Symplectic duality between complex domains

In this paper after extending the denition of symplectic duality (given in [3] for bounded symmetric domains ) to arbitrary complex domains of Cn centered at the origin we generalize some of the results proved in [3] and [4] to those domain

Kahler manifolds and their relatives

Let M1 and M2 be two K¨ahler manifolds. We call M1 and M2 relatives if they share a non-trivial K¨ahler submanifold S, namely, if there exist two holomorphic and isometric immersions (K¨ahler immersions) h1 : S → M1 and h2 : S → M2. Moreover, two K¨ahler manifolds M1 and M2 are said to be weakly relatives if there exist two locally isometric (not necessarily holomorphic) K¨ahler manifolds S1 and S2 which admit two K¨ahler immersions into M1 and M2 respectively. The notions introduced are not equivalent (cf. Example 2.3). Our main results in this paper are Theorem 1.2 and Theorem 1.4. In the first theorem we show that a complex bounded domain D ⊂ Cn with its Bergman metric and a projective K¨ahler manifold (i.e. a projective manifold endowed with the restriction of the Fubini-Study metric) are not relatives. In the second theorem we prove that a Hermitian symmetric space of noncompact type and a projective K¨ahler manifold are not weakly relatives. Notice that the proof of the second result does not follows trivially from the first one. We also remark that the above results are of local nature, i.e. no assumptions are used about the compactness or completeness of the manifolds involve

Geometric phase accumulation-based effects in the quantum dynamics of an anisotropically trapped ion

New physical effects in the dynamics of an ion confined in an anisotropic two-dimensional Paul trap are reported. The link between the occurrence of such manifestations and the accumulation of geometric phase stemming from the intrinsic or controlled lack of symmetry in the trap is brought to light. The possibility of observing in laboratory these anisotropy-based phenomena is briefly discussed.Comment: 10 pages. Acta Physica Hungarica B 200

Quasi-saddles as relevant points of the potential energy surface in the dynamics of supercooled liquids

The supercooled dynamics of a Lennard-Jones model liquid is numerically investigated studying relevant points of the potential energy surface, i.e. the minima of the square gradient of total potential energy $V$. The main findings are: ({\it i}) the number of negative curvatures $n$ of these sampled points appears to extrapolate to zero at the mode coupling critical temperature $T_c$; ({\it ii}) the temperature behavior of $n(T)$ has a close relationship with the temperature behavior of the diffusivity; ({\it iii}) the potential energy landscape shows an high regularity in the distances among the relevant points and in their energy location. Finally we discuss a model of the landscape, previously introduced by Madan and Keyes [J. Chem. Phys. {\bf 98}, 3342 (1993)], able to reproduce the previous findings.Comment: To be published in J. Chem. Phy

Dissipation and entanglement dynamics for two interacting qubits coupled to independent reservoirs

We derive the master equation of a system of two coupled qubits by taking into account their interaction with two independent bosonic baths. Important features of the dynamics are brought to light, such as the structure of the stationary state at general temperatures and the behaviour of the entanglement at zero temperature, showing the phenomena of sudden death and sudden birth as well as the presence of stationary entanglement for long times. The model here presented is quite versatile and can be of interest in the study of both Josephson junction architectures and cavity-QED.Comment: 14 pages, 3 figures, submitted to Journal of Physics A: Mathematical and Theoretica

Zeno Dynamics and High-Temperature Master Equations Beyond Secular Approximation

Complete positivity of a class of maps generated by master equations derived beyond the secular approximation is discussed. The connection between such class of evolutions and physical properties of the system is analyzed in depth. It is also shown that under suitable hypotheses a Zeno dynamics can be induced because of the high temperature of the bath.Comment: 9 pages, 2 figure

Non-Markovian dissipative dynamics of two coupled qubits in independent reservoirs: a comparison between exact solutions and master equation approaches

The reduced dynamics of two interacting qubits coupled to two independent bosonic baths is investigated. The one-excitation dynamics is derived and compared with that based on the resolution of appropriate non-Markovian master equations. The Nakajima-Zwanzig and the time-convolutionless projection operator techniques are exploited to provide a description of the non-Markovian features of the dynamics of the two-qubits system. The validity of such approximate methods and their range of validity in correspondence to different choices of the parameters describing the system are brought to light.Comment: 6 pages, 3 figures. Submitted to PR

GHZGHZ state generation of three Josephson qubits in presence of bosonic baths

We analyze an entangling protocol to generate tripartite Greenberger-Horne-Zeilinger states in a system consisting of three superconducting qubits with pairwise coupling. The dynamics of the open quantum system is investigated by taking into account the interaction of each qubit with an independent bosonic bath with an ohmic spectral structure. To this end a microscopic master equation is constructed and exactly solved. We find that the protocol here discussed is stable against decoherence and dissipation due to the presence of the external baths.Comment: 16 pages and 4 figure

The bisymplectomorphism group of a bounded symmetric domain

An Hermitian bounded symmetric domain in a complex vector space, given in its circled realization, is endowed with two natural symplectic forms: the flat form and the hyperbolic form. In a similar way, the ambient vector space is also endowed with two natural symplectic forms: the Fubini-Study form and the flat form. It has been shown in arXiv:math.DG/0603141 that there exists a diffeomorphism from the domain to the ambient vector space which puts in correspondence the above pair of forms. This phenomenon is called symplectic duality for Hermitian non compact symmetric spaces. In this article, we first give a different and simpler proof of this fact. Then, in order to measure the non uniqueness of this symplectic duality map, we determine the group of bisymplectomorphisms of a bounded symmetric domain, that is, the group of diffeomorphisms which preserve simultaneously the hyperbolic and the flat symplectic form. This group is the direct product of the compact Lie group of linear automorphisms with an infinite-dimensional Abelian group. This result appears as a kind of Schwarz lemma.Comment: 19 pages. Version 2: minor correction