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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Preparation of Schrödinger cat states in noncommutative space

Ying Wu, Xiaoxue Yang

Published Mar 24, 2006• 31 citations
Publisher

Abstract

We show that there exist a large class of new nontrivial exactly solvable models with their potentials depending only on one spatial variable on a 2D noncommutative plane and show that their general solutions are Schr\"odinger cat states. This feature not only establishes an interesting connection between quantum mechanics on noncommutative spaces and quantum information science but also shed light on proposing new schemes based on the quantum entanglement to test the noncommutativity.

DOI
10.1103/physrevd.73.067701

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We show that there exist a large class of new nontrivial exactly solvable models with their potentials depending only on one spatial variable on a 2D noncommutative plane and show that their general solutions are Schr"odinger cat states. This feature not only establishes an interesting connection between quantum mechanics on noncommutative spaces and quantum information science but also shed light on proposing new schemes based on the quantum entanglement to test the noncommutativity.
Schr\"odinger cat states, which are superpositions of macroscopically distinct states, are potentially critical resources for upcoming quantum information technologies. In this paper we introduce a scheme to generate entangled Schr\"odinger cat states in a nonrelativistic electric dipole system situated on a two-dimensional plane, along with an external potential and a uniform strong magnetic field perpendicular to the plane. Additionally, our findings demonstrate that this setup can lead to the phenomenon of collapse and revival of entanglement for a specific range of our model parameters.
We introduce new representations to formulate quantum mechanics on noncommutative coordinate space, which explicitly display entanglement properties between degrees of freedom of different coordinate components and hence could be called entangled state representations. Furthermore, we derive unitary transformations between the new representations and the ordinary one used in noncommutative quantum mechanics (NCQM) and obtain eigenfunctions of some basic operators in these representations. To show the potential applications of the entangled state representations, a two-dimensional harmonic oscillator on the noncommutative plane with both coordinate-coordinate and momentum-momentum couplings is exactly solved.
Nonclassicality is an interesting property of light having applications in many different contexts of quantum optics, quantum information and computation. Nonclassical states produce substantial amount of reduced noise in optical communications. Furthermore, they often behave as sources of entangled quantum states, which are the most elementary requirement for quantum teleportation. We study various nonclassical properties of coherent states and Schrödinger cat states in a setting of noncommutative space resulting from the generalized uncertainty relation, first, in a complete analytical fashion and, later, by computing their entanglement entropies, which in turn provide supporting arguments behind our analytical results. By using standard theoretical frameworks, they are shown to produce considerably improved squeezing and nonclassicality and, hence, significantly higher amount of entanglement in comparison to the usual quantum mechanical models. Both the nonclassicality and the entanglement can be enhanced further by increasing the noncommutativity of the underlying space. In addition, we find as a by-product some rare explicit minimum uncertainty quadrature and number squeezed states, i.e., ideal squeezed states.
We study several classical-like properties of qqq-deformed nonlinear coherent states as well as nonclassical behaviors of qqq-deformed version of the Schr\"odinger cat states in noncommutative space. Coherent states in qqq-deformed space are found to be minimum uncertainty states together with the squeezed photon distributions unlike the ordinary systems, where the photon distributions are always Poissonian. Several advantages of utilizing cat states in noncommutative space over the standard quantum mechanical spaces have been reported here. For instance, the qqq-deformed parameter has been utilized to improve the squeezing of the quadrature beyond the ordinary case. Most importantly, the parameter provides an extra degree of freedom by which we achieve both quadrature squeezed and number squeezed cat states at the same time in a single system, which is impossible to achieve from ordinary cat states.
Dirac talked about q-numbers versus c-numbers. Quantum observables are q-number variables that generally do not commute among themselves. He was proposing to have a generalized form of numbers as elements of a noncommutative algebra. That was Dirac's appreciation of the mathematical properties of the physical quantities as presented in Heisenberg's new quantum theory. After all, the familiar real, or complex, number system only came into existence through the history of mathematics. Values of physical quantities having a commutative product is an assumption that is not compatible with quantum physics. The revolutionary idea of Heisenberg and Dirac was pulled back to a much more conservative setting by the work of Schrödinger, followed by Born and Bohr. What Bohr missed is that the real number values we obtained from our measurements are only a consequence of the design of the kind of experiments and our using real numbers to calibrate the output scales of our apparatus. It is only our modeling of the information obtained about the physical quantities rather than what Nature dictates. We have proposed an explicit notion of definite noncommutative values of observables that gives a picture of quantum mechanics as realistic as the classical theory. In this article, we illustrate how matrices can be taken as noncommutative (q-)numbers serving as the values of physical quantities, each to be seen as a piece of quantum information. Our main task is to clarify the subtle issues involved in setting up a conventional scheme assigning matrices as values to the physical quantities.
We propose a new kind of Schrödinger cat state introduced as a superposition of spin coherent states in the framework of noncommutative spaces. We analyze the nonclassical features for these noncommutative deformed states in terms of the main physical parameters. The physical importance of deformed states is that they provide a convenient description of a large set of laser systems. As an application, we develop the Jaynes–Cummings model by considering the interaction among atoms and cat state fields associated to deformed spin algebras. In this context, we show the dynamical behavior of the nonlocal correlation and nonclassical properties in these quantum systems.