Efficient $2^3S$ positronium production by stimulated decay from the $3^3P$ level

TL;DR

This paper demonstrates a method to significantly enhance the production of $2^3S$ positronium atoms by stimulating the decay from the $3^3P$ level using a broadband IR laser, surpassing natural spontaneous decay limitations.

Contribution

The study introduces an experimental technique to increase $2^3S$ positronium production via stimulated decay, achieving over three times the spontaneous decay efficiency.

Findings

Maximum relative increase of 3.1 times in $2^3S$ production.

Optimal IR laser pulse intensity for stimulation identified.

Enhanced positronium production surpasses natural spontaneous decay rates.

Abstract

We investigate experimentally the possibility of enhancing the production of $2^3S$ positronium atoms by driving the $1^3S$-$3^3P$ and $3^3P$-$2^3S$ transitions, overcoming the natural branching ratio limitation of spontaneous decay from $3^3P$ to $2^3S$. The decay of $3^3P$ positronium atoms towards the $2^3S$ level has been effciently stimulated by a 1312.2nm broadband IR laser pulse. The dependence of the stimulating transition efficiency on the intensity of the IR pulse has been measured to find the optimal enhancement conditions. A maximum relative increase of $ \times (3.1 \pm 1.0) $ in the $2^3S$ production efficiency, with respect to the case where only spontaneous decay is present, was obtained.

Full text

††preprint: APS/123-QED

Positronium (Ps) is the neutral matter-antimatter bound state of an electron ($\mathrm{e^{-}}$) and a positron ($\mathrm{e^{+}}$). Ps has two distinct ground states: the singlet $1^{1}\text{S}$ (para-Ps), annihilating into two $\gamma$-rays with a lifetime of $0.125\text{\,}\mathrm{ns}$, and the triplet $1^{3}\text{S}$ (ortho-Ps), annihilating into three $\gamma$-rays with a lifetime of $142\text{\,}\mathrm{ns}$ Rich (1981). Ps, being a purely leptonic two-body system, is well-known for offering an ideal testing ground for high-precision Quantum Electrodynamics (QED) calculations Karshenboim (2005). Among the many precision experiments, the most accurate were recently conducted using two-photon Doppler-free laser spectroscopy of the $1^{3}\text{S}$–$2^{3}\text{S}$ transition Cassidy (2018). The $2^{3}\text{S}$ level has an extended lifetime of $1142\text{\,}\mathrm{ns}$ in vacuum. This is due to its optical metastability: single-photon radiative decays to $1^{3}\text{S}$ are prohibited by the electric dipole selection rules and the reduced overlap between the positron and the electron wave-functions increases its annihilation lifetime by a factor of eight Charlton and Humberston (2001). On top of its high-precision spectroscopy applications, $2^{3}\text{S}$ Ps is one of the few notable candidate systems being considered for measuring the gravitational interaction between matter and antimatter Oberthaler (2002), together with Ps in long-lived Rydberg states Mills and Leventhal (2002); Cassidy and Hogan (2014), antihydrogen Amole et al. (2013); S. Aghion et al. (2014) (AEgIS collaboration); P. Perez et al. (2015) (GBAR collaboration) and muonium Antognini et al. (2018). Moreover, the metastable $2^{3}\text{S}$ Ps has a very low electrical polarizability, thus being scarcely sensitive to stray electric fields Wall et al. (2015), and is a good candidate for atom interferometry, provided that a beam with sufficiently low divergence and high intensity is demonstrated Oberthaler (2002). Furthermore, an intense source of polarized $2^{3}\text{S}$ atoms has been recently shown to be of extreme usefulness to achieve Bose-Einstein condensation of Ps Zhang et al. (2019).

