Symmetries and Supersymmetries in Trapped Ion Hamiltonian Models

Trapped ions provide an interesting physical scenario wherein fundamental aspects of quantum mechanics may be tested and useful applications in quantum computation may be realized [1]. Generally speaking, a time dependent quadrupolar electric field is responsible for a charged particle dynamics which may be effectively described as the motion of a point into a quadratic potential [2]. When the harmonically confined particle is an ion, the system we obtain possesses both bosonic and fermionic degrees of freedom, the first corresponding to the ion center of mass motion, the second describing the atomic internal state, i.e. the electronic configuration. In most of the cases, in trapped ion dynamics only two atomic levels are involved, making the Pauli operator two-level system description effective. Once the ion has been confined, acting upon it via laser field it is possible to implement vibronic couplings (i.e. involving both bosonic and fermionic degrees of freedom) practically at will [3]. Unfortunately, sometimes, the vibronic coupling problem is not solvable even if implementable. The study of symmetries plays a central role in the analysis of trapped ion dynamical problems. Indeed, in most of the cases, symmetries provide an elegant and effective way for finding Hamiltonian eigensolutions. In this paper we describe a trapped ion physical situation described by a Hamiltonian model possessing interesting and useful symmetries. In detail, conservation of total excitation number and rotational invariance are found. Moreover a Second Order Hidden Supersymmetry is discovered. Symmetry based diagonalization provides the possibility of writing down the time evolution concerning simple initial conditions. Interesting nonclassical behaviors like the generation of GHZ [7] states are predicted.