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G. Morandi

Publications and source records attributed to G. Morandi.

At least 19 recordsLinked to original sources
arXiv2013

Null Phase Curves and Manifolds in Geometric Phase Theory

Bargmann invariants and null phase curves are known to be important ingredients in understanding the essential nature of the geometric phase in quantum mechanics. Null phase manifolds in quantum-mechanical ray spaces are submanifolds made up entirely of null phase curves, and so are equally important for geometric phase considerations. It is shown that the complete characterization of null phase manifolds involves both the Riemannian metric structure and the symplectic structure of ray space in equal measure, which thus brings together these two aspects in a natural manner.

quant-ph↗
arXiv2010

Entanglement and nonclassicality for multi-mode radiation field states

Nonclassicality in the sense of quantum optics is a prerequisite for entanglement in multi-mode radiation states. In this work we bring out the possibilities of passing from the former to the latter, via action of classicality preserving systems like beamsplitters, in a transparent manner. For single mode states, a complete description of nonclassicality is available via the classical theory of moments, as a set of necessary and sufficient conditions on the photon number distribution. We show that when the mode is coupled to an ancilla in any coherent state, and the system is then acted upon by a beamsplitter, these conditions turn exactly into signatures of NPT entanglement of the output state. Since the classical moment problem does not generalize to two or more modes, we turn in these cases to other familiar sufficient but not necessary conditions for nonclassicality, namely the Mandel parameter criterion and its extensions. We generalize the Mandel matrix from one-mode states to the two-mode situation, leading to a natural classification of states with varying levels of nonclassicality. For two--mode states we present a single test that can, if successful, simultaneously show nonclassicality as well as NPT entanglement. We also develop a test for NPT entanglement after beamsplitter action on a nonclassical state, tracing carefully the way in which it goes beyond the Mandel nonclassicality test. The result of three--mode beamsplitter action after coupling to an ancilla in the ground state is treated in the same spirit. The concept of genuine tripartite entanglement, and scalar measures of nonclassicality at the Mandel level for two-mode systems, are discussed. Numerous examples illustrating all these concepts are presented.

quant-ph↗
arXiv2009

The Hamilton--Jacobi Theory and the Analogy between Classical and Quantum Mechanics

We review here some conventional as well as less conventional aspects of the time-independent and time-dependent Hamilton-Jacobi (HJ) theory and of its connections with Quantum Mechanics. Less conventional aspects involve the HJ theory on the tangent bundle of a configuration manifold, the quantum HJ theory, HJ problems for general differential operators and the HJ problem for Lie groups.

math-ph↗
arXiv2009

Effective mapping of spin-1 chains onto integrable fermionic models. A study of string and Neel correlation functions

We derive the dominant contribution to the large-distance decay of correlation functions for a spin chain model that exhibits both Haldane and Neel phases in its ground state phase diagram. The analytic results are obtained by means of an approximate mapping between a spin-1 anisotropic Hamiltonian onto a fermionic model of noninteracting Bogolioubov quasiparticles related in turn to the XY spin-1/2 chain in a transverse field. This approach allows us to express the spin-1 string operators in terms of fermionic operators so that the dominant contribution to the string correlators at large distances can be computed using the technique of Toeplitz determinants. As expected, we find long-range string order both in the longitudinal and in the transverse channel in the Haldane phase, while in the Neel phase only the longitudinal order survives. In this way, the long-range string order can be explicitly related to the components of the magnetization of the XY model. Moreover, apart from the critical line, where the decay is algebraic, we find that in the gapped phases the decay is governed by an exponential tail multiplied by algebraic factors. As regards the usual two points correlation functions, we show that the longitudinal one behaves in a 'dual' fashion with respect to the transverse string correlator, namely both the asymptotic values and the decay laws exchange when the transition line is crossed. For the transverse spin-spin correlator, we find a finite characteristic length which is an unexpected feature at the critical point. We also comment briefly the entanglement features of the original system versus those of the effective model. The goodness of the approximation and the analytical predictions are checked versus density-matrix renormalization group calculations.

