Observing the collapse of super-Bloch oscillations in strong-driving photonic temporal lattices

. Super-Bloch oscillations (SBOs) are amplified versions of direct current (dc)-driving Bloch oscillations realized under the detuned dc-and alternating current (ac)-driving electric fields. A unique feature of SBOs is the coherent oscillation inhibition via the ac-driving renormalization effect, which is dubbed as the collapse of SBOs. However, previous experimental studies on SBOs have only been limited to the weak ac-driving regime, and the collapse of SBOs has not been observed. Here, by harnessing a synthetic temporal lattice in fiber-loop systems, we push the ac-field into a strong-driving regime and observe the collapse of SBOs, which manifests as the oscillation-trajectory localization at specific ac-driving amplitudes and oscillation-direction flip by crossing collapse points. By adopting arbitrary-wave ac-driving fields, we also realize generalized SBOs with engineered collapse conditions. Finally, we exploit the oscillation-direction flip features to design tunable temporal beam routers and splitters. We initiate and demonstrate the collapse of SBOs, which may feature applications in coherent wave localization control for optical communications and signal processing.

In this work, by utilizing a synthetic temporal lattice based on a coupled fiber-loop circuit, we can synthesize an ac-driving field with arbitrary value and successfully achieve the collapse of SBOs in the strong-driving regime.6][47] Among others, synthetic temporal lattices created from two coupled fiber loops have drawn extensive attention due to their advantages in introducing and controlling effective gauge fields therein through external modulations.A plethora of discrete-wave phenomena that are difficult to realize in the spatial lattices have been observed in these temporal lattices, ranging from parity-time symmetry, 37 non-Hermitian skin effect, 38,48 to the measurement of Berry curvature. 36In particular, a variety of electric-field-driving effects have also been demonstrated based on this platform, including dc-driving BOs, 30,32 ac-driving dynamic localizations, 40,43 and Landau-Zener tunneling. 44Here, by combining both a dcdriving and a nearly detuned ac-driving electric field in the synthetic temporal lattice, we successfully achieve SBOs up into the strong-driving regime.Particularly, in this regime, we observe the features of vanishing oscillation amplitude and a flip of initial oscillation direction at specific driving amplitudes, showing the clear signatures of SBO collapse.The characteristic rapid swing features of SBOs and the collapse of SBOs have also been analyzed from the Fourier spectrum of oscillation patterns.By generalizing SBOs from the sinusoidal driving to an arbitrarywave driving format, we also observe the generalized SBOs with tunable collapse conditions.Finally, we exploit the oscillation direction flip feature to design tunable temporal beam routers and splitters.The study may find applications in temporal pulse reshaping and coherent oscillation control used for optical communications and signal processing.

