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European Journal of Physics

Simulating a one-half spin with two coupled pendula: the free Larmor precession

Valentin Leroy, J.-C. Bacri, Thierry Hocquet, Martin Devaud

Published Sep 27, 2006• 8 citations
Publisher

Abstract

It is shown that, with a simple two-pendulum device, one can experimentally simulate the behaviour of a one-half spin. After a quick overflight of the two-degree-of-freedom harmonic oscillator's properties, we show that, in the quasi-degeneracy limit, this system is formally equivalent to a one-half spin: its physical states can be described by state vectors belonging to a two-dimensional Hilbert space; its free motion obeys a Schrödinger equation. An -isomorphous representation space is used to illustrate this movement in terms of Larmor precession.

DOI
10.1088/0143-0807/27/6/011

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It is shown that, with a simple two-pendulum device, one can experimentally simulate the behaviour of a one-half spin. After a quick overflight of the two-degree-of-freedom harmonic oscillator's properties, we show that, in the quasi-degeneracy limit, this system is formally equivalent to a one-half spin: its physical states can be described by state vectors belonging to a two-dimensional Hilbert space; its free motion obeys a Schrödinger equation. An -isomorphous representation space is used to illustrate this movement in terms of Larmor precession.
It was already shown that, with a simplest classical device, namely two coupled pendula, one can simulate the movement of a free quantum two-level system, as for instance a one-half spin in a static magnetic field. We now show that, with a hardly more sophisticated device, one can simulate the latter system undergoing a sinusoidal perturbation. We present an experimental mechanical illustration of the transition of the system between its two eigenstates, known as the Rabi precession. Abstract theoretical concepts of absorption and stimulated emission of energy are thus provided with an intuitive support.
We propose a classical analogy to illustrate the up ↔ down flip of a one-half spin submitted to a static and a rotating magnetic field, in the standard magnetic resonance configuration. Surprising though it may seem, it is possible to design a rather simple device-namely a set of two coupled pendula-to simulate and visualize the Larmor and Rabi precessions of the latter spin. In this analogy, the static field is associated with the angular frequency detuning between both eigenmodes of the two-degree-of-freedom harmonic oscillator constituted by the two pendula, whereas the rotating field is simulated by means of a parametric modulation of the length of one of the pendula. Under the circumstances, the up or down spin states are represented by either eigenmodes, and consequently the spin-flip is put in concrete form by the coupled pendula being driven from one eigenmode to the other. In the present paper, we experimentally demonstrate the possibility of triggering such a flip using a mechanical analogy of the so-called rapid adiabatic passage technique.
It has already been shown that, with the simplest classical device, namely two coupled pendula, one can simulate the movement of a free quantum two-level system, as for instance a one-half spin in a static magnetic field. We now show that, with a hardly more sophisticated device, one can simulate the latter system undergoing outer perturbation. In the case of sinusoidal excitation, we present an experimental mechanical illustration of the transition of the system between its two eigenstates, known as the Rabi precession. Abstract theoretical concepts of absorption and stimulated emission of energy are thus provided with intuitive support.
In a Wilberforce pendulum, two mechanical oscillators are coupled: one pertains to the longitudinal (tension) motion and the other to the rotational (twisting) motion. It is shown that the longitudinal magnetic moment of circular currents, and similarly the magnetic moment of a spin-chain, can exhibit a Wilberforce-like vibration. The longitudinal oscillation is related to the Langevin diamagnetism, while the twisting motion is superimposed on the magnetic moment and spin precession. The calculations show that the coupling term is nonlinear in this (longitudinal) vibrating and (magnetic moment) precession system. By increasing the strength of the coupling we arrive at a spectrum, where further vibrational modes can be associated with the rotation of the precession. This means that the extent of the change in coherence can be demonstrated. Since the coupling strength can be different due to local effects, this can be an important factor from the point of view of signal propagation and in preserving signal shapes. The amount specifying the dissipation is introduced to express the degree of deviation. A relationship exists between the parameter characteristic of the coupling strength and how its quantity influences decoherence and dissipation.
Abstract We present two novel solutions of real Hilbert state quaternionic quantum mechanics . Firstly, we observe that the angular momentum operator admits two different classes of physically non-equivalent free particles. As a second result, we study the Larmor precession to observe that it has a quaternionic solution where a novel phenomenological interpretation is possible, as well as a different spin is possible, and these results may encourage experimental and theoretical investigations of the quaternionic theory.
We propose an intuitive introduction to some ideas of quantum mechanics, by means of a toy-model: a chain of coupled classical pendula. Quasi-particles are associated with the eigenmodes of the chain, ruled by a Schrödinger-like motion equation when using the classical Glauber variables. The set of the physical states of the system is shown to be a Hilbert C-vector space, in which observables can be defined. A continuous description of the chain leads to a Klein-Gordon motion equation. At last, a 3D extension is proposed, accompanied with a few speculative remarks, inter alia concerning the link with Special Relativity and the covariance of the Schrödinger-like equation.