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arXiv:1511.04114 · The Journal of Chemical Physics

Rotational dynamics of a diatomic molecular ion in a Paul trap

A. Hashemloo, Claude M. Dion

Published Nov 25, 2015• 8 citations
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Abstract

We present models for a heteronuclear diatomic molecular ion in a linear Paul trap in a rigid-rotor approximation, one purely classical and the other where the center-of-mass motion is treated classically, while rotational motion is quantized. We study the rotational dynamics and their influence on the motion of the center-of-mass, in the presence of the coupling between the permanent dipole moment of the ion and the trapping electric field. We show that the presence of the permanent dipole moment affects the trajectory of the ion and that it departs from the Mathieu equation solution found for atomic ions. For the case of quantum rotations, we also evidence the effect of the above-mentioned coupling on the rotational states of the ion.

DOI
10.1063/1.4936425

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We present models for a heteronuclear diatomic molecular ion in a linear Paul trap in a rigid-rotor approximation, one purely classical and the other where the center-of-mass motion is treated classically, while rotational motion is quantized. We study the rotational dynamics and their influence on the motion of the center-of-mass, in the presence of the coupling between the permanent dipole moment of the ion and the trapping electric field. We show that the presence of the permanent dipole moment affects the trajectory of the ion and that it departs from the Mathieu equation solution found for atomic ions. For the case of quantum rotations, we also evidence the effect of the above-mentioned coupling on the rotational states of the ion.
This thesis explores the dynamics of a heteronuclear diatomic molecular ion, possessing a permanent electric dipole moment, µ, which is trapped in a linear Paul trap and can interact with an off-resonance laser field. To build our model we use the rigid-rotor approximation, where the dynamics of the molecular ion are limited to its translational and rotational motions of the center-of-mass. These dynamics are investigated by carrying out suitable numerical calculations. To introduce our numerical methods, we divide our research topic into two different subjects. First, we ignore the rotational dynamics of the ion by assuming µ = 0. By this assumption, the system resembles an atomic ion, which mainly exhibits translational motion for its center of the mass when exposed to an external trapping field. To study this translational behavior, we implement full-quantum numerical simulations, in which a wave function is attributed to the ion. Finally, we study the quantum dynamics of the mentioned wave packet and we compare our results with those obtained classically. In the latter case, we keep the permanent dipole moment of the ion and we study the probable effects of the interaction between the dipole moment and the trapping electric field, on both the translational and the rotational dynamics of the trapped molecular ion. In order to study these dynamics, we implement both classical and semi-classical numerical simulations. In the classical method, the rotational and the translational motions of the center of mass of the ion are obtained via classical equations of motion. On the other hand, in the semi-classical method, while the translational motion of the center-of-mass is still obtained classically, the rotation is treated full-quantum mechanically by considering the rotational wave function of the ion. In the semi-classical approach, we mainly study the probable couplings between the rotational states of the molecular ion, due to the interaction of the permanent dipole moment with the trapping electric field. In the end, we also present a semi-classical model, where the trapped molecular ion interacts with an off-resonance laser field.
Abstract We discuss the trapping of heteronuclear diatomic molecules prepared in their electronic and vibrational ground states. We tune and shape the trapping potential for bosonic polar molecules in superpositions of rotational states by dressing rotational excitations with a static sextupole electric field. The translational motion of a molecule is treated classically. We examine the Hamiltonian which governs the center of mass dynamics. The effective potential has a global minimum that provides the trapping ability of this trap. The first term of its Taylor series expansion, corresponding to the quadratic Stark shifts, results in the integrable potential. In terms of cylindrical coordinates the center of mass Hamiltonian splits into axial and radial parts. Corresponding trajectories are parameterized by elliptic functions. At low electric fields, the non-approximated Hamiltonian is treated as a small perturbation of the mentioned integrable system described by Kolmogorov–Arnold–Moser theory. The applicability of this approximation is discussed and illustrated using the Poincaré cross-section method. We present results of numerical simulations illustrating the trapping and confinement of a polar molecule in the trap.
Using numerical simulations of the time-dependent Schrödinger equation, we study the full quantum dynamics of the motion of an atomic ion in a linear Paul trap. Such a trap is based on a time-varying, periodic electric field and hence corresponds to a time-dependent potential for the ion, which we model exactly. We compare the center-of-mass motion with that obtained from classical equations of motion, as well as to results based on a time-independent effective potential. We also study the oscillations of the width of the ion’s wave packet, including close to the border between stable (bounded) and unstable (unbounded) trajectories. Our results confirm that the center-of-mass motion always follows the classical trajectory, that the width of the wave packet is bounded for trapping within the stability region, and therefore that the classical trapping criterion is fully applicable to quantum motion.
Abstract A model of rotational potential that represents an analog of the pseudopotential for localization of a model of diatomic particle with rigid bond in a plane quadrupole radio-frequency trap is considered. It is shown that the effective rotational potential can be used to describe dynamics of various diatomic particles with different centers of mass and charges. Comparison of the model of pseudopotential for localization of a single ion and the proposed model of the effective rotational potential for diatomic structure is used to determine additional positions of quasi-equilibrium for the center of mass of the diatomic particle and orientation angle of the molecule. Additional positions of quasi-equilibrium substantially affect dynamics of charged structure localized in the field of a plane quadrupole ion trap.
When molecular ions are trapped together with atomic ions in a Paul trap, their dipole moment couples the molecular rotation to the joint vibrational motion of the particles in the trap. To leverage this coupling for quantum control, a rotational transition in the molecule should be resonant with one of the ion crystal vibrational modes. Focusing on the example of singly charged thorium fluoride molecular ions, cotrapped with two ytterbium ions, we determine the conditions for resonant dipole-phonon coupling, fully accounting for the molecular hyperfine structure. We identify several choices for resonant coupling and discuss its detection using sideband-resolved laser spectroscopy and measurements of decoherence.
We investigate rotational state changes in a single collision of diatomic molecular ions, polar or apolar, with an atomic ion. Rotational state changes may occur since the angular degree of freedom of the molecular ions interacts with the electric field due to the atomic ion. Thanks to the very different time and energy scales of translational and rotational motion, we may treat the collision classically and describe only the rotations quantum mechanically. We first investigate a number of example systems numerically and then derive closed-form approximations for the rotational excitation per collision, depending on the scattering energy and the molecular parameters. These findings provide the basis for estimating the accumulated rotational excitation in sympathetic cooling of molecular ions by laser-cooled atomic ions [arXiv:2410.22458 ] which involves many single collisions.