Hyperfine Interact (2009) 193:321–327 DOI 10.1007/s10751-009-0018-5

Efficient Rydberg positronium laser excitation for antihydrogen production in a magnetic field M. G. Giammarchi · (AEGIS Collaboration)

Published online: 2 September 2009 © Springer Science + Business Media B.V. 2009

Abstract Antihydrogen production by charge exchange reaction between positronium (Ps) and antiprotons requires an efficient excitation of Ps atoms up to high-n levels (Rydberg levels). We propose a two-step laser light excitation, the first from ground to n = 3 and the second from this level to a Rydberg level n > 15. In this study it is assumed that a Ps cloud is produced by positrons hitting a target converter located in a Penning-Malmberg trap within a uniform ∼ 1 T magnetic field. We model the optical transition structure by taking into account Doppler and motional Stark effects. The predicted efficiency for population deposition in high n states is of ∼ 30%. Keywords Rydberg positronium laser excitation · Antihydrogen production · Magnetic fields

AEGIS Collaboration: A. S. Belov, G. Bonomi, I. Boscolo, N. Brambilla, R. S. Brusa, V. M. Byakov, L. Cabaret, C. Canali, C. Carraro, F. Castelli, S. Cialdi, M. de Combarieu, D. Comparat, G. Consolati, N. Djourelov, M. Doser, G. Drobychev, A. Dupasquier, D. Fabris, R. Ferragut, G. Ferrari, A. Fischer, A. Fontana, P. Forget, L. Formaro, M. Lunardon, A. Gervasini, S. N. Gninenko, G. Gribakin, R. Heyne, A. Kellerbauer, D. Krasnicky, V. Lagomarsino, G. Manuzio, S. Mariazzi, V. A. Matveev, S. Moretto, C. Morhard, G. Nebbia, P. Nedelec, M. K. Oberthaler, P. Pari, V. Petracek, M. Prevedelli, I. Y. Al-Qaradawi, F. Quasso, O. Rohne, S. Pesente, A. Rotondi, S. Stapnes, D. Sillou, S. V. Stepanov, H. H. Stroke, G. Testera, G. M. Tino, A. Vairo, G. Viesti, F. Villa, H. Walters, U. Warring, S. Zavatarelli, A. Zenoni, D. S. Zvezhinskij. M. G. Giammarchi (B) Istituto Nazionale di Fisica Nucleare, Via Celoria 16, 20133 Milan, Italy e-mail: [email protected]

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1 Introduction Some fundamental questions of modern physics relevant to unification of gravity with the other fundamental interactions, models involving vector and scalar gravitons, matter–antimatter symmetry (CPT) can be enlightened via experiments with antimatter [1]. Another important question involves testing the validity of the equivalence principle of General Relativity, in particular measuring the equality of the inertial and gravitational mass, i.e. measuring gravitational acceleration on cold atoms [2]. All these experiments are performed on matter systems; there are no direct measurements on the validity of the equivalence principle for antimatter. In addition, some quantum gravity models predict possible violations of this principle for antimatter [3]. The experiment AEGIS (Antimatter Experiment: Gravity, Interferometry, Spectroscopy) [4] at the CERN Antiproton Decelerator (AD), has been proposed to directly measure the Earth gravitational acceleration g¯ on antimatter. The AEGIS experimental program features the production of a cold and collimated antihydrogen beam flying horizontally with a velocity around 100 m/s for a path length of ∼1 m. The gravitational acceleration will be obtained by detecting the beam vertical deflection using a classical Moiré deflectometer [5]. The cold antihydrogen beam is produced in highly excited (Rydberg) states, with a pulsed production scheme, and subsequently accelerated in the horizontal direction using inhomogeneous electric fields. The production of cold antihydrogen bunches (N H¯ ∼ 104 per bunch) occurs in the charge transfer of a cloud of Rydberg excited Ps with cold antiproton p¯ by means of the reaction Ps∗ + p → H + e− [6]. Ps atoms and antiprotons are prepared and manipulated in two parallel Penning-Malmberg traps with magnetic field, installed inside a < 1 K cryostat. Here we will focus on the Rydberg excitation of the Ps atomic cloud, leaving the description of the AEGIS experimental setup to the references [7]. For this experimental programme an efficient H¯ formation is required. The number of produced antihydrogen atoms in the charge exchange reaction is expressed with obvious notation as N H¯ = ρ N Ps N p¯ σ/A where ρ is the overlap factor between the trapped p¯ and the moving Ps cloud with transverse area A. Since the cross section σ depends upon the fourth power of the principal quantum number n of the excited Ps [6] (σ ∝ n4 πa20 , where a0 is the Bohr radius), n can be chosen to be in the range n = 20 ÷ 30, avoiding higher n values to reduce the ionization losses due to stray fields and dipole-dipole interactions. Positronium excitation to these high-n levels can either be obtained via collisions [6] or photon excitation [8]. In this paper we propose a direct Ps excitation by two simultaneous laser pulses within a time length of a few ns. Positronium atoms are expected to be produced at the surface of a target converter [9] by positron implantation at kinetic energies ranging from several 100 eV to a few keV. In the AEGIS design, the Ps exiting the target surface forms a cloud with a transverse area of ∼1 mm diameter, at a maximum equivalent temperature of 100 K, and immersed in a ∼ 1 T magnetic field [4]. Ps atom resonances will then be broadened by Doppler effect because of random velocities v ∼ 105 m/s. Moreover the sublevels of a Rydberg state will be split and mixed in energy by the Zeeman and motional Stark effect. Because of these effects the transition will be from a level to a broadened Rydberg level-band.

