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Europhysics Letters (EPL)

Motional quantum metrology in a Penning trap

Javier Cerrillo, Daniel Rodríguez

Published May 1, 2021• 9 citations
Publisher

Abstract

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.

DOI
10.1209/0295-5075/134/38001

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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.

•Feb 1, 2013•1 citation
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 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.
Two-ion crystals constitute a platform for investigations of quantum nature that can be extended to any ion species or charged particle provided one of the ions in the crystal can be directly laser cooled and manipulated with laser radiation. This paper presents the formation of two-ion crystals for quantum metrology in a 7-tesla open-ring Penning trap. 40Ca+^{40}\mathrm{Ca}^{+}40Ca+ ions are produced either internally by photoionization or externally in a (Paul-trap) source, transported through the strong magnetic field gradient of the superconducting solenoid, and captured in-flight with a mean kinetic energy of a few electronvolts with respect to the minimum of the Penning-trap potential well. Laser cooling of the two-ion crystal in a strong magnetic field towards reaching the quantum regime is also presented, with particular emphasis on the cooling of the radial modes.
Two-ion crystals constitute a platform for investigations of quantum nature that can be extended to any ion \nspecies or charged particle provided one of the ions in the crystal can be directly laser cooled and manipulated \nwith laser radiation. This paper presents the formation of two-ion crystals for quantum metrology in a 7-tesla \nopen-ring Penning trap. 40Ca+ ions are produced either internally by photoionization or externally in a (Paul-trap) \nsource, transported through the strong magnetic field gradient of the superconducting solenoid, and captured \nin-flight with a mean kinetic energy of a few electronvolts with respect to the minimum of the Penning-trap \npotential well. Laser cooling of the two-ion crystal in a strong magnetic field towards reaching the quantum \nregime is also presented, with particular emphasis on the cooling of the radial modes.
Laser-cooled trapped ions are one of the main systems used in experiments in the fields of quantum optics and quantum information. There are two commonly-used types of ion trap, the radiofrequency trap and the Penning trap. In general, the RF trap has been more widely used in quantum information experiments, meaning that the set of experimental tools for conducting these types of experiments in a Penning trap is currently less complete. The aim of the work presented in this thesis is to increase this array of tools. Our experiment uses 40Ca+ ions in a Penning trap alongside lasers. Spectroscopy has been performed on single Doppler cooled ions, and on ions that have been prepared close to their ground state of motion using a multiple-stage sideband cooling technique resulting in an average phonon occupation number of $\nzbar = 0.029\pm 0.011$ from an initial $\nzbar \approx 24$. This corresponds to a quantum ground state population of 97%97\%97%, which is close to the sideband cooling limit in our system. An axial heating rate measurement of $\dot{\bar{n}}_z=\SI{0.56\pm0.52}{\per\second}$ has also been obtained, which is the lowest recorded to date, due in part to the large dimensions of our trap. Additionally improved control of the Penning trap radial motion has been demonstrated and the average phonon occupation number of the modified cyclotron motion has been reduced from $\ncbar\approx150$ after Doppler cooling to $\ncbar=0.6$ after sideband cooling. Some preliminary single-ion coherent operations have also been performed. In the future we hope that this system of ground state cooled ions in a Penning trap can be used to generate highly-entangled states in two or more ions, with a view to performing simple quantum error-correcting codes and eventually an analogue quantum simulation of spin systems.
For 35 years, laser-cooled trapped ions have been at the frontier of progress in quantum computing, quantum simulation and precision measurement, and remain one of the most valuable tools in these fields to this day. Most of these experiments are predicated upon or benefit from the ability to place ions in the motional quantum ground state, a technique that was first demonstrated in radiofrequency ion traps 25 years ago. For a range of crucial experiments that are impossible to conduct in radiofrequency traps or are not well suited to this architecture, Penning traps provide an important alternative. However, the performance of Penning traps had been limited by the fact that ground-state cooling was yet to be achieved in such a system. This thesis reports the first demonstration of resolved-sideband cooling in a Penning trap, for 40Ca+ions cooled with light at 729-nm, achieving a ground state occupation of 99% in one dimension. Demonstrations of the coherent manipulations possible at this level of confinement are presented. The ion heating rate is measured and although higher than might be expected given the unusually large ion-electrode distance remains amongst the lowest reported in any trap to date. Achieving this result required the development of a number of new experimental systems and major upgrades to the stability and reliability of the experiment, the details of which are also given. The thesis also presents theoretical work into the use of two-dimensional Coulomb crystals in a Penning trap as a resource for quantum information. Using the symmetries of the crystal, we find that it is possible to engineer complex entangled states, specifically two small quantum error correcting codes, using a very small number of global entangling pulses. Efficient entanglement protocols such as these are vital for the implementation of useful quantum error correction.
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.