Limits of evaporative cooling of trapped antiproton plasmas: the net-heating budget, the lowest attainable temperature, and the demonstration of sub-Kelvin nonneutral antiproton plasmas

Forced evaporative cooling of a magnetically confined nonneutral antiproton plasma was demonstrated by the ALPHA collaboration in 2010, reaching about 9 K from roughly 45,000 trapped antiprotons and validating a rate-equation model whose net heating term $P$ was estimated only from expansion-driven Joule heating. The source paper itself left open four linked issues: the complete quantitative origin of $P$ and its dependence on trap electronics and geometry; the ultimate lowest temperature attainable by evaporation in a given apparatus (expected to be limited by electronic noise on the confining electrodes and by the precision with which the trapping potential can be set); the characterization of evaporation dynamics with the superposed magnetostatic (minimum-B) atom-trap field energized; and the achievement of sub-Kelvin nonneutral antiproton plasmas. Later work advanced adjacent goals — adiabatic cooling of antiprotons to 3.5 K, identification of anomalous microwave-driven heating of cryogenic lepton plasmas, sympathetic cooling of positrons by laser-cooled Be$^+$ ions, laser cooling of neutral antihydrogen, and single-particle cooling of one proton to 170 mK — none of which settles the stated problem.

Background

A Penning-Malmberg trap confines a cloud of charged particles axially with electrostatic wells formed by stacked cylindrical electrodes and radially with a strong solenoidal magnetic field. Such a cloud of like charges is a 'nonneutral plasma': it has a collective self-potential (space charge) in addition to the applied trap potential, and it interacts with the electrodes through image charges and through noise on the electrode-driving electronics. Low-energy antiprotons (delivered by CERN's Antiproton Decelerator) are caught and accumulated this way, because the antiproton cannot be laser cooled directly, and cold antiprotons are the raw ingredient for synthesizing antihydrogen, the simplest pure antimatter bound state, whose spectroscopy and gravitational behaviour test CPT symmetry (the Charge-Parity-Time invariance requiring particle and antiparticle properties to coincide) and the weak equivalence principle.

Before 2010 the standard way to cool trapped antiprotons was electron cooling: the antiprotons are mixed with electrons that radiate away energy via cyclotron emission and cool the antiprotons by collisions; the electrons must then be removed, which leaves the antiprotons at temperatures of order a few hundred to a thousand kelvin (the 2010 experiment started from $(1040\pm 45)$ K). In 'Evaporative Cooling of Antiprotons to Cryogenic Temperatures' (Phys. Rev. Lett. 105, 013003 (2010)) the ALPHA collaboration instead applied forced evaporative cooling — long established for neutral atoms but never before realized in a cold nonneutral plasma — by lowering the depth of one side of the confining well so that the most energetic antiprotons escape and the remainder rethermalize at a lower temperature. Starting from about 45,000 antiprotons at a well depth of roughly 1500 mV, linear ramps to final on-axis well depths as low as $(10\pm 4)$ mV produced measured temperatures of $(9\pm 4)$ K with $(6\pm 1)\%$ of the particles surviving (a 1 s ramp destroyed the cloud, because the evaporation time then becomes shorter than the rethermalization time). The escaping-particle energy spectrum was fitted to an exponential (Boltzmann) tail to extract the temperature, with a particle-in-cell simulation indicating a roughly 16% overestimate to be corrected for. The evolution was modelled with coupled rate equations for the trapped number $N$ and temperature $T$:

$$\frac{dN}{dt} = -\frac{N}{\tau_{\rm ev}} - \gamma N,$$ $$\frac{dT}{dt} = -\alpha \frac{T}{\tau_{\rm ev}} + P,$$

where $\tau_{\rm ev}$ is the evaporation time scale (related to the Coulomb collision relaxation time through the dimensionless barrier height $\eta$, the well depth in units of $k_B T$), $\alpha$ the excess energy removed per evaporating particle, $\gamma$ the residual-gas annihilation loss rate, and $P$ a net heating term. $P$ was not fitted: it was estimated as Joule heating released by the observed radial expansion of the cloud (radius growing from 0.6 mm to about 3 mm), with the expansion tied to particle loss by conservation of canonical angular momentum (the quantity $L_z = mr v_\theta + qBr^2/2$, which the solenoidal field keeps nearly constant per particle, so losing particles forces the remaining cloud to larger radius). Including the plasma self-potential (about 1.5 µV per antiproton, i.e. ~30 mV for 20,000 antiprotons in a 108 mV well) was essential to reproduce the measured particle numbers and evaporation time scales. The same apparatus superimposes an octupole-based minimum-B magnetic neutral trap (a Ioffe-Prichard-type field that confines antihydrogen atoms but breaks the azimuthal symmetry of the Penning trap).

The paper states that its measurements 'cannot exclude, but only limit, other contributions to $P$' beyond expansion-driven Joule heating — electronic noise from the electrode driving circuits being one plausible candidate; that 'electronic noise in the electrode driving circuit and the ability to precisely set the confining potential will most likely limit the lowest temperature achievable in any given apparatus'; that only 'preliminary studies' exist on running the evaporation with the octupole atom-trap field energized; and that 'perhaps even sub-Kelvin plasmas, foreseen to be a prerequisite for proposed measurements of gravitational forces on antimatter, can be achieved' — a target proposed, not demonstrated, since sub-Kelvin antiprotons would greatly increase the fraction of synthesized antihydrogen below the ~0.5 K depth of a ground-state magnetic atom trap.

Problem Statement

Establish the low-temperature limits of forced evaporative cooling of magnetically confined nonneutral antiproton plasmas. Specifically, for trapped antiproton clouds prepared as in the 2010 ALPHA demonstration (of order $10^4$ particles, number densities of order $10^6$-$10^7\ \mathrm{cm}^{-3}$, in a cryogenic Penning-Malmberg trap): (1) quantitatively determine each contribution to the net heating rate $P$ of the rate-equation model that is significant at the stated energy/temperature precision — including expansion-driven Joule heating, electronic noise from the electrode driving circuits, patch potentials and potential-setting precision, and radiative or microwave coupling to warm trap surfaces — and state the bound below which all remaining apparatus-dependent terms are shown negligible, together with their dependence on trap electronics and geometry, replacing the 2010 estimate in which $P$ was fixed only by the measured radial expansion and other contributions could be neither excluded nor quantified; (2) determine the quantitative ultimate lowest temperature attainable by evaporative cooling in a given apparatus, including which of the candidate limits (electrode-drive noise, confining-potential precision) actually binds, and the ramp protocols and the scaling of particle-number loss versus final temperature required to reach temperatures below the ~0.5 K magnetic trap depth relevant to antihydrogen; (3) characterize the evaporation dynamics and limits when the superposed multipole (Ioffe-type) magnetostatic neutral-trap field is energized, i.e., when the azimuthal symmetry of the Penning trap is broken; and (4) experimentally demonstrate a sub-Kelvin nonneutral antiproton plasma, or establish experimentally and quantitatively why it cannot be reached. These four linked questions jointly bear on establishing the lowest attainable temperature and the heating physics of evaporatively cooled antiproton plasmas.

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