Tunable Bands in 1D Fractional Quantum Media Demonstrate Lévy Index Control of Energy Dispersion

Fractional calculus, a mathematical framework for describing systems with long-range interactions, now extends into the realm of quantum mechanics, offering new ways to understand and control electron behaviour in materials. Brenden R. Guyette, Joshua M. Lewis, and Lincoln D. Carr, all from the Quantum Engineering Program at the Colorado School of Mines, demonstrate how applying fractional calculus to the Schrödinger equation allows precise tuning of energy bands in one-dimensional quantum systems. Their work reveals that by manipulating a parameter known as the Lévy index, it becomes possible to not only reshape the energy landscape for electrons, but also to induce band inversion, a phenomenon where the usual ordering of energy levels flips. This control over band structure opens exciting possibilities for designing novel electronic devices and exploring new quantum phenomena, potentially leading to advances in areas like valleytronics, where electron ‘valleys’ act as information carriers, and materials with highly tunable sensitivity to their physical geometry.

Fractional Quantum Media and Tunable Bands

Scientists are applying fractional calculus, a branch of mathematics dealing with non-integer derivatives, to explore quantum systems exhibiting behaviours between traditional quantum mechanics and completely random interactions. They investigate how these fractional systems behave in one dimension, specifically focusing on the fractional Gross-Pitaevskii equation, which describes the behaviour of quantum particles. This research demonstrates that by adjusting the order of the fractional derivative, scientists can precisely control the bandwidth and central frequency of energy bands within the material, offering a new way to manipulate quantum states. The fractional derivative introduces long-range interactions, leading to spatially extended wave functions and modified energy spectra, paving the way for designing quantum devices with tailored properties for advanced quantum information processing and sensing.

Researchers extend the standard Schrödinger equation to its fractional form, allowing them to investigate how a parameter called the Lévy index, denoted as q, governs the formation and inversion of energy bands. This approach provides a pathway to engineer novel physical behaviours and device functionalities by tuning q in periodic quantum systems. The team solves the fractional Schrödinger equation for periodic rectangular potentials, varying the potential height, barrier thickness, and well width, using an imaginary-time evolution method to simulate the system’s behaviour.

Fractional Schrödinger Equation Controls Band Structure

This research explores the use of fractional derivatives within the Schrödinger equation, motivated by the idea that fractional dynamics can better describe complex systems where traditional quantum mechanics falls short. The central goal is to demonstrate that the fractional order, represented by ‘q’, provides a new and controllable parameter for manipulating the electronic band structure of one-dimensional periodic quantum systems. This is significant because band structure dictates many material properties, such as conductivity and optical behaviour.

The research investigates how ‘q’ affects the formation of valleys in the band structure, relevant to valleytronics, which uses valleys as information carriers, and how it alters the effective mass of electrons, with lower effective mass leading to higher mobility and faster devices. The team’s findings reveal a distinct shift in behaviour depending on the value of ‘q’. For values of ‘q’ greater than 2, the band topology undergoes a significant change, with a symmetry-breaking inversion creating valleys in the band structure. These valleys can be tuned by both ‘q’ and the geometry of the periodic potential, opening possibilities for valleytronic devices where information is encoded and processed using these valleys. The ability to control valley position with ‘q’ is a key advantage, and the research establishes quantitative relationships between ‘q’, the effective mass, and the valley position, providing a framework for designing materials with specific properties.

Lévy Index Tunes Band Inversion and Qubits

This work extends fractional calculus to the study of particle behaviour within periodic potentials, revealing how the Lévy index governs energy band formation and inversion. Researchers solved the fractional Schrödinger equation for varying rectangular potentials using an imaginary-time evolution algorithm, supplemented by Gaussian process regression to map energy dispersion. Analysis demonstrates a distinct shift in band structure at a Lévy index of two, separating regimes with differing characteristics. For Lévy indices greater than two, band topology changes through symmetry-breaking inversion, creating a Bloch-momentum qubit tunable by both the index and potential geometry.

Conversely, for indices less than two, the ground band sharpens around zero momentum, resulting in a monotonic decrease in effective mass with increasing index. The rate of effective mass decrease depends on the geometry of the potential, with the strongest effect observed when the Lévy index approaches two and the weakest as it nears one. Across all tested potentials, the effective mass saturates at an index of one, reaching approximately 0. 15 with minimal variance. These findings establish the Lévy index as a practical control parameter for one-dimensional periodic quantum media, offering a means to reshape band curvature and topology. While this study focused on one-dimensional systems, researchers suggest that higher-dimensional systems may exhibit even richer dispersions, and future work could explore experimental platforms that realize Lévy-flight transport to probe these predicted band transformations, and time-domain studies of the Bloch-momentum qubit to assess coherence and potential device applications.