Rydberg positronium laser excitation for antihydrogen production Fig. 1 Laser system for the Ps excitation (see text). The predicted energy flow is also shown

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Nd:YAG 200 mJ 4 ns

180 mJ

2ω

10 mJ

615 nm

2ω

ω+2ω

Dye laser

PPLN 4 cm

OPG

205 nm 200 μJ

LiNb03 crystal

OPA

1670 nm 1 mJ

10 mJ

2 The laser system for positronium excitation We propose to photo-excite the Ps to the Rydberg band by using a two-step excitation from the ground state to the n = 3 state (λ = 205 nm), and from the n = 3 (10 ns long-lived) state to high-n levels (λ around 1670 nm). The proposed laser system (Fig. 1) is designed to give the necessary saturation fluence as discussed in the following. An optically pumped Dye–prism laser coupled to a third harmonic generator provides the 205 nm photons for the first transition. The laser system for the second pulse is composed by the cascade of an OPG (Optical Parametric Generator) to generate the required frequency and an OPA (Optical Parametric Amplifier) to provide the required amount of energy. The system has flexibility both in spectral bandwidth and power, allowing the selection of the final Ps excited energy, starting from n = 15. The ionization probability, proportional to the total energy of the pulse, remains limited below 0.3% in the proposed operative conditions (and a maximum value of n ∼30). A Q-switched Nd:YAG laser pulse of 200 mJ and 4 ns drives both the Dye and the OPG–OPA systems. Most of the energy of the Nd:YAG laser, about 180 mJ, is conveyed along the first branch (the upper part of Fig. 1) and is up-converted to the 532 nm second harmonic for pumping a 615 nm Dye laser. The bandwidth of this laser has to be sufficiently large to cover the Doppler bandwidth of the 1 → 3 transition (nearly 0.04 nm in the case of 100 K). The output radiation from the Dye laser is then up-converted with a sequence of a second and third harmonic crystals. This system is designed to deliver up to 200 μJ at 205 nm. The expected linewidth will be ≥ 0.05 nm. The OPG system performs the down-conversion of the 1064 nm pulsed radiation of the Nd:YAG, thereby producing a radiation in the frequency range 1600 ÷ 1700 nm with a ∼3 nm bandwidth. It consists of a Periodically Poled Lithium Niobate (PPLN) crystal [10] with a period of 30.25 μm. The crystal has a very high non linear coefficient (def f  14 pm/V) and, operates in the Quasi Phase

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Matching (QPM) condition converting a 1064 nm photon in two new photons: a signal photon with λ ∈ [1600, 1700] nm and an idler photon with λ ∈ [2600, 3000] nm. In this process [11] the idler and signal frequencies can be fine tuned by controlling the crystal temperature around 200 Celsius. The parametric amplification of the OPG radiation pulse is achieved with an OPA system based on a LiNbO3 crystal of 10 mm length and 5 × 5 mm2 transverse area. This device transforms Nd:YAG pump photons into signal photons by a stimulated down-conversion process; the pulse can be amplified up to a final energy of 1 mJ.

