You have the impression that your world is three-dimensional, but if you look well around yourself, you will notice that the branches of the trees or their roots actually have many voids in between. These structures are called fractals, and it turns out that they are not really three-dimensional, nor two-dimensional, but some fractional value in between. The same happens if you look inside your own body: your lungs, your circulatory system, and even your neuronal system all have this porous-like structure with fractal dimension. This occurs because these configurations are very convenient for exchanging matter or information.
Fractals do not only occur in nature and in the human body, but they are known also in art, via the works of Escher and Pollock, or in technology, where they are used to build antennas or to store solar energy. In mathematics, they emerge as beautiful patterns that repeat themselves ad infinitum, and generate the impossible: a structure that has a finite area, but an infinite perimeter. In addition, fractals are not only static patterns that appear in space, but they can also be dynamical: our heart beating is neither periodic nor chaotic, but fractal. And if it stops being fractal, it is a sign that you are very ill!
One of the most important open questions concerning humans is how our consciousness gets established. Long before winning the 2020 Nobel Prize for his prediction of black holes, physicist Roger Penrose worked together with anesthesiologist Stuart Hameroff to propose the orchestrated objective reduction (Orch OR) theory in the early 90’s. In this theory, they claim that consciousness should obey the rules of quantum mechanics, the theory that determines how very small particles, such as the electrons, move around. However, quantum mechanical laws are usually observable only at very low temperatures or for extremely small particles. Since our body works at room temperature, the proposal by Penrose and Hameroff was met with incredulity. According to them, consciousness originates from microtubules inside neurons, where quantum coherence emerges, and may occur at various scales in a fractal-like brain hierarchy. The theory generated much controversy, with strong opposers and persuaded supporters, but the issue has not been settled.
Instead of entering into the polemic, Cristiane Morais Smith, a theoretical physics professor at Utrecht University, decided to join forces with experimental colleagues from China (group of Prof. XianMin Jin) to create an artificial fractal on a chip, composed of waveguides. By injecting light at the border of the fractal structure and performing many measurements, they could visualize how the quantum of light – photons – diffuse in the fractal. Their findings reveal that the quantum dynamics in fractals is very different from the classical one, and future measurements in the human body can be compared to their results to definitely decide whether consciousness is a classical, or rather a quantum phenomenon.
By investigating quantum transport in these artificially designed fractal structures, they might be making the first steps towards the long journey that will be the unification of physics, mathematics and biology, to reach a deeper understanding of the human body and its functioning, as well as to inspire designs of more efficient quantum algorithms.
Classical and quantum fractals
he human fascination for fractals dates back centuries or even millennia, as we see fractal structures known nowadays as a Sierpinski gasket used in decorative art in churches. Nonetheless, it was only in the last century that mathematicians faced the difficult task of classifying these structures. In the 80’s and 90’s, the foundational work of Mandelbrot triggered enormous activity in the field. Many theoretical physicists, such as the Nobel Prize winner Pierre Gilles de Gennes, devoted years of research to understand what he named “The ant in the labyrinth problem”, which describes how a particle diffuses in a fractal structure. However, those were classical fractals.
This century, the task is to understand quantum fractals. In pioneering research last year involving a scanning tunneling microscope (STM), Morais Smith and collaborators showed that the wavefunction describing electrons in a Sierpinski gasket fractal has the Hausdorff dimension d = 1.58. This work, published in Nature Physics 15, 127 (2019) has attracted enormous interest from the scientific community, as well as from the press: the research was featured in Physics Today and several European scientific magazines; a YouTube movie from Seeker has, at the time of writing, 812,000 views. However, STM techniques can only describe equilibrium properties and cannot see particles in action.
ere, we go a huge step beyond and using state-of-the-art photonics experiments, we reveal the quantum dynamics in fractals. By injecting photons in waveguide arrays arranged in a fractal shape, we are able to follow its motion and understand the quantum dynamics in fractals with unprecedented details (see Figure). We built and investigated three types of fractal structures to reveal not only the influence of different fractal or Hausdorff dimensions, but also how the geometry (lacunarity and connectivity) influences the quantum transport in fractals with the same Hausdorff dimension.
Connecting physics to mathematics and biology
Our work could have profound implications in physics, mathematics, biology, and computer science. For physics, it represents the first experimental realization of continuous-time quantum walks in a fractal. Our experimental and numerical results show that the quantum behavior is qualitatively and quantitatively different compared to classical behavior. For mathematics, our results provide a solid background to test concepts like lacunarity and Hausdorff dimension, as well as their interplay. In biology, it connects to the fractality of the human body, and might reveal at which level quantum mechanics influences transport in human bodies.
/Cristiane Morais Smith
“Shining light on quantum transport in fractal networks”, by Xiao-Yun Xu, Xiao-Wei Wang, Dan-Yang Chen, C. Morais Smith, and Xian-Min Jin, Nature Photonics 2021 July 2021, https://doi.org/10.1038/s41566-021-00845-4
The Conversation in England has interviewed Cristiane Morais Smith about the latest research, and the result is a huge and long discussion on their website.
Physics Today 72, 1, 14 (2019) Quantum corral herds surface electrons into a fractal lattice - The method, based on scanning tunneling microscopy, lets researchers explore quantum mechanics in geometries not found in nature. https://doi.org/10.1063/PT.3.4105