Momentum Gap Research Articles

Spatiotemporal Bragg gratings forming inside a nonlinear dispersive medium.

We show that a spatiotemporal Bragg grating can be created inside a nonlinear dispersive medium (such as silica fibers) by launching a periodic train of pump pulses that travel as fundamental solitons. We develop a theoretical model and use it to find the band structure of such gratings. We study the interaction of a probe pulse with the Bragg grating, both within and outside of momentum gaps. We also show that a photonic analog of the Anderson localization is possible when a disorder is introduced into a spatiotemporal Bragg grating.

Enhanced scattering from an almost-periodic optical temporal slab

We showcase the impact of almost-periodicity on the parametric amplification associated with the first-order momentum gap in photonic time-crystals with time-varying permittivity. Utilizing a vectorial coupled-wave theory approach, we rigorously analyze the scattering by a temporal slab of the considered medium. We pinpoint a critical regime wherein flaws in material tuning paradoxically enhance amplification due to the coupling of fewer, broader modes, resulting in a higher and broader pulselike amplification envelope. Additionally, we demonstrate that the intensity reflectances of time-reversed waves corresponding to secondary “Bragg” resonances achieve remarkably high levels of subharmonic parametric amplification, with the epsilon-near-zero regime serving as a preferred candidate for experimental implementation. Our counterintuitive findings highlight the potential of intentionally leveraging modulation desynchronization and impurities in the temporal unit cell of photonic time-crystals to enhance both the level and the bandwidth of amplification. Published by the American Physical Society 2024

Inverse design of topological photonic time crystals via deep learning

Photonic time crystals are a new kind of photonic system in modern optical physics, leading to devices with new properties in time. However, so far, it is still a challenge to design photonic time crystals with specific topological states due to the complex relations between time crystal structures and topological properties. Here, we propose a deep-learning-based approach to address this challenge. In a photonic time crystal with time inversion symmetry, each band separated by momentum gaps can have a non-zero quantized Berry phase. We show that the neural network can learn the relationship between time crystal structures and Berry phases, and then determine the crystal structures of photonic time crystals based on given Berry phase properties. Our work shows a new way of applying machine learning to the inverse design of time-varying optical systems and has potential extensions to other fields, such as time-varying phononic devices.

Bandgap engineering and amplification in photonic time crystals

Recently, there has been a growing interest in time-varying photonic media owing to their significant potential in the field of wave manipulation. Here, we explore the exotic characteristics of wave amplification in a photonic time crystal (PTC) made of a spatially homogeneous medium where the refractive index varies periodically in time. Based on qualitative and quantitative analysis of the amplification, we show that the amplification not only depends on the choice of wave vector of a propagating light but also attains different values in different bandgaps. Our approach further extends towards achieving the minimum amount of variation of permittivity required to open momentum gaps to facilitate the phase-dependent amplification of PTCs. Further, we investigate the impact of permittivity variation and choice of number of temporal unit cells to truncate a PTC to mimic the properties of infinite PTC and offer new opportunities to manipulate and control the amplification of light for applications including highly tunable PTC lasers and devices.

Waveform shaping in photonic time-crystals

This paper reports on the waveform shaped by a finite duration photonic time-crystal with its permittivity and permeability periodically varying in time. A Gaussian-modulated sinusoidal pulse is incident onto this photonic time-crystal to evaluate the backward- and forward-scattering waveforms. An analytical formulation, utilizing a cascade of temporal transfer matrices and the inverse fast Fourier transform, was employed to conduct time-domain waveform computations. Interestingly, the dispersion diagram of the temporal unit cell, which displays a momentum gap characterized by a complex effective angular frequency, plays a crucial role in shaping the incident waveform. Specifically, the presence of momentum gaps in the spectrum of the incident pulse determines the frequencies of the generated oscillation modes.

De Haas-van Alphen spectroscopy and magnetic breakdown in moiré graphene.

