2.1.

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,^30–44 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 ${\varphi }_u(m)$ and ${\varphi }_v(m)$ along the leftward and rightward paths. The pulse dynamics in the lattice is governed by the following coupled-mode equations:

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 ${\varphi }_u(m)$ and ${\varphi }_v(m)$. (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 ${\varphi }_u(m)$ and ${\varphi }_v(m)$ in short and long loops. (c) Effective time-averaging band structure of the synthetic temporal lattice at $E_{\omega }=1.8$, 3.8, and 5.3. The dashed curve is the original band structure without electric-field driving, where $\delta$ is the frequency detuning between the dc- and ac-driving electric field.

To induce SBOs, we need to apply simultaneously a detuned dc and ac electric field in the lattice, $E_{\text{eff}}(m)=\alpha +E_{\mathrm{ac}}\sin ({\omega }_{\mathrm{ac}}m+\phi )$, where $\alpha$ is the magnitude of dc electric field, corresponding to the BO frequency ${\omega }_{\mathrm{BOs}}=\alpha$, and $E_{\mathrm{ac}}$, ${\omega }_{\mathrm{ac}}$, and $\phi$ 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), $\alpha =N{\omega }_{\mathrm{ac}}$, 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.^7,8 Under specific ac-driving amplitude satisfying $J_0(E_{\omega })=0$, destructive suppression of lattice’s hopping occurs, leading to the dynamic localization effect, where $E_{\omega }=E_{\mathrm{ac}}/{\omega }_{\mathrm{ac}}$ is the normalized ac-driving amplitude denoting the amplitude-to-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., $\alpha =N{\omega }_{\mathrm{ac}}+\delta$, where $\delta$ is the frequency detuning. Under the dc and ac electric fields, the effective vector potential evolves as $A_{\text{eff}}(m)=-{\int }_0^m{E}_{\text{eff}}(m)\text{d}m=-\alpha m+E_{\omega }\cos ({\omega }_{ac}m+\phi )$, 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 $A_{\text{eff}}(m)$ by applying opposite phase modulations into the two fiber loops, ${\varphi }_u(m)=-A_{\mathrm{eff}}(m)$ and ${\varphi }_v(m)=A_{\mathrm{eff}}(m)$.^25,49,50 The vector potential is assumed to vary slowly with $m$, so that Zener tunneling between the two quasi-energy bands is negligible. Moreover, both ${\omega }_{\mathrm{ac}}$ and $\alpha$ are assumed to be fractional integers than $2\pi$ to ensure periodicity of the dynamics after a full cycle. The instantaneous Bloch momentum is given by

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 $\delta$ 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.^10,12 In 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

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 $J_N(E_{\omega })$. Specifically, as $E_{\omega }$ takes each root of $J_N$ function, the oscillation amplitude will vanish, $A_{\text{SBOs}}=0$, 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 $⟨\mathrm{\Delta }{n}_{\pm }(m)⟩$ as $E_{\omega }$ 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 $⟨\mathrm{\Delta }{n}_{\pm }(m)⟩$ 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),

A larger standard deviation means that the spectrum contains more high-frequency swing components, indicating that SBOs contain more rapid swing oscillation details.

2.2.

Experimental Realization of SBOs

To comprehensively study SBOs, we build two coupled fiber-loop 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 $\sim 100\mathrm{ns}$. 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 $2{\varphi }_u(m)$ is implemented into the short loop by a PM, corresponding to the opposite phase modulations $\pm {\varphi }_u(m)$ 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 $\sim 5\mathrm{km}$ and a length difference of $\sim 30\mathrm{m}$, and the coupling ratio of the central OC is fixed at 75:25, which corresponds to $\beta =\pi /3$. 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 high-power 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.

Fig. 2

Experimental setup. Panels (a) and (b) denote the long and short loops, respectively. All optical and electric components are as follows: PC, polarization controller; MZM, Mach–Zehnder modulator; OS, optical switch; OC, optical coupler; SMF, single-mode fiber; EDFA, erbium-doped fiber amplifier; VOA, variable optical attenuator; BPF, bandpass filter; AOM, acoustic optical modulator; PD, photodiode; OSC, oscilloscope; PBS, polarization beam splitter; PM, phase modulator; AWG, arbitrary waveform generator.

