Visual observation of optical Floquet-Bloch oscillations

Bloch oscillations, an important transport phenomenon, have extensively been studied in static systems but remain largely unexplored in Floquet systems. Here, we propose a new type of Bloch oscillations, namely the"Floquet-Bloch oscillations,"which refer to rescaled Bloch oscillations with a period of extended least common multiple of the modulation and Bloch periods. We report the first visual observation of such Floquet-Bloch oscillations in femtosecond-laser-written waveguide arrays by using waveguide fluorescence microscopy. These Floquet-Bloch oscillations exhibit exotic properties, such as fractal spectrum and fractional Floquet tunneling. This new transport mechanism offers an intriguing method of wave manipulation, which has significant applications in coherent quantum transport.


INTRODUCTION
As a fundamental phenomenon of coherent quantum motion, Bloch oscillations, the oscillatory motion of a quantum particle with a BO period ɅBO, were first predicted by Bloch and Zener in the context of crystal under a constant electric field (1,2).Nevertheless, Bloch oscillations have never been experimentally observed in natural crystals owing to electron-phonon interactions.
SBOs refer to rescaled BOs with super large oscillation amplitude and period, where the BO period ɅBO (or its integer multiple) is slightly detuned from the modulation period ɅFL, i.e., ɅFL ~ NɅBO.
While BOs in Floquet systems with several specific cases have been investigated, a general phenomenon concerning BOs in Floquet systems and the corresponding experimental observation remain largely elusive.
In this article, we explore the optical Bloch oscillations in Floquet systems and draw two essential conclusions: (1) Floquet lattice with a period ɅFL in a linear tilted potential leading to BOs with a period ɅBO can also be mapped onto another Floquet lattice with a period ɅFBO of the extended least common multiple (LCM) of ɅFL and ɅBO; (2) When ɅFL ≠ NɅBO, the Floquet-Bloch oscillations occur with a FBO period of LCM(ɅFL, ɅBO); when ɅFL = NɅBO, spreading usually occurs.We emphasize that all the above conclusions are tenable for arbitrary Floquet engineering with a rational ratio of ɅBO/ɅFL.Therefore, Floquet-Bloch oscillations are a unified phenomenon of the existing Bloch oscillations, namely super-Bloch oscillations (ɅFL ~ NɅBO) and quasi-Bloch oscillations (ɅBO = NɅFL).
We experimentally verified our prediction in one-dimensional curved waveguide arrays fabricated with femtosecond laser writing technology.With waveguide fluorescence microscopy, we directly visualized the breathing and oscillatory motions of Floquet-Bloch oscillations.We provided a detailed analysis of FBOs, and investigated fractal spectrum and fractional Floquet tunneling.More specifically, we found that the FBO period ɅFBO is the Thomae's function (a fractal spectrum) of the ratio ɅBO/ɅFL, and several peaks of such a fractal spectrum were experimentally confirmed.In addition, the modulation-induced rescaling of the FBO amplitude depends largely on the ratio ɅBO/ɅFL, which refers to fractional Floquet tunneling.By varying the amplitude of harmonic modulation, we experimentally demonstrated that such rescaling of FBO amplitude follows a linear combination of fractional-order Anger and Weber functions.Our demonstration provides a promising method for controlling wave transport in photonics with potential applications in self-imaging, optical communication, and photonic quantum simulations.

Theory of Bloch oscillations in a Floquet lattice
Here, we employ a femtosecond-laser-written waveguide array in a fused silica substrate (Corning 7980) as an experimental platform for visualizing optical BOs in a Floquet systems (39)(40)(41).As depicted in Fig. 1A, we first considered a curved photonic lattice that consists of identical waveguides with waveguide spacing d.In the transverse direction x, the center of each waveguide core varies along the longitudinal direction z by following a combined trajectory according to x0(z) = xBO(z) + xFL(z), where xBO(z) is the circular bending term with a bend radius R (R >> z) and xFL(z) = M(z) is the periodic bending term with a modulation period ɅFL and modulation function M(z) that satisfies M(z) = M(z + ɅFL).In the case of paraxial propagation along the longitudinal direction z, the envelope ψ(x, y, z) of the optical field guided in this photonic lattice at operating wavelength λ is governed by the Schrödinger-type equation: where is the Laplacian operator in the transverse plane, n0 ~ 1.46 is the refractive index of the substrate, k0 = 2πn0/λ is the wave number, and Δn(x, y, z) = n(x, y, z) -n0 is the femtosecond-laser-induced refractive-index increase (Δn > 0) that defines the entire photonic lattice.By considering a reference coordinate frame where the waveguides are straight in the z̃ direction, namely: x = x + x 0 (z) , y = y , and z̃ = z , the paraxial equation in the transformed coordinates can be expressed as with ψ = ψ(x , y , z)exp - Figure 1C displays the cross-sectional microscope image of a fabricated sample.Each waveguide in our sample supports a well-confined fundamental mode, allowing the application of nearest-neighbor tight-binding approximation, so the propagation of guided light can be described by the following set of coupled equations: where am is the amplitude of guided mode |m⟩ in the m th waveguide and c0 is the coupling constant between the nearest-neighbor waveguides.In the absence of force F(z), i.
and the

