[en] We discuss a system of nonlinear Kerr-like oscillator pumped linearly by an external, single-mode electromagnetic field. Assuming that the system was initially in the vacuum state, we compare its quantum and classical dynamics. To find correspondences between the quantum and classical models the method based on the time-evolution of the fidelity of quantum states is applied. (author)

[en] We discuss a system of nonlinear coupler pumped linearly by an external, single-mode electromagnetic field. The coupler referred as to a non-linear quantum scissor comprises two Kerr-like nonlinear oscillators coupled to each other through a nonlinear coupling. We show that for some values of the parameters describing the system, its evolution remains closed within a finite set of n-photon states corresponding to the two modes of the oscillators. As a consequence, the coupler discussed in the paper can behave as a qubit-qutrit system. Moreover, we show that the system dynamics can lead to the generation of Bell-like states. (author)

[en] We discuss quantum properties of displaced Kerr states, in particular the periodic behaviour of the mean values of various quantum parameters describing our model. Thus, we introduce an operator evolution approach that justifies our conclusions concerning the periodic behaviour of the system. (author)

[en] The properties of the entangled pure states in phase space are analysed using the classical entropy introduced by Wehrl. The general entropy inequalities for non-entangled pure states are derived, which are violated by any entangled state. To measure the strength of intermode correlations in phase space the parameters related to these inequalities are proposed. As an example we study the correlations between amplitudes and phases for two-mode Fock states. We find that the amplitudes as well as phases of different modes are correlated. It is also shown that the degree of the intermode correlation strongly depends on the photon number difference in two-mode Fock states. (author)

[en] We discuss quantum states generated in a Mach-Zehnder interferometer with a Kerr medium in one of its arms. These states are referred to as displaced Kerr states. Contrary to the former approaches, we describe a nonlinear Kerr medium not only by the third-order(two-photon) but also by higher-order (k-photon, k=3,4,...) nonlinear oscillators. We refer to the states generated in the system with higher optical Kerr nonlinearity as higher-order displaced Kerr states or multiphoton displaced Kerr states. We investigate the quantum-statistical properties of these states showing their dependence on the degree of the Kerr nonlinearity. (authors)

[en] We discuss a system comprising two nonlinear (Kerr-like) oscillators coupled mutually by a nonlinear interaction. The system is excited by an external coherent field that is resonant to the frequency of one of the oscillators. We show that the coupler evolution can be closed within a finite set of n-photon states, analogously as in the nonlinear quantum scissors model. Moreover, for this type of evolution our system can be treated as a Bell-like state generator. Thanks to the nonlinear nature of both oscillators and their internal coupling, these states can be generated even if the system exhibits its energy dissipating nature, contrary to systems with linear couplings

[en] We discuss a system comprising an anharmonic oscillator permanently excited by a series of ultra-short coherent pulses. Assuming that the system was initially in the vacuum state we investigate and compare its classical and quantum dynamics. Moreover, we compare the picture based on the single classical trajectory with the 'averaged' one. (Authors)

[en] We study the dynamics of an anharmonic oscillator driven by a train of pulses. The cumulant expansion and quantum evolution operator approaches are presented and compared. The modifications introduced by quantum mechanics into the dynamics of classical systems which manifest chaos are a problem of great importance. It is known that quantization modifies the dynamics of classical system is usually studied by means of the equation for the Wigner function derived from the quantum Liouville equation. In Wigner's formulation of quantum mechanics we treat a quantum system in a 'classical way' including all their quantum features. And what is more, we can contrast the quantum and classical dynamics within the framework of one formalism. The problem is, that the equations for the Wigner functions are mathematically cumbersome and their analytic solutions for most nonlinear systems are unknown. However, instead of the equation for the Wigner function we can use the set of equations for statistical moments generated by our equation for the Wigner function. It is obvious that in this approach a quantum system is governed by an infinite set of equations. Therefore, for numerical reasons the set of equations for statistical moments has to be truncated at a finite number, which means approximating it. It is known that first cumulant approximation represents the classical dynamics. The second cumulant approximation adds the first quantum corrections to the classical dynamics. In this paper we compare some aspects of the cumulant method and the method used by Leonski and Tanas to study an anharmonic oscillator driven by a train of pulses. The Kerr oscillator model is the same ad that is discussed in an earlier paper albeit without the damping mechanism

[en] We study a nonlinear oscillator interacting with a one-mode cavity field. We assume, that the cavity is periodically kicked by a series of ultra-short coherent pulses. We show that for a special choice of parameters the system evolution is restricted to a finite set of n-photon states. In consequence, the mean energy of the cavity remains finite despite the fact that the cavity is continuously pumped. We study the properties of the cavity field showing that the field exhibits nonclassical features. (Authors)

[en] Two nonlinear Kerr oscillators mutually coupled by parametric pumping are studied as a source of states entangled in photon numbers. Temporal evolution of entanglement quantified by negativity shows the effects of sudden death and birth of entanglement. Entanglement is preserved even in asymptotic states under certain conditions. The role of reservoirs at finite temperature in entanglement evolution is elucidated. Relation between generation of entangled states and violation of Cauchy-Schwartz inequality for oscillator intensities is found.