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Great news! - Our latest research on Branched Flow was published in PNAS

Abstract: We report unexpected classical and quantum dynamics of a wave propagating in a periodic potential in high Brillouin zones. Branched flow appears at wavelengths shorter than the typical length scale of the ordered periodic structure and for energies above the potential barrier. The strongest branches remain stable indefinitely and may create linear dynamical channels, wherein waves are not confined directly by potential walls as electrons in ordinary wires but rather, indirectly, and more subtly by dynamical stability. We term these superwires since they are associated with a superlattice.

Significance: Waves propagating through random media can accumulate in strong branches, intensifying fluctuations and powerful phenomena such as tsunamis. However, branched flow is not restricted to the large scale, and here, we find surprisingly that branched flow is not restricted to random media. We show that quantum waves living in the high Brillouin zones of periodic potentials also branch. Moreover, some of these branches do not decay as in random media but remain robust indefinitely, creating dynamically stable channels that we call superwires. The waves in these stable branches have enough energy to surmount the channel potential and go elsewhere, but classically, nonlinear dynamics keeps them confined within the channel. These results have direct experimental consequences for superlattices and optical systems.

Great News! The Blochbusters Alvar, Rick, Anton and Esa published a new paper on Branched Flow in the peer-reviewed Journal PNAS. 

Did this picture spark your interest? Check out our new article on PNAS and learn all about the science behind this image!

Here you can see that phonon modes awake as temperature rises.

Atoms of a frozen crystal at zero temperature are believed to remain still in their lattice positions. However, as temperature rises, atoms start to vibrate and therefore phonons are created. First, the long-wave phonons appear, since they do not require much energy. Then, for higher temperatures, short-wave phonons accumulate on top of the longer modes. Finally, as temperature increases over the Debye temperature all modes are activated, so no new features appear and simply the population of phonons increases. As a consequence, the peaks grow higher and the valleys get deeper.

Technical description: Using the expression of the deformation potential, we fixed all the parameters (including random phases) and only varied the temperature. All the frames were plotted with the same colour bar revealing how, as the temperature increases, new details appear and the peaks grow higher.

Description:  Magnetic focussing and branched flow compete in shaping the flow of electrons emitted form a quantum point contact into a two dimensional semiconductor crystal subjected to a magnetic field.

Technical details: Simulation of quasiclassical trajectories of electrons in a perpendicular magnetic field in a model potential of the disorder in a semiconductor nanostructure (upper half) and in an idealized model without disorder (lower half).

Read on related research:

>Maryenko D, et al. (2012), "How branching can change the conductance of ballistic semiconductor devices", Physical Review B., May, 2012. Vol. 85(19), pp. 195329.

>Aidala, K. E. et al. (2007), ” Imaging magnetic focusing of coherent electron waves”. Nature Phys 3, 464–468.

Description:  The interplay of a regular and a random potential creating the precursor of a branched flow. The phenomenon is analogous to the images "RANDOM REFLECTIONS" and "REGULAR BUT STRANGE" which you can find above!

Technical details: The picture shows the ray density in a kick-drift model with (2+1) dimensions after a single kick. The two  dimensional kick potential is generated by the product of two sinusoidal functions, a Gaussian and a Gaussian random field. 

Description:  Regular cousin of a 3-dimensional branched flow. These staggering forms are created by 5 deflections from a regular and localized potential instead of the random potentials creating an “ordinary” branched flow.

Technical details: The picture shows the ray density in a kick-drift model with (2+1) dimensions after 5 kicks. Each kick is the impulse generated by the same 2-dimensional regular potential created by the product of two sinusoidal functions and a Gaussian.

We called this image “Green 3”, green for Green’s function. A snapshot of 25 quantum explosions, with 25 waves that were tiny dots a moment ago now blowing up because of the Uncertainty Principle. Starting out as dots, each was very certain where it was, which in quantum mechanics means you will have little idea how fast you are going or in what direction.  So a moment after they were set free, the 25 quantum waves are caught spreading out, and fast.  The thing is, the landscape around each dot is irregular, and differently so for each one, so the waves bend in weird ways. There is another beautiful piece of physics built into this picture: if you look at each wave explosion, the farther you go out from the center, the shorter the wave lengths become. This is because the faster parts of the wave have travelled farther from the center, and in quantum mechanics, faster waves have shorter wavelength.  The initial uncertainty in speed is revealed in a large range of positions, with speeds revealed by position later.

