Bi-Hamiltonian representation of St\"{a}ckel systems

It is shown that a linear separation relations are fundamental objects for integration by quadratures of St\"{a}ckel separable Liouville integrable systems (the so-called St\"{a}ckel systems). These relations are further employed for the classification of St\"{a}ckel systems. Moreover, we prove that {\em any} St\"{a}ckel separable Liouville integrable system can be lifted to a bi-Hamiltonian system of Gel'fand-Zakharevich type. In conjunction with other known result this implies that the existence of bi-Hamiltonian representation of Liouville integrable systems is a necessary condition for St\"{a}ckel separability.Comment: To appear in Physical Review

From St\"{a}ckel systems to integrable hierarchies of PDE's: Benenti class of separation relations

We propose a general scheme of constructing of soliton hierarchies from finite dimensional St\"{a}ckel systems and related separation relations. In particular, we concentrate on the simplest class of separation relations, called Benenti class, i.e. certain St\"{a}ckel systems with quadratic in momenta integrals of motion.Comment: 24 page

Integrability, Stäckel spaces, and rational potentials

For a variety of classical mechanical systems embeddable into flat space with Cartesian coordinates {xi} and for which the Hamilton–Jacobi equation can be solved via separation of variables in a particular curvalinear system {uj}, we answer the following question. When is the separable potential function v expressible as a polynomial (or as a rational function) in the defining coordinates {xi}? Many examples are given

Statistical properties of eigenvalues for an operating quantum computer with static imperfections

We investigate the transition to quantum chaos, induced by static imperfections, for an operating quantum computer that simulates efficiently a dynamical quantum system, the sawtooth map. For the different dynamical regimes of the map, we discuss the quantum chaos border induced by static imperfections by analyzing the statistical properties of the quantum computer eigenvalues. For small imperfection strengths the level spacing statistics is close to the case of quasi-integrable systems while above the border it is described by the random matrix theory. We have found that the border drops exponentially with the number of qubits, both in the ergodic and quasi-integrable dynamical regimes of the map characterized by a complex phase space structure. On the contrary, the regime with integrable map dynamics remains more stable against static imperfections since in this case the border drops only algebraically with the number of qubits.Comment: 9 pages, 10 figure

Comment on "Coherent Ratchets in Driven Bose-Einstein Condensates"

C. E. Creffield and F. Sols (Phys. Rev. Lett. 103, 200601 (2009)) recently reported finite, directed time-averaged ratchet current, for a noninteracting quantum particle in a periodic potential even when time-reversal symmetry holds. As we explain in this Comment, this result is incorrect, that is, time-reversal symmetry implies a vanishing current.Comment: revised versio

Maximal superintegrability of Benenti systems

For a class of Hamiltonian systems naturally arising in the modern theory of separation of variables, we establish their maximal superintegrability by explicitly constructing the additional integrals of motion.Comment: 5 pages, LaTeX 2e, to appear in J. Phys. A: Math. Ge

Entanglement transitions in random definite particle states

Entanglement within qubits are studied for the subspace of definite particle states or definite number of up spins. A transition from an algebraic decay of entanglement within two qubits with the total number $N$ of qubits, to an exponential one when the number of particles is increased from two to three is studied in detail. In particular the probability that the concurrence is non-zero is calculated using statistical methods and shown to agree with numerical simulations. Further entanglement within a block of $m$ qubits is studied using the log-negativity measure which indicates that a transition from algebraic to exponential decay occurs when the number of particles exceeds $m$. Several algebraic exponents for the decay of the log-negativity are analytically calculated. The transition is shown to be possibly connected with the changes in the density of states of the reduced density matrix, which has a divergence at the zero eigenvalue when the entanglement decays algebraically.Comment: Substantially added content (now 24 pages, 5 figures) with a discussion of the possible mechanism for the transition. One additional author in this version that is accepted for publication in Phys. Rev.

Global Superdiffusion of Weak Chaos

A class of kicked rotors is introduced, exhibiting accelerator-mode islands (AIs) and {\em global} superdiffusion for {\em arbitrarily weak} chaos. The corresponding standard maps are shown to be exactly related to generalized web maps taken modulo an ``oblique cylinder''. Then, in a case that the web-map orbit structure is periodic in the phase plane, the AIs are essentially {\em normal} web islands folded back into the cylinder. As a consequence, chaotic orbits sticking around the AI boundary are accelerated {\em only} when they traverse tiny {\em ``acceleration spots''}. This leads to chaotic flights having a quasiregular {\em steplike} structure. The global weak-chaos superdiffusion is thus basically different in nature from the strong-chaos one in the usual standard and web maps.Comment: REVTEX, 4 Figures: fig1.jpg, fig2.ps, fig3.ps, fig4.p

Thermodynamic Bounds on Efficiency for Systems with Broken Time-reversal Symmetry

We show that for systems with broken time-reversal symmetry the maximum efficiency and the efficiency at maximum power are both determined by two parameters: a "figure of merit" and an asymmetry parameter. In contrast to the time-symmetric case, the figure of merit is bounded from above; nevertheless the Carnot efficiency can be reached at lower and lower values of the figure of merit and far from the so-called strong coupling condition as the asymmetry parameter increases. Moreover, the Curzon-Ahlborn limit for efficiency at maximum power can be overcome within linear response. Finally, always within linear response, it is allowed to have simultaneously Carnot efficiency and non-zero power.Comment: Final version, 4 pages, 3 figure

Extended phase space for a spinning particle

Extended phase space of an elementary (relativistic) system is introduced in the spirit of the Souriau's definition of the `space of motions' for such system. Our formulation is generally applicable to any homogeneous space-time (e.g. de Sitter) and also to Poisson actions. Calculations concerning the Minkowski case for non-zero spin particles show an intriguing alternative: we should either accept two-dimensional trajectories or (Poisson) noncommuting space-time coordinates.Comment: 12 pages, late