Quantum oscillations originating from the quantization of electron cyclotron orbits provide sensitive diagnostics of electron bands and interactions. We report on nanoscale imaging of the thermodynamic magnetization oscillations caused by the de Haas-van Alphen effect in moiré graphene. Scanning by means of superconducting quantum interference device (SQUID)-on-tip in Bernal bilayer graphene crystal axis-aligned to hexagonal boron nitride reveals large magnetization oscillations with amplitudes reaching 500 Bohr magneton per electron in weak magnetic fields, unexpectedly low frequencies, and high sensitivity to superlattice filling fraction. The oscillations allow us to reconstruct the complex band structure, revealing narrow moiré bands with multiple overlapping Fermi surfaces separated by unusually small momentum gaps. We identified sets of oscillations that violate the textbook Onsager Fermi surface sum rule, signaling formation of broad-band particle-hole superposition states induced by coherent magnetic breakdown.

Propagation of Dirac waves through various temporal interfaces, slabs, and crystals

We investigate the influence of the temporal variations of various medium parameters on the propagation of Dirac-type waves in materials where the quasiparticles are described by a generalized version of the pseudospin-1/2 Dirac equation. Our considerations also include the propagation of electromagnetic waves in metamaterials with the Dirac-type dispersion. We focus on the variations of the scalar and vector potentials, mass, Fermi velocity, and tilt velocity describing the Dirac cone tilt. We derive the scattering coefficients associated with the temporal interfaces and slabs analytically and find that the temporal scattering is caused by the changes of the mass, Fermi velocity, and vector potential, but does not arise from the changes of the scalar potential and tilt velocity. We also explore the conditions under which the temporal Brewster effect and total interband transition occur and calculate the change in total wave energy. We examine bilayer Dirac temporal crystals where parameters switch between two different sets of values periodically and prove that these systems do not have momentum gaps. Finally, we assess the potential for observing these temporal scattering effects in experiments.

Superluminal k-Gap Solitons in Nonlinear Photonic Time Crystals.

We propose superluminal solitons residing in the momentum gap (k gap) of nonlinear photonic time crystals. These gap solitons are structured as plane waves in space while being periodically self-reconstructing wave packets in time. The solitons emerge from modes with infinite group velocity causing superluminal evolution, which is the opposite of the stationary nature of the analogous Bragg gap soliton residing at the edge of an energy gap (or a spatial gap) with zero group velocity. We explore the faster-than-light pulsed propagation of these k-gap solitons in view of Einstein's causality by introducing a truncated input seed as a precursor of a signal velocity forerunner, and find that the superluminal propagation of k-gap solitons does not break causality.

Unidirectional scattering with spatial homogeneity using correlated photonic time disorder

Recently, there has been increasing interest in the temporal degree of freedom in photonics due to its analogy with spatial axes, causality and open-system characteristics. In particular, the temporal analogues of photonic crystals have allowed the design of momentum gaps and their extension to topological and non-Hermitian photonics. Although recent studies have also revealed the effect of broken discrete time-translational symmetry in view of the temporal analogy of spatial Anderson localization, the broad intermediate regime between time order and time uncorrelated disorder has not been examined. Here we theoretically investigate the inverse design of photonic time disorder to achieve optical functionalities in spatially homogeneous platforms. By developing the structure factor and order metric using causal Green’s functions for disorder in the time domain, we propose an engineered time scatterer, which provides unidirectional scattering with controlled scattering amplitudes. We also show that the order-to-disorder transition in the time domain allows the manipulation of scattering bandwidths, which makes resonance-free temporal colour filtering possible. Our work could advance optical functionalities without spatial patterning.

Transverse Goldstone mode in holographic fluids with broken translations

In this paper we investigate the low energy shear modes in fluid systems with spontaneously broken translations by a specific holographic model. In absence of momentum relaxation, we find that there exist two decoupled gapless modes in the transverse channel, one of which is purely diffusive and the other corresponds to vortex like excitations. The diffusive mode is associated with the conservation of momentum and the vortex mode can be viewed as the Goldstone mode of the spontaneous symmetry breaking. Switching on an external source which breaks the translations explicitly but weakly, the would-be gapless modes both get relaxed and acquire a tiny mass gap. Finally, in the strong momentum relaxation regime, we find a (pseudo-)diffusive-to-sound crossover that is set by a momentum gap.