The freedom we have to display data is enormous, and we do it differently both for artistic reasons and to highlight different aspects of the science. Branched flow, as seen elsewhere on Blochbusters, is a ubiquitous wave and particle phenomenon coming from travel over a smooth but (typically) random landscape. It can also happen traversing a smooth but periodic landscape, as in a crystal.  To learn more about this topic click here. 

Description: The picture shows the density of classical trajectories in a square lattice. Despite the periodicity of the potential, the chaotic dynamics can generate branched flow: aperiodic structures arise from regular lattices. In random potentials branches decay exponentially fast from the origin, but in periodic potentials the stable branches (pink tones) can remain stable indefinitely.

Technical details: We ran classical trajectories from the center with the same initial energy and a uniform distribution of angles. The potential in which they evolve is made by bumps arranged in a square lattice. As usual, we employed a symplectic integrator and represented the density of trajectories. Quantum simulations in superlattices (lattice constant much larger than the wavelength) display similar patterns.

Authors: Alvar Daza, Eric J. Heller, Anton M. Graf, Esa Räsänen.

Waves propagating through weakly refracting media typically accumulate in strong branches. Tsunamis provide a powerful example of this branched flow in the large scale, but this general wave phenomenon affects the quantum realm too. So far, branched flow had only been studied in random media, but here we report for the first time its occurrence in periodic potentials.

One of the major differences of this research with respect to standard investigation of periodic systems is the methodology. Periodic systems such as crystals are usually studied by means of periodic functions, that is, Bloch waves. However, we study the evolution of wave packets in real space by directly integrating the time-dependent Schrödinger equation numerically. Also, we focus into a peculiar regime, where wavelengths are short compared to the potential scale. This would be unphysical for typical crystals, but modern superlattices (such as twisted graphene bilayers) operate precisely in this regime.

Branched flow in periodic potentials comes along with a special feature: dynamically stable channels, which we call superwires (because its relevance to superlattices). In random media, branches can carry high flux density for long distances, but they eventually die. In regular lattices, some branches can remain indefinitely stable. Unlike energetic channels, waves in these stable branches have enough energy to surmount the potential and go elsewhere, but its dynamics keep them confined within the channel. These results have direct experimental consequences for superlattices and optical systems, and they also help understand the role of nonlinear dynamics in branched flow.

Find the full article here. 

Description: This image can inspire many different things, but technically, this is an electron’s Green function evolving under the action of a periodic potential. In a glimpse, it shows many interesting features. The central part corresponds to the slower components of the electron wave, which lag behind undulating and creating strange patterns. The fast components, where the alternation of blue and yellow shows shorter wavelengths, have spread their branches far from the epicenter. Despite the underlying periodic potential, the electron wave functions display a rich variety of branched flow patterns. Of course, it is possible to assemble such patterns with Bloch waves, but certainly, it is hard to imagine all the complexity that can arise in periodic lattices if one is restricted to periodic waves.

Technical details: The picture was made integrating the the time-dependent Schrödinger equation with a Gaussian wavepacket with zero initial momentum evolving under a triangular potential. The wave lengths involved are shorter than the lattice constant of the potential (high Brillouin zones). The split-operator method was used for the integration, and here we show only the real part of the complex wave function. If we made an energy Fourier transform of this picture (fixed wavelength), we would obtain an eigenfunction.

Title: Branched Flow and Applications

Speaker: Eric J. Heller

Duration: 58 min.

Abstract: In classical and quantum phase space flow, there exists a regime of great physical relevance that is belatedly but rapidly generating a new field. In evolution under smooth, random, weakly deflecting but persistent perturbations, a remarkable regime develops, called branched flow. Lying between the first cusp catastrophes at the outset, leading to fully chaotic statistical flow much later, lies the visually beautiful regime of branched flow.

It applies to tsunami wave propagation, freak wave formation, light propagation, cosmic microwaves arriving from pulsars, electron flow in metals and devices, sound propagation in the atmosphere and oceans, the large scale structure of the universe, and much more. The mathematical structure of this flow is only partially understood, involving exponential instability coexisting with "accidental" stability. The flow is qualitatively universal, but this has not been quantified. Many questions arise, including the scale(s) of the random medium, and the time evolution of manifolds and "fuzzy" manifolds in phase space. The classical-quantum (ray-wave) correspondence in this flow is only partially understood. This talk will be an introduction to the phenomenon, both visual and mathematical, emphasizing unanswered questions