Theoretical Model of SBOs in Photonic Temporal Lattices
The synthetic temporal lattice is constructed by mapping from the pulse evolution in two coupled fiber loops, [30][31][32][33][34][35][36][37][38][39][40][41][42][43][44] as shown schematically in Figs.1(a) and 1(b).Here, the pulse circulation in the short/long loops corresponds to the leftward/rightward hopping in the links, and pulse interference at the central coupler is associated with light scattering at each node in the lattice.Then, the circulation times and pulse relative positions can be mapped into the longitudinal evolution step "m" and transverse lattice site n.By further incorporating phase modulators (PMs) into the two loops, we can introduce the additional phase shifts of ϕ u ðmÞ and ϕ v ðmÞ along the leftward and rightward paths.The pulse dynamics in the lattice is governed by the following coupled-mode equations: where u m n and v m n represent the complex amplitudes of pulses in the short and long loops, and sin 2 ðβÞ∕ cos 2 ðβÞ is the splitting ratio of the central optical coupler (OC) with β ∈ ½0; π∕2.In the absence of the phase shifts, the eigen Bloch mode supported in the lattice is ðu m n ; v m n Þ T ¼ ðU; VÞ T expðiknÞ expðiθmÞ, where ðU; VÞ T is the eigenvector and k and θ are the transverse Bloch momentum and longitudinal propagation constant, respectively.By plugging the Bloch mode into Eq.( 1), we can obtain the band structure θ AE ðkÞ ¼ AEarccos½cosðβÞ cosðkÞ, where "±" denotes upper and lower bands.Usually for β → π∕2, by applying Taylor expansion and ignoring high-order terms, we can obtain the approximated band structure, 37,42,43 To induce SBOs, we need to apply simultaneously a detuned dc and ac electric field in the lattice, E eff ðmÞ ¼ α þ E ac sinðω ac m þ φÞ, where α is the magnitude of dc electric field, corresponding to the BO frequency ω BOs ¼ α, and E ac , ω ac , and φ are the driving amplitude, frequency, and initial phase of ac electric field, respectively.As the dc and ac fields satisfy the resonance condition (Stark resonance), α ¼ Nω ac , where N is an integer denoting the resonance order, the effect of directional transport rather than SBOs will occur.The packet will undergo aperiodic transport instead of periodic SBO motion in the lattice. 7,8Under specific ac-driving amplitude satisfying J 0 ðE ω Þ ¼ 0, destructive suppression of lattice's hopping occurs, leading to the dynamic localization effect, where E ω ¼ E ac ∕ω ac is the normalized ac-driving amplitude denoting the amplitudeto-frequency ratio.Based on the resonance condition, SBOs can be further induced by introducing a slight detuning between the ac-driving and BO frequencies, i.e., α ¼ Nω ac þ δ, where δ is the frequency detuning.Under the dc and ac electric fields, the effective vector potential evolves as φÞ, which consists of a linearly varying and a sinusoidally oscillating term.In our synthetic mesh lattice, both dc and ac fields are simultaneously introduced by creating a time-varying vector potential A eff ðmÞ by applying opposite phase modulations into the two fiber loops, ϕ u ðmÞ ¼ −A eff ðmÞ and ϕ v ðmÞ ¼ A eff ðmÞ. 25,49,50he vector potential is assumed to vary slowly with m, so that Zener tunneling between the two quasi-energy bands is negligible.Moreover, both ω ac and α are assumed to be fractional integers than 2π to ensure periodicity of the dynamics after a full cycle.The instantaneous Bloch momentum is given by where k is the initial Bloch momentum.Usually, the frequency detuning δ is a very tiny quantity, which means that the highfrequency driving condition is always satisfied, i.e., ω ac ≫ δ.Under this condition, the instantaneous Bloch momentum evolves at two different time scales: a very rapid one induced by ω ac and a slow one induced by δ, such that the term "δm" in Eq. ( 3) can be considered constant in one ac-driving period M ac ¼ 2π∕ω ac .In the spirit of Magnus expansion, we can use a time-averaging band structure to approximate the rigorous band structure θ AE ½kðmÞ (see Note 1 in the Supplementary Material for derivation), where J N ðE ω Þ is the Nth-order Bessel function of the first kind.By comparing it with Eq. ( 2), we can find that the physical effect of the ac-driving field on the effective band structure is to renormalize the width of the two bands by a renormalization factor J N ðE ω Þ.The effective band structures for several E ω ¼ 1.8, 3.8, and 5.3 with N ¼ 1, ω ac ¼ π∕30, and φ ¼ π∕2 are shown in Fig. 1(c).One can see that the band collapses to a totally flat band as E ω is chosen as the root of As we will demonstrate below, this band collapse condition is just responsible for the collapse of SBOs.
From the effective band structure, we can further obtain the averaging group velocity for a Bloch-wave packet with a central Bloch momentum k, During a long evolution time, i.e., with multiple ac-driving periods, the effect of frequency detuning δ is to make the Bloch momentum linearly sweep across the Brillouin zone, which is analogous to the role of dc field in the situation of BOs.Due to the periodic nature of the band structure, the linear sweeping of Bloch momentum will also induce periodic oscillation of the packet in the lattice, which is dubbed the effect of SBOs. 10,12n particular, SBOs can be described as an oscillatory averaging trajectory of the packet's center of mass in the (n; m) plane, In an analogy to BOs, SBOs also exhibit a cosine-shaped averaging oscillation trajectory where the oscillation amplitude, frequency (period), and initial phase are given by for the upper band and φ SBOs;− ¼ φ SBOs;þ þ π for the lower band.Similar to BOs where both the oscillation amplitude and period are inversely proportional to the dc-driving field, A BOs ¼ j cosðβÞ∕αj; M BOs ¼ j2π∕αj, the oscillation amplitude and period of SBOs are both inversely proportional to frequency detuning δ.Since jδj ≪ jαj, SBOs exhibit much larger oscillation amplitude and longer oscillation period compared with those of BOs, manifesting themselves as very giant oscillatory motions.As indicated by Eqs.(8a) and (8b), both SBO oscillation amplitude A SBOs and period M SBOs increase as the frequency detuning jδj decreases.In the limit case of a vanishing frequency detuning jδj ¼ 0, both oscillation amplitude and period become divergent, i.e., A SBOs ¼ ∞; M SBOs ¼ ∞, where SBOs degrade into the directional transport effect realized as the Stark resonance condition is satisfied. 8,9So the directional transport is the limit case of SBOs with divergent oscillation amplitude and period under vanishing detuning jδj ¼ 0. In addition, the initial oscillation phase of SBOs is proportional to the incident Bloch momentum k, which is also determined by the resonant order N as well as the ac-driving phase φ, as described in Eq. ( 8c).An essential difference between SBOs and BOs is that the oscillation amplitude of SBOs can be further modified by the ac-driving amplitude through the renormalization factor J N ðE ω Þ.Specifically, as E ω takes each root of J N function, the oscillation amplitude will vanish, A SBOs ¼ 0, which is dubbed the collapse of SBOs.Note that at the collapse point, the effective band structure collapses into flat bands, as is clearly shown in Fig. 1(c).The presence of oscillation amplitude collapse is a clear signature of SBOs, which does not occur in BOs.Another interesting feature accompanying the collapse of SBOs is the change of the sign "±" for hΔn AE ðmÞi as E ω passes the collapse point, which manifests as the flip of oscillation direction.In this sense, we can also control the initial oscillation direction by harnessing the collapse of SBOs.
Although the above analysis based on averaging oscillation trajectory hΔn AE ðmÞi can capture the main oscillation feature of SBOs, the more rapid swing oscillation details within one ac-driving period are lost.Also, for previous studies on SBOs in cold-atomic systems, 10,17 these rapid swing oscillation details have not been observed, mainly due to the limited time resolution in measurements.Here, our measurements can resolve the wave-packet dynamics at each explicit evolution step, which allows us to inspect these rapid swing oscillation details.In the following, we will study the oscillation details by checking the Fourier spectrum of the SBO trajectory.As we will show below, the occurrence of SBO collapse can also be probed from the information on the Fourier spectrum.With this aim, we start from the packet's instantaneous group velocity obtained from Eq. (2), from which we can obtain the rigorous, instantaneous oscillation trajectory Δn AE ðmÞ ¼ R m 0 v g;AE ðmÞdm, which consists of rapid oscillation details around the SBO trajectory, Δn AE ðmÞ ¼ hΔn AE ðmÞi þ Δn swing ðmÞ.A more detailed analysis of the instantaneous group velocities and trajectories is present in Note 2 in the Supplementary Material.In terms of Fourier analysis, the spectrum of rigorous oscillation trajectory can be decomposed into the superposition of the SBO frequency ω SBOs and a continuum of higher swing spectrum components, where AðωÞ ¼ F ½Δn swing ðmÞ is the Fourier transform of the rapid swing trajectory.From the Fourier spectrum, we can further extract the ratio of the power spectrum for SBOs, and R swing ¼ 1 − R SBOs for the swing spectrum components.Meanwhile, we adopt the standard deviation to estimate the spectrum distribution of swing components beyond α, A larger standard deviation means that the spectrum contains more high-frequency swing components, indicating that SBOs contain more rapid swing oscillation details.

