In this paper an invariant of motion for Hamiltonian systems is introduced: the Casimir companion. For systems with simple dynamical algebras (e.g., coupled spins, harmonic oscillators) our invariant is useful in problems that consider adiabatically varying the parameters in the Hamiltonian. In particular, it has proved useful in optimal control of changes in these parameters. The Casimir companion also allows simple calculation of the entropy of nonequilibrium ensembles.

Reversible adiabatic processes connecting thermal equilibrium states are usually considered to be infinitely slow. Recently, fast reversible adiabatic processes for quantum systems have been discussed. Here we present time-optimal processes for a paradigmatic ensemble of two interacting spin-1 2 systems in an external magnetic field, which previously had been employed as working fluid in a quantum refrigerator. These processes are realized by appropriate bang-bang or quasi-bang-bang controls of the external magnetic field. Explicit control protocols including the necessary times for a transition connecting thermal equilibrium states depending on the limiting conditions on the magnetic field strength are presented.

The entropy production paradox for anomalous diffusion processes describes a phenomenon where one-parameter families of dynamical equations, falling between the diffusion and wave equations, have entropy production rates (Shannon, Tsallis or Renyi) that increase toward the wave equation limit unexpectedly. Moreover, also surprisingly, the entropy does not order the bridging regime between diffusion and waves at all. However, it has been found that relative entropies, with an appropriately chosen reference distribution, do. Relative entropies, thus, provide a physically sensible way of setting which process is "nearer" to pure diffusion than another, placing pure wave propagation, desirably, "furthest" from pure diffusion. We examine here the time behavior of the relative entropies under the evolution dynamics of the underlying one-parameter family of dynamical equations based on space-fractional derivatives.

We discuss fastest effectively adiabatic transitions (FEATs) for a collection of noninteracting harmonic oscillators with shared controllable real frequencies. The construction of such transitions is presented for given initial and final equilibrium states, and the dependence of the minimum time control on the interval of achievable frequencies is discussed. While the FEAT times and associated FEAT processes are important in their own right as optimal controls, the FEAT time is an added feature which provides a measure of the quality of a shortcut to adiabaticity (STA). The FEAT time is evaluated for a previously reported experiment, wherein a cloud of Rb atoms is cooled following a STA recipe that took about twice as long as the FEAT speed limit, a time efficiency of 50%.

The spectral properties of a dense exciton gas are discussed using an exciton self-energy in second collisional Born approximation with respect to exciton-exciton interaction. The shift of the continuum edge and the 1s exciton as well as the increase of the damping of the 1s exciton are calculated linearly in the exciton density. The distribution of 1s excitons is assumed to be either a macro-occupation of the k = 0 state or a Boltzmann distribution with effective temperature varying between 10 and 60 K. A weakening of the level shift is obtained with increasing temperature. I n the case of macro-occupation damping is calculated self-consistently resulting in a sublinear dependence on density.Die Spektraleigenschaften eines dichten Exzitonengases werden bei Benutzung einer ExzitonSelbstenergie in zweiter Bornscher Nflherung in bezug auf die Exziton-Exziton-Wechselwirkung diskutiert. Sowohl die Shifts der Kontinuumskante und des 1s-Exzitons als auch das Anwaohsen der Dflmpfung des 1s-Exzitons werden linear in der Exziton-Dichte berechnet. Die Verteilung des 1s-Exzitons wird entweder als Makrobesetzung des k = 0-Zustandes oder als Boltzmann-Verteilung mit einer effektiven Temperatur, die zwischen 10 und 60 K variiert wird, angesetzt. Der Shift wird mit zunehmender Temperatur abgeschwacht. I m Falle der Makrobesetzung wird die Dampfung selbstkonsistent berechnet, was zu einer sublinearen AbhSngigkeit von der Dichte fuhrt.