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Journal of Modern Optics

Dynamical properties of a trapped two-level ion interacting with a single-mode quantized field in the nonlinear regime

Navid Yazdanpanah, Mohammad Kazem Tavassoly

Published Dec 3, 2014• 9 citations
Publisher

Abstract

Following the path of Bužek et al. in [Phys. Rev. A 56 2352 1997], we investigate the full intensity-dependent interaction between a motional two-level ion which is trapped in a f-deformed oscillator (so possessing vibrational modes too) with a single-mode nonlinear quantized field in the Lamb-Dicke limit. After finding the explicit form of the entire ion-field state vector in a general framework, we particularly restrict ourselves to an appropriate nonlinearity function (describing intensity-dependent interaction) corresponding to the center-of-mass motion of a trapped ion which explicitly depends on the Lamb-Dicke parameter as well as another well-known nonlinearity function in the form n. Then, we evaluate the effects of two different initial quantized field states as well as the considered nonlinearity function (via varying the Lamb-Dicke parameter) on a few physical properties of the system. For this purpose, we pay attention to the amount of entanglement, mean number of vibrational quanta (phonons), population inversion of the two-level ion and Q-distribution function. Finally, by comparing our results with the previously considered “linear” one which is pointed in the above Ref., we establish that, by tuning the Lamb-Dicke parameter, one can adjust the mentioned nonclassical properties of the system appropriately.

