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Acta Physica Sinica

Evolution properties of the field quantum entropy in the nonlinear Jaynes-Cummings model of a trapped ion

Liu Wang-Yun, BI Si-wen, Xi-bo Dou, 中国科学院西安光学精密机械研究所瞬态光学与光子技术国家重点实验室,西安 710119

Published Jan 1, 2010• 3 citations

Abstract

The time evolution properties of the field quantum entropy in the system of a trapped ion interacting resonantly with a standing-wave laser field is studied by utilizing the Von Neumann reduced quantum entropy theory, and our attention focuses on the discussion of the influence of the Lamb-Dick parameter, the position of the ion in the standing-wave laser field and the initial state of the trapped ion on the evolution properties of the field quantum entropy. The results obtained from the numerical calculation indicate that: the value of the Lamb-Dick parameter effect the oscillation frequency and amplitude of the quantum entanglement between the trapped ion and the standing-wave laser field, the larger the Lamb-Dick parameter is, the weaker the average entanglement level between the ion and the field will be. When moving the tapped ion from the node of the standing-wave laser to the loop, the vibration frequency of the quantum entanglement between the field and the ion becomes slow gradually, and the entanglement degree gets weaker and weaker. With the decrease of the probability of the trapped ion being in the excited state, the quantum entanglement between the trapped ion and the stanging-wave laser field shows the tendency of increase first and then decrease. These properties have certain reference value for the preparation of entangled states and for the quantum communications with the thapped ion, and so on.

DOI: 10.7498/aps.59.1780

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The time evolution properties of the field quantum entropy in the system of a trapped ion interacting resonantly with a standing-wave laser field is studied by utilizing the Von Neumann reduced quantum entropy theory, and our attention focuses on the discussion of the influence of the Lamb-Dick parameter, the position of the ion in the standing-wave laser field and the initial state of the trapped ion on the evolution properties of the field quantum entropy. The results obtained from the numerical calculation indicate that: the value of the Lamb-Dick parameter effect the oscillation frequency and amplitude of the quantum entanglement between the trapped ion and the standing-wave laser field, the larger the Lamb-Dick parameter is, the weaker the average entanglement level between the ion and the field will be. When moving the tapped ion from the node of the standing-wave laser to the loop, the vibration frequency of the quantum entanglement between the field and the ion becomes slow gradually, and the entanglement degree gets weaker and weaker. With the decrease of the probability of the trapped ion being in the excited state, the quantum entanglement between the trapped ion and the stanging-wave laser field shows the tendency of increase first and then decrease. These properties have certain reference value for the preparation of entangled states and for the quantum communications with the thapped ion, and so on.

Starting from the Jaynes-Cummings (J-C) model describing interaction of a trapped ion with a standing-wave laser field, we derive the interaction Hamiltonian on the nonlinear J-C model of the trapped ion excitated resonantly with 1-quantum vibrational mode. After solving the time-dependent states of this model,the entropy evolution of the ion are investigated numerically related with Lamb-Dick parameter,the position of the ion in the standing-wave laser field and the initial number of phonon. It is shown that the entropy evolution exhibit nonperiodicity vibration. With the increasing of the parameter, it will increase the frequency of the vibration,therefore the less time of disentanglement between the internal and external degrees of freedom of a trapped ion. When moving the ion from the node of the standing-wave laser to the antinode,the frequency of the vibration becomes less,the longer time of the disentanglement between the internal and external degrees of freedom. With the more initial number of phonon , the entropy evolution of the ion is similar to the field entropy evolution of Jaynes-Cummings model, and the vibration frequency become less gradually.

Starting from the Jaynes-Cummings (J-C) model describing the interaction of a trapped ion with a standing-wave laser field, we derive the interaction Hamiltonian on the nonlinear J-C model of the trapped ion excitated resonantly with 1-quantum vibrational mode. After solving the time-dependent states of this model, we study the evolution of population inversion for the ion numerically. It is shown that the period of the quantum collapse and revival of the population inversionis related with the Lamb-Dick parameter and the position of the ion in the standing-wave laser field. With the increasing of the parameter, it will decrease the period of the collapse and revival. When moving the ion from the node of the standing-wave laser to the antinode, the period of the collapse and revival becomes longer.

By performing a unitary transformation, we transform the Hamiltonian of the trapped ion in any position of standing wave to that of the normal Jaynes-Cummings model in ionic bare basis and we obtain a general evolution operator of the trapped ion system. We study the quantum entropic dynamics of the phonons and trapped ion. Our results show that, when the trapped ion is located at the node of standing wave, the quantum entropic dynamics of phonons and trapped ion are the same as the one of the field in the Jaynes-Cummings model. When the trapped ion deviates from the node of standing wave, the entropies of the phonons and ion keep their maximum value except at the initial stage, and the phonons and trapped ion become extremely entangled.

Starting from the Jaynes-Cummings(J-C) model describing the interaction of a trapped ion with a standing wave laser field,we derive the interaction Hamiltonian on the nonlinear J-C model of the trapped ion excited resonantly with single quantum vibrational mode.After solving the time dependent states of this model,we study the ion centroid motion of squeezing effects by utilizing the ion numerical model.It is shown that the ion squeezing effects is related with the Lamb-Dicke parameter and the position of the ion in the standing wave laser field.

Trapped and laser-cooled ions are increasingly used for a variety of modern high-precision experiments, frequency standard applications and quantum information processing. Therefore, in this communication we present a comprehensive analysis of the pattern of information entropy arising in the time evolution of an ion interacting with a laser field. A general analytic approach is proposed for a three-level trapped-ion system in the presence of the time-dependent couplings. By working out an exact analytic solution, we conclusively analyse the general properties of the von Neumann entropy and quantum information entropy. It is shown that the information entropy is affected strongly by the time-dependent coupling and exhibits long time periodic oscillations. This feature attributed to the fact that in the time-dependent region Rabi oscillation is time dependent. Using parameters corresponding to a specific three-level ionic system, a single beryllium ion in a RF-(Paul) trap, we obtain illustrative examples of some novel aspects of this system in the dynamical evolution. Our results establish an explicit relation between the exact information entropy and the entanglement between the multi-level ion and the laser field. We show that different nonclassical effects arise in the dynamics of the ionic population inversion, depending on the initial states of the vibrational motion/field and on the values of Lamb–Dicke parameter η.