3 Modeling Ps excitation from n = 1 to high-n levels A simple theoretical model of two-step Ps excitation was developed to calculate saturation fluence and useful bandwidth [12]. The spectral profile of the two laser pulses is characterized by a Gaussian function whose width Δλ L will be matched to a selected Rydberg level-bandwidth around a definite n state. The laser linewidths have a coherence time Δtcoh = λ2 /c Δλ L , where λ is the central wavelength of the relevant transition. This parameter is much shorter than the 5-ns duration of the laser pulses, hence we are operating with a complete incoherent excitation for both transitions. The detailed structure of the optical transitions to high-n energy levels of Ps is dominated by the Doppler effect and by the motional Stark effect. Their energy contribution is larger than hyperfine and spin-orbit splitting in the experimental conditions. In the case of the Ps atom, Zeeman effect only mixes ortho and para Ps states with m S = 0 without affecting orbital quantum numbers [13, 14]. This interaction energy contribution amounts to ±1.2 × 10−4 eV for B = 1 T (much lower than the actual Doppler broadening, see below), while the energy of m S = ±1 states are unchanged. The level mixing leads to the well known enhancement of the average annihilation rate of the Ps ground state n = 1 [15], leaving in fact only the ortho-Ps states with m S = ±1 surviving in the Ps cloud expanding from the converter. From the observation that the electric dipole selection rules for optical transitions impose conservation of spin quantum number, and that the broadband characteristics of the laser overlaps the Zeeman splitting, we may conclude that in first approximation this effect does not play any role in the transition. Both Doppler and Stark effects have the same temperature dependence. In the following calculations we select for definiteness a Ps equivalent temperature of 100 K and an initial cloud spot of 1 mm diameter. For the first excitation step the Doppler linewidth Δλ D turns out to be around 4 × 10−2 nm. A motional Stark electric field E = v × B is induced by the Ps motion within the relatively strong magnetic field B ∼ 1 T [16, 17] . This effect splits in energy the sub-levels of the state n = 3 and leads to some mixing of quantum numbers m and , due to the breaking of the axial symmetry of moving Ps atoms. The maximum broadening due to this effect is evaluated from the usual theory of the Stark effect:  ΔE S = 6 e a0 n (n − 1) |E(v⊥ )| = 6 e a0 n (n − 1) B k B T/m (1)

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Fig. 2 Doppler (a) and Stark (c) line-broadenings as a function of the principal quantum number n for the transition 3 → n. The dashed line (b) shows the energy distance (in nm) between adjacent unperturbed n states. The dotted vertical line is the ionization limit for the lowest sublevel. The useful range for Ps Rydberg excitation is indicated

 where v⊥ = k B T/m is the Ps thermal transverse velocity. Correspondingly, Δλ S  ΔE S λ2 /2π c  1.8 × 10−3 nm, which is negligible with respect to the Doppler broadening. Therefore we conclude that the width of the transition 1 → 3 is dominated by the Doppler broadening, and the laser linewidth must be provided accordingly. Since Ps excitation is incoherent, the saturation fluence can be calculated with a rate equation model [12], matching the resonant laser linewidth to the Doppler broadening and assuming for definiteness linear laser polarization along the magnetic field axis. Accordingly:  c2 2 π 3 Δλ D Fsat (1 → 3) = · 2 = 93.3 μJ/cm2 (2) B1→3 ln 2 λ13 where B1→3 is the relevant Einstein absorption coefficient. This can be translated into an energy requirement assuming 30 ns of free expansion from the target (with an initial beam diameter of 1 mm). A transverse area of around 6 mm2 can be covered with laser beam waist of 1.4 mm. The saturation of the excitation is then reached when the laser pulse energy is greater than 5.6 μJ, a fraction of the energy delivered by the proposed system (see Fig. 1). In the n = 3 → high-n transition, the Doppler broadening is practically independent from n (and  0.35 nm at 100 K), whereas the Stark motional effect (1) is significantly bigger, as depicted in Fig. 2. As a consequence, the motional Stark electric field becomes the dominant characteristic of the transition. Degenerate high-n levels become fans or manifolds of their n2 sub-levels with a complete mixing of their m and  substates, while the mixing between n-levels in Ps atoms does not occur to a good extent [18, 19]. Since the splitting between adjacent unperturbed energy levels is given by ΔEn  13.6/n3 eV (also shown in Fig. 2), for n > 16 the bandwidth filled by the sublevels relative to an n state becomes greater than the interval between two adjacent n-levels causing sublevels interleaving. The useful range for n is determined by both the cross section behaviour for H¯ formation (requiring n ≥ 15) and the possible atom ionization in the region of level mixing. The minimum Stark electric field Emin inducing high ionization probability