Houston function
where is the corresponding Floquet dispersion that provides the effective transport properties over a period ɅFBO.Under the single band approximation, the Floquet dispersion is expressed as where Equation (5) implies that there are two possibilities for optical Bloch oscillations in a Floquet system.When ɅFL ≠ NɅBO, a complete cancellation of all orders of diffraction = 0 results in flat Floquet dispersion ε(kx) ≡ 0, indicating that the state experiences a periodic motion and returns to the initial state after propagating a period ɅFBO.We call this phenomenon "Floquet-Bloch oscillations," because it is a combined phenomenon of Floquet engineering and Bloch oscillation.
is in general no longer flat and the state experiences spreading.We emphasize that the above conclusions are valid for an arbitrary modulation function M(z).In this connection, the existing BOs under harmonic modulation, namely QBOs (ɅBO = NɅFL) and SBOs (ɅFL ~ NɅBO), are the specific cases of FBOs.

Visual observation of Bloch oscillations in a Floquet lattice
To illustrate the similarity and difference between Floquet-Bloch oscillations and the existing Bloch oscillations, we employed a harmonic modulation M(z) = Acos(2πz/ɅFL) (see Fig. 1D) with modulation amplitude A. Without loss of generality, we considered four specific scenarios that correspond to typical BOs (A = 0), QBOs (ɅBO/ɅFL = 3), SBO-like oscillations (ɅBO/ɅFL = 4/3), and spreading (ɅBO/ɅFL = 1).The corresponding shifts of the transverse Bloch momentum k x (z) = 1F, where the harmonic modulation contributes a sub-oscillation to the states with Bloch-momentum-oscillation amplitude (2πAk0)/ɅFL.
In the latter three scenarios, we considered the modulation amplitude A = A0ɅFL/ɅBO so that the sub-oscillation amplitude was normalized to (2πA0k0)/ɅBO.To experimentally verify our prediction, we fabricated a set of 90-mm-long samples composed of 31 identical waveguides with a waveguide spacing d = 16 μm.With such a waveguide spacing d, the coupling coefficient between straight waveguides c0 ~ 1.45 cm −1 was experimentally characterized.These waveguides follow the combined trajectories having a bend radius R = 110.8cm (corresponding to ɅBO ~ 30 mm) and the modulation period ɅFL = 10, 22.5, and 30 mm (corresponding to the ratios ɅBO/ɅFL = 3, 4/3, and 1, respectively).With the considered modulation period, A0 = 18 μm was chosen to reduce the associated radiation losses of waveguides.Further details of the fabrication processes are provided in the Supplementary Materials.
Similar to the existing Bloch oscillations, Floquet-Bloch oscillations exhibit a breathing and an oscillatory motion under a single-site excitation and a broad-beam excitation, respectively.In the following experiments, we implemented visible-light excitation (λ = 633 nm) and directly visualized both the breathing modes and oscillating modes of Floquet-Bloch oscillations by using waveguide fluorescence microscopy (see the Supplementary Materials).A coordinate transformation that maps circular arcs into straight lines was applied to digitally process the fluorescence image so that the light evolution could be visualized more intuitively.
First, we focus on the breathing modes under a single-site excitation.The narrow excitation in the real space corresponds to a broad excitation of Bloch modes in the reciprocal space, resulting in strongly diffracting wave packets.To quantify the diffraction of wave packets for the single-site excitation, we define the variance of excitation at the distance z in such a discrete system as The light is initially excited in the central waveguide resulting in a vanishing variance σ 2 (0) = 0, and a rise of the variance indicates that the light experiences broadening.Under the single-site excitation, the experimental results, respective simulations, and extracted variances σ 2 (z) for the scenarios considered in Fig. 1F are summarized in Fig. 2, where Figs. 2 (A to C), (D to F), (G to I), (J to H) corresponding to typical BOs, QBOs, SBO-like oscillations, and spreading, respectively.
