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:

We introduce a class of quantum heat engines which consists of two-energy-eigenstate systems, the simplest of quantum mechanical systems, undergoing quantum adiabatic processes and energy exchanges with heat baths, respectively, at different stages of a cycle. Armed with this class of heat engines and some interpretation of heat transferred and work performed at the quantum level, we are able to clarify some important aspects of the second law of thermodynamics. In particular, it is not sufficient to have the heat source hotter than the sink, but there must be a minimum temperature difference between the hotter source and the cooler sink before any work can be extracted through the engines. The size of this minimum temperature difference is dictated by that of the energy gaps of the quantum engines involved. Our new quantum heat engines also offer a practical way, as an alternative to Szilard's engine, to physically realise Maxwell's daemon. Inspired and motivated by the Rabi oscillations, we further introduce some modifications to the quantum heat engines with single-mode cavities in order to, while respecting the second law, extract more work from the heat baths than is otherwise possible in thermal equilibria. Some of the results above are also generalisable to quantum heat engines of an infinite number of energy levels including 1-D simple harmonic oscillators and 1-D infinite square wells.

Abstract. We propose the use of periodic arrays of permanent magnetic films for producing magnetic lattices of microtraps for confining, manipulating and controlling small clouds of ultracold atoms and quantum degenerate gases. Using analytical expressions and numerical calculations we show that periodic arrays of magnetic films can produce one-dimensional (1D) and two-dimensional (2D) magnetic lattices with non-zero potential minima, allowing ultracold atoms to be trapped without losses due to spin flips. In particular, we show that two crossed layers of periodic arrays of parallel rectangular magnets plus bias fields, or a single layer of periodic arrays of square-shaped magnets with three different thicknesses plus bias fields, can produce 2D magnetic lattices of microtraps having nonzero potential minima and controllable trap depth. For arrays with micron-scale periodicity, the magnetic microtraps can have very large trap depths (∼0.5 mK for the realistic parameters chosen for the 2D lattice) and very tight confinement.

We employ quantum mechanical principles in the computability exploration of the class of classically noncomputable Hilbert's tenth problem which is equivalent to the Turing halting problem in Computer Science. The Quantum Adiabatic Theorem enables us to establish a connection between the solution for this class of problems and the asymptotic behaviour of solutions of a particular type of time-dependent Schrödinger equations. We then present some preliminary numerical simulation results for the quantum adiabatic processes corresponding to various Diophantine equations.

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