$2^{3}\text{S}$ Ps sources have been demonstrated via RF transition from laser-excited $2^{3}\text{P}$ Ps in a weak magnetic field Mills et al. (1975), via two-photon Doppler-free $1^{3}\text{S}$–$2^{3}\text{S}$ laser excitation Chu et al. (1984); Fee et al. (1993) and more recently via single-photon excitation of $1^{3}\text{S}$ to $2^{3}\text{P}$ in a rapidly switching electric field Alonso et al. (2017) and via single-photon excitation of $1^{3}\text{S}$ to $3^{3}\text{P}$ with radiative decay to $2^{3}\text{S}$ in an electric field S. Aghion et al. (2018) (AEgIS collaboration) and in absence of electric field C. Amsler et al. (2019) (AEgIS collaboration). This last method in particular showed that it is possible to build an almost-monochromatic $2^{3}\text{S}$ Ps source with a selected and tunable velocity distribution in the $10^{4}\,$\mathrm{m}\text{,}{\mathrm{s}}^{-1}$$ range, with an overall efficiency between $0.7-1.4\%$ according to the selected velocity C. Amsler et al. (2019) (AEgIS collaboration).

In the present work, following this experimental line, we investigate the possibility of stimulating the $3^{3}\text{P}$–$2^{3}\text{S}$ transition to increase the overall $2^{3}\text{S}$ production efficiency. Indeed, laser-excited $3^{3}\text{P}$ Ps can spontaneously decay radiatively to $2^{3}\text{S}$ via the dipole-allowed $3^{3}\text{P}$–$2^{3}\text{S}$ transition (rate $A_{23}=2\pi\times 1.1\cdot 10^{7}\,\,${\mathrm{s}}^{-1}) with $\sim\mkern-4.0mu10\%$ measured branching efficiency S. Aghion et al. (2018) ([AEgIS collaboration](#bib.bib18)); C. Amsler et al. (2019) ([AEgIS collaboration](#bib.bib19)), limited by the competition with the more efficient spontaneous decay channel $3^{3}\text{P}$–$1^{3}\text{S}$ ($A_{13}=2\pi\times 8.4\cdot 10^{7}\,\,${\mathrm{s}}^{-1}). Increasing the $3^{3}\text{P}$–$2^{3}\text{S}$ transition rate, and thus the branching efficiency of the $2^{3}\text{S}$ decay, is possible through stimulated emission.

A straightforward way to stimulate the $3^{3}\text{P}$–$2^{3}\text{S}$ transition consists in introducing a synchronized broadband IR laser pulse at $1312.2\text{\,}\mathrm{nm}$ Oberthaler (2002) on top of the pulsed UV laser at $\sim$205\text{,}\mathrm{nm} used for $1^{3}\text{S}$–$3^{3}\text{P}$ excitation S. Aghion et al. (2016) ([AEgIS collaboration](#bib.bib20)). In our experiment, such a laser pulse could be obtained from the same optical setup producing the UV beam, which is described in Cialdi et al. ([2011](#bib.bib21)); Caravita ([2013](#bib.bib22)). In this setup, a sequence of non-linear optical conversion processes are used to generate the $205\text{\,}\mathrm{nm}$ wavelength, and in particular an optical parametric generation (OPG) crystal yields as a byproduct a broadband, amplified idler beam at $1314\text{\,}\mathrm{nm}$. Both $205\text{\,}\mathrm{nm}$ and $1314\text{\,}\mathrm{nm}$ wavelengths could be tuned by varying the temperature set-point of the OPG crystal around its nominal working point of $T_{1}=$175.6\text{\,}\mathrm{\SIUnitSymbolCelsius} (Fig. 1). This temperature value, chosen for the UV wavelength to fall on the $1^{3}\text{S}$–$3^{3}\text{P}$ resonance ($\lambda_{3}=$205.045\text{,}\mathrm{nm}), is ineffective for a stimulated emission experiment as the bandwidth of the IR does not arrive to cover the $1312.2\text{\,}\mathrm{nm}$ wavelength with sufficient optical power to efficiently stimulate the $3^{3}\text{P}$–$2^{3}\text{S}$ transition. Hence, a different set-point $T_{2}=$172.4\text{\,}\mathrm{\SIUnitSymbolCelsius} was selected to gain enough power at $1312.2\text{\,}\mathrm{nm}$, while mantaining an acceptable detuning $\delta\nu_{3}$ of the UV beam from the $1^{3}\text{S}$–$3^{3}\text{P}$ resonance frequency, to minimize the unavoidable reduction in excitation efficiency. The set-point setting accuracy was limited to $0.5\text{\,}\mathrm{\SIUnitSymbolCelsius}$ by the TC200 temperature controller. At $T_{2}$, the induced $1^{3}\text{S}$–$3^{3}\text{P}$ detuning was ${\delta\nu}_{3}\approx\mkern-4.0mu$410\text{,}\mathrm{GHz}, corresponding to $205.103\text{\,}\mathrm{nm}$ and $0.9\sigma$ of the Doppler distribution of our Ps source ($\sigma_{\nu}\approx\mkern-4.0mu$470\text{\,}\mathrm{GHz} S. Aghion et al. (2016) (AEgIS collaboration)). A reduction in the $1^{3}\text{S}$–$3^{3}\text{P}$ excitation efficiency of about $\sim 50\%$ was expected S. Aghion et al. (2016) (AEgIS collaboration), due to the Doppler selection of a Ps atoms distribution with $\lambda_{3}{\delta\nu}_{3}\approx 0.85\cdot 10^{5}\,$\mathrm{m}\text{,}{\mathrm{s}}^{-1}$$ average velocity in the direction parallel to the laser beam.