Without modulation (A = 0), Figs. 2 (A and B) displays the light evolution that corresponds to typical BOs, where the measured BO period ~30 mm is consistent with its theoretical value ɅBO = Rλ/(n0d).The light first broadens until it propagates half of the BO period and then focuses into the central waveguide again at the BO period, as σ 2 reaches its maximum at z ~ 15 mm and then decreases to zero at z ~ 30 mm (see Fig. 2C).When the modulation is introduced, Bloch oscillations in the Floquet lattice exhibit diverse transport properties as expected, where the ratio ɅBO/ɅFL makes a significant difference.For ɅBO/ɅFL = 3, the FBOs are observed and reduce into QBOs, where the FBO period ɅFBO is equal to the BO period ɅBO (see Figs. 2D and 2E).The QBOs pattern is basically similar to that of typical BOs, except that light experiences additional sub-oscillations, as σ 2 oscillates with dual periods (see Fig. 2F).For ɅBO/ɅFL = 4/3, the FBOs exhibit their similarity to SBOs, where the FBO period ɅFBO ~ 90 mm is much longer than the BO period ɅBO (see Figs. 2G and 2H).In addition to the extended FBO period, we also observed dramatic broadening of the light, as the maximum of σ 2 is far larger than that of typical BOs (see Fig. 2I).For ɅBO/ɅFL = 1, the evolution of light propagating from 0 to ɅFBO/2 cannot be cancelled with that propagating from ɅFBO/2 to ɅFBO.As a result, the condition for FBOs is destroyed and spreading occurs, where light exhibits ballistic spreading and is no longer localized (see Figs. 2J and 2K).The typical discrete diffraction pattern accompanied by oscillations is observed, as σ 2 oscillates around the gray-dashed where B1 is the first-order Bessel function (see Fig. 2L).
Next, we focus on the oscillation modes under a broad-beam excitation.The broad-beam excitation in the real space corresponds to a narrow excitation in the reciprocal space.In this case, the group velocity of beam motion in the lattices can be expressed as Vgroup(z) = −dβ(z)/dkx(z) = 2dc0sin(kx(z)d), and the transverse displacement Δx(z) of beam center is determined by Δx(z) = Here we define the weighted-average position of excitation at the distance z in such a discrete system as The excitation is located at the center of the lattice, i.e., x(0) = 0.During propagation, a rise (drop) of x(z) indicates that the light shifts toward the x (−x) direction.Here, we launched a 7-waveguidewide Gaussian beam at normal incidence to the edge of the substrate.This corresponds to a narrow spectrum centered at kx(0) = 0 in the reciprocal space.Under the broad excitation, the experimental results, respective simulations, and extracted trajectories of the beam x(z) (white dashed lines) for the scenarios considered in Fig. 1F are summarized in Fig. 3, where Fig. 3 (A and B), (C and D), (E and F), and (G and H) correspond to BOs, QBOs, SBOs-like oscillations, and spreading, respectively.Without modulation (A = 0), Fig. 3 (A and B) display the light evolution that corresponds to typical BOs, where the broad beam undergoes a sinusoidal oscillation with the BO period ɅBO.Similar to the breathing motion discussed previously, the oscillating motion exhibits diverse transport properties when the modulation is introduced.For ɅBO/ɅFL = 3, Fig. 3 (C and D) display the light evolution that corresponds to QBOs, where the trajectory of the broad beam follows a doubly oscillating function.The broad beam evolves along the x direction and returns to the initial position after propagating any multiple of the BO period ɅBO ~ 30 mm.For ɅBO/ɅFL = 4/3, Fig. 3 (E and F) display the light evolution that corresponds to SBOs-like oscillations, where the trajectory of the broad beam follows a giant doubly oscillating function with an extended period of ~90 mm.The maximal displacement of the broad beam for SBOs-like oscillations is observed at half of the FBO period, i.e., z ~ 45 mm.For ɅBO/ɅFL = 1, Fig. 3 (G and H) display the light evolution that corresponds to spreading.Although the trajectory of the broad beam follows an oscillating function, beam broadening is observed during propagation.As a result, the beam does not return to the initial state of excitation and the condition for FBOs is destroyed.
For both single-site and broad-beam excitations, the visual observations of fluorescence images and quantitative analyses have excellent agreement with the respective simulation results.
Therefore, our waveguide arrays are capable of accurately revealing BOs in Floquet lattices.

