Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2112.02203 · International Journal of Modern Physics A

Simulation of motion of many ions in a linear Paul trap

Martin Oral, Ondřej Čı́p, L. Slodička

Published Dec 30, 2019• 2 citations
arXivHTMLPDF

Abstract

The quadrupole linear Paul trap is one of the key instruments in building highly stable atomic clocks. However, a frequency reference based on a single trapped ion is limited in stability due to the time needed for the interrogation cycle which cannot be further shortened. A promising strategy is the utilization of multiple trapped ions. The ions of the same kind then repulse each other with the Coulomb force, which is countered by the ponderomotive force of the time depended field in the trap. A few ions form a chain along the axis of a linear Paul trap. Adding more ions (a few tens or hundreds) gives rise to Coulomb crystals. We created an efficient simulation code which calculates the motion of such collections of ions in quasi-static radiofrequency fields of real linear quadrupole traps (including the micromotion). We include a model for laser cooling of the ions. The simulation tool can be used to study the formation and the dynamics of Coulomb crystals under conditions corresponding to various experimental setups.

DOI
10.1142/s0217751x1942003x

Related papers

Simulating Trapped Ion Crystals as Potential Superradiant Emitters

L. Iqbal

Superradiant lasers have emerged as a promising platform for optical frequency references due to their potential to achieve exceptionally narrow linewidths with reduced sensitivity to cavity noise. Trapped ion Coulomb crystals provide an attractive emitting medium due to their long coherence times, great environmental isolation and precise control over their motional dynamics. However, scaling trapped ion systems to the large ion numbers required for strong collective emission introduces new challenges, including crystal stability and position-dependent decoherence mechanisms. This work presents a numerical framework for modelling large Coulomb crystals confined within a linear Paul trap and assessing their suitability for superradiant lasing. A numerical simulation was developed to generate equilibrium crystal configurations, followed by a normal mode analysis to verify crystal’s dynamical stability and characterise the collective motional spectrum. The dependence of the crystal properties on the trapping parameters was examined, while the damping coefficient and integration timestep used in the numerical relaxation procedure were systematically optimised to obtain equilibrium configurations. The decoherence mechanisms expected to influence superradiant operation, including RF micromotion and electric quadrupole shifts, were also investigated as functions of ion number and trapping conditions. The simulations can produce accurate models of mechanically stable Coulomb crystals with up to 10,000 ions under appropriate relaxation conditions. They also show how the collective motional spectrum evolves across the trapping parameter space where they were also compared with the theoretical lasing threshold to assess the suitability of the trapping conditions for superradiant operation. Although the largest micromotion amplitudes increase with crystal size, a significant fraction of the ions remain confined close to the RF null, suggesting that large ion numbers may still be compatible with superradiant operation. The developed framework also provides a means of quantifying the quadrupole shift, indicating that the resulting frequency shifts are relatively small for the trapping parameters considered here, although further optimisation of the operating conditions is required to fully assess their influence on superradiant perfomance. Overall, the computational suite developed here establishes a versatile platform for studies of large Coulomb ion crystals and provides the tools to determine whether superradiant lasing thresholds can be reached with trapped ion crystals.

•Jan 1, 2026•0 citations

Dynamics of charged particles in a Paul radio-frequency quadrupole trap

John D. Prestage, Ann Katherine Williams, Lute Maleki +2 more

We have carried out a molecular-dynamics simulation of hundreds of ions confined in a Paul trap. The simulation includes the trapped particles' micromotion and interparticle Coulomb interactions. We have implemented a random walk in velocity to bring the secular motion to a given temperature which we have numerically measured. When the coupling \ensuremath{\Gamma} is large the ions form concentric shells which undergo a quadrupole oscillation at the rf frequency, while the ions within a shell form a 2D hexagonal lattice. Ion clouds at 5 mK show no rf heating for ${\mathit{q}}_{\mathit{z}}$\ensuremath{\lesssim}0.6, whereas rapid heating is seen for ${\mathit{q}}_{\mathit{z}}$=0.8.

Physical Review Letters•Jun 10, 1991•65 citations

Motional states of laser cooled Yb ions in an optimized radiofrequency trap

Aleksandr Borisenko, Ilia V. Zalivako, Ilya A. Semerikov +3 more

Abstract Efficient cooling of trapped ions is one of the key problems in the development of optical clocks and quantum sensors, in quantum computations, and many other fields. We numerically analyze the motion of linear Yb + crystals and optimized trapping potential to achieve a uniform spatial distribution of ions. Using a compact robust laser system we demonstrate Doppler cooling of a 174 Yb + isotope in a quadrupole Paul trap. This is an important step towards laser cooling of 171 Yb + ions, which will be used in single-ion optical clocks and for the realization of high-fidelity quantum gates.

