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New Journal of Physics

Optical simulation of quantum mechanics on the Möbius strip, Klein’s bottle and other manifolds, and Talbot effect

Jakub Bělín, Tomáš Tyc, S. A. R. Horsley

Published Feb 9, 2021• 8 citations
Publisher

Abstract

Abstract We analyse the evolution of the wavefunction of a quantum particle propagating on several compact manifolds, including the Klein bottle, Möbius strip and projective plane. We find analytically the stationary states and the energy spectrum and show that the wavefunction exhibits perfect revivals. Using the orbifold structure of the discussed manifolds, we establish the relation of wave evolution on the manifolds to Fresnel diffraction and consequently to the Talbot effect. This connection provides a novel method of optical simulation of the quantum motion on compact manifolds. We discuss some novel phenomena as well as the effects of topology on the properties of the waves on the manifolds.

DOI
10.1088/1367-2630/abe445

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Abstract We analyse the evolution of the wavefunction of a quantum particle propagating on several compact manifolds, including the Klein bottle, Möbius strip and projective plane. We find analytically the stationary states and the energy spectrum and show that the wavefunction exhibits perfect revivals. Using the orbifold structure of the discussed manifolds, we establish the relation of wave evolution on the manifolds to Fresnel diffraction and consequently to the Talbot effect. This connection provides a novel method of optical simulation of the quantum motion on compact manifolds. We discuss some novel phenomena as well as the effects of topology on the properties of the waves on the manifolds.
We analyze the propagation of waves on the surface of the simplest Platonic solid, the regular tetrahedron. We find that the time evolution of a quantum particle confined to such a surface is equivalent to the diffraction pattern from a certain grating profile and can therefore be investigated in a simple diffraction experiment. The quantum particle exhibits perfect quantum state revivals, which translate into the fractional Talbot effect in the diffraction pattern. We present the results of such an experiment and compare them with numerical simulations.
In this work, we discuss the topological transformation of quantum dynamics by showing the wave dynamics of a quantum particle on different types of topological structures in various dimensions from the fundamental polygons of the corresponding universal covering spaces. This is not the view from different perspectives of an observer who simply uses different coordinate systems to describe the same physical phenomenon but rather possible geometric and topological structures that quantum particles are endowed with when they are identified with differentiable manifolds that are embedded or immersed in Euclidean spaces of higher dimension. We present our discussions in the form of Bohr model in one, two and three dimensions using linear wave equations. In one dimension, the fundamental polygon is an interval and the universal covering space is the straight line and in this case the standing wave on a finite string is transformed into the standing wave on a circle which can be applied into the Bohr model of the hydrogen atom. In two dimensions, the fundamental polygon is a square and the universal covering space is the plane and in this case, the standing wave on the square is transformed into the standing wave on different surfaces that can be formed by gluing opposite sides of the square, which include a 2-sphere, a 2-torus, a Klein bottle and a projective plane. In three dimensions, the fundamental polygon is a cube and the universal covering space is the three-dimensional Euclidean space. It is shown that a 3-torus and the manifold K × S1 defined as the product of a Klein bottle and a circle can be constructed by gluing opposite faces of a cube. Therefore, in three-dimensions, the standing wave on a cube is transformed into the standing wave on a 3-torus or on the manifold K × S1. We also suggest that the mathematical degeneracy may play an important role in quantum dynamics and be associated with the concept of wavefunction collapse in quantum mechanics.
In this work we discuss the topological transformation of quantum dynamics by showing the wave dynamics of a quantum particle on different types of topological structures in various dimensions from the fundamental polygons of the corresponding universal covering spaces. This is not the view from different perspectives of an observer who simply uses different coordinate systems to describe the same physical phenomenon but rather possible geometric and topological structures that quantum particles are endowed with when they are identified with differentiable manifolds that are embedded or immersed in Euclidean spaces of higher dimension. We present our discussions in the form of Bohr model in one, two and three dimensions using linear wave equations. In one dimension, the fundamental polygon is an interval and the universal covering space is the straight line and in this case the standing wave on a finite string is transformed into the standing wave on a circle which can be applied into the Bohr model of the hydrogen atom. In two dimensions, the fundamental polygon is a square and the universal covering space is the plane and in this case the standing wave on the square is transformed into the standing wave on different surfaces that can be formed by gluing opposite sides of the square, which include a 2-sphere, a 2-torus, a Klein bottle and a projective plane. In three dimensions, the fundamental polygon is a cube and the universal covering space is the three-dimensional Euclidean space. It is shown that a 3-torus and the manifold K×S^1 defined as the product of a Klein bottle and a circle can be constructed by gluing opposite faces of a cube therefore in three-dimensions the standing wave on a cube is transformed into the standing wave on a 3-torus or on the manifold K×S^1. We also suggest that the mathematical degeneracy may play an important role in quantum dynamics and be associated with the concept of wavefunction collapse in quantum mechanics.
Any quantum (Schrödinger) wavefunction on a sphere is necessarily periodic in time. The corresponding statement down one dimension, for a circular line instead, is the quantum version of the 'Talbot effect' for a diffraction grating in paraxial optics (which is fully analogous to quantum mechanics). In the circle case the 'revival' of any initial wavefunction at the period, or 'Talbot time', is accompanied by a kind of partial revival at any rational fraction of the period, increasing in complexity for less simple fractions. In particular, any piecewise constant initial wavefunction is again piecewise constant at such times. By contrast, in the sphere case, the simplest piecewise constant wave, constant on hemispheres is shown not to retain its piecewise constancy at rational fractions of the period, but instead, rather strikingly, to develop infinities at calculable locations. The calculation requires the uniform asymptotic form of the Legendre polynomials together with the Poisson sum formula leading to Gauss sums.
We introduce a geometrical framework to construct a large class of time-dependent quantum systems in which the position of a classical particle moving autonomously on a smooth connected manifold is used to steer a quantum Hamiltonian over time. This results in quantum drives with structured temporal profiles and properties dependent on the local and global nature of the underlying choice of manifold. We show that our construction recovers the well-known classes of periodically driven and quasiperiodically driven quantum systems, but also unveils fundamentally new classes of quantum dynamics: by utilizing a compact two-dimensional hyperbolic Bolza surface and a nonorientable Klein-bottle surface, we demonstrate examples of a hyperbolically driven quantum system and a nonorientably driven quantum system respectively. Furthermore, we demonstrate that these driven systems exhibit unusual quantized dynamical responses reflecting their different underlying topologies, under the condition of being fully gapped and in the adiabatic limit, and which have interpretations as quantized crystalline electromagnetic responses in certain exotic effective tight-binding lattice models. We envision geometric quantum driving as a general framework to chart the landscape of time-dependent quantum systems and investigate the universal phase structures they exhibit, as well as a useful tool to enhance the capabilities of modern day quantum simulators.
The evolution of the free-particle wave function in one dimension is the same as scalar Fresnel diffraction from a one-dimensional structure. Quantum mechanics courses often explore the propagation of Gaussian wave packets, but the diffractionlike mathematics is sufficiently tractable to investigate the propagation of other wave packets, both numerically and analytically. More importantly, the diffraction analogy facilitates the development of an intuitive understanding of the role that the initial phase plays in free-particle wave packet evolution. This article considers some of the effects of the initial phase function on the subsequent evolution of free-particle wave packets in the position representation. These considerations reinforce the idea that the classical mechanics limit embodied in the correspondence principle and formalized in the Ehrenfest theorem is necessarily an incomplete representation of quantum behavior.