Quantum manifestations of order and chaos in the Paul trap

Moore, M G; Blümel, R.

doi:10.1103/physreva.48.3082

Cited by 23 publications

(18 citation statements)

References 39 publications

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“…We have found an extreme sensitivity of the transition amplitudes between energy eigenstates to small variations of the coefficients of the Hamiltonian. It is worth noticing in this regard that the system of two trapped ions interacting via the true Coulomb potential exhibits under certain conditions a chaotic behaviour [41,43,44]. Of course, one cannot expect any chaos, in the strict sense of this word, in the framework of the one-dimensional exactly solvable model.…”

Section: Discussionmentioning

confidence: 99%

Quantum singular oscillator as a model of a two-ion trap: An amplification of transition probabilities due to small-time variations of the binding potential

1998

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Following the paper by M. Combescure [Ann. Phys. (NY) 204, 113 (1990)], we apply the quantum singular time dependent oscillator model to describe the relative one dimensional motion of two ions in a trap. We argue that the model can be justified for low energy excited states with the quantum numbers n ≪ n max ∼ 100, provided that the dimensionless constant characterizing the strength of the repulsive potential is large enough, g * ∼ 10 5 . Time dependent Gaussian-like wave packets generalizing odd coherent states of the harmonic oscillator, and excitation number eigenstates are constructed. We show that the relative motion of the ions, in contradistinction to its center of mass counterpart, is extremely sensitive to the time dependence of the binding harmonic potential, since the large value of g * results in a significant amplification of the transition probabilities between energy eigenstate even for slow time variations of the frequency.

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Section: Discussionmentioning

confidence: 99%

Quantum singular oscillator as a model of a two-ion trap: An amplification of transition probabilities due to small-time variations of the binding potential

1998

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“…ν denotes the scaled axial represents the second secular frequency [29]. We emphasize that both λ and ν are strictly and |ν| > 0 (ν = 0), as stated in [30].…”

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confidence: 99%

“…The equations of the relative motion corresponding to the Hamiltonian function 204 described by eq. (24) can be cast into [29,30]:…”

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confidence: 99%

Investigations on Dynamical Stability in 3D Quadrupole Ion Traps

Mihalcea¹,

Lynch²

2021

Preprint

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of two coupled oscillators, restricted to the case of two ions confined in a 3D Radio-Frequency (RF) trap. A well known model is used to characterize system dynamics, depending on two control parameters: the axial angular moment and the ratio between the radial and axial trap secular frequencies. We extend this model by using the Hessian matrix of the potential function to better characterize stability and critical points of the system. Then, we investigate semi-classical stability for many-body systems consisting of N identical ions and supply a method to identify the critical points. Finally, the model is applied to the case of a 3D Quadrupole Ion Trap (QIT) with axial symmetry for which the associated Hamilton function is obtained. An alternative approach is used to provide insight into the associated many-body dynamics in combined (Paul and Penning) traps, with applications such as stable trapping of antimatter. Ion distribution within the trap can be illustrated by standard numerical programming, using the Hamilton function. The approach is useful in generalizing the parameters of different types of traps in a unified manner.

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“…The simplest non-trivial model to describe the dynamic behaviour is the Hamilton 147 function of the relative motion of two levitated ions that interact via the Coulomb force 148 in a 3D QIT that exhibits axial symmetry, under the time-independent approximation 149 (autonomous Hamiltonian) [16][17][18]. The paper uses this well established model, which 150 we extend.…”

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“…ν denotes the scaled axial 218 (z) component of the angular momentum L z and it is a constant of motion, while µ z 219 represents the second secular frequency [16]. We emphasize that both λ and ν are strictly 25) is integrable and even separable, excluding the case when λ = 1/2 222 and |ν| > 0 (ν = 0), as stated in [17].…”

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confidence: 99%

Investigations on Dynamical Stability in 3D Quadrupole Ion Traps

Mihalcea¹,

Lynch²

2021

Preprint

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We firstly discuss classical stability for a dynamical system of two ions levitated in a 3D Radio-Frequency (RF) trap, assimilated with two coupled oscillators. We obtain the solutions of the coupled system of equations that characterizes the associated dynamics. In addition, we supply the modes of oscillation and demonstrate the weak coupling condition is inappropriate in practice, while for collective modes of motion (and strong coupling) only a peak of the mass can be detected. Phase portraits and power spectra are employed to illustrate how the trajectory executes quasiperiodic motion on the surface of torus, namely a Kolmogorov-Arnold-Moser (KAM) torus. In an attempt to better describe dynamical stability of the system, we introduce a model that characterizes dynamical stability and the critical points based on the Hessian matrix approach. The model is then applied to investigate quantum dynamics for many-body systems consisting of identical ions, levitated in 2D and 3D ion traps. Finally, the same model is applied to the case of a combined 3D Quadrupole Ion Trap (QIT) with axial symmetry, for which we obtain the associated Hamilton function. The ion distribution can be described by means of numerical modeling, based on the Hamilton function we assign to the system. The approach we introduce is effective to infer the parameters of distinct types of traps by applying a unitary and coherent method, and especially for identifying equilibrium configurations, of large interest for ion crystals or quantum logic.

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Quantum manifestations of order and chaos in the Paul trap

Cited by 23 publications

References 39 publications

Quantum singular oscillator as a model of a two-ion trap: An amplification of transition probabilities due to small-time variations of the binding potential

Quantum singular oscillator as a model of a two-ion trap: An amplification of transition probabilities due to small-time variations of the binding potential

Investigations on Dynamical Stability in 3D Quadrupole Ion Traps

Investigations on Dynamical Stability in 3D Quadrupole Ion Traps

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