Quantum gears: a simple mechanical system in the quantum regime

With the rapid rise in interest in microand nano-mechanical and electromechanical devices it will soon be necessary to consider the quantum aspects of the behaviour of such mechanical devices in the same way as the development of microand nano-scale electronic devices forces us to consider quantum finite size and interference effects as well as single electron effects (Drexler 1992, Cleland & Roukes 1999, Cleland 2002). As a model system to investigate the sort of effects which the onset of quantum behaviour could give rise to, a system of gears is considered. Such a system is a vital component of most classical “machines” but one for which it is easy to anticipate different behaviour in the quantum system. Classically the angular velocities of the gears are locked together by the action of the teeth, whereas in the quantum system the angular momenta of the individual gears will be quantised in a way which is not necessarily consistent with the expected classical solution. The system of 2 gears serves as a model system for any nano–machine in which circular (or spherical components) are constrained to rotate at angular frequencies which have a fixed ratio to each other, but are also subject to quantisation of their angular momenta. One could consider, for example, anything involving different sized wheels moving along a surface, such as a “penny–farthing” bicycle. For 2 gears with n1 and n2 teeth their angular velocities, ω1 and ω2, are locked together as n1ω1 + n2ω2 = 0. On the other hand each of them is subject to an angular momentum quantisation, L = Iω = m~ where I is the moment of inertia and m is an integer. By eliminating the 2 ω’s we obtain a constraint on the behaviour of the pair of gears