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arXiv:quant-ph/0012098 · Physical Review A

Dynamical stability of an ion in a linear trap as a solid-state problem of electron localization

G. P. Berman, A. R. Bishop, Daniel F. V. James, Richard Hughes, D. I. Kamenev

Published Oct 5, 2001• 3 citations
arXivHTMLPDF

Abstract

When an ion confined in a linear ion trap interacts with a coherent laser field, the internal degrees of freedom, related to the electron transitions, couple to the vibrational degree of freedom of the ion. As a result of this interaction, the quantum dynamics of the vibrational degree of freedom becomes complicated, and in some ranges of parameters even chaotic. We analyze the vibrational ion dynamics using a formal analogy with the solid-state problem of electron localization. In particular, we show how the resonant approximation used in analysis of the ion dynamics, leads to a transition from a two-dimensional (2D) to a one-dimensional problem (1D) of electron localization. The localization length in the solid-state problem is estimated in cases of weak and strong interaction between the sites of the 2D cell by using the methods of resonance perturbation theory, common in analysis of 1D time-dependent dynamical systems. We show that the localization length can be used as an indicator of the effective temperature of the trapped ion, which can be experimentally measured.

DOI
10.1103/physreva.64.053406

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When an ion confined in a linear ion trap interacts with a coherent laser field, the internal degrees of freedom, related to the electron transitions, couple to the vibrational degree of freedom of the ion. As a result of this interaction, the quantum dynamics of the vibrational degree of freedom becomes complicated, and in some ranges of parameters even chaotic. We analyze the vibrational ion dynamics using a formal analogy with the solid-state problem of electron localization. In particular, we show how the resonant approximation used in analysis of the ion dynamics, leads to a transition from a two-dimensional (2D) to a one-dimensional problem (1D) of electron localization. The localization length in the solid-state problem is estimated in cases of weak and strong interaction between the sites of the 2D cell by using the methods of resonance perturbation theory, common in analysis of 1D time-dependent dynamical systems. We show that the localization length can be used as an indicator of the effective temperature of the trapped ion, which can be experimentally measured.
We describe the transition to quantum chaos of an ion trapped in a linear ion trap and interacting with two laser fields. Under the conditions of adiabatic illumination of the upper level of the ion, and when the frequencies of the two laser beams are slightly different, the system is reduced to a quantum linear oscillator interacting with a monochromatic wave. The property of localization over the quantum resonance cells is proposed to exploit in order to facilitate the process of measurement of the probability distribution of an ion on the vibrational levels. In the regime of strong chaos the time-averaged values of the energy and dispersion of energy are computed and compared with the corresponding classical quantities for different values of the perturbation amplitude. In the exact resonance case, the classical analog of the system possesses an infinite inhomogeneous stochastic web. We analyze the quantum dynamics inside the inhomogeneous web. It is shown that the quantum system mimics on average the dynamics of the corresponding classical system. Formation of the quantum resonance cells is illustrated in the case of a finite detuning from the exact resonance, and under increasing of the wave amplitude. The parameters of the model and the initial conditions are close to the real physical situation which can be realized in the system of cold trapped ion perturbed by two lasers fields with close frequencies. (c) 2000 American Institute of Physics.
The phenomenon of localization manifests itself in many quantum mechanical systems ranging from the localization of light in a random medium via Anderson localization of an electronic wave to the motion of atoms in time-dependent laser fields. In all these cases the underlying classical system is chaotic and shows diffusion as a function of time. In contrast, the quantum mechanical counterpart has a localized wave function whose width is governed by the classical diffusion and Planck’s constant. In this paper we show that there exists an additional quantum parameter that controls the localization length. In the system of a two-level ion stored in a Paul trap and interacting with a standing wave it is the detuning between the transition frequency and the laser field. We also discuss the effect of decoherence in form of spontaneous emission. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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.
Actually it is possible to trap single ions in Paul or Penning traps, charged particles are confined in space via electromagnetic fields. A trapped ion can have a lot vibrational energy which is undesirable in some applications. In this thesis, starting from the Hamiltonian of a trapped ion in a harmonic potential, it is shown that is possible to get Jaynes-Cummings and anti- Jaynes- Cummings¬like transitions in the intermediate intensity regime, that is, adjusting the intensity of the laser that induce the coupling between the intemal and vibrational states of the ion proportionally to the vibrational frequency of the ion; these transitions are useful because make possible to extract the vibrational energy. In our treatment we consider that the ion and the laser are in resonance what eliminate the need of using tuned lasers.\nAlso, the problem of a trapped ion in a trap with time-dependent parameters interacting with a laser field is analyzed. By using a set of time-dependent unitary transformations is shown that this system is equivalent to the interaction between a quantized field and a two level system with time dependent parameters. The Hamiltonian is linearized in such a way that can be solved with methods that are found in some papers and that involve time-dependent parameters. The linearization is free of approximations and assumptions on the parameters of the system as are, for instance, the Lamb-Dicke parameter, the time-dependency of the frequency trap and the detuning, with what we can obtain the best solution for this kind of sytem.
Atoms and ions confined with electric and optical fields form the basis of many current quantum simulation and computing platforms. When excited to high-lying Rydberg states, long-ranged dipole interactions emerge which strongly couple the electronic and vibrational degrees of freedom through state-dependent forces. This vibronic coupling and the ensuing hybridization of internal and external degrees of freedom manifest through clear signatures in the many-body spectrum. We illustrate this by considering the case of two trapped Rydberg ions, for which the interaction between the relative vibrations and Rydberg states realizes a quantum Rabi model. We proceed to demonstrate that the aforementioned hybridization can be probed by radio frequency spectroscopy and discuss observable spectral signatures at finite temperatures and for larger ion crystals.
We propose a scheme for preparation of entangled coherent states for the motion of an ion in a two-dimensional anisotropic trap. In the scheme, the ion is driven by four laser beams along different directions in the ion trap plane, resulting in carrier excitation and couplings between the internal and external degrees of freedom. When the total quantum number of the vibrational modes initially has a definite parity, the competition between the unitary dynamics and spontaneous emission will force the system to evolve to a steady state, where the vibrational modes are in a two-mode cat state. We show that the method can be extended to realization of entangled coherent states for three vibrational modes of an ion in a three-dimensional anisotropic trap.