Product of Volterra Type Integral and Composition Operators on Weighted Fock Spaces

We characterize the bounded, compact, and Schatten class product of Volterra type integral and composition operators acting between weighted Fock spaces. Our results are expressed in terms of certain Berezin type integral transforms on the complex plane ?. We also estimate the norms and essential norms of these operators in terms of the integral transforms. All our results are valid for weighted composition operators when acting between the class of weighted Fock spaces considered.     

Schatten class weighted composition operators on weighted Fock spaces

We describe the Schatten class weighted composition operators on Fock–Sobolev spaces and a large class of weighted Fock spaces, where the weights defining such spaces are radial, decay at least as fast as the classical Gaussian weight, and satisfy certain mild smoothness condition. To prove our main results, we characterize the Schatten class membership of the Toeplitz operators T _μ induced by nonnegative measures μ on the complex space ${\mathbb{C}^n}$ .     

Role of phase fluctuation and dephasing in the enhancing continuous variable entanglement of a two-photon coherent beat laser

A steady state analysis of the nonclassical features and statistical properties of the cavity radiation of a two-photon coherent beat laser is presented. Results show that the degree of two-mode squeezing, detectable entanglement and intensity of the cavity radiation can increase with the deviation of the phase fluctuations of the laser employed in preparing the atoms, but decrease with the increasing rate at which the induced coherence superposition decays. Although it is found that varying the phase fluctuations and dephasing can lead to modification in the quantum features and statistical properties of the radiation, it does not alter the similarity in the nature of the degree of entanglement detectable by the criteria following from Duan-Giedke-Cirac-Zoller and logarithmic negativity in a perceivable manner. Since the intensity and quantum features can be readily enhanced, this system is expected to be a viable source of a strong robust entangled (squeezed) light under various conditions. Moreover, comparison of the mean number of photon pairs with intensity difference shows that the chance of inciting a two-photon process can be enhanced by changing the rate of dephasing and phase fluctuations.     

On Toeplitz Operators Between Fock Spaces

We study some mapping properties of Toeplitz operators T _μ associated with nonnegative Borel measures μ on the complex space ${\mathbb{C}^n}$ . In particular, we describe the bounded and compact properties of T _μ acting between Fock spaces in terms of the objects t-Berezin transforms, averaging functions, and averaging sequences of μ. We also obtain an asymptotic estimate for the norms of the operators. The results extend and complete a recent work of Z. Hu and X. Lv when both the smallest and the largest Banach–Fock spaces are taken into account.     

Isolated and essentially isolated Volterra-type integral operators on generalized Fock spaces

We studied some topological properties of the space of bounded Volterra-type integral operators on generalized Fock spaces with weight function growing faster than the classical Gaussian function. In particular, the connected and path-connected components, isolated and essentially isolated points of the space are described. The result shows that while the space contains all the non-compact operators as its isolated points, it fails to admit essentially isolated points.     

Path connected components of the space of Volterra-type integral operators

We study the topological structure of the space of Volterra-type integral operators on Fock spaces endowed with the operator norm. We prove that the space has the same connected and path connected components which is the set of all those compact integral operators acting on the spaces. We also obtain a characterization of isolated points of the space of the operators and show that there exists no essentially isolated Volterra-type integral operator.     

Continuous variable quantum discord in a correlated emission laser

A detailed analysis of the nonclassical features of the cavity radiation of a nondegenerate correlated emission laser is presented. Closed mathematical relation that describes the quantum discord in terms of the involved auto- and cross-correlations is provided. It is found that the quantum discord increases with the linear gain coefficient and initially prepared atomic coherent superposition. Although the overall trend of the forward and backward quantum discords turns out to be similar, it is also unequivocally observed that the forward quantum discord is greater or equal to the backward quantum discord. Moreover, comparison of the quantum discord with entanglement shows that there is a very wide range of parameters for which an overlap in capturing the nonclassical nature of the radiation can be observed. However, when the atoms are initially prepared in a maximum coherent superposition, the quantum discord is found to capture the quantumness even when Duan–Giedke–Cirac–Zoller entanglement criterion failed.     

Two-mode squeezing in a coherently driven degenerate parametric down conversion

Squeezing properties of a two-mode radiation produced by a process of driven degenerate parametric down conversion, when the cavity is coupled to two independent squeezed vacuum reservoirs employing the linearization procedure, are analyzed. The two-mode cavity and output radiations exhibit considerable squeezing even when the oscillator is coupled to a vacuum reservoir. One of the effects of coupling the cavity to the squeezed vacuum reservoirs is to increase the degree of squeezing exponentially. For the output radiation the correlation of the quadrature operators evaluated at different times also contributes to the squeezing, which is the reason for quenching of the overall noise in one of the quadrature components of the fundamental mode.     

A study into asymmetric Nicholas cyclisation reactions

Three systematic approaches have been employed to investigate asymmetric Nicholas reactions. We found that the use of a chiral N-enoyl derivative provided acceptable levels of selectivity for an intermolecular Nicholas reaction, however, we were unable to identify an auxiliary that could be utilized in an asymmetric conjugate addition and a tandem inter/intramolecular series of Nicholas reactions. The use of chiral pool non-racemic propargyl alcohols, derived from citronellal, provided enhanced levels of selectivity. As a result of these studies we developed a series of Nicholas cyclisations derived from chiral non-racemic salicylaldehyde derivatives. These underwent an extremely rapid and highly efficient cyclisation, under Nicholas conditions, to afford a range of benzopyrans. The adjacent stereogenic centres appear to be formed with high levels of stereocontrol.