Abstract
One of the goals of synthesizing and trapping antihydrogen is to study the validity of charge–parity–time symmetry through precision spectroscopy on the antiatoms, but the trapping yield achieved in recent experiments must be significantly improved before this can be realized. Antihydrogen atoms are commonly produced by mixing antiprotons and positrons stored in a nested PenningMalmberg trap, which was achieved in ALPHA by an autoresonant excitation of the antiprotons, injecting them into the positron plasma. In this work, a hybrid numerical model is developed to simulate antiproton and positron dynamics during the mixing process. The simulation is benchmarked against other numerical and analytic models, as well as experimental measurements. The autoresonant injection scheme and an alternative scheme are compared numerically over a range of plasma parameters which can be reached in current and upcoming antihydrogen experiments, and the latter scheme is seen to offer significant improvement in trapping yield as the number of available antiprotons increases.
This work was supported by CNPq, FINEP/RENAFAE (Brazil); ISF (Israel); FNU (Denmark); VR (Sweden); NSERC, NRC/TRIUMF, AITF, FQRNT (Canada); DOE, LBNL LDRD, NSF (USA); and EPSRC, the Royal Society and the Leverhulme Trust (UK). We are grateful for the efforts of the CERN AD team, without which the experimental data presented here could not have been taken.
I. INTRODUCTION
II. BASIC THEORY OF AUTORESONANT EXCITATION
III. NUMERICAL MODEL
IV. COMPARISONS WITH NUMERICAL AND ANALYTIC MODELS
A. Timeresolved AR excitation
B. Perturbation amplitude threshold for pickup
V. COMPARISONS WITH EXPERIMENT
A. Final antiproton energy distribution versus drive amplitude and stopping frequency
B. Injection ratio versus stopping frequency
VI. INJECTION LIMITS
VII. INCREMENTAL INJECTION
VIII. CONCLUSIONS AND OUTLOOK
Key Topics
 Positrons
 71.0
 Numerical modeling
 20.0
 Chirping
 15.0
 Electrodes
 15.0
 Phase space methods
 15.0
Figures
Potentials and geometry for measuring the AR excitation of an antiproton bunch. (a) The physical setup, with the electrode marked pink connected to the AR signal. (b) The external potential created by the electrodes at . (c) A closeup of b, emphasizing the effect of various antiproton space charges. (d) The perturbation created at when 1 V is applied to the AR electrode. The potentials used in the PPM Vlasov model are deduced by solving the 2D Poisson equation with physically accurate boundary conditions. Those in the spectral Vlasov solver are analytic fits up to . The external potential in the analytic model fits up to , and the perturbation to .
Potentials and geometry for measuring the AR excitation of an antiproton bunch. (a) The physical setup, with the electrode marked pink connected to the AR signal. (b) The external potential created by the electrodes at . (c) A closeup of b, emphasizing the effect of various antiproton space charges. (d) The perturbation created at when 1 V is applied to the AR electrode. The potentials used in the PPM Vlasov model are deduced by solving the 2D Poisson equation with physically accurate boundary conditions. Those in the spectral Vlasov solver are analytic fits up to . The external potential in the analytic model fits up to , and the perturbation to .
Time evolution of (a) the energy and (b) phase angle of the antiproton distribution, as predicted by different numerical and analytic models. The phase difference is defined as , where is the phase angle of the center of charge of the distribution, and the phase angle of the AR perturbation.
Time evolution of (a) the energy and (b) phase angle of the antiproton distribution, as predicted by different numerical and analytic models. The phase difference is defined as , where is the phase angle of the center of charge of the distribution, and the phase angle of the AR perturbation.
Critical perturbation amplitude for varying chirp rates. The prediction of the analytic model and the results from the single particle and PPM Vlasov model are compared.
Critical perturbation amplitude for varying chirp rates. The prediction of the analytic model and the results from the single particle and PPM Vlasov model are compared.