cond-mat.stat-mech↗
arXiv2007

Alternative linear structures for classical and quantum systems

The possibility of deforming the (associative or Lie) product to obtain alternative descriptions for a given classical or quantum system has been considered in many papers. Here we discuss the possibility of obtaining some novel alternative descriptions by changing the linear structure instead. In particular we show how it is possible to construct alternative linear structures on the tangent bundle TQ of some classical configuration space Q that can be considered as "adapted" to the given dynamical system. This fact opens the possibility to use the Weyl scheme to quantize the system in different non equivalent ways, "evading", so to speak, the von Neumann uniqueness theorem.

quant-ph↗
arXiv2007

Ray space Riccati evolution and geometric phases for N-level quantum systems

We present a simple derivation of the matrix Riccati equations governing the reduced dynamics as one descends from the group U(N) describing the Schroedinger evolution of an N-level quantum system to the various coset spaces, Grassmanian manifolds, associated with it. The special case pertaining to the geometric phase in N-level systems is described in detail. Further, we show how the matrix Riccati equation thus obtained can be reformulated as an equation describing Hamiltonian evolution in a classical phase space and establish correspondences between the two descriptions.

quant-ph↗
arXiv2006

Stable particles in anisotropic spin-1 chains

Motivated by field-theoretic predictions we investigate the stable excitations that exist in two characteristic gapped phases of a spin-1 model with Ising-like and single-ion anisotropies. The sine-Gordon theory indicates a region close to the phase boundary where a stable breather exists besides the stable particles, that form the Haldane triplet at the Heisenberg isotropic point. The numerical data, obtained by means of the Density Matrix Renormalization Group, confirm this picture in the so-called large-D phase for which we give also a quantitative analysis of the bound states using standard perturbation theory. However, the situation turns out to be considerably more intricate in the Haldane phase where, to the best of our data, we do not observe stable breathers contrarily to what could be expected from the sine-Gordon model, but rather only the three modes predicted by a novel anisotropic extension of the Non-Linear Sigma Model studied here by means of a saddle-point approximation.

cond-mat.str-el↗
arXiv2006

Alternative linear structures associated with regular Lagrangians. Weyl quantization and the Von Neumann uniqueness theorem

We discuss how the existence of a regular Lagrangian description on the tangent bundle $TQ$ of some configuration space $Q$ allows for the construction of a linear structure on $TQ$ that can be considered as "adapted" to the given dynamical system. The fact then that many dynamical systems admit alternative Lagrangian descriptions opens the possibility to use the Weyl scheme to quantize the system in different non equivalent ways, "evading", so to speak, the von Neumann uniqueness theorem.

math-ph↗
arXiv2005

Local Measures of Entanglement and Critical Exponents at Quantum Phase Transitions

We discuss on general grounds some local indicators of entanglement, that have been proposed recently for the study and classification of quantum phase transitions. In particular, we focus on the capability of entanglement in detecting quantum critical points and related exponents. We show that the singularities observed in all local measures of entanglement are a consequence of the scaling hypothesis. In particular, as every non-trivial local observable is expected to be singular at criticality, we single out the most relevant one (in the renormalization group sense) as the best-suited for finite-size scaling analysis. The proposed method is checked on a couple of one-dimensional spin systems. The present analysis shows that the singular behaviour of local measures of entanglement is fully encompassed in the usual statistical mechanics framework.

quant-ph↗
arXiv2005

Phase-space descriptions of operators and the Wigner distribution in quantum mechanics I. A Dirac inspired view

Drawing inspiration from Dirac's work on functions of non commuting observables, we develop a fresh approach to phase space descriptions of operators and the Wigner distribution in quantum mechanics. The construction presented here is marked by its economy, naturalness and more importantly, by its potential for extensions and generalisations to situations where the underlying configuration space is non Cartesian.

