arXiv subjects

Joshua M. Lewis

Publications and source records attributed to Joshua M. Lewis.

4 recordsLinked to original sources

arXiv2026

Tunable Bands in 1D Fractional Quantum Media

Fractional calculus has become an essential framework in geophysics, optics, and biological systems to capture long-range correlations and anomalous transport. In this article, we extend this framework to explore a particle in a periodic potential, where the Schrodinger equation is extended to its fractional form. This allows us to study how the Levy index $q$ governs the formation and inversion of energy bands, offering a pathway to engineer new physical behaviors by tuning $q$ in periodic quantum systems. We solve the fractional Schrodinger equation (FSE) for periodic rectangular potentials of varying height $V_0$, barrier thickness $L$, and well width $W$ using an imaginary-time evolution algorithm, supplemented by Gaussian process regression. This analysis reveals a qualitative shift in the system's band structure at $q = 2$, separating into distinct regimes of dispersion for $q > 2$ and $q < 2$. For $q > 2$, the energy bands invert as symmetric minima emerge within the first Brillouin zone and shift from $k = 0$ toward $k=\pm π/a$ with increasing $q$. These degenerate minima define a Bloch-momentum qubit, suggesting an analog to valley degrees of freedom in valleytronics. The $q$ at which inversion completes scales as $q \propto V_0^{-0.28\pm0.05}$, $q \propto L^{-0.35\pm0.08}$, and $q \propto W^{-0.49\pm0.06}$ when varying parameters individually, indicating a tunable sensitivity to potential geometry. In contrast, for $q < 2$, the ground band hardens around $k = 0$, with a dispersion following $C|k|^q+E_0$ near $k = 0$. This suggests an effective mass of 0 for $1<q<2$ at the band's lowest energy state. These results demonstrate that the Levy index serves as a tunable degree of freedom in quantum periodic systems, capable of driving band inversion, modulating the band gap, and reshaping carrier dynamics through effective-mass control.

quant-ph↗

arXiv2025

Propagation of Spin Waves in Doubly Periodic Magnonic Crystals

Towards the development of strategies for tailoring spin-wave band gaps in magnonic crystals, this work examines the band gap properties in a one-dimensional magnonic crystal with double periodicity. A long and narrow yttrium iron garnet (YIG) thin film strip is etched with an array of transverse groove lines separated by alternating distances, where the second distance is twice the first. This double periodicity in the magnonic crystal translates into dissimilar band gaps in the frequency domain, with the third and sixth band gaps being more pronounced than others. These band gaps are more pronounced because the corresponding wavenumbers simultaneously satisfy the Bragg scattering conditions for the periods equal to the two groove separations as well as their sum. Experimental observations are reproduced by numerical simulations. Together, the experimental and numerical results demonstrate how multiple periodicities could be an effective design parameter for creating magnonic crystals with desired band gaps.

cond-mat.mes-hall↗

arXiv2025

Classical and Quantum Phase Transitions in Multiscale Media: Universality and Critical Exponents in the Fractional Ising Model

Until now multiscale quantum problems have appeared to be out of reach at the many-body level relevant to strongly correlated materials and current quantum information devices. In fact, they can be modeled with $q$-th order fractional derivatives, as we demonstrate in this work, treating classical and quantum phase transitions in a fractional Ising model for $0 < q \leq 2$ ($q = 2$ is the usual Ising model). We show that fractional derivatives not only enable continuous tuning of critical exponents such as $ν$, $δ$, and $η$, but also define the Hausdorff dimension $H_D$ of the system tied geometrically to the anomalous dimension $η$. We discover that for classical systems, $H_D$ is precisely equal to the fractional order $q$. In contrast, for quantum systems, $H_D$ deviates from this direct equivalence, scaling more gradually, driven by additional degrees of freedom introduced by quantum fluctuations. These results reveal how fractional derivatives fundamentally modify the fractal geometry of many-body interactions, directly influencing the universal symmetries of the system and overcoming traditional dimensional restrictions on phase transitions. Specifically, we find that for $q < 1$ in the classical regime and $q < 2$ in the quantum regime, fractional interactions allow phase transitions in one dimension. This work establishes fractional derivatives as a powerful tool for engineering critical behavior, offering new insights into the geometry of multiscale systems and opening avenues for exploring tunable quantum materials on NISQ devices.

quant-ph↗

arXiv2024

Exploring Multiscale Quantum Media: High-Precision Efficient Numerical Solution of the Fractional Schrödinger equation, Eigenfunctions with Physical Potentials, and Fractionally-Enhanced Quantum Tunneling

Fractional evolution equations lack generally accessible and well-converged codes excepting anomalous diffusion. A particular equation of strong interest to the growing intersection of applied mathematics and quantum information science and technology is the fractional Schrödinger equation, which describes sub-and super-dispersive behavior of quantum wavefunctions induced by multiscale media. We derive a computationally efficient sixth-order split-step numerical method to converge the eigenfunctions of the FSE to arbitrary numerical precision for arbitrary fractional order derivative. We demonstrate applications of this code to machine precision for classic quantum problems such as the finite well and harmonic oscillator, which take surprising twists due to the non-local nature of the fractional derivative. For example, the evanescent wave tails in the finite well take a Mittag-Leffer-like form which decay much slower than the well-known exponential from integer-order derivative wave theories, enhancing penetration into the barrier and therefore quantum tunneling rates. We call this effect \emph{fractionally enhanced quantum tunneling}. This work includes an open source code for communities from quantum experimentalists to applied mathematicians to easily and efficiently explore the solutions of the fractional Schrödinger equation in a wide variety of practical potentials for potential realization in quantum tunneling enhancement and other quantum applications.

quant-ph↗