Mathias Michel

Modeling heat transport through completely positive maps.

We investigate heat transport in a spin-1/2 Heisenberg chain, coupled locally to independent thermal baths of different temperature. The analysis is carried out within the framework of the theory of open systems by means of appropriate quantum master equations. The standard microscopic derivation of the weak-coupling Lindblad equation in the secular approximation is considered, and shown to be inadequate for the description of stationary nonequilibrium properties like a nonvanishing energy current. Furthermore, we derive an alternative master equation that is capable of describing a stationary energy current and, at the same time, leads to a completely positive dynamical map. This paves the way for efficient numerical investigations of heat transport in larger systems based on Monte Carlo wave function techniques.

Non-Markovian quantum dynamics: correlated projection superoperators and Hilbert space averaging.

The time-convolutionless (TCL) projection operator technique allows a systematic analysis of the non-Markovian quantum dynamics of open systems. We present a class of projection superoperators that project the states of the total system onto certain correlated system-environment states. It is shown that the application of the TCL technique to this class of correlated superoperators enables the nonperturbative treatment of the dynamics of system-environment models for which the standard approach fails in any finite order of the coupling strength. We demonstrate further that the correlated superoperators correspond to the idea of a best guess of conditional quantum expectations, which is determined by a suitable Hilbert-space average. The general approach is illustrated by means of the model of a spin that interacts through randomly distributed couplings with a finite reservoir consisting of two energy bands. Extensive numerical simulations of the full Schrödinger equation of the model reveal the power and efficiency of the method.

Driven spin systems as quantum thermodynamic machines: Fundamental limits

We show that coupled two-level systems like qubits studied in quantum-information processing can be used as a thermodynamic machine. At least three qubits or spins are necessary and they must be arranged in a chain. The system is interfaced between two split baths and the working spin in the middle is externally driven. The machine performs Carnot-type cycles and is able to work as a heat pump or engine depending on the temperature difference of the baths, DeltaT , and the energy difference in the spin system, DeltaE . It can be shown that the efficiency is a function of DeltaT and DeltaE .

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.

Thermalization of quantum systems by finite baths

We consider a discrete quantum system coupled to a finite bath, which may consist of only one particle, in contrast to the standard baths which usually consist of continua of oscillators, spins, etc. We find that such finite baths may nevertheless equilibrate the system though not necessarily in the way predicted by standard open-system techniques. This behavior results regardless of the initial state being correlated or not.

Transport in open spin chains: A Monte Carlo wave-function approach

We investigate energy transport in several two-level atom or spin-1/2 models by a direct coupling to heat baths of different temperatures. The analysis is carried out on the basis of a recently derived quantum master equation which describes the nonequilibrium properties of internally weakly coupled systems appropriately. For the computation of the stationary state of the dynamical equations, we employ a Monte Carlo wave-function approach. The analysis directly indicates normal diffusive or ballistic transport in finite models and hints toward an extrapolation of the transport behavior of infinite models.

Cyclic cooling algorithm

We introduce a scheme to perform the cooling algorithm, first presented by Boykin et al. in 2002, for an arbitrary number of times on the same set of qbits. We achieve this goal by adding an additional SWAP gate and a bath contact to the algorithm. This way one qbit may repeatedly be cooled without adding additional qbits to the system. By using a product Liouville space to model the bath contact we calculate the density matrix of the system after a given number of applications of the algorithm.

Small quantum networks operating as quantum thermodynamic machines

We show that a 3-qubit system as studied for quantum information purposes can alternatively be used as a thermodynamic machine when externally driven in finite time and interfaced between two split baths. The spins are arranged in a chain where the working spin in the middle exercises Carnot cycles the area of which defines the exchanged work. The cycle orientation (sign of the exchanged work) flips as the difference of bath temperatures goes through a critical value.

MICROSCOPIC QUANTUM MECHANICAL FOUNDATION OF FOURIER'S LAW

Besides the growing interest in old concepts such as temperature and entropy at the nanoscale, theories of relaxation and transport have recently regained a lot of attention. With the electronic circuits and computer chips getting smaller and smaller, a fresh look on the equilibrium and nonequilibrium thermodynamics at small length scales far below the thermodynamic limit, i.e. on the theoretical understanding of original macroscopic processes, e.g. transport of energy, heat, charge, mass, magnetization, etc., should be appropriate. Only from the foundations of a theory its limits of applicability may be inferred. This review tries to give an overview on the background and recent developments in the field of nonequilibrium quantum thermodynamics, focusing on the transport of heat in small quantum systems.