Experimental Realization of SBOs
To comprehensively study SBOs, we build two coupled fiberloop experimental platforms, as shown in Fig. 2. The initial optical pulse is prepared from the 1555 nm continuous wave laser beam by passing it through a Mach-Zehnder modulator (MZM), which generates a pulse with a width of ∼100 ns.Then, the pulse is injected into the long loop through an optical switch (OS) and circulated in the double-loop circuit by the central OC.To construct the artificial dc-and ac-driving electric field essential for the SBOs, the phase modulation 2ϕ u ðmÞ is implemented into the short loop by a PM, corresponding to the opposite phase modulations AEϕ u ðmÞ in the two loops, as required in the previous text.The evolution of pulse-train intensity is recorded at each step with photodiodes (PDs) and oscilloscopes (OSCs) by coupling the circulating pulses out of the loops.In detail, the two loops have an average length of ∼5 km and a length difference of ∼30 m, and the coupling ratio of the central OC is fixed at 75:25, which corresponds to β ¼ π∕3.Specifically, the erbium-doped fiber amplifiers (EDFAs) are inserted in the two loops to compensate for losses during pulse circulation.To overcome the transient response noise, a highpower 1538 nm continuous-wave control light is introduced before the EDFA and a bandpass filter (BPF) is used after the EDFA to remove the control light and the spontaneous emission noise.In addition, the polarization controllers (PCs) and polarization beam splitter (PBS) are utilized to control the polarization states of pulses.The acoustic optical modulators (AOMs) serve as intensity modulators in the two loops, which can absorb the optical signals after hundreds of circulations.All modulators in our setup, including MZM, AOM, and PM, are driven by the AWGs, which can be flexibly controlled and reconfigured in real time and are advantageous for the synthesis of arbitrary driving waveforms to realize generalized SBOs.
Then, we experimentally study the initial oscillation phase of SBOs. Figure 3(c) shows the initial oscillation phases φ SBOs versus the incident Bloch momentum k for two different ac-driving amplitudes E ω ¼ 1.8 and 5.3 between the collapse points.The oscillation phases show linear dependences on k, i.e., φ SBOs ¼ k − π by choosing N ¼ 1 and φ ¼ π∕2, also in agreement with the theoretical analysis in Eq. (8c).Note that there is a flip of oscillation direction as a π-phase jump as E ω crosses the SBO collapse point, which is the direct consequence of sign change of J 1 ðE ω Þ function.This π-phase jump feature can also be verified by the field evolutions in Figs.3(d) and 3(g), which show opposite initial oscillation directions with φ SBOs ¼ −0.47π and −0.03π for k ¼ π∕2 under E ω ¼ 1.8 and 5.3, respectively.
Next, we investigate the rapid swing oscillation details by calculating the Fourier spectrum of the measured packet's oscillation trajectory.Figure 4(a) shows the power ratio R SBOs of SBO frequency ω SBOs relative to all spectrum components as a function of the ac-driving amplitude.One can see a dip in the ratio R SBOs ¼ 0 for E ω ¼ 3.8, showing the clear signature of the occurrence of SBO collapse.By contrast, as E ω ¼ 3.8 is chosen far away from the collapse point, such as for E ω ¼ 1.8 and E ω ¼ 5.3, R SBOs approaches unity, meaning that SBOs are dominated by the slowly varying averaging trajectory and that the portion of rapid swing details is very tiny.Figure 4(b) shows the standard deviation σðωÞ of the Fourier spectrum for the rapid swing frequency components, which manifests as a monotonical increase with E ω , indicating that the presence of SBO collapse does not influence the spectral distribution for the fast-swing frequency components.Figures 4(c)-4(e) show the corresponding Fourier spectra for the measured SBO trajectories for E ω ¼ 1.8, 3.8, and 5.3.We can find that the peak at ω ¼ ω SBOs disappears in the spectrum for E ω ¼ 3.8, which can further validate the occurrence of the collapse of SBOs, while for E ω ¼ 1.8 and 5.3, the spectrum reaches the maximum at ω ¼ ω SBOs , meaning that the SBOs dominate by choosing far away from the SBO collapse point.