Fractal spectrum and fractional Floquet tunneling
We also made a quantitative analysis of FBOs.Firstly, we studied the dependence of ɅBO/ɅFBO on ɅBO/ɅFL.As shown in Fig. 4A, the theoretically predicted FBO period ɅFBO = LCM(ɅBO, ɅFL) determines that the FBO period spectrum follows the Thomae's function.One may find that the Thomae's function is a fractal structure composed of infinite discrete peaks, where the patterns exhibit self-similarity at increasingly smaller scales.Owing to limited sample lengths, we fabricated a set of samples with ɅBO/ɅFBO ≥ 1/6, fixed ɅBO = 30 mm, and varied ɅFL from 15 to 30 mm.As expected, we experimentally verified several peaks of such a fractal spectrum by fitting the measured and simulated variance σ 2 (z) under single-site excitation (see the Supplementary Materials).This fractal spectrum demonstrates the relationship between FBOs and SBOs.When ɅBO/ɅFL approaches 1, the Thomae's function can be approximated to a continuous linear function, implying that the FBOs reduce into SBOs with a period ɅFBO = ɅFLɅBO/(ɅBO − ɅFL).Moreover, the existence of FBOs is confirmed even for a large ɅBO/ɅFL, which goes far beyond SBOs.In agreement with theoretical prediction, such a spectrum reveals the fractal nature of the FBOs.
Secondly, under single-site excitations, we defined the FBO amplitude as σ 2 (ɅFBO/2) and studied the dependence of FBO amplitude on modulation amplitude A. We found that the harmonic modulation leads to a rescaling of FBO amplitude following the square of a linear combination of the Anger function J v

DISCUSSION
In summary, we report the first visual observation of optical BOs in a Floquet lattice and the investigation of Floquet-Bloch oscillations.In addition to the above-discussed cases with a harmonic modulation, we emphasize that Floquet-Bloch oscillations occur for arbitrary Floquet engineering M(z) and the corresponding experimental results are provided in the Supplementary Materials.The visual observation of Floquet-Bloch oscillations is a key to understanding the underlying transport mechanism, which has a significant impact on both fundamental research and practical applications.For fundamental research, our theoretical and experimental work enables the exploration of a branch of fundamental phenomena involving FBOs, such as the interplay between FBOs and binary lattices (16), non-Hermitian lattices (42), and optical nonlinearity (43).
For practical applications, the demonstrated manipulation of optical waves can be implemented in synthetic dimensions of time (44), frequency (45), and angular momenta (46,47), leading to applications in high-efficiency frequency conversion and signal processing.

ik0
2π z̃x (z)x -ik0 4π ∫ [ z̃x (τ)] τ z̃ and F(z) = -n ∂ z2 0 (z) .The additional term F(z) is determined by the combined trajectory and can be separated into two terms, i.e., F(z) = F BO + F FL , with FBO = n0/R and F FL = -n ∂ z2 M(z).Equation(2) indicates that the propagation of low-power light in the proposed lattice is analogous to the temporal evolution of noninteracting electrons in a periodic potential subject to an electric field, where the spatial coordinate z̃ acts as time t, F(z) plays the role of the electric field force F(t), and the refractive index profile [Δn(x , y , z) + F(z)x ] is related to the sign-reversed driven potential -V(t, x ).As sketched in Fig.1B, the effective potential -(Δn(x , y , z) + F FL x ) refers to a Floquet lattice, and the linear potential gradient FBO gives rise to BOs with a period ɅBO = λR/(n0d).As a result, our proposed scheme provides an experimental realization of optical Bloch oscillations in a Floquet lattice.

