263 research outputs found
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
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
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
A 'User-Friendly' Approach to the Dynamical Equations of Non-Holonomic Systems
Two effective methods for writing the dynamical equations for non-holonomic systems are illustrated. They are based on the two types of representation of the constraints: by parametric equations or by implicit equations. They can be applied to linear as well as to non-linear constraints. Only the basic notions of vector calculus on Euclidean 3-space and on tangent bundles are needed. Elementary examples are illustrated
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
Dynamical localization simulated on a few qubits quantum computer
We show that a quantum computer operating with a small number of qubits can
simulate the dynamical localization of classical chaos in a system described by
the quantum sawtooth map model. The dynamics of the system is computed
efficiently up to a time , and then the localization length
can be obtained with accuracy by means of order computer runs,
followed by coarse grained projective measurements on the computational basis.
We also show that in the presence of static imperfections a reliable
computation of the localization length is possible without error correction up
to an imperfection threshold which drops polynomially with the number of
qubits.Comment: 8 pages, 8 figure
Driven cofactor systems and Hamilton-Jacobi separability
This is a continuation of the work initiated in a previous paper on so-called
driven cofactor systems, which are partially decoupling second-order
differential equations of a special kind. The main purpose in that paper was to
obtain an intrinsic, geometrical characterization of such systems, and to
explain the basic underlying concepts in a brief note. In the present paper we
address the more intricate part of the theory. It involves in the first place
understanding all details of an algorithmic construction of quadratic integrals
and their involutivity. It secondly requires explaining the subtle way in which
suitably constructed canonical transformations reduce the Hamilton-Jacobi
problem of the (a priori time-dependent) driven part of the system into that of
an equivalent autonomous system of St\"ackel type
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 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 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 .
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.
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