Distribution of local entropy in the Hilbert space of bi-partite quantum systems: Origin of Jaynes' principle

For a closed bi-partite quantum system partitioned into system proper and environment we interprete the microcanonical and the canonical condition as constraints for the interaction between those two subsystems. In both cases the possible pure-state trajectories are confined to certain regions in Hilbert space. We show that in a properly defined thermodynamical limit almost all states within those accessible regions represent states of some maximum local entropy. For the microcanonical condition this dominant state still depends on the initial state; for the canonical condition it coincides with that defined by Jaynes' principle. It is these states which thermodynamical systems should generically evolve into.Comment: Submitted to Physical Review

Entanglement and the factorization-approximation

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 Schroedinger 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.Comment: Accepted for publication in European Physical Journal

Relaxation into equilibrium under pure Schr\"odinger dynamics

We consider bipartite quantum systems that are described completely by a state vector $|\Psi(t)>$ and the fully deterministic Schr\"odinger equation. Under weak constraints and without any artificially introduced decoherence or irreversibility, the smaller of the two subsystems shows thermodynamic behaviour like relaxation into an equilibrium, maximization of entropy and the emergence of the Boltzmann energy distribution. This generic behaviour results from entanglement.Comment: 5 pages, 9 figure

Scaling behavior of interactions in a modular quantum system and the existence of local temperature

We consider a quantum system of fixed size consisting of a regular chain of $n$-level subsystems, where $n$ is finite. Forming groups of $N$ subsystems each, we show that the strength of interaction between the groups scales with $N^{- 1/2}$. As a consequence, if the total system is in a thermal state with inverse temperature $\beta$, a sufficient condition for subgroups of size $N$ to be approximately in a thermal state with the same temperature is $\sqrt{N} \gg \beta \bar{\delta E}$, where $\bar{\delta E}$ is the width of the occupied level spectrum of the total system. These scaling properties indicate on what scale local temperatures may be meaningfully defined as intensive variables. This question is particularly relevant for non-equilibrium scenarios such as heat conduction etc.Comment: 7 pages, accepted for publication in Europhysics Letter

Model Studies on the Quantum Jarzynski Relation

We study the quantum Jarzynski relation for driven quantum models embedded in various environments. We do so by generalizing a proof presented by Mukamel [Phys. Rev. Lett 90, 170604 (2003)] for closed quantum systems. In this way, we are able to prove that the Jarzynski relation also holds for a bipartite system with microcanonical coupling. Furthermore, we show that, under the assumption that the interaction energy remains constant during the whole process, the relation is valid even for canonical coupling. The same follows for open quantum systems at high initial temperatures up to third order of the inverse temperature. Our analytical study is complemented by a numerical investigation of a special model system.Comment: 7 figure

Non Thermal Equilibrium States of Closed Bipartite Systems

We investigate a two-level system in resonant contact with a larger environment. The environment typically is in a canonical state with a given temperature initially. Depending on the precise spectral structure of the environment and the type of coupling between both systems, the smaller part may relax to a canonical state with the same temperature as the environment (i.e. thermal relaxation) or to some other quasi equilibrium state (non thermal relaxation). The type of the (quasi) equilibrium state can be related to the distribution of certain properties of the energy eigenvectors of the total system. We examine these distributions for several abstract and concrete (spin environment) Hamiltonian systems, the significant aspect of these distributions can be related to the relative strength of local and interaction parts of the Hamiltonian.Comment: RevTeX, 8 pages, 13 figure

Local effective dynamics of quantum systems: A generalized approach to work and heat

By computing the local energy expectation values with respect to some local measurement basis we show that for any quantum system there are two fundamentally different contributions: changes in energy that do not alter the local von Neumann entropy and changes that do. We identify the former as work and the latter as heat. Since our derivation makes no assumptions on the system Hamiltonian or its state, the result is valid even for states arbitrarily far from equilibrium. Examples are discussed ranging from the classical limit to purely quantum mechanical scenarios, i.e. where the Hamiltonian and the density operator do not commute.Comment: 5 pages, 1 figure, published versio

On conjectures and problems of Ruzsa concerning difference graphs of S-units

Given a finite nonempty set of primes S, we build a graph $\mathcal{G}$ with vertex set $\mathbb{Q}$ by connecting x and y if the prime divisors of both the numerator and denominator of x-y are from S. In this paper we resolve two conjectures posed by Ruzsa concerning the possible sizes of induced nondegenerate cycles of $\mathcal{G}$, and also a problem of Ruzsa concerning the existence of subgraphs of $\mathcal{G}$ which are not induced subgraphs.Comment: 15 page

Cavity-induced temperature control of a two-level system

We consider a two-level atom interacting with a single mode of the electromagnetic field in a cavity within the Jaynes-Cummings model. Initially, the atom is thermal while the cavity is in a coherent state. The atom interacts with the cavity field for a fixed time. After removing the atom from the cavity and applying a laser pulse the atom will be in a thermal state again. Depending on the interaction time with the cavity field the final temperature can be varied over a large range. We discuss how this method can be used to cool the internal degrees of freedom of atoms and create heat baths suitable for studying thermodynamics at the nanoscale