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at the lowest energy E = En − 3ea0 n (n − 1) |Emin | of the level fan is calculated as [18, 19] e 1 |Emin | = . (3) 2 9 n4 16 π ε0 a0 Hence, for B = 1 T and for the reference temperature of 100 K, the ionization starts affecting part of the level fan for n > 27. This ionization limit, and the useful range for n, are indicated in Fig. 2. In the n > 16 range, for a laser linewidth Δλ L greater than Δλ D and smaller than Δλ S , the number of interacting Ps levels per unit bandwidth is practically independent from the Ps transverse velocity, due to the strong sublevels interleaving. The transition saturation fluence, with linear laser polarization along the B axis, can be estimated as [12]: Fsat (3 → n) 

c × 13.6 eV  0.98 mJ/cm2 , B3→n  n3

(4)

which is roughly constant over the 16–27 useful range for n. For the expanding Ps cloud of 6 mm2 transverse area, the laser pulse energy needed for saturation of the Rydberg excitation is of 60 μJ (a fraction of the energy delivered by the proposed system, see Fig. 1. The results found in (2) and (4) give the minimum saturation energies for the two transitions. However, because of the time constraints dictated by the Ps cloud expansion and the n = 3 level non-negligible spontaneous emission, the real Rydberg excitation will be performed with near simultaneous laser pulses. In this conditions the excitation dynamics involves all the three levels of the two-step transition. To determine the efficiency of the excitation, we have developed a simplified dynamical simulations considering transitions from (n, l, m) = (1, 0, 0) to the state (3, 1, 0) and from this state to the final states (n , 2, 0) and (n , 0, 0), assuming linear laser polarization as discussed before. In simulations we have considered the total cross section of the transition from the lower level to the upper band of levels substantially equal to the cross section of the transition between the two levels connected by electric dipole selection rules. The resonant Ps excitation is described with a model of multilevel Bloch system of equations, derived from a density matrix formulation [20], including population losses due to spontaneous decay and photoionization, for both excited states. Since the laser pulses are substantially incoherent, the phase of the light in our model is taken as a “random walk” with the step equal to the coherence time. In the simulation, we have modeled the 1 → 3 and the 3 → 25 (specifically (1, 0, 0) → (3, 1, 0) → (25, 2, 0)) transitions. For both pulses we have used a fluence F(t) of about twice the saturation fluences of the relative transitions, to compensate for population losses. Their characteristics are: (1) time length 4 ns, power 12.6 μJ and spectral width Δλ = 0.045 nm, (2) time length 2 ns, power 128 μJ and spectral width Δλ = 0.72 nm (two times the Doppler bandwidth), respectively. The final excitation probability comes from an averaging process over many simulation outputs. The calculation shows that a fraction of about 30% of Ps atoms are excited to the Rydberg state. This result does not change by irradiating with larger laser fluences or considering the slightly less effective transition to the state (25, 0, 0).

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4 Conclusion We propose a new laser system tailored to the task of exciting Ps atoms to Rydberg states in a constant 1 T magnetic fields in the frame and in the experimental conditions of the AEGIS experiment. The system is designed to perform the transition 1 → 3 → n to Rydberg excited states with a predicted population deposition of 30%.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

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