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

Physical Review A

Quantum motion of two trapped ions in one dimension

Jiaju Yin, Juha Javanainen

Published May 1, 1995• 23 citations
Publisher

Abstract

We develop models for the motion of two trapped ions in one dimension, including both harmonic trapping forces and the Coulomb interaction between the ions. Mathematical and physical consistency dictates that the divergence of the Coulomb interaction at small distances be tempered with a soft core. A split-operator method is used to analyze numerically the energies and eigenstates of the motion of the two ions. Classification and closed-form expressions are also presented for the energies and the eigenstates in the limits when the equilibrium distance due to the Coulomb repulsion is either much smaller or much larger than the amplitude of the zero-point motion of the ions. Implications of the results for the properties of two identical ions with Bose-Einstein or Fermi-Dirac statistics are discussed.

DOI
10.1103/physreva.51.3959

Related papers

Optical Trapping of Ion Coulomb Crystals

Julian Schmidt, Alexander Lambrecht, Pascal Weckesser +3 more

The electronic and motional degrees of freedom of trapped ions can be controlled and coherently coupled on the level of individual quanta.Assembling complex quantum systems ion by ion while keeping this unique level of control remains a challenging task.For many applications, linear chains of ions in conventional traps are ideally suited to address this problem.However, driven motion due to the magnetic or radio-frequency electric trapping fields sometimes limits the performance in one dimension and severely affects the extension to higher-dimensional systems.Here, we report on the trapping of multiple barium ions in a single-beam optical dipole trap without radio-frequency or additional magnetic fields.We study the persistence of order in ensembles of up to six ions within the optical trap, measure their temperature, and conclude that the ions form a linear chain, commonly called a one-dimensional Coulomb crystal.As a proof-of-concept demonstration, we access the collective motion and perform spectrometry of the normal modes in the optical trap.Our system provides a platform that is free of driven motion and combines advantages of optical trapping, such as state-dependent confinement and nanoscale potentials, with the desirable properties of crystals of trapped ions, such as long-range interactions featuring collective motion.Starting with small numbers of ions, it has been proposed that these properties would allow the experimental study of many-body physics and the onset of structural quantum phase transitions between one-and two-dimensional crystals.

arXiv:1712.08385•Physical Review X•May 1, 2018•46 citations

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

V. V. Dodonov, Vladimir I. Man’ko, Luigi Rosa

Following the paper by Combescure [Ann. Phys. (N.Y.) 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 $n\ensuremath{\ll}{n}_{\mathrm{max}}\ensuremath{\sim}100$, provided the dimensionless constant characterizing the strength of the repulsive potential is large enough ${g}_{*}\ensuremath{\sim}{10}^{5}$. 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.

arXiv:quant-ph/9712028•Physical Review A•Apr 1, 1998•36 citations

Three-dimensional non-approximate Coulomb interaction between two trapped quantum particles

Nicolás Z. Lizama, Sebastian Camilo Carrasco, Jose Antonio Rogan +1 more

The two trapped quantum particles interacting problem is generalized to three dimensions, and the exact Coulomb potential is used. The system is solved by expanding the wavefunction in terms of the isotropic harmonic oscillator eigenfunctions and Hydrogen atom eigenfunctions independently, showing that each one results in a prime approximation for different domains of the normalized coupling constant of the relative interactions, suggesting that the combination of the basis is enough to build a well-suited base for the non-approximate problem. The results are compared to previous works that use a model of approximate finite-rage soft-core interaction model of the problem to give insights into the many-body states of strongly correlated ultracold bosons and fermions. We conclude that the proposed three-dimensional approach facilitates the distinction between bosons and fermions while the solutions given by the expansions better define the behavior of the particles for repulsive potentials. In addition, we discuss the substantial differences between our work and the previous approximate model.

Scientific Reports•Oct 24, 2023•1 citation

Coherence-Like States of Two Coulomb-Correlated Ions Confined in a Paul Trap

Hui Li, Hai Wen-Hua, Chen Wen-Qin +1 more

We report the n × n ' coherence-like state solutions in the cases of n , n ' = 1,2,... for the system including two Coulomb-correlated ions confined in a one-dimensional Paul trap with a time-dependent harmonic potential. One of the n ' exact solutions of the centre-of-mass motion describes a generalized coherent state. For a small driving strength the n approximate solutions of relative motion are constructed, which describe the coherent oscillations of the two ions around the classical equilibrium position.

Chinese Physics Letters•Apr 1, 2007•2 citations

Energy eigenstates of two ions in a two-dimensional paul trap

Wu Yun-Wen, Hai Wen-Hua, 湖南吉首大学物理系，吉首 416000;湖南师范大学物理系，长沙 410081

Taking into consideration the Coulomb interaction between two ions trapped in the coplanar Paul trap,the exact solutions of the Schrdinger equations of the two-ions system are obtained.The energy level degeneracy for center-of mass is obtained using the exact solutions. We also calculate the equilibrium distance of the ions and several state functions of the low energy levels. The radial probability distribution graph for the center of mass is plotted using a computer program.