Generalized SBOs under an Arbitrary-Wave ac-Driving Field
While all previous studies on SBOs have been focused on the simplest sinusoidal ac-driving case, here we will show that SBOs can still persist even under an arbitrary-wave ac-driving field, which is dubbed generalized SBOs.Interestingly, generalized SBOs manifest different renormalization factors in the oscillation amplitude compared with the Bessel function factor for the sinusoidal driving case, which will lead to different collapse conditions.Below, we will choose two exemplified arbitrarywave ac-driving fields with rectangular and triangular waveforms for experimental demonstrations.
For rectangular wave driving, the waveform in one ac-driving period m ∈ ½0; M ac is given by By applying similar procedures to the sinusoidal driving case in Eqs. ( 3)-( 6), we can also calculate the time-averaging trajectory for the generalized SBOs (see Note 3 in the Supplementary Material, for detailed derivation), where the renormalization factor is given by Similarly, for a triangular-waveform driving we can also obtain a generic averaging oscillation trajectory of Eq. ( 14), with only fðE ω Þ replaced by where CðxÞ ¼ R x 0 cosðt 2 Þdt and SðxÞ ¼ For the two cases, the oscillation amplitude, frequency (period), and initial phase can be extracted from Eq. ( 14), By comparing Eqs.(18a)-(18c) with the sinusoidal-wave driving case in Eqs.(8a)-(8c), we can find that applying different driving waveforms does not change the oscillation frequency of SBOs but can modify the oscillation amplitude via a generic renormalization factor fðE ω Þ, which are given by J N ðE ω Þ and Eqs. ( 15) and ( 17) for the three waveforms.Meanwhile, the initial oscillation phase is still proportional to the incident Bloch momentum k.Likewise, as the driving amplitude E ω takes the roots of the fðE ω Þ function, the oscillation amplitude will also vanish, which can be referred to as the collapse of generalized SBOs.In particular, for rectangular-wave driving, to achieve fðE ω Þ ¼ 0, one requires πðE ω þ NÞ∕2 ¼ pπ (p is an integer) and E ω − N ≠ 0, which leads to E ω ¼ 2p − N, and p ≠ N.For example, for N ¼ 1, the collapse of SBOs occurs at an odd value of E ω ¼ 2p − 1 ¼ 3, 5; …; etc.
The theoretical analysis has also been verified by our experiments.Figure 5  the rectangular-and triangular-driving waveforms can achieve SBO collapses at weaker and stronger driving amplitudes.In Figs.5(c)-5(e), we display the measured packet evolutions for the three cases by choosing the first collapse point of E ω ¼ 3 for the rectangular-wave case.One can see that A SBOs only vanish for rectangular-wave driving but are nonzero with A SBOs ¼ 12.6 and 8.3 for the other two cases, which verifies the above theoretical analysis.