Laser Physics•Jul 26, 2019•3 citations

Linear Paul trap design for an optical clock with Coulomb crystals

Norbert Herschbach, Karsten Pyka, J. Keller +1 more

We report on the design of a segmented linear Paul trap for optical clock applications using trapped ion Coulomb crystals. For an optical clock with an improved short-term stability and a fractional frequency uncertainty of 10−18, we propose 115In+ ions sympathetically cooled by 172Yb+. We discuss the systematic frequency shifts of such a frequency standard. In particular, we elaborate on high-precision calculations of the electric radiofrequency field of the ion trap using the finite element method. These calculations are used to find a scalable design with minimized excess micromotion of the ions at a level at which the corresponding second-order Doppler shift contributes less than 10−18 to the relative uncertainty of the frequency standard.

•Aug 17, 2016•34 citations

Characterization of ion Coulomb crystals in a linear Paul trap

K. Okada, M. Wada, Toshiyuki Takayanagi +2 more

We describe a simple and fast method for simulating observed images of ion Coulomb crystals. In doing so, cold elastic collisions between Coulomb crystals and virtual very light atoms are implemented in a molecular dynamics (MD) simulation code. Such an approach reproduces the observed images of Coulomb crystals by obtaining density plots of the statistics of existence of each ion. The simple method has the advantage of short computing time in comparison with previous calculation methods. As a demonstration of the simulation, the formation of a planar Coulomb crystal with a small number of ions has been investigated in detail in a linear ion trap both experimentally and by simulation. However, also large Coulomb crystals including up to 1400 ions have been photographed and simulated to extract the secular temperature and the number of ions. For medium-sized crystals, a comparison between experiments and calculations has been performed. Moreover, an MD simulation of the sympathetic cooling of small molecular ions was performed in order to test the possibility of extracting the temperature and the number of refrigerated molecular ions from crystal images of laser-cooled ions. Such information is basic to studying ultracold ion-molecule reactions using ion Coulomb crystals including sympathetically cooled molecular ions.

Physical Review A•Jan 28, 2010•52 citations

Modes of oscillation in radiofrequency Paul traps

Haggai Landa, Michael Drewsen, Benni Reznik +1 more

We examine the time-dependent dynamics of ion crystals in radiofrequency traps. The problem of stable trapping of general three-dimensional crystals is considered and the validity of the pseudopotential approximation is discussed. We analytically derive the micromotion amplitude of the ions, rigorously proving well-known experimental observations. We use a recently proposed method to find the modes that diagonalize the linearized time-dependent dynamical problem. This allows one to obtain explicitly the ('Floquet–Lyapunov') transformation to coordinates of decoupled linear oscillators. We demonstrate the utility of the method by analyzing the modes of a small 'peculiar' crystal in a linear Paul trap. The calculations can be readily generalized to multispecies ion crystals in general multipole traps, and time-dependent quantum wavefunctions of ion oscillations in such traps can be obtained.