Spatiotemporal photonic crystals

We study light propagation in spatiotemporal photonic crystals: dielectric media that vary periodically in both space and time. While photonic crystals (spatially periodic media) are well understood, the combination of periodic change in both time and space poses considerable challenges and requires new analysis methods. We find that the band structure of such systems contains energy gaps, momentum gaps, and mixed energy–momentum gaps in which both energy and momentum may attain complex values. We identify the unique interplay between the exponential growth induced by temporal modulation and the exponential decay caused by spatial modulation, and how these can completely counteract one another. Under proper conditions, these two opposing forces are exactly matched, causing the mixed energy–momentum gap to collapse to a single point, which is an exceptional point known from non-Hermitian dynamics. Such spatiotemporal photonic crystals possess unique properties that could pave the way to new ways of controlling the propagation of light.

Experimental Implementation of Wave Propagation in Disordered Time-Varying Media.

Here, we study and implement the temporal analog in time disordered sytems. A spatially homogeneous medium is endowed with a time structure composed of randomly distributed temporal interfaces. This is achieved through electrostriction between water surface and an electrode. The wave field observed is the result of the interferences between reflected and refracted waves on the interfaces. Although no eigenmode can be associated with the wave field, several common features between space and time emerge. The waves grow exponentially depending on the disorder level in agreement with a 2D matrix evolution model such as in the spatial case. The relative position of the momentum gap appearing in the time modulated systems plays a central role in the wave field evolution. When tuning the excitation to compensate for the damping, transient waves, localized in time, appear on the liquid surface. They result from a particular history of the multiple interferences produced by a specific sequence of time boundaries.

Does the momentum gap explain momentum in Taiwan?

Huang (2021) proposes the momentum gap, which is the difference in the formation-period returns between past winners and losers, as a strong predictor of momentum returns. This study extends Huang's (2021) analyses to the Taiwan stock market by proposing that the composition of market participants and the imposition of price limits in this market have opposite impacts on the momentum gap. Our results indicate that the momentum gap fails to predict momentum returns in Taiwan, a phenomenon that is in line with our conjecture that the imposition of price limits mitigates the effectiveness of the momentum gap.

Light emission by free electrons in photonic time-crystals

Photonic time-crystals (PTCs) are spatially homogeneous media whose electromagnetic susceptibility varies periodically in time, causing temporal reflections and refractions for any wave propagating within the medium. The time-reflected and time-refracted waves interfere, giving rise to Floquet modes with momentum bands separated by momentum gaps (rather than energy bands and energy gaps, as in photonic crystals). Here, we present a study on the emission of radiation by free electrons in PTCs. We show that a free electron moving in a PTC spontaneously emits radiation, and when associated with momentum-gap modes, the electron emission process is exponentially amplified by the modulation of the refractive index. Moreover, under strong electron-photon coupling, the quantum formulation reveals that the spontaneous emission into the PTC bandgap experiences destructive quantum interference with the emission of the electron into the PTC band modes, leading to suppression of the interdependent emission. Free-electron physics in PTCs offers a platform for studying a plethora of exciting phenomena, such as radiating dipoles moving at relativistic speeds and highly efficient quantum interactions with free electrons.

Parametric amplification interactions in time-periodic media: coupled waves theory

Waves and their interactions with time-periodic media have recently been of theoretical interest for their unusual properties. It has been revealed that such interactions lead to momentum gaps at ω = N Ω / 2 , where ω is the wave frequency, Ω is the modulation frequency, and N ∈ Z . While waves can propagate at the boundaries of the momentum gaps, it has been shown that waves exhibit parametric amplification in the momentum gaps due to the active nature of such temporally dynamic media. In this paper, the coupled waves theory is used for the modeling of the parametric amplification phenomena in time-periodic optical media. Simple formulas are derived using the 0th and first Fourier components of θ ( t ) = 1 / ε ( t ) , approximating the results of the cumbersome stability analysis of the associated hypergeometric equation of the electric displacement field and providing the conditions to define the wave parameters, which can fall in the momentum gap, resulting in parametric amplification or allow propagating modes. Numerical results are compared with the Hill’s determinant of the Floquet theory and FDTD simulations showing that the coupled waves theory can efficiently model the first and second momentum gaps. Further extensions can be made to higher-order gaps, by considering higher-order harmonics and following the same steps of this analysis.