Applications in Beam Routing and Splitting Using SBO Collapse
In this section, we will exploit the collapse of SBOs to realize the temporal beam routing and splitting applications.Figure 6(a) shows the theoretical and measured packet oscillation displacements hΔn AE ðmÞi as a function of the driving amplitude E ω for upper and lower band excitations, which exhibit the flip of oscillation directions by crossing the collapse point of SBOs at E ω ¼ 3.8.To achieve beam routing, we just need to switch between these collapse points by choosing a weaker or stronger driving amplitude.As depicted in Figs.6(b) and 6(c), by choosing E ω ¼ 1.8 and E ω ¼ 5.3 and exciting only from the upper band at k ¼ π∕2, we can achieve the rightward and leftward beam routing with hΔn AE ðmÞi > 0 and hΔn AE ðmÞi < 0, respectively.The routing directions can also be exchanged by exciting from the lower band.Furthermore, if we simultaneously excite the upper and lower bands, we can also realize the temporal beam splitting.Figures 6(d) and 6(e) show the measured beam evolutions for the case of E ω ¼ 1.8 and E ω ¼ 5.3, where the excitation power ratio of upper and lower bands is fixed at 65/35.Interestingly, the power ratios of the two split beams can be switched with each other by switching between weak-and strong-driving regimes to cross the collapse point.

Conclusion
We have experimentally demonstrated the collapse of SBOs in the strong-driving regime based on a photonic temporal lattice system.For a sinusoidal ac-driving, we have shown that as the amplitude-to-frequency ratio of the ac-driving field takes the root of the first-order Bessel function, the SBO collapse occurs, manifesting as a complete inhibition of oscillation with a vanishing oscillation amplitude as well as the flip of the initial oscillation direction by crossing the collapse point.Fourier analysis for the instantaneous oscillation trajectory was performed to probe the occurrence of SBO collapse.By replacing the sinusoidal ac-driving with arbitrary-wave ac-driving fields, we also achieved generalized SBOs, which possess different renormalization factors for the oscillation amplitude and hence the distinct collapse conditions.Finally, by switching the driving amplitude between SBO collapse points, we demonstrated experimentally the rightward and leftward beam routing.Thanks to the dual-band nature of the temporal mesh lattice, the temporal beam splitting functionality was also demonstrated by the multiband excitation of the lattice.Our work reported on the first experimental observation of SBO collapse and extended SBOs to the arbitrary-wave ac-driving situations.This paradigm may find potential applications in temporal pulse reshaping and coherent oscillation control for optical communications and signal processing.

