Antihydrogen 1S–2S: A Trapping Problem Wearing the Costume of Spectroscopy — Epoche C2
Purpose of the note A common statement is that comparing matter with antimatter is hard because antimatter annihilates on contact with matter. Annihilation is not the difficulty. Charged antiparticles have been held in Penning traps — electrode structures that confine charges with static electric and magnetic fields — for months at a time, and antiprotons never touch a wall. The difficulty appears at the moment the antiproton and the positron combine, because the resulting antihydrogen atom is neutral and can no longer be held by electric fields. From that instant the experiment is limited by a magnetic well roughly half a kelvin deep, and everything else follows. This note traces the working sequence and shows where the precision is actually lost. SI units are used throughout, as in the source publications: frequencies in Hz, magnetic fields in tesla, temperatures in kelvin. What is being tested, and why it is a sharp test The CPT theorem, proved by Lüders and Pauli in the 1950s, states that any local, Lorentz-invariant quantum field theory with a Hermitian Hamiltonian is invariant under the combined operation of charge conjugation, parity inversion and time reversal. One corollary is that antihydrogen and hydrogen must have identical energy level structures — not similar, identical. The 1S–2S transition in hydrogen has been measured at $f_{1S-2S} = 2\,466\,061\,413\,187\,035$ Hz with an uncertainty near 10 Hz. Any measured difference in antihydrogen is therefore a direct violation, with no model-dependent subtraction in between. That is what makes an absolute frequency comparison stronger than a comparison of differences. The transition is also unusually forgiving. It is driven by two counter-propagating photons of 243 nm, so the first-order Doppler shift cancels between them; a single-photon transition at the equivalent 121.6 nm would not permit this. The 2S state decays only by two-photon emission, with a lifetime of 122 ms, so the natural linewidth is $\Gamma/2\pi = 1/(2\pi \times 0.122\ \mathrm{s}) = 1.3$ Hz. Relative to the transition frequency that is $1.3/2.466\times10^{15} = 5.3\times10^{-16}$. Hold that number; it is the benchmark against which the achieved precision should be judged. Procedure Make antiprotons. A 26 GeV proton beam strikes a metal target. The Antiproton Decelerator delivers antiprotons at 100 MeV/c, which for the antiproton mass of 938.3 MeV/c$^2$ is a kinetic energy of $\sqrt{100^2 + 938.3^2} - 938.3 = 5.3$ MeV. ELENA reduces this further to 100 keV. Catch and cool. A thin degrader foil slows a small fraction below a few keV, low enough to be caught in a Penning trap. The antiprotons are then cooled by contact with a cloud of electrons, which themselves cool by emitting cyclotron radiation in the several-tesla field. Prepare positrons. Positrons from a radioactive source are accumulated and cooled to the same tens of kelvin, forming a non-neutral plasma held in an adjacent trap region. Mix. The two plasmas are brought together. The dominant channel is three-body recombination, $\bar{p} + e^+ + e^+ \to \bar{H} + e^+$, in which a second positron carries off the excess energy. The rate coefficient scales as $n_{e^+}^2 T_{e^+}^{-9/2}$, where $n_{e^+}$ is the positron density and $T_{e^+}$ the positron temperature; the steep temperature exponent is why cooling matters so much. Halving $T_{e^+}$ from 40 K to 20 K multiplies the rate by $2^{4.5} = 22.6$. Trap what you can. The neutral antihydrogen atom has a magnetic moment of essentially one Bohr magneton, $\mu_B = 9.274\times10^{-24}$ J T$^{-1}$, carried by the positron spin, so it can be held in a magnetic field minimum if it is in a low-field-seeking state. The ALPHA trap has a field difference of about 0.8 T between the centre and the wall, giving a well depth $U = \mu_B \Delta B = 7.4\times10^{-24}$ J, or in temperature units $U/k_B = 0.54$ K. Interrogate. Illuminate the trapped atoms with 243 nm light in a build-up cavity. An atom excited to 2S is then either photoionised by a further 243 nm photon — the photon energy $1239.8/243 = 5.10$ eV exceeds the 2S binding energy of $13.606/4 = 3.40$ eV, so one photon suffices — or spin-flipped through 2P. Either way it leaves the trap. Count losses. Atoms that leave annihilate on the electrode wall, producing charged pions whose tracks reconstruct to a vertex in a silicon detector. The measurement is thus a disappearance experiment: compare the number of annihilation vertices with the laser on resonance and detuned. Where the precision is lost Step 5 is the bottleneck, and it can be quantified from the numbers already given. Antihydrogen forms with roughly the kinetic energy of the parent plasmas. For a three-dimensional Maxwell–Boltzmann distribution at temperature $T_{e^+}$, the fraction of atoms with kinetic energy below the well depth $U$ is, for $U \ll k_BT_{e^+}$, $$P(E<U) \;\simeq\; \frac{4}{3\sqrt{\pi}}\left(\frac{U}{k_BT_{e^+}}\right)^{3/2},$$ which is just the small-argument limit of the incomplete gamma function that integrates the Maxwell–Boltzmann speed distribution, the $E^{3/2}$ coming from the volume of momentum space below the cut. With $U/k_B = 0.54$ K and $T_{e^+} = 20$ K the argument is $0.027$, and $P = 0.752 \times 0.027^{3/2} = 3.3\times10^{-3}$. About one atom in 300 is retained; the rest fly out and annihilate within microseconds. This is the whole reason experiments count atoms individually and accumulate them over hundreds of mixing cycles. Field inhomogeneity is the second constraint, and it is worth stating why it is not fatal. The bare Zeeman scale is $\mu_B/h = 9.274\times10^{-24}/6.626\times10^{-34} = 1.40\times10^{10}$ Hz T$^{-1}$, that is 14.0 GHz per tesla. Against the transition frequency this is $1.40\times10^{10}/2.466\times10^{15} = 5.7\times10^{-6}$ per tesla — more than six orders of magnitude larger than the precision sought, since $5.7\times10^{-6}/2\times10^{-12} = 2.8\times10^{6}$. The measurement is possible only because