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Chinese Physics B

Entropy, non-classical property of the trapped ion system driven by running-wave lasers

Xiang Liu, 方卯发

Published Jun 12, 2005• 0 citations
Publisher

Abstract

We study the dynamical behaviour of the ionic inversion in a trapped ion system in running-wave lasers,and we discuss the properties of entanglement between the ion and phonons and the statistics of the vibrational phonon field.It is shown that the appearance of the super-revivals of the ion inversion is dependent on the initial parameters of the vibrational phonon field.The Schredinger cat and sub-Poissonican statistics of the phonon field can be also exhibited in this system for suitable initial values.

DOI
10.1088/1009-1963/11/9/014

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We study the dynamical behaviour of the ionic inversion in a trapped ion system in running-wave lasers,and we discuss the properties of entanglement between the ion and phonons and the statistics of the vibrational phonon field.It is shown that the appearance of the super-revivals of the ion inversion is dependent on the initial parameters of the vibrational phonon field.The Schredinger cat and sub-Poissonican statistics of the phonon field can be also exhibited in this system for suitable initial values.
We study the dynamical behaviour of the ionic inversion in a trapped ion system in running-wave lasers and we discuss the properties of entanglement between the ion and phonons and the statistics of the vibrational phonon field. It is shown that the appearance of the super-revivals of the ion inversion is dependent on the initial parameters of the vibrational phonon field. The Schrödinger cat and sub-Poissonian statistics of the phonon field can be also exhibited in this system for suitable initial values.
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.
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.
In this paper we use the quasi-mutual entropy, which does not depend on quantum channel, as a measure of information content of ionic state due to ion-laser interaction in a single trapped ion. It is found that the quasi-mutual entropy is a good measure of the entanglement degree of the ionic state at any time. We examine numerically the population inversion, the photon number distribution and degree of entanglement due to quasi-mutual entropy for a single trapped ion interacting with a laser held. During the transition from collapse to revival and vice-versa we have a minimum degree of entanglement value. Successive revival peaks show a lowering of the local maximum points indicating a dissipative irreversible change in the ionic state. We propose a generation of Bell-type states having a simple initial state preparation of a single trapped ion.
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.
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.