A direct way to observe an enhancement in the $2^{3}\text{S}$ signal due to the action of the stimulating laser is to compare the Ps annihilation time distribution with the UV laser only to that measured with both UV and IR pulses. The experimental methodology was the same used in previous works S. Aghion et al. (2016) (AEgIS collaboration); S. Aghion et al. (2018) (AEgIS collaboration); C. Amsler et al. (2019) (AEgIS collaboration). Bursts of  $10^{7}$ $\mathrm{e^{+}}$, $7\text{\,}\mathrm{ns}$ in time length S. Aghion et al. (2015) (AEgIS collaboration), were guided by a $25\text{\,}\mathrm{mT}$ magnetic field, focused by an electric field of about $300\text{\,}\mathrm{V}\text{\,}{\mathrm{cm}}^{-1}$ and implanted at $3.3\text{\,}\mathrm{keV}$ in a circular spot of $\sim$ $3\text{\,}\mathrm{mm}$ full-width at half maximum (FWHM) into a $\mathrm{e^{+}}$/Ps converter held at room temperature. The converter is constitued by a Si(111) p-type crystal with nanochannels produced via electrochemical etching and oxidized in air at $100\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for $2\text{\,}\mathrm{h}$ Mariazzi et al. (2010a). Ps formed in the converter out-diffuses back into vacuum through the nanochannels with an efficiency up to  35% loosing a fraction of its formation energy by collisions with the channel walls Mariazzi et al. (2010b). A fraction of the emitted cloud was subsequently conveyed to $2^{3}\text{S}$ either by the $205\text{\,}\mathrm{nm}$ UV beam alone (i.e. through spontaneous decay from $3^{3}\text{P}$, as in S. Aghion et al. (2018) (AEgIS collaboration); C. Amsler et al. (2019) (AEgIS collaboration)) or by a combination of the UV and IR laser pulses (i.e. through stimulated decay from $3^{3}\text{P}$). Both beams were linearly polarized perpendicularly to the target, with a nearly Gaussian temporal profile with a FWHM of $1.5\text{\,}\mathrm{ns}$ (UV) and $4.0\text{\,}\mathrm{ns}$ (IR), and a nearly-Gaussian spectral profile with bandwidths $\sigma^{\mathrm{UV}}\approx 2\pi\times$120\text{,}\mathrm{GHz} (UV) and $\sigma^{\mathrm{IR}}\approx 2\pi\times$440\text{\,}\mathrm{GHz} (Fig. 1). The energy of the two pulses was $(53\pm 5)\,$\mathrm{\SIUnitSymbolMicro J} for the UV and $(405\pm 10)\,$\mathrm{\SIUnitSymbolMicro J} for the IR (at the entrance viewport of the experimental chamber). The UV spot was nearly Gaussian with $(7.0\pm 0.7)\,$\mathrm{mm} FWHM both in horizontal and vertical directions, while the IR spot was uniform in power and slightly astigmatic, $(12.9\pm 0.5)\times(17.2\pm 0.5)\,$\mathrm{mm} FWHM in the horizontal and vertical directions, due to a geometrical cut of an optical element. Two spurious light backgrounds at $532\text{\,}\mathrm{nm}$ and $894\text{\,}\mathrm{nm}$ were found superimposed to the $1312\text{\,}\mathrm{nm}$ beam with the same spot size, caused by the non-ideality of the dichroic mirrors used to separate the beams, with energies $\approx$80\text{,}\mathrm{\SIUnitSymbolMicro J} and $\approx$100\text{\,}\mathrm{\SIUnitSymbolMicro J} respectively. Photo-ionisation of $2^{3}\text{S}$ or $3^{3}\text{P}$ due to these backgrounds was negligible due to their low intensities and the small photo-ionisation cross-sections ($10^{-16}\div 10^{-17}\,${\mathrm{cm}}^{2}). As the presence of an electric field would have shorten considerably the optical lifetime of $2^{3}\text{S}$ Ps C. Amsler et al. (2019) ([AEgIS collaboration](#bib.bib19)), a fast HV switch with a rise time $<$15\text{\,}\mathrm{ns} was used to disable the guiding electric field of the focusing electrode (see Fig. 2.a) $\sim$5\text{,}\mathrm{ns} after $\mathrm{e^{+}}$ implantation C. Amsler et al. (2019) ([AEgIS collaboration](#bib.bib19)), such that the field was negligible when the excitation lasers were shot ($\sim$20\text{\,}\mathrm{ns} after $\mathrm{e^{+}}$ implantation).