The final energy of an antiproton bunch after various AR perturbations, as measured in the experiment and predicted by the single particle and the PPM Vlasov models. (a) The final antiproton energy after AR perturbations of various amplitudes and a fixed stopping frequency of 360 kHz. (b) The final antiproton energy after AR perturbations of various stopping frequencies and a fixed amplitude of 0.15 V. (c) The energy distribution of an antiproton bunch after a typical AR perturbation in a—the delta function for the single particle result indicates the inability of the model to simulate a distribution. The experimental data have been corrected for systematics—see main text.
The final energy of an antiproton bunch after various AR perturbations, as measured in the experiment and predicted by the single particle and the PPM Vlasov models. (a) The final antiproton energy after AR perturbations of various amplitudes and a fixed stopping frequency of 360 kHz. (b) The final antiproton energy after AR perturbations of various stopping frequencies and a fixed amplitude of 0.15 V. (c) The energy distribution of an antiproton bunch after a typical AR perturbation in a—the delta function for the single particle result indicates the inability of the model to simulate a distribution. The experimental data have been corrected for systematics—see main text.
Potentials and geometry for injecting antiprotons into a positron plasma. (a) The physical setup of the experiment, with the pink electrode connected to the AR signal generator. (b) The external potential created by the electrodes at , and the effect of the positron space charge. (c) A closeup of b, showing the effect of the antiproton space charge.
Potentials and geometry for injecting antiprotons into a positron plasma. (a) The physical setup of the experiment, with the pink electrode connected to the AR signal generator. (b) The external potential created by the electrodes at , and the effect of the positron space charge. (c) A closeup of b, showing the effect of the antiproton space charge.
(a) The simulated distribution in speed of injected antiprotons as they travel across the positron plasma, conditioned on the radius. The blue dotted curve shows a reference thermal distribution of antiprotons at 800 K, which has the same area under the curve as the curve. The total number of injected antiproton is 7400 (out of the 16000 initial antiprotons). (b) Simulated antiproton distributions at various t during an AR perturbation. The AR chirp starts at . The contours are lines of constant total energy, and increase by 0.25 eV (2900 K) per contour. At each time, the phase space at is displayed, together with the charge density.
(a) The simulated distribution in speed of injected antiprotons as they travel across the positron plasma, conditioned on the radius. The blue dotted curve shows a reference thermal distribution of antiprotons at 800 K, which has the same area under the curve as the curve. The total number of injected antiproton is 7400 (out of the 16000 initial antiprotons). (b) Simulated antiproton distributions at various t during an AR perturbation. The AR chirp starts at . The contours are lines of constant total energy, and increase by 0.25 eV (2900 K) per contour. At each time, the phase space at is displayed, together with the charge density.
(a) The simulated fraction of antiprotons injected into the positron plasma conditioned on their injected KE, using AR perturbations of various stopping frequencies. (b) Same as a, except the curves are conditioned on the radius. (c) The number of antiprotons from experiment that successfully inject into the positron plasma and form antihydrogen atoms, divided by the estimated initial number of antiprotons, at various stopping frequencies. The error bars indicate the statistical error of the experimental measurement, and do not include the detector calibration uncertainty ( ) which is systematic to all the data points.
(a) The simulated fraction of antiprotons injected into the positron plasma conditioned on their injected KE, using AR perturbations of various stopping frequencies. (b) Same as a, except the curves are conditioned on the radius. (c) The number of antiprotons from experiment that successfully inject into the positron plasma and form antihydrogen atoms, divided by the estimated initial number of antiprotons, at various stopping frequencies. The error bars indicate the statistical error of the experimental measurement, and do not include the detector calibration uncertainty ( ) which is systematic to all the data points.
Contours showing the fraction of antiprotons injected by an AR perturbation into the positron plasma with KE below the indicated value on each subfigure, as a function of the initial antiproton number and temperature. Each antiproton bunch with a specific initial number and temperature is injected using the optimal AR perturbation that leads to the highest injection ratio at KE —i.e., these contours reflect the bestcase capability of a conventional AR perturbation.
