The Second Law, Maxwell's Demon, and Work Derivable from Quantum Heat Engines

Abstract

With a class of quantum heat engines which consists of two-energy-eigenstate systems undergoing, respectively, quantum adiabatic processes and energy exchanges with heat baths at different stages of a cycle, we are able to clarify some important aspects of the second law of thermodynamics. The quantum heat engines also offer a practical way, as an alternative to Szilard's engine, to physically realise Maxwell's daemon. While respecting the second law on the average, they are also capable of extracting more work from the heat baths than is otherwise possible in thermal equilibrium.Present technology allows for the probing and realisation of quantum mechanical systems of mesoscopic and even macroscopic sizes (like those of superconductors, Bose-Einstein condensates, ...), which can also be restricted to a relatively small number of energy states. It is thus important to study these quantum systems directly in relation to the second law of thermodynamics, which has been applicable to composite systems. We will pursue below this path without assuming for the systems anything extra and beyond the principles of quantum mechanics [18]. Our study is part of a growing body of investigations into quantum heat engines [1,2,3,4,5,6]. Explicitly, the only principles we will need are those of the Schrödinger equation, the Born probability interpretation of the wavefunctions and the von Neumann measurement postulate [7]. In particular, we will not exclude, but will make full use of, any exceptional initial conditions, as long as they are realisable physically. However, without a better understanding of the emergence of classicality from quantum mechanics, we will have to assume the thermal equilibrium Gibbs distributions for the heat baths which are coupled to the quantum systems. This assumption is extra to those of quantum mechanics.The expectation value of the measured energy of a quantum system is U = E = i p i E i , in which E i are the energy levels and p i are the corresponding occupation probabilities. Infinitesimally,from which we make the following identifications for infinitesimal heat transferreddQ and work donedW ,Thus, equation (1) is just an expression of the first law, dU =dQ +dW . These identifications concur with the fact that work done on or by a system can only be performed through a change in the generalised coordinates of the system, which in turn gives rise to a change in the distribution of the energy levels [8].Our quantum heat engines are just two-level quantum systems, which are the quantum version of the Otto engines [5]. (They are readily extendable to systems of many discrete energy levels.) They could be realised with coherent macroscopic quantum systems like, for instance, a Bose-Einstein condensate confined to the bottom two energy levels of a trapping potential. The exact cyclicity will be enforced to ensure that upon completing a cycle all the output products of the engines are clearly displayed without any hidden effect.A cycle of the quantum heat engine has four stages:

Implications of Coupling in Quantum Thermodynamic Machines

“…When the working medium of the Otto cycle is classical ideal gas, the efficiency of the system is written as η = 1 − (V 1 /V 2 ) γ−1 , where V 1 and V 2 are initial and final volumes (V 1 < V 2 ) of the adiabatic expansion process and γ = C p /C v , is the ratio of the specific heats [33]. Similar to the classical cycle, the quantum Otto cycle consists of two adiabatic processes and two thermalization processes [34,35]. The system exchanges heat with the bath during the thermalization processes and the work is done when the system undergoes adiabatic processes.…”

“…Otto cycle with a single harmonic oscillator (or a spin system) constituting the working medium of the engine is studied in different works [27,30,34,35,38,39]. Here we briefly review this.…”

Entropy production and thermodynamic power of the squeezed thermal reservoir

“…1(b)] as in traditional quantum Otto cycles [54][55][56], while the isentropic expansion is allowed to unsqueeze the mode, which in turn will allow us to exploit the full power of the squeezed thermal reservoir.…”

Coupled quantum Otto cycle

“…When the working medium is a few-level quantum system, new lines of enquiry open up due to additional features like discreteness of states, quantum correlations, quantum coherence and so on [1][2][3][4][5][6][7]. Many models have served to investigate the validity of second law of thermodynamics in the quantum regime [8,9].…”