Spectroscopy of antiprotonic helium atoms and its contribution to the fundamental physical constants.

Antiprotonic helium atom, a metastable neutral system consisting of an antiproton, an electron and a helium nucleus, was serendipitously discovered, and has been studied at CERN's antiproton decelerator facility. Its transition frequencies have recently been measured to nine digits of precision by laser spectroscopy. By comparing these experimental results with three-body QED calculations, the antiproton-to-electron massratio was determined as 1836.152674(5). This result contributed to the CODATA recommended values of the fundamental physical constants

At CERN’s antiproton decelerator (AD), the ASACUSA (atomic spectroscopy and collisions using slow antiprotons) collaboration, a Japan-Austria-Hungary-Italy-Denmark-Germany group led by the author, has been working on the precision laser spectroscopy of antiprotonic helium atoms since 2000.[1)],[2)] By comparing the measured laser transition frequencies with three-body quantum electrodynamics (QED) calculations,[3)],[4)] we have determined the antiproton-to-electron mass ratio m/ to the level of 2 × 10−9.[5)] Together with the atomic transition frequencies in hydrogen and deuterium, our results (assuming CPT symmetry, i.e., the equality of particle-antiparticle masses) yield information on the Rydberg constant, the proton and deuteron charge radii, and the relative atomic mass of the electron, in the CODATA recommended values of the fundamental physical constants 2006.[6)]

In this article, we describe how a series of nuclear physics experiments ultimately led both to a more precise knowledge of the fundamental physics constants and to testing the CPT symmetry theorem.

History

It was in late 80’s at the 12-GeV proton synchrotron at KEK, Japan, when we first discovered an intriguing anomaly. Our aim then was to search for a sigma hypernucleus ∑4He, [7)] a four-body system comprising a ∑ hyperon and a three-nucleon core, produced in the 4He(stopped K−, π−) reaction. The experiment was done by stopping kaons in a liquid helium target, and by measuring the momenta of outgoing pions using a magnetic spectrometer.

Figure 1 shows a momentum spectrum of negatively-charged particles measured by the magnetic spectrometer.[8)] It is in this spectrum we found the abovementioned anomaly.

Fig. 1

Momentum spectrum of negatively charged particles emitted after K− stopping in liquid helium. Only the K− weak-decay peaks are visible with delayed timing selection. From ref. [8].

The peak at 255 MeV/c is due to the formation of Λ4He hypernuclei, and the bump around 170 MeV/c corresponds to the ∑ hypernuclear formation. The dominant peak at 235 MeV/c and another peak at 205 MeV/c are due respectively to K− → μ−ν̄ (aka Kμ2) and K − → π−π0 (aka K2) weak decays. The observation of Kμ2 and K2 peaks was totally un-expected, because the K− cascade time, the time difference between the K− atomic capture and the K− nuclear absorption, is known to be quite short, (indeed as short as ≈ ps as is also the case for π− and p̄ ), while the kaon weak-decay lifetime is 12 ns. Thus, there is no way for the slow weak decay to compete with the fast cascade to produce such large peaks, which accounted for some 3% of the stopped K−, unless such kaons are somehow “trapped” during the cascade process.

A natural question to ask was if similar “cascade trapping” can be observed in other exotic helium atoms. We did find a similar anomaly in the case of π−-helium,[9)] but the most spectacular was the discovery of p̄ longevity in helium.[10)] Shown in Fig. 2 is the p̄ annihilation time spectrum, i.e., the distribution of time difference between the p̄ arrival in the target and the p̄ annihilation. The peak at t = 0, containing about 97% of the events, is due to the normal, prompt annihilation. This however is followed by an anomalous delayed component. The data taken on a liquid nitrogen target just had a prompt peak. We thus serendipitously found that about 3% of antiprotons trapped in liquid helium survived with a mean lifetime of 3μs. This was the beginning of our long series of experiments on antiprotonic helium.

Fig. 2

The antiproton annihilation time spectrum measured by stopping antiprotons in liquid helium at KEK. From ref. [10].

Metastability of antiprotonic helium

Normally, when an antiproton is stopped in matter, it annihilates on a nucleus within picoseconds, leaving no time for high-precision spectroscopy during any preceding atomic cascade. The above described exception is now known to occur only in helium, and has been shown to be due to the formation of the antiprotonic helium atom (hereafter denoted p̄ He+), a naturally-occurring antiproton trap in which an antiproton can be “stored” for several microseconds. This anomaly had been in fact theoretically predicted by Condo and Russell more than 40 years ago,[11)],[12)] but had never been observed before our work. The p̄4He+ atoms have the following remarkable features:

The atom has a metastable lifetime in excess of a microsecond. This longevity occurs when the antiproton occupies a near-circular orbit having a large n(~ 38) and also large L(≿35).

