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Rapid adiabatic passage with two coupled pendula

Thierry Hocquet, Hugo Cavet, Mathéo Aksil, Martin Devaud

Published Mar 30, 2022• 0 citations
OpenAlex

Abstract

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.

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When a Rydberg atom is slowly swept through a dipole allowed resonance with an oscillating field adiabatic rapid passage (ARP) from one state to the other can occur. By measuring the time resolved momentum of a Rydberg electron with a half cycle pulse we demonstrate that during ARP the classical oscillating dipole due to the orbital motion of the electron becomes phase locked to the microwave field driving the resonance. Depending on the direction of the sweep through resonance, the atom is in one or the other of the two Floquet eigenstates in which the electron is phase locked in or $\ensuremath{\pi}$ out of phase with the driving field. In contrast, in Rabi oscillations the electron motion oscillates between leading and lagging the driving microwave field.

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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 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.
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 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.
We discuss two mechanical systems in which the approximate equations of motion are the same as the rapid adiabatic passage of a two-state quantum-mechanical system, such as that which occurs with quasistatic interactions and from the quasiperiodic action of a chirped-frequency laser pulse. We discuss a pair of masses on springs, weakly coupled by another spring, one of whose spring constants varies slowly, and a pair of coupled pendula, where the length of one pendulum slowly changes. In each example an approximation analogous to the rotating-wave approximation used in the corresponding quantum system brings the second-order macroscopic equations of motion into first-order form, and a slow variation in a system parameter characteristic leads to adiabatic change and rapid adiabatic passage.
Neutron resonance spin echo phenomena, produced by resonance coils with adiabatic passage of the neutron spin, are investigated experimentally and theoretically. The adiabatic passage of the neutron spin through a resonance coil requires a specific configuration of the magnetic field. The solution of the Schr\"odinger equation, obtained for the required configuration, shows the probability of a spin-flip process; it also shows the phase shift, which the neutron experienced, when it flies through it. The precession phase inside the coil consists of three contributions in the rotating frame approach. The first, biggest contribution is the phase of the rotating frame ${\ensuremath{\omega}}_{0}\ensuremath{\tau}.$ The second is the precession phase of the neutron spin in the rotating frame since it follows adiabatically the effective field as seen in this frame. The third, smallest contribution is Berry's phase since the magnetic field rotates over an angle approaching $\ensuremath{\pi}$ in this rotating frame. This rotation is followed adiabatically by the neutron spin. Finally, the amplitude of the interference pattern and the phase shift between the neutron-spin states are derived for a system consisting of two such flippers. The theoretical consideration is experimentally confirmed.
When a Rydberg atom is slowly swept through a dipole allowed resonance with an oscillating field adiabatic rapid passage (ARP) from one state to the other can occur. By measuring the time resolved momentum of a Rydberg electron with a half cycle pulse we demonstrate that during ARP the classical oscillating dipole due to the orbital motion of the electron becomes phase locked to the microwave field driving the resonance. Depending on the direction of the sweep through resonance, the atom is in one or the other of the two Floquet eigenstates in which the electron is phase locked in or $\ensuremath{\pi}$ out of phase with the driving field. In contrast, in Rabi oscillations the electron motion oscillates between leading and lagging the driving microwave field.