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Translational dynamics of diatomic molecule in magnetic quadrupole trap

Yurij Yaremko, Maria Przybylska, Andrzej J. Maciejewski

Abstract

We study the translational motions of homonuclear diatomic molecules prepared in their ${}^3Σ$ electronic states, deeply bound vibrational states, and rotational states of well-defined parity. The trapping potential arises due to the interaction of the total spin of electrons and orbital angular momentum of nuclei with the trap's quadrupole magnetic field. The translational motion of a molecule is treated classically. We examine the Hamilton equations that govern the center of mass dynamics numerically and analytically. Using data of a hydrogen molecule at the ground vibrational state, we present global dynamics using the Poincaré section method and various types of trajectories: periodic, quasi-periodic and chaotic. We prove that the Hamiltonian system governing this motion is non-integrable. The particle's orbits are confined to a bound region of space that grows with energy, but for small energies (< 1.8 K), the motion is restricted to a processing chamber (a few centimetres). Solutions of equations of motion occurring on the symmetry axis and the horizontal plane are expressed in terms of Jacobi elliptic functions.

Translational dynamics of diatomic molecule in magnetic quadrupole trap

Abstract

We study the translational motions of homonuclear diatomic molecules prepared in their electronic states, deeply bound vibrational states, and rotational states of well-defined parity. The trapping potential arises due to the interaction of the total spin of electrons and orbital angular momentum of nuclei with the trap's quadrupole magnetic field. The translational motion of a molecule is treated classically. We examine the Hamilton equations that govern the center of mass dynamics numerically and analytically. Using data of a hydrogen molecule at the ground vibrational state, we present global dynamics using the Poincaré section method and various types of trajectories: periodic, quasi-periodic and chaotic. We prove that the Hamiltonian system governing this motion is non-integrable. The particle's orbits are confined to a bound region of space that grows with energy, but for small energies (< 1.8 K), the motion is restricted to a processing chamber (a few centimetres). Solutions of equations of motion occurring on the symmetry axis and the horizontal plane are expressed in terms of Jacobi elliptic functions.
Paper Structure (14 sections, 1 theorem, 101 equations, 15 figures, 1 table)

This paper contains 14 sections, 1 theorem, 101 equations, 15 figures, 1 table.

Table of Contents

  1. Introduction
  2. Zeeman effect
  3. Spin Zeeman effect
  4. Linear Zeeman effect
  5. Quadratic Zeeman effect
  6. Depth of the trap
  7. Numerical simulations
  8. Integrability analysis and solutions on invariant submanifolds
  9. Conclusions
  10. Hydrogen molecule in ${}^3\Sigma$ electronic state
  11. Differential Galois obstructions for integrability of homogeneous potentials
  12. Turning points
  13. Turning points in the manifold $z=p_z=0$
  14. Turning points in the manifold $x=y=p_x=p_y=0$

Key Result

Theorem 1

Assume that an algebraic homogeneous potenial $V(\boldsymbol{q})$, $\boldsymbol{q}$ of degree $k\in\mathbb{Q}\setminus\{0\}$ defined by minimal polynomial $F(\boldsymbol{q},u)$ satisfies the following conditions: Then

Figures (15)

  • Figure 1: The Poincaré section for parameters $\sigma=0.502723$ and $\delta=0.0000179305$, $p_{\varphi}=\frac{1}{100}$ for different values of energy $h$. Cross-section plane $z=0$ and the orientation of the trajectories $p_z>0$. Energy and temperature values in the subfigures: (a) $h=0.03$ (0.4201 K); (b) $h=0.0449$ (0.6288 K); (c) $h=0.057$ (0.7983 K); (d) $h=0.065$ (0.9103 K).
  • Figure 2: The Poincaré section for parameters $\sigma=0.502723$ and $\delta=0.0000179305$, $p_{\varphi}=\frac{1}{100}$ and the energy value $h=0.125$ (1.7507 K). Cross-section plane $z=0$ and the orientation of the trajectories $p_z>0$.
  • Figure 3: Magnification of chaotic region near intersection point $CH$ in Fig. \ref{['fig:J10Mm10_en_0d125_arrows']}.
  • Figure 4: Evolution of the center of mass vector $\mathbf{x}$ and the time of integration $\tau$ for selected periodic orbits $P_1,P_2$ and $P_3$ in Fig.\ref{['fig:J10Mm10_en_0d125_arrows']}. Subfigures: (a) orbit $P_1$ in plane $(r,z)$, $\tau=700$; (b) orbit $P_1$ in space, $\tau=700$; (c) orbit $P_2$ in plane $(r,z)$, $\tau=700$; (d) orbit $P_2$ in space, $\tau=700$; (e) orbit $P_3$ in plane $(r,z)$, $\tau=700$; (f) orbit $P_3$ in space, $\tau=700$.
  • Figure 5: Evolution of the center of mass vector $\mathbf{x}$ and the time of integration $\tau$ for selected quasi-periodic orbits $Q_1,Q_2$ and $Q_3$ in Fig. \ref{['fig:J10Mm10_en_0d125_arrows']}. Subfigures: (a) orbit $Q_1$ in plane $(r,z)$, $\tau=600$; (b) orbit $Q_1$ in space, $\tau=1200$; (c) orbit $Q_2$ in plane $(r,z)$, $\tau=500$; (d) orbit $Q_2$ in space, $\tau=900$; (e) orbit $Q_3$ in plane $(r,z)$, $\tau=1000$; (f) orbit $Q_3$ in space, $\tau=1000$.
  • ...and 10 more figures

Theorems & Definitions (1)

  • Theorem 1