The p̄ 4He+ atoms can be abundantly produced just by stopping antiprotons in a helium target. Then, about 3% of the stopped antiprotons automatically become trapped in the metastable states.

We usually use low-temperature (T ~ 10 K) helium gas as the target. The p̄ 4He+ atoms that are produced collide with the surrounding helium atoms and are thermalized. Therefore, the antiprotonic helium atoms are already cold and are well suited for high-precision spectroscopy (i.e., small Doppler width).

We have demonstrated already that we can perform laser spectroscopy of p̄ 4He+. Note that we are not changing the electronic state as in ordinary laser spectroscopy, but are inducing transitions between different antiproton orbits.

Figure 3 shows an energy level diagram of p̄ 4He+. The levels indicated by the continuous lines have metastable (≿1 μs) lifetimes and de-excite radiatively, while the levels shown by wavy lines are short lived (≾10 ns) and de-excite by Auger transitions to antiprotonic helium ion states (shown by dotted lines). Since the ionic states are hydrogenic, Stark collisions quickly induce antiproton annihilation on the helium nucleus, also indicated in Fig. 3.

Fig. 3

Level diagram of p̄ 4He+ in relation to that of p̄ 4He++. The continuous and wavy bars stand for metastable and short-lived states, respectively, and the dotted lines are for L-degenerate ionized states.

Principle of p̄ 4He+ laser spectroscopy

Laser spectroscopy of p̄ 4He+ works as follows: As shown in Fig. 3, there is a boundary between metastable states and short-lived states. For example, (n, L) = (35, 33) is metastable, while (n, L) = (34, 32), which can be reached from (35, 33) by an E1 transition, is short lived. Thus, if we use a laser (λ = 372.6 nm in this particular case) to induce a transition from (35, 33) to (34, 32), (and of course if an antiproton happens to be occupying the (35, 33) level at the time of laser ignition), the antiproton is de-excited to the short-lived state, which then Auger decays to an ionic (n, L) = (30, 29) state within ≾10 ns. The ionic state is then promptly destroyed by Stark collisions, leading to the nuclear absorption or annihilation of the antiproton. Adding all these together, we expect to see a sharp increase in the p̄ annihilation rate in coincidence with a resonant laser pulse, as shown in Fig. 4. We measure the intensity of the laser-induced annihilation spike as a function of laser detuning to obtain the transition frequency νexp.

Fig. 4

Laser resonance of the (39, 35) → (38, 34) transition in p̄4 He+. Left: Observed time spectra of delayed annihilation of antiprotons with laser irradiation of various wavelengths near 597.2 nm. Upper right: Enlarged time profile of the resonance spike. Lower right: Normalized peak count versus wavelength in the resonance region. From ref. [13].

The corresponding theoretical values νth are obtained by first solving a non-relativistic three-body Schrödinger equation using a complex-coordinate rotation (CCR) variational method,[23)] and then perturbatively applying relativistic and quantum-electrodynamical corrections. In the coordinate system centered on the helium nucleus (atomic base), the non-relativistic Hamiltonian in atomic units can be written as,

wherein the indices 1, 2 and 3 respectively correspond to He, p̄ and e−.

Intuitively, the transition frequency from the state (n, L) to (n′, L′) can be written similarly to the well-known hydrogen formula,

where R is the Rydberg constant, c is the speed of light, M*/m is the reduced antiproton-to-electron mass ratio and Zeff is the effective nuclear charge. Here, Zeff would be 2 for the antiprotonic helium ion (p̄He++), but is <2 in the case of p̄He+ due to the nuclear-charge shielding by the remaining electron, and is evaluated in the three-body calculation. This shows how m/me can be deduced from the precision measurement of νexp with the help of three-body calculations.

Progress over the years

Figures 5 and 6 show our progress over the years. The first series of measurements, conducted at CERN’s low energy antiproton ring (LEAR, closed down at the end of 1996) were of exploratory nature, in which we established the formation and structure of the antiprotonic helium atoms.[11)] Our first successful observation of the (n, L) = (39, 35) → (38, 34) laser resonance of p̄ 4He+ at λ = 597.3 nm[13)] triggered theoretical efforts to perform high-precision three-body QED calculations, which in turn helped us guide the search for new transitions.

Fig. 5

Limit on the possible differences between the antiproton mass m (and charge Q ) and those of the proton’s measured over the years, derived from p̄ 4He+ laser spectroscopy. The experimental techniques used to improve the precision are indicated.

