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arXiv:1907.08045 · Physical Review A

Dynamics of an unbalanced two-ion crystal in a Penning trap for application in optical mass spectrometry

M. J. Gutiérrez, J. Berrocal, Francisco Domínguez, Íñigo Arrazola, M. Block, E. Solano, Daniel Rodríguez

Published Dec 10, 2019• 18 citations
arXivHTMLPDF

Abstract

In this paper, the dynamics of an unbalanced two-ion crystal comprising the ``target'' and the ``sensor'' ions confined in a Penning trap along the magnetic-field axis has been studied. First, the low amplitude regime is addressed. In this regime, the overall potential including the Coulomb repulsion between the ions can be considered harmonic and the axial, magnetron, and reduced-cyclotron modes split up into the so-called stretch and common modes, that are generalizations of the well-known ``breathing'' and ``center-of-mass'' motions of a balanced crystal made of two ions. By using optical detection to measure the frequencies of the modes of the crystal, and of the sensor ion on its own, in the quantum regime of motion, it will be possible to determine the target ion's free-cyclotron frequency. The nonharmonicity of the Coulomb interaction is also discussed since this causes large systematic effects, which are minimized due to the high sensitivity of the optical detection method when the crystal is cooled to the ground state of motion in the Penning trap.

DOI
10.1103/physreva.100.063415

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In this paper, the dynamics of an unbalanced two-ion crystal comprising the target'' and the sensor'' ions confined in a Penning trap along the magnetic-field axis has been studied. First, the low amplitude regime is addressed. In this regime, the overall potential including the Coulomb repulsion between the ions can be considered harmonic and the axial, magnetron, and reduced-cyclotron modes split up into the so-called stretch and common modes, that are generalizations of the well-known breathing'' and center-of-mass'' motions of a balanced crystal made of two ions. By using optical detection to measure the frequencies of the modes of the crystal, and of the sensor ion on its own, in the quantum regime of motion, it will be possible to determine the target ion's free-cyclotron frequency. The nonharmonicity of the Coulomb interaction is also discussed since this causes large systematic effects, which are minimized due to the high sensitivity of the optical detection method when the crystal is cooled to the ground state of motion in the Penning trap.
Laser cooled ions in a Penning trap can be isolated from the environment by placing them in vacuum and only interacting with them through optical and RF fields. The number of trapped particles can be varied from a single ion up to thousands. Confinement is provided by a static homogeneous magnetic field and a quadrupole electric potential. In the natural frame of the ions, this appears as a 3D simple harmonic potential. Therefore three dimensional structures can be formed in the absence of any additional RF field which may lead to heating as is the case with RF traps. There are 3N different motional modes for N particles. I present an analysis of the motion of a single particle showing that the energy levels for all three modes are equally spaced. I also describe the interaction between a trapped two level atom and an optical field. During my time in the lab the laser and computer control of the experiment has been significantly improved. In addition, an existing trap was modified to provide greater optical access and fluorescence collection. This allowed the vibrational levels superimposed on the internal states of a single 40Ca+ ion to be resolved via a narrow linewidth, electric quadrupole transition. This is the first observation of magnetron and modified cyclotron sidebands on an optical transition. When more than one calcium ion is laser cooled, and their temperature reduced below 5mK, they form a Coulomb crystal. The locations of the ions minimise the total potential energy which is comprised of the Coulomb repulsion and trap potential. The fluorescence collection optics have been arranged to resolve individual ions in these crystals. Information about the motion of the ions is deduced by comparing photos from the experiment to numerical simulations. Previously, only two ions have ever been aligned along the magnetic field in a Penning trap. I present strings of up to 29 particles and suggest the only limitation, apart from the electrode structure, is the overlap of the laser beams with the ions.
In traditional mechanics, harmonic oscillators can be used to measure force, acceleration, or rotation. Herein, we describe a quantum harmonic oscillator based on a Penning-trapped calcium ion crystal. Similar to traditional oscillators, the Coriolis force-induced axial oscillation amplitude is precisely measured to determine the input velocity. We show that the magnetron motion can be controlled through the rotating wall driving and treated as the driving oscillator. The Coriolis force couples with the magnetron motion and induces vibration in the axial direction or the z direction. The center of mass motion of the ion crystal in the axial direction could be precisely detected by the entanglement between the spins of the ions and the harmonic motion through lasers. The frequency of the magnetron motion needs to meet that of the axial motion under certain conditions, and, thus, the axial motion could be tuned to the resonance peak for maximum detection signal. We provided the parameter spaces for the matching of the magnetron frequencies with those of the axial frequencies. The measurement sensitivity was calculated in detail, and results show that a rotation angular velocity of 3.0×10−9rad/s/Hz could be achieved with 10 000 ions. Amplitude sensing could reach a sensitivity of 0.4 pm/Hz. With spin squeezing, the sensitivity could be further improved.
We propose to measure the six motional frequencies of an unbalanced two-ion crystal in a Penning trap using a scheme that has been demonstrated in a quantum-metrology experiment with a single ion in a linear Paul trap. To study the feasibility, we derive the quantum Hamiltonian for the unbalanced crystal formed by a target and a sensor ion ( 40 Ca , and deduce characteristic coordinates , and , associated to the amplitudes of the displacement operator , and , respectively, when a dipolar time-varying field is applied. The final state after excitation is read out using rapid adiabatic passage. We also present the status of the 7 tesla open-ring Penning trap where first experiments on the 232 Th Ca crystal, cooled to the ground state, are foreseen in the short-term future. Finally, we underline possible experiments with this unique platform.
In traditional mechanics, harmonic oscillators can be used to measure force, acceleration, or rotation. Herein, we describe a quantum harmonic oscillator based on a penning trapped calcium ion crystal. Similar to traditional oscillators, the Coriolis force induced axial oscillation amplitude is precisely measured to determine the input velocity. We show that the magnetron motion can be controlled through the rotating wall driving and treated as the driving oscillator. The Coriolis force couples with the magnetron motion and induces vibration in the axial direction or the zzz direction. The center of mass motion of the ion crystal in the axial direction could be precisely detected by the entanglement between the spins of the ions and the harmonic motion through lasers. The frequency of the magnetron motion needs to meet that of the axial motion under certain conditions and thus the axial motion could be tuned to the resonance peak for maximum detection signal. We gave the parameter spaces for the meeting of the magnetron frequencies to that of the axial frequencies. The measurement sensitivity was calculated in details and results show that rotation angular velocity of 3.0×10−9rad/s/Hz3.0\times10^{-9}rad/s/\sqrt{Hz}3.0×10−9rad/s/Hz​ could achieve with 10000 ions. Amplitude sensing could reach sensitivity of 0.4pm/Hzpm/\sqrt{Hz}pm/Hz​. With spin squeezing, the sensitivity could be further improved.
We have studied the dynamics of one and two laser-cooled trapped 40Ca+ ions by applying electric fields of different nature along the axial direction of the trap, namely, driving the motion with a harmonic dipolar field, or with white noise. These two types of driving induce distinct motional states of the axial modes: a coherent oscillation with the dipolar field, or an enhanced Brownian motion due to an additional contribution to the heating rate from the electric noise. In both scenarios, the sensitivity of an isolated ion and a laser-cooled two-ion crystal has been evaluated and compared. The analysis and understanding of this dynamics is important towards the implementation of a novel Penning trap mass-spectroscopy technique based on optical detection, aiming at improving precision and sensitivity.