reaching the first interface (m = 21), the pulse bifurcates into two modes with zero group velocity (Fig. 2g), meaning they are “frozen” in space. These modes either increase or decrease exponentially in the second period; the total energy of the modes exhibits exponential growth (Fig. 2i), in contrast to the conserved energy in the first case. At the second temporal interface, the mode with amplifying energy dominates, further splitting into time-refracted and time-reflected modes (Fig. 2g, h). Thus, only two pulses are observed after the pulse traverses the temporal slab, which markedly differs from the first case. Note that the amplification of the momentum-gap modes bears some relation with the parametric amplification38, particularly in photonic time crystals. However, they are fundamentally different because the former (latter) is essentially a non-resonant (resonant) phenomenon27.

Temporal topological boundary states from momentum-gap topology

Next, we perform experiments to study the temporal topological boundary states at the interface between two lattices with distinct k-gap topology (Fig. 3a). The k-gap topology can be understood from an effective momentum operator18 near θ = π (see Supplementary Note 5)

$${H}_{{{{\boldsymbol{\delta }}}}Q}({{\delta }}{{\theta }})=-{m}_{{{{\gamma }}}_{0}}{{{\sigma }}}_{x}+\frac{{{\delta }}{{\theta }}}{{{\alpha }}}{{{\sigma }}}_{z}$$
(4)

where δQ is the Bloch momentum deviated from Q = π, δθ is the quasienergy deviated from θ = π, and \({m}_{{{{\gamma }}}_{0}}={{s}}{{i}}{{n}}{{h}}({{{\gamma }}}_{0})\).α = \(\sqrt{2}\)/2, corresponding to the group velocity at δQ = 0 in the lattice without gain and loss. Interestingly, Eq. (4) is analogue to the well-known 1D massive Dirac Hamiltonian39; Here, however, the eigenvalue is momentum rather than energy as in the usual Hamiltonian. The bandstructure defined by Eq. (4) can be viewed as a momentum bandstructure. Besides, when γ0 switches from negative to positive, a momentum band inversion occurs around θ = π, where two mutually orthogonal states exchange their positions (Fig. 3b). This is similar to the band inversion of spatial topological photonic crystals30. Similar to the Jackiw-Rebbi solution at a mass domain wall, when placing two media together with opposite mass terms \({m}_{{{{\gamma }}}_{0}}\), a zero-momentum topological state is localised at the temporal boundary (see Supplementary Note 6).

Fig. 3: Robust temporal topological boundary states from k-gap topology.
Fig. 3: Robust temporal topological boundary states from k-gap topology.
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a Schematic diagram of a topological temporal interface in the synthetic lattice. b Bandstructures of two lattices with distinct k-gap topology. The insets show two eigenvectors at θ = π. In the region I (γ0 > 0), the eigenvector \({[\bar{U},\bar{V}]}^{T}\) at the left (right) band is \({[-1/\sqrt{2},1/\sqrt{2}]}^{T}\) (\({[1/\sqrt{2},1/\sqrt{2}]}^{T}\)), while in the region II (γ0 < 0), the eigenvectors are inverted for two bands. c–f Experimental observation of temporal topological edge states. Parameter γ0 as a function of time (c), the measured pulse propagation (d) and the calculated result (e). Measured light energy (f). In the region I (m < 30), γ0 = −0.23, In the region II (m > 30), γ0 = 0.23. In the entire process, β = π/4. g–n Temporal topological boundary with disorder around the boundary (g–j) and in the bulk (k–n). The blue shadow area represents the time duration when the disorder is applied.

For our case (γ0 is relatively small), the nonzero γ0 mainly affects the eigenstates near θ = π, while the eigenstates far away from θ = π remain almost unchanged. For this consideration, we compare the local geometric phases40 between lattices with opposite \({m}_{{{{\gamma }}}_{0}}\) around θ = π, which can fully characterise the global topology of the band (see Supplementary Note 7). To do so, we perform parallel transport on the band lower than the k gap (δQ < 0), which yields a vanishing Berry connection everywhere on the momentum band. We choose the eigenstate at a low quasienergy limit, [1,0]T, as an initial state with no phase difference. For the final state at a high quasienergy limit [0,1]T, the phase is 0 for a positive \({m}_{{{{\gamma }}}_{0}}\), in contrast to π for a negative \({m}_{{{{\gamma }}}_{0}}\) (see Supplementary Note 5). This implies the distinct k-gap topology determined by the sign of \({m}_{{{{\gamma }}}_{0}}\). Furthermore, by letting Q be complex-valued and θ to be real-valued, we can get continuous momentum bands. Thus, topological invariants (w) on the momentum bands are well defined, which characterise the global accumulation of geometrical phase along θ. We find that, w = 0, when γ0 > 0, and, w = 1, when γ0 < 0 (see more details in Supplementary Note 7).