Figure 3(a) shows the measured SBO oscillation amplitude $A_{\text{SBOs}}$ as the function of the ac-driving amplitude $E_{\omega }$ and the inverse of frequency detuning $1/\delta$. The other parameters are chosen as ${\omega }_{\mathrm{ac}}=\pi /30$, $\phi =\pi /2$, and $\alpha =N{\omega }_{\mathrm{ac}}+\delta$, where $N=1$. The input Bloch wave packet is excited from the upper band with central carrier Bloch momentum $k=\pi /2$. To meet the high-frequency ac-driving condition, we choose $\delta =\pi /90$, $\pi /120$, and $\pi /150$ in our experiment, which satisfy ${\omega }_{\mathrm{ac}}=3\mathit{\delta }-5\mathit{\delta }$. For a fixed frequency detuning $\delta =\pi /150$, the oscillation amplitude $A_{\text{SBOs}}$ increases in the weak-driving regime and decreases in the strong-driven regime, which precisely follows the $J_1(E_{\omega })$ function, as given by the theoretical prediction in Eq. (8a). The measured packet intensity evolutions for four different ac-driving amplitudes $E_{\omega }=1.8$, 3, 3.8, and 5.3 cases are shown in Figs. 3(d)–3(g), which possess $A_{\mathrm{SBOs} }=14.3$, 8.2, 0, and 8.3. Note that for $E_{\omega }=3.8$, i.e., the first root of $J_1(E_{\omega })$ function is reached, the oscillation amplitude vanishes (denoted by green arrow), clearly showing the collapse of SBOs. On the other hand, for a fixed $E_{\omega }$, we can get $A_{\text{SBOs}}=8.3$ for $\delta =\pi /150$ and $A_{\text{SBOs}}=4.9$ for $\delta =\pi /90$, as shown in Figs. 3(g) and 3(h), also verifying the inverse dependence of $A_{\text{SBOs}}$ on $\mathit{\delta }$ as given in Eq. (8a). The oscillation period $M_{\text{SBOs}}$ versus $1/\delta$ is depicted in Fig. 3(b), which is also inversely proportional to $\mathit{\delta }$ described in Eq. (8b). This inverse dependence law can also be evidenced in Figs. 3(g) and 3(h), which show $M_{\text{SBOs}}=289$ for $\delta =\pi /150$ and $M_{\text{SBOs}}=178$ for $\delta =\pi /90$.

Fig. 3

Simulated and measured results of SBOs. (a) SBO oscillation amplitude A_SBOs as a function of the ac-driving amplitude $E_{\omega }$ and the inverse frequency detuning $1/\mathit{\delta }$. The golden spheres represent the measured results. (b) SBO oscillation period M_SBOs as a function of the inverse frequency detuning $1/\delta$. The curve and squares denote the calculated and measured results, respectively. The inset figure shows $M_{\text{SBOs}}$ as a function of $\mathit{\delta }$. (c) Initial oscillation phase of SBOs versus the initial Bloch momentum $k$ for $E_{\omega }=1.8$, $N=1$, $\phi =\pi /2$, and $\delta =\pi /150$. The solid curves and spheres denote the theoretical and experimental results, respectively. (d)–(g) Measured pulse intensity evolutions for $E_{\omega }=1.8$, 3, 3.8, and 5.3 under $\delta =\pi /150$. The white solid curves denote the averaging SBO oscillation trajectories $⟨\mathrm{\Delta }{n}_{\pm }(m)⟩$ obtained by fitting from the experimental results using the cosine function. The blue and green lines denote the SBO oscillation amplitude $A_{\text{SBOs}}$ and period $M_{\text{SBOs}}$, respectively. (h) Experimental pulse intensity evolution for $E_{\omega }=5.3$ and $\delta =\pi /90$.

