S Lakshmi Bala

In recent years, the performance of different entanglement indicators obtained directly from tomograms has been assessed in continuous-variable and hybrid quantum systems. In this paper, we carry out this task in the case of spin systems. We compute the entanglement indicators from actual experimental data obtained from three liquid-state nuclear magnetic resonance (NMR) experiments and compare them with standard entanglement measures calculated from the corresponding density matrices, both experimentally reconstructed and numerically computed. The gross features of entanglement dynamics and spin squeezing properties are found to be reproduced by these entanglement indicators. However, the extent to which these indicators and spin squeezing track the entanglement during time evolution of the multipartite systems in the NMR experiments is very sensitive to the precise nature and strength of interactions as well as the manner in which the full system is partitioned into subsystems. We also use the IBM quantum computer to implement equivalent circuits that capture the dynamics of the multipartite system in one of the NMR experiments and carry out a similar comparative assessment of the performance of tomographic indicators. This exercise shows that these indicators can estimate the degree of entanglement without necessitating detailed state reconstruction procedures, establishing the advantage of the tomographic approach.

The performance of bipartite entanglement indicators obtained directly from tomograms is compared with that of standard entanglement measures, in time-evolving hybrid quantum systems and spin systems. Bipartite and tripartite models are judiciously chosen so as to exhibit interesting effects such as entanglement sudden death and entanglement collapse to a constant nonzero value over significant time intervals during evolution. For specific initial states, the reliability of the tomographic indicators is investigated numerically. The IBM Q platform has been invoked to examine how well the equivalent circuits of specific model systems fare, to understand how experimental losses affect the entanglement. Using data obtained from a set of NMR experiments, we have examined the viability of tomographic indicators in monitoring bipartite entanglement, and find that they perform as satisfactorily as standard measures and monotones. As a trailer to the next chapter on the dynamics of quantum observables, we show that in a tripartite model examined in this chapter, an intensity-dependent bifurcation cascade appears at short times, and is captured both by the indicators and by the mean photon number.

In this review we present the salient features of dynamical chaos in classical gauge theories with spatially homogeneous fields. The chaotic behavour displayed by both abelian and non-abelian gauge theories and the effect of the Higgs term in both cases are discussed. The role of the Chern-Simons term in these theories is examined in detail. Whereas, in the abelian case, the pure Chern-Simons-Higgs system is integrable, addition of the Maxwell term renders the system chaotic. In contrast, the non-ablian Chern-Simons-Higgs systems is chaotic both in the presence and the absence of the Yang-Mills term. We support our conclusions with numerical studies on plots of phase trajectories and Lyapunov exponents. Analytical tests of integrability such as the Painlevé criterion are carried out for these theories. The role of the various terms in the Hamiltonians for the abelian Higgs, Yang-Mills-Higgs and Yang-Mills-Chern-Simons-Higgs systems with spatially homogeneous fields, in determining the nature of order-disorder transitions is highlighted, and the effects are shown to be counter-intuitive in the last-named system.

The master equation for dichotomous diffusion (DD) (the integral of a random telegraph process) is the well-known telegrapher's equation, which is converted to the Klein-Gordon equation by a simple transformation. After a brief recapitulation of the solution and of the analogy between DD and the Dirac equation in one spatial dimension, we consider velocity-biased DD. The corresponding master equation and its solution are presented. It is shown that these may be interpreted physically in terms of a Lorentz transformation to a frame moving with a boost velocity equal to the mean drift velocity of the diffusing particle. The modifications that arise in the connection with the Dirac equation are also exhibited. The correspondence between the rest mass of the Dirac particle and the frequency of direction reversal in the DD is shown to be modified precisely by the time dilatation correction to the latter quantity. © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.

The quantum-classical divide and the limitations of using classical tools to study quantum systems have received attention for several decades. We examine specific aspects of this broad theme, primarily using quantum optics as the platform for our investigation. In this introductory chapter, we outline nonclassical effects such as wave packet revivals, squeezing and entanglement. We define several optical states, and give a pedagogical illustration of how these states differ in their dynamics. We then indicate two aspects studied in the sequel. First, the extraction of nonclassical properties directly from tomograms that are readily available from experiments, bypassing detailed state reconstruction. Tomograms, being probability distributions in contrast to the reconstructed Wigner functions, provide useful pointers to assess nonclassicality. Standard entanglement indicators defined using the density matrix are introduced, with reference to which we compare the performance of indicators obtained directly from optical and qubit tomograms. Second, we demonstrate the usefulness of applying time series and network analysis (originating from classical dynamical systems theory) to the dynamics of quantum observables.

Using a generic Hamiltonian that models wave packet propagation in a Kerr-like medium, matter wave field dynamics in Bose-Einstein condensation, etc., we show that distinctive signatures of wave packet revivals and fractional revivals are displayed by the time evolution of the expectation values of appropriate observables, enabling selective identification of different fractional revivals. © 2004 Elsevier B.V.

The dynamics of tripartite systems of field-atom interactions and optomechanics is investigated through relevant tomograms. Quadrature and tomographic entropic squeezing and entanglement properties are examined . Entanglement collapses to constant nonzero values over significant time intervals.

We introduce various bipartite entanglement indicators that are defined entirely from optical and spin tomograms, circumventing reconstruction of the density matrix. Some of these indicators are adapted from the manner in which the distance between two classical probability distributions is estimated. We also indicate how spin squeezing can be related to the tomographic entanglement indicators. A comparison is made of the performance of these tomographic indicators with standard entanglement measures based on the density matrix, with two objectives: to gauge the utility of specific indicators in different situations both in bipartite continuous variable systems and in hybrid quantum systems; and to assess how far ideas from classical probability theory and its role in information processing can be used to understand quantum entanglement. These investigations are conducted specifically in systems in which, with the variation of an appropriate parameter, some of the energy levels move close to each other and then move apart, avoiding level crossing. We summarize known results pointing out that standard quantum entanglement measures extremize at avoided energy-level crossings. Hence such systems provide an ideal platform for our investigations on the utility and limitations of tomographic entanglement indicators.

The motion of a particle moving under the influence of a central force is a fundamental paradigm in dynamics. The problem of planetary motion, specifically the derivation of Kepler’s laws motivated Newton’s monumental work, Principia Mathematica, effectively signalling the start of modern physics. Today, the central force problem stands as a basic lesson in dynamics. In this article, we discuss the classical central force problem in a general number of spatial dimensions n, as an instructive illustration of important aspects such as integrability, super-integrability and dynamical symmetry. The investigation is also in line with the realisation that it is useful to treat the number of dimensions as a variable parameter in physical problems. The dependence of various quantities on the spatial dimensionality leads to a proper perspective of the problems concerned. We consider, first, the orbital angular momentum (AM) in n dimensions, and discuss in some detail the role it plays in the integrability of the central force problem. We then consider an important super-integrable case, the Kepler problem, in n dimensions. The existence of an additional vector constant of the motion (COM) over and above the AM makes this problem maximally super-integrable. We discuss the significance of these COMs as generators of the dynamical symmetry group of the Hamiltonian. This group, the rotation group in n + 1 dimensions, is larger than the kinematical symmetry group for a general central force, namely, the rotation group in n dimensions.

We consider the SU(4) Skyrme model with explicit chiral and flavour symmetry-breaking terms. Using the masses of the 15-plet pseudoscalar mesons as the input, we calculate the masses of the 20-plet baryons. The baryon masses predicted by this model agree with results based on quark model to about 15%. We find that the generalized Gell-Mann Okubo mass relation is very well satisfied. © 1988 Indian Academy of Sciences.