Vanishing efficiency of speeded-up quantum Otto engines

Abstract

We assess the energy cost of shortcuts to adiabatic expansions or compressions of a harmonic oscillator, the power strokes of a quantum Otto engine. Difficulties to identify the cost stem from the interplay between different parts of the total system (the primary system —the particle— and the control system) and definitions of work (exclusive and inclusive). While attention is usually paid to the inclusive work of the microscopic primary system, we identify the energy cost as the exclusive work of the total system, which, for a clear-cut scale disparity between a microscopic primary system and a macroscopic control system, coincides with the exclusive work for the control system alone. We redefine the "engine efficiency" taking into account this cost. Our working horse model is an engine based on an ion in a Paul trap with power strokes designed via shortcuts to adiabaticity. Opposite to the paradigm of slow-cycle reversible engines with vanishing power and maximal efficiency, this fast-cycle engine increases the microscopic power at the price of a vanishing efficiency. The Paul trap fixes the gauge for the primary system, resulting in a counterintuitive evolution of its inclusive power and internal energy. Conditions for which inclusive power of the primary system and exclusive power control system are proportional are found.