Fig. 6

(a): Frequency of the p̄ 4He+ transition (n, L) = (37, 35) → (38, 34) measured by the ASACUSA collaboration over the years. (b): m/m recommended by the CODATA98 and 2002 compilations, and m/m.

The most important outcome of the LEAR era was the study of the density-dependent shifts and widths of the transition frequencies.[14)] Since the p̄ 4He+ atoms were produced by stopping antiprotons in a helium gas target of temperature T ≈ 5 K and pressure P ≈ 0.5 bars, or atomic density of ρ ≈ 1021 atoms per cm3, the atoms undergo many collisions against ordinary helium atoms, producing density-linear (as well as state-dependent) shift of the transition frequencies of ≈ 500 MHz. By taking data at several different target densities in the range 0.1–3 × 1021 cm−3, and extrapolating to the zero target density, we were able to determine two transition frequencies of p̄ 4He+ ((n, L) = (39, 35) → (38, 34) and (37, 34) → (36, 33)) with precisions δν/ν of 0.5 – 1 × 10−6.[14)]

In our first experiment at the CERN AD, we measured six transition frequencies of p̄ 4He+ to δν/ν = 1 – 10 × 10−7.[15)] The essential difference between the LEAR and the AD experiments was the time structure of the antiproton beam. At LEAR, antiprotons were slowly extracted from the LEAR ring, so that laser was fired for each p̄ 4He+-candidate event which occurred randomly with a mean rate of some 300 Hz. In contrast, the AD provides a short pulse of ≈ 100-ns wide containing some 3 × 107 p̄ s, repeated every 100 s. A single laser pulse in this case irradiates some 106 metastable atoms. The conventional event-by-event collection of antiproton-annihilation events used at LEAR is impossible with the pulsed beam at AD. We thus developed an entirely new detection scheme (Fig. 7) based on analogue waveform (delayed-annihilation time spectra (DATS)) recording of Čerenkov counter(s) viewed by gateable photomultipliers. Here, because of the velocity-selectivity of the Čerenkov radiation, the Čerenkov counters are sensitive to the energetic pions ejected from the antiproton annihilation events. The Good control of systematic errors by stabilizing the laser frequency, intensity, the antiproton-beam position, etc., over a long measuring time of ≈ 8 hours was essential to achieve high precision.

Fig. 7

Antiproton beam, helium gas target and antiproton annihilation detectors. From ref. [16].

In the first measurement at AD, the zero-density extrapolation procedure was still unavoidable (see Fig. 8), but this was largely eliminated by constructing a radio-frequency quadrupole decelerator (RFQD), an “inverse linac” which decelerates the 5.3 MeV antiprotons ejected from the AD ring to <100 keV with an efficiency of 20–25%.[17)] Using the RFQD (Fig. 9), the antiprotons can be stopped in a much lower-density target having ρ ≈ 1017 cm−3. In 2003, we determined 7 transition frequencies of p̄ 4He+ and 6 of p̄3He+, with errors of δν/ν ≈ 0.5 – 2 × 10−7.[18)]

Fig. 8

(left) Antiprotonic helium laser resonance profiles: four transitions in p̄ 4He+ and two in p̄3He+. (right) The dependence of the transition frequencies on the helium target density. From ref. [18].

Fig. 9

Experimental layout of the radio-frequency quadrupole decelerator, and the laser spectroscopy experiment of p̄ He+. Drawing not to scale.[17)]

Pulse-amplified cw lasers and an optical frequency comb

Having eliminated the collisional shift, the line width of the laser and its frequency calibration remained as the largest contributory factors to the errors on measured values of the resonant frequencies.

Only pulsed lasers can provide the megawatt-scale intensities needed here to induce the p̄He + transitions. However, fluctuations in their frequency and line width and the difficulty of calibrating the wide range of p̄ He+ wavelengths (from infrared to ultra-violet) have limited our experimental precision. We circumvented these problems by basing our experiments on a continuous-wave (cw) laser whose frequency νcw could be stabilized with a precision <4 × 10−10 against an optical comb (Fig. 10).[19)]–[21)] Its intensity was then amplified by a factor 106 to produce a pulsed laser beam of frequency νpl ~ νcw with an accuracy and resolution 1–2 orders of magnitude higher than before. The profile of the (n, ℓ) = (36, 34) → (37, 33) resonance in p̄ 3He+ is shown in Fig. 11(a). It contains

Fig. 10

Pulse-amplified cw laser stabilized against an optical frequency comb. From ref. [5].

