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European Journal of Physics

Simulating a one-half spin with two coupled pendula: II. The parametrically induced Rabi precession

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

Published Nov 26, 2009• 1 citation
Publisher

Abstract

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.

DOI
10.1088/0143-0807/31/1/015

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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.
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 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.
Abstract The behavior of a magnetic dipole in magnetic fields is of interest in many systems including magnetic resonance imaging. The explanation presented in this chapter combines quantum physics and electricity and magnetism. The chapter begins with an explanation of magnetic dipole Larmor precession in static magnetic fields. The discussion then shifts to time-dependent magnetic fields and how they lead to transitions between the dipole’s energy states. The calculations are carried out both by using the time dependence of energy eigenstates and by solving the differential equations describing the quantum state vector amplitudes. The results are expressed in terms of the Rabi frequency and the Rabi formula. For the special cases of π and π/2 pulses of the oscillating magnetic field, the results show how those pulses implement quantum gates for the dipole system.
Quantum systems driven by strong oscillating fields are the source of many interesting physical phenomena. In this work, we experimentally study the dynamics of a two-level system of a single spin driven in the strong-driving regime where the rotating-wave approximation is not valid. This two-level system is a subsystem of a single nitrogen-vacancy (NV) center in diamond coupled to a first-shell 13C^{13}\mathrm{C}13C nuclear spin at a level anticrossing point. This near degeneracy occurs in the ${m}_{s}=\ifmmode\pm\else\textpm\fi{}1$ manifold of the electron spin when the energy level splitting between the ${m}_{s}=\ensuremath{-}1$ and +1+1+1 states due to the static magnetic field is $\ensuremath{\approx}127$ MHz and thus equal to the splitting due to the 13C^{13}\mathrm{C}13C hyperfine interaction. The transition frequency of this electron spin two-level system in a static magnetic field of 28.9 G is 1.7 MHz and it can be driven only by the component of the radio-frequency (RF) field along the NV symmetry axis. Electron spin Rabi frequencies in this system can reach tens of MHz even for moderate RF powers. The simple sinusoidal Rabi oscillations that occur when the amplitude of the driving field is small compared to the transition frequency evolve into complex patterns when the driving field amplitude is comparable to or greater than the energy level splitting. We observe that the system oscillates faster than the amplitude of the driving field and the response of the system shows multiple frequencies.
One of the simplest nontrivial problems involving the atom–field interaction is the coupling of a two-level atom with a single mode of the electromagnetic field. A two-level atom description is valid if the two atomic levels involved are resonant or nearly resonant with the driving field, while all other levels are highly detuned. Under certain realistic approximations, it is possible to reduce this problem to a form which can be solved exactly; allowing essential features of the atom-field interaction to be extracted. In this chapter we present a semiclassical theory of the interaction of a single two-level atom with a single mode of the field in which the atom is treated as a quantum two-level system and the field is treated classically. A fully quantum mechanical theory will be presented in Chapter 6. A two-level atom is formally analogous to a spin-1/2 system with two possible states. In the dipole approximation, when the field wavelength is larger than the atomic size, the atom–field interaction problem is mathematically equivalent to a spin-1/2 particle interacting with a time-dependent magnetic field. Just as the spin-1/2 system undergoes the so-called Rabi oscillations between the spin-up and spin-down states under the action of an oscillating magnetic field, the two-level atom also undergoes optical Rabi oscillations under the action of the driving electromagnetic field. These oscillations are damped if the atomic levels decay. An understanding of this simple model of the atom–field interaction enables us to consider more complicated problems involving an ensemble of atoms interacting with the field.