1 √N∑
e., for straight waveguide arrays, inserting a Bloch function ψ m,k x = e ik x dm |m m yields the single-band dispersion β(kx) = 2c0cos(kxd) (blue line in Fig. 1E), where β(kx) denotes the quasienergy and kx denotes the transverse Bloch momentum.According to the generalized acceleration theory (29), the presence of force F(z) leads to a shift of the transverse Bloch momentum k x is the reconstructed solution (see the Supplementary Materials).When PɅBO = QɅFL (Q, P are mutually prime integers), the extended least common multiple (LCM) of ɅBO and ɅFL is defined as LCM(ɅBO, ɅFL) = PɅBO = QɅFL, and β[kx(z)] is a z-periodic function with a period ɅFBO = LCM(ɅFL, ɅBO) (see the Supplementary Materials).Consequently, the integral of β[kx(z)] can be expressed as a sum of a linear function and a periodic function, i.e., ∫ β[k x (τ)]dτ z 0 = ε(k x )z + P(z) with P(z) = P(z + ɅFBO).As a result, the entire lattice can be mapped onto another Floquet lattice, since the Houston function can be reduced to Floquet states as order v = ɅFL/ɅBO, which refers to fractional Floquet tunneling (see the Supplementary Materials).Note that the rescaling of FBO amplitude depends largely on ɅBO/ɅFL, which provides a flexible way to manipulate the light.Figure4Bdisplays two examples of such Floquet tunneling, including QBOs (red line, ɅBO/ɅFL = 3) and SBOs-like oscillations (blue line, ɅBO/ɅFL = 4/3).Each curve is normalized to unity at its maximum.For the QBOs, the theoretically predicted FBO amplitude has a characteristic 2cos(π/3)E 1/3 -A/ɅFL.By contrast, the Floquet tunneling for the SBOs-like oscillations exhibits a different behavior, where the FBO amplitude has a characteristic 8J 3A/ɅFL.To verify our prediction, we fabricated two sets of samples with a varied modulation amplitude A and extracted the corresponding variance σ 2 (z) from the measured fluorescence images.As expected, one finds that the measured FBO amplitude has excellent agreement with its theoretical prediction.For the QBOs, with increasing amplitude A the FBO amplitude decreases before it reaches zero, indicating that the introduction of harmonic modulation will not broaden the FBO amplitude compared with the typical BOs (A = 0).For the SBOs-like oscillations, with increasing amplitude A the FBO amplitude first increases to its maximum around A = 22.5 μm and then decreases.The detailed experimental results are provided in the Supplementary Materials.

Fig. 1 .
Fig. 1.Photonic implementation and generalized acceleration theory.(A) Schematic of a onedimensional lattice composed of evanescently coupled waveguides with combined bending trajectory.(B) Schematic of a reduced Floquet lattice in the transformed coordinate frame.(C) Cross-sectional optical microscope image of the fabricated sample.(D) Top-view optical microscope image of the fabricated sample with a harmonic modulation.(E) Representation of F(z) induced wave vector shift according to the generalized acceleration theory.(F) z-dependent shift of the transverse Bloch momentum for several specific cases corresponding to typical BOs (A = 0, blue solid line), QBOs (ɅBO = 3ɅFL, orange dashed line), SBO-like oscillations (3ɅBO = 4ɅFL, red dashed line), and spreading (ɅBO = ɅFL, gray solid line).

Fig. 2 .
Fig. 2. Experimental visualization, simulation, and variance of the breathing modes for single-site excitation.(top) Fluorescence microscopy images of the wave evolution in curved waveguide arrays with a fixed circular bend radius R = 110.8cm (corresponding to ɅBO = 30 mm). (A) A = 0, corresponding to typical BOs; (D) A = 6 μm and ɅFL = 10 mm, corresponding to QBO; (G) A = 13.5 μm and ɅFL = 22.5 mm, corresponding to SBO-like oscillations; (J) A = 18 μm and ɅFL = 30 mm, corresponding to spreading.(middle) Simulated the wave evolution corresponding to those in (top).(bottom) Corresponding variances σ 2 of the measured (top) and simulated (middle) light evolution as a function of the propagation distance z.

Fig. 3 .
Fig. 3. Experimental visualization and simulation of the oscillating modes for broad-beam excitation.(top) Fluorescence microscopy images of the wave evolution in curved waveguides arrays with a fixed circular bend radius R = 110.8cm (corresponding to ɅBO = 30 mm). (A) A = 0, corresponding to typical BOs; (C) A = 6 μm and ɅFL = 10 mm, corresponding to QBOs; (E) A = 13.5 μm and ɅFL = 22.5 mm, corresponding to SBO-like oscillations; (G) A = 18 μm and ɅFL = 30 mm, corresponding to spreading.(bottom) Simulated wave evolution corresponding to those in (top).The trajectories of the beam x(z) extracted from the measured (top) and simulated (bottom) light evolution are marked as white dashed lines.

Fig. 4 .
Fig. 4. Fractal spectrum and fractional Floquet tunneling of FBOs.(A) Theoretical (blue stems) and measured (red dots) ratio ɅBO/ɅFBO as a function of the ratio ɅBO/ɅFL.The inset is a close-up spectrum at a finer scale, which show the property of self-similarity of this spectrum.(B) Normalized theoretical (lines) and measured (dots) FBO amplitude σ 2 (ɅFBO/2) as a function of the ratio A/ɅFL.