The time distribution of the annihilation $\gamma$ rays due to the implanted $\mathrm{e^{+}}$ and decaying/annihilating Ps was measured with a $20\times 25\,$\mathrm{mm} lead tungstate ($\mathrm{PbWO_{4}}$) scintillator Cassidy and Mills ([2007](#bib.bib26)) coupled to a Hamamatsu R11265-100 photomultiplier tube (PMT), placed $40\text{\,}\mathrm{mm}$ above the Ps converter. The signal from the PMT was 50%-50% split and digitized using two channels of a HD4096 Teledyne LeCroy 2.5Gs oscilloscope set at high ($100\text{\,}\mathrm{m}\mathrm{V}\mathrm{/}\mathrm{d}\mathrm{i}\mathrm{v}$) and low ($1\text{\,}\mathrm{V}\mathrm{/}\mathrm{d}\mathrm{i}\mathrm{v}$) gains to further extend the linear dynamic range of the digitizer. The data of the two channels were joined to form the so-called single-shot positron annihilation lifetime spectroscopy (SSPALS) spectra Cassidy et al. ([2006](#bib.bib27)), whose average is proportional to the amount of Ps/$\mathrm{e^{+}}$ annihilating per unit time. The total lifetime of $2^{3}\text{S}$ Ps in absence of an electric field ($1142\text{\,}\mathrm{ns}$) is longer than that of $1^{3}\text{S}$ ($142\text{\,}\mathrm{ns}$) and of all other populated sublevels in the $n=1-3$ manifolds. Thus, an increase in the delayed annihilations (either due to in-flight or to pick-off annihilations as the Ps atoms hit the chamber walls) for $t\gg$142\text{\,}\mathrm{ns} ($t$ being the time elapsed from $\mathrm{e^{+}}$ implantation) can be directly related to the amount of $2^{3}\text{S}$ Ps (as in C. Amsler et al. (2019) (AEgIS collaboration)). Relative time-dependent variations between two SSPALS spectra families with different laser configurations were quantified using the parameter $S(t)=(A_{1}(t)-A_{2}(t))/A_{1}(t)$ where $A_{1}(t)$, $A_{2}(t)$ are the (averaged) integrated areas below single SSPALS shots of the two families in a selected time window centered in $t$. In this definition $A_{1}(t)$ has the role of the reference area. An alternating measurement scheme between the two families was used to minimize the effect of time drifts of the experimental conditions. Contributions of eventual residual long-time drifts in the $S$ calculation were further reduced by normalizing the shots in each family to a second-order polynomial fit of their value versus time (detrending technique, see S. Aghion et al. (2018) (AEgIS collaboration) App.).