quant-ph↗
arXiv2005

Phase-space descriptions of operators and the Wigner distribution in quantum mechanics II. The finite dimensional case

A complete solution to the problem of setting up Wigner distribution for N-level quantum systems is presented. The scheme makes use of some of the ideas introduced by Dirac in the course of defining functions of noncommuting observables and works uniformly for all N. Further, the construction developed here has the virtue of being essentially input-free in that it merely requires finding a square root of a certain N^2 x N^2 complex symmetric matrix, a task which, as is shown, can always be accomplished analytically. As an illustration, the case of a single qubit is considered in some detail and it is shown that one recovers the result of Feynman and Wootters for this case without recourse to any auxiliary constructs.

quant-ph↗
arXiv2005

Wigner distributions for finite dimensional quantum systems: An algebraic approach

We discuss questions pertaining to the definition of `momentum', `momentum space', `phase space', and `Wigner distributions'; for finite dimensional quantum systems. For such systems, where traditional concepts of `momenta' established for continuum situations offer little help, we propose a physically reasonable and mathematically tangible definition and use it for the purpose of setting up Wigner distributions in a purely algebraic manner. It is found that the point of view adopted here is limited to odd dimensional systems only. The mathematical reasons which force this situation are examined in detail.

quant-ph↗
arXiv2005

Particle Content of the Nonlinear Sigma Model with Theta-Term: a Lattice Model Investigation

Using new as well as known results on dimerized quantum spin chains with frustration, we are able to infer some properties on the low-energy spectrum of the O(3) Nonlinear Sigma Model with a topological theta-term. In particular, for sufficiently strong coupling, we find a range of values of theta where a singlet bound state is stable under the triplet continuum. On the basis of these results, we propose a new renormalization group flow diagram for the Nonlinear Sigma Model with theta-term.

cond-mat.stat-mech↗
arXiv2003

Geometric phase for mixed states: a differential geometric approach

A new definition and interpretation of geometric phase for mixed state cyclic unitary evolution in quantum mechanics are presented. The pure state case is formulated in a framework involving three selected Principal Fibre Bundles, and the well known Kostant-Kirillov-Souriau symplectic structure on (co) adjoint orbits associated with Lie groups. It is shown that this framework generalises in a natural and simple manner to the mixed state case. For simplicity, only the case of rank two mixed state density matrices is considered in detail. The extensions of the ideas of Null Phase Curves and Pancharatnam lifts from pure to mixed states are also presented.

quant-ph↗
arXiv2003

Low-Dimensional Spin Systems: Hidden Symmetries, Conformal Field Theories and Numerical Checks

We review here some general properties of antiferromagnetic Heisenberg spin chains, emphasizing and discussing the role of hidden symmetries in the classification of the various phases of the models. We present also some recent results that have been obtained with a combined use of Conformal Field Theory and of numerical Density Matrix Renormalization Group techniques.

cond-mat.stat-mech↗
arXiv2002

Continuum in the Excitation Spectrum of the S=1 Compound CsNiCl_3

Recent neutron scattering experiments on CsNiCl_3 reveal some features which are not well described by the nonlinear sigma model nor by numerical simulations on isolated S=1 spin chains. In particular, in real systems the intensity of the continuum of multiparticle excitations, at T=6K, is about 5 times greater than predicted. Also the gap is slightly higher and the correlation length is smaller. We propose a theoretical scenario where the interchain interaction is approximated by a staggered magnetic field, yielding to a correct prediction of the observed quantities.

cond-mat↗
arXiv2002

Alternative Structures and Bihamiltonian Systems

In the study of bi-Hamiltonian systems (both classical and quantum) one starts with a given dynamics and looks for all alternative Hamiltonian descriptions it admits.In this paper we start with two compatible Hermitian structures (the quantum analog of two compatible classical Poisson brackets) and look for all the dynamical systems which turn out to be bi-Hamiltonian with respect to them.

quant-ph↗