Then, we experimentally study the initial oscillation phase of SBOs. Figure 3(c) shows the initial oscillation phases ${\phi }_{\mathrm{SBOs}}$ versus the incident Bloch momentum $k$ for two different ac-driving amplitudes $E_{\omega }=1.8$ and 5.3 between the collapse points. The oscillation phases show linear dependences on $k$, i.e., ${\phi }_{\mathrm{SBOs} }=k-\pi$ by choosing $N=1$ and $\phi =\pi /2$, also in agreement with the theoretical analysis in Eq. (8c). Note that there is a flip of oscillation direction as a $\pi$-phase jump as $E_{\omega }$ crosses the SBO collapse point, which is the direct consequence of sign change of $J_1(E_{\omega })$ function. This $\pi$-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 ${\phi }_{\mathrm{SBO}s}=-0.47\pi$ and $-0.03\pi$ for $k=\pi /2$ under $E_{\omega }=1.8$ 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 $R_{\text{SBOs}}$ of SBO frequency ${\omega }_{\mathrm{SBOs}}$ relative to all spectrum components as a function of the ac-driving amplitude. One can see a dip in the ratio $R_{\text{SBOs}}=0$ for $E_{\omega }=3.8$, showing the clear signature of the occurrence of SBO collapse. By contrast, as $E_{\omega }=3.8$ is chosen far away from the collapse point, such as for $E_{\omega }=1.8$ and $E_{\omega }=5.3$, $R_{\text{SBOs}}$ 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 $\sigma (\omega )$ of the Fourier spectrum for the rapid swing frequency components, which manifests as a monotonical increase with $E_{\omega }$, 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 $E_{\omega }=1.8$, 3.8, and 5.3. We can find that the peak at ${\omega }_{ }={\omega }_{\mathrm{SBOs}}$ disappears in the spectrum for $E_{\omega }=3.8$, which can further validate the occurrence of the collapse of SBOs, while for $E_{\omega }=1.8$ and 5.3, the spectrum reaches the maximum at ${\omega }_{ }={\omega }_{\mathrm{SBOs}}$, meaning that the SBOs dominate by choosing far away from the SBO collapse point.

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 $E_{\omega }$. The solid curve and spheres denote the theoretical and experimental results, respectively. (b) The standard deviation of the Fourier spectrum for $\omega >\alpha$ varying with $E_{\omega }$. (c)–(e) Fourier spectra of measured SBO trajectories at $E_{\omega }=1.8$, 3.8, and 5.3.

2.3.

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 arbitrary-wave ac-driving fields with rectangular and triangular waveforms for experimental demonstrations.

For rectangular wave driving, the waveform in one ac-driving period $m\in [0,M_{ac}]$ is given by

By applying similar procedures to the sinusoidal driving case in Eqs. (36), we can also calculate the time-averaging trajectory for the generalized SBOs (see Note 3 in the Supplementary Material, for detailed derivation),

Similarly, for a triangular-waveform driving

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 $f(E_{\omega })$, which are given by $J_N(E_{\omega })$ 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 $E_{\omega }$ takes the roots of the $f(E_{\omega })$ 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 $f(E_{\omega })=0$, one requires $\pi (E_{\omega }+N)/2=p\pi$ ($p$ is an integer) and $E_{\omega }-N\ne 0$, which leads to $E_{\omega }=2p-N$, and $p\ne N$. For example, for $N=1$, the collapse of SBOs occurs at an odd value of $E_{\omega }=2p-1=\mathrm{3,}5,\dots ,\mathrm{etc}$.

The theoretical analysis has also been verified by our experiments. Figure 5(b) shows the measured A_SBOs versus $E_{\omega }$ under the three driving waveforms, all of which can vanish with $A_{\text{SBOs}}=0$ under certain driving amplitude $E_{\omega }$ but showing different collapse positions. Compared with the sinusoidal case, 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 $E_{\omega }=3$ for the rectangular-wave case. One can see that $A_{\text{SBOs}}$ only vanish for rectangular-wave driving but are nonzero with $A_{\text{SBOs}}=12.6$ and 8.3 for the other two cases, which verifies the above theoretical analysis.

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 $A_{\text{SBOs}}$ versus the ac-driving amplitude $E_{\omega }$ 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 rectangular-wave driving of $E_{\omega }=3$.

2.4.

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 $⟨\mathrm{\Delta }{n}_{\pm }(m)⟩$ as a function of the driving amplitude $E_{\omega }$ for upper and lower band excitations, which exhibit the flip of oscillation directions by crossing the collapse point of SBOs at $E_{\omega }=3.8$. 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 $E_{\omega }=1.8$ and $E_{\omega }=5.3$ and exciting only from the upper band at $k=\pi /2$, we can achieve the rightward and leftward beam routing with $⟨\mathrm{\Delta }{n}_{\pm }(m)⟩>0$ and $⟨\mathrm{\Delta }{n}_{\pm }(m)⟩<0$, 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 $E_{\omega }=1.8$ and $E_{\omega }=5.3$, 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.

Fig. 6

Application of beam routing and splitting based on SBO collapse. (a) Packet oscillation displacements $⟨\mathrm{\Delta }{n}_{\pm }(m)⟩$ as a function of the driving amplitude $E_{\omega }$ 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 $E_{\omega }=1.8$ 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 $E_{\omega }=1.8$ and 5.3, respectively.