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Acta Physica Sinica

Generations of coherent states of single trapped cold ion beyond the Lamb-Dicke limit

Miao Zhang, Jia Huan-Yu, 西南交通大学现代物理研究所，成都 610031

Published Jan 1, 2008• 9 citations
Publisher

Abstract

Based on the quantum dynamics of laser-ion interaction beyond the Lamb-Dicke limit，we discuss how to superpose a series of vibrational number states from motional ground state by sequentially applying laser beams. The Q function，Wigner distribution function，and the nonclassical properties such as squeezing effect and anti-bunching effect of the generated superposition states are investigated in detail by numerical method. It is shown that，by controlling the durations of the applied laser pulses，these quantum states could well approach to the well-known coherent states.

DOI
10.7498/aps.57.880

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Based on the quantum dynamics of laser-ion interaction beyond the Lamb-Dicke limit，we discuss how to superpose a series of vibrational number states from motional ground state by sequentially applying laser beams. The Q function，Wigner distribution function，and the nonclassical properties such as squeezing effect and anti-bunching effect of the generated superposition states are investigated in detail by numerical method. It is shown that，by controlling the durations of the applied laser pulses，these quantum states could well approach to the well-known coherent states.
Based on the quantum dynamics of laser-ion interaction beyond the Lamd_Dicke limit, we discuss how to superpose a series of vibrational number states of a single trapped ion from the motional ground state. It is shown that, by controlling the durations and phases of the sequentially applied laser pulses, these superposed quantum states could well approach to the various target quantum states, e.g., squeezed coherent states, squeezed odd/even coherent states and squeezed vacuum states, etc.Their experimental realizabilities are also discussed.
Based on the conditional quantum dynamics of laser-ion interactions, we propose an efficient theoretical scheme to deterministically generate quantum pure states of a single trapped cold ion without performing the Lamb-Dicke approximation. An arbitrary quantum state can be created by sequentially using a series of classical laser pulses with selected frequencies, initial phases and durations. As special examples, we further show how to create or approximate several typical macroscopic quantum states, such as the phase state and the even/odd coherent states. Unlike previous schemes operating in the Lamb-Dicke regime, the present one does well for an arbitrary-strength coupling between the internal and external degrees of freedom of the ion. The experimental realizability of this approach is also discussed.
The effects of higher-order Lamb–Dicke approximations (LDAs), on engineering motional quantum states of a single trapped cold ion, are discussed. By measuring the internally electronic states of a single trapped cold ion, which is driven by two classical laser beams tuning respectively to the first upper and lower sidebands, we show that the externally macroscopic vibrational states superposed by either even or odd number states are generated. Under the LDA (i.e., only the first order effect of LD parameter η is considered), these states just are the usual even and odd coherent states. For relative large values of η, we showed that the effects of higher-order LDAs, i.e., terms relating to ηk, k ≥ 2, are not negligible. They might significantly enhance certain nonclassical properties, e.g., squeezing and antibunching effects of the generated macroscopic quantum states.
In coherent state representation and in the absence of rotating wave approximation, the interaction between a standing wave laser and a trapped ultracold ion within the Lamb-Dicke limit, expanded to the second-order term of the Lamb-Dicke parameter, can be solved analytically by series expansion technique. It can be seen that, for some particular detunings, the eigenenergies of the system can be exactly obtained and the eigenfunctions, presented as series forms, are a superposition of displaced Fock states and displaced squeezed states. Making use of our solution, some particular nonclassical states could be prepared.
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.