First, a set of reference spontaneous $2^{3}\text{S}$ Ps production measurements was acquired on the detuned OPG set-point ($T_{2}$) to measure the $S(t)=(A^{\mathrm{off}}(t)-A^{\mathrm{UV}}(t))/A^{\mathrm{off}}(t)$ parameter in the absence of stimulated emission in the conditions selected for the following experiments. The measured $S(t)$ curve was compared to one acquired at the on-resonance set-point ($T_{1}$) (see Fig. 2), which was well characterized previously C. Amsler et al. (2019) (AEgIS collaboration). $A^{\mathrm{UV}}$ and $A^{\mathrm{off}}$ correspond to the integrated averaged areas with and without the UV laser. Time windows of $300\text{\,}\mathrm{ns}$ width with steps of $50\text{\,}\mathrm{ns}$ were used to calculate $S(t)$. All measurements were obtained alternating shots with the UV laser to the same number without it ($\sim 200$ shots for $T_{1}$, $\sim 300$ shots for $T_{2}$). The peak of $2^{3}\text{S}$ annihilations on the chamber walls is more evident in the on-resonance ($T_{1}$) measurement (between $500\text{\,}\mathrm{ns}$ and $750\text{\,}\mathrm{ns}$) than in the detuned ($T_{2}$) measurement, where it is lower and smeared to longer times, in agreement with a reduced $2^{3}\text{S}$ production efficiency and longer atoms’ traveled distances, because of the Doppler selection operated by the detuned UV beam that selects atoms traveling longer trajectories.

The experimental $S(t)$ curves were fitted using a previously-developed Monte Carlo (MC) model S. Aghion et al. (2018) (AEgIS collaboration); C. Amsler et al. (2019) (AEgIS collaboration). This MC calculates the atoms’ flight trajectories (starting from our Ps source velocity distribution S. Aghion et al. (2016) (AEgIS collaboration)) and their excitation dynamics by simultaneously integrating the center-of-mass equations of motion and optical rate equations for the internal level dynamics. The annihilation position/time distribution for $1^{3}\text{S}$ and $2^{3}\text{S}$ Ps in flight and by collisions with the chamber walls S. Aghion et al. (2018) (AEgIS collaboration) are calculated. The only fitting parameter was the $1^{3}\text{S}$–$3^{3}\text{P}$ excitation efficiency $\eta_{3}$, while the $2^{3}\text{S}$ branching $\eta_{m}$ ($=0.097$) and quenching $\eta_{q}$ ($=0.17$) efficiencies were set according to past measurements (with the same laser delay of $20\text{\,}\mathrm{ns}$, see C. Amsler et al. (2019) (AEgIS collaboration)). Different Doppler selections in the two set-points were now constrained to the measured UV detunings ($0\text{\,}\mathrm{GHz}$ for $T_{1}$ and $410\text{\,}\mathrm{GHz}$ for $T_{2}$).

The fit (Fig. 2, solid lines) yielded $\eta_{3}=(12.4\pm 0.7)\%$ for $T_{1}$ and $\eta_{3}=(5.2\pm 0.4)\,\%$ for $T_{2}$, corresponding to a $\approx\mkern-4.0mu58\,\%$ reduction in the excitation efficiency as a consequence of the UV laser detuning, in agreement with the expected $\sim\mkern-4.0mu50\%$ S. Aghion et al. (2016) (AEgIS collaboration). The annihilations’ position distributions resulting from the fit (Fig. 2.a) emphasize the difference in the atoms’ flight trajectories in the chamber geometry in the two different detuning conditions.

