Laser Excitation of Muonic 1S Hydrogen Hyperfine Transition: Effects of Multi-pass Cell Interference

Abstract

Calculating the laser-induced transition probability by using the fluence distribution that neglects interference effects (e.g., by employing ray-tracing methods) can lead to an overestimation of this probability, as it underestimates saturation effects. In this paper, we investigate how interference effects in the multi-pass cell, used to enhance the laser fluence, affect the laser-induced transition probability between hyperfine levels in muonic hydrogen, a bound system of a negative muon and a proton. To avoid complications related to the exact knowledge of the intra-cavity field, we develop a simple model that estimates the maximal possible interference effects for given laser and multi-pass cell parameters, thereby providing an upper bound for the resulting decrease in transition probability relative to the case where these effects are neglected. A numerical evaluation of this upper bound for muonic hydrogen shows that, under our experimental conditions, such effects can be safely neglected. Nonetheless, the methodology presented here could be applied to estimate the impact of interference effects on the laser-induced transition probability in other experiments involving coherent light in multi-pass systems.

Figures (6)

• Figure 1: Schematic (not to scale) of the experimental setup for the CREMA HFS experiment. The dashed lines represent the dimensions of the muon stopping volume and the solid line represents the cell's diameter, $D$.
• Figure 2: (a) 3D rendering of the toroidal multi-pass cell with an example of a ray-tracing simulation. The laser pulse enters through a 0.5 mm-thick slit at a small angle relative to the x-axis. To illustrate the beam path, the first few reflections inside the cell are shown in a different colour. (b) Spatial distribution of fluence along the $z$-axis (muon beam axis) of the toroidal cell, obtained from ray-tracing simulations, and respective transition probability (laser excitation followed by collisional de-excitation) for an in-coupled pulse of 1 mJ energy and $R=0.992$. The transition probability is calculated for a $\mu$p atom in the $F = 0$ state assuming a laser bandwidth of 100 MHz, a pulse length of 50 ns, a target temperature of 22 K and a target pressure of 0.6 bar. (c) Same as (b) but for the $x$-axis. Figure and caption reproduced from Nuber2023.
• Figure 3: Simplified model of the multi-pass cell used to estimate the maximal possible interference effects. The model is one-dimensional, consisting of two reflecting surfaces with reflectivity $R$ separated by a distance $D$. A laser pulse undergoes multiple reflections between the surfaces, interfering with itself. The pulse length is not drawn to scale; in the cases relevant to this paper and for the considered HFS experiment, its spatial extent far exceeds the mirror separation $D$. The $\mu$p location is indicated by the blue shaded region.
• Figure 4: Diagram of the HFS sub-levels showing the laser-induced transition and the collisional de-excitation, which produces a $\mu$p atom with approximately 100 meV of kinetic energy amaro_2022. 5 meV corresponds to the kinetic energy of $\mu$p atoms thermalized at a temperature of 22 K.
• Figure 5: Fluence distributions for a fixed average fluence of $\overline{\mathcal{F}} = 100$ J/cm$^2$: (Left) for various values of $D$ at fixed $R = 0.995$; (Right) for various values of $R$ at fixed $D=15$ cm. The dashed vertical line represents the delta-like fluence distribution for the case without interference. In both cases we assume $\tau=50$ ns.