According to topological physics, when two subsystems with topologically different bandgaps are placed together, topological boundary states are formed at the interfaces. In our case, the topological boundary states appear at a temporal interface between subsystems with distinct k-gap topology. For experimental verification, a step-like temporal modulation γ0 is applied (as depicted in Fig. 3c), abruptly altering the system’s k-gap topology, creating a topological temporal interface at m = 30 (Fig. 3c). At m = 0, a Gaussian-shaped wave packet with a momentum residing in the k gap is excited. Before reaching the temporal interface, the pulse energy increases exponentially, as demonstrated in the previous section. After passing the temporal interface, the energy exhibits an exponential decrease, forming an energy peak localised at the interface (Fig. 3d–f). Interestingly, if these two time periods are considered separately, the pulse propagating in each of them should increase exponentially as time progresses. However, the energy decays in both temporal directions at the interface, which is completely counterintuitive. Upon comparing the characteristics of our temporal topological boundary states with the spatial equivalents in photonic crystals, striking differences emerge: In spatial scenarios, the topological edge states are confined around the interface between two regions, supporting only evanescent waves that decay in space. Conversely, in temporal scenarios like ours, the topological boundary states reside between two regions, each supporting modes growing exponentially in time and being able to propagate in space.

Further, topological temporal boundary states exhibit marked robustness against disorders akin to their spatial counterparts. To verify the robustness, we first perturb the temporal interface by adding several temporal layers with random γ0 ∈ [−0.23, 0.23] around the interface (Fig. 3g). Notably, we can identify a clear energy peak around the interface (Fig. 3h–j). We then perturb the temporal bulk by introducing spatially uniform disorders δγ0 ∈ [−γ0/5, γ0/5] (as illustrated in Fig. 3k) into the temporal bulk across all lattice sites. The energy peak at the topological interface can still be spotted (Fig. 3l–n). To conclude, the persistence of temporal topological interface states remains evident whether the disorder is introduced at the boundary or permeates the bulk. These results indicate the intrinsic robustness of these states against considerable disorder, highlighting their topological protection—a hallmark of topological systems.

Discussion

To summarise, we experimentally confirm the topological properties of the k gap by directly observing temporal topological boundary states in a large-scale optical temporal synthetic lattice. Moreover, the propagation dynamics of the k-gap modes through a temporal slab are observed, particularly elucidating their time-refraction and time-reflection behaviour at temporal boundaries. Those phenomena are otherwise extremely challenging (if not impossible) to observe in real optical materials. Our work thus establishes an ideal platform using light to explore the unconventional physics associated with k gaps, encompassing wave dynamics and topological attributes. While our study predominantly focuses on the temporal dimension, an intriguing avenue for future exploration involves incorporating one or more spatial dimensions to form topological phases within a comprehensive space-time framework. Additionally, there exists potential for incorporating nonlinearity33 and disorder36 into our fibre-loop systems, leading to an intriguing synergy between k-gap topology, nonlinearity, and disorder.

Methods

Experimental setup

The input light beam is generated by a distributed feedback laser at 1550 nm and later truncated by an acousto-optical modulator with 50 dB extinction ratio, resulting in a pulse with 50 ns temporal width. Upon encountering the central variable optical coupler, the incoming pulses split and, after several round trips, multipath interference occurs between the emerging sub-pulses. Single mode fibres extend the propagation time for each loop to about 50 μs. Adding a fibre optic patch cable to one loop introduces a 273 ns time difference between the two loops. Erbium-doped fibre amplifiers compensate for the suppression loss of the Mach–Zehnder intensity modulator and other losses, including insertion and absorption losses. A 1535 nm pilot light is coupled before the erbium-doped fibre amplifier to suppress transients and ensure energy stability within the loop. An optical tuneable filter follows the erbium-doped fibre amplifier to remove out-of-band amplified spontaneous emission. A polarization beam splitter filters out orthogonal polarized light, maintaining a single polarization orientation within the loop. An isolator ensures unidirectional light propagation, preventing backscatter. The mechanical polarization controllers offer the requisite adjustments to the light’s polarization. Optical pulses are detected using 50:50 optical couplers that couple the light to photodetectors, converting the optical signals into electrical signals. These signals are then sampled by a digital storage oscilloscope.