Subsequently, the $1312.2\text{\,}\mathrm{nm}$ IR pulse was introduced to stimulate the $3^{3}\text{P}$–$2^{3}\text{S}$ transition. A first measurement campaign to optimize the IR laser $2^{3}\text{S}$ production efficiency was carried out by progressively attenuating its energy from the nominal $405\text{\,}\mathrm{\SIUnitSymbolMicro J}$ with a set of graded neutral density filters. The idea behind this optimization search is that the desired stimulated emission from the $3^{3}\text{P}$ level competes dynamically with the repumping of this level by absorption from the $2^{3}\text{S}$. This mainly depends on the pulse energy (mantaining the other pulse parameters, in particular the FWHM temporal profile). One expects that on our nanosecond time scale, while a low energy can induce only a low population gain on the $2^{3}\text{S}$ level, a too high energy subtracts population from that previously efficiently excited by the rising part of the pulse. The optimization measurements were acquired alternating ($\sim 200$) shots with both UV and IR lasers to shots with UV laser only. Their associated $S(t)=(A^{\mathrm{UV}}(t)-A^{\mathrm{UV+IR}}(t))/A^{\mathrm{UV}}(t)$ reflected the relative signal changes only caused by the IR, i.e. isolating the effects of this laser pulse (resonant with the $3^{3}\text{P}$–$2^{3}\text{S}$ transition) on the $2^{3}\text{S}$ Ps population. The resulting $S(t)$ curves (an example of which - with $5\text{\,}\mathrm{dB}$ attenuation - is shown in Fig. 3, inset) exhibit an excess of annihilations in the region around $550\text{\,}\mathrm{ns}$ and $1050\text{\,}\mathrm{ns}$, where the impact of $2^{3}\text{S}$ atoms onto the chamber walls is also observed in the reference measurement ($T_{2}$ in Fig. 2). A convenient way to show the enhancement in the $2^{3}\text{S}$ production is to consider a total $\bar{S}$ parameter calculated in the wide time window 550-1050 $\mathrm{ns}$, normalized on the total parameter $\bar{S}$ of the reference measurement (calculated in the same time window). The results are reported in Fig. 3 as a function of the $1312.2\text{\,}\mathrm{nm}$ laser pulse attenuation. The effect of the $1312.2\text{\,}\mathrm{nm}$ laser is compatible with zero when the beam is sent with full power. Then the effect progressively increases as the laser energy is decreased to reach a maximum at $5\text{\,}\mathrm{dB}$ attenuation ($E^{peak}\approx$126\text{,}\mathrm{\SIUnitSymbolMicro J}$$) to slowly decrease again at even higher attenuations.

As expected, with the maximum pulse energy a depopulation effect is observed. As the laser energy lowers, a positive gain in the $2^{3}\text{S}$ population progressively increases, reaching a maximum around the $5\text{\,}\mathrm{dB}$ attenuation (where $E_{peak}\sim$126\text{,}\mathrm{\SIUnitSymbolMicro J}$$), to slowly decrease again at even higher attenuations. These measurements indicate that, selecting the $1312.2\text{\,}\mathrm{nm}$ laser pulse energy at $E_{peak}$, an increase of $(190\pm 90)\%$ (corresponding to a $\times(2.9\pm 0.9)$ gain efficiency) in the amount of produced $2^{3}\text{S}$ with respect to the reference was obtained.

A final measurement campaign was conducted to directly evaluate the $2^{3}\text{S}$ branching efficiency achieved by setting the $1312.2\text{\,}\mathrm{nm}$ laser to the optimal energy. The set of measurements was acquired alternating ($\sim\mkern-4.0mu200$) shots with both UV and IR lasers to shots with both lasers off. The resulting $S(t)=(A^{\mathrm{off}}(t)-A^{\mathrm{UV+IR}}(t))/A^{\mathrm{off}}(t)$ curve, compared to the reference $2^{3}\text{S}$ curve obtained in the same temperature set-point $T_{2}$ and in the absence of the IR enhancement laser (previously plotted in Fig. 2), is shown in Fig. 4. The $2^{3}\text{S}$ branching efficiency was evaluated by fitting $-S(t)$ with the Monte Carlo model. The fit was now performed letting $\eta_{m}$ vary, while keeping all other parameters fixed to those determined in the reference measurement. The found value for the branching efficiency was $\eta^{\mathrm{UV}}_{m}=0.297\pm 0.019$. This value, compared to the branching efficiency previously estimated from the measurement with the UV laser alone with the same laser delay ($\eta^{\mathrm{UV}}_{m}=(0.097\pm 0.027)$ C. Amsler et al. (2019) (AEgIS collaboration)), leads to a $2^{3}\text{S}$ Ps enhancement of $\times(3.1\pm 1.0)$, in perfect agreement with the IR pulse optimization measurement.