Contours showing the fraction of antiprotons injected by an AR perturbation into the positron plasma with KE below the indicated value on each subfigure, as a function of the initial antiproton number and temperature. Each antiproton bunch with a specific initial number and temperature is injected using the optimal AR perturbation that leads to the highest injection ratio at KE —i.e., these contours reflect the bestcase capability of a conventional AR perturbation.
The simulated, conditional fraction of antiprotons injected into the positron plasma using incremental injection with different stopping frequencies, and two initial antiproton numbers. The rightmost frequency (325 kHz) corresponds to an AR chirp of zero length, with the chirp length increasing towards the left of the horizontal axis. (a) The fraction of antiprotons injected, out of an initial 16 000, conditioned on their KE in the positron plasma. (b) Same as a, except that the ratios are conditioned on radius. (c) The fraction of antiprotons injected, out of an initial 160 000, conditioned on their KE in the positron plasma. (d) Same as c, except that the ratios are conditioned on radius.
The simulated, conditional fraction of antiprotons injected into the positron plasma using incremental injection with different stopping frequencies, and two initial antiproton numbers. The rightmost frequency (325 kHz) corresponds to an AR chirp of zero length, with the chirp length increasing towards the left of the horizontal axis. (a) The fraction of antiprotons injected, out of an initial 16 000, conditioned on their KE in the positron plasma. (b) Same as a, except that the ratios are conditioned on radius. (c) The fraction of antiprotons injected, out of an initial 160 000, conditioned on their KE in the positron plasma. (d) Same as c, except that the ratios are conditioned on radius.
The external potential seen by the antiprotons during a linear ramp of the AR electrode shown in Fig. 5 . The numbers displayed in each subfigure are the electrode's voltage and the number of remaining positrons, the rest being lost to evaporative escape.
The external potential seen by the antiprotons during a linear ramp of the AR electrode shown in Fig. 5 . The numbers displayed in each subfigure are the electrode's voltage and the number of remaining positrons, the rest being lost to evaporative escape.
Contours showing the fraction of antiprotons injected into the positron plasma after a pure linear ramp, against the initial antiproton number and temperature. The four figures show the fraction of antiprotons injected at a KE below the indicated value. Each antiproton bunch with a specific initial number and temperature is injected using the optimal linear ramp depth that leads to the highest injection ratio at KE smaller than .
Contours showing the fraction of antiprotons injected into the positron plasma after a pure linear ramp, against the initial antiproton number and temperature. The four figures show the fraction of antiprotons injected at a KE below the indicated value. Each antiproton bunch with a specific initial number and temperature is injected using the optimal linear ramp depth that leads to the highest injection ratio at KE smaller than .
Tables
Typical plasma conditions and parameters just before injection manipulations. The uncertainties in particle numbers refer to the shottoshot fluctuation of the species. The plasma dimensions are defined by the region enclosed by the equidensity contour in r–z space that encloses 90% of the total material, and the density is defined by the average therein. The Debye lengths and plasma oscillation periods are derived from this average density. The mean free time is the mean time between effective collisions estimated from the nonmagnetized Coulomb collision model.
Typical plasma conditions and parameters just before injection manipulations. The uncertainties in particle numbers refer to the shottoshot fluctuation of the species. The plasma dimensions are defined by the region enclosed by the equidensity contour in r–z space that encloses 90% of the total material, and the density is defined by the average therein. The Debye lengths and plasma oscillation periods are derived from this average density. The mean free time is the mean time between effective collisions estimated from the nonmagnetized Coulomb collision model.
Injection performance of some representative plasma parameters, taken from Figs. 8 and 11 . The “Injected, ” row gives the number of antiprotons injected into the positron plasma at below 10 K. The “Injected, T fit” row gives the temperature fit of the KE distribution of injected antiprotons.
Injection performance of some representative plasma parameters, taken from Figs. 8 and 11 . The “Injected, ” row gives the number of antiprotons injected into the positron plasma at below 10 K. The “Injected, T fit” row gives the temperature fit of the KE distribution of injected antiprotons.
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