The J-PARC E34 experiment is progressing in realization to measure the
muon anomalous magnetic moment, aμ= (g-2)/2, and electric dipole
moment (EDM). Its unique, innovative experimental approach based on a
small-emittance muon beam is indispensable to reveal new physics
beyond the standard model. Such a muon beam can be obtained by
accelerating muons thermalized in the form of muonium atom (μ+e-) in
silica aerogel at room temperature, having subsequent
laser-ionization. A slab of silica aerogel with laser ablation on its
face has been developed for the muonium production target, giving the
muonium yield of 3.4 x 10^-3 per incident muon projected in the
laser-ionization region of the experiment. Optimization of the
ablation structure is on-going to achieve a higher yield, which
improves the experimental sensitivity directly. Collaborations with
other projects that utilize thermal muons and muonium atoms result in
the synergy to develop high-power laser systems for muonium
ionization. The current status on the development of thermal muon
source will be presented having the above-mentioned activities.