DOI
10.1080/09500340.2014.986547

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Following the path of Bužek et al. in [Phys. Rev. A 56 2352 1997], we investigate the full intensity-dependent interaction between a motional two-level ion which is trapped in a f-deformed oscillator (so possessing vibrational modes too) with a single-mode nonlinear quantized field in the Lamb-Dicke limit. After finding the explicit form of the entire ion-field state vector in a general framework, we particularly restrict ourselves to an appropriate nonlinearity function (describing intensity-dependent interaction) corresponding to the center-of-mass motion of a trapped ion which explicitly depends on the Lamb-Dicke parameter as well as another well-known nonlinearity function in the form n. Then, we evaluate the effects of two different initial quantized field states as well as the considered nonlinearity function (via varying the Lamb-Dicke parameter) on a few physical properties of the system. For this purpose, we pay attention to the amount of entanglement, mean number of vibrational quanta (phonons), population inversion of the two-level ion and Q-distribution function. Finally, by comparing our results with the previously considered “linear” one which is pointed in the above Ref., we establish that, by tuning the Lamb-Dicke parameter, one can adjust the mentioned nonclassical properties of the system appropriately.
In this paper, the dynamical properties of a trapped three-level ion which interacts with a two-mode quantized field in the presence of a Kerr medium are studied. The ion-field coupling is also considered to be intensity-dependent (nonlinear interaction). In this respect, the Λ, V and ladder-type three-level ions are considered. Via solving the time-dependent Schörodinger equation, the state vectors of the corresponding ion-field systems are analytically obtained after choosing special initial conditions. In the continuation, the mean phonon number as a criterion of trapping energy, the population inversion as a criterion of exchanging energy (between the ion and the field) and the linear entropy as a criterion of entanglement between the trapped ion and the field are studied. To get more general insight about the interaction conditions, the effect of intensity-dependent ion-field coupling (nonlinearity regime) on the dynamics of system is investigated. Finally, by comparing the obtained results for the three configurations of trapped three-level ions with the previously considered two-level one, we conclude that the degree of entanglement for three-level ions is more than the two-level ion. Also, among the considered trapped ions, the ladder-type ion possesses the most amount of entanglement with the quantized field, whereas the Λ-type possesses the least. We also conclude that, the general temporal behaviour of the other considered quantities (mean phonon number and population inversion) are the same for all three types of trapped three-level ions.
To achieve more accuracy in a system containing N trapped two-level ions interacting with a single-mode quantized field, in this paper we will consider the dipole–dipole interaction (DDI) between the trapped ions. In more detail, we first suggest an appropriate form of a quantized DDI Hamiltonian between two trapped ions that has arisen from the spontaneous emission of the excited one. Accordingly, we have considered the spontaneous emission as the origin of dissipation in our proposed model. Then, the case in which the trapped ions interact with a single-mode quantized field in the Lamb–Dicke regime as well as the first vibrational sideband is considered, in which the DDI between the trapped ions is also taken into account. Using the state vector, which is analytically obtained for the whole system under our consideration, we evaluate the amount of entanglement between the field and the two trapped ions by utilizing the linear entropy. We observe that the entanglement depends on the distance between the two trapped ions; i.e., decreasing the distance led to decrement in the amount of entanglement. In the continuation, we extend the proposed model at first to three (N=3) and then to more trapped ions (N>3), and the most general entangled state vectors related to the outlined systems are obtained, for which the same scenario (for two trapped ions) is repeated. Comparing the latter cases (N≥3) with the previous one (N=2), we conclude that the entanglement is sensitive to the number of interacting trapped ions in addition to the distances between them in such a way that by increasing the number of trapped ions, the degree of entanglement between them and the field is increased. These results help one to control the entanglement between the field and the trapped ions in a more accurate manner by tuning the distances between the trapped ions and their numbers.
We investigate the dynamics of a cold trapped ion coupled to the quantized field inside a high-finesse cavity, considering exact resonance between the ionic internal levels and the field (carrier transition). We derive an intensity-dependent Hamiltonian in which terms proportional to the square of the Lamb-Dicke parameter $(\ensuremath{\eta})$ are retained. We show that different nonclassical effects arise in the dynamics of the ionic population inversion, depending on the initial states of the vibrational motion and field and on the values of $\ensuremath{\eta}.$
We analyze the quantum motion of a cold, trapped two-level ion interacting with a quantized light field in a single-mode cavity. We show that in the nonclassical Lamb-Dicke limit the time evolution of the vibrational mode representing the quantized motion of the center of mass of the trapped ion is very sensitive to the quantum statistics of the light field. We also show that the system under consideration may evolve into the maximally entangled three-particle Greenberger-Horne-Zeilinger state. We briefly discuss the dynamics of a cluster of two-level ions trapped in a cavity and interacting with a quantized light field.
We investigate quantum dynamical properties of a trapped three-level ion interacting with two laser beams in $\ensuremath{\Lambda}$ configuration. A unitary transformation method is developed to study the interaction of the ion with its vibrational phonons, quanta of ion's own center-of-mass motion. Under certain conditions on laser parameters, this interaction is shown to be unitarily equivalent to two-phonon cascade transitions. Complicated temporal behaviors of level populations and mean number of phonons are described clearly by identifying dynamical variables of the cascade model as building blocks. Furthermore, analyzing quantum states of vibrational phonons by Husimi-QQQ function, we find that at times, determined by the underlying cascade dynamics, two- and three-component macroscopic quantum superposition states can be obtained depending on the Lamb-Dicke parameter and the initial conditions of the system. A wide range of initial conditions and experimental parameters are discussed using both exact and analytical solutions. Alternative routes to reach the target states are found.
The dynamics of a two-level atom driven by a single laser beam and three-level atom (Lambda configuration) irradiated by two laser beams are studied taking into account of the quantized center-of-mass motion of the atom. It is shown that the trapped atom system under appropriate resonance condition exhibits the large time-scale revivals when the index of the vibrational sideband responsible for the atomic electronic transition is greater than unity. The revival times are shown to be dependent on the initial number of vibrational excitations and the magnitude of the Lamb-Dicke parameter. The sub-Poissonian statistics in vibrational quantum number is observed at certain time intervals. The minimum time of interaction for which the squeezed states of motional quadrature are generated is found to be decreasing with the increase in the Lamb-Dicke parameter.