Guenter Mahler

Existence of Temperature on the Nanoscale

We consider a regular chain of quantum particles with nearest neighbor interactions in a canonical state with temperature T. We analyze the conditions under which the state factors into a product of canonical density matrices with respect to groups of n particles each and under which these groups have the same temperature T. In quantum mechanics the minimum group size n(min) depends on the temperature T, contrary to the classical case. We apply our analysis to a harmonic chain and find that n(min)=const for temperatures above the Debye temperature and n(min) proportional to T(-3) below.

Work extremum principle: structure and function of quantum heat engines.

We consider a class of quantum heat engines consisting of two subsystems interacting with a work-source and coupled to two separate baths at different temperatures Th>Tc. The purpose of the engine is to extract work due to the temperature difference. Its dynamics is not restricted to the near equilibrium regime. The engine structure is determined by maximizing the extracted work under various constraints. When this maximization is carried out at finite power, the engine dynamics is described by well-defined temperatures and satisfies the local version of the second law. In addition, its efficiency is bounded from below by the Curzon-Ahlborn value 1-radical Tc/Th and from above by the Carnot value 1-(Tc/Th). The latter is reached-at finite power--for a macroscopic engine, while the former is achieved in the equilibrium limit Th-->Tc . The efficiency that maximizes the power is strictly larger than the Curzon-Ahloborn value. When the work is maximized at a zero power, even a small (few-level) engine extracts work right at the Carnot efficiency.

Suppression of arbitrary internal coupling in a quantum register

For the implementation of a quantum computer it is necessary to exercise complete control over the Hamiltonian of the used physical system. For NMR quantum computing the effectively acting Hamiltonian can be manipulated via pulse sequences. Here we examine a register consisting of N selectively addressable spins with pairwise coupling between each spin pair. We show that complete decoupling of the spins is possible independent of the particular form of the spin-spin interaction. The proposed method based on orthogonal arrays is efficient in the sense that the effort regarding time and amount of pulses increases only polynomially with the size N of the register. However, the effect of external control errors in terms of inaccurate control pulses eventually limits the achievable precision.

Entanglement and the factorization-approximation

Abstract:For a bi-partite quantum system defined in a finite dimensional Hilbert-space we investigate in what sense entanglement change and interactions imply each other. For this purpose we introduce an entanglement-operator, which is then shown to represent a non-conserved property for any bi-partite system and any type of interaction. This general relation does not exclude the existence of special initial product states, for which the entanglement remains small over some period of time, despite interactions. For this case we derive an approximation to the full Schrödinger-equation, which allows the treatment of the composite systems in terms of product states. The induced error is estimated. In this factorization-approximation one subsystem appears as an effective potential for the other. A pertinent example is the Jaynes-Cummings model, which then reduces to the semi-classical rotating wave approximation.

Fourier's Law from Schrodinger Dynamics

We consider a class of one-dimensional chains of weakly coupled many level systems. We present a theory which predicts energy diffusion within these chains for almost all initial states, if some concrete conditions on their Hamiltonians are met. By numerically solving the time dependent Schrödinger equation, we verify this prediction. Close to equilibrium we analyze this behavior in terms of heat conduction and compute the respective coefficient directly from the theory.

Fourier's Law confirmed for a class of small quantum systems

Abstract.Within the Lindblad formalism we consider an interacting spin chain coupled locally to heat baths. We investigate the dependence of the energy transport on the type of interaction in the system as well as on the overall interaction strength. For a large class of couplings we find a normal heat conduction and confirm Fouriers Law. In a fully quantum mechanical approach linear transport behavior appears to be generic even for small quantum systems.

Spectral Densities and Partition Functions of Modular Quantum ystems as Derived from a Central Limit Theorem

Using a central limit theorem for arrays of interacting quantum systems, we give analytical expressions for the density of states and the partition function at finite temperature of such a system, which are valid in the limit of infinite number of subsystems. Even for only small numbers of subsystems we find good accordance with some known, exact results.

Thermodynamic limits of dynamic cooling

We study dynamic cooling, where an externally driven two-level system is cooled via reservoir, a quantum system with initial canonical equilibrium state. We obtain explicitly the minimal possible temperature T(min)>0 reachable for the two-level system. The minimization goes over all unitary dynamic processes operating on the system and reservoir and over the reservoir energy spectrum. The minimal work needed to reach T(min) grows as 1/T(min). This work cost can be significantly reduced, though, if one is satisfied by temperatures slightly above T(min). Our results on T(min)>0 prove unattainability of the absolute zero temperature without ambiguities that surround its derivation from the entropic version of the third law. We also study cooling via a reservoir consisting of N≫1 identical spins. Here we show that T(min)∝1/N and find the maximal cooling compatible with the minimal work determined by the free energy. Finally we discuss cooling by reservoir with an initially microcanonic state and show that although a purely microcanonic state can yield the zero temperature, the unattainability is recovered when taking into account imperfections in preparing the microcanonic state.

Quantum Brownian motion and the second law of thermodynamics

Abstract.We consider a single harmonic oscillator coupled to a bath at zero temperature. As is well-known, the oscillator then has a higher average energy than that given by its ground state. Here we show analytically that for a damping model with arbitrarily discrete distribution of bath modes and damping models with continuous distributions of bath modes with cut-off frequencies, this excess energy is less than the work needed to couple the system to the bath, therefore, the quantum second law is not violated. On the other hand, the second law may be violated for bath modes without cut-off frequencies, which are, however, physically unrealistic models.

Uncertainty rescued: Bohr's complementarity for composite systems

Abstract Generalized uncertainty relations may depend not only on the commutator relation of two observables considered, but also on mutual correlations, in particular, on entanglement. The equivalence between the uncertainty relation and Bohrs complementarity thus holds in a much broader sense than anticipated.