This work discusses quantum states defined in a finite-dimensional Hilbert space. In particular, after the presentation of some of them and their basic properties the work concentrates on the group of the quantum optical models that can be referred to as quantum optical scissors. Such "devices" can generate on their outputs states that are finite-dimensional, and simultaneously use for such preparation quantum states that are defined in the infinity-dimensional space. The work concentrates on two groups of models: the first one, comprising linear elements and the second one -models for which optical, Kerr-like nonlinear elements were applied. IntroductionProblems of quantum optical states engineering have attracted remarkable interest in last years. Various concepts of such states and methods of their production and manipulation have been presented in numerous papers. They have diverse applications in atomic and molecular, solid state and nano-systems physics, and also in the quantum information theory. The latter have recently given a stimulating pulse for the investigation of the states defined in finite-dimensional Hilbert space. However, one should keep in mind that the general idea of such states was born much earlier. In particular, Radclife (Radcilffe, 1971) and Arecchi et.al (Arecchi, Courtens, Gilmore, and Thomas, 1972) proposed the atomic (or spin) coherent states definition for the optical models involving atomic systems interacting with transverse 1 arXiv:1312.0118v1 [quant-ph] 30 Nov 2013 electromagnetic field. Those states are finite-dimensional analogues of the coherent states proposed by Glauber (Glauber, 1963b,a) and play a crucial role in the quantum optics theory. Finite-dimensional coherent states have been discussed in various aspects -for example see (the references quoted therein.Another milestone in the development of the idea of finite-dimensional states was the proposal of the phase-states given by Pegg and Barnett in Barnett, 1988, 1989). The key idea of the definition they proposed is to calculate the state and all physical quantities in the (s + 1)-dimensional space and then, to take the limit s → ∞. These states are not the subject of this paper and we shall concentrate on the quantum scissors systems, however, they are interesting enough to be mentioned here. For instance, in (Vogel, Akulin, and Schleich, 1993) the model involving atoms injected into a cavity was discussed as a potential source of the phase-states. The Pegg-Burnett formalism is important as a method of defining other finite-dimensional states. As it will be presented, some of the finite-dimensional states are defined in an analogous way to the phase-states, i.e. it shall be assumed that the space is finite-dimensional and within such a space all operators and desired states are defined. Such states we will referred to as finite dimensional states. Another approach to the states definiton which is presented in this paper, is similar to that proposed in Zhou, 1993, 1994). For this case the expansion of the discussed...

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 states 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.

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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.

Inspired by the recent experiment of Hamsen et al. [Phys. Rev. Lett. 118, 133604 (2017)], which demonstrated two-photon blockade in a driven nonlinear system (composed of a harmonic cavity with a driven atom), we show that two-photon blockade and other nonstandard types of photonblockade and photon-induced tunneling can be generated in a driven harmonic cavity without an atom or any other kind of nonlinearity, but instead coupled to a nonlinear (i.e., squeezed) reservoir. We also simulate these single-and two-photon effects with squeezed coherent states and displaced squeezed thermal states.

We analyse the entanglement dynamics in a nonlinear Kerr-like coupler interacting with external environment. Whenever the reservoir is in a thermal vacuum state the entanglement (measured by concurrence for a two-qubit system) exhibits regular oscillations of decreasing amplitude. In contrast, for thermal reservoirs we can observe dark periods in concurrence oscillations (which can be called a sudden death of the entanglement) and the entanglement rebuild (which can be named the sudden birth of entanglement). We show that these features can be observed when we deal with 2-qubit system as well as qubitqutrit system.

We discuss a model comprising a chain of three Kerr-like nonlinear oscillators pumped by two modes of external coherent field. We show that the system can be treated as nonlinear quantum scissors and behave as a three-qubit model. For such situation different types of tripartite entangled states can be generated, even when damping effects are present in the system. Some amount of such entanglement can survive even in a long-time limit. The flow of bipartite entanglement between subsystems of the model and relations among first-, second-order correlations and the entanglement are discussed.

We deal with a system comprising a nonlinear (Kerr-like) oscillator excited by a series of ultra-short external pulses. We introduce the fidelity-based entropic parameter that can be used as an indicator of quantum chaos. Moreover, we propose to use the fidelity-like parameter comprising the information about the mean number of photons in the system. We shall concentrate on the long-time behaviour of the parameters discussed, showing that for deep chaos cases the quantum fidelities behave chaotically in the classical sense despite their strictly quantum character.