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Acta Physica Sinica

PERIODIC ENTANGLEMENT BETWEEN THE INTERNAL AND EXTERANL DEGREE OF FREEDOM OF A TRAPPED ION IN A STANDING WAVE LASER

Mao-Fa Fang, LIU XIANG, 湖南师范大学理学院物理系,长沙410081

Published Jan 1, 2001• 1 citation
Publisher

Abstract

We have investigated the quantum entropy and entanglement of the internal and external degrees of freedom of a trapped ion in a standing wave laser by transforming the Hamiltonian to that in the Jaynes-Cummings system for the bare basis of the trapped ion under the Lamb-Dicke limit. Under the off-resonant condition, we have found the periodic entanglement between the internal and external degrees of freedom of a trapped ion.

DOI
10.7498/aps.50.2363

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We have investigated the quantum entropy and entanglement of the internal and external degrees of freedom of a trapped ion in a standing wave laser by transforming the Hamiltonian to that in the Jaynes-Cummings system for the bare basis of the trapped ion under the Lamb-Dicke limit. Under the off-resonant condition, we have found the periodic entanglement between the internal and external degrees of freedom of a trapped ion.
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.
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.
We have investigated the position-dependent dynamics of a trapped ion in a standing wave laser by transforming it to the Jaynes-Cummings-type system under the Lamb-Dicke limit. A variety of novel phenomena are exhibited, e.g. periodic collapse and revival features and long-time scaled revivals of the ionic inversion, depending on its position in the standing wave. Our result provides a way of producing a system equivalent to the two-photon Jaynes-Cummings model in the trapped ion system, with its exact periodicities.
We have investigated the position-dependent dynamics of a trapped ion in a standing wave laser by transforming it to the Jaynes-Cummings-type system under the Lamb-Dicke limit.A variety of novel phenomena are exhibited.e.g.periodic collapse and revival features and long-time scaled revivals of the ionic inversion,depending on its position in the standing wave.Our result provides a way of producing a system equivalent to the two-photon Jaynes-Cummings model in the trapped ion system,with its exact periodicities.
We unitarily transform the Hamiltonian of a trapped ion, situated at any position of a standing-wave laser field, to that for the normal Jaynes-Cummings model in the bare ionic basis, and we obtain the general evolution operator of the trapped ion system. We show the existence of super-revivals of the ionic inversion if the trapped ion is not located at a node of the standing wave, and we examine the degree of entanglement between the vibrational phonons and the trapped ion. For some parameter values, we find chaotic behavior.
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.