Entangled State Reconstruction of an Electron in the Penning Trap

Entanglement [1] is one of the essential and most puzzling features of quantum mechanics. It is also the basic ingredient and resource of quantum informa­ tion processing: quantum teleportation [2, 3], quan­ tum dense coding [4], quantum cryptography [5], and quantum computation [6] are just a few examples of the possibilities offered by sharing quantum en­ tanglement. Recently, for example, it has been made possible to entangle four trapped ions [7]. The quantum system we want to study here is a sin­ gle electron trapped in a Penning trap [8]. This system allows the measurement o f fundamental physical con­ stants with striking accuracy and is reasonably free of decoherence and dissipation [9]. As a consequence, the determination (or the reconstruction [10]) of the entangled quantum state of the trapped electron is a decisive issue, especially in the light of the fact that such a system has been already proposed as a candi­ date for quantum information processing [11]. In this paper we propose to reconstruct the full entangled state (combined cyclotron and spin state) of an electron in a Penning trap by using a modified version of quantum state tomography. Previous pro­ posals [12] need the a priori knowledge of the spin state and therefore are not able to deal with entangled states. Our method, on the contrary, can measure the full (entangled) pure state of the two relevant degrees of freedom of the electron. In order to reach this scope, our method takes advantage of the magnetic bottle configuration to perform simultaneous measurements of the cyclotron excitation and of the 2 component of the spin as a function of the phase of an applied driving electromagnetic field. The complete structure of the cyclotron-spin quantum state is then obtained with the help of a tomographic reconstruction from the measured data. The present paper is organized as follows: In Sect. 2 we describe the basic physics o f an electron trapped in a Penning trap, while in Sect. 3 we present our reconstruction procedure. In Sects 4 and 5 we con­ centrate on the measurement of the spin and on the tomographic reconstruction of the cyclotron states, respectively. We finally discuss the results o f our nu­ merical simulations in Section 6.