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arXiv:1505.06013 · Open Systems & Information Dynamics

Many Unstable Particles from an Open Quantum System's Perspective

Kordian A. Smoliński

Published Aug 7, 2014• 5 citations
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Abstract

We postulate a master equation, written in the language of creation and annihilation operators, as a candidate for unambiguous quantum mechanical description of unstable particles. We have found Kraus representation for the evolution driven by this master equation and study its properties. Both Schrödinger and Heisenberg picture of the system evolution are presented. We show that the resulting time evolution leads to exponential decay law. Moreover, we analyse mixing of particle flavours and show that it can lead to flavour oscillation phenomenon.

DOI
10.1142/s1230161214500036

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We postulate a master equation, written in the language of creation and annihilation operators, as a candidate for unambiguous quantum mechanical description of unstable particles. We have found Kraus representation for the evolution driven by this master equation and study its properties. Both Schrödinger and Heisenberg picture of the system evolution are presented. We show that the resulting time evolution leads to exponential decay law. Moreover, we analyse mixing of particle flavours and show that it can lead to flavour oscillation phenomenon.
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Master equations in the Lindblad form describe evolution of open quantum systems that are completely positive and simultaneously have a semigroup property. We analyze the possibility to derive this type of master equations from an intrinsically discrete dynamics that is modelled as a sequence of collisions between a given quantum system (a qubit) with particles that form the environment. In order to illustrate our approach we analyze in detail how the process of an exponential decay and the process of decoherence can be derived from a collision-like model in which particular collisions are described by SWAP and controlled-NOT interactions, respectively.
Decays of heavier neutrino mass eigenstates into lighter ones, while very slow in the Standard Model, can be significantly enhanced in scenarios with more than three neutrino flavours, or in models with new ultra-light particles such as Majorons. A full theoretical description is challenging due to the intricate interplay between oscillations and decay, interference between different decay channels, and the possibility of multi-step decay cascades. In this paper, we develop a fully general description of arbitrarily complex systems of oscillating and decaying neutrinos using methods from the theory of open quantum systems. Notably, we demonstrate how such systems can be implemented using the Lindblad master equation, the Liouvillian superoperator, as well as Kraus operators. The last two methods eschew the need for solving a differential equation, thereby showing superior numerical performance.
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We introduce a dynamical evolution operator for dealing with unstable physical process, such as scattering resonances, photon emission, decoherence and particle decay. With that aim, we use the formalism of rigged Hilbert space and represent the time evolution of quantum observables in the Heisenberg picture, in such a way that time evolution is nonunitary. This allows to describe observables that are initially noncommutative, but become commutative after time evolution. In other words, a non-abelian algebra of relevant observables becomes abelian when times goes to infinity. We finally present some relevant examples.
We have shown that it is possible to formulate the consistent and probability-preserving description of the CP\mathit{CP}CP-symmetry-violating evolution of a system of decaying particles. This has been done within the framework of quantum mechanics of open systems. This approach allows the description of both the exponential decay and flavor oscillations. We have solved explicitly the Kossakowski-Lindblad master equation for a system of particles with violated CP\mathit{CP}CP symmetry and found the evolution of any observable bilinear in creation and annihilation operators. The choice of a concrete observable can be done by the proper choice of initial conditions for the system of differential equations. We have calculated the evolution as well as mean values of the observables most interesting from the physical point of view, and we have found their lowest order difference with the CP\mathit{CP}CP-preserved values.