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Revista Mexicana de Física

Eigenestados exactos y estados entrelazados de un ión atrapado oscilando en dos dimensiones

I. Ramos-Prieto, H. M. Moya-Cessa

Published Jan 1, 2017• 2 citations
Publisher

Abstract

We show that it is possible to find exact eigenstates in the interaction between a trapped ion oscillating in two dimensions and they will be entangled states. We also show that the Hamiltonian that describes this interaction may be mapped to a quantum Rabi interaction Hamiltonian (for a two-level atom and two quantized fields) by means of a unitary transformation.

DOI
10.31349/revmexfis.63.2.194

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We show that it is possible to find exact eigenstates in the interaction between a trapped ion oscillating in two dimensions and they will be entangled states. We also show that the Hamiltonian that describes this interaction may be mapped to a quantum Rabi interaction Hamiltonian (for a two-level atom and two quantized fields) by means of a unitary transformation.
The quantum nondemolition measurement is applied to a two-dimensional (2D) trapped-ion model in which two laser beams drive the corresponding vibrational motions and are carrier resonant with the two-level system of the ion. The information about the ionic vibrational energy can be detected by the occupation probability of the internal electronic level. The substantial difference of the 2D model from the one-dimensional one is that two orthogonal beams have a fixed phase shift instead of statistical independence. As a result, the atomic Rabi oscillation is involved in the coherent superposition of two sub-Rabi oscillations induced by the corresponding driving beams. This means that, in the bidimensional case, the phase difference between the two laser beams plays the role of an adjustable external parameter which allows us to optimize the measurement scheme itself, in terms of precision and sensitivity. As in the one-dimensional case, our proposal leads to interesting applications as, for example, the cooling and preparation of arbitrary bimodal Fock states. The model can be considered as a cooling method for the trapped ion from the vibrational thermal state to the vacuum state. Due to the coherent superposition of two sub-Rabi oscillations, the Rabi frequency degeneration and offset may occur in this model. This provides the possibility of generating various entangled superposition of Fock states, the pair coherent state, and entangled pair coherent states.
Abstract We show that perfect revivals of Rabi oscillations are possible, under certain conditions, in the population inversion of a trapped ion. Based on this property, we find that Schrödinger cat states of the atomic motion are naturally generated by the unitary dynamics. Using a pair of symmetric and antisymmetric Schrödinger cat states of motion, together with an electronic excited or ground state, we find that the interaction leads to four orthonormal maximally entangled states or hybrid Bell states as they are encoded in two partitions of different nature. We also study a quadratic Kerr-type evolution that is possible for short interaction times.
We propose a scheme to generate a particular kind of tripartite entangled states using a single two-level trapped ion coupled to a quantized monomode electromagnetic field within an optical cavity. The resulting state involves motional Schr\\"odinger-cat states of the trapped ion, its internal electronic states, and Fock states of the cavity field. By means of a measurement of the internal state of the ion, it is also possible to generate either entangled bipartite states involving light and motion or single motional Schr\\"odinger-cat states. Our proposal is based on a previously introduced intensity dependent Hamiltonian [Phys. Rev. A 66, 063403 (2002)] for a trapped ion in a cavity.