Fig. 11

Frequency profiles (laser detuning plotted against the laser-induced annihilation signal intensity of the transition (a) (36, 34) → (37, 33) in p̄3He+ and (b) (37, 35) → (38, 34) in p̄4He+.

eight intense lines (indicated by four arrows in Fig. 11(a), where each arrow in fact represents a closely-spaced doublet) corresponding to E1 transitions involving no spin-flip between the eight hyper-fine substates (i.e., the coupling of the electron spin, the antiproton orbital angular momentum and the antiproton spin) of states (36, 34) and (37, 33) and

12 weak lines wherein one of the constituent particles flips its spin. Only the two peaks separated by 1.8 GHz that arise from the interaction between the orbital angular momentum of the antiproton and electron spin could be resolved, however, due to the 400 MHz Doppler broadening caused by the motion of the p̄ 3He+ thermalized to T = 10 K.

The spin-averaged transition frequency νexp was determined by fitting this profile with the theoretical line shape (solid line) obtained from the optical Bloch equations which describe the evolution of the p̄He + state populations during laser irradiation. The νexp values of p̄ 4He+ resonances (Fig. 11(b)), which contain four intense, non-spin-flip lines and four weak, spin-flip ones, were similarly obtained. The ac Stark shifts caused by the laser interacting with p̄He + are estimated to be

Weighing the antiproton

The νexp values thus obtained are compared with theoretical values νth[3)],[4)],[22)],[23)] (see Fig. 12), which included QED and nuclear-size (Δνnuc = 2–4 MHz) effects, and used the 2002 CODATA recommended values for fundamental constants, including m/m = m/m = 1836.15267261(85), m4He/m = 7294.2995363(32) and m3He/m = 5495.885269(11). Theory also provided coefficients for dνth/d(m/m). These we used to determine the antiproton-to-electron mass ratio as the value m/m = 1836.152674(5) by minimizing the sum ∑[νth(m/m) − νexp]2/σexp2 over the 12 transitions.[5)] Here, the experimental (1σ) error σexp = 4–15 MHz was the quadratic sum of the statistical (3–13 MHz) and systematic ones arising from the chirp (2–4 MHz), collisional shifts (0.1–2 MHz) and the harmonic generation (1–2 MHz). The error 5 on the last digit of m/m is the quadratic sum of 4 (the minimization error) and the systematic ones 3 (arising from σsyst) and 2 (from σth). This result was included in the evaluation of the CODATA 2006 fundamental physical constants. Note that m, m4He and m3He are more precisely known in the atomic mass unit (u) rather than in the atomic units (au) (e.g., m in u was known in CODATA 2002 with a relative standard uncertainty of 1.3 × 10−10 while the proton mass in the atomic units m/m is known with a relative standard uncertainty of 4.4 × 10−10). Our experiment therefore contributes mainly to the determination of m/m, or more precisely, m in the atomic mass unit.

Fig. 12

Experimental νexp (circles with 1σ errors) vs theoretical νth (triangles[23)] and squares[3)]) transition frequencies. From ref. [5].

CPT constraints on the p̄ mass and charge

Let us now examine to what extent we can constrain the mass m and charge Q of the antiproton, if we let both p̄ mass and charge tovary with respect to those of the proton (m and Q).

Näively, since the p̄He+ binding energies (and hence transition frequencies) are roughly proportional to m Q2, we expect to see a correlation between the fractional mass change δm/m vs δQ/Q such as

The 90-%-confidence level contour in the δm − δQ plane indeed shows that the CPT bounds on charge and mass obtained by the antiprotonic helium atoms are correlated along the δm/m ≈ −2δQ/Q direction, so that it is not possible to individually constrain δm/m and δQ/Q. By combining the p̄He+ results with the proton-to-antiproton cyclotron frequency comparison of the TRAP group at CERN LEAR,[24)] a much tighter bound on the equality of antiproton mass and charge with those of the proton of 2 × 10−9 (90-% confidence level), as shown in Fig. 13.

Fig. 13

A 90-%-confidence limit contour on the antiproton mass and charge deduced from the p̄He + laser spectroscopy is shown.

Summary and future prospects

Antiprotonic helium atoms (p̄He + ≡e− − p̄ − He++), serendipitously discovered at KEK (1991) and subsequently studied (1992–1996) at CERN’s low energy antiproton ring (LEAR), have been further studied (1999−) using high-precision laser-spectroscopic methods at CERN’s Antiproton Decelerator facility (AD). Experimental methods were successively developed to measure the laser transition frequencies between (n, ℓ) and (n ± 1, ℓ − 1) states of p̄He + (n ~ 40 and ℓ ≾n − 1 are the principal and orbital quantum numbers of the antiproton) to progressively higher precisions, from the initial LEAR value of 3 parts in 106, to the most recent value of 9 parts in 109.