In conclusion, we experimentally demonstrated the possibility to efficiently stimulate the $3^{3}\text{P}$–$2^{3}\text{S}$ transition of Ps by employing a pulsed, broadband $1312.2\text{\,}\mathrm{nm}$ laser. The highest enhancement efficiency was found by tuning the intensity of the IR laser pulse inducing the $3^{3}\text{P}$–$2^{3}\text{S}$ transition. In these optimal conditions of our experiment, due to the pulsed excitation dynamics, the relative enhancement of the $2^{3}\text{S}$ Ps atoms production from the excited $2^{3}\text{S}$ population, with respect to the reference of the spontaneous emission decay, was measured to be $\times(3.1\pm 1.0)$. This corresponds to a branching efficiency of $2^{3}\text{S}$ production from the $3^{3}\text{P}$ level that can be up to $\sim 30\%$. Anyway, the overall $2^{3}\text{S}$ Ps excitation efficiency was limited by the present technical restriction in producing both pulsed laser beams with the correct wavelengths and energies using a single optical parametric generation stage. The variation in the temperature set-point of the laser generation crystal, necessary to output the correct $1312.2\text{\,}\mathrm{nm}$ wavelength with sufficient energy, induced an UV detuning of about $410\text{\,}\mathrm{GHz}$ and a reduction in the amount of excited $3^{3}\text{P}$ of $\approx\mkern-4.0mu58\,\%$.

In a future realization of this experiment, the present technical limitations could be overcome by separating the two UV and IR laser lines, i.e. having independent non-linear optical generation and amplification stages. An advantage of having the two laser wavelengths independent from each other would be to retain the full tunability characteristics of our $2^{3}\text{S}$ source C. Amsler et al. (2019) (AEgIS collaboration), while conveying $\sim 30\%$ of what could be excited to $3^{3}\text{P}$ to $2^{3}\text{S}$. Furthermore, if one accepts to sacrifice the mentioned velocity selection, a further increase up to a factor of five of the overall excitation efficiency could be obtained by enlarging the UV laser bandwidth to cover efficiently the Ps Doppler profile. Finally, the first laser spectroscopy of the $3^{3}\text{P}$–$2^{3}\text{S}$ transition would become feasible. A laser system with independent UV and IR laser lines is currently under development to take full advantage of this $1^{3}\text{S}$–$3^{3}\text{P}$–$2^{3}\text{S}$ stimulated excitation scheme in view of future measurements on a beam $2^{3}\text{S}$ Ps.

The authors are grateful to Dr. S. Cialdi for the original development of the $1^{3}\text{S}$-$3^{3}\text{P}$ laser. This work was supported by Istituto Nazionale di Fisica Nucleare; the CERN Fellowship programme and the CERN Doctoral student programme; the Swiss National Science Foundation Ambizione Grant (No. 154833); a Deutsche Forschungsgemeinschaft research grant; an excellence initiative of Heidelberg University; Marie Sklodowska-Curie Innovative Training Network Fellowship of the European Commission’s Horizon 2020 Programme (No. 721559 AVA); European Research Council under the European Unions Seventh Framework Program FP7/2007-2013 (Grants Nos. 291242 and 277762); European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement ANGRAM No. 748826; Austrian Ministry for Science, Research, and Economy; Research Council of Norway; Bergen Research Foundation; John Templeton Foundation; Ministry of Education and Science of the Russian Federation and Russian Academy of Sciences and the European Social Fund within the framework of realizing the project, in support of intersectoral mobility and the European Social Fund within the framework of realizing Research infrastructure for experiments at CERN, LM2015058.