Generalized qubits of the vibrational motion of a trapped ion

Abstract

We present a method to generate qubits of the vibrational motion of an ion. The method is developed in the non-rotating wave approximation regime, therefore we consider regimes where the dynamics has not been studied. Because the solutions are valid for a more extended range of parameters we call them generalized qubits.PACS numbers: 32.80.Qk, 42.50.Vk Nonclassical states of the center-of-mass motion of a trapped ion have played an important role because of the potential practical applications such as precision spectroscopy [1] [6,7], number states, specific superpositions of them, and in particular, robust to noise (spontaneous emmision) qubits have been proposed [8]. In theoretical and experimental studies of a laser interacting with a single trapped ion it has been usually considered the case in which it may be modeled as a Jaynes-Cummings interaction [5,9,10], then exhibiting the peculiar features of this model like collapses and revivals [11], and the generation of nonclassical states common to such a model or (multi-photon) generalizations of it [12][13][14]. In treating this system usually two rotating wave approximations (RWA's) are done (the first related to the laser [optical] frequency and the second to the vibrational frequency of the ion), to remove counter-propagating terms of the Hamiltonian (that can not be treated analytically). Approximations on the Lamb-Dickke parameter, η, are usually done, considering it much smaller than unity. Additionaly, other approximations are done, based on the intensity of the laser shining on the trapped ion: the low-excitation regime Ω ≪ ν and the strong-excitation regime Ω ≫ ν [15], with Ω being the intensity of the field, and ν the vibrational frequency of the ion.Recently there has been an alternative approach to the study of this dynamics [16]. In this approach a unitary transformation is used in order to linearise the ion-laser Hamiltonian. This transformation has been also used to propose schemes for realising faster logic gates for quantum information processing [3]. Under this unitary transformation the Hamiltonian takes exactly the form (note that not an effective form) of the Jaynes-Cummings Hamiltonian plus an extra term (an atomic driving term). In such a case a RWA may be done [16] in order to have an analytical solution for this problem, but it brings with it a new condition: Ω of the order of 2ν (note however that this is a new regime) and η still much less than unity. Later, another transformation [17] was used to diagonalize the linearized ion-laser Hamiltonian, without further conditions on Ω or η. This allowed the diagonalization of the Hamiltonian only in the ion basis. Exact eigenstates of the ion-laser Hamiltonian, i.e. trapping states for this system have been found [18], but because they do not form a basis, a complete (exact) solution may be found only for such states (eigenstates).In this contribution we consider the complete Hamiltonian for the ion-laser interaction, linearise it as in [16] and further unitarily transform it, witho...

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