Nucleon momentum gap in asymmetric nuclear matter

For more than half a century, nucleons were considered to move continuously in the nuclei. Recent electron-scattering experiments indicate about 20--25% of nucleons in heavier nuclei are involved in the neutron-proton short-range correlations. Nucleons in the nuclei thus have abnormal behavior unlike those in the noninteraction fermi gas. In the neutron-rich nuclei, around the Fermi momentum, the neutron-proton short-range correlations lead to a momentum gap in the proton momentum distribution, which is circumstantially supported by the pionic experimental data. Likewise, there should also be a neutron momentum gap in the proton-rich nuclei. Nucleon momentum gap is thought to have profound and extensive implications in the studies ranging from particle physics to neutron stars as well as ultracold atomic gases.

The Momentum Gap and Return Predictability

Abstract The formation period return difference between past winners and losers, which I call the momentum gap, negatively predicts momentum profits. I document this for the U.S. stock market and find consistent results across 21 major international markets. A one-standard-deviation increase in the momentum gap predicts a 1.25$\%$ decrease in the monthly momentum return after controlling for existing predictors. This predictability extends up to 5 years for static momentum portfolios, consistent with time-varying investor biases. Following the simple real-time strategy of investing in momentum only when the momentum gap is below the 80th percentile delivers a Sharpe ratio of 0.78.

Spatiotemporal plane wave expansion method for arbitrary space-time periodic photonic media.

We have derived a systematic method to calculate the photonic band structures and mode field profiles of arbitrary space-time periodic media by adopting the plane wave expansion method and extending to the space-time domain. We have applied the proposed method to a photonic crystal with time periodic permittivity, i.e., the Floquet photonic crystal, and showed that the method efficiently predicts driving-induced opening of frequency and momentum gaps and breaking of mirror symmetry in the photonic band structures. This method enables systematic investigation of various optical phenomena in space-time periodic media, such as nonreciprocal propagation of light, parametric processes, and photonic Floquet topological phases.

Probing Floquet modes in a time periodic system with time defects using Faraday instability

Time crystals are systems whose properties are periodically modulated in time. Among these, Floquet time crystals exhibit momentum gaps in their band structures, analogous to energy gaps in spatial crystals. Recently, time defects with a -shift in the time modulation have been introduced theoretically as temporal analogues of spatial topological defects associated with localized edge modes. Here we perform experiments in a time periodic system using Faraday instability, a parametric excitation of a liquid bath by vertical shaking. Although time defects also trigger an exponentially decaying wave, we show that the analogy does not hold due to temporal causality and lack of energy conservation. However, these time defects provide an original way to explore momentum gaps and reveal their overdamped modes.

A theoretical study on quantum dynamics of energy transfer for HF colliding with D2

The cross sections and rate constants of pure rotational and ro-vibrational inelastic collision processes for the D2 –HF system were investigated by employing the recently developed coupled-states approximation including the nearest neighbor Coriolis couplings, based on our newly constructed potential energy surface. The state-to-state cross sections and rate constants were found to follow the usual energy and angular momentum gap laws for pure rotational and ro-vibrational energy transfer processes. The calculated rate constant of 1.18 × 10-10 cm3 mol−1 s−1 for D2 (v1 =0) + HF(v2 =0, j2 =13) is very close to the experimental value of (1.2 ± 0.2) × 10-10 cm3 mol−1 s−1 . The calculated vibrational-resolved rate constants are in reasonably consistent to the available experimental results and follow the similar trend with temperature.