Fig. 1
Fig. 1 Principle of SBOs in electric-field-driven synthetic temporal lattices.(a) Two fiber loops with slightly different lengths are connected by an OC to construct the temporal lattice.The incorporated PMs in the short and long loops can introduce the step-dependent phase shifts of ϕ u ðmÞ and ϕ v ðmÞ.(b) Schematic of the synthetic temporal lattice mapped from the pulse evolution in two coupled fiber loops in panel (a) and the sketch of SBO trajectory denoted by the red curve.The purple curves denote the required dc-and ac-driving electric fields to induce SBOs, which are created simultaneously by imposing opposite phase shifts of ϕ u ðmÞ and ϕ v ðmÞ in short and long loops.(c) Effective time-averaging band structure of the synthetic temporal lattice at E ω ¼ 1.8, 3.8, and 5.3.The dashed curve is the original band structure without electric-field driving, where δ is the frequency detuning between the dc-and ac-driving electric field.

Fig. 3
Fig. 3 Simulated and measured results of SBOs.(a) SBO oscillation amplitude A SBOs as a function of the ac-driving amplitude E ω and the inverse frequency detuning 1∕δ.The golden spheres represent the measured results.(b) SBO oscillation period M SBOs as a function of the inverse frequency detuning 1∕δ.The curve and squares denote the calculated and measured results, respectively.The inset figure shows M SBOs as a function of δ.(c) Initial oscillation phase of SBOs versus the initial Bloch momentum k for E ω ¼ 1.8, N ¼ 1, φ ¼ π∕2, and δ ¼ π∕150.The solid curves and spheres denote the theoretical and experimental results, respectively.(d)-(g) Measured pulse intensity evolutions for E ω ¼ 1.8, 3, 3.8, and 5.3 under δ ¼ π∕150.The white solid curves denote the averaging SBO oscillation trajectories hΔn AE ðmÞi obtained by fitting from the experimental results using the cosine function.The blue and green lines denote the SBO oscillation amplitude A SBOs and period M SBOs , respectively.(h) Experimental pulse intensity evolution for E ω ¼ 5.3 and δ ¼ π∕90.
(b) shows the measured A SBOs versus E ω under the three driving waveforms, all of which can vanish with A SBOs ¼ 0 under certain driving amplitude E ω but showing different collapse positions.Compared with the sinusoidal case,

Fig. 4
Fig. 4 Fourier spectrum of SBOs.(a) The power ratio of SBOs with respect to all Fourier spectrum components as a function of the ac-driving amplitude E ω .The solid curve and spheres denote the theoretical and experimental results, respectively.(b) The standard deviation of the Fourier spectrum for ω > α varying with E ω .(c)-(e) Fourier spectra of measured SBO trajectories at E ω ¼ 1.8, 3.8, and 5.3.

Fig. 5
Fig. 5 Generalized SBOs under arbitrary-wave ac-driving fields.(a) Schematic of the sinusoidal-, rectangular-, and triangular-wave ac-driving electric fields.(b) SBO oscillation amplitude A SBOs versus the ac-driving amplitude E ω under the sinusoidal-, rectangular-, and triangular-wave driving.The solid curves and spheres denote the theoretical and experimental results, respectively.(c)-(e) Measured pulse intensity evolutions under sinusoidal-, rectangular-, and triangular-wave driving, respectively.The ac-driving amplitude is taken as the collapse point for the rectangularwave driving of E ω ¼ 3.

Fig. 6
Fig. 6 Application of beam routing and splitting based on SBO collapse.(a) Packet oscillation displacements hΔn AE ðmÞi as a function of the driving amplitude E ω for upper and lower band excitations.The solid curves and spheres represent the theoretical and experimental results, respectively.(b), (c) Measured pulse intensity evolutions for the upper band excitation at E ω ¼ 1.8 and 5.3, respectively.(d), (e) Measured pulse intensity evolutions for the simultaneous excitation of upper and lower bands with the power ratio of 65/35 under the ac-driving amplitudes of E ω ¼ 1.8 and 5.3, respectively.