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Chaos An Interdisciplinary Journal of Nonlinear Science

Quantum chaos of an ion trapped in a linear ion trap

G. P. Berman, Daniel F. V. James, D. I. Kamenev

Published Jun 1, 2000• 3 citations

Abstract

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.

DOI: 10.1063/1.166504

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A system of N two-level atoms in a resonator is considered interacting with a resonant eigenmode field and with an external coherent field, with a frequency slightly different from the frequency of the atomic transition. A model Hamiltonian is constructed for describing the slow quantum dynamics of the system, and a set of closed c-number equations for time-dependent quantum expectation values is derived in the boson and spin coherent states. In the region of parameters for developed chaos in the semiclassical limit (when the radiation field is considered classically) we show that the semiclassical approximation is violated by quantum effects at the time scale ${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\Elzxh}}}$\ensuremath{\sim}lnN. We also show that quantum correlations grow exponentially during the time interval 0${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\Elzxh}}}$. An additional averaging over the initial density matrix is considered. The connection of our model with the Dicke model [Phys. Rev. 93, 99 (1956)] is demonstrated, and the possiblity of measuring the quantum chaos effects considered in this paper is discussed.

A classically chaotic system consisting of a Paul trapped ion and a sequences of standing laser pulses is treated quantum-mechanically. Under the circumstance of time-dependence, we derive the transition probability from the ion's motional state to ', and find, in the first-order approximation, the classically chaotic character disappears. Theoretical analysis and numerical calculations show that by regulating the phase parameter we can control the transition probability. When reaches some specific values, the transition from the state to ' is forbidden and, for some laser periods, resonance occurs, which leads to the corresponding transitions between different motional states. The time-evolution of an initial motional state just over one period is also studied in detail.

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.