Quantum Scissors – Finite-Dimensional States Engineering

Abstract

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...

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