The first major advance was brought about by using of pulsed beams of antiprotons in high precision measurements of the p̄He + transition frequencies. Under this new experiment mode, a single laser pulse of high spectral purity induce transitions in ~103–104 p̄He + atoms simultaneously. This allowed large numbers of p̄He + to be produced and studied at a higher rate and precision than was possible using continuous antiproton beams. The next advance was introduced by the construction of the radio-frequency quadrupole decelerator (RFQD), a post-decelerator to the AD. The RFQD made it possible to stop antiprotons in a helium gas target at very low densities of ρ ~ 1017 cm−3, some 104 times lower than those previously used, and to study p̄He+ atoms under near-vacuum conditions. This eliminated the need for the zero-density extrapolation, a large source of errors in previous measurements, thereby improving the experimental precision by nearly an order of magnitude. Another order of magnitude improvement was recently achieved by using a femtosecond optical frequency comb and continuous-wave pulse-amplified laser. The new laser system made it possible to reduce systematic uncertainties in absolute frequency determination to a few-MHz level, better than the statistical uncertainties of 3–13 MHz.

This progress in experimental technique was matched by similar advances in the theoretical methods employed in high-precision three-body QED calculations. Methods have been developed to obtain complex eigenvalues of the three-body Hamiltonian to 1-MHz-scale precisions, and QED corrections have been applied up to the order of α5.

Comparisons of experimental νexp and theoretical νth frequencies for twelve transitions, seven in p̄ 4He+ and five in p̄ 3He+, yielded an antiproton-to-electron mass ratio of m/m = 1826.152674(5), which agrees with the known proton-to-electron mass ratio at the level of ~2 × 10−9. The experiment also set a limit on any CPT-violating difference between the antiproton and proton charges and masses, (Q − |Q|)/Q ~ (m − m)/m < 2 × 10−9 to a 90% confidence level. If on the other hand we assume the validity of CPT invariance, the m/m result can be taken to be equal to m/m. The present result has in fact contributed to the improvement of the relative standard uncertainty in the proton-electron mass ratio from 0.46 ppb (2002 CODATA values)[25)] to 0.43 ppb (2006 CODATA values).[6)]

Further improvements in the experimental precision would be possible if the thermal Doppler broadening could be reduced. In 2007, we succeeded to achieve a higher frequency precision by performing a two-photon (36, 34) → (34, 32) resonance (by using two counter-propagating beams of λ1 ≈ 372 nm and λ2 ≈ 417 nm), which canceled the first order Doppler width. A preliminary analysis shows that m/m can be measured with a sub-ppb (<10−9) precision (frequency precision of ≈1 MHz) using this method.

In order to further improve, the use of pulsed lasers must be abandoned. So far, in order to induce strong-enough transitions for the very dilute antiprotonic helium atoms, the use of MW-class pulsed lasers has been inevitable. However, such pulsed lasers have inherent problems of frequency chirp and ac Stark shift, which are now the main source of systematic errors at the level of ≈1 MHz. We have therefore commenced the development of a new Doppler-free spectroscopy method using cw lasers. If our attempts succeed, it should be possible to improve the frequency precision by more than a factor 10.

However, in order to achieve a better determination of m/m, we also need improved theoretical predictions. We now rely heavily on Korobov’s calculated values which claim to have frequency precisions of ~1 MHz for most of the transitions, but we need at least two theoretical predictions which agree with each other within quoted numerical errors. If both experiment and theory are improved, it appears possible to determine m/m as good as or even better than m/m, and to contribute to the better determination of a fundamental constant.

Table 1

QED corrections to the (39, 35) → (38, 34) transition in p̄4He+[3)]

ΔEnr	=	501 972 347.9
ΔErc	=	−27 526.1
ΔEae	=	233.3
ΔEse	=	3 818.1
ΔEvp	=	−122.5
ΔEkin	=	37.3
ΔEexch	=	−34.7
ΔErecoil (3)	=	0.8
ΔEtwo-loop	=	0.9
ΔEnuc	=	2.4
ΔEα4	=	−2.6
ΔEtotal	=	501 948 754.9(1.3)(0.5)

E is the non-relativistic variational calculation result, to which all other corrections are applied perturbatively. E is the relativistic correction to the electron, E is the electron’s anomalous magnetic moment correction, E is the 1-loop self energy term, E is the 1-loop vacuum polarization, E is the relativistic correction to the heavy particles (p̄ and helium nucleus), E and E(3) are the first- and third-order transverse photon exchange corrections, E is the two-loop QED corrections, E is the finite nuclear-charge distribution correction, and E is the α4-order QED correction.