Acta Physica Sinica•Jan 1, 2006•2 citations

Two quantum particles trapped in three dimensions harmonic oscillator and interacting via finite range soft-core interaction

Muhammad Adnan Shahzad

We study the exactly solvable quantum system of two particles confined in a three-dimensional harmonic trap and interacting via finite-range soft-core interaction by means of the separation of variables and ansatz method. Supposing the solution in the form of ansatz $Ψ(r)=r^{a}e^{-λr^2} ψ(r)$ we transform the time independent Schrödinger equation into Kummer's differential equation whose solution are given in the form of confluent hypergeometric function. We also discuss that in the absence of central force potential, the quantum system map into the problem of two quantum particle trapped in one-dimension harmonic oscillator and interacting through finite distance soft-core potential. In such special case the eigen value equation become the Weber's differential equation and its solution are also given in the form of confluent hypergeometric function.

arXiv:1906.03688•Jun 9, 2019•0 citations
We develop models for the motion of two trapped ions in one dimension, including both harmonic trapping forces and the Coulomb interaction between the ions. Mathematical and physical consistency dictates that the divergence of the Coulomb interaction at small distances be tempered with a soft core. A split-operator method is used to analyze numerically the energies and eigenstates of the motion of the two ions. Classification and closed-form expressions are also presented for the energies and the eigenstates in the limits when the equilibrium distance due to the Coulomb repulsion is either much smaller or much larger than the amplitude of the zero-point motion of the ions. Implications of the results for the properties of two identical ions with Bose-Einstein or Fermi-Dirac statistics are discussed.
The electronic and motional degrees of freedom of trapped ions can be controlled and coherently coupled on the level of individual quanta.Assembling complex quantum systems ion by ion while keeping this unique level of control remains a challenging task.For many applications, linear chains of ions in conventional traps are ideally suited to address this problem.However, driven motion due to the magnetic or radio-frequency electric trapping fields sometimes limits the performance in one dimension and severely affects the extension to higher-dimensional systems.Here, we report on the trapping of multiple barium ions in a single-beam optical dipole trap without radio-frequency or additional magnetic fields.We study the persistence of order in ensembles of up to six ions within the optical trap, measure their temperature, and conclude that the ions form a linear chain, commonly called a one-dimensional Coulomb crystal.As a proof-of-concept demonstration, we access the collective motion and perform spectrometry of the normal modes in the optical trap.Our system provides a platform that is free of driven motion and combines advantages of optical trapping, such as state-dependent confinement and nanoscale potentials, with the desirable properties of crystals of trapped ions, such as long-range interactions featuring collective motion.Starting with small numbers of ions, it has been proposed that these properties would allow the experimental study of many-body physics and the onset of structural quantum phase transitions between one-and two-dimensional crystals.
Following the paper by Combescure [Ann. Phys. (N.Y.) 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 $n\ensuremath{\ll}{n}_{\mathrm{max}}\ensuremath{\sim}100$, provided the dimensionless constant characterizing the strength of the repulsive potential is large enough ${g}_{*}\ensuremath{\sim}{10}^{5}$. 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∗{g}_{*}g∗​ results in a significant amplification of the transition probabilities between energy eigenstate even for slow time variations of the frequency.
The two trapped quantum particles interacting problem is generalized to three dimensions, and the exact Coulomb potential is used. The system is solved by expanding the wavefunction in terms of the isotropic harmonic oscillator eigenfunctions and Hydrogen atom eigenfunctions independently, showing that each one results in a prime approximation for different domains of the normalized coupling constant of the relative interactions, suggesting that the combination of the basis is enough to build a well-suited base for the non-approximate problem. The results are compared to previous works that use a model of approximate finite-rage soft-core interaction model of the problem to give insights into the many-body states of strongly correlated ultracold bosons and fermions. We conclude that the proposed three-dimensional approach facilitates the distinction between bosons and fermions while the solutions given by the expansions better define the behavior of the particles for repulsive potentials. In addition, we discuss the substantial differences between our work and the previous approximate model.
We report the n × n ' coherence-like state solutions in the cases of n , n ' = 1,2,... for the system including two Coulomb-correlated ions confined in a one-dimensional Paul trap with a time-dependent harmonic potential. One of the n ' exact solutions of the centre-of-mass motion describes a generalized coherent state. For a small driving strength the n approximate solutions of relative motion are constructed, which describe the coherent oscillations of the two ions around the classical equilibrium position.
We study the exactly solvable quantum system of two particles confined in a three-dimensional harmonic trap and interacting via finite-range soft-core interaction by means of the separation of variables and ansatz method. Supposing the solution in the form of ansatz Ψ(r)=rae−λr2ψ(r)Ψ(r)=r^{a}e^{-λr^2} ψ(r)Ψ(r)=rae−λr2ψ(r) we transform the time independent Schrödinger equation into Kummer's differential equation whose solution are given in the form of confluent hypergeometric function. We also discuss that in the absence of central force potential, the quantum system map into the problem of two quantum particle trapped in one-dimension harmonic oscillator and interacting through finite distance soft-core potential. In such special case the eigen value equation become the Weber's differential equation and its solution are also given in the form of confluent hypergeometric function.