arXiv:1206.4006•New Journal of Physics•Sep 13, 2012•65 citations
The quadrupole linear Paul trap is one of the key instruments in building highly stable atomic clocks. However, a frequency reference based on a single trapped ion is limited in stability due to the time needed for the interrogation cycle which cannot be further shortened. A promising strategy is the utilization of multiple trapped ions. The ions of the same kind then repulse each other with the Coulomb force, which is countered by the ponderomotive force of the time depended field in the trap. A few ions form a chain along the axis of a linear Paul trap. Adding more ions (a few tens or hundreds) gives rise to Coulomb crystals. We created an efficient simulation code which calculates the motion of such collections of ions in quasi-static radiofrequency fields of real linear quadrupole traps (including the micromotion). We include a model for laser cooling of the ions. The simulation tool can be used to study the formation and the dynamics of Coulomb crystals under conditions corresponding to various experimental setups.
Superradiant lasers have emerged as a promising platform for optical frequency references due to their potential to achieve exceptionally narrow linewidths with reduced sensitivity to cavity noise. Trapped ion Coulomb crystals provide an attractive emitting medium due to their long coherence times, great environmental isolation and precise control over their motional dynamics. However, scaling trapped ion systems to the large ion numbers required for strong collective emission introduces new challenges, including crystal stability and position-dependent decoherence mechanisms. This work presents a numerical framework for modelling large Coulomb crystals confined within a linear Paul trap and assessing their suitability for superradiant lasing. A numerical simulation was developed to generate equilibrium crystal configurations, followed by a normal mode analysis to verify crystal’s dynamical stability and characterise the collective motional spectrum. The dependence of the crystal properties on the trapping parameters was examined, while the damping coefficient and integration timestep used in the numerical relaxation procedure were systematically optimised to obtain equilibrium configurations. The decoherence mechanisms expected to influence superradiant operation, including RF micromotion and electric quadrupole shifts, were also investigated as functions of ion number and trapping conditions. The simulations can produce accurate models of mechanically stable Coulomb crystals with up to 10,000 ions under appropriate relaxation conditions. They also show how the collective motional spectrum evolves across the trapping parameter space where they were also compared with the theoretical lasing threshold to assess the suitability of the trapping conditions for superradiant operation. Although the largest micromotion amplitudes increase with crystal size, a significant fraction of the ions remain confined close to the RF null, suggesting that large ion numbers may still be compatible with superradiant operation. The developed framework also provides a means of quantifying the quadrupole shift, indicating that the resulting frequency shifts are relatively small for the trapping parameters considered here, although further optimisation of the operating conditions is required to fully assess their influence on superradiant perfomance. Overall, the computational suite developed here establishes a versatile platform for studies of large Coulomb ion crystals and provides the tools to determine whether superradiant lasing thresholds can be reached with trapped ion crystals.
We have carried out a molecular-dynamics simulation of hundreds of ions confined in a Paul trap. The simulation includes the trapped particles' micromotion and interparticle Coulomb interactions. We have implemented a random walk in velocity to bring the secular motion to a given temperature which we have numerically measured. When the coupling \ensuremath{\Gamma} is large the ions form concentric shells which undergo a quadrupole oscillation at the rf frequency, while the ions within a shell form a 2D hexagonal lattice. Ion clouds at 5 mK show no rf heating for qz{\mathit{q}}_{\mathit{z}}qz​\ensuremath{\lesssim}0.6, whereas rapid heating is seen for qz{\mathit{q}}_{\mathit{z}}qz​=0.8.
Abstract Efficient cooling of trapped ions is one of the key problems in the development of optical clocks and quantum sensors, in quantum computations, and many other fields. We numerically analyze the motion of linear Yb + crystals and optimized trapping potential to achieve a uniform spatial distribution of ions. Using a compact robust laser system we demonstrate Doppler cooling of a 174 Yb + isotope in a quadrupole Paul trap. This is an important step towards laser cooling of 171 Yb + ions, which will be used in single-ion optical clocks and for the realization of high-fidelity quantum gates.
We report on the design of a segmented linear Paul trap for optical clock applications using trapped ion Coulomb crystals. For an optical clock with an improved short-term stability and a fractional frequency uncertainty of 10−18, we propose 115In+ ions sympathetically cooled by 172Yb+. We discuss the systematic frequency shifts of such a frequency standard. In particular, we elaborate on high-precision calculations of the electric radiofrequency field of the ion trap using the finite element method. These calculations are used to find a scalable design with minimized excess micromotion of the ions at a level at which the corresponding second-order Doppler shift contributes less than 10−18 to the relative uncertainty of the frequency standard.
We describe a simple and fast method for simulating observed images of ion Coulomb crystals. In doing so, cold elastic collisions between Coulomb crystals and virtual very light atoms are implemented in a molecular dynamics (MD) simulation code. Such an approach reproduces the observed images of Coulomb crystals by obtaining density plots of the statistics of existence of each ion. The simple method has the advantage of short computing time in comparison with previous calculation methods. As a demonstration of the simulation, the formation of a planar Coulomb crystal with a small number of ions has been investigated in detail in a linear ion trap both experimentally and by simulation. However, also large Coulomb crystals including up to 1400 ions have been photographed and simulated to extract the secular temperature and the number of ions. For medium-sized crystals, a comparison between experiments and calculations has been performed. Moreover, an MD simulation of the sympathetic cooling of small molecular ions was performed in order to test the possibility of extracting the temperature and the number of refrigerated molecular ions from crystal images of laser-cooled ions. Such information is basic to studying ultracold ion-molecule reactions using ion Coulomb crystals including sympathetically cooled molecular ions.
We examine the time-dependent dynamics of ion crystals in radiofrequency traps. The problem of stable trapping of general three-dimensional crystals is considered and the validity of the pseudopotential approximation is discussed. We analytically derive the micromotion amplitude of the ions, rigorously proving well-known experimental observations. We use a recently proposed method to find the modes that diagonalize the linearized time-dependent dynamical problem. This allows one to obtain explicitly the ('Floquet–Lyapunov') transformation to coordinates of decoupled linear oscillators. We demonstrate the utility of the method by analyzing the modes of a small 'peculiar' crystal in a linear Paul trap. The calculations can be readily generalized to multispecies ion crystals in general multipole traps, and time-dependent quantum